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José A. González, Hernando Quevedo, Marcelo Salgado and Daniel Sudarsky\
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title: Local Constraints on the Oscillating G Model
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Submission to: [**Physical Review D15 (Brief Reports)**]{}
Introduction
============
The success of the standard cosmological model in describing the evolution of our Universe, beginning with the era of nucleosynthesis until the present state, has been confronted with serious difficulties resulting from the analysis of cosmological data. At large cosmological scales we find two main problems which are not dealt within the framework of the standard (old) cosmological model. The first one concerns the cosmological dark matter problem according to which the luminous matter (baryonic matter and radiation) content of the Universe represents only a small fraction of the total matter content. In fact, the inflationary models predicted that the total energy density $\Omega = 1$, with $\Omega$ given in terms of the critical energy density [@dm3]. This prediction has recently been given further support by observational data resulting from the recent cosmic microwave background (CMB) experiments like Boomerang and Maxima and the high red-shift supernovae (SNIa) measurements [@flat], leading to the conclusion that the average energy-density of the Universe is indeed near the critical value. Obviously, these observations have increased the importance of the dark matter problem for the understanding of our Universe.
The second problem is related to the observations that indicate a periodicity of 128$h^{-1}$ Mpc (where $h$ is the Hubble parameter in units of 100 km s$^{-1}$ Mpc$^{-1}$) in the galaxy number distribution, observed in deep pencil beams [@Broad; @Szalay] in the north and south poles of our galaxy. This shocking discovery would, in its simplest interpretation, indicate that galaxies in the Universe are situated on the surface of concentric spheres with the center situated in our own galaxy. This is in complete contradiction with the basis of modern cosmology: the cosmological principle of homogeneity and isotropy of the Universe. It has been argued [@Szalay], and it seems to be the pervading view among researchers in the field, that this periodicity could be the result of the appearance of an intrinsic length scale in the distribution of matter. However, we have shown [@per] that this explanation is not really satisfactory as there are scenarios of this type that result in a negligible probability for such observation to be obtained in a particular direction.
In a series of works [@ssq1; @ssq2; @ssq3; @ssq4] we have investigated an alternative model based on a massive scalar field which is non-minimally coupled to gravity. The oscillation of the scalar field in cosmic time results in a time-dependent effective gravitational constant. We have shown that this model leads to predictions which are in good agreement with most of the observational data. In fact, although this model was originally proposed [@Morik] to explain the observed periodicity in the galaxy number distribution, we have shown that it was possible to adjudicate most of the energy density of the Universe to the oscillating massive scalar field which, therefore, could be regarded as candidate for the non-baryonic nature of the cosmological dark energy. That is, this model is able to explain simultaneously both, the problem of the cosmological dark energy and the problem of the periodicity in the galaxy number distribution. We have checked that the model satisfies some of the cosmological constraints. More precisely, we have seen that the model reproduces correctly the primordial nucleosynthesis of $^4$He, and is consistent with the present value of the energy density of baryonic matter and the age of the Universe. In this work, we will analyze the additional constraints following from local observations, namely, the Viking experiments [@viking], which impose bounds on the rate of change in time of the effective gravitational constant and on the effective Brans-Dicke parameter.
In a previous work [@ssq5] we have shown that all but one of the free parameters entering the model are fixed by the cosmological analysis and that with these values it was not possible to satisfy the Brans-Dicke bound. In this work, we analyze the possibility of overcoming this problem by relaxing the single condition freely imposed in our previous cosmological studies. We will show that even with this relaxation it is not possible to satisfy the local constraints and the periodicity observations simultaneously. This result indicates that either the behavior of the scalar field in the presence of local inhomogeneities is different from its behavior at large scales [@ssq5] or that a modified model would be necessary if we want to explain in a unified way the apparent galactic periodicity and the cosmological dark energy.
Constraints on the oscillating $G$ model
========================================
The dynamics of oscillating $G$ model is described by the Lagrangian: = ([ 116G\_0]{} + \^2) R - , \[lag\] where $G_0$ is Newton’s gravitational constant, $\xi$ stands for the non-minimally coupling constant, $R$ is the scalar curvature, $\phi$ is the scalar field and $V(\phi)$ is a scalar potential which in its simplest form is taken as the harmonic potential $V=m^2 \phi^2$, with $m$ the mass of the scalar field. If we consider a time-dependent scalar field, the non-minimal coupling results in a time-dependent effective gravitational constant $G_{\rm eff} =G_0(1+ 16\pi G_0 \xi \phi^2)^{-1}$. The central feature of the oscillating $G$ model is that oscillations in the expectation value of $\phi$ induce oscillations in $G_{\rm eff}$ and this leads to oscillations in the Hubble parameter $H$ which manifest themselves in the redshift measurements of distant points of the Universe. In turn, the redshift oscillations give rise to an apparent variation in the density of galaxies. Consequently, a temporal oscillation of the redshift can be mistakenly interpreted as a real spatial periodicity in the galaxy number distribution. This was used in previous works [@ssq1; @ssq2; @ssq3; @ssq4] to explain the observed periodicity of 128$h^{-1}$Mpc in the distribution of galaxies in our Universe. To this end, we analyzed the Friedman-Robertson-Walker cosmology with a combination of two non-interacting perfect fluids (radiation and baryonic matter). From the field equations we obtain the following expression for the total effective energy density of the the system (see [@ssq1; @ssq3]): \_[tot]{} = [11+ 16\_0\^2]{} , \[omtot\] where \_0 = [ddt]{}|\_[today]{} , t = t H\_0 ,= [H\_0]{} , m\^2 = [43]{} \^2 . In the above equation, a subscript “0" stands for the value of the corresponding quantity at present time $t=t_0$. The frequency of oscillation $\omega =
m\sqrt{3/4\pi}$ is determined by the period of 128$h^{-1}$Mpc observed in the pencil beam surveys and turns out to be $\omega \approx 147 H_0$. Here $\Omega_{\rm matt}=
\Omega_{\rm bar} + \Omega_{\rm rad}$. Notice that in Eq.(\[omtot\]) and for the present analysis we can neglect the contribution of the photon energy density $\Omega_{\rm rad}$ because the observations of the cosmic microwave background radiation of 2.725 K implies that $\Omega_{\rm rad} \approx 10^{-3}\Omega_{\rm bar}$[@footnote]. Furthermore, the value of $\Omega_{\rm bar}$ must lie within the range $[0.01, 0.02]h^{-2}$ determined by the abundance of the light elements other than $^4$He [@copi]. Finally, for the total energy density we take the value $\Omega_{\rm tot}=1$ in accordance with the standard inflationary model and with the recent CMB and SNIa observations [@flat]. Consequently, Eq.(\[omtot\]) can be interpreted as a constraint relating the initial cosmological values of the scalar field, $\phi_0$ and $\dot{\widetilde\phi}_0$, and the coupling parameter $\xi$. Another, in some sense, more realistic approach would be to identify $\Omega_{\rm matt}$ with the total amount of clumped matter in our Universe which would include besides the baryonic component also the so called Cold Dark Matter, leading us to take $\Omega_{\rm matt}\sim 0.3$. However, we will see that even this drastic change of view does not alter our conclusions in a significant way.
A further constraint is imposed by the observed redshift-galaxy-count amplitude ${\cal A}_0\geq {\cal O}(0.5)$, which for the oscillating $G$ model can be approximated by the expression [@CritStein] \_0 = [16]{}( \^2\_0\^2 + \_0\^2) . \[amp\] Here we are considering the additional term $\dot{\widetilde\phi}_0^2$ which was set to zero in previous analysis because we want to remove all the arbitrarily imposed conditions on the model in order to examine whether all the constraints can be solved simultaneously. Since the values of ${\cal A}_0$ and $\widetilde\omega$ are fixed by the pencil beam observations, Eq.(\[amp\]) represents a constraint between the values of $\phi_0$, $\dot{\widetilde\phi}_0$ and $\xi$.
We call Eqs.(\[omtot\]) and (\[amp\]) the global constraints of the oscillating $G$ model because the values of ${\cal A}_0$ and $\widetilde\omega$ are fixed by the large scale observations of the galactic periodicity, and the values of $\Omega_{\rm tot}$ and $\Omega_{\rm mat}$ are the result of global cosmological observations.
On the other hand, the Solar System local observations impose an upper bound on the variation of the gravitational constant $|\dot G/(GH)|\leq 0.3 h^{-1}$ [@CritStein]. For the oscillating $G$ model this yields = [G\_[eff]{}G\_[eff]{} H]{} = - [32\_0\_0 1 +16\_0\^2 ]{} , ||0.3 \[beta\] It is well known that scalar-tensor models of the kind defined by the Lagrangian (\[lag\]) can be transformed by means of a conformal transformation into an effective Brans-Dicke theory. Then, such models can be characterized by an effective Brans-Dicke parameter $\omega_{\rm BD}^{\rm eff}$ which must satisfy the lower bound imposed by the Viking experiments [@viking], $\omega_{\rm BD}^{\rm eff}> 3000$. In the case of the oscillating $G$ model we obtain \_[BD]{}\^[eff]{} = [1+16\_0\^2 128\^2 \_0\^2]{} , \[bd\] a constraint that relates $\phi_0$ with $\xi$.
Now we proceed to the analysis of the global constraints (\[omtot\]) and (\[amp\]) and the local constraints (\[beta\]) and (\[bd\]). In our previous cosmological studies, we were able to satisfy simultaneously the total energy constraint (\[omtot\]) as well as the nucleosynthesis and age constraints, together with the constraints for the amplitude (\[amp\]) and the variation of the effective gravitational constant (\[beta\]) by setting $\dot{\widetilde\phi}_0 = 0$. In fact, in this case the constraint (\[beta\]) is automatically satisfied ($\beta=0$), whereas the constraints (\[omtot\]) and (\[amp\]), together with the “plateau hypothesis" [@ssq3] that ensures a successful nucleosynthesis, fix the values of the remaining parameters $\phi_0$ ($\sim 10^{-3}$) and $\xi$ ($\sim 6)$. The evolution of the model with these conditions result in a value for the age of the Universe compatible with the standard bounds [@age].
However, as we have shown in [@ssq5], with these values the oscillating $G$ model is unable to satisfy the Brans-Dicke limit (\[bd\]) with $\omega_{\rm BD}^{\rm eff} > 3000$ (or even the less severe bound $\omega_{\rm BD}^{\rm eff} > 500$). The simplest possibility to overcome this problem is to relax the condition $\dot{\widetilde\phi}_0=0$ within the range allowed by the constraints (\[amp\]) and (\[beta\]). To this end, we replace the values of $\phi_0$, $\dot{\widetilde\phi}_0$ and $\xi$ following from the constraints (\[amp\]), (\[beta\]) and (\[bd\]) into the total energy constraint (\[omtot\]). Then we obtain \[ncon\] f(\_[BD]{}\^[eff]{},,[A]{}\_0 )= 1-\_[mat]{} - + = 0 , with a=\^2 + 4\^2 , b=-\^2+2\_0 +2 , a constraint that, for a specific value of $\Omega_{\rm mat}$, determines the amplitude in terms of the effective Brans-Dicke parameter and the parameter $\beta$ (recall that the frequency $\widetilde\omega$ has been fixed by the period of oscillation). Notice that the initial values $\phi_0$ and $\dot{\widetilde\phi}_0$ do not appear at all in Eq.(\[ncon\]). In order to investigate the constraint (\[ncon\]) in a systematic way we have to solve the algebraic equation (\[ncon\]) as ${\cal A}_0= {\cal A}_0(\beta)$. Actually this is equivalent to solve the differential equation $df(\beta,{\cal A}_0(\beta))/d\beta=0$ (i.e., the resulting differential equation $d{\cal A}_0/d\beta = F({\cal A}_0,\beta)$) subject to the boundary values $(\beta^i, {\cal A}_0^i)$ such that $f(\beta^i,{\cal A}_0^i)=0$, for a fixed $\omega_{\rm BD}^{\rm eff}$ and $\Omega_{\rm mat}$. For instance, for $\dot{\widetilde\phi}_0=0$, and $\omega_{\rm BD}^{\rm eff} = 3000$, $\Omega_{\rm mat}= 0.0236$ the pair $(\beta^i =0, {\cal A}_0^i\approx 0.022)$ satisfies the constraint (\[ncon\]) as well as the remaining conditions (except of course the order of magnitude in the bound on ${\cal A}_0$). The result of this calculation is plotted in Figure 1 for two different values of $\Omega_{\rm mat}$ within the range allowed by observations. We see that the range of values $(\beta,{\cal A}_0)$ that satisfy $f(\beta,{\cal A}_0)=0$ is extremely narrow and that all of the values for the amplitude within this range are situated well below the lower bound ${\cal A}_0 \geq {\cal O}(0.5)$ imposed by the redshift-galaxy-count observations. The conclusion is that the Brans-Dicke local constraint is not compatible with the observed value for the oscillation amplitude. Further numerical analysis of the constraint (\[ncon\]) show that an increase of the matter density $\Omega_{\rm mat}$ or of the effective Brans-Dicke parameter leads to even lower values for the amplitude.
We conclude that the relaxation of the original condition $\dot{\widetilde\phi}_0=0$ does not allow the oscillating $G$ model to satisfy simultaneously both global and local constraints, and that we have to look for further generalizations of this model if we want to consider it as a candidate to explain the apparent galactic periodicity simultaneously with the nature of the non-baryonic dark matter content in the Universe. Needless is to say that had the model succeeded in these tests, then it would be necessary to confront the oscillating $G$ model to further tests in light of the recent CMB and SNIa observations.
Finally, it is worthwhile to emphasize that the generalized view on the problem of the galactic periodicity is that perhaps there is no problem at all and that such a “periodicity” is only the result of an excess of power at some characteristic length scales. While this could be the case, the simplest analysis on this matter shows that the existence of a characteristic distance in the large scale distribution is not enough to explain such observations [@per], and therefore serious doubts arise in taking such a comfortable position. Clearly, the observation of galactic periodicity or lack thereoff in directions other than those corresponding to the north and south galactic poles will put an end to the controversy. On the other hand, if the existence of such periodicity in a large number of directions were to be confirmed we would be in the uncomfortable situation of having no model to account for it, and we would need to resort to variations on the oscillating $G$ model presented here as the only type of scenario capable of explaining such observations within the context of the cosmological principle.
.3cm [**Acknowledgments**]{}
We acknowledge partial support from DGAPA-UNAM Project No. IN121298 and from CONACyT Projects 32551-E and 32272-E.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In this article, we generalize some works of Bertolini-Darmon and Vatsal on anticyclotomic $L$-functions attached to modular forms of weight two to higher weight case. We construct a class of anticyclotomic $L$-functions for ordinary modular forms and derive the functional equation and the interpolation formula at all critical specializations. Moreover, we prove results on the vanishing of $\mu$-invariant of these $L$-functions and the non-vanishing of central $L$-values with anticyclotomic twists.'
address:
- ' Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto, 606-8502, Japan. '
- ' The Hakubi Center for Advanced Research, Kyoto University, Yoshida-Ushinomiya-cho, Sakyo-ku, Kyoto, 606-8302, Japan '
- |
Department of Mathematics \
National Taiwan University \
No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
author:
- Masataka Chida
- 'Ming-Lun Hsieh'
bibliography:
- 'mybib.bib'
date: 'January 23, 2013'
title: 'Special values of anticyclotomic L-functions for modular forms'
---
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We construct analytical and numerical vortex solutions for an extended Skyrme-Faddeev model in a $(3+1)$ dimensional Minkowski space-time. The extension is obtained by adding to the Lagrangian a quartic term, which is the square of the kinetic term, and a potential which breaks the $SO(3)$ symmetry down to $SO(2)$. The construction makes use of an ansatz, invariant under the joint action of the internal $SO(2)$ and three commuting $U(1)$ subgroups of the Poincaré group, and which reduces the equations of motion to an ordinary differential equation for a profile function depending on the distance to the $x^3$-axis. The vortices have finite energy per unit length, and have waves propagating along them with the speed of light. The analytical vortices are obtained for special choice of potentials, and the numerical ones are constructed using the Successive Over Relaxation method for more general potentials. The spectrum of solutions is analyzed in detail, specially its dependence upon special combinations of coupling constants.'
---
[**Vortices in the extended Skyrme-Faddeev model**]{}
L. A. Ferreira$^{a,}$[^1]; J. Jäykkä$^{b,}$[^2]; Nobuyuki Sawado$^{c,}$[^3]; Kouichi Toda$^{d,}$[^4];
.2in $^{a}$ Instituto de Física de São Carlos; IFSC/USP;\
Universidade de São Paulo - USP\
Caixa Postal 369, CEP 13560-970, São Carlos-SP, Brazil\
$^{b}$ School of Mathematics, University of Leeds\
LS2 9JT Leeds, United Kingdom\
$^c$ Department of Physics, Tokyo University of Science,\
Noda, Chiba 278-8510, Japan\
$^d$ Department of Mathematical Physics, Toyama Prefectural University,\
Kurokawa 5180, Imizu, Toyama, 939-0398, Japan
Introduction {#sec:intro}
============
The so-called Skyrme-Faddeev model was introduced in the seventies [@sf] as a generalization to $(3+1)$ dimensions of the $O(3)$ non-linear sigma model in $(2+1)$ dimensions [@bp]. The Skyrme term, quartic in derivatives of the field, balances the quadratic kinetic term and according to Derrick’s theorem, allows the existence of stable solutions with non-trivial Hopf topological charges. Due to the highly non-linear character of the model and the lack of symmetries, the first soliton solutions were only constructed in the late nineties using numerical methods [@glad; @solfn; @sutcliffe; @hietarinta]. Since then the interest in the model has increased considerably and it has found applications in many areas of physics due mainly to the knotted character of the solutions [@babaev]. The numerical efforts in the construction of the solutions have improved our understanding of the properties of the model [@improve] and even the scattering of knotted solitons has been investigated [@hietarinta-scatter]. One of the aspects of the model that has attracted considerable attention has been its connection with gauge theories. Faddeev and Niemi have conjectured that it might describe the low energy limit of the pure $SU(2)$ Yang-Mills theory [@fn]. They based their argument on a decomposition of the physical degrees of freedom of the $SU(2)$ connection, proposed in the eighties by Cho [@chofn], and involving a triplet of scalar fields ${\vec n}$ taking values on the sphere $S^2$ (${\vec n}^2=1$). The conjecture, which is quite controversial [@wipf], states that the low energy effective action of the $SU(2)$ Yang-Mills theory is the Skyrme-Faddeev action, and the knotted solitons would describe glueballs or even vacuum configurations. The fact that the Skyrme-Faddeev model has an $O(3)$ symmetry, and so possesses Goldstone boson excitations, is one of the many difficulties facing the conjecture, and some modifications of it were in fact proposed [@newfaddeev]. Any check of such type of conjectures is of course very difficult to perform since it must involve non-perturbative calculations in the strong coupling regime of the Yang-Mills theory. However, Gies [@gies] has calculated the Wilsonian one loop effective action for the pure $SU(2)$ Yang-Mills theory assuming Cho’s decomposition, and found that the Skyrme-Faddeev action is indeed part of it, but additional quartic terms in the derivatives of the triplet ${\vec n}$ are unavoidable. In fact, the first numerical Hopf solitons were first constructed for the Skyrme-Faddeev model modified by a quartic term [@glad] which is the square of the kinetic term. However, the soliton solutions in [@glad] were constructed for a sector of the theory where the signs of the coupling constants disagree with those indicated by Gies’ calculations. The addition of quartic terms has the drawback of making the Lagrangian dependent on terms which are quartic in time derivatives and so the energy is not positive definite. However, as a quantum field theory the Skyrme-Faddeev model is not renormalizable by power counting and has to be considered as a low energy effective theory. In addition, under the Wilsonian renormalization group flow the square of the kinetic term is as unavoidable as the Skyrme quartic term. Therefore, it is quite natural to investigate the properties of the Skyrme-Faddeev model with such modifications.
In this paper we consider an extended Skyrme-Faddeev model defined by the Lagrangian $$\begin{aligned}
{\cal L} = M^2\, \partial_{\mu} {\vec n}\cdot\partial^{\mu} {\vec n}
-\frac{1}{e^2} \, \(\partial_{\mu}{\vec n} \wedge
\partial_{\nu}{\vec n}\)^2 + \frac{\beta}{2}\,
\left(\partial_{\mu} {\vec n}\cdot\partial^{\mu} {\vec n}\right)^2- V\(n_3\)
\lab{action}\end{aligned}$$ where ${\vec n}$ is a triplet of real scalar fields taking values on the sphere $S^2$, $M$ is a coupling constant with dimension of $\({\rm length}\)^{-1}$, $e^2$ and $\beta$ are dimensionless coupling constants, and the potential is a functional of the third component $n_3$ of the triplet ${\vec n}$. Note that the potential breaks the $O(3)$ symmetry of the original Skyrme-Faddeev down to $O(2)$, the group of rotations on the plane $n_1\, n_2$, and so eliminating two of the three Goldstone boson degrees of freedom. In this paper the main role of potential is to stabilize the vortex solutions.
The first exact vortex solutions for the theory were constructed in [@vortexlaf] for the case where the potential vanishes, and by exploring the integrability properties of a submodel of . In order to describe those exact vortex solutions it is better to perform the stereographic projection of the target space $S^2$ onto the plane parameterized by the complex scalar field $u$ and related to ${\vec
n}$ by $$\begin{aligned}
{\vec n} = \(u+u^*,-i\(u-u^*\),\u2 -1\)/\(1+\u2\)
\lab{udef}\end{aligned}$$ It was shown in [@vortexlaf] that the field configurations of the form $$\begin{aligned}
u\equiv u\(z,y\)\qquad \quad u^*\equiv u^*\( z^*,y\) \qquad \quad {\rm for} \qquad \quad
\beta\,e^2=1\qquad \quad V=0 \lab{exactclass}\end{aligned}$$ are exact solutions of , where $z=x^1+i\,\varepsilon_1\,x^2$ and $y=x^3-\varepsilon_2\,x^0$, with $\varepsilon_a=\pm 1$, $a=1,2$, and $x^{\mu}$, $\mu=0,1,2,3$, are the Cartesian coordinates of the Minkowski space-time. Despite the fact that constitutes a very large class of solutions, no finite energy solutions were found within it. If the dependence of the $u$ field upon the variable $y$ is in the form of phases like $e^{i\,k\,y}$, then one finds solutions with finite energy per unit of length along the $x^3$-axis. The simplest solution is of the form $u=z^n\,e^{i\,k\,y}$, with $n$ integer, and it corresponds to a vortex parallel to the $x^3$-axis and with waves traveling along it with the speed of light. More general solutions of the class were constructed in [@newsf], including multi-vortices separated from each other and all parallel to the $x^3$-axis. The ideas of [@vortexlaf] were generalized to an extended Skyrme-Faddeev defined on the target space $CP^N$, possessing $N-1$ complex scalar fields $u_i$, and the class of solutions constructed is like , where the fields $u_i$’s are arbitrary functions of $z$ and $y$ [@CPNvortex]. Note that the solutions are not solutions of the original Skyrme-Faddeev model, since that corresponds to $\beta=0$, and requires the condition $\beta\,e^2=1$. If one takes the limit $\beta\rightarrow 0$ and $\frac{1}{e^2}\rightarrow 0$ with keeping the product $\beta\,e^2$ constant and equal to unity, one observes that reduces to the $CP^1$ model (if $V=0$). Therefore, the configurations are also solutions of the four dimensional $CP^1$ model. The ideas of [@CPNvortex] were then used to construct multi vortex solutions for the the four dimensional $CP^N$ model [@fkz1; @fkz2]. Aproximate vortex solutions for the pure Skyrme-Faddeev model, without the potential and $\beta$ terms in , were constructed in [@kundu].
The static energy density (${\cal H}_{{\rm static}}=-{\cal L}$) associated to is positive definite if $V>0$, $M^2>0$, $e^2>0$ and $\beta<0$. That is the sector explored in [@glad] and where Hopf soliton solutions were first constructed (for $V=0$). In addition, that is also the sector explored in [@Sawado:2005wa] but with additional terms involving second derivatives of the ${\vec n}$ field, and where Hopf solitons were also constructed. The static energy density of is also positive definite for $V>0$ if $$\begin{aligned}
M^2>0\,; \qquad e^2<0\, ; \qquad \beta <0 \, ; \qquad \beta\, e^2\geq 1
\lab{nicesector}\end{aligned}$$ That is the sector that agrees with the signature of the terms in the one loop effective action calculated in [@gies] and that we will consider in this paper. Static Hopf solitons were constructed in [@sawadohopfions; @todahopfions] for the sector (with $V=0$) and their quantum excitations, including comparison with glueball spectrum, were considered in [@quantumhopfions]. An interesting feature of the Hopf solitons constructed in [@sawadohopfions] is that they shrink in size and then disappear as $\beta\,e^2\rightarrow 1$, which is exactly the point where the vortex solution of the class exists.
The aim of the present paper is to investigate if vortex solutions for the model continue to exist when the condition $\beta\, e^2=1$ is relaxed, and so if they co-exist with the Hopf solitons in [@sawadohopfions]. We also aim at the study of their properties and stability. The idea is to keep the solutions as close to those of the class as possible. In order to do that, we follow the ideas of [@babelon] and implement an ansatz based on the $O(2)$ internal symmetry given by the transformations $u\rightarrow e^{i\alpha}\,u$, together with three commuting transformations of the Poincaré group given by rotations on the plane $x^1\, x^2$, and translations in the directions $x^0$ and $x^3$. We impose the field configurations to be invariant under the diagonal subgroups of the tensor product of the internal $O(2)$ group with each one of the three commuting one parameter subgroups of the Poincaré group. The resulting ansatz is given by $$\begin{aligned}
u\equiv f\(\rho\)\,e^{i\,\(n\,\vp+\lambda\, z+k\, \tau\)}
\lab{ansatz}\end{aligned}$$ where $n$ is an integer for $u$ to be single valued, and $\lambda$, $k$ are real dimensionless parameters, and where we have used the dimensionless polar coordinates $\(\rho, \varphi, z, \tau\)$, defined by $$\begin{aligned}
x^0=c\, t= r_0\,\tau\qquad \;\;
x^1=r_0\,\rho\, \cos \vp\qquad\;\;
x^2=r_0\,\rho\, \sin \vp\qquad\;\;
x^3= r_0\,z
\lab{coord}\end{aligned}$$ and where we have introduced a length scale $r_0$ given by $$\begin{aligned}
r_0^2=-\frac{4}{M^2\,e^2}
\lab{r0def}\end{aligned}$$ which is positive since we are dealing with $e^2<0$ (see ). Note that when $\lambda=\pm k$ and $f\sim \rho^{\pm n}$, the configurations are of the type . The ansatz is in fact a generalization to $(3+1)$ dimensions of the ansatz used in the Baby Skyrme models [@wojtekpotential; @joaquinansatz].
The types of potential we will consider in this part are of the form $$\begin{aligned}
V\(n_3\)\equiv \frac{\mu^2}{2}\,\(1+n_3\)^{2-\frac{2}{N}}\,\(1-n_3\)^{2+\frac{2}{N}}
\lab{potdef}\end{aligned}$$ where $N$ is a non-vanishing integer, $\mu$ a real coupling constant. It is interesting to note that when the integer $n$ of has the same modulus as $N$ of , one obtains analytical solutions of the form $$\begin{aligned}
u(\rho, \varphi, z, \tau)=\Bigl(\frac{\rho}{a}\Bigr)^{N}e^{i[\varepsilon\,N\varphi+k(z+\tau)]}
\lab{zcsolutionintro}\end{aligned}$$ with $\varepsilon=\pm 1$, and where $a=\mid N\mid\left[\frac{(-e^2)\,(\beta e^2-1)M^4}{4\,\mu^2}\right]^{1/4}$. Such exact solutions are valid for all values of the coupling constants. In particular, for the case $\beta=0$, are exact solutions of the theory $$\begin{aligned}
{\cal L} = M^2\, \partial_{\mu} {\vec n}\cdot\partial^{\mu} {\vec n}
-\frac{1}{e^2} \, \(\partial_{\mu}{\vec n} \wedge
\partial_{\nu}{\vec n}\)^2 - \frac{\mu^2}{2}\,\(1+n_3\)^{2-\frac{2}{N}}\,\(1-n_3\)^{2+\frac{2}{N}}
\lab{sfactionwithpot}\end{aligned}$$ which is the proper Skyrme-Faddeev model in the presence of a potential. In this case we have $a=\mid N\mid\left[\frac{e^2\, M^4}{4\,\mu^2}\right]^{1/4}$, and $\mu^2 \,e^2>0$.
In some cases we have been unable to find a numerical solution which is expected in the analytical set, i.e. . Reasons for this are presented below. Apart from those cases, we have checked numerically the existence of the above solution. The numerical simulations are performed using a standard technique for a differential equation, the Successive Over Relaxation (SOR) method. In order to further confirm the accuracy and correctness of the SOR code, some of the results were reproduced by an independent code using Newton’s method, giving typical differences of the order of less than $10^{-4}$.
The paper is organized as follows. In the next section we briefly describe the extended Skyrme-Faddeev model. The equations of motion are also introduced in Sec. \[sec:vortex\]. We discuss the Hamiltonian density of the model in Sec. \[sec:hamiltonian\]. The method and the solutions of the integrable sector of the present model are discussed in Sec. \[sec:integrable\]. In Sec. \[sec:numerics\], we show the numerical solutions. Sec. \[sec:application\] is devoted to note briefly potential physical applications of our solutions. A brief summary is presented in Sec. \[sec:summary\].
The model {#sec:vortex}
=========
In terms of the complex scalar field $u$ introduced in the Lagrangian becomes
$$\begin{aligned}
{\cal L}=
4\,M^2\,\frac{\partial_{\mu}u\;\partial^{\mu}u^*}{\(1+\u2\)^2} +
\frac{8}{e^2}\left[
\frac{\(\partial_{\mu}u\)^2\(\partial_{\nu}u^*\)^2}{\(1+\u2\)^4}+
\(\beta\,e^2-1\)\,\frac{\(\partial_{\mu}u\;\partial^{\mu}u^*\)^2}{\(1+\u2\)^4}
\right]- V\(\mid u\mid^2\)
\lab{actionu}\end{aligned}$$
where we have used the fact that $n_3$ is a functional of $\mid u\mid^2$ only, and so is the potential. The Euler-Lagrange equations following from , or , reads $$\begin{aligned}
\(1+\u2\)\, \partial^{\mu}{\cal K}_{\mu}-2\,u^{*}\,{\cal K}_{\mu}\,
\partial^{\mu} u=-\frac{u}{4}\,\(1+\u2\)^3\,V^{\prime}
\lab{eqmot}\end{aligned}$$ where $V^{\prime}=\frac{\partial \,V}{\partial \u2}$, and $$\begin{aligned}
{\cal K}_{\mu}\equiv M^2\, \partial_{\mu}u
+\frac{4}{e^2}\,\frac{
\left[\(\partial_{\nu}u\partial^{\nu} u\)
\partial_{\mu}u^{*}+\(\beta\,e^2-1\)\,\(\partial_{\nu}u\,\partial^{\nu}u^{*}\)\,
\partial_{\mu} u\right]}{\(1+\u2\)^2}
\lab{kdef}\end{aligned}$$ We point out that the theory possesses an integrable sector defined by the condition $$\begin{aligned}
\(\partial_{\mu}u\)^2=0
\lab{eikonal}\end{aligned}$$ Such condition was first discovered in the context of the $CP^1$ model using the generalized zero curvature condition for integrable theories in any dimension [@afs1], and then applied to many models with target space being the sphere $S^2$, or $CP^1$ (see [@afs2] for a review). It leads to an infinite number of local conserved currents. Indeed, together with the equations of motion imply the conservation of the infinity of currents given by $$\begin{aligned}
J_{\mu}^G\equiv {\cal K}_{\mu}\, \frac{\delta G}{\delta u}-{\cal K}_{\mu}^*\, \frac{\delta G}{\delta u^*}
\lab{infinitecurr}\end{aligned}$$ where $G$ is any functional of $\mid u\mid^2$ only. For the case where the potential vanishes, the set of conserved currents is considerably enlarged since $G$ can be an arbitrary functional of $u$ and $u^*$, but not of their derivatives. If in addition to the condition one takes $V=0$ and $\beta\,e^2=1$, then the equations of motion reduce to $\partial^2u=0$. It is in that integrable sector that the solutions lie, and were studied in [@vortexlaf]. For theories defined by Lagrangians which are functionals of the Skyrme term only (pullback of the area form of the sphere) the currents of the form are Noether currents associated to the area preserving diffeomorphisms of $S^2$ [@razumov]. It is possible to define conditions weaker than that lead to integrable theories associated to Abelian subgroups of the group of the area preserving diffeomorphisms [@joaquinabelian].
Substituting the ansatz into the equations of motion we get an ordinary differential equation for the profile function $f$ as $$\begin{aligned}
\frac{1}{\rho}\,\partial_{\rho}\left[\rho\,\frac{f^{\prime}}{f}\,R\right]-
\frac{\(1-f^2\)}{\(1+f^2\)}\,S\,\left[\Lambda-\(\frac{f^{\prime}}{f}\)^2\right]=
\frac{r_0^2}{4\,M^2}\(1+f^2\)^2\, \frac{\partial\, V}{\partial\mid u\mid^2}
\lab{eqforf}\end{aligned}$$ where the primes denote derivatives w.r.t. $\rho$, and where $$\begin{aligned}
\Lambda&= \lambda^2-k^2+\frac{n^2}{\rho^2}\nonu\\
S&= 1+\beta\,e^2\,\frac{f^2}{\(1+f^2\)^2}\,
\left[\Lambda+\(\frac{f^{\prime}}{f}\)^2\right]
\lab{deflambdasr}\\
R&= 1+\frac{f^2}{\(1+f^2\)^2}\, \left[\(\beta\,e^2-2\)\,\Lambda+\beta\,e^2\,
\(\frac{f^{\prime}}{f}\)^2\right]\nonu \lab{eqforf2}\end{aligned}$$ With the choice of potential given in we get that $$\begin{aligned}
\frac{r_0^2}{4\,M^2}\(1+f^2\)^2\, \frac{\partial\, V}{\partial\mid u\mid^2}=
2\,\frac{r_0^2\,\mu^2}{M^2}\left[\frac{\(2- \frac{2}{ N}\)
\(f^{2}\)^{\(1-\frac{2}{ N}\)}-\(2+ \frac{2}{ N}\)\(f^{2}\)^{\(2-\frac{2}{ N}\)}}{\(1+f^2\)^{3}}\right]\end{aligned}$$ We look for solutions satisfying the following boundary conditions $$\begin{aligned}
\vec{n} \rightarrow
\begin{cases}
(0,0,-1) & \text{for } \rho \rightarrow 0 \\
(0,0,+1) & \text{for } \rho \rightarrow \infty
\end{cases}\end{aligned}$$ which imply that the profile function should satisfy $$\begin{aligned}
f\rightarrow 0 \qquad {\rm for} ~~\rho \rightarrow 0 \qquad\qquad {\rm and} \qquad\qquad
f\rightarrow \infty \qquad {\rm for} ~~\rho \rightarrow \infty\end{aligned}$$ Let us then assume the following behavior of the profile function $$\begin{aligned}
f =
\begin{cases}
\alpha_0\,\rho^{s_1}\,\bigr(1+\alpha_1\,\rho+\alpha_2\,\rho^2\ldots\bigl) &
\text{for }\rho \rightarrow 0\\
\beta_0\,\rho^{s_2}\,\bigr(1+\beta_1\,\frac{1}{\rho}+\beta_2\,\frac{1}{\rho^2}\ldots\bigl) &
\text{for } \rho \rightarrow \infty
\end{cases}
\lab{limitbehavior}\end{aligned}$$ where $s_i>0$, $i=1,2$. Substituting that into the equation one gets the behavior for small $\rho$ implies that $$\begin{aligned}
s_1^2=n^2 \end{aligned}$$ where $n$ is the integer in the ansatz . The behavior of for large $\rho$ implies that the relation between $n^2$ and $s_2^2$ depends upon the form of the potential, and $$\begin{aligned}
\lambda^2-k^2 =
\begin{cases}
\begin{alignedat}{2}
&8\,\frac{r_0^2\,\mu^2}{M^2} &\qquad& \text{for $N=-1$}
\\
&-2\,\frac{r_0^2\,\mu^2}{M^2} && \text{\rm for $N=-2$}
\\
&0 && \text{for all other $N$}
\end{alignedat}
\end{cases}
$$ where $N$ is the integer appearing in the potential , and $\lambda$, $k$ are the parameters of the ansatz . Therefore, except for the cases $N=-1$ and $N=-2$, the waves along the vortex have to travel with the speed of light since the dependency upon $x^3$ and $x^0$ has to be of the form $x^3\pm c\,t$. For the dimensionfull constants $L:=r_0\lambda$, and $K:=r_0k$, the velocity is defined as $$\begin{aligned}
\frac{Kc}{L}=
\begin{cases}
\frac{Kc}{\sqrt{K^2+8\mu^2/M^2}}<c \quad & \text{for $N=-1$} \\
\frac{Kc}{\sqrt{K^2-2\mu^2/M^2}}>c \quad & \text{for $N=-2$}
\end{cases}\end{aligned}$$ Therefore, the mode is tachyonic for $N=-2$. $N=-1$ is not tachyonic, but the energy diverges from the boundary behavior of the potential. In the following analysis, we will concentrate on the analysis for $N\geqq 1$ (thus $\lambda^2=k^2$).
The Energy {#sec:hamiltonian}
==========
The Hamiltonian density associated to is not positive definite due to the quartic terms in time derivatives. We shall arrange the Legendre transform of each term in to make explicit such non positive contributions, and write the Hamiltonian density as (see [@bonfim] for details) $$\begin{aligned}
\begin{split}
{\cal H} &= 4\, M^2\, \frac{\left[\mid{\dot u}\mid^2+{\vec \nabla}u\cdot
{\vec\nabla}u^*\right]}{\(1+\u2\)^2} -\frac{24}{e^2}\,\frac{\({\vec \nabla}u\)^2\,\(
{\vec \nabla}u^*\)^2}{\(1+\u2\)^4} \,\left[ \(\frac{2}{3}\)^2-F^2 \right]
\\
&-24\,\frac{\(\beta\,e^2-1\)}{e^2}\,\frac{\left[\mid{\dot u}\mid^2+ \frac{1}{3}\,{\vec
\nabla}u\cdot {\vec\nabla}u^*\right] \left[ {\vec \nabla}u\cdot
{\vec\nabla}u^*-\mid{\dot u}\mid^2\right]}{\(1+\u2\)^4} +V\(\mid u\mid^2\)
\lab{energy}
\end{split}\end{aligned}$$ where ${\dot u}$ denotes the $x^0$-derivative of $u$, and ${\vec \nabla}u$ its spatial gradient, and where we have denoted $$\begin{aligned}
\frac{{\dot u}^2}{\({\vec \nabla}u\)^2}\equiv \frac{1}{3} + F\, e^{i\,\Phi}
\lab{fphidef}\end{aligned}$$ with $F>0$ and $0\leq\Phi\leq 2\pi$, being functions of the space-time coordinates. Note that ${\cal H}$ given in is positive definite for static configurations and for the range of parameters given in .
Using the ansatz and the coordinates we get ${\dot
u}=i\,k\,u/r_0$. The metric on the spatial sub-manifold is given by $ds^2=r_0^2\(d\rho^2+ \rho^2\,d\varphi^2+dz^2\)$, and so $$\begin{aligned}
\({\vec \nabla}u\)^2=\frac{u^2}{r_0^2}\,\(\Omega_{-}-\lambda^2\)\qquad\qquad
{\vec \nabla}u\cdot {\vec \nabla}u^*=\frac{f^2}{r_0^2}\,\(\Omega_{+}+\lambda^2\)\end{aligned}$$ where $$\begin{aligned}
\Omega_{\pm}=\(\frac{f^{\prime}}{f}\)^2\pm\frac{n^2}{\rho^2}
\lab{omegadef}\end{aligned}$$ In addition one gets that $\frac{{\dot u}^2}{\({\vec \nabla}u\)^2}=\frac{1}{3}+F\,
e^{i\,\Phi}=-\frac{k^2}{\(\Omega_{-}\lambda^2\)}$, and since it is real it follows that $\Phi=0$ or $\pi$. Therefore, $ \(\frac{2}{3}\)^2 -F^2
=\(\Omega_{-}-\lambda^2-3k^2\)\(\Omega_{-}+k^2-\lambda^2\)/3\(\Omega_{-}-\lambda^2\)^2$. So, the Hamiltonian density becomes $$\begin{aligned}
\lab{energy2}
\begin{split}
{\cal H} &= \frac{4}{r_0^4}\,\biggl[M^2\,r_0^2\, \frac{f^2}{\(1+f^2\)^2}\,
\(\Omega_{+}+\lambda^2+k^2\) \\
&+2\,\frac{f^4}{\(1+f^2\)^4}
\biggl\{-
\frac{1}{e^2}\(\Omega_{-}-\lambda^2-3k^2\)\(\Omega_{-}+k^2-\lambda^2\)
\\
&-\frac{\(\beta\,e^2-1\)}{e^2}\,
\(\Omega_{+}+\lambda^2+3k^2\)\(\Omega_{+}+\lambda^2-k^2\) +\mu^2\,r_0^4\,
\(f^2\)^{-\frac{2}{N}} \biggr\}
\biggr]
\end{split}\end{aligned}$$
The integrable sector {#sec:integrable}
=====================
It is interesting to note that - with a special choice of the potential have an analytical solution for each topological charge $n$. In fact, solutions of the integrable equation also become the solutions of the present model. For $\lambda^2=k^2$, the solutions can be written of the form $$\begin{aligned}
u(\rho, \varphi, z, \tau)=\Bigl(\frac{\rho}{a}\Bigr)^{n}e^{i[\varepsilon\,n\varphi+k(z+\tau)]}
\lab{zcsolution}\end{aligned}$$ where $\varepsilon=\pm 1$, and $a$ is a dimensionless constant to be fixed by the equations of motion. Substituting this into the equation -, we get $$\begin{aligned}
(\beta e^2-1)\frac{4n^3}{a^4}\Bigl\{1+\Bigl(\frac{\rho}{a}\Bigr)^{2n}\Bigr\}^{-3}\Bigl[(n-1)
\Bigl(\frac{\rho}{a}\Bigr)^{2n-4}-(n+1)\Bigl(\frac{\rho}{a}\Bigr)^{4n-4}\Bigr] \nonumber \\
=\frac{2r_0^2\mu^2}{M^2}\Bigl\{1+\Bigl(\frac{\rho}{a}\Bigr)^{2N}\Bigr\}^{-3}
\Bigl[(2-\frac{2}{N})\Bigl(\frac{\rho}{a}\Bigr)^{2N-4}
-(2+\frac{2}{N})\Bigl(\frac{\rho}{a}\Bigr)^{4N-4}\Bigr]\end{aligned}$$ The constant $a$ determines the scale of the vortex and the equation is satisfied if $n=N$ and $$\begin{aligned}
a=\mid n\mid\left[\frac{M^2(\beta e^2-1)}{r_0^2\mu^2}\right]^{1/4}=
\mid n\mid\left[\frac{(-e^2)\,(\beta e^2-1)M^4}{4\,\mu^2}\right]^{1/4}
\lab{conditiona}\end{aligned}$$ Thus, for all possible values of $\beta e^2$ we have analytical solutions. All those solutions satisfy the condition . Clearly the class of solutions contain the special solution at $\beta e^2=1$ found previously in [@vortexlaf] if we take a proper limit of vanishing potential, i.e. $\beta e^2\rightarrow 1$ and $\mu^2\rightarrow 0$, with $\frac{\beta e^2-1}{\mu^2}={\rm constant}$. Also, apparently we have no solution at $\beta e^2 \neq 1$ without any potential because the scale goes to infinity. Note that the case $\beta =0$ is particularly interesting since it corresponds to the proper Skyrme-Faddeev model (without the extra quartic term) in the presence of a potential. Therefore, the configurations are exact solutions of the theory (for $n=N$), with $a=\mid N\mid\left[\frac{e^2\, M^4}{4\,\mu^2}\right]^{1/4}$, and $\mu^2 \,e^2>0$.
As we mentioned in Sec.\[sec:vortex\], for the sector satisfying , the model possesses the infinite set of conserved currents . In particular, choosing a form of $G=-4i(1+|u|^2)^{-1}$, one gets of the Noether current for the symmetry of an arbitrary angle $\alpha$, i.e., $u \to e^{i\alpha}u$ $$\begin{aligned}
J_\mu=-4iM^2\frac{u\partial_\mu u^*-u^*\partial_\mu u}{(1+|u|^2)^2}
-i\frac{8}{e^2}(\beta e^2-1)\frac{2(\partial_\nu u\partial^\nu u^*)(\partial_\mu u^*u-u^*\partial_\mu u)}{(1+|u|^2)^4}
\lab{noether}\end{aligned}$$ For the solution , we can evaluate the charge per unit length for the solution $$\begin{aligned}
Q=\int dx_1dx_2 J_0=-8\pi M^2 ka^2r_0
\Bigl[I(n)+\frac{n}{6}\frac{1}{a^2}(\beta e^2-1)\Bigr]\end{aligned}$$ where $I(n)=\frac{1}{n}\Gamma(1+\frac{1}{n})\Gamma(1-\frac{1}{n})$, with $\Gamma$ being the Euler’s Gamma function. Here we used an integral formula [@gradshteyn] $$\begin{aligned}
\int^\infty_0 \frac{x^{\mu-1}dx}{(p+qx^\nu)^{m+1}}
=\frac{1}{\nu p^{m+1}}\Bigl(\frac{p}{q}\Bigr)^{\frac{\mu}{\nu}}
\frac{\Gamma(\frac{\mu}{\nu})\Gamma(1+m-\frac{\mu}{\nu})}{\Gamma(1+m)}\end{aligned}$$
For the Hamiltonian , we perform the similar computations. As a result, we get the energy per unit length by the integration on the $x_1x_2$ plane. For $n=1$, the energy of the static vortex is (in units of $4M^2$) $$\begin{aligned}
E_{static}=2\pi+\frac{4\pi}{3}\frac{1}{a^2}(\beta e^2-1)
\label{stt_energy_ana}\end{aligned}$$ and for $n\geq 2$ they are $$\begin{aligned}
E_{static}=2\pi n+\frac{2\pi}{3}\frac{1}{a^2}(\beta e^2-1)(n^2-1) I(n)
\label{stt_energy_ana2}\end{aligned}$$ Note that the first term is proportional to the topological charge. The energy per unit of length of the time-dependent vortex diverges for $n=1$, and for $n\geqq 2$ we obtain $$\begin{aligned}
&&E_{wave}
=2\pi n+\frac{2\pi}{3}\frac{1}{a^2}(\beta e^2-1)(n^2-1) I(n)
+k^2\Bigl[2\pi a^2I(n)+\frac{2\pi}{3}(\beta e^2-1)n\Bigr]
\label{tt_energy_ana}\end{aligned}$$ For the limit of $\beta e^2\to 1, \mu^2\to 0$ with keeping $a^2=n^2\sqrt{M^2(\beta
e^2-1)/r_0^2\mu^2}$ finite, we obtain the energy per unit of length found previously in [@vortexlaf]. The energy monotonically grows as $k^2$ increases. Interestingly, the static vortex has a minimum at $\beta e^2=1.0$ and/or $\mu^2=0.0$ but for the time dependent vortex there is a minimum of the energy for fixed $\beta e^2$ and $k^2$ and finite $\mu^2$. The solutions are confirmed numerically in the subsequent section.
The numerical analysis {#sec:numerics}
======================
Although the ansatz is given in terms of the polar coordinates, for the numerical analysis it is more convenient to use a new radial coordinate $y$, defined by $\rho=\sqrt{\frac{1-y}{y}}$. Accordingly, we adopt a function $g$ called the profile function, instead of using $f$, i.e., $f(\rho)=\sqrt{\frac{1-g(y)}{g(y)}}$.
The equation can be promptly rewritten as $$\begin{aligned}
&&\frac{d}{dy}\biggl[\frac{y(1-y)}{g(1-g)}g'R\biggr]
+\Bigl(g-\frac{1}{2}\Bigr)\frac{S}{y(1-y)}\biggl\{\Omega-\biggl(\frac{y(1-y)}{g(1-g)}g'\biggr)^2\biggr\} \nonumber \\
&&\hspace{2cm}=-\frac{1}{y^2}\frac{r_0^2\mu^2}{M^2}(1-g)^{1-\frac{2}{N}}g^{1+\frac{2}{N}}\Bigl\{4g-2\Bigl(1+\frac{1}{N}\Bigr)\Bigr\}
\lab{eqforg}\end{aligned}$$ where the primes at this time indicate derivatives w.r.t.$y$ and where $$\begin{aligned}
\Omega&=(\lambda^2-k^2)\frac{1-y}{y}+n^2 \\
S&=1+\beta e^2g(1-g)\frac{y}{1-y}\biggl\{\Omega+\biggl(\frac{y(1-y)}{g(1-g)}g'\biggr)^2\biggr\}\\
R&=1+g(1-g)\frac{y}{1-y}\biggl\{(\beta e^2-2)\Omega+\beta
e^2\biggl(\frac{y(1-y)}{g(1-g)}g'\biggr)^2\biggr\} \lab{eqforg2}\end{aligned}$$
The energy in the unit of $4M^2$ per unit length for the time-dependent vortex can be estimated in terms of following four parts of integrals of the dimensionless Hamiltonian $H:={\cal H}/4M^2$ $$\begin{aligned}
E&=&2\pi \int^\infty_0 \rho d\rho H(\rho)=E_2+E_4^{(1)}+(\beta e^2-1)E_4^{(2)}+\frac{r_0^2\mu^2}{M^2}E_0
\lab{energy_split}\end{aligned}$$ in which the components are defined as $$\begin{aligned}
&&E_2=\pi\int^1_0\frac{dy}{y(1-y)}\biggl\{(k^2+\lambda^2)\frac{1-y}{y}+n^2+\biggl(\frac{y(1-y)}{g(1-g)}g'\biggr)^2\biggr\}g(1-g)
\lab{ecomp2}\\
&&E_4^{(1)}=\pi\int^1_0\frac{dy}{2(1-y)^2}\biggl\{(3k^2+\lambda^2)\frac{1-y}{y}+n^2-\biggl(\frac{y(1-y)}{g(1-g)}g'\biggr)^2\biggr\}\nonumber \\
&&\hspace{3cm}\times \biggl\{(k^2-\lambda^2)\frac{1-y}{y}+n^2-\biggl(\frac{y(1-y)}{g(1-g)}g'\biggr)^2\biggr\}\bigl(g(1-g)\bigr)^2
\lab{ecomp41}\\
&&E_4^{(2)}=\pi\int^1_0\frac{dy}{2(1-y)^2}\biggl\{(3k^2+\lambda^2)\frac{1-y}{y}+n^2+\biggl(\frac{y(1-y)}{g(1-g)}g'\biggr)^2\biggr\}\nonumber \\
&&\hspace{3cm}\times \biggl\{(k^2-\lambda^2)\frac{1-y}{y}+n^2+\biggl(\frac{y(1-y)}{g(1-g)}g'\biggr)^2\biggr\}\bigl(g(1-g)\bigr)^2
\lab{ecomp42}\\
&&E_0=2\pi\int^1_0\frac{dy}{y^2} g^{2+\frac{2}{N}}(1-g)^{2-\frac{2}{N}}
\lab{ecompp}\end{aligned}$$ For the integrable sector, we should choose $N=n$.
![\[profiles1\_v04\]The $n=1$ profile $g(y)$ and the corresponding Hamiltonian density of the real space $H(\rho)$ of $k^2=0.0$ for the constant $r_0^2\mu^2/M^2=1.0$.](profile1_v04.eps "fig:"){width="8.1cm"} ![\[profiles1\_v04\]The $n=1$ profile $g(y)$ and the corresponding Hamiltonian density of the real space $H(\rho)$ of $k^2=0.0$ for the constant $r_0^2\mu^2/M^2=1.0$.](hdensity1_v04.eps "fig:"){width="8cm"}
![\[energy1\_v04\]The static energy and its components corresponding to the solutions of Fig.\[profiles1\_v04\].](energy1_v04a.eps "fig:"){width="10cm"}\
\
\
![\[energy1\_v04\]The static energy and its components corresponding to the solutions of Fig.\[profiles1\_v04\].](energy1_v04.eps "fig:"){width="16cm"}
Generally speaking, vortex is an object in three spatial dimensions, thus we have explored solutions in three spatial dimensions of . In the three spatial dimensions, the 4th order terms in the Lagrangian (including the Skyrme term) successfully avoid the non-existence theorem of static and finite energy solutions by Derrick. However, the equation of the ansatz is the same as an equation of corresponding static two spatial dimensions. This means $z$ component has no essential contribution to the stability. In fact, the Derrick’s theorem for two spatial dimensions implies that the contribution to the energy per unit length from quartic terms and the potential must be equal, namely $$\begin{aligned}
E_4^{(1)}+(\beta e^2-1)E_4^{(2)}=\frac{r_0^2\mu^2}{M^2}E_0
\label{derrick}\end{aligned}$$ Since the solutions of the integrable sector satisfy $E_4^{(1)}=0$, putting together with $\beta e^2=1$ and $\mu^2=0$, one can confirm that the solution without the potential found in [@vortexlaf] satisfies the above condition automatically. This fact indicates that the Derrick’s argument to the energy per unit length also works well outside of the integrable sector.
The definition of the scale parameter $a$ given in indicates the existence of the analytical solution for the same sign of $\beta e^2-1$ and $\mu^2$. In our previous study of Hopfions on the extended Skyrme-Faddeev model, we confirmed numerically the solutions exist only for $\beta e^2>1$ [@sawadohopfions], so we begin our analysis with the case of $\beta e^2>1$. We shall give comments for the possibility of finding solution of $\beta e^2<1$ in the next subsection.
The solutions of the integrable sector {#subsec:integrable}
--------------------------------------
The analytic profiles can be written in the coordinate $y$ as $$\begin{aligned}
g(y) =
\begin{cases}
\displaystyle\frac{a^2y}{a^2y+1-y} ~~~~~~~~~~~~ &\text{for ~$n=1$} \\
\displaystyle\frac{a^4y^2}{a^4y^2+(1-y)^2} ~~~~~~~~~~~&\text{for ~$n=2$}
\end{cases}
\lab{anasol}\end{aligned}$$ where $a$ is determined via . Apparently are solutions of . Next, we will see whether the solutions appear or not when we numerically solve without any constraint. Also, for the obtained solutions we will check the zero curvature condition .
Since is an ordinary second order differential equation, of course there are several methods to investigate. However, it is easily noticed that the equation may exhibit singular-like behavior at the boundary because of the term $g(1-g)$ of the denominator. Once the computation contains a small numerical error, the equation quickly diverges. The numerical method which can safely solve such a difficulty is well-known, the SOR method. Essentially we have solved the following diffusion equation for a field $\tilde{g}(y,t)$ $$\begin{aligned}
\frac{\partial \tilde{g}}{\partial t}=\omega{\cal A}\Bigl[\tilde{g}, \; \frac{\partial
\tilde{g}}{\partial y}, \; \frac{\partial^2 \tilde{g}}{\partial y^2}\Bigr]\end{aligned}$$ in which we employ as ${\cal A}$. Here $\omega$ is called as a relaxation factor which is usually chosen $\omega= 1.0\sim 2.0$. After a huge number of iteration steps, the field is relaxed to the static one, i.e, $\tilde{g}(y,t)\to g(y)$, which we are finding.
![ \[profiles2\_v13\] The $n=2$ profile $g(y)$ and the corresponding Hamiltonian density of the real space $H(\rho)$ of $k^2=0.0$ for the constant $r_0^2\mu^2/M^2=1.0$.](profile2_v13.eps "fig:"){width="8.1cm"} ![ \[profiles2\_v13\] The $n=2$ profile $g(y)$ and the corresponding Hamiltonian density of the real space $H(\rho)$ of $k^2=0.0$ for the constant $r_0^2\mu^2/M^2=1.0$.](hdensity2_v13.eps "fig:"){width="8cm"}
![ \[energy2\_v13\] The static energy and its components corresponding to the solutions of Fig.\[profiles2\_v13\].](energy2_v13a.eps "fig:"){width="10cm"}\
\
\
![ \[energy2\_v13\] The static energy and its components corresponding to the solutions of Fig.\[profiles2\_v13\].](energy2_v13.eps "fig:"){width="16cm"}
![ \[profiles3\_v48\] The $n=3$ profile $g(y)$ and the corresponding Hamiltonian density of the real space $H(\rho)$ of $k^2=0.0$. for the constant $r_0^2\mu^2/M^2=1.0$.](profile3_v48.eps "fig:"){width="8cm"} ![ \[profiles3\_v48\] The $n=3$ profile $g(y)$ and the corresponding Hamiltonian density of the real space $H(\rho)$ of $k^2=0.0$. for the constant $r_0^2\mu^2/M^2=1.0$.](hdensity3_v48.eps "fig:"){width="8cm"}
{width="10cm"}\
\
\
{width="16cm"}
{width="8cm"} {width="7.2cm"}
The first case is $n=1$. From , the explicit form of the potential is $$\begin{aligned}
V_{n=1}=\frac{\mu^2}{2}(1-n_3)^4\end{aligned}$$ In Fig.\[profiles1\_v04\] we present the numerical solution $g(y)$ and the corresponding Hamiltonian density $H(\rho)$ for $n=1$. Fig.\[energy1\_v04\] is the energy per unit length and its components for several values of $\beta e^2$ with fixed $\mu$. The function $g(y)$ in Fig.\[profiles1\_v04\] perfectly agrees with the analytical solution . We shall give a few comments for the components of the energy. For the integrable solution, the topological contribution of the energy, i.e., $E_2$ should be a constant. Also, $E_4^{(1)}$ is exactly zero for the integrable solution. The value of the component $E_4^{(1)}$ in the numerical solution is not exactly zero, but compatible with zero within the numerical precision. Note that the plot seems to blow up for the vicinity of $\beta e^2=1.0$, but the value is still up to order $\sim 10^{-8}$, so it is still negligible. This clearly means that our numerical solutions satisfy the zero curvature condition and thus belong to the integrable sector. These numerical errors are probably originated in the finite number of the mesh points. In a usual case, we used the number $N_{\rm mesh}=1000$. When we employ a larger number, the value of $E_2$ should be converge to the constant, i.e., $2\pi$.
For the $n=2$, form of the potential is $$\begin{aligned}
V_{n=2}=\frac{\mu^2}{2}(1+n_3)(1-n_3)^3\end{aligned}$$ thus the potential is zero at both the origin and the infinity. Fig.\[profiles2\_v13\] is the profile function and the Hamiltonian density for $n=2$. Again the numerical profile and the analytical one coalesce. Contrary to the case of $n=1$, the density has annular shape . Fig.\[energy2\_v13\] is the energy per unit length and its components for several values of $\beta e^2$ and fixed $\mu^2$. Again we confirmed that the value of the component $E_1^{(4)}$ is regarded as zero within the numerical uncertainty.
{width="12cm"}\
{width="12cm"}
For $n=3$, form of the potential is $$\begin{aligned}
V_{n=3}=\frac{\mu^2}{2}(1+n_3)^{4/3}(1-n_3)^{8/3}\end{aligned}$$ thus again the potential is zero at both the origin and the infinity. Fig.\[profiles3\_v48\] is the profile function and the Hamiltonian density for $n=3$. As is easily seen the radius of the annulus is larger than that of $n=2$. Fig.\[energy3\_v48\] is the energy per unit length and its components for several values of $\beta e^2$ and fixed $\mu^2$. In this case, we face a numerical difficulty. During the computation by the SOR method, the solution $g$ tends to oscillate around the true value and sometimes it accidentally goes below zero at the vicinity of the origin, and then the computation fails because of the term $g^{1+2/n}$ in . In order to avoid it, we employ a finer mesh, i.e., the number is at least $N_{\rm mesh}=3000$.
Until now, we have examined in the case of $\beta e^2>1$. The formalism leading to suggests that the choice $\beta e^2<1$ and $\mu^2<0$ might also be possible and the scale is now defined as $$\begin{aligned}
a=|n| \biggl[\frac{M^2(1-\beta e^2)}{-r_0^2\mu^2}\biggr]^{1/4} \quad \text{for $\quad \beta e^2<1,\;\mu^2<0$}\end{aligned}$$ The result is plotted in Fig.\[profile\_minus\]. However, existence of such solution seems dubious; the energy turns negative at a critical value of $\beta e^2$ thus the solution has no energy lower bound. Also, numerically the change of sign of the potential in the equation of motion quickly breaks the computation.
The solutions outside of the integrable sector {#subsec:nonintegrable}
----------------------------------------------
Although we have obtained the analytical solutions for a special form of the potential , we have many options for choice of the potential. The potentials which we employed in this paper essentially belong to a class of the generalized Baby-Skyrmion (BS) potential formally written as $$\begin{aligned}
V_{\rm BS}=\frac{\mu^2}{2}v^{\alpha}_\gamma, \; \qquad \qquad v^\alpha_\gamma=(1+n_3)^\alpha(1-n_3)^\gamma
\label{potential}\end{aligned}$$ For $\alpha=0$, the potential is called (the class of) the [*old*]{}-BS potential which was introduced in [@Leese:1989gi], while $\alpha \neq 0$ is (the class of) the [*new*]{}-BS potential [@Kudryavtsev:1997nw]. Our case corresponds to $\alpha=2-2/n,\gamma=2+2/n$. Note that the possible choice of the potential is certainly restricted by the analysis of the limiting behavior for several potentials. The results are summarized in Table \[table1\].
We can obtain many numerical solutions for the several types of the potentials. We show the result of $n=2$ for the potentials $v^0_2,v^0_4,v^{4/3}_{8/3}$; of course these are not of the form of the analytical solution. Fig.\[energy2\_pots\] presents the energies and the component $E_4^{(1)}$ for these potentials. For $n=2$, the [*old*]{}-BS potentials give higher total energy than the new one. This indicates that the same class of potentials gives the similar energy and then, for $n=2$ the energy of the [*new*]{} type potential $v^{4/3}_{8/3}$ is closest to the integrable sector, which is also plotted in Fig. \[energy2\_pots\] for reference.
$v^\alpha_\gamma$ $n=1$ $n=2$ $n=3$
------------------------------------------ ------------ ------------ ------------
[*old*]{}-BS: $1-n_3$ $\times$ $\times$ $\times$
$(1-n_3)^2$ $\times$ $\bigcirc$ $\bigcirc$
$(1-n_3)^3$ $\bigcirc$ $\bigcirc$ $\bigcirc$
$(1-n_3)^4$ $\bigcirc$ $\bigcirc$ $\bigcirc$
[*new*]{}-BS:$(1+n_3)(1-n_3)$ $\times$ $\times$ $\times$
$(1+n_3)(1-n_3)^3$ $\bigcirc$ $\bigcirc$ $\bigcirc$
$(1+n_3)^\frac{4}{3}(1-n_3)^\frac{8}{3}$ $\bigcirc$ $\bigcirc$ $\bigcirc$
: The analysis of the limiting behavior of the solutions at both $y=0,1$ for several choice of the potential, and $\bigcirc$ ($\times$) which indicates that there exist (no) solutions.[]{data-label="table1"}
Potential physical applications of the solutions {#sec:application}
================================================
Since the model was proposed in the context of Wilsonian renormalization group argument of the $SU(2)$ Yang-Mills theory, we expect that the vortex solutions constructed in this paper should describe some features of strong coupling regimes, such as the dual superconductor picture [@Nambu:1974zg]. Apart from that, vortices appear in several areas of physics. The Nielsen-Olesen (NO) vortices in the Abelian Higgs model [@Nielsen:1973cs] were applied for type II superconductors (SC) and later they have extensively been studied in the context of cosmology, i.e., the cosmic string [@Hindmarsh:1994re] and the brane-world scenario [@Giovannini:2001hh]. The model has a close relationship with the standard electroweak theory, specially when one considers the case of a global $SU(2)$ and a local $U(1)$ breaking into a global $U(1)$, where the model reduces to an Abelian Higgs model with two charged scalar fields [@Achucarro:1999it]. It is interesting to note that the vortices of such model carry the so-called longitudinal electromagnetic currents [@Forgacs:2006pm; @Volkov:2006ug].
In we give the Noether current associated to a global $U(1)$, i.e., $u\to e^{i\alpha}u$, and so one can straightforwardly compute the longitudinal current in the integrable/nonintegrable sector. In Fig.\[longitudinalcurrent\], we plot the typical results of the transverse spatial structure of the polar component of the current in the case of the integrable sector. (Using and , one can easily see that the radial component of the current is always zero.) Note that for higher winding numbers as well for unit winding number the solutions exhibit the pipe-like structure, which was observed in the analysis of [@Chernodub:2010sg].
Our model enjoys a symmetry breaking of the type $O(3)_{\rm global} \to O(2)_{\rm global}$ which is similar to $SU(2)_{\rm global}\otimes U(1)_{\rm local}\to U(1)_{\rm global}$. A notable difference between the NO vortices and ours is that the gauge degrees of freedom are absent in our model. If one wishes to discuss the existence of the gauge field in type II SC, however, the gauging of the model according to [@Gladikowski:1995sc] should work. Confinement or squeezing of the magnetic field into type II SC should be realized in terms of the localization of the gauge field into our vortices.
{width="12cm"}\
Summary {#sec:summary}
=======
We have studied vortex solutions of the extended Skyrme-Faddeev model especially for the outside of the integrable constraint $\beta e^2=1$. In order to find the solutions, we introduced potentials of the extension of the Baby-Skyrmion type. We found several analytical solutions of the model. We also confirmed the existence of the solutions in terms of the numerical analysis. By using the standard SOR method, we obtained the axially symmetric solutions for charge up to $n=3$ with several form of the potential for various value of the model parameters.
In this work, we imposed the axial symmetry to the solution ansatz. However, solutions with lower symmetry, such as $\mathbb{Z}_2$-symmetry were found by a numerical simulation for the Baby-Skyrme model [@Hen:2007in]. It would be interesting to investigate whether such deformed solutions appear in the extended Skyrme-Faddeev model. Furthermore, full 3D simulations of the model will certainly clarify the detailed structure of the vortices. The analysis implementing these issues will be discussed in a forthcoming paper.
[**Acknowledgments**]{}
We are grateful to the anonymous referee for a careful reading of the manuscript and for valuable remarks on the physical applications. We would like to thank Wojtek Zakrzewski and Paweł Klimas for many useful discussions. NS would like to thank the kind hospitality at Instituto de Física de São Carlos, Universidade de São Paulo. He also acknowledges the financial support of FAPESP (Brazil). JJ would also like to thank the UK Engineering and Physical Sciences Research Council for support. LAF is partially supported by CNPq-Brazil.
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[^1]: e-mail: [[email protected]]{}
[^2]: e-mail: [[email protected]]{}
[^3]: e-mail: [[email protected]]{}
[^4]: e-mail: [[email protected]]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Toeplitz CAR flows are a class of $E_0$-semigroups including the first type III example constructed by R. T. Powers. We show that the Toeplitz CAR flows contain uncountably many mutually non cocycle conjugate $E_0$-semigroups of type III. We also generalize the type III criterion for Toeplitz CAR flows employed by Powers (and later refined by W. Arveson), and show that Toeplitz CAR flows are always either of type I or type III.'
address:
- |
Department of Mathematics\
Graduate School of Science\
Kyoto University\
Sakyo-ku, Kyoto 606-8502\
Japan
- |
Chennai Mathematical Institute\
Siruseri 603103, India.
author:
- Masaki Izumi
- 'R. SRINIVASAN'
title: Toeplitz CAR flows and type I factorizations
---
[^1]
Introduction {#intro}
============
The famous E. Wigner’s theorem establishes that any one parameter group of automorphisms $\{\alpha_t: t \in {\mathbb{R}}\}$ on $B(H)$, the algebra of all bounded operators on a separable Hilbert space $H$, is described by a strongly continuous one-parameter unitary group $\{U_t\}$, through the relation $$\alpha_t(X)={\operatorname{Ad}}(U_t)(X)=U_tXU_t^*,~~\forall ~~X \in B(H).$$ An analogous statement of Wigner’s theorem for an $E_0$-semigroup, a continuous semigroup of unit preserving endomorphisms of $B(H)$, would be that the semigroup is completely determined by the set of all intertwining semigroup of isometries. That is the $E_0$-semigroup $\{\alpha_t: t \in (0,\infty)\}$ is completely described, up to cocycle conjugacy, by the set of all $C_0$-semigroups of isometries $\{U_t\}$, satisfying $$\alpha_t(X)U_t = U_tX, ~~\forall ~~X \in B(H).$$ A subclass of $E_0$-semigroups, where this analogy is indeed true, are called as type I $E_0$-semigroups. But due to the existence of type II and type III $E_0$-semigroups in abundance, it is well known by now that such an analogy does not hold for $E_0$-semigroups in general.
In [@PO0], Powers raised the question whether such an intertwining semigroup of isometries always exists for any given $E_0$-semigroup. Later in 1987, he answered this question (see [@Po1]) in negative, by constructing an $E_0$-semigroup without any intertwining semigroup of isometries. This is the first example of what is called as a type III $E_0$-semigroup. For quite some time this was the only known example of a type III $E_0$-semigroup, even though it was conjectured that there are uncountably many type III $E_0$-semigroups, which are mutually non cocycle conjugate. In 2000, B. Tsirelson constructed a one-parameter family of nonisomorphic product systems of type III (see [@T1]). Using previous results of Arveson [@Ar], this leads to the existence of uncountably many $E_0$-semigroups of type III, which are mutually non cocycle conjugate. Since then there has been a flurry of activity along this direction (see [@pdct], [@I] and [@IS]).
In this paper, we turn our attention to the first example of a type III $E_0$-semigroup produced by Powers, which can be constructed on the type I factor obtained through the GNS construction of the CAR algebra corresponding to a (non-vacuum) quasi-free state. Although his purpose in [@Po1] is to construct a single type III example, his construction is rather general, and it could produce several $E_0$-semigroups, by varying the associated quasi-free states. However, it is not at all clear whether they contain more than one cocycle conjugacy classes of type III $E_0$-semigroups. As is emphasized in Arveson’s book [@Arv Chapter 13], the 2-point function of Powers’ quasi-free state is given by a Toeplitz operator whose symbol is a matrix-valued function with a very subtle property. Arveson clarified the role of the Toeplitz operator in Powers’ construction, and gave the most general form of the symbols for which the same construction works. We refer to the $E_0$-semigroups obtained in this way as the *Toeplitz CAR-flows*. Arveson also made a refinement of a sufficient condition obtained by Powers for the Toeplitz CAR flows to be of type III.
One of our main purposes in this paper is to show that there exist uncountably many cocycle conjugacy classes of type III Toeplitz CAR flows. More precisely, we explicitly give a one parameter family of symbols, including Powers’ one, that give rise to mutually non cocycle conjugate type III examples. We also generalize Powers and Arveson’s type III criterion mentioned above, and give a necessary and sufficient condition in full generality, which solves Arveson’s problem raised in [@Arv p.417]. In particular, our result says that Toeplitz CAR flows are always either of type I or of type III, which is a CAR version of the same result obtained in [@pdct] (see also [@I1] and [@IS]) for product systems arising from sum systems, or equivalently, generalized CCR flows.
As in our previous work [@IS], we employ the local von Neumann algebras of an $E_0$-semigroup as a classification invariant. In [@IS], we computed the type of the von Neumann algebras corresponding to bounded open subsets of $(0,\infty)$ for a class of generalized CCR flows. The key fact in our previous computation is that the von Neumann algebras in question always arise from quasi-free representations of the Weyl algebra. Since an analogous statement does not seem to be true in the case of Toeplitz CAR flows (even if the usual twisting operation in the duality for the CAR algebra is taken into account), we have to take an alternative approach. For this reason, we use the notion of a type I factorization, introduced by H. Araki and J. Woods [@AW], consisting of the local von Neumann algebras corresponding to a countable partition of a finite interval. For each fixed such partition, whether the associated type I factorization is a complete atomic Boolean algebra of type I factors or not is a cocycle conjugacy invariant of type III $E_0$-semigroups.
Part of this work was done when the first named author visited the University of Rome “Tor Vergata", and he would like to thank Roberto Longo and his colleagues for their hospitality. The second named author would like to thank the ‘CMI-TCS Academic Initiative’ for the travel support to visit University of Kyoto, during which a part of this work was done.
Preliminaries. {#pre}
==============
We use the following notation throughout the paper.
For a family of von Neumann algebras $\{{\mathcal{M}}_{\lambda}\}_{\lambda\in \Lambda}$ acting on the same Hilbert space $H$, we denote by $\bigvee_{\lambda\in \Lambda}{\mathcal{M}}_\lambda$ the von Neumann algebra generated by their union $\bigcup_{\lambda\in \Lambda}{\mathcal{M}}_\lambda$. We will always denote by $1$ either the identity element in a $C^*$-algebra or the identity operator on a Hilbert space. When we need to specify the $C^*$-algebra ${\mathfrak{A}}$ or the Hilbert space $H$, we use the symbols $1_{\mathfrak{A}}$ or $1_H$ respectively.
For a bounded positive operator $A$ on a Hilbert space $H$, we denote by ${\operatorname{tr}}(A)$ the usual trace of $A$, which could be infinite. For $X\in B(H)$, we denote its Hilbert-Schmidt norm by $\|X\|_{\mathrm{H.S.}}={\operatorname{tr}}(X^*X)^{1/2}$.
For a tempered distribution $f$ on ${\mathbb{R}}$, we denote by $\hat{f}$ the Fourier transform of $f$ with normalization $$\hat{f}(p)=\int_{\mathbb{R}}f(x)e^{-ipx}dx,\quad f\in L^1({\mathbb{R}}).$$ For an open set $O\subset {\mathbb{R}}$, we denote by ${\mathcal{D}}(O)$ the set of smooth functions with compact support. For a measurable set $E\subset {\mathbb{R}}$, we denote by $|E|$ and $\chi_E$ its Lebesgue measure and its characteristic function respectively.
$E_0$-semigroups and product systems
------------------------------------
We briefly recall basics of $E_0$-semigroups and product systems. The reader is referred to Arveson’s monograph [@Arv] for details.
Let $H$ be a separable Hilbert space. A family of unital $*$-endomorphisms $\alpha=\{\alpha_t\}_{t \geq 0}$ of $B(H)$ is an $E_0$-semigroup if
- The semigroup relation $\alpha_s \circ \alpha_t = \alpha_{s+t}$ holds for $\forall~s,t \in (0,\infty)$ and $\alpha_0={\operatorname{id}}$.
- The map $t \mapsto \langle \alpha_t(X)\xi,\eta\rangle$ is continuous for every fixed $X \in B(H),~~\xi,\eta \in H$.
For an $E_0$-semigroup $\alpha=\{\alpha_t\}_{t \geq 0}$ and positive $t$, we set $${\mathcal{E}}_\alpha(t)=\{T \in B(H); \alpha_t(X) T = TX, ~ \forall ~X \in
B(H)\},$$ which is a Hilbert space with the inner product $\langle T, S \rangle 1_{H}= S^*T$. The system of Hilbert spaces ${\mathcal{E}}_\alpha=\{{\mathcal{E}}_\alpha(t)\}_{t>0}$ satisfies the following axioms of a product system:
\[productsystem\] A product system of Hilbert spaces is a one parameter family of separable complex Hilbert spaces $E=\{E(t)\}_{t>0}$, together with unitary operators $$U_{s,t} : E(s) \otimes E(t) \rightarrow E(s+t)~ \mbox{for}~ s, t \in (0,\infty),$$ satisfying the following two axioms of associativity and measurability.
- (Associativity) For any $s_1, s_2, s_3 \in (0,\infty)$, $$U_{s_1, s_2 + s_3}( 1_{E(s_1)} \otimes U_{s_2 ,s_3})=
U_{s_1+ s_2 , s_3}( U_{s_1 ,s_2} \otimes 1_{E(s_3)}).$$
- (Measurability) There exists a countable set $E^0$ of sections $$(0,\infty)\ni t \rightarrow h_t \in E(t)$$ such that $ t \mapsto
\langle h_t, h_t^\prime\rangle$ is measurable for any two $h, h^\prime \in E^0$, and the set $\{h_t; h \in E^0\}$ is total in $E(t)$, for each $ t \in (0,\infty)$. Further it is also assumed that the map $(s,t) \mapsto \langle U_{s,t}(h_s \otimes h_t), h^\prime_{s+t} \rangle$ is measurable for any two $h , h^\prime \in E^0$.
Two product systems $(\{E(t)\}, \{U_{s,t}\})$ and $(\{E^\prime(t)\}, \{U^\prime_{s,t}\})$ are said to be isomorphic if there exists a unitary operator $V_t:E(t) \rightarrow E(t)^\prime$, for each $t \in (0,\infty)$ satisfying $$V_{s+t}U_{s,t}= U_{s,t}^\prime (V_s \otimes V_t).$$
Arveson showed that every product system is isomorphic to a product system arising from an $E_0$-semigroup, and that two $E_0$-semigroups $\alpha$ and $\beta$ are cocycle conjugate if and only if the corresponding product systems ${\mathcal{E}}_\alpha$ and ${\mathcal{E}}_\beta$ are isomorphic.
For a fixed positive number $a$ and for $0\leq s\leq t\leq a$, we define the local von Neumann algebra ${\mathcal{A}}^E_a(s,t)\subset B(E(a))$ for the interval $(s,t)$ by $${\mathcal{A}}_a^E(s,t)=U_{s,t-s,a-t}({\mathbb{C}}1_{E(s)}\otimes B(E(t-s))\otimes {\mathbb{C}}1_{E(a-t)}){U_{s,t-s,a-t}}^*,$$ where $U_{s,t-s,a-t}=U_{t,a-t}(U_{s,t-s}\otimes 1_{E_{a-t}})=U_{s,a-s}(1_{E_s}\otimes U_{t-s,a-t})$. For any open subset $O\subset [0,a]$, we set ${\mathcal{A}}^E_a(O)=\bigvee_{I\subset O}{\mathcal{A}}^E(I)$, where $I$ runs over all intervals contained in $O$. When $a=1$, we simply write ${\mathcal{A}}^E(s,t)$ for ${\mathcal{A}}^E_a(s,t)$. When $E={\mathcal{E}}_\alpha$, we often identify $B(E(a))$ with $B(H)\cap\alpha_a(B(H))'$. When we need to distinguished them, we denote by $\sigma$ the isomorphism from $B(H)\cap\alpha_a(B(H))'$ onto ${\mathcal{A}}^E_a(0,a)$ given by the left multiplication. By this identification, the inclusion ${\mathcal{A}}^E_a(s,t)\subset B(E(a))$ is identified with $$\alpha_s(B(H)\cap \alpha_{t-s}(B(H))')\subset B(H)\cap \alpha_a(B(H))'.$$
In what follows, we often omit $U_{s,t}$, and simply write $xy$ instead of $U_{s,t}(x\otimes y)$ if there is no possibility of confusion.
\[unit\] A unit for a product system $E$ is a non-zero section $$u=\{u_t\in E_t;t > 0 \},$$ such that the map $t \mapsto \langle u_t, h_t\rangle$ is measurable for any $h \in E^0$ and $$u_s u_t= u_{s+t}, \quad \forall s,t \in (0,\infty).$$
In order to avoid possible confusion, we refer to the condition $\|x\|=1$ for a vector $x\in E(t)$ as “normalized" instead of “unit" throughout the paper. An intertwining $C_0$-semigroup of isometries of an $E_0$-semigroup $\alpha$ is naturally identified with a normalized unit for ${\mathcal{E}}_\alpha$.
A product system ($E_0$-semigroup) is said to be of type I, if units exist for the product system and they generate the product system, i.e. for any fixed $t \in (0,\infty)$, the set $$\{u^1_{t_1}u^2_{t_2}
\cdots u^n_{t_n}; \sum_{i=1}^n t_i = t, u^i \in {\mathcal{U}}_E\},$$ is a total set in $E_t$, where ${\mathcal{U}}_E$ is the set of all units. It is of type II if units exist but they do not generate the product system. An $E_0$-semigroup is said to be spatial if it is either of type I or type II. We say a product system to be of type III, or unitless, if there does not exist any unit.
Type I product systems are further classified into type I$_n$, $n=1,2,\cdots,\infty$, according to their indices $n$. There exists only one isomorphism class of type $I_n$ product systems.
We recall V. Liebscher’s useful criterion [@L Corollary 7.7] for isomorphic product systems in terms of the local von Neumann algebras.
\[isomorphic\] Let $E$ and $F$ be product systems. If there is an isomorphism $\rho$ from $B(E(1))$ onto $B(F(1))$ such that $\rho({\mathcal{A}}^E(0,t))={\mathcal{A}}^F(0,t)$ for $t$ in a dense subset of $(0,1)$, then $E$ and $F$ are isomorphic.
Type I factorizations {#invariant for type III}
---------------------
In this subsection, we introduce a new classification invariant for type III product systems using the notion of type I factorizations introduced by Araki and Woods [@AW]. Throughout this subsection, every index set is assumed to be countable, and every Hilbert space is assumed to be separable.
Let $H$ be a Hilbert space. We say that a family of type I subfactors $\{{\mathcal{M}}_{\lambda}\}_{\lambda\in \Lambda}$ of $B(H)$ is a *type I factorization* of $B(H)$ if
- ${\mathcal{M}}_\lambda\subset {\mathcal{M}}_\mu'$ for any $\lambda,\mu\in \Lambda$ with $\lambda\neq \mu$,
- $B(H)=\bigvee_{\lambda\in \Lambda}{\mathcal{M}}_\lambda$.
We say that a type I factorization $\{{\mathcal{M}}_{\lambda}\}_{\lambda\in \Lambda}$ is a *complete atomic Boolean algebra of type I factors* (abbreviated as *CABATIF*) if for any subset $\Gamma\subset \Lambda$, the von Neumann algebra $\bigvee_{\lambda\in \Gamma}{\mathcal{M}}_{\lambda}$ is a type I factor.
Two type I factorizations $\{{\mathcal{M}}_{\lambda}\}_{\lambda\in \Lambda}$ of $B(H)$ and $\{{\mathcal{N}}_{\mu}\}_{\mu\in \Lambda'}$ of $B(H')$ are said to be unitarily equivalent if there exist a unitary $U$ from $H$ onto $H'$ and a bijection $\sigma:\Lambda\rightarrow \Lambda'$ such that $U{\mathcal{M}}_\lambda U^*={\mathcal{N}}_{\sigma(\lambda)}$.
\[invariant for E\] Let $E$ be a product system, and let $\{a_n\}_{n=0}^\infty$ be a strictly increasing sequence of non-negative numbers starting from $0$ and converging to $a<\infty$. Then $\{{\mathcal{A}}_a^E(a_n,a_{n+1})\}_{n=0}^\infty$ is a type I factorization of $B(E(a))$ because $$B(E(a))=\bigvee_{0<t<a}{\mathcal{A}}^E_a(0,t)$$ holds (see [@Arv Proposition 4.2.1]). For a fixed sequence as above, the unitary equivalence class of the type I factorization $\{{\mathcal{A}}_a^E(a_n,a_{n+1})\}_{n=0}^\infty$ is an isomorphism invariant of the product system $E$. In particular, whether it is a CABATIF or not will be used to distinguish concrete type III examples in Section \[examples\]. As we will see now, this invariant may be useful only in the type III case.
When $\{{\mathcal{M}}_\lambda\}_{\lambda\in \Lambda}$ is a type I factorization of $B(H)$, we say that a non-zero vector $\xi$ is *factorizable* if for any $\lambda$, there exists a minimal projection $p_\lambda$ of ${\mathcal{M}}_\lambda$ such that $p_\lambda\xi=\xi$.
Araki and Woods characterized a CABATIF as a type I factorization with a factorizable vector. Since we need a more precise statement, we briefly recall basics of the incomplete tensor product space (abbreviated as ITPS) now.
Let $\{(H_\lambda,\xi_\lambda)\}_{\lambda\in \Lambda}$ be a family of Hilbert spaces $H_\lambda$ with normalized vectors $\xi_\lambda\in H_\lambda$. Let ${\mathcal{F}}(\Lambda)$ be the set of all finite subsets of $\Lambda$, which is a directed set with respect to the inclusion relation. For $F_1,F_2\in {\mathcal{F}}(\Lambda)$ with $F_1\subset F_2$, we introduce an isometric embedding $V_{F_1,F_2}$ from $\bigotimes_{\lambda\in F_1}H_\lambda$ into $\bigotimes_{\lambda\in F_2}H_\lambda$ by $$V_{F_1,F_2}\eta= \eta\otimes (\bigotimes_{\mu\in F_2\setminus F_1}\xi_{\mu}),\quad \eta\in \bigotimes_{\lambda\in F_1}H_\lambda.$$ Then the ITPS $$H=\bigotimes_{\lambda\in \Lambda}{}^{(\otimes \xi_{\lambda})} H_\lambda$$ of the Hilbert spaces $\{H_\lambda\}_{\lambda\in \Lambda}$, with respect to the reference vectors $\{ \xi_\lambda \}_{\lambda \in \Lambda}$, is the completion of the direct limit of the directed family $\{\bigotimes _{\lambda\in F}H_\lambda\}_{F\in {\mathcal{F}}(\Lambda)}$. When there is no possibility of confusion, we omit the superscript $(\otimes \xi_{\lambda})$ for simplicity. We denote by $V_{F,\infty}$ the canonical embedding of $\bigotimes_{\lambda\in F}H_\lambda$ into $H$.
The product vector $\xi=\bigotimes_{\lambda\in \Lambda} \xi_\lambda\in H$ is understood as $V_{F,\infty}\bigotimes_{\lambda\in F}\xi_\lambda$, which does not depend on $F\in {\mathcal{F}}(\Lambda)$. More generally, if $\{\eta_\lambda\}_{\lambda\in\Lambda}$, $\eta_\lambda\in H_\lambda$, is a family of vectors such that $0<\prod_{\lambda\in \Lambda}\|\eta_\lambda\|<\infty$, and $$\sum_{\lambda\in \Lambda}|{\langle{\eta_\lambda,\xi_\lambda}\rangle}-1|<\infty,$$ then the net $\{V_{F,\infty}\bigotimes_{\lambda\in F} \eta_\lambda\}_{F\in {\mathcal{F}}(\Lambda)}$ converges in $H$. The product vector $\eta=\bigotimes_{\lambda\in \Lambda} \eta_\lambda$ is defined as its limit. Two product vectors $\eta=\bigotimes_{\lambda\in \Lambda}\eta_\lambda$ and $\zeta=\bigotimes_{\lambda\in \Lambda}\zeta_\lambda$ satisfy $${\langle{\eta,\zeta}\rangle}=\prod_{\lambda\in \Lambda}{\langle{\eta_\lambda,\zeta_\lambda}\rangle}.$$
For a subset $\Lambda_1\subset \Lambda$, we often identify $\bigotimes_{\lambda\in \Lambda}H_\lambda$ with $$(\bigotimes_{\lambda\in \Lambda_1}H_\lambda)\otimes(\bigotimes_{\mu\in \Lambda\setminus\Lambda_1}H_\mu)$$ in a canonical way. When $\Lambda_1$ consists of only one point $\lambda$, we set $${\mathcal{M}}_\lambda:=B(H_\lambda)\otimes {\mathbb{C}}1_{\bigotimes_{\mu\neq \lambda}H_\mu}\subset B(H).$$ Then $\{{\mathcal{M}}_\lambda\}_{\lambda\in \Lambda}$ is a CABATIF. Any type I factorization unitarily equivalent to this $\{{\mathcal{M}}_\lambda\}_{\lambda\in \Lambda}$ is said to be a *tensor product factorization*. Note that there is only one tensor product factorization, up to unitary equivalence, with each constituent type I factor infinite dimensional.
One can find the following theorem in [@AW Lemma 4.3, Theorem 4.1].
\[ArakiWoods\] A type I factorization is a CABATIF if and only if it has a factorizable vector. When this condition holds, then it is a tensor product factorization.
When a product system $E$ has a unit, then it gives a factorizable vector of the type I factorization $\{{\mathcal{A}}_a^E(a_n,a_{n+1})\}_{n=0}^\infty$ in Example \[invariant for E\], which is necessarily a CABATIF thanks to Theorem \[ArakiWoods\].
We use the following lemma in Section \[factorization\].
\[fixed product vector\] Let $H=\bigotimes_{\lambda\in \Lambda}H_\lambda$ be the ITPS of Hilbert spaces $\{H_\lambda\}_{\lambda\in \Lambda}$ with respect to reference vectors $\{\xi_\lambda\}_{\lambda\in \Lambda}$, and let $$\rho_\lambda:B(H_\lambda)\ni X\mapsto X\otimes 1_{\bigotimes_{\mu\neq \lambda}H_\mu}\in {\mathcal{M}}_\lambda$$ be the canonical isomorphism. Let $R\in B(H)$ be a self-adjoint unitary such that $R{\mathcal{M}}_\lambda R^*={\mathcal{M}}_\lambda$ for all $\lambda\in \Lambda$. Then there exist self-adjoint unitaries $R_\lambda\in B(H_\lambda)$ and a product vector $\eta=\bigotimes_{\lambda\in \Lambda}\eta_\lambda$ such that
- for $\forall \lambda\in \Lambda$ and $\forall X\in {\mathcal{M}}_\lambda$, $$\rho_\lambda(R_\lambda) X \rho_\lambda(R_\lambda^*)=RXR^*,$$
- $R_\lambda \eta_\lambda=\eta_\lambda$ for $\forall \lambda\in \Lambda$,
- either $R\eta=\eta$ or $R\eta=-\eta$.
Since the restriction of ${\operatorname{Ad}}R$ to ${\mathcal{M}}_\lambda$ is an automorphism of period two and ${\mathcal{M}}_\lambda$ is a type I factor, there exist self-adjoint unitaries $R_\lambda\in B(H_\lambda)$ such that $$\rho_\lambda(R_\lambda)X\rho_{\lambda}(R_\lambda^*)=RXR^*,\quad \forall X\in {\mathcal{M}}_\lambda.$$ By replacing $R_\lambda$ with $-R_\lambda$ if necessary, we may assume ${\langle{R_\lambda \xi_\lambda,\xi_\lambda}\rangle}\geq 0$ for $\forall \lambda\in \Lambda$. Let $\xi=\bigotimes_{\lambda\in \Lambda}\xi_\lambda$, and let $p_\lambda\in {\mathcal{M}}_\lambda$ be the minimal projection satisfying $p_\lambda \xi=\xi$. Then $q_\lambda=Rp_\lambda R^*$ is a minimal projection of ${\mathcal{M}}_\lambda$ satisfying $q_\lambda R\xi=R\xi$, and so $R\xi$ is a factorizable vector. The proof of [@AW Lemma 3.2] shows that there exist a complex number $c$ of modulus 1 and normalized vectors $\zeta_\lambda\in H_\lambda$ such that $R\xi=c\bigotimes_{\lambda\in \Lambda} \zeta_\lambda$ and ${\langle{\zeta_\lambda,\xi_\lambda}\rangle}\geq 0$ for $\forall \lambda\in \Lambda$. Since $q_\lambda=\rho_\lambda(R_\lambda) p_\lambda \rho_\lambda(R_\lambda^*)$, the normalized vector $\zeta_\lambda$ is a scalar multiple of $R_\lambda\xi_\lambda$. Let $$\Lambda_0=\{\lambda\in\Lambda;\; {\langle{\zeta_\lambda,\xi_\lambda}\rangle}=0 \},$$ and $\Lambda_1=\Lambda\setminus \Lambda_0$. Then $\Lambda_0$ is a finite set. The two conditions ${\langle{R_\lambda \xi_\lambda,\xi_\lambda}\rangle}\geq 0$ and ${\langle{\zeta_\lambda,\xi_\lambda}\rangle}\geq 0$ imply that for any $\lambda\in \Lambda_1$, we actually have $R_\lambda\xi_\lambda=\zeta_\lambda$. Let $Q_\lambda$ be the spectral projection of $R_\lambda$ corresponding to eigenvalue 1. Then since $R_\lambda=2Q_\lambda-1$, we have ${\langle{\zeta_\lambda,\xi_\lambda}\rangle}=2{\langle{Q_\lambda\xi_\lambda,\xi_\lambda}\rangle}-1$.
For $\lambda\in \Lambda_0$, by replacing $R_\lambda$ with $-R_\lambda$ if necessary, we can find a normalized vector $\eta_\lambda\in H_\lambda$ satisfying $R_\lambda\eta_\lambda=\eta_\lambda$. For $\lambda\in \Lambda_1$, we set $\eta_\lambda=Q_\lambda\xi_\lambda$. Then $$\|\eta_\lambda\|^2={\langle{\eta_\lambda,\xi_\lambda}\rangle}=\frac{1+{\langle{\zeta_\lambda,\xi_\lambda}\rangle}}{2}.$$ This shows that the product vector $\bigotimes_{\lambda\in \Lambda}\eta_\lambda\in H$ exists and $R_\lambda \eta_\lambda=\eta_\lambda$.
It only remains to show (iii). Let $e_\lambda\in B(H_\lambda)$ be the projection onto ${\mathbb{C}}\eta_\lambda$. Then the proof of [@AW Lemma 3.2] shows that the net $\{\prod_{\lambda\in F}\rho_{\lambda}(e_\lambda)\}_{F\in {\mathcal{F}}(\Lambda)}$ strongly converges to the projection $e\in B(H)$ onto ${\mathbb{C}}\eta$. Since $ReR^*=e$ and $R$ is a self-adjoint unitary, we get either $R\eta=\eta$ or $R\eta=-\eta$.
Quasi-free representations of the CAR algebra
---------------------------------------------
We recall some of the well-known results about quasi-free representations of the algebra of canonical anticommutation relations (called as CAR algebra).
Let $K$ be a complex Hilbert space. We denote by ${\mathfrak{A}}(K)$ the CAR algebra over $K$, which is the universal $C^*$-algebra generated by $\{a(x); x \in K\}$, determined by the linear map $x \mapsto a(x)$ satisfying the CAR relations: $$\begin{aligned}
a(x)a(y) +a(y)a(x)& = & 0, \\
a(x)a(y)^* +a(y)^*a(x) & = & \langle x,y\rangle1, \end{aligned}$$ for all $x,y \in K$. Since ${\mathfrak{A}}(K)$ is known to be simple, any set of operators satisfying the CAR relations generates a $C^*$-algebra canonically isomorphic to ${\mathfrak{A}}(K)$.
For any state $\varphi$ of ${\mathfrak{A}}(K)$, there exists a unique positive contraction $A\in B(K)$ satisfying $\varphi(a(f)a(g)^*)={\langle{Af,g}\rangle}$ for $\forall f,g\in K$. We call $A$ the covariance operator (or 2-point function) of $\varphi$.
A *quasi-free state* $\omega_A$ on ${\mathfrak{A}}(K)$, associated with a positive contraction $A \in B(K)$, is the state whose $(n,m)$-point functions are determined by its 2-point function as $$\omega_A(a(x_n) \cdots a(x_1)a(y_1)^* \cdots a(y_m)^* ) = \delta_{n,m} \det (\langle Ax_i, y_j\rangle) ,$$ where $\det(\cdot)$ denotes the determinant of a matrix. Given a positive contraction, it is a fact that such a state always exists and is uniquely determined by the above relation. This is usually called as the gauge invariant quasi-free state (or generalized free state). Since we will be dealing only with gauge invariant quasi-free states, we just call them as quasi-free states.
We denote by $(H_A, \pi_A, \Omega_A)$ the GNS triple associated with a quasi-free state $\omega_A$ on ${\mathfrak{A}}(K)$, and set ${\mathcal{M}}_A:=\pi_A({\mathfrak{A}}(K))''$. We call $\pi_A$ the *quasi-free representation* associated with $A$.
Recall that two representations $\pi_1$ and $\pi_2$ of a $C^*$-algebra ${\mathfrak{A}}$ are said to be quasi-equivalent if there is a $*$-isomorphism of von Neumann algebras $$\theta:\pi_1({\mathfrak{A}})^{\prime\prime} \longmapsto \pi_2({\mathfrak{A}})^{\prime \prime}$$ satisfying $\theta(\pi_1(X))=\pi_2(X)$ for all $X \in {\mathfrak{A}}$. Two states are said to be quasi-equivalent if their GNS representations are quasi-equivalent.
We now summarize standard results on quasi-free states. For the proofs, the reader is referred to [@Arv Chapter13], [@Po1 Section II], and references therein.
\[qfstate\] Let $K$ be a Hilbert space, let $P\in B(K)$ be a projection, and let $A,B\in B(K)$ be positive contractions.
- Every quasi-free state $\omega_A$ of the CAR algebra ${\mathfrak{A}}(K)$ is a factor state, that is, the von Neumann algebra ${\mathcal{M}}_A$ is a factor.
- The restriction of the GNS representation $\pi_A$ to ${\mathfrak{A}}(PK)$ is quasi-equivalent to the GNS representation $\pi_{PAP}$ of ${\mathfrak{A}}(PK)$, where $PAP$ is regarded as a positive contraction of $PK$.
- The quasi-free state $\omega_A$ is of type I if and only if ${\operatorname{tr}}(A -A^2) < \infty.$
- The two quasi-free states $\omega_A$ and $\omega_B$ are quasi-equivalent if and only if both operators $A^{1/2} - B^{1/2}$ and $(1-A)^{1/2} -(1-B)^{1/2}$ are Hilbert-Schmidt.
- The two quasi-free states $\omega_A$ and $\omega_P$ are quasi-equivalent if and only if $${\operatorname{tr}}\big(P(1-A)P+(1-P)A(1-P)\big)<\infty.$$
We frequently use the following criterion, which is more or less (v) above.
\[typeI\] Let $A,B\in B(K)$ be positive contractions. We assume that $\omega_B$ is a type I state. Then the two quasi-free states $\omega_A$ and $\omega_B$ are quasi-equivalent if and only if $${\operatorname{tr}}\big(B(1-A)B+(1-B)A(1-B)\big)<\infty.$$
Let $P$ be the spectral projection of $B$ corresponding to the interval $[1/2,1]$. Since $\omega_B$ is a type I state, Theorem \[qfstate\],(iii),(iv) imply that $P-B$ is a trace class operator, and $\omega_P$ and $\omega_B$ are quasi-equivalent. Thus $\omega_A$ and $\omega_B$ are quasi-equivalent if and only if $\omega_A$ and $\omega_P$ are quasi-equivalent, which is further equivalent to $${\operatorname{tr}}\big(P(1-A)P+(1-P)A(1-P)\big)<\infty,$$ thanks to Theorem \[qfstate\],(v). Now the statement follows from the fact that $P-B$ is a trace class operator.
Let $\gamma$ be the period two automorphism of ${\mathfrak{A}}(K)$ determined by $\gamma(a(f))=-a(f)$ for $\forall f\in K$. Since any quasi-free state $\omega_A$ is invariant under $\gamma$, the automorphism $\gamma$ extends to a period two automorphism $\overline{\gamma}$ of the von Neumann algebra ${\mathcal{M}}_A$. For a ${\mathbb{Z}}/2{\mathbb{Z}}$-grading of ${\mathfrak{A}}(K)$ (respectively ${\mathcal{M}}_A$), we always refer to the one coming from $\gamma$ (respectively $\overline{\gamma}$). When there is no possibility of confusion, we abuse the notation and use the same symbol $\gamma$ for $\overline{\gamma}$.
Let $\omega_A$ be a type I state. Then since every automorphism of a type I factor is inner, there exists a self-adjoint unitary $R^A\in \pi_A({\mathfrak{A}})''$ satisfying ${\operatorname{Ad}}R^A(X)=\gamma(X)$ for all $X\in {\mathcal{M}}_A$. The operator $R^A$ is uniquely determined up to a multiple of $-1$. In the same way, for every closed subspace $L\subset K$ such that the restriction of $\pi_A$ to ${\mathfrak{A}}(L)$ is of type I, there exists a self-adjoint unitary $R^A_L\in \pi_A({\mathfrak{A}}(L))''$ satisfying ${\operatorname{Ad}}R^A_L(X)=\gamma(X)$ for all $ X\in \pi_A({\mathfrak{A}}(L))''$. For each $L$, we fix such $R^A_L$, which itself is an even operator with respect to $\gamma$. When $L_1$ and $L_2$ are mutually orthogonal closed subspaces of $K$ satisfying the above condition, then we have $$R^A_{L_1\oplus L_2}=\epsilon_{L_1,L_2}R^A_{L_1}R^A_{L_2}=\epsilon_{L_1,L_2}R^A_{L_2}R^A_{L_1},$$ where $\epsilon_{L_1,L_2}\in \{1,-1\}$.
When $\omega_A$ is of type I, the family of operators $\{i\pi_A(a(f))R^A;\;f\in K \}$ also satisfies the CAR relation. We denote by $\pi^t_A$ the representation of ${\mathfrak{A}}(K)$ determined by $\pi^t_A(a(f))=i\pi_A(a(f))R^A$ for all $f\in K$, and call it the twisted representation associated with $\omega_A$. Note that the two representations $\pi_A$ and $\pi_A^t$ coincides on the even part of ${\mathfrak{A}}(K)$.
\[commutant\] Let $\omega_A$ be a type I quasi-free state of ${\mathfrak{A}}(K)$.
- For any subspace $L\subset K$, $${\mathcal{M}}_A\cap \pi_A({\mathfrak{A}}(L))'=\pi_A^t({\mathfrak{A}}(L^\perp))''.$$
- Let $U=\frac{1}{\sqrt{2}}(1-iR^A)\in {\mathcal{M}}_A$. Then $$U\pi_A(X)U^*=\pi_A^t(X)$$ holds for all $X\in {\mathfrak{A}}(K)$.
\(i) Let $Q$ be the spectral projection of $A$ corresponding to the interval $[1/2,1]$. Then $\pi_A$ and $\pi_Q$ are quasi-equivalent, and we may assume that $A$ is a projection for the proof by replacing $A$ with $Q$ if necessary. Now the statement follows from the twisted duality theorem [@F Theorem 2.4].
\(ii) This follows from a direct computation (or [@F Proposition 2.3]).
As in [@Po1], we also need to use a few facts about general factor states of ${\mathfrak{A}}(K)$.
\[covariance operator\] Let $A$ be the covariance operator of a state $\varphi$ of ${\mathfrak{A}}(K)$. Then,
- If $A$ is a projection, then $\varphi$ is the pure state $\omega_A$.
- If $\varphi$ is quasi-equivalent to a quasi-free state $\omega_B$, then $A-B$ is compact.
\(i) See, for example, [@Ara Lemma 4.3].
\(ii) The statement follows from [@Arv Theorem 13.1.3].
Toeplitz CAR flows
------------------
Let $V$ be an isometry of a Hilbert space $K$. Then we have an endomorphism $\rho$ of ${\mathfrak{A}}(K)$ determined by $\rho(a(f))=a(Vf)$ for all $ f\in K$. For a positive contraction $A$, the composition $\pi_A\circ \rho$ gives a representation of ${\mathfrak{A}}(K)$, which is quasi-equivalent to $\pi_{V^*AV}$ thanks to Theorem \[qfstate\],(ii). Thus if both $A^{1/2}-(V^*AV)^{1/2}$ and $(1-A)^{1/2}-(1-V^*AV)^{1/2}$ are Hilbert-Schmidt operators, then $\rho$ extends to an endomorphism of the von Neumann algebra ${\mathcal{M}}_A$. In particular, if $A$ satisfies ${\operatorname{tr}}(A-A^2)<\infty$ and $\{V_t\}_{t\geq 0}$ is a strongly continuous semigroup of isometries on $K$ satisfying the above condition for $V_t$ in place of $V$, then we get an $E_0$-semigroup.
In what follows, we assume $K=L^2((0,\infty),{\mathbb{C}}^N)$, and that $\{S_t\}_{t\geq 0}$ is the shift semigroup $$S_tf(x)=\left\{
\begin{array}{ll}
0 , &\quad 0<x\leq t, \\
f(x-t) , &\quad t<x.
\end{array}
\right.$$ In his attempt to clarify Powers’ construction [@Po1] of the first example of a type III $E_0$-semigroup, Arveson [@Arv Section 13.3] determined the most general form of a positive contraction $A\in B(K)$ satisfying ${\operatorname{tr}}(A-A^2)<\infty$ and $S_t^*AS_t=A$ for all $t$, which we state now.
We regard $K$ as a closed subspace of ${\tilde{K}}:=L^2({\mathbb{R}},{\mathbb{C}}^N)$, and we denote by $P_+$ the projection from ${\tilde{K}}$ onto $K$. We often identify $B(K)$ with $P_{+}B({\tilde{K}})P_+$.
We denote by $M_N({\mathbb{C}})$ the $N$ by $N$ matrix algebra. For $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$, we define the corresponding Fourier multiplier $C_\Phi\in B({\tilde{K}})$ by $$\hat{(C_\Phi f)}(p)=\Phi(p)\hat{f}(p).$$ Then the Toeplitz operator $T_\Phi\in B(K)$ and the Hankel operator $H_\Phi\in B(K,K^\perp)$ with the symbol $\Phi$ are defined by $$T_\Phi f=P_+C_\Phi f,\quad f\in K,$$ $$H_\Phi f=(1_{{\tilde{K}}}-P_+)C_\Phi f,\quad f\in K.$$
\[symbol\] Let $K=L^2((0,\infty),{\mathbb{C}}^N)$. A positive contraction $A\in B(K)$ satisfies ${\operatorname{tr}}(A-A^2)<\infty$ and $S_t^*AS_t=A$ if and only if there exists a projection $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ satisfying the following two conditions:
- $A=T_\Phi$,
- the Hankel operator $H_\Phi$ is Hilbert-Schmidt.
We call the symbol $\Phi$ satisfying the condition of Theorem \[symbol\] as *admissible*.
Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be a projection. Arveson briefly mentioned in [@Arv p.401], without giving a proof, that the condition (ii) of Theorem \[symbol\] holds if and only if the Fourier transform $\hat{\Phi}(x)$ (in distribution sense) restricted to ${\mathbb{R}}\setminus \{0\}$ is locally square integrable and $$\sup_{\delta>0}\int_{|x|>\delta}|x|{\operatorname{tr}}(|\hat{\Phi}(x)|^2)dx<\infty.$$ He also observed that any admissible symbol is necessarily quasi-continuous, though he used only the fact that $H_\Phi$ is a compact operator. Now, first we figure out the most suitable function space for the admissible symbols without using the Fourier transform, and then we give a proof to the above characterization in terms of the Fourier transform. We will see similarity between admissible symbols and logarithm of spectral density functions of off-white noises discussed in [@T2].
We denote by ${\mathbb{T}}$ the unit circle in ${\mathbb{C}}$. Let $U$ be the unitary from $L^2({\mathbb{R}})$ onto $L^2({\mathbb{T}},\frac{dt}{2\pi})$ induced by the change of variables $$e^{it}=-\frac{p+i}{p-i},$$ (since the Fourier transform $\hat{f}(p)$ of $f\in K$ has analytic continuation to the *lower* half-plane, we need a conformal transformation between the unit disk and the lower half-plane). Let $F$ be the unitaries associated with the Fourier transform. Then the Hankel operator $H_\Phi$ is transformed to the Hankel operator $H_\phi$ for ${\mathbb{T}}$ by $UF$, where $\Phi$ and $\phi$ are related by $\phi(e^{it})=\Phi(p)$ (see for example, [@T2 Section 3]). Let $\phi_{ij}(p)$ be the matrix element of $\phi(p)$. Since $\phi(e^{it})$ is a projection, the Hankel operator $H_\phi$ is Hilbert-Schmidt if and only if $H_{\phi_{ij}}$ and $H_{\overline{\phi_{ij}}}$ are Hilbert-Schmidt for all $i\leq j$.
It is easy to see that the Hankel operators $H_h$ and $H_{\overline{h}}$ for $h\in L^\infty({\mathbb{T}})$ are Hilbert-Schmidt if and only if $h$ is in the Sobolev space $W^{1/2}_2({\mathbb{T}})$, that is $$\sum_{n\in {\mathbb{Z}}}|n| |\hat{h}(n)|^2<\infty,$$ where $\hat{h}(n)$ is the Fourier coefficient $$\hat{h}(n)=\frac{1}{2\pi}\int_0^{2\pi}h(e^{it})e^{-int}dt.$$ This is further equivalent to the condition that $h$ belongs to the Besov space $B_{2,2}^{1/2}({\mathbb{T}})$ because $$\begin{aligned}
\int_0^{2\pi}\int_0^{2\pi}\frac{|h(e^{is})-h(e^{it})|^2}{|e^{is}-e^{it}|^2}dsdt
&=&\int_0^{2\pi}\int_0^{2\pi}\frac{|h(e^{i(s+t)})-h(e^{it})|^2}{|e^{is}-1|^2}dtds\\
&=&2\pi\int_0^{2\pi}\sum_{n\in {\mathbb{Z}}}\frac{|(e^{ins}-1)\hat{h}(n)|^2}{|e^{is}-1|^2}ds\\
&=&4\pi^2\sum_{n\in {\mathbb{Z}}}|n| |\hat{h}(n)|^2.\end{aligned}$$ As was done in [@T2 Section 3], we can translate this condition back into that for functions on ${\mathbb{R}}$. Now we see that the Hankel operator $H_\Phi$ with a projection $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ is Hilbert-Schmidt if and only if $$\int_{{\mathbb{R}}^2}\frac{{\operatorname{tr}}(|\Phi(p)-\Phi(q)|^2)}{|p-q|^2}dpdq<\infty.$$
Although the following lemma may be found in the literature of Besov spaces, for the reader’s convenience, we give a proof to the first part. (i) and (ii) are essentially due to Tsirelson [@T2 Proposition 3,6].
\[Besov\] Let $\psi(p)$ be a measurable function on ${\mathbb{R}}$ giving a tempered distribution, and let $0<\mu\leq 1$. Then the following two conditions are equivalent:
- The function $\psi$ satisfies $$\int_{{\mathbb{R}}^2}\frac{|\psi(p)-\psi(q)|^2}{|p-q|^{1+\mu}}dpdq<\infty.$$
- There exists a measurable function $\hat{\psi}_0(x)$ on ${\mathbb{R}}$ such that $$\int_{\mathbb{R}}|x|^\mu|\hat{\psi}_0(x)|^2dx<\infty,$$ and $x\hat{\psi}(x)=x\hat{\psi}_0(x)$ as distributions.
Moreover,
- If $\psi$ satisfies the conditions $(1)$,$(2)$, then $$\int_{\mathbb{R}}|\psi(2p)-\psi(p)|^2\frac{dp}{|p|^\mu}<\infty.$$
- If $\psi$ is an even differentiable function satisfying $$\int_0^\infty|\psi'(p)|^2 |p|^{2-\mu}dp<\infty,$$ then $\psi$ satisfies the conditions $(1)$,$(2)$.
Assume that (1) holds. Since the condition (1) is written as $$\int_{{\mathbb{R}}^2}\frac{|\psi(p+q)-\psi(q)|^2}{|p|^{1+\mu}}dqdp<\infty,$$ the function $q\mapsto \psi(p+q)-\psi(q)$ is square integrable for almost all $p\in {\mathbb{R}}$, and so is the distribution $(e^{ipx}-1)\hat{\psi}(x)$ by the Plancherel theorem. This shows that the restriction of $\hat{\psi}$ to ${\mathcal{D}}({\mathbb{R}}\setminus \{0\})$ is given by a locally square integrable function on ${\mathbb{R}}\setminus\{0\}$, say $\hat{\psi}_0(x)$, and that for almost all $p\in {\mathbb{R}}$, the equation $$\label{distribution}
(e^{ipx}-1)\hat{\psi}(x)=(e^{ipx}-1)\hat{\psi}_0(x)$$ holds as distributions in the variable $x$. In the above, we regards $\hat{\psi}_0(x)$ as a measurable function on ${\mathbb{R}}$ by setting $\hat{\psi}_0(0)=0$. Now the Plancherel formula implies $$\begin{aligned}
\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|\psi(p+q)-\psi(q)|^2}{|p|^{1+\mu}}dqdp
&=&\frac{1}{2\pi} \int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|(e^{ipx}-1)\hat{\psi}_0(x)|^2}{|p|^{1+\mu}}dxdp\\
&=& \frac{2}{\pi} \int_{\mathbb{R}}\int_{\mathbb{R}}\frac{\sin^2\frac{px}{2}|\hat{\psi}_0(x)|^2}{|p|^{1+\mu}}dpdx \\
&=&\frac{2^{1-\mu}}{\pi}\int_{\mathbb{R}}\frac{\sin^2r}{|r|^{1+\mu}}dr\int_{\mathbb{R}}|x|^\mu|\hat{\psi}_0(x)|^2dx. \end{aligned}$$ This implies the convergence of the integral in (2), which shows that $x\hat{\psi}_0(x)$ is a tempered distribution. Since the support of $x\hat{\psi}(x)-x\hat{\psi}_0(x)$ is contained in $\{0\}$, we have $$x\hat{\psi}(x)-x\hat{\psi}_0(x)=\sum_{k=0}^nc_k\delta_0^{(k)}(x),$$ where $c_k\in {\mathbb{C}}$ and $\delta_0$ is the Dirac mass at 0. We choose $p\neq 0$ such that (\[distribution\]) holds, and set $$h(x)=\left\{
\begin{array}{ll}
\frac{e^{ipx}-1}{x} , &\quad x\neq 0, \\
ip , &\quad x=0
\end{array}
\right..$$ Then $$0=(e^{ipx}-1)(\hat{\psi}(x)-\hat{\psi}_0(x))=h(x)(x\hat{\psi}(x)-x\hat{\psi}_0(x))=\sum_{k=0}^nc_kh(x)\delta_0^{(k)}(x).$$ It is routine work to show $c_k=0$ for all $k$ from this and $h(0)\neq 0$, and we get (2).
By tracing back the same computation as above, we can also show the implication from (2) to (1).
\(i) and (ii) are essentially [@T2 Proposition 3,6].
Summarizing our argument so far, we get
\[Sobolev\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be a projection. Then the following three conditions are equivalent:
- The symbol $\Phi$ is admissible.
- $$\int_{{\mathbb{R}}^2}\frac{{\operatorname{tr}}(|\Phi(p)-\Phi(q)|^2)}{|p-q|^2}dpdq<\infty.$$
- There exists a $M_N({\mathbb{C}})$-valued measurable function $\hat{\Phi}_0(x)$ on ${\mathbb{R}}$ such that $$\int_{\mathbb{R}}|x|{\operatorname{tr}}(|\hat{\Phi}_0(x)|^2)dx<\infty,$$ and $x\hat{\Phi}(x)=x\hat{\Phi}_0(x)$ as $M_N({\mathbb{C}})$-valued distributions.
Moreover,
- If $\Phi$ is admissible, then $$\int_{\mathbb{R}}{\operatorname{tr}}(|\Phi(2p)-\Phi(p)|^2)\frac{dp}{|p|}<\infty.$$
- If $\Phi$ is an even differentiable function satisfying $$\int_0^\infty {\operatorname{tr}}(|\Phi'(p)|^2)pdp<\infty,$$ then $\Phi$ is admissible.
For an admissible symbol $\Phi$, we call $\hat{\Phi}_0$ in Theorem \[Sobolev\],(3) the *regular part of $\hat{\Phi}$*. It is not clear whether $\hat{\Phi}_0$ gives a distribution on ${\mathbb{R}}$ in general. However, when it is the case, e.g. $\hat{\Phi}_0\in L^1({\mathbb{R}})\otimes M_N({\mathbb{C}})$, then we have $\hat{\Phi}=\hat{\Phi}_0+\delta_0\otimes Q$ for some $Q\in M_N({\mathbb{C}})$.
\[Toeplitz\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol, and let $A=T_\Phi$. We denote by $\alpha^\Phi=\{\alpha^\Phi_t\}_{t\geq 0}$ the $E_0$-semigroup acting on the type I factor ${\mathcal{M}}_A$ determined by $$\alpha^\Phi_t(\pi_A(a(f)))=\pi_A(a(S_tf)),\quad \forall f\in K.$$ We call $\alpha^\Phi$ the *Toeplitz CAR flow* associated with the symbol $\Phi$.
For a Toeplitz CAR flow $\alpha^\Phi$, we simply denote ${\mathcal{E}}_\Phi:={\mathcal{E}}_{\alpha^\Phi}$ and ${\mathcal{A}}^\Phi_a(I):={\mathcal{A}}^{{\mathcal{E}}_\Phi}_a(I)$.
When $\Phi\in M_N({\mathbb{C}})$ is a constant projection, the corresponding Toeplitz CAR flow is nothing but the CAR flow of index $N$, which gives the unique cocycle conjugacy class of type I$_N$ $E_0$-semigroups.
\[typical examples\] Powers’ first example of a type III $E_0$-semigroup is the Toeplitz CAR flow associated with the symbol $$\Phi(p)=\frac{1}{2}\left(
\begin{array}{cc}
1 &e^{i\theta(p)} \\
e^{-i\theta(p)} &1
\end{array}
\right),$$ where $\theta(p)=(1+p^2)^{-1/5}$. More generally, if $\theta(p)$ is a real differentiable function satisfying $\theta(-p)=\theta(p)$ for $\forall p\in {\mathbb{R}}$ and $$\int_0^\infty |\theta'(p)|^2pdp<\infty,$$ then Theorem \[Sobolev\] shows that the symbol $\Phi$ as above is admissible. In Section \[examples\], we will show that for $0<\nu\leq 1/4$, the symbols $\Phi_\nu$, given by $\theta_\nu(p)=(1+p^2)^{-\nu}$ in place of $\theta(p)$ above, give rise to mutually non cocycle conjugate type III $E_0$-semigroups.
We summarize a few facts frequently used in this paper in the next lemma. For a measurable subset $E\subset {\mathbb{R}}$, we set $K_E=L^2(E,{\mathbb{C}}^N)$. We denote by $P_E$ the projection from ${\tilde{K}}$ onto $K_E$. When $I\subset (0,\infty)$, we often regard $P_I$ as an element of $B(K)$. For simplicity, we write $K_t=K_{(0,t)}$ and $P_t=P_{(0,t)}$ for $t>0$.
\[interval\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol, and let $\hat{\Phi}_0$ be the regular part of $\hat{\Phi}$. We set $A=T_\Phi$.
- The relative commutant ${\mathcal{M}}_A\cap \alpha^\Phi_t({\mathcal{M}}_A)'$ is $\pi_A^t({\mathfrak{A}}(K_t))''$.
- Let $I$ and $J$ be mutually disjoint two open sets in ${\mathbb{R}}$. We assume that $I$ and $J$ have only finitely many connected components. Then $P_JC_\Phi P_I$ is a Hilbert-Schmidt operator with Hilbert-Schmidt norm $$\|P_JC_\Phi P_I\|_{\mathrm{H.S.}}^2=\frac{1}{4\pi^2}\int_{\mathbb{R}}|(J+t)\cap I|{\operatorname{tr}}(|\hat{\Phi}_0(t)|^2)dt.$$
- Let $I\subset (0,\infty)$ be an open (finite or infinite) interval. Then the restriction of $\pi_{A}$ to ${\mathfrak{A}}(K_I)$ is of type I, and the commutator $[C_\Phi,P_I]$ is Hilbert-Schmidt.
\(i) The statement follows Lemma \[commutant\],(i).
\(ii) Let $f\in {\mathcal{D}}(I,{\mathbb{C}}^N)$ and $g\in {\mathcal{D}}(J,{\mathbb{C}}^N)$. Then $${\langle{C_\Phi f,g}\rangle}=\sum_{i,j=1}^N\frac{1}{2\pi}\int_{{\mathbb{R}}}\Phi(p)_{ij}\hat{f_j}(p)\overline{\hat{g_i}(p)}dp
=\sum_{i,j=1}^N\frac{1}{2\pi}\int_{{\mathbb{R}}}\Phi(p)_{ij}\widehat{f_j*g_i^{\#}}(p)dp,$$ where $g_i^{\#}(x)=\overline{g_i(-x)}$. Since $\widehat{f_j*g_i^{\#}}\in {\mathcal{D}}({\mathbb{R}}\setminus \{0\})$, we get $$\begin{aligned}
{\langle{C_\Phi f,g}\rangle}&=&\sum_{i,j=1}^N\frac{1}{2\pi}\int_{{\mathbb{R}}}\hat{\Phi}_0(x)_{ij}f_j*g_i^{\#}(x)dx\\
&=&\sum_{i,j=1}^N\frac{1}{2\pi}\int_{{\mathbb{R}}^2}\hat{\Phi}_0(y-x)_{ij}f_j(y)\overline{g_i(x)}dxdy.\end{aligned}$$ Since $\chi_J(x)\chi_I(y)\hat{\Phi}_0(y-x)$ is square integrable (as we will see below), the operator $P_JC_\Phi P_I$ is Hilbert-Schmidt, and its Hilbert-Schmidt norm is $$\begin{aligned}
\frac{1}{4\pi^2}\int_{{\mathbb{R}}^2}\chi_J(x)\chi_I(y){\operatorname{tr}}(|\hat{\Phi}_0(y-x)|^2)dxdy
&=&\frac{1}{4\pi^2}\int_{\mathbb{R}}|(J+t)\cap I|{\operatorname{tr}}(|\hat{\Phi}_0(t)|^2)dt\\
&<&\infty,\end{aligned}$$ where we use Theorem \[Sobolev\],(3).
\(iii) Applying (ii) to $I$ and $J={\mathbb{R}}\setminus \overline{I}$, we see that $(1_{{\tilde{K}}}-P_I)C_\Phi P_I$ is Hilbert-Schmidt. This and Theorem \[qfstate\],(ii),(iii) show the first statement. Since $$[C_\Phi,P_I]=(1_{{\tilde{K}}}-P_I)C_\Phi P_I-P_IC_\Phi (1_{{\tilde{K}}}-P_I),$$ the commutator $[C_\Phi,P_I]$ is Hilbert-Schmidt.
A dichotomy theorem
===================
Based on Powers’ argument in [@Po1], Arveson proved the following type III criterion in [@Arv Theorem 13.6.1]:
\[P-A\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol having the limit $$\Phi(\infty):=\lim_{|p|\to\infty}\Phi(p).$$ If the Toeplitz CAR flow $\alpha^\Phi$ is spatial, then $$\int_{\mathbb{R}}{\operatorname{tr}}(|\Phi(p)-\Phi(\infty)|^2)dp<\infty.$$
The purpose of this section is to generalize Theorem \[P-A\], and to show the following dichotomy theorem, which can be considered as an analogue of [@pdct Theorem 39].
\[dichotomy\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol. Then the following conditions are equivalent:
- The Toeplitz CAR flow $\alpha^\Phi$ is of type I$_N$.
- The Toeplitz CAR flow $\alpha^\Phi$ is spatial.
- There exists a projection $Q\in M_N({\mathbb{C}})$ satisfying $$\int_{\mathbb{R}}{\operatorname{tr}}(|\Phi(p)-Q|^2)dp<\infty.$$
In particular, every Toeplitz CAR flow is either of type I or type III.
The implication from (i) to (ii) is trivial. That from (ii) to (iii) is a generalization of Theorem \[P-A\]. Although we follow the same strategy as in the proof of Theorem \[P-A\], we will make a significant simplification of the argument using Arveson’s classification of type I product systems (see Lemma \[even unit\] below), which allows us to obtain the statement of this form. Since $\alpha^Q$ with a constant projection $Q\in M_N({\mathbb{C}})$ is of type I$_N$, the implication from (iii) to (i) follows from a $L^2$-perturbation theorem stated below, which can be considered as an analogue of [@IS Theorem 7.4,(1)].
\[L2\] Let $\Phi, \Psi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be admissible symbols. If $$\int_{\mathbb{R}}{\operatorname{tr}}(|\Phi(p)-\Psi(p)|^2)dp<\infty,$$ then $\alpha^\Phi$ and $\alpha^\Psi$ are cocycle conjugate.
We first give a representation theoretical consequence of the above square integrability condition.
\[local equivalence\] Let $\Phi, \Psi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be admissible symbols. We set $A=T_\Phi$ and $B=T_\Psi$. Then $$\int_{\mathbb{R}}{\operatorname{tr}}(|\Phi(p)-\Psi(p)|^2)dp<\infty,$$ if and only if for any (some) non-degenerate finite interval $I\subset (0,\infty)$, the two quasi-free states $\omega_{P_IAP_I}$ and $\omega_{P_IBP_I}$ of ${\mathfrak{A}}(K_I)$ are quasi-equivalent.
Thanks to Lemma \[typeI\], the two states $\omega_{P_IAP_I}$ and $\omega_{P_IBP_I}$ are quasi-equivalent if and only if the following quantity is finite: $$\begin{aligned}
\lefteqn{{\operatorname{tr}}\big(P_IC_\Phi P_I C_{1-\Psi}P_I C_\Phi P_I+P_IC_{1-\Phi}P_IC_{\Psi}P_IC_{1-\Phi}P_I\big)
} \\
&=&{\operatorname{tr}}\big(C_{1-\Psi}(P_IC_\Phi P_I)^2 C_{1-\Psi}+C_{\Psi}(P_IC_{1-\Phi}P_I)^2C_{\Psi}\big).\end{aligned}$$ Since $P_IAP_I-(P_IAP_I)^2$ and $P_IBP_I-(P_IBP_I)^2$ are trace class operators (see Theorem \[qfstate\],(iii) and Lemma \[interval\],(iii)), we can replace $(P_IC_\Phi P_I)^2$ with $P_IC_\Phi P_I$ and $(P_IC_{1-\Phi}P_I)^2$ with $P_IC_{1-\Phi}P_I$ in the above formula, and we get $${\operatorname{tr}}\big(C_{1-\Psi}P_IC_\Phi P_IC_{1-\Psi}+C_{\Psi}P_IC_{1-\Phi}P_IC_{\Psi}\big)
=\|C_\Phi P_IC_{1-\Psi}\|_{{\mathrm{H.S.}}}^2+\|C_{1-\Phi}P_IC_{\Psi}\|_{{\mathrm{H.S.}}}^2.$$ Since the commutators $[C_{1-\Psi},P_I]$ and $[C_\Phi,P_I]$ are Hilbert-Schmidt (see Lemma \[interval\],(iii)), the right-hand side is finite if and only if $$\|C_{\Phi(1-\Psi)}P_I\|_{{\mathrm{H.S.}}}^2+\|C_{(1-\Phi)\Psi}P_I\|_{{\mathrm{H.S.}}}^2$$ is finite. [@Arv Proposition 13.4.1] shows that this is equal to $$\frac{|I|}{2\pi}\int_{\mathbb{R}}{\operatorname{tr}}(|\Phi(p)-\Psi(p)|^2)dp,$$ and we get the statement.
Assume that $\Phi-\Psi$ is square integrable. Let $A=T_\Phi$ and $B=T_\Psi$. We will apply Theorem \[isomorphic\] to $E={\mathcal{E}}_\Phi$ and $F={\mathcal{E}}_\Psi$, and show that $\alpha^\Phi$ and $\alpha^\Psi$ are cocycle conjugate.
Thanks to Lemma \[local equivalence\], the two representations $\pi_A$ and $\pi_B$ are quasi-equivalent when they are restricted to ${\mathfrak{A}}(K_1)$. This implies that there exists an isomorphism $\rho_0$ from $\pi_A({\mathfrak{A}}(K_1))''$ onto $\pi_B({\mathfrak{A}}(K_1))''$ satisfying $\rho_0(\pi_A(a(f)))=\pi_B(a(f))$ for $\forall f\in K_1$. Since $\rho_0$ preserves the grading, we may assume $\rho_0(R^A_{K_1})=R^B_{K_1}$ by replacing $R^B_{K_1}$ with $-R^B_{K_1}$ if necessary. We may also assume $R^A=R^A_{K_1}R^A_{K_{(1,\infty)}}$ and $R^B=R^B_{K_1}R^B_{K_{(1,\infty)}}$.
We claim that $\rho_0$ extends to an isomorphism $\rho_1$ from $(\pi_A({\mathfrak{A}}(K_1))\cup\{R^A\})''$ onto $(\pi_B({\mathfrak{A}}(K_1))\cup\{R^B\})''$ satisfying $\rho_1(R^A)=R^B$. Indeed, since $R^A_{K_{(1,\infty)}}$ commutes with $\pi_A({\mathfrak{A}}(K_1))$, we have $$(\pi_A({\mathfrak{A}}(K_1))\cup\{R^A\})''=\frac{1+R^A_{K_{(1,\infty)}}}{2}\pi_A({\mathfrak{A}}(K_1))''\oplus \frac{1-R^A_{K_{(1,\infty)}}}{2}
\pi_A({\mathfrak{A}}(K_1))''.$$ For the same reason, $$(\pi_B({\mathfrak{A}}(K_1))\cup\{R^B\})''=\frac{1+R^B_{K_{(1,\infty)}}}{2}\pi_B({\mathfrak{A}}(K_1))''\oplus \frac{1-R^B_{K_{(1,\infty)}}}{2}
\pi_B({\mathfrak{A}}(K_1))'',$$ and so $\rho_0$ extends to $\rho_1$ satisfying $\rho_1(R^A_{K_{(1,\infty)}})=\rho_1(R^B_{K_{(1,\infty)}})$. In consequence, we have $\rho_1(R^A)=R^B$.
Let $\rho$ be the restriction of $\rho_1$ to ${\mathcal{M}}_A\cap \alpha^\Phi_1({\mathcal{M}}_A)'$, which is identified with $B({\mathcal{E}}_\Phi(1))$. Thanks to Lemma \[commutant\],(i), it is generated by $\{\pi_A(a(f))R^A;\; f\in K_1\}.$ Then the image of $\rho$ is generated by $\{\pi_B(a(f))R^B;\; f\in K_1\},$ and so it is ${\mathcal{M}}_B\cap \alpha^{\Psi}_1({\mathcal{M}}_B)'$, which is identified with $B({\mathcal{E}}_\Psi(1))$. In the same way, we can see that $\rho$ satisfies $\rho({\mathcal{A}}^\Phi(0,s))={\mathcal{A}}^\Psi(0,s)$ for any $0\leq s\leq 1$. Thus we get the statement from Theorem \[isomorphic\].
Now we start the proof of the implication (ii) $\Rightarrow$ (iii) in Theorem \[dichotomy\]. Recall that $\gamma$ is the grading automorphism $\gamma(\pi_A(a(f)))=-\pi_A(a(f))$.
\[even unit\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol. If $\alpha^\Phi$ is spatial, then there exists a unit $V=\{V_t\}_{t>0}$ for $\alpha^{\Phi}$ satisfying $\gamma(V_t)=V_t$ for $\forall t>0$.
Since $\gamma$ commutes with $\alpha^\Phi_t$ for $\forall t>0$, it induces an automorphism of the corresponding product system ${\mathcal{E}}_\Phi$. When ${\mathcal{E}}_\Phi$ is of type II$_0$, it is easy to show the statement, and so we assume the index of ${\mathcal{E}}_\Phi$ is not $0$. Let $E$ be the subproduct system of ${\mathcal{E}}_{\alpha^{\Phi}}$ generated by the units, and let $\beta$ be the automorphism of $E$ induced by $\gamma$. Then the statement follows from the following claim: for any period two automorphism $\beta$ of any type I product system $E$, there exists a unit of $E$ fixed by $\beta$. Note that the type I product systems are completely classified, and the action of ${\operatorname{Aut}}(E)$ on the set of units ${\mathcal{U}}_E$ is well-known (see [@Ar Section 3.8]).
Let $L$ be a Hilbert space whose dimension is the same as the index of $E$, and let ${\mathcal{U}}(L)$ be the unitary group of $L$. Then ${\operatorname{Aut}}(E)$ is identified with $G_L={\mathbb{R}}\times L \times {\mathcal{U}}(L)$ having the group operation $$(\lambda,\xi,U)(\mu,\eta,V)=(\lambda+\mu+{\mathrm{Im}\,}{\langle{\xi,U\eta}\rangle},\xi+U\eta,UV).$$ The set ${\mathcal{U}}_E$ together with the ${\operatorname{Aut}}(E)$-action on it is identified with ${\mathbb{C}}\times L$ with the $G_L$-action $$(\lambda,\xi,U)\cdot(a,\eta)=(a+i\lambda-\frac{\|\xi\|^2}{2}-{\langle{U\eta,\xi}\rangle},\xi+U\eta).$$ Any element $g\in G_L$ of order two is of the form $g=(0,\xi,U)$ with $U^2=1$ and $U\xi=-\xi$. Now we can see that $(0,\frac{1}{2}\xi)\in {\mathbb{C}}\times L$ is fixed by $g$.
The following lemma is a slight generalization of [@Po1 Lemma 4.5] and [@Arv Lemma 13.6.5]. For later use, we will show a little stronger statement than we need in this section.
\[making a state\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol and $A=T_\Phi$. If $V\in {\mathcal{E}}_\Phi(t)$ is a normalized vector satisfying $\gamma(V)=\pm V$, then there exists a pure $\gamma$-invariant state $\varphi$ of ${\mathfrak{A}}(K_t)$ such that $V^*\pi_A(X)V=\varphi(X)1$ for any $X\in {\mathfrak{A}}(K_t)$.
Throughout the proof, the symbol $a^\dagger(f)$ means either $a(f)$ or $a(f)^*$. Let $f_1,f_2, \cdots, f_n\in K_t$, and $X=a^\dagger(f_1)a^\dagger(f_2)\cdots a^\dagger(f_n)$. Then for any $g\in K$, we have $$\begin{aligned}
V^*\pi_A(X)V\pi_A(a^\dagger(g))&=&V^*\pi_A(Xa^\dagger(S_tg))V=(-1)^nV^*\pi_A(a^\dagger(S_tg)X)V\\
&=&(-1)^n\pi_A(a^\dagger(g))V^*\pi_A(X)V.\end{aligned}$$ If $n$ is even, this shows that $V^*\pi_A(X)V$ is in the center $Z({\mathcal{M}}_A)$ of ${\mathcal{M}}_A$, and so it is a scalar. If $n$ if odd, the operator $R^AV^*\pi_A(X)V$ is a scalar for the same reason, and on the other hand, it is an odd operator with respect to $\gamma$. Thus $V^*\pi_A(X)V=0$, which shows that there exists a $\gamma$-invariant state $\varphi$ such that $V^*\pi_A(X)V=\varphi(X)1$ for all $X\in {\mathfrak{A}}(K_t)$.
It only remains to show that $\varphi$ is pure. Recall that the twisted representation $\pi_A^t$ is defined by $\pi_A^t(a(f))=i\pi_A(a(f))R^A$, and ${\mathcal{M}}_A\cap \alpha^\Phi_t({\mathcal{M}}_A)'=\pi_A^t({\mathfrak{A}}(K_t))''$. We denote by $\pi$ the irreducible representation of ${\mathfrak{A}}(K_t)$ on ${\mathcal{E}}_\Phi(t)$ given by $\pi(X)=\sigma(\pi_A^t(X))$ on ${\mathcal{E}}_\Phi(t)$, where $\sigma(Y)$ denotes the left multiplication of $Y$. Then the pure state of ${\mathfrak{A}}(K_t)$ given by $X\mapsto {\langle{\pi(X)V,V}\rangle}=V^*\pi_A^t(X)V$ coincides with $\varphi$ because both $\varphi$ and this state are $\gamma$-invariant, and $\pi_A$ and $\pi_A^t$ coincide on the even part of ${\mathfrak{A}}(K_t)$.
Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol, and let $A=T_\Phi$. Assume that $\alpha^\Phi$ is spatial. Then Lemma \[even unit\] shows that there exists a normalized unit $V=\{V_t\}_{t\geq 0}$ satisfying $\gamma(V_t)=V_t$ for all $t$. Let $\varphi$ be the state of ${\mathfrak{A}}(K_1)$ defined by $\varphi(X)={\langle{\pi_A(X)V_1\Omega_A,V_1\Omega_A}\rangle}$ for $X\in {\mathfrak{A}}(K_1)$, and let $B\in B(K_1)$ the covariance operator for $\varphi$. Then Lemma \[making a state\] shows that $V_t^*\pi_A(a(f))V_t=0$ for any $f\in K_t$. We claim that there exists a positive contraction $Q\in L^\infty((0,1))\otimes M_N({\mathbb{C}})$ such that $B$ is the multiplication operator of $Q$. To prove the claim, it suffices to show that $B$ commutes with $P_t$ for all $0< t<1$. Indeed, if $f\in K_t$ and $g\in K_{(t,1)}$, then $$V_1^*\pi_A(a(f)a(g)^*)V_1=V_{1-t}^*V_t^*\pi_A(a(f))V_t\pi_A(a(S^*_tg)^*)V_{1-t}=0.$$ Thus we get $P_{(t,1)}BP_t=0$, and the claim is shown.
Note that $\varphi$ is quasi-equivalent to $\omega_{P_1AP_1}$. We claim that $B$ is a projection. Let ${\mathbb{K}}(K_1)$ be the set of compact operators of $K_1$, and let $q:B(K_1)\to B(K_1)/{\mathbb{K}}(K_1)$ be the quotient map. Then thanks to Lemma \[covariance operator\],(ii), we have $q(P_1AP_1)=q(B)$. Since $\omega_{P_1AP_1}$ is a type I state, we have $q(P_1AP_1)^2=q(P_1AP_1)$, and so $B-B^2$ is a compact operator. This is possible only if $Q(x)$ is a projection for almost every $x\in (0,1)$, and so $B$ is a projection.
Since $B$ is a projection, Lemma \[covariance operator\],(i) implies $\varphi=\omega_B$. Since $\omega_{P_1AP_1}$ and $\omega_B$ are quasi-equivalent, Theorem \[qfstate\],(v) implies $$\|C_\Phi(P_1-B)\|_{\mathrm{H.S.}}^2+\|C_{1-\Phi}B\|_{\mathrm{H.S.}}^2={\operatorname{tr}}\big((P_1-B)C_\Phi(P_1-B)+BC_{1-\Phi}B\big)<\infty.$$ A similar computation as in [@Arv Proposition 13.4.1] shows that the left-hand side is $$\frac{1}{2\pi}\int_0^1\int_{\mathbb{R}}{\operatorname{tr}}(|\Phi(p)-Q(x)|^2)dpdx.$$ Thus the integral $$\int_{\mathbb{R}}{\operatorname{tr}}(|\Phi(p)-Q(x)|^2)dp$$ is finite for almost every $x\in (0,1)$, and the proof is finished.
\[nonconverging\] Let $\theta(p)$ be a real smooth function satisfying $\theta(-p)=\theta(p)$ for all $ p\in {\mathbb{R}}$ and $\theta(p)=\log(\log |p|)$ (or $\theta(p)=\log^\alpha|p|$ with $0<\alpha<1/2$) for large $|p|$. Then $\Phi$ associated with $\theta$ in Example \[typical examples\] is an admissible symbol without having limit at infinity. While Theorem \[P-A\] does not apply to such $\Phi$, now we know from Theorem \[dichotomy\] that the Toeplitz CAR flow $\alpha^{\Phi}$ is of type III.
Type I factorizations associated with Toeplitz CAR flows {#factorization}
========================================================
Thanks to Theorem \[dichotomy\], we have a complete understanding of spatial Toeplitz CAR flows now. The purpose of this section is to calculate the invariant we introduced in Subsection \[invariant for type III\] in the case of type III Toeplitz CAR flows.
\[CABATIF\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol, and let $\{a_n\}_{n=0}^\infty$ be a strictly increasing sequence of non-negative numbers such that $a_0=0$ and it converges to a finite number $a$. Let $I_n=(a_n,a_{n+1})$ and $O=\bigcup_{n=0}^\infty I_{2n}$.
- If $$\sum_{n=0}^\infty\|(1_{{\tilde{K}}}-P_{I_n})C_\Phi P_{I_n}\|_{\mathrm{H.S.}}^2<\infty,$$ then $\{{\mathcal{A}}^\Phi_a(I_n)\}_{n=1}^\infty$ is a CABATIF.
- If $\{{\mathcal{A}}^\Phi_a(I_n)\}_{n=0}^\infty$ is a CABATIF, then $\|(1_{{\tilde{K}}}-P_O)C_\Phi P_O\|_{\mathrm{H.S.}}^2<\infty$.
We prepare a few facts used in the proof of (i) first.
\[PQ\] Let $H$ be a Hilbert space, and let $P,Q\in B(H)$ be projections. Then $$\|(1-P)QP\|_{\mathrm{H.S.}}=\|(1-Q)PQ\|_{\mathrm{H.S.}}.$$
There is a decomposition of $H$ into closed subspaces (each subspace could possibly be $\{0\}$) $$H=H_1\oplus H_2\oplus H_3\oplus H_4\otimes {\mathbb{C}}^2\oplus H_5$$ such that the two projections are expressed as $$P=1_{H_1}\oplus 1_{H_2}\oplus 0\oplus
\left(\begin{array}{cc}
1_{H_4} &0 \\
0 &0
\end{array}\right)
\oplus 0,$$ $$Q=1_{H_1}\oplus 0\oplus 1_{H_3}\oplus
\left(\begin{array}{cc}
c^2 &cs \\
cs &s^2
\end{array}\right)
\oplus 0,$$ where $c$ and $s$ are non-singular positive contractions satisfying $c^2+s^2=1_{H_4}$ (see [@Ta p.308]). Then we have $$\|(1-P)QP\|_{\mathrm{H.S.}}^2
=\|\left(
\begin{array}{cc}
0 &0 \\
cs &0
\end{array}
\right)\|_{\mathrm{H.S.}}^2
={\operatorname{tr}}(c^2s^2),$$ $$\begin{aligned}
\|(1-Q)PQ\|_{\mathrm{H.S.}}^2&=&
\|\left(
\begin{array}{cc}
s^2 &-cs \\
-cs &c^2
\end{array}
\right)
\left(
\begin{array}{cc}
1 &0 \\
0 &0
\end{array}
\right)
\left(
\begin{array}{cc}
c^2 &cs \\
cs &s^2
\end{array}
\right)
\|_{\mathrm{H.S.}}^2 \\
&=&\|\left(
\begin{array}{cc}
c^2s^2 &cs^3 \\
-c^3s &-c^2s^2
\end{array}
\right)\|_{\mathrm{H.S.}}^2
={\operatorname{tr}}\big(2c^4s^4+c^2s^6+c^6s^2\big)\\
&=&{\operatorname{tr}}\big(c^2s^2(c^2+s^2)^2\big)
={\operatorname{tr}}(c^2s^2). \end{aligned}$$
\[qe and type I\] Let the notation be as in Theorem \[CABATIF\], and let $A=T_\Phi$. We set $$B=\sum_{n=0}^\infty P_{I_n}AP_{I_n}+P_{(a,\infty)}AP_{(a,\infty)}.$$ Then the following conditions are equivalent:
- The assumption of Theorem \[CABATIF\],(i) holds.
- The quasi-free state $\omega_B$ is of type I.
- The two quasi-free states $\omega_A$ and $\omega_B$ are quasi-equivalent.
Theorem \[qfstate\],(iii) and Lemma \[interval\],(iii) imply that (i) and (ii) are equivalent. We show the equivalence of (i) and (iii). We set $I_{-1}=(a,\infty)$. Since $\omega_A$ is of type I, Lemma \[typeI\] shows that $\omega_A$ and $\omega_B$ are quasi-equivalent if and only if the following quantity is finite: $$\begin{aligned}
\lefteqn{{\operatorname{tr}}\big(A(1_K-B)A+(1_K-A)B(1_K-A)\big)}\\
&=&\sum_{n=-1}^\infty {\operatorname{tr}}\big(P_+C_\Phi P_{I_n}C_{1-\Phi}P_{I_n}C_\Phi P_+ + P_+C_{1-\Phi}P_{I_n}C_\Phi P_{I_n}C_{1-\Phi}P_+\big) \\
&=&\sum_{n=-1}^\infty (\| C_{1-\Phi}P_{I_n}C_\Phi P_+\|_{\mathrm{H.S.}}^2+\|C_\Phi P_{I_n}C_{1-\Phi}P_+\|_{\mathrm{H.S.}}^2).\end{aligned}$$ Note that since $$\begin{aligned}
\lefteqn{
\sum_{n=-1}^\infty (\| C_{1-\Phi}P_{I_n}C_\Phi (1_{{\tilde{K}}}-P_+)\|_{\mathrm{H.S.}}^2+\|C_\Phi P_{I_n}C_{1-\Phi}(1_{{\tilde{K}}}-P_+)\|_{\mathrm{H.S.}}^2)}\\
&\leq&\sum_{n=-1}^\infty{\operatorname{tr}}\big((1_{{\tilde{K}}}-P_+)C_\Phi P_{I_n}C_\Phi (1_{{\tilde{K}}}-P_+)
+(1_{{\tilde{K}}}-P_+)C_{1-\Phi}P_{I_n}C_{1-\Phi}(1_{{\tilde{K}}}-P_+)\big)\\
&=&
\|P_+ C_\Phi (1_{{\tilde{K}}}-P_+)\|_{\mathrm{H.S.}}^2+\|P_+ C_{1-\Phi}(1_{{\tilde{K}}}-P_+)\|_{\mathrm{H.S.}}^2\\
&=&2\|P_+ C_\Phi (1_{{\tilde{K}}}-P_+)\|_{\mathrm{H.S.}}^2<\infty,\end{aligned}$$ the above quantity is finite if and only if $$\sum_{n=-1}^\infty \| C_{1-\Phi}P_{I_n}C_\Phi \|_{\mathrm{H.S.}}^2<\infty.$$ Thanks to Lemma \[PQ\], this is equivalent to $$\sum_{n=-1}^\infty \| (1_{{\tilde{K}}}-P_{I_n})C_\Phi P_{I_n}\|_{\mathrm{H.S.}}^2<\infty.$$ Since $\| (1_{{\tilde{K}}}-P_{I_{-1}})C_\Phi P_{I_{-1}}\|_{\mathrm{H.S.}}^2<\infty$, we conclude that (i) is equivalent to (iii).
Assume that the assumption of Theorem \[CABATIF\],(i) holds. It suffices to show that for any strictly increasing sequence of non-negative integers $\{n_{m}\}_{m=0}^\infty$, the von Neumann algebra ${\mathcal{A}}^\Phi_a(E):=\bigvee_{m=0}^\infty {\mathcal{A}}^\Phi_a(I_{n_m})$ is a type I factor, where $E=\bigcup_{m=0}^\infty I_{n_m}$. Note that ${\mathcal{A}}^\Phi_a(E)$ is always a factor (see [@IS Remark 8.2]). We may assume $n_0=0$ without loss of generality. Identifying $B({\mathcal{E}}^\Phi(a))$ with ${\mathcal{M}}_A\cap \alpha^\Phi_a({\mathcal{M}}_A)'$, we see that it suffices to show the factor $$\bigvee_{m=0}^\infty \alpha^\Phi_{a_{n_m}}({\mathcal{M}}_A\cap \alpha^\Phi_{a_{n_m+1}-a_{n_m}}({\mathcal{M}}_A)')$$ is of type I. Recall that we have ${\mathcal{M}}_A\cap \alpha^\Phi_{t}({\mathcal{M}}_A)'=\pi_A^t({\mathfrak{A}}(K_t))''$, where $\pi_A^t(a(f))=i\pi_A(a(f))R^A$. Since $$\alpha^\Phi_t(R^A)=\pm R^A_{K_{(t,\infty)}}=\pm \epsilon_{K_t,K_{(t,\infty)}}R^AR^A_{K_t},$$ we get $$\alpha^\Phi_{a_{n_m}}({\mathcal{M}}_A\cap \alpha^\Phi_{a_{n_m+1}-a_{n_m}}({\mathcal{M}}_A)')
=\{\pi_A^t(a(f))R^A_{K_{a_{n_m}}},R^A_{K_{a_{n_m}}}\pi_A^t(a(f))^*;\; f\in K_{I_{n_m}}\}''.$$ Thanks to Lemma \[commutant\],(ii), it suffices to show that the factor $$\bigvee_{m=0}^\infty\{\pi_A(a(f))R^A_{K_{a_{n_m}}},R^A_{K_{a_{n_m}}}\pi_A(a(f))^*;\; f\in K_{I_{n_m}}\}''$$ is of type I.
Let $${\mathcal{N}}_m:=\bigvee_{k=0}^m\{\pi_A(a(f))R^A_{K_{a_{n_k}}},R^A_{K_{a_{n_k}}}\pi_A(a(f))^*;\; f\in K_{I_{n_k}}\}'' ,$$ and let $J_m=\bigcup_{k=1}^m(a_{n_{k-1}+1},a_{n_k})$. Since $$R^A_{K_{a_{n_m}}}=\pm R^A_{K_{J_m}}\prod_{k=0}^{m-1} R^A_{K_{I_{n_k}}},$$ and $$R^A_{I_{n_k}}\in \{\pi_A(a(f))R^A_{K_{J_k}},\; R^A_{K_{J_k}}\pi_A(a(f))^*;\; f\in K_{I_{n_k}}\}'',$$ we can show $${\mathcal{N}}_m=\bigvee_{k=0}^m\{\pi_A(a(f))R^A_{K_{J_k}},R^A_{K_{J_k}}\pi_A(a(f))^*;\; f\in K_{I_{n_k}}\}''$$ by induction, where we use the convention $R^A_{K_{J_0}}=1$. Thus to prove the statement, it suffices to show that the factor $$\bigvee_{m=0}^\infty\{\pi_A(a(f))R^A_{K_{J_m}},R^A_{K_{J_m}}\pi_A(a(f))^*;\; f\in K_{I_{n_m}}\}''$$ is of type I.
Let $B$ be as in Lemma \[qe and type I\]. Since $\pi_A$ and $\pi_B$ are quasi-equivalent, there exists an isomorphism $\theta$ from ${\mathcal{M}}_A$ onto ${\mathcal{M}}_B$ satisfying $\theta(\pi_A(f))=\theta(\pi_B(a(f)))$ for any $f\in K$. Since $\theta$ preserves the grading, we may assume $\theta(R^A_{K_I})=R^B_{K_I}$ for any interval $I\subset (0,\infty)$. Thus to prove the statement, it suffices to show that the factor $${\mathcal{N}}:=\bigvee_{m=0}^\infty\{\pi_B(a(f))R^B_{K_{J_m}},R^B_{K_{J_m}}\pi_B(a(f))^*;\; f\in K_{I_{n_m}}\}''$$ is of type I.
Since $J_m$ is disjoint from $E$, the self-adjoint unitary $R^B_{K_{J_m}}$ commutes with any $\pi_B(a(f))$ with $f\in K_E$. Thus ${\mathcal{N}}$ is generated by the factor representation $\pi$ of ${\mathfrak{A}}(K_E)$ determined by $\pi(a(f))=\pi_B(a(f))R^B_{J_m}$ for $f\in K_{I_{n_m}}$. Let $\omega$ be the state of ${\mathfrak{A}}(K_E)$ defined by $\omega(X):={\langle{\pi(X)\Omega_B,\Omega_B}\rangle}$ for $X\in {\mathfrak{A}}(K_E)$. Since $\pi$ is a factor representation, the GNS representation of $\omega$ is quasi-equivalent to $\pi$.
We claim that $\omega$ coincides with $\omega_{P_EBP_E}$. Let $X_i\in {\mathfrak{A}}(K_{I_{n_i}})$, $i=0,1,\cdots, m$ be of the form $$X_i=a^\dagger(f^i_1)a^\dagger(f^i_2)\cdots a^\dagger(f^i_{l_i}),\quad$$ with $f^{i}_j\in K_{I_{n_i}}$, where $a^\dagger(f)$ means either $a(f)$ or $a(f)^*$. Then we have $\pi(X_i)=\pi_B(X_i){R^B_{K_{J_i}}}^{l_i},$ and $$\omega(X_1X_2\cdots X_m)={\langle{\pi_B(X_1X_2\cdots X_m)Y\Omega_B,\Omega_B}\rangle}$$ where $Y$ is an element in the even part of $\pi_B({\mathfrak{A}}(K_E^\perp))''$. Since $B$ commutes with $P_{I_n}$ for any $n$, if one of $l_1,l_2,\cdots, l_m$ is odd, then approximating $Y$ by polynomials of $\pi_B(a^\dagger(f))$ with $f\in K_E^\perp$, we see that the right-hand side is 0 (consider the contributing 2-point functions). When $l_1,l_2,\cdots,l_m$ are all even, we have $$\omega(X_1X_2\cdots X_m)={\langle{\pi_B(X_1X_2\cdots X_m)\Omega_B,\Omega_B}\rangle}=\omega_B(X_1X_2\cdots X_m),$$ which shows $\omega=\omega_{P_EBP_E}$. Thus to prove the statement, it suffices to show that $\omega_{P_EBP_E}$ is of type I.
Since $P_E$ commutes with $B$, we get $${\operatorname{tr}}\big(P_EBP_E-(P_EBP_E)^2\big)={\operatorname{tr}}\big(P_E(B-B^2)\big)\leq {\operatorname{tr}}(B-B^2).$$ Now the statement follows from Theorem \[qfstate\],(iii) and Lemma \[qe and type I\].
We proceed to the proof of Theorem \[CABATIF\],(ii).
\[product state\] Let $L_n$, $n=0,1,\cdots,$ be Hilbert spaces, and let $L=\bigoplus_{n=0}^\infty L_n$. Assume that $\varphi$ is a $\gamma$-invariant state of ${\mathfrak{A}}(L)$ satisfying the following two conditions:
- For any natural number $n$ and $X_i\in {\mathfrak{A}}(L_i)$, $i=0,1,\cdots,n$, $$\varphi(X_1X_2\cdots X_n)=\varphi(X_1)\varphi(X_2)\cdots \varphi(X_n).$$
- The restriction $\varphi_n$ of $\varphi$ to ${\mathfrak{A}}(L_n)$ is a pure state for any $n$.
Then $\varphi$ is a pure state.
Let $(H_n,\pi_n,\Omega_n)$ be the GNS triple of $\varphi_n$, and let $H=\bigotimes_{n=0}^\infty{}^{(\otimes \Omega_n)}H_n$ be the ITPS of the Hilbert spaces $\{H_n\}_{n=0}^\infty$ with respect to the reference vectors $\{\Omega_n\}_{n=0}^\infty$. We set $\Omega=\bigotimes_{n=0}^\infty \Omega_n$. Since $\varphi_n$ is a $\gamma$-invariant state of ${\mathfrak{A}}(L_n)$, there exists a self-adjoint unitary $R_n\in B(H_n)$ satisfying $R_n\pi_n(X)\Omega_n=\pi_n(\gamma(X))$ for all $X\in {\mathfrak{A}}(L_n)$. We introduce a representation $\pi$ of ${\mathfrak{A}}(L)$ on $H$ by setting $\pi(a(f))$ for $f\in L_n$ as $$\pi(a(f))=\left\{
\begin{array}{ll}
\pi_0(a(f))\otimes 1_{\bigotimes_{k=1}^\infty H_k} , &\quad n=0 \\
R_0\otimes R_1\otimes \cdots \otimes R_{n-1}\otimes \pi_n(a(f))\otimes 1_{\bigotimes_{k=n+1}^\infty H_k},
&\quad n>0
\end{array}
\right..$$ Then $\pi$ is irreducible, and the pure state $\psi$ of ${\mathfrak{A}}(L)$ defined by $\psi(X)={\langle{\pi(X)\Omega,\Omega}\rangle}$ satisfies the two conditions (i) and (ii). Moreover, the restriction of $\psi$ to ${\mathfrak{A}}(L_n)$ coincides with $\varphi_n$. Since $\{\varphi_n\}_{n=0}^\infty$ and the condition (i) uniquely determine $\varphi$, we conclude that $\varphi=\psi$ and it is a pure state.
\[decomposable vector\] If the assumption of Theorem \[CABATIF\],(ii) holds, then there exist normalized vectors $V\in {\mathcal{E}}_\Phi(a)$, $V_n\in {\mathcal{E}}_\Phi(a_{n+1}-a_n)$, and $W_n\in {\mathcal{E}}_\Phi(a-a_{n+1})$, $n=0,1,2,\cdots,$ such that $V$ is factorized as $V=V_0V_1V_2\cdots V_nW_n$, and $\gamma(V_n)=\pm V_n$, $\gamma(W_n)=\pm W_{n}$ for any non-negative integer $n$.
Assume that $\{{\mathcal{A}}^\Phi_a(I_n)\}_{n=0}^\infty$ is a CABATIF. Then thanks to Theorem \[ArakiWoods\], there exists a sequence of Hilbert spaces with normalized vectors $\{(H_n,\xi_n)\}_{n=0}^\infty$ and unitary $U$ from ITPS $H:=\bigotimes_{n=0}^\infty {}^{\otimes \xi_n}H_n$ onto ${\mathcal{E}}_\Phi(a)$ such that $U{\mathcal{M}}_nU^*={\mathcal{A}}^\Phi_a(I_n)$, where $${\mathcal{M}}_n=B(H_n)\otimes {\mathbb{C}}1_{\bigotimes_{m\neq n}H_m}.$$
For $X\in {\mathcal{M}}_A\cap \alpha^\Phi_a({\mathcal{M}}_A)'$, we denote by $\sigma(X)\in B({\mathcal{E}}_\Phi(a))$ the corresponding left multiplication operator. We claim that for any $0<t<a$, there exists $\epsilon_t\in \{1,-1\}$ such that $\gamma(X)=\epsilon_t\sigma(R^A_{K_t})X$ for any $X\in {\mathcal{E}}^\Phi(t)$. Indeed, let $\epsilon_t$ be the constant determined by $\alpha^\Phi_t(R^A)=\epsilon_t R^A_{K_t}R^A$. Then $$\gamma(X)=R^AX{R^A}^*=R^A\alpha^\Phi_t(R^A)^*X=\epsilon_t R^A_{K_t}X,$$ which shows the claim.
The claim (or Lemma \[commutant\]) implies that for any $X\in {\mathcal{M}}_A\cap \alpha^\Phi_a({\mathcal{M}}_A)'$ we have $R^A_{K_a}X{R^A_{K_a}}^*=\gamma(X)$. Thus $\sigma(R^A_{K_a})$ is a self-adjoint unitary satisfying $$\sigma(R^A_{K_a}){\mathcal{A}}^\Phi_a(I_n)\sigma(R^A_{K_a})^*={\mathcal{A}}^\Phi_a(I_n).$$ For the same reason, the operator $\sigma(R^A_{K_{I_n}})$ is a self-adjoint unitary in ${\mathcal{A}}^\Phi_a(I_n)$ satisfying $$\label{eq in proof of (ii)}
\sigma(R^A_{K_a})X\sigma(R^A_{K_a})^*=\sigma(R^A_{K_{I_n}})X\sigma(R^A_{K_{I_n}})^*,\quad \forall X\in {\mathcal{A}}^E_a(I_n).$$
Applying Lemma \[fixed product vector\] to the self-adjoint unitary $R=U^*\sigma(R^A_{K_a})U\in B(H)$, we get a product vector $\eta=\bigotimes_{n=1}^\infty\eta_n\in H$ and self-adjoint unitaries $R_n\in B(H_n)$ satisfying the three conditions in the conclusion of Lemma \[fixed product vector\]. We may assume $\|\eta_n\|=1$ by normalizing each $\eta_n$. We set $V:=U\eta$. Then we have $\gamma(V)=\epsilon_a \sigma(R^A_{K_a})V=\pm V$.
Let $e_n\in {\mathcal{M}}_n$ be the minimal projection satisfying $e_n\eta=\eta$ for all $n$, and set $f_n=Ue_nU^*$, which is a minimal projection of ${\mathcal{A}}^\Phi_a(I_n)$. Then we have $f_nV=V$ for all $n$. For each $n$, we can choose a normalized vector $V_n\in {\mathcal{E}}_\Phi(a_{n+1}-a_n)$ so that for any $X\in {\mathcal{E}}_\Phi(a_n)$ and $Y\in {\mathcal{E}}_\Phi(a-a_{n+1})$ we have $f_n(XV_nY)=XV_nY$. Since the self-adjoint unitary $U\rho_n(R_n)U^*\in {\mathcal{A}}^\Phi_a(I_n)$ satisfies the same equation as (\[eq in proof of (ii)\]) in place of $\sigma(R^A_{K_{I_n}})$, we have either $U\rho_n(R_n)U^*=\sigma(R^A_{K_{I_n}})$ or $UR_nU^*=-\sigma(R^A_{K_{I_n}})$. Thus $\rho_n(R_n)e_n\rho_n(R_n)^*=e_n$ implies $\sigma(R^A_{K_{I_n}})f_n\sigma(R^A_{K_{I_n}})^*=f_n$. Since $f_n$ is a minimal projection of ${\mathcal{A}}^\Phi_a(I_n)$ and $\sigma(R^A_{K_{I_n}})\in {\mathcal{A}}^\Phi_a(I_n)$ is a self-adjoint unitary, this shows that $XV_nY$ is an eigenvector of $\sigma(R^A_{K_{I_n}})$ and $R^A_{K_{I_n}}XV_nY=\pm XV_nY$. On the other hand, since $\alpha^\Phi_{a_n}(R^A_{a_{n+1}-a_n})=\pm R^A_{I_n}$ and $$R^A_{K_{I_n}}XV_nY=X({\alpha^\Phi_{a_n}}^{-1}(R^A_{I_n})V_n)Y,$$ we see that $V_n$ is an eigenvector of $\sigma(R^A_{a_{n+1}-a_n})$. Thus we get $\gamma(V_n)=\pm V_n$. Letting $W_n=(V_1V_2\cdots V_n)^*V$, we finish the proof.
Since $\|(1_{{\tilde{K}}}-P_O)C_\Phi P_O\|_{\mathrm{H.S.}}^2={\operatorname{tr}}\big(P_OAP_O-(P_OAP_O)^2\big)$, it suffices to show that the restriction of $\pi_A$ to ${\mathfrak{A}}(K_O)$ is a type I representation thanks to Theorem \[qfstate\](ii),(iii).
Let $L_n=K_{a_{2n+1}-a_{2n}}$, and let $L=\bigoplus_{n=0}^\infty L_n$. We denote by $\pi$ the representation of ${\mathfrak{A}}(L)$ on $H_A$ determined by $$\pi(a(f))=\pi_A(a(S_{a_{2n}}f))=\alpha^\Phi_{a_{2n}}(\pi_A(a(f))),\quad f\in L_n.$$ Since $\pi({\mathfrak{A}}(L))=\pi_A({\mathfrak{A}}(K_O))$, it suffices to show that $\pi$ is a type I representation. Let $V$, $V_n$, and $W_n$ be the normalized vectors obtained in Lemma \[decomposable vector\]. We set $\varphi(X)={\langle{\pi(X)V\Omega_A,V\Omega_A}\rangle}$ for $X\in {\mathfrak{A}}(L)$. Then $\varphi$ is a state of ${\mathfrak{A}}(L)$ whose GNS representation is quasi-equivalent to $\pi$. We show that $\varphi$ is pure using Lemma \[product state\].
Lemma \[making a state\] shows that there exists a $\gamma$-invariant pure state $\varphi_n$ of ${\mathfrak{A}}(L_n)$ satisfying $V_n^*\pi_A(X)V_n=\varphi_n(X)1$ for $\forall X\in {\mathfrak{A}}(L_n)$. Let $X_i\in {\mathfrak{A}}(L_i)$, $i=1,2,\cdots,n$. Then $$\begin{aligned}
\lefteqn{V^*\pi_A(X_0X_1\cdots X_n)V}\\
&=&W_{2n}^*V_{2n}^*\cdots V_1^*V_0^*\pi_A(X_0)\alpha^\Phi_{a_2}(\pi_A(X_1))\cdots \alpha^\Phi_{a_{2n}}(\pi_A(X_n))V_0V_1\cdots V_{2n}W_{2n}\\
&=&W_{2n}^*V_{2n}^*\cdots V_1^*V_0^*\pi_A(X_0)V_0V_1\pi_A(X_1)\cdots \alpha^\Phi_{a_{2n}-a_2}(\pi_A(X_n))V_2\cdots V_{2n}W_{2n} \\
&=&\varphi_0(X_0)W_{2n}^*V_{2n}^*\cdots V_2^*\pi_A(X_1)\cdots \alpha^\Phi_{a_{2n}-a_2}(\pi_A(X_n))V_2\cdots V_{2n}W_{2n} \\
&=&\varphi_0(X_0)\varphi_1(X_1)W_{2n}^*V_{2n}^*\cdots V_4^*\pi_A(X_2)\cdots \alpha^\Phi_{a_{2n}-a_4}(\pi_A(X_n))V_4\cdots V_{2n}W_{2n}\\
&=&\cdots=\varphi_0(X_0)\varphi_1(X_1)\cdots \varphi_n(X_n). \end{aligned}$$ Thus Lemma \[product state\] shows that $\varphi$ is a pure state, and in consequence, $\pi$ is a type I representation.
In order to apply Theorem \[CABATIF\] to concrete examples, we state the assumptions of Theorem \[CABATIF\] in terms of the regular part $\hat{\Phi}_0$ of the Fourier transform $\hat{\Phi}$.
\[integral\] Let the notation be as in Theorem \[CABATIF\]. Assume that $\Phi$ is an even function. Then
- The assumption of Theorem \[CABATIF\],(i) holds if and only if $$\int_0^\infty \sum_{n=0}^\infty\min\{x,|I_n|\}{\operatorname{tr}}(|\hat{\Phi}_0(x)|^2)dx<\infty.$$
- The assumption of Theorem \[CABATIF\],(ii) holds if and only if $$\int_0^\infty |O\ominus (O+x)|{\operatorname{tr}}(|\hat{\Phi}_0(x)|^2)dx<\infty,$$ where $O\ominus (O+x)$ is the symmetric difference of $O$ and $O$ translated by $x$.
\(i) The statement follows from $|(I_n\setminus (I_n+t))|=\min\{|t|,|I_n|\}$ and Lemma \[interval\],(ii).
\(ii) We set $J_{-1}:=(-\infty,0)$, $J_0:=(a,\infty)$, and $J_n=I_{2n-1}$ for $n\in {\mathbb{N}}$. Then $$\|(1_{{\tilde{K}}}-P_O)C_\Phi P_O\|_{\mathrm{H.S.}}^2=\sum_{m=-1}^\infty\sum_{n=0}^\infty\|P_{J_m}C_\Phi P_{I_{2n}}\|_{H.S.}^2.$$ The statement follows from this and Lemma \[interval\],(ii).
Lemma \[PQ\] implies $\|(1_{{\tilde{K}}}-P_E)C_\Phi P_E\|_{\mathrm{H.S.}}^2=\|C_{1-\Phi}P_EC_\Phi\|_{\mathrm{H.S.}}^2$. Thus by using Fourier transform, we can also get the following criteria, though we do not use them in this paper.
Let the notation be as in Theorem \[CABATIF\]. Then
- The assumption of Theorem \[CABATIF\],(i) holds if and only if $$\int_{{\mathbb{R}}^2}\frac{{\operatorname{tr}}(|\Phi(p)-\Phi(q)|^2)}{|p-q|^2} \sum_{n=0}^\infty\sin^2\frac{|I_n|(p-q)}{2}dpdq<\infty.$$
- The assumption of Theorem \[CABATIF\],(ii) holds if and only if $$\int_{{\mathbb{R}}^2}{\operatorname{tr}}(|\Phi(p)-\Phi(q)|^2)|\hat{\chi_O}(p-q)|^2dpdq<\infty.$$
Examples
========
Applying Theorem \[CABATIF\] to concrete sequences, we get the following theorem, which provides us with a computable invariant for type III Toeplitz CAR flows.
\[Sobolev 2\] Let $\Phi\in L^\infty({\mathbb{R}})\otimes M_N({\mathbb{C}})$ be an admissible symbol satisfying $\Phi(p)=\Phi(-p)$ for all $p\in {\mathbb{R}}$, and let $0<\mu<1$. We set $a_0=0$, $$a_n=\sum_{k=1}^n\frac{1}{k^{1/(1-\mu)}},\quad n\in {\mathbb{N}},$$ and $a=\lim_{n\to \infty}a_n$. Then the following three conditions are equivalent:
- The type I factorization $\{{\mathcal{A}}^{\Phi}_a(a_n,a_{n+1})\}_{n=0}^\infty$ is a CABATIF.
- $$\int_0^\infty\int_0^\infty\frac{{\operatorname{tr}}(|\Phi(p)-\Phi(q)|^2)}{|p-q|^{1+\mu}}dpdq<\infty.$$
- $$\int_0^\infty x^\mu {\operatorname{tr}}(|\hat{\Phi}_0(x)|^2)dx<\infty.$$
Moreover,
- If $\{{\mathcal{A}}^{\Phi}_a(a_n,a_{n+1})\}_{n=0}^\infty$ is a CABATIF, then $$\int_0^\infty{\operatorname{tr}}(|\Phi(2p)-\Phi(p)|^2) \frac{dp}{p^{\mu}}<\infty.$$
- If $\Phi$ is differentiable and $$\int_0^\infty {\operatorname{tr}}(|\Phi'(p)|^2)p^{2-\mu}dp<\infty,$$ then $\{{\mathcal{A}}^{\Phi}_a(a_n,a_{n+1})\}_{n=0}^\infty$ is a CABATIF.
The statement follows from Lemma \[Besov\], Lemma \[integral\], and Lemma \[Oestimate\] below applied to $h(x)=x^{\mu-1}$.
The following lemma is more or less [@IS Lemma 8.6].
\[Oestimate\] Let $h(x)$ be a non-negative strictly decreasing continuous function on $(0,\infty)$ satisfying $\lim_{x\to +0}h(x)=\infty$, $\lim_{x\to \infty}h(x)=0$, and $$\int_0^1h(x)dx<\infty.$$ We set $a_0=0$, $$a_n=\sum_{k=1}^nh^{-1}(k),\quad n\in {\mathbb{N}},$$ $I_n=(a_n,a_{n+1})$, and $O=\bigcup_{n=0}^\infty I_{2n}$. Then the sequence $\{a_n\}_{n=0}^\infty$ converges, and $$x(h(x)-1)\leq |O\ominus (O+x)|\leq 2\sum_{n=0}^\infty \min\{x,|I_n|\}\leq 2\int_0^x h(t)dt,\quad \forall x>0.$$
Note that we have $$\sum_{k=n+1}^\infty h^{-1}(k)\leq \int_0^{h^{-1}(n+1)}h(t)dt-nh^{-1}(n+1),$$ and in particular, the sequence $\{a_n\}_{n=0}^\infty$ converges. Since $\min\{x,|I_n|\}=|I_n\setminus (I_n\pm x)|$, the middle inequality follows from the definition of $O$.
For fixed $x>0$, we take the unique non-negative integer $n$ satisfying $h^{-1}(n+1)<x\leq h^{-1}(n)$ (or equivalently, $n\leq h(x)<n+1$). Then $$\begin{aligned}
\sum_{k=0}^\infty \min\{x,|I_k|\}&=&\sum_{k=0}^{n-1} x+\sum_{k=n}^\infty |I_k|=nx+\sum_{k=n}^\infty h^{-1}(k+1)\\
&\leq&\int_0^{h^{-1}(n+1)}h(t)dt+n(x-h^{-1}(n+1))\\
&\leq&\int_0^xh(t)dt. \end{aligned}$$ When $n$ is even, counting only contribution from $\{I_{2k}\}_{k=0}^{(n-2)/2}$, we get $$|(U+x)\setminus U|\geq \frac{n}{2}x.$$ In a similar way, we get $$|U\setminus (U+x)|\geq \frac{n}{2}x,$$ and so $$|U\ominus (U+x)|\geq nx\geq (h(x)-1)x.$$ When $n$ is odd, we have $|(U+x)\setminus U|\geq \frac{n+1}{2}x$ and $|U\setminus (U+x)|\geq \frac{n+1}{2}x$ in a similar way, which shows $|U\ominus (U+x)|\geq xh(x)$.
Now we apply Theorem \[Sobolev 2\] to concrete examples.
\[uncountable\] For $\nu>0$, let $\theta_\nu(p)=(1+p^2)^{-\nu}$, and let $$\Phi_\nu(p)=\frac{1}{2}\left(
\begin{array}{cc}
1 &e^{i\theta_\nu(p)} \\
e^{-i\theta_\nu(p)} &1
\end{array}
\right).$$ Then $\Phi_\nu$ is admissible. Let $\alpha^\nu:=\alpha^{\Phi_\nu}$ be the corresponding Toeplitz CAR flow.
- If $\nu>1/4$, then $\alpha^\nu$ is of type I$_2$.
- If $0<\nu\leq 1/4$, then $\alpha^\nu$ is of type III.
- If $0<\nu_1<\nu_2\leq 1/4$, then $\alpha^{\nu_1}$ and $\alpha^{\nu_2}$ are not cocycle conjugate.
The fact that $\Phi_\nu$ is admissible follows from Theorem \[Sobolev\],(ii). (i) and (ii) follow from Theorem \[dichotomy\]. To show (iii), we choose $\mu$ in the interval $(1-4\nu_2,1-4\nu_1)$, which satisfies $0<\mu<1$. Applying Theorem \[Sobolev 2\],(i),(ii) to this $\mu$ and $\Phi=\Phi_{\nu_i}$, $i=1,2$, we see that $\{{\mathcal{A}}^{\Phi_{\nu_2}}_a(a_n,a_{n+1})\}_{n=0}^\infty$ is a CABATIF, while $\{{\mathcal{A}}^{\Phi_{\nu_1}}_a(a_n,a_{n+1})\}_{n=0}^\infty$ is not. Therefor $\alpha^{\nu_1}$ and $\alpha^{\nu_2}$ are not cocycle conjugate.
Let $\Phi$ be as in Example \[nonconverging\], and let $\mu$ and $\{a_n\}_{n=0}^\infty$ be as in Theorem \[Sobolev 2\]. Then Theorem \[Sobolev 2\],(i) implies that $\{{\mathcal{A}}^\Phi_a(a_n,a_{n+1})\}_{n=0}^\infty$ is not a CABATIF for any $0<\mu<1$. This shows that $\alpha^\Phi$ is not cocycle conjugate to $\alpha^\nu$ for any $\nu$.
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B. V. Rajarama Bhat and R. Srinivasan, *On product systems arising from sum systems.* Infinite dimensional analysis and related topics, Vol. 8, Number 1, March 2005.
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M. Izumi and R. Srinivasan, *Generalized CCR flows.* Comm. Math. Phys. **281** (2008), 529–571.
V. Liebscher, *Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces.* to appear in Mem. Amer. Math. Soc. arXiv:math/0306365.
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R. T. Powers, *A nonspatial continuous semigroup of $*$-endomorphisms of $B(H)$.* Publ. Res. Inst. Math. Sci. **23** (1987), 1053-1069.
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G. L. Price, B. M. Baker, P. E. T. Jorgensen and P. S. Muhly, (Editors), [*Advances in Quantum Dynamics*]{}. (South Hadley, MA, 2002) Contemp. Math. 335, Amer. Math. Society, Providence, RI (2003).
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[^1]: Supported in part by the Grant-in-Aid for Scientific Research (C) 19540214, JSPS
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Kuramoto model is a classical model used in the study of spontaneous synchronizations in networks of coupled oscillators. In this model, frequency synchronization configurations can be formulated as complex solutions to a system of algebraic equations. Recently, upper bounds to the number of frequency synchronization configurations in cycle networks of $N$ oscillators were calculated under the assumption of generic non-uniform coupling. In this paper, we refine these results for the special cases of uniform coupling. In particular, we show that when, and only when, $N$ is divisible by 4, the upper bound for the number of synchronization configurations in the uniform coupling cases is significantly less than the bound in the non-uniform coupling cases. This result also establishes an explicit formula for the gap between the birationally invariant intersection index and the Bernshtein-Kushnirenko-Khovanskii bound for the underlying algebraic equations.'
address: 'Department of Mathematics, Auburn University Montgomery, Montgomery, Alabama'
author:
- Tianran Chen
- Evgeniia Korchevskaia
bibliography:
- 'library.bib'
title: On the root count of algebraic Kuramoto equations in cycle networks with uniform coupling
---
Introduction
============
The spontaneous synchronization of oscillators is a ubiquitous phenomenon that appear naturally in many seemingly independent biological, mechanical, and electrical systems. One of the classical models in the study of synchronization is the Kuramoto model [@Kuramoto1975Self], which describes the dynamics of a network of coupled oscillators. The Kuramoto model has been extensively studied in the recent decades and remains an enduring subject for the modeling of synchronization phenomena arising from the areas of science and engineering. Originally, the Kuramoto model had been applied to infinite complete networks (with all-to-all coupling). In order to adapt the model to complex topologies, numerous reformulations of the Kuramoto model have been introduced and studied, both analytically and numerically. See, e.g., review articles [@ARENAS200893; @dorfler_synchronization_2014; @RODRIGUES20161]. One of such generalizations of the Kuramoto model investigates synchronizations within cycle networks (i.e., ring-like networks) [@DelabaysColettaJacquod2016Multistability; @Denes2019Pattern; @Ha2012Basin; @ManikTimmeWitthaut2017Cycle; @Rogge2004Stability; @Roy2012Synchronized; @Xi2017Synchronization]. From an algebraic view point, the upper bound on the number of frequency synchronization configurations (including complex configurations) that can exist in a cycle network of $N$ oscillators with generic non-uniform coupling is shown to be $N \binom{N-1}{ \lfloor (N-1)/2 \rfloor }$ [@ChenDavisMehta2018Counting; @dal2019faces]. The main contribution of this paper is a significant refinement of this upper bound for the case of cycle networks with *uniform coupling*. We show that if $N$ is not divisible by 4, despite being a very special case, the generic number of complex synchronization configurations is the same as in the cases with generic non-uniform coupling. On the other hand, if $N$ is divisible by 4, then the generic synchronization configuration count is significantly lower than the count in the case of non-uniform coupling. This result also quantifies the gap between the birationally invariant intersection index of a family of rational functions over the toric variety $({\mathbb{C}}^*)^n$ and the Bernshtein-Kushnirenko-Khovanskii bound [@Bernshtein1975Number; @Khovanskii1978Newton; @Kushnirenko1976Polyedres] of a generic algebraic system in this family.
This paper is organized as follows. In , we review the Kuramoto equations and their algebraic formulation and state the root counting problems which this paper focuses on. describes the construction of the adjacency polytope and explores its geometric properties which are central to our main arguments. Then in , we establish the main result which is the generic root count for Kuramoto equations arising from cycle networks with uniform coupling. The conclusion follows in .
This project grew out of authors’ discussion with Anton Leykin and Josephine Yu in 2017 and with Anders Jensen and Yue Ren in 2018 on closely related problems. The authors also thank Rob Davis for kindly sharing his insights into the geometric structures of adjacency polytopes.
Preliminaries and problem statements {#sec:Review and problem statement}
====================================
Kuramoto model with uniform coupling and synchronization equations {#sec: Kuramoto model}
------------------------------------------------------------------
[r]{}[0.25]{}
\(0) at ( 0,1.8) [0]{}; (1) at (-0.9,0.9) [1]{}; (2) at (0,0) [2]{}; (3) at (0.9,0.9) [3]{}; (1) edge (0); (1) edge (2); (3) edge (2); (3) edge (0);
The network of $N$ coupled nonlinear oscillators can be modeled by an undirected graph $G$ with vertices $V(G)$ and edges $E(G)$ representing the oscillators and their connections respectively. In addition, each oscillator $i$ has its natural frequency $\omega_i$, while a nonzero constant $K$ quantifies the coupling strength between two oscillators. In this paper, we focus exclusively on the cases of cycle graphs of $N$ vertices. That is, we only consider $G = C_N$ with $V(C_N) = \{0,1,\dots,n\}$, $n=N-1$ and $E(C_N) = \{ \{0,1\}, \{1,2\},\dots,\{i,i+1\},\dots,\{n-1,n\},\{n,0\} \}$. For example, shows a cycle network of 4 oscillators. We also assume that the oscillators are non-homogeneous, i.e., $\omega_0,\dots,\omega_n$ are distinct, but the coupling is uniform, i.e., the strength of the coupling along any edge is $K$. Under these assumptions, the dynamics of the system is described by the Kuramoto model with the governing equations $$\label{equ:kuramoto-sin}
\frac{d \theta_i}{dt}=\omega_i - K \sum_{j \in \mathcal{N}_{C_N}(i)} \sin(\theta_i - \theta_j)
\quad\text{for } i = 0,\dots,n,$$ where $\theta_i$ is the phase angle of the $i$-th oscillator and $\mathcal{N}_{C_N}(i)$ is the set of adjacent nodes of node $i$ in the $C_N$. Frequency synchronization configurations (simply synchronization configurations, hereafter) are defined to be configurations of $(\theta_0,\dots,\theta_n)$ for which all oscillators are tuned to have the exact same angular velocity. That is, there is a single constant $c$ such that $\frac{d\theta_i}{dt} = c$ for $i=0,\dots,n$. By adopting a proper rotational frame of reference, we can further assume $\theta_0 = 0$ and $c = 0$. Then the synchronization configurations are defined by the equilibrium conditions $\frac{d \theta_i}{dt}=0$ for $i=1,\dots,n$. That is, they are solutions to the system of transcendental equations $$\label{equ:kuramoto-sin=0}
0 =
\omega_i - K \sum_{j \in \mathcal{N}_{C_N}(i)} \sin(\theta_i - \theta_j)
\quad\text{for } i = 1,\dots,n.$$ Throughout this paper, we will use the following notations. Let $f = \sum_{{\mathbf{a}}\in S} c_{{\boldsymbol{\alpha}}} {\mathbf{x}}^ {\mathbf{a}}$ denote a Laurent polynomial in $n$ variables ${\mathbf{x}}= (x_1,\dots,x_n)$ with coefficients in ${\mathbb{C}}$, where the finite set $S \subset {\mathbb{Z}}^n$, known as its support, collects the exponents and ${\mathbf{x}}^{\mathbf{a}}= x_1^{a_1} \cdots x_n^{a_n}$ represents the monomial with exponent ${\mathbf{a}}= (a_1,\dots,a_n)^\top$. The Newton polytope of $f$ is the set $\operatorname{Newt}(f):=\operatorname{conv}(S) \subset {\mathbb{R}}^n$. With respect to a nonzero vector ${\boldsymbol{\alpha}}\in {\mathbb{R}}^n$, the initial form of $f$ is $\operatorname{init}_{{\boldsymbol{\alpha}}}f := \sum_{{\mathbf{a}}\in (S)_{\mathbf{v}}} c_{{\boldsymbol{\alpha}}} {\mathbf{x}}^ {\mathbf{a}}$, where $(S)_{\mathbf{v}}$ is the subset of $S$ on which the linear functional ${\langle \, {\mathbf{v}}\,,\, \cdot \, \rangle}$ is minimized over $S$. For a system ${\mathbf{f}}= (f_1,\dots,f_n)^\top$ of Laurent polynomials, the initial system with respect to ${\boldsymbol{\alpha}}\in {\mathbb{R}}^n$ is $\operatorname{init}_{{\boldsymbol{\alpha}}} {\mathbf{f}}:= (\operatorname{init}_{{\boldsymbol{\alpha}}}f_1, \dots , \operatorname{init}_{{\boldsymbol{\alpha}}} f_n)^\top$. While considering the root count of the system ${\mathbf{f}}$, we will make use of the Bernshtein’s theorem [@Bernshtein1975Number], which states that the generic root count of the system $f$ in $({\mathbb{C}}^*)^n = ({\mathbb{C}}\setminus \{0\})^n$ is given by the mixed volume of the Newton polytopes $\operatorname{Newt}(f_i)$, $i=1,\dots, n$. This count is also known as the Bernshtein-Kushnirenko-Khovanskii (BKK) bound [@Bernshtein1975Number; @Kushnirenko1976Polyedres; @Khovanskii1978Newton].
The central question of finding the maximum number of synchronization configurations for a cycle network of oscillators with uniform coupling is equivalent to the root counting question of the system . To leverage the power of root counting results from algebraic geometry, the transcendental equations can be reformulated into an algebraic system via the change of variables $x_i = e^{{\mathbf{i}}\theta_i}$ for $i = 0,\dots,n$ where ${\mathbf{i}}= \sqrt{-1}$ and $x_0 = e^{{\mathbf{i}}0} = 1$ corresponds to the fixed phase angle of the reference oscillator. Then $\sin(\theta_i - \theta_j) = \frac{1}{2 {\mathbf{i}}}(\frac{x_i}{x_j} - \frac{x_j}{x_i})$, and is transformed into a system of $n$ Laurent polynomial equations $\mathbf{f} = (f_1,\dots,f_n)^\top = {\mathbf{0}}$ in the $n$ complex variables ${\mathbf{x}}= (x_1,\dots,x_n)$ given by $$ \label{equ:kuramoto-alg}
f_{i}(x_1,\dots,x_n) =
\omega_i - a \sum_{j \in \mathcal{N}_{C_N}(i)}
\left(
\frac{x_i}{x_j} - \frac{x_j}{x_i}
\right)
= 0
\quad \text{for } i = 1,\dots,n,$$ where $a = \frac{K}{2{\mathbf{i}}}$. This system captures all synchronization configurations in a way that the real solutions to correspond to the complex solutions of with each $|x_i| = |e^{{\mathbf{i}}\theta}| = 1$, i.e., solutions on the real torus $(S^1)^n$.
Problem statements
------------------
Counting solutions of an algebraic system on real torus $(S^1)^n$ is a notoriously difficult problem. Using Morse inequalities and the Betti numbers of the real torus, lower bounds on the generic solution count was established by Baillieul and Byrnes in certain cases [@Baillieul1982] (e.g. the cases of homogeneous oscillators with nondegenerate synchronization states). An upper bound for the number of solutions is also established to be ${2N-2}\choose{N-1}$ in the same papers by bounding the total number of complex solutions to . This upper bound does not take into consideration the graph topology and coupling coefficients and only depends on the number of oscillators $N$. Following this approach of counting complex solutions, later works (e.g. root counting results given by Guo and Salam [@Guo1990] and Molzahn, Mehta, and Niemerg [@MolzahnMehtaNiemerg2016Toward]) suggest that sparse networks tend to have less synchronization configurations. These observations have motivated a study on the tighter upper bound for the number of synchronization configurations in cycle networks with non-uniform coupling coefficients [@Chen2019Unmixing; @ChenDavisMehta2018Counting] where a sharp upper bound (counting complex synchronization configurations) is shown to be $N \binom{N-1}{ \lfloor (N-1)/2 \rfloor }$. In this paper we provide a significant refinement of the root count for the cases of cycle networks with uniform coupling.
An important result from the intersection theory is that, over the field of complex numbers, the maximal behavior is also, in a sense, the generic behavior (e.g. the Theorem of Bertini [@SommeseWampler2005Numerical]). The key question we aim to answer is therefore the following generic root count question.
\[prb:one\] Given a cycle graph $C_N$ of $N$ nodes and generic choices of parameters ${\boldsymbol{\omega}}= (\omega_1,\dots,\omega_n)$ and $a$, what is the total number of isolated complex solutions to the algebraic Kuramoto system ?
This question can also be stated in terms of birationally invariant intersection index. Since the $i$-th Laurent polynomial in is a linear combination of $1$ and $\ell_i := \sum_{j \in \mathcal{N}_{C_N}(i)} ( x_i / x_j - x_j / x_i)$ with generic coefficient. It can be considered as a generic element in the vector space of rational functions spanned by $\{ 1, \ell_i \}$. We are therefore interested in the intersection index of $n$ generic elements from these vector spaces in $({\mathbb{C}}^*)^n$. This is precisely the birationally invariant intersection index [@KavehKhovanskii2010Mixed].
\[prb:two\] Given a cycle graph $C_N$ of $N$ nodes, let $$L_i = \operatorname{span}
\left\{
1,
\sum_{j \in \mathcal{N}_{C_N}(i)} \left( \frac{x_i}{x_j} - \frac{x_j}{x_i} \right)
\right\}$$ be the ${\mathbb{C}}$-vector space spanned by two rational functions for each $i=1,\dots,n$. What is the intersection index $[\, L_1,\dots,L_n \,]$ ?
This intersection index is less than or equal to the BKK bound for the same set of equations. In this paper, we show that there is a gap between the intersection index described above and the BKK bound if and only if $N$ is divisible by 4. Indeed, in this case, $[L_1,\dots,L_n]$ is significantly smaller than the BKK bound in the sense that the ratio of the two goes to zero as $N \to \infty$.
Adjacency polytope {#sec: Adjacency polytope}
==================
[r]{}[0.45]{}
\[scale=0.28\] (0,4)– (8,2); (8,2)– (4,-4); (4,-4)– (-4,-2); (-4,-2)– (0,4); (-4,4)– (4,2); (4,2)– (8,2); (4,2)– (0,-4); (-8,-2)– (-4,-2); (-4,-2)– (0,4); (0,4)– (-4,4); (-4,4)– (-8,-2); (0,4)– (-4,-2); (-8,-2)– (0,-4); (0,-4)– (4,-4); (4,-4)– (-4,-2); (-4,-2)– (-8,-2);
(-5,4.6) node [(0,-1,1)]{}; (0.8,4.6) node [(0,0,1)]{}; (-9.7,-1.5) node [(1,-1,0)]{}; (-1.9,-1.5) node [(1,0,0)]{}; (-1.2,-4.6) node [(0,0,-1)]{}; (5,-4.6) node [(0,1,-1)]{}; (8.8,2.8) node [(-1,1,0)]{}; (1.8, 1.6) node [(-1,0,0)]{};
(-8,-2) – (-4,4) – (0,4) – (-4,-2) – cycle;
Recent studies suggest that the range of possible synchronization configurations is strongly tied to the network topology [@Bronski2016Graph; @MolzahnMehtaNiemerg2016Toward]. A promising approach to elucidate this connection [@Chen2019Directed; @ChenDavisMehta2018Counting; @Chen2019Unmixing] makes use of a construction known as an “adjacency polytope”. This method allows us to encode the network topology and provides valuable insights into the algebraic structure of the Kuramoto equations. The adjacency polytope constructed in this context coincides with the symmetric edge polytope introduced earlier in the study of the roots of Ehrhart polynomials [@Matsui2011Roots]. The geometric structure of adjacency polytopes has been instrumental in the study of generic root count of the algebraic Kuramoto equations in the case of cycle networks with generic non-uniform coupling [@ChenDavisMehta2018Counting]. In this paper, we extract more refined initial form information from the polytopes and establish a sharper bound on the generic root count in the case of uniform coupling.
For the cycle graph $C_N$, its *adjacency polytope* is defined as $$\label{equ:AP}
\nabla_{C_N} = \operatorname{conv}\left\{
\pm ({\mathbf{e}}_i - {\mathbf{e}}_j)
\right\}_{\{i,j\} \in E(C_N)},$$ where ${\mathbf{e}}_i \in {\mathbb{R}}^n$ is the column vector with 1 on the $i$-th position and zero elsewhere for $i=1,\dots,n$, and ${\mathbf{e}}_0 = {\mathbf{0}}$. The polytope $\nabla_{C_N}$ is a full-dimensional centrally symmetric lattice polytope. It is originally constructed as the Newton polytope of the randomized system ${\mathbf{f}}^{R} := R \cdot {\mathbf{f}}$, which is created from a nonsingular $n \times n$ matrix $R = [ r_{ij} ]$ with generic entries. Here, components of ${\mathbf{f}}^R = (f_1^R,\dots,f_n^R)$ are of the form $$f_{k}^R =
c_k -
\sum_{\{i,j\} \in E(C_N),i<j} a_{ijk}^R
\left(
\frac{x_i}{x_j} - \frac{x_j}{x_i}
\right)
\quad\text{for } k = 1,\dots,n,$$ where ${\mathbf{c}}= (c_1,\dots,c_n)^\top = R \, {\boldsymbol{\omega}}$, $E(C_N)$ is the edge set of $C_N$, and $a^R_{ijk} = a (r_{ki} - r_{kj})$. Each component in ${\mathbf{f}}^R$ has the same set of terms, and the (unmixed) Newton polytope of ${\mathbf{f}}^R$, $\operatorname{Newt}({\mathbf{f}}^R)$, is precisely the adjacency polytope $\nabla_{C_N}$. Since $R$ is nonsingular, ${\mathbf{f}}$ and ${\mathbf{f}}^R = R \cdot {\mathbf{f}}$ share the same zero set. Yet, the randomization simplifies the algebraic structure of the problem and allows us to utilize the known results concerning the triangulation of $\nabla_{C_N}$ [@ChenDavis2018Toric; @ChenDavisMehta2018Counting]. The normalized volume of $\nabla_{C_N}$, also known as *adjacency polytope bound*, is the BKK bound for both the algebraic Kuramoto system and the randomized system ${\mathbf{f}}^R$.
In the following discussion, an important role is given to the facets ($(n-1)$-dimensional faces) of $\nabla_{C_N}$. As described in Ref. [@ChenDavisMehta2018Counting], when $N$ is odd, $\nabla_{C_N}$ is unimodularly equivalent to the *del Pezzo polytope* [@Nill2005Classification]. The number of facets of $\nabla_{C_N}$ is $N {{N-1}\choose{(N-1)/2}}$. Each facet is simplicial, unimodular, and given by $$\begin{split}
\label{def_oddfacet}
\operatorname{conv}\Biggl\{
\lambda_j ({\mathbf{e}}_i - {\mathbf{e}}_j)
\,\Bigm|\,
\{i,j\} \in E(C_N) \setminus \bigl\{\{p,q\}\bigr\} \text{ for some } \{p,q\} \in E(C_N) , \\ \lambda_1,\dots, \lambda_{q-1}, \lambda_{q+1}, \dots, \lambda_{n+1} \in \{ \pm 1 \},\;
\sum_{j=1}^{q-1} \lambda_j +\sum_{j=q+1}^{n+1} \lambda_j =0
\Biggr\}.
\end{split}$$
When $N$ is even, $\nabla_{C_N}$ has $N\choose{N/2}$ facets [@ChenDavis2018Toric; @dal2019faces]. Each facet $F$ is defined by exactly $N = n + 1$ vertices and is of the form $$\label{def_evenfacet}
\operatorname{conv}\Biggl\{
\lambda_j ({\mathbf{e}}_i - {\mathbf{e}}_j)
\, \Bigm| \, \{i,j\} \in E(C_N), \,
\lambda_1,\dots \lambda_{n+1} \in \{ \pm 1 \},\;
\sum_{j=1}^{n+1} \lambda_j =0
\Biggr\}.$$
\[example-c4\] (Running example, 4-cycle). Our reference example throughout this paper will be a a cycle network with $N=4$ coupled oscillators. See Figure \[fig: C4 network\]. Synchronization configurations of this network are characterized by the Algebraic Kuramoto equations $$\begin{aligned}
\label{ex: C-4 synchr.system}
\omega_1 - a\left(\frac{x_1}{x_0}-\frac{x_0}{x_1}+\frac{x_1}{x_2}-\frac{x_2}{x_1}\right)=0, \nonumber\\
\omega_2 - a\left(\frac{x_2}{x_1}-\frac{x_1}{x_2}+\frac{x_2}{x_3}-\frac{x_3}{x_2}\right)=0,\\
\omega_3 - a\left(\frac{x_3}{x_2}-\frac{x_2}{x_3}+\frac{x_3}{x_0}-\frac{x_0}{x_3}\right)=0. \nonumber
\end{aligned}$$ The adjacency polytope associated with this network is a parallelepiped as illustrated on Figure \[fig: adj polytope\]. Its six facets are $$\begin{aligned}
\label{facets 4-cycle}
&\operatorname{conv}\left\{
\left[ \begin{smallmatrix} 1\\0\\0 \end{smallmatrix}
\right],
\left[\begin{smallmatrix} \phantom{-}1\\-1\nonumber\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\{-}1\\\phantom{-}1 \end{smallmatrix} \right],
\left[\begin{smallmatrix} 0\\0\\1 \end{smallmatrix}
\right] \right\},
&&\operatorname{conv}\left\{
\left[\begin{smallmatrix} 1\\0\\0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} {-}1\\\phantom{-}1\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\\phantom{-}1\\{-}1 \end{smallmatrix}\right],
\left[\begin{smallmatrix} 0\\0\\1 \end{smallmatrix}\right]
\right\},\\
&\operatorname{conv}\left\{
\left[\begin{smallmatrix} 1\\0\\0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}1\\{-}1\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\\phantom{-}1\\{-}1 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\\phantom{-}0\\{-}1 \end{smallmatrix}\right]
\right\},
&&\operatorname{conv}\left\{
\left[\begin{smallmatrix} {-}1\\\phantom{-}0\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} {-}1\\\phantom{-}1\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\{-}1\\\phantom{-}1 \end{smallmatrix}\right],
\left[\begin{smallmatrix} 0\\0\\1 \end{smallmatrix}\right]
\right\},\\
&\operatorname{conv}\left\{
\left[\begin{smallmatrix} {-}1\\\phantom{-}0\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} {-}1\\\phantom{-}1\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\\phantom{-}1\\{-}1 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\\phantom{-}0\\{-}1 \end{smallmatrix}\right]
\right\},
&&\operatorname{conv}\left\{
\left[\begin{smallmatrix} {-}1\\\phantom{-}0\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}1\\{-}1\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\{-}1\\\phantom{-}1 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\\phantom{-}0\\{-}1 \end{smallmatrix}\right]
\right\}.\nonumber\end{aligned}$$ The normalized volume of each facet is 2, and the adjacency polytope bound is 12. This also agrees with the BKK bound of . We will show, however, that the generic root count under the uniform coupling condition, i.e., the intersection index $[L_1,L_2,L_3]$, is only 6.
We next prove some important properties of the facets of $\nabla_G$. For a facet $F$ of $\nabla_G$, we define its *facet matrix* to be a matrix $V$ whose columns are the vertices of $F$, arranged in such a way, that vertex $\lambda_k ({\mathbf{e}}_i - {\mathbf{e}}_j)$ corresponds to the $i$-th column of $V$. Let us denote by $V^*$ the reduced row echelon form of $V$. We will refer to $V^*$ as a *reduced facet matrix*. Finally, we define a *facet reduction matrix* $Q$ as an $(N-1) \times (N-1)$ matrix that satisfies $QV=V^*$.
\[Lemma: reduction matrix Q\] For a facet $F$ of $\nabla_G$ with its facet matrix $V$ and reduced facet matrix $V^*$, the facet reduction matrix $Q$ is a unimodular integer matrix.
If $N$ is odd, then $V$ is a square unimodular matrix with integer entries. Its reduced row echelon form is therefore the identity matrix, and $Q = V^{-1}$ is a unimodular integer matrix.
If $N$ is even, employing the description of the facets given in , we write each facet matrix $V$ as $$V=\begin{bmatrix}
-1 & 1 & & \\
& \ddots & \ddots & \\
& & -1 & 1 \\
\end{bmatrix}
\begin{bmatrix}
\lambda_1 & & \\
& \ddots & \\
& & \lambda_N \\
\end{bmatrix}.$$ We define an $(N-1) \times (N-1)$ unimodular (integer) matrix $$Q \coloneqq
\begin{bmatrix}
-\lambda_{1} & & & \\
& -\lambda_{2} & & \\
& & \ddots & \\
& & & -\lambda_{N-1} \\
\end{bmatrix}
\begin{bmatrix}
1 & 1 & \dots & 1 \\
& 1 & \dots & 1 \\
& & \ddots & \vdots \\
& & & 1 \\
\end{bmatrix},$$ which has determinant $\pm 1$ since $\lambda_k = \pm 1$. Via a direct computation, we observe that $$\label{V*}
QV = \begin{bmatrix}
1 & & & & -\lambda_{1} \lambda_{N} \\
& 1 & & & -\lambda_{2} \lambda_{N} \\
& & \ddots & & \vdots \\
& & & 1 & -\lambda_{N-1} \lambda_{N} \\
\end{bmatrix}=V^*.$$ Therefore, the integer matrix $Q$ is the facet reduction matrix.
It has been mentioned in Ref. [@ChenDavis2018Toric] that facets of $\nabla_G$ are unimodularly equivalent to one another. In , we provide an equivalent proof in terms of facet matrices.
\[prop.unimod.equiv\] Let $F$ and $F'$ be two facets of the adjacency polytope $\nabla_{C_N}$, and let $V$ and $V'$ be their corresponding facet matrices. Then there exist a unimodular $(N-1) \times (N-1)$ matrix $U$ and a $N \times N$ permutation matrix $P$ such that $UVP=V'$.
(Odd N) For odd $N$, $V$ is square and unimodular, and therefore $V^{-1}$ is an integer matrix. We let $U = V' V^{-1}$, and let $P=I_{N-1}$ be the $(N-1) \times (N-1)$ identity matrix, then $UVP=V'$.
(Even $N$) For even $N$, let $(\lambda_{1}, \dots \lambda_{N}), (\lambda_1',\dots,\lambda_N') \in \{-1,1\}^N$ be the indices of the $F$ and $F'$ respectively. These indices satisfy the conditions $\sum_{i=1}^N \lambda_i = 0$ and $\sum_{i=1}^N \lambda_i' = 0$. Therefore exactly half of the entries in each collection of indices are positive. Let $Q$ and $Q'$ be the facet reduction matrices of $F$ and $F'$ respectively. As stated in , the corresponding reduced facet matrices are given by $$\begin{aligned}
V^* = QV &=
\left[
\begin{smallmatrix}
1 & & & & -\lambda_{1} \lambda_{N} \\
& 1 & & & -\lambda_{2} \lambda_{N} \\
& & \ddots & & \vdots \\
& & & 1 & -\lambda_{N-1} \lambda_{N} \\
\end{smallmatrix}
\right]
&&\text{and}&
(V')^* = Q'V' &=
\left[
\begin{smallmatrix}
1 & & & & -\lambda_{1}' \lambda_{N}' \\
& 1 & & & -\lambda_{2}' \lambda_{N}' \\
& & \ddots & & \vdots \\
& & & 1 & -\lambda_{N-1}' \lambda_{N}' \\
\end{smallmatrix}
\right].\end{aligned}$$ We observe that if $\lambda_N$ is positive, then the list $\lambda_1, \dots, \lambda_{N-1}$ contains $\frac{N}{2}-1$ positive and $\frac{N}{2}$ negative entries. If $\lambda_N$ is negative, then the list $\lambda_1. \dots ,\lambda_{N-1}$ contains $\frac{N}{2}-1$ negative and $\frac{N}{2}$ positive entries. It follows that the last columns of both $V^*$ and $(V')^*$ have exactly $\frac{N}{2}$ entries equal $1$ and $\frac{N}{2}-1$ entries equal $-1$. Therefore $V^*$ and $(V')^*$ are equal up to a permutation of the entries in the last column. In other words, there exist permutations matrices $L$ and $P$ of sizes $(N-1) \times (N-1)$ and $N \times N$ respectively such that $$L V^* P=(V')^*.$$ Let $U = (Q')^{-1} L Q$, which is unimodular since $Q'$, $L$, and $Q$ are all unimodular, then we have $U V P = V'$, as desired.
Face and facet subsystems
-------------------------
Faces of the adjacensy polytope $\nabla_{C_N}$ give rise to face and facet subsystems that form the foundation of our root counting argument. Let $F$ be a positive-dimensional face of $\nabla_{C_N}$, then the *face subsystem* induced by $F$ is the system given by $$f_{F,k}^R =
c_k -
\sum_{({\mathbf{e}}_i - {\mathbf{e}}_j) \in F} (a_{ijk} - a_{jik}) \, \frac{x_i}{x_j}
\quad\text{ for } k=1,\dots,n,$$ which consists of all the terms in the algebraic Kuramoto system with exponents vectors in $F$ together with the constant terms. Using the compact vector exponent notation, we can write the system as $${\mathbf{f}}_F^R =
{\mathbf{c}}- a \sum_{({\mathbf{e}}_i - {\mathbf{e}}_j) \in F}
(R_i - R_j) \, ({\mathbf{x}}^{({\mathbf{e}}_i - {\mathbf{e}}_j)})^\top
\label{eq: def. facet subsystem}$$ where $R_k$ is the $k$-th column of $R$ if $k \ne 0$ and the zero vector otherwise. If $F$ is a facet, we call ${\mathbf{f}}_F^R$ a *facet subsystem*. Facet subsystems correspond to facet subnetworks investigated in Ref. [@Chen2019Directed].
Counting roots {#sec: counting roots}
==============
The main goal here is to provide a refined generic root count in the special case of uniform coupling, which will be the answer for both and . This will be done via analysis of the roots of the much simpler face subsystems described above. Throughout this section, we assume the choices of the natural frequencies $\omega_i$’s and the complex coupling coefficient $a$ to be generic. This can be interpreted in terms of Zariski topology within the space of all possible coefficients. We say that a property holds for a generic choice of coefficients, if there is a nonempty Zariski open set of the coefficients for which this property holds. From a probabilistic point of view, “genericity" implies that if $a$ and ${\boldsymbol{\omega}}$ are selected at random, then the property holds with *probability one*. Now, following a standard “generic smoothness” argument, we show that the solution set of such a face subsystem consists of nonsingular points.
Let $C_N$ be a cycle graph of $N$ nodes, and let $F$ be a face of $\nabla_{C_N}$. For generic choices of ${\mathbf{c}}\in {\mathbb{C}}^n$ and $a \in {\mathbb{C}}^*$, the complex zero set of the face subsystem ${\mathbf{f}}_F^R$ is either empty or consists of nonsingular isolated points.
This statement follows directly from the properties of a generic member of a linear system of divisors away from the base locus. Here, we include a short proof for completeness.
The face subsystem ${\mathbf{f}}_F^R$ is a linear combination of the system of Laurent polynomials $$\{ 1 \} \cup
\{ x_i x_j^{-1} \}_{({\mathbf{e}}_i - {\mathbf{e}}_j) \in F}$$ with coefficients that are the images of ${\mathbf{c}}\in {\mathbb{C}}^n$ and $aR \in {\mathbb{C}}^n \times {\mathbb{C}}^n$ under a nonsingular linear transformation. Note that the base locus of this system in $({\mathbb{C}}^*)^n$ is empty. Then by Bertini’s theorem, there exists a Zariski open set $U$ of ${\mathbb{C}}^n \times {\mathbb{C}}^n \times {\mathbb{C}}^n$ such that $({\mathbf{c}},aR) \in U$ implies that the zero set of ${\mathbf{f}}_F^R$ in $({\mathbb{C}}^*)^n$ is either empty or 0-dimensional and nonsingular.
Since we require $a \in {\mathbb{C}}^*$ and $R$ nonsingular, by assumption, the inverse image of $U$ in ${\mathbb{C}}^* \times {\mathbb{C}}^n \times {\mathbb{C}}^n$ remains Zariski open. Therefore, for generic choices of ${\mathbf{c}}\in {\mathbb{C}}^n$ and $a \in {\mathbb{C}}^*$, the zero set of the facet subsystem ${\mathbf{f}}_F^R$ consists of nonsingular isolated points.
By considering the entire polytope $\nabla_{C_N}$ as a face, the above lemma implies that the generic solution set to the algebraic Kuramoto equation consists of nonsingular isolated points.
\[cor:f-count\] For generic choices of ${\mathbf{c}}\in {\mathbb{C}}^n$ and $a \in {\mathbb{C}}^*$, the complex zero set of ${\mathbf{f}}$ consists of nonsingular isolated points, and the total number is a constant which is independent from ${\mathbf{c}}$ and $a$.
We establish the root count for ${\mathbf{f}}$ by studying the root count of individual facet subsystems, since the complex zeros of ${\mathbf{f}}$ are one-to-one correspondence with complex zeros of all facet subsystems as show in Ref. [@Chen2019Directed]. Here, we restate the result in the current context.
\[thm:facet-sum\] For generic choices of ${\mathbf{c}}\in {\mathbb{C}}^n$ and $a \in {\mathbb{C}}^*$, the complex zeros of the algebraic Kuramoto system ${\mathbf{f}}$ on $C_N$ with uniform coupling are all isolated and nonsingular, and the total number is exactly the sum of the complex root count of all the facet subsystems ${\mathbf{f}}_F^R$ over all facets $F \in \mathcal{F}(\nabla_{C_N})$.
The root counting question ( and ) is now reduced to computing the root count of each facet subsystem. The root count for facet subsystems associated with a cycle network with *non-uniform coupling* is established in Ref. [@ChenDavisMehta2018Counting]. However, under the additional condition of *uniform coupling* the generic root count could be strictly less. We analyze this gap using Bernshtein’s second theorem, which states that the actual ${\mathbb{C}}^*$-root count for a system of $n$ polynomial in $n$ variables is strictly less than the BKK bound if and only if there is an initial system that has a nontrivial solution in $({\mathbb{C}}^*)^n$.
Let $\mathbf{f} = (f_1,\dots,f_n)^\top$ be a Laurent polynomial system in $n$ complex variables with Newton polytopes $P_1,\dots,P_n$. If an initial system $\operatorname{init}_{{\boldsymbol{\alpha}}} \mathbf{f}$ has no roots in $({\mathbb{C}}^*)^n$ for any ${\boldsymbol{\alpha}}\neq {\mathbf{0}}$, then all roots of $\mathbf{f}$ in $({\mathbb{C}}^*)^n$ are isolated and their number, counting multiplicity, equals the mixed volume of $P_1, \dots, P_n$. If an initial system $\operatorname{init}_{{\boldsymbol{\alpha}}} \mathbf{f}$ has a root in $({\mathbb{C}}^*)^n$ for some ${\boldsymbol{\alpha}}\neq {\mathbf{0}}$, then the number of isolated roots of the system $\mathbf{f}$ in $({\mathbb{C}}^*)^n$ counted according to multiplicity, is strictly smaller than the mixed volume of $P_1, \dots, P_n$, given this mixed volume is nonzero.
The following theorem provides the condition under which an initial system we are interest in has a nontrivial solution.
\[prop.div.4\] Given a facet subsystem ${\mathbf{f}}_F^R$, an initial system $\operatorname{init}_{{\boldsymbol{\alpha}}} {\mathbf{f}}_F^R$ for ${\boldsymbol{\alpha}}\ne {\mathbf{0}}$ has a zero in $({\mathbb{C}}^*)^n$ if and only if $N$ is divisible by 4 and ${\boldsymbol{\alpha}}$ is the inner normal vector of the facet $F$ of $\nabla_{C_N}$.
We shall first consider an initial system induced by a facet $F$ of $\nabla_{C_N}$ listed in and . Let ${\boldsymbol{\alpha}}$ be an inner normal vector of $F$ in $\nabla_{C_N}$, then the induced initial form is $$\operatorname{init}_{\alpha} {\mathbf{f}}_F^R =
- a \sum_{({\mathbf{e}}_i - {\mathbf{e}}_j) \in F}
(R_i - R_j) \, ({\mathbf{x}}^{({\mathbf{e}}_i - {\mathbf{e}}_j)})^\top$$ which is equivalent to the system $$\label{equ: unmixed init.system}
-\frac{1}{a}R^{-1} \operatorname{init}_{\alpha} {\mathbf{f}}_F^R =
\sum_{({\mathbf{e}}_i - {\mathbf{e}}_j) \in F}
({\mathbf{e}}_i - {\mathbf{e}}_j) \, ({\mathbf{x}}^{({\mathbf{e}}_i - {\mathbf{e}}_j)})^\top
= V \, ({\mathbf{x}}^{V})^\top,$$ where $V$ is the facet matrix.
(Odd N) If $N$ is odd, then with the biholomorphic change of variables ${\mathbf{y}}= {\mathbf{x}}^{V}$, the system is equivalent to $$V^{-1}V \left(({\mathbf{y}}^{-V})^V)\right)^\top={\mathbf{y}}^\top,$$ as far as their zero sets in $({\mathbb{C}}^*)^n$ are concerned. It is easy to see, however, that the system above does not have solutions in $({\mathbb{C}}^*)^n$.
(Even N) If $N$ is even, then the reduced facet matrix of $F$, $$V^* = QV =
\begin{bmatrix}
{\mathbf{e}}_1 & \cdots &{\mathbf{e}}_n & {\mathbf{h}}\end{bmatrix},$$ has the last column ${\mathbf{h}}$ with $\frac{N}{2}$ entries equal $1$ and $\frac{N}{2}-1$ entries equal $-1$ (See proof of for details). Then via the biholomorphic change of variables ${\mathbf{x}}={\mathbf{y}}^Q$, the above system is equivalent to $$QV((({\mathbf{y}})^Q)^V)^\top = V^* ({\mathbf{y}}^{V^*})^\top.$$ That is, the initial system $\operatorname{init}_{\alpha} {\mathbf{f}}_F^R = {\mathbf{0}}$ has a ${\mathbb{C}}^*$-solution if and only if $$\label{equ:init*}
V^* ({\mathbf{y}}^{V^*})^\top = {\mathbf{0}}$$ has a ${\mathbb{C}}^*$-solution. We now show this is possible if and only if $N$ is divisible by 4.
Assume $N$ is divisible by 4, we shall show has a ${\mathbb{C}}^*$-solution. In fact, ${\mathbf{h}}$ is a solution. Let ${\mathbf{y}}= {\mathbf{h}}^\top = (h_1,\dots,h_n) \in ({\mathbb{C}}^*)^n$, then $$\begin{aligned}
V^*({\mathbf{y}}^{V^*})^\top = V^* \, (({\mathbf{h}}^\top)^{V^*})^\top =
V^* \,
\begin{bmatrix}
h_1 \\
\vdots \\
h_n \\
(h_1)^{h_1}\cdots(h_n)^{h_n}
\end{bmatrix}.
\label{eq:init h}\end{aligned}$$ Recall that $h_i \in \{+1,-1\}$, so $h_i^{h_i}=h_i$ for each $i=1,\dots,n$. Since $N$ is divisible by 4, and since ${\mathbf{h}}$ has exactly $\frac{N}{2}-1$ entries equal $-1$ while the rest are 1’s the last entry in $({\mathbf{h}}^\top)^{V^*}$ is $$(h_1)^{h_1}\cdots(h_n)^{h_n} =
h_1\cdots h_n=(-1)^{\frac{N}{2}-1}
= -1.$$
So, becomes $$\begin{aligned}
V^* \, ({\mathbf{h}}^\top)^{V^*}= \begin{bmatrix}
1 & & &h_1\\
&\ddots& &\vdots\\
& &1 &h_n
\end{bmatrix}
\begin{bmatrix}
h_1 \\
\vdots \\
h_n \\
-1
\end{bmatrix}
=
\begin{bmatrix}
h_1 - h_1 \\
\vdots \\
h_n - h_n
\end{bmatrix}
={\mathbf{0}}.
\label{eq:init 4}\end{aligned}$$ That is, ${\mathbf{h}}\in ({\mathbb{C}}^*)^n$ is a solution to . In fact, since is homogeneous, $\lambda {\mathbf{h}}\in ({\mathbb{C}}^*)^n$ for any $\lambda \in {\mathbb{C}}^*$ will also be solution. Moreover, for any other solution ${\mathbf{y}}$, equation is equivalent to $$\begin{aligned}
\begin{bmatrix}
1 & & \\
&\ddots& \\
& &1
\end{bmatrix}
\begin{bmatrix}
y_1 \\
\vdots \\
y_n
\end{bmatrix}
=-(y_1)^{h_1}\cdots(y_n)^{h_n} {\mathbf{h}}. \end{aligned}$$ Hence, all the solutions of are of the form form $\lambda {\mathbf{h}}\in ({\mathbb{C}}^*)^n$, $\lambda \in {\mathbb{C}}^*$. Consequently, the initial system $\operatorname{init}_{\alpha} {\mathbf{f}}_F^R$ has a 1-dimensional zero set in $({\mathbb{C}}^*)^n$.
Now assume $N$ is even but not divisible by 4. Suppose ${\mathbf{x}}\in ({\mathbb{C}}^*)^n$ is a zero of the initial system $\operatorname{init}_{{\boldsymbol{\alpha}}} {\mathbf{f}}_F^R$, then ${\mathbf{y}}= {\mathbf{x}}^{Q^{-1}} \in ({\mathbb{C}}^*)^n$ is zero of , i.e., $$\begin{aligned}
V^* \, {\mathbf{y}}^{V^*} =
\begin{bmatrix}
1 & & &h_1 \\
&\ddots& &\vdots \\
& &1 &h_n
\end{bmatrix}
\begin{bmatrix}
y_1 \\
\vdots \\
y_n \\
(y_1)^{h_1}\cdots(y_n)^{h_n}
\end{bmatrix}
={\mathbf{0}}.
\label{eq:init_not4 }\end{aligned}$$ We have $y_i=-h_i (y_1)^{h_1}\cdots(y_n)^{h_n}$, $i=1,\dots , n$, which implies $$y_1^{h_1}\cdots y_n^{h_n}=(-1)^{h_1+\dots +h_n} h_1 \cdots h_n \left((y_1)^{h_1}\cdots(y_n)^{h_n}\right)^{h_1+\dots+ h_n}.$$ Recall that $h_1+\dots+ h_n=1$ and $h_1 \cdots h_n=(-1)^{\frac{N}{2}-1}$. Hence, the equation above is equivalent to $$y_1^{h_1} \cdots y_n^{h_n} =
(-1)^{\frac{N}{2}} \cdot y_1^{h_1}\cdots y_n^{h_n} =
- y_1^{h_1}\cdots y_n^{h_n}$$ since $N$ is not divisible by 4. This equation implies that $y_1^{h_1} \cdots y_n^{h_n} = 0$, i.e., $y_k = 0$ for at least one $k \in \{1,\dots,n\}$, contradicting with the assumption that ${\mathbf{y}}= {\mathbf{x}}^{E^{-1}} \in ({\mathbb{C}}^*)^n$. Therefore we can conclude that the initial system $\operatorname{init}_{{\boldsymbol{\alpha}}} {\mathbf{f}}_F^R$ has no zeros in $({\mathbb{C}}^*)^n$.
We now show all other initial systems of ${\mathbf{f}}_F^R$ have no zeros in $({\mathbb{C}}^*)^n$ for any $N$. Consider an initial system $\operatorname{init}_{{\boldsymbol{\alpha}}} {\mathbf{f}}_F^R$ induced by a nonzero vector ${\boldsymbol{\alpha}}\in {\mathbb{R}}^n$ for which $(\operatorname{Newt}({\mathbf{f}}_F^R))_{{\boldsymbol{\alpha}}} \ne F$. If $\operatorname{init}_{{\boldsymbol{\alpha}}} {\mathbf{f}}_F^R$ does not involves the constant term, then this system is equivalent to $$W ({\mathbf{x}}^W)^\top = {\mathbf{0}},$$ where $W$ consists of less than $N-1$ columns of $V$ when $N$ is odd and less than $N$ columns of $V$ when $N$ is even. Consequently, $W$ has full column rank. By the transformation via its Moore-Penrose inverse, we get $${\mathbf{0}}= W^+ {\mathbf{0}}= W^+ W({\mathbf{x}}^W)^\top = {\mathbf{x}}^W$$ which implies ${\mathbf{x}}\not\in ({\mathbb{C}}^*)^n$. If $\operatorname{init}_{{\boldsymbol{\alpha}}} {\mathbf{f}}_F^R$ involves the constant term, by Generalized Sard’s theorem, its zero set must be 0-dimensional for generic choices of the constant terms ${\mathbf{c}}\in {\mathbb{C}}^n$. But its Newton polytope is of lower dimension, so by the Bernshtein-Kushnirenko-Khovanskii Theorem [@Bernshtein1975Number; @Khovanskii1978Newton; @Kushnirenko1976Polyedres], its zero set in $({\mathbb{C}}^*)^n$ must be empty.
This result shows that for $N$ not divisible by 4, no initial system of a facet subsystem has a nontrivial ${\mathbb{C}}^*$-solution. Therefore, the root count for the facet subsystem agrees with the BKK bound. Combining with the root count results established in Ref. [@ChenDavisMehta2018Counting], the root count in cases where $N$ is not divisible by 4 can be derived immediately.
If $N$ is odd, a facet subsystem ${\mathbf{f}}_F^R$, with generic choices of ${\mathbf{c}}$ and $a$, has a unique ${\mathbb{C}}^*$-solution.
If $N$ is even but not divisible by 4, then the number of ${\mathbb{C}}^*$-solutions to a facet subsystem ${\mathbf{f}}_F^R$, with generic choices of ${\mathbf{c}}$ and $a$, is $\frac{N}{2}$.
For the cases where $N$ is divisible by 4, we have identified the unique initial system of a given facet subsystem that has a nontrivial ${\mathbb{C}}^*$-solution. Consequently, the root count, even under the assumption of generic ${\mathbf{c}}$ and $a$, is strictly less than the BKK bound or the adjacency polytope bound. We compute the exact root count below.
\[cor: number of subsystem solutions div. by 4\] If $N$ is divisible by 4, then the number of ${\mathbb{C}}^*$-solutions to a facet subsystem ${\mathbf{f}}_F^R$ is $\frac{N}{2} - 1$.
Let $V$ be the facet matrix associated with $F$, and let $V^*$ and $Q$ be the corresponding reduced facet matrix and facet reduction matrix respectively. Recall that the facet system is given by $${\mathbf{f}}_F^R =
{\mathbf{c}}- a \sum_{({\mathbf{e}}_i - {\mathbf{e}}_j) \in F}
(R_i - R_j) \, ({\mathbf{x}}^{({\mathbf{e}}_i - {\mathbf{e}}_j)})^\top
=
{\mathbf{c}}- a R V ({\mathbf{x}}^V)^\top,$$ where $V$ is the facet matrix associated with $F$. By the root counting result established in Ref. [@ChenDavisMehta2018Counting], Proposition 12, the BKK bound for this system is $\frac{N}{2}$, i.e., this system has at most $\frac{N}{2}$ zeros in $({\mathbb{C}}^*)^n$, and this upper bound is attainable for generic choices coefficients (i.e., all coefficients are chosen generically and independently). Here we show that due to the special algebraic relations among the coefficients, the actual number of zeros in $({\mathbb{C}}^*)^n$ is $\frac{N}{2} - 1$.
Via the unimodular change of variables ${\mathbf{x}}= {\mathbf{y}}^{Q}$, the above facet system can be transformed into $$\label{equ:facet-system-y}
{\mathbf{c}}- a R V ({\mathbf{y}}^{QV})^\top =
{\mathbf{c}}- a R V ({\mathbf{y}}^{V^*})^\top.$$ Since the change of variables preserves the number of zeros in $({\mathbb{C}}^*)^n$, it is thus sufficient to count the number of zeros of this system instead.
All zeros of in $({\mathbb{C}}^*)^n$ are isolated and simple, and under this transformation, the only initial system with nontrivial ${\mathbb{C}}^*$-solution is the initial system defined by the vector ${\boldsymbol{\alpha}}= (-1,\dots,-1)$. Therefore the only zeros outside $({\mathbb{C}}^*)^n$ are at infinity, i.e., $\mathbb{CP}^n \setminus {\mathbb{C}}^n$, which we can compute explicitly by considering the homogenization of . At infinity, the system is equivalent to $$-aRV ({\mathbf{y}}^{V^*})^\top = {\mathbf{0}},$$ which is, in turn, is equivalent to $$V^* ({\mathbf{y}}^{V^*})^\top = {\mathbf{0}}.$$ This system coincide with the initial system in the proof of , which has a unique nonsingular solution in $\mathbb{CP}^n$. In other words, has only one simple zero at infinity. Its root count in $({\mathbb{C}}^*)^n$ is therefore one less than the BKK bound, i.e., the root count is $\frac{N}{2} - 1$.
(Running example, 4-cycle). To illustrate the result of Corollary \[cor: number of subsystem solutions div. by 4\], we consider the facet $$\operatorname{conv}\left\{
\left[\begin{smallmatrix} 1\\0\\0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}1\\-1\\\phantom{-}0 \end{smallmatrix}\right],
\left[\begin{smallmatrix} \phantom{-}0\\-1\\\phantom{-}1 \end{smallmatrix}\right],
\left[\begin{smallmatrix} 0\\0\\1 \end{smallmatrix} \right]
\right\}$$ of the adjacency polytope of a 4-cycle network (the shaded facet in Figure \[sec: Adjacency polytope\]). With a direct computation, we can verify that the normalized volume of this facet is 2. This is the BKK bound of the facet subsystem. Therefore, if all coefficients were chosen randomly, we expect the facet subsystem to have 2 solutions in $({\mathbb{C}}^*)^3$. However, due to the special algebraic relations among the coefficients, as a result of the uniform-coupling requirement, this facet subsystem has only one solution. Indeed, with direct symbolic computation, we can compute the unique solution which is given by $$\begin{aligned}
x_0 &= 1, &
x_1 &= \frac{1}{a} \, \frac{(\omega_1+\omega_2+\omega_3)\omega_1}{\omega_1+\omega_3},\\
x_2 &= \frac{(\omega_1+\omega_2+\omega_3)\omega_3}{\omega_1+\omega_3}, &
x_3 &= \frac{1}{a} \, \frac{(\omega_2+\omega_3)(\omega_1+\omega_2+\omega_3)}{\omega_1+2\omega_2+\omega_3}.
\end{aligned}$$ The same argument can be applied to all 6 facet subsystems (corresponding to the 6 facets listed in . We can thus conclude that under the generic uniform-coupling assumption, the algebraic Kuramoto equations for 4-cycle graph has exactly 6 complex solutions even though its BKK bound is 12.
Combining the above corollaries, , and the total number facets of $\nabla_{C_N}$, we get the following generic root count for under the assumption of generic natural frequency and generic but uniform coupling coefficients.
Given a cycle network of $N$ oscillators with uniform coupling and generically chosen complex constants $a,\omega_i,\dots,\omega_n$, the number of isolated complex solutions to the system is $$\left\{
\begin{aligned}
&N \binom{N-1}{ \lfloor (N-1) / 2 \rfloor } && \text{if $N$ is not divisible by 4} \\[2ex]
&(N-2) \binom{N-1}{N/2-1} && \text{if $N$ is divisible by 4}.
\end{aligned}
\right.$$
These are also the answers to , i.e., they are the birationally invariant intersection indices $[L_1,\dots,L_n]$ for the family of vectors spaces of rational functions over the toric variety $({\mathbb{C}}^*)^n$. Note that this intersection index is strictly less than the BKK bound when $N$ is divisible by 4.
In the cases where $N$ is divisible by 4, the gap between the birationally invariant intersection index and the BKK bound is $2 \binom{N-1}{N/2-1} = \binom{N}{N/2}$, which grows exponentially as $N \to \infty$.
Concluding remarks {#sec: conclusion}
==================
The Kuramoto model is one of the most widely studied models for describing the pervasive phenomenon of spontaneous synchronization in networks of coupled oscillators. In this model, frequency synchronization configurations can be formulated as complex solutions to a system of algebraic equations. Under the assumption of generic natural frequencies and generic non-uniform coupling strength, the upper bounds to the number of frequency synchronization configurations in cycle networks of $N$ oscillators were computed in a recent work [@ChenDavisMehta2018Counting]. This paper provides a sharper upper bound for the special cases of networks with uniform coupling. The uniform coupling assumption imposes an algebraic condition on the coefficients of the algebraic Kuramoto equations and potentially reduces the maximum number of solutions. We have established the exact condition under which the maximum root count is lower and quantified the gap. In particular, if $N$ is not divisible by 4, then the maximum complex root count of the Kuramoto equations remains the same with or without the uniform coupling assumption. On the other hand, if $N$ is divisible by 4, the uniform coupling assumption significantly lowers the maximum root count. Indeed, the gap between the bounds is $\binom{N}{N/2}$ which grows exponentially as $N \to \infty$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We present a detailed theory for finite-frequency conductivities Re$%
[\sigma_{\alpha\beta}(\omega)]$ of quantum Hall stripes, which form at Landau level $N\geq 2$ close to half filling, in the presence of weak Gaussian disorder. We use an effective elastic theory to describe the low-energy dynamics of the stripes with the dynamical matrix being determined through matching the density-density correlation function obtained in the microscopic time-dependent Hartree-Fock approximation. We then apply replicas and the Gaussian variational method to deal with the disorder. Within this method, a set of saddle point equations for the retarded self energies are obtained, which are solved numerically to get Re$%
[\sigma_{\alpha\beta}(\omega)]$. We find a quantum depinning transition as $%
\Delta\nu$, the fractional part of the filling factor, approaches a critical value $\Delta\nu_c$ from below. For $\Delta\nu<\Delta\nu_c$, the pinned state is realized in a replica symmetry breaking (RSB) solution, and the frequency-dependent conductivities in both the directions perpendicular and parallel to the stripes show resonant peaks. These peaks shift to zero frequency as $\Delta\nu\rightarrow \Delta\nu_c$. For $\Delta\nu\ge\Delta%
\nu_c $, we find a *partial RSB (PRSB)* solution in which there is RSB perpendicular to the stripes, but replica symmetry along the stripes, leading to free sliding along the stripe direction. The quantum depinning transition is in the Kosterlitz-Thouless universality class. The result is consistent with a previous renormalization group analysis.
author:
- 'Mei-Rong Li$^{1}$, H.A. Fertig$^{2,3}$, R. Côté$^{1}$, Hangmo Yi$^{4}$'
title: Dynamical Conductivity of Disordered Quantum Hall Stripes
---
Introduction
============
Charge density waves (CDWs) may form in many correlated electronic systems when the Coulomb interaction dominates over the kinetic energy. For a two-dimensional electron gas in a perpendicular magnetic field, the quantization of the kinetic energy into Landau levels can enhance this possibility [stripestheory]{}. Each Landau level is highly degenerate, with the number of states equal to the number of magnetic flux quanta passing through the system. If the field and corresponding degeneracy is sufficiently large, the low-energy physics of the system may then be dominated by electrons in the highest partially occupied ($N$th) Landau level, with the other electrons essentially renormalizing the effective Coulomb interaction in this level [@AG]. In this situation, the kinetic energy is quenched and the system arranges itself in order to minimize the interactions. The Hartree-Fock approximation [@stripestheory] predicts the formation of CDW ground states for $N\geq 2$. These CDWs evolve from Wigner crystals to bubble states [@stripestheory] as the partial Landau-level filling factor $\Delta \nu $ increases. For $\Delta
\nu \gtrsim 0.4$, the bubble states give way to stripe states (also called unidirectional CDWs.) This Hartree-Fock result is corroborated by density matrix renormalization group calculations [@DMRG] and exact diagonalization studies [@exactdiag]. DC transport experiments indeed observe highly anisotropic, and apparently metallic, conductivity [stripesexpr]{} near half-filling of higher Landau levels ($N\geq 2$). This is likely due to the formation of stripe states.
Because the stripe state breaks translational symmetry in only one direction, it has the symmetry of a smectic liquid crystal [@fradkin], and as in that system supports a set of gapless phonon modes [@CF]. These modes are present because the stripe state lacks any restoring force when a single stripe slides with respect to the others. In the context of an electron system, the smectic state can be thought of as a self-organized array of Luttinger liquids [fradkin,edgestatemodel]{}, a state of fermions that does not obey Landau Fermi liquid theory and (so far) is known to exist only in one dimension [@voit]. One way of viewing the Luttinger liquid is in terms of a one dimensional crystal that has lost long-range order due to quantum fluctuations [@kolomeisky]. This idea is easily generalized to the case of stripes [@fertig99], and suggests that the low-energy degrees of freedom for the system may be described in terms of a displacement field. This will be the basic language for our study.
It has long been recognized that disorder can pin a CDW and render it insulating [@FL1; @Larkin]. (Similarly, Luttinger liquids may be pinned by impurities in spite of their liquid-like correlations [@KF].) In this situation, the real part of the zero wavevector, finite-frequency conductivity $\sigma _{\alpha \beta }(\omega )$ vanishes as $\omega
\rightarrow 0$, and has a resonance at higher $\omega $ with a peak (or pinning) frequency and width that are determined by the effective restoring force due to the disorder [@FL1]. Such behavior has indeed been observed in high Landau levels for $\Delta \nu $ sufficiently far away from 1/2 so that one expects the electrons to be organized into bubble states[muwave]{}. As $\Delta \nu $ is increased, there is a general trend for the pinning frequency to decrease, eventually becoming lost in the noise as the filling factor approaches the value at which the DC conductivity becomes anisotropic and metallic [@stripesexpr]. Experiments to better resolve the dynamical conductivity as the stripe phase is entered are currently underway [@Florida].
A fundamental question that arises in this context is whether the apparent metallic behavior seen in DC transport experiments represents the true zero temperature behavior in the stripe state. While current data suggests the diagonal conductivities $\sigma _{xx}$ and $\sigma _{yy}$ saturate to finite values at low temperatures, presumably such experiments can answer this fundamental question unambiguously only by reaching significantly lower temperatures than are currently available. The possibility that near half-filling the stripes may not be fully pinned is extremely intriguing because, if this is indeed the case, then the electrons have avoided becoming localized and the resulting anisotropic metal cannot be a Fermi liquid. Thus this state could well be a higher dimensional analogue of a Luttinger liquid. Developing and understanding the results of experiments beyond DC transport – such as measurements of the dynamical conductivity [@muwave] – then take on an additional significance.
One possible route to metallic behavior for the stripe system could be via a depinning transition. In principle this could happen in a one dimensional Luttinger liquid, if one could continuously tune the interactions from repulsive to attractive [@KF]. Because the stripe system has a larger parameter space needed to describe its elastic properties than the single stiffness that describes a one-dimensional solid (specifically, one needs to estimate the dynamical matrix along a line in the Brillouin zone, as we discuss below), it is possible to arrive at this depinning transition even when the bare interaction parameters among the electrons are purely repulsive [@fertig99]. The question of whether stripes may become depinned must be answered via a detailed calculation of the stripe elasticity, and depending on how one estimates this, different answers are possible [@YFC; @edgestatemodel], as we discuss in more detail below.
In what follows, we will adopt an approach that models the quantum Hall (QH) stripe system as an array of one dimensional, quantum disordered solids as shown in Fig. \[stripeselastic\]. We estimate the dynamical matrix of this system by matching the elastic theory to the results of a microscopic calculation within the time-dependent Hartree-Fock approximation (TDHFA). This approach was taken by some of us [@YFC] in a perturbative renormalization group (RG) calculation, which demonstrated that a quantum depinning transition can occur as $\Delta\nu
\rightarrow \Delta\nu_c$ from below, with the critical partial filling $%
\Delta\nu_c$ depending on the details of the system: Landau level index, layer thickness, disorder strength, etc. Our goal is to compute the dynamical conductivity as the system passes through the transition, to identify signatures that would indicate that the system has passed into an unpinned stripe state. A brief summary of our major results has been published elsewhere [@LFCY]. In this article, we provide details of the calculations as well as some further results.
Our general approach to this problem is to use replicas [@MPV] and the Gaussian variational method (GVM), as was first introduced by Mézard and Parisi for elastic manifolds [@MP] and then further developed by Giamarchi and Le Doussal and their coworkers in applications to a variety of condensed matter systems [@GLD96; @GLD95; @GO; @CGLD]. In the QH systems, this method was used by Chitra [*et al.*]{} [@CGLD] for pinned Wigner crystals in the $N=0$ Landau level, and by Orignac and Chitra [@OC] for stripes, in the latter case using a different set of approximations than us and yielding very different results than those described below. Because of the strong fluctuations inherent in the depinning transition, we have found that one cannot correctly solve for the dynamical conductivity using the semiclassical approximation to the saddle point equations (SPE’s), which we review below, that has given this approach its attractive simplicity. By relaxing this approximation we will see that in the depinned state the dynamical conductivity can have a surprising power-law frequency dependence, and a discontinuous behavior at the transition that is analogous to the universal stiffness jump that occurs at a Kosterlitz-Thouless (KT) transition [@nelson]. Moreover, we will see that the solution to the SPE’s that yield this behavior have an unusual structure involving breaking the replica symmetry for motion perpendicular to the stripes, while preserving it parallel to the stripes. This *partial replica symmetry breaking* (PRSB) indicates that the stripes may be pinned for perpendicular motion while free to slide relative to one another. This qualitative behavior was anticipated by the perturbative RG study [YFC]{}.
As discussed above, when the system is pinned there are resonant peaks which appear in $\sigma _{xx}$ and $\sigma
_{yy}$. (We choose the $\hat{x}$ direction to be perpendicular to the stripes, and the $\hat{y}$ direction to be parallel to it as shown in Fig. \[stripeselastic\]. Of course, the two diagonal conductivities are not the same due to the anisotropy of the stripe state.) The peaks drop to zero frequency as $%
\Delta \nu _{c}$ is approached from below, with their weights increasing for motion along the stripes, and decreasing for motion perpendicular to them. As the transition is approached, the resonance peaks become increasingly asymmetric. Upon crossing the transition, $\sigma _{yy}$ develops a $\delta $-function at $\omega =0$, indicating superconducting behavior, while $\sigma _{xx}$ rises from zero as a power of $\omega $. This unusual behavior is a result of the power-law correlations associated with the Luttinger liquid-like behavior of the unpinned state. We note that our $\omega =0$ results are not consistent with the DC conductivity results seen in experiments, although preliminary experimental results for finite frequency do bear some resemblance to our predictions [@Florida]. We will comment below on what is missing from our model that we believe leads to this discrepancy.
This paper is organized as follows. In Sec. II we review the procedure for determining the dynamical matrix in the elastic model. This is followed by a review in Sec. III of the qualitative effects of disorder within the RG analysis. In Sec. IV, we review the replica and GVM which leads to a set of saddle point equations (SPE’s) for the self energy. Solutions of the SPE’s and the result for the conductivities are presented in Sec. V, focusing on the pinned state for $\Delta \nu <\Delta \nu _{c}$, and in Sec. VI, which is devoted to the depinned state for $\Delta \nu >\Delta \nu _{c}$. We discuss the nature of the depinning transition in Sec. VII, and conclude in Sec. VIII. There are four appendices: the first summarizes the Hartree-Fock (HF) and the TDHFA formalisms, the result of which is used for determination of the dynamical matrix; the second gives a derivation for the inversion rules needed for hierarchical matrices of the type dealt with in this paper; the third discusses analytic continuation of the dynamical conductivity from imaginary time to real time; and the forth discusses another possible solution to the SPE’s that has unphysical properties.
Elastic model of QH stripes
===========================
Elastic action
--------------
In our approach, low energy distortions from the mean-field state are described by an elastic model, with displacement fields $u_{x}\left( \mathbf{%
r}\right) $ and $u_{y}\left( \mathbf{r}\right) $ representing the effective dynamical variables of the QH stripes. Fig. \[stripeselastic\] shows schematically the one-dimensional arrays modelling the stripes. They obey single Landau level dynamics [@kubo] $\left[ u_{x}(\mathbf{R}),u_{y}(\mathbf{R}^{\prime })%
\right] =il_{B}^{2}\delta_{\mathbf{R},\mathbf{R}^{\prime }} $, where $l_{B}=\sqrt{\hbar c/eB}$ is the magnetic length. In the pure limit, the Euclidean action of the elastic model may be written as (throughout this work, we use the unit $k_{B}=\hbar =1$) $$S_{0}={\frac{1}{2T}}\sum_{\mathbf{q},\omega _{n}}\sum_{\alpha ,\beta
=x,y}u_{\alpha }\left( \mathbf{q},\omega _{n}\right) \,G_{\alpha \beta
}^{(0)-1}\left( \mathbf{q},\omega _{n}\,\right) u_{\beta }\left( -\mathbf{q}%
,-\omega _{n}\right) , \label{pureS0}$$where $T$ is the temperature, $\omega _{n}\left( =2\pi n/T\right) $ the bosonic Matsubara frequency, and $$G^{(0)}_{\alpha \beta}\left( \mathbf{q},i\omega _{n}\right)
=\frac{l_{B}^{4}}{\left( \omega_{n}^{2}+\omega _{\mathbf{q}}^{2}\right) }\left(
\begin{array}{cc}
D_{yy}\left( \mathbf{q}\right) & \frac{\omega _{n}}{l_{B}^{2}}-D_{xy}\left(
\mathbf{q}\right) \\
-\frac{\omega _{n}}{l_{B}^{2}}-D_{yx}\left( \mathbf{q}\right) & D_{xx}\left(
\mathbf{q}\right)%
\end{array}%
\right)_{\alpha \beta} \label{GF0}$$is the unperturbed Green’s function of the displacement fields with $D_{\alpha\beta}(\q)$ being the dynamical matrix and $$\omega _{\mathbf{q}}=l_{B}^{2}\sqrt{D_{xx}\left( \mathbf{q}\right)
D_{yy}\left( \mathbf{q}\right) -D_{xy}^{2}\left( \mathbf{q}\right) }
\label{eigenmode}$$being a general expression for phonon modes of a charged elastic system in a strong magnetic field (magnetophonon modes). As always for a Gaussian theory, the correlation function may be expressed in terms of the Green’s function via $$G_{\alpha \beta }^{(0)}\left( \mathbf{q},\omega _{n}\right)
=\int_{0}^{1/T}d\tau e^{i\omega _{n}\tau }\,\left\langle \mathcal{T}_{\tau
}u_{\alpha }\left( \mathbf{q},\tau \right) u_{\beta }\left( -\mathbf{%
q},0\right) \right\rangle _{S_{0}},$$where $\left\langle \cdots \right\rangle _{S_{0}}$ denotes an average over the displacement fields with the usual weighting factor $e^{-S_{0}}$, and $\mathcal{T}_{\tau }$ is the imaginary time ordering operator.
(200,140)
Because of inversion and reflection symmetries, and the fact that the dynamical matrix elements in real space are real, we have $$\begin{aligned}
&&D_{\alpha \beta }\left( \mathbf{q}\right) =D_{\beta \alpha }\left( -%
\mathbf{q}\right) , \\
&&D_{xy}\left( \mathbf{q}\right) =D_{yx}\left( \mathbf{q}\right) , \\
&&D_{xy}\left( q_{x},q_{y}\right) =-D_{xy}\left( -q_{x},q_{y}\right)
=-D_{xy}\left( q_{x},-q_{y}\right) ,\end{aligned}$$so that the unperturbed Green’s function has the symmetries $$\begin{aligned}
&&G_{\alpha \alpha }^{(0)}\left( \mathbf{q},\omega _{n}\right) =G_{\alpha
\alpha }^{(0)}\left( -\mathbf{q},\omega _{n}\right) =G_{\alpha \alpha
}^{(0)}\left( \mathbf{q},-\omega _{n}\right) =G_{\alpha \alpha }^{(0)}\left(
-\mathbf{q},-\omega_{n}\right) , \label{symmetry1} \\
&&G_{xy}^{(0)}\left( \mathbf{q},\omega _{n}\right) =G_{yx}^{(0)}\left(
\mathbf{q},-\omega_{n} \right), \label{symmetry2} \\
&&G_{xy}^{(0)}\left( q_{x},q_{y},\omega _{n}\right) =G_{xy}^{(0)}\left(
-q_{x},-q_{y},\omega _{n}\right) =-G_{xy}^{(0)}\left( -q_{x},q_{y},\omega
_{n}\right) =-G_{xy}^{(0)}\left( q_{x},-q_{y},\omega _{n}\right) .
\label{symmetry3}\end{aligned}$$
To perform quantitative calculations, it is necessary to produce estimates of the dynamical matrix elements $D_{\alpha \beta }(\mathbf{q})$ for the QH stripe states. We do this with a matching procedure that uses results from microscopic TDHFA computations. Below we briefly review this matching procedure.
Relation between $G^{(0)}_{\alpha\beta}
\left( \mathbf{q},\protect\omega \right) $ and guiding-center density-density correlation function
--------------------------------------------------------------------------------------------------
In a classical model, each site of the crystal is occupied by an electron whose charge density is specified by a form factor $f\left( \mathbf{r}%
\right) $ (with $\int d\mathbf{r}f\left( \mathbf{r}\right) =1\,$). In the absence of any fluctuations these electrons will lie on the oblique Bravais lattice as shown in Fig. \[stripeselastic\]. Fluctuations around this reference state are given in terms of the displacement fields $\mathbf{u}(\mathbf{R})$. The time-dependent electronic density is then written as $$n\left( \mathbf{r},t\right) =\sum_{\mathbf{R}}f\left( \mathbf{r}-\mathbf{R}-%
\mathbf{u}\left( \mathbf{R},t\right) \right) . \label{21_5}$$The Fourier transform of this density is given by$$n(\mathbf{q},t)=\int d\mathbf{r}e^{-i\mathbf{q}\cdot \mathbf{r}}n\left(
\mathbf{r},t\right) \approx f(\mathbf{q})\delta _{\mathbf{q},\mathbf{K}%
}-if(\mathbf{q})\sqrt{N_{s}}\mathbf{q}\cdot \mathbf{u}(\mathbf{q}),
\label{1_17}$$where $N_{s}$ is the number of crystal sites or electrons and $\K$ is a reciprocal lattice vector. The form factor $%
f(\mathbf{r})$ is real and has inversion symmetry so that $f(\mathbf{q})$ is real.
The fact that the density fluctuations are related to the displacement field via $$\delta n(\mathbf{q}+\mathbf{K},t)\approx -if(\mathbf{q+K})\sqrt{N_{s}}\left(
\mathbf{q+K}\right) \cdot \mathbf{u}(\mathbf{q})$$(with $\q$ a vector in the first Brillouin zone of the reciprocal lattice) implies that we can relate the displacement Green’s function $G^{(0)}_{\alpha\beta}\left( \mathbf{q},\protect\omega \right) $ to the density-density correlation function $\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( n,n\right) }\left( \mathbf{q}%
,\tau \right)$ (introduced in Appendix \[HFappendix\]) through $$\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( n,n\right) }\left( \mathbf{q}%
,\tau \right) =-N_{s}f(\mathbf{q+K})f\left( \mathbf{q}+\mathbf{K}^{\prime
}\right) \left[ \left( \mathbf{q+K}\right) \cdot \widehat{G}^{(0)}\left( \mathbf{q}%
,\tau \right) \cdot \left( \mathbf{q+K}^{\prime }\right) \right] .
\label{chiGF}$$Here $\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( n,n\right) }
\left( \mathbf{q},\tau \right)$ is a quantity that we compute in the microscopic TDHFA [@CF]. In Appendix \[HFappendix\], we summarize the HF and TDHF formalisms. Eq. (\[21\_13\]) there will be used for the determination of the dynamical matrix. Substituting Eq. (\[GF0\]) in Eq. (\[chiGF\]) yields $$\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( n,n\right) }\left( \mathbf{q}%
,i\omega _{n}\right) =-\frac{N_{s}l_{B}^{4}}{\left( \omega _{n}^{2}+\omega _{%
\mathbf{q}}^{2}\right) }\left[ \Gamma _{1}+\Gamma _{2}\frac{\omega _{n}}{%
l_{B}^{2}}\right] f\left( \mathbf{q}+\mathbf{K}\right) f\left( \mathbf{q}+%
\mathbf{K}^{\prime }\right) , \label{chiwn}$$with the definitions $$\Gamma _{1}=-\left( \mathbf{q}+\mathbf{K}\right) \times \overleftrightarrow{D%
}\left( \mathbf{q}\right) \times \left( \mathbf{q}+\mathbf{K}^{\prime
}\right) ,$$and $$\Gamma _{2}=\left( \mathbf{q}+\mathbf{K}\right) \times \left( \mathbf{q}+%
\mathbf{K}^{\prime }\right) .$$The two-dimensional vector product in the last two equations stands for $%
\mathbf{a}\times \mathbf{b}=a_{x}b_{y}-a_{y}b_{x}.$
The analytical continuation of $\chi _{\mathbf{K},\mathbf{K}^{\prime
}}^{\left( n,n\right) }\left( \mathbf{q},i\omega _{n}\right) $ in Eq. (\[chiwn\]) results in $$\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( n,n\right) }\left( \mathbf{q}%
,\omega \right) =-N_{s}l_{B}^{4}\left[ \frac{Z}{\omega +i\delta +\omega _{%
\mathbf{q}}}-\frac{Z^{\ast }}{\omega +i\delta -\omega _{\mathbf{q}}}\right]
f\left( \mathbf{q}+\mathbf{K}\right) f\left( \mathbf{q}+\mathbf{K}^{\prime
}\right) , \label{21_8}$$where $$Z=\frac{\Gamma _{1}}{2\omega _{\mathbf{k}}}-i\frac{\Gamma _{2}}{2l_{B}^{2}}.$$
We can now request that Eq. (\[21\_8\]) be equivalent to Eq. (\[21\_13\]) in Appendix \[HFappendix\] in order to obtain the dynamical matrix. This requires that $$\Delta \nu \, l_{B}^{4}Z^{\ast }f\left( \mathbf{q}+\mathbf{K}\right) f\left(
\mathbf{q}+\mathbf{K}^{\prime }\right) =F\left( \mathbf{q}+\mathbf{K}\right)
F\left( \mathbf{q}+\mathbf{K}^{\prime }\right) W_{i}\left( \mathbf{q}+%
\mathbf{K},\mathbf{q}+\mathbf{K}^{\prime }\right) , \label{21_10}$$where $W_{i}\left( \mathbf{q}+\mathbf{K},\mathbf{q}+\mathbf{K}^{\prime
}\right) $ is the weight associated with the magnetophonon frequency $%
\varepsilon _{i}$ in the TDHFA response and $\Delta \nu =N/N_{\varphi }$ is the filling factor of the partially filled level. The magnetophonon frequency $\varepsilon _{i}$ is found, at small wavevector $\mathbf{q}$, by locating the eigenvalue $\varepsilon _{i}$ of the matrix $M$ defined in Eq. (\[eqnm\]) with the biggest weight $W_{i}\left( \mathbf{q}+\mathbf{K},%
\mathbf{q}+\mathbf{K}\right) $ in the [*diagonal*]{} response function $%
\chi _{\mathbf{K},\mathbf{K}}^{\left( n,n\right) }\left( \mathbf{q},\omega
\right) .$
A careful examination shows that, because $\omega _{\mathbf{q}}$ is given by the determinant of the matrix $\hat{D}$, the quantity $\Gamma _{1}/2\omega _{%
\mathbf{q}}$ is unchanged if all the components of the dynamical matrix are multiplied by some constant. Eq. (\[21\_10\]) is thus indeterminate. To avoid this, we replace $\omega _{\mathbf{q}}$ by $\varepsilon _{i}$ in this equation. Our final equation is then$$f\left( \mathbf{q}+\mathbf{K}\right) f\left( \mathbf{q}+\mathbf{K}^{\prime
}\right) \left[ l_{B}^{2} \Gamma _{1}+i\varepsilon _{i}\Gamma _{2}\right]
=\frac{%
2\varepsilon _{i}}{\Delta \nu l_{B}^{2}}F\left( \mathbf{q}+\mathbf{K}\right)
F\left( \mathbf{%
q}+\mathbf{K}^{\prime }\right) W_{i}\left( \mathbf{q}+\mathbf{K},\mathbf{q}+%
\mathbf{K}^{\prime }\right) . \label{1_14}$$With this equation, we can determine the 3 components of the dynamical matrix as well as the form factors $f\left( \mathbf{q}+\mathbf{K}\right) .$
Matching procedure
------------------
At this point, it is worthwhile remarking that, in the HFA, there is an extremely small energy difference (of order $10^{-6}$ $e^{2}/\kappa l_{B}$) between the energies of the stripe crystals with in-phase and out-of-phase modulation on adjacent stripes. As a result, the magnetophonon dispersion in the TDHFA has a very small gap in the perpendicular direction. This interstripe locking energy is, however, not accessible within our numerical accuracy so that our calculated magnetophonon dispersion is that appropriate for a smectic. In particular, it contains a line of gapless modes for $%
q_{x}\neq 0,q_{y}=0$. Because of this nodal line, we need to fit the dynamical matrix for small $q_{y}$ i.e., for long wavelengths along the stripes, and for all values of $q_{x}$ in the Brillouin zone. Indeed, these low-energy modes play a crucial role in determining the effects of both quantum and thermal fluctuations on the system.
We choose to solve Eq. (\[1\_14\]) for the shortest three reciprocal lattice vectors: $\mathbf{K,K}%
^{\prime }=(0,0),(0,\pm K_{y0})$ where $K_{y0}=2\pi /a_{y}$ with $a_y$ being the lattice constant along the stripes direction (see Fig. \[stripeselastic\]). For each $%
\mathbf{q}$, the TDHFA calculation provides ten independent numbers, nine in the $3\times 3$ Hermitian matrix $W_{i}\left( \mathbf{q}+\mathbf{K},\mathbf{q%
}+\mathbf{K}^{\prime }\right) $ and one in $\varepsilon _{i}$. We use six of them to determine $D_{xx}$, $D_{xy}$, $D_{yy}$, and the three real parameters $f(\mathbf{q}+\mathbf{K})$. The rest may be used to check the consistency of the numerical procedure. The final result [@YFC] indicates that the matching is very accurate.
(250,200)
A typical result for $D_{yy}(\mathbf{q})$ as a function of $q_{y}$ at small $%
q_{y}$ and $q_{x}=K_{x0}/2$ is shown in Fig. \[Dyy\]. Clearly $D_{yy}(%
\mathbf{q})$ is quadratic in $q_{y}$. Indeed, based on symmetry considerations [@edgestatemodel; @CF], the low energy sector of $\mathbf{D}%
(\mathbf{q})$ should have the form for small $q_{y}$: $$\begin{aligned}
&&D_{xx}\left( \mathbf{q}\right) \simeq d_{xx}\left( q_{x}\right) +\kappa
_{b}q_{y}^{4}, \label{smecticDxx} \\
&&D_{xy}\left( \mathbf{q}\right) \simeq d_{xy}\left( q_{x}\right) q_{y},
\label{smecticDxy} \\
&&D_{yy}\left( \mathbf{q}\right) \simeq d_{yy}\left( q_{x}\right) q_{y}^{2},
\label{smecticDyy}\end{aligned}$$where $\kappa _{b}$ is the bending coefficient. The absence of a quadratic $%
q_{y}$ term in $D_{xx}$ follows from rotational symmetry and is the major difference between a smectic and a crystal dynamical matrix. In our calculation below, we will use this smectic form, determining $d_{xx}\left( q_{x}\right) $, $d_{xy}\left( q_{x}\right) $ and $%
d_{yy}\left( q_{x}\right) $ on a grid of $q_{x}$ points numerically. The $\kappa_{b}q_{y}^{4}$ in Eq. (\[smecticDxx\]) reflects the bending energy of the stripes. In practice, this term merely plays the role of high-$q_y$ cutoff and thus $\kappa_{b}$ is chosen for convenience to be 2 in our numerical calculation.
Inserting Eqs. (\[smecticDxx\]-\[smecticDyy\]) into Eq. (\[eigenmode\]) yields $$\omega _{\mathbf{q}}\simeq l_{B}^{2}\sqrt{d_{xx}\left( q_{x}\right)
d_{yy}\left( q_{x}\right) -d_{xy}^{2}\left( q_{x}\right) }\;q_{y}$$ for small $q_y$.
Once the Green’s function has been determined, we can easily compute the conductivity. Since the electric current is carried by the charge, the current density can be expressed as $$\mathbf{j}\left( \mathbf{q},\tau \right) =ie
{\frac{d\mathbf{u}\left( \mathbf{q},\tau \right) }{d\tau }}.$$The conductivity is then determined by the Kubo formula to be $$\sigma _{\alpha \beta }\left( \omega \right) =-{\frac{1}{\omega \,a_x a_y}}\,
\left[\int_{0}^{1/T}d\tau e^{i\omega _{n}\tau }\left\langle j_{\alpha }\left(
\mathbf{q}=0,\tau \right) j_{\beta }\left( \mathbf{q}=0,0\right)
\right\rangle \right] _{i\omega _{n}\rightarrow \omega +i0^{+}}=-{\frac{e^{2}%
}{a_{x}a_{y}}}\,i\omega \,G_{\alpha \beta }^{\mathrm{ret}}\left( \mathbf{q}%
=0,\omega \right) . \label{conductivity0}$$where $a_x$ is the distance between the centers of two neighboring stripes (see Fig. \[stripeselastic\]). It is easy to check that in the pure limit, the electromagnetic response of the system is purely transverse. Calculating $G_{\alpha \beta}^{\mathrm{ret}}$ in the presence of disorder is our next (and indeed most important) task.
Qualitative effect of disorder
==============================
Modeling the disorder
---------------------
We assume that the disorder can be modeled as a Gaussian random potential $V(%
\mathbf{r})$. The disorder action reads $$S_{\mathrm{imp}}=\int d\mathbf{r}\,\int_{0}^{1/T}d\tau \,V\left( \mathbf{r}%
\,\right) n\left( \mathbf{r},\tau \right) \label{1_15}$$where $V\left( \mathbf{r}\right) $ has the following Gaussian distribution function $$P\left( V\right) =\exp \left[ -{\frac{1}{2}} \int d\r_1 \int d\r_2
V\left( \mathbf{r}_{1}\right) \Gamma ^{-1}\left( \mathbf{r}_{1}-\mathbf{%
r}_{2}\right) V\left( \mathbf{r}_{2}\right) \right] ,$$with $$\Gamma \left( \mathbf{r}_{1}-\mathbf{r}_{2}\right) =\overline{V\left(
\mathbf{r}_{1}\right) V\left( \mathbf{r}_{2}\right) }=V_{0}^{2}\,a_{x}%
\,a_{y}\,\delta \left( \mathbf{r}-\mathbf{r}^{\prime }\right) .$$Here the overline denotes average over disorder: $$\overline{A}={\frac{\int \mathcal{D}VP\left( V\right) A}{\int \mathcal{D}%
VP\left( V\right) }}.$$The electron density operator $n\left( \mathbf{r},\tau \right) $ in Eq. ([\[1\_15\]]{}) must, in order to capture the possibility of pinning by disorder, be approximated more accurately than was needed in the matching procedure discussed in the preceding section. Following Giamarchi and Le Doussal [@GLD95], under the assumption of small $\nabla {\mathbf{u}}(\mathbf{r})$ (which is justified for weak disorder) we write $$n(\mathbf{q},\tau )\simeq f(\mathbf{q})\left[ N_s-i\sqrt{N_s}
\mathbf{q}\cdot \mathbf{u}%
\left( \mathbf{q},\tau \right) +\sum_{\mathbf{K}\neq 0}\int d\mathbf{r}\,e^{i%
\mathbf{K}\cdot \left[ \mathbf{r}-\mathbf{u}(\mathbf{r},\tau )\right] -i%
\mathbf{q}\cdot \mathbf{r}}\right] . \label{1_19}$$This differs from our approximation in Eq. (\[1\_17\]) essentially via the last term which captures the short wavelength oscillations in the charge density and allows pinning by impurities. In employing Eq. \[1\_19\], since only the last term can actually lead to pinning [@GLD95], we will drop the first two terms upon substitution into Eq. \[1\_15\]. Moreover, in the reciprocal lattice sum we retain only the smallest non-trivial wavevectors, so that in what follows (unless otherwise specified) $\sum_{%
\mathbf{K}\neq 0}$ really means sum over $\mathbf{K}=\left( \pm
K_{x0},0\right) ,\left( 0,\pm K_{y0}\right) ,\left( \pm K_{x0},\pm
K_{y0}\right) $, where $K_{x0}=2\pi /a_{x}$. These simplifications, we will see, allow us to compute the Green’s function in a relatively straightforward manner while retaining the essential physics of pinning so that our results are qualitatively correct. The major effect of these approximations is to replace the soft cutoff in wavevector that would enter through the form factor with a hard one in the reciprocal lattice sum. With these approximations, the impurity action with which we work is $$S_{\mathrm{imp}}^{\prime }=n_{0}\int d\mathbf{r}\,d\tau \,V\left( \mathbf{r}%
\right) \,\sum_{\mathbf{K}\neq 0}e^{i\mathbf{K}\cdot \left[ \mathbf{r}-%
\mathbf{u}(\mathbf{r},\tau )\right] }, \label{impurityaction}$$ where $n_0=1/a_xa_y$.
Review of the RG analysis
-------------------------
Before proceeding with our replica analysis, we review the highlights of the perturbative RG analysis previously undertaken by some of us [@YFC] to set the stage for our expectations for the results. In the RG approach, one performs momentum shell integrals for large (absolute values of) frequency and $q_{y}$, rescales the lengths and times to keep the cutoffs fixed, and then examines how the parameters of the theory evolve under this transformation. The power of this approach is that it may be carried out perturbatively in the disorder, allowing one to avoid the subtleties that arise from the employment of replicas or other methods needed to handle disorder averages when $V\left( \mathbf{r}\right) $ remains in the exponent.
Another useful feature of the RG approach is that it allows one to look at the contributions to the impurity action individually. Specifically, one must modify Eq. (\[impurityaction\]) to read $$S_{\mathrm{imp}}^{\prime }=n_{0}\int d\mathbf{r}\,d\tau \,V\left( \mathbf{r}%
\right) \,\sum_{\mathbf{K}\neq 0}\Delta _{\mathbf{K}}\left( \ell \right) e^{i%
\mathbf{K}\cdot \left[ \mathbf{r}-\mathbf{u}\left( \mathbf{r},\tau \right) %
\right] }, \label{RGflow}$$where $l$ is the standard scaling variable and $\Delta _{\mathbf{K}}\left( \ell =0\right) =1$. The behavior of $\Delta
_{\mathbf{K}}\left( \ell \right) $ is different depending on whether $%
\mathbf{K}$ is parallel or perpendicular to the stripes. For $\mathbf{K}$ *parallel* to the stripes, one finds $${\frac{d\Delta _{\mathbf{K}}\left( \ell \right) }{d\ell }}=\left( {\frac{%
1-\gamma _{\mathbf{K}}}{2}}\right) \Delta _{\mathbf{K}},$$with $\gamma _{\mathbf{K}}$ increasing for increasing $K$, and taking the value $$\gamma _{\mathbf{K}}={\frac{a_{x}l_{B}^{2}}{a_{y}}}\sum_{q_{x}}d{q_{x}}{%
\frac{d_{xx}\left( q_{x}\right) }{\sqrt{d_{xx}\left( q_{x}\right)
d_{yy}\left( q_{x}\right) -d_{xy}^{2}\left( q_{x}\right) }}}-2,
\label{gammaRG}$$for $\K=K_{y0}\hat{y}$, i.e., for the shortest wavevector parallel to the stripes. The form of Eq. (\[RGflow\]) indicates that the stripes can undergo a quantum phase transition, from one in which they are pinned for motion parallel to the stripes ($\Delta _{\mathbf{K}=(2\pi /a_{y})\hat{y}}$ relevant) to one in which they are unpinned ($\Delta _{\mathbf{K}=\left(
2\pi /a_{y}\right) \hat{y}}$ irrelevant) and free to slide. As can be seen from Eq. (\[gammaRG\]), which state the system ends up in depends in detail on the elastic stiffness of the stripes. For the $N=3$ Landau level, using the same matching procedure as we described above, it was found that the stripes undergo a quantum depinning transition around $\Delta \nu \approx
0.43$ for very weak disorder, with the unpinned state occurring for the larger values of $\Delta \nu $. From the form of Eq. (\[RGflow\]), one can see the depinning occurs via a KT transition [@YFC].
The RG analysis is more complicated if ${\mathbf{K}}$ is perpendicular to the stripes. In this case, for any ${\mathbf{K}}=(K_{x},0)$, the free energy $F\equiv -\ln \int {\mathcal{D}}{\mathbf{u}}\exp \left(
-S_{0}-S_{imp}^{\prime }\right) $ diverges at low temperatures as $T^{-2/5}$ for any $\Delta \nu $. This indicates that pinning *perpendicular* to the stripes is always relevant. Our interpretation of this is that the stripes will be trapped in channels; however, they are still free to move along the channels so that this would not spoil the phase transition described above.
The perturbative RG thus leads us to expect a quantum phase transition from a pinned to an unpinned state as $\Delta\nu$ increases towards 1/2. We will see that the replica analysis discussed below bears out this expectation, and gives results very much in harmony with those of the perturbative RG.
Before closing this section, we believe it is important to point out that different methods for estimating the dynamical matrix $D_{\alpha \beta }$ will lead to different values of $\gamma _{\mathbf{K}}$, and may ultimately lead to different conclusions regarding whether there is a depinning transition in this system. Specifically, calculations based on edge state models for the low energy states of the stripes [@edgestatemodel; @OC] lead to estimates in which the stripes are always in the pinned phase for all $%
\Delta \nu\neq 1/2$. This difference does *not* come as a result of a fundamental difference in the assumed degrees of freedom for the underlying low-energy model; indeed, one may show the edge state and disordered solid models can be mapped onto one another [@edgestatemodel; @Fogler1]. The difference arises purely as a result of the different estimates one arrives at for the dynamical matrix using the two different approaches.
A convincing argument has been made [@edgestatemodel] in the context of the edge state model that the stripes should be in the pinned state provided the system preserves particle-hole symmetry at $\Delta\nu=1/2$. This is not the situation for the model we have adopted: by modeling the stripes as quantum disordered crystals, we assume the system is isomorphic to one in which the system is composed of point particles, which does not have this symmetry. This is natural for our starting point, the modulated stripe HF ground state. These states are highly reminiscent of a collection of electrons in wavepackets, and it is natural to suppose the low-energy fluctuations will consist of displacements of these wavepackets. Moreover, the HF groundstates from which we start *spontaneously* breaks particle-hole symmetry at $%
\Delta\nu=1/2$, arriving at a lower energy state than the uniform, particle-hole symmetric one. Although the sliding fluctuations modify the density to one where the particle-hole symmetry breaking may not be immediately apparent, one does expect to see the broken symmetry in pair correlation functions. Since our estimates of the dynamical matrix elements are taken from the density-density response function, which is closely related to the pair correlation function, it is not surprising that our final result does not respect the limit set by particle-hole symmetry.
An interesting aspect of our approach is that it predicts, in the clean limit, that there will be *two* smectic states, a particle-like one, and a hole-like one, at $\Delta\nu=1/2$. The transition between them as a function of $\Delta\nu$ will presumably be first order. While a direct experimental confirmation of this is difficult, the predictions we make in the present study – a depinning transition, and a dynamical conductivity whose form is characteristic of a depinned state – offer a falsifiable test of whether the QH smectic actually breaks particle-hole symmetry: should experiments show that the dynamical conductivity unambiguously displays behavior associated with the depinned state, then it is most likely that the QH smectic indeed spontaneously breaks particle-hole symmetry at $\Delta\nu=1/2$.
Beyond perturbation theory: Replicas and the GVM
================================================
When a perturbation is relevant in an RG analysis, it is necessary to develop some method for approximating the action to which the system is flowing in order to compute properties of the system. For a pinned elastic system, replicas combined with the Gaussian variational method (GVM) make this possible. In this section, we briefly introduce this method, and go on to discuss some aspects of its application to the stripe system. A fuller discussion may be found in Refs. .
Basic equations
---------------
The fundamental idea of the GVM is to replace a complicated action $S$ with a variational action $S_{\mathrm{var}}$ that is *quadratic*, with coefficients chosen to best match the original problem. This is accomplished by minimizing a free energy [@GO] $$F_{\mathrm{var}}=F_{0}+T\left[ \left\langle S\right\rangle _{S_{\mathrm{var}%
}}-\left\langle S_{\mathrm{var}}\right\rangle _{S_{\mathrm{var}}}\right] ,
\label{var_prin}$$where $S_{\mathrm{var}}$ is the quadratic variational action, $F_{0}$ is the free energy associated with that action, and here $\left\langle \cdots
\right\rangle _{S_{\mathrm{var}}}$ indicates a functional integral over displacements, with $S_{\mathrm{var}}$ as a weighting. For our problem, we would like to disorder average $F_{\mathrm{var}}$, a difficult task because the disorder potential $V$ enters $F_{\mathrm{var}}$ in a complicated and analytically intractable way. A standard method for dealing with this is the replica trick [@dotsenko], in which one creates $n$ copies of the original action, computes the replicated partition function $Z^{n}$ , and then takes the $n\rightarrow 0$ limit. The identity $F=\lim_{n\rightarrow
0}\left( 1-Z^{n}\right) /n$ connects the disorder-averaged, replicated partition function to the free energy. In practice, one first replicates both the Gaussian variational free energy and the original action, performs the disorder average on $Z^{n}$, and then applies Eq. \[var\_prin\] to the resulting replicated effective action, taking the $n\rightarrow 0$ limit only after finding the equations that come from minimizing $F_{\mathrm{var}}$.
Following this program, the effective replicated action after disorder averaging is defined by $$\exp \left( -S_{\mathrm{eff}}\right) ={\frac{1}{\int DV\,P\left( V\right) }}%
\int DV\,P\left( V\right) \exp \left\{ -\sum_{a=1}^{n}\left[ S_{0}^{(a)}+S_{%
\mathrm{imp}}^{^{\prime }(a)}\right] \right\} ,$$which yields $$\begin{aligned}
&&S_{\mathrm{eff}}=S_{0}^{(\mathrm{eff})}+S_{\mathrm{imp}}^{\mathrm{(eff)}},
\label{Seff} \\
&&S_{0}^{(\mathrm{eff})}={\frac{1}{2T}}\sum_{a=1}^{n}
\sum_{\mathbf{q},\omega _{n}}\sum_{\alpha ,\beta
=x,y}u^{a}_{\alpha }\left( \mathbf{q},\omega _{n}\right) \,G_{\alpha \beta
}^{(0)-1}\left( \mathbf{q},\omega _{n}\,\right) u^{a}_{\beta }\left( -\mathbf{q}%
,-\omega _{n}\right) , \label{S0eff} \\
&&S_{\mathrm{imp}}^{\mathrm{(eff)}}\simeq -v_{\mathrm{imp}%
}\sum_{a,b=1}^{n}\,\int_{0}^{1/T}d\tau _{1}\int_{0}^{1/T}d\tau _{2}\,\int d%
\mathbf{r}\sum_{\mathbf{K}\neq 0}\cos \left[ \mathbf{K}\cdot \left[ \mathbf{u%
}^{a}(\mathbf{r},\tau _{1})-\mathbf{u}^{b}(\mathbf{r},\tau _{2})\right] %
\right] , \label{Simpeff}\end{aligned}$$where $v_{\mathrm{imp}}=V^2_0 a^2_x a^2_y n^2_0$, and $a,b$ are replica indices that run from $1$ to $n$. In obtaining the last line of Eq. (\[Simpeff\]) we have neglected some rapidly oscillating terms.
In the pure limit the action is diagonal in the replica indices. Disorder averaging introduces coupling among the replicas through the impurity coupling $S_{\mathrm{imp}}^{\mathrm{(eff)}}$ in Eq. (\[Simpeff\]). This coupling is non-Gaussian, so we next apply the GVM. We introduce the Gaussian variational action $S_{\mathrm{var}}$ which takes the form $$S_{\mathrm{var}}={\frac{1}{2T}}\sum_{\mathbf{q},\omega _{n}}u_{\alpha
}^{a}\left( \mathbf{q},\omega _{n}\right) \,\left( G^{-1}\right) _{\alpha
\beta }^{ab}\left( \mathbf{q},\omega _{n}\right) \,u_{\beta }^{b}\left( -%
\mathbf{q},-\omega _{n}\right) ,$$where $G_{\alpha \beta }^{ab}\left( \mathbf{q},\omega _{n}\right) $ is the displacement Green’s function, $$G_{\alpha \beta }^{ab}\left( \mathbf{q},\omega _{n}\right)
=\int_{0}^{1/T}d\tau \left\langle T_{\tau }u_{\alpha }^{a}\left( \mathbf{q}%
,\tau \right) u_{\beta }^{b}\left( -\mathbf{q},0\right) \right\rangle _{S_{%
\mathrm{var}}}.$$This quantity is to be determined through minimization of the free energy. It is convenient to write it in terms of the bare Green’s function via $$\left( G^{-1}\right) _{\alpha \beta }^{ab}\left( \mathbf{q},\omega
_{n}\right) =G_{\alpha \beta }^{(0)-1}\left( \mathbf{q},\omega _{n}\right)
\,\delta _{ab}-\zeta _{\alpha \beta }^{ab}\left( \omega _{n}\right) ,$$where $\zeta _{\alpha \beta }^{ab}\left( \omega _{n}\right) $ is the element of the variational self-energy matrix $\hat{\zeta}$ (here and hereafter the “hat” indicates that the quantity is a $2\times 2$ matrix). Note that there is no $\mathbf{q}$ dependence in $\hat{\zeta}$ because we have chosen our impurity action to be local in space; this will become clear when we find the saddle point equations below. Note also the obvious symmetries $G^{ab}=G^{ba}$ and $\zeta
^{ab}=\zeta ^{ba}$.
Substituting $S=S_{\mathrm{eff}}$ into Eq. (\[var\_prin\]) and performing the functional integrals, one finds $$F_{\mathrm{var}}=F_{0}+T\left[ \left\langle S_{0}^{\mathrm{(eff)}%
}\right\rangle _{S_{\mathrm{var}}}+\left\langle S_{\mathrm{imp}}^{\mathrm{%
(eff)}}\right\rangle _{S_{\mathrm{var}}}-\left\langle S_{\mathrm{var}%
}\right\rangle _{S_{\mathrm{var}}}\right] ,$$where $$\begin{aligned}
&&F_{0}=-{\frac{1}{2}}T\,\mathrm{Tr}\,\mathrm{ln}\,\widehat{G}+\mathrm{const.%
}, \label{F0} \\
&&\left\langle S_{0}^{\mathrm{(eff)}}-S_{\mathrm{var}}\right\rangle _{S_{%
\mathrm{var}}}={\frac{1}{2}}\sum_{\mathbf{q},\omega
_{n}}\sum_{a,b=1}^{n}\,\sum_{\alpha ,\beta =x,y}\left[ G_{\alpha \beta
}^{(0)-1}\left( \mathbf{q},\omega _{n}\right) \delta _{ab}-(G^{-1})_{\alpha
\beta }^{ab}\left( \mathbf{q},\omega _{n}\right) \right] G_{\alpha \beta
}^{ba}(\mathbf{q},\omega _{n}), \\
&&\left\langle S_{\mathrm{imp}}^{\mathrm{(eff)}}\right\rangle _{S_{\mathrm{var}%
}} =-{\frac{v_{\mathrm{imp}}}{T}}\sum_{a,b=1}^{n}\sum_{\mathbf{K}\neq
0}\int_{0}^{1/T}d\tau \exp \left[ -{\frac{1}{2}}\sum_{\alpha \beta
}K_{\alpha }K_{\beta }\,B_{\alpha \beta }^{ab}\left( \tau \right) \right] ,\end{aligned}$$with $$B_{\alpha \beta }^{ab}(\tau )=\left\langle T_{\tau }[u_{\alpha }^{a}(%
\mathbf{r},\tau )-u_{\beta }^{b}(\mathbf{r},0)]^{2}\right\rangle _{S_{%
\mathrm{var}}}=T\sum_{\mathbf{q},\omega _{n}}\left[ G_{\alpha \beta }^{aa}(%
\mathbf{q},\omega _{n})+G_{\alpha \beta }^{bb}(\mathbf{q},\omega _{n})-2\cos
(\omega _{n}\tau )G_{\alpha \beta }^{ab}(\mathbf{q},\omega _{n})\right] .
\label{Bab}$$
Saddle point equations
----------------------
Equation (\[var\_prin\]) next needs to be extremized, which is accomplished by taking derivatives with respect to the matrix elements of $G$, $\partial F_{%
\mathrm{var}}/\partial \hat{G}=0$. The resulting saddle point equations (SPE’s) are most easily expressed in terms of the self-energy matrix as [GLD96,GLD95]{} $$\begin{aligned}
&&\zeta _{\alpha \beta }^{aa}(\omega _{n})=4v_{\mathrm{imp}%
}\int_{0}^{1/T}d\tau \,\left\{ \left( 1-\cos \omega _{n}\tau \right)
\,V_{\alpha \beta }^{\prime }\left[ B^{aa}(\tau )\right] +\sum_{b\neq
a}V_{\alpha \beta }^{\prime }\left[ B^{ab}(\tau )\right] \right\} ,
\label{zetaaa} \\
&&\zeta _{\alpha \beta }^{a(b\neq a)}(\omega _{n})=-4v_{\mathrm{imp}%
}\int_{0}^{1/T}d\tau \,\cos \omega _{n}\tau \,V_{\alpha \beta }^{\prime }%
\left[ B^{ab}(\tau) \right], \label{zetaab}\end{aligned}$$where $$V_{\alpha \beta }^{\prime }\left[ B^{ab}(\tau )\right] =\sum_{\mathbf{K}\neq
0}K_{\alpha }K_{\beta }\,\exp \left[ -{\frac{1}{2}}\sum_{\mu \nu =x,y}K_{\mu
}K_{\nu }B_{\mu \nu }^{ab}(\tau )\right] .$$
It is apparent at this point that the self-energy has no $\mathbf{q}$ dependence. Moreover, if we assume that reflection symmetry for the stripe system is not spontaneously broken after disorder averaging, it is clear that the solutions of interest to Eqs. (\[zetaaa\]) and (\[zetaab\]) will satisfy $\zeta _{xy}^{ab}=0$. Our task will be to find the self-energy matrix elements that are diagonal in the spatial indices.
It is now convenient to take $n\rightarrow 0$ limit. In doing so, the replica indices are taken to be continuous rather than integral, and they are taken from running from $1$ to $n$ to running from $1$ to $0$. An important aspect of taking this limit is that one *assumes* the self-energy and Green’s function matrices may be written in a hierarchical form [@MPV; @dotsenko]. In the limit $n\rightarrow 0$ such matrices are characterized by diagonal and off-diagonal terms, which may be written as $$\begin{aligned}
&&\zeta _{\alpha \alpha }^{aa}\rightarrow \tilde{\zeta}_{\alpha }, \\
&&\zeta _{\alpha \alpha }^{ab(\neq a)}\rightarrow \zeta _{\alpha }(u),\;\;\;%
\mathrm{for}\;\;0\leq u\leq 1.\end{aligned}$$Similarly, $G_{\alpha \beta }^{aa}\rightarrow \widetilde{G}_{\alpha \beta }$, $G_{\alpha \beta }^{ab(\neq a)}\rightarrow G_{\alpha \beta }(u)$ ($0\leq
u\leq 1$). Since the disorder potential $V(\mathbf{r})$ is time independent, a further simplification one finds is that the off-diagonal replica components $\zeta _{\alpha \alpha }^{ab(\neq a)}$ and $G_{\alpha \beta
}^{ab(\neq a)}$ are $\tau $ independent, [@GLD96; @GLD95; @CGLD] so that $\widehat{G}(\mathbf{q},\omega _{n},u)$ and $\hat{\zeta}(\mathbf{q},\omega _{n},u)$ are different than zero only for $%
\omega _{n}=0$: $$\begin{aligned}
&&\widehat{G}(\mathbf{q},\omega _{n},u)=\widehat{G}(\mathbf{q},u)\,\delta
_{\omega _{n},0}, \label{nown1} \\
&&\hat{\zeta}(\omega _{n},u)=\hat{\zeta}(u)\,\delta _{\omega _{n},0}.
\label{nown2}\end{aligned}$$The SPE’s (\[zetaaa\]) and (\[zetaab\]) now may be written as $$\begin{aligned}
&&\tilde{\zeta}_{\alpha }(\omega _{n})=\int_{0}^{1}du\,\zeta _{\alpha
}(u)+4v_{\mathrm{imp}}\int_{0}^{1/T}d\tau \left( 1-\cos \left( \omega
_{n}\tau \right) \right) \,V_{\alpha \alpha }^{\prime }\left[ \widetilde{B}%
(\tau )\right] , \label{zetatilde} \\
&&\zeta _{\alpha }(u)=-{\frac{4v_{\mathrm{imp}}}{T}}V_{\alpha \alpha
}^{\prime }\left[ B(u)\right] , \label{zetau}\end{aligned}$$where, from Eq. (\[Bab\]), $$\begin{aligned}
&&\widetilde{B}_{\mu \mu }(\tau )=2T\sum_{\mathbf{q},\omega _{n}}\left(
1-\cos (\omega _{n}\tau )\right) \widetilde{G}_{\mu \mu }(\mathbf{q},\omega
_{n}), \label{Btilde} \\
&&B_{\mu \mu }(u)=2T\sum_{\mathbf{q}}\left\{ \left[ \sum_{\omega _{n}}%
\widetilde{G}_{\mu \mu }(\mathbf{q},\omega _{n})\right] -G_{\mu \mu }(%
\mathbf{q},u)\right\} \notag \\
&&\;\;\;\;\;\;\;\;=2T\sum_{\mathbf{q}}\left\{ \left[ \widetilde{G}_{\mu \mu
}(\mathbf{q},\omega _{n}=0)-G_{\mu \mu }(\mathbf{q},u)\right] +\sum_{\omega
_{n}\neq 0}\widetilde{G}_{\mu \mu }(\mathbf{q},\omega _{n})\right\} .
\label{Bu}\end{aligned}$$Note that Eq. (\[zetatilde\]) also gives us $$\tilde{\zeta}_{\alpha }(\omega _{n}=0)=\int_{0}^{1}du\zeta _{\alpha }(u).$$
To solve the Eqs. (\[zetatilde\]) and (\[zetau\]) we must know the relation between $\widetilde{G}(\omega _{n})$, $G(u)$ and $\tilde{\zeta}%
(\omega _{n})$ and $\zeta (u)$. Eqs. (\[nown1\]) and (\[nown2\]) indicate that $$\widetilde{G}_{\mu \mu }(\mathbf{q},\omega _{n}\neq 0)=\left[ \widehat{G}%
^{(0)-1}(\mathbf{q},\omega _{n})-\hat{\tilde{\zeta}}(\omega _{n})\right]
_{\mu \mu }^{-1}. \label{tildeG}$$The quantities $\widehat{\widetilde{G}}(\mathbf{q},\omega _{n}=0)$ and $\hat{%
G}(u)$ are related to $\hat{\tilde{\zeta}}(\omega _{n}=0)$ and $\hat{\zeta}%
(u)$ through inversion rules that generalize the inversion of an $n\times n$ hierarchical matrix to the $n\rightarrow 0$ limit. The inversion rules for a simple hierarchical matrix are well-known [@MP; @dotsenko], and their generalization to a situation in which the elements of the hierarchical matrix are proportional to the unit matrix – which would be the case for our matrices if the elastic system were isotropic – is trivial. However, in our case the entries of the hierarchical matrix are $2\times 2$ matrices with a non-trivial structure. Moreover, the perturbative RG indicates we should expect the pinning properties perpendicular and parallel to the stripes to be different, and we need to generalize the inversion rules to allow for this possibility. With some work, the most general inversion rules for our situation can be derived analytically, and we present this derivation in Appendix \[inversionrules\]. According to Eq. ([inversion3]{}), the Green’s functions are related to the self-energy by $$\widehat{\widetilde{G}}(\mathbf{q},\omega _{n}=0)-\widehat{G}(\mathbf{q},u)=%
\left[ \widehat{D}(\mathbf{q})-\hat{\tilde{\zeta}}(\omega _{n}=0)+\hat{\zeta}%
(u)\right] ^{-1}+\int_{u}^{1}dv\left[ \widehat{D}(\mathbf{q})+\left[ \hat{%
\zeta}\right] (v)\right] ^{-1}\cdot \hat{\zeta}^{\prime }(v)\cdot \left[
\widehat{D}(\mathbf{q})+\left[ \hat{\zeta}\right] (v)\right] ^{-1},
\label{Ginversion}$$where $\hat{\zeta}^{\prime }(v)=d\hat{\zeta}(v)/dv$, and $$\left[ \hat{\zeta}\right] (u)=u\,\hat{\zeta}(u)-\int_{0}^{u}dv\,\hat{\zeta}%
(v). \label{[zetau]}$$ Once we have obtained the self-energy, we can compute the finite-frequency conductivities in Eq. (\[conductivity0\]) by analytically continuing to real frequency in Eq. (\[tildeG\]), so that $$\begin{aligned}
\widetilde{G}_{\mu \mu }^{\mathrm{ret}}(\mathbf{q},\omega \neq 0)&=&\left[
\widehat{G}_{\mathrm{ret}}^{(0)-1}(\mathbf{q},\omega )-\hat{\tilde{\zeta}}^{%
\mathrm{ret}}(\omega )\right] _{\mu \mu }^{-1}. \label{Gomega}\end{aligned}$$Inserting Eq. (\[Gomega\]) into Eq. (\[conductivity0\]) we arrive at the longitudinal conductivity $$\sigma _{\alpha \alpha }(\omega )={\frac{e^{2}}{a_{x}a_{y}}}{\frac{i\omega
\tilde{\zeta}_{\bar{\alpha}}^{\mathrm{ret}}(\omega )}{\tilde{\zeta}_{x}^{%
\mathrm{ret}}(\omega )\tilde{\zeta}_{y}^{\mathrm{ret}}(\omega )-\omega
^{2}/l_{B}^{4}},} \label{longitudinalconductivity}$$where $\bar{\alpha}=y\,(x)$ for $\alpha =x\,(y)$.
To obtain $\tilde{\zeta}_{\alpha }^{\mathrm{ret}}(\omega )$, we analytically continue Eq. (\[zetatilde\]). As shown in Appendix [Appendixanalycon]{}, this results in the equation (for $T=0$) $$\tilde{\zeta}_{\alpha }^{\mathrm{ret}}(\omega )=e_{\alpha }-4v_{\mathrm{imp}%
}\sum_{\mathbf{K}\neq 0}K_{\alpha }^{2}\int_{0}^{\infty }dt\;(e^{i\omega
t}-1)\mathrm{Im}\left[ I(t,K_{x})I\left( t,K_{y}\right) \right] ,
\label{SPEs}$$where $$\begin{aligned}
&&I\left( t,K_{\mu }\right) =\mathrm{\exp }\left[ -{\frac{K_{\mu }^{2}}{\pi }%
}\int_{0}^{\infty }df\,A_{\mu }(f)\left( 1-e^{itf}\right) \right] ,
\label{Ialpha} \\
&&A_{\mu }(f)=\sum_{q}\mathrm{Im}\left[ \widetilde{G}_{\mu \mu }^{\mathrm{ret%
}}(\mathbf{q},f)\right] , \label{spectralfunction}\\
&&e_{\alpha }=\tilde{\zeta}_{\alpha }^{\mathrm{ret}}(0^{+})=\int_{0}^{1}du\,%
\zeta _{\alpha }(u)-4v_{\mathrm{imp}}\sum_{\mathbf{K}\neq 0}K_{\alpha
}^{2}\int_{0}^{\infty }dt\;\mathrm{Im}\left[ I(t,K_{x})I\left(
t,K_{y}\right) \right] \notag \\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-2v_{\mathrm{imp}}\sum_{\mathbf{K}\neq
0}K_{\alpha }^{2}\int_{0}^{1/T}d\tau \exp \left[ -{\frac{1}{2}}\sum_{\mu
=x,y}K_{\mu }^{2}\widetilde{B}_{\mu \mu }(\tau )\right] . \label{e0}\end{aligned}$$As we proceed with our analysis, it is helpful to keep in mind that $A_{\mu
}(f)$ is a spectral function, and that $e_{\alpha }\neq 0$ is an energy offset that in a pinned state opens a gap in the phonon spectrum, as discussed more fully below. Note also that Eq. (\[SPEs\]) indicates that $%
\tilde{\zeta}_{\alpha }^{\mathrm{ret}}(\omega )$ at each $\omega $ point depends on the whole spectrum of $A_{\alpha }(f)$, and this will be complications that for this analysis cannot be avoided, as is the case for other pinned systems [@GLD96; @GLD95]. Since quantum fluctuations play a crucial role in this system, it is useful for us to define effective Debye-Waller factors via $$W(\mathbf{K})={\frac{1}{\pi }}\sum_{\mu =x,y}K_{\mu }^{2}\int_{0}^{\infty
}df\,A_{\mu }(f).$$These quantities are a measure of the mean square displacements in units of the lattice constants, and when large they indicate that quantum fluctuations cannot be ignored in computing the dynamical conductivity. Clearly this will be the case in the vicinity of the quantum depinning transition. On the other hand, if $W(\mathbf{K})$ are small for all $\mathbf{%
K}$, one may expand the exponential function on the right-hand side of Eq. (\[Ialpha\]) and keep only the leading order term. Eqs. (\[SPEs\]) then become greatly simplified, taking the form $$\tilde{\zeta}_{\alpha }^{\mathrm{ret}}(\omega )=e_{\alpha }+2v_{\mathrm{imp}%
}\sum_{\mathbf{K}\neq 0}K_{\alpha }^{2}\sum_{\mu =x,y}K_{\mu }^{2}\sum_{%
\mathbf{q}}\left[\tilde{G}_{\mu \mu }^{\mathrm{ret}}(\mathbf{q},\omega )
- \tilde{G}_{\mu \mu }^{\mathrm{ret}}(\mathbf{q},\omega=0^+) \right ].
\label{SPEsc}$$This is called the semiclassical approximation (SCA) [@GLD96; @GLD95], and it presents a powerful simplification when it is valid. In particular one sees that Eq. (\[SPEsc\]) is local in the frequency, so that $\tilde{\zeta}%
_{\alpha }^{\mathrm{ret}}(\omega )$ may be determined one frequency at a time. Unfortunately, the SCA is not valid in our present problem, and we are forced to solving the full SPE’s (\[SPEs\]) numerically. We will see however that the solutions have several interesting properties that give clear signatures of the depinned phase and the transition leading to it.
Replica symmetric (RS) solution vs replica symmetry breaking (RSB) solution
---------------------------------------------------------------------------
Eq. (\[SPEs\]) shows that the replica diagonal self energy $\tilde{\zeta}^{%
\mathrm{ret}}_{\alpha}(\omega)$ depends on the off-diagonal terms $%
\zeta_{\alpha}(u)$ through the constants $e_\alpha$. It is instructive to first examine the possible structure of $\zeta_{\alpha}(u)$. If $%
\zeta_{\alpha}(u)$ is a constant in $u$, the symmetry of permutation of the replica indices is kept and the solution is “replica symmetric” (RS). On the other hand, when $\zeta_{\alpha}(u)$ varies with $u$, the solution displays replica symmetry breaking (RSB).
For many low-dimensional systems ($d\leq 2$), the appropriate solution to the SPE’s is of the RSB type. Often there is a simple one-step RSB solution, with $\zeta (u)$ piecewise constant, but stepping up or down at a single point $u_{c}$ ($0<u_{c}<1$). It follows from Eq. (\[\[zetau\]\]) that $\left[ \zeta _{\alpha }\right]
(u_{c})\neq 0$ in the RSB state. On the other hand, $\left[ \zeta _{\alpha }%
\right] (u_{c})=0$ for the RS solution.
Following Refs. and , one can establish a close relation between $\left[ \zeta _{\alpha }\right] (u_{c})$ and $e_{\alpha }$. By making use of $\left[ \zeta _{\alpha }\right] (u_{c})$, Eq. (\[zetatilde\]) can be rewritten as $$\tilde{\zeta}_{\alpha }(\omega _{n}\neq 0)=\left[ \zeta _{\alpha }\right]
(u_{c})-4v_{\mathrm{imp}}\int_{0}^{1/T}d\tau \left( 1-\cos \left( \omega
_{n}\tau \right) \right) \,\left\{ V_{\alpha \alpha }^{\prime }[\widetilde{B}%
(\tau )]-V_{\alpha \alpha }^{\prime }[B(u_{c})]\right\} . \label{spe1}$$Here, substituting $V_{\alpha \alpha }^{\prime }\left[ B(u_{c})\right] $ from $V_{\alpha \alpha }^{\prime }\left[ \widetilde{B}(\tau )\right] $ guarantees that as $T\rightarrow 0$ the second term of the right-hand side of Eq. (\[spe1\]) vanishes at $\omega _{n}\rightarrow 0$. Comparing Eqs. (\[spe1\]) and (\[SPEs\]) one immediately concludes that $$e_{\alpha }=-\left[ \zeta _{\alpha }\right] (u_{c}). \label{e01}$$So $e_{\alpha }=0$ in the RS state and $e_{\alpha }\neq 0$ in the RSB state. The two constants $e_{x}$ and $e_{y}$ have significant physical meanings. They may be regarded as a measure of the strength of pinning by the disorder potential and are roughly speaking proportional to the gap in the low-energy magnetophonon modes. If $e_{\alpha }=0$, the phonon spectrum is gapless at ${%
\mathbf{q}}=0$ indicating the system can slide as a whole without energy cost, and is not pinned. Thus an RS solution is expected in the unpinned state. If $e_{\alpha }\neq 0$ as in the RSB solution, a gap opens up in the low-energy magnetophonon modes, uniform sliding cannot be achieved at zero energy, and the system is pinned by disorder.
We will show in Sec. V that for $\Delta\nu<\Delta\nu_c$, both $e_x$ and $e_y$ are nonzero and the stripes are thus fully pinned. As $\Delta\nu\rightarrow
\Delta\nu_c$, $e_y\rightarrow 0$, indicating a quantum depinning transition. The solution to the SPE’s is RS for motion along the stripes but RSB for motion perpendicular to the stripes. We call this type of solution to the SPE’s a *partial* RSB state. The detailed behavior of the system in this state will be explored in Sec. VI.
Results for pinned state: RSB solution
======================================
We begin by examining solutions of the SPE’s for which the QH stripes are fully pinned by disorder. According to the RG result reviewed in Sec. III B, this corresponds to $\Delta\nu <\Delta\nu_c$ in which the disorder is relevant. In this case both $e_x$ and $e_y$ are nonzero. We begin by discussing constraints on $e_x$ and $e_y$ which determine their values, allowing us to solve the SPE’s without explicitly solving for $%
\zeta_{\alpha}(u)$ and $\tilde{\zeta}_{\alpha}$. We then present numerical results for the conductivity.
Two constraints for $e_{x}$ and $e_{y}$
---------------------------------------
The SPE’s (\[SPEs\]) become a set of closed equations if $e_{x}$ and $%
e_{y} $ are known. Formally, $e_{x}$ and $e_{y}$ need to be determined self-consistently by solving Eq. (\[zetau\]). The SPE’s in fact have a family of solutions (parameterized by $u_{c}$), and determining which is best generically would be determined by minimization of the free energy. In the case of spatial dimension $d>2$, $u_{c}$ determined this way leads to Re$%
\left[ \sigma (\omega )\right] \sim \omega ^{2}$ at small $\omega $. This is consistent with arguments by Mott as well as some exact solutions [exactsolution]{} (up to a logarithmic correction). However, for $d\leq 2$, this approach can yield an unphysical result in which, in the pinned state, the conductivity shows a true gap: Re$\left[ \sigma \left( \omega \right) %
\right] $ vanishes below some gap frequency. Alternatively, one may *impose* the condition Re$\left[ \sigma (\omega )\right] \sim \omega ^{2}$ at small $\omega $. It is known that the doing so generates an equation that may be understood as imposing a marginal stability on the so-called replicon mode [@GLD96]. Although this point is not fully understood, it is a common procedure that leads to physically reasonable results, and we will adopt it by imposing the condition Re$\left[ \sigma _{\alpha \alpha }(\omega
)\right] \sim \omega ^{2}$ at small $\omega $ in the pinned state. From Eq. (\[longitudinalconductivity\]), this is equivalent to Im$\left[ \zeta
_{\alpha }^{\mathrm{ret}}(\omega )\right] \sim \omega $. Note that this guarantees the magnetophonon mode density of state vanishes at zero frequency, as one should expect for a pinned system.
To obtain the explicit condition leading to Im$\left[ \zeta _{\alpha }^{%
\mathrm{ret}}(\omega )\right] \sim \omega $, we expand the SPE’s, Eqs. (\[SPEs\]), for small-$\omega $. The integral over $t$ now is dominated by the large $t$ region. Therefore, the term $\int_{0}^{\infty }df\,A_{\mu
}(f)e^{ift}$ in the argument of the exponential function in Eq. (\[Ialpha\]) must be small due to the rapidly oscillating nature of $e^{ift}$, leading to $$I(t,K_{x})I(t,K_{y})\simeq e^{-W(\mathbf{K})}\left[ 1+{\frac{1}{\pi }}%
\sum_{\mu =x,y}K_{\mu }^{2}\int_{0}^{\infty }df\,A_{\mu }(f)e^{itf}\right] .$$The SPE’s (\[SPEs\]) at small $\omega $ become $$\tilde{\zeta}_{\alpha }^{\mathrm{ret}}(\omega )\simeq e_{\alpha }+2\sum_{%
\mathbf{K}\neq 0}v(\mathbf{K})\,K_{\alpha }^{2}\sum_{\mu =x,y}K_{\mu
}^{2}\sum_{q} \left[\tilde{G}_{\mu \mu }^{\mathrm{ret}}(\mathbf{q},\omega )
- \tilde{G}_{\mu \mu }^{\mathrm{ret}}(\mathbf{q},\omega=0^+)\right ],
\label{smallw}$$where $v(\mathbf{K})=v_{\mathrm{imp}}e^{-W(\mathbf{K})}$. This is very similar to the SPEs (\[SPEsc\]) within the SCA except that $v_{\mathrm{imp}%
}$ in Eq. (\[SPEsc\]) is now replaced by $v(\mathbf{K})$. Apparently, when $W(\mathbf{K})\ll 1$ the semiclassical approximation is valid, and Eq. ([smallw]{}) reduces to Eq. (\[SPEsc\]).
At small $\omega $, we write $$\begin{aligned}
&&\mathrm{Re}\left[ \tilde{\zeta}_{\alpha }^{\mathrm{ret}}\right] \simeq
e_{\alpha }, \label{rezetasmallwpin} \\
&&\mathrm{Im}\left[ \tilde{\zeta}_{\alpha }^{\mathrm{ret}}\right] \simeq
\beta _{\alpha }\omega , \label{imzetasmallwpin}\end{aligned}$$and correspondingly $\sum_{\mathbf{q}}\tilde{G}_{\mu \mu }^{\mathrm{ret}}(%
\mathbf{q},\omega )\simeq G_{\mu 0}+\sum_{\alpha =x,y}g_{\mu \alpha }\beta
_{\alpha }\omega $. The condition for nonvanishing $\beta _{\alpha }$ from Eq. (\[smallw\]) becomes $$(U_{xx}-1)(U_{yy}-1)-U_{yx}U_{xy}=0, \label{constraint1}$$where $$U_{\mu \nu }=2\sum_{\mathbf{K}\neq 0}K_{\nu }^{2}\,v(\mathbf{K})\sum_{\alpha
=x,y}K_{\alpha }^{2}g_{\alpha \mu }. \label{U}$$Eq. (\[constraint1\]) is our first constraint for $e_{x}$ and $e_{y}$.
The second constraint follows from the assumption of a *one-step* RSB solution in which $\zeta _{\alpha }(u<u_{c})=0$. Eqs. ([e01]{}) and (\[\[zetau\]\]) immediately yield $$e_{\alpha }=-u_{c}\zeta _{\alpha }\left( u_{c}\right) . \label{ea}$$Inserting Eq. (\[ea\]) into Eq. (\[Ginversion\]) and noting $\zeta
_{\alpha }^{\prime }(v)=0$ for $v\geq u_{c}$ we get $$\widehat{\widetilde{G}}(\mathbf{q},n=0)-\widehat{G}(\mathbf{q},u_{c})=\left[
\widehat{D}(\mathbf{q})+\hat{e}\right] ^{-1}, \label{dG}$$where the elements of the matrix $\hat{e}$ are $e_{\alpha \beta }=e_{\alpha
}\delta _{\alpha \beta }$. Substituting Eq. (\[dG\]) in (\[Bu\]) and making use of $\widetilde{G}_{\mu \mu }(\mathbf{q},\omega _{n}\rightarrow 0)=%
\left[ \widehat{D}(\mathbf{q})+\hat{e}\right] _{\mu \mu }^{-1}$ results in the equation $$B_{\mu \mu }(u_{c})={\frac{2}{\pi }}\int_{0}^{\infty }dfA_{\mu }(f),
\label{Buc}$$which can be inserted in Eq. (\[zetau\]) to give $$\zeta _{\alpha }(u_{c})={\frac{4v_{imp}}{T}}\sum_{\mathbf{K}\neq 0}K_{\alpha
}^{2}e^{-W(\mathbf{K})}. \label{zetauc}$$Eqs. (\[zetauc\]) and (\[ea\]) lead to the ratio $${\frac{e_{y}}{e_{x}}}={\frac{\sum_{\mathbf{K}\neq 0}K_{y}^{2}\,e^{-W(\mathbf{%
K})}}{\sum_{\mathbf{K}\neq 0}K_{x}^{2}\,e^{-W(\mathbf{K})}}}.
\label{constraint2}$$This shows that the pinning of the stripes will generically be anisotropic, and serves as our second constraint for $e_{\alpha }$.
The appearance of the Debye-Waller factors $W(\mathbf{K})$ in Eq. ([constraint2]{}) has a significant impact: they are responsible for the change of behavior in $\tilde{\zeta}^{\mathrm{ret}}_{\alpha}(\omega)$ across the depinning transition. As we shall see, whenever $K_y\neq 0$, $W(\mathbf{K})$ increases as $\Delta\nu\rightarrow \Delta\nu_c$ from below, and it eventually diverges at $\Delta\nu_c$ leading to a suppression of $e_y$. We will discuss this in detail below. We stress that this behavior cannot be captured by the semiclassical approximation.
Numerical results
-----------------
We are now in a position to solve the problem numerically. For a given pair of $e_{x}$ and $e_{y}$, we use an iterative method to solve for $\tilde{\zeta%
}_{\alpha }^{\mathrm{ret}}(\omega )$ from the SPE’s (\[SPEs\]). (Typically 20-30 iterations lead to a good convergence.) The computed $%
\tilde{\zeta}_{\alpha }^{\mathrm{ret}}(\omega )$ are then inserted in the two constraint equations (\[constraint1\]) and (\[constraint2\]) to generate new values of $e_{\alpha }$, and the entire process is repeated until we reach self-consistency. We work in the $N=3$ Landau level, although different Landau indices should give similar results. All our calculations are obtained for a disorder level $v_{\mathrm{imp}}=0.0005e^{4}/l_{B}^{2}$. This is likely to be somewhat larger than experimental values, but we choose it for numerical convenience [@footnotedisorderlevel]. We do not expect our results to qualitatively change for smaller disorder strengths. We note that the bending term in Eq. (\[smecticDxx\]) plays an important role of eliminating an artificial ultraviolet divergence at large $q_{y}$, but beyond this has little effect. We choose $\kappa _{b}=2$ for all the fillings since this leads to a relatively fast convergence of the SPE’s, although we believe the value should be somewhat smaller (of order 1).
(200,200)
Results for $e_x$ and $e_y$ as functions of the partial filling in the pinned state are shown in Fig. \[figpeak\] (a). The quantity $e_x$ is a weak function of $\Delta\nu$, but $e_y$ decreases with increasing $\Delta\nu$, and eventually vanishes at $\Delta\nu=\Delta\nu_c\simeq 0.459$. This is the consequence of a divergence in $W(K_x;K_y\neq 0)$ at $%
\Delta\nu=\Delta\nu_c$. Note $\Delta\nu_c$ is somewhat larger than what was found in the perturbative RG [@YFC]. This is due to the non-vanishing disorder strength; as $v_{\mathrm{imp}}$ decreases, $\Delta\nu_c$ decreases to the value found in Ref. .
The dynamical conductivities perpendicular to the stripes in a pinned (RSB) phase are presented in Fig. \[figsgmxpin\]. For $\Delta \nu $ well below $%
\Delta \nu _{c}\approx 0.459$, Re$\left[ \sigma _{xx}\left( \omega \right) %
\right] $ has a pinning peak whose lineshape is qualitatively similar to what is found using the SCA. [@CGLD]. The prominent behavior visible in Fig. \[figsgmxpin\] is a monotonic decrease of the peak frequency $\Omega
_{px}$ with growing $\Delta \nu $, and its eventual collapse as the depinning transition is approached. The peak frequency behavior is more clearly shown in Fig. \[figpeak\] (b). Notice the lineshape becomes increasingly asymmetric as the transition is approached. Experimental observations so far seem to be consistent with this[@muwave; @Florida].
(250,200)
The real part of the conductivity along the stripes Re$\left[ \sigma
_{yy}\left( \omega \right) \right] $ is shown in Fig. \[figsgmypin\]. It also presents a pinning peak whose frequency $\Omega _{py}$ falls down with increasing $\Delta \nu $ as shown in Fig. \[figpeak\] (b). But the observed peak lineshape is more interesting than that of Re$\left[ \sigma
_{xx}\left( \omega \right) \right] $. Below the peak frequency $\Omega _{py}$, in the range $e_{y}<\omega <\Omega _{py}$ the conductivity appears to tend toward a non-vanishing value when $\Delta \nu $ is sufficiently below $%
\Delta \nu _{c}$; only for $\omega $ well below this range does one find Re$%
\left[ \sigma _{yy}(\omega )\right] $ decreasing. The reason for this is that the quantity $e_{y}$ turns out to be rather small \[as shown in Fig. [figpeak]{} (a)\] due to a large Debye-Waller factor, and in this frequency range the system displays a behavior similar to an incoherent metal response [@kohn]. We discuss this in more detail for the depinned (PRSB) phase below. For $\omega \ll e_{y}$, Re$\left[ \sigma _{yy}(\omega )\right] $ vanishes quadratically with $\omega $ (not visible on the scale of Fig. [figsgmypin]{}), as required for a pinned state. As $\Delta \nu \rightarrow
\Delta \nu _{c}$, we eventually reach a situation in which $e_{y}$ and $%
\Omega _{py}$ are of similar order, in which case the pinning peak sharpens and grows quite large. This peak continuously evolves into a $\delta $-function at zero frequency as the system enters into the PRSB state, so that the transition from pinned to depinned behavior is very continuous.
(250,200)
Interestingly, as shown in Fig. \[figpeak\], $e_{y}\rightarrow 0$ governs the vanishing of both $\Omega _{px}$ and $\Omega _{py}$. To understand this, we note that there are two gapless collective modes in the absence of the magnetic field; the magnetic field mixes them into two other modes, one of which is at a high value (order of $\hbar \omega _{c}$), leaving the other (magnetophonon) mode as the only gapless one. It is this single mode that responds to the electric field, albeit in an anisotropic manner in the $\hat{%
x}$ and $\hat{y}$ directions. Technically, at $\omega \sim \Omega
_{px},\Omega _{py}$, $\zeta _{\alpha }^{\mathrm{ret}}(\omega )$ obeys Eqs. (\[rezetasmallwpin\]-\[imzetasmallwpin\]), and the longitudinal conductivities in Eq. (\[longitudinalconductivity\]) become $$\mathrm{Re}\left[ \sigma _{\alpha \alpha }(\omega )\right] \simeq {\frac{%
e^{2}}{a_{x}a_{y}}}\;\omega ^{2}{\frac{e_{\bar{\alpha}}(e_{x}\beta
_{y}+e_{y}\beta _{x})+\beta _{\bar{\alpha}}\left[ \left( 1+\beta _{x}\beta
_{y}\right) \omega ^{2}-e_{x}e_{y})\right] }{\left[ \left( 1+\beta _{x}\beta
_{y}\right) \omega ^{2}-e_{x}e_{y}\right] ^{2}+\omega ^{2}\left[ e_{x}\beta
_{y}+e_{y}\beta _{x}\right] ^{2}}}. \label{sigma}$$From this we can extract $$\Omega _{px}\sim \Omega _{py}\sim \sqrt{\frac{e_{x}e_{y}}{1+\beta _{x}\beta
_{y}}}.$$
Results for depinned state: Partial RSB (PRSB) solution
=======================================================
For $\Delta\nu\ge \Delta\nu_c$ the state is characterized by $e_x\ne 0$ but $%
e_y=0$. As discussed in Sec. IV.C, this corresponds to a RSB solution for $%
\zeta_x(u)$ but a RS solution for $\zeta_y(u)$. We call this the *PRSB* state. This state has various interesting properties that we will present below.
Power law behavior for $\tilde{\protect\zeta}_{y}^{\mathrm{ret}}(%
\protect\omega )$
------------------------------------------------------------------
From the results of the perturbative RG, we expect the stripes to remain pinned for motion in the $\hat{x}$ direction even as the stripes become depinned for motion in the $\hat{y}$ direction. We therefore assume that in the PRSB state, the small-$\omega $ asymptotic behavior of $\tilde{\zeta}%
_{x}^{\mathrm{ret}}(\omega )$ remains the same as that in the RSB state \[Eqs. (\[rezetasmallwpin\]-\[imzetasmallwpin\])\], and this turns out to yield a self-consistent solution. However, $\tilde{\zeta}_{y}^{\mathrm{ret}%
}(\omega )$ is qualitatively different in the depinned state. To see this explicitly we examine the SPEs (\[SPEs\]) for $\tilde{\zeta}_{y}^{\mathrm{%
ret}}(\omega )$, $$\tilde{\zeta}_{y}^{\mathrm{ret}}(\omega )=-4v_{\mathrm{imp}}\left\{
2K_{y0}^{2}\int_{0}^{\infty }dt\;\left( e^{i\omega t}-1\right) \mathrm{Im}%
\left[ I(t,K_{y0})\right] +4K_{y0}^{2}\int_{0}^{\infty }dt\;(e^{i\omega t}-1)%
\mathrm{Im} \left[I(t,K_{y0})I(t,K_{x0})\right]\right\} . \label{SPEPRSB}$$There is a self-consistent solution for this equation in which $\tilde{\zeta}%
_{y}^{\mathrm{ret}}(\omega )$ has an anomalous power law behavior at low frequencies, $$\begin{aligned}
&&\mathrm{Re}\left[ \mathrm{~}\tilde{\zeta}_{y}^{\mathrm{ret}}(\omega )%
\right] \simeq \alpha _{y}\omega ^{2}, \label{rezetasmallwdepin} \\
&&\mathrm{Im~}\left[ \tilde{\zeta}_{y}^{\mathrm{ret}}(\omega )\right] \simeq
\beta _{y}\omega ^{\gamma +1}. \label{imzetasmallwdepin}\end{aligned}$$This solution is only valid when $$\gamma \geq 1, \label{inequality}$$where $\gamma $ is defined in Eq. (\[gamma\]) below. Note that at small $%
\omega $, $\tilde{\zeta}_{y}^{\mathrm{ret}}(\omega )$ is dominated by large-$%
t$ behavior of the integrands in Eq. (\[SPEPRSB\]), and this in turn depends on small-$f$ asymptotics of $A_{\alpha }(f)\left( 1-e^{itf}\right) $ in $I(t,K_{\alpha 0})$ \[see Eq. (\[Ialpha\])\]. Making use of Eqs. ([rezetasmallwpin]{}-\[imzetasmallwpin\]) and (\[rezetasmallwdepin\]-[imzetasmallwdepin]{}), and of the smectic form of the dynamical matrix $%
\widehat{D}(\mathbf{q})$ in Eqs. (\[smecticDxx\]-\[smecticDyy\]) (with the bending term neglected), we obtain $$A_{y}(f)\simeq {\frac{a_{x}a_{y}}{(2\pi )^{2}}}\int dq_{x}\left[
d_{xx}(q_{x})-e_{x}\right] \,\mathrm{Im}\,\left[ \int_{-\infty }^{\infty }{%
\frac{dq_{y}}{g_{x}(q_{x})q_{y}^{2}-(1+\alpha _{y})\left( f+i0^{+}\right)
^{2}}}\right] \simeq {\frac{\pi c_{v}}{f}}, \label{Ay}$$where $$g_{x}(q_{x})=(1+\alpha _{y})\,\left\{ \left[ d_{xx}(q_{x})-e_{x}\right]
d_{yy}(q_{x})-d_{xy}^{2}(q_{x})\right\} ,$$and $$c_{v}={\frac{a_{x}a_{y}l_{B}^{2}}{(2\pi )^{2}}}\int d{q_{x}}{\frac{%
d_{xx}-e_{x}}{\sqrt{g_{x}(q_{x})}}}. \label{cv}$$The quantity $A_{y}$ bears a singular $1/f$ term which is responsible for the unusual feature of $\tilde{\zeta}_{y}^{\mathrm{ret}}(\omega )$. In contrast, $A_{x}(f)$ can be easily shown to converge to a constant at small $%
f$. Therefore, in Eq. (\[SPEPRSB\]), the large $t$ behavior of $%
I(t,K_{y0})I(t,K_{x0})$ is dominated by $I(t,K_{y0})$, and the two terms within the braces have qualitatively the same small-$\omega $ behavior. Substituting Eq. (\[Ay\]) in Eq. (\[Ialpha\]) leads to $$I(t,K_{y0})\sim (1+it\Lambda _{\omega })^{\gamma +2}, \label{Iy1}$$where $$\gamma ={\frac{K_{y0}^{2}c_{v}}{\pi }}-2. \label{gamma}$$Here $\Lambda _{\omega }$ is a high-energy cutoff of order the magnetophonon band width. Inserting Eq. (\[Iy1\]) into Eq. (\[SPEPRSB\]) and keeping only the leading-order terms in $\omega $ we produce Eqs. ([rezetasmallwdepin]{}-\[imzetasmallwdepin\]) provided Eq. (\[inequality\]) is met. For larger values of $\gamma $, the solution is not self-consistent, and one must revert to the full RSB (pinned) solution.
Equations (\[rezetasmallwdepin\]-\[inequality\]) are the criteria for the existence of a PRSB solution. The inequality (\[inequality\]) defines a critical value $\gamma_c=1$. Since $\gamma$ in Eq. (\[gamma\]) increases monotonically with $\Delta\nu$, this critical value corresponds to a critical filling $\Delta\nu_c$. Our numerical result shown below in Sec. VI C indicates that $\Delta\nu_c$ obtained this way matches nicely with the critical filling defined in the RSB state through the collapse of the pinning peaks. One can also see that in the vanishing disorder limit ($e_x,
\alpha_y\rightarrow 0$), $\gamma$ reduces to $\gamma_0$ defined in Eq. ([gammaRG]{}) that occurs in the RG flow equation (\[RGflow\]), and the condition (\[inequality\]) matches the RG condition for the irrelevance of the disorder. Technically, the reason these coincide originates from the similar ways in which the Green’s function enters in the SPE (\[SPEPRSB\]) and in the calculation of the scaling dimension of the impurity term in the RG analysis. The minor difference is that the GVM includes the renormalization of the Green’s function by disorder, while the RG analysis, being perturbative, uses the Green’s function for the pure system.
In both the RSB and the PRSB states, Im$\left[ \tilde{\zeta}_{y}^{\mathrm{ret%
}}(\omega )\right] $ shows power-law behavior Im$\left[ \tilde{\zeta}_{y}^{%
\mathrm{ret}}(\omega )\right] \sim \omega ^{\gamma _{\zeta y}}$ \[Eqs. ([imzetasmallwpin]{}) and (\[imzetasmallwdepin\])\], although the exponent is fixed at $1$ for the RSB state. Plotting $\gamma _{\zeta y}$ as a function of the partial filling in Fig. \[figexponents\] (a), we see that $\gamma
_{\zeta y}$ *jumps* from $1$ in the RSB state to $2$ in the PRSB state. This jump arises from an underlying jump in the low-frequency exponent in Re$%
\left[ \tilde{\zeta}_{y}^{\mathrm{ret}}(\omega )\right] $ (from $0$ in the RSB state \[Eq. (\[rezetasmallwpin\])\] to $2$ in the PRSB state \[Eq. ([rezetasmallwdepin]{})\]). Such jumps are typical for a KT-type phase transition. In the next subsection we will show the corresponding jumps in the low-frequency exponents in conductivities.
(250,220)
Anomalous low-frequency exponents for conductivities
----------------------------------------------------
The unusual low-frequency exponents of $\tilde{\zeta}_{y}^{\mathrm{ret}%
}(\omega )$ directly affect the low-frequency behavior of the conductivities. Inserting Eqs. (\[rezetasmallwpin\]-\[imzetasmallwpin\]) and (\[rezetasmallwdepin\]-\[imzetasmallwdepin\]) in Eq. ([longitudinalconductivity]{}) we find that at small $\omega $, $$\begin{aligned}
&&\mathrm{Re}\,\left[ \sigma _{yy}(\omega )\right] \simeq e^{2}\left[
s_{y0}\,\delta (\omega )+s_{y1}\omega ^{\gamma -2}+s_{y2}\right] ,
\label{sgmysmallwPRSB} \\
&&\mathrm{Re}\left[ \,\sigma _{xx}(\omega )\right] \simeq e^{2}\left[
s_{x1}\,\omega ^{\gamma }+s_{x2}\,\omega ^{2}\right] ,
\label{sgmxsmallwPRSB}\end{aligned}$$where $s_{y0}=\Delta \nu e_{x}/2(1-e_{x}\alpha _{y})$, $s_{y1}=e_{x}^{2}%
\beta _{y}\Delta \nu /2\pi $, $s_{y2}=\beta _{x}(1+e_{x}\alpha _{y})\Delta
\nu /2\pi $, $s_{x1}=\beta _{y}(1+e_{x}\alpha _{y})\Delta \nu /2\pi $, $%
s_{x2}=\alpha _{y}^{2}\beta _{x}\Delta \nu /2\pi $. The most significant feature in Re$\left[ \sigma _{yy}(\omega )\right] $ lies in the $\delta $ peak at $\omega =0$ [@footnotedeltapeak]. Physically, this means that the PRSB phase is a superconducting state, and the system manages to find an effective *free* path to slide along the stripes. By contrast, Re$\left[ \sigma_{xx}(\omega \rightarrow 0)\right] =0$, implying the system is insulating for motion perpendicular to the stripes. This suggests that the PRSB state has *infinite* anisotropy in the DC conductivity. This is not observed in DC transport experiments [@stripesexpr], and we comment in Sec. VIII on what is missing from our model that we believes leads to this discrepancy.
The other two terms in Eq. (\[sgmysmallwPRSB\]) and the terms in Eq. ([sgmxsmallwPRSB]{}) imply an incoherent contribution at $\omega \neq 0$. Interestingly, these terms compete with each other in determining the low-frequency exponents of the conductivities, leading to a second transition. For $1\leq \gamma <2$, the second term in Eq. ([sgmysmallwPRSB]{}) and the first term in Eq. (\[sgmxsmallwPRSB\]) dominate, so that Re$\left[ \sigma _{yy}(\omega )\right] \sim \omega ^{-(2-\gamma )}$ which diverges as $\omega \rightarrow 0$ [@missed]. This is a very unusual finite frequency response, which arises from the form of the Green’s function in the PRSB state and so appears to be specific to this system just after the depinning transition. The response perpendicular to the stripes is insulating but also anomalous, Re$\left[ \sigma _{xx}\left( \omega \right) %
\right] \sim \omega ^{\gamma }$ with $\gamma $ non-integer. Since $\gamma $ increases with the filling $\Delta \nu $, the low-frequency exponents of Re$%
\left[ \sigma _{yy}(\omega )\right] $ and Re$\left[ \sigma _{xx}(\omega )%
\right] $ evolve (continuously) from $\Delta \nu _{c}$ (for which $\gamma =1$) to a second critical filling $\Delta \nu'_{c}$ (for which $\gamma =2$). As $\Delta \nu $ further increases from $\Delta \nu'_{c}$, $\gamma $ becomes larger than $2$, and the third term in Eq. (\[sgmysmallwPRSB\]) and the second term in Eq. (\[sgmxsmallwPRSB\]) dominate the low frequency behavior. Consequently, Re$\left[ \sigma _{yy}(\omega )\right] \sim const.$ for small but non-vanishing $\omega$, which is a standard finite frequency response for a superconductor, sometimes called incoherent metallic behavior [@kohn]. Furthermore, Re$\left[ \sigma
_{xx}\left( \omega \right) \right] \sim \omega ^{2}$ which is similar to the behavior in the fully pinned state. Thus, at $\Delta \nu'_{c}$, the system experiences a second transition in which the finite-frequency behavior of the stripes changes. The conductivities thus have a very unusual low-frequency behavior for a small window of filling factors, $\Delta \nu
_{c}<\Delta \nu <\Delta \nu'_{c}$. Interestingly, such changes in power-law behavior above a KT transition is known to occur in other contexts [sudbo]{}. The qualitative result of the low-frequency exponents $\gamma
_{\sigma \alpha }$ of Re$\left[ \sigma _{\alpha \alpha }\left( \omega
\right) \right] $ discussed here can be seen in Fig. \[figexponents\] (b). The numerical values of $\Delta \nu _{c}$ and $\Delta \nu'_{c}$ for our calculations will be discussed in the next subsection. As also shown in Fig. \[figexponents\] (b), both $\gamma _{\sigma x}$ and $\gamma _{\sigma y}$ jump at the depinning transition point $\Delta \nu _{c}$.
In practice, the visibility of the various terms in Eqs. ([sgmysmallwPRSB]{}) and (\[sgmxsmallwPRSB\]) depends on the relative size of the coefficients of each term, which we discuss in the next subsection.
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Constraint for $e_x$ and numerical result for the conductivities
----------------------------------------------------------------
To obtain quantitative results for the conductivities in the PRSB state, we need to numerically solve the SPEs (Eqs. \[SPEs\]) together with the constraint for $e_{x}$. This constraint can be obtained, under the assumption of the existence of linear-$\omega $ term in $\tilde{\zeta}_{x}^{%
\mathrm{ret}}(\omega )$ at small $\omega $, from the first constraint Eq. (\[constraint1\]) in the pinned state by examining the limit $%
e_{y}\rightarrow 0$. It is easy to find that $g_{xx}\sim const.$, $%
g_{xy}\sim |e_{y}|^{1/2}$, and $g_{yy}\sim |e_{y}|^{3/2}$. On the other hand, $e^{-W(\pm K_{x0},0)}$ tends to a non-vanishing constant as $%
e_{y}\rightarrow 0$, while both $e^{-W(0,\pm K_{y0})}$ and $e^{-W(\pm K_{x0},\pm K_{y0})}$ scale as $%
|e_{y}|^{-(\gamma +2)/2}$, where $\gamma $ was defined in Eq. (\[gamma\]). Plugging these into Eq. (\[U\]) yields $U_{xx}\sim const.$, $U_{xy}\sim
|e_{y}|^{-1/2}$, $U_{yx}\sim |e_{y}|^{(\gamma +1)/2}$, and $$U_{yy}\sim |e_{y}|^{(\gamma -1)/2}. \label{Uyy}$$Eq. (\[constraint1\]) thus becomes $$\left( U_{xx}-1\right) \left( c_{0}|e_{y}|^{(\gamma -1)/2}-1\right) \sim
|e_{y}|^{\gamma /2}, \label{constraintPRSB0}$$where $c_{0}$ is a constant whose precise value is irrelevant for our discussion. In the PRSB state, $\gamma >1$ and $e_{y}=0$, so that the constraint becomes $$U_{xx}-1=0. \label{constraintPRSB}$$In Appendix \[solution2\], we discuss another way of satisfying Eq. ([constraintPRSB0]{}) which leads to an unphysical solution.
-------------------------------------------------------------------------------------------------------
$\Delta\nu$ $e_x$ $\beta_x$ $\alpha_y$ $\beta_y$ $% $s_{y2}$ $s_{x1}$ $s_{x2}$
s_{y1}$
------------- --------- ----------- ------------ ----------- --------- ---------- ---------- ----------
0.46 -0.0037 2.26 0.68 2.2 2.2e-6 0.165 0.161 0.076
0.463 -0.0037 1.96 0.34 2.6 2.6e-6 0.143 0.192 0.016
0.466 -0.0037 1.98 0.18 2.9 2.9e-6 0.146 0.215 0.0049
0.47 -0.0035 1.97 0.10 3.4 3.1e-6 0.147 0.252 0.0014
0.48 -0.0034 1.85 0.02 5.0 4.4e-6 0.141 0.382 4e-5
-------------------------------------------------------------------------------------------------------
: Table of the coefficients of the leading order terms of the self energy and conductivities in the PRSB state at various filling.
We have carried out numerical calculations of the exponents and conductivities for the fillings $\Delta \nu =0.46,0.463,0.466,0.47,0.48$, all of which are in the PRSB state. Results for $\gamma $ are shown as squares in Fig. \[figgamma\]. Clearly, $\gamma $ increases monotonically with filling factor. Notice that $\gamma $ at $\Delta \nu =0.46$ is very close to the critical value $1$, and by an extrapolation we conclude that $%
\Delta \nu _{c}\simeq 0.459$. By comparing with Fig.\[figpeak\] in the RSB state, we find that $\Delta \nu _{c}$ agrees with that from the RSB state, yielding a non-trivial check on our numerics. In Fig. \[figgamma\] we also plot $\gamma _{0}$ (circles) which is computed from Eq. (\[gammaRG\]) in the pure limit. The critical $\gamma _{0}$ results in a critical filling at the vanishing disorder limit $\Delta \nu _{c}^{(0)}\simeq 0.432$. The result of $\Delta \nu _{c}>\Delta \nu _{c}^{(0)}$ reflects the fact that a stronger disorder strength makes pinning more likely and so increases $%
\Delta \nu _{c}$. As we mentioned before, the disorder level we choose is most likely larger than the experimental situation. We expect that in the experimental parameter regime the critical filling for the quantum depinning transition for $N=3$ is some value between $0.432$ and $0.459$.
The numerically computed values of $\gamma_{\zeta y}$, $\gamma_{\sigma x}$ and $\gamma_{\sigma y}$ are shown as circles in Fig. (\[figexponents\]). We find $\Delta\nu'_{c}\simeq 0.467$.
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Our numerical result also confirms the low-energy behavior of the self-energy Eqs. (\[rezetasmallwpin\]-\[imzetasmallwpin\]) and ([rezetasmallwdepin]{}-\[imzetasmallwdepin\]). In Tab. 1, we present the coefficients of the leading terms of the self energy. It is clear that $e_x$ is nearly a constant, and $\beta_x$ and $\beta_y$ are also moderate functions of $\Delta\nu$. But $\alpha_y$ increases drastically as $\Delta\nu$ approaches $\Delta\nu_c$ from above. We will comment on this in Sec. VII.
It is important to note that the coefficients we find for $s_{y1}$ are numerically very small (see Table I), so that the anomalous divergence near $%
\omega =0$ can only be visible at very small frequencies. This suggests that the divergence may be in practice difficult to observe, and indeed it is beyond the numerical accuracy of our calculations too because the frequency grid required would be much finer than can practically be achieved. For frequencies of order $\omega >10^{-5}$ we find that the anomalous divergence cannot be seen (for the parameters of our calculation) and the finite frequency response appears to be that of an incoherent metal. Interestingly we find that the incoherent contribution to the dynamical conductivity becomes sharply peaked for $\omega <.001$, but this levels off to a constant when the frequency becomes small enough.
Numerical results for the conductivities perpendicular to the stripes at various fillings are shown for a fairly large frequency range in Fig. [sgmxdepin]{}. The low-energy pinning mode is absent, and there is instead a broad peak at high frequencies. We would like to remark that the result at such high energy scales should be taken with a grain of salt, as our elastic model only reproduces the excitation spectrum of the quantum Hall stripes for low energies. The peak may be interpreted as being due to a maximum in the phonon density of states that occurs in the elastic model, and is *not* a pinning peak. For much lower frequencies where our computation is accurate, Re$\left[
\sigma _{xx}(\omega )\right] $ shows the power law behavior as expected. For $\Delta \nu =0.47$ and $0.48$ which are larger than $\Delta \nu'_{c}$, the anticipated Re$\left[ \sigma _{xx}(\omega )\right] \simeq \omega ^{2}$ is not visible within our numerical accuracy. As shown in Table I, the coefficient of the $\omega ^{2}$ term, $S_{x2}$, is much smaller than that of the $\omega ^{\gamma }$ term, $S_{x1}$, for $\Delta \nu =0.47$ and $0.48$, again requiring a very fine $\omega $ grid to observe. Thus in practice one may observe the anomalous power law dependence over a relatively large range of filling factors.
Quantum depinning transition - KT universality class
====================================================
In the previous section, we observed jumps in the low-frequency exponents of $\tilde{\zeta}_{y}^{\mathrm{ret}},$ Re$\left[ \sigma _{xx}\left( \omega
\right) \right] $ and Re$\left[ \sigma _{yy}\left( \omega \right) \right] $ at the quantum depinning transition point. In this section, we discuss the connection of these jumps with the universal jump in the superfluid stiffness and the critical exponent of correlation functions of the KT transition [@nelson].
The depinning transition we have found is of the KT form, as is clear from the perturbative RG analysis [@YFC]. Inserting the smectic form of $%
D_{yy}(\mathbf{q})$ in Eq. (\[smecticDyy\]) into the action (\[S0eff\]) one can see that $d_{yy}(q_x)$ acts as an effective stiffness along the stripes direction. The action in Eqs. (\[Seff\])-(\[Simpeff\]) for the stripes phase then can be viewed as a generalized quantum sine-Gordon model: for $v_{\mathrm{imp}}=0$, the action behaves as a collection of 1+1 dimensional elastic systems, one for each $q_x$; the impurity term couples these systems. As is well-known, the two-dimensional classical sine-Gordon model supports a roughening transition [@chaikin], which formally is closely related to a smectic-to-crystal transition, and is a dual description of the KT vortex unbinding transition [@chaikin].
An interesting aspect of our system is that, in the pure limit, there is no term that is quadratic in $\omega $ in either diagonal component of the Green’s function, so that there is no analogue of $d_{yy}(q_{x})$ in the time direction. However, such a term *is* generated in the self-energy as a result of the variational method when the disorder is present, even in the depinned state. Writing Re$\left[ \tilde{\zeta}_{y}^{%
\mathrm{ret}}(\omega )\right] \simeq \alpha _{y}\omega ^{2}$ for small $%
\omega $, we plot $\alpha _{y}$ as a function of partial filling factor $\Delta \nu $ above the transition in Fig. \[rezyPRSB\]. One can see the sharp increase as the transition is approached. Such an increase is consistent with the usual RG for the roughening transition, for which the stiffness increases in the RG flows, although one needs to go to higher order in perturbation theory than was undertaken in Ref. to see this. We note finally that $%
\alpha _{y}$ *cannot* increase indefinitely: as it increases, the value of $\gamma $ (Eq. (\[gamma\])) decreases, eventually crossing the critical value and forcing the system into the fully pinned state. In this state, Re$\left[ \tilde{\zeta}_{y}^{\mathrm{ret}}(\omega )\right] \simeq
\alpha _{y}\omega ^{2}$ no longer vanishes as $q_{y},\omega \rightarrow 0$, but rather goes to a constant. This can be roughly interpreted as a system with an infinite stiffness, so that one may associate the transition with a jump in $\alpha _{y}$ from its critical value to infinity.
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Conclusion
==========
In this paper, we have used replicas and the Gaussian variational method to calculate the finite-frequency conductivity of QH stripes in order to see its detailed behavior in the vicinity of the quantum depinning transition. The low-energy degrees of freedom of the QH stripes are described within an effective elastic model that is characterized by a dynamical matrix which is determined by matching to microscopic TDHFA calculation. Our results show that in the pinned state for $\Delta \nu <\Delta \nu _{c}$, the system is in an RSB state, and the conductivities have resonant peaks for excitation both parallel and perpendicular to the stripes. As $\Delta \nu $ approaches $%
\Delta \nu _{c}$ from below, a Debye-Waller factor $W(K_{x};K_{y}\neq 0)$ increases and eventually diverges at $\Delta \nu =\Delta \nu _{c}$, resulting in a vanishing pinning energy $e_{y}$ for motion along the stripes. For $\Delta \nu >\Delta \nu _{c}$, the system enters a new state with *partial* replica symmetry breaking (PRSB), in which the solution has RSB perpendicular to the stripes, but is replica symmetric along them. In this state Re$\left[ \sigma _{yy}(\omega )\right] $ has a superconducting response at zero frequency and an anomalous power law behavior for both Re$\left[ \sigma _{xx}(\omega )\right] $ and Re$\left[
\sigma _{yy}(\omega )\right] $ for $\Delta \nu $ just above the critical value. Moreover, there are jumps in the low-frequency exponents of both the self-energy and conductivities at the transition point, as one might expect for a KT transition.
We conclude by discussing a prominent discrepancy between our results and those of existing experiments. In DC transport, one observes metallic behavior with [finite]{} anisotropy rather than the infinite one found in the PRSB state. We believe the missing ingredients from our model are processes allowing hopping of electrons between stripes. These processes are very difficult to incorporate into an elastic model. It is clear that, if relevant in the RG sense, such processes can broaden the $\delta $-function response to yield anisotropic metallic behavior. Our results should apply at frequency scales above this broadening. Indeed, microwave absorption experiments become quite challenging at low frequencies, and it is unclear whether existing measurements of the dynamical conductivity can access the low frequency conductivity in the unpinned state, whether or not it is broadened. In any case, it is interesting to speculate that a true $\delta $-function response might be accessible in structured environments where barriers between stripes may suppress electron hopping among stripes [Endo]{}, or that there may be analogous states for layered 2+1 dimensional classical systems of long string-like objects, which has been shown [fertig99]{} to be closely related to the two-dimensional quantum stripe problem.
The authors are especially grateful to R. Lewis, L. Engel, and Y. Chen for many stimulating discussions about this problem, and for showing us their experimental data prior to publication. We are also indebted to T. Giamarchi, G. Murthy, E. Orignac, E. Poisson, and A.H. MacDonald for useful discussions and suggestions. This work was supported by a NSF Grant No. DMR-0414290, by a grant from the Fonds Québécois de la recherche sur la nature et les technologies and a grant from the Natural Sciences and Engineering Research Council of Canada, and by a grant from SKORE-A program.
Summary of HF and TDHF formalisms {#HFappendix}
=================================
In a previous work[@CF], two of us have shown that, in the HFA, the smectic state (as in the edge state model[@edgestatemodel]) is unstable with respect to density modulations along the direction of the stripes. The ground state of the two-dimensional electron gas near half filling of the higher Landau levels is instead an anisotropic two-dimensional Wigner crystal with basis vectors $\mathbf{R}_{1}=\left( 0,a_{y}\right) $ and $\mathbf{R}_{2}=\left( a_{x},a_{y}/2\right) $. (One can also see this crystal as an array of 1D Wigner crystals with out-of-phase modulations on adjacent 1D crystals). The electronic density of this crystal is fully determined by the Fourier components of the electronic density $\left\{ \left\langle
n\left( \mathbf{K}\right) \right\rangle \right\} $ where $\mathbf{K}$ is a reciprocal lattice vector of the oblique lattice shown in Fig. \[stripeselastic\].
In our analysis, the Hilbert space is restricted to that of the partially filled Landau level. It is then convenient to define a density of orbit centers or guiding-center density $%
\left\langle \rho \left( \mathbf{K}\right) \right\rangle $ which is related to the electronic density by the equation$$\left\langle n\left( \mathbf{K}\right) \right\rangle =N_{\phi }F_{N}\left(
\mathbf{K}\right) \left\langle \rho \left( \mathbf{K}\right) \right\rangle ,$$where $N_{\varphi }$ is the Landau-level degeneracy and $$F_{N}\left( \mathbf{K}\right) =e^{-K^{2}l_{B}^{2}/4}L_{N}^{0}\left( \frac{%
K^{2}l_{B}^{2}}{2}\right) ,$$( $L_{N}^{0}\left( x\right) $ is a generalized Laguerre polynomial) is a form factor for an electron in Landau level $N.$ The $\left\langle \rho
\left( \mathbf{K}\right) \right\rangle ^{\prime }s$ can be computed[@CF] by solving the HF equation of motion for the single particle Green’s function$${\cal G}\left( \mathbf{K,}\tau \right) =-\frac{1}{N_{\phi }}\sum_{X,X^{\prime }}e^{-%
\frac{i}{2}K_{x}\left( X+X^{\prime }\right) }\delta _{X,X^{\prime
}-K_{y}l_{B}^{2}}\left\langle {\mathcal{T}}_{\tau }c_{X}\left( \tau \right)
c_{X^{\prime }}^{\dagger }\left( 0\right) \right\rangle , \label{1_3}$$with $$\left\langle \rho \left( \mathbf{K}\right) \right\rangle ={\cal G}
\left( \mathbf{K,} \tau =0^{-}\right) .$$In Eq. (\[1\_3\]), $c_{X}\left( c_{X}^{\dagger }\right) $ is the destruction(creation) operator for an electron in Landau level $N$ with guiding-center $X$ in the Landau gauge.
From the set of $\left\langle \rho \left( \mathbf{K}\right) \right\rangle
^{\prime }s$ computed in the HFA, one can derive the dynamical density-density correlation function$$\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( \rho ,\rho \right) }\left(
\mathbf{q},\tau \right) =-N_{\varphi }\left\langle {\mathcal{T}}_{\tau }%
\widetilde{\rho }\left( \mathbf{q}+\mathbf{K},\tau \right) \widetilde{\rho }%
\left( -\mathbf{q}-\mathbf{K}^{\prime },0\right) \right\rangle , \label{1_4}$$in the TDHFA [@CF]. In Eq. (\[1\_4\]), $\mathbf{q}$ is a vector restricted to the first Brillouin zone of the stripe crystal and $\widetilde{\rho }\equiv \rho -\left\langle \rho
\right\rangle $. By following the poles of $\chi^{\left( \rho ,\rho \right) }$ with non-vanishing weight as the wavevector $\mathbf{q}$ is varied in the Brillouin zone of the reciprocal lattice, we get the dispersion relation of the phonon and higher-energy collective modes of the stripe state. The equation of motion for $\chi _{%
\mathbf{K},\mathbf{K}^{\prime }}^{\left( \rho ,\rho \right) }\left( \mathbf{q%
},\tau \right) $, in the TDHFA, is given by $$\sum_{\mathbf{K}^{\prime \prime }}\left[ i\omega _{n}\delta _{\mathbf{K},%
\mathbf{K}^{\prime }}-M_{\mathbf{K},\mathbf{K}^{\prime \prime }}\left(
\mathbf{q}\right) \right] \chi _{\mathbf{K}^{\prime \prime },\mathbf{K}%
^{\prime }}^{\left( \rho ,\rho \right) }\left( \mathbf{q},i\omega
_{n}\right) =B_{\mathbf{K},\mathbf{K}^{\prime }}\left( \mathbf{q}\right) ,
\label{7p1}$$where $\omega _{n}$ is a Matsubara bosonic frequency and the matrices $M_{%
\mathbf{K},\mathbf{K}^{\prime }}$ and $B_{\mathbf{K},\mathbf{K}^{\prime }}$ are defined by$$\begin{aligned}
M_{\mathbf{K},\mathbf{K}^{\prime }}\left( \mathbf{q}\right) &=&-2i\left(
\frac{e^{2}}{\kappa l_{B}}\right) \left\langle \rho \left( \mathbf{K-K}%
^{\prime }\right) \right\rangle \label{rene1} \\
&&\times \sin \left[ \frac{\left( \mathbf{q}+\mathbf{K}\right) \times \left(
\mathbf{q}+\mathbf{K}^{\prime }\right) l_{B}^{2}}{2}\right] \left[
H_{N}\left( \mathbf{K}-\mathbf{K}^{\prime }\right) -X_{N}\left( \mathbf{K-K}%
^{\prime }\right) -H_{N}\left( \mathbf{q}+\mathbf{K}^{\prime }\right)
+X_{N}\left( \mathbf{q}+\mathbf{K}^{\prime }\right) \right] \notag\end{aligned}$$and$$B_{\mathbf{K},\mathbf{K}^{\prime }}\left( \mathbf{k}\right) =2i\sin \left[
\frac{\left( \mathbf{q}+\mathbf{K}\right) \times \left( \mathbf{q}+\mathbf{K}%
^{\prime }\right) l_{B}^{2}}{2}\right] \left\langle \rho \left( \mathbf{K-K}%
^{\prime }\right) \right\rangle$$respectively. (Here $\mathbf{a}\times \mathbf{b}$ stands for $a_xb_y-a_yb_x$.)
In Eq. (\[rene1\]), $H_{N}\left( \mathbf{q}\right) $ and $X_{N}\left(
\mathbf{q}\right) $ are the HF interactions in Landau level $N$:$$\begin{aligned}
H_{N}\left( \mathbf{q}\right) &=&\left( \frac{e^{2}}{\kappa l_{B}}\right)
\frac{1}{ql_{B}}e^{\frac{-q^{2}l_{B}^{2}}{2}}\left[ L_{N}^{0}\left( \frac{%
q^{2}l_{B}^{2}}{2}\right) \right] ^{2}, \\
X_{N}\left( \mathbf{q}\right) &=&\left( \frac{e^{2}}{\kappa l_{B}}\right)
\sqrt{2}\int_{0}^{\infty }dx\,e^{-x^{2}}\left[ L_{N}^{0}\left( x^{2}\right) %
\right] ^{2}J_{0}\left( \sqrt{2}xql_{B}\right) .\end{aligned}$$
To solve for $\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( \rho ,\rho
\right) }\left( \mathbf{q},i\omega _{n}\right) $, we diagonalize the matrix $%
M_{\mathbf{K},\mathbf{K}^{\prime \prime }}\left( \mathbf{q}\right) $ by the transformation
$$M=CEC^{-1}, \label{eqnm}$$
where $C$ is the matrix of the eigenvectors of $M$ and $E_{i,j}=\varepsilon
_{j}\delta _{i,j}$ is the diagonal matrix of its eigenvalues. The analytic continuation of $\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( \rho ,\rho
\right) }\left( \mathbf{q},i\omega _{n}\right) $ is given by $$\begin{aligned}
\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( \rho ,\rho \right) }\left(
\mathbf{q},\omega \right) &=&\sum_{j,k}\frac{C_{\mathbf{K},j}\left( \mathbf{q%
}\right) \left[ C\left( \mathbf{q}\right) ^{-1}\right] _{j,k}B_{k,\mathbf{K}%
^{\prime }}\left( \mathbf{q}\right) }{\omega +i\delta -\varepsilon
_{j}\left( \mathbf{q}\right) } \label{1_12} \\
&\equiv &\sum_{i}\frac{W_{i}\left( \mathbf{q}+\mathbf{K},\mathbf{q}+\mathbf{K%
}^{\prime }\right) }{\omega +i\delta -\varepsilon _{i}},\end{aligned}$$where $W_{i}\left( \mathbf{q}+\mathbf{K},\mathbf{q}+\mathbf{K}^{\prime
}\right) $ is the weight of the pole $\varepsilon _{i}$ in the response function. The true density response function is simply $$\chi _{\mathbf{K},\mathbf{K}^{\prime }}^{\left( n,n\right) }\left( \mathbf{q}%
,\omega \right) =N_{\phi }\sum_{i}\frac{F_{N}\left( \mathbf{q}+\mathbf{K}%
\right) W_{i}\left( \mathbf{q}+\mathbf{K},\mathbf{q}+\mathbf{K}^{\prime
}\right) F_{N}\left( \mathbf{q}+\mathbf{K}^{\prime }\right) }{\omega
+i\delta -\varepsilon _{i}}. \label{21_13}$$
Inversion rules for matrices {#inversionrules}
============================
The inversion rules for hierarchical matrices in the $n \rightarrow 0$ limit for the case where the entries are scalars may be found in Ref. [@MP]. In this appendix we generalize these inversion rules for the situation when the entries are themselves $n_0 \times n_0$ matrices, with our problem corresponding to $n_0=2$.
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In the replica method, we introduce $n$ replicas of the system and thus deal with$\left( n_{0}n\right) \times \left( n_{0}n\right) $ matrices. In the RSB states, in order to invert matrices in the limit of $n\rightarrow 0$ analytically, we follow the scalar case to assume that the $\left(
n_{0}n\right) \times \left( n_{0}n\right) $ matrices have a hierarchical structure. This may be described by a set of integers $m_{0}(=n),m_{1},%
\cdots ,m_{k},m_{k+1}(=1)$ where $m_{i}/m_{i+1}$ is also an integer. Such matrices may be constructed by introducing $k+2$ block matrices $\underline{M}^{(i)}$ ($i=0,\cdots ,k+1$) all of size $n\times n$. These are defined such that their elements are $1$ within the $m_{i}$ blocks along the diagonal and $0$ elsewhere. These matrices can be used as a basis for a group, so that any $n_{0}n$ by $n_{0}n$ hierarchical matrix $\underline{\underline{A}}$ can be expressed as $$\underline{\underline{A}}=\tilde{\hat{a}}\otimes \underline{1}+\sum_{i=0}^{k}%
\hat{a}_{i}\otimes \lbrack \underline{M}^{(i)}-\underline{M}^{(i+1)}].$$where $\tilde{\hat{a}}$ and $\hat{a}_{i}$ are $n_{0}\times n_{0}$ matrices. It is easy to check that $$(\hat{a}_{i}\otimes \underline{M}^{(j)})\cdot (\hat{a}_{l}\otimes \underline{%
M}^{(s)})=(\hat{a}_{i}\cdot \hat{a}_{l})\otimes (\underline{M}^{(j)}\cdot
\underline{M}^{(s)}).$$This means that $\underline{\underline{A}}$ is characterized by $k+2$ $n_{0}$ by $n_{0}$ matrices $\tilde{\hat{a}}$ and $\hat{a_{i}}$ ($i=0,\cdots ,k$). In fact, $\underline{\underline{A}}$ is completely parametrized by its topmost row $$\begin{aligned}
&&\tilde{\hat{a}}\;\;\;\underbrace{\hat{a_{k}}\,\cdots \,\hat{a_{k}}}\;\;\;%
\underbrace{\hat{a_{k-1}}\,\cdots \,\hat{a_{k-1}}}\;\;\;\cdots \cdots \;\;\;%
\underbrace{\hat{a_{0}}\,\cdots \,\hat{a_{0}}}. \\
&&~~~~~~~~~m_{k}~~~~~~~~~~~~~m_{k-1}~~~~~~~~~~~~~~~~~~~~~m_{0} \notag\end{aligned}$$We can then define $$\hat{a}(u)=\left\{
\begin{array}{ll}
\hat{a_{0}}\;\; & \mathrm{for}\;n-m_{1}<u<n \\
& \vdots \;\;\;\; \\
\hat{a_{k}}\;\; & \mathrm{for}\;1<u<m_{k}%
\end{array}%
\right.$$to parameterize the off-diagonal element matrices.
We assume that the matrix $\underline{\underline{A}}$ has an inverse matrix $%
\underline{\underline{B}}$ which because of the group properties should also be a hierarchical matrix, and thus is characterized by $\tilde{\hat{b}}$ and $\hat{b_{i}}$ ($i=1,\cdots ,k$). If we multiply two matrices $\underline{%
\underline{A}}$ and $\underline{\underline{B}}$ and call the product $%
\underline{\underline{C}}$, it may be written as $$\underline{\underline{C}}=\underline{\underline{A}}\cdot \underline{%
\underline{B}}=\tilde{\hat{c}}\otimes \underline{1}+\sum_{i=0}^{k}\hat{c_{i}}%
\otimes \lbrack \underline{M}^{(i)}-\underline{M}^{(i+1)}],$$with $$\begin{aligned}
&&\tilde{\hat{c}}=\tilde{\hat{a}}\cdot \tilde{\hat{b}}%
-\sum_{i=0}^{k}(m_{i+1}-m_{i})\hat{a_{i}}\cdot \hat{b_{i}}, \label{tildec0}
\\
&&\hat{c_{i}}=\hat{a_{i}}\cdot \tilde{\hat{b}}-m_{i}\hat{a_{i}}\cdot \hat{%
b_{i}}+\sum_{j=i+1}^{k}(m_{j}-m_{j+1})(\hat{a_{i}}\cdot \hat{b_{j}}+\hat{%
a_{j}}\cdot \hat{b_{i}})-\sum_{j=0}^{i}(m_{j+1}-m_{j})\hat{a_{j}}\cdot \hat{%
b_{j}}. \label{ci}\end{aligned}$$
Now we are in the position to analytically continue the hierarchical matrix to $n\rightarrow 0$. We first analytically continue $\hat{a}(u)$ to be defined for $u\in \lbrack 1,n]$ and then take the limit $n\rightarrow 0$. The limit $n\rightarrow 0$ then suggests that the hierarchical matrix $%
\underline{\underline{A}}$ is specified by a diagonal-element matrix $\tilde{%
\hat{a}}$ and a matrix function $\hat{a}(u)$ for $u\in \lbrack 0,1]$. The matrix $\underline{\underline{B}}$ can be analytically continued in the same way. Eqs. (\[tildec0\]-\[ci\]) therefore become $$\begin{aligned}
&&\tilde{\hat{c}}=\tilde{\hat{a}}\cdot \tilde{\hat{b}}-\int_{0}^{1}du\,\hat{a%
}(u)\cdot \hat{b}(u), \label{tildec} \\
&&\hat{c}(u)=(\tilde{\hat{a}}-\left\langle \hat{a}\right\rangle )\cdot \hat{b%
}(u)+\hat{a}(u)\cdot (\tilde{\hat{b}}-\langle \hat{b}\rangle )-\int_{0}^{u}dv%
\left[ \hat{a}(u)-\hat{a}(v)\right] \cdot \left[ \hat{b}(u)-\hat{b}(v)\right]
, \label{cu}\end{aligned}$$where $\left\langle \hat{a}\right\rangle =\int_{0}^{1}dv\,\hat{a}(v)$. Since $\underline{\underline{B}}$ is the inverse matrix of $\underline{\underline{A%
}}$, we require $$\tilde{\hat{c}}=\hat{1},\;\;\;\;\hat{c}(u)=\hat{0}. \label{inverse}$$Differentiating Eq. (\[cu\]) with respect to $u$ and using Eq. ([inverse]{}) leads to $$\left\{ \tilde{\hat{a}}-\left\langle \hat{a}\right\rangle -\left[ \hat{a}%
\right] (u)\right\} \cdot \hat{b}^{\prime }(u)+\hat{a}^{\prime }(u)\cdot
\left\{ \tilde{\hat{b}}-\langle \hat{b}\rangle -[\hat{b}](u)\right\} =\hat{0}%
, \label{p1}$$where $\left[ \hat{a}\right] (u)=\int_{0}^{u}dv\left[ \hat{a}(u)-\hat{a}(v)%
\right] $, and $\hat{a}^{\prime }(u)=d\hat{a}(u)/du$. By making use of $([%
\hat{a}](u))^{\prime }=u\hat{a}^{\prime }(u)$, Eq. (\[p1\]) becomes $$\left\{ \tilde{\hat{a}}-\left\langle \hat{a}\right\rangle -\left[ \hat{a}%
\right] (u)\right\} \cdot \left\{ \tilde{\hat{b}}-\langle \hat{b}\rangle -[%
\hat{b}](u)\right\} =const. \label{p2}$$To determine the constant matrix in Eq. (\[p2\]), we examine Eqs. ([tildec]{}) and (\[cu\]) at $u=1$ and get $$\left[ \tilde{\hat{a}}-\hat{a}(1)\right] \cdot \left[ \tilde{\hat{b}}-\hat{b}%
(1)\right] =\hat{1}. \label{p3}$$So $const.=\hat{1}$, and Eq. (\[p2\]) gives $$\left\{ \tilde{\hat{b}}-\langle \hat{b}\rangle -[\hat{b}](u)\right\}
=\left\{ \tilde{\hat{a}}-\langle \hat{a}\rangle -[\hat{a}](u)\right\} ^{-1},$$which can be inserted into Eq. (\[p1\]) to produce one of the inversion rules $$\hat{b}(u)-\hat{b}(v)=\int_{u}^{v}dy\,\left\{ \tilde{\hat{a}}-\langle \hat{a}%
\rangle -[\hat{a}](y)\right\} ^{-1}\cdot \hat{a}^{\prime }(y)\cdot \left\{
\tilde{\hat{a}}-\langle \hat{a}\rangle -[\hat{a}](y)\right\} ^{-1}.
\label{inversion2}$$This is very similar to Eq. (AII.5) in Ref. . Eqs. ([inversion2]{}) and (\[p3\]) lead to $$\tilde{\hat{b}}-\hat{b}(u)=\left[ \tilde{\hat{a}}-\hat{a}(1)\right]
^{-1}-\int_{u}^{1}dv\,\left\{ \tilde{\hat{a}}-\langle \hat{a}\rangle -[\hat{a%
}](v)\right\} ^{-1}\cdot \hat{a}^{\prime }(v)\cdot \left\{ \tilde{\hat{a}}%
-\langle \hat{a}\rangle -[\hat{a}](v)\right\} ^{-1}. \label{inversion3}$$This is the inversion rule we have used in our work \[see Eq. ([Ginversion]{}) in the text\].
For completeness, we also show, without giving the details of the derivation, some other inversion rules as well as the formula for $%
\lim_{n\rightarrow 0}\left( {\frac{1}{n}}\mathrm{Tr}\,\mathrm{ln}\,%
\underline{\underline{A}}\right) $ which appears in the expression of free energy: $$\begin{aligned}
&&\tilde{\hat{b}}=\left( \tilde{\hat{a}}-\langle \hat{a}\rangle \right)
^{-1}\cdot \left\{ \hat{1}-\int_{0}^{1}{\frac{du}{u^{2}}}\,[\hat{a}](u)\cdot
\left( \tilde{\hat{a}}-\langle \hat{a}\rangle -[\hat{a}](u)\right) ^{-1}-%
\hat{a}(0)\cdot \left( \tilde{\hat{a}}-\langle \hat{a}\rangle \right)
^{-1}\right\} , \label{inversion1} \\
&&\hat{b}(u)=-\left( \tilde{\hat{a}}-\langle \hat{a}\rangle \right)
^{-1}\cdot \left\{ \hat{a}(0)\cdot \left( \tilde{\hat{a}}-\langle \hat{a}%
\rangle \right) ^{-1}+{\frac{1}{u}}\,[\hat{a}](u)\cdot \left( \tilde{\hat{a}}%
-\langle \hat{a}\rangle -[\hat{a}](u)\right) ^{-1}\right\} \notag \\
&&\lim_{n\rightarrow 0}\left( {\frac{1}{n}}\mathrm{Tr}\,\mathrm{ln}\,%
\underline{\underline{A}}\right) =\mathrm{ln}\,\mathrm{det}\,\left( \tilde{%
\hat{a}}-\langle \hat{a}\rangle \right) +\mathrm{Tr}\,\left[ \hat{a}(0)\cdot
\left( \tilde{\hat{a}}-\langle \hat{a}\rangle \right) ^{-1}\right]
-\int_{0}^{1}{\frac{du}{u^{2}}}\;\mathrm{ln}\left[ {\frac{\mathrm{det}\left(
\tilde{\hat{a}}-\langle \hat{a}\rangle -[\hat{a}](u)\right) }{\mathrm{det}%
\left( \tilde{\hat{a}}-\langle \hat{a}\rangle \right) }}\right] .\end{aligned}$$
(250,320)
SPE’s for the retarded self energy {#Appendixanalycon}
==================================
In this Appendix, we analytically continue the SPE’s (\[zetatilde\]) for the Matsubara self energy in order to derive the SPE’s (\[SPEs\]) for the retarded self energy. We rewrite Eq. (\[zetatilde\]) as $$\tilde{\zeta}_{\alpha }(\omega _{n}) =
\int_{0}^{1}du\,\zeta _{\alpha}(u)+4v_{\mathrm{imp}}\int_{0}^{1/T}
d\tau V_{\alpha \alpha }^{\prime }\left[ \widetilde{B}(\tau )\right]
- 4 v_{\mathrm{imp}} J(\omega _{n}),
\label{zetaJ}$$where $J(\omega _{n})$ is the Fourier transform in Matsubara frequencies of $$J_{0}(\tau )=\exp \left[-{1\over 2}\sum_{\mu }K_{\mu }^{2}
\widetilde{B}_{\mu \mu }(\tau ) \right] \label{Jtau}$$with $\widetilde{B}_{\mu \mu }(\tau )$ being defined in Eq. (\[Btilde\]). Obviously, $J_{0}(\tau )$ is a Matsubara correlation function, and its corresponding real-time ordered correlation function reads $$\tilde{J}_{0}(t) = i J_{0}(\tau \rightarrow it) ={\cal \theta}(t) J_{1}(t)
+ {\cal \theta}(-t) J_{2}(t), \label{Jt}$$ where $J_{1}(t)=\tilde{J}_{0}(t>0)$, $J_{2}(t)=\tilde{J}_{0}(t<0)$ with the relation $J_{2}(-t)/i = (J_{2}(t)/i)^*$, and ${\cal \theta}(t)$ is the step function. The retarded function becomes $$J^{\rm ret}_{0}(t) = {\cal \theta}(t) [J_{1}(t) - J_{2}(t)]. \label{Jret}$$
Using the Nambu representation $$\widetilde{G}_{\mu \nu }(\mathbf{q},\omega _{n})=-{\frac{1}{\pi }}%
\int_{-\infty }^{\infty }df \, {\frac{A_\mu(f)}{i\omega _{n}-f}} ,$$where $A_\mu(f)$ is defined in Eq. (\[spectralfunction\]), we find that $\widetilde{B}_{\mu \mu }(\tau )$ in Eq. (\[Btilde\]) becomes $$\widetilde{B}_{\mu \mu }(\tau )={\frac{1}{\pi }}\int_{0}^{\infty }dfA_{\mu
}(f)\left[ T \sum_{\omega_n}(1-\cos \omega_n \tau) {2f\over \omega_n^2+f^2}
\right] .$$ We can then easily sum over the Matsubara frequency in the above equation to get $$\widetilde{B}_{\mu \mu }(\tau )={\frac{1}{\pi }}\int_{0}^{\infty }dfA_{\mu
}(f)\left[ 1-e^{-|\tau |f}+\frac{2\left[ 1-\cosh (f\tau )\right] }{e^{u/T}-1}%
\right] . \label{BB}$$Apparently, at $T=0$, the last term inside the parentheses in Eq. (\[BB\]) vanishes. Inserting Eq. (\[BB\]) into Eq. (\[Jtau\]) and following the procedure described in Eqs. (\[Jt\]) and (\[Jret\]), we find that at $T=0$, $$J^{\rm ret}_{0}(t) = i \, {\cal \theta}(t) \, {\rm Im} \,
\exp \left[-\sum_{\mu }{K_{\mu }^{2}\over {\pi }}\int_{0}^{\infty }dfA_{\mu
}(f)\left( 1-e^{-|\tau |f} \right) \right].
\label{Jret1}$$ From Eqs. (\[zetaJ\]) and (\[Jret1\]) and noting that $J^{\rm ret}_{0}(\omega)= \int^{-\infty}_\infty dt e^{i\omega t}
J^{\rm ret}_{0}(t)$, we immediately obtain Eq. (\[SPEs\]).
Unphysical solution of the SPE’s {#solution2}
================================
In our numerical search, we also notice the existence of another solution which we present here and argue is unphysical. Fig. \[unphyssolution\] shows the result of the conductivities from this solution which corresponds to a much smaller disorder level. Both Re$\left[ \sigma _{xx}\left( \omega
\right) \right] $ and Re$\left[ \sigma _{yy}\left( \omega \right) \right] $ show the pinning behavior and the peak frequencies move in as also shown in the inset of Fig. \[unphyssolution\] (b). However, unlike in the solution we presented in the text, no quantum depinning transition occurs.
Interestingly, this solution displays a peak move-in behavior that is reminiscent of what is seen in the physical solution. This is the result of a decreasing $e_y$, due to the increasing $W(K_x;K_y\neq 0)$ with the partial filling $\Delta\nu$. However, at small $e_y$, unlike in the other solution, the constraint (\[constraintPRSB0\]) is not satisfied through $U_{xx}=1$, but instead through $U_{yy}=1$. This can be seen from Eq. (\[Uyy\]), according to which $U_{yy}$ will rapidly decrease from very large values to very small values right near $\gamma=1$. This means that near this value $%
U_{yy}$ must pass through one, satisfying the constraint. In this solution, $%
\gamma$ remains very close to one over a range of filling factors, and does so by making $|e_x|$ very large, even for small $v_{\mathrm{imp}}$. This implies an unphysically large pinning for sliding perpendicular to the stripes. Because of this, and the close agreement between the other solution and the perturbative RG results, we ignore this solution to the SPE’s as physically unreasonable.
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R. Lewis, L. Engel and Y. Chen, private communication.
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M.-R. Li, H.A. Fertig, R. Côté, and H. Yi, Phys. Rev. Lett. **92**, 186804 (2004).
M. Mézard, G. Parisi, and M. Virasoro, *Spin Glass Theory and Beyond* (World Scientific, Singapore, 1987).
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To use discrete fast-Fourier transformation to compute the integrals over $\omega$ and $t$, we have to set a typical stepsize in frequency to be much smaller than $e_x$ and $e_y$ and the pinning peak widths. These quantities are dependent on the disorder level. On the other hand, the frequency stepsize is inversely proportional to the large-$t$ cutoff. As the depinning transition is approached and in the depinned state that we will discuss in the next section, to see all the interesting features at small $\omega$ we need to use very large cutoff. We thus need to compromise on the disorder level in order to capture the interesting physics at low frequencies.
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We want to mention that the weight of this delta function term is not computed accurately in the current method since this term has to be obtained along with the $\omega_n=0$ mode of $\tilde{\zeta}$.
The anomalous finite frequency response for $\sigma
_{yy}(\omega )$ was not noticed in our early analysis of these results, and so was not reported in Ref. .
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We know from previous work [@clark] that non topological solitons, Q balls, evaporate into fermions. All the constructions we used to find evaporation rate were dased on the fact that no fermions would move towards the Q ball. All these constructions left an opened question that is : what happens when a fermion interacts with a Q ball. We shall answer this question in this work by using the constructions done to compute evaporation rates. We shall also obtain a new approach to compute evaporation rates.'
author:
- 'Stephen S. Clark'
title: 'Interactions of Q-balls and matter.'
---
Introduction
============
A scalar field theory with an unbroken continuous global symmetry admits a remarkable class of solutions, non-topological solitons or Q-Balls. These solutions are spherically symmetric non-dissipative solutions to the classical field equations [@Qballscoleman; @Qballscohen; @Qballsnew; @clark]. These solutions are constructed using the fact that they are absolute minima of the energy, this property enshures the stability of Q-balls versus decay into scalars. The addition of a coupling to fermions will result in Q-balls becoming unstable versus decay into fermions [@Qballscohen; @Qballsnew; @clark]. The result of this instability is Q-ball evaporation.
Q-ball evaporation admits an absolute upper bound resulting from the fact that all the energy ranges are finate. The fact that energy ranges are finate comes from the mixing of positive and negative frequency terms. We could ask, what would happen if the fermions interacting with the Q-ball have an energy lying outside the evaporation range? This is the question we propose to answer in this work. We know from previous work [@clark] that we can use two methods to obtain a solution to the equations of motion describing a fermion interacting with a Q-ball. Our starting point will be the solutions expressed in terms of partial waves, that we shall study outside the evaporation range. This partial wave superposition has the reflection and transmission amplitudes as expansion coefficients. So we shall study the diffusion of a fermion on a Q ball. We shall then apply the same calculations to the total energy spectrum to see to the interference of both phemomenoms (evaporation and diffusion).
In fact we expect the difficulties to appear when the energy of the interacting fermion lies inside the evaporation range for this reason we shall start with fermions having their energy bigger than the evaporation upper bound. In this situation expect that the only wave having a non-trivial behaviour would be one going through the Q ball. The second regime where the energy of the interacting fermion lies inside the evaporation range will lead to some new results.
The work will be organised as follows we shall first give a brief overview of the method to construct the solutions. We shell then study these solutions outside the evaporation range and then we shall try to look at the total energy range.
Preliminaries
=============
The details of all the following calculations can be found in previous work [@Qballscohen; @Qballsnew; @clark] and we shall only give here a brief overview of the results and methods used. The major idea to solve this problem is considering a Q ball localized in space, so the fermionic field equations are free ones outside the zone where we have Yukawa coupling (interaction of Q ball and fermions). For the two cases we shall consider, massless and massive fermions, we shall need to match the solutions outside to the solutions inside the Q ball. The matching coefficients will give the reflection and transmission amplitudes we shall use in the problem.
Solutions to the equations of motion, massless case
---------------------------------------------------
Writing down the Lagrangian of a massless fermion having a Yukawa interaction with a scalar field gives in one spatial dimension, $$\begin{aligned}
{{\mathcal L}}_{ferm.}=i\bar{\psi}\sigma^\mu{{\partial_\mu}}\psi+(g\phi\bar{\psi}^C\psi+h.c),\end{aligned}$$ Equations of motion for the two components of the $\Psi$ field are : $$\begin{aligned}
{\begin{array}}{c}(i{\partial}_0+i{\partial}_z)\psi_1-g\phi\psi_2^{\star}=0, \\
(i{\partial}_0-i{\partial}_z)\psi_2^{\star}-g\phi^\star\psi_1=0. {\end{array}}\end{aligned}$$ and $\phi=\phi_0{{\mathrm{e}}^{-i{\omega}_0t}}$ in the zone from $-l$ to $+l$ and zero everywhere else. Using the ansatz : $$\begin{aligned}
{\left(}{\begin{array}}{c} \psi_1 \\ \psi_2^{\star} {\end{array}}{\right)}={\left(}{\begin{array}}{cc} {{\mathrm{e}}^{-i\frac{{\omega}_0}{2}t}} & 0\\
0 & {{\mathrm{e}}^{i\frac{{\omega}_0}{2}t}} {\end{array}}{\right)}{\left(}{\begin{array}}{c} A \\
B{\end{array}}{\right)}{{\mathrm{e}}^{-i{\epsilon}t+i(k+\frac{{\omega}_0}{2})}},\label{ansatz}\end{aligned}$$ the equations of motion are reduced to the following $2\times2$ linear system $$\begin{aligned}
{\left(}{\begin{array}}{cc} k-{\epsilon}& M \\ M & -(k+{\epsilon}){\end{array}}{\right)}{\left(}{\begin{array}}{c} A \\ B {\end{array}}{\right)}=0 . \nonumber\end{aligned}$$ The determinant of the system gives $k=\pm\sqrt{{\epsilon}^2-M^2}\equiv \pm k_{\epsilon}$. Solving for the two cases $k=+k_{\epsilon}$ and $k=-k_{\epsilon}$, we obtain the solution inside the Q-Ball: $$\begin{aligned}
\Psi_Q={\left(}{\begin{array}}{c}\psi_1\\ \psi_2^\star{\end{array}}{\right)}=
A{\left(}{\begin{array}}{c} 1 \\ \frac{k_{\epsilon}+{\epsilon}}{M} {\end{array}}{\right)}{{\mathrm{e}}^{-ik_{\epsilon}z}} + B{\left(}{\begin{array}}{c} \frac{k_{\epsilon}+{\epsilon}}{M} \\ 1 {\end{array}}{\right)}{{\mathrm{e}}^{ik_{\epsilon}z}},
\label{inside}\end{aligned}$$ where $M=g\phi_0$, $g$ is the coupling constant and $\phi_0$ the value of the scalar field. The second part is the solution when $\phi_0=0$ (outside the Q-Ball) it is, $$\begin{aligned}
\Psi={\left(}{\begin{array}}{c} \psi_1 \\ \psi_2^\star {\end{array}}{\right)}=
{{\mathrm{e}}^{-i{\epsilon}t}}{\left(}{\begin{array}}{c} C_1^{L,R}{{\mathrm{e}}^{i{\epsilon}z}} \\ C_2^{L,R}{{\mathrm{e}}^{-i{\epsilon}z}}{\end{array}}{\right)}, \label{coeflibre}\end{aligned}$$ where superscripts $L,R$ indicate the left and right side of the Q-Ball. In order to solve Dirac’s equation everywhere the solution needs to be continuous in space. Space continuity gives at $z=-l$ : $$\begin{aligned}
C_1^L=A{{\mathrm{e}}^{i(k_{\epsilon}+{\epsilon})l}}+B\alpha_{\epsilon}{{\mathrm{e}}^{-i(k_{\epsilon}-{\epsilon})l}}, \nonumber \\
C_2^L=A\alpha_{\epsilon}{{\mathrm{e}}^{i(k_{\epsilon}-{\epsilon})l}}+B{{\mathrm{e}}^{-i(k_{\epsilon}+{\epsilon})l}}, \nonumber\end{aligned}$$ and at $z=+l$, $$\begin{aligned}
C_1^R=A{{\mathrm{e}}^{-i(k_{\epsilon}+{\epsilon})l}}+B\alpha_{\epsilon}{{\mathrm{e}}^{+i(k_{\epsilon}-{\epsilon})l}}, \nonumber\\
C_2^R=A\alpha_{\epsilon}{{\mathrm{e}}^{-i(k_{\epsilon}-{\epsilon})l}}+B{{\mathrm{e}}^{+i(k_{\epsilon}+{\epsilon})l}}. \nonumber\end{aligned}$$ These two relations will give the matrix linking the coefficients on the left to those on the right.
Solution to the motion equations, massive case.
-----------------------------------------------
Using the same Lagrangian as for the massless case and adding a Dirac mass coupling for massive fermions, gives the fermionic Lagrangian : $$\begin{aligned}
{{\mathcal L}}=\bar{\psi}i{\gamma}^\mu{{\partial_\mu}}\psi+g(\bar{\psi}^C\psi\phi+h.c.)+M_D(x)\bar{\psi}\psi.\end{aligned}$$ The equations of motion using two component $\psi$-field, in $1\oplus1$-dimensions are $$\begin{aligned}
{\begin{array}}{c}(i{\partial}_t+i{\partial}_z)\psi_1-M{{\mathrm{e}}^{-i\frac{{\omega}_0}{2}t}}\psi_2^\star+M_D\psi_2=0, \\
(i{\partial}_t-i{\partial}_z)\psi_2+M{{\mathrm{e}}^{-i\frac{{\omega}_0}{2}t}}\psi_1^\star+M_D\psi_1=0. {\end{array}}\label{mass1}\end{aligned}$$ These equations can be solved if we use fields with four degrees of freedom. Solving this system will give us the solution inside the Q-Ball, which is the first step. To solve these equations of motion we use the following ansatz $$\begin{aligned}
{\begin{array}}{c}\psi_1=f_1(z){{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}+f_2(z){{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}, \\
\psi_2=g_1(z){{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}+g_2(z){{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}. {\end{array}}\label{mass2}\end{aligned}$$ Due to the separation of time components this ansatz will lead to four equations. Taking, $$\begin{aligned}
f_1(z)=A{{\mathrm{e}}^{ipz}}, f_2^\star(z)=B{{\mathrm{e}}^{ipz}}, g_1(z)=C{{\mathrm{e}}^{ipz}} g_2^\star(z)=D{{\mathrm{e}}^{ipz}}. \end{aligned}$$ and after some re-arrangement of the equations we obtain : $$\begin{aligned}
{\begin{array}}{c}-{\epsilon}_-f_1-g_2^\star+M_Dg_1=pf_1, \\
{\epsilon}_-g_1-f_2^\star-M_Df_1=pg_1, \\
-{\epsilon}_+f_2^\star+g_1-M_Dg_2^\star=pf_2^\star, \\
{\epsilon}_+g_2^\star+f_1+M_Df_2^\star=pg_2^\star,{\end{array}}\end{aligned}$$ where ${\epsilon}_-={\epsilon}-\frac{{\omega}_0}{2}$ and ${\epsilon}_+={\epsilon}+\frac{{\omega}_0}{2}$. This arrangement has the advantage that we can now write the $\psi$-field in terms of four component spinors as : $$\begin{aligned}
\Psi={\left(}{\begin{array}}{c}{\left(}{\begin{array}}{c}f_1\\g_1{\end{array}}{\right)}\\
{\left(}{\begin{array}}{c}f_2^\star\\g_2^\star{\end{array}}{\right)}{\end{array}}{\right)}.\end{aligned}$$ The equations of motion become in matrix form, $$\begin{aligned}
{\left(}{\begin{array}}{cccc} -{\epsilon}_- & M_D & 0 & -1 \\ -M_D & {\epsilon}_-& -1 & 0 \\
0& 1 & -{\epsilon}_+& -M_D \\ 1 & 0 & M_D & {\epsilon}_+{\end{array}}{\right)}{\left(}{\begin{array}}{c} C_1 \\C_2 \\ C_3 \\ C_4 {\end{array}}{\right)}=
Mp{\left(}{\begin{array}}{c} C_1 \\ C_2 \\ C_3 \\ C_4 {\end{array}}{\right)}.\label{qballmatrix1}\end{aligned}$$ All the parameters are normalised by $M$. Inside the Q-ball the time dependent solution is : $$\begin{aligned}
\Psi=\sum_{j=1}^{4}C_j{{\mathrm{e}}^{i({\epsilon}-{\omega})t}}v_{p_j}^{up}{{\mathrm{e}}^{i\bar{p}_jz}}+C^\star_j{{\mathrm{e}}^{-i({\epsilon}+{\omega})t}}(v_{p_j}^{down}{{\mathrm{e}}^{i\bar{p}_jz}})^\star,\end{aligned}$$ where the $up$ superscript stands for the first two components of the eigenvectors, while the $down$ one indicates we take the two last components of the eigenvectors of the inner motion matrix. The time dependent solution outside the Q-ball is once more given by, $$\begin{aligned}
\Psi=\sum_{j=1}^{4}(A_j,B_j){{\mathrm{e}}^{i({\epsilon}-{\omega})t}}u_{p_j}^{up}{{\mathrm{e}}^{i\bar{p}_jz}}+(A_j^\star,B_j^\star){{\mathrm{e}}^{-i({\epsilon}+{\omega})t}}(u_{p_j}^{down}{{\mathrm{e}}^{i\bar{p}_jz}})^\star,\end{aligned}$$ where we this time use the eigenvectors of the outside motion matrix ($M_D=0$). The $A$’s $B$’s and $C$’s are expension coefficients. The matching rules for these coefficients will give the reflection and transmission coefficients.
Interactions of Q-balls and matter.
===================================
We know from previous work [@clark] that we can use two equivalent methods to solve the problem of particle creation from Q-balls. These two methods allowed us to obtain the exact quantysed solutions describing fermions being produced by a Q-ball. The method we shall use now will consist in using the total solutions outside the evaporation range ${\epsilon}\notin[-\frac{{\omega}_0}{2}+M_D;\frac{{\omega}_0}{2}+M_D]$. The solutions we constructed are superpositions of partial waves containing positive and negative movers.
Interaction of Q ball and matter, massless case.
------------------------------------------------
We now study the diffusion processes on a Q ball. In the first part of the construction shall not consider any evaporation for Q balls. This simplification will allow us in the first place to build and study the diffusion of a fermion on a Q ball that we expect to be very similar as diffusion on a potential barrier. The simplest aproach will be to consider all possible transmition trough the Q ball, that is we might have a transmitted particle or anti-particle due to the energy shift (the $\frac{{\omega}_0}{2}$ factor in the time dependent solution).
For simplicity we shall consider the construction where there is no incoming anti-particle.
We start by using the solution outside the Q-Ball for the simple massles case we have: $$\begin{aligned}
\Psi={\left(}{\begin{array}}{c} \psi_1 \\ \psi_2 {\end{array}}{\right)}=
{\left(}{\begin{array}}{c} C_1^{L}{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}+\frac{{\omega}_0}{2})z}} \\
(C_2^{L})^\star{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})z}}{\end{array}}{\right)},\end{aligned}$$ we have also a relation for coefficients on the right and on the left given by, $$\begin{aligned}
{\left(}{\begin{array}}{c} C_1^R \\ C_2^R {\end{array}}{\right)}=\frac{1}{1-\alpha^2}{\left(}{\begin{array}}{cc} {{\mathrm{e}}^{-2ik_{\epsilon}l}}-\alpha_{\epsilon}^2 {{\mathrm{e}}^{2ik_{\epsilon}l}} &
-\alpha_{\epsilon}{{\mathrm{e}}^{-2ik_{\epsilon}l}}+\alpha_{\epsilon}{{\mathrm{e}}^{2ik_{\epsilon}l}} \\\alpha_{\epsilon}{{\mathrm{e}}^{-2ik_{\epsilon}l}}-\alpha_{\epsilon}{{\mathrm{e}}^{2ik_{\epsilon}l}} &
{{\mathrm{e}}^{2ik_{\epsilon}l}}-\alpha_{\epsilon}^2 {{\mathrm{e}}^{-2ik_{\epsilon}l}} {\end{array}}{\right)}{\left(}{\begin{array}}{c} C_1^L \\ C_2^L {\end{array}}{\right)}. \nonumber\end{aligned}$$ For the calculation of evaporation rate we considered that both components where the same particles what we do now is consider both componets to be different particles (doing so we can construct the normal field containing both particles and anti-particles). we shall consider as a first study : $$\begin{aligned}
{\epsilon}+\frac{{\omega}_0}{2}\geq0\rightarrow{\epsilon}\geq-\frac{{\omega}_0}{2} \nonumber \\
{\epsilon}-\frac{{\omega}_0}{2}\geq0\rightarrow{\epsilon}\geq\frac{{\omega}_0}{2} \nonumber, \end{aligned}$$ it means we have the same type of particles on the letf hand side moving both of them towards the Q-Ball. Since we we have chosen the parameters to be the incident amplitudes we can choose that there is no incident anti-particle, $C_2^L=0$. Using the matrix relations we find, $$\begin{aligned}
C_1^R=\frac{{{\mathrm{e}}^{-2ik_{\epsilon}l}}-\alpha_{\epsilon}^2{{\mathrm{e}}^{2ik_{\epsilon}l}}}{1-\alpha_{\epsilon}^2}=T_{pp} \nonumber, \\
C_2^R=\frac{\alpha_{\epsilon}{{\mathrm{e}}^{-2ik_{\epsilon}l}}-\alpha_{\epsilon}{{\mathrm{e}}^{2ik_{\epsilon}l}}}{1-\alpha_{\epsilon}^2}=T_{p\bar{p}} \nonumber,\end{aligned}$$ The transformation amplitude we are looking for is the $C_2^R$ coefficient. As long as $k_{\epsilon}$ is complex (see chap three), we can easily see that this amplitude decreases as the Q-Ball’s size increases. Here we considered the range where ${\epsilon}\geq\frac{{\omega}_0}{2}$ and the solution is, $$\begin{aligned}
\Psi_L=\int_{\frac{{\omega}_0}{2}}^{\infty}d{\epsilon}{\left(}{\begin{array}}{c}A({\epsilon}){{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}+\frac{{\omega}_0}{2})t}}\\0{\end{array}}{\right)}\nonumber \\
\Psi_R=\int_{\frac{{\omega}_0}{2}}^{\infty}d{\epsilon}{\left(}{\begin{array}}{c}A({\epsilon})T_{pp}{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}+\frac{{\omega}_0}{2})t}}\\
A^\star({\epsilon})T_{p\bar{p}}^\star{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{\end{array}}{\right)}, \nonumber\end{aligned}$$ where the $L$ and $R$ subscripts indicate on which side of the Q-Ball we are. At this point the $A$ expansion factor is considered to be contiunous so it can be upgraded to an operator later on. It is not the only range where transformation occurs we also have the other range ${\epsilon}\leq-\frac{{\omega}_0}{2}$, with this time incident particles coming from the right hand side of Q-Ball, we have $$\begin{aligned}
{\left(}{\begin{array}}{c} C_1^R \\ C_2^R {\end{array}}{\right)}=\frac{1}{1-\alpha^2}{\left(}{\begin{array}}{cc} {{\mathrm{e}}^{-2ik_{\epsilon}l}}-\alpha_{\epsilon}^2 {{\mathrm{e}}^{2ik_{\epsilon}l}} &
-\alpha_{\epsilon}{{\mathrm{e}}^{-2ik_{\epsilon}l}}+\alpha_{\epsilon}{{\mathrm{e}}^{2ik_{\epsilon}l}} \\\alpha_{\epsilon}{{\mathrm{e}}^{-2ik_{\epsilon}l}}-\alpha_{\epsilon}{{\mathrm{e}}^{2ik_{\epsilon}l}} &
{{\mathrm{e}}^{2ik_{\epsilon}l}}-\alpha_{\epsilon}^2 {{\mathrm{e}}^{-2ik_{\epsilon}l}} {\end{array}}{\right)}{\left(}{\begin{array}}{c} C_1^L \\ C_2^L {\end{array}}{\right)}. \nonumber\end{aligned}$$ we have this time $C^1_R=0$, leading to the same solution for the transmisson amplitudes, $$\begin{aligned}
\tilde{T}_{pp}=\frac{{{\mathrm{e}}^{-2ik_{\epsilon}l}}-\alpha_{\epsilon}^2{{\mathrm{e}}^{2ik_{\epsilon}l}}}{1-\alpha_{\epsilon}^2}=T_{pp} \nonumber, \\
\tilde{T}_{p\bar{p}}=\frac{\alpha_{\epsilon}{{\mathrm{e}}^{-2ik_{\epsilon}l}}-\alpha_{\epsilon}{{\mathrm{e}}^{2ik_{\epsilon}l}}}{1-\alpha_{\epsilon}^2}=T_{p\bar{p}} \nonumber,\end{aligned}$$ even if the amplitudes are the same, wich is a effect of symmetry, we shall use tilde symbols to distinguish both contributions. In fact we would like to keep in mind that this second contribution comes from waves having negative energy. These two contributions will mix together to build the total solution. We have this time for solution, $$\begin{aligned}
\Psi_L=\int^{-\frac{{\omega}_0}{2}}_{-\infty}d{\epsilon}{\left(}{\begin{array}}{c}B({\epsilon})\tilde{T}_{p\bar{p}}{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}+\frac{{\omega}_0}{2})z}}\\
B^\star({\epsilon})\tilde{T}_{pp}^\star{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})z}}{\end{array}}{\right)}\nonumber \\
\Psi_R=\int^{-\frac{{\omega}_0}{2}}_{-\infty}d{\epsilon}{\left(}{\begin{array}}{c}0\\
B^\star({\epsilon}){{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{\end{array}}{\right)}.\end{aligned}$$ Let us take a look at the pieces on the left, $$\begin{aligned}
\Psi_L&=&\int_{\frac{{\omega}_0}{2}}^{\infty}d{\epsilon}u_1{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}+\frac{{\omega}_0}{2})z}}A({\epsilon})+
\int^{-\frac{{\omega}_0}{2}}_{-\infty}d{\epsilon}u_1{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}+\frac{{\omega}_0}{2})z}}\tilde{T}_{p\bar{p}}B({\epsilon}) \nonumber \\
&+&\int^{-\frac{{\omega}_0}{2}}_{-\infty}d{\epsilon}u_2{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})z}}\tilde{T}_{pp}B^\star({\epsilon}), \nonumber\end{aligned}$$ where $u_1={\left(}{\begin{array}}{c}1\\0{\end{array}}{\right)}$ and $u_2={\left(}{\begin{array}}{c}0\\1{\end{array}}{\right)}$. We change the varible ${\epsilon}\rightarrow-{\epsilon}$ in the integrals on the negative range to obtain, $$\begin{aligned}
\Psi_L&=&\int_{\frac{{\omega}_0}{2}}^{\infty}d{\epsilon}u_1{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}+\frac{{\omega}_0}{2})z}}A({\epsilon})+
\int_{\frac{{\omega}_0}{2}}^{\infty}d{\epsilon}u_1{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{-i({\epsilon}-\frac{{\omega}_0}{2})z}}\tilde{T}_{p\bar{p}}(-{\epsilon})B(-{\epsilon}) \nonumber \\
&+&\int_{\frac{{\omega}_0}{2}}^{\infty}d{\epsilon}u_2{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})z}}\tilde{T}_{pp}(-{\epsilon})B^\star(-{\epsilon}). \nonumber\end{aligned}$$ The next thing to do is change the variable ${\epsilon}+\frac{{\omega}_0}{2}={\epsilon}'$ and ${\epsilon}-\frac{{\omega}_0}{2}={\epsilon}'$ to obtain, $$\begin{aligned}
\Psi_L&=&\int_{{\omega}_0}^{\infty}d{\epsilon}u_1{{\mathrm{e}}^{-i{\epsilon}'t}}{{\mathrm{e}}^{i{\epsilon}'z}}A({\epsilon}'-\frac{{\omega}_0}{2})+
\int_{0}^{\infty}d{\epsilon}u_1{{\mathrm{e}}^{i{\epsilon}'t}}{{\mathrm{e}}^{-i{\epsilon}'z}}\tilde{T}_{p\bar{p}}(-\frac{{\omega}_0}{2}-{\epsilon})B(-\frac{{\omega}_0}{2}-{\epsilon}) \nonumber \\
&+&\int_{{\omega}_0}^{\infty}d{\epsilon}u_2{{\mathrm{e}}^{-i{\epsilon}'t}}{{\mathrm{e}}^{-i{\epsilon}'z}}\tilde{T}_{pp}(-{\epsilon}'+\frac{{\omega}_0}{2})B^\star(-{\epsilon}'+\frac{{\omega}_0}{2}). \nonumber\end{aligned}$$ Due to the energy shift comming from the Q ball we have two different bounds on the integration ${\epsilon}\in[{\omega}_0,\infty[$ and ${\epsilon}\in[0,\infty[$. These two ranges will be of great interest, since each one can be identified to a different particle. Before we construct the quantised solution we need to do the same transformations to the solution on right-hand side. $$\begin{aligned}
\Psi_R&=&\int_{\frac{{\omega}_0}{2}}^{\infty}d{\epsilon}u_1{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}+\frac{{\omega}_0}{2})z}}T_{pp}A({\epsilon})+
\int^{\frac{{\omega}_0}{2}}_{\infty}d{\epsilon}u_2{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})z}}T_{p\bar{p}}^\star A^\star({\epsilon}) \nonumber \\
&+&\int^{-\frac{{\omega}_0}{2}}_{-\infty}d{\epsilon}u_2{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})z}}B^\star({\epsilon}), \nonumber\end{aligned}$$ after the variable changes and the sign change we obtain, $$\begin{aligned}
\Psi_R&=&\int_{{\omega}_0}^{\infty}d{\epsilon}u_2{{\mathrm{e}}^{-i{\epsilon}'t}}{{\mathrm{e}}^{i{\epsilon}'z}}B(-{\epsilon}'+\frac{{\omega}_0}{2})+
\int_{0}^{\infty}d{\epsilon}u_2{{\mathrm{e}}^{i{\epsilon}'t}}{{\mathrm{e}}^{-i{\epsilon}'z}}T_{p\bar{p}}^\star(\frac{{\omega}_0}{2}+{\epsilon})A(\frac{{\omega}_0}{2}+{\epsilon}) \nonumber \\
&+&\int_{{\omega}_0}^{\infty}d{\epsilon}u_1{{\mathrm{e}}^{-i{\epsilon}'t}}{{\mathrm{e}}^{-i{\epsilon}'z}}T_{pp}({\epsilon}'-\frac{{\omega}_0}{2})A({\epsilon}'-\frac{{\omega}_0}{2}). \nonumber\end{aligned}$$ We see here that the $B(-{\epsilon}'+\frac{{\omega}_0}{2})$ and $A({\epsilon}'-\frac{{\omega}_0}{2})$ are in the same integration range and therefor will corespond to the same particles. We now have both solutions one on each side of the Q-Ball, the last step of construction will be quantization. Quantization will be done identifying the parts coresponding to particles and the parts coresponding to anti-particles. It is easy to identify the particles and the anti-particles since the integration domain is well known and definied we have, $$\begin{aligned}
\Psi_L&=&\int_{{\omega}_0}^{\infty}d{\epsilon}{{\mathrm{e}}^{-i{\epsilon}t}}\{{{\mathrm{e}}^{i{\epsilon}z}}u_1+{{\mathrm{e}}^{-i{\epsilon}z}}\tilde{T}_{pp}(\frac{{\omega}_0}{2}-{\epsilon})u_2\}a_l({\epsilon}-\frac{{\omega}_0}{2}) \nonumber \\
&+&\int_{0}^{\infty}d{\epsilon}{{\mathrm{e}}^{i{\epsilon}t}}\{{{\mathrm{e}}^{-i{\epsilon}z}}\tilde{T}_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})u_1\}b_l^\dagger({\epsilon}+\frac{{\omega}_0}{2}) \nonumber\end{aligned}$$ for the solution on the left and $$\begin{aligned}
\Psi_R&=&\int_{{\omega}_0}^{\infty}d{\epsilon}{{\mathrm{e}}^{-i{\epsilon}t}}\{{{\mathrm{e}}^{-i{\epsilon}z}}u_2a_r(\frac{{\omega}_0}{2}-{\epsilon})+{{\mathrm{e}}^{i{\epsilon}z}}T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})u_1
a_r^\dagger(\frac{{\omega}_0}{2}-{\epsilon})\} \nonumber \\
&+&\int_{0}^{\infty}d{\epsilon}{{\mathrm{e}}^{i{\epsilon}t}}\{{{\mathrm{e}}^{i{\epsilon}z}}T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})u_1\}b_r^\dagger({\epsilon}+\frac{{\omega}_0}{2}), \nonumber\end{aligned}$$ for the solution on the right hand side of the Q-Ball. During the construction we used, $$\begin{aligned}
B(k)=a(k)\theta(k)+b^\dagger(k)\theta(-k). \nonumber\end{aligned}$$ We keep for the moment the $l$ and $r$ subscript on the operators to identify the side of the Q ball they correspond. During this construction we did not normalise any solution. The normalisation of solutions is a simple task [@clark], and will not be needed in the rest of the discussion. We just need to keep in mind that all the waves packets are normalised in the way that the sum of the reflected amplitude and the transmitted one is equal to one.
Now that we have the total solution in terms of creation and anihilisation operators we can compute anything we want, by applying the proper operator valued observable. We can see that the energy of the transmited wave is the same as the energy of the incoming wave, while the energy of the reflected anti-particle is shifted by a factor of ${\omega}_0$. This shift in energy, one component shifted downwards and the other one shifted upwards is the reason why we found finite evaporation ranges. This property is due to the Yukawa coupling inside the Q-Ball. We can choose a state where there is no incoming particles on the right so our transmission coeficients can be identified to reflection ones. In fact this construction can be used in the full energy range. The difficulty will be to seperate the phenomenons linked to diffusion and those linked to evaporation.
Diffusion on a Q ball
---------------------
Now that we have the quantised solutions we can try to study diffusion on a Q ball. We need to use the total solution having no incident wave on the right-hand side of the Q ball. We write, $$\begin{aligned}
\Psi_L&=&\int_{{\omega}_0}^{\infty}d{\epsilon}{{\mathrm{e}}^{-i{\epsilon}t}}\{{{\mathrm{e}}^{i{\epsilon}z}}u_1+{{\mathrm{e}}^{-i{\epsilon}z}}\tilde{T}_{pp}(\frac{{\omega}_0}{2}-{\epsilon})u_2\}a_l({\epsilon}-\frac{{\omega}_0}{2}) \nonumber \\
&+&\int_{0}^{\infty}d{\epsilon}{{\mathrm{e}}^{i{\epsilon}t}}\{{{\mathrm{e}}^{-i{\epsilon}z}}\tilde{T}_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})u_1\}b_l^\dagger({\epsilon}+\frac{{\omega}_0}{2}) \nonumber \\
&+&\int_{{\omega}_0}^{\infty}d{\epsilon}{{\mathrm{e}}^{-i{\epsilon}t}}\{{{\mathrm{e}}^{i{\epsilon}z}}T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})u_1a_r^\dagger(\frac{{\omega}_0}{2}-{\epsilon})\} \nonumber \\
&+&\int_{0}^{\infty}d{\epsilon}{{\mathrm{e}}^{i{\epsilon}t}}\{{{\mathrm{e}}^{i{\epsilon}z}}T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})u_1\}b_r^\dagger({\epsilon}+\frac{{\omega}_0}{2}), \nonumber\end{aligned}$$ the construction of the Bogolubov transformation linking the far past, the incident wave, to the far future reads : $$\begin{aligned}
c_{out}&=&T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})a_l({\epsilon}-\frac{{\omega}_0}{2})+T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})b_l^\dagger({\epsilon}+\frac{{\omega}_0}{2}) \nonumber \\
&+&T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})a_r^\dagger(\frac{{\omega}_0}{2}-{\epsilon})+T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})b_r^\dagger({\epsilon}+\frac{{\omega}_0}{2})\nonumber.\end{aligned}$$ We here distinguish two energy ranges, if ${\epsilon}\geq\frac{{\omega}_0}{2}$ the $a_r^\dagger(\frac{{\omega}_0}{2}-{\epsilon})$ operator becomes $a_r({\epsilon}-\frac{{\omega}_0}{2})$, while if ${\epsilon}\leq\frac{{\omega}_0}{2}$ it is the $a_l({\epsilon}-\frac{{\omega}_0}{2})$ becomming $a_l^\dagger(\frac{{\omega}_0}{2}-{\epsilon})$. As we said before we had to separate both regimes in order to see what type of diffusion we have.
First we shall consider an infinite Q ball, so the right-hand side does not exist, and we have $$\begin{aligned}
c_{out}&=&T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})a_l({\epsilon}-\frac{{\omega}_0}{2})+T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})b_l^\dagger({\epsilon}+\frac{{\omega}_0}{2}), \nonumber \\\end{aligned}$$ when we constructed the solution the incident particle had energy bigger than ${\omega}_0$. So in this range ${\epsilon}-\frac{{\omega}_0}{2}$ is allways bigger than zero, and the $a$ operator remains an anihilation operator. We have now in the final state (in the far future), $$\begin{aligned}
_{out}<0|c^\dagger c|0>_{out}=|T_{p\bar{p}}|^2<\bar{p}|\bar{p}>.\end{aligned}$$ This simply is a particle being reflected into an anti-particle to conserve helicity. To study the rest of the possibilities we shall use a small sized Qball.
The first possibility stands for ${\epsilon}\geq{\omega}_0/2$ we need to change the type of the second $a$ operator, $a_r^\dagger(\frac{{\omega}_0}{2}-{\epsilon})$ becomes $a_r({\epsilon}-\frac{{\omega}_0}{2})$ : $$\begin{aligned}
c_{out}&=&T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})a_l({\epsilon}-\frac{{\omega}_0}{2})+T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})b_l^\dagger({\epsilon}+\frac{{\omega}_0}{2}) \nonumber \\
&+&T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})a_r({\epsilon}-\frac{{\omega}_0}{2})+T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})b_r^\dagger({\epsilon}+\frac{{\omega}_0}{2})\nonumber.\end{aligned}$$ Applying this to the vaccum state in the far future we have, $$\begin{aligned}
_{out}<0|c^\dagger c|0>_{out}=|\tilde{T}_{p\bar{p}}|^2{_l<\bar{p}|\bar{p}>_l}+|T_{p\bar{p}}|^2{_r<\bar{p}|\bar{p}>_r}.\end{aligned}$$
The second possibility will be for a particle having energy lower than ${\omega}_0/2$ we have to change the type of the first $a$, $a_l({\epsilon}-\frac{{\omega}_0}{2})$ becomes $a_l^\dagger(\frac{{\omega}_0}{2}-{\epsilon})$. We write : $$\begin{aligned}
c_{out}&=&T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})a_l^\dagger(\frac{{\omega}_0}{2}-{\epsilon})+T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})b_l^\dagger({\epsilon}+\frac{{\omega}_0}{2}) \nonumber \\
&+&T_{pp}(\frac{{\omega}_0}{2}-{\epsilon})a_r^\dagger(\frac{{\omega}_0}{2}-{\epsilon})+T_{p\bar{p}}(\frac{{\omega}_0}{2}+{\epsilon})b_r^\dagger({\epsilon}+\frac{{\omega}_0}{2})\nonumber.\end{aligned}$$ This time we have four particles in the final state, it is the combination of evaporation and diffusion.
The third possiblity would be for incident fermions having their energy in the range ${\epsilon}\in[{\omega}_0/2;{\omega}_0]$, we shall not study this range since we only considered incident fermions with enrgy bigger than ${\omega}_0$ or smaller than ${\omega}_0/2$. But this range does not introduce any new or difficult construction. These construction can also be used to compute evaporation rate we just need to separate both constructions and then identify wich contribution is diffusion and wich one is evaporation.
Diffusion on a Q ball massive case.
-----------------------------------
Starting with the solution for a Q ball interacting with a fermion we have, [$$\begin{aligned}
\Psi&=&\int_{M_D+\frac{{\omega}_0}{2}}^\infty d{\epsilon}\{{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}[{{\mathrm{e}}^{i\bar{p}_1x}}u_{\bar{p_1}}^{up}+r_{11}{{\mathrm{e}}^{-i\bar{p}_1x}}u_{-\bar{p_1}}^{up}]b^\dagger_{\bar{p}_1}
+{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}[r_{12}^\star{{\mathrm{e}}^{i\bar{p}_2^\star x}}(u_{-\bar{p_2}}^{down})^\star]b_{\bar{p}_1}\}, \nonumber\\
&+&\int_{M_D-\frac{{\omega}_0}{2}}^\infty d{\epsilon}\{{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}[r_{21}{{\mathrm{e}}^{-i\bar{p}_1x}}u_{-\bar{p_1}}^{up}]a^\dagger_{\bar{p}_2}
+{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}[{{\mathrm{e}}^{-i\bar{p}_2^\star x}}u_{\bar{p_2}}^\star+r_{22}^\star{{\mathrm{e}}^{i\bar{p}_2^\star x}}(u_{-\bar{p_2}}^{down})^\star]
a_{\bar{p}_2}\},
\nonumber\\
&+&\int_{M_D+\frac{{\omega}_0}{2}}^\infty d{\epsilon}\{{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}[\tilde{t}_{11}{{\mathrm{e}}^{-i\bar{p}_1x}}u_{-\bar{p_1}}^{up}]b^\dagger_{-\bar{p}_1}
+{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}[\tilde{t}^\star_{12}{{\mathrm{e}}^{i\bar{p}_2^\star x}}(u_{-\bar{p_2}}^{down})^\star]b_{-\bar{p}_1}\}, \nonumber\\
&+&\int_{M_D-\frac{{\omega}_0}{2}}^\infty d{\epsilon}\{{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}[\tilde{t}_{21}{{\mathrm{e}}^{-i\bar{p}_1x}}u_{-\bar{p_1}}^{up}]a^\dagger_{-\bar{p}_2}
+{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}[\tilde{t}^\star_{22}{{\mathrm{e}}^{i\bar{p}_2^\star x}}(u_{-\bar{p_2}}^{down})^\star]a_{-\bar{p}_2}\},\nonumber\end{aligned}$$ ]{} as before this solution has two different integration ranges one for each type of particles. We shall now only consider the range where ${\epsilon}\geq M_D+\frac{{\omega}_0}{2}$ in this range we only have diffusion and to not need to study the interaction of both, diffusion and evaporation. We have , [$$\begin{aligned}
\Psi&=&\int_{M_D+\frac{{\omega}_0}{2}}^\infty d{\epsilon}\{{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}[{{\mathrm{e}}^{i\bar{p}_1x}}u_{\bar{p_1}}^{up}+r_{11}{{\mathrm{e}}^{-i\bar{p}_1x}}u_{-\bar{p_1}}^{up}]b^\dagger_{\bar{p}_1}
+{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}[r_{12}^\star{{\mathrm{e}}^{i\bar{p}_2^\star x}}(u_{-\bar{p_2}}^{down})^\star]b_{\bar{p}_1}\}, \nonumber\\
&+&\int_{M_D+\frac{{\omega}_0}{2}}^\infty d{\epsilon}\{{{\mathrm{e}}^{i({\epsilon}-\frac{{\omega}_0}{2})t}}[r_{21}{{\mathrm{e}}^{-i\bar{p}_1x}}u_{-\bar{p_1}}^{up}]a^\dagger_{\bar{p}_2}
+{{\mathrm{e}}^{-i({\epsilon}+\frac{{\omega}_0}{2})t}}[{{\mathrm{e}}^{-i\bar{p}_2^\star x}}u_{\bar{p_2}}^\star+r_{22}^\star{{\mathrm{e}}^{i\bar{p}_2^\star x}}(u_{-\bar{p_2}}^{down})^\star]
a_{\bar{p}_2}\},
\nonumber\end{aligned}$$ ]{} the only reason why we used a very big Q ball is that we have less terms to deal with. With this solution we can as in the previous chapter construct a bogolubov transformation considering that in the far future only out going waves survies. We have this time : $$\begin{aligned}
r_{11}b^\dagger_{\bar{p}_1}+r_{12}^\star b_{\bar{p}_1}+r_{21}a^\dagger_{\bar{p}_2}+r_{22}^\star a_{\bar{p}_2}=c_{out}.\end{aligned}$$ If we want $c_{out}$ to have the same anti-commutation relations as $a$ and $b$ we need to normalise the operators with : $$\begin{aligned}
a=\frac{a}{|r_{22}|^2+|r_{21}|^2}, \\
b=\frac{b}{|r_{12}|^2+|r_{11}|^2},\end{aligned}$$ There is no obvious reasons why $|r_{12}|^2=|r_{21}|^2$ and $|r_{11}|^2=|r_{22}|^2$. These equalities will depend on the type of Q ball we study, they depend on the way the Q ball mixes the particles and anti-particle energies. We can use this $c$ operator to compute the number of particles in the final state when the original one was the vacuum state, $$\begin{aligned}
_{out}<0|c^\dagger c|0>_{out}&=&|r_{21}|^2_{in}<0|aa^\dagger|0>_{in}+|r_{11}|^2_{in}<0|bb^\dagger|0>_{in}, \nonumber \\
&=&|r_{21}|^2<p|p>+|r_{11}|^2<ap|ap>.\end{aligned}$$ We have this time, when fermions are massive, a particle and an anti-particle in the final state. Those two reflected particules will be part of the state combining evaporation and diffusion. This combination of evaporation and diffusion is more complicated to study since we need to use the full solution and be carefull with the energy ranges and the type of operator linked to them.
Conclusions.
============
As expected the interaction of Q balls and fermions can be separated into two different cases. The first one stands for interaction between the Q ball and fermions having their energy lying outside the evaporation rate. In this case we demonstrated that the interaction of Q balls and matter is nothing but standard diffusion. If we have an incident fermion we have a reflected and transmitted anti-fermion. The transmitted particles has its energy shifted by a ${\omega}_0/2$ factor due to interaction with the Q ball.
The second case happens when the fermion interacting with the Q ball has its energy lying inside the evaporation range. In this case we have a superposition of two phenomenoms the first one is diffusion while the second one is evaporation. It seems that both phenomenoms to not interfear together and be considered separately. This fact is important in the way that it provides a new approach to compute Q ball evaporation rates. This new approach would be to study the diffusion of a particle on the Q ball’s surface and find the two ranges where we have diffusion and both diffusion and evaporation. Then we just need to substract both amplitudes to obtain evaporation range.
The interaction with massive fermions does not introduce any new facts. The calculations becomes more complicated but the separation into two ranges stands the same way, exept this time the evaporation range is different since it depends on the fermion mass. In this case the other difference comes from the fact that we do not use helicity conservation to find the reflected waves.
[50]{}
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|
{
"pile_set_name": "ArXiv"
}
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---
author:
- 'Vasileios Tzoumas,$^{1,2}$ Luca Carlone,$^{2}$ George J. Pappas,$^{1}$ Ali Jadbabaie$^{2}$ [^1] [^2] [^3]'
bibliography:
- 'references.bib'
---
[^1]: $^{1}$The authors are with the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104 USA (email: [{pappagsg, vtzoumas}@seas.upenn.edu]{}).
[^2]: $^{2}$The authors are with the Institute for Data, Systems and Society, and the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (email: [{jadbabai, lcarlone, vtzoumas}@mit.edu]{}).
[^3]: This work was supported in part by TerraSwarm, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA, and in part by AFOSR Complex Networks Program.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In 1935, Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics. EPR also postulated the existence of more fundamental theory where physical reality of any system would be completely describe by the variables/states of that fundamental theory. This variable is commonly called hidden variable and the theory is called hidden variable theory (HVT). In 1964, John Bell proposed an empirically verifiable criterion to test for the existence of these HVTs. He derived an inequality, which must be satisfied by any theory that fulfill the conditions of *locality* and *reality*. He also showed that quantum mechanics, as it violates this inequality, is incompatible with any *local-realistic* theory. Later it has been shown that Bell’s inequality can be derived from different set of assumptions and it also find applications in useful information theoretic protocols. In this review we will discuss various foundational as well as information theoretic implications of Bell’s inequality. We will also discuss about some restricted nonlocal feature of quantum nonlocality and elaborate the role of Uncertainty principle and Complementarity principle in explaining this feature.'
author:
- Guruprasad Kar
- Manik Banik
title: 'Several foundational and information theoretic implications of Bell’s theorem'
---
Introduction
============
Quantum mechanics (QM) is, certainly, the most successful theory to describe physical phenomena at very small scales. Apart from gravity, it provides completely correct mathematical description for all natural phenomena, stating from the structure of atoms, the rules of chemistry and properties of condensed matter to nuclear structure and the physics of elementary particles. However, from a fundamental point of view different concepts of this theory departs in various ways from that of classical physics. Undoubtedly the most debatable *non classical* concept is the existence of *nonlocal* correlations among spatially separated quantum systems. John Bell, a Northern Irish physicist, in 1964, established a seminal result, that certain quantum correlations, unlike all other correlations in the Universe, cannot arise from any local cause [@Bell; @Gottfried]. This result is commonly known as ‘Bell’s theorem’. For the last $50$ years, this theorem remains at the center of almost all *metaphysical* debates of Quantum foundation and during the last two decades it also finds applications in various information theoretic tasks.
In his paper Bell addressed a long standing question introduced by Einstein along with his two colleagues Podolsky and Rosen (EPR) at the very early days of quantum theory [@EPR]. From historical perspective, Einstein was the most prominent opponent of the ‘Copenhagen interpretation’, according to which quantum system is completely described by a wave function belonging to an Hilbert space associated with the system [@Copenhagen]. He believed that the fundamental theory of the nature must be deterministic. It was the intrinsic randomness of QM (lack of reality) which bothered him the most. His dissatisfaction with QM, in describing the natural phenomena, comes clear from one of his famous comment “*God does not play dice with the universe*”. In his view, though QM provide the correct description of physical phenomena it could not be the complete one. In the groundbreaking $1935$ EPR paper this philosophical idea has been condensed into a physical argument. In this paper the authors first described a necessary condition of completeness for a physical theory:
1. [**Condition of completeness**]{}: *Every element of the physical reality must have a counter part in the physical theory*.
Once the the condition of completeness is given, naturally the question arises: ‘what are the elements of the physical reality?’ EPR did not provide a comprehensive definition of reality, but, they described a sufficient condition of it:
1. [**Physical reality**]{}: *If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding lo this physical quantity.*
Invoking, then, the well established idea from special theory of relativity, that nothing can travel faster than light, EPR succeed to demonstrate that “*wave function does not provide a complete description of the physical reality*”. EPR ended there paper in the following fashion “*....we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible*”. EPR work immediately gave rise to intense debates among physicists as well as philosophers about the foundational status of QM. Within few days, an important critique appeared by N. Bohr, a pioneer proponent of the Copenhagen interpretation, against the EPR argument [@Bohr]. But, for a long period of time the debate was only a subject of theoretical interest, no empirical testable method was known to resolve this debate. John Bell took the challenge whether some experimental criteria can be derived to address this debate and ultimately in $1964$ he succeeded to derive it as an powerful theorem. This theorem tells that there exist quantum correlations which are incompatible with conjunction of two classical notion, namely, *locality* and *reality*. However, later it has been shown that the same Bell’s theorem can be derived under different sets of assumptions. Violation of this theorem implies different consequences depending on under which set of assumptions the theorem is derived.
This paper is organized in the following manner. In Section (\[sec2\]) we first discuss operational and ontological framework for any physical theory. We then discuss the Bell’s original derivation of his famous inequality under the assumptions of *locality* and *reality*. We also discuss about the quantum mechanical violation of this inequality. In Section (\[sec3\]) we discuss that Bell’s inequality (BI) can be derived under the assumption of joint measurability and relativistic causality or no signaling principle. In Section (\[sec4\]) we analyze that the same BI can be derived under some operational assumptions. We also discuss that as a novel implication of this derivation BI finds application in an important information theoretic task, namely device independent randomness certification. In Section (\[sec5\]) we discuss another important feature of quantum correlation namely the limited violation of BI in QM compared to the violation allowed by no-signaling principle. We discuss the role of uncertainty and complementarity principle in explaining this feature.
Bell’s inequality {#sec2}
=================
Bell realized that to understand the source of conflicts in QM, it is essential to know where and how our classical concepts and intuitions start to fail in describing the quantum phenomena. He succeeded to derive an experimentally testable condition obtained under some *metaphysical* assumptions which follow from the classical world view. For the first time, in the history of science, there was a concrete suggestion to test the correctness of different world view from measurement results. This area of study is called ‘*experimental metaphysics*’, a term coined by Abner Shimony [@Shimony]. Before discussing various assumptions under which BI is derived, we briefly describe the *operational* and *ontological* framework for any physical theory.
Operational interpretation of a theory
--------------------------------------
In an operational interpretation of a physical theory, the primitive elements are preparation procedures, transformation procedures, and measurement procedures, which can be viewed as lists of instructions to be implemented in the laboratory. The goal of an operational theory is merely to specify the probabilities $p(k|M,P,T)$ of different outcomes $k\in\mathcal{K}_M$ that may result from a measurement procedure $M\in\mathcal{M}$ given a particular preparation procedure $P\in\mathcal{P}$, and a particular transformation procedure $T\in\mathcal{T}$; where $\mathcal{M}$, $\mathcal{P}$ and $\mathcal{T}$ respectively denote the sets of measurement procedures, preparation procedures and transformation procedures; $\mathcal{K}_M$ denotes the set of measurement results for the measurement M. When there is no transformation procedure, we simply have $p(k|M,P)$. The only restrictions on $\{p(k|M,P)\}_{k\in\mathcal{K}_M}$ is that all of them are non negative and $\sum_{k\in\mathcal{K}_M}p(k|M,P)=1$ $\forall$ M, P. As an example, in an operational formulation of quantum theory, every preparation P is associated with a density operator $\rho$ on Hilbert space, and every measurement M is associated with a positive operator valued measure (POVM) $\{E_k|~E_k\ge0~\forall~k~\mbox{and}~\sum_kE_k=I\}$. The probability of obtaining outcome $k$ is given by the generalized Born rule, $p(k|M,P)=\mbox{Tr}(E_k\rho)$. An operational theory does not tell anything about *physical state* of the system. So one can construct an ontological model of an operational theory (for details of this framework, we refer to [@Spek05; @Rudolph]).
Ontological model for a theory
------------------------------
In an ontological model of an operational theory the primitives of description are the actual state of affairs of the system. A preparation procedure is assumed to prepare a system with certain properties and a measurement procedure is assumed to reveal something about those properties. A complete specification of the properties of a system is referred to as the ontic state of that system. The knowledge of the ontic state may remain unknown to the observer. For this reason it is also called hidden variable and the model is called hidden variable theory (HVT).
In an ontological model for quantum theory, a particular preparation method $P_{\psi}$ which prepares the quantum state $|\psi\rangle$, actually puts the system into some ontic state $\lambda\in\Lambda$, $\Lambda$ denotes the ontic state space. An observer who knows the preparation $P_{\psi}$ may nonetheless have incomplete knowledge of $\lambda$. Thus, in general, an ontological model associates a probability distribution $\mu(\lambda|P_{\psi})$ with preparation $P_{\psi}$ of $|\psi\rangle$. $\mu(\lambda| P_{\psi})$ is called the *epistemic state* as it encodes observer’s *epistemic ignorance* about the state of the system. It must satisfy $$\label{epis}
\int_{\Lambda}\mu(\lambda|P_{\psi})d\lambda=1~~
\forall~|\psi\rangle~\mbox{and}~P_{\psi}.$$ Similarly, the model may be such that the ontic state $\lambda$ determines only the probability $\xi(k|\lambda,M)$, of different outcomes $k$ for the measurement method $M$. However, when the model is deterministic one, $\xi(k|\lambda,M)\in \{0,1\}$. The response functions $\xi(k|\lambda,M)\in[0,1]$, should satisfy $$\label{response}
\sum_{k\in\mathcal{K}_M}\xi(k|\lambda,M)=1~~\forall~~\lambda,~~M.$$ As the model is required to reproduce the observed frequencies (quantum predictions) hence the following must also be satisfied $$\label{reproduce}
\int_{\Lambda} \xi(\phi|M,\lambda)\mu(\lambda|P_{\psi}) d\lambda = |\langle\phi|\psi\rangle|^2.$$ This requirement is called ‘quantum reproducibility condition’.
Bell’s locality-reality theorem
-------------------------------
It is a statement about certain type of correlations established between two spatially separated observers, say Alice and Bob. Each of the observers hold some physical system which may have interacted previously. Denote the preparation state of the composite system as $P$. Both of them perform measurement on there respective subsystem and observe the measurement result. Let, Alice can choose to perform any of two possible measurements $x\in\{0,1\}\equiv X$. Also consider that for each measurement there are two possible outcomes denoted as $a\in\{+1,-1\}\equiv A$. Similarly, Bob measurements are denoted as $y\in\{0,1\}\equiv Y$ and the outcome as $b\in\{+1,-1\}\equiv B$. Now for any pair of Alice’s and Bob’s measurement settings the expectation value is calculated as $$\label{expect}
\langle xy\rangle_P=\sum_{a,b=+1}^{-1}ab~p(a,b|x,y,P),$$ where the subindex $\langle *\rangle_P$ denote that expectation is calculated on the preparation state $P\in\mathcal{P}$. $\{p(a,b|x,y,P)\}_{x\in X,y\in Y}^{a\in A, b\in B}$ are the observed frequency in the operational theory, where $p(a,b|x,y,P)$ denote the joint conditional probability of obtaining outcome ‘$a$’ by Alice and outcome ‘$b$’ by Bob when they perform measurements ‘$x$’ and ‘$y$’ respectively. For this operational experiment one can consider a ontological model as described previously. Denoting the ontological variable (hidden variable) as $\lambda$, the conditional joint probability reads as $P(a,b|x,y,P,\lambda)$. To satisfy the reproducibility condition (\[reproduce\]) we have $$\label{reproduce1}
p(a,b|x,y,P)=\int_{\lambda\in\Lambda}d\lambda\rho(\lambda)p(a,b|x,y,P,\lambda).$$ The assumptions under which Bell derived his result state certain properties of the ontological theory.
***Definition 1***: An ontological model is said to satisfy locality iff $$\begin{aligned}
p(a|x,y,P,\lambda)=p(a|x,P,\lambda);~\forall~a,x,y\nonumber\\
p(b|x,y,P,\lambda)=p(b|y,P,\lambda);~\forall~b,x,y.\end{aligned}$$\[locality\]
***Definition 2***: An ontological model is said to deterministic iff it satisfy determinism iff $$\label{determinism}
p(a,b|x,y,P,\lambda)\in\{0,1\};~\forall~a,b,x,y.$$ This implies that $a=a(x,y,P,\lambda)$ and $b=b(x,y,P,\lambda)$. It is straightforward to check that if any model satisfy locality and determinism then it also satisfy the following factorisability relation $$\label{facto}
p(a,b|x,y,P,\lambda)=p(a|x,P,\lambda)P(b|y,P,\lambda).$$
**Theorem 1** (Bell): Any theory which satisfy the factorisability condition or in other words satisfy locality and determinism must satisfy the following inequality: $$\label{BI}
|\langle x_0y_0\rangle+\langle x_0y_1\rangle+\langle x_1y_0\rangle-\langle x_1y_1\rangle|\le 2.$$ Proof of the above theorem is straightforward and we omit the proof (for the proof we refer the papers [@Bell; @CHSH]). It is important to note that another implicit assumption, namely the assumption of *measurement independence*, has been used in the derivation of Bell’s inequality (BI) [@Freewill]. The assumption of measurement independence contravenes to depend the distribution of the ontic variable on the Alice’s and Bob’s measurement settings, i.e., $\rho(\lambda|x,y)=\rho(\lambda)$. In other word, applying Bay’s theorem, we can say that Alice’s (Bob’s) choices of measurement setting is independent of the ontic variables, i.e., Alice and Bob can choose their measurement setting freely.
Interestingly, quantum statistics violates this inequality. In the following we discuss regarding the violation of BI in QM.
Violation of BI by quantum correlation
--------------------------------------
Consider two spin-half particle prepared in the state $|\psi^-\rangle_{12}=\frac{1}{\sqrt{2}}(|0\rangle_1\otimes|1\rangle_2-|1\rangle_1\otimes|2\rangle_2)$, where sub-indices have been used to denote first and second particle respectively, and $|0\rangle$ ($|1\rangle$) denotes the up (down) eigenstate of the Pauli $\sigma_z$ operator. The state $|\psi^-\rangle_{12}$ is called singlet state and it belongs to the tensor product Hilbert space $\mathbb{C}^2\otimes\mathbb{C}^2$. This state has an interesting property that it can not be written as mixer of product states of the composite system and hence it is entangled [@Werner; @Horodecki]. If Alice perform spin measurement along $\hat{n}$ direction and Bob perform along $\hat{m}$ direction on their respective particle of singlet state then according to QM the joint expectation reads as: $$\label{singlet}
\langle\hat{n}\hat{m}\rangle=_{12}\langle\psi^-|\hat{n}.\vec{\sigma}\otimes\hat{m}.\vec{\sigma}|\psi^-\rangle_{12}=-\hat{n}.\hat{m}=-\cos(\theta_{nm}).$$ Let Alice and Bob fix there measurement setting as following: $\cos(\theta_{n_0m_0})=\cos(\theta_{n_0m_1})=\cos(\theta_{n_1m_0})=\frac{\pi}{4}$ and $\cos(\theta_{n_1m_1})=\frac{3\pi}{4}$. With these measurement settings on singlet sate the Bell inequality expression (i.e. left hand side of Eq.(\[BI\])) turns out to be $2\sqrt{2}>2$ (loophole free experimental verification of nonlocal nature of QM is an active area of research till date [@experiment]).
It is interesting question to ask what conclusion one can draw from BI violation. As in the previous section we have discussed that BI is derived under the assumption of locality and determinism, so when a theory violates BI we conclude that the description of the theory can not be replaced by an local deterministic ontological model. Since QM violates BI, it can not have a local deterministic description and hence it is called *nonlocal*. Note that to show the violation of BI in QM the measurements that have been chosen on Alice’s (Bob’s) side are incompatible *i.e* not jointly measurable. In the following section we discuss the connection between incompatibility of observables and BI violation.
Joint measurability and nonlocality {#sec3}
===================================
To show the violation of BI in QM we have used two feature of quantum theory, i.e., we have considered (i) entangled system and (ii) incompatible measurement. Naturally the question arise whether these two feature are generic requirement for BI violation in QM. It has been shown that by performing arbitrary local measurements on a separable state it is not possible to establish any form of quantum nonlocality [@Werner]. On the other hand, if we want to address the question of incompatible observable, we have to put more general the notion of incompatibility. There are several notions of incompatibility. Two such notions are non-commutativity and non existence of joint measurement [@Heinosaari; @Busch]. For projective measurement (PVM) these two notions are identical. Andersson *et al.* have shown that BI can be derived under the assumption of existence of joint measurement and no signaling or signal locality condition [@Andersson]. It is important to note that if joint measurement exists on both side then the four probability distribution exists and the BI follows according to Fine’s result [@Fine]. In the following we briefly sketch the proof of Andersson *et al.*
Consider two measurements $A_1$ and $A_2$ on Alice side and two measurements $B_1$ and $B_2$ on Bob’s side. Denote $p[v(A_1)=v(A_2);B]$ be the probability that $A_1$ and $A_2$ have same measurement outcome when Bob’s measurement is $B$. We have $$p[v(A_1)=v(A_2);B]=p[v(A_1)=v(A_2)=v(B)]+p[v(A_1)=v(A_2)=-v(B)].$$ Positivity condition of the probability implies, $$p[v(A_1)=v(A_2)=v(B)]+p[v(A_1)=v(A_2)=-v(B)]$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~\ge|p[v(A_1)=v(A_2)=v(B)]-p[v(A_1)=v(A_2)=-v(B)]|.$$ Writing the expectation value as $$E(A,B)=p[v(A)=v(B)]-p[v(A)=-v(B)]$$ And putting $B=B_1$ we have $$p[v(A_1)=v(A_2);B_1]\ge\frac{1}{2}|E(A_1,B_1)+E(A_1,B_2)|$$ Similar argument gives $$p[v(A_1)=-v(A_2);B_2]\ge\frac{1}{2}|E(A_1,B_2)-E(A_2,B_2)|$$ Adding the above two inequalities we have $$p[v(A_1)=v(A_2);B_1]+p[v(A_1)=-v(A_2);B_2]$$ $$~~~~~~~~~~~~~~~~~~~\ge\frac{1}{2}|E(A_1,B_1)+E(A_1,B_2)|+|E(A_1,B_2)-E(A_2,B_2)|.$$ The no signaling condition implies $$p[v(A_1)=-v(A_2);B_2]=p[v(A_1)=-v(A_2);B_1].$$ using the normalization of probability, $$p[v(A_1)=v(A_2);B_1]+p[v(A_1)=-v(A_2);B_1]=1$$ finally we get $$|E(A_1,B_1)+E(A_1,B_2)|+|E(A_1,B_2)-E(A_2,B_2)|\le 2.$$
However, for positive-operator-valued-measures (POVMs) non existence of joint measurability and non-commutativity are not equivalent notions. From now on incompatibility would mean the impossibility of joint measurement. So, at this point one can asks the following two questions: (1) do all entangled states demonstrate nonlocality? (2) Can all pair of incompatible measurements on both sides exhibit nonlocality? It has been shown that though all pure entanglement states violates BI [@Gisin], there exist mixed entangled states which have LHV model for all PVMs [@Werner] and also for all POVMs [@Barrett], which means there exist local entangled states or in other words, nonlocality and entanglement are two different concepts. If two incompatible measurements are considered, each with binary outcome, then Wolf et al. have shown that it can always lead to violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) [@CHSH] inequality [@Wolf]. This result has established a connection between (im)possibility of joint measurability and nonlocality. However, for general situation i.e., for arbitrary number of POVMs with arbitrarily many outcomes the relation between joint measurability and nonlocality is not yet established. The question is important as pairwise joint measurability, in general, does not imply full joint measurability. We discuss two different such examples.
\(a) ***Orthogonal spin measurement***: Consider the set of three dichotomic POVMs, acting on $\mathbb{C}^2$, given by the following positive operators $$\label{ortho}
M_{0|j}(\eta)=\frac{1}{2}(\mathbf{1}+\eta \sigma_j)$$ for $j = x, y, z$, where $\sigma_x$, $\sigma_y$, $\sigma_z$ are the Pauli matrices, and $0 \le \eta \le 1$. The operator corresponding to other outcome is $M_{1|j}(\eta)=\mathbf{1}-M_{0|j}(\eta)$. The triple of measurements are pairwise jointly measurable iff $\eta\le\frac{1}{\sqrt{2}}\approx 0.707$, but triple-wise jointly measurable iff $\eta\le\frac{1}{\sqrt{3}}\approx 0.577$. Hence in the range $\frac{1}{\sqrt{3}}\le\eta\le\frac{1}{\sqrt{2}}$ the the set $\{M_{a|j}(\eta)\}$ forms a hollow triangle [@Heinosaari; @Liang].
\(b) ***Trine spin measurement***: Consider another three dichotomic POVMs, acting on $\mathbb{C}^2$, given by $$\label{trin}
M_{0|j}(\eta)=\frac{1}{2}(\mathbf{1}+\eta \hat{n}_j.\vec{\sigma}),$$ where, the three vectors $\{\hat{n}_j|j=1,2,3\}$ are equally separated in a plane, i.e. separated by a trine or an angle of $2\pi/3$. This triple of measurements are pairwise jointly measurable if $\eta\le \sqrt{3}-1\approx0.73205$, but triple-wise jointly measurable only if $\eta\le\frac{2}{3}$ [@Liang].
The above two example shows that the connection between joint measurability and nonlocality for more than two measurement scenario, even with each measurement being dichotomic, is not a trivial extension of that of two measurement scenario. In two recent works partial but very useful progress has been obtained [@Quintino; @Uola]. The authors in [@Quintino] showed that (non) joint measurability can be viewed as equivalent to EPR steering, a strictly weaker type of nonlocality than Bell’s nonlocality [@Schrodinger; @Wiseman]. Specifically, the authors in [@Quintino] have shown that for any set of POVMs that are incompatible (i.e. not jointly measurable), one can find an entangled state, such that the resulting statistics violates a steering inequality. They further conjectured that (im)possibility of joint measurability and Bell nonlocality are inequivalent. A conclusive answer requires proof (or disproof) of this conjecture.
Other than these foundational implications, in the last decade various device-independent [@Scarani] information theoretic protocols like cryptography [@cry1], randomness certification [@rand1; @rand2], Hilbert Spaces dimension witness [@Brunner] have made use of nonlocality arguments [@nonlocality]. In the following we discuss one of such protocol, namely, device independent randomness certification. Before that we discuss another derivation of BI based on some *operational* assumptions.
Bell’s inequality from operational assumptions {#sec4}
==============================================
In the previous section we have seen that BI can be derived under the conjunction of two assumption, namely, *locality* and *determinism*. Both these assumptions assert some *ontological* properties. As QM violates BI, it can not have a local-deterministic ontological description. However, there are models that violate locality but maintain determinism (Bohmian mechanics [@Bohm] is an example), and models that maintain locality but violate determinism (standard operational quantum theory is an example). Recently, Cavalcanti and Wiseman have derived BI from another set of assumptions, namely, *signal locality* and *predictability* [@Cavalcanti]. Unlike the Bell’s original assumptions (i.e. locality and determinism), these assumptions are purely operational, that is, they refer to operational quantities only. Whereas the assumption of signal locality describe the impossibility to send signals faster than light, predictability assumes that one can predict the outcomes of all possible measurements to be performed on a system.
To derive BI from the conjunction of the assumptions signal locality and predictability it is sufficient to show that the joint probabilities of experimental outcomes given by a model is factorisable, i.e., satisfy the Eq.(\[facto\]).
***Definition 3***: A model is said to satisfy signal locality iff $$\begin{aligned}
\label{s-locality}
p(a|x,y,P)=p(a|x,P);~\forall~a,x,y\nonumber\\
p(b|x,y,P)=p(b|y,P);~\forall~b,x,y.\end{aligned}$$
***Definition 4***: A model is said to be predictable, or to satisfy predictability iff $$\label{predict}
p(a,b|x,y,P)\in\{0,1\};~\forall~a,b,x,y.$$ This implies that $a=a(x,y,P)$ and $b=b(x,y,P)$.
**Lemma 1** (Cavalcanti & Wiseman): signal locality $\wedge$ predictability $\Rightarrow$ BI.
*Proof*: The definition of predictability implies that $$p(a,b|x,y,P,\lambda)=p(a,b|x,y,P)$$ as $p(a,b|x,y,P)\in\{0,1\}$, conditioning on $\lambda$ can not alter it. Using Bay’s rule of conditional probability $$p(a,b|x,y,P)=p(a|b,x,y,P)p(b|x,y,P)$$ Predictability also implies that $p(a|b,x,y,P)=p(a|x,y,P)$. Hence $$p(a,b|x,y,P,\lambda)=p(a|x,y,P)p(b|x,y,P)$$ Applying the signal locality condition $$p(a,b|x,y,P,\lambda)=p(a|x,P)p(b|y,P)$$ Conditioning on the ontic variable $\lambda$, we reached at the desired factorisable condition given by $$p(a,b|x,y,P,\lambda)=p(a|x,P,\lambda)p(b|y,P,\lambda)$$
The above derivation gives authority in drawing remarkable conclusion from BI violation. Since signal locality is an empirically testable (and well-tested) consequence of relativity, violation of BI would imply unpredictability. This understanding actually plays the fundamental role in an important information theoretic protocol, namely, device independent randomness certification, a pioneer work done by Pironio et al. [@rand1]. Before describing the work of Pironio *et al.* we first briefly discuss the framework of device independent scenario.
Device independent scenario
---------------------------
The necessity of device independent technique in quantum information processing tasks was pointed out by A. Acín *et al.* [@Masanes]. They observed that in the BB84 quantum key distribution (QKD) protocol [@BB84] the security proof was based on
- the validity of the quantum formalism,
- the legitimate partners perfectly know how their correlation is established, e.g., they know the dimensions of the Hilbert space describing their quantum systems.
The security proofs of QKD schemes exploit the well-established quantum features such as the no-cloning theorem, or the monogamy (i.e., nonshareability) of of certain quantum correlations. But if there is no restriction on the Hilbert space dimension, the security proof breaks down. In the noise-free case, the BB84 correlation satisfies $p(a=b|x=y)=1$ and $p(a=b|x=y)=1/2$. If such correlation results from measurements in the $x$ and $z$ bases on qubit pairs, then the state of these two qubits is necessarily maximally entangled and security follows from monogamy of entanglement. However, the authors, in [@Barrett], pointed out that the same correlation can also be reproduced by the following four-qubit state: $$\rho_{AB}=\frac{1}{4}(|00\rangle\langle 00|_z+|11\rangle\langle 11|_z)\otimes(|00\rangle\langle 00|_x+|11\rangle\langle 11|_x)$$ Here, Alice holds the first and third qubit. Whenever she measures in the $z$ ($x$) basis, she is actually measuring the first (third) qubit in this basis. The same happens for Bob, with the second and fourth qubit. It is easy to verify that the correlation is precisely same as ideal BB84 case. As the state is separable, so a secure key cannot be established. Thus key distribution in these schemes is successful when assumption (ii) is satisfied.
In device independent scenario one does not have detail knowledge about the experimental apparatus, i.e., the experimental set up is like a black-box as shown in Fig(\[pic1\]). On each side, the experimental device is depicted like a box with some knobs. A knob with different positions on each device, denoted respectively by $A_i$ and $B_j$, allows Alice and Bob to change the parameters of each measuring apparatus.
![Black-box description of an experimental set up used in device independent protocol.[]{data-label="pic1"}](pic.png){height="5cm" width="7cm"}
Each measurement performed by Alice and Bob has $d$ possible outcomes. Finally, the frequencies $P(A_i=m,B_j=n)$ of occurrence of a given pair of outcomes for each pair of measurements have to be collected. Then it has to be checked whether the correlation satisfy some established physical principles. Only if it fulfills the required conditions, the correlation can be used for secure protocols. In the following we discuss such a device independent task, namely, device independent randomness certification.
DI randomness certification from BI
-----------------------------------
In DI randomness certification scenario one (Say Alice) has a private place which is completely inaccessible from outside i.e., no illegitimate system may enter in this place. From a cryptographic point of view assumption of such private place is admissible. Alice choses classical inputs $x\in X$ and $y\in Y$ with probability distributions $\mathcal{P}_X(x)$ and $\mathcal{P}_Y(y)$, respectively, and sends them to two measurement devices ($\mathcal{MD}1$ and $\mathcal{MD}2$ respectively) through some secure classical communication channels. The inputs prescribe the measurement devices to perform some POVM $\{M_{a|x}~|~M_{a|x}\ge 0~\forall a;~\sum_{a}M_{a|x}=\mathbb{I}_{\mathcal{H}_A}\}$ and $\{M_{b|y}~|~M_{b|y}\ge 0~\forall b;~\sum_{b}M_{b|y}=\mathbb{I}_{\mathcal{H}_B}\}$ on some quantum state $\rho$, shared between the two devices. Once the inputs are received, no classical communication between the measurement devices $\mathcal{MD}1$ and $\mathcal{MD}2$ is allowed. Alice collects the input-output statistics $P(AB|XY)=\{p(ab|xy)\}$.
![Setup for DI randomness certification. Classical inputs are sent from Alice’s private place to the measurement devices ($\mathcal{MD}1$ and $\mathcal{MD}2$) through secure classical channels. Classical communication is not allowed between two measurement devices.[]{data-label="fig1"}](fig1.pdf){height="7cm" width="9cm"}
If the input-output statistics violate the Bell expression $I$, then min-entropy associated with the output is non zero and thus certifiable randomness is obtained. In this case the randomness is certified by the Bell’s theorem. As the allowed correlations satisfy signal locality, BI violation implies that operational statistic must be unpredictable and thus randomness can be obtained without knowing the detail of the experimental device. The setup for DI randomness certification is depicted in Fig.\[fig1\]. To obtain a lower bound in the min-entropy corresponding to a given Bell violation $I$ which does not depend on the input pair one has to first optimize the guessing probability over the all input pair $(x,y)$. If this optimized value is denoted as $G^*$ then $-\log_2G^*$ denote the minimum amount of random bits associated with Bell violation $I$.
Let Alice is interested in the amount of minimum randomness obtained in NS theory; which mean that any correlation satisfying NS condition is allowed to share between the measurement devices. In that case, the following optimization problem need to be solved: $$\begin{aligned}
\label{rand_ns}
p^*_{ns}(ab|xy)&=&~~~~~~\mbox{max}~~~~~~p(ab|xy)\nonumber\\
&&\mbox{subject~to}~~\sum_{abxy}c_{abxy}p(ab|xy)=I\nonumber\\
&& P(AB|XY)\in \mathcal{P}^{NS}. \end{aligned}$$ An input-output probability distribution would lie in $\mathcal{P}^{NS}$ if the following conditions are satisfied:
- $p(ab|xy)\ge 0$ $\forall~~a,b,x,y$
- $\sum_{a,b}p(ab|xy)=1$ $\forall~~x,y$
- $\sum_{b}p(ab|xy)=\sum_{b}p(ab|xy')$ $\forall~~x,y,y'$
- $\sum_{a}p(ab|xy)=\sum_{a}p(ab|x'y)$ $\forall~~x,x',y$
where the first and second conditions, respectively, denote positivity and normalization of a probability distribution, the last two conditions are known as NS condition. The minimum amount of random bits corresponding to Bell violation $I$ in NS framework is therefore $H_\infty(AB|XY)=-\log_2\max_{ab}p_{ns}^*(ab|xy)$.
To obtain the minimum randomness in quantum theory one has to perform the following optimization problem: $$\begin{aligned}
\label{rand_quantum}
p^*_q(ab|xy)&=&~~~~~~\mbox{max}~~~~~~p(ab|xy)\nonumber\\
&&\mbox{subject~to}~~\sum_{abxy}c_{abxy}p(ab|xy)=I\nonumber\\
&&p(ab|xy)=\mbox{tr}[M_{a|x}\otimes M_{b|y} \rho]\end{aligned}$$ where the optimization is performed over all states and all POVMs defined over Hilbert space of arbitrary dimension. The second condition is to ensure that the obtained correlation is quantum one. Adapting a straightforward way of technique introduced in [@Navascues], one can efficiently check whether a given correlation can be obtained via quantum means or not. The minimum random bits obtained in quantum theory corresponding to BI violation $I$ is thus quantified as $H_\infty(AB|XY)=-\log_2\max_{ab}p_q^*(ab|xy)$. From the analysis of Ref.[@rand1] it turns out that the minimum amount of randomness corresponding to a Bell violation $I$ obtained from quantum mechanics is higher than that obtained under consideration of NS framework.
Nonlocal correlation finds applications in important information theoretic tasks and hence is considered as useful resource. Quantifying the amount of nonlocality in a given correlation is, therefore, important from practical point of view. The amount of BI violation, beyond the *local-realistic* bound, gives a natural measure of nonlocality. This measure also has a operational interpretation as it gives the success probability of winning a nonlocal game (eg. for the two-input two-output scenario the well known XOR game). Though the optimal success probability in QM is strictly greater than classical theory, it must be restricted in comparison to more general correlations compatible with relativistic causality. In the following we discuss this nontrivial aspect of quantum nonlocal correlations.
Nonlocality in quantum theory: Cirel’son bound {#sec5}
==============================================
Popescu and Rohrlich first gave an example of a correlation which is more nonlocal than quantum correlations but still satisfy the relativistic causality or no signaling (NS) principle. There correlation reads as: $$\begin{aligned}
\label{PR}
p(ab|xy)&=&\frac{1}{2}~~\mbox{if}~~a\oplus b=xy,\nonumber\\
&=& 0~~\mbox{otherwise}.\end{aligned}$$
While in a generalized non-signaling theory Bell expression (left hand side of Eq.(\[BI\])) can reach the maximum algebraic value $4$ i(t is easy to verify that the correlation of Eq.(\[PR\]) achieves the maximal value $4$) in QM this value is restricted by $2\sqrt{2}$. We have already seen that performing suitable dichotomic measurements on singlet state the value $2\sqrt{2}$ can be achieved. However, Cirel’son showed that sharing any bipartite state and performing any pairs of dichotomic observables on each part the maximum value of Bell expression can not go beyond $2\sqrt{2}$ [@Cirelson]. This value is known as *Cirel’son bound*.
So, on the one hand QM contains astonishing nonlocal correlation, on the other hand, surprisingly, the nonlocality in QM is restricted is some sense. In the recent past researchers have tried to find out physical principles that can explain this restricted nonlocal feature of QM. Various interesting results have been derived in this regard. First successful attempt was made by W. van Dam [@van-Dam]. The result is further generalized by Brassard *et al.*, who showed that in any world in which communication complexity is nontrivial, there is a bound on how much nature can be nonlocal [@Brassard]. Pawlowski *et al.* introduced a new information principle, namely Information Causality (IC), which says that information that Bob can gain about a data set with Alice, by using all his local resources (which may be correlated with Alice’s resources) and a classical communication from her, is bounded by the information volume of the communication [@Pawlowski]. The authors further showed that for any theory, satisfying IC, the Bell quantity can not be larger than $2\sqrt{2}$. on the other hand, Navascues *et al.* postulated that any post-quantum theory should recover classical physics in the macroscopic limit [@Navascues]. Using this mechanism they were able to derive nontrivial bound on the strength of correlations between distant observers in any physical theory. Recently, another principle, namely, Local orthogonality [@Fritz] or Exclusivity [@Cabello1; @Amaral], an intrinsically multi-partite principle, have been used for the same purpose. Furthermore it has been shown that this principle also exactly singles out the Cirel’son bound [@Cabello2]. Recently, there is a different kind of development where Uncertainty principle and Complementarity principle have been shown to be related to the nonlocality of a theory in a fundamental way. In the following we discuss these two aspects [@Oppenheim; @Banik].
Uncertainty, Steering and Nonlocality
-------------------------------------
In a recent paper Oppenheim and Wehener have shown that QM cannot be more nonlocal with measurements that respect the uncertainty principle [@Oppenheim]. In fact they have proved that degree of nonlocality of any theory is determined by two factors: the strength of the uncertainty principle and the strength of a property called “steering”. Important to note that they have used a new mathematical form, namely the *fine grained uncertainty*, to establish their result.
[**Fine grained uncertainty**]{}: Heisenberg’s uncertainty principle tells that there are incompatible measurements whose results cannot be simultaneously predicted with certainty [@Heisenberg]. Mathematically, Heisenberg expressed this fact in term of a nonzero lower bound on the product of the standard deviation of the concerned incompatible measurements. There are more modern approach to express this impossibility in term of entropic measures [@Entropic]. Oppenheim *el al.* realized that entropic functions are, however, a rather coarse way of measuring the uncertainty and they have introduced the ‘fine grained’ form of the uncertainty principle. Let $p(x^{(t)}|t)_{\sigma}$ denote the probability that we obtain outcome $x^{(t)}$ when performing a measurement labeled $t\in\mathcal{T}$ when the system is prepared in the state $\sigma$ and let $\mathbf{x}=(x^{(1)},x^{(2)},...,x^{(n)})\in\mathcal{B}^{\times n}$, with $n=|\mathcal{T}|$, denote the combination of possible outcomes. Then for a fixed set of measurements and a probability distribution $\mathcal{D}=\{p(t)\}_t$ the set of inequalities $$\label{fgu}
U=\left\lbrace \sum_{t=1}^{n}p(t)p(x^{(t)}|t)_{\sigma}\le\zeta_{\mathbf{x}}~
|~\forall~\mathbf{x}\in\mathcal{B}^{\times n}\right\rbrace$$ describe a fine-grained uncertainty relation. This relations tell that whenever $\zeta_{\mathbf{x}}<1$ one cannot obtain a measurement outcome with certainty for all measurements. The quantity $$\zeta_\mathbf{x}=\max_{\sigma}\sum_{t=1}^{n}p(t)p(x^{(t)}|t)_{\sigma}$$ characterizes the ‘amount of uncertainty’ in a particular physical theory. Here, maximization is taken over all states allowed on a particular system. As for example, in quantum theory $\zeta_\mathbf{x}^Q=\frac{1}{2}(1+\frac{1}{\sqrt{2}})$, on the other hand in classical theory as well as in the box theory (marginal correlation for PR correlation or in other word for g-bit) $\zeta_\mathbf{x}^C=\zeta_\mathbf{x}^{box}=1$.
[**Steering**]{}: Steering denotes the power of remotely preparing different ensembles of a system’s state at one place from another spatially separated place. In quantum theory this concept was first introduced by Schrödinger [@Schro]. We know that a mixed state $\sigma$ of a quantum system can be decomposed in many different ways as a convex sum: $$\sigma=\sum_jp_j\sigma_j,$$ where $\sigma_j$’s are density operators and $\{p_j\}_j$ denote probability distribution. This can be written as an ensemble representation $\mathcal{E}=\{p_j,\sigma_j\}_a$ of the state $\sigma$. If Alice and Bob share a pure entangled state $\sigma_{AB}$ with the reduced state $\sigma_B=\mbox{tr}_A(\sigma_{AB})=\sigma$ on Bob’s side then corresponding to every ensemble $\mathcal{E}$ there exists a measurement on Alice’s system that allows her to prepare the ensemble [@gisin; @hjw]. It is important to note that this steering phenomena does not violate no-signaling principle as the unconditional state on Bob’s side remains same for Alice’s different measurements. As an precise example, consider that Alice and Bob share the singlet state $|\psi^-\rangle_{AB}$. Consider two different ensembles $\{\frac{1}{2},\frac{1}{2},|0_z\rangle\langle0_z|,|1_z\rangle\langle1_z|\}$ and $\{\frac{1}{2},\frac{1}{2},|0_x\rangle\langle0_x|,|1_x\rangle\langle1_x|\}$ for Bob’s reduced state. Alice can prepare these two ensemble by performing measurement on her particle along $z$-direction and $x$-direction, respectively. However, it has been later observed that steering is not a strict quantum phenomenon, rather there exist broad class of theories which exhibit this *non classical* feature [@Oppenheim; @Barnum; @Spekkens].
With these two ideas, Oppenheim *et al.* succeeded to establish that for any physical theory, the strength of nonlocal correlations is determined by a trade off between two aspects: steerability and uncertainty. In the following we discuss this trade off for some interesting class of theories.
1. *Box world*: Here the power of steering is compatible to no signaling principle and there is no uncertainty in this theory. As a result the Bell expression in this theory obtained it’s algebraic maximum value $4$.
2. *Q. theory*: Here, also, the ability to steer is only limited by the no-signaling principle, i.e., maximal. But, due to presence of uncertainty the nonlocal power of this theory is restricted by Cirel’son bound.
3. *Classical theory*: In classical theory there is no restriction in knowing the values of different observables, simultaneously, or in other word we have no uncertainty relations on the full set of deterministic states. But, this theory shows no nonlocal property as steering is absent in this theory.
4. *Local theory*: On the other hand there exists local hypothetical theories, which have perfect steering, but only due to a high degree of uncertainty they do not exhibit nonlocal behavior. Spekkens toy-bit model is one interesting example of such a theory [@Spekkens]. It has steering comparable to that in QM and Box world. Had there been no uncertainty, nonlocality would have been equal to that of Box world. However, it can be shown that $\xi_{\mathbf{X}}^{toy}=\frac{1}{2}$ and hence the theory turns out to be local one.
It is important to note that there is no difference among the theories described in (c) and (d), so far their nonlocal behavior is concerned. However, the other properties of these theories are completely different.
Complementarity and degree of nonlocality
-----------------------------------------
One of the original versions of the complementarity principle tells that there are observables in quantum mechanics that do not admit unambiguous joint measurement. Examples are position and momentum [@davice; @prugo; @bu], spin measurement in different directions [@bu; @kraus], path and interference inn the double slit experiment [@scully; @woot], etc. With the introduction of the generalized measurement i.e. positive operator-valued measure (POVM), it was shown that observables which do not admit perfect joint measurement, may allow joint measurement if the measurements are made sufficiently fuzzy [@busch]. The optimal value of unsharpness that guarantees joint measurement of all possible pairs of dichotomic observables can be considered as the degree of complementarity and it has been shown that it determines the degree of nonlocality of the theory [@Banik].
[**General framework**]{}: Consider a generalized probability theory [@barrett] where any state of the system is described by an element $\omega$ of $\Omega$, the convex state-space of the system. $\Omega$ may be considered as a convex subset of a real vector space. By convexity of the state-space $\Omega$ we mean that any probabilistic mixture of any two states $\omega_1, \omega_2\in \Omega$, will describe a physical state of the system. An observable $\verb"A"$ (with the corresponding outcome set $\{\verb"A"_{j}~ : j \in J\}$) is an affine map from $\Omega$ into the set of probability distributions on the outcome set. A measurement of an observable $\verb"A" \equiv \{\verb"A"_j~|~\sum_jp^{\omega}_{\verb"A"_j}=1~~\forall~\omega\in\Omega\}$, performed on the system, allows us to gain information about the state $\omega$ of the physical system. The measurement of $\verb"A"$ consists of various outcomes $\verb"A"_{j}$ with $p^{\omega}_{\verb"A"_j}$ being the probability of getting outcome $\verb"A"_j$, given the state $\omega$. Let $\Gamma$ be the set of all observables with two measurement outcomes ( $j = + 1, - 1$ ), say ‘yes’$(=+1)$ and ‘no’$(=-1)$. If $\verb"A"\in\Gamma$ is such a kind of two-outcome observable, then the average value of $\verb"A"$ on a state $\omega$ is given by $$\langle \verb"A"\rangle_\omega = p^\omega_{\verb"A"_{yes}}-p^\omega_{\verb"A"_{no}}.$$ Given a two-outcome observable $\verb"A"\equiv\{\verb"A"_{yes},\verb"A"_{no}|~p^{\omega}_{\verb"A"_{yes}}
+p^{\omega}_{\verb"A"_{no}}=1~~\forall~\omega\in\Omega\}$, we define a fuzzy or unsharp observable, again with binary outcomes $\verb"A"^{(\lambda)}\equiv\{\verb"A"^{(\lambda)}_{yes},\verb"A"^{(\lambda)}_{no}~|~p^{\omega}_{\verb"A"^{(\lambda)}_{yes}}
+p^{\omega}_{\verb"A"^{(\lambda)}_{no}}=1~~\forall~\omega\in\Omega\}$, with ‘unsharpness parameter’ $\lambda\in(0,1]$, where $p^{\omega}_{\verb"A"^{(\lambda)}_{yes(no)}}$ is the probability of getting the outcome $\verb"A"^{(\lambda)}_{yes(no)}$ in the measurement of $\verb"A"^{(\lambda)}$ with the result ‘yes’ (‘no’). The probabilities $p^{\omega}_{\verb"A"^{(\lambda)}_{yes(no)}}$ are smooth versions of the probabilities of their original counterparts in the following way: $$p^\omega_{\verb"A"^{(\lambda)}_{yes}}=\left(\frac{1+\lambda}{2}\right)p^\omega_{\verb"A"_{yes}}+\left(\frac{1-\lambda}{2}\right)p^\omega_{\verb"A"_{no}},$$ for all $\omega\in\Omega$. We denote the set of all unsharp observables with binary outcomes for a given $\lambda$ by $\Gamma^{(\lambda)}$. For any $\verb"A"^{(\lambda)}\in\Gamma^{(\lambda)}$ the average value of $\verb"A"^{(\lambda)}$ on a given state $\omega\in\Omega$ can be calculated as: $$\langle \verb"A"^{(\lambda)}\rangle_\omega = p^\omega_{\verb"A"^{(\lambda)}_{yes}}-p^\omega_{\verb"A"^{(\lambda)}_{no}}= \lambda \langle \verb"A"\rangle_\omega.$$ Given a state $\omega\in\Omega$ and two observables $\verb"A"_1\equiv\{\verb"A"_{1j}~|~\sum_jp^{\omega}_{\verb"A"_{1j}}=1~~\forall~\omega\in\Omega\}$ and $\verb"A"_2\equiv\{\verb"A"_{2k}~|~\sum_kp^{\omega}_{\verb"A"_{2k}}=1~~\forall~\omega\in\Omega\}$, we say that joint measurement of $\verb"A"_1$ and $\verb"A"_2$ exists if there exists a joint probability distribution $\{p^{\omega}_{\verb"A"_{1j},\verb"A"_{2k}}|\sum_{j,k}p^{\omega}_{\verb"A"_{1j},\verb"A"_{2k}}=1\}$ satisfying the following conditions: $$\begin{aligned}
\sum_{k}p^{\omega}_{\verb"A"_{1j},\verb"A"_{2k}}=p^{\omega}_{\verb"A"_{1j}},~~\forall~j,\nonumber\\
\sum_{j}p^{\omega}_{\verb"A"_{1j},\verb"A"_{2k}}=p^{\omega}_{\verb"A"_{1k}},~~\forall~k,\end{aligned}$$ whatever be the choice of $\omega\in\Omega$. For our purpose, we will concentrate only on the existence of the joint measurement of two two-outcome observables $\verb"A"_1,\verb"A"_2\in\Gamma$. Given a physical theory it is not justifiable to demand that joint measurement should exist for any pair of $\verb"A"_1,\verb"A"_2\in\Gamma$, although in the classical world, it is always possible to construct a joint measurement observable. On the other hand, there are certain observables in quantum theory which can not be jointly measured jointly.
[**Degree of complementarity**]{}: It may be possible that observables which are not jointly measurable in a theory, may admit joint measurement for their unsharp counterparts within that theory. For two given observables, the values of unsharp parameter that make joint measurement possible, depend on the observables. Let $\lambda_{opt}$ denotes the optimum (maximum) value of the unsharp parameter $\lambda$ that guarantees the existence of joint measurement for $all$ possible pairs of dichotomic observables $\verb"A"^{(\lambda)}_1,\verb"A"^{(\lambda)}_2\in\Gamma^{(\lambda)}$. $\lambda_{opt}$ can then be considered as a property of that particular theory. It is obvious from the definition that joint measurement must exists for any two $\verb"A"^{(\lambda)}_1,\verb"A"^{(\lambda)}_2\in\Gamma^{(\lambda)}$, where $\lambda\leq\lambda_{opt}$. The value of $\lambda_{opt}$ measures the degree of complementarity of the theory in the sense that as $\lambda_{opt}$ decreases, the corresponding theory has more complementarity. Of course, finding the value of $\lambda_{opt}$ for a theory will depend on the details of the mathematical structure of the theory.
[**Bound on nonlocality**]{}: Let us now consider the case of a composite system consisting of two subsystems with associated state spaces ${\Omega}_1$ and ${\Omega}_2$ respectively (in a no-signaling probabilistic theory). The state space of the composite system is defined to be ${\Omega}_1 \otimes {\Omega}_2$, which is again a convex subset of a real vector space, whereas, an observable $\verb"A"_{12}$ is an affine map (with outcome space $\{\verb"A"_{12}^{(j)}~ :~ j \in J\}$) from ${\Omega}_1 \otimes {\Omega}_2$ into the set of all probability distributions on the outcome space [@barrett]. Given any observable $\verb"A"_{1}$ for the first subsystem (with outcome space $\{\verb"A"_{1j} : j \in J_1\}$) and any observable $\verb"A"_{2}$ for the second subsystem (with outcome space $\{\verb"A"_{2k} : k \in J_2\}$), here, for our purpose, we consider only observables of the form $\verb"A"_{12} = \{\verb"A"_{12}^{(jk)} : p^{\eta}_{\verb"A"_{12}^{(jk)}} = p^{\eta}_{\verb"A"_{1j}, \verb"A"_{2k}}~ {\rm for}~ {\rm all}~ (j, k) \in J_1 \times J_2~ {\rm and}~ {\rm for}~ {\rm all}~ \eta \in {\Omega}_1 \otimes {\Omega}_2\}$ where $p^{\eta}_{\verb"A"_{1j}, \verb"A"_{2k}}$ is the probability of getting the result $(j, k)$ when measurement of $\verb"A"_{1}$ and $\verb"A"_{2}$ are performed on the joint state $\eta$. Thus, when we take the unsharp version $\verb"A"^{(\lambda)}$ of a dichotomic observable for the first subsystem and a dichotomic observable $\verb"B"$ for the second subsystem then, for any state $\eta \in {\Omega}_1 \otimes {\Omega}_2$, $p^{\eta}_{\verb"A"^{(\lambda)}_{yes}, \verb"B"_{yes}}$ will be the unsharp version of the probability $p^{\eta}_{\verb"A"_{yes}, \verb"B"_{yes}}$, [*i.e.*]{}, $p^{\eta}_{\verb"A"^{(\lambda)}_{yes}, \verb"B"_{yes}} = (1/2 + {\lambda}/2)p^{\eta}_{\verb"A"_{yes}, \verb"B"_{yes}} + (1/2 - {\lambda}/2)p^{\eta}_{\verb"A"_{no}, \verb"B"_{yes}}$, etc.
**Theorem 2**: Consider a composite system composed of two subsystem with state spaces $\Omega_1$ and $\Omega_2$ respectively in a no-signaling probabilistic theory. For any pair of dichotomic observabless $\verb"A"_1,\verb"A"_2\in\Gamma_1$ on the first system and dichotomic observables $\verb"B"_1,\verb"B"_2\in\Gamma_2$ on the second system with the joint state $\eta\in\Omega_1 \otimes \Omega_2$, we have the following inequality: $$\label{comple}
|\langle \verb"A"_1\verb"B"_1\rangle_\eta+\langle \verb"A"_1\verb"B"_2\rangle_\eta+\langle \verb"A"_2\verb"B"_1\rangle_\eta-\langle \verb"A"_2\verb"B"_2\rangle_\eta|\leq\frac{2}{\lambda_{opt}},$$ where $\lambda_{opt}$ has the meaning as described above.
From the expression of inequality (\[comple\]) it is clear that the amount of Bell violation is upper bounded by the unsharp parameter $\lambda_{opt}$, which is a characteristic of complementarity of that particular physical theory. As for example, in classical theory joint measurement of any two dichotomic observables is possible which means $\lambda_{opt}=1$. Contrary to this, we will show that, in quantum mechanics, the value of $\lambda_{opt}$ is $\frac{1}{\sqrt{2}}$ and this has been proved as a theorem in Ref[@Banik].
**Theorem 3**: Given any $d$-dimensional quantum system, joint measurement for unsharp versions of any two dichotomic observables $\overline{\mathcal{M}}_1$ and $\overline{\mathcal{M}}_2$ of the system is possible with the largest allowed value of the unsharpness parameter $\lambda_{opt}=\frac{1}{\sqrt{2}}$.
**Outline of proof**: First we describe the condition of joint measurability of unsharp versions of two dichotomic projection valued measurements (PVM). Let $\mathcal{M}_j\equiv\{\wp_j=|\psi_j\rangle\langle\psi_j|,~ \mathcal{I}-\wp_j=|\psi^\bot_j\rangle\langle\psi^\bot_j|\}$ (for $j=1,2)$ be two dichotomic PVMs, where $|\psi_j\rangle$, $|\psi^\bot_j\rangle$ are normalized pure states such that $\langle\psi_j|\psi^\bot_j\rangle=0$. The unsharp version of $\mathcal{M}_j$ be denoted as $\mathcal{M}^{(\lambda)}_j\equiv \{\wp_j^{({\lambda})} \equiv
\frac{(1 + {\lambda})}{2}{\wp_j} + \frac{(1 - {\lambda})}{2}({\mathcal{I}} - {\wp_j}), ({\mathcal{I}} - {\wp_j})^{({\lambda})} \equiv \frac{(1- {\lambda})}{2}{\wp_j} + \frac{(1 + {\lambda})}{2}({\mathcal{I}} - {\wp_j})\}$. Joint measurement of $\mathcal{M}^{(\lambda)}_j$’s is possible iff there there exists a POVM $\mathcal{M}^{(\lambda)}_{12}\equiv\{G_{++},G_{+-},G_{-+},G_{--}\}$ such that each $G_{ij}$ is a positive operator satisfying the following properties: $$\begin{aligned}
G_{++}+G_{+-}+G_{-+}+G_{--}=\mathcal{I};\nonumber\\
G_{++}+G_{+-}=\wp^{(\lambda)}_1~;G_{-+}+G_{--}=(\mathcal{I}-\wp_1)^{(\lambda)};\nonumber\\
G_{++}+G_{-+}=\wp_2^{(\lambda)}~;~G_{+-}+G_{--}=(\mathcal{I}-\wp_2)^{(\lambda)}.\end{aligned}$$ In the measurement of the POVM $\mathcal{M}^{(\lambda)}_{12}$ if $G_{++}$ ‘clicks’, we say that both $\wp^{(\lambda)}_1$ as well as $\wp^{(\lambda)}_2$ have been ‘clicked’, and so on.
P. Busch have shown that for the qubit system, the POVM $\mathcal{M}^{(\lambda)}_{12}$ will exist for all possible pairs of unsharp measurement iff $0< \lambda \leq \frac{1}{\sqrt{2}}$ [@busch]. Thus the maximum allowed value of the unsharpness parameter $\lambda_{opt}$ is $\frac{1}{\sqrt{2}}$. In the case of PVM, the factor $\frac{1+\lambda}{2}$ ($\frac{1-\lambda}{2}$) has been interpreted as degree of reality (unsharpness) [@busch]. Consider $d$ dimensional Hilbert space $\mathcal{H}$ and also consider two dichotomic PVMs, $\{P,\mathbf{1}-P\}$ and $\{Q,\mathbf{1}-Q\}$. There is an useful result in linear algebra, which guarantees the existence of an orthonormal basis such that $\mathcal{H}=\bigoplus_{\alpha=1}^k\mathcal{H}_{\alpha}$, with $\mathcal{H}_{\alpha}$ at most two dimensional [@Halmos; @Masanes]. Therefore we have: $$\begin{aligned}
P=\oplus_{\alpha=1}^kP^{\alpha};~~~\mathbf{1}-P=\oplus_{\alpha=1}^k(\mathbf{1}-P)^{\alpha};\nonumber\\
Q=\oplus_{\alpha=1}^kQ^{\alpha};~~~\mathbf{1}-Q=\oplus_{\alpha=1}^k(\mathbf{1}-Q)^{\alpha},\end{aligned}$$ where $P^{\alpha},(\mathbf{1}-P)^{\alpha},Q^{\alpha},(\mathbf{1}-Q)^{\alpha}$ are null, one dimensional or two dimensional projectors on $\mathcal{H}_{\alpha}$. Using this result we extended Busch’s result for any pair of two outcomes observables and reached at the Theorem 3 (see [@Banik] for detail).
As an immediate consequence of Theorem 2 and Theorem 3, it follows that the amount of nonlocality of quantum theory respects the Cirel’son bound. According to Theorem 2, in a generalized probability theory the optimal BI violation is restricted to $\frac{2}{\lambda_{opt}}$. However, Theorem 2 can not determine whether this value will be achieved (or not) in a particular theory. In a recent work, Stevens and Busch have provided sufficient criteria for achieving this bound [@Stevens].
**Theorem 4** (Stevens and Busch): In any probabilistic model of a system A that supports uniform universal steering, the Cirel’son bound is given by the tight inequality that can be saturated $$\mathbb{B}\le \frac{2}{\lambda_{opt}}.$$
In the Ref.[@Stevens] the authors have provided a simple nonclassical, nonquantum example, namely, *squit* or Box theory (which is the marginal state space of the PR correlation). The two-dimensional state space of squit is given by a square, denoted as $\square$, which contains all the points $(x,y,1)$ with $-1\le x+y\le 1$, $-1\le x-y\le 1$. The collection of effects on $\square$ is a convex set $\mathcal{E}(\square)$ in the ordered linear space $\mathcal{A}(\square)$ of affine functionals on $\square$, i.e. $$\mathcal{E}(\square):=\{e\in\mathcal{A}(\square)|0\le e(\omega)\le 1, ~\forall~\omega\in\square\}.$$
The squit leads to maximally incompatible effects in the sense that it leads to the smallest possible value of $\lambda_{opt}^{squit}=\frac{1}{2}$. As the correlated system (PR correlation) satisfy the sufficient criterion stated in Theorem 4, hence this theory achieves the maximal BI violation $4$.
Conclusion
==========
Discovery of Bell’s theorem is sometime considered as the most profound discovery of physical science of the last Century. Apart from its implication concerning world views about properties of physical world, the violation of Bell’s inequality by quantum theory finds real application in information theory and communication tasks. In this review, we present various derivation of Bell’s inequality with the aim of understanding the various implications of the violation of this inequality. We also discuss the recent discoveries of some universal principles (in the sense of encompassing all physical theories) and discuss how they can reproduce bound on Bell’s inequality violation which exactly matches with the quantum bound. Finally we discuss the recent development regarding the relation of nonlocality with uncertainty and complementarity principles where uncertainty along with steering capacity determines the amount of nonlocality and complementarity provides an upper bound on nonlocality of the theory.
Acknowledgments {#acknowledgments .unnumbered}
===============
We gratefully acknowledge discussions with Sibashis Ghosh and Md. Rajjak Gazi.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present new HST optical imagery as well as new UV and IR spectroscopic data obtained with the Hubble and Spitzer Space Telescopes, respectively, of the halo planetary nebula DdDm-1. For the first time we present a resolved image of this object which indicates that the morphology of DdDm-1 can be described as two orthogonal elliptical components in the central part surrounded by an extended halo. The extent of the emission is somewhat larger than was previously reported in the literature. We combine the spectral data with our own previously published optical measurements to derive nebular abundances of He, C, N, O, Ne, Si, S, Cl, Ar, and Fe. Our abundance determinations include the use of the newly developed program ELSA for obtaining abundances directly from emission line strengths along with detailed photoionization models to render a robust set of abundances for this object. The metallicity, as gauged by oxygen, is found to be 0.46 dex below the solar value, confirming DdDm-1’s status as a halo PN. In addition, we find that Si and Fe are markedly underabundant, suggesting their depletion onto dust. The very low (but uncertain) C/O ratio suggests that the chemistry of the nebula should be consistent with an oxygen-rich environment. We find that the sulfur abundance of DdDm-1 is only slightly below the value expected based upon the normal lockstep behavior between S and O observed in H II regions and blue compact galaxies. The central star effective temperature and luminosity are estimated to be 55,000 K and 1000 L$_{\odot}$, respectively, implying an initial progenitor mass of $<$1 M$_{\odot}$. Finally, we report on a new radial velocity determination from echelle observations.'
author:
- 'R.B.C. Henry'
- 'K.B. Kwitter'
- 'R.J. Dufour'
- 'J.N. Skinner'
title: 'A MULTIWAVELENGTH ANALYSIS OF THE HALO PLANETARY NEBULA DdDm-1'
---
INTRODUCTION
============
Planetary nebulae (PNe) have long served as useful tracers of galactic interstellar abundances of elements such as O, Ne, S, Cl, and Ar, alpha elements whose abundances are expected to evolve in lockstep due to their common synthesis sites in massive stars. In a recent study of S, Cl, and Ar abundances in over 80 type II PNe located mainly in the northern Galactic hemisphere, @HKB04 discovered that while Ne and O abundances track each other closely, S abundances in a large fraction of objects fall significantly below the values expected for their O abundances and exhibit a large amount of scatter as well. These authors also showed that the same pattern is present in data in the large southern survey by @KB94.
@HKB04 suggested that this tendency of PNe to display a S deficit, a situation they dubbed the “sulfur anomaly”, was probably due to the underestimation for many objects of the ionization correction factor used to account for unobserved ions of S when determining the total gas-phase elemental abundance from the optically observable ions of S$^+$ and S$^{+2}$. However, measurements of S$^{+3}$ abundances by @D03 and the results from numerous ISO projects summarized in @PBS06 using the \[S IV\] 10.5$\mu$m line appear to rule out this explanation, although it is probably too early to draw definite conclusions. Instead, @PBS06 suggested that the sulfur anomaly could result from the depletion of S onto dust due to the formation of sulfides such as MgS and FeS. Sulfide formation is expected to occur most readily in carbon rich environments, i.e., in objects where C/O $>$ 1.
The goal of the project described in this paper is to determine and study the gas phase abundances of the halo PN DdDm-1, an object for which we present new optical imagery as well as spectroscopic measurements of important UV and IR emission lines. DdDm-1 was chosen primarily because of the large amount of data both new (presented here) and previously published for this object. Of the many catalogued PNe in our Galaxy, only about 12 are believed to be located in the halo. Abundance studies of these objects can be used to determine the chemical composition of the halo and can provide additional insight into the evolution of this region of the Milky Way. These objects have consistently been shown to possess sub-solar levels of metals \[@TPRP81, @PTPR92, @HHM97, @D03\]. DdDm-1 has been included in many large abundance surveys but has only been studied in detail in a few instances, e.g. @BC84, @D03, and @WCP05. Here we focus on it exclusively.
In this paper we present new optical HST imagery, new UV data taken with the FOS on HST, and new IR data taken with the IRS on the Spitzer Space Telescope. The imagery allows us for the first time to resolve the nebula and study its morphology. Then combining the spectral data with our own previously published optical data, we compute the abundances of He, C, N, O, Ne, Si, S, Cl, Ar, and Fe using both empirical and numerical methods to ensure robust results. For the empirical method we employ the program ELSA [@J07]; the program Cloudy [@Fer98] is used for the numerical method. The numerical approach also allows us to infer the effective temperature and luminosity of the central star. Since the Spitzer data allow us to determine the abundance of S$^{+3}$, the primary unobservable ion when only optical measurements are available, we evaluate the accuracy of the sulfur ionization correction factor for DdDm-1. With a new sulfur abundance in hand, we then assess the S deficit of DdDm-1, as well as check on the consistency of the deficit’s magnitude with the value of C/O derived from our new UV measurements. Finally, we present new echelle data for DdDm-1 and determine its radial velocity.
The paper is organized as follows. In Section 2, we discuss our observations; 3 contains a description of our procedure for computing abundances; in 4, we discuss our results and we give a summary and conclusions in 5.
OBSERVATIONS
============
Optical Imagining
-----------------
In Figure \[image1\] we present what we believe to be the first resolved image of DdDm-1 in the literature taken from a 1993 WFPC1 image made with the Hubble Space Telescope. \[Recently, @WCP05 obtained an image of DdDm-1 which, according to the authors, was not well-resolved.\] The nebula was exposed for 40 seconds through the F675W filter, which covers H$\alpha$ and nearby emission lines. In our analysis we used the processed WPC1 image from STScI (W1j00201T), which was corrected for CCD artifacts and noise. Cosmic-rays were removed using a combination of a median filter and direct inspection. A 60$\times$60 array ($6\arcsec\times6\arcsec$) centered on the nebula was extracted for our analysis. Then, the image was rebinned into a 120$\times$120 pixel array using a cubic spline interpolation, to give $0.05\arcsec$ pixels. A theoretical point-spread-function was generated using the STScI [*Tiny Tim*]{} software appropriate to the location of the nebula on the WFC1, the filter, and the observation date. This PSF was also rebinned into 0.05$\arcsec$ pixels as per the nebula image. The STSDAS “Lucy" software in the restoration package was then used to deconvolve the DdDm-1 image with this PSF. Our results appeared best for about 35 iterations. We then rotated the images by 105$^{\circ}$ counterclockwise to align them with the equatorial coordinate system. The final image is $3\arcsec\times3\arcsec$, with north up and east to the left. Figure \[image2\] shows the same image as the one in Fig. \[image1\] but with contours added. The latter were generated using the [*disconlab*]{} software in IRAF.
From the deconvolved image and contoured overlay in Figs. \[image1\] and \[image2\], the structure of DdDm-1 is seen to be elliptical with two central components possibly surrounded by an extended (and nearly circular) halo. Evidence for a central star is also seen in the center of the brighter inner ellipse, which has a major axis of $\sim$0.50$\arcsec$ along a PA = 65$^o$ and minor axis of $\sim$0.35$\arcsec$. This is surrounded by a fainter outer elliptical nebula with a major axis of $\sim$1.1$\arcsec$ and minor axis $\sim$0.95$\arcsec$ which is extended approximately orthogonal to the major axis of the inner ellipse. These are somewhat larger dimensions for DdDm-1 than previously reported in @Acker92. Indeed, a logarithmic stretch of our processed image suggests the halo of DdDm-1 may have a diameter of upwards of $\sim$1.75$\arcsec$ (but difficult to define due to the HST spherical aberration scattered light problems at that time).
Spectrophotometry
-----------------
Here we report spectroscopically observed emission line fluxes in the UV, optical, and IR spectral regions. The previously unpublished UV observations were obtained with the FOS on HST, while the optical data were measured with the Goldcam on the 2.1m telescope at KPNO and reported by @KH98. We also present new IR measurements obtained with the Spitzer Space Telescope with the IRS. Finally, we report on radial velocity measurements obtained with the echelle spectrograph on the 4m telescope at KPNO. These sets of observations are described separately in detail below.
Our complete list of emission line measurements is presented in Table \[fluxes\], where the first column lists the line wavelength and identification, the second column contains the relevant f value of the reddening function, and the third and fourth columns list the raw and dereddened fluxes, respectively. Columns 5 and 6 list two sets of model-predicted line strengths pertaining to the discussion in [§]{}3.2. Lines identified with bold-faced type in column 1 are used in the initial abundance analysis described in [§ ]{}3.1. Because the angular size of DdDm-1 is smaller than the size of all slits employed in the observations, we are confident that the entire flux was observed in each line within each spectral range. Thus, no adjustments were necessary in order to place all line strengths on the same scale. In the following subsections we describe individually the observations obtained within the three spectral regions. Finally, in performing the dereddening calculations we used the reddening functions of @S79 for the UV, @SM79 for the optical, and @I05 and @RL85 shortward and longward of 8 $\mu$m in the IR, respectively.
### Ultraviolet Data
DdDm-1 was observed with the Faint Object Spectrograph on the Hubble Space Telescope during 1995 October 5 as part of the Cycle 5 program GO6031. Five “H-series” gratings were used, covering the spectral range 1087-6817Å. The observations were made through the 0.9 arcsec circular aperture (post-COSTAR) with a peak-up centering the central star in the aperture. Figure \[uv\] shows the ultraviolet spectrum from 1700Å to 3250Å obtained by splicing together the archival spectra taken with the G190H(1140 sec), and G270H(480 sec) gratings. Not shown is the G130H(2270 sec) spectrum which exhibits only a strong continuum and no obvious emission lines. Cospatial optical wavelength spectra were also obtained with the G400H(90 sec) and G570H(60 sec) gratings, which permitted scaling of the UV lines to the H I Balmer lines, enabling an accurate tie-in between the UV spectra with the ground-based optical and Spitzer IR spectra in this study. All of the FOS spectra that were analyzed have been recalibrated by the POA-CALFOS pipeline developed by the ST-ECF in 2002 [@Alexov02].
The strongest emission lines in the far-UV are dielectronic recombination pairs of O III\] $\lambda$1663, and Si III\] $\lambda\lambda$1882,92 (resolved and unusually strong relative to C III\]). In the mid-UV, the dominant emission lines are the C II\] $\lambda\lambda$2325 multiplet, \[O II\] $\lambda$2470, and Mg II $\lambda\lambda$2795, 2803. Table \[fluxes\] gives the measured strengths of the important UV abundance diagnostic lines measured from these spectra. Fluxes of the UV lines were measured from gaussian fits to their profiles. We further note that the errors in the UV line strengths are purely statistical (the square root of their FWHM times the rms fluctuations of the nearby continuum) and do not include possible errors in the FOS calibrations or extinction.
### Ground-based Optical Data
The optical data were obtained at Kitt Peak National Observatory in May 1996, using the 2.1m telescope and Goldcam CCD spectrograph. The spectral range from 3700-9600 Å was covered in two parts, with overlap from 5700-6800 Å. The total blue integration time was 600 s and the red was 2400 s. The data were reduced with standard IRAF[^1] routines. Further details of the observations and reductions can be found in @KH98. The merged spectrum from KPNO is shown in Fig. \[optical\]. We point out that our measurement of the flux in H$\beta$ agrees closely with the value reported by @D03.
### Infrared Data
DdDm-1 was observed with the Infrared Spectrograph (IRS)[^2] [@H04] on the Spitzer Space Telescope in June 2006. We used the Short Low (SL 1 and SL2), the Short-High (SH) and the Long-High (LH) modules, giving coverage from 5.2-37.2 $\mu$m. Details of the observations are given in Table \[coverage\].
Spectra were extracted using SPICE, a Java tool available from the Spitzer Science Center website. Since DdDm-1 has an angular diameter of 0.6" [@Acker92], i.e. smaller than the spatial resolution of all the IRS modules, it was extracted as a point source, and we presume that we have detected all of the nebular flux. This is confirmed by the measured fluxes relative to H$\beta$ of the strongest observed H transitions (9-7 at 11.3 $\mu$m and 7-6 at 12.4 $\mu$m), which agree, within the measurement uncertainties, with the values predicted for DdDm-1’s temperature and density by the models presented in Table \[fluxes\] and discussed in [§ ]{}3.
Orders were trimmed and merged and line fluxes for DdDm-1 were measured with SMART[^3] [@Higdon04], which produces fluxes and uncertainty estimates for each line from its line-fitting routine. SMART was also able to fit a thermal continuum to the SH-LH spectrum, obtaining a temperature of 125 K, typical of thermal dust emission. Figs. \[sl1\] and \[lh\] show the IRS spectra.
Echelle Data and the Radial Velocity of DdDm-1
----------------------------------------------
DdDm-1 was observed in June 2002 using the echelle spectrograph on the Mayall 4-meter telescope at Kitt Peak. We used the T2KB 2048x2048 pixel CCD, binned 2x2. We observed in two configurations in order to cover the full spectral range: the blue configuration, which spans wavelengths between approximately 4300 Å and 7200 Å, and the red configuration, with a spectral range between 6500 Å and 9600 Å. Total exposure time was 6600 s. We used the 79-63$^\circ$ echelle grating and the 226-2 cross disperser on all nights. The reductions were done using the [*echelle*]{} package in IRAF. Though accurate flux calibration among the observed echelle orders proved impossible, we were able to extract kinematic information from the observations.
Observed and measured wavelengths for a selection of lines in the spectrum are given in Table \[radvel\]; based on these measurements we find the radial velocity of DdDm-1 to be -300.9 $\pm$ 1.4 km s$^{-1}$. We used the [*rvcorrect*]{} task in the [*astutil*]{} package in IRAF to correct for the earth’s rotation and orbital motion at the time of the observations and found a correction of -9.4 km s$^{-1}$, giving a heliocentric radial velocity of -310.3 $\pm$ 1.4 km s$^{-1}$. The radial velocity of DdDm-1 has been measured by @BC84 to be -304$\pm$ 20 km s$^{-1}$; and by @WCP05 as -317 $\pm$ 13 km s$^{-1}$; our value agrees very well with both of these, and is better constrained.
We take the opportunity here to mention that the echelle spectrum contains many forbidden iron lines that appear split. We will address the issue of DdDm-1’s expansion and ramifications for its morphology in a future paper.
ANALYSIS
========
We performed both an empirical and a numerical analysis of DdDm-1, using the line strengths reported in the previous section and listed in Table \[fluxes\]. We first employed the abundance software package ELSA [@J07], a new C program based upon a 5-level atom routine, to derive empirical electron temperatures and densities as well as ionic and elemental abundances of numerous elements. These same abundances were then used as input for detailed photoionization model calculations of DdDm-1 using the program Cloudy [@Fer98] version 07.02.00. The purpose of this numerical work was to further refine the empirical abundances to produce our final abundance set as well as to derive information about central star properties. These two steps are discussed separately below. This dual-phase approach generated a final set of elemental abundances for DdDm-1 which is very reliable and allows us to compare results from numerical and empirical methods. We now describe each of the steps in detail.
Empirical Analysis\[ea\]
------------------------
In this step, strengths of many of the emission lines in Table \[fluxes\] were entered as input in ELSA. The program then derived electron temperature and density estimates based upon temperature-sensitive or density-sensitive line sets and then calculated ionic abundances. Finally, total elemental abundances were determined through the use of ionization correction factors which account for the contributions of unobserved ions to the total. Results for electron temperatures and densities, ionic abundances, and elemental abundances derived using the empirical method are presented in Tables \[td\], \[ions\], and \[elements\], respectively.
In Table \[td\] we report the values of five electron temperatures from diagnostic emission line ratios of \[O III\], \[N II\], \[O II\], \[S II\], and \[S III\], and three values of electron density, \[S II\], \[Cl III\], and \[S III\]. (The emission lines used to calculate these values are provided in a footnote to the table.) Note that for the \[S III\] temperature the $\lambda$9532 line strength was used, since the line strength ratio of $\lambda$9532/$\lambda$9069 line exceeded the theoretical value, thereby indicating that $\lambda$9069 emission has been partially absorbed in the Earth’s atmosphere.
Columns 3-6 list temperature and density values derived by @C87, @BC84, @HKB04, and @WLB05. All of the \[O III\] temperatures agree closely with our new value. However, there is nearly a 2000 K range in the \[N II\] temperatures with larger variations still for the \[O II\], \[S II\], and \[S III\] temperatures, although note that our values for the first two are very consistent with those computed by @WLB05. At the same time all three of the electron densities which we derived agree nicely with one another as well as with the values inferred by earlier studies.
Empirical ionic abundances based upon our temperature and density values in Table \[td\] are shown in Table \[ions\]. For each ion indicated in column 1 we list the electron temperature in column 2 that was used to determine the ion abundance given in column 3. At the same time, the \[S II\] density was used for all ionic abundance calculations. Uncertainties were determined by adding in quadrature the individual contributions to uncertainty made by such things as temperature and density uncertainties as well as uncertainties in the line strengths themselves. For most ions we provide several abundance values, where each is based upon the emission line whose wavelength is indicated in parentheses in column 1. When more than one abundance is computed for an ion, the last value is a weighted mean of the values marked with an asterisk (\*), where the weight is related to the uncertainty assigned to each of the individual values for that ion. The last entry for each element is the value of the ionization correction factor that was used to compute the total elemental abundance. The ICFs were determined using the relations provided in @KH01.
For comparison purposes we have included results from @C87 and @BC84 for ions and emission lines provided in those two studies. \[An additional comparison in the cases of Ne and S will be made with the results of @D03 below.\] For the major ions there appear to be no significant discrepancies among the three studies. The situation is nearly the same for the ICFs, although the moderate variance among derived values for N suggests that the abundance of this major element may be in dispute.
Numerical Analysis\[na\]
------------------------
We next employed the program Cloudy [@Fer98], version 07.02.01 to calculate detailed photoionization models of DdDm-1 with the goal of refining our empirically derived abundances as well as inferring information about the central star properties. Cloudy uses a trial set of input parameters, whose values are determined by the investigator, and predicts emission line strengths and physical conditions for the nebula. In refining the empirical abundances, then, we followed the procedure described and used by @KH98 to study DdDm-1 previously, a routine which has proven to produce a set of robust results. The steps in the procedure are as follows:
1. Calculate a photoionization model whose output line strengths closely match the observed ones of the real nebula.
2. Derive a set of empirical abundances for the model nebula using the output line strengths and ELSA.
3. Use the model empirical abundances from 2., along with the model input abundances, to determine a correction factor for each element, where the correction factor is the ratio of the model input abundance to the model empirical abundance.
4. Multiply the empirical abundance values determined in [§]{}\[ea\] by the relevant correction factors to obtain a final set of abundances.
In calculating the model in step 1, our empirically derived abundances from Table \[elements\] and electron density from Table \[td\] were used to set the input parameter values in the first model. The predicted emission line strengths of that model were then compared with their observed counterparts, the parameter values adjusted accordingly, and a new model calculated. This process was repeated until a suitable match between theory and observation was obtained. Note that the images of DdDm-1 presented above support our use of a relatively simple density distribution for our modeling exercise, since they suggest a smooth distribution of matter and a symmetrical shape. We present the results for two successful models, 18 and 32, in Table \[fluxes\].
For model 18 the input central star flux was taken from the H-Ni grid of synthetic central star fluxes by @R02, which was calculated by assuming non-LTE hydrostatic conditions, line-blanketing, and plane-parallel geometry. The luminosity of the central star was taken to be 1000 L$_{\odot}$. The model was radiation bounded with a constant total density of 4000 cm$^{-3}$ throughout the nebula, consistent with the smooth appearance of the nebula in Figs. \[image1\] and \[image2\] and the electron density reported in Table \[td\]; the filling factor was unity. Model 18 was successful in reproducing most of the important lines (shown in bold) in Table \[fluxes\]. However, the predicted strength of \[O III\] $\lambda$4363 was somewhat higher than the observed value, while the \[S II\] $\lambda\lambda$6716,6731 line strengths were also overpredicted by the model. Numerous models were run in an effort to reduce these particular problems, but improvements in these lines came only with serious damage to other predicted line strengths; in the case of sulfur this often meant poor matches with the \[S III\] and \[S IV\] lines. An example is our attempt to reduce \[S II\] emission by truncating the nebula, i.e. making it matter bounded with the intention of reducing the volume of gas in the outer region of the nebula where large amounts of \[S II\] are produced. However, this led to a significant reduction of emission from ions such as \[O II\] and \[N II\], making the altered model an untenable solution.
Model 32 differs from 18 most significantly in the character of the central star. For this model we used a blackbody spectrum of T$_{eff}$=40,000K with a bolometric luminosity of 10$^5$ L$_{\odot}$. The other major difference was that model 32 had a filling factor of 0.5. Otherwise the density was unchanged and the abundances were very similar to those employed in model 18. For this model, the prediction of \[O III\] $\lambda$4363 is slightly better, although this time it is below rather than above the observed value. In addition, the predicted strength of \[O III\] $\lambda$5007 is below the observed level. Model 32 was primarily an attempt to achieve improved agreement in the \[S II\] lines over that found in model 18. However, in doing so the level of agreement in the near IR lines of \[S III\] and \[S IV\] at 10.5$\mu$m became worse.
The main parameters for models 18 and 32 are summarized in Table \[parameters\]. For the remainder of the analysis we will use model 18, since it includes the use of a realistic central star model spectrum, and its predicted line strengths satisfactorily match most of the important observed line strengths. In addition, this model’s H$\beta$ luminosity closely matches the observed value. The fact that this model does not reproduce a few line strengths exactly is not a problem here, since we are using the model results primarily to derive a correction factor (see points 3 and 4 above).
DISCUSSION
==========
Adopted Abundances of DdDm-1
----------------------------
The correction factors (based upon model 18) and our final abundance results for DdDm-1 are presented in column 3 of Table \[final\]. Final elemental abundances relative to H were determined by adding the log of the correction factor in column 2 to the associated value in Table \[elements\]. The uncertainties given in column 3 are statistical and based upon error propagation results calculated by ELSA from estimated line strength errors. Columns 4-8 of Table \[final\] show values for comparison purposes from @C87, @BC84, and @WLB05 for DdDm-1, @AGS05 for the sun, and @E98 for the Orion Nebula, respectively. The last column provides a comparison of our abundances in column 3 with solar abundances, using the usual bracket notation defined in the footnote. Note that since neither Si nor Fe is currently included in ELSA, our final values for these elements correspond to the model input values required to reproduce several of the measured line strengths of these elements.
The fact that all of the correction factor values in column 2 exceed unity suggests that the empirical method of abundance determinations tends to underestimate the abundance of each element. This is possibly caused by insufficient corrections for unobserved ionization stages by the ionization correction factors used in the empirical method. Furthermore, the magnitude of the offsets is on the order of 0.10-0.15 dex, or roughly the size of the uncertainties which we established for the final abundances.
There is good agreement among the four studies of DdDm-1 represented in Table \[final\] for He/H, O/H, Ne/H, Si/H, and S/H, where all ratios but the first one represent alpha elements, and therefore their abundances are expected to exhibit lockstep behavior. We note that our value for O/H tends to be slightly higher than the others, although even this difference is likely explained by uncertainties. At the same time our value for Fe/H agrees reasonably well with that published by @C87.
While our N/H abundance is within a factor of 2 of those found in the other two studies, C/H is markedly discrepant among the four determinations, where the range exceeds two orders of magnitude. Our value is more than a factor of two lower than the recently measured level published by @WLB05. We note that @BC84 determined their C abundance using the recombination line C II $\lambda$4267, and it is often the case that abundances derived from recombination lines yield significantly higher values than abundances determined from collisionally excited lines ([@WLB05]. Unlike Barker & Cudworth, we did not detect 4267 in our spectrum, nor apparently did @WLB05. However, an upper limit for the $\lambda$4267 line strength in our spectrum is 0.03 (H$\beta$=100) or about 1/10 the strength reported by @BC84. The correction for reddening is insignificant. Running this value through ELSA produces an upper limit on the abundance ratio of C$^{+2}$/H$^+$ of 3.1E-5, or about 1/10 the level of this ratio determined by @BC84. In any case, our C/H was inferred directly from our photoionization models, using the 1909 line as a constraint, and thus we consider it somewhat uncertain. In terms of C/O, our result and that of @C87 and @WLB05 suggest that DdDm-1 is a C-poor (or O-rich) system (C/O$<$1), while @BC84’s value implies a C-rich (O-poor) system (C/O$>$1).
@D03 used infrared and optical line measurements acquired at the IRTF and the 2.7m telescope at McDonald Observatory, respectively, to study abundances of S and Ne in DdDm-1. Table \[dinerstein\] provides a comparison of their results with ours. The agreement between the two groups is remarkably good, with the exception of the factor of 4 discrepancy in the case of Ne$^+$/H$^+$. We are currently unable to explain this disagreement, as our measured strength of \[Ne II\] 12.8$\mu$m agrees closely with the value reported by @D03, as do our derived electron temperature and density values. We also employed a collision strength of 0.318 [@GMB01] which agrees closely with their value of 0.306 taken from @JK87. On the other hand, the close agreement in the case of sulfur is strong evidence that the ionization stages above S$^{+3}$ in DdDm-1 are relatively unpopulated.
Fig. \[abun\_sun\] presents a comparison of the elemental abundances of DdDm-1 and three other well-studied halo PNe, BB1, H4-1, and K648 \[each was analyzed by @HKB04, from which the abundances in the figure are taken\], all normalized to solar values from @AGS05. For clarity, uncertainties are not plotted; they generally have values of 0.10 dex or less.
We can see very clearly that all four PNe are metal-poor, since the offsets for all of the alpha elements (O through Ar) are negative by significant amounts. However, it is interesting that for any one object these offsets do not have the same value for the all of the alpha elements as would be expected from nucleosynthesis theory[^4]. For example we see in column 8 of Table \[final\] that for DdDm-1 these values vary from -0.25 for Ar/H to to -1.36 for Si/H. Some of the variation can certainly be explained by uncertainty, particularly in the cases of Ar and Cl where line strengths are weak and only one or two ionization states have observable lines. It is likely that some of the underabundance of Si in particular is the result of dust depletion, as this element is highly refractory [@SS96]. In fact its offset from solar is essentially identical to that of Fe, another refractory element. However, dust cannot explain the large offset differences between O and Ne for H4-1 and BB1, each long known for their unusual Ne abundances relative to O. In addition there are large differences in S offsets for BB1, H4-1, and K648, perhaps related to the S anomaly discussed below. Finally, the Ar offset is well above its expected value for K648 but well below the expected one for BB1. It is unlikely that any of these peculiar offsets can be explained by dust formation. Rather they may be related to the increased scatter often seen in low metallicity halo stars \[see the data compilation in Fig. 1.2 of @M03\]. It is hoped that with future discoveries of additional halo PNe this situation will become better understood.
We also see in Fig. \[abun\_sun\] that C/H and C/O are much lower in DdDm-1 than in the other three halo PNe, with significantly subsolar values for both ratios. In contrast, the C/O offset is positive for the other three objects, with a very high value in the case of BB1.
As alpha elements, S and O are expected to evolve in lockstep, as we explained above. @HKB04 found that while this is true when H II regions and blue compact galaxies are used to probe the abundances of these two elements, the expectation is often unmet in the case of PNe[^5] These authors found marked scatter in S abundances for PNe of roughly the same O abundance. In addition, S abundances were regularly determined to be below the expected level for a given O abundance by 0.3 dex on average. This unexpected finding was referred to as the sulfur anomaly by @HKB04. In the specific case of DdDm-1, however, @HKB04 found that its S abundance was close to the value expected from its O abundance.
We now revisit this situation with our updated S and O abundances obtained here. Employing our O abundance in Table \[elements\] along with a least squares fit to measurements of 12+log(S/H) versus 12+log(O/H) in H II regions in M101 [@KBG03] and blue compact galaxies [@IT99] to estimate the expected S abundance[^6], we obtain 12+log(S/H)=6.60 as the expected value for S. Our corresponding measured value is 6.47, yielding a sulfur deficit of 6.60-6.47=0.13, which is close to the uncertainty in the S abundance and less than the typical value of 0.3 for the sulfur deficit found by @HKB04. We conclude that the S and O abundances associated with DdDm-1 are consistent with the expected lockstep behavior for these two elements.
It has been suggested by @PBS06 that the sulfur anomaly found by @HKB04 could be the result of dust formation. In particular, S may be removed by the formation of compounds such as MgS and FeS in those PNe exhibiting large S deficits. In fact this would be expected to occur more readily in C-rich environments, where sulfide formation is favored. Interestingly, the low C abundance which we find for DdDm-1, with C/O$<$1, implies that oxygen-rich chemistry exists in the nebula of DdDm-1 and that dust composition should be dominated by silicates and other oxygen-rich species and not sulfides. Thus, if the sulfur anomaly is indeed related to sulfide formation, then the small S deficit that we observe in DdDm-1 would be expected as the result of its low value for C/O.
Further evidence for the low C/O ratio in DdDm-1 is the absence of polycyclic aromatic hydrocarbon (PAH) emission bands in the IRS SL spectra (Fig. \[sl1\]).Ê The IDL routine PAHFIT [@SD07] was applied to the SL spectra, and no evidence was found of the PAH emission features near 3.3, 6.2, 7.7, 8.7, and 11.3 microns, which are strong in the IR spectra of many PNe [@CB05].Ê In their study of ISO spectra of 43 PNe, Cohen and Barlow found that 17 objects exhibited strong PAH emission and that the 7.7 and 11.3 micron PAH band strengths relative to the total infrared luminosity are correlated with the nebular C/O ratio.Ê As is evident from our IRS spectra in Figures \[sl1\] and \[lh\], all of the emission features seen in DdDm-1 are nebular lines with no broad PAH emission evident.Ê However, a strong IR continuum, apparently due to warm dust emission, is evident.Ê Blackbody fits to the LH spectrum using SMART give a good fit for a temperature of 125(+/- 7) K.Ê This is comparable to other PNe studied by ISO and Spitzer.Ê Finally, no significant silicate absorption bands at 9.7 and 18 microns are visible in the IR spectra.Ê These findings further support our result for a low C/O ratio in DdDm-1, as well as the liklihood that Si/O is low as well.
Central Star Properties
-----------------------
The central star temperature which we infer for DdDm-1 is 55,000 K with a luminosity of 10$^3$ L$_{\odot}$, based upon our preferred model 18 (see Table \[parameters\]). Recall that model 18 employed a stellar spectrum from @R02 which was calculated by assuming realistic conditions of high gravity and low metallicity for the central star. Our effective temperature is somewhat higher than the value of 40,000 ($\pm$5,000) K determined by @PTPR92 and 45,600 K estimated from models by @HHM97. This situation may reflect our use of model stellar fluxes calculated specifically for a low metallicity regime.
These derived central star properties along with the O abundance for DdDm-1 can be compared with AGB star model tracks calculated by @VW94 in order to infer a progenitor mass. Data in their Fig. 7 suggests that the central star of DdDm-1 is a He-burning object which had a main sequence mass of $<$1 M$_{\odot}$. This relatively low mass is consistent with the idea that DdDm-1 is associated with an old stellar population such as that found in the halo.
SUMMARY & CONCLUSIONS
=====================
We have reported on new IR and UV spectra of the halo planetary nebula DdDm-1 obtained with the Spitzer Space Telescope with the IRS and Hubble Space Telescope with the FOS, respectively. By combining these new data with existing optical measurements, the nebular abundances of He, C, N, O, Ne, Si, S, Cl, Ar, and Fe were determined. The abundance analysis included the computation of detailed photoionization models which made use of an input stellar atmosphere whose characteristics were consistent with a central star of low metallicity. We also present new echelle data for DdDm-1. We have found the following:
1. We present what we believe to be the first resolved image of DdDm-1. Reconstructed imagery taken with the HST in 1993 indicated that the morphology of DdDm-1 can be described as two orthogonal elliptical components in the central part surrounded by an extended halo. The extent of the emission is somewhat larger than was previously reported in the literature.
2. We present new UV, IR, and echelle data, where the first two sets were acquired using HST and Spitzer, respectively.
3. Our new determinations of the Si and Fe abundances indicate that their levels are far below those expected from the metallicity of DdDm-1, possibly indicating that their gas phase abundances have been depleted by dust formation.
4. We determine that C/O $<$ 1, although our C/H abundance is uncertain. However, a C/O ratio below unity suggests that the chemistry of the nebula is O-rich in character. Thus, sulfides should be absent any dust that has formed in the environment of DdDm-1.
5. We find that the abundance of S$^{+3}$ which we determined directly from our new \[S IV\] 10.5$\mu$m measurement agrees closely with another modern one by @D03 and is consistent with the level predicted by the value of the ICF obtained when only optical lines of \[S II\] and \[S III\] are used. The small abundance of S$^{+3}$ that we derive indicates that DdDm-1 is a relatively low excitation nebula when compared with other PNe.
6. Our total gas-phase S abundance for DdDm-1 is consistent with the value expected from its O abundance, under the assumption of lockstep behavior between these two elements. Thus DdDm-1 has a negligible S deficit. On the other hand, if the large S deficits observed in other PNe are indeed due to sulfide formation in a C-rich environment as others have suggested, then the small S deficit of DdDm-1 is entirely consistent with its C-poor properties.
7. For the central star we find that T$_{eff}$=55,000 K and L=1000 L$_{\odot}$. Comparing the star with theoretical model tracks suggests that it is a He-burning object with a mass of less than 1 M$_{\odot}$.
8. The heliocentric radial velocity of DdDm-1 is -310.3 $\pm$ 1.4 km s$^{-1}$.
All four authors thank their respective institutions for travel support for attending IAU Symposium 234 on planetary nebulae in spring, 2006, where we had the opportunity to meet and discuss various problems surrounding this and related studies of target objects. RJD and KBK wish to thank Rice graduate student Greg Brunner for applying PAHFIT and SMART blackbody analyses to our Spitzer IRS spectra. RBCH acknowledges fruitful discussions with Angela Speck about dust formation in PNe. KBK thanks R. Gruendl and Y.-H. Chu for considerable advice and assistance with the echelle observations. RBCH is also grateful for partial support of his research by NSF grant AST 03-07118 to the University of Oklahoma.
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[lcr@lr@lcc]{} He II $\lambda$1640 & 1.136 & 2.21 &: & 2.58 & $\pm$1.56: & 0.040 & 0.123\
O III\] $\lambda$1662 & 1.129 & 5.76 & & 6.72 & $\pm$3.58 & 5.18 & 2.01\
N III\] $\lambda$1750 & 1.119 & 1.10 &:: & 1.28 & $\pm$0.93:: & 18.4 & 8.0\
Si III\] $\lambda$1887 & 1.200 & 4.95 & & 5.84 & $\pm$3.27 & 16.1 & 9.85\
[**C III\] $\lambda$1909**]{} & 1.229 & 11.0 & & 13.0 & $\pm$7.44 & 8.21 & 7.60\
$\lambda$2470 & 1.025 & 9.58 & & 11.0 & $\pm$5.43 & 12.1 & 10.9\
[**$\lambda$3727**]{} & 0.292 & 103 & & 107 & $\pm$24 & 116 & 111\
He II + H11 $\lambda$3770 & 0.280 & 3.57 & & 3.71 & $\pm$0.81 & 3.90 & 4.04\
He II + H10 $\lambda$3797 & 0.272 & 4.50 & & 4.67 & $\pm$1.01 & 5.22 & 5.39\
He II + H9 $\lambda$3835 & 0.262 & 6.76 & & 7.01 & $\pm$1.49 & 7.23 & 7.44\
[**$\lambda$3869**]{} & 0.252 & 29.4 & & 30.4 & $\pm$6.40 & 32.3 & 21.3\
He I + H8 $\lambda$3889 & 0.247 & 19.0 & & 19.6 & $\pm$4.09 & 18.7 & 12.7\
$\lambda$3968 & 0.225 & 11.3 & & 11.7 & $\pm$5.58 & 9.73 & 6.42\
H$\epsilon$ $\lambda$3970 & 0.224 & 15.6 & & 16.0 & & 15.81 & 16.2\
He I + $\lambda$4008 & 0.214 & 0.591 & & 0.609 & $\pm$0.120 & &\
He I + He II $\lambda$4026 & 0.209 & 2.12 & & 2.18 & $\pm$0.43 & 2.23 & 2.12\
$\lambda$4046 & 0.203 & 0.102 &:: & 0.105 & $\pm$0.055:: & &\
$\lambda$4071 & 0.196 & 2.67 & & 2.74 & $\pm$0.53 & 3.79 & 1.62\
H$\delta$ $\lambda$4101 & 0.188 & 25.4 & & 26.1 & $\pm$4.95 & 25.8 & 26.3\
He I $\lambda$4121 & 0.183 & 0.227 &:: & 0.233 & $\pm$0.122:: & 0.273 & 0.242\
He I $\lambda$4144 & 0.177 & 0.280 &:: & 0.287 & $\pm$0.150:: & 0.321 & 0.303\
C III $\lambda$4167 & 0.170 & 0.104 & & 0.106 & $\pm$0.020 & &\
H$\gamma$ $\lambda$4340 & 0.124 & 47.7 & & 48.5 & $\pm$8.32 & 47.0 & 47.2\
[**$\lambda$4363**]{} & 0.118 & 5.07 & & 5.15 & $\pm$0.88 & 8.03 & 3.87\
He I $\lambda$4388 & 0.112 & 0.607 & & 0.616 & $\pm$0.104 & 0.581 & 0.552\
He I $\lambda$4472 & 0.090 & 4.91 & & 4.97 & $\pm$0.81& 4.81 & 4.57\
$\lambda$4606 & 0.056 & 0.139 &:: & 0.140 & $\pm$0.072:: & 0.126 & 0.121\
$\lambda$4658 & 0.043 & 2.38 & & 2.39 & $\pm$0.36 & 2.14 & 2.05\
$\lambda$4702 & 0.032 & 0.793 & & 0.796 & $\pm$0.118 & 0.733 & 0.705\
He I + $\lambda$4711 & 0.030 & 0.793 & & 0.796 & $\pm$0.118 & 1.03 & 0.981\
$\lambda$4734 & 0.024 & 0.312 & & 0.313 & $\pm$0.046 & 0.247 & 0.239\
$\lambda$4740 & 0.023 & 0.166 & & 0.167 & $\pm$0.024 & 0.193 & 0.134\
$\lambda$4755 & 0.019 & 0.398 & & 0.400 & $\pm$0.058 & 0.391 & 0.375\
$\lambda$4770 & 0.015 & 0.280 & & 0.281 & $\pm$0.041 & 0.246 & 0.236\
$\lambda$4778 & 0.013 & 0.149 & & 0.149 & $\pm$0.021 & 0.117 & 0.114\
H$\beta$ $\lambda$4861 & 0.000 & 100 & & 100 & & 100 & 100\
$\lambda$4881 & -0.012 & 0.989 & & 0.987 & $\pm$0.137 & 0.694 & 0.683\
He I $\lambda$4922 & -0.021 & 1.20 & & 1.20 & $\pm$0.16 & 1.23 & 1.16\
$\lambda$4959 & -0.030 & 150 & & 149 & $\pm$20 & 163 & 108\
[**$\lambda$5007**]{} & -0.042 & 458 & & 455 & $\pm$61 & 490 & 326\
Si II $\lambda$5056 & -0.053 & 5.20(-2) &:: & 5.16 & $\pm$2.62(-2):: & &\
$\lambda$5085 & -0.060 & 0.436 &: & 0.432 & $\pm$0.135: & 0.050 & 0.049\
$\lambda$5159 & -0.077 & 0.717 & & 0.709 & $\pm$0.092 & &\
$\lambda$5192 & -0.085 & 0.914 & & 0.903 & $\pm$0.116 & 0.167 & 0.104\
$\lambda$5199 & -0.086 & 0.388 & & 0.383 & $\pm$0.049 & 0.427 & 0.048\
$\lambda$5270 & -0.102 & 1.18 & & 1.16 & $\pm$0.15 & 1.14 & 1.11\
$\lambda$5518 & -0.157 & 0.217 &: & 0.212 & $\pm$0.066: & 0.161 & 0.128\
$\lambda$5538 & -0.161 & 0.271 &: & 0.265 & $\pm$0.082: & 0.190 & 0.152\
$\lambda$5755 & -0.207 & 1.43 & & 1.39 & $\pm$0.17 & 1.67 & 1.44\
[**He I $\lambda$5876**]{} & -0.231 & 14.6 & & 14.1 & $\pm$1.76 & 14.2 & 12.9\
O I $\lambda$6046 & -0.265 & 0.103 &:: & 9.93 & $\pm$5.03(-2):: & &\
$\lambda$6300 & -0.313 & 2.47 & & 2.37 & $\pm$0.32 & 2.95 & 0.425\
+ He II $\lambda$6312 & -0.315 & 1.84 & & 1.76 & $\pm$0.24 & 2.55 & 1.21\
Si II $\lambda$6347 & -0.322 & 8.45(-2) & & 8.09 & $\pm$1.09(-2) & &\
$\lambda$6364 & -0.325 & 0.861 & & 0.824 & $\pm$0.112 & 0.940 & 0.136\
$\lambda$6548 & -0.358 & 17.0 & & 16.2 & $\pm$2.29 & 20.7 & 18.8\
H$\alpha$ $\lambda$6563 & -0.360 & 296 & & 282 & $\pm$1 & 289 & 287\
[**$\lambda$6584**]{} & -0.364 & 54.0 & & 51.4 & $\pm$7.34 & 61.0 & 55.4\
He I $\lambda$6678 & -0.380 & 3.84 & & 3.65 & $\pm$0.53 & 3.30 & 3.08\
[**$\lambda$6716**]{} & -0.387 & 3.02 & & 2.86 & $\pm$0.42 & 6.73 & 2.00\
[**$\lambda$6731**]{} & -0.389 & 5.13 & & 4.86 & $\pm$0.72 & 10.8 & 3.38\
$\lambda$7006 & -0.433 & 9.33(-2) & & 8.79 & $\pm$1.40(-2) & &\
He I $\lambda$7065 & -0.443 & 8.01 & & 7.54 & $\pm$1.22 & 8.98 & 10.8\
$\lambda$7136 & -0.453 & 8.10 & & 7.61 & $\pm$1.25 & 10.5 & 8.15\
$\lambda$7155 & -0.456 & 9.95(-2) &: & 9.35 & $\pm$3.06(-2): & &\
+ $\lambda$7170 & -0.458 & 3.59(-2) &: & 3.37 & $\pm$1.10(-2): & 0.006 & 0.004\
O I $\lambda$7255 & -0.471 & 9.52(-2) & & 8.92 & $\pm$1.51(-2) & &\
He I $\lambda$7281 & -0.475 & 0.927 & & 0.869 & $\pm$0.148 & 1.02 & 0.854\
[**$\lambda$7324**]{} & -0.481 & 16.4 & & 15.3 & $\pm$2.63 & 14.4 & 13.0\
$\lambda$7378 & -0.489 & 4.85(-2) &: & 4.54 & $\pm$1.51(-2): & &\
$\lambda$7388 & -0.490 & 2.25(-2) & & 2.10 & $\pm$0.37(-2) & &\
C III $\lambda$7578 & -0.516 & 5.30(-3) & & 4.94 & $\pm$0.90(-3) & &\
$\lambda$7531 & -0.510 & 1.73(-2) &: & 1.62 & $\pm$0.54(-2): & 0.101 & 0.084\
$\lambda$7751 & -0.539 & 1.95 & & 1.81 & $\pm$0.34 & 2.53 & 1.97\
$\lambda$7890 & -0.556 & 8.25(-2) &: & 7.65 & $\pm$2.62(-2): & &\
P16 $\lambda$8467 & -0.618 & 0.394 & & 0.362 & $\pm$0.077 & 0.496 & 0.540\
$\lambda$8481 & -0.620 & 0.134 & & 0.123 & $\pm$0.026 & 0.008 & 0.006\
+ P15 $\lambda$8501 & -0.622 & 0.525 & & 0.482 & $\pm$0.103 & 0.584 & 0.630\
P14 $\lambda$8545 & -0.626 & 0.620 & & 0.569 & $\pm$0.123 & 0.512 & 0.537\
$\lambda$8579 & -0.629 & 0.142 & & 0.130 & $\pm$0.028 & 0.112 & 0.076\
P13 $\lambda$8598 & -0.631 & 0.706 & & 0.648 & $\pm$0.141 & 0.634 & 0.665\
$\lambda$8617 & -0.633 & 0.131 & & 0.120 & $\pm$0.026 & 0.288 & 0.077\
P12 $\lambda$8665 & -0.637 & 0.899 &:: & 0.824 & $\pm$0.442:: & 0.794 & 0.831\
P11 $\lambda$8750 & -0.644 & 1.14 & & 1.04 & $\pm$0.23 & 1.01 & 1.06\
P10 $\lambda$8863 & -0.654 & 1.41 & & 1.29 & $\pm$0.29 & 1.32 & 1.37\
P9 $\lambda$9015 & -0.666 & 1.99 & & 1.82 & $\pm$0.42 & 1.76 & 1.83\
[**$\lambda$9069**]{} & -0.670 & 20.7 & & 18.9 & $\pm$4.35 & 20.3 & 11.8\
P8 $\lambda$9228 & -0.610 & 3.40 & & 3.13 & $\pm$0.66 & 2.43 & 2.52\
[**$\lambda$9532**]{} & -0.632 & 52.4 & & 48.0 & $\pm$10.47 & 50.3 & 29.3\
P7 $\lambda$9546 & -0.633 & 2.95 & & 2.71 & $\pm$0.59 & 3.49 & 3.61\
HI 9-6 $\lambda$5.91$\mu$m & -0.988 & 0.410 & & 0.358 & $\pm$0.123 & 0.071 & 0.076\
$\lambda$6.99$\mu$m & -0.990 & 0.770 & & 0.672 & $\pm$0.232 & 0.288 & 0.532\
HI 6-5 $\lambda$7.45$\mu$m & -0.990 & 4.48 & & 3.91 & $\pm$1.35 & 0.148 & 0.160\
$\lambda$8.99$\mu$m & -0.959 & 5.25 & & 4.60 & $\pm$1.53 & 5.41 & 4.82\
[**$\lambda$10.52$\mu$m**]{} & -0.959 & 10.1 & & 8.86 & $\pm$2.95 & 7.59 & 3.52\
HI 9-7 $\lambda$11.31$\mu$m & -0.970 & 0.338 & & 0.296 & $\pm$0.100 & 0.269 & 0.291\
HI 7-6 $\lambda$12.37$\mu$m & -0.980 & 0.996 & & 0.871 & $\pm$0.297 & 0.839 & 0.904\
$\lambda$12.80$\mu$m & -0.983 & 7.85 & & 6.86 & $\pm$2.35 & 1.72 & 5.24\
$\lambda$15.50$\mu$m & -0.985 & 29.6 & & 25.9 & $\pm$8.87 & 16.7 & 14.7\
$\lambda$18.70$\mu$m & -0.981 & 13.3 & & 11.6 & $\pm$3.97 & 17.5 & 11.5\
$\lambda$22.93$\mu$m & -0.987 & 1.84 & & 1.61 & $\pm$0.55 & 1.34 & 1.53\
$\lambda$33.48$\mu$m & -0.993 & 5.74 & & 5.01 & $\pm$1.73 & 6.13 & 3.81\
$\lambda$34.81$\mu$m & -0.993 & 0.700 & & 0.611 & $\pm$0.211 & 0.762 & 0.229\
$\lambda$36.01$\mu$m & -0.993 & 3.50 & & 3.05 & $\pm$1.06 & 1.39 & 1.21\
c & & 0.06 &&&&&\
H$\alpha$/H$\beta$ && 2.82 &&&&&\
log F$_{H\beta}$ & & -11.78 & & & &-11.63 & -9.71\
[lcc]{} SL2 & 5.2-8.7 & 960\
SL1 & 7.4-14.5 & 240\
SH & 9.9-19.6 & 240\
LH & 18.7-37.2 & 3600
[lccc]{}
& 4358.81 & 4363.23 & -303.9\
H$\gamma$ & 4336.12 & 4340.47 & -300.7\
He I & 4467.01 & 4471.5 & -301.2\
H$\beta$ & 4856.45 & 4861.33 & -301.2\
& 4953.95 & 4958.91 & -300.1\
& 5001.84 & 5006.84 & -299.6\
& 5748.84 & 5754.6 & -300.3\
He I & 5869.76 & 5875.66 & -301.2\
& 6541.51 & 6548.1 & -301.9\
H$\alpha$ & 6556.21 & 6562.77 & -299.9\
& 6576.85 & 6583.5 & -303.0\
& 6709.75 & 6716.44 & -298.8\
& 6724.09 & 6730.82 & -300.0\
He I & 7058.15 & 7065.25 & -301.5\
average measured &&& -301$\pm$1.4\
correction to heliocentric &&& -9.4\
[**final radial velocity**]{} &&& -310.3 $\pm$1.4\
[lccccc]{} T$_{[O III]}$ & 12060($\pm$690) & 11800($\pm$800) & 12100($\pm$600) & 11700 & 12300\
T$_{[N II]}$ & 12940$\pm$1236) & 11000($\pm$1200) & 12800($\pm$1000) & 11400 & 12980\
T$_{[O II]}$ & 15830($\pm$9500) & &11000($\pm$2000) & 10100 & 16110\
T$_{[S II]}$ & 15650($\pm$13760) & & 18000($\pm$6000) & 7900 & 13850\
T$_{[S III]}$ & 12950($\pm$1392) & & 11500($\pm$3000) & 12700 &\
N$_e$$_{[S II]}$ & 4092($\pm$2491) & 4400 & 3200($\pm$1300) & 4000 &\
N$_e$$_{[Cl III]}$ & 3973($\pm$4427) & & & &\
N$_e$$_{[S III]}$ & 3293($\pm$743) & & & &\
[lcccc]{} He$^{+}$ & \[O III\] & 9.34$\pm$1.28(-2) & 0.11 & 0.084\
He$^{+2}$ & \[O III\] & 3.53$\pm$2.14(-4) & $<$5.0(-4) & $<$3.0(-4)\
icf(He) & & 1.00 & &\
\
O$^{0}$(6300) & \[N II\] & 2.05$\pm$0.60(-6) & 4.6(-6) &\
O$^{0}$(6363) & \[N II\] & 2.23$\pm$0.65(-6) & &\
O$^{0}$ & wm & 2.10$\pm$0.60(-6) & &\
O$^{+}$(3727) & \[N II\] & 2.64$\pm$1.10(-5) & 4.7(-5) & 3.0(-5)\
O$^{+}$(7325) & \[N II\] & 3.23$\pm$1.36(-5) & &\
O$^{+}$ & wm & 2.72$\pm$1.00(-5) & &\
O$^{+2}$(5007) & \[O III\] & 8.88$\pm$2.27(-5) & 8.96(-5) & 7.6(-5)\
O$^{+2}$(4959) & \[O III\] & 8.41$\pm$1.72(-5) & &\
O$^{+2}$(4363) & \[O III\] & 8.88$\pm$2.27(-5) & &\
O$^{+2}$ & wm & 8.76$\pm$2.08(-5) & &\
icf(O) & & 1.00 & 1.0 & 1.0\
\
N$^{+}$(6584) & \[N II\] & 6.10$\pm$1.61(-6) & 8.00(-6) & 5.6(-6)\
N$^{+}$(6548) & \[N II\] & 5.65$\pm$1.33(-6) & &\
N$^{+}$(5755) & \[N II\] & 6.10$\pm$1.61(-6) & &\
N$^{+}$ & wm & 5.99$\pm$1.51(-6) & &\
N$^{+2}$(1751) & \[O III\] & 4.08$\pm$2.92(-6) & &\
icf(N) & & 4.22 & 2.95 & 3.53\
\
C$^{+2}$(1909) & \[O III\] & 8.53$\pm$4.63(-6) & $<$6.9(-6) & 4.5(-4)\
icf(C) & & 1.31 & & 1.5\
\
Ne$^{+}$(12.8$\mu m$) & \[O III\] & 8.46$\pm$3.05(-6) & &\
Ne$^{+2}$(3869) & \[O III\] & 1.54$\pm$0.37(-5) & 1.3(-5) & 1.5(-5)\
Ne$^{+2}$(3967) & \[O III\] & 1.96$\pm$0.89(-5) & &\
Ne$^{+2}$(15.5$\mu m$) & \[O III\] & 1.58$\pm$0.57(-5) & &\
Ne$^{+2}$(36.0$\mu m$) & \[O III\] & 2.28$\pm$0.84(-5) & &\
Ne$^{+2}$ & wm & 1.60$\pm$0.33(-5) & &\
icf(Ne) & & 1.31 & 1.5 & 1.39\
\
S$^{+}$ (6717+6731)& \[N II\] & 1.83$\pm$0.71(-7) & 2.7(-7) & 1.6(-7)\
S$^{+}$(6716) & \[N II\] & 1.83$\pm$0.71(-7) & &\
S$^{+}$(6731) & \[N II\] & 1.83$\pm$0.70(-7) & &\
S$^{+}$ & wm & 1.50$\pm$1.64(-7) & &\
S$^{+2}$(9532) & \[S III\] & 1.73$\pm$0.55(-6) & &\
S$^{+2}$(6312) & \[S III\] & 1.73$\pm$0.55(-6) & 2.56(-6) & 2.3(-6)\
S$^{+2}$(18.7$\mu m$) & \[S III\] & 1.38$\pm$0.55(-6) & &\
S$^{+2}$(33.4$\mu m$) & \[S III\] & 1.59$\pm$1.05(-6) & &\
S$^{+2}$ & wm & 1.66$\pm$0.54(-6) & &\
S$^{+3}$(10.5$\mu m$) & \[O III\] & 2.10$\pm$0.78(-7) & &\
icf(S) & & 1.17 & 1.2 & 1.17\
\
Ar$^{+}$(7.0$\mu m$) & \[N II\] & 5.28$\pm$1.93(-8) & &\
Ar$^{+2}$(7135) & \[O III\] & 4.72$\pm$1.11(-7) & 4.9(-7) & 3.2(-7)\
Ar$^{+2}$(7751) & \[O III\] & 4.66$\pm$1.21(-7) & &\
Ar$^{+2}$(9.0$\mu m$) & \[O III\] & 4.36$\pm$1.54(-7) & &\
Ar$^{+2}$ & wm & 4.59$\pm$1.19(-7) & &\
Ar$^{+3}$(4740) & \[O III\] & 1.85$\pm$0.36(-8) & $<$4.0(-8) &\
Ar$^{+4}$(7005) & \[O III\] & 1.18$\pm$0.29(-8) & &\
icf(Ar) & & 1.31 & 1.5 & 1.5\
\
Cl$^{+}$(8578) & \[N II\] & 8.62$\pm$2.40(-9) & &\
Cl$^{+2}$(5517) & \[S III\] & 1.81$\pm$0.75(-8) & &\
Cl$^{+2}$(5537) & \[S III\] & 1.79$\pm$0.64(-8) & &\
Cl$^{+2}$ & wm & 1.80$\pm$0.54(-8) & &\
icf(Cl) & & 1.00 & &\
\
[lc]{} He/H & 10.97\
O/H & 8.06\
N/H & 7.40\
C/H & 7.05\
Ne/H & 7.32\
S/H & 6.33\
S(w/\[S IV\])/H & 6.31\
Ar/H & 5.81\
Cl/H & 4.42\
[lcc]{} He/H & 11.0 & 11.0\
C/H & 6.55 & 6.85\
N/H & 7.56 & 7.56\
O/H & 8.06 & 8.06\
Ne/H & 7.16 & 7.30\
S/H & 6.45 & 6.25\
Cl/H & 4.25 & 4.25\
Ar/H & 5.81 & 5.81\
Fe/H & 6.11 & 6.11\
$T_{eff}$ (K) & 55,000 & 40,000\
log L/L$_{\odot}$ (ergs s$^{-1}$) & 3.0 & 5.0\
Total Density (cm$^{-3}$) & 3980 & 3980\
Radius (pc) & 0.032 & 0.032\
Filling Factor & 1 & 0.5\
[lcccccccc]{} He/H & 1.16 & 11.03$\pm$.056 & 11.00 & 11.02 & 10.95 & 10.93 & 10.99 & +0.10\
C/H & & 6.55$\pm$.20 & 7.14 & 8.83 & 6.91 & 8.39 & 8.39 & -1.84\
N/H & 1.64 & 7.62$\pm$.11 & 7.40 & 7.30 & 7.31 & 7.78 & 7.78 & -0.16\
O/H & 1.40 & 8.20$\pm$.085 & 8.15 & 8.04 & 8.05 & 8.66 & 8.63 & -0.46\
Ne/H & 1.13 & 7.37$\pm$.080 & 7.30 & 7.32 & 7.24 & 7.84 & 7.89 & -0.47\
Si/H & & 6.15$\pm$.13 & 6.30 & & & 7.51 & & -1.36\
S/H & 1.34 & 6.46$\pm$.12 & 6.46 & 6.46 & 6.35 & 7.14 & 7.17 & -0.68\
Cl/H & 1.23 & 4.51$\pm$.10 & & & 4.69 & 5.50 & 5.33 & -0.99\
Ar/H & 1.32 & 5.93$\pm$.096 & & 5.68 & 5.16 & 6.18 & 6.80 & -0.25\
Fe/H & & 6.10$\pm$.053 & 6.32 & & & 7.45 & 6.41 & -1.35\
C/O & & -1.65$\pm$.20 & -1.01 & +0.79 & -1.14 & -0.27 & -0.24 & -1.38\
N/O & & -0.58$\pm$.10 & -0.75 & -0.74 & -0.74 & -0.88 & -0.85 & +0.30\
O/Fe & & +2.1$\pm$.14 & +1.83 & & & +1.21 & +2.22 & +0.89\
[lcc]{} S$^+$/H$^+$ & $2.2 \times 10^{-7}$ & $1.5 \times 10^{-7}$\
S$^{+2}$/H$^+$ & $1.63 \times 10^{-6}$ & $1.66 \times 10^{-6}$\
S$^{+3}$/H$^+$ & $2.3 \times 10^{-7}$ & $2.1 \times 10^{-7}$\
Total S/H & $2.1 \times 10^{-6}$ & $2.9 \times 10^{-6}$\
Ne$^+$/H$^+$ & $2.1 \times 10^{-6}$ & $8.46 \times 10^{-6}$\
Ne$^{+2}$/H$^+$ & $1.17 \times 10^{-5}$ & $1.6 \times 10^{-5}$\
Total Ne/H & $1.4 \times 10^{-5}$ & $2.3 \times 10^{-5}$
![Grey-scaled surface brightness image of DdDm-1 from an archival HST WFPC1 F675W 40 sec exposure taken in 1993. North is up and east is to the left, with image dimensions of 3$\arcsec$ x 3$\arcsec$. The image quality has been partly restored using Lucy-Richardson techniques based on a theoretical PSF for the camera/filter used. Details of the processing and morphology are given in the text.[]{data-label="image1"}](f1.eps){width="4in"}
![Same image as in Fig. \[image1\] but now overlaid with contours of surface brightness and the FK5 coordinate system.[]{data-label="image2"}](f2.eps){width="4in"}
![Plot of the ultraviolet spectrum of DdDm-1 from archival HST FOS data. The original spectra (with the G190H and G270H gratings) have been smoothed by a 3-point boxcar.[]{data-label="uv"}](f3.eps){width="6in"}
![Merged spectrum of DdDm-1 from KPNO observations. Note the coverage from \[O II\] $\lambda$3727 to \[S III\] $\lambda\lambda$9069,9532.[]{data-label="optical"}](f4.eps){width="6in"}
![IRS merged SL2-SL1 spectrum[]{data-label="sl1"}](f5.eps){width="4in"}
![IRS merged SH-LH spectrum[]{data-label="lh"}](f6.eps){width="4in"}
![Plot of \[X\] versus element ratio for DdDm-1 (red), BB1 (green), H4-1 (orange), K648 (violet), where \[X\] is the logarithmic value normalized to solar of a ratio on the horizontal axis.[]{data-label="abun_sun"}](f7.eps){width="4in"}
[^1]: IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation.
[^2]: The IRS was a collaborative venture between Cornell University and Ball Aerospace Corporation funded by NASA through the Jet Propulsion Laboratory and Ames Research Center.
[^3]: SMART was developed by the IRS Team at Cornell University and is available through the Spitzer Science Center at Caltech.
[^4]: Since alpha elements result from He burning processes, one expects them to track each other, and therefore the offsets should be roughly the same for any one object.
[^5]: See also the discussion of Ne and S abundances in H II regions in @LBS07.
[^6]: 12+log(S/H)=0.888\[12+log(O/H)\] - 0.683
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present , a compiler which enables to encode finite domain constraint problems to CNF. Using both eases the encoding process for the user and also performs transformations to simplify constraints and optimize their encoding to CNF. These optimizations are based primarily on equi-propagation and on partial evaluation, and also on the idea that a given constraint may have various possible CNF encodings. Often, the better encoding choice is made after constraint simplification. is written in Prolog and integrates directly with a SAT solver through a suitable Prolog interface. We demonstrate that constraint simplification is often highly beneficial when solving hard finite domain constraint problems. A implementation is available with this paper.'
author:
- |
AMIT METODIMICHAEL CODISH\
Department of Computer Science, Ben-Gurion University, Israel
title: |
Compiling Finite Domain\
Constraints to SAT with
---
\[firstpage\]
SAT encoding, FD constraints, Equi-propagation, partial evaluation.
Introduction
============
In recent years, Boolean SAT solving techniques have improved dramatically. Today’s SAT solvers are considerably faster and able to manage larger instances than yesterday’s. Moreover, encoding and modeling techniques are better understood and increasingly innovative. SAT is currently applied to solve a wide variety of hard and practical combinatorial problems, often outperforming dedicated algorithms. The general idea is to encode a (typically, NP) hard problem instance, $\mu$, to a Boolean formula, $\varphi_\mu$, such that the solutions of $\mu$ correspond to the satisfying assignments of $\varphi_\mu$. Given the encoding, a SAT solver is then applied to solve $\mu$.
Tailgating the success of SAT technology are a variety of tools which can be applied to specify and then compile problem instances to corresponding SAT instances. For example, introduce , a logic-based specification language which allows to specify combinatorial problems in a declarative way. At the core of this system is a compiler which translates specifications to CNF formula. The general objective of such tools is to facilitate the process of providing high-level descriptions of how the (constraint) problem at hand is to be solved. Typically, a constraint based modeling language is introduced and used to model instances. Drawing on the analogy to programming languages, given such a description, a compiler then provides a low-level executable for the underlying machine. Namely, in our context, a formula for the underlying SAT or SMT solver. One obstacle when seeking to optimize CNF encodings derived from high-level descriptions, is that CNF encodings are *“bit-level”* representations and do not maintain *“word-level”* information. For example, from a CNF encoding one cannot know that certain bits originate from the same integer value in the original constraint. This limits the ability to apply optimizations which rely on such word-level information.
We mention two relevant tools. Sugar [@sugar2009], is a SAT-based constraint solver. To solve a finite domain linear constraint satisfaction problem it is first encoded to a CNF formula by Sugar, and then solved using the MiniSat solver [@minisat2003]. is like Sugar, but applies optimizations. Sugar is the first system which demonstrates the advantage in adopting the, so-called, unary order-encoding to represent integers. We follow suite, and introduce additional novel encoding techniques that take advantage of, previously unobserved, properties of the order-encoding. MiniZinc [@miniZinc2007], is a constraint modeling language which is compiled by a variety of solvers to the low-level target language FlatZinc for which there exist many solvers. It creates a standard for the source language (which we follow loosely). is like FlatZinc, but with a focus on a subset of the language relevant for finite domain constraint problems.
We present a tool, [(****en-Gurion University ****qui-propagation ****ncoder)]{} which translates models in a constraint based modeling language, similar to Sugar and FlatZinc, to CNF. Conceptually, maintains two representations for each constraint in a model so that each constraint is also viewed as a Boolean function. Partial evaluation, and other word-level techniques, drive simplification through the constraint part; whereas, equi-propagation [@Metodi2011], and other bit-level techniques, drive simplification through the Boolean part. Finally, an encoding technique is selected for a constraint, depending on its context, to derive a CNF.
The name, “” refers both to the constraint language as well as to its compiler to CNF. is not a constraint solver, but can be applied in combination with a SAT solver to solve finite domain constraint problems. We report on our experience with applications which indicates that using , like any compiler, has two main advantages. On the one hand, it facilitates the process of programming (or modeling). On the other hand, given a program (a model), it simplifies the corresponding CNF which, in many cases, is faster to solve than with other approaches. The tool integrates with SWI Prolog and can be downloaded from [@bee2012].
Representing Integers
=====================
A fundamental design choice when encoding finite domain constraints concerns the representation of integer variables. surveys several of the possible choices (the *direct-*, *support-* and *log-* *encodings*) and introduces the *log-support encoding*. We focus in this paper on the use of unary representations and primarily on the, so-called, *order-encoding* (see e.g. [@baker; @BailleuxB03]) which has many nice properties when applied to small finite domains. We describe the setting where all integer variables are represented in the order-encoding except for those involved in a global “all-different” constraint which take a dual representation with channeling between the order-encoding and the *direct encoding*. This choice derives from the observation by that the direct-encoding is superior when encoding the all-different constraint.
Let bit vector $X=[x_1,\ldots,x_n]$ represent a finite domain integer variable. In the *order-encoding*, $X$ constitutes a monotonic non-increasing Boolean sequence. Bit $x_i$ is interpreted as $X\geq
i$. For example, the value 3 in the interval $[0,5]$ is represented in 5 bits as $[1,1,1,0,0]$. In the *direct-encoding*, $X$ constitutes a characteristic function (exactly one bit takes value 1) and $x_i$ is interpreted as stating $X= i-1$. For example, the value 3 in the interval $[0,5]$ is represented in 6 bits as $[0,0,0,1,0,0]$.
An important property of a Boolean representation for finite domain integers is the ability to represent changes in the set of values a variable can take. It is well-known that the order-encoding facilitates the propagation of bounds. Consider an integer variable $X=[x_1,\ldots,x_n]$ with values in the interval $[0,n]$. To restrict $X$ to take values in the range $[a,b]$ (for $1\leq a\leq b\leq n$), it is sufficient to assign $x_{a}=1$ and $x_{b+1}=0$ (if $b<n$). The variables $x_{a'}$ and $x_{b'}$ for $0\geq a'> a$ and $b<b'\leq n$ are then determined true and false, respectively, by *unit propagation*. For example, given $X=[x_1,\ldots,x_9]$, assigning $x_3=1$ and $x_6=0$ propagates to give $X=[1,1,1,x_4,x_5,0,0,0,0]$, signifying that $dom(X)=\{3,4,5\}$. This property is exploited in Sugar [@sugar2009] which also applies the order-encoding.
We observe, and apply in , an additional property of the order-encoding: its ability to specify that a variable cannot take a specific value $0\leq v\leq n$ in its domain by equating two variables: $x_{v}=x_{v+1}$. This indicates that the order-encoding is well-suited not only to propagate lower and upper bounds, but also to represent integer variables with an arbitrary, finite set, domain. For example, given $X=[x_1,\ldots,x_9]$, equating $x_2=x_3$ imposes that $X\neq 2$. Likewise $x_5=x_6$ and $x_7=x_8$ impose that $X\neq 5$ and $X\neq 7$. Applying these equalities to $X$ gives, $X=[x_1,\underline{x_2,x_2},x_4,\underline{x_5,x_5},\underline{x_7,x_7},x_9]$, signifying that $dom(X)=\{0,1,3,4,6,8,9\}$.
The order-encoding has many additional nice features that are exploited in to simplify constraints and their encodings to CNF. To illustrate one, consider a constraint of the form $\mathtt{A+B=5}$ where `A` and `B` are integer values in the range between 0 and 5 represented in the order-encoding. At the bit level we have: $\mathtt{A=[a_1,\ldots,a_5]}$ and $\mathtt{B=[b_1,\ldots,b_5]}$. The constraint is satisfied precisely when $\mathtt{B=[\neg a_5,\ldots,\neg a_1]}$. Instead of encoding the constraint to CNF, we substitute the bits $\mathtt{b_1,\ldots,b_5}$ by the literals $\mathtt{\neg a_5,\ldots,\neg a_1}$, and remove the constraint. In Prolog, this is implemented as a unification and does not generate any clauses in the encoding.
Constraints in
===============
Boolean constants “${\mathit{true}}$” and “${\mathit{false}}$” are viewed as (integer) values “1” and “0”. Constraints are represented as (a list of) Prolog terms. Boolean and integer variables are represented as Prolog variables, which may be instantiated when simplifying constraints. Table \[tab:beeStntax\] introduces the syntax for (a simplified subset of) . In the table, $\mathtt{X}$ and $\mathtt{Xs}$ (possibly with subscripts) denote a literal (a Boolean variable or its negation) and a vector of literals, $\mathtt{I}$ (possibly with subscript) denotes an integer variable, and $\mathtt{c}$ (possibly with subscript) denotes an integer constant.
[rlll]{}\
(1) &$\mathtt{new\_bool(X)}$ &&declare Boolean `X`\
(2) &$\mathtt{new\_int(I,c_1,c_2)}$ & & declare integer `I`, $\mathtt{c_1\leq I\leq c_2}$\
(3) & $\mathtt{ordered([X_1,\ldots,X_n])}$ & & $\mathtt{X_1\geq X_2\geq\cdots\geq X_n}$ (on Booleans)\
\
(4) & $\mathtt{bool\_eq(X_1,X_2)}$ or $\mathtt{bool\_eq(X_1,-X_2)}$& $\mathtt{}$& $\mathtt{X_1 = X_2}$ or $\mathtt{X_1 = \neg X_2}$\
(5) & $\mathtt{bool\_array\_op([X_1,\ldots,X_n])}$ & $\mathtt{}$& $\mathtt{X_1 ~op~ X_2 \cdots op~ X_n}$\
(6) & $\mathtt{bool\_array\_op\_reif([X_1,\ldots,X_n],~X)}$ & $\mathtt{}$& $\mathtt{X_1 ~op~ X_2 \cdots op~ X_n\Leftrightarrow X}$\
(7) & $\mathtt{bool\_op\_reif(X_1,X_2,~X)}$ & $\mathtt{}$& $\mathtt{X_1 ~op~ X_2\Leftrightarrow X}$\
(8) & $\mathtt{bool\_array\_lex(Xs_1,Xs_2)}$ &$\mathtt{}$& $\mathtt{Xs_1}$ precedes $\mathtt{Xs_2}$ in the lex order\
\
\
(9) & $\mathtt{int\_rel(I_1,I_2)}$ & $\mathtt{}$& $\mathtt{I_1 ~rel~ I_2}$\
(10) & $\mathtt{int\_rel\_reif(I_1,I_2,~X)}$ & $\mathtt{}$& $\mathtt{I_1 ~rel~ I_2 \Leftrightarrow X}$\
(11) &$\mathtt{int\_op(I_1,I_2,~I)}$ & $\mathtt{}$& $\mathtt{I_1 ~op~ I_2 = I}$\
(12) &$\mathtt{int\_array\_op'([I_1,\ldots,I_n],~I)}$ & $\mathtt{}$& $\mathtt{I_1 ~op'\cdots op'~ I_n = I}$\
\
(13) & $\mathtt{allDiff([I_1,\ldots,I_n])}$ & & $\mathtt{\bigwedge_{i<j}I_i \neq I_j}$\
(14) & $\mathtt{bool\_array\_sum\_rel([X_1,\ldots,X_n],~I)}$ & $\mathtt{}$& $\mathtt{(\Sigma ~X_i)~ rel~ I}$\
(15) & $\mathtt{comparator(X_1,X_2,X_3,X_4)}$ & & $\mathtt{sort([X_1,X_2])=[X_3,X_4]}$\
On the right column of the table are brief explanations regarding the constraints. The table introduces 15 constraint templates. Constraints (1-2) are about variable declarations: Booleans and integers. Constraint (3) signifies that a bit sequence is monotonic non-increasing, and is used to specify that an integer variable is in the order-encoding. Constraints (4-7) are about Boolean (and reified Boolean) statements. The cases for $\mathtt{bool\_array\_or([X_1,\ldots,X_n])}$ and $\mathtt{bool\_array\_xor([X_1,\ldots,X_n])}$ facilitate the specification of clauses and of `xor` clauses (supported in the CryptoMiniSAT solver [@Crypto]). Constraint (8) specifies that two bit-vectors are ordered lexicographically. Constraints (9-12) are about integer relations and operations. Constraints (13-14) are the all-different constraint on integers and the cardinality constraint on Booleans. Constraint (15) specifies that sorting a bit pair $\mathtt{[X_1,X_2]}$ (decreasing order) results in the pair $\mathtt{[X_3,X_4]}$. This is a basic building block for the construction of sorting networks [@Batcher68] used to encode cardinality constraints during compilation as described in [@AsinNOR11] and in [@DBLP:conf/lpar/CodishZ10].
An Example Application: magic graph labeling {#sec:magic}
=============================================
We illustrate the application of to solve a graph labeling problem. A typical application has the form depicted as Figure \[fig:generic\] where the predicate `solve/2` takes a problem `Instance` and provides a `Solution`. The specifics of the application are in the call to `encode/3` which given the `Instance` generates the `Constraints` that solve it together with a `Map` relating instance variables with constraint variables. The calls to `compile/2` and `sat/1` compile the constraints to a `CNF` and solve it applying a SAT solver. If the instance has a solution, the SAT solver binds the constraint variables accordingly. Then, the call to `decode/2`, using the `Map`, provides a `Solution` in terms of the instance variables. The definitions of `encode/3` and `decode/3` are application dependent and provided by the user. The predicates `compile/2` and `sat/1` provide the interface to and the underlying SAT solver.
:- use\_module(bee\_compiler, \[compile/2\]).\
:- use\_module(sat\_solver, \[sat/1\]).\
\
solve(Instance, Solution) :-\
encode(Instance, Map, Constraints),\
compile(Constraints, CNF),\
sat(CNF),\
decode(Map, Solution).\
Graph labeling is about finding an assignment of integers to the vertices and edges of a graph subject to certain conditions. Graph labelings were introduced in the 60’s and hundreds of papers on a wide variety of related problems have been published since then. See for example the survey by with more than 1200 references. Graph labelings have many applications. For instance in radars, xray crystallography, coding theory, etc.
We focus here on the vertex-magic total labeling (VMTL) problem where one should find for the graph $G=(V,E)$ a labeling that is a one-to-one map $V\cup E \rightarrow \{1,2,\ldots,|V|+|E|\}$ with the property that the sum of the labels of a vertex and its incident edges is a constant $K$ independent of the choice of vertex. A problem instance takes the form $vmtl(G,K)$ specifying the graph $G$ and a constant $K$. The query $\mathtt{solve(vmtl(G,K), Solution)}$ poses the question: “Does there exist a vmtl labeling for $G$ with magic constant $K$?” It binds $\mathtt{Solution}$ to indicate such a labeling if one exists, or to “unsat” otherwise.
------------------------------------------------------------
An Instance The Graph The Map
------------------------------------------------------------
$\begin{array}{l}
\mathtt{Instance = vmtl(G,K),}\\
\mathtt{G=(V,E),}\\
\mathtt{V=[1,2,3,4],}\\
\mathtt{E=[(1,2),(1,3),}\\
\qquad \mathtt{(2,3),(3,4)], }\\
\mathtt{K=14}
\end{array}$$\begin{array}{l}
\xymatrix@C=9pt@R=15pt{
& 4\ar@{-}[d] & \\
& 3\ar@{-}[dl]\ar@{-}[dr] & \\
2\ar@{-}[rr]&&1 }
\end{array}$ $\mathtt{M=} \left[\begin{array}{ll}
\mathtt{((1,2),~E_1),} & \mathtt{(1,~V_1),} \\
\mathtt{((1,3),~E_2),} & \mathtt{(2,~V_2),} \\
\mathtt{((2,3),~E_3),} & \mathtt{(3,~V_3),} \\
\mathtt{((3,4),~E_4),} & \mathtt{(4,~V_4)} \\
\end{array} \right]$
The Constraints
$\mathtt{Cs=}
\left[\begin{array}{lll}
\mathtt{new\_int(V_{1}, 1, 8),} &
\mathtt{new\_int(E_{1}, 1, 8),} &
\mathtt{int\_array\_plus([V_1,E_1,E_2], K),}
\\
\mathtt{new\_int(V_{2}, 1, 8),} &
\mathtt{new\_int(E_{2}, 1, 8),} &
\mathtt{int\_array\_plus([V_2,E_1,E_3], K),}
\\
\mathtt{new\_int(V_{3}, 1, 8),} &
\mathtt{new\_int(E_{3}, 1, 8),} &
\mathtt{int\_array\_plus([V_3,E_2,E_3,E_4], K),}
\\
\mathtt{new\_int(V_{4}, 1, 8),} &
\mathtt{new\_int(E_{4}, 1, 8),} &
\mathtt{int\_array\_plus([V_4,E_4], K),}
\\
\mathtt{new\_int(K, 14, 14),} &
\multicolumn{2}{l}{
\mathtt{allDiff([V_1,V_2,V_3,V_4,E_1,E_2,E_3,E_4])}}
\end{array}\right]$
------------------------------------------------------------
Figure \[fig:vmtl-instance\] illustrates an example problem instance together with the constraints, `Cs` and the map, `M`, generated by the `encode/3` predicate for this instance. The constraints introduce integer variables for the vertices and edges, specify that these variables take “all different” values, and specify that the labels for each vertex with its incident edges sum to $\mathtt{K}$. Solving the constraints from Figure \[fig:vmtl-instance\] for the example VMTL instance binds the Map, `M`, as follows, indicating a solution: $$\mathtt{M=}{\small \left[\begin{array}{ll}
\mathtt{((1,2),~[1,1,1,1,1,1,1,0]),} & \mathtt{(1,~[1,1,1,1,0,0,0,0]),} \\
\mathtt{((1,3),~[1,1,1,0,0,0,0,0]),} & \mathtt{(2,~[1,1,1,1,1,0,0,0]),} \\
\mathtt{((2,3),~[1,1,0,0,0,0,0,0]),} & \mathtt{(3,~[1,0,0,0,0,0,0,0]),} \\
\mathtt{((3,4),~[1,1,1,1,1,1,1,1]),} & \mathtt{(4,~[1,1,1,1,1,1,0,0])} \\
\end{array} \right]}$$ In Section \[results\] we report that using enables us to solve interesting instances of the VMTL problem not previously solvable by other techniques.
Compiling to CNF {#sec:compiling}
=================
The compilation of a constraint model to a CNF using goes through three phases. In the first phase, (unary) bit blasting, integer variables (and constants) are represented as bit vectors in the order-encoding. Now all constraints are about Boolean variables. The second phase, the main loop of the compiler, is about constraint simplification. Three types of actions are applied: equi-propagation, partial evaluation, and decomposition of constraints. These are specified as a set of transitions which we write in the form $c_1\overset{\theta}{\longmapsto}c_2$ to specify that constraint $c_1$ reduces to constraint $c_2$ generating the (possibly empty) substitution $\theta$. Simplification is applied repeatedly until no rule is applicable. In the third, and final phase, simplified constraints are encoded to CNF. We elaborate below. To simplify the presentation, we assume that integer variables are represented in a positive interval starting from $0$. As later detailed in Section \[sec:implementation\] there is no such limitation in .
####
Each integer variable declaration $\mathtt{new\_int(I,c_1,c_2)}$ triggers a unification $\mathtt{I=[1,\dots,1,X_{c_1+1},\ldots,X_{c_2}]}$ and introduces a constraint $\mathtt{ordered(I)}$ to specify that the bits representing $\mathtt{I}$ are in the order-encoding. To illustrate bit-blasting, consider again the VMTL example detailed in Figure \[fig:vmtl-instance\]. Each variable in the `Map` occurs in a $\mathtt{new\_int}$ declaration. So the following unifications are performed: $${\small\begin{array}{ll}
\mathtt{V_{1}=[1,V_{1,2},V_{1,3},V_{1,4},V_{1,5},V_{1,6},V_{1,7},V_{1,8}]}, &
\mathtt{E_{1}=[1,E_{1,2},E_{1,3},E_{1,4},E_{1,5},E_{1,6},E_{1,7},E_{1,8}]}, \\
\mathtt{V_{2}=[1,V_{2,2},V_{2,3},V_{2,4},V_{2,5},V_{2,6},V_{2,7},V_{2,8}]}, &
\mathtt{E_{2}=[1,E_{2,2},E_{2,3},E_{2,4},E_{2,5},E_{2,6},E_{2,7},E_{2,8}]}, \\
\mathtt{V_{3}=[1,V_{3,2},V_{3,3},V_{3,4},V_{3,5},V_{3,6},V_{3,7},V_{3,8}]}, &
\mathtt{E_{3}=[1,E_{3,2},E_{3,3},E_{3,4},E_{3,5},E_{3,6},E_{3,7},E_{3,8}]}, \\
\mathtt{V_{4}=[1,V_{4,2},V_{4,3},V_{4,4},V_{4,5},V_{4,6},V_{4,7},V_{4,8}]}, &
\mathtt{E_{4}=[1,E_{4,2},E_{4,3},E_{4,4},E_{4,5},E_{4,6},E_{4,7},E_{4,8}]},\\
\mathtt{K=~[1,1,1,1,1,1,1,1,1,1,1,1,1,1]}&
\end{array}}$$ Integer variables occurring in an `allDiff` constraint are bit-blasted twice: first, in the order-encoding, when declared, as explained above, and second, in the direct encoding, when processing the `allDiff` constraint, as described below.
####
is about detecting situations in which a small number of constraints imply an equality of the form $X=L$ where $X$ is a Boolean variable and $L$ is a Boolean literal or constant. In this case $X$ becomes redundant and can be replaced by $L$ in all constraints. In we consider as candidates for equi-propagation, individual constraints together with constraints specifying that their integer variables are in the order-encoding. If $X=L$ is such an equality, then equi-propagation is implemented by unifying $X$ and $L$. This unification applies to all occurrences of $X$ and in this sense “propagates” to other constraints involving $X$. Once equi-propagation detects such an equation, this may trigger further equi-propagation from other constraints. For example, consider the constraint $\mathtt{int\_neq(I_1,I_2)}$ where $\mathtt{I_1=[x_1,x_2,x_3,x_4}]$ and $\mathtt{I_2=[1,1,0,0}]$. We propagate that $\mathtt{(x_2=x_3)}$ because $$\left(
\begin{array}{l}
\mathtt{I_1=[x_1,x_2,x_3,x_4] ~\wedge~ I_2=[1,1,0,0] ~\wedge}\\
\mathtt{int\_neq(I_1,I_2) ~\wedge~ ordered(I_1)}
\end{array}
\right)\models \mathtt{(x_2=x_3)}.$$ To see why, consider that $\mathtt{ordered(I_1)}$ implies that $\mathtt{x_2\geq x_3}$. Furthermore, also $\mathtt{x_2\leq x_3}$ as otherwise $\mathtt{x_2=1}$ and $\mathtt{x_3=0}$ which implies that $\mathtt{I_1=[1,1,0,0}]$, contradicting $\mathtt{int\_neq(I_1,I_2)}$.
In , equi-propagation is implemented by a collection of ad-hoc transition rules for each type of constraint. While this approach is not complete — there are equations implied by a constraint that will not detect — the implementation is fast, and works well in practice. An alternative approach is to implement equi-propagation, using BDD’s, as described in [@Metodi2011]. This approach, though complete, is slower and not included in the current release of . The following are two of the simplification (equi-propagation) rules of that apply to $\mathtt{int\_neq}$ constraints:
$\mathtt{neq_1}:$
: applies when one of the (order-encoding) integers in the relation is a constant and $\theta=\{X_1=X_2\}$: $$\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_2,\hspace{-3mm}& \ldots{]} \\
{[}\ldots,\hspace{-3mm}&1, \hspace{-3mm}& 0,\hspace{-3mm}& \ldots{]}
\end{array}
\right)
\overset{\theta}{\longmapsto}
\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_1,\hspace{-3mm}& \ldots{]} \\
{[}\ldots,\hspace{-3mm}&1, \hspace{-3mm}& 0,\hspace{-3mm}& \ldots{]}
\end{array}
\right)$$
$\mathtt{neq_2}:$
: applies when the integers share common variables as in the rule template and $\theta=\{X_1=X_2\}$: $$\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_2,\hspace{-3mm}&\ldots{]}, \\
{[}\ldots,\hspace{-3mm}&\neg X_2,\hspace{-3mm}&\neg X_1,\hspace{-3mm}&\ldots{]}
\end{array}
\right)
\overset{\theta}{\longmapsto}
\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_1,\hspace{-3mm}&\ldots{]}, \\
{[}\ldots,\hspace{-3mm}&\neg X_1,\hspace{-3mm}&\neg X_1,\hspace{-3mm}&\ldots{]}
\end{array}
\right)$$
For the rule $\mathtt{neq_1}$, observe that after applying this rule the constraint obtained is a tautology. Hence it is subsequently removed by one of the other “partial evaluation” rules. For the rule $\mathtt{neq_2}$, to see why the equation $X_1=X_2$ is implied by the constraint (on the left side of the rule), consider all possible truth values for the variables $X_1$ and $X_2$: (a) If $X_1=0$ and $X_2=1$ then both integers in the relation take the form $[\ldots,0,1,\ldots]$ violating their specification as `ordered`, so this is not possible. (b) If $X_1=1$ and $X_2=0$ then both numbers take the form $[1,\ldots,1,0,\ldots,0]$ and are equal, violating the $\mathtt{neq}$ constraint. The only possible bindings for $X_1$ and $X_2$ are those where $X_1=X_2$. The template expressed in rule $\mathtt{neq_2}$ is not contrived. It comes up frequently as a result of applying other equi-propagation rules.
####
is about simplifying constraints in view of variables that are (partially) instantiated, either because of information from the constraint model or else due to equi-propagation. Typical cases include constant elimination and elimination of tautologies. The following are some of ’s partial evaluation rules that apply to $\mathtt{int\_neq}$ constraints ($\epsilon$ denotes the empty substitution).
$\mathtt{neq_3}:$
: applies to remove replicated variables: $$\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_1,\hspace{-3mm}& \ldots{]} \\
{[}\ldots,\hspace{-3mm}&Y_1,\hspace{-3mm}&Y_1,\hspace{-3mm}& \ldots{]}
\end{array}
\right)
\overset{\epsilon}{\longmapsto}
\mathtt{int\_neq}\left(
\begin{array}{ccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}& \ldots{]} \\
{[}\ldots,\hspace{-3mm}&Y_1,\hspace{-3mm}& \ldots{]}
\end{array}
\right)$$
$\mathtt{neq_4}:$
: applies to remove leading 1 bits (there is a similar rule for trailing 0’s): $$\mathtt{int\_neq}([1,1,X_3,\ldots], [Y_1,Y_2,Y_3,\ldots])
\overset{\epsilon}{\longmapsto}
\mathtt{int\_neq}([1,X_3,\ldots],[Y_2,Y_3,\ldots])$$
We now detail three of the simplification rules (equi-propagation and partial evaluation) that apply to a constraint of the form `int_plus(A,B,C)` where we assume for simplicity of presentation (the tool supports the general case) that $\mathtt{A=[A_1,\ldots,A_n]}$, $\mathtt{B=[B_1,\ldots,B_m]}$, and $\mathtt{C=[C_1,\ldots,C_{n+m}]}$. We denote by $\mathtt{min(I)}$ (or $\mathtt{max(I)}$) the minimal (or maximal) value that integer variable `I` can take, determined by the number of leading ones (or trailing zeros) in its bit representation.
Rule $\mathtt{plus_{1}}$ is standard propagation for interval arithmetics. Rule $\mathtt{plus_2}$ removes redundant bits (assigned values through $\mathtt{plus_{1}}$). Rules $\mathtt{plus_{3(a)}}$ and $\mathtt{plus_{3(b)}}$ remove constraints and may seem contrived: `3(a)` assumes that $\mathtt{m=0}$ and `3(b)` assumes that $\mathtt{n=m}$ and that $\mathtt{C}$ represents the (same) constant $\mathtt{n}$. However, in the general case, when $\mathtt{n}$, $\mathtt{m}$ are arbitrary and constant `C` is represented in $\mathtt{m+n}$ bits, then application of the other rules will reduce the constraint to one of these special cases.
$\mathtt{plus_1}:$
: applies to propagate bounds: $
\mathtt{int\_plus(A,B,C)}
\overset{\theta}{\longmapsto}
\mathtt{int\_plus(A,B,C)}
$ where\
$\theta=\left\{
\begin{array}{ll}
C_{max\{min(C), min(A)+min(B)\}} = 1, & C_{min\{max(C), max(A)+max(B)\}+1} = 0,\\
A_{max\{min(A), min(C)-max(B)\}} = 1, & A_{min\{max(A), max(C)-min(B)\}+1} = 0, \\
B_{max\{min(B), min(C)-max(A)\}} = 1, & B_{min\{max(B), max(C)-min(A)\}+1} = 0
\end{array}\right\}
$
$\mathtt{plus_{2}}:$
: applies to remove leading 1’s (there are similar rules for trailing 0’s and for the case when the 1’s or 0’s are on $\mathtt{[B_1,\ldots,B_m]}$): $$\mathtt{int\_plus}\left(\begin{array}{l}
\mathtt{[1,A_2,\ldots,A_{n}], }\\
\mathtt{[B_1,\ldots,B_m],}\\
\mathtt{{[}1,C_2,\ldots,C_{n+m}{]}}
\end{array}\right)
\overset{\epsilon}{\longmapsto}
\mathtt{int\_plus}\left(\begin{array}{l}
\mathtt{[A_2,\ldots,A_{n}], }\\
\mathtt{[B_1,\ldots,B_m],}\\
\mathtt{{[}C_2,\ldots,C_{n+m}{]}}
\end{array}\right)$$
$\mathtt{plus_{3(a)}}:$
: applies when `A` or `B` is the empty bit list and $\theta = {\left\{~C_{i}=A_i \left|
\begin{array}{l}1 \leq i \leq
n\end{array}
\right. \right\}}$ $$\mathtt{int\_plus([A_1,\ldots,A_n], [~], [C_1,\ldots,C_n])}
\overset{\theta}{\longmapsto} none$$
$\mathtt{plus_{3(b)}}:$
: applies when `C` is a constant `n` and $\theta = {\left\{~A_{i}=\neg B_{n-i+1} \left|
\begin{array}{l}1 \leq i
\leq n\end{array}
\right. \right\}}$ $$\mathtt{int\_plus([A_1,\ldots,A_n], [B_1,\ldots,B_n], [1,\ldots1,0,\ldots,0])}
\overset{\theta}{\longmapsto}
none$$
We illustrate the simplification of a `int_plus` constraint by the following example.
\[ex:plus14\] Consider constraint $\mathtt{int\_plus(A,B,C)}$ where $A$ and $B$ are integer variables with domain $[1..8]$ and $C$ is the constant 14 represented in 16 bits. Constraint simplification follows the steps:
------------------------------------------------------------------------
$\xrightarrow{\mathtt{plus_1}}$ $\xrightarrow{\mathtt{plus_{2}}}$
$\xrightarrow{\mathtt{plus_{2}}}$ $\xrightarrow{\mathtt{plus_{3(b)}}}$
------------------------------------------------------------------------
After constraint simplification variables `A` and `B` take the form: $\mathtt{[1, 1, 1,
1, 1, 1, A_{7}, A_{8}]}$ and $\mathtt{[1, 1, 1, 1, 1,
1,\neg A_{8}, \neg A_{7}]}$ (and nothing is left to encode to CNF).
####
is about replacing complex constraints (for example about arrays) with simpler constraints (for example about array elements). Consider, for instance, the constraint $\mathtt{int\_array\_plus(As,Sum)}$. It is decomposed to a list of $\mathtt{int\_plus}$ constraints applying a straightforward divide and conquer recursive definition. At the base case, if `As=[A]` then the constraint is replaced by `int_eq(A,Sum)`, or if $\mathtt{As=[A_1,A_2]}$ then it is replaced by $\mathtt{int\_plus(A_1,A_2,Sum)}$. In the general case `As` is split into two halves, then constraints are generated to sum these halves, and then an additional $\mathtt{int\_plus}$ constraint is introduced to sum the two sums.
As another example, consider the $\mathtt{int\_plus(A_1,A_2,A)}$ constraint. One approach, supported by , decomposes the constraint as an odd-even merger (from the context of odd-even sorting networks) [@Batcher68]. Here, the sorted sequences of bits $\mathtt{A_1}$ and $\mathtt{A_2}$ are merged to obtain their sum $\mathtt{A}$. This results in a model with $O(n\log n)$ `comparator` constraints (and later in an encoding with $O(n\log n)$ clauses). Another approach, also supported in , does not decompose the constraint but encodes it directly to a CNF of size $O(n^2)$, as in the context of so-called totalizers [@BailleuxB03]. A hybrid approach, leaves the choice to , depending on the size of the domains of the variables involved. Finally, we note that the user can configure to fix the way it compiles this constraint (and others).
####
is the last phase and applies to all remaining simplified constraints. The encoding of constraints to CNF is standard and similar to the encodings in Sugar [@sugar2009].
####
are about the cardinality of sets of Boolean variables and are specified by the template $\mathtt{bool\_array\_sum\_rel([X_1,\ldots,X_n],~I)}$. Cardinality constraints are normalized, see e.g., [@EenS06], so we only consider $\mathtt{rel\in\{leq,eq\}}$. Partial evaluation rules for cardinality constraints are the obvious. For example, in the special case when `I` is a constant:
$\mathtt{bool\_array\_sum\_leq([X_1,X_2,1,X_4],~3)\mapsto
bool\_array\_sum\_leq([X_1,X_2,X_4],~2)}$
$\mathtt{bool\_array\_sum\_leq([X_1,X_2,0,X_4],~3)\mapsto
bool\_array\_sum\_leq([X_1,X_2,X_4],~3)}$
$\mathtt{bool\_array\_sum\_leq([X_1,X_2,-X_1,X_4],~3)\mapsto
bool\_array\_sum\_leq([X_2,X_4],~2)}$
The special case, when `I` is the constant 1 is called the “at-most-one” constraint and it has been studied extensively (for a recent survey see [@Frisch+Giannaros/10/SAT]). In , we support two different encodings for this case (the user can choose). The first is the standard “pairwise” encoding which specifies a clause $(\neg
x_i\vee\neg x_j)$ for each pair of Boolean variables $x_i$ and $x_j$. This encoding introduces $O(n^2)$ clauses and is sometimes too large. The second, is a more compact encoding which follows the approach described in [@NewAtMostOne]. In the general case (when $\mathtt{I}>1$) the constraint is decomposed, much the same as the $\mathtt{int\_array\_plus}$ constraint, to a network of $\mathtt{int\_plus}$ constraints.
####
specifies that a set of integer variables take all different values. Although we adopt the order-encoding for integer variables, it is well accepted that for these constraints the direct encoding is superior [@direct4allDiff]. For this reason, in , when processing the constraint, a dual representation is chosen. When integer variable $\mathtt{I}$, occurring in an `allDiff` constraint, is declared, it was unified with its unary representation in the order-encoding: $\mathtt{I=[x_1,\ldots,x_n]}$. In addition, we associate `I` with a new bit-blast, $\mathtt{[d_0,\ldots,d_{n}]}$, in the direct encoding. We introduce for each such `I` a channeling formula to capture the relation between its two representations. $$\mathtt{channel([x_1,\ldots,x_n],[d_0,\ldots,d_{n}])=
\left(\begin{array}{r} d_0 = \neg x_1 \\
\wedge~ d_n = x_n
\end{array}\right) \wedge
\bigwedge_{i=1}^{n-1}(d_i\leftrightarrow x_{i}\wedge\neg x_{i+1})}$$
During constraint simplification, the $\mathtt{allDiff([I_1,\ldots,I_n])}$ constraint is viewed as a bit matrix where each row consists of the bits $\mathtt{[d_{i0},\ldots,d_{im}]}$ for $\mathtt{I_i}$ in the direct encoding. The element $d_{ij}$ is true iff $I_i$ takes the value $j$. The $j^{th}$ column specifies which of the $I_i$ take the value $j$ and hence, at most one variable in a column may take the value true. distinguishes the special case when $\mathtt{[I_1,\ldots,I_n]}$ must take precisely $n$ different values. In this case the constraint is about “permutation”. We denote this by a flag (\*) as in $\mathtt{allDiff^*([I_1,\ldots,I_n])}$. In this case, exactly one bit in each column of the representation must take the value true.
To simplify an `allDiff` constraint, applies simplification rules to the implicit cardinality constraints on the columns and also two specific `allDiff` rules. The first is essentially the usual domain consistent propagator [@regin] focusing on Hall sets of size 2. The second rule applies only to an $\mathtt{allDiff^*}$ constraint which is about permutation. We denote the values that $\mathtt{I_i}$ can take as $\mathtt{dom(I_i)}$.
$\mathtt{allDiff_1}:$
: when $\mathtt{dom(I_1) = dom(I_2)=\{v_1,v_2\}}$: $$\mathtt{allDiff([I_1,I_2,I_3,\ldots,I_n])}
\overset{\theta}{\longmapsto}
\mathtt{allDiff([I_3,\ldots,I_n])}$$ where $\theta = \bigcup_{3\leq i\leq n}
\{d_{i,v_1}=0, d_{i,v_2}=0\} \cup
\{d_{1,v_1}= -d_{2,v_1}, d_{1,v_2}= -d_{2,v_2} \}$.
$\mathtt{allDiff_2}:$
: when $\mathtt{\{v_1,v_2\}\subseteq dom(I_1)\cap dom(I_2)}$, and for $i\geq 3$, $\mathtt{\{v_1,v_2\}\cap dom(I_i)=\emptyset}$ $$\mathtt{allDiff^*([I_1,\ldots,I_n])}
\overset{\theta}{\longmapsto}
\mathtt{allDiff^*([I_1,\ldots,I_n])}$$ where $\theta = \bigcup_{j\neq v_1, j\neq v_2}\{d_{1j}=0, d_{2,j}=0\}$.
To illustrate the two rules for `allDiff` consider the following.
Consider an `allDiff` constraint on 5 integer variables taking values in the interval $[0,7]$ where the first two can take only values 0 and 1. So, they are a Hall set of size two and rule $\mathtt{allDiff_1}$ applies. We present the simplification step on the order encoding representation (though it is triggered through the direct encoding representation): $$\small
\mathtt{allDiff}\left(
\begin{array}{cccc}
{[}X_{1,1},\hspace{-3mm}&0,\hspace{-3mm}& \ldots,\hspace{-3mm}&0{]} \\
{[}X_{2,1},\hspace{-3mm}&0,\hspace{-3mm}& \ldots,\hspace{-3mm}&0{]} \\
{[}X_{3,1},\hspace{-3mm}&X_{3,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{3,7}{]} \\
{[}X_{4,1},\hspace{-3mm}&X_{4,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{4,7}{]} \\
{[}X_{5,1},\hspace{-3mm}&X_{5,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{5,7}{]}
\end{array}
\right)
\overset{\theta}{\longmapsto}
\mathtt{allDiff}\left(
\begin{array}{ccc}
{[}1,X_{3,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{3,7}{]} \\
{[}1,X_{4,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{4,7}{]} \\
{[}1,X_{5,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{5,7}{]}
\end{array}
\right)$$ where $\theta = \{ X_{1,1}=\neg X_{2,1} , X_{3,1}=0 , X_{4,1}=0 ,
X_{5,1}=0\}$.
Now consider a setting where an `allDiff` constraint is about 5 variables that can take 5 values (permutation) and the first two are the only two that can take values 0 and 1. So rule $\mathtt{allDiff_2}$ applies. We present the simplification step on the order encoding representation (though it is triggered through the direct encoding representation): $$\small
\mathtt{allDiff^*}\left(
\begin{array}{lccr}
{[}X_{1,1},\hspace{-3mm}&X_{1,2},\hspace{-3mm}&X_{1,3},\hspace{-3mm}&X_{1,4}{]} \\
{[}X_{2,1},\hspace{-3mm}&X_{2,2},\hspace{-3mm}&X_{2,3},\hspace{-3mm}&X_{2,4}{]} \\
{[}1,\hspace{-3mm}&1,\hspace{-3mm}&X_{3,3},\hspace{-3mm}&X_{3,4}{]} \\
{[}1,\hspace{-3mm}&1,\hspace{-3mm}&X_{4,3},\hspace{-3mm}&X_{4,4}{]} \\
{[}1,\hspace{-3mm}&1,\hspace{-3mm}&X_{5,3},\hspace{-3mm}&X_{5,4}{]}
\end{array}
\right)
\overset{\theta}{\longmapsto}
\mathtt{allDiff^*}\left(
\begin{array}{lccr}
{[}X_{1,1},\hspace{-3mm}&0,\hspace{-3mm}&0,\hspace{-3mm}&0{]} \\
{[}X_{2,1},\hspace{-3mm}&0,\hspace{-3mm}&0,\hspace{-3mm}&0{]} \\
{[}1,\hspace{-0mm}&1,\hspace{-0mm}&X_{3,3},\hspace{-3mm}&X_{3,4}{]} \\
{[}1,\hspace{-0mm}&1,\hspace{-0mm}&X_{4,3},\hspace{-3mm}&X_{4,4}{]} \\
{[}1,\hspace{-0mm}&1,\hspace{-0mm}&X_{5,3},\hspace{-3mm}&X_{5,4}{]}
\end{array}
\right)$$ where $\theta = \{ X_{1,2},\ldots,X_{1,4} = 0 , X_{2,2},\ldots,X_{2,4} = 0\}$.
When no further simplification rules apply the `allDiff` constraint is decomposed to the corresponding cardinality constraints on the columns of its bit matrix representation.
Constraint simplification in the VMTL example
=============================================
Consider again the VMTL example and the constraints from Figure \[fig:vmtl-instance\]. We focus on three constraints and follow the steps made when compiling these (we write “14” as short for $\mathtt{[\underbrace{1,1,\ldots,1}_{14}]}$).
$\begin{array}{ll}
(1) & \mathtt{int\_array\_plus([V_4,E_4], 14)} \\
(2) & \mathtt{allDiff([V_1,V_2,V_3,V_4,E_1,E_2,E_3,E_4]),} \\
(3) & \mathtt{int\_array\_plus([V_3,E_2,E_3,E_4], 14),}
\end{array}$
In the first steps, constraint (1) is decomposed to an `int_plus` constraint which has the same form as the constraint in Example \[ex:plus14\]. So, we have the bindings $\mathtt{V_4 = [1,1,1,1,1,1,V_{4,7}, V_{4,8}]}$ and $\mathtt{E_4 = [1,1,1,1,1,1,\neg V_{4,8},\neg V_{4,7}]}$. Now, consider the `allDiff` constraint (2). determines that this constraint is about permutation (8 integer variables with 8 different values in the range \[1,8\]). The simplification rules for `allDiff` detect that $\mathtt{\{V_4,E_4\}}$ must take together the two values 6 and 8 (using a simplification rule similar to $\mathtt{neq_2}$) triggerring the substitution $\mathtt{\{V_{4,7} =
V_{4,8}\}}$. Now rule $\mathtt{allDiff_1}$ detects a Hall set $\mathtt{\{V_4,E_4\}}$ of size two: $$\small \mathtt{allDiff([V_1,V_2,V_3,V_4,E_1,E_2,E_3,E_4])}
\xrightarrow{\theta}
\mathtt{allDiff([V_1,V_2,V_3,E_1,E_2,E_3])}$$ where $\theta$ is the unification that imposes $\mathtt{V_1,V_2,V_3,E_1,E_2,E_3 \neq 6,8}$. So we have the following bindings (where the impact of $\theta$ is underlined): $$\small \begin{array}{ll}
\mathtt{V_{1}=
[1,V_{1,2},V_{1,3},V_{1,4},V_{1,5},\underline{V_{1,7},V_{1,7}},0]} \qquad&
\mathtt{E_{1}=
[1,E_{1,2},E_{1,3},E_{1,4},E_{1,5},\underline{E_{1,7},E_{1,7}},0]} \\
\mathtt{V_{2}=
[1,V_{2,2},V_{2,3},V_{2,4},V_{2,5},\underline{V_{2,7},V_{2,7}},0]} &
\mathtt{E_{2}=
[1,E_{2,2},E_{2,3},E_{2,4},E_{2,5},\underline{E_{2,7},E_{2,7}},0]} \\
\mathtt{V_{3}=
[1,V_{3,2},V_{3,3},V_{3,4},V_{3,5},\underline{V_{3,7},V_{3,7}},0]} &
\mathtt{E_{3}=
[1,E_{3,2},E_{3,3},E_{3,4},E_{3,5},\underline{E_{3,7},E_{3,7}},0]} \\
\mathtt{V_{4}=
[1,1,1,1,1,1,V_{4,7},V_{4,7}]} &
\mathtt{E_{4}=
[1,1,1,1,1,1,\neg V_{4,7},\neg V_{4,7}]}
\end{array}$$
Consider now the constraint (3). Equi-propagation (because of bounds) dictates that $\mathtt{max(V_1)=max(V_2)=max(V_3)=5}$, so this constraint then simplifies as follows:
$$\small
\fbox{$\begin{array}{l}
\mathtt{int\_array\_plus([}\\
~~\mathtt{[1,V_{3,2},V_{3,3},V_{3,4},V_{3,5},0,0,0]},\\
~~\mathtt{[1,E_{2,2},E_{2,3},E_{2,4},E_{2,5},0,0,0]},\\
~~\mathtt{[1,E_{3,2},E_{3,3},E_{3,4},E_{3,5},0,0,0]},\\
~~\mathtt{[1,1,1,1,1,1,\neg V_{4,7},\neg V_{4,7}]},~14~
\mathtt{])}
\end{array}$}
\longmapsto
\fbox{$\begin{array}{l}
\mathtt{int\_array\_plus([}\\
~~\mathtt{[V_{3,2},V_{3,3},V_{3,4},V_{3,5}]},\\
~~\mathtt{[E_{2,2},E_{2,3},E_{2,4},E_{2,5}]},\\
~~\mathtt{[E_{3,2},E_{3,3},E_{3,4},E_{3,5}]},\\
~~\mathtt{[\neg V_{4,7},\neg V_{4,7}]},~5~
\mathtt{])}
\end{array}$}$$ After applying simplification and decomposition rules on all the constraints from Figure \[fig:vmtl-instance\] until no further rules can be applyed, the constraints will be encoded to CNF. The generated CNF contains 301 clauses and 48 Boolean variables. Compiling the same set of constraints from Figure \[fig:vmtl-instance\] without applying simplification rules generates a larger CNF which contains 642 clauses and 97 Boolean variables.
Another Example Application: DNA word problem
==============================================
The DNA word problem (Problem `033` of CSPLib) seeks the largest parameter $n$, such that there exists a set $S$ of $n$ eight-letter words over the alphabet $\Sigma=\{A,C,G,T\}$ with the following properties: **(1)** Each word in $S$ has exactly 4 symbols from $\{C,G\}$; **(2)** Each pair of distinct words in $S$ differ in at least 4 positions; and **(3)** For every $x,y\in S$: $x^R$ (the reverse of $x$) and $y^C$ (the word obtained by replacing each $A$ by $T$, each $C$ by $G$, and vice versa) differ in at least 4 positions.
In [@dnaWordPaper], the authors present a strategy to solve this problem where the four letters are modeled by bit-pairs ${\langle t,m \rangle}$. Each eight-letter word can then be viewed as the combination of a *“t-part”*, ${\langle t_1,\ldots,t_8 \rangle}$, which is a bit-vector, and a *“m-part”*, ${\langle m_1,\ldots,m_8 \rangle}$, also a bit-vector. Building on the approach described in [@dnaWordPaper], we pose conditions on sets of *“t-parts”* and *“m-parts”*, $T$ and $M$, so that their Cartesian product $S=T\times M$ will satisfy the requirements of the original problem. From the three conditions below, $T$ is required to satisfy (1$'$) and (2$'$), and $M$ is required to satisfy (2$'$) and (3$'$). For a set of bit-vectors $V$, the conditions are: **(1$'$)** Each bit-vector in $V$ sums to 4; **(2$'$)** Each pair of distinct bit-vectors in $V$ differ in at least 4 positions; and **(3$'$**) For each pair of bit-vectors (not necessarily distinct) $u,v\in V$, $u^R$ (the reverse of $u$) and $v^C$ (the complement of $v$) differ in at least 4 positions. This is equivalent to requiring that $(u^r)^c$ differs from $v$ in at least 4 positions.
It is this strategy that we model in our encoding. An instance takes the form $\mathtt{dna(n_1,n_2)}$ signifying the numbers of bit-vectors, $n_1$ and $n_2$ in the sets $T$ and $M$. Without loss of generality, we impose, to remove symmetries, that $T$ and $M$ are lexicographically ordered. A solution is the Cartesian product $S=T\times M$. In Section \[results\] we report that using enables us to solve interesting instances of the problem not previously solvable by other techniques.
Implementation {#sec:implementation}
==============
is implemented in (SWI) Prolog and can be applied in conjunction with the CryptoMiniSAT solver [@Crypto] through a Prolog interface [@satPearl]. can be downloaded from [@bee2012] where one can find also the examples from this paper and others. The distribution includes also a solver, which we call Bumble, which enables to specify a model as an input file and solve it. The output is a set of bindings to the declared variables in the model.
In , Boolean variables are represented as Prolog variables. The negation of `X` is represented as `-X`. The truth values, ${\mathit{true}}$ and ${\mathit{false}}$, are denoted `1` and `-1`. Integer variables (including negative range values) are represented in the order-encoding. When processing (bit-blasting) a declaration $\mathtt{new\_int(I,Min,Max)}$, Prolog variable `I` is unified with the tuple `(Min,Max,Bits,LastBit)` where `Min` and `Max` are constants indicating the interval domain of `I`, `Bits` is a list of $\mathtt{(Max-Min)}$ variables, and `LastBit` is the last variable of `Bits`. This representation is more concise than the one assumed for simplicity in the previous sections and it also supports negative numbers. Maintaining direct access to the last bit in the representation (we already can access the first bit through the list `Bits`) facilitates a (constant time) check if the lower and upper bound values of a variable has changed. This way we can more efficiently determine when (certain) simplification rules apply. We make a few notes: (1) Integer variables must be declared before use; (2) allows the use of constants in constraints instead of declaring them as integer variables (for example $\mathtt{int\_gt(I,5)}$ represents a declaration $\mathtt{new\_int(I',5,5)}$ together with the constraint $\mathtt{int\_gt(I,I')}$); (3) integer variables can be negated.
maintains constraints as a Prolog list (of terms). Each type of constraint is associated with corresponding rules for simplification, decomposition, and encoding to CNF. After bit-blasting, constraints are first simplified (equi-propagation and partial evaluation) using these rules until no further rules apply. During this process, if a pair of literals is equated (e.g. as in `X=Y, X=-Y, X=1, X=-1`), then they are unified, thus propagating the effect to other constraints. After constraint simplification, some constraints are decomposed, and this process repeats. We end up with a set of “basic” constraints (which cannot be further decomposed or simplified). These are then encoded to CNF.
Experiments {#results}
===========
We report on our experience in applying . To appreciate the ease in its use, and for further details, the reader is encouraged to view the example encodings available with the tool [@bee2012]. All experiments run on an Intel Core 2 Duo E8400 3.00GHz CPU with 4GB memory under Linux (Ubuntu lucid, kernel 2.6.32-24-generic). is written in Prolog and run using SWI Prolog v6.0.2 64-bits. Comparisons with Sugar (v1.15.0) are based on the use of identical constraint models, apply the same SAT solver (CryptoMiniSat v2.5.1), and run on the same machine. For all of the tables describing experiments, columns indicate:
- compile time (seconds)
- number of CNF clauses
<!-- -->
- number of CNF variables
- SAT solving time (seconds)
We first focus on the impact of the dual representation for [`allDiff`]{} constraints. We report on the application of to Quasi-group completion problems (QCP), proposed by as a constraint satisfaction benchmark, where the model is a conjunction of [`allDiff`]{} constraints.
####
[lc|rrrr|rrrr|rrr]{} & & &\
& &comp &clauses & vars & sat & comp &clauses& vars & sat & clauses & vars & sat\
25-264-0 & sat & 0.23 & 6509 & 1317 & 0.33 & 0.36 & 33224 & 887 & 8.95 & 126733 & 10770 & 34.20\
25-264-1 & sat & 0.20 & 7475 & 1508 & 3.29 & 0.30 & 34323 & 917 & 97.50 & 127222 & 10798 & 13.93\
25-264-2 & sat & 0.21 & 6531 & 1329 & 0.07 & 0.30 & 35238 & 905 & 2.46 & 127062 & 10787 & 8.06\
25-264-3 & sat & 0.21 & 6819 & 1374 & 0.83 & 0.29 & 32457 & 899 & 18.52 & 127757 & 10827 & 44.03\
25-264-4 & sat & 0.21 & 7082 & 1431 & 0.34 & 0.29 & 32825 & 897 & 19.08 & 126777 & 10779 & 85.92\
25-264-5 & sat & 0.21 & 7055 & 1431 & 3.12 & 0.30 & 33590 & 897 & 46.15 & 126973 & 10784 & 41.04\
25-264-6 & sat & 0.21 & 7712 & 1551 & 0.34 & 0.33 & 39015 & 932 & 69.81 & 128354 & 10850 & 12.67\
25-264-7 & sat & 0.21 & 7428 & 1496 & 0.13 & 0.30 & 36580 & 937 & 19.93 & 127106 & 10794 & 7.01\
25-264-8 & sat & 0.21 & 6603 & 1335 & 0.18 & 0.27 & 31561 & 896 & 10.32 & 124153 & 10687 & 9.69\
25-264-9 & sat & 0.21 & 6784 & 1350 & 0.19 & 0.27 & 35404 & 903 & 34.08 & 128423 & 10853 & 38.80\
25-264-10 & unsat & 0.21 & 6491 & 1296 & 0.04 & 0.30 & 33321 & 930 & 10.92 & 126999 & 10785 & 57.75\
25-264-11 & unsat & 0.12 & 1 & 0 & 0.00 & 0.28 & 37912 & 955 & 0.09 & 125373 & 10744 & 0.47\
25-264-12 & unsat & 0.16 & 1 & 0 & 0.00 & 0.29 & 39135 & 984 & 0.08 & 127539 & 10815 & 0.57\
25-264-13 & unsat & 0.12 & 1 & 0 & 0.00 & 0.29 & 35048 & 944 & 0.09 & 127026 & 10786 & 0.56\
25-264-14 & unsat & 0.23 & 5984 & 1210 & 0.07 & 0.28 & 31093 & 885 & 11.60 & 126628 & 10771 & 15.93\
Total & & & & & 8.93 & & & & 349.58 & & & 370.63\
We consider 15 instances from the 2008 CSP competition[^1]. Table \[tab:qcpExpr\] considers three settings: with its dual encoding for [`allDiff`]{} constraints, using only the order encoding (equivalent to using $\mathtt{int\_neq}$ constraints instead of [`allDiff`]{}), and Sugar. The results indicate that: (1) Application of using the dual representation for [`allDiff`]{} is 38 times faster and produces 20 times less clauses (in average) than when using the order-encoding alone (despite the need to maintain two encodings); (2) Without the dual representation, solving encodings generated by is only slightly faster but generates CNF encodings 4 times smaller (on average) than those generated by Sugar. Observe that 3 instances are found unsatisfiable by (indicated by a CNF with a single clause and no variables). We comment that Sugar preprocessing times are higher than those of and not indicated in the table.
To further appreciate the impact of the tool we describe results for three additional applications which shift the state-of-the-art with respect to what could previously be solved. The experiments clearly illustrate that decreases the size of CNF encodings as well as the subsequent SAT solving time.
####
In [@MacDougall2002] the authors conjecture that the $n$ vertex complete graph, $K_n$, for $n\geq 5$ has a vertex magic total labeling with magic constants for specific range of values of $k$, determined by $n$. This conjecture is proved correct for all odd $n$ and verified by brute force for $n=6$. We address the cases for $n=8$ and $n=10$ which involve 15 instances (different values of $K$) for $n=8$, and 23 (different values of $K$) for $n=10$. Starting from the simple constraint model (illustrated by the example in Figure \[fig:vmtl-instance\]), we add additional constraints to exploit that the graphs are symmetric: (1) We assume that the edge with the smallest label is $e_{1,2}$; (2) We assume that the labels of the edges incident to $v_1$ are ordered and hence introduce constraints $e_{1,2} < e_{1,3} < \cdots <
e_{1,n}$; (3) We assume that the label of edge $e_{1,3}$ is smaller than the labels of the edges incident to $v_2$ (except $e_{1,2}$) and introduce constraints accordingly. In this setting can solve all except 2 instances with a 4 hour timeout and Sugar can solve all except 4.
Table \[tab:k8\] depicts results for the 10 hardest instances for $K_8$ and the 20 hardest for $K_{10}$ with a 4 hour time-out. compilation times are on the order of 0.5 sec/instance for $K_8$ and 2.5 sec/instance for $K_{10}$. Sugar encoding times are slightly larger. The instances are indicated by the magic constant, $k$; the columns for and Sugar indicate SAT solving times (in seconds). The bottom two lines indicate average encoding sizes (numbers of clauses and variables).
[c|c|r|r]{} $K_8$& $k$ & &\
&143 & 1.26 & 2.87\
&142 & 10.14 & 1.62\
&141 & 7.64 & 2.94\
&140 & 14.68 & 6.46\
&139 & 25.60 & 6.67\
&138 & 12.99 & 2.80\
&137 & 22.91 & 298.58\
&136 & 14.46 & 251.82\
&135 & 298.54 & 182.90\
&134 & 331.80 & $\infty$\
\
&248& 402\
&5688&9370\
[c|c|r|r]{} $K_{10}$&k & &\
&277 & 5.31 & 9.25\
&276 & 7.11 & 9.91\
&275 & 13.57 & 19.63\
&274 & 4.93 & 9.24\
&273 & 45.94 & 9.03\
&272 & 22.74 & 86.45\
&271 & 7.35 & 9.49\
&270 & 6.03 & 55.94\
&269 & 5.20 & 11.05\
&268 & 94.44 & 424.89\
\
\
\
[r|r|r]{} k & &\
267 & 88.51 & 175.70\
266 & 229.80 & 247.56\
265 & 1335.31 & 259.45\
264 & 486.09 & 513.61\
263 & 236.68 & 648.43\
262 & 1843.70 & 6429.25\
261 & 2771.60 & 7872.76\
260 & 4873.99 & $\infty$\
259 & $\infty$ & $\infty$\
258 & $\infty$ & $\infty$\
\
&1229&1966\
&15529&25688\
The results indicate that the Sugar encodings are (in average) about 60% larger, while the average SAT solving time for the encodings is about 2 times faster (average excluding instances where Sugar times-out).
To address the two VMTL instances not solvable using the models described above ($K_{10}$ with magic labels 259 and 258), we partition the problem fixing the values of $e_{1,2}$ and $e_{1,3}$ and maintaining all of the other constraints. Analysis of the symmetry breaking constraints indicates that this results in 198 new instances for each of the two cases. The original VMTL instance is solved if any one of of these 198 instances is solved. So, we solve them in parallel. Fixing $e_{1,2}$ and $e_{1,3}$ “fuels” the compiler so the encodings are considerably smaller. The instance for $k= 259$ is solved in 1379.50 seconds where $e_{1,2}=1$ and $e_{1,3}=6$. The compilation time is 2.09 seconds and the encoding consists in 1056107 clauses and 14143 variables.
To the best of our knowledge, the hard instances from this suite are beyond the reach of all previous approaches to program the search for magic labels. The SAT based approach presented in [@Jaeger2010] cannot handle these.[^2] The comparison with Sugar indicates the impact of the compiler.
####
provide a comparison of several state-of-the-art solvers applied to the DNA word problem with a variety of encoding techniques. Their best reported result is a solution with 87 DNA words, obtained in 554 seconds, using an OPL [@opl] model with lexicographic order to break symmetry. In [@dnaWordPaper], the authors report a solution composed from two pairs of (t-part and m-part) sets ${\langle T_1,M_1 \rangle}$ and ${\langle T_2,M_2 \rangle}$ where $|T_1|=6$, $|M_1|=16$, $|T_2|=2$, $|M_2|=6$. This forms a set $S$ with $(6\times 16)+(2\times 6)=108$ DNA words. Marc van Dongen reports a larger solution with 112 words.[^3] Using , we find, in a fraction of a second, a template of size 14 and a map of size 8. This provides a solution of size $14\times 8=112$ to the DNA word problem. Running Comet (v2.0.1) we find a 112 word solution in about 10 seconds using a model by Håkan Kjellerstrand.[^4] We also prove that there does not exist a template of size 15 (0.15 seconds), nor a map of size 9 (4.47 seconds). These facts were unknown prior to . Proving that there is no solution to the DNA word problem with more than 112 words, not via the two part t-m strategy, is still an open problem.
####
(MBD) is an artificial intelligence based approach that aims to cope with the, so-called, diagnosis problem (e.g. [@Reiter87]). In [@MBD], we (with other researchers) focus on a notion of minimal cardinality MBD and apply to model and solve the instances of a standard MBD benchmark. Experimental evidence (see [@MBD]), indicates that our approach is superior to all existing algorithms for minimal cardinality MBD. We determine, for the first time, minimal cardinality diagnoses for the entire standard benchmark. Prior attempts to apply SAT for MBD (for example, by and where a MaxSAT solver is used) indicate that SAT solvers perform poorly on the standard benchmarks. So, really makes the difference.
Conclusion
==========
We introduce , a compiler to encode finite domain constraints to CNF. A key design point is to apply bit-level techniques, locally as prescribed by the word-level constraints in a model. Optimizations are based on equi-propagation and partial evaluation. Implemented in Prolog, compilation times are typically small (measured in seconds) even for instances which result in several millions of CNF clauses. On the other hand, the reduction in SAT solving time can be larger in orders of magnitude.
It is well-understood that making a CNF smaller is not the ultimate goal: often smaller CNF’s are harder to solve. Indeed, one often introduces redundancies to improve SAT encodings: so removing them is counter productive. Our experience is that reduces the size of an encoding in a way that is productive for the subsequent SAT solving. In particular, by removing variables that can be determined “at compile time” to be definitely equal (or definitely different) in any solution.
The simplification rules illustrated in Section \[sec:compiling\] apply standard constraint programming techniques (i.e. to reduce variable domains). However, equi-propagation is more powerful. It focuses, in general, in specializing the bit-level representation of the constraints in view of equations implied by the constraints. In this way it captures many of the well-known constraint programming preprocessing techniques, and more.
Future work will investigate: how to strengthen the implementation of equi-propagation using BDD’s and SAT solving techniques, how to improve the compiler implementation using better data-structures for the constraint store (for example applying a CHR based approach for the simplification rules), and how to enhance the underlying constraint language.
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[^1]: <http://www.cril.univ-artois.fr/CPAI08/>
[^2]: Personal communication (Gerold Jäger), March 2012.
[^3]: See <http://www.cs.st-andrews.ac.uk/~ianm/CSPLib/>.
[^4]: See <http://www.hakank.org/comet/word_design_dna1.co>.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three-fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank-based selection model, the Aiello-Chung-Lu models, the generalized linear preference model, directed scale-free graphs, the Cooper-Frieze model, and random unordered increasing $k$-trees. Our results shed light on why the small-world phenomenon is observed in so many real-world graphs.'
author:
- |
Abbas Mehrabian[^1]\
[Department of Combinatorics and Optimization]{}\
[University of Waterloo]{}\
[`[email protected]`]{}
bibliography:
- 'webgraph.bib'
title: |
Justifying the small-world phenomenon\
via random recursive trees
---
Introduction
============
‘Small-world phenomenon’ refers to a striking pattern observed in numerous real-world graphs: most pairs of vertices are connected by a path whose length is considerably smaller than the size of the graph. Travers and Milgram [@milgram] in 1969 conducted an experiment in which participants were asked to reach a target person by sending a chain letter. The average length of all completed chains was found to be 6.2, an amazingly small number, hence the phrase ‘six degrees of separation.’ The Webgraph is a directed graph whose vertices are the static web pages, and there is an edge joining two vertices if there is a hyperlink in the first page pointing to the second page. Broder, Kumar, Maghoul, Raghavan, Rajagopalan, Stata, Tomkins, and Wiener [@Broderetal] in 1999 crawled about 200 million web pages and found that the expected shortest-directed-path distance between two random web pages (when a path exists at all) is 16.18; this figure is 6.83 in the corresponding underlying undirected graph.
Backstrom, Boldi, Rosa, Ugander, and Vigna [@four_degrees_separation] studied the Facebook graph in May 2011, which had about 721 million vertices. The vertices of this graph are people, and two of them are joined by an edge if they are friends on Facebook. The diameter of the giant component of this graph was found to be 41, and the average distance between reachable pairs was found to be around 4.74. For other examples, see, e.g., Tables 1 and 2 in [@tables], Table 8.1 in [@newman_book] or Table 4 in [@google+].
Another fascinating observation on many real-world graphs is that their degree sequences are heavy-tailed and almost obey a power law. As Erdős-Rényi random graphs do not satisfy this property, in recent years a great deal of research has been built around defining new probabilistic models, aiming at capturing the aforementioned and other properties of real-world graphs (see, e.g., Bonato [@Bonato Chapter 4] or Chakrabarti and Faloutsos [@chakrabarti_book Part II]). Lots of models have been defined so far, yet very few rigorously analysed.
The *diameter* of an undirected graph is the maximum shortest-path distance between any two vertices. It is a well known metric quantifying how ‘small-world’ the graph is; informally speaking, it measures how quickly one can get from one ‘end’ of the graph to the other. The diameter is related to various processes, e.g. it is within a constant factor of the memory complexity of the depth-first search algorithm. Also, it is a natural lower bound for the mixing time of any random walk ([@mixing_times Section 7.1]) and the broadcast time of the graph ([@broadcast_times Section 3]). Another well studied metric is the *average diameter* of a graph, which is the expected value of the shortest-path distance between two random vertices. Despite the fact that these are two of the most studied parameters of a network, for several models introduced in the literature the degree sequence has been proved to be power-law, but no sublinear upper bound for the diameter or average diameter is known.
We fill in this gap by presenting a new technique for establishing upper bounds for diameters of certain random graph models, and demonstrate it by proving logarithmic upper bounds for the diameters of a variety of models, including the following well known ones: the forest fire model [@forest_fire_journal], the copying model [@copying_model_def], the PageRank-based selection model [@pagerank_model_journal], the Aiello-Chung-Lu models [@acl_model], the generalized linear preference model [@bu_towsley], directed scale-free graphs [@directed_scale_free], the Cooper-Frieze model [@cooper-frieze-model], and random unordered increasing $k$-trees [@randomktree_def]. This means that in each of these models, for *every* pair $(u,v)$ of vertices there exists a very short $(u,v)$-path, a path connecting $u$ and $v$ whose length is logarithmic in the number of vertices. This implies, in particular, that the average diameters of these models are logarithmic. We also prove polylogarithmic upper bounds for the diameter of the preferential attachment model with random initial degrees [@randominitial] in the case that the initial degrees’ distribution has an exponential decay. Prior to this work no sublinear upper bound was known even for the average diameter of any of these models. (This claim can quickly be verified by looking at Table 8.2 from the recent monograph [@chakrabarti_book], or [@Chakrabarti Table III], or the table in [@web_survey p. 162]: each cited table contains a summary of known results on the diameter and other properties of several real-world network models.)
This is the first paper that proves logarithmic upper bounds for such a wide range of random graph models. Our results shed light on why the small-world phenomenon is observed in so many real-world graphs. At their core, our arguments are based on the fact that in all models considered here, there is a sort of ‘rough uniformity’ for the (random) destination of each new link. Thus, we may expect that for any growing network in which the endpoints of new links are chosen according to a probability distribution that is ‘not too biased,’ i.e. does not greatly favour some vertices over others, the diameter grows at most logarithmically. We believe this is the primary reason that most real-world graphs are small-world.
From a wider perspective, it would be appealing to have a mathematical theory for characterizing those evolving random graphs which have logarithmic diameters. This paper is a fundamental step in building this theory. The technique developed here gives unified simple proofs for known results, provides lots of new ones, and will help in proving many of the forthcoming network models are small-world. We hope this theory will be developed further to cover other network models, e.g. spatial models [@spatial_survey], as well.
Our technique and outline of the paper
--------------------------------------
We study *evolving models* (also called *on-line* or *dynamic* models), i.e. the graph changes over time according to pre-defined probabilistic rules, and we are interested in the long-term structure of this evolving graph. We assume that in discrete time-steps new vertices and edges appear in the graph, but no deletion occurs. The goal is to show that the evolving graph at time $n$ has diameter $O(\log n)$ *asymptotically almost surely (a.a.s.)*, that is, with probability tending to 1 as $n$ goes to infinity. In all models considered here, the Chernoff bound implies that the number of vertices at time $n$ is $\Omega(n)$ a.a.s., hence we will conclude that a.a.s. the evolving graph has diameter $O(\log n)$ when it has $n$ vertices.
Let us informally explain our technique. In this section when we write a certain graph/tree has a logarithmic diameter/height, we mean its diameter/height has a logarithmic upper bound. An important object in this paper is a *random recursive tree*, defined as follows: there exists a single node at time 0, and in every time-step $t=1,2,\dots$, a new node is born and is joined to a uniformly at random (*u.a.r.*) node of the current tree. It is known that when this tree has $n$ nodes, a.a.s. its height is $\Theta(\log n)$ [@random_recursive_trees]. The technique consists of two main steps: first, we build a *coupling* between our evolving random graph and some variant of a random recursive tree in such a way that the diameter of the graph is dominated by a linear function of the height of the tree, and then we prove that a.a.s. the tree has a logarithmic height. The second step is usually straightforward (see Lemma \[lem:multichildren\] for an example) and the tricky part is defining the ‘coupled’ tree. Let us give some examples.
To distinguish between a vertex of the graph and that of the tree, the latter is referred to as a ‘node’. For models studied in Section \[sec:uniformly\], namely the forest fire model [@forest_fire_journal], the copying model [@copying_model_def], and the PageRank-based selection model [@pagerank_model_journal], the coupled tree is a random recursive tree with weighted edges, which has the same node set as the vertex set of the graph. Let us assume that the initial graph has one vertex, so the tree starts with a single node corresponding to this initial vertex. These models evolve as follows: in every time-step a new vertex, say $v$, is born and is joined to some random vertices, say $w_1,\dots,w_d$, in the existing graph in such a way that for each $j$, vertex $w_j$ has a short distance to a u.a.r. vertex $x_j$ of the existing graph. We let the coupled tree evolve as follows: a new node $v$ is born and is joined to node $x_1$ in the existing tree, and the weight of the edge $vx_1$ in the tree is set to be the distance between $v$ and $x_1$ in the graph. Then by induction, the distance in the graph between the initial vertex and $v$ is at most the weighted distance in the tree between the initial node and node $v$. Moreover, by construction, the tree evolves as a weighted random recursive tree. Finally, examining the distribution of the weights carefully, we prove that a.a.s. the obtained evolving tree has a logarithmic weighted height.
We remark that in the argument outlined above, we may *ignore* the other neighbours $w_2,\dots,w_d$ of the new vertex; only the first edge $vw_1$ is effectively used for bounding the diameter. This is a repeating phenomenon in our arguments. An interesting implication is that one can quickly and locally build a spanning tree with logarithmic diameter as the graph evolves. This might have algorithmic applications.
In Section \[sec:pref\] we study models that incorporate preferential attachment. As a simple example, consider the following evolving rule: in every time-step, a vertex is chosen using preferential attachment, i.e. the probability of choosing a specific vertex is proportional to its degree, then a new vertex is born and is joined to the chosen vertex. It is easy to observe that sampling a vertex using preferential attachment can be done by choosing a u.a.r. endpoint of a u.a.r. edge of the graph. Using this sampling procedure, the evolving rule can be re-stated as follows: in every time-step, an edge $e$ is sampled u.a.r., then a random endpoint $w$ of $e$ is chosen, then a new vertex $v$ is born and is joined to $w$. One of the main novel ideas in this paper is introducing *edge trees* and employing them in this context. An *edge* tree is a tree whose nodes correspond to the *edges* of the evolving graph. We couple the evolving graph with an edge tree, and let the edge tree evolve as follows: in the corresponding time-step a new node $vw$ is born and is joined to a u.a.r. node $e$. Clearly, the edge tree indeed grows like a random recursive tree as the graph evolves, hence its height can be easily bounded. Moreover, the constructed coupling implies that the graph’s diameter is dominated by a linear function of the tree’s height, so we conclude that a.a.s. the graph has logarithmic diameter. If a reader wants to read only one theorem from this paper, Theorem \[thm:pref\] should be the one, which formalizes and generalizes this idea and illustrates the crux of our technique without having too much details. This theorem states that Model \[def\_pref\], a generic model based on the preferential attachment scheme, has logarithmic diameter; the Aiello-Chung-Lu models [@acl_model] and the generalized linear preference model [@bu_towsley] are then proved to be special cases of this model.
In the generalized linear preference model, the probability of choosing a specific vertex is proportional to a linear function, say $ax + b$, of the vertex’s degree $x$. Assuming $a$ and $b$ are even positive integers, we handle this by putting $a$ multiple edges corresponding to each edge, and putting $b/2$ loops at each vertex. Then choosing a u.a.r. endpoint of a u.a.r. edge in the new graph corresponds to sampling according to the linear function of the degrees in the old graph. See Theorem \[thm:remark\] for details. At the end of Section \[sec:pref\], we also analyse the ‘preferential attachment with random initial degrees’ model [@randominitial], and show that if the initial degrees’ distribution has an exponential decay, then a.a.s. the generated graph has a polylogarithmic diameter. This is straightforward to prove using the developed machinery, see Theorem \[thm:parid\].
In Section \[sec:directed\] we study the ‘directed scale-free graphs’ [@directed_scale_free]. The diameter of a directed graph is defined as that of the underlying undirected graph (we follow [@forest_fire_journal] in this regard). When constructing a graph using this model, one may sample vertices according to linear functions of either out-degrees or in-degrees, and the two functions have different constant terms. To cope with this, we introduce ‘[headless]{}’ and ‘[tailless]{}’ edges. These are dummy edges in the graph that do not play any role in connecting the vertices, but they appear in the tree and their job is just to adjust the selection probabilities. Details can be found in Theorem \[thm:directed\], which states that a generalized version of directed scale-free graphs has logarithmic diameter.
In Section \[combine\] we study the Cooper-Frieze model [@cooper-frieze-model], which is the most general evolving model known to have a power-law degree sequence. In this model, the neighbours of a new vertex can be chosen either according to degrees or uniformly at random. For dealing with this intricacy, we couple with a tree having two types of nodes: some correspond to the vertices, and the others correspond to the edges of the graph. A multi-typed random recursive tree is obtained, in which at every time-step a new node is born and is joined to a node chosen u.a.r. from all nodes of a certain type. We prove that a.a.s this tree has a logarithmic height, and by using the coupling’s definition we conclude that a.a.s the Cooper-Frieze model has a logarithmic diameter (see Theorem \[thm:cooper-frieze\]). For this model, proving that the tree has logarithmic height is actually the harder step.
Finally, in Section \[sec:other\] we prove logarithmic upper bounds for three further models: graphs generated by the pegging process [@pegging_def], random unordered increasing $k$-trees [@randomktree_def], and random $k$-Apollonian networks [@high_RANs]. For the first and last of these, it is already known that a.a.s. the diameter is $O(\log n)$, but our approach gives a shorter proof.
Related work
------------
Surprisingly few results are known about the diameters of evolving random graph models. Chung and Lu [@coupling_online_offline] defined an evolving (online) and a non-evolving (offline) model. They state that ‘The online model is obviously much harder to analyze than the offline model’, and hence analyse the former by coupling it with the latter, which had been analysed before. The difficulty of analysing evolving models over non-evolving ones arises perhaps due to the dependencies between edges in the former models.
The evolving model that has attracted the most attention is the *linear preference model*, in which in every step a new vertex is born and is joined to a fixed number of old vertices. This is done in such a way that the probability of joining to a given vertex is proportional to a linear function of its degree. A logarithmic upper bound has been proved for the diameter of this model, and sharper results are known in various special cases [@random_recursive_trees; @diameter_preferential_attachment; @diameters_pa; @vander]. See the remark before Theorem \[thm:remark\] for details. When the new vertex is joined to exactly one vertex in the existing graph (so the resulting evolving graph is always a tree), a general technique based on branching processes is developed by Bhamidi [@bhamidi], using which he proved the diameter of a variety of preferential attachment trees is a.a.s $\Theta(\log n)$.
Chung and Lu [@coupling_online_offline] used couplings with a non-evolving random graph model to prove that the diameter of a certain growth-deletion model is $\Theta(\log n)$. On one hand, their model is more general than the models we consider, as they allow vertex and edge deletions, but on the other hand, their result holds for graphs with at least $\omega (n \log n)$ edges whereas our results covers graphs with $O(n)$ edges, too. Moreover, their proof is quite technical and uses general martingale inequalities. See the remark before Theorem \[thm:remark\] for details.
Other evolving models whose diameters have been studied include the Fabrikant-Koutsoupias-Papadimitriou model [@fkp_not_powerlaw], protean graphs [@protean_diameter], the geometric preferential attachment model [@geometric2; @geometric_hybrid], the spatial preferred attachment model [@spa_typical], random Apollonian networks [@we_rans_abstract; @CF13; @istvan], and random surfer Webgraphs [@we_surfer]. See [@inhom Section 14] and [@vander] for collections of results on diameters of non-evolving models.
Some of the above papers estimate the diameter up to constant factors, or even up to $1+o(1)$ factors. Our approach gives logarithmic upper bounds that are perhaps not tight, but on the positive side, it is applicable to a broad variety of models, including those incorporating preferential attachment. Another advantage of our technique is simplicity: all proofs given here are elementary and fairly short, and the only probabilistic tools we use are couplings and the Chernoff bound. The third advantage of our technique is that the constant factor it gives (hidden in the $O(\log n)$ notion) is typically small: for all the models studied here, the constant is at most 20.
Let us emphasize that we are concerned with upper bounds only and no lower bound for the diameter is proved in this paper. We believe that for all considered models, at least in the special case when the evolving graph is always a tree, the diameter is $\Theta(\log n)$.
Notation
--------
In this paper graphs can be directed or undirected, but all trees are undirected. The distance between two vertices is the number of edges in the shortest path connecting them. If the graph is directed, the direction of edges is ignored when calculating the distance. The *diameter* of a graph is the maximum distance between any two vertices. We will work with (weakly) connected graphs only, so the diameter is always well defined. Graphs may have parallel edges and loops (note that adding these does not change the diameter). All considered graphs are finite and rooted, i.e. there is a special vertex which is called the root. The *depth* of a vertex is its distance to the root, and the *height* of a graph is the maximum depth of its vertices. Clearly the diameter is at most twice the height, and we always bound the diameter by bounding the height. The depth of vertex $v$ in graph $G$ is denoted by ${\operatorname{depth}}(v,G)$. All logarithms are in the natural base. Let us denote ${{\mathbb{N}}}=\{1,2,\dots\}$, ${{\mathbb{N}_0}}=\{0,1,2,\dots\}$, and $[n]=\{1,2,\dots,n\}$.
A *growing graph* is a sequence $(G_t)_{t=0}^{\infty}$ of random graphs such that $G_t$ is a subgraph of $G_{t+1}$ for all $t\in{{\mathbb{N}_0}}$. We always assume that $G_0$ has size $O(1)$. A *growing tree* is defined similarly. This sequence can be thought of as a graph ‘growing’ as time passes, and $G_t$ is the state of the graph at time $t$. We write informal sentences such as ‘at time $t$, a new vertex is born and is joined to a random vertex of the existing graph,’ which formally means ‘$G_t$ is obtained from $G_{t-1}$ by adding a new vertex and joining it to a random vertex of $G_{t-1}$.’
Basic technique {#sec:uniformly}
===============
The following lemma exemplifies proving a variant of a random recursive tree has logarithmic height. The argument here is inspired by a proof in Frieze and Tsourakakis [@first]. We will use a simple inequality: let $a_1,\dots,a_m$ be positive numbers, and let $h\in[m]$. Then observe that $$\sum_{1 \le t_1 < \dots < t_h\le m}\ \left( \prod_{k=1}^{h}a_{t_k} \right)
< \frac{1}{h!}
\left(\sum_{i=1}^{m}a_i\right)^{h} \:.$$
\[lem:multichildren\] Let $(A_t)_{t\in{{\mathbb{N}}}}$ be a sequence of ${{\mathbb{N}}}$-valued random variables. Consider a growing tree $(T_t)_{t=0}^{\infty}$ as follows. $T_0$ is arbitrary. At each time-step $t\in{{\mathbb{N}}}$, a random vector $(W_1,W_2,\dots,W_{A_t}) \in V(T_{t-1})^{A_t}$ is chosen in such a way that for each $i\in[A_t]$ and each $v\in V(T_{t-1})$, the marginal probability $\p{W_i=v}$ equals $|V(T_{t-1})|^{-1}$. In other words, each $W_i$ is a node of $T_{t-1}$ sampled uniformly; however, the $W_j$’s may be correlated. Then $A_t$ new nodes $v_1,\dots,v_{A_t}$ are born and $v_i$ is joined to $W_i$ for each $i\in[A_t]$. Let $\ell=\ell(n),u=u(n)$ be positive integers such that $\ell\le A_t \le u$ for all $t\in[n]$. Then the height of $T_n$ is a.a.s. at most $(u/\ell) e\log n + 2ue+O(1)$.
Note that we do not require any independence for $(A_t)_{t\in{{\mathbb{N}}}}$. In particular they can be correlated and depend on the past and the future of the process.
Let $n_0=|V(T_0)|$. For a given integer $h = h(n)$, let us bound the probability that $T_n$ has a node at depth exactly $h+n_0$. Given a sequence $1 \le t_1 < t_2<\dots < t_h\le n$, the probability that there exists a path $v_{t_1}v_{t_2}\dots v_{t_h}$ in $T_n$ such that $v_{t_j}$ is born at time $t_j$ is at most $$u^{h} \prod_{k=2}^{h}\frac{1}{n_0+ \ell\cdot(t_k-1)} \:,$$ since there are at most $u^{h}$ choices for $(v_{t_1},\dots,v_{t_h})$, and for each $k=2,3,\dots,h$, when $v_{t_k}$ is born, there are at least $n_0+ \ell\cdot(t_k-1)$ nodes available for it to join to. By the union bound, the probability that $T_n$ has a node at depth $h+n_0$ is at most $$\begin{aligned}
u^{h}\sum_{1 \le t_1 < \dots < t_h\le n}\ \left( \prod_{k=2}^{h}\frac{1}{n_0+ \ell\cdot(t_k-1)}\right)
& < \frac{u^{h}}{h!}\left(1 + \sum_{j=1}^{n-1}\frac{1}{n_0+\ell j}\right)^{h} \\
& < \left(\frac{u e}{h}
\cdot \left( \frac{\log n}{\ell} + 2\right) \right)^h
\bigg/\sqrt{2\pi h} \:,\end{aligned}$$ where we have used Stirling’s formula and the inequality $1 + \frac12 + \frac13 + \dots + \frac{1}{n-1} < 1 +\log n$. Putting $h \ge (u/\ell) e\log n + 2ue$ makes this probability $o(1)$. Hence a.a.s. the height of $T_n$ is at most $(u/\ell) e\log n + 2ue + n_0$, as required.
Given $p\in(0,1]$, let ${\operatorname{Geo}}(p)$ denote a geometric random variable with parameter $p$; namely $\p{{\operatorname{Geo}}(p) = k} = (1-p)^kp$ for every $k\in{{\mathbb{N}_0}}$. The first model we study is the *basic forest fire model* of Leskovec, Kleinberg, and Faloutsos [@forest_fire_journal Section 4.2.1].
\[def:forest\_fire\] Let $p,q\in[0,1]$ be arbitrary. We build a growing directed graph as follows. $G_0$ is an arbitrary weakly connected directed graph. At each time-step $t\in{{\mathbb{N}}}$, a new vertex $v$ is born and edges are created from it to the existing graph using the following process.
1. All vertices are marked ‘unvisited.’ An *ambassador vertex* $W$ is sampled uniformly from the existing graph.
2. Vertex $v$ is joined to $W$ and $W$ is marked as ‘visited.’
3. We independently generate two random variables $X={\operatorname{Geo}}(p)$ and $Y={\operatorname{Geo}}(q)$. We randomly select $X$ unvisited out-neighbours and $Y$ unvisited in-neighbours of $W$. If not enough unvisited in-neighbours or out-neighbours are available, we select as many as we can. Let $W_1,\dots,W_{Z}$ denote these vertices.
4. Vertex $v$ is joined to $W_1, \dots, W_{ Z}$, then we apply steps 2–4 recursively to each of $W_1, \dots, W_{Z }$.
Consider $(G_t)_{t=0}^{\infty}$ generated by Model \[def:forest\_fire\]. A.a.s. for every vertex $v$ of $G_n$ there exists a directed path of length at most $e\log n + O(1)$ connecting $v$ to some vertex of $G_0$. In particular, a.a.s. the diameter of $G_n$ is at most $2e\log n + O(1)$.
We define a growing tree $(T_t)_{t=0}^{\infty}$ in such a way that $T_t$ is a spanning tree of $G_t$ for all $t\in{{\mathbb{N}_0}}$: $T_0$ is an arbitrary spanning tree of $G_0$. For every $t\in{{\mathbb{N}}}$, if $v$ is the vertex born at time $t$ and $w$ is the corresponding ambassador vertex, then $v$ is joined only to $w$ in $T_t$. By Lemma \[lem:multichildren\], a.a.s. the height of $T_n$ is at most $e\log n + O(1)$.
We next study the *linear growth copying model* of Kumar, Raghavan, Rajagopalan, Sivakumar, Tomkins, and Upfal [@copying_model_def Section 2.1].
\[def:copying\] Let $p\in[0,1]$ and $d\in{{\mathbb{N}}}$. We build a growing directed graph in which every vertex has out-degree $d$, and there is a fixed ordering of these $d$ edges. $G_0$ is an arbitrary weakly connected directed graph with all vertices having out-degree $d$. In each time-step $t\in{{\mathbb{N}}}$ a new vertex $v$ is born and $d$ outgoing edges from $v$ to the existing graph are added, as described below. An ambassador vertex $W$ is sampled uniformly from the existing vertices. For $i\in[d]$, the head of the $i$-th outgoing edge of $v$ is chosen as follows: with probability $p$, it is a random vertex of the existing graph sampled uniformly, and with probability $1-p$ it is the head of the $i$-th outgoing edge of $W$, in which case we say $v$ has *copied* the $i$-th outgoing edge of $W$.
A.a.s. the diameter of $G_n$ defined in Model \[def:copying\] is at most $4e\log n + O(1)$.
We inductively define a growing tree $(T_t)_{t=0}^{\infty}$ in such a way that the node set of $T_t$ equals the vertex set of $G_t$ for all $t$. We prove by induction that for each $v\in V(G_t)$, ${\operatorname{depth}}(v,G_t) \le 2 {\operatorname{depth}}(v,T_t)$. Let $T_0$ be a breadth-first search tree of $G_0$, rooted at the root of $G_0$. For each $t\in{{\mathbb{N}}}$, let $v$ be the vertex born at time $t$, and let $w$ be the corresponding ambassador vertex. We consider two cases:
Case 1. $v$ copies at least one outgoing edge of $w$.
: In this case, we join $v$ to $w$ in $T_t$. Since $v$ and $w$ have distance 2 in $G_t$, ${\operatorname{depth}}(v,G_t)\le {\operatorname{depth}}(w,G_t)+2$, so by the induction hypothesis for $w$, $${\operatorname{depth}}(v,G_t)
\le {\operatorname{depth}}(w,G_t) + 2
\le 2 {\operatorname{depth}}(w,T_t) + 2
= 2 {\operatorname{depth}}(v,T_t)
\:,$$ as required.
Case 2. $v$ does not copy any outgoing edge of $w$.
: Let $x$ denote the head of the first outgoing edge of $v$. In this case, we join $v$ to $x$ in $T_t$. Using the induction hypothesis for $x$, $$\begin{aligned}
{\operatorname{depth}}(v,G_t)
\le {\operatorname{depth}}(x,G_t) + 1
& \le 2 {\operatorname{depth}}(x,T_t) + 1 \\
& < 2 {\operatorname{depth}}(x,T_t) + 2
= 2 {\operatorname{depth}}(v,T_t)
\:,\end{aligned}$$ as required.
Notice that in either case, node $v$ is joined to a node of $T_{t-1}$ sampled uniformly. By Lemma \[lem:multichildren\], a.a.s. the height of $T_n$ is at most $e\log n + O(1)$, so a.a.s. the diameter of $G_n$ is at most $4e \log n + O(1)$.
Sampling neighbours using PageRank {#sec:pagerank}
----------------------------------
In this section we study a model in which the neighbours of each new vertex are chosen according to the PageRank distribution. We recall the definition of PageRank.
\[def:pagerank\] Let $q\in[0,1]$ and let $G$ be a directed graph. *PageRank* is the unique probability distribution $\pi_q : V(G) \to [0,1]$ that satisfies $$\label{eq:pagerank}
\pi_q (v) = \frac{1-q}{|V(G)|} + q \sum_{u\in V(G)} \frac{\pi_q(u) \cdot \#(uv)}{{\operatorname{out-deg}}(u)} \:.$$ Here $\#(uv)$ denotes the number of copies of the directed edge $uv$ in the graph (which is zero if there is no edge from $u$ to $v$), and ${\operatorname{out-deg}}(u)$ denotes the out-degree of $u$.
PageRank is used as a ranking mechanism in Google [@google]. More details and applications can be found in [@pagerank_deep].
\[def:pagerankmodel\] Let $p_a,p_b,p_c$ be nonnegative numbers summing to 1, let $q\in[0,1]$ and $d\in{{\mathbb{N}}}$. We build a growing directed graph $(G_t)_{t=0}^{\infty}$ in which every vertex has out-degree $d$. $G_0$ is a weakly connected directed graph with all vertices having out-degree $d$. In each time-step $t\in{{\mathbb{N}}}$, a new vertex is born and $d$ outgoing edges from it to the existing graph are added. The heads of the new edges are chosen independently. For choosing the head of each edge, we perform one of the following operations, independently of previous choices.
(a) With probability $p_a$, the head is a vertex sampled uniformly from the existing graph.
(b) With probability $p_b$, it is the head of an edge sampled uniformly from the existing graph.
(c) With probability $p_c$, it is a vertex sampled from the existing graph using $\pi_q$.
Model \[def:pagerankmodel\] is defined by Pandurangan, Raghavan, and Upfal [@pagerank_model_journal Section 2]. They call it the *hybrid selection model*. For the special case $p_b=0$, which is referred to as the *PageRank-based selection model*, it has been proved using a different argument that a.a.s. the diameter is $O(\log n)$ [@we_surfer]. For bounding the diameter of Model \[def:pagerankmodel\] we will need a lemma.
\[lem:cm\] Assume that $q<1$. There exists a random variable $L$ such that the head of each new edge in Model \[def:pagerankmodel\] can be obtained by sampling a vertex $W$ uniformly from the existing graph and performing a simple random walk of length $L$ starting from $W$. Moreover, $L$ is stochastically smaller than $1+{\operatorname{Geo}}(1-q)$.
We claim that $$L = \begin{cases}
0 & \mathrm{with\ probability\ } p_a \:,\\
1 & \mathrm{with\ probability\ } p_b \:,\\
{\operatorname{Geo}}(1-q) & \mathrm{with\ probability\ } p_c \:.
\end{cases}$$ If we sample a vertex uniformly and perform a random walk of length 1, then since all vertices have the same out-degree, the last vertex of the walk is the head of a uniformly sampled edge.
So it suffices to show that if we sample a vertex uniformly and perform a random walk of length ${\operatorname{Geo}}(1-q)$, the last vertex of the walk has distribution $\pi_{q}$. This was first observed in [@pagerank_random_surfer]. Let ${\tau}\in[0,1]^{V(G)}$ denote the probability distribution of the last vertex, let $\mathcal{P}$ denote the probability transition matrix of the simple random walk, and let $\sigma = \big[1/|V(G)|,1/|V(G)|,\dots,1/|V(G)|\big]^T$ be the uniform distribution. Then we have $$\tau
= \sum_{k=0}^{\infty} q^k (1-q) \mathcal{P}^k \sigma
= (1-q) \sigma +
q\mathcal{P}
\left(\sum_{k=1}^{\infty} q^{k-1} (1-q) \mathcal{P}^{k-1} \sigma\right)
=
(1-q) \sigma
+
q\mathcal{P} \tau \:.$$ Comparing with (\[eq:pagerank\]) and noting that the stationary distribution of an Ergodic Markov chain is unique, we find that $\tau=\pi_{q}$, as required.
[The Chernoff bound]{} Given $n\in{{\mathbb{N}_0}}$ and $p\in[0,1]$, let ${\operatorname{Bin}}(n,p)$ denote a binomial random variable with parameters $n$ and $p$. We refer to the following inequality, valid for every $\varepsilon\ge0$, as the *Chernoff bound*. See Motwani and Raghavan [@rand_algs Theorem 4.2] for a proof. $$\p{{\operatorname{Bin}}(n,p)<(1-\varepsilon)np} \le \exp(-\varepsilon^2 np / 2)\:.$$
If $q<1$, then a.a.s. the diameter of $G_n$ defined in Model \[def:pagerankmodel\] is at most $18 \log n /(1-q)$.
A *weighted tree* is a tree with nonnegative weights assigned to the edges. The *weighted depth* of a node $v$ is defined as the sum of the weights of the edges connecting $v$ to the root. We define a growing weighted tree $(T_t)_{t=0}^{\infty}$ such that for all $t$, the node set of $T_t$ equals the vertex set of $G_t$. We prove by induction that the depth of each vertex in $G_t$ is at most its weighted depth in $T_t$. Let $T_0$ be a breadth-first search tree of $G_0$ rooted at the root of $G_0$, and let all edges of $T_0$ have unit weights. Assume that when obtaining $G_t$ from $G_{t-1}$, the heads of the new edges are chosen using the procedure described in Lemma \[lem:cm\]. For every $t\in{{\mathbb{N}}}$, if $v$ is the vertex born at time $t$, and $w$ and $l$ are the first sampled vertex and length of the first random walk taken, respectively, then $v$ is joined only to $w$ in $T_t$ and the weight of the edge $vw$ is set to $l+1$. Note that the edge weights are mutually independent. Since the distance between $v$ and $w$ in $G_t$ is at most $l+1$, by induction the weighted depth of $v$ in $T_t$ is at most the depth of $v$ in $G_t$. We show that a.a.s. the weighted height of $T_n$ is at most $ 9 \cdot \log n /(1-q)$, and this completes the proof.
By Lemma \[lem:multichildren\], a.a.s. the (unweighted) height of $T_n$ is less than $ 1.001e\log n$. We prove that any given node at depth at most $ 1.001e\log n$ of $T_n$ has weighted depth at most $ 9(\log n)/(1-q)$ with probability $1-o(1/n)$, and then the union bound completes the proof. Let $v$ be a node of $T_n$ at depth $h$, where $h \le 1.001 e \log n$. By Lemma \[lem:cm\], the weighted depth of $v$ equals the sum of $h$ independent random variables, each stochastically smaller than $2+{\operatorname{Geo}}(1-q)$. The probability that the sum of $h$ independent random variables distributed as $2+{\operatorname{Geo}}(1-p)$ is greater than $9\log n / (1-q)$ is $$\p{{\operatorname{Bin}}\left(\frac{9\log n}{1-q} - h , 1-q\right) < h}\:.$$ Since $h \le 1.001 e \log n$, using the Chernoff bound we infer that this probability is less than $
\exp\left(-0.566^2 \times 6.27 (\log n)/2\right)
< n^{-1.004}
$.
Incorporating preferential attachment: edge trees {#sec:pref}
=================================================
In this section we study models incorporating preferential attachment. We first define a model that has a lot of flexibility (Model \[def\_pref\]) and prove it has logarithmic diameter. Then we reduce Models \[def\_acld\], \[def\_butowsley\], and \[def\_aclc\] to this model.
\[def\_pref\] Let $(A_t,B_t)_{t=1}^{\infty}$ be sequences of ${{\mathbb{N}_0}}$-valued random variables. Consider a growing undirected graph $(G_t)_{t=0}^{\infty}$ as follows. $G_0$ is an arbitrary connected graph with at least one edge. At each time-step $t\in{{\mathbb{N}}}$, $G_t$ is obtained from $G_{t-1}$ by doing a vertex operation and an edge operation, as defined below.
In a *vertex operation*, if $A_t>0$, a new vertex is born and $A_t$ edges are added in the following manner: we sample an edge uniformly from $G_{t-1}$, choose one of its endpoints arbitrarily, and join it to the new vertex. For the other $A_t - 1$ new edges, one endpoint is the new vertex, and the other endpoint is arbitrary (can be the new vertex as well). If $A_t=0$ then the vertex operation does nothing.
In an *edge operation*, we independently sample $B_t$ edges uniformly from $G_{t-1}$ and we choose an arbitrary endpoint of each sampled edge. Then we add $B_t$ new edges, joining these vertices to arbitrary vertices of $G_{t-1}$.
Note that we do not require any independence for $(A_t,B_t)_{t\in{{\mathbb{N}}}}$. In particular they can be correlated and depend on the past and the future of the process.
A novel idea in this paper is introducing edge trees: these are trees coupled with graphs whose nodes correspond to the edges of the graph. The following theorem demonstrates their usage.
\[thm:pref\] Let $\ell=\ell(n),u=u(n) \in {{\mathbb{N}}}$ be such that $\ell\le A_t + B_t \le u$ for every $t\in{{\mathbb{N}}}$. A.a.s. the graph $G_n$ generated by Model \[def\_pref\] has diameter at most $ 4e(u/\ell) \log n + 8eu + O(1)$.
We define the *depth* of an edge $xy$ as $1+\min\{{\operatorname{depth}}(x),{\operatorname{depth}}(y)\}$. We inductively define a growing tree $(T_t)_{t=0}^{\infty}$ such that for all $t\in{{\mathbb{N}_0}}$, $V(T_t) = E(G_t) \cup \{{\aleph}\}$. Here ${\aleph}$ denotes the root of $T_t$, which has depth 0. We prove by induction that for all $e\in E(G_t)$, ${\operatorname{depth}}(e, G_t) \le 2 {\operatorname{depth}}(e,T_t)$. Let $H$ be the graph obtained from $G_0$ by adding an edge labelled ${\aleph}$ incident to its root. Let $T_0$ be a breadth-first tree of the line graph of $H$ rooted at ${\aleph}$. (The *line graph* of a graph $H$ is a graph whose vertices are the edges of $H$, and two edges are adjacent if they have a common endpoint.) Note that ${\operatorname{depth}}({\aleph},T_0)=0$ and ${\operatorname{depth}}(e,T_0)={\operatorname{depth}}(e,G_0)$ for every $e\in E(G_0)$.
Given $T_{t-1}$, we define $T_t$ and prove the inductive step. First, consider a vertex operation with $A_t>0$. Let $v$ be the new vertex, $e_1$ be the sampled edge, and $w_1$ be the chosen endpoint of $e_1$. Notice that ${\operatorname{depth}}(v,G_t) \le {\operatorname{depth}}(e_1,G_t)+1$. In $T_t$, we join the $A_t$ edges incident with $v$ to $e_1$. For any such edge $e$ we have $$\begin{aligned}
{\operatorname{depth}}(e,G_t)
\le {\operatorname{depth}}(v,G_t) + 1
\le {\operatorname{depth}}(e_1,G_t) + 2
\le 2 {\operatorname{depth}}(e_1,T_t) + 2
= 2 {\operatorname{depth}}(e,T_t),\end{aligned}$$ where we have used the inductive hypothesis for $e_1$ in the third inequality.
Second, consider an edge operation. Let $e_1,\dots,e_{B_t}$ be the sampled edges, and let $w_1,w_2,\dots,w_{B_t}$ be the chosen endpoints. For each $j\in[B_t]$, in $G_t$ we join $w_j$ to some vertex of $G_{t-1}$, say $x_j$. In $T_t$, we join the new edge $w_j x_j$ to $e_j$. We have $$\begin{aligned}
{\operatorname{depth}}(w_j x_j,G_t)
\le {\operatorname{depth}}(w_j,G_t) + 1
& \le {\operatorname{depth}}(e_j,G_t) + 1\\
& \le 2 {\operatorname{depth}}(e_j,T_t) + 1
= 2 {\operatorname{depth}}(w_j x_j,T_t) - 1 \:,\end{aligned}$$ where we have used the fact that $e_j$ is incident to $w_j$ in the second inequality, and the inductive hypothesis for $e_j$ in the third inequality.
Hence for all $e\in E(G_t)$, ${\operatorname{depth}}(e, G_t) \le 2 {\operatorname{depth}}(e,T_t)$. On the other hand, examining the construction of $(T_t)_{t\in{{\mathbb{N}_0}}}$ and using Lemma \[lem:multichildren\], we find that a.a.s. the height of $T_n$ is at most $ (u/\ell) e\log n + 2ue+O(1)$. This implies that a.a.s. the diameter of $G_n$ is at most $4(u/\ell) e\log n + 8ue+O(1)$.
For an undirected graph $G$ and a real number $\delta$, we define the function $\rho_{\delta}:V(G)\to\mathbb{R}$ as $$\rho_{\delta}(v) = \frac{\deg(v)+\delta}{\sum_{u\in V(G)} (\deg(u)+\delta)} \:.$$ Here $\deg(v)$ denotes the degree of vertex $v$, and a loop is counted twice. Note that if $\delta>-1$ then $\rho_{\delta}$ is a probability distribution.
Observe that to sample a vertex using $\rho_0$, one can sample an edge uniformly and then choose one of its endpoints uniformly. Most of our arguments are based on this crucial fact, and this is the reason for introducing edge trees.
\[def\_acld\] Let $\{X_t: t\in{{\mathbb{N}}}\}$ be a sequence of ${{\mathbb{N}}}$-valued random variables, and let $\{Y_t,Z_t : t\in{{\mathbb{N}}}\}$ be sequences of ${{\mathbb{N}_0}}$-valued random variables. We consider a growing undirected graph $(G_t)_{t=0}^{\infty}$ as follows. $G_0$ is an arbitrary connected graph with at least one edge. At each time-step $t\in{{\mathbb{N}}}$, $G_t$ is obtained from $G_{t-1}$ by performing the following three operations.
1. We sample $X_t$ vertices $N_1,\dots,N_{X_t}$ independently using $\rho_0$.
2. We sample $2Z_t$ vertices $W_1,W'_1,W_2,W'_2,\dots,W_{Z_t},W'_{Z_t}$ independently using $\rho_0$.
3. We add a new vertex $v$ and add the edges $W_1W'_1,\dots,W_{Z_t}W'_{Z_t}$, $vN_1,\dots,vN_{X_t}$. We also add $Y_t$ loops at $v$.
Model \[def\_acld\] is a generalization of a model defined by Aiello, Chung, and Lu [@acl_model Section 2.1, Model D], which has bounded $X_t,Y_t,Z_t$. The following theorem implies that a.a.s. the latter model has diameter $O(\log n)$.
Let $\ell=\ell(n),u=u(n)$ be positive integers such that $X_t>0$ and $\ell \le X_t+Y_t+Z_t\le u$ for all $t\in{{\mathbb{N}}}$. A.a.s. the diameter of $G_n$ generated by Model \[def\_acld\] is at most $4e(u/\ell) \log n + 8eu+O(1)$.
We claim that $(G_t)_{t=0}^{\infty}$ grows as described in Model \[def\_pref\]. Sampling a vertex using $\rho_{0}$ corresponds to choosing a random endpoint of a random edge. The three operations of Model \[def\_acld\] correspond to applying a vertex operation with $A_{t} = X_t+Y_t$ and an edge operation with $B_t=Z_t$. By Theorem \[thm:pref\], a.a.s. the diameter of ${G}_{n}$ is at most $4e(u/\ell) \log n + 8eu+O(1)$.
We analyse another model by reducing it to Model \[def\_pref\].
\[def\_butowsley\] Let $\delta \in (-1,\infty)$, $p\in[0,1]$ and let $(X_t)_{t\in{{\mathbb{N}}}}$ be a sequence of ${{\mathbb{N}}}$-valued random variables. We consider a growing undirected graph $(G_t)_{t=0}^{\infty}$ as follows. $G_0$ is an arbitrary connected graph with at least one edge. At each time-step $t\in{{\mathbb{N}}}$, we apply exactly one of the following operations: operation (a) with probability $p$ and operation (b) with probability $1-p$.
(a) We sample $X_t$ vertices independently using $\rho_{\delta}$, then we add a new vertex $v$ and join it to the sampled vertices.
(b) We sample $2X_t$ vertices $W_1,W'_1,\dots,W_{X_t},W'_{X_t}$ independently using $\rho_{\delta}$, then we add the edges $W_1W'_1, \dots,W_{X_t}W'_{X_t}$.
Model \[def\_butowsley\] is a generalization of the *generalized linear preference model* of Bu and Towsley [@bu_towsley], in which $X_t=d$ for all $t$, where $d$ is a fixed positive integer. Theorem \[thm:remark\] below gives that if $\delta$ is rational and nonnegative then a.a.s. the generalized linear preference model has diameter at most $(4+2\delta/d)e\log n + O(1)$.
Model \[def\_butowsley\] with $p=1$ and $X_t$ being a constant independent of $t$ and $n$ is called the *linear preference model*, whose diameter has been studied extensively. Assume that $X_t=d$ for all $t$, where $d\in{{\mathbb{N}}}$ is fixed. If $d=1$ and $\delta\ge0$, Pittel [@random_recursive_trees] showed the diameter is $\Theta(\log n)$. If $d>1$ and $\delta\in(-d,0)$, the diameter is $\Theta(\log \log n)$ as proved by Dommers, van der Hofstad, and Hooghiemstra [@diameters_pa; @vander]. If $d>1$ and $\delta=0$, the diameter is $\Theta (\log n / \log \log n)$, see Bollob[á]{}s and Riordan [@diameter_preferential_attachment]. Finally, if $d>1$ and $\delta>0$, the diameter is $\Theta(\log n)$ [@diameters_pa; @vander].
Chung and Lu [@coupling_online_offline] studied a variation of Model \[def\_butowsley\] with the following differences: the process is conditioned on generating a graph with no multiple edges or loops; $X_t=d$ for all $t$, where $d$ may depend on $n$; there are two additional operations: in the first one, a vertex is sampled uniformly and deleted, and in the second one, $X_t$ edges are sampled uniformly and deleted. They proved that if $d > \log ^{1+\Omega(1)} n$, then a.a.s. the evolving graph has diameter $\Theta(\log n)$, where $n$ is the number of vertices.
\[thm:remark\] Suppose that $\delta=r/s$, where $r\in{{\mathbb{N}_0}}$ and $s\in{{\mathbb{N}}}$, and suppose that $\ell=\ell(n),u=u(n)\in{{\mathbb{N}}}$ are such that $\ell \le X_t\le u$ for all $t$. A.a.s. the diameter of $G_n$ generated by Model \[def\_butowsley\] is at most $4e(u/\ell + \delta/(2\ell)) \log n + O(u)$.
For $t\in{{\mathbb{N}_0}}$, let $\widehat{G}_{t}$ be the graph obtained from $G_t$ by copying each edge $2s-1$ times, and adding $r$ loops at each vertex. So $\widehat{G}_{t}$ has $2s |E(G_t)| + r |V(G_t)|$ edges. Note that the diameters of $G_t$ and $\widehat{G}_{t}$ are the same. We claim that $(\widehat{G}_t)_{t=0}^{\infty}$ grows as described in Model \[def\_pref\]. First, sampling a vertex of $G_{t-1}$ using $\rho_{\delta}$ corresponds to choosing a random endpoint of a random edge of $\widehat{G}_{t-1}$. Second, applying operation (a) corresponds to applying only a vertex operation with $A_{t}=2sX_t+r$. Second, applying operation (b) corresponds to applying only an edge operation with $B_{t} = 2sX_t$. By Theorem \[thm:pref\], a.a.s the diameter of $\widehat{G}_{n}$ is at most $4e(u/\ell + \delta/(2\ell)) \log n + (16esu + 8er)+O(1)$, completing the proof.
We now analyse the preferential attachment with random initial degrees (PARID) model of Deijfen, van den Esker, van der Hofstad, and Hooghiemstra [@randominitial Section 1.1] by reducing it to Model \[def\_butowsley\].
\[def\_parid\] Let $\{X_t: t\in{{\mathbb{N}}}\}$ be a sequence of i.i.d. ${{\mathbb{N}}}$-valued random variables and let $\delta$ be a fixed number such that almost surely $X_1 + \delta > 0$. We consider a growing undirected graph $(G_t)_{t=0}^{\infty}$ as follows. $G_0$ is an arbitrary connected graph with at least one edge. At each time-step $t\in{{\mathbb{N}}}$, $G_t$ is obtained from $G_{t-1}$ by sampling $X_t$ vertices $N_1,\dots,N_{X_t}$ independently using $\rho_{\delta}$ and adding one new vertex $v$ and $X_t$ new edges $vN_1,\dots,vN_{X_t}$.
Note that $X_t$ is the (random) initial degree of the vertex born at time $t$. The following theorem implies that if the initial degrees’ distribution in the PARID model has an exponential decay (e.g. if it is the Poisson or the geometric distribution), and $\delta$ is positive and rational, then a.a.s. the generated graph has a polylogarithmic diameter.
\[thm:parid\] Assume that $\delta$ is a positive rational number and that $\ell=\ell(n)$ and $u=u(n)$ are positive integers such that $\p{X_1 \notin[\ell, u]} = o(1/n)$. A.a.s. the diameter of $G_n$ generated by Model \[def\_parid\] is at most $4e(u/\ell + \delta/(2\ell)) \log n + O(u)$.
Since $\p{X_1 \notin[\ell, u]} = o(1/n)$ and the $X_i$ are i.i.d., a.a.s. we have $\ell \le X_t \le u$ for all $t\in[n]$. The rest of the proof is the same as that of Theorem \[thm:remark\], where all operations are of type (a).
A directed model
----------------
In this section we study a directed analogous of Model \[def\_acld\], which is also a generalization of a model of Aiello et al. [@acl_model]. Sampling probabilities in this model depend on vertices’ out-degrees and in-degrees, as defined below.
For a directed graph $G$ and a real number $\delta$, we define the functions $\rho_{\delta}^{out},
\rho_{\delta}^{in}:V(G)\to\mathbb{R}$ as $$\rho^{out}_{\delta}(v) = \frac{{\operatorname{out-deg}}(v)+\delta}{\sum_{u\in V(G)} ({\operatorname{out-deg}}(u)+\delta)}$$ and $$\rho^{in}_{\delta}(v) = \frac{{\operatorname{in-deg}}(v)+\delta}{\sum_{u\in V(G)} ({\operatorname{in-deg}}(u)+\delta)} \:.$$ Here ${\operatorname{out-deg}}(v)$ and ${\operatorname{in-deg}}(v)$ denote the out-degree and the in-degree of vertex $v$, respectively.
\[def\_aclc\] Let $\{X_t,Y_t,Z_t,Q_t : t\in{{\mathbb{N}}}\}$ be sequences of ${{\mathbb{N}_0}}$-valued random variables satisfying $X_t+Y_t>0$ for all $t$. We consider a growing directed graph $(G_t)_{t=0}^{\infty}$ as follows. $G_0$ is an arbitrary weakly connected directed graph with at least one edge. At each time-step $t\in{{\mathbb{N}}}$, we perform the following operations:
1. We sample $X_t$ vertices $x_1,\dots,x_{X_t}$ independently using $\rho_0^{out}$ and $Y_t$ vertices $y_1,\dots,y_{Y_t}$ independently using $\rho_0^{ in}$.
2. We sample $Z_t$ vertices $w_1,w_2,\dots,w_{Z_t}$ independently using $\rho_0^{out}$, and we sample $Z_t$ vertices $w'_1,w'_2,\dots,w'_{Z_t}$ independently using $\rho_0^{in}$,
3. We add a new vertex $v$, and then we add the directed edges $w_1w'_1,\dots,w_{Z_t}w'_{Z_t}$, $x_1v,\dots,x_{X_t}v$, $vY_1,\dots,vY_{Y_t}$. We also add $Q_t$ loops at $v$.
Model \[def\_aclc\] generalizes of [@acl_model Section 2.1, Model C], which has bounded $X_t,Y_t,Z_t,Q_t$. The following theorem implies that a.a.s. the diameter of the latter model is $O(\log n)$.
Let $\ell=\ell(n),u=u(n)$ be positive integers such that $\ell \le X_t+Y_t+Z_t+Q_t\le u$ for all $t\in{{\mathbb{N}}}$. A.a.s. the diameter of $G_n$ generated by Model \[def\_aclc\] is at most $4 e(u/\ell) \log n + 8eu+O(1)$.
We claim that the underlying undirected graph of $(G_t)_{t=0}^{\infty}$ grows as described in Model \[def\_pref\]. Sampling a vertex using $\rho_{0}^{out}$ and $\rho_{0}^{in}$ correspond to choosing the tail and the head of a random edge, respectively. The operations of Model \[def\_acld\] correspond to applying a vertex operation with $A_t = X_t+Y_t+Q_t$ and an edge operation with $E_t = Z_t$. By Theorem \[thm:pref\], a.a.s. the diameter of $G_{n}$ is at most $4 e(u/\ell) \log n + 8eu+O(1)$, as required.
Directed scale-free graphs: dummy edges {#sec:directed}
=======================================
We study two directed models in this section. In contrast to the previous directed model (Model \[def\_aclc\]), in models considered here, the constant term in the definition of attachment probabilities ($\delta$ in Model \[def\_butowsley\]) can be different for in-degrees and out-degrees. We handle this issue by introducing dummy edges whose role is just to adjust the attachment probabilities (similar to, but more complicated than, what we did in the proof of Theorem \[thm:remark\]). As in Section \[sec:pref\], we first define a general model (Model \[def\_directed\]) with a lot of flexibility and prove that a.a.s. it has a logarithmic diameter, and then reduce Model \[def\_directed\_scale\_free\] (which is a generalization of the so-called ‘directed scale-free graphs’) to that.
In a directed graph, each edge has a tail and a head. A *generalized directed graph* is a directed graph some of whose edges do not have a head or a tail. Edges of such a graph are of three type: *tailless* edges have a head but do not have a tail, *headless* edges have a tail but do not have a head, and *proper* edges have a tail and a head. A *headed* edge is one that is not headless, and a *tailed* edges is one that is not tailless.
The following model is a directed analogous of Model \[def\_pref\].
\[def\_directed\] Let $(A_t,B_t,C_t,D_t,E_t)_{t=1}^{\infty}$ be sequences of ${{\mathbb{N}_0}}$-valued random variables. We consider a growing generalized directed graph $(G_t)_{t=0}^{\infty}$ as follows. $G_0$ is an arbitrary weakly connected generalized directed graph with at least one edge. At each time-step $t\in{{\mathbb{N}}}$, $G_t$ is obtained from $G_{t-1}$ by performing a vertex operation and an edge operation, as defined below.
In a *vertex operation*, if $A_t+B_t>0$, a new vertex $v$ is born and $A_t+B_t+C_t+D_t$ edges are added in the following manner:
Case 1:
: If $A_t>0$, we sample a headed edge from $G_{t-1}$ uniformly and add a proper edge from $v$ to its head. Then we add $A_t - 1$ new proper edges, tailed at $v$ and headed at arbitrary vertices of $G_{t-1}$. Then $B_t$ proper edges are added, tailed at arbitrary vertices of $G_{t-1}$ and headed at $v$. Then $C_t$ headless edges tailed at $v$, and $D_t$ tailless edges headed at $v$ are added.
Case 2:
: If $A_t=0$, we sample a tailed edge from $G_{t-1}$ uniformly and add a proper edge from its tail to $v$. Then $B_t-1$ new proper edges are added from arbitrary vertices of $G_{t-1}$ to $v$. Then $C_t$ headless edges tailed at $v$, and $D_t$ tailless edges headed at $v$ are added.
If $A_t+B_t=0$, then we do nothing in the vertex operation.
In an *edge operation*, we independently sample $E_t$ tailed edges from $G_{t-1}$ uniformly, then we add $E_t$ proper edges, joining the tails of the sampled edges to arbitrary vertices of $G_{t-1}$.
Note that we do not require any independence for $(A_t,B_t,C_t,D_t,E_t)_{t\in{{\mathbb{N}}}}$. In particular they can be correlated and can depend on the past and the future of the process.
\[thm:directed\] Let $\ell=\ell(n),u=u(n) \in {{\mathbb{N}}}$ be such that $\ell\le A_t + B_t+E_t$ and $A_t + B_t +C_t+D_t+E_t\le u$ for every $t\in{{\mathbb{N}}}$. A.a.s. the graph $G_n$ generated by Model \[def\_directed\] has diameter at most $4e(u/\ell) \log n + 8eu + O(1)$.
The argument is similar to that of Theorem \[thm:pref\]. We define the depth of a headless edge as one plus the depth of its tail, and the depth of a tailless edge as one plus the depth of its head, and the [depth]{} of a proper edge $uv$ as $1+\min\{{\operatorname{depth}}(u),{\operatorname{depth}}(v)\}$. We inductively define a growing undirected tree $(T_t)_{t=0}^{\infty}$ such that for all $t\in{{\mathbb{N}_0}}$, $V(T_t) = E(G_t) \cup \{{\aleph}\}$. Here ${\aleph}$ denotes the root of $T_t$, which has depth 0. We prove by induction that for all $e\in E(G_t)$, ${\operatorname{depth}}(e, G_t) \le 2 {\operatorname{depth}}(e,T_t)$. Let $H$ be the graph obtained from the underlying undirected graph of $G_0$ by adding an edge labelled ${\aleph}$ incident to its root. Let $T_0$ be a breadth-first tree of the line graph of $H$ rooted at ${\aleph}$. Note that ${\operatorname{depth}}({\aleph},T_0)=0$ and ${\operatorname{depth}}(e,T_0)={\operatorname{depth}}(e,G_0)$ for every $e\in E(G_0)$.
Given $T_{t-1}$, we define $T_t$ and prove the inductive step. First, consider a vertex operation, Case 1. Let $v$ be the new vertex and $e_1$ be the sampled headed edge. Notice that ${\operatorname{depth}}(v,G_t) \le {\operatorname{depth}}(e_1,G_t)+1$. In $T_t$, we join the $A_t+B_t+C_t+D_t$ new nodes (new edges of $G_t$) to $e_1$. For any such edge $e$ we have $$\begin{aligned}
{\operatorname{depth}}(e,G_t)
\le {\operatorname{depth}}(v,G_t) + 1
\le {\operatorname{depth}}(e_1,G_t) + 2
\le 2 {\operatorname{depth}}(e_1,T_t) + 2
= 2 {\operatorname{depth}}(e,T_t),\end{aligned}$$ where we have used the inductive hypothesis for $e_1$ in the third inequality.
Second, consider a vertex operation, Case 2. Let $v$ be the new vertex and let $e_1$ be the sampled tailed edge. Notice that ${\operatorname{depth}}(v,G_t) \le {\operatorname{depth}}(e_1,G_t)+1$. In $T_t$, we join the $B_t+C_t+D_t$ new nodes (new edges of $G_t$) to $e_1$. For any such edge $e$ we have $$\begin{aligned}
{\operatorname{depth}}(e,G_t)
\le {\operatorname{depth}}(v,G_t) + 1
\le {\operatorname{depth}}(e_1,G_t) + 2
\le 2 {\operatorname{depth}}(e_1,T_t) + 2
= 2 {\operatorname{depth}}(e,T_t).\end{aligned}$$
Third, consider an edge operation. Let $e_1,\dots,e_{E_t}$ be the sampled tailed edges, and denote by $w_1,w_2,\dots,w_{E_t}$ their tails. For each $j\in[E_t]$, in $G_t$ we join $w_j$ to a vertex of $G_{t-1}$, say $x_j$. In $T_t$, we join the new node $w_j x_j$ to $e_j$. We have $$\begin{aligned}
{\operatorname{depth}}(w_j x_j,G_t)
\le {\operatorname{depth}}(w_j,G_t) + 1
& \le {\operatorname{depth}}(e_j,G_t) + 1 \\
& \le 2 {\operatorname{depth}}(e_j,T_t) + 1
= 2 {\operatorname{depth}}(w_j x_j,T_t) - 1 \:,\end{aligned}$$ where we have used the fact that $w_j$ is incident with $e_j$ for the second inequality, and the inductive hypothesis for $e_j$ in the third inequality. Hence for all $e\in E(G_t)$, we have ${\operatorname{depth}}(e, G_t) \le 2 {\operatorname{depth}}(e,T_t)$, as required. To complete the proof, it suffices to show that a.a.s. the height of $T_n$ is at most $(u/\ell) e\log n + 2ue+O(1)$.
The argument is similar to that for Lemma \[lem:multichildren\]. Note that at any time $t$, graph $G_t$ has at least $|V(T_0)| +\ell t$ proper edges. Let $n_0=|V(T_0)|$. For a given $h = h(n)$, we bound the probability that $T_n$ has a node at depth exactly $h+n_0$. Given a sequence $1 \le t_1 < \dots < t_h\le n$, the probability that there exists a path $v_{t_1}v_{t_2}\dots v_{t_h}$ in $T_n$ with $v_{t_j}$ born at time $t_j$ is at most $$u^h \prod_{k=2}^{h}\frac{1}{n_0+ \ell \cdot (t_k-1)} \:,$$ since there are at most $u^h$ choices for $(v_{t_1},\dots,v_{t_h})$, and for each $k=2,\dots,h$, when $v_{t_k}$ is born, there are at least $n_0+ \ell \cdot (t_k-1)$ nodes available for it to join to (corresponding to the proper edges of $G_{t_k-1}$). By the union bound, the probability that $G_n$ has a node at depth $h+n_0$ is at most $$\begin{aligned}
u^h \sum_{1 \le t_1 <t_2< \dots < t_h\le n}\left( \prod_{k=2}^{h}\frac{1}{n_0+\ell\cdot(t_k-1)}\right)
& < \frac{u^h}{h!}\left(1+\sum_{j=1}^{n-1}\frac{1}{n_0+\ell j}\right)^h \\
&
< \left(\frac{u e}{h}
\cdot \left( \frac{\log n}{\ell} + 2\right) \right)^h
\bigg/\sqrt{2\pi h} \:.\end{aligned}$$ Putting $h \ge (u/\ell) e\log n + 2ue$ makes this probability $o(1)$. Hence a.a.s. the height of $T_n$ is less than $ (u/\ell) e\log n + 2ue+O(1)$, as required.
The following model is a directed analogous of Model \[def\_butowsley\].
\[def\_directed\_scale\_free\] Let $p_a,p_b,p_c$ be nonnegative numbers summing to 1, and let $\alpha,\beta \in [0,\infty)$. Let $(X_t)_{t\in{{\mathbb{N}}}}$ be a sequence of ${{\mathbb{N}}}$-valued random variables. We consider a growing directed graph $(G_t)_{t=0}^{\infty}$ as follows. $G_0$ is an arbitrary weakly connected directed graph. At each time-step $t\in{{\mathbb{N}}}$, we perform exactly one of the following three operations, with probabilities $p_a,p_b$, and $p_c$, respectively.
1. We sample $X_t$ vertices from the existing graph, independently using $\rho_{\alpha}^{in}$. Then we add a new vertex and join it to the sampled vertices.
2. We sample $X_t$ vertices from the existing graph, independently using $\rho_{\beta}^{out}$. Then we add a new vertex and join the sampled vertices to it.
3. We sample $X_t$ vertices $w_1,\dots,w_{X_t}$ independently using $\rho_{\beta}^{out}$, and we sample $X_t$ vertices $w'_1,\dots,w'_{X_t}$ independently using $\rho_{\alpha}^{in}$. Then we add the edges $w_1w'_1$, $\dots$, $w_{X_t}w'_{X_t}$.
Model \[def\_directed\_scale\_free\] is a generalization of *directed scale-free graphs* of Bollob[á]{}s, Borgs, Chayes, and Riordan [@directed_scale_free Section 2], which has $X_t=1$ for all $t$. The following theorem implies that if $\alpha$ and $\beta$ are rational, then a.a.s. the diameter of the latter model is at most $4 e (1+\alpha+\beta) \log n + O(1)$.
Suppose that $\alpha=r/s$ and $\beta=q/s$ with $r,q\in{{\mathbb{N}_0}}$ and $s\in{{\mathbb{N}}}$. Also suppose that $\ell=\ell(n),u=u(n)\in{{\mathbb{N}}}$ are such that $\ell \le X_t\le u$ for all $t$. A.a.s. the diameter of $G_n$ generated by Model \[def\_directed\_scale\_free\] is at most $4 e (u+\alpha+\beta) \log n /\ell + O(u)$.
For $t\in{{\mathbb{N}_0}}$, let $\widehat{G}_{t}$ be the generalized directed graph obtained from $G_t$ by copying each edge $s-1$ times, adding $r$ tailless edge at each vertex, and adding $q$ headless edges at each vertex. So $\widehat{G}_{t}$ has $s |E(G_t)| + (r+q) |V(G_t)|$ edges. Note that the diameters of $G_t$ and $\widehat{G}_{t}$ are the same. We claim that $(\widehat{G}_t)_{t=0}^{\infty}$ grows as described in Model \[def\_directed\]. First, sampling a vertex of $G_t$ using $\rho_{\beta}^{out}$ or $\rho_{\alpha}^{in}$ correspond to choosing the tail or the head of a uniformly random tailed or headed edge of $\widehat{G}_t$, respectively. Second, applying operation (a) corresponds to applying only a vertex operation with $A_{t}=sX_{t},B_{t}=0,C_t=q,D_t=r$. Third, applying operation (b) corresponds to applying only a vertex operation with $A_{t}=0,B_t=sX_{t},C_t=q,D_t=r$. Fourth, applying operation (c) corresponds to applying only an edge operation with $E_t=sX_t$. By Theorem \[thm:directed\], a.a.s the diameter of $\widehat{G}_{n}$ is at most $4 e (u+\alpha+\beta) \log n /\ell + 8e(su+q+r)+O(1)$, completing the proof.
The Cooper-Frieze model: multi-typed edge trees {#combine}
===============================================
In this section we study an undirected model that combines uniform and preferential attachment when choosing the neighbours of a new vertex.
\[def\_cf\] Let $p_a,\dots,p_f$ be nonnegative numbers summing to 1 and satisfying $p_a+p_b>0$, and let $(X_t)_{t\in{{\mathbb{N}}}}$ be a sequence of ${{\mathbb{N}}}$-valued random variables. We consider a growing undirected graph $(G_t)_{t=0}^{\infty}$ as follows. $G_0$ is an arbitrary connected graph. At each time-step $t\in{{\mathbb{N}}}$, we perform exactly one of the following six operations, with probabilities $p_a,\dots,p_f$ and independently of previous choices.
(a) $X_t$ vertices are sampled uniformly, then a new vertex is born and is joined to the sampled vertices.
(b) $X_t$ vertices are sampled using $\rho_0$, then a new vertex is born and is joined to the sampled vertices.
(c) $X_t+1$ vertices are sampled uniformly. Then $X_t$ edges are added joining the first sampled vertex to the others.
(d) A vertex is sampled uniformly and $X_t$ vertices are sampled using $\rho_0$. Then $X_t$ edges are added joining the first sampled vertex to the others.
(e) A vertex is sampled using $\rho_0$ and $X_t$ vertices are sampled uniformly. Then $X_t$ edges are added joining the first sampled vertex to the others.
(f) $X_t+1$ vertices are sampled using $\rho_0$. Then $X_t$ edges are added joining the first sampled vertex to the others.
Note that each operation increases the number of edges by $X_t$. Again, we do not require any independence for $(X_t)_{t\in{{\mathbb{N}}}}$.
Model \[def\_cf\] is a generalization of a model defined by Cooper and Frieze [@cooper-frieze-model Section 2], in which the random variables $X_t$ are bounded. The following theorem implies that a.a.s. the diameter of the latter model is $O(\log n)$.
\[thm:cooper-frieze\] Let $q = p_a + p_b$ and let $\ell=\ell(n),u=u(n)$ be positive integers such that $\ell \le X_t \le u$ for all $t$. A.a.s. the diameter of $G_n$ generated by Model \[def\_cf\] is at most $4(u/\ell+11/q) e\log n + 8e(u/\ell)+O(1)$.
As before, we define a growing tree whose height multiplied by 2 dominates the height of $(G_t)$, and then we upper bound the tree’s height. The main difference with Theorem \[thm:pref\] is that in some operations we may sample the *vertices* of the graph. In a growing tree, when a new vertex $v$ is born and is joined to a vertex $w$ of the existing tree, we say $w$ is the *parent* of $v$, and that $w$ is *given birth* to $v$.
We inductively define a growing tree $(T_t)_{t=0}^{\infty}$ such that $V(T_t) = V(G_t)\cup E(G_t)$ for all $t$, and we prove that ${\operatorname{depth}}(f,G_t)\le2{\operatorname{depth}}(f,T_t)$ for each vertex or edge $f$ of $G_t$. A node of $T_t$ is called a *V-node* or an *E-node* if it corresponds to a vertex or an edge of $G_t$, respectively. We may assume $T_0$ has been defined (for instance, we can build it by taking a breadth-first search tree of $G_0$ and joining all the E-nodes to its deepest V-node) and we describe the growth of $T_{t-1}$ to $T_t$ corresponding to each operation.
(a) Let $w$ be the first sampled vertex. In $T_t$ we join all new nodes (corresponding to the new vertex and the new edges in $G_t$) to $w$. In this case, a V-node of $T_{t-1}$ has been sampled uniformly and is given birth to one V-node and $X_t$ E-nodes.
(b) For sampling a vertex using $\rho_0$, we sample a random edge and then choose a random endpoint of it. Let $e$ be the first sampled edge. In $T_t$ we join all new nodes (corresponding to the new vertex and the new edges in $G_t$) to $e$. In this case, an E-node of $T_{t-1}$ has been sampled uniformly and is given birth to one V-node and $X_t$ E-nodes.
(c) Let $w$ be the first sampled vertex. In $T_t$ we join all new nodes (corresponding to the new edges in $G_t$) to $w$. In this case, a V-node of $T_{t-1}$ has been sampled uniformly and is given birth to $X_t$ E-nodes.
(d) For sampling a vertex using $\rho_0$, we sample a random edge and then choose a random endpoint of it. Let $e$ be the first sampled edge. In $T_t$ we join all new nodes (corresponding to the new edges in $G_t$) to $e$. In this case, an E-node of $T_{t-1}$ has been sampled uniformly and is given birth to $X_t$ E-nodes.
Similar to the proof of Theorem \[thm:pref\], an inductive argument gives ${\operatorname{depth}}(f,G_t)\le2{\operatorname{depth}}(f,T_t)$ for each vertex or edge $f$ of $G_t$. Hence, showing that a.a.s. the height of $T_n$ is at most $(u/\ell+11/q) e\log n + 2e(u/\ell)+O(1)$ completes the proof.
For $t\in{{\mathbb{N}_0}}$, let $L(t)$ denote the number of V-nodes of $T_t$. Let $n_0=|V(T_0)|$ and $m_0=(9/q)\log n$. Note that $L(t) = n_0 + {\operatorname{Bin}}(t,q)$. Using the Chernoff bound and the union bound, a.a.s we have $L(t) \ge tq/2$ for all $m_0 \le t \le n$. We condition on an arbitrary vector $(L(1),\dots,L(n))=(g(1),\dots,g(n))$ for which this event happens.
For a given integer $h = h(n)$, we bound the probability that $T_n$ has a vertex at depth exactly $n_0+h$. Given a sequence $1 \le t_1 < \dots < t_h\le n$, the probability that there exists a path $v_{t_1}v_{t_2}\dots v_{t_h}$ in $T_n$ such that $v_{t_j}$ is born at time $t_j$ is at most $$\prod_{k=2}^{h}\left(\frac{u}{n_0+ \ell\cdot(t_k-1)} +\frac{1}{g(t_k-1)}\right)\:,$$ since for each $k=h,h-1,\dots,3,2$, if $v_{t_k}$ wants to choose an E-node as its parent, there are at least $n_0+ \ell\cdot(t_k-1)$ E-nodes available for it to join to, and at most $u$ of them were born at time $t_{k-1}$; and if $v_{t_k}$ wants to choose a V-node as its parent, there are at least $g(t_k-1)$ V-nodes available for it to join to, and at most one of them was born at time $t_{k-1}$. By the union bound, the probability that $T_n$ has a vertex at depth $h+n_0$ is at most $$\label{eq:prob}
\sum_{1 \le t_1 < \dots < t_h\le n}\left( \prod_{k=2}^{h}\left(\frac{u}{n_0+ \ell\cdot(t_k-1)} +\frac{1}{g(t_k-1)}\right)\right)
< \frac{1}{h!}
\left(1 + \sum_{j=1}^{n-1}\frac{u}{n_0+\ell j}+
\sum_{j=1}^{n-1}\frac{1}{g(j)}\right)^h \:.$$ We have $$\sum_{j=1}^{n-1}\frac{u}{n_0+\ell j}
< \frac{u}{\ell}
\sum_{j=1}^{n-1}\frac{1}{j}
< (u/\ell)(1 + \log n)\:,$$ and $$\sum_{j=1}^{n-1}\frac{1}{g(j)}
=
\sum_{j=1}^{m_0-1}\frac{1}{g(j)}
+
\sum_{j=m_0}^{n-1}\frac{1}{g(j)}
\le
m_0 +
\sum_{j=m_0}^{n-1}\frac{2}{qj}
<
\frac{11}{q}\:\log n \:.$$ Setting $h\ge(u/\ell+11/q) e\log n + 2e(u/\ell)$ makes the right hand side of (\[eq:prob\]) become $o(1)$, as required.
Further models {#sec:other}
==============
We first mention a model whose diameter is known to be logarithmic, but our approach gives a shorter proof. The *pegging process*, defined by Gao and Wormald [@pegging_def], is parametrized by $d\in {{\mathbb{N}}}$. Here we define the process for $d=3$ only, see [@pegging_def Section 2] for the definition for $d>3$. Consider a growing undirected graph $(G_t)_{t=0}^{\infty}$ starting from a connected $3$-regular $G_0$ and growing as follows. In every time-step, a pair $(e,f)$ of distinct edges is sampled uniformly from the existing graph. Assume that $e=ab$ and $f=cd$. Then two new vertices $e'$ and $f'$ are born, the edges $e$ and $f$ are deleted, and the edges $ae', be', cf', df'$, and $e'f'$ are added. Note that if the original graph is 3-regular then the new graph is also 3-regular, so $G_t$ is $3$-regular for all $t$. Gerke, Steger, and Wormald [@pegging Theorem 1.1] proved that for every $d$, a.a.s. $G_n$ has diameter $O(\log n)$. Using the techniques of Section \[sec:pref\], it can be shown that for every $d$, a.a.s. its diameter is at most $4e \log n + O(1)$. We give the proof for the case $d=3$ here, which is much shorter than the 5-page proof in [@pegging], and provides a small explicit constant.
Let $d\ge 3$ be fixed. A.a.s. the diameter of the graph $G_n$ generated by pegging process is at most $4e\log n + O(1)$.
We give the proof for $d=3$, and the proof can easily be extended to $d>3$. Define the *depth* of an edge $xy$ as $1+\min\{{\operatorname{depth}}(x),{\operatorname{depth}}(y)\}$. We inductively define a growing tree $(T_t)_{t=0}^{\infty}$ such that for all $t\in{{\mathbb{N}_0}}$, $V(T_t) = E(G_t) \cup \{{\aleph}\}$. Here ${\aleph}$ denotes the root of $T_t$, which has depth 0. We prove by induction that for all $e\in E(G_t)$, ${\operatorname{depth}}(e, G_t) \le 2 {\operatorname{depth}}(e,T_t)$. Let $H$ be the graph obtained from $G_0$ by adding an edge labelled ${\aleph}$ incident to its root. Let $T_0$ be a breadth-first tree of the line graph of $H$ rooted at ${\aleph}$. Note that ${\operatorname{depth}}({\aleph},T_0)=0$ and ${\operatorname{depth}}(e,T_0)={\operatorname{depth}}(e,G_0)$ for every $e\in E(G_0)$.
Given $T_{t-1}$, we define $T_t$ and prove the inductive step. Assume that in time-step $t$, the pair $(e,f)=(ab,cd)$ of edges is chosen from $G_{t-1}$. By symmetry, we may assume that ${\operatorname{depth}}(a,G_{t-1}) \le {\operatorname{depth}}(b,G_{t-1})$ and ${\operatorname{depth}}(d,G_{t-1}) \le {\operatorname{depth}}(c,G_{t-1})$. Then two new vertices $e'$ and $f'$ are born, the edges $ab$ and $cd$ are deleted, and the edges $ae', be', cf', df'$, and $e'f'$ are added. To obtain $T_t$ from $T_{t-1}$, we replace the nodes $ab$ and $cd$ with $ae'$ and $df'$, respectively. Also, we join the other new edges $be'$, $e'f'$, and $cf'$ to $ae'$. Observe that $$\begin{gathered}
{\operatorname{depth}}(ae',G_t) \le 1 + {\operatorname{depth}}(a,G_t)
= {\operatorname{depth}}(ab,G_{t-1}) \le 2 {\operatorname{depth}}(ab,T_{t-1})
= 2{\operatorname{depth}}(ae',T_{t}),\\
{\operatorname{depth}}(df',G_t) \le 1 + {\operatorname{depth}}(d,G_t)
= {\operatorname{depth}}(cd,G_{t-1}) \le 2 {\operatorname{depth}}(cd,T_{t-1})
= 2{\operatorname{depth}}(df',T_{t}),\\
{\operatorname{depth}}(be',G_t) \le 1 + {\operatorname{depth}}(ae',G_t)
\le 1 + 2 {\operatorname{depth}}(ae', T_t)
< 2 {\operatorname{depth}}(be',T_t) \:,\\
{\operatorname{depth}}(f'e',G_t) \le 1 + {\operatorname{depth}}(ae',G_t)
\le 1 + 2 {\operatorname{depth}}(ae', T_t)
< 2 {\operatorname{depth}}(f'e',T_t) \:,\\
{\operatorname{depth}}(cf',G_t) \le 2 + {\operatorname{depth}}(ae',G_t)
\le 2 + 2 {\operatorname{depth}}(ae', T_t)
= 2 {\operatorname{depth}}(cf',T_t) \:.\end{gathered}$$ Hence for all $e\in E(G_t)$, ${\operatorname{depth}}(e, G_t) \le 2 {\operatorname{depth}}(e,T_t)$. On the other hand, examining the construction of $(T_t)_{t\in{{\mathbb{N}_0}}}$ and using Lemma \[lem:multichildren\], we find that a.a.s. the height of $T_n$ is at most $e\log n +O(1)$. This implies that a.a.s. the diameter of $G_n$ is at most $4 e\log n +O(1)$.
Next we mention two closely related models for which we can easily prove logarithmic bounds using our technique. Let $k>1$ be a positive integer. A *random unordered increasing $k$-tree*, defined by Gao [@randomktree_def], is built from a $k$-clique by applying the following operation $n$ times: in every time-step, a $k$-clique of the existing graph is chosen uniformly at random, a new vertex is born and is joined to all vertices of the chosen $k$-clique.[^2] *Random $k$-Apollonian networks* [@high_RANs] have a similar construction, the only difference being that once a $k$-clique is chosen in some time-step, it will never be chosen in the future. Cooper and Frieze [@CF13 Theorem 2] and independently, Kolossváry, Komjáty and Vágó [@istvan Theorem 2.2] have recently proved that random $k$-Apollonian networks have diameter $\Theta(\log n)$.
Here we prove that a.a.s. the diameter of a random unordered increasing $k$-tree is at most $2e \log n + O(1)$, and that a.a.s. the diameter of a random $k$-Apollonian network is at most $2ek \log n/(k-1) + O(1)$. For the proof for random $k$-Apollonian networks we need the following variant of Lemma \[lem:multichildren\].
\[lem:multichildren2\] For a tree $T$, let $\mathcal{L}(T)$ denote its set of leaves. Let $(A_t)_{t\in{{\mathbb{N}}}}$ be a sequence of ${{\mathbb{N}}}$-valued random variables. Consider a growing tree $(T_t)_{t=0}^{\infty}$ as follows. $T_0$ is arbitrary. At each time-step $t\in{{\mathbb{N}}}$, a random vector $(W_1,W_2,\dots,W_{A_t}) \in \mathcal{L}(T_{t-1})^{A_t}$ is chosen in such a way that for each $i\in[A_t]$ and each $v\in \mathcal{L}(T_{t-1})$, the marginal probability $\p{W_i=v}$ equals $|\mathcal{L}(T_{t-1})|^{-1}$. In other words, each $W_i$ is a leaf of $T_{t-1}$ sampled uniformly; however, the $W_j$’s may be correlated. Then $A_t$ new nodes $v_1,\dots,v_{A_t}$ are born and $v_i$ is joined to $W_i$ for each $i\in[A_t]$. Let $\ell=\ell(n)$ and $u=u(n)$ be positive integers such that $1<\ell\le A_t \le u$ for all $t\in[n]$. Then the height of $T_n$ is a.a.s. at most $ u e\log n /(\ell-1) + 2ue+O(1)$.
Let $n_0=|V(T_0)|$. For a given integer $h = h(n)$, let us bound the probability that $T_n$ has a node at depth exactly $h+n_0$. Given a sequence $1 \le t_1 < t_2<\dots < t_h\le n$, the probability that there exists a path $v_{t_1}v_{t_2}\dots v_{t_h}$ in $T_n$ such that $v_{t_j}$ is born at time $t_j$ is at most $$u^{h} \prod_{k=2}^{h}\frac{1}{n_0+ (\ell-1)(t_k-1)} \:,$$ since for each $k=2,3,\dots,h$, when $v_{t_k}$ is born, there are at least $n_0+ (\ell-1)(t_k-1)$ leaves available for it to join to. By the union bound, the probability that $T_n$ has a node at depth $h+n_0$ is at most $$\begin{aligned}
u^{h}\sum_{1 \le t_1 < \dots < t_h\le n}\ \left( \prod_{k=2}^{h}\frac{1}{n_0+ (\ell-1)(t_k-1)}\right)
& < \frac{u^{h}}{h!}\left(1 + \sum_{j=1}^{n-1}\frac{1}{n_0+(\ell-1) j}\right)^{h} \\
& < \left(\frac{u e}{h}
\cdot \left( \frac{\log n}{\ell-1} + 2\right) \right)^h
\bigg/\sqrt{2\pi h} \:.\end{aligned}$$ Putting $h \ge u e\log n /(\ell-1)+ 2ue$ makes this probability $o(1)$. Hence a.a.s. the height of $T_n$ is at most $ u e\log n /(\ell-1) + 2ue + n_0$, as required.
A.a.s. the diameter of an $(n+k)$-vertex random unordered increasing $k$-tree is at most $2e \log n + O(1)$, and the diameter of an $(n+k)$-vertex random $k$-Apollonian network is at most $2ek \log n/(k-1) + O(1)$.
We define the *depth* of a $k$-clique as the maximum depth of its vertices. Let the first $k$ vertices have depth zero. We couple with a growing tree whose nodes corresponds to the $k$-cliques of the growing graph. Whenever in the graph a new vertex is born and is joined to the vertices of a $k$-clique, in the tree the chosen $k$-clique gives birth to $k$ new children. By induction, the graph’s height is always less than or equal to the tree’s height.
For the tree corresponding to a random unordered increasing $k$-tree, in every step a node is chosen uniformly at random and gives birth to $k$ new children, hence its height is bounded by $e \log n + O(1)$ by Lemma \[lem:multichildren\]. This gives an upper bound of $2e \log n + O(1)$ for the diameter of the corresponding graph.
For the tree corresponding to a random $k$-Apollonian network, in every step a leaf is chosen uniformly at random and gives birth to $k$ new children, hence its height is bounded by $e k \log n /(k-1) + O(1)$ by Lemma \[lem:multichildren2\]. This gives an upper bound of $2e k\log n /(k-1) + O(1)$ for the diameter of the corresponding graph.
[^1]: Supported by the Vanier Canada Graduate Scholarships program. Most of this work was done while the author was visiting Monash University, Australia.
[^2]: The resulting graph is named a *random $k$-tree* in [@randomktree_def]. However, since a different model for generating $k$-trees has been defined in [@CU10] and is also called a random $k$-tree, we used the name ‘random unordered increasing $k$-tree’ here to avoid any confusion. This terminology is from [@ktreenames].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, a novel Full Reference method is proposed for image quality assessment, using the combination of two separate metrics to measure the perceptually distinct impact of detail losses and of spurious details. To this purpose, the gradient of the impaired image is locally decomposed as a predicted version of the original gradient, plus a gradient residual. It is assumed that the detail attenuation identifies the detail loss, whereas the gradient residuals describe the spurious details. It turns out that the perceptual impact of detail losses is roughly linear with the loss of the positional Fisher information, while the perceptual impact of the spurious details is roughly proportional to a logarithmic measure of the signal to residual ratio. The affine combination of these two metrics forms a new index strongly correlated with the empirical Differential Mean Opinion Score (DMOS) for a significant class of image impairments, as verified for three independent popular databases. The method allowed alignment and merging of DMOS data coming from these different databases to a common DMOS scale by affine transformations. Unexpectedly, the DMOS scale setting is possible by the analysis of a single image affected by additive noise.'
author:
- 'Elio D. Di Claudio[^1][^2] and Giovanni Jacovitti[^3][^4]'
bibliography:
- 'IEEEabrv.bib'
- 'imageprocessing.bib'
- 'arrayprocessing.bib'
title: A Detail Based Method for Linear Full Reference Image Quality Prediction
---
Full Reference image quality assessment, detail analysis, linear quality metric, Fisher information, linear prediction, VICOM, image gradient.
Introduction {#section:Introduction}
============
interest on the quality assessment of images grows with the volume of visual communications and the advance of imaging technologies, pushing the development of effective techniques for the prediction of the average image quality judged by the human users, hereinafter referred to as the *subjective quality*.
The subjective quality is measured by averaging the scores assigned by a panel of human observers following specific protocols and is usually rated on the so-called Mean Opinion Score (MOS) scale, or on the Differential MOS (DMOS) scale, defined as the MOS difference between the *reference* (original) and the *test* (impaired) images.
However, measurement of subjective quality is unpractical for routine and large scale image quality assessment (IQA). Therefore, the subjective quality is usually predicted by means of *objective* IQA algorithms, founded on abstract models of the human observer.
IQA methods operate in three basic modes. In the Full Reference (FR) mode they quantify the differences between the reference and test images. In the Reduced Reference (RR) mode, the comparison is limited to partial representations of the images. Finally, in the No Reference (NR) mode, selected features extracted from the test image alone are compared to ideal targets [@MOORTY11; @MITTAL12; @SAAD12].
Surveys of several IQA methods are available in literature (e.g., [@CHANDLER13]). Here, to place the presented method in a precise conceptual frame, a concise classification of the IQA methods is provided on the basis of the underlying modeling of the human observer.
*Psycho-physical models* quantify the differences between original and test images by emulating the signal processing of the early stages of the e Human Visual System (HVS) [@WINKLER99; @CHANDLER07], supposed typical for all subjects. The generalization capability of these methods is limited by the influence of higher-level mental factors, such as emotion, education, past experience, etc. [@CHANDLER06; @CAVANAGH11; @VU08].
*Cognitive models* [@MARR10] look at the image quality scoring as a *computational process* taking place in the human mind [@LARSON10] and assume that the subjective quality degradation is strongly correlated with some measure of *visual information loss*. The Peak Signal to Noise Ratio (PSNR) method is perhaps the simplest cognitive method, which identifies the information loss with a reduction of the signal to noise ratio (SNR), where the difference between the original and the test images is interpreted as noise.
The Structural Similarity (SSIM) method measures the so-called structural information, defined through the local normalized Mean Square Error (MSE) between the test and the reference images [@WANG04]. Variants of the SSIM include Multi-Scale analysis (MS-SSIM) [@WANG03] or image gradient analysis, adopted in the Feature Similarity (FSIM) [@ZHANG11] and in the Gradient Magnitude Similarity Deviation (GMSD) [@XUE14] methods.
The Visual Information (VIF) method measures the loss of Shannon mutual information [@SHEIKH06], while the Virtual Cognitive Method (VICOM) measures the loss of the Fisher information about the localization of patterns [@CAPODIFERRO12].
As far as cognitive NR methods are concerned, they measure the loss of information as the distance of some statistical features from their nominal values [@SIMONCELLI01; @ZHAI12].
The generalization capability of cognitive models depends upon the coverage of adopted information measure. For instance, the PSNR is sensitive to any kind of impairment, whereas the VIF [@SHEIKH06] is specialized for impairments modeled by linear distortion plus additive noise.
*Behavioral models* are automatic learning machines trained by samples to give DMOS estimates directly from the images [@PINSON04; @LI12] or from a set of selected features [@LIU17]. If behavioral models are applied for known impairments, they may assume a very simple form [@WANG00; @SAZZAD08; @ZHU12], but in general they must include an automatic classification of the type of impairment. In these cases, the generalization is a critical issue. Neural nets can tightly fit the DMOS for specific image data sets using many adjustable weights, but the effects of different impairment factors are hard to predict.
Shortcomings of existing methods {#sec:ShortcomingsOfExistingMethods}
--------------------------------
One typical problem of most FR IQA methods is their unequal sensitivity with respect to the covered impairments [@SHEIKH06; @CAPODIFERRO12]. To compensate for this drawback, some methods combine values of the same metric calculated at different resolutions [@WANG03], relying on the fact that blur effects vanish when resolution shrinks, whereas noise and artifacts are only attenuated. Another compensation technique exploits the inhomogeneous spatial distribution of errors generated by smoothing and additive effects, measured by the variance of the local MSE [@XUE14].
A more systematic solution to this problem lies in the combination of different metrics specialized for different impairments. This approach was proposed in [@MARTENS96], where two metrics were used to measure the amounts of blur and additive noise. This objective was also pursued in [@PINSON04] for video. In [@CAPODIFERRO12] a categorical index was paired to the positional Fisher information. Recently, a similar approach was followed in [@IEREMEIEV16].
Another issue overlooked by most existing IQA methods is that the relationship between metrics and the empirical DMOS scale is strongly non-linear [@CAPODIFERRO12; @GU13]. As a matter of fact, the Spearman Rank Order Correlation Coefficient (SROCC) [@HUBER81], often employed to compare the performance of different metrics, is insensitive to linearity issues. On the other hand, the linearity of the DMOS estimates is essential in applications, since the quality must be quantified at the end on the DMOS scale.
To circumvent this problem, it is customary to *linearize* the metrics versus the empirical DMOS scale using ad hoc non-linear parametric (*logistic*) functions [@SHEIKH06B], as suggested also by the Video Quality Expert Group [@VQEG03]. However, unequal sensitivity of metrics to different impairments cannot be compensated by linearization and produces irreducible DMOS fitting errors.
Moreover, the linearization involves the non-linear optimization of several parameters, that critically depend on the empirical data set. The generalization capability across different data sets degrades linearly with the number of adjustable parameters [@AKAIKE74; @BURNHAM04], regardless of the training procedure [@STONE77], and may lead to under-fitting or over-fitting effects, strong statistical scattering of sample fitting parameters and local minima issues.
These are serious obstacles for the effectiveness of IQA metrics. As a matter of fact, default values for linearization parameters are hard to find in the technical literature. In conclusion, the intrinsic linearity of metrics with the DMOS appears as a highly desirable feature for IQA [@OKARMA10; @SKUROWSKY14].
The proposed approach {#sec:TheProposedApproach}
---------------------
The objective of the method presented in this paper is to solve the problems of unequal sensitivity to different impairments and of the a posteriori parametric linearization. Generally speaking, the approach followed here consists of the combination of different metrics, tailored to different impairments. Specifically, it stems from the consideration that most existing IQA methods treat image detail loss and spurious details in the same way. In other words, they do not distinguish between impairments caused by the loss of visual structures (*depriving errors*) or by the appearance of artifacts (*meddling errors*). However, this is at odds with common evidence, since detail losses and spurious details have a very different visual appearance.
Therefore, it is reasonable to measure their effects on subjective quality using different metrics. Previous attempts in this direction appeared in [@LUBIN97; @CAPODIFERRO10B], where point-wise increments and decrements of the gradient magnitude were discriminated. In [@LI11] detail losses and additive impairments were separated using a restored version of the test image as a watershed.
Herein, this discrimination is operated through a Least Squares (LS) decomposition of the gradient field of the test image into two components: a component *linearly predicted* from the gradient field of the original image, and the *residual, unpredictable* gradient, The detail loss is then identified by the attenuation of the predicted gradient into a small observation window. Likewise, the presence of spurious detail is identified by the gradient residual observed into a small window. This modeling is suggested by the orthogonality of the residual gradient with respect to the predicted gradients [@HAYKIN96].
The second step is the definition of two metrics, representing the *perceptual impact* of detail losses and spurious details respectively, each one as linear as possible with respect to the empirical DMOS scores. To this purpose, the contributions to the overall perceptual impact coming from detail losses are quantified by the loss of the detail positional information [@NERI04; @CAPODIFERRO12], proportional to the square root of the gradient energy attenuation into a detail window. The perceptual impact of spurious details is quantified by a logarithmic measure of the ratio between the original gradient energy and the residual gradient energy, pooled over the image.
The metrics are considered as the coordinates of a two-dimensional ($2$-D) space following the VICOM [@CAPODIFERRO12] scheme. The VICOM is structured as a set of computational layers, starting from the extraction of gradients from the reference and the test images, up to the computation of multiple features that form the coordinates of the virtual cognitive state space. Each point of this space is then mapped to a corresponding DMOS estimate by a parametric function trained on experimental data [@CAPODIFERRO12]. For this reason, the method presented here is referred to as *Detail VICOM* (D-VICOM). As a reference for the reader, a block scheme of the D-VICOM is displayed later in Fig. \[fig:DVICOMdfg\].
At this point, the visual diversity of detail losses and spurious details is again invoked to argue that their individual perceptual impacts will contribute independently to determine the overall subjective quality loss. It follows that the mapping function from the state space to the DMOS estimate boils down to a simple *affine* transformation.
The effectiveness of this new model was preliminarily tested on the LIVE DBR2 [@LIVEDBR2] database, where the method exhibited superior performance with respect to competing methods. Then, it was verified on two other independent databases (TID2008 [@PONOMARENKO09] and CSIQ [@MADPAGE10]) for the classes of impairment common to the three databases.
The outstanding fitting and cross-fitting of the D-VICOM estimates up to an affine transformation of the DMOS scale reveal that these heterogeneous experimental data are highly consistent under the D-VICOM paradigm. This consistency allowed to align and fuse these data into a unique large database, called the SUPERQUARTET.
Rather unexpectedly, it is also concluded that it is possible to define a *database independent* D-VICOM (ID-VICOM) quality estimator, specifying only conventional DMOS values of an original image and of an its noisy version.
This paper is organized as follows. In Sect. \[section:Detail analysis\] the decomposition of the gradient of the test image is described, and the results are illustrated by means of visual examples. In Sect. \[section:Linear metrics\] the metrics are defined and their affine combination is proposed for DMOS estimation.
In Sect. \[section:Experiments and results\] the DMOS estimates are statistically tested using the LIVE DBR2 [@LIVEDBR2] and then verified with the data contained in the TID2008 [@PONOMARENKO09] and the CSIQ [@MADPAGE10]. The performance of the linear D-VICOM and of some competing methods after logistic linearization is compared on the SUPERQUARTET. In Sect. \[section:DirectDMOSScaleSetting\] it is shown that the D-VICOM can be quickly calibrated without training. In Sect. \[section:Computational budget\] the computational costs are discussed and in Sect. \[section:Conclusion\] the merits of the D-VICOM are finally underlined.
Detail analysis {#section:Detail analysis}
===============
The detail analysis is performed on the luminance components $I\left( {\bf{p}} \right)$ of an impaired test image and $\tilde I\left( {\bf{p}} \right)$ of the reference image, that are gray-scale functions of the generic pixel position ${\bf{p}} = \left( {{x_1},{x_2}} \right)$.
Image gradients were theoretically supported as a relevant feature for IQA in [@CAPODIFERRO12] and adopted by recent methods [@XUE14]. For the proposed analysis, it is convenient to consider the *Gaussian smoothed complex gradients* $y\left( {\bf{p}} \right)$ and $\tilde y\left( {\bf{p}} \right)$, extracted by the following operator of scale $s$ and unit energy [@CAPODIFERRO12] $${h_0}\left( {\bf{p}},s \right) = \frac{1}{{s \sqrt \pi }}{e^{ - \frac{{{x_1}^2 + {x_2}^2}}{{2{s^2}}}}}\sqrt {\frac{{{x_1}^2 + {x_2 }^2}}{{{\sigma ^2}}}} \, {e^{j\arg \left( {{{{x_1 + j x_2}}}} \right)}}
\label{eqn:CGG}$$ as
[l]{} y ( [**[p]{}**]{} ) = I( [**[p]{}**]{} ) ( [[**[p]{}**]{};s]{} )\
y( [**[p]{}**]{} ) = I( [**[p]{}**]{} ) ( [[**[p]{}**]{};s]{} ) \[eqn:imageCGG\]
where $\star$ denotes $2$-D convolution and $\arg$ is the principal phase angle of the complex argument. For $s = 1$ pixel or slightly more, the frequency response of (\[eqn:CGG\]) well approximates the Contrast Sensitivity Function (CSF) of the HVS front end [@LI11; @mannos74], at nominal viewing distance [@CAPODIFERRO12]. The operator summarizes the horizontal and vertical filters commonly used for gradient approximation [@XUE14] and is *steerable*, i.e., has the same frequency response for any pattern orientation [@DICLAUDIO10].
The test image gradient $y\left( {\bf{p}} \right)$ is decomposed for any $\bf p$ as the sum of a *linearly distorted* (smoothed) version $\hat y \left( {\bf{p}} \right)$ of the original gradient $\tilde y \left( {\bf{p}} \right)$ (identified as the *distorted* detail) and of an *unpredictable* gradient error $\nu \left( {\bf{p}} \right)$ (considered as a *spurious* detail): $$y\left( {\bf{p}} \right) = \hat y\left( {\bf{p}} \right) + \nu \left( {\bf{p}} \right) \; .
\label{eqn:reproduced detail}$$
The distorted $\hat y\left( {\bf{p}} \right)$ is further modeled by a linear combination of the original gradient $\tilde y \left( {\bf{p}} \right)$ and of a pair of its directionally filtered versions $$\hat y\left( {\bf{p}} \right) = {b_{\bf{p}}}\left( 0 \right)\tilde y\left( {\bf{p}} \right) + {b_{\bf{p}}}\left( 1 \right){\tilde y_1}\left( {\bf{p}} \right) + {b_{\bf{p}}}\left( 2 \right){\tilde y_2}\left( {\bf{p}} \right)
\label{eqn:blur operator}$$ where ${b_{\bf{p}}}\left( 0 \right)$, ${b_{\bf{p}}}\left( 1 \right)$ and ${b_{\bf{p}}}\left( 2 \right)$ are real valued coefficients depending on the position ${\bf{p}}$, and ${\tilde y_1}\left( {\bf{p}} \right)$ and ${\tilde y_2}\left( {\bf{p}} \right)$ are calculated as
[l]{} [y\_1]{}( [**[p]{}**]{} ) = y( [**[p]{}**]{} ) h\_1( [[**[p]{}**]{};s]{} )\
[y\_2]{}( [**[p]{}**]{} ) = y( [**[p]{}**]{} ) h\_2( [[**[p]{}**]{};s]{} ) . \[eqn:side\_gradients\]
The impulse responses $h_1\left( {{\bf{p}};s} \right)$ and $h_2\left( {{\bf{p}};s} \right)$ are second order normalized Gaussian derivatives (i.e., Hermite-Gauss) functions of the same scale $s$ as ${h_0}\left( {\bf{p}},s \right)$, defined as $${h_1}\left( {{\bf{p}};s} \right) = \frac{{2\frac{{x_1^2}}{{{s^2}}} - 1}}{{s\sqrt {2\pi } }}{e^{ - \frac{{x_1^2}}{{2{s^2}}}}}
\label{eqn:h_1}$$ $${h_2}\left( {{\bf{p}};s} \right) = \frac{{2\frac{{x_2^2}}{{{s^2}}} - 1}}{{s\sqrt {2\pi } }}{e^{ - \frac{{x_2^2}}{{2{s^2}}}}}
\label{eqn:h_2}$$ and are introduced to adequately model a local astigmatic blur [@NERI04; @DICLAUDIO10].
The coefficients ${b_{\bf{p}}}\left( k \right)$ for $k=0,1,2$ are identified by a regularized LS system [@golub89], which minimizes the local error energy
[c]{} ( [**[p]{}**]{} ) = \_[**[q]{}**]{} [w[[( [**[q]{}**]{} )]{}\^2]{}[[| [( [[**p**]{}+[**q**]{}]{} ) ]{} |]{}\^2]{}]{} \[eqn:residual\_error\]
within a spot centered on the analyzed point $\bf p$. To minimize the interference from adjacent points, the squared errors are weighted by a Gaussian window $w\left( {\bf{q}} \right) \propto {e^{ - \frac{{{{\left| {\bf{q}} \right|}^2}}}{{4 s_w^2}}}}$ of spread $s_w \sqrt{2}$, scaled so that $\sum\limits_{\bf{q}} {w{{\left( {\bf{q}} \right)}^2}} = 1$. In particular, the smallest $s_w = 1$ pixel compatible with the number of estimated parameters within the spot was adopted [^5].
The overall cost function is $$J\left( {\bf{p}} \right) = \varepsilon \left( {\bf{p}} \right) + \xi\left[ {\sum\limits_{k = 0}^2 {b_{\bf{p}}^2\left( k \right)} } \right]
\label{eqn:regularized_LScost}$$ where the penalty term $\xi = 1$ was chosen for $256$ gray-level images to regularize [@golub89] the sample $b_{\bf{p}} \left( {k} \right)$ if the original gradient magnitude is small, without significantly biasing the LS parameter estimate. The LS solution is computationally efficient and its statistical properties are well known [@HAYKIN96; @tarpey00] (see also the Appendix).
To put into evidence how the proposed decomposition is actually correlated with the visual findings, in Fig. \[fig1\] the *gradient attenuation map* $$l\left( {\bf{p}} \right) = 1 - \frac{{\left| {\hat y\left( {\bf{p}} \right)} \right| + V}}{{\left| {\tilde y\left( {\bf{p}} \right)} \right| + V}}\;,
\label{eqn:detail_loss}$$ an indicator of detail loss[^6], and the residual gradient magnitude map $\rho\left( {\bf{p}} \right) = \left| {\nu \left( {\bf{p}} \right)} \right|$, an indicator of the spurious detail presence, are displayed for blurred, noisy, JPEG and JPEG2000 compressed images.
Maps are enhanced to reveal details well below the visibility threshold. The noisy image is characterized by random gradient residuals and, conversely, the blurred image is characterized by diffuse gradient attenuation and negligible residuals. The coded images, that are affected by both lost and spurious details, exhibit mostly complementary patterns of gradient attenuation and gradient residuals.
{width="7in"}
Metrics {#section:Linear metrics}
=======
Detail loss metric {#section:DetailLossMetric}
------------------
The perceptual impact of detail loss was not directly analyzed in the past. The closest problem considered in literature was the FR rating of noiseless images blurred by optical devices, characterized by their Modulation Transfer Function [@granger72; @carlson80; @barten90].
In this work the impact of the detail loss on the subjective quality rating is analyzed from a different viewpoint. In [@CAPODIFERRO12] it was stressed that the subjective degradation of an image should be strictly correlated with the accuracy of pattern *localization*, since it plays an essential role for distance estimation in binocular vision. Since the HVS perception of spatial displacements is basically linear [@stevens62], it is argued that the subjective degradation should be proportional to the increase of the position uncertainty of the observed distorted patterns with respect to the original ones.
Now, accepting that the performance of vision mechanisms is near optimal, it follows that the visual localization accuracy of a detail is measured by the Fisher information (FI) about its position. In [@NERI04; @CAPODIFERRO12] the *positional FI* of a portion of image extracted by a window $w{{\left( {\bf{q}} \right)}^2}$ centered on $\bf p$, in the presence of additive Gaussian white noise of variance $\sigma _N^2$, was calculated as $$FI({\bf{p}}) = \sigma _N^{ - 2}\left[ {\sum\limits_{\bf{q}} {w{{\left( {\bf{q}} \right)}^2}{{\left| {g\left( {{\bf{p}} + {\bf{q}}} \right)} \right|}^2}} } \right]
\label{eqn:PFI}$$ where $g\left( {\bf{p}} \right)$ is the *true* image gradient.
It is deduced that the positional accuracies of the original and test images, respectively characterized by the gradients $\tilde g\left( {\bf{p}} \right)$ and $\hat g\left( {\bf{p}} \right)$, are given by $$\sqrt {\tilde{FI}({\bf{p}})} = \sigma _N^{ - 1}\sqrt {{{\sum\limits_{\bf{q}} {w{{\left( {\bf{q}} \right)}^2}\left| {\tilde g\left( {{\bf{p}} + {\bf{q}}} \right)} \right|} }^2}}
\label{eqn:PositionalAccuracyOriginal}$$ $$\sqrt {\hat{FI}({\bf{p}})} = \sigma _N^{ - 1}\sqrt {{{\sum\limits_{\bf{q}} {w{{\left( {\bf{q}} \right)}^2}\left| {\hat g\left( {{\bf{p}} + {\bf{q}}} \right)} \right|} }^2}}
\label{eqn:PositionalAccuracyDistorted}$$ for the same noise variance $\sigma _N^2$.
Using the *smoothed* gradients, the loss of positional accuracy due to *detail loss only* is estimated as $$e^{(0)} = \frac{{\sum\limits_{\bf{p}} {\sqrt {\hat \lambda \left( {\bf{p}} \right)} } }}{{\sum\limits_{\bf{p}} {\sqrt {\tilde \lambda \left( {\bf{p}} \right)} } }}
\label{eqn:detaillossquality}$$ after posing $$\tilde \lambda \left( {\bf{p}} \right) = \sum\limits_{\bf{q}} {w{{\left( {\bf{q}} \right)}^2}{{\left| {\tilde y\left( {{\bf{q}} + {\bf{p}}} \right)} \right|}^2}} \label{eqn:tilde_lambda}$$ $$\hat \lambda \left( {\bf{p}} \right) = \left[ {\sum\limits_{\bf{q}} {w{{\left( {\bf{q}} \right)}^2}{{\left| {\hat y\left( {{\bf{q}} + {\bf{p}}} \right)} \right|}^2}} } \right] - \alpha \hat \mu \left( {\bf{p}} \right)\label{eqn:hat_lambda}$$ where $$\hat \mu \left( {\bf{p}} \right) = \sum\limits_q {w{{\left( {\bf{q}} \right)}^2}{{\left| {\nu \left( {{\bf{p}} + {\bf{q}}} \right)} \right|}^2}}
\label{eqn:Mu}$$ and the coefficient $\alpha = 0.56$, valid for $s_w = 1$, takes into account the worst case spurious noise prediction in , as described in Appendix. The same window $w{{\left( {\bf{q}} \right)}^2}$ is used for , and , so that the effective area interested for detail loss evaluation is about four times the prediction spot, due to the convolutional spread.
In $\hat \lambda \left( {\bf{p}} \right)$ is clipped within the admissible interval $\left[ {0,\tilde \lambda \left( {\bf{p}} \right)} \right]$ to increase its robustness to estimation errors.
However, expression has to be refined considering that:
- the FI is defined using the *ideal* gradient, which amplifies the spectral components of the image proportionally to the spatial frequency. However, the magnitude response of the smoothed gradient operator decays at high spatial frequencies. Since the detail loss is mostly characterized by the attenuation of high frequency components, the gradient loss is actually under-estimated by the smoothed gradients and . Therefore, the FI calculated from the smoothed gradient is under-estimated in spots characterized by a rich high frequency content. In this work, the compensation of the FI estimate was obtained in a simple way by raising the magnitude of and to an exponent $\gamma > 1$. The optimal value of $\gamma$ is hard to predict and must be empirically identified. Empirical DMOS values of the three independent databases employed in this work are well explained by setting $\gamma = 1.5$ (see Fig. \[fig4\]);
- the pooling should be extended only to the set of points ${\bf{p}} \in \left\{ P \right\}$ where the gradient prediction by is reliable. In particular, the spots characterized by severe ill-conditioning due to unbalanced energy of the LS equations [@HUBER81] are excised from the pooling set. In particular, the spots almost exactly centered on strong edges are discarded, according to the heuristic rule $$P = \left\{ {{\bf{p}}:\tilde y\left( {\bf{p}} \right) < 0.3\mathop {\max }\limits_{\bf{p}} \left[ {\left| {\tilde y\left( {\bf{p}} \right)} \right|} \right]} \right\} \; .
\label{eqn:poolingset}$$
- To enhance the independence between metrics, full weight should be given only to the points where spurious details are negligible, indicating that the model is really accurate, while the weight should be reduced everywhere detail loss overlaps with spurious details.
Then, the proposed estimate of the perceptual impact due to details loss only is $$e = \frac{{\sum\limits_{{\bf{p}} \in \left\{ P \right\}} {{\rho ^ - }\left( {\bf{p}} \right)\hat \lambda {{\left( {\bf{p}} \right)}^{{\gamma \mathord{\left/
{\vphantom {\gamma 2}} \right.
\kern-\nulldelimiterspace} 2}}} + \upsilon } }}{{\sum\limits_{{\bf{p}} \in \left\{ P \right\}} {{\rho ^ - }\left( {\bf{p}} \right)\tilde \lambda {{\left( {\bf{p}} \right)}^{{\gamma \mathord{\left/
{\vphantom {\gamma 2}} \right.
\kern-\nulldelimiterspace} 2}}} + \upsilon } }}
\label{eqn:detaillossfinal}$$ where $\upsilon = 0.1$ is a small regularizing constant, $\gamma = 1.5$ and $${\rho ^ - }\left( {\bf{p}} \right) = {\rm{ }}\left\{ {\begin{array}{*{20}{l}}
1&{{\rm{if}}\quad \hat \mu \left( {\bf{p}} \right) < 0.01\tilde \lambda \left( {\bf{p}} \right)\;}\\
{0.25}&{{\rm{elsewhere}}}
\end{array}} \right.
\label{eqn:PoolingWeights}$$ is a non critical weighting factor. Finally, the DMOS component attributed to pure detail loss is estimated as $${d^ - } = 1 - e \;.
\label{eqn:dminus}$$
Spurious detail metric {#section:DetailadditionMetric}
----------------------
The spurious details are substantially meaningless for the observer, so that their localization accuracy should not be relevant as in the case of detail loss. It is rather plausible that the annoyance produced by spurious details is similar to that caused by random noise. Since for images affected by additive noise the DMOS is quite well estimated by the PSNR logarithmic index, it is assumed that the perceptual impact caused by spurious details is logarithmically related to the ratio between the average energy ${{\tilde \lambda }_{av}}$ of the original details and the average energy ${{\hat \mu }_{av}}$ of the lost details, defined as
[rl]{} [\_[av]{}]{} & =\
[\_[av]{}]{} & = . \[eqn:average\_quantities\]
Imposing that the subjective quality estimate is zero for diverging noise, let us define the quantity $$f = \ln \left( {1 + c \frac{{{{\tilde \lambda }_{av}}}}{{{{\hat \mu }_{av} + \sigma _V^2}}}} \right)
\label{eqn:logscore}$$ where the constant $\sigma _V^2 = 20$ accounts for gradient noise variance which starts to impair the quality for white noise corrupted images [@SHEIKH06], and the positive constant $c$ sets the working point of the logarithmic law. It has been found that the empirical DMOS values are well explained by the non-critical value $c = 0.1$ (see Fig. \[fig4\]).
Finally, the proposed estimate of the perceptual impact attributed to spurious details only is calculated as[^7] $$t = \frac{f}{{{f_0}}} = \frac{{\ln \left( {1 + c\frac{{{{\tilde \lambda }_{av}}}}{{{{\hat \mu }_{av} + \sigma _V^2}}}} \right)}}{{\ln \left( {1 + c \frac{{{{\tilde \lambda }_{av}}}}{{\sigma _V^2}}} \right)}}
\label{eqn:detailadditionquality}$$ where $t = 1$ is imposed for identical test and reference images. Thus the estimate of the DMOS component attributed to spurious details is $${d^ + } = 1 - t \;.
\label{eqn:detaillossmetric}$$
Metric combination {#section:MetricCombination}
------------------
Following the VICOM scheme [@CAPODIFERRO12], the overall DMOS prediction, referred to as $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}
\over D} $, is calculated by combining ${d^ - }$ and ${d^ + }$ through a general *two-dimensional* parametric function $$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}
\over D} = f\left( {{d^ - },{d^ + };{\bf{a}}} \right)
\label{eqn:MetricsCombination}$$ where $\bf a$ is the fitting coefficient vector.
The basic assumption of this paper is that detail losses and spurious details are distinctly perceived, as evident by common experience. For a Taylor series expansion of around any point $\left( {d_0^ - ,d_0^ + } \right)$, this hypothesis is expressed by $$\frac{{{\partial ^{m + n}}f\left( {{d^ - },{d^ + };{\bf{a}}} \right)}}{{\partial {{\left( {{d^ - }} \right)}^m}\partial {{\left( {{d^ + }} \right)}^n}}} \approx 0
\label{eqn:cross_sensitivities}$$ for $m,n \ge 1$, leading to the decoupled form $$f\left( {{d^ - },{d^ + };{\bf{a}}} \right) = {f^ - }\left( {{d^ - };{{\bf{a}}^ - }} \right) + {f^ + }\left( {{d^ + };{{\bf{a}}^ + }} \right) \;.
\label{eqn:decoupled_f}$$
In addition, if the marginal metrics ${f^ - }\left( {{d^ - };{{\bf{a}}^ - }} \right) $ and ${f^ + }\left( {{d^ + };{{\bf{a}}^ + }} \right) $ are affine functions of ${d^ - }$ and ${d^ + }$, respectively, the approximation boils down to $$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}
\over D} = {a_0} + a_1^ - {d^ - } + a_1^ + {d^ + }
\label{eqn:LinearMetricsCombination}$$ where the constant $a_0$ accounts for possible non-zero DMOS scores assigned to the original images during experimental sessions, while $a_1^-$ and $a_1^+$ compensate for the different DMOS sensitivity with respect to ${d^ - }$ and ${d^ + }$. The last point will be discussed in detail in Sect. \[subsection:AnInvarianceProperty\].
The form of is deemed valid for impairments coming from a mixture of detail loss and spurious detail addition over small spots and distributed all over the test image, as it happens for instance in coding and error correction applications. Other impairments, such as luminance and contrast changes, chromatic aberrations, strong and isolated artifacts and low-frequency, correlated noise, are not directly covered by the present D-VICOM model.
The D-VICOM decomposition scheme is loosely related to the Most Apparent Distortion (MAD) scheme, based on a HVS model [@LARSON10]. The basic difference is that the MAD uses two distinct metrics for high and low quality regions (instead of lost and spurious details), according to a visibility map.
Experiments and results {#section:Experiments and results}
=======================
The linearity and the uniformity of the *scatterplot* of the IQA metric versus the empirical DMOS (see, for instance, Fig. \[fig3\] in the sequel) of different databases is the ultimate requirement in applications. To achieve this goal, IQA metrics generally need a linearization through a parametric function [@SHEIKH06B]. The linearized scatterplot should be tightly concentrated around the diagonal.
MSE and LCC are good indicators of the scatterplot concentration [@SHEIKH06B], if a wide number of test images, uniformly distributed across the DMOS range, is available, the outliers [@HUBER81] of the empirical DMOS have been properly excised and the fitting parameter estimates are not critically influenced by a specific image content [@tarpey00]. On the other hand, the proper training of the D-VICOM requires the presence in the data set of a class of mixtures covering from essentially spurious detail addition to essentially lost details.
For these reasons, the statistical analysis of the proposed DMOS estimator was performed on large image databases, containing test images affected by different impairments. Specifically, the LIVE DBR2 [@LIVEDBR2], the TID2008 [@PONOMARENKO09] and the CSIQ [@LARSON10] databases were chosen. These independent databases contain annotated DMOS values obtained through different experimental settings and share four common impairment classes: additive white noise, Gaussian blur, JPEG and JPEG2000 coding.
The statistical performance of D-VICOM and other competing estimators is illustrated in the sequel, listing the number of the adjustable parameters of the DMOS fitting function. Except for the classical VICOM [@CAPODIFERRO12], linearization was performed by a five parameter monotonic logistic function [@SHEIKH06B; @VQEG03], trained by the BRLS modified Newton algorithm [@parisi96] over several runs, to minimize the risk of trapping into local minima.
Ordinary LS fitting was used to directly optimize the RMSE. Since we are mainly interested in the linearization parameters, it is worth noting that ordinary LS fitting coefficients are unbiased under the reasonable assumptions of zero mean empirical DMOS distribution, even in the case of different variance across test images [@HAYKIN96].
All metrics were computed by on-line available software [@LIVEDBR2; @fsimpage13; @gmsdpage14; @MADPAGE10; @msssimpage11]. A preliminary statistical analysis was performed on the full LIVE DBR2 [@LIVEDBR2], which includes impairment classes [@SHEIKH06] all covered by the D-VICOM model. Then, the statistical analysis was focused on the classes of impairment common to all three databases.
LIVE DBR2 analysis {#subsection:LIVEDBR2Analysis}
------------------
LS fitting of onto the empirical LIVE DBR2 [@LIVEDBR2] DMOS scores yields the following estimator: $$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}
\over D} = 7.8 + 71. 2\, {d^ - } + 47.0 \,{d^ + } \;.
\label{eqn:LIVEDVICOM}$$
Following the VICOM scheme [@CAPODIFERRO12], the *virtual cognitive state* of each distorted image is defined by the pair $\left( {d^ -, d^ + }\right)$. The two-dimensional mapping of the cognitive states constitutes the *cognitive chart*, depicted in Fig. \[fig2\] for the LIVE DBR2 [@LIVEDBR2]. The relationship between the cognitive state associated to each test image and the DMOS estimated by can be read on the superimposed iso-metric DMOS lines, which are rectilinear. They provide an immediate visualization of the quality level of the whole data set.
As expected, blurred images, mostly characterized by lost details, and noisy images, mostly characterized by spurious details, cluster in the proximity of the coordinate axes. This empirical verification supports the fundamental assumption made in this paper about the distinct perception of detail loss and spurious details.
Nevertheless, at very high noise levels, gray-scale saturation induces detail loss, and, for very strong blur, halos generate spurious details. A nice surprise is that the metrics $d ^ -$ and $d ^ +$ remain admirably complementary, even in these extreme regions.
JPEG and JPEG2000 images cluster into distinct chart regions, pointing out the different management of detail losses and artifacts operated by these compression techniques [@CAPODIFERRO12]. Fast fading states reflect a prevalence of spurious details for light impairments and of detail loss for strong impairments.
![The cognitive chart for the LIVE DBR2 [@LIVEDBR2]. The iso-metric lines for the D-VICOM regression are superimposed.[]{data-label="fig2"}](Quality2014dbr2vcsFig2.png){width="3.2in"}
In Table \[table:VICOM statistics\] the RMSE, the SROCC and the LCC of the D-VICOM are compared to the ones of popular methods, after logistic linearization [@SHEIKH06B].
[@|c|c|c|c|c|]{} Model & Parameter no. & RMSE & SROCC & LCC\
PSNR & $5$ & $13.40$ & $0.876$ & $0.872$\
MS-SSIM & $5$ & $8.638$ & $0.951$ & $0.949$\
VIF & $5$ & $7.853$ & $0.964$ & $0.958$\
VICOM & $5$ & $6.686$ & $0.970$ & $0.969$\
FSIM & $5$ & $7.693$ & $0.963$ & $0.960$\
MAD & $5$ & $6.802$ & $0.968$ & $0.969$\
GMSD & $5$ & $7.625$ & $0.963$ & $0.960$\
D-VICOM & $\bf 3$ & $\bf{6.371}$ & $\bf{0.975}$ & $\bf{0.972}$\
\[table:VICOM statistics\]
![Top row: Scatterplot of the D-VICOM estimate versus DMOS for the LIVE DBR2. $RMSE = 6.371$, $SROCC = 0.975$, $LCC = 0.972$. Middle row: Scatterplot of the MAD [@LARSON10] scores versus DMOS for the LIVE DBR2 before logistic linearization. Bottom row: Scatterplot of the MAD estimate versus DMOS after logistic linearization. $RMSE = 6.802$, $SROCC = 0.968$, $LCC = 0.969$.[]{data-label="fig3"}](Quality2014liveDvicomFig3.png "fig:"){width="3.2in"} ![Top row: Scatterplot of the D-VICOM estimate versus DMOS for the LIVE DBR2. $RMSE = 6.371$, $SROCC = 0.975$, $LCC = 0.972$. Middle row: Scatterplot of the MAD [@LARSON10] scores versus DMOS for the LIVE DBR2 before logistic linearization. Bottom row: Scatterplot of the MAD estimate versus DMOS after logistic linearization. $RMSE = 6.802$, $SROCC = 0.968$, $LCC = 0.969$.[]{data-label="fig3"}](Quality2014LivemadrawFig3b.png "fig:"){width="3.2in"} ![Top row: Scatterplot of the D-VICOM estimate versus DMOS for the LIVE DBR2. $RMSE = 6.371$, $SROCC = 0.975$, $LCC = 0.972$. Middle row: Scatterplot of the MAD [@LARSON10] scores versus DMOS for the LIVE DBR2 before logistic linearization. Bottom row: Scatterplot of the MAD estimate versus DMOS after logistic linearization. $RMSE = 6.802$, $SROCC = 0.968$, $LCC = 0.969$.[]{data-label="fig3"}](Quality2014LivemadFig3c.png "fig:"){width="3.2in"}
In Fig. \[fig3\] the scatterplot of the D-VICOM is compared to the scatterplot of the seemingly closest uni-dimensional competitor, i.e., the MAD estimator [@LARSON10], before and after logistic linearization. The linearity of the D-VICOM is comparable to the linearity of the MAD after logistic transformation. The DMOS scale goes beyond the conventional upper limit of $100$, because the LIVE DBR2 scores were *realigned* among different impairments [@SHEIKH06B; @LIVEDBR2].
Fig. \[fig4\] puts into evidence the negligible marginal sensitivity of the D-VICOM RMSE with respect to $\gamma$, $c$ and $\sigma_V^2$ around the values ($\gamma = 1.5$, $c = 0.1$ and $\sigma_V^2 = 20$) adopted as constants in our model.
![Plots showing the negligible marginal sensitivity of the RMSE of the LIVE DBR2 DMOS estimates with respect to the values of (from top to bottom) $\gamma$, $c$ and $\sigma_V^2$, posed as constants in the D-VICOM model in Sect. \[section:Linear metrics\]. In each plot, the other two quantities are kept fixed at the nominal values ($\gamma = 1.5$, $c = 0.1$ and $\sigma_V^2 = 20$). []{data-label="fig4"}](Quality2014LiveSensFig4.png){width="3.2in"}
Performance on equivalent subsets of TID2008 and CSIQ databases {#subsection:Performance on equivalent subsets of TID2008 and CSIQ databases}
---------------------------------------------------------------
To validate previous results, the same analysis was extended to the similar subsets of the TID2008 and CSIQ databases. In particular, the five image subsets of TID2008 (white Gaussian noise, Gaussian blur, JPEG and JPEG2000 coding and JPEG2000 transmission errors) already used in [@CAPODIFERRO12] and four subsets (white Gaussian noise, Gaussian blur, JPEG and JPEG2000) of the CSIQ were chosen. Fitting results are summarized in Tables \[table:TID2008fivesetstatistics\] and \[table:CSIQfoursetstatistics\]. TID2008 MOS scores were translated into DMOS scores for uniformity.
[@|c|c|c|c|c|c|]{} Model & Parameter & RMSE & SROCC & LCC & LCC\
& no. & & & & (LIVE DBR2)\
PSNR & $5$ & $9.774$ & $0.840$ & $0.801$ & $0.809$\
MS-SSIM & $5$ & $7.895$ & $0.893$ & $0.876$ & $0.858$\
VIF & $5$ & $6.800$ & $0.908$ & $0.910$ & $0.917$\
VICOM & $5$ & $6.436$ & $0.929$ & $0.919$ & $0.876$\
FSIM & $5$ & $6.673$ & $0.938$ & $0.913$ & $0.905$\
MAD & $5$ & $6.005$ & $0.921$ & $0.930$ & $0.914$\
GMSD & $5$ & $7.200$ & $0.932$ & $ 0.898$ & $0.894$\
D-VICOM & $\bf 3$ & $\bf 5.279$ & $\bf 0.956$ & $\bf 0.946$ & $\bf 0.942$\
\[table:TID2008fivesetstatistics\]
[@|c|c|c|c|c|c|]{} Model & Parameter & RMSE & SROCC & LCC & LCC\
& no. & & & & (LIVE DBR2)\
PSNR & $5$ & $0.1466$ & $0.922$ & $0.855$ & $0.891$\
MS-SSIM & $5$ & $0.0915$ & $0.953$ & $0.946$ & $0.952$\
VIF & $5$ & $0.0722$ & $0.919$ & $0.924$ & $0.960$\
FSIM & $5$ & $0.1017$ & $0.962$ & $0.934$ & $0.963$\
MAD & $5$ & $0.0667$ & $0.967$ & $ 0.972$ & $\bf 0.973$\
GMSD & $5$ & $0.0701$ & $0.969$ & $0.969$ & $0.969$\
D-VICOM & $\bf 3$ & $\bf 0.0646$ & $\bf 0.972$ & $\bf 0.974$ & $ \bf 0.973$\
\[table:CSIQfoursetstatistics\]
For these subsets, the accuracy of the D-VICOM is still the best one. This is also visible from the comparison between the scatterplots of the D-VICOM and of the linearized MAD (the seemingly most performing competitor) reported in Fig. \[fig8\].
{width="6.4in"}
Statistical analysis {#subsection:Statistical analysis}
--------------------
Returning back to the Tables \[table:TID2008fivesetstatistics\] and \[table:CSIQfoursetstatistics\], the rightmost column contains the LCC obtained by applying the regression coefficients computed for the the largest and most complete data set in our analysis (the LIVE DBR2) on the target subset to assess the *generalization* capabilities of IQA metrics. Considering that the LCC is insensitive to affine metric and/or DMOS transformations[^8], results support the generalization power of good IQA metrics and, at the same time, show that D-VICOM is uniformly the most performing subjective quality predictor in this *cross-validation* exercise, achieving very close results between the two fitting scenarios.
Moderate non Gaussianity and local additive effects of DMOS errors are expected features [@SHEIKH06B], due to the low number of human observers involved in subjective tests and the generally non-robust [@HUBER81] protocols for score averaging and outlier rejection. Therefore it is useful to check the leave-one-out cross validation (LOOCV) RMSE, i.e., the square root of the *Predicted Sum of Squares* (PRESS) [@tarpey00], scaled by the subset size, since it measures the *expected RMSE* for a test image added to the data set. The LOOCV RMSE is *analytically* computable for D-VICOM, depending on the projection (*hat*) matrix [@HUBER81; @tarpey00] of the LS system matrix generated by , and yields $6.395$ for the LIVE DRB2, $5.316$ for the TID2008 subset and $0.0650$ for the CSIQ subset. All these values are *extremely* close to the corresponding fitting RMSE, confirming the completeness and the DMOS stability of tried subsets, as well as the good conditioning of the LS fittings.
For the other metrics, the empirical LOOCV RMSE computation is unpractical, but, assuming Gaussianity and homoscedasticity of the DMOS errors within each subset, it can be estimated for large samples by the Akaike Information Criterion (AIC) [@AKAIKE74; @STONE77], which can be simplified for the generic $k$th IQA metric as $$AIC_k = 2N \ln \left({RMSE_k}\right) + 2 \left({P_k+1}\right)
\label{eqn:aic}$$ for a subset size $N$ and $P_k$ adjustable parameters[^9]. AIC application does not require nested models [@STONE77], is summable across databases for overall evaluation and admits a significance test through the analysis of the (raw) AIC weights [@BURNHAM04] ${\Delta _k} = {e^{ - 0.5\left[ {AI{C_k} - \mathop {\min }\limits_l \left( {AI{C_l}} \right)} \right]}} $. In particular, $\Delta_k < 0.1$ (i.e., AIC scores different by more than $4$ nats) indicate high confidence ($\approx 95\%$) in the model with the smallest AIC [@BURNHAM04].
For the three analyzed subsets, the AIC and ${\Delta_k}$ scores are listed in Table \[table:DBR2TID5CSIQ4statistics2\]. The AIC significance test is passed by the D-VICOM in all cases. In the same table, the residual kurtosis and the $95$th percentile ($95p$) of residual magnitudes provide additional insight about the fitting error properties.
However, the kurtosis and $95p$ values do not indicate clear differences or worrisome issues across metrics and subsets, with a moderate non Gaussianity of the residuals and an acceptable $p95 \approx 2{\mkern 1mu} \cdot RMSE$.
On the other hand, the small typical LCC ($<0.5$) between the residual vectors of the best metrics here (the D-VICOM and the MAD) suggests that there is a substantial excess variance not explained by the models. So a comparison of the LS error variances by F-test for *independent* Gaussian variables [@snedecor89] is useful.
In this case, the $95\%$ one-sided confidence level about the significance of the D-VICOM RMSE improvement is attained whenever the RMSE of a competing IQA metric exceeds $6.767$ for the LIVE DBR2 database, $5.684$ for the TID2008 subset and $0.0691$ for the CSIQ subset. This confidence level is always attained by the D-VICOM, except against the classical VICOM for the LIVE DBR2 and the MAD for the CSIQ subset.
[@|c|c|c|c|c|c|c|c|c|c|c|c|c|]{} & & & &\
Model & LIVE & TID5 & CSIQ4 & LIVE & TID5 & CSIQ4 & LIVE & TID5 & CSIQ4 & LIVE & TID5 & CSIQ4\
PSNR & $3.06$ & $4.37$ & $3.48$ & $27.12$ & $21.48$ & $0.28$ & $4051.7$ & $2291.7$ & $-2292.1$ & $0.00$ & $0.00$ & $0.00$\
MS-SSIM & $3.74$ & $2.95$ & $4.32$ & $17.33$ & $15.24$ & $0.18$ & $3371.3$ & $2078.2$ & $-2857.7$ & $0.00$ & $0.00$ & $0.00$\
VIF & $2.83$ & $3.47$ & $5.84$ & $15.17$ & $13.74$ & $0.15$ & $3222.9$ & $1928.9$ & $-3142.0$ & $0.00$ & $0.00$ & $0.00$\
FSIM & $4.47$ & $3.99$ & $3.58$ & $15.25$ & $13.56$ & $0.19$ & $3190.8$ & $1910.1$ & $-2730.9$ & $0.00$ & $0.00$ & $0.00$\
MAD & $4.29$ & $3.16$ & $3.67$ & $13.94$ & $12.36$ & $0.13$ & $2999.0$ & $1804.6$ & $-3237.1$ & $0.00$ & $0.00$ & $0.00$\
GMSD & $5.35$ & $4.38$ & $4.78$ & $15.35$ & $15.96$ & $0.15$ & $3177.0$ & $1986.1$ & $-3177.4$ & $0.00$ & $0.00$ & $0.00$\
D-VICOM & $3.49$ & $4.22$ & $3.90$ & $12.84$ & $10.90$ & $0.13$ & $2893.0$ & $1671.7$ & $-3279.4$ & $1.00$ & $1.00$ & $1.00$\
\[table:DBR2TID5CSIQ4statistics2\]
Un-covered impairments {#subsection:Un-covered impairments}
----------------------
To meet the possible, legitimate curiosity of the reader, in Tables \[table:TID2008statistics\] and \[table:CSIQstatistics\] the statistical performance indexes are reported for the whole TID2008 and CSIQ archives, that contain impairments not explicitly modeled by the D-VICOM scheme. The average accuracy of D-VICOM is still acceptable, even if theoretically unsupported.
[@|c|c|c|c|c|]{} Model & Parameter no. & RMSE & SROCC & LCC\
PSNR & $5$ & $12.74$ & $0.553$ & $0.519$\
MS-SSIM & $5$ & $7.968$ & $0.854$ & $0.845$\
VIF & $5$ & $9.169$ & $0.750$ & $0.789$\
FSIM & $5$ & $7.256$ & $0.881$ & $0.874$\
MAD & $5$ & $8.278$ & $0.835$ & $0.832$\
GMSD & $5$ & $\bf 7.156$ & $\bf 0.891$ & $\bf 0.877$\
D-VICOM & $\bf 3$ & $7.923$ & $0.834$ & $0.837$\
\[table:TID2008statistics\]
[@|c|c|c|c|c|]{} Model & Parameter no. & RMSE & SROCC & LCC\
PSNR & $5$ & $0.1719$ & $0.806$ & $0.756$\
MS-SSIM & $5$ & $0.1275$ & $0.910$ & $0.874$\
VIF & $5$ & $0.1004$ & $0.919$ & $0.924$\
FSIM & $5$ & $0.1197$ & $0.922$ & $0.890$\
MAD & $5$ & $\bf 0.0804$ & $0.949$ & $\bf 0.952$\
GMSD & $5$ & $0.0824$ & $\bf 0.957$ & $0.949$\
D-VICOM & $\bf 3$ & $0.0961$ & $0.937$ & $0.931$\
\[table:CSIQstatistics\]
An invariance property {#subsection:AnInvarianceProperty}
----------------------
Consistency aspects of human perception of quality among different people are relevant for IQA, as discussed in [@wajid14]. The analysis performed so far neglected the relative perceptual sensitivity of lost and spurious details on the DMOS. In the sequel, an invariance property is deduced from the following *equivalence argument*: a subject has to indicate the blur strength which yields the same subjective quality loss of a fixed amount of noise added to the same image. There is not any apparent reason why the response should change among different people with normal vision capability under the same viewing conditions, at least in an average sense.
In a formal setting, it is assumed that two generic databases $A$ and $B$, characterized by the D-VICOM coefficient sets $\left\{ {{a_{0A}},a_{1A}^ - ,a_{1A}^ + } \right\}$ and $\left\{ {{a_{0B}},a_{1B}^ - ,a_{1B}^ + } \right\}$, share the same original images and that a generic blurred image contained in $A$ and $B$ has DMOS $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over D}_A $ and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over D}_B $ respectively. Since blur essentially does not add spurious details, $d ^ + \approx 0$.
Moreover, let us imagine that in both databases there exists an *equivalent* image contaminated by additive noise and characterized by the same DMOS values $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over D}_A $ and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over D}_B $. Since white noise essentially does not introduce detail loss, $d ^ - \approx 0$.
Inserting these values into yields
[rl]{} \_A = [a\_[0A]{}]{} + a\_[1A]{} \^ - d \^ - & = [a\_[0A]{}]{} + a\_[1A]{}\^+ d \^ +\
\_B = [a\_[0B]{}]{} + a\_[1B]{} \^ - d \^ - & = [a\_[0B]{}]{} + a\_[1B]{}\^+ d \^ + . \[eqn:blurnoise\]
Simplifying, it turns out that the ratio $$r = \frac{{{d^ + }}}{{{d^ - }}} = \frac{{a_{1A}^ - }}{{a_{1A}^ + }} = \frac{{a_{1B}^ - }}{{a_{1B}^ + }}
\label{eqn:UniversalRatio}$$ should not change across databases. On the basis of this equivalence argument, the D-VICOM is rewritten as $$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}
\over D} = {a_0} + a_1^ + \left( {{d^ + } + r{d^ - }} \right)
\label{eqn:D-VICOMtwinpar}$$ where $r$ is kept as a *constant*. This argument can be extended to the actual cases where images are different among databases, stating that the D-VICOM estimator can be tuned to any empirical database by adjusting only the pair $\left( {{a_0},a_1^ + } \right)$, i.e., the estimator *offset* and *slope*. This result confirms at a glance the excellent D-VICOM LCC cross-validation results of Sect. \[subsection:Performance on equivalent subsets of TID2008 and CSIQ databases\].
Statistical identification of $r$ started from a coarse guess obtained by the regression coefficients of each *quartet* of distortions common to LIVE DBR2, TID2008 and CSIQ databases. Then an iterative LS optimization of the pairs $\left( {{a_0},a_1^ + } \right)$ of each quartet in and of a unique $r$ on the *joint* database (seven parameters in total) was performed by minimizing the sum of the squared DMOS residuals of the three subsets[^10]. The cost function achieved a broad minimum at $r=1.64$, so that the D-VICOM estimator assumes the *operative* form $$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}
\over D} = {a_0} + a_1^ + \left( {{d^ + } + 1.64{d^ - }} \right)
\label{eqn:ID-VICOMtwinpar}$$ characterized by only *two* parameters, offset ($a_0$) and slope ($a ^ + $), to be tuned to the desired DMOS scale. Table \[table:TwoparDVICOMstatistics\] shows that the statistical performance of the two-parameter D-VICOM remains substantially unchanged, as expected.
The overall flow chart of the D-VICOM is shown in Fig. \[fig:DVICOMdfg\].
![The flow chart of the D-VICOM which shows the computational layers.[]{data-label="fig:DVICOMdfg"}](DVICOMdfg.png){width="2.8in"}
To explore the impairment coverage of , Table \[table:TwoparDVICOMstatisticsTIDdetail\] lists the statistical indexes of the D-VICOM, the MAD, and the GMSD (i.e., the most robust metric from Table \[table:CSIQstatistics\]) on the $17$ image subsets of the TID2008 [@PONOMARENKO09], trained on the five impairment subset previously used, deemed *safe* for ordinary LS fitting.
The RMSE strongly varies among subsets and metrics, complicating the suitability check of IQA metrics on specific impairment sets. However, AIC minimization, under the proper statistical assumptions, remains useful for a quick significance test [@BURNHAM04]. Assuming in this scenario zero mean Gaussian fitting errors with a different $RMSE_{mk}$ for the $k$th metric and the $m$th subset, containing $N_m$ images, and $P_k$ fitting parameters, an upper bound to the $AIC_k$ for the wider set $M = \left\{ {{m_1},{m_2}, \ldots } \right\}$ is: $$AI{C_k} \le \sum\limits_{m \in M} 2{\left[ {{N_m}\ln \left( {RMS{E_{mk}}} \right) + 1} \right]} + 2{P_k} \;.
\label{eqn:compoundAIC}$$
The AIC verification effort is still combinatorial, but from Table \[table:TwoparDVICOMstatisticsTIDdetail\] the D-VICOM emerges as the best overall choice as regards the fitting RMSE (and therefore the AIC), except for low frequency impairments (quantization noise, block distortions, contrast and intensity changes), barely observable by the gradient operator , but exhibits a surprising robustness to impulse distortion.
[@|c|c|c|c|c|]{} Subset & RMSE & SROCC & LCC & AIC\
LIVE DBR2 (full) & $6.415$ & $0.975$ & $0.972$ & $2901.8$\
TID2008 (5 dist. subset) & $5.344$ & $0.955$ & $0.945$ & $1682.0$\
TID2008 (full) & $7.946$ & $0.830$ & $0.846$ & $7053.1$\
CSIQ (4 dist. subset) & $0.0648$ & $0.973$ & $0.973$ & $-3277.7$\
CSIQ (full) & $0.1065$ & $0.918$ & $0.914$ & $-3783.0$\
\[table:TwoparDVICOMstatistics\]
[|c|c|c|c|c|c|c|c|c|c|c|]{} & & &\
No. & Subset name [@PONOMARENKO09] & RMSE & SROCC & LCC & RMSE & SROCC & LCC & RMSE & SROCC & LCC\
1 & Additive Gaussian noise & $\bf 3.971$ & $0.877$ & $0.856$ & $6.380$ & $0.838$ & $0.816$ & $4.053$ & $\bf 0.918$ & $\bf 0.897$\
2 & Additive noise in color... & $3.394$ & $0.884$ & $0.884$ & $3.881$ & $0.827$ & $0.823$ & $\bf 2.882$ & $\bf 0.898$ & $\bf 0.896$\
3 & Spatially correlated noise & $\bf 5.069$ & $0.887$ & $0.877$ & $5.659$ & $0.868$ & $0.860$ & $6.960$ & $\bf 0.913$ & $\bf 0.916$\
4 & Masked noise & $\bf 4.925$ & $\bf 0.851$ & $\bf 0.867$ & $5.094$ & $0.734$ & $0.760$ & $6.072$ & $0.709$ & $0.569$\
5 & High frequency noise & $\bf 4.266$ & $0.907$ & $\bf 0.936$ & $8.530$ & $0.887$ & $0.894$ & $4.448$ & $\bf 0.919$ & $0.928$\
6 & Impulse noise & $\bf 7.562$ & $\bf 0.823$ & $\bf 0.802$ & $14.28$ & $0.065$ & $0.042$ & $7.602$ & $0.661$ & $0.627$\
7 & Quantization noise & $10.627$ & $0.838$ & $0.786$ & $13.29$ & $0.819$ & $0.800$ & $\bf 4.536$ & $\bf 0.890$ & $\bf 0.882$\
8 & Gaussian blur & $\bf 4.710$ & $\bf 0.954$ & $\bf 0.944$ & $5.116$ & $0.926$ & $0.934$ & $6.161$ & $0.897$ & $0.894$\
9 & Image denoising & $6.243$ & $0.948$ & $0.955$ & $\bf 5.687$ & $0.943$ & $0.961$ & $6.332$ & $\bf 0.975$ & $\bf 0.978$\
10 & JPEG compression & $5.651$ & $0.935$ & $0.956$ & $6.444$ & $0.927$ & $0.949$ & $\bf 4.629$ & $\bf 0.952$ & $\bf 0.985$\
11 & JPEG2000 compression & $6.734$ & $0.969$ & $0.964$ & $\bf 5.540$ & $0.971$ & $0.974$ & $10.30$ & $\bf 0.980$ & $\bf 0.978$\
12 & JPEG transmission errors & $9.911$ & $\bf 0.867$ & $\bf 0.860$ & $\bf 7.766$ & $0.863$ & $0.854$ & $9.402$ & $0.862$ & $0.852$\
13 & JPEG2000 transmission errors & $\bf 5.252$ & $0.874$ & $\bf 0.875$ & $6.417$ & $0.839$ & $0.830$ & $8.793$ & $\bf 0.883$ & $0.868$\
14 & Non eccentricity pattern noise & $\bf 8.859$ & $0.746$ & $0.742$ & $13.71$ & $\bf 0.829$ & $\bf 0.825$ & $12.65$ & $0.760$ & $0.756$\
15 & Local block-wise distortions... & $17.61$ & $0.640$ & $0.650$ & $11.07$ & $0.798$ & $0.801$ & $\bf 3.880$ & $\bf 0.897$ & $\bf 0.900$\
16 & Mean shift (intensity shift) & $8.642$ & $0.475$ & $0.529$ & $8.789$ & $0.521$ & $0.576$ & $\bf 5.160$ & $\bf 0.650$ & $\bf 0.677$\
17 & Contrast change & $13.30$ & $\bf 0.676$ & $\bf 0.727$ & $19.75$ & $0.239$ & $0.215$ & $\bf 11.87$ & $0.465$ & $0.542$\
& $\bf 5.344$ & $\bf 0.955$ & $\bf 0.945$ & $6.005$ & $0.921$ & $0.930$ &$7.200$ &$0.932$ & $0.898$\
& $8.296$ & $0.830$ & $0.846$ & $9.631$ & $0.835$ & $0.830$ & $\bf 7.366$ & $\bf 0.891$ & $\bf 0.877$\
The SUPERQUARTET {#subsection:TheSUPERQUARTET}
----------------
The empirical DMOS values of the TID2008 and of the CISQ databases were mapped onto the LIVE DBR2 DMOS scale using the slope/offset pairs estimated in the previous step, forming a unique large database called the *SUPERQUARTET*[^11], viewed as the LIVE DBR2 populated by compatible samples migrated from the TID2008 and the CSIQ databases, after proper realignment of the DMOS values to cope with protocol differences.
The D-VICOM and a set of popular FR quality estimators, linearized by a logistic function [@SHEIKH06B], were applied to the SUPERQUARTET. Since the DMOS scale is unique (i.e., the LIVE DBR2 one), all estimators were tuned using a single set of parameters (five for the linearization of uni-dimensional IQA estimators and one offset and slope pair for the D-VICOM). Table \[table:SUPERQUARTETstatistics\] lists the RMSE, the SROCC, the LCC and theAIC scores. In Fig. \[fig5\] the cognitive chart of the SUPERQUARTET is displayed. The state clusters of all databases remarkably overlap. In Fig. \[fig6\] the scatterplots of the SUPERQUARTET DMOS values versus the optimal estimates of D-VICOM and MAD [@LARSON10], the seemingly best competitor from Table \[table:SUPERQUARTETstatistics\], are shown.
![The cognitive chart for the SUPERQUARTET. The iso-metric lines for the D-VICOM regression are superimposed.[]{data-label="fig5"}](Quality2014superquartetVCSFig5.png){width="3.2in"}
[@|c|c|c|c|c|c|]{} Model & Parameter no. & RMSE & SROCC & LCC & AIC\
PSNR & $5$ & $14.34$ & $0.892$ & $0.840$ & $8714.9$\
MS-SSIM & $5$ & $8.984$ & $0.949$ & $0.941$ & $7186.7$\
VIF & $5$ & $8.052$ & $0.960$ & $0.953$ & $6828.8$\
FSIM & $5$ & $7.555$ & $0.966$ & $0.958$ & $6620.6$\
MAD & $5$ & $7.305$ & $0.966$ & $0.962$ & $6510.6$\
GMSD & $5$ & $7.627$ & $0.968$ & $0.958$ & $6651.6$\
D-VICOM & $2$ & $\bf{6.299}$ & $\bf{0.975}$ & $\bf{0.971}$ & $\bf 6020.4$\
\[table:SUPERQUARTETstatistics\]
![Top: The two parameter D-VICOM scatterplot of the SUPERQUARTET, realigned on the LIVE DBR2 scale. $RMSE = 6.299$, $SROCC = 0.975$ and $LCC = 0.971$; Bottom: The MAD scatterplot over the SUPERQUARTET after the linearization by the five parameter logistic. $RMSE = 7.305$, $SROCC = 0.966$ and $LCC = 0.962$. []{data-label="fig6"}](Quality2014DvicomsuperquartetFig6a.png "fig:"){width="3.2in"} ![Top: The two parameter D-VICOM scatterplot of the SUPERQUARTET, realigned on the LIVE DBR2 scale. $RMSE = 6.299$, $SROCC = 0.975$ and $LCC = 0.971$; Bottom: The MAD scatterplot over the SUPERQUARTET after the linearization by the five parameter logistic. $RMSE = 7.305$, $SROCC = 0.966$ and $LCC = 0.962$. []{data-label="fig6"}](Quality2014MADsuperquartetFig6b.png "fig:"){width="3.2in"}
In turn, the excellent cross-fitting of the LIVE DBR2, the TID2008 and the CSIQ data, up to a simple affine transformation, demonstrates the remarkable statistical homogeneity of these independent experiments under the D-VICOM paradigm.
In conclusion, the following quality estimator $${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}
\over D} } = 8.0 + 45.0 \left( {{d^ + } + 1.64{d^ - }} \right)
\label{eqn:SuperquartetDVICOM}$$ is proposed as a conventional *calibration free* metric, adjusted to the LIVE DBR2 DMOS scale for the covered impairments. This estimator, hereinafter called *ID-VICOM*, when applied to the *entire* LIVE DBR2, yielded $RMSE = 6.431$, $SROCC = 0.975$ and LCC = $0.972$, nearly identical to the top performance of the three-parameter D-VICOM in Table \[table:VICOM statistics\], as well as $RMSE = 6.050$, $SROCC = 0.977$ and $LCC = 0.974$ on the joint TID+CSIQ realigned quartets.
Direct DMOS scale setting {#section:DirectDMOSScaleSetting}
=========================
However, the LIVE DBR2 DMOS scale adopted for ID-VICOM might not be adequate in some applications. It is possible to set any desired scale by conveniently specifying another *offset* and *slope* pair, using a *single* test image. To this purpose, the offset of the DMOS scale is fixed to the desired value for perfect images[^12], say ${a_0} = {a_{0U}}$, so that $${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}
\over D}} \approx {a_{0U}} + a_{1U}^ + \left( {{d^ + } + 1.64{d^ - }} \right) \;.
\label{eqn:conventionalIDVicom}$$
![Plot of the white noise standard deviation $\sigma_N$ versus $\sqrt{\hat \mu _{av}}$ for the LIVE DBR2 white noise image subset.[]{data-label="fig9"}](Quality2014awgnFig9.png){width="3.2in"}
The slope $a_{1U}^ + $ can be determined from a single image $I_N$ affected by additive white noise of known variance $\sigma_N^2$. In fact, from the diagram of Fig. \[fig9\], where the values of $\sqrt{\hat \mu _{av}}$ are plotted versus $\sigma_N$ for the LIVE DBR2 [@LIVEDBR2] noisy images, we see that the value of ${\hat \mu _{av}}$ is reliably linked to the noise variance. Moreover ${\tilde \lambda _{av}}$ is calculated from the noise free image. Therefore, $d ^ +$ is obtained from and . Assigning now a specific DMOS value to $I_N$, the value of $d ^ -$ is read on the cognitive chart of Fig. \[fig5\] and $a_{1U}^ + $ is finally determined from .
Such a huge generalization power of ID-VICOM starting from the quality of a single image is surprising at glance, but it is a necessary consequence of the following concurrent facts:
- the pronounced linearity of the metrics with the target impairments;
- the statistical stability of the additive white noise variance estimate through ${\hat \mu _{av}}$;
- the precision of the locus traced by images affected by white noise in the cognitive chart.
Computational budget {#section:Computational budget}
====================
The major computational burden of D-VICOM lies in the gradient extraction and in the solution of , which amounts to $15$ real-valued $2$-D fast correlations/convolution across the image, plus the solution of a linear system of normal equations [@golub89] of size three for each image point. With a negligible performance cost, it is also possible to solve on a point grid decimated by two on each axis and to linearly interpolate the computed coefficient sets $\left\{ {{b_{\bf{p}}}\left( k \right)} \right\}$ in the remaining points, thus reducing the prediction cost by about $75 \%$.
The average computational time of the D-VICOM MATLAB code with $s = s_w = 1$, running on PC equipped by an Intel Core i7-3370K $3.5$ GHz CPU, was $12.5$ s ($4.7$ s for the fast version[^13]) on a $1024 \times 768$ pixel image. For the sake of comparison, the MAD ran at the same $12.5$ s, the VIF at $5.1$ s, the FSIM at $1.34$ s, the MS-SSIM at $0.96$ s and the GMSD at only $60$ ms.
Conclusion {#section:Conclusion}
==========
The intricate problem of forcing a unique metric to linearly fit subjective scores caused by heterogeneous impairments was circumvented by decomposing it into two easier problems of determining partial metrics linearly related to two well discernible error causes: detail loss and spurious detail addition.
On the basis of the approximate linearity of metrics and of the assumed perceptual properties, it is shown that it is possible to fit these metrics to subjective DMOS ratings by adjusting just the two parameters of an affine transformation. This makes possible to simply merge empirical results coming from different databases for the class of impairments covered by the D-VICOM model.
Last but not least, it is possible to determine a conventional ID-VICOM index, after fixing the DMOS value for one original image and a noisy version of the same image.
In conclusion, the main merits of the D-VICOM estimator are:
- *a priori* connotation of the impairments covered by the method;
- outstanding accuracy over a wide quality range;
- simple and accurate merging of empirical DMOS from different databases;
- analytic diagnostic capabilities through the maps of the gradient attenuation, of the gradient residuals and of the cognitive states;
- choice of fast calibration based on a single sample image.
Let us finally remark that the D-VICOM quality predictor is not prone to the natural image hypothesis, which supports many existing methods. Moreover, the D-VICOM could be extended to other classes of impairments by adding appropriate metrics. In particular, the D-VICOM is suitable for detail displacement measurements, required in video and $3$-D quality assessment.
Appendix: Noise effects compensation in {#appendix:NoiseEffectsCompensation .unnumbered}
========================================
The LS system which computes the estimate ${{\bf{\hat b}}_{\bf{p}}}$ of ${{\bf{ b}}_{\bf{p}}}$ at each point $\bf p$ has the general form:
$$\left[ {\begin{array}{*{20}{c}}
{{\bf{W}}{{{\bf{\tilde Y}}}_r}}\\
{{\bf{W}}{{{\bf{\tilde Y}}}_i}}\\
{\sqrt C {{\bf{I}}_3}}
\end{array}} \right]{{\bf{\hat b}}_{\bf{p}}} \cong \left[ {\begin{array}{*{20}{c}}
{{\bf{W}}{{\bf{y}}_r}}\\
{{\bf{W}}{{\bf{y}}_i}}\\
{\bf{0}}
\end{array}} \right] \approx \left[ {\begin{array}{*{20}{c}}
{{\bf{W}}\left( {{{{\bf{\tilde Y}}}_r}{{\bf{b}}_{\bf{p}}} + {{\bf{n}}_r}} \right)}\\
{{\bf{W}}\left( {{{{\bf{\tilde Y}}}_i}{{\bf{b}}_{\bf{p}}} + {{\bf{n}}_i}} \right)}\\
{\bf{0}}
\end{array}} \right]
\label{eqn:LSSystem}$$
where $ {\bf W} $ is a diagonal matrix holding the analysis window weights $w \left({\bf q}\right) $, ${\bf{\tilde Y}}_r$ and ${\bf{\tilde Y}}_i$ are matrices built from the real and the imaginary parts of (filtered) original gradients, ${\bf{\tilde y}}_r$ and ${\bf{\tilde y}}_i$ are vectors built from the real and the imaginary parts of the test image gradients, herein assumed corrupted by the additive noise vectors ${\bf{n}}_r$ and ${\bf{n}}_i$ having zero mean, i.i.d. entries of variance ${{\sigma _N^2} \mathord{\left/ {\vphantom {{\sigma _N^2} 2}} \right. \kern-\nulldelimiterspace} 2}$.
Under above hypotheses and neglecting the small influence of the regularizing constant $\xi$, the predicted target ${\bf{\hat y}}$ can be written as [@HAYKIN96] $${\bf{\hat y}} \approx {\bf{A}}{{\bf{A}}^T}\left[ {\begin{array}{*{20}{c}}
{{\bf{W}}\left( {{{\bf{Y}}_r}{{\bf{b}}_{\bf{P}}} + {{\bf{n}}_r}} \right)}\\
{{\bf{W}}\left( {{{\bf{Y}}_i}{{\bf{b}}_{\bf{P}}} + {{\bf{n}}_i}} \right)}\\
{\bf{0}}
\end{array}} \right]
\label{eqn:predictedimage}$$ where $\bf A$ is a three-column orthogonal matrix with the same span as the system matrix. Therefore some noise is added by ${{\bf{\hat b}}_{\bf{p}}}$ to the true predicted gradient with variance $${\hat \sigma ^2} = \frac{{\sigma _N^2}}{2}{\mathop{\rm trace}\nolimits} \left( {{{\bf{A}}^T}\left[ {\begin{array}{*{20}{c}}
{{{\bf{W}}^2}}&{\bf{0}}&{\bf{0}}\\
{\bf{0}}&{{{\bf{W}}^2}}&{\bf{0}}\\
{\bf{0}}&{\bf{0}}&{\bf{0}}
\end{array}} \right]{\bf{A}}} \right)
\label{eqn:PredictedNoise}$$ which unduly increases the estimated $\hat \lambda \left( {\bf{p}} \right)$ in .
Under the same assumptions, the expected residual energy $E\left[ {J} \right]$ of is $$E\left[ {J} \right] = \sigma _N^2 \rm{trace}\left( {{{\bf{W}}^2}} \right) - {\hat \sigma ^2} \; .
\label{eqn:ResidualVariance}$$
The exact calculus of ${\hat \sigma ^2}$ at each $\bf p$ would require a costly full QR solution of [@golub89]. Following a worst case approach and setting $\rm{trace}\left( {{{\bf{W}}^2}} \right) = 1$, as done throughout the paper, the bound ${\hat \sigma ^2} \le \sigma _N^2{\rm{ }}\left( {w_1^2 + {{w_2^2} \mathord{\left/ {\vphantom {{w_2^2} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)$ holds by the Courant Fischer min-max theorem [@golub89], where $w_1$ and $w_2$ respectively are the largest and the second largest samples of $w \left({\bf q}\right) $. Combining this bound with , we get $${\hat \sigma ^2} \le \frac{{w_1^2 + {{w_2^2} \mathord{\left/
{\vphantom {{w_2^2} 2}} \right.
\kern-\nulldelimiterspace} 2}}}{{1 - \left( {w_1^2 + {{w_2^2} \mathord{\left/
{\vphantom {{w_2^2} 2}} \right.
\kern-\nulldelimiterspace} 2}} \right)}}E\left[ J \right]{\rm{ }}\mathop {\rm{ = }}\limits^{def} \alpha E\left[ J \right]
\label{eqn:AlphaCalculus}$$ whose estimate $\alpha \mu \left({\bf p}\right)$ is subtracted from the energy of the predicted gradient in .
This worst case assumption is justified by the fact that most Gaussian-windowed gradient energy is concentrated in few equations of . The efficacy of the proposed correction is demonstrated by the statistical stability of $d ^ -$ for the white noise corrupted images in Figs. \[fig2\] and \[fig5\], at the cost of a negligible positive bias.
[^1]: Copyright ©2010 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].
[^2]: Contact author: Elio D. Di Claudio, Dept. of Information Engineering, Electronics and Telecommunications (DIET), University of Rome “La Sapienza,” Via Eudossiana 18, I-00184 Rome, Italy. Ph.: +39-06-44585490, Fax: +39-06-44585632, e-mail [email protected].
[^3]: Giovanni Jacovitti, Dept. of Information Engineering, Electronics and Telecommunications (DIET), University of Rome “La Sapienza,” Via Eudossiana 18, I-00184 Rome, Italy. Ph.: +39-06-44585838, Fax: +39-06-44585632, e-mail [email protected].
[^4]: EDICS No.: SMR-HPM.
[^5]: For $s_w = 1$ the effective spot has a diameter larger than four pixels. So more than $18$ effective real valued equations are available for each $\bf p$ to estimate the three real $b_{\bf{p}} \left( k \right)$ plus the local error variance.
[^6]: The constant $V = 20$ was inserted for display regularization purposes.
[^7]: It is worth noting that $\mathop {\lim }\limits_{{{\tilde \lambda }_{av}} \to 0} \left( t \right) = \frac{{\sigma _V^2}}{{{{\hat \mu }_{av} + \sigma _V^2}}}$ is always finite.
[^8]: The SROCC magnitude of uni-dimensional metrics is invariant for monotonic transformations. The SROCC is a *robust* version of the LCC [@HUBER81] and therefore it is equally insensitive to true outliers, non-linear fitting and not well explained scores.
[^9]: The RMSE is a further nuisance parameter for LS regression.
[^10]: CSIQ DMOS values were up-scaled by $100$ to balance residual energies.
[^11]: A similar realignment of different databases was already performed in [@wolf02] for video sequences, on a space of seven features. The entire SUPERQUARTET realigned DMOS database is freely available upon request to the contact author. It contains $634$ impaired images from the LIVE DBR2, $400$ from the TID2008 and $600$ from the CSIQ, for a total of $1634$ images.
[^12]: The sample DMOS is usually affected by equivocation between original and slightly distorted images.
[^13]: The basic and fast versions of the D-VICOM are freely available upon request from the contact author.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We analyze the potential of the CERN Large Hadron Collider running at 7 TeV to search for deviations from the Standard Model predictions for the triple gauge boson coupling $ZW^+W^-$ assuming an integrated luminosity of 1 fb$^{-1}$. We show that the study of $W^+W^-$ and $W^\pm Z$ productions, followed by the leptonic decay of the weak gauge bosons can improve the present sensitivity on the anomalous couplings $\Delta g_1^Z$, $\Delta \kappa_Z$, $\lambda_Z$, $g_4^Z$, and $\tilde{\lambda}_Z$ at the $2\sigma$ level.'
address:
- 'Instituto de Física, Universidade de São Paulo, São Paulo – SP, Brazil.'
- 'Departament d’Estructura i Constituents de la Matèria and ICC-UB, Universitat de Barcelona, 647 Diagonal, E-08028 Barcelona, Spain'
- 'Institució Catalana de Recerca i Estudis Avançats (ICREA), Departament d’Estructura i Constituents de la Matèria and ICC-UB, Universitat de Barcelona, 647 Diagonal, E-08028 Barcelona, Spain'
- 'C.N. Yang Institute for Theoretical Physics, SUNY at Stony Brook, Stony Brook, NY 11794-3840, USA'
author:
- 'O. J. P. Éboli'
- 'J. Gonzalez-Fraile'
- 'M. C. Gonzalez-Garcia'
title: 'Scrutinizing the $ZW^+W^-$ vertex at the Large Hadron Collider at 7 TeV'
---
95.30.Cq
Recently the CERN Large Hadron Collider (LHC) started a run with center-of-mass energy of 7 TeV and plans to accumulate an integrated luminosity of $\simeq 1$ fb$^{-1}$. This new high energy frontier allow us to further test the Standard Model (SM), as well as, to check for its possible extensions. In particular, within the framework of the SM, the structure of the trilinear and quartic vector–boson couplings is completely determined by the $SU(2)_L \times U(1)_Y$ gauge symmetry. Thus the study of these interactions can either lead to an additional confirmation of the model or give some hint on the existence of new phenomena at a higher scale [@anomalous]. The triple gauge–boson vertices (TGV’s) have been probed directly at the Tevatron [@tevatron] and LEP [@lep] through the production of vector–boson pairs and the experimental results agree with the SM predictions, see Table \[tab:bounds\]. Moreover, TGV’s contribute at the one–loop level to the $Z$ physics and consequently they can also be indirectly constrained by precision electroweak data [@indirect]. At the LHC, the TGV’s will be subject to a more severe scrutiny via the production of electroweak gauge boson pairs, [*e.g.*]{} $W \gamma$ and $WZ$. Running at 14 TeV center-of-mass energy and with 30–100 $fb^{-1}$ integrated luminosity it will probe these couplings at the few percentage level; see Ref.[@Dobbs:2005ev] for a recent update.
In this work we assess the potential of the LHC already running at 7 TeV to probe deviations from the SM prediction for the $ZW^+W^-$ interaction through the reactions $$\begin{aligned}
p p &&\to W^+ W^- \to \ell^+ \ell^{\prime -} {/\!\!\!E}_T
\label{ppww}
\\
p p && \to W^\pm Z \to \ell^{\prime\pm} \ell^{+}\ell^{-} {/\!\!\!E}_T
\label{ppwz}\end{aligned}$$ where $\ell^{(\prime)} = e$ or $\mu$.
The most general form of the $ZW^+W^-$ vertex compatible with Lorentz invariance is given by the effective Lagrangian [@Hagiwara:1986vm] $$\begin{aligned}
{\mathcal L}_{\text{eff}} /g_{WWZ} = && + i g_1^Z \left (
W^\dagger_{\mu\nu} W^\mu Z^\nu - W^\dagger_\mu W^{\mu\nu} Z_\nu \right )
+ i \kappa_Z W^\dagger_\mu W_\nu Z^{\mu\nu}
\nonumber
\\
&&
+ i \frac{\lambda_Z}{M^2_W} W^\dagger_{\rho \mu} W^\mu_\nu Z^{\nu \rho}
+ g_5^Z \epsilon^{\mu\nu\rho\sigma} (W^\dagger_\mu \partial_\rho W_\nu -
\partial_\rho W^\dagger_\mu W_\nu ) Z_\sigma
\label{leff}
\\
&&
- g_4^Z W^\dagger_\mu W_\nu (\partial^\mu Z^\nu + \partial^\nu Z^\mu)
+ i \tilde{\kappa}_Z W^\dagger_\mu W_\nu \tilde{Z}^{\mu\nu}
+ i \frac{\tilde{\lambda}_Z}{M_W^2} W^\dagger_{\sigma\mu} W^\mu_\nu
\tilde{Z}^{\nu\sigma}
\nonumber\end{aligned}$$ where $Z^{\mu\nu} = \partial^\mu Z^\nu -\partial^\nu Z^\mu$ and $\tilde{Z}^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma}Z_{\rho\sigma}$. $g_{WWZ} = -e \cot \theta_W$ and $\theta_W$ is the weak mixing angle. The couplings $g_1^Z$, $\kappa_Z$ and $\lambda_Z$ are C and P conserving, $\tilde{\kappa}_Z$ and $\tilde{\lambda}^Z$ are P odd and violate CP, while $g_4^Z$ violates C and CP and $g_5^Z$ violates C and P but is CP conserving. In the SM $g_1^Z = \kappa_Z =1$ and $\lambda_Z
= g_4^Z =g_5^Z = \tilde{\kappa}_Z = \tilde{\lambda}_Z = 0$.
couplings PDG bounds indirect limits Unit. $W^+ W^-$ Unit. $W^\pm Z$
--------------------- ---------------------------- --------------------- ----------------- -----------------
$\Delta g_1^Z$ $-0.016^{+0.022}_{-0.019}$ $[-0.051~,~0.0092]$ 2.7 0.22
$\Delta \kappa_Z$ $-0.076^{+0.059}_{-0.056}$ $[-0.050~,~0.0039]$ 0.22 3.5
$\lambda_Z$ $-0.088^{+0.060}_{-0.057}$ $[-0.061~,~0.10]$ 0.15 0.14
$g_5^Z$ $-0.07 \pm 0.09$ $[-0.085~,~0.049]$ 2.7 1.7
$g_4^Z$ $-0.30\pm0.17$ — 2.7 0.22
$\tilde{\kappa}_Z$ $-0.12^{+0.06}_{-0.04}$ — 2.7 3.5
$\tilde{\lambda}_Z$ $-0.09\pm0.07$ — 0.15 0.14
: Available limits on the anomalous TGV couplings. The first column contains a compilation of the direct searches performed by the Particle Data Group [@Amsler:2008zzb]. The indirect bounds are presented in the second column [@indirect] where the entries not evaluated in the literature are marked as —. The third and fourth columns contain the bounds derived from the processes $q q \to W^+W^-$ and $W^\pm Z$ [@unitarity] imposing that unitarity is satisfied for energies below 2 TeV.[]{data-label="tab:bounds"}
0.3cm
In presence of these anomalous couplings the cross sections for the processes $p p \to \ell^+ \ell^{\prime -} {/\!\!\!E}_T$ and $p p \to \ell^\pm \ell^{\prime +}\ell^{\prime -} {/\!\!\!E}_T$ take the form $$\sigma=\sigma_{\text{SM}}+ \sum_{i} \sigma_{\text{int}}^i ~g^i_{\text{ano}}
+\sum_{i, j \geq i}\sigma_{\text{ano}}^{ij} ~g_{\text{ano}}^i
~g_{\text{ano}}^j \;\; ,
\label{def:sigma}$$ where $\sigma_{\text{SM}}$, $\sigma_{\text{int}}^i$, and $\sigma_{\text{ano}}^{ij}$ are, respectively, the SM contribution, the interference between the SM and the anomalous contribution, and the pure anomalous ones. For the CP violating couplings $\sigma_{\text{int}}^i$ vanishes.
SM contributions to $p p \to \ell^+ \ell^{\prime -} {/\!\!\!E}_T$ include electroweak (EW) processes leading to this final state – such us $W^+W^-$ production or $ZZ$ production with one $Z$ decaying in charged leptons and the other in neutrinos – and $t\bar{t}$ production with the top quarks decaying semi-leptonicaly. For $p p \to \ell^\pm \ell^{\prime +}\ell^{\prime
-} {/\!\!\!E}_T$ the main SM backgrounds are the EW production of $W^\pm Z$ pairs and $ZZ$ production with the subsequent decays of the $Z$’s into leptons when one charged lepton escapes detection. An additional background comes from $t \bar{t}$ production if the semi-leptonic decay of a $b$ gives rise to an isolated charged lepton.
The signal and backgrounds were simulated at the parton level with full tree level matrix elements generated with the package MadEvent [@madevent] conveniently modified to include the anomalous TGV’s. We employed CTEQ6L parton distribution functions [@cteq6l] throughout. We took the electroweak parameters to be $\alpha_{em} = 1/132.51$, $m_Z = 91.188$ GeV, $m_W = 80.419$ GeV, and $\sin^2 \theta_W = 0.222$, which was obtained imposing the tree level relation $\cos \theta_W = m_W/m_Z$. We simulated experimental resolutions by smearing the energies (but not directions) of all final state charged leptons with a Gaussian error $\Delta(E)/E =
0.02/\sqrt{E}$. We also included in our analysis a 90% lepton detection efficiency.
We began our analysis of processes (\[ppww\]) and (\[ppwz\]) by imposing some basic acceptance cuts for the charged leptons and missing energy $$p_T^\ell \ge 10 \hbox{ GeV} \;\;\;,\;\;\;
|\eta_\ell| < 2.5 \;\;\;,\;\;\;
\Delta R_{\ell\ell} \ge 0.4 \;\;\;,\;\;\; {/\!\!\!p}_T\geq 10\; {\rm GeV}
\label{basiccuts}$$ where $\eta_\ell$ is the charged lepton pseudo-rapidity.
For $p p \to \ell^+ \ell^- {/\!\!\!E}_T$ events with the two leptons of the same flavor we further required that the lepton pair invariant mass ($M_{\ell\ell}$) is not compatible with a $Z$ production, [ *i.e.*]{} $$| M_{\ell\ell} - M_Z | > 10 \hbox{ GeV} \; .
\label{zveto}$$ Furthermore the top quark pairs are a potentially large background due to its production by strong interactions. To further suppress these events we vetoed the presence of central jets with $$p_T^j > 20 \hbox{ GeV} \;\;\;\;\hbox{ and }\;\;\;\;
| \eta_j| < 3 \; .
\label{jetveto}$$
For $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ in the case with only a pair of same flavor different sign leptons (this is, $\ell^\prime \neq \ell$) we demanded that the invariant mass of the equal flavor lepton pair is compatible with the $Z$ mass, [*i.e.*]{} $$|M_{\ell \ell} - M_Z| < 10 \hbox{ GeV} \;,
\label{isaz}$$
The presence of just one neutrino in the final state of this channel permits the reconstruction of its momentum by imposing the transverse momentum conservation and requiring that the invariant mass of the third lepton and the neutrino is the $W$ mass $$M_{\ell^\prime \nu} = M_W \;.
\label{isaw}$$ This procedure exhibits a twofold ambiguity on the neutrino longitudinal momentum. In our analysis we kept only events that possess a solution to the neutrino momentum.
Conversely when the three leptons have the same flavor we demanded that one opposite sign lepton pair satisfies (\[isaz\]) and the third lepton and the missing transverse momentum reconstructs a $W$ as in (\[isaw\]). We further required that the invariant mass of the third lepton and the lepton of opposite charge used to reconstruct the $Z$ is not compatible with a $Z$, therefore complying with (\[zveto\]). The top pair background to $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ after cuts and (\[isaz\])–(\[isaw\]) (plus (\[zveto\]) for $\ell'=\ell$) is already very suppressed since it requires that one of the isolated leptons originates from a $b$ quark semi-leptonic decay. Vetoing any central jet activity, as in Eq. (\[jetveto\]) renders the $t
\bar{t}$ cross section negligible.
[|c|c||c|c|c|c||c|c|c|]{} & &\
\
$l^+\nu_ll'^-\nu_{l'}$ & $t \bar{t}$ & $\Delta g_1^Z$ & $\Delta \kappa_Z$ & $\lambda_Z$ & $g_5^Z$ & $g_4^Z$ & $\tilde{\kappa}_Z$ & $\tilde{\lambda}_Z$\
824. & 11.1 & 254. & 2540. & 5750. & 163. & 219. & 412. & 6030.\
& & -55.7 & -166. & -22.1 & 15.1 & 68.8 & -89.2 & 152.\
\
$\ell^+\ell^-\ell^{\prime \pm} \nu$ & $ZZ$ & $\Delta g_1^Z$ & $\Delta \kappa_Z$ & $\lambda_Z$ & $g_5^Z$ & $g_4^Z$ & $\tilde{\kappa}_Z$ & $\tilde{\lambda}_Z$\
63.0 & 2.32 & 1280. & 65.4 & 2290. & 391. & 1020. & 77.6 & 2390.\
& & -106. & -21.3 & -24.3 & -7.2 & -20.2 & -2.2 & -10.0\
0.2cm
We present in Table \[tab:resww\] the cross sections of the SM backgrounds and anomalous contributions to process $pp \to
\ell^+\ell^{\prime -} {/\!\!\!E}_T$ after the cuts (\[basiccuts\])—(\[jetveto\]) and $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ after the cuts (\[basiccuts\]) and (\[jetveto\])–(\[isaw\]) (plus (\[zveto\]) for $\ell'=\ell$). For $pp \to \ell^+\ell^{\prime -} {/\!\!\!E}_T$ the cut (\[jetveto\]) is very important to tame the dangerous $t\bar{t}$ background whose cross section is 3.9 pb when we remove this cut. For simplicity we have only considered one non-vanishing anomalous vertex at a time. This simplifying hypothesis can be consistently made when the integration of the heavy degrees of freedom associated to a new physics leads to scenario where the $SU(2)_L \times U(1)_Y$ symmetry is realized non-linearly in the low energy effective Lagrangian. If the low energy remains of the new physics are described by a linear realization of the $SU(2)_L \times U(1)_Y$ there will be relations between the anomalous TGV’s; see, for instance, reference [@linear].
The normalized spectra of the SM and the anomalous contributions for some relevant kinematic variables are displayed in Figure \[fig:distri\]. For $pp \to \ell^+\ell^{\prime -} {/\!\!\!E}_T$ we have defined the transverse mass $M_T^{WW}$ as: $$M_T^{WW}
= \biggl[ \left( \sqrt{(p_T^{\ell^+\ell^{\prime -}})^2 + m^2_{\ell^+ \ell^{\prime -}}}
+ \sqrt{{/\!\!\!p_T}^2 + m^2_{\ell^+\ell^{\prime -} }} \right)^2 \biggr .
\biggl . - (\vec{p}_T^{~\ell^+\ell^{\prime -}} + \vec{{/\!\!\!p_T}} )^2 \biggr]^{1/2}
\label{mtww}$$ where $\vec{{/\!\!\!p_T}}$ is the missing transverse momentum vector, $\vec{p}_T^{~\ell^+\ell^{\prime -}}$ is the transverse momentum of the pair $\ell^+ \ell^{\prime -}$ and $m_{\ell^+\ell^{\prime -}}$ is the $\ell^+ \ell^{\prime -}$ invariant mass.
For $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ we define $p_{TZ}$ as the transverse momentum of the opposite sign equal flavor leptons verifying (\[isaz\]). Furthermore, it is possible to reconstruct the neutrino momentum, and consequently, we can evaluate the total $\ell\ell\ell\nu$ invariant mass (which we label $M_{WZ}$) that takes two possible values for each event. In the lower right panel of Fig. \[fig:distri\] we show the distribution in this variable where each of the solutions have been given weight $1/2$.
Figure \[fig:distri\] illustrates the well-known fact that the anomalous contributions enhance the cross section at higher collision energies (eventually leading to perturbative unitarity violation) and that this behaviour can be well traced by either $p_{T \ell}^{max}$, $M_{\ell\ell}$, $M^T_{WW}$, $p_{TZ}$ or $M_{WZ}$ respectively.
![Normalized spectra for some relevant kinematic variables for the SM and some of the anomalous TGV’s for $pp \to
\ell^+\ell^{\prime -} {/\!\!\!E}_T$ (upper panels) and $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ (lower panels). The upper panels show the distribution in transverse momentum of the hardest lepton (left panel), dilepton invariant mass (central panel) and reconstructed $WW$ transverse invariant mass (right panel). The lower panels show the transverse momentum of the $Z$ (left panel), hardest lepton transverse momentum (central panel), and reconstructed $WZ$ transverse invariant mass (right panel).[]{data-label="fig:distri"}](dist_ww "fig:"){width="90.00000%"} ![Normalized spectra for some relevant kinematic variables for the SM and some of the anomalous TGV’s for $pp \to
\ell^+\ell^{\prime -} {/\!\!\!E}_T$ (upper panels) and $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ (lower panels). The upper panels show the distribution in transverse momentum of the hardest lepton (left panel), dilepton invariant mass (central panel) and reconstructed $WW$ transverse invariant mass (right panel). The lower panels show the transverse momentum of the $Z$ (left panel), hardest lepton transverse momentum (central panel), and reconstructed $WZ$ transverse invariant mass (right panel).[]{data-label="fig:distri"}](dist_wz "fig:"){width="90.00000%"}
In order to extract the attainable sensitivity on anomalous TGV we analyzed for each kinematic variable shown in Fig. \[fig:distri\] the choice of cut that maximizes the sensitivity for deviations in the TGV’s. Given the limited statistics of the 7 TeV LHC run we do not attempt to make a fit to the distributions and use instead as unique variable the total number of observed events above a certain minimum cut for each of the variables. In each case we assumed that the total number of observed events is the one predicted by the SM at integrated luminosity of 1 fb$^{-1}$. The corresponding statistical uncertainty were obtained using Poisson or Gaussian statistics depending on whether the expected number of SM events was smaller or larger than 20. We performed our analysis of the channels $pp \to
\ell^+\ell^{\prime -} {/\!\!\!E}_T$ and $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ independently.
We depict in Figure \[fig:boundww\] the achievable $2\sigma$ limits from $pp \to \ell^+\ell^{\prime -} {/\!\!\!E}_T$ on some of the anomalous TGV as a function of the the minimum cut on the maximum transverse momentum of the leptons (left panels), dilepton invariant mass (central panels) and minimum reconstructed transverse invariant mass ($M_{WW}^T$) in the right panels. As we can see from this figure the couplings $\Delta \kappa_Z$, $\tilde{\lambda}_Z$ and $\lambda_Z$ possess a mild dependence on the kinematic cut while $g_4^Z$ and $\tilde{\kappa}_Z$ experience larger changes with the variation of the cuts. This can be understood from Fig. \[fig:distri\] that shows that $g_4^Z$ and $\tilde{\kappa}_Z$ distributions decrease much faster that the ones for $\tilde{\lambda}_Z$. We find that maximum sensitivity for any of the anomalous TGV is obtained from a minimum cut in transverse momentum of the hardest lepton with the optimum cut ranging between 50–200 GeV depending on the anomalous coupling considered. The corresponding attainable 2$\sigma$ bounds are listed in Table \[tab:wwlimits\]. Our results show that this channel can tighten the present direct bounds on $\Delta \kappa_Z$, $\lambda_Z$, $g_4^Z$, and $\tilde{\lambda}_Z$.
![Dependence of the upper (top panels) and lower (lower panels) $2\sigma$ bounds achievable from the study of $pp \to
\ell^+\ell^{\prime -} {/\!\!\!E}_T$ as a function of the cut on the minimum value on the hardest lepton transverse momentum (left panels), the dilepton invariant mass (central panels), and the reconstructed $WW$ transverse invariant mass (right panels). The dashed curves correspond to $\tilde\lambda_Z$ and are almost indistinguishable from the full blue lines corresponding to $\lambda_Z$ . The presently allowed $2\sigma$ ranges are indicated by the vertical lines in the left panels.[]{data-label="fig:boundww"}](2sig_ww){width="85.00000%"}
--------------------- ------------------------- ------------------------- ------------------------ ------------------------
No form factor $\Lambda=3$ TeV No form factor $\Lambda=3$ TeV
$\Delta g_1^Z$ $[-0.33\;,\; 0.56]$ $[-0.35\;,\; 0.59]$ $[-0.055 \;,\; 0.094]$ $[-0.061 \;,\; 0.11]$
$\Delta \kappa_Z$ $[-0.088 \;,\;0.11]$ $[-0.10 \;,\;0.14]$ $[-0.27 \;,\; 0.55]$ $[-0.29 \;,\; 0.61]$
$\lambda_Z$ $[-0.055 \;,\; 0.056]$ $[-0.074 \;,\; 0.075]$ $[-0.051\;,\; 0.054]$ $[-0.060\;,\; 0.064]$
$g_5^Z$ $[-0.53 \;,\; 0.51]$ $[-0.56 \;,\; 0.55]$ $[-0.18 \;,\; 0.19]$ $[-0.19 \;,\; 0.20]$
$g_4^Z$ $[-0.48 \;,\; 0.48]$ $[-0.51 \;,\; 0.51]$ $[-0.080 \;,\; 0.080]$ $[-0.091 \;,\; 0.091]$
$\tilde{\kappa}_Z$ $[-0.38 \;,\;0.38]$ $[-0.39 \;,\;0.39]$ $[-0.40 \;,\; 0.40]$ $[-0.42 \;,\; 0.42]$
$\tilde{\lambda}_Z$ $[-0.055 \;,\; 0.055]$ $[-0.074 \;,\; 0.074]$ $[-0.053 \;,\; 0.053]$ $[-0.062 \;,\; 0.062]$
--------------------- ------------------------- ------------------------- ------------------------ ------------------------
: Attainable $2\sigma$ bounds on anomalous TGV at the LHC at 7 TeV with 1 fb$^{-1}$.[]{data-label="tab:wwlimits"}
0.2cm
Correspondingly we show in Figure \[fig:boundwz\] the bounds from $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ as a function of the minimum cut in either the $Z$ transverse momentum (left panels), the hardest lepton transverse momentum (central panels), and the reconstructed $WZ$ invariant mass (right panels). We find that a minimum cut in either of the transverse momentum variables (that of the $Z$ or the hardest lepton $p_T$) leads to the best sensitivity. The corresponding attainable 2$\sigma$ bounds are also listed in Table \[tab:wwlimits\] that shows that this channel can improve the present direct constraints on the couplings $\Delta
g_1^Z$, $\lambda_Z$, $g_4^Z$ and $\tilde{\lambda}_Z$.
![ Dependence of the upper (top panels) and lower (lower panels) $2\sigma$ bounds from $p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T$ a function of the cut on the minimum value on the $Z$ transverse momentum (left panels), the hardest lepton transverse momentum (central panels), and the reconstructed $WZ$ invariant mass (right panels). The dashed curves correspond to $\tilde\lambda_Z$ and are almost indistinguishable from the full blue lines corresponding to $\lambda_Z$ . The presently allowed $2\sigma$ ranges are indicated by the vertical lines in the left panels.[]{data-label="fig:boundwz"}](2sig_wz){width="85.00000%"}
So far we have applied the same type of analysis to CP conserving or CP violating couplings. For these last ones their CP breaking nature can be addressed constructing some CP-odd or $\hat T$-odd observable by weighting the events with the sign of the relevant cross product of the measured momenta. For example, following Refs.[@Han; @Kum] we can define $$\begin{aligned}
\Xi_\pm &\equiv&
\mbox{sign}
\left[\left(\vec{p}_{\ell^+}-\vec{p}_{\ell^{\prime -}}\right)^z\right]\,
\mbox{sign}\left(\vec{p}_{\ell^+} \times\vec{p}_{\ell^{\prime -}}\right)^z
\;\;\; \;\;\; \;\; \,
\hfill
{\rm for}\;
p p \to \ell^{+}\ell^{\prime -} {/\!\!\!E}_T
\, , \label{ha} \\
\Xi_\pm &\equiv&\mbox{sign}(p^z_Z)\, \mbox{sign}(p_{\ell^\prime}\times p_Z)^z
\;\;\; \;\;\; \;\;\; \;\;\; \;\;\;
\;\;\; \;\;\; \;\;\; \;\;\; \;\;
\hfill {\rm for }\; p p \to \ell^{\prime \pm} \ell^{+}\ell^{-} {/\!\!\!E}_T
\, ,
\label{ku}\end{aligned}$$ where $z$ is the collision axis. The CP-violating couplings give a non-vanishing contribution to the sign-weighted cross section $$g^i_{\text{ano}} \, \Delta \sigma^{i}_{\text{ano}}
\equiv \int d\sigma \,\Xi_\pm \; .
\label{deltasigCP}$$ We present in Table \[tab:resww\] the values of the corresponding sign-weighted cross sections. The resulting number of sign-weighted events has to be compared with the statistical fluctuations from the SM events (which are sign symmetric). We find that given the existing bounds on $\tilde{\kappa}_Z$, $\tilde{\lambda}^Z$ and $g_4^Z$, the study of these events at the 7 TeV run of LHC is not precise enough to provide information on the CP properties of the anomalous couplings.
It is well known that the introduction of anomalous couplings spoils delicate cancellations in scattering amplitudes, leading to their growth with energy and, eventually, to unitarity violation above a certain scale $\Lambda$. The way to cure this problem that is being used in the literature is to introduce an energy dependent form factor that dumps the anomalous scattering amplitude growth at high energy, such as $$\frac{1}{(1 + \frac{\hat{s}}{\Lambda^2})^2}
\label{eq:formfactor}$$ where $\sqrt{\hat{s}}$ is the center–of–mass energy of the $WW$ or $WZ$ pair. Here we advocate that the need to introduce a form factor at the 7 TeV run of LHC is marginal because the center-of-mass energy for the contributing sub-process in (\[ppww\]) and (\[ppwz\]) is $\lesssim
2$ TeV, and the unitarity bounds on the anomalous TGV steaming from these processes are much weaker than the ones that we obtain; see the fourth and fifth columns of Table \[tab:bounds\]. In principle one may worry about the corresponding unitarity violation in longitudinal $VV$ ($V=W^\pm$ or $Z$) scattering which can lead to stronger bounds on the TGV since they can lead to a scattering amplitude which grows as $\hat s^2$. However, the actual energy behaviour of the scattering amplitude in longitudinal gauge boson scattering depends strongly on the assumptions about the quartic gauge boson couplings [@unitWW]. In particular, if there is a mechanism relating the quartic and triple anomalous contributions the $VV$ scattering unitarity bounds turn out to be similar to the ones in reference [@unitarity]. Altogether we find that within the bounds that we derive, unitarity is held up to $\sqrt{\hat s} \simeq 3$ TeV. As a final consistency check we derive the bounds obtained if a form factor (\[eq:formfactor\]) was included with $\Lambda=3$ TeV. We show in Table \[tab:wwlimits\] the changes in the $2\sigma$ sensitivity.
Our analysis leaves some room for improvement. For instance, we considered only one kinematic distribution to extract the bounds, leaving out the possibility of optimizing the analysis for joint distributions or a binned maximum likelihood fit. Moreover, our calculations were carried out at the parton level with the lowest order perturbation theory. Certainly a full Monte Carlo analysis taking into account detector simulation, as well as, NLO QCD [@qcdnlo] and EW [@ewnlo] is in order. Although QCD NLO corrections are potentially dangerous due to changes in $p_T$ distributions, our jet veto cut (\[jetveto\]) is enough to guarantee that the attainable limits are not significantly altered [@qcdnlo; @anonlo]. In brief, we anticipate that our results should give a fair estimate of the LHC potential to study the $ZW^+W^-$ vertex.
Summarizing, we have shown that the study of the processes (\[ppww\]) and (\[ppwz\]) at the LHC with a center–of–mass energy of 7 TeV and an integrated luminosity of 1 fb$^{-1}$ can improve the presently available direct limits on the $Z$ anomalous couplings $\Delta g_1^Z$, $\Delta \kappa_Z$, $\lambda_Z$, $g_4^Z$, and $\tilde{\lambda}_Z$. Due to the small integrated luminosity predicted for this initial run, the limits on these couplings will be only slightly more stringent than the present available ones. Nevertheless, the more precise results on $\Delta g_1^Z$ and $\lambda_Z$ will start to compete with the indirect limits coming from precision measurements; see Table \[tab:bounds\].
0.3cm This work is supported by USA-NSF grant PHY-0653342, by Spanish grants from MICINN 2007-66665-C02-01, the INFN-MICINN agreement program ACI2009-1038, consolider-ingenio 2010 program CSD-2008-0037 and by CUR Generalitat de Catalunya grant 2009SGR502. and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo à Pesquisa de Estado de São Paulo (FAPESP).
[99]{} For a review see: H. Aihara [*et al.*]{}, [ *Anomalous gauge boson interactions*]{} in Electroweak Symmetry Breaking and New Physics at the TeV Scale, edited by T. Barklow, S. Dawson, H. Haber and J. Seigrist, (World Scientific,
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present the effective low-energy theory for interacting 1D quantum wires subject to Rashba spin-orbit coupling. Under a one-loop renormalization group scheme including all allowed interaction processes for not too weak Rashba coupling, we show that electron-electron backscattering is an irrelevant perturbation. Therefore no gap arises and electronic transport is described by a modified Luttinger liquid theory. As an application of the theory, we discuss the RKKY interaction between two magnetic impurities. Interactions are shown to induce a slower power-law decay of the RKKY range function than the usual 1D noninteracting $\cos(2k_F x)/|x|$ law. Moreover, in the noninteracting Rashba wire, the spin-orbit coupling causes a twisted (anisotropic) range function with several different spatial oscillation periods. In the interacting case, we show that one special oscillation period leads to the slowest decay, and therefore dominates the RKKY interaction for large separation.'
author:
- 'Andreas Schulz$^1$, Alessandro De Martino$^2$, Philip Ingenhoven$^{1,3}$, and Reinhold Egger$^1$'
title: 'Low-energy theory and RKKY interaction for interacting quantum wires with Rashba spin-orbit coupling'
---
Introduction {#sec1}
============
Spin transport in one-dimensional (1D) quantum wires continues to be a topic of much interest in solid-state and nanoscale physics, offering interesting fundamental questions as well as technological applications.[@fabian] Of particular interest to this field is the spintronic field effect transistor (spin-FET) proposal by Datta and Das,[@datta] where a gate-tunable Rashba spin-orbit interaction (SOI) of strength $\alpha$ allows for a purely electrical manipulation of the spin-dependent current. While the Rashba SOI arises from a structural inversion asymmetry[@rashba; @winkler; @winklerbook] of the two- dimensional electron gas (2DEG) in semiconductor devices hosting the quantum wire, additional sources for SOI can be present. In particular, for bulk inversion asymmetric materials, the Dresselhaus SOI (of strength $\beta$) should also be taken into account. By tuning the Rashba SOI (via gate voltages) to the special point $\alpha=\beta$, the spin-FET was predicted to show a remarkable insensitivity to disorder,[@schliemann] see also Ref. . On top of these two, additional (though generally weaker) contributions may arise from the electric confinement fields forming the quantum wire. In this paper, we focus on the case of Rashba SOI and disregard all other SOI terms. This limit can be realized experimentally by applying sufficiently strong backgate voltages,[@exp1; @exp1b; @grundler; @exp2] which create a large interfacial electric field and hence a significant and tunable Rashba SOI coupling $\alpha$. The model studied below may also be relevant to 1D electron surface states of self-assembled gold chains.[@schaefer]
The noninteracting theory of such a “Rashba quantum wire” has been discussed in the literature, and is summarized in Sec. \[sec2\] below. We here discuss electron-electron (e-e) interaction effects in the 1D limit, where only the lowest (spinful) band is occupied. The bandstructure at low energy scales is then characterized by two velocities,[@tsvelik] $$\label{vab}
v_{A,B} = v_F ( 1\pm \delta ), \quad \delta(\alpha) \propto \alpha^4 .$$ These reduce to a single Fermi velocity $v_F$ in the absence of Rashba SOI ($\delta=0$ for $\alpha=0$), but they will be different for $\alpha\ne 0$, reflecting the broken spin $SU(2)$ invariance in a spin-orbit coupled system. The small-$\alpha$ dependence $\delta\propto \alpha^4$ follows for the model below and has also been reported in Ref. . Therefore, the velocity splitting (\[vab\]) is typically weak. While a similar velocity splitting also happens in a magnetic Zeeman field (without SOI),[@zeeman] the underlying physics is different since time-reversal symmetry is not broken by SOI.
The bandstructure of a single-channel quantum wire with Rashba SOI should be obtained by taking into account at least the lowest two (spinful) subbands, since a restriction to the lowest subband alone would eliminate spin relaxation.[@governale02; @governale04; @kaneko] The problem in this truncated Hilbert space can be readily diagonalized, and yields two pairs of energy bands. When describing a single-channel quantum wire, one then keeps only the lower pair of these energy bands. We mention in passing that bandstructure effects in the presence of both Rashba SOI and magnetic fields have also been studied.[@pershin; @schapers; @pereira; @debald; @serra] In addition, the possibility of a spatial modulation of the Rashba coupling was discussed,[@wang] but such phenomena will not be further considered here. Finally, disorder effects were addressed in Refs. .
For 1D quantum wires, it is well known that the inclusion of e-e interactions leads to a breakdown of Fermi liquid theory, and often implies Luttinger liquid (LL) behavior. This non-Fermi liquid state of matter has a number of interesting features, including the phenomenon of spin-charge separation.[@gogolin] Motivated mainly by the question of how the Rashba spin precession and Datta-Das oscillations in spin-dependent transport are affected by e-e interactions, Rashba SOI effects on electronic transport in interacting quantum wires have been studied in recent papers. In effect, however, all those works only took e-e forward scattering processes into account. Because of the Rashba SOI, one obtains a modified LL phase with broken spin-charge separation, leading to a drastic influence on observables such as the spectral function or the tunneling density of states. Moroz *et al.* argued that e-e backscattering processes are irrelevant in the renormalization group (RG) sense, and hence can be omitted in a low-energy theory.[@moroz3; @moroz4] Unfortunately, their theory relies on an incorrect spin assignment of the subbands, which then invalidates several aspects of their treatment of interaction processes.
The possibility that e-e backscattering processes become relevant (in the RG sense) in a Rashba quantum wire was raised in Ref. , where a spin gap was found under a weak-coupling two-loop RG scheme. If valid, this result has important consequences for the physics of such systems, and would drive them into a spin-density-wave type state. To establish the spin gap, Ref. starts from a strict 1D single-band model and assumes both $\alpha$ and the e-e interaction as weak coupling constants flowing under the RG. Our approach below is different in that we include the Rashba coupling $\alpha$ from the outset in the single-particle sector, i.e., in a nonperturbative manner. We then consider the one-loop RG flow of all possible interaction couplings allowed by momentum conservation (for not too small $\alpha$). This is an important difference to the scheme of Ref. , since the Rashba SOI eliminates certain interaction processes which become momentum-nonconserving. This mechanism is captured by our approach. The one-loop RG flow then turns out to be equivalent to a Kosterlitz-Thouless flow, and for the initial values realized in this problem, e-e backscattering processes are always irrelevant. Our conclusion is therefore that no spin gap arises because of SOI, and a modified LL picture is always sufficient. We mention in passing that in the presence of a magnetic field (which we do not consider), a spin gap can be present because of spin-nonconserving e-e “Cooper” scattering processes; the effects of e-e forward scattering in Rashba wires with magnetic field were studied as well.[@yu; @lee; @cheng; @thierry] Below, we also provide estimates for the *renormalized* couplings entering the modified LL theory, see Eq. (\[gfix\]) below. When taking bare (instead of renormalized) couplings, we recover previous results.[@governale04] Note that the SOI in carbon nanotubes[@prlale] or graphene ribbons[@sandler] leads to a similar yet different LL description. In particular, for (achiral) carbon nanotubes, the leading SOI does not break spin-charge separation.[@prlale] We here only discuss Rashba SOI effects in semiconductor quantum wires in the absence of magnetic fields.
We apply our formalism to a study of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction[@RKKY; @RKKYkittel] between two spin-$1/2$ magnetic impurities, ${\bm \Sigma}_{1,2}$, separated by a distance $x$. The RKKY interaction is mediated by the conduction electrons in the quantum wire which are exchange-coupled (with coupling $J$) to the impurity spins. In the absence of both the e-e interaction and the SOI, one finds an isotropic exchange (Heisenberg) Hamiltonian, $$\label{rkkybasic}
H_{\rm RKKY} = - J^2 F_{\rm ex}(x) \ {\bm \Sigma}_1\cdot {\bm \Sigma}_2,
\quad
F_{\rm ex}(x)\propto \frac{ \cos(2k_F x)}{|x|},$$ where the $2k_F$-oscillatory RKKY range function $F_{\rm ex}(x)$ is specified for the 1D case. When the spin $SU(2)$ symmetry is broken by the SOI, spin precession sets in and the RKKY interaction is generally of a more complicated (twisted) form. For a noninteracting Rashba quantum wire, it has indeed been established[@bruno; @lyu; @simonin] that the RKKY interaction becomes anisotropic and thus has a tensorial character. It can always be decomposed into an exchange (scalar) part, a Dzyaloshinsky-Moriya-like (vector) interaction, and an Ising- like (traceless symmetric tensor) coupling. On the other hand, in the presence of e-e interactions but without SOI, the range function has been shown[@RKKYreinh] to exhibit a slow power-law decay, $F_{\rm ex}(x)\propto\cos(2k_F x) |x|^{-\eta}$, with an interaction- dependent exponent $\eta<1$. The RKKY interaction in interacting quantum wires with SOI has not been studied before.
For the benefit of the focussed reader, we briefly summarize the main results of our analyis. The effective low-energy theory of an interacting Rashba quantum wire is given in Eq. (\[tll\]), with the velocities (\[veloc\]) and the dimensionless interaction parameters (\[llpar\]). Previous theories did not fully account for the e-e backscattering processes, and the conspiracy of these processes with the broken $SU(2)$ invariance due to spin-orbit effects leads to $K_s<1$ in Eq. (\[llpar\]). This in turn implies novel effects in the RKKY interaction of an interacting Rashba wire. In particular, the power-law decay exponent in an interacting Rashba wire, see Eq. (\[etaB\]), depends explicitly on both the interaction strength and on the Rashba coupling.
The structure of the remainder of this paper is as follows. In Sec. \[sec2\], we discuss the bandstructure. Interaction processes and the one-loop RG scheme are discussed in Sec. \[sec3\], while the LL description is provided in Sec. \[sec4\]. The RKKY interaction mediated by an interacting Rashba quantum wire is then studied in Sec. \[sec5\]. Finally, we offer some conclusions in Sec. \[sec6\]. Technical details can be found in the Appendix. Throughout the paper we use units where $\hbar=1$.
Single-particle description {#sec2}
===========================
We consider a quantum wire electrostatically confined in the $z$- direction within the 2DEG ($xz$-plane) by a harmonic potential, $V_c(z)= m\omega^2 z^2/2$, where $m$ is the effective mass. The noninteracting problem is then defined by the single-particle Hamiltonian[@moroz1; @moroz2; @governale02; @rashba; @loss2] $$\label{h0}
H_{\rm sp} = \frac{1}{2m} \left(p_x^2+p_z^2\right)+ V_c(z) +
\alpha \left( \sigma_z p_x - \sigma_x p_z\right),$$ where $\alpha$ is the Rashba coupling and the Pauli matrices $\sigma_{x,z}$ act in spin space. For $\alpha=0$, the transverse problem is diagonal in terms of the familiar 1D harmonic oscillator eigenstates (Hermite functions) $H_n(z)$, with $n=0,1,2,\ldots$ labeling the subbands (channels). Eigenstates of Eq. (\[h0\]) have conserved longitudinal momentum $p_x=k$, and with the $z$-direction as spin quantization axis, $\sigma_z|\sigma
\rangle=
\sigma|\sigma\rangle$ with $\sigma=\uparrow,\downarrow=\pm$, the $\sigma_x p_z$ term implies mixing of adjacent subbands with associated spin flips. Retaining only the lowest ($n=0$) subband from the outset thus excludes spin relaxation. We follow Ref. and keep the two lowest bands, $n=0$ and $n=1$. The higher subbands $n\geq 2$ yield only tiny corrections, which can in principle be included as in Ref. . The resulting $4\times 4$ matrix representing $H_{\rm sp}$ in this truncated Hilbert space is readily diagonalized and yields four energy bands. We choose the Fermi energy such that only the lower two bands, labeled by $s=\pm$, are occupied, and arrive at a reduced two-band model, where the quantum number $s=\pm$ replaces the spin quantum number. The dispersion relation is $$\label{spectrum}
E_{s}(k)=\omega + \frac{k^2}{2m} -
\sqrt{\left(\frac{\omega}{2}+s\alpha k\right)^{2}+
\frac{m\omega \alpha^2}{2}},$$ with eigenfunctions $\sim e^{ikx}\phi_{k,s}(z)$. The resulting asymmetric energy bands (\[spectrum\]) are shown in Fig. \[f1\]. The transverse spinors (in spin space) are given by $$\begin{aligned}
\label{eigenfunction}
\phi_{k,+}(z)&=&
\left( \begin{array}{c} i \cos [\theta_+(k)] H_1(z) \\ \sin[\theta_+(k)]
H_0(z) \end{array}\right), \\ \nonumber
\phi_{k,-}(z)&=&
\left( \begin{array}{c} \sin [\theta_-(k)] H_0(z) \\ i\cos[\theta_-(k)]
H_1(z) \end{array}\right),\end{aligned}$$ with $k$-dependent spin rotation angles (we take $0\leq \theta_s(k) \leq \pi/2$) $$\label{theta}
\theta_s(k) = \frac12 \cot^{-1}\left( \frac{- 2 s k- \omega/\alpha}{
\sqrt{2m\omega} }\right) =\theta_{-s}(-k).$$ As a result of subband mixing, the two spinor components of $\phi_{k,s}(z)$ carry a different $z$-dependence. They are therefore not just the result of a $SU(2)$ rotation. For $\alpha=0$, we recover $\theta_s=\pi/2$, corresponding to the usual spin up and down eigenstates, with $H_0(z)$ as transverse wavefunction; the $s=+$ ($s=-$) component then describes the $\sigma=\downarrow$ ($\sigma=\uparrow$) spin eigenstate. However, for $\alpha\ne 0$, a peculiar implication of the Rashba SOI follows. From Eq. (\[theta\]) we have $\lim_{k\to \pm \infty}
\theta_s(k)=(1\pm s)\pi/4,$ such that both $s=\pm$ states have (approximately) spin $\sigma=
\downarrow$ for $k\to \infty$ but $\sigma=\uparrow$ for $k\to -\infty$; the product of spin and chirality thus always approaches $\sigma {\rm sgn}(k)=-1$. Moreover, under the time-reversal transformation, ${\cal T}=i\sigma_y {\cal C}$ with the complex conjugation operator ${\cal C}$, the two subbands are exchanged, $$\label{trs}
e^{-ikx} \phi_{-k,-s}(z) = s {\cal T}[e^{ikx} \phi_{k,s}(z)] ,
\quad E_{-s}(-k) = E_s(k).$$ Time-reversal symmetry, preserved in the truncated description, makes this two-band model of a Rashba quantum wire qualitatively different from Zeeman-spin-split models.[@zeeman]
In the next step, since we are interested in the low-energy physics, we linearize the dispersion relation around the Fermi points $\pm k_F^
{(A,B)}$, see Fig. \[f1\], which results in two velocities $v_A$ and $v_B$, see Eq. (\[vab\]). The linearization of the dispersion relation of multi-band quantum wires around the Fermi level is known to be an excellent approximation for weak e-e interactions.[@gogolin] Explicit values for $\delta$ in Eq. (\[vab\]) can be derived from Eq. (\[spectrum\]), and we find $\delta(\alpha)
\propto
\alpha^4$ for $\alpha\to 0$, in accordance with previous estimates.[@haus1] We mention that $\delta\alt 0.1$ has been estimated for typical geometries in Ref. . The transverse spinors $\phi_{ks}(z)$, Eq. (\[eigenfunction\]), entering the low-energy description can be taken at $k=\pm k_F^{(A,B)}$, where the spin rotation angle (\[theta\]) only assumes one of the two values $$\label{angleab}
\theta_A= \theta_+\left ( k_F^{(A)}\right ),\quad \theta_B=\theta_-
\left (
k_F^{(B)} \right ) .$$ The electron field operator $\Psi(x,z)$ for the linearized two-band model with $\nu=A,B=+,-$ can then be expressed in terms of 1D fermionic field operators $\psi_{\nu,r}(x)$, where $r=R,L=+,-$ labels right- and left-movers, $$\label{expand}
\Psi(x,z) = \sum_{\nu,r=\pm} e^{i r k_F^{(\nu)}x } \
\phi_{r k_F^{(\nu)} , s= \nu r }(z) \
\psi_{\nu,r}(x),$$ with $\phi_{k,s}(z)$ specified in Eq. (\[eigenfunction\]). Note that in the left-moving sector, band indices have been interchanged according to the labeling in Fig. \[f1\].
In this way, the noninteracting second-quantized Hamiltonian takes the standard form for two inequivalent species of 1D massless Dirac fermions with different velocities, $$\label{dirac}
H_0 = -i \sum_{\nu,r=\pm} r v_\nu \int dx\ \psi^\dagger_{\nu,r}
\partial_x
\psi^{}_{\nu,r}.$$ The velocity difference implies the breaking of the spin $SU(2)$ symmetry, a direct consequence of SOI. For $\alpha=0$, the index $\nu$ coincides with the spin quantum number $\sigma$ for left-movers and with $-\sigma$ for right-movers, and the above formulation reduces to the usual Hamiltonian for a spinful single-channel quantum wire.
Interaction effects {#sec3}
===================
Let us now include e-e interactions in such a single-channel disorder-free Rashba quantum wire. With the expansion (\[expand\]) and ${\bf r}=(x,z)$, the second-quantized two-body Hamiltonian $$H_I=\frac12\int d{\bf r}_1 d{\bf r}_2 \ \Psi^\dagger({\bf r}_1)
\Psi^\dagger({\bf r}_2) V({\bf r}_1-{\bf r}_2) \Psi({\bf r}_2) \Psi
({\bf r}_1)$$ leads to 1D interaction processes. We here assume that the e-e interaction potential $V({\bf r}_1-{\bf r}_2)$ is externally screened, allowing to describe the 1D interactions as effectively local. Following standard arguments, for weak e-e interactions, going beyond this approximation at most leads to irrelevant corrections.[@footnote] We then obtain the local 1D interaction Hamiltonian[@starykh] $$\label{1dint}
H_I =\frac{1}{2} \sum_{ \{ \nu_i, r_i \} } V_{ \{ \nu_i, r_i\} }
\int dx \ \psi^\dagger_{\nu_1,r_1} \psi^\dagger_{\nu_2,r_2}
\psi^{}_{\nu_3,r_3} \psi^{}_{\nu_4,r_4} ,$$ where the summation runs over all quantum numbers $\nu_1,\ldots,\nu_4$ and $r_1,\ldots,r_4$ subject to momentum conservation, $$\label{consmom}
r_1 k_F^{(\nu_1)} +r_2 k_F^{(\nu_2)}= r_3 k_F^{(\nu_3)} +r_4 k_F^
{(\nu_4)} .$$ With the momentum transfer $q= r_1 k_F^{(\nu_1)}-r_4 k_F^{(\nu_4)}$ and the partial Fourier transform $$\label{pot}
\tilde V(q;z)=\int dx \ e^{-iqx} V(x,z)$$ of the interaction potential, the interaction matrix elements in Eq. (\[1dint\]) are given by $$\begin{aligned}
V_{\{ \nu_i,r_i\}} &=& \int dz_1 dz_2 \ \tilde V(q;z_1-z_2)
\nonumber \\
&\times& \ \left[ \phi^\dagger_{r_1 k_F^{(\nu_1)} , \nu_1 r_1 } \cdot
\phi_{r_4 k_F^{(\nu_4)} , \nu_4 r_4 }\right](z_1) \nonumber \\ &\times&
\left [ \phi^\dagger_{r_2 k_F^{(\nu_2)} , \nu_2 r_2 } \cdot
\phi_{r_3 k_F^{(\nu_3)} , \nu_3 r_3 }\right](z_2). \label{intmat}\end{aligned}$$ Since the Rashba SOI produces a splitting of the Fermi momenta for the two bands, $\left| k^{(A)}_F-k_F^{(B)}\right| \simeq 2\alpha m$, the condition (\[consmom\]) eliminates one important interaction process available for $\alpha=0$, namely interband backscattering (see below). This is a distinct SOI effect besides the broken spin $SU(2)$ invariance. Obtaining the complete “g-ology” classification of all possible interaction processes allowed for $\alpha\ne 0$ is then a straightforward exercise. The corresponding values of the interaction matrix elements are generally difficult to evaluate explicitly, but in the most important case of a thin wire, $$\label{thinwire}
d \gg \frac{1}{\sqrt{m\omega}},$$ where $d$ is the screening length (representing, e.g., the distance to a backgate), analytical expressions can be obtained.[@footnoteDelta] To simplify the analysis and allow for analytical progress, we therefore employ the thin-wire approximation (\[thinwire\]) in what follows. In that case, we can neglect the $z$ dependence in Eq. (\[pot\]). Going beyond this approximation would only imply slightly modified values for the e-e interaction couplings used below. Using the identity $$\begin{aligned}
\label{ident}
&& \int dz \ \left[ \phi^\dagger_{r k_F^{(\nu)} , \nu r } \cdot
\phi_{r' k_F^{(\nu')} , \nu' r' }\right](z) = \\ \nonumber
&& =
\delta_{\nu\nu'}\delta_{rr'}+\cos(\theta_A-\theta_B)\delta_{\nu,-\nu'}
\delta_{r,-r'},\end{aligned}$$ where the angles $\theta_{A,B}$ were specified in Eq. (\[angleab\]), only two different values $W_0$ and $W_1$ for the matrix elements in Eq. (\[intmat\]) emerge. These nonzero matrix elements are $$\begin{aligned}
\nonumber
&& V_{ \nu r, \nu' r', \nu' r', \nu r} \equiv W_0 = \tilde V(q=0) , \\
\label{g01} && V_{ \nu r, \nu' r', -\nu' -r', -\nu -r} \equiv W_1 \\
\nonumber && = \cos^2(\theta_A-\theta_B) \ \tilde V\left(
q=k_F^{(A)}+k_F^{(B)}\right) .\end{aligned}$$
We then introduce 1D chiral fermion densities $\rho_{\nu r}(x)= \ :\psi^\dagger_{\nu r} \psi^{}_{\nu r}:$, where the colons indicate normal-ordering. The interacting 1D Hamiltonian is $H=H_0+H_I$ with Eq. (\[dirac\]) and $$\begin{aligned}
\nonumber
H_I &=& \frac{1}{2} \sum_{\nu \nu', r r'} \int dx \Bigl(
[ g_{2\parallel\nu} \delta_{\nu,\nu'} + g_{2\perp}
\delta_{\nu,-\nu'}]\delta_{r,-r'} \\ & +& \label{1dint1}
[ g_{4\parallel\nu} \delta_{\nu,\nu'} +
g_{4\perp} \delta_{\nu,-\nu'} ]
\delta_{r,r'} \Bigr) \ \rho_{\nu r} \rho_{\nu' r'}
\\ \nonumber &+& \frac{g_f}{2} \sum_{\nu r} \int dx \
\psi_{\nu r}^\dagger \psi_{\nu,- r}^\dagger
\psi_{-\nu r}^{} \psi_{-\nu, -r}^{}.\end{aligned}$$ The e-e interaction couplings are denoted in analogy to the standard $g$-ology, whereby the $g_4$ ($g_2$) processes describe forward scattering of 1D fermions with equal (opposite) chirality $r=R,L=+,-$, and the labels $\parallel$, $\perp$, and $f$ denote intraband, interband, and band flip processes, respectively. Since the bands $\nu=A,B=+,-$ are inequivalent, we keep track of the band index in the intraband couplings. The $g_f$ term corresponds to intraband backscattering with band flip. The interband backscattering without band flip is strongly suppressed since it does not conserve total momentum[@footnote2] and is neglected in the following. For $\alpha=0$, the $g_{4,\parallel/\perp}$ couplings coincide with the usual ones[@gogolin] for spinful electrons, while $g_f$ reduces to $g_{1\perp}$ and $g_{2,\parallel/\perp}\to g_{2,\perp/\parallel}$ due to our exchange of band indices in the left-moving sector. According to Eq. (\[g01\]), the bare values of these coupling constants are $$\begin{aligned}
&& g_{4\parallel\nu} = g_{4\perp}=g_{2\parallel\nu}= W_0, \nonumber \\
&& g_{2\perp} =W_0-W_1,\quad g_{f}=W_1.
\label{init}\end{aligned}$$ The equality of the intraband coupling constants for the two bands is a consequence of the thin-wire approximation, which also eliminates certain exchange matrix elements.
The Hamiltonian $H_0+H_I$ then corresponds to a specific realization of a general asymmetric two band-model, where the one-loop RG equations are known.[@starykh; @muttalib] Using RG invariants, we arrive after some algebra at the two-dimensional Kosterlitz-Thouless RG flow equations, $$\label{ktrg}
\frac{d\bar g_2}{dl} = -\bar g_{f}^2,\quad
\frac{d\bar g_{f}}{dl} = - \bar g_{f} \bar g_2,$$ for the rescaled couplings $$\begin{aligned}
\label{scaling}
\bar g_2 &=& \frac{g_{2\parallel A}}{2\pi v_A} +
\frac{g_{2\parallel B}}{2\pi v_B} - \frac{g_{2\perp}}{\pi v_F} ,\\
\nonumber
\bar g_{f} &=& \sqrt{\frac{1+\gamma}{2}}\frac{g_{f}}{\pi v_F} ,\end{aligned}$$ where we use the dimensionless constant $$\label{gamma}
\gamma=\frac{v_F^2}{v_A v_B}=\frac{1}{1-\delta^2} \geq 1.$$ As usual, the $g_4$ couplings do not contribute to the one-loop RG equations. The initial values of the couplings can be read off from Eq. (\[init\]), $$\begin{aligned}
\bar g_2(l=0) &=& \frac{(\gamma-1) W_0+W_1}{\pi v_F}, \nonumber \\
\bar g_{f}(l=0) &=& \sqrt{\frac{1+\gamma}{2}}
\frac{W_1}{\pi v_F} .\label{init2}\end{aligned}$$ The solution of Eq. (\[ktrg\]) is textbook material,[@gogolin] and $\bar g_f$ is known to be marginally irrelevant for all initial conditions with $|\bar g_{f}(0)|\leq \bar g_2(0)$. Using Eqs. (\[g01\]) and (\[init2\]), this implies with $\gamma\simeq 1+\delta^2$ the condition $$\label{finalinit}
\tilde V(0) \ge \frac{1}{4} \cos^2(\theta_A-\theta_B) \ \tilde V
\left(k_F^{(A)}+k_F^{(B)}\right),$$ which is satisfied for all physically relevant repulsive e-e interaction potentials. As a consequence, intraband backscattering processes with band flip, described by the coupling $\bar g_{f}$, are *always marginally irrelevant*, i.e., they flow to zero coupling as the energy scale is reduced, $\bar g_{f}^*= \bar g_{f}(l\to \infty)=0$. Therefore *no gap arises*, and a modified LL model is the appropriate low-energy theory. We mention in passing that even if we neglect the velocity difference in Eq. (\[vab\]), no spin gap is expected in a Rashba wire, i.e., the broken $SU(2)$ invariance in our model is not required to establish the absence of a gap.
The above RG procedure also allows us to extract *renormalized couplings* entering the low-energy LL description. The fixed-point value $\bar g^*_2=\bar g_2(l\to \infty)$ now depends on the Rashba SOI through $\gamma$ in Eq. (\[gamma\]). With the interaction matrix elements $W_{0,1}$ in Eq. (\[g01\]), it is given by $$\label{gfix}
\bar g_2^* = \frac{ \sqrt{ [(\gamma-1)W_0+W_1]^2- (\gamma+1)W_1^2/2 }}
{\pi v_F}.$$ For $\alpha=0$, we have $\gamma=1$ and therefore $\bar g_2^*=0$. The Rashba SOI produces the nonzero fixed-point value (\[gfix\]), reflecting the broken $SU(2)$ symmetry.
Luttinger liquid description {#sec4}
============================
In this section, we describe the resulting effective low-energy Luttinger liquid (LL) theory of an interacting single-channel Rashba wire. Employing Abelian bosonization,[@gogolin] we introduce a boson field and its conjugate momentum for each band $\nu=A,B=+,-$. It is useful to switch to symmetric (“charge”), $\Phi_c(x)$ and $\Pi_c(x)=-\partial_x\Theta_c(x)$, and antisymmetric (“spin” for $\alpha=0$), $\Phi_s(x)$ and $\Pi_s(x)=-\partial_x\Theta_s$, linear combinations of these fields and their momenta. The dual fields $\Phi$ and $\Theta$ then allow to express the electron operator from Eq. (\[expand\]) and the “bosonization dictionary,” $$\begin{aligned}
\label{bosonize}
\Psi(x,z) &=& \sum_{\nu,r} \phi_{r k_F^{(\nu)} , \nu r }(z)
\frac{\eta_{\nu r}}{\sqrt{2\pi a}} \\ &\times& \nonumber
e^{i r k_F^{(\nu)}x +i\sqrt{\pi/2 } [r\Phi_c+\Theta_c+\nu r\Phi_s+
\nu \Theta_s] } ,\end{aligned}$$ where $a$ is a small cutoff length and $\eta_{\nu r}$ are the standard Klein factors.[@gogolin; @RKKYreinh; @conbos] (To recover the conventional expression for $\alpha=0$, due to our convention for the band indices in the left-moving sector, one should replace $\Phi_s,\Theta_s \rightarrow -\Theta_s, - \Phi_s$.) Using the identity (\[ident\]), we can now express the 1D charge and spin densities, $$\label{1ddensities}
\rho(x)=\int dz \, \Psi^\dagger \Psi, \quad
{\bm S}(x)=\int dz \, \Psi^\dagger \frac{\bm \sigma}{2} \Psi,$$ in bosonized form. The (somewhat lengthy) result can be found in Appendix \[appa\].
The low-energy Hamiltonian is then taken with the fixed-point values for the interaction constants, i.e., backscattering processes are disregarded and only appear via the renormalized value of $\bar g_2^*$ in Eq. (\[gfix\]). Following standard steps, the kinetic term $H_0$ and the forward scattering processes then lead to the exactly solvable Gaussian field theory of a modified (extended) Luttinger liquid, $$\begin{aligned}
\label{tll}
H&=& \sum_{j=c,s} \frac{v_j}{2} \int dx \ \left(
K_j \Pi_j^2 + \frac{1}{K_j}(\partial_x \Phi_j)^2 \right) \\
\nonumber &+& v_\lambda \int dx \ \left(
K_\lambda \Pi_c \Pi_s + \frac{1}{K_\lambda}
(\partial_x \Phi_c) (\partial_x\Phi_s) \right).\end{aligned}$$ Using the notations $\bar g_4=W_0/\pi v_F$ and $$\begin{aligned}
y_\delta &=& \frac{g^*_{2\parallel A} - g^*_{2\parallel B}}{4\pi v_F},\\
y_{\pm} &=& \frac{ g^*_{2\parallel A} +
g^*_{2\parallel B} \pm 2 g^*_{2\perp} }{4\pi v_F },\end{aligned}$$ where explicit (but lengthy) expressions for the fixed-point values $g^*_{2\parallel A/B}$ and $g^*_{2\perp}$ can be straightforwardly obtained from Eqs. (\[scaling\]) and (\[gfix\]), the renormalized velocities appearing in Eq. (\[tll\]) are $$\begin{aligned}
\nonumber
v_c &=& v_F \sqrt{(1+ \bar g_{4})^2 - y_{+}^{2}} \\ \nonumber
&\simeq& v_F \sqrt{\left(1+ \frac{W_0}{\pi v_F}\right)^2-\left(
\frac{2W_0-W_1}{2\pi v_F}\right)^2 }, \\ \label{veloc}
v_s &=& v_F \sqrt{1 - y_{-}^{2} } \simeq v_F,\\ \nonumber
v_\lambda &=& v_F \sqrt{\delta^2 - y_\delta^2} \simeq v_F\delta
\sqrt{1-\left( \frac{W_1}{4\pi v_F} \right)^2 } .\end{aligned}$$ In the respective second equalities, we have specified the leading terms in $|\delta|\ll 1$, since the SOI-induced relative velocity asymmetry $\delta$ is small even for rather large $\alpha$, see Eq. (\[vab\]). The corrections to the quoted expressions are of ${\cal O}(\delta^2)$ and are negligible in practice. It is noteworthy that the “spin” velocity $v_s$ is *not* renormalized for a Rashba wire, although it is well-known that $v_s$ will be renormalized due to $W_1$ for $\alpha=0$.[@gogolin] This difference can be traced to our thin-wire approximation (\[thinwire\]). When releasing this approximation, there will be a renormalization in general. Finally, the dimensionless LL interaction parameters in Eq. (\[tll\]) are given by $$\begin{aligned}
\nonumber
K_c &=&\sqrt{\frac{1+ \bar g_{4} - y_{+} }{1+ \bar g_{4} + y_{+} }}
\simeq \sqrt{\frac{2\pi v_F+W_1}{2\pi v_F+4W_0-W_1}} , \\
\label{llpar}
K_s &=&\sqrt{\frac{1 - y_{-} }{1+ y_{-} }} \simeq
1- \frac{\sqrt{W_0 W_1}}{\sqrt{2}\ \pi v_F} |\delta| , \\ \nonumber
K_\lambda &=& \sqrt{\frac{\delta - y_\delta}{\delta + y_\delta}} \simeq
\sqrt{\frac{4\pi v_F+W_1}{4\pi v_F-W_1}},\end{aligned}$$ where the second equalities again hold up to contributions of ${\cal O}(\delta^2)$. When the $2k_F$ component of the interaction potential $W_1=0$, see Eq. (\[g01\]), we obtain $K_s=K_\lambda=1$, and thus recover the theory of Ref. . The broken spin $SU(2)$ symmetry is reflected in $K_s<1$ when both $\delta\ne 0$ and $W_1\ne 0$.
Since we arrived at a Gaussian field theory, Eq. (\[tll\]), all low-energy correlation functions can now be computed analytically without further approximation. The linear algebra problem needed for this diagonalization is discussed in App. \[appa\].
RKKY interaction {#sec5}
================
Following our discussion in Sec. \[sec1\], we now investigate the combined effects of the Rashba SOI and the e-e interaction on the RKKY range function. We include the exchange coupling, $H'=J \sum_{i=1,2} {\bm \Sigma}_i \cdot {\bm S}(x_i)$, of the 1D conduction electron spin density ${\bm S}(x)$ to localized spin-1/2 magnetic impurities, separated by $x=x_1-x_2$. The RKKY interaction $H_{\rm RKKY}$, describing spin-spin interactions between the two magnetic impurities, is then obtained by perturbation theory in $J$.[@RKKYkittel] In the simplest 1D case (no SOI, no interactions), it is given by Eq. (\[rkkybasic\]). In the general case, one can always express it in the form $$\label{RkkyHam}
H_{\rm RKKY} = - J^2 \sum_{a,b} F^{ab}(x) \Sigma^a_1 \Sigma^b_2,$$ with the range function now appearing as a tensor ($\beta=1/k_B T$ for temperature $T$), $$\label{range}
F^{ab}(x) = \int_0^\beta d\tau \ \chi^{ab}(x,\tau).$$ Here, the imaginary-time ($\tau$) spin-spin correlation function appears, $$\label{spinspin}
\chi^{ab}(x,\tau) = \langle S^a(x,\tau) S^b(0,0) \rangle .$$ The 1D spin densities $S^a(x)$ (with $a=x,y,z$) were defined in Eq. (\[1ddensities\]), and their bosonized expression is given in App. \[appa\], which then allows to compute the correlation functions (\[spinspin\]) using the unperturbed ($J=0$) LL model (\[tll\]). The range function thus effectively coincides with the static space-dependent spin susceptibility tensor. When spin $SU(2)$ symmetry is realized, $\chi^{ab}(x)=\delta^{ab}F_
{\rm ex}(x)$, and one recovers Eq. (\[rkkybasic\]), but in general this tensor is not diagonal. For a LL without Rashba SOI, $F_{\rm ex}(x)$ is as in Eq. (\[rkkybasic\]) but with a slow power-law decay.[@RKKYreinh]
If spin $SU(2)$ symmetry is broken, general arguments imply that Eq. (\[RkkyHam\]) can be decomposed into three terms, namely (i) an isotropic exchange scalar coupling, (ii) a Dzyaloshinsky-Moriya (DM) vector term, and (iii) an Ising-like interaction, $$\begin{aligned}
\nonumber
H_{\rm RKKY}/J^2 &=& - F_{\rm ex}(x) {\bm \Sigma}_1 \cdot {\bm \Sigma}
_2 -
{\bm F}_{\rm DM}(x) \cdot \left( {\bm \Sigma}_1 \times {\bm \Sigma}_2
\right)\\
\label{RKKYgen}
&-& \sum_{a,b} F^{ab}_{\rm Ising}(x) \Sigma^a_1 \Sigma^b_2 ,\end{aligned}$$ where $F_{\rm ex}(x) = \frac{1}{3}\sum_a F^{aa}(x)$. The DM vector has the components $$F_{\rm DM}^c(x) = \frac{1}{2}\sum_{a,b} \epsilon^{cab} F^{ab}(x),$$ and the Ising-like tensor $$F_{\rm Ising}^{ab}(x)= \frac{1}{2}\left(
F^{ab} + F^{ba} -\frac{2}{3} \sum_c F^{cc}
\delta^{ab} \right)(x)$$ is symmetric and traceless. For a 1D noninteracting quantum wire with Rashba SOI, the “twisted” RKKY Hamiltonian (\[RKKYgen\]) has recently been discussed,[@bruno; @lyu; @simonin] and all range functions appearing in Eq. (\[RKKYgen\]) were shown to decay $\propto |x|^{-1}$, as expected for a noninteracting system. Moreover, it has been emphasized[@lyu] that there are different spatial oscillation periods, reflecting the presence of different Fermi momenta $k_F^{(A,B)}$ in a Rashba quantum wire.
Let us then consider the extended LL model (\[tll\]), which includes the effects of both the e-e interaction and the Rashba SOI. The correlation functions (\[spinspin\]) obey $\chi^{ba}(x,\tau)=\chi^{ab}(-x,-\tau)$, and since we find $\chi^{xz}=\chi^{yz}=0$, the anisotropy acts only in the $xy$-plane. The four nonzero correlators are specified in App. \[appa\], where only the long-ranged $2k_F$ oscillatory terms are kept. These are the relevant correlations determining the RKKY interaction in the interacting quantum wire. We note that in the noninteracting case, there is also a “slow” oscillatory component, corresponding to a contribution to the RKKY range function $\propto
\cos\left[ \left(k_F^{(A)}-k_F^{(B)}\right)x\right]/|x|$. Remarkably, we find that this $1/x$ decay law is not changed by interactions. However, we will show below that interactions cause a slower decay of certain “fast” oscillatory terms, e.g., the contribution $\propto \cos(2k_F^{(B)}x)$. We therefore do not further discuss the “slow” oscillatory terms in what follows.
Collecting everything, we find the various range functions in Eq. (\[RKKYgen\]) for the interacting case, $$\begin{aligned}
\nonumber
F_{\rm ex}(x) &=&\frac{1}{6}\sum_\nu \Bigl[
\left(1+\cos^2(2\theta_\nu) \right) \cos\left (2k_F^{(\nu)} x\right)
F_\nu^{(1)}(x) \\
\nonumber
&+& \cos^2 (\theta_A+\theta_B)
\cos\left[(k^{(A)}_F+k_F^{(B)})x
\right] F_\nu^{(2)}(x) \Bigr], \\ \label{rangell}
{\bm F}_{\rm DM}(x) &=& \hat e_z \sum_\nu \frac{\nu}{2}
\cos (2\theta_\nu) \sin \left(2k_F^{(\nu)}x\right) F^{(1)}_\nu(x),\\
\nonumber
F^{ab}_{\rm Ising}(x) &=& \left[ \frac12 \sum_\nu G_\nu^a(x) - F_{ex}(x)
\right] \delta^{ab} ,\end{aligned}$$ with the auxiliary vector $${\bm G}_\nu = \left( \begin{array}{c}
\cos\left(2k_F^{(\nu)} x\right) F_\nu^{(1)}(x) \\
\cos^2(2\theta_\nu) \cos\left(2k_F^{(\nu)} x\right) F_\nu^{(1)}
(x) \\
\cos^2(\theta_A+\theta_B)
\cos\left[(k^{(A)}_F+k_F^{(B)})x
\right] F_\nu^{(2)}(x) \end{array}\right).$$ The functions $F_\nu^{(1,2)}(x)$ follow by integration over $\tau$ from $\tilde F_\nu^{(1,2)}(x,\tau)$, see Eqs. (\[func1\]) and (\[func2\]) in App. \[appa\]. This implies the respective decay laws for $a\ll |x|\ll v_F/k_B T$, $$\begin{aligned}
\label{asymp}
F_\nu^{(1)}(x) &\propto& |a/x|^{-1+K_c+K_s+2\nu (1-K_c/K^2_\lambda)
\frac{v_\lambda K_\lambda}{v_c+v_s} },\\
F_\nu^{(2)}(x) &\propto& |a/x|^{-1+K_c+1/K_s}. \nonumber\end{aligned}$$ All those exponents approach unity in the noninteracting limit, in accordance with previous results.[@bruno; @lyu] Moreover, in the absence of SOI ($\alpha=\delta=0$), Eq. (\[asymp\]) reproduces the known $|x|^{-K_c}$ decay law for the RKKY interaction in a conventional LL.[@RKKYreinh]
Since $K_s<1$ for an interacting Rashba wire with $\delta\ne 0$, see Eq. (\[llpar\]), we conclude that $F_\nu^{(1)}$ with $\nu=B$, corresponding to the slower velocity $v_B=v_F(1-\delta),$ leads to the slowest decay of the RKKY interaction. For large distance $x$, the RKKY interaction is therefore dominated by the $2k_F^{(B)}$ oscillatory part, and all range functions decay $\propto |x|^{-\eta_B}$ with the exponent $$\label{etaB}
\eta_B = K_c + K_s - 1 - 2 \left(1-\frac{K_c}{K_\lambda^2}\right)
\frac{v_\lambda K_\lambda}{v_c+v_s} < 1.$$ This exponent depends both on the e-e interaction potential and on the Rashba coupling $\alpha$. The latter dependence also implies that electric fields are able to change the power-law decay of the RKKY interaction in a Rashba wire. The DM vector coupling also illustrates that the SOI is able to effectively induce off-diagonal couplings in spin space, reminiscent of spin precession effects. Also these RKKY couplings are $2k_F^{(B)}$ oscillatory and show a power-law decay with the exponent (\[etaB\]).
Discussion {#sec6}
==========
In this paper, we have presented a careful derivation of the low-energy Hamiltonian of a homogeneous 1D quantum wire with not too weak Rashba spin-orbit interactions. We have studied the simplest case (no magnetic field, no disorder, single-channel limit), and in particular analyzed the possibility for a spin gap to occur because of electron-electron backscattering processes. The initial values for the coupling constants entering the one-loop RG equations were determined, and for rather general conditions, they are such that backscattering is marginally irrelevant and no spin gap opens. The resulting low-energy theory is a modified Luttinger liquid, Eq. (\[tll\]), which is a Gaussian field theory formulated in terms of the boson fields $\Phi_c(x)$ and $\Phi_s(x)$ (and their dual fields). In this state, spin-charge separation is violated due to the Rashba coupling, but the theory still admits exact results for essentially all low-energy correlation functions.
Based on our bosonized expressions for the 1D charge and spin density, the frequency dependence of various susceptibilities of interest, e.g., charge- or spin-density wave correlations, can then be computed. As the calculation closely mirrors the one in Refs. , we do not repeat it here. One can then infer a “phase diagram” from the study of the dominant susceptibilities. According to our calculations, due to a conspiracy of the Rashba SOI and the e-e interaction, spin-density-wave correlations in the $xy$ plane are always dominant for repulsive interactions.
We have studied the RKKY interaction between two magnetic impurities in such an interacting 1D Rashba quantum wire. On general grounds, the RKKY interaction can be decomposed into an exchange term, a DM vector term, and a traceless symmetric tensor interaction. For a noninteracting wire, the corresponding three range functions have several spatial oscillation periods with a common overall decay $\propto |x|^{-1}$. We have shown that interactions modify this picture. The dominant contribution (characterized by the slowest power-law decay) to the RKKY range function is now $2k_F^{(B)}$ oscillatory for all three terms, with the same exponent $\eta_B<1$, see Eq. (\[etaB\]). This exponent depends both on the interaction strength and on the Rashba coupling. This raises the intriguing possibility to tune the power-law exponent $\eta_B$ governing the RKKY interaction by an electric field, since $\alpha$ is tunable via a backgate voltage. We stress again that interactions imply that a single spatial oscillation period (wavelength $\pi/k_F^{(B)}$) becomes dominant, in contrast to the noninteracting situation where several competing wavelengths are expected.
The above formulation also holds promise for future calculations of spin transport in the presence of both interactions and Rashba spin-orbit couplings, and possibly with disorder. Under a perturbative treatment of impurity backscattering, otherwise exact statements are possible even out of equilibrium. We hope that our work will motivate further studies along this line.
We wish to thank W. Häusler and U. Zülicke for helpful discussions. This work was supported by the SFB TR 12 of the DFG, and by the ESF network INSTANS.
Bosonization for the extended Luttinger liquid {#appa}
==============================================
In this appendix, we provide some technical details related to the evaluation of the spin-spin correlation function under the extended Luttinger theory (\[tll\]). The exact calculation of such correlations is possible within the bosonization framework, and requires a diagonalization of Eq. (\[tll\]).
The 1D charge and spin densities (\[1ddensities\]) can be written as the sum of slow and fast (oscillatory) contributions. Using Eq. (\[ident\]), the bosonized form for the 1D charge density is $$\begin{aligned}
\rho(x) &=& \sqrt{\frac{2}{\pi}} \ \partial_x \Phi_c -\frac{2i}{\pi a}
\eta_{AR} \eta_{AL} \cos(\theta_A-\theta_B) \\ &\times&
\sin\left[ \left(k_F^{(A)}+k_F^{(B)}\right) x+\sqrt{2\pi} \Phi_c \right]
\cos(\sqrt{2\pi} \Theta_s).\end{aligned}$$ Similarly, using the identity $$\begin{aligned}
&& \int dz\, \left[ \phi^\dagger_{r k_F^{(\nu)} , \nu r } \
{\bm \sigma} \, \phi_{r' k_F^{(\nu')} , \nu' r' } \right] (z)
= \\
&& \delta_{r,r'} \left(\begin{array}{c}
\cos\left(\theta_{A}-\theta_{B}\right) \delta_{\nu,-\nu'} \\
-i\nu r \,\cos\left(\theta_{A}+\theta_{B}\right) \delta_{\nu,-\nu'} \\
\nu r\,\cos\left(2\,\theta_{\nu}\right) \delta_{\nu,\nu'}
\end{array} \right) + \\
&&+ \delta_{r,-r'} \left( \begin{array}{c}
\delta_{\nu,\nu'}\\ -i\nu r \,
\cos\left(2\,\theta_{\nu} \right) \delta_{\nu,\nu'} \\
\nu r \,\cos\left(\theta_{A}+\theta_{B}\right) \delta_{\nu,-\nu'}
\end{array} \right),\end{aligned}$$ the 1D spin density vector has the components
$$\begin{aligned}
S^x(x) &=& -i\frac{\eta_{AR}\eta_{BR}}{\pi a}
\cos\left(\theta_A-\theta_B\right)
\cos \left[ \left(k_F^{(A)}-k_F^{(B)}\right) x +\sqrt{2\pi} \Phi_s
\right]
\sin (\sqrt{2\pi} \Theta_s) \\
&-& i\frac{\eta_{AR}\eta_{AL}}{\pi a}
\cos \left[ \left(k_F^{(A)} + k_F^{(B)}\right) x +\sqrt{2\pi} \Phi_c
\right]
\sin \left[ \left(k_F^{(A)}-k_F^{(B)}\right) x +\sqrt{2\pi} \Phi_s
\right] ,\end{aligned}$$
$$\begin{aligned}
S^y(x) &=& i
\frac{\eta_{AR}\eta_{BR}}{\pi a} \cos\left(\theta_A + \theta_B\right)
\sin \left[ \left(k_F^{(A)}-k_F^{(B)}\right) x +\sqrt{2\pi} \Phi_s
\right]
\sin( \sqrt{2\pi} \Theta_s ) \\ &-& i
\sum_{\nu=A,B=+,-} \nu \frac{\eta_{\nu R} \eta_{\nu L}}{2\pi a}
\cos( 2\theta_\nu ) \cos \left[2 k_F^{(\nu)} x +\sqrt{2\pi}
\left( \Phi_c+\nu \Phi_s\right) \right] ,\end{aligned}$$
$$\begin{aligned}
S^z(x) &=& \frac{1}{\sqrt{8\pi}} \left[
\left(\cos 2\theta_A +\cos 2\theta_B \right)
\partial_x \Theta_s +
\left( \cos 2\theta_A -\cos 2\theta_B \right)
\partial_x \Theta_c \right] \\ \nonumber
&-&i \frac{\eta_{AR}\eta_{BL}}{\pi a} \cos(\theta_A+\theta_B)
\cos \left[ \left(k_F^{(A)}+k_F^{(B)} \right) x +\sqrt{2\pi} \Phi_c
\right]
\sin (\sqrt{2\pi}\Phi_s ).\end{aligned}$$
Note that while $\partial_x \Phi_c$ is proportional to the (slow part of the) charge density, the (slow) spin density is determined by both $c$ and $s$ sectors.
Next we specify the nonzero components of the imaginary-time spin-spin correlation function $\chi^{ab}(x,\tau)$, see Eq. (\[spinspin\]). Using the above bosonized expressions, some algebra yields $$\chi^{xx}(x,\tau)=
\sum_\nu \frac{\cos\left ( 2k_F^{(\nu)}x \right )}{2(2\pi a)^2}
\tilde F^{(1)}_\nu(x,\tau),$$ $$\chi^{yy}(x,\tau)=
\sum_\nu\frac{ \cos^2(2\theta_\nu) \cos \left(2k_F^{(\nu)}x\right)}{2
(2\pi a)^2}
\tilde F^{(1)}_\nu(x,\tau),$$ $$\begin{aligned}
\chi^{zz}(x,\tau)&=&
\sum_{\nu r} \frac{ \cos^2(\theta_A+\theta_B) }{2 (2\pi a)^2} \\
&& \times \cos
\left[\left(k^{(A)}_F+k^{(B)}_F\right)x\right] \tilde
F^{(2)}_{\nu}(x,\tau),\end{aligned}$$ and $$\chi^{xy}(x,\tau)= \sum_\nu
\frac{ \nu \cos( 2\theta_\nu ) \sin \left(2k_F^{(\nu)}x\right)}{2(2
\pi a)^2}
\tilde F^{(1)}_\nu(x,\tau).$$ Here, the functions $\tilde F^{(1,2)}_{\nu=A,B=+,-}(x,\tau)$ are given by
$$\tilde F^{(1)}_\nu(x,\tau) = \prod_{j=1,2} \left |
\frac{\beta u_j}{\pi a} \sin \left(\frac{ \pi (u_j \tau - i x ) }
{ \beta u_j } \right)
\right |^{ - \left( \Gamma^{(j)}_{\Phi_c\Phi_c} +
\Gamma^{(j)}_{\Phi_s \Phi_s} + 2 \nu \Gamma^{(j)}_{\Phi_c\Phi_s}
\right ) }$$ and $$\tilde F^{(2)}_{\nu} (x,\tau) = \prod_{j=1,2}
\left | \frac{\beta u_j}{\pi a} \sin
\left( \frac{\pi(u_j\tau-ix)}{\beta u_j} \right)
\right |^{- \left( \Gamma^{(j)}_{\Phi_c\Phi_c} +
\Gamma^{(j)}_{\Theta_s \Theta_s}\right ) } \left[
\frac{ \sin \left( \frac{\pi (u_j \tau + i x ) }{ \beta u_j} \right) }
{ \sin \left(\frac{\pi (u_j \tau - i x) }{ \beta u_j} \right)}
\right]^{\nu \Gamma^{(j)}_{\Phi_c\Theta_s}}.$$
The dimensionless numbers $\Gamma^{(j)}$ appearing in the exponents follow from the straightforward (but lengthy) diagonalization of the extended LL Hamiltonian (\[tll\]), where the $u_{j}$ are the velocities of the corresponding normal modes. With the velocities (\[veloc\]) and the dimensionless Luttinger parameters (\[llpar\]), the result of this linear algebra problem can be written as follows. The normal-mode velocities $u_1$ and $u_2$ are $$\begin{aligned}
&& 2u_{j=1,2}^2 = v_c^2+v_s^2+2v_\lambda^2
- (-1)^j \Bigl [ (v_c^2-v_s^2)^2 + \\
&& +4v_\lambda^2\left[ v_c v_s \left( \frac{K_\lambda^2}{K_c K_s}
+ \frac{K_c K_s}{K_\lambda^2}\right) + v_c^2+v_s^2 \right] \Bigr]^{1/2},\end{aligned}$$ and the exponents $\Gamma^{(j=1,2)}$ appearing in $\tilde F^{(1,2)}_\nu(x,\tau)$ are given by $$\Gamma_{\Phi_c\Phi_c}^{(j)} =
\frac{(-1)^j K_c v_c}{u_j(u_1^2-u_2^2)} \left(v_s^2-u_j^2-
\frac{K_\lambda^2 v_\lambda^2 v_s}{K_c K_s v_c} \right) ,$$ $$\Gamma_{\Phi_s\Phi_s}^{(j)} =
\frac{(-1)^j K_s v_s}{u_j(u_1^2-u_2^2)} \left(v_c^2-u_j^2-
\frac{K_\lambda^2 v_\lambda^2 v_c}{K_c K_s v_s} \right) ,$$ $$\Gamma_{\Phi_c\Phi_s}^{(j)} =
\frac{(-1)^j K_\lambda v_\lambda}{u_j(u_1^2-u_2^2)} \left(v_
\lambda^2-u_j^2-
\frac{K_c K_s v_s v_c}{K_\lambda^2 } \right) ,$$ $$\Gamma_{\Theta_s\Theta_s}^{(j)} =
\frac{(-1)^j v_s}{K_s u_j(u_1^2-u_2^2)} \left(v_c^2-u_j^2-
\frac{K_c K_s v_\lambda^2 v_c}{K_\lambda^2 v_s} \right) ,$$ $$\Gamma_{\Phi_c\Theta_s}^{(j)} =
\frac{(-1)^j v_\lambda}{u_1^2-u_2^2} \left(
\frac{K_\lambda}{K_s} v_s + \frac{K_c}{K_\lambda}v_c\right).$$
Since $|\delta|\ll 1$, we now employ the simplified expressions for the velocities in Eq. (\[veloc\]) and the Luttinger liquid parameters in Eq. (\[llpar\]), which are valid up to ${\cal O}(\delta^2)$ corrections. In the interacting case, this yields for the normal-mode velocities simply $u_1=v_c$ and $u_2=v_s$. (In the noninteracting limit, the above equation instead yields $u_1=v_A$ and $u_2=v_B$, see Eq. (\[vab\]).) Moreover, the exponents $\Gamma^{(j)}$ simplify to $$\Gamma^{(1)}_{\Phi_c\Phi_c} = K_c ,\quad
\Gamma^{(2)}_{\Phi_c\Phi_c} = \Gamma^{(1)}_{\Phi_s\Phi_s} =
\Gamma^{(1)}_{\Theta_s\Theta_s} = 0 ,$$ $$\Gamma^{(2)}_{\Phi_s\Phi_s} = K_s , \quad
\Gamma^{(2)}_{\Theta_s\Theta_s} = 1/K_s ,$$ $$\Gamma^{(1)}_{\Phi_c\Phi_s} = \frac{v_\lambda}{v_c^2-v_s^2}
( K_\lambda v_c + K_c v_s/K_\lambda ),$$ $$\Gamma^{(2)}_{\Phi_c\Phi_s} = -\frac{v_\lambda}{v_c^2-v_s^2}
( K_\lambda v_s + K_c v_c/K_\lambda ),$$ $$\Gamma^{(1,2)}_{\Phi_c\Theta_s} = \pm \Gamma^{(2)}_{\Phi_c\Phi_s}.$$ Collecting everything and taking the zero-temperature limit, the functions $\tilde F^{(1,2)}_{\nu=\pm}(x,\tau)$ take the form $$\begin{aligned}
\nonumber
&& \tilde F^{(1)}_\nu(x,\tau)=
\left|\frac{v_c \tau-ix}{ a}\right|^{ -K_c-2\nu v_\lambda\frac{K_\lambda
v_c+K_c v_s/K_\lambda}{v_c^2-v_s^2} } \\ &\times& \label{func1}
\left|\frac{v_s \tau-ix}{ a}\right|^{-K_s+2\nu v_\lambda
\frac{K_\lambda v_s+K_c v_c/K_\lambda}{v_c^2-v_s^2} } ,\end{aligned}$$ and $$\begin{aligned}
\label{func2}
&& \tilde F^{(2)}_\nu(x,\tau)= \left|\frac{v_c \tau-ix}{a}\right|^{-K_c}
\left|\frac{v_s \tau-ix}{ a}\right|^{-1/K_s} \\ \nonumber
&\times& \left( \frac{(v_s \tau-ix)(v_c\tau+ix)}{(v_s\tau+ix)(v_c\tau-
ix)}
\right)^{-\nu \frac{v_\lambda (K_\lambda v_s+K_cv_c/K_\lambda)}{v_c^2-
v_s^2} }.\end{aligned}$$ The known form of the spin-spin correlations in a LL with $\alpha=0$ is recovered by putting $v_\lambda \propto \delta=0$.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We derive an explicit form of the quadratic-in-fermions Dirac action on the M5 brane for an arbitrary on-shell background of 11D supergravity with non-vanishing fluxes and in presence of a chiral 2-form on M5. This action may be used to generalize the conditions for which the non-perturbative superpotential can be generated in M/string theory. We also derive the Dirac action with bulk fluxes on the M2 brane.'
---
SU-ITP-05/01\
DFPD-05/TH/02\
\
[**Dirac Action on M5 and M2 Branes with Bulk Fluxes**]{}
[**Renata Kallosh$^{1}$ and Dmitri Sorokin$^{2}$** ]{}\
[**]{}
$^1$ Institute for Theoretical Physics, Department of Physics
Stanford University, Stanford, CA 94305
[[email protected]]{}
$^2$ INFN Sezione di Padova ${\&}$ Dipartimento di Fisica “Galileo Galilei",
Università degli Studi di Padova, 35131, Padova, Italy
*[[email protected]]{}*
January 2005\
Introduction
============
Our purpose is to derive the quadratic action for fermions on the M5 brane in a background of 11D supergravity with generic fluxes. Although the action of M5 in supergravity background [@Bandos:1997ui] as well as the equations of motion for all fields on M5 surface are known [@Howe:1997fb; @Bandos:1997gm; @Sorokin:1999jx], an explicit dependence on 4-form fields of 11D supergravity in the Dirac action on M5 has not been given yet.
The motivation for this work is two-fold. First of all to find an explicit form of a Dirac operator in presence of generic fluxes for the fundamental brane objects may have its own merit, especially if the answer is reasonably simple and allows to understand implications for the physics of the branes in such backgrounds. For the M2 and Dp branes such studies have already been undertaken in the literature. For instance, the coupling of D3 brane fermions to type IIB supergravity fluxes was studied in [[@Grana:2002tu]]{}. And in [@Marolf:2003vf; @Marolf:2003ye] Dp brane actions in generic supergravity backgrounds were derived in the quadratic approximation for fermions (in the Green–Schwarz form without gauge fixing worldvolume reparametrization and kappa-symmetry) from a corresponding M2 brane action [@Grisaru:2000ij] using T–duality [^1]. In this paper we present the results for the M5 brane and also for the M2 brane both in the Green–Schwarz and purely worldvolume form. This will allow one to investigate, in particular, various matrix models in backgrounds which are not necessarily of an $AdS\times S$-type but more general ones.
Secondly, significant part of the motivation for this work was the recent interest in non-perturbative corrections to M/string theory. It has been argued by Witten in [@Witten:1996bn] a while ago that the $SO(2)$ symmetry, which is a subgroup of the structure group $SO(5)$ may be the exact symmetry of the M5 brane action, no background fluxes were considered at that time. This was a basis for establishing a very powerful theorem about the conditions when non-perturbative superpotentials can be generated in M/string theory.
The analysis is based on algebraic geometry and leads to the statement that the compactification four-fold must admit divisors of arithmetic genus one, $\chi_D \equiv \sum (-1)^{n} h_n=1$. This indicates that in type IIB compactifications there can be [*non-vanishing corrections to the superpotential*]{} coming from Euclidean D3 branes. In the presence of such instantons, there is a correction to the superpotential which at large volume yields a term $
W_{\rm inst} = T(z_i) \exp(2\pi i \rho)
$ where $T(z_i)$ is a one-loop determinant, and the leading exponential dependence comes from the action of a Euclidean D3 brane wrapping a four-cycle in the compactified manifold. The non-perturbative corrections to the superpotential are absent, according to [@Witten:1996bn], when $\chi_D \neq 1$ (in absence of fluxes).
The presence of such non-perturbative corrections to the superpotential plays a crucial role in the stabilization of the volume modulus, as was shown in the simplest KKLT model with one Kähler modulus [@Kachru:2003aw] and in a general class of models with many Kähler moduli in [@Denef:2004dm]. In particular, in all models of Denef, Douglas and Florea in [@Denef:2004dm] the choice of compactification manifolds was always satisfying the restriction that $\chi_D=1$ and a significant effort was made to find them.
An analysis of some of these issues was performed in [@Robbins:2004hx] where the role of the 4-flux in the generation of instanton corrections has been discussed. In particular it was shown that the total flux through the divisor must vanish.
It was suggested by Gorlich, Kachru, Tripathy and Trivedi in [@Gorlich:2004qm] and argued for particular examples of compactification that in the presence of fixed fluxes the $U(1)$ symmetry of the fermionic action of the M5 brane might be broken. This in turn leads to a possibility of generating non-perturbative superpotentials in models with divisors on a four–fold of an arithmetic genus $\chi_D\geq 1$.
There are two possible points of view on $U(1)$ symmetry in presence of fluxes. These correspond to two well known aspects of symmetries of the background functional in field theory [@DeWitt:1967ub]. One is related to the so-called “quantum gauge transformations” and the second one is related to the “background gauge transformations” accompanied by the corresponding transformations of the quantum fields (variables of integration). In our case when the $U(1)$ symmetry acts as “quantum gauge transformations” only the fermions on the M5 brane are transformed but background fluxes are kept fixed. In presence of fixed background fluxes the corresponding $U(1)$ symmetry on the M5 brane may be broken. On the other hand, the $U(1)$ symmetry acting as the “background gauge transformations” in our case means that the fluxes, if they are vectors or tensors in the structure group, transform simultaneously with the action of $U(1)$ on fermions. In this sense the classical Dirac action on the brane has an unbroken $U(1)$ symmetry in presence of the background fluxes.
In the setting used in [@Witten:1996bn] we find it useful to consider fluxes transforming together with fermions under the $U(1)$–symmetry so that the classical invariance of the M5 brane action takes place. This symmetry of the classical theory may be broken by one-loop anomalies. However, with account of the inflow from the bulk these anomalies are expected to be canceled [@Witten:1996bn] since the $U(1)$ is just a part of diffeomorphisms and the theory is expected to be exactly invariant.
This all development is suggestive that the exact dependence of the Dirac operator on M5 with fluxes may help to understand the situation in a completely general setting since with account of bulk fluxes the algebraic properties of the M5 brane Dirac operator will change. To find the corresponding changes we need to find an explicit contribution of bulk fluxes to the Dirac operator on the M5 brane. We may proceed in two ways.
1\. First we may look for the fermion action in a target space covariant Green-Schwarz form where the corresponding anti-commuting fields are worldvolume scalars and target space spinors of $Spin (1,10)$. For the reader who just wants to see the effect of the flux $F_{\underline a \underline b \underline c
\underline d}$ on M5 and M2 fermions, here is the simplified form (without tensor field couplings on the M5 brane) of the Dirac equation for the 16-component ‘kappa-projected’ spinor in the background with fluxes. Note that in the approximation without tensor fields Dirac equations have the same form for M5 and M2 $$\label{Diracflux1}
\Gamma^{a}(\nabla_{a} + T_{a}{}^{\underline
a \underline b \underline c \underline d}\,F_{\underline a
\underline b \underline c \underline d}) \theta_- =0\,.$$ Here $T_{a}{}^{\underline a \underline b \underline c \underline
d}$ stands for a product of $\gamma$-matrices (the detailed notation is introduced below).
2\. We may look for the action for the world-surface fermions transforming in $Spin(1.5)\times Spin (5)$. This form is more useful for the future studies of the instanton effects from the Euclidean M5 brane wrapped on a six–cycle of a Calabi–Yau 4–fold [@Witten:1996bn]. We will need eventually to perform an analytic continuation to Euclidean space with spinors in $Spin(6)\times Spin (5)$. Here again is the simplified form of the Dirac equation for the chiral spinor on the M5 brane $$\label{Diracflux2}
\left[{\tilde \gamma}^{a} \,\nabla_a
+{1\over{24}}\,\left(\gamma^{ijk}\,
\tilde\gamma^{a}\,F_{aijk}-\gamma^i\,
\tilde\gamma^{abc}\,F_{abci}\right)\,\theta\right]^{\alpha}_q
=0\,,$$ see notation and details in the paper. An analogous equation is also given for the M2 brane below.
The derivation of the explicit form of the quadratic–in–fermions Dirac action on the M5 brane for an arbitrary on-shell background of 11D supergravity with non-vanishing fluxes and tensor fields on the M5 brane and on the M2 brane and of the corresponding equations of motion is explained below.
Since we are looking for a quadratic part of the action in a non-trivial background we may simply look at fermionic equations of motion which in a general form can be obtained either using superembedding techniques [@Howe:1997fb] or directly from the M5–brane action [@Bandos:1997ui; @Bandos:1997gm]. We will follow notation of the review paper [@Sorokin:1999jx].
In Sec. 2 we derive the Green-Schwarz type fermionic equations of the M5 brane in a flux background for space-time spinors being world-volume scalars. In Sec. 3 we derive Dirac equations for the world-volume spinors in a flux background. In both cases $\kappa$-symmetry is effectively gauge-fixed, so that the relevant fermion is 16-component. In Sec. 4 we present the Dirac action on M5 in a flux background with examples of how $U(1)$ symmetry acts on the fluxes. Dirac equations and the corresponding Lagrangian with bulk fluxes on the M2 brane are given in Sec. 5. In conclusion some final comments are made. Appendix contains some useful technical details on gamma-matrices and Lorentz spinor harmonics.
Green-Schwarz type fermionic equations of the M5 brane
======================================================
We start with the M5 brane fermionic equation in the Green–Schwarz form $$\label{5.2.18}
{1\over 2}m^{ba}E_a^{\underline\beta} \left[E_b^{\underline
a}\Gamma_{\underline a}(1-\bar\Gamma)\right]
_{\underline\beta\underline\alpha} =0\,.$$ In eq. (\[5.2.18\]) $$\label{pb}
E_a^{\underline\beta}=e_a^m(\xi)\,\partial_m\,Z^{\underline
M}(\xi)\,E_{\underline M}^{\underline\beta}(Z)\,,\quad
E_a^{\underline a}=e_a^m(\xi)\,\partial_m\, Z^{\underline
M}(\xi)\,E_{\underline M}^{\underline a}(Z)$$ are the pullbacks on the M5 brane worldvolume, parametrized by the coordinates $\xi^m$ $(m=0,1,\cdots,5)$, of the $D=11$ supergravity supervielbeins $$\label{sv}
E^{\underline A}(Z)=dZ^{\underline M}\,E^{\underline
A}_{\underline M}=(E^{\underline\alpha}\,, E^{\underline a}),$$ where $Z^{\underline M}=(x^{\underline m},\,\theta^{\underline\mu}$) (${\underline m}=0,1,\cdots,10;~{\underline \mu}=1,\cdots,32)$ are local coordinates of curved $D=11$ superspace [^2].
$e^m_a(\xi)$ is the inverse vielbein on the M5 brane worldvolume associated with the induced worldvolume metric $g_{mn}=\partial_m\,Z^{\underline M}\,E_{\underline M}^{\underline
a}\,\partial_n\,Z^{\underline N}\,E_{\underline N}^{\underline
b}\,\eta_{\underline {ab}}$. As so, $E^{~\underline a}_a$ satisfy the orthogonality condition $$\label{oc}
E^{~\underline a}_a\,E^{~\underline
b}_b\,\eta_{\underline{ab}}=\eta_{ab}\,,$$ where $\eta_{\underline{ab}}$ and $\eta_{ab}$ are respectively $D=11$ and $d=6$ Minkowski metric.
Finally, the matrix $m^{ab}=m^{ba}=\eta^{ab}-2\,h^{a}{}_{cd}h^{bcd}$ describes the interaction of the fermionic (and bosonic) worldvolume fields with the self–dual world–volume tensor $h_{abc}={1\over{3!}}\epsilon_{abcdef}\,h^{def}$. The tensor $h_{abc}$ is related via the nonlinear equation $$\label{h}
4(m^{-1})_a^{~d}\,h_{dbc}=H_{abc}\,$$ to the field strength $H^{(3)}={1\over
{3!}}\,e^c\,e^b\,e^a\,H_{abc}=d\,b^{(2)}(\xi)-A^{(3)}$ of the M5 brane chiral 2–form gauge field $b^{(2)}(\xi)$ extended with the pull back of the 3–form gauge potential $A^{(3)}$ of $D=11$ supergravity whose field strength $$\label{fda}
F^{(4)}=dA^{(3)}= {i\over 2}\,E^{\underline a}E^{\underline b}\bar
E\Gamma_{\underline{ba}}E+ {1\over{4!}}E^{\underline
a_4}\cdots\,E^{\underline a_1}F_{\underline a_1\cdots \underline
a_4}\,$$ generates the background fluxes. $\Gamma^{\underline
a}_{\underline\alpha\underline\beta}$ are $D=11$ gamma–matrices in the Majorana representation and $\bar \Gamma$ is their antisymmetrized product $$\label{5.2.19}
\bar\Gamma={1\over {6!}}\epsilon^{a_1\cdots \, a_6}
\Gamma_{a_1\cdots\,a_6}+{1\over 3}h^{abc}\Gamma_{abc}, \quad
\Gamma_a=E_a^{\underline a}\Gamma_{\underline a},$$ such that $\bar\Gamma^2=1$.
Equation (\[5.2.18\]) is invariant under the $\kappa$–symmetry transformations $$\label{4.49}
\delta_\kappa\, Z^{\underline M}\,E_{\underline M}^{\underline
\alpha}= {1\over 2}\,(1+\bar\Gamma)^{\underline
\alpha}_{~\underline\beta}\, \kappa^{\underline\beta}\,, \quad
\delta_\kappa Z^{\underline M}E_{\underline M}^{\underline a}=0\,.$$ Kappa–symmetry allows one to eliminate half of the M5 brane fermionic degrees of freedom. To see this one may notice that the right hand side of (\[5.2.18\]) is annihilated by the $\kappa$–symmetry projector $ {1\over
2}(1+\bar\Gamma)_{~~\underline\beta}^{\underline\alpha}$.
To extract the explicit dependence on fluxes in the Dirac equation on the M5 brane in the linear approximation for the fermions we have to evaluate the ingredients in this equation in the corresponding approximation. We look at $$E_m^{\underline \beta}=\partial_mZ^{\underline M}E_{\underline M}
^{\underline \beta}= \partial_mZ^{\underline \mu}E_{\underline
\mu} ^{\underline \beta}+ \partial_mZ^{\underline m}E_{\underline
m} ^{\underline \beta}\,. \label{vielbein}$$ We now define $$\label{tha}
\theta^{\underline \beta}\equiv Z^{\underline \mu}E_{\underline \mu}
^{\underline \beta}$$ and rewrite eq. (\[vielbein\]) in the following form $$E_m^{\underline \beta}=\partial_m \theta ^{\underline \beta}-
Z^{\underline \mu} \partial_m E_{\underline \mu} ^{\underline
\beta}+ \partial_mZ^{\underline m}E_{\underline m} ^{\underline
\beta} \label{vielbein1}$$ The first term is already at the linear level in $\theta$ and provides the free Dirac equation in the flat $D=11$ background. Without coupling to the bosonic fields and to the tensor field on the brane we would find, for $m^{ba}=\delta^{ba}$, $e_b^m=
\delta_b^m$ and $E_n^{\underline a}= \delta_n^a$, $$E_{ a}^{\underline\beta}[ \Gamma^{ a}(1-\bar\Gamma)]_{\underline\beta\underline\alpha}
= \partial_{ a}\theta^{\underline\beta}[ \Gamma^{ a}(1-\bar\Gamma)]_{\underline\beta\underline\alpha}
=0,$$ If we introduce the chiral spinor $\theta_-=
{1\over2}(1-\bar\Gamma)\theta$, which is manifestly invariant under $\kappa$–symmetry transformations (\[4.49\]), we find a simple Dirac equation $$\Gamma^{ a}\partial_{ a}\theta_-=0$$ This equation in turn easily transfers into a free Dirac equation for a chiral spinor on the M5 brane. However, here we would like to take care of corrections due to background geometry and in particular bulk fluxes. We therefore should evaluate the remaining terms in eq. (\[vielbein1\]).
In the Wess–Zumino gauge the term $\partial_m E_{\underline \mu}
^{\underline \beta}$, that has to be evaluated at the zero order in $\theta$ since it is multiplied by $\theta^{\underline \mu}$, vanishes. The third term in eq. (\[vielbein1\]) provides us with the information we are looking for. The vector–spinor vielbein $E_{\underline m} ^{\underline \beta}$ in $(11|32)$ superspace starts with gravitino $\psi_{\underline m}^{\underline \beta}(x)$, which we shall not take into account below restricting ourselves to pure bosonic $D=11$ supergravity backgrounds. The term linear in $\theta$ was calculated long time ago in [@Cremmer:1980ru]: this term is proportional to the rhs of the gravitino supersymmetry transformation excluding the term with the space–time derivative acting on the supersymmetry parameter[^3]. In notation of [@Sorokin:1999jx] $$\label{wzg}
E_{\underline m}^{\underline \beta}=
\psi_{\underline m}^{\underline \beta}
+( \Omega_{\underline m \underline \alpha }^{\underline \beta}+
T_{\underline m \underline \alpha}^{\underline \beta})\theta^{\underline \alpha}+...$$ where the term linear in $\theta$ includes a $Spin(1,10)$ connection $$\Omega_{\underline m \underline \alpha }^{\underline \beta}=
({1\over 4} \Omega_{\underline m}^{ \underline a \underline
b}\,\Gamma_{\underline a \underline b })_{~\underline \alpha
}^{\underline \beta}$$ and the flux–dependent superspace torsion term $$T_{\underline m \underline \alpha }^{\underline \beta}=
(T_{\underline m}^{~\underline{ a b c d}}F_{\underline a \underline
b \underline c \underline d})^{\underline \beta}{}_{\underline
\alpha }\,.$$ Here $$\label{T}
T^{\underline a \underline b \underline c \underline d \underline
e} \equiv {1\over 288} (\Gamma^{\underline a \underline b
\underline c \underline d \underline e} - 8 \eta^{\underline
a[\underline b}\Gamma^{ \underline c \underline d \underline e]}
)\,.$$ Thus we get $$\label{Dirac}
m^{ba}\,e_b^m\,e^n_a \,\left (\partial_m\, \theta ^{\underline
\beta}+( \Omega_{ m \underline \alpha }^{\underline \beta}+
T_{ m \underline \alpha}^{\underline \beta})\,\theta^{\underline \alpha}\right)
\left[E_n^{\underline a}\,\Gamma_{\underline
a}\,(1-\bar\Gamma)\right] _{\underline\beta\underline\gamma} =0,$$
In the absence of 11D fluxes $F_{\underline a \underline b
\underline c \underline d}$ and of the M5 brane chiral gauge field, and in the approximation in which the fermionic equation is linear in $\theta$, it is simply the Dirac equation with a metric compatible spin connection and an $SO(5)$ gauge group connection encoded in the covariant derivative $\nabla_a$ $$\label{Diracfree}
\Gamma^{a} \nabla_a \theta_- =0\,.$$ When fluxes are present we find that $$\label{Diracflux}
m^{ab}\,\Gamma_{b}\,( \nabla_{a} + T_{a}{}^{\underline a \underline
b \underline c \underline d}\,F_{\underline a \underline b
\underline c \underline d}) \theta_- =0\,.$$ Equation (\[Diracflux\]) is of a Green–Schwarz type in the sense that the fermionic field and the fluxes carry the target superspace vector and spinor indices. To reduce it to an equation which describes the dynamics of the M5 brane fermionic modes in the effective $6d$ worldvolume field theory one can, for example, impose on the worldvolume scalars the physical (static) gauge, which fixes worldvolume reparametrization invariance, and eliminate half of the fermionic modes by gauge fixing $\kappa$–symmetry. Such a gauge fixing breaks $D=11$ local $Spin(1,10)$ symmetry down to its subgroup $Spin(1,5)\times
Spin(5)$, where $Spin(1,5)$ is the $6d $ worldvolume ’Lorentz’ symmetry and $Spin(5)\sim USp(4)\sim SO(5)$ is the internal R–symmetry of the effective chiral (2,0) $d=6$ supersymmetric worldvolume field theory. This method was used in [@Claus:1997cq] where the free action for the tensor multiplets on the worldvolume of M5 brane was derived.
Alternatively, one can get the same worldvolume fermion equation in a simpler way, without breaking $D=11$ Lorentz invariance, by singling out the irreducible $\kappa$–invariant part of $\theta^{\underline\alpha}$ using the method of Lorentz harmonics which is part of the superembedding approach (see [@BPSTV; @Sorokin:1999jx]) for a review and references). In the next section we shall use the latter method to derive the purely worldvolume fermionic equation with fluxes.
Dirac equation on M5 surface
============================
Here we start with another form of the fermionic equation $$\label{fe}
\tilde \gamma_b^{\alpha\beta}\,m^{ba}\,E_a^{\underline\beta}\,
v_{\underline\beta,\beta q}=0\,,$$ which is related to eq. (\[5.2.18\]) by a certain transformation [@Howe:1997fb; @Sorokin:1999jx].
In eq. (\[fe\]) $\tilde \gamma_b^{\alpha\beta}$ and $\gamma^{a}_{\alpha\beta}$ are antisymmetric $d=6$ $Spin(1,5)$ ${\gamma}$–matrices having the properties described in eqs. (\[5.2.03\])–(\[5.2.003\]) of the Appendix, and $v_{\underline\beta,\beta q}(\xi)$ are half of the components of the $Spin(1,10)$ matrix (called Lorentz spinor harmonics) $$\label{5.2.04}
v_{\underline\alpha}^{~\underline\beta}=
(v_{\underline\alpha}^{~\alpha p},v_{\underline\alpha,\beta
q})\,,\quad C^{\underline\alpha\underline\gamma}\,
v_{\underline\alpha}^{~\underline\beta}\,v_{\underline\gamma}^{~\underline\delta}
=C^{\underline\beta\underline\delta} =\left(
\begin{array}{cc}
0 & \delta^\alpha_\beta \delta_q^{p}\\
-\delta_\gamma^\delta\delta^r_{s} & 0
\end{array}
\right)\,.$$ In (\[5.2.04\]) the $Spin(1,10)$ index ${}^{\underline\beta}$ is split into the two pairs $^{\alpha p}$ and ${}_{\beta q}$ of indices of $Spin(1,5)\times Spin(5)$ which is the symmetry of the M5 brane worldvolume theory. The corresponding realization of the $D=11$ $\Gamma$–matrices is given in the Appendix, eqs. (\[5.2.1\])–(\[5.2.C\]). Note that the upper and lower $Spin(1,5)$ indices $^{\alpha}$ and ${}_{\beta}$ correspond to inequivalent chiral spinor representations of $Spin(1,5)$, and there is no a $6d$ charge conjugation matrix which would raise and lower these indices.
The Lorentz harmonics (\[5.2.04\]) are auxiliary worldvolume fields. They are related to the pullback $E^{\underline
A}_a(Z(\xi))$ of the $D=11$ supervielbein (\[sv\]) by the equations (\[4.36\])–(\[lvh\]) of the Appendix. In particular, $$\label{0}
E^{~\underline\alpha}_a\,v_{\underline \alpha}^{~\alpha p}=0\,.$$
We can use the spinor harmonics (\[5.2.04\]) to convert the target space spinor field (\[tha\]) into a pair of chiral and anti–chiral worldvolume spinors $$\label{cs}
\theta^{\underline \beta}v_{\underline\beta\beta q}\equiv
\theta_{\beta q}\,, \qquad \theta^{\underline
\beta}v_{\underline\beta}^{~\alpha p}\equiv \theta^{\alpha p}\,.$$ Because of $\kappa$–invariance (which is reflected in the orthogonality condition (\[0\])) the anti–chiral spinor field $\theta^{\alpha p}$ does not appear in the fermionic equation (\[fe\]). In other words, one can use local $\kappa$–symmetry transformations to put $\theta^{\alpha p}$ to zero.
Then (\[fe\]) takes the form similar to (\[Diracflux\]) but with $d=6$ worldvolume matrix $\tilde\gamma_b$ instead of pulled back $D=11$ matrix $\Gamma_b$ $$\label{fe1}
[\tilde\gamma_b\,m^{ba}\,(\nabla_{a} + T_{a}{}^{\underline a
\underline b \underline c \underline d}F_{\underline a \underline
b \underline c \underline d})\theta]^{\alpha}_ q=0\,.$$ Here the covariant derivative is the derivative with account of metric compatible spin connection for $Spin(1,5)\times Spin(5)$ structure group (see [@Sorokin:1999jx] for details).
Using the relations (\[5.2.1\])–(\[lvh\]) of the Appendix we can rewrite (\[fe1\]) in a purely worldvolume form $$\label{wfe}
{\tilde \gamma}^{\alpha\beta}_b\,m^{ba}\nabla_a\, \theta_{\beta
q}+{1\over{24}}\,\left[\left(\gamma^{ijk}\,
\tilde\gamma^{b}\,(2\delta^a_b-m^{~a}_b)\,F_{aijk}+\gamma^i\,
\tilde\gamma^{bcd}\,(
2\delta^a_b-3m^{~a}_b)F_{acdi}\right)\,\theta\right]^{\alpha}_q
=0\,,$$ where the indices $i,j,k=1,2,3,4,5$ correspond to the target space directions transversal to the M5 brane.
If we ignore the 3-form $h$ contribution, we find that in our approximation eq. (\[wfe\]) reduces to $$\label{wfe-h}
{\tilde \gamma}^{a\,\alpha\beta} \,\nabla_a\, \theta_{\beta
q}+{1\over{24}}\,\left[\left(\,\gamma^{ijk}\,
\tilde\gamma^{a}\,F_{aijk}-\,\gamma^i\,
\tilde\gamma^{abc}\,F_{abci}\right)\,\theta\right]^{\alpha}_q
=0\,.$$
Dirac action and examples of flux transforming under $U(1)$ symmetry
=====================================================================
The fermion Lagrangian which produces the equations (\[wfe\]) is $$\label{m5l}
L^{M5}_f={1\over 2}\,\theta\,\left[{\tilde
\gamma}_b\,m^{ba}\nabla_a\, +{1\over{24}}\,\left(\gamma^{ijk}\,
\tilde\gamma^{d}\,(2\delta^a_d-m^{~a}_d)\,F_{aijk}+\gamma^i\,
\tilde\gamma^{bcd}\,(2\delta^a_d-3m^{~a}_d)F_{abci}\right)\right]\,\theta\,.$$
Let us note that via $m^{ab}$, which contains the [*self–dual*]{} field $h_{abc}$ defined in (\[h\]), the M5 brane fermions couple [*directly*]{}, though non–minimally, to the pull back of the 3–form [*flux potential*]{} $A_{abc}$. In the presence of the worldvolume flux $h_{abc}$ both the self–dual and anti–self–dual worldvolume parts of the flux $F_{abci}$ appear in the fermion Lagrangian, while if we neglect the contribution of $h_{abc}$ (so that $m^{~a}_{b}=\delta^{~a}_{b}$), the flux $F_{abci}$ should be [*anti–self–dual*]{} on the M5 worldvolume, since $\tilde\gamma^{abc}$ is [*self–dual*]{} (see eq. (\[5.2.0003\]) of the Appendix).
The action in presence of fluxes is invariant under the $SO(5)$ structure group transformations with the flux $F_{aijk}$ transforming as a 3d rank antisymmetric tensor and the $F_{abci}$ transforming as an $SO(5)$ vector.
We may split the $SO(5)$ index into those of $SO(3)$ and $SO(2)$, namely $i= \mu,\nu, \lambda , I, J$. This corresponds to splitting 5 directions normal to the brane into ${\bf R}^3$ for the 3 directions $i= \mu,\nu, \lambda$ and the remaining two directions $i= I, J$ will correspond to $SO(2)\sim U(1)$. We will be interested in the situation that $F_{aijk}$ has only $F_{a\mu\nu\lambda}$ components and $F_{abci}$ has only $F_{abcI}$ components. This would correspond to the 4-fold compactification of M-theory with 8 coordinates $a=1,..., 6, I=1,2$ to a 3-dimensional space with 3 coordinates $\mu,\nu, \lambda$.
The action is invariant under $SO(3) \times SO(2)$ symmetry with the flux $F_{abcI}$ transforming as a vector under $SO(2)$ symmetry.
Now we will look at an example which is more specific in the context of M theory, orientifolds and G-flux vacua [@Dasgupta:1999ss], [@Gorlich:2004qm]. In M-theory one starts with $X=K3_1\times K3_2$ four-fold in the presence of the flux $F_4$. In type IIB this is an orientifold $K3\times {T^2\over
Z_2}$. This $K3$ is $K3_1$ in M-theory. The second $K3$ which is called $K3_2$ is elliptically fibered. Thus we have an M-theory on a Calabi-Yau four-fold ${\bf R}^3 \times X$. We will take an Euclidean signature both in the target as well as on M5. In this case the complex divisor $D$ on which the five-brane is wrapped is a six-dimensional cycle in $X$. The 4-flux related to a 3-form flux in type IIB theory $G_3= F_3-\phi H_3$ (where $F_3$ and $H_3$ are respectively RR and NS 3-forms and $\phi$ is a complex axion–dilaton of IIB theory) can be chosen as follows $$F_4= -{1\over \phi-\bar \phi}\, G_3 \wedge d\bar z_2+ {1\over
\phi-\bar \phi} \bar G_3\, \wedge d z_2 \label{4form}\,,$$ where $dz_2$ and $d\bar z_2 $ are holomorphic and anti-holomorphic differentials along the elliptically fibered torus in $K3_2$.
The examples of supersymmetric background fluxes studied in [@Dasgupta:1999ss] and [@Gorlich:2004qm] require that $F_4$ has two legs along $K3_1$ and two legs along $K3_2$. This means that $H_3$ and $F_3$ have 2 legs in $K3_1$ and one leg in $K3_2$, in direction with coordinates $z_1, \bar z_1$. In our split of the 11-dimensional Euclidean space into 6+5, the six coordinates of the M5 have to be in $D$. The 5 directions normal to the brane include ${\bf R}^3$ for the 3 directions. The remaining 2 directions, normal to $D$ are related to the $SO(2)\sim U(1)$ symmetry, which is a rotation in $z_1, \bar z_1$ plane. One of the examples studied in [@Dasgupta:1999ss] and [@Gorlich:2004qm] is $$F_4 = C \Omega \wedge d\bar z_1 \wedge d\bar z_2+ \bar C \bar
\Omega \wedge dz_1\wedge d z_2 \label{example}\,,$$ where $C$ is a complex constant and where $\Omega$ ($\bar \Omega$) is a holomorphic (anti-holomorphic) two-form on $K3_1$. The flux has an $F_{z_1}$ as well as an $F_{\bar z_1}$ component which under the $U(1)$ transformation with a parameter $\varphi$ acquire the phase $e^{i\varphi} F_{z_1}$ and $e^{-i\varphi} F_{\bar z_1}$, respectively. If we would consider the fixed vacuum expectation value of the flux we would see that it violates the $U(1)$ symmetry. However, in the context in which the background flux transforms under $U(1)$ we have the following situation. The flux transforms under $U(1)$ if $C$ transforms as $e^{-i\varphi} C$ and $\bar C$ as $e^{i\varphi}
\bar C$. This leaves us with the Dirac action on M5 in presence of the background fluxes which is invariant under $U(1)$.
The Dirac equation and action on the M2 brane in the presence of $D=11$ fluxes
==============================================================================
For completeness, here we present the Green–Schwarz and purely worldvolume form of the Dirac operator on an M2 brane coupled to a $D=11$ supergravity gauge field flux $F_{\underline{abcd}}$. Its derivation can be carried out in the same way as for the M5 brane. In the Green–Schwarz form the M2 brane fermionic equation [@Bergshoeff:1987cm] is $$\label{gsm2}
\eta^{ab}\,E^{\underline\beta}_a\,\left[E^{\underline
a}_b\Gamma_{\underline
a}\,(1-\bar\Gamma)\right]_{\underline{\beta\alpha}}=0\,,$$ where $\bar\Gamma={1\over{3!}}\,\epsilon^{abc}\,\Gamma_{abc},~\bar\Gamma^2=1$, $\Gamma_a=E^{\underline\beta}_a\,\Gamma_{\underline a}$ and $a=0,1,2$ are the worldvolume tangent space indices.
Using eqs. (\[wzg\])–(\[T\]) we find that in the linear approximation in $\theta$ eq. (\[gsm2\]) reduces to $$\label{Diracfluxm2}
\Gamma^{a}\,( \nabla_{a} + T_{a}{}^{\underline a \underline b
\underline c \underline d}\,F_{\underline a \underline b \underline
c \underline d})\, \theta_- =0\,,$$ where $\theta_-={1\over2}(1-\bar\Gamma)\theta$, being ‘kappa–projected’, has 16 independent components. This form of equation can also be extracted from the more general answer in [@deWit:1998tk] or from the M2 brane quadratic action of [@Marolf:2003vf; @Marolf:2003ye].
To get the purely worldvolume form of the M2 brane Dirac operator with fluxes, one starts from the $M2$ brane fermionic equation in the superembedding formulation [@BPSTV; @Sorokin:1999jx] $$\label{m2f}
\gamma^{a\alpha\beta}\,E_a^{\underline\beta}\,
v_{\underline\beta,\,\beta q'}=0\,,$$ where now $\gamma^{a}_{\alpha\beta}=\gamma^{a}_{\beta\alpha}$ are $d=3$ $M2$ worldvolume symmetric gamma–matrices whose spinor indices $\alpha,\beta=1,2$ are raised and lowered by the antisymmetric unit matrices $\epsilon^{\alpha\beta}=\epsilon_{\alpha\beta}$, $q'=1,\cdots,\,8$ is the index of a spinor representation of the $SO(8)$ group of transformations of $d=8$ target space directions transversal to the $M2$ brane and $v_{\underline\beta,\beta q'}$ are half of the components of the $Spin(1,10)$ spinor Lorentz harmonics. Then, using eqs. (\[wzg\])–(\[T\]) and expressions of Section 5.1 of [@Sorokin:1999jx], one can reduce eq. (\[m2f\]) to a form analogous to that for the M5 brane $$\label{m2ff}
\gamma^{a\,\alpha\beta} \,\nabla_a\, \theta_{\beta
q'}+{1\over{96}}\,\left[\left(\gamma^{ijkl}\,
F_{ijkl}-6\,\epsilon^{abc}\,\gamma_c\,\gamma^{ij}
\,F_{abij}\right)\,\theta\right]^{\alpha}_{q'} =0\,.$$ The corresponding $M2$ brane worldvolume Lagrangian is $$\label{m2l}
L^{M2}_f={1\over 2}\, \theta\,\left[\gamma^{a} \,\nabla_a\,
+{1\over{96}}\,\left(\,\gamma^{ijkl}\, F_{ijkl}-6
\,\epsilon^{abc}\,\gamma_c\,\gamma^{ij}
\,F_{abij}\right)\right]\,\theta\,.$$ In eqs. (\[m2ff\]) and (\[m2l\]) $\gamma^{ijkl}_{q'p'}=\gamma^{ijkl}_{p'q'}$ and $\gamma^{ij}_{q'p'}=-\gamma^{ij}_{p'q'}$ are antisymmetric products of $SO(8)$ gamma matrices $\tilde\gamma^i_{q'p}$ and $\gamma^i_{pr'}$ with the indices $i,j,k,l=1,\cdots\,,8$ labeling the vector representation of $SO(8)$ and the index $p=1,\cdots\,,8$ corresponding to the second spinor representation of $SO(8)$ different from that labeled by $q'$ (see e.g. [@BPSTV; @Sorokin:1999jx] for details). Note that there is no difference between upper and lower $SO(8)$ indices since they all are raised and lowered by the unit symmetric matrices $\delta^{ij}$, $\delta^{pq}$ and $\delta^{p'q'}$. The gamma matrices $\gamma^i_{pr'}=\tilde\gamma^i_{r'p}$, such that $\gamma^i_{pr'}\tilde\gamma^j_{r'q}+\gamma^j_{pr'}\tilde\gamma^i_{r'q}=\delta^{ij}\delta_{pq}$, imply well known triality of the three inequivalent 8–dimensional fundamental representations of $SO(8)$.
It is interesting to note that in the presence of fluxes there is a kind of anomalous magnetic moment coupling of the worldvolume fermions to the field strengths of the fluxes. In the most straightforward way this anomalous magnetic moment coupling is seen in the Dirac equation for the M2 brane (\[m2ff\]). In the last term of this equation $F_{abij}$ can be regarded as a 2–form field strength on the 3d worldvolume with $i,j$ being the indices of an internal local symmetry group. When $i,j=1,2$ take only the values of $SO(2)$, we see that this is nothing but the anomalous magnetic moment coupling of 3d fermions to an electromagnetic field strength in a 3d field theory. A similar interpretation may also be given to flux terms in the Dirac operator on the M5 brane. For instance, the first flux term in (\[m5l\]) can be rewritten as $\gamma^{ijk}\,
\tilde\gamma^{a}(2\delta^d_a-m^{~d}_a)\,F_{dijk}=\tilde\gamma^{a}\,F_{a}^{ij}\,\gamma_{ij}$ (where $F_a^{ij}={1\over
{3!}}\,\varepsilon^{ij\,i_1j_1k_1}\,(2\delta^a_d-m^{~a}_d)\,F_{d\,i_1j_1k_1}$ and $\gamma_{ij}={1\over
{3!}}\,\varepsilon_{ij\,i_1j_1k_1}\,\gamma^{i_1j_1k_1}$). One can notice that the ‘anomalous magnetic moment’ term $F_{a}^{ij}\,\gamma_{ij}$ has the form similar to that of the $SO(5)$ connection $\Omega^{ij}_a\,\gamma_{ij}$ which enters the covariant derivative of the M5 brane Dirac operator (\[m5l\]).
Conclusion
==========
Thus we have derived here the explicit dependence on fluxes in the fermionic action on the M5 brane in a generic background. We have shown that there are two types of fluxes which enter the Dirac action. One of them transforms as an antisymmetric rank 3 tensor and another one as a vector of the R–symmetry group $SO(5)$ of the M5-brane, and in particular of its SO(2) subgroup. This poses a question: what happens in general with the condition $\chi_D=1$ derived in [@Witten:1996bn] and studied more recently in [@Robbins:2004hx] and [@Gorlich:2004qm]. In examples of compactification shown in [@Gorlich:2004qm] the answer was that $\chi_D\geq 1$ might provide the non-vanishing superpotential. It is not yet known how to generalize Witten’s analysis for the Dirac operator in eq. (\[wfe\]). This equation is a generalization of a simple Dirac equation used in [@Witten:1996bn] under the condition that there is no background flux, $F_4=0$ and there is no chiral 2-form on M5, $b_2=0$. When the background fluxes and world-volume 2-forms are present, in general, the analysis of the instanton corrections has to be redone. The immediate reason for this is the fact that background fluxes are necessarily required for stabilization of the dilaton-axion and complex structure moduli [@GKP].
The importance of this topic has to do with the fact that the only known way at present in which string theory and higher-dimensional supergravities may, possibly, address the current cosmological observations, require stabilization of moduli. The most difficult part, stabilization of Kähler moduli, is based on non-perturbative instanton corrections to the superpotential discussed here. Clarification of the restrictions on compactification manifolds which might provide such non-perturbative superpotentials might lead to a significant progress in string cosmology.
0.5cm
We are grateful to F. Denef and B. Florea who suggested to look for the Dirac operator with fluxes on M5 in the context of instanton corrections, and to E. Bergshoeff, I. Bandos, J. Gomis, S. Gukov, S. Kachru, G. Moore, A. Kashani-Poor, and S. Sethi, A. Tomasiello, S. Trivedi and A. Van Proeyen for clarifying discussions. The work of R. K. was supported by NSF grant 0244728. The work of D. S. was supported by the EU MRTN-CT-2004-005104 grant ‘Forces Universe’, and by the MIUR contract no. 2003023852.
Appendix {#appendix .unnumbered}
========
In our notation the underlined indices correspond to $D=11$ target superspace and not underlined ones correspond to the M5 brane worldvolume. The indices from the beginning of the Latin and Greek alphabet are vector and spinor tangent (super)space indices, while the indices from the middle of the Latin and Greek alphabet are that of local curved coordinates. The letters $i,j,k$ and $p,q,r,s$ stand, respectively for vector and spinor $Spin(5)$ indices.
We use the form of the $D=11$ $\Gamma$–matrices and of the charge conjugation matrices $C_{\underline\alpha\underline\beta}=C^{\underline\alpha\underline\beta}$ which reflects the embedding of the M5 brane $6d$ worldvolume into $N=1$, $D=11$ superspace $$\label{5.2.1}
\Gamma^a_{\underline\alpha\underline\beta}=\left(
\begin{array}{cc}
\gamma^a_{\alpha\beta}C_{pq} & 0\\
0 & \tilde\gamma^{a\alpha\beta}C^{pq}
\end{array}
\right)\,, \quad a=0,1,...,5, \quad \alpha,\beta=1,2,3,4,$$ $$\label{5.2.2}
\Gamma^i_{\underline\alpha\underline\beta}=\left(
\begin{array}{cc}
0 & \delta^\alpha_\beta(\gamma^{i})_q^{~p}\\
-\delta_\alpha^\beta(\gamma^i)^q_{~p} & 0
\end{array}
\right)\,, \quad i=1,...,5\,,\quad q,p=1,2,3,4\,,$$ $$\label{5.2.C}
C_{\underline\alpha\underline\beta}=C^{\underline\alpha\underline\beta}
=\left(
\begin{array}{cc}
0 & \delta^\alpha_\beta \delta_q^{p}\\
-\delta_\alpha^\beta\delta^q_{p} & 0
\end{array}
\right)\,.$$ In (\[5.2.2\]) $(\gamma^i)^q_{~p}=C^{qs}(\gamma^{i})_s^{~r}C_{rp}$ are $USp(4)\sim SO(5)$ gamma–matrices and $C^{qs}=C_{qs}$ are charge conjugation matrices. The matrices $(\gamma^{i})_{qp}=(\gamma^{i})_q^{~r}C_{rp}$ and $C^{qs}$ are antisymmetric.
$\tilde \gamma_b^{\alpha\beta}$ and $\gamma^{a}_{\alpha\beta}$ are antisymmetric $d=6$ $Spin(1,5)$ ${\gamma}$–matrices having the following properties $$\label{5.2.03}
\gamma^a_{\alpha\gamma}\tilde\gamma^{b\gamma\beta}+
\gamma^b_{\alpha\gamma}\tilde\gamma^{a\gamma\beta}
=2\delta^{\beta}_\alpha\eta^{ab}, \qquad {\rm
tr}(\gamma^a\tilde\gamma^b)=4\eta^{ab}, \quad
\gamma_{a\alpha\beta}\gamma^a_{\gamma\delta}
=-2\epsilon_{\alpha\beta\gamma\delta}\,,$$ $$\label{5.2.003}
\gamma^{abc}_{\alpha\beta}= \gamma^{abc}_{\beta\alpha}\equiv
(\gamma^{[a}\,\tilde\gamma^{b} \,\gamma^{c]})_{\alpha\beta}
=-{1\over 6}\epsilon^{abcdef}(\gamma_{def})_{\alpha\beta}\,,$$ $$\label{5.2.0003}
\tilde\gamma^{abc\,\alpha\beta}=
\tilde\gamma^{abc\,\beta\alpha}\equiv
(\tilde\gamma^{[a}\,\gamma^{b}\,
\tilde\gamma^{c]})^{\alpha\beta}={1\over
6}\epsilon^{abcdef}\,\tilde\gamma_{def}^{\alpha\beta}\,,$$
The Lorentz spinor harmonics (\[5.2.04\]) $$v_{\underline\alpha}^{~\underline\beta}=
(v_{\underline\alpha}^{~\alpha p},v_{\underline\alpha,\beta
q})\,,\qquad C^{\underline\alpha\underline\gamma}\,
v_{\underline\alpha}^{~\underline\beta}\,v_{\underline\gamma}^{~\underline\delta}
=C^{\underline\beta\underline\delta}\,, \qquad
v^{\underline{\alpha}}{}_{\underline{\gamma}}
=C^{\underline\alpha\underline\alpha'}\,v_{\underline\alpha'}{}^{\underline\gamma'}\,
C_{\underline\gamma'\underline\gamma}\,$$ are auxiliary worldvolume fields. They are related to the pullback $E^{\underline
A}_a(Z(\xi))$ of the $D=11$ supervielbein (\[sv\]) by the following equations $$\begin{aligned}
\label{4.36}
E^{~\underline\alpha}_a\,v_{\underline \alpha}^{~\alpha p}=0\,,
\qquad
\Gamma^{a}_{\underline{\gamma}\underline{\delta}}\,
E^{~\underline{a}}_{{a}}+\Gamma^{i}_{\underline{\gamma}\underline{\delta}}\,u_i^{~\underline
a}\equiv\Gamma^{\underline
b}_{\underline{\gamma}\underline{\delta}}\,u_{\underline
b}^{~\underline a}
= v^{\underline{\alpha}}{}_{\underline{\gamma}}\,
\Gamma^{{\underline a}}_{\underline{\alpha}\underline{\beta}}\,
v^{\underline{\beta}}{}_{\underline{\delta}}\,,\end{aligned}$$ where $u_i^{~\underline a}(\xi)$ are a set of five $D=11$ Lorentz vectors with indices ($i=1,\cdots,\,5$) belonging to the vector representation of $Spin(5)$. $u_i^{~\underline a}(\xi)$ are defined to be orthogonal to $E^{~\underline{a}}_{{a}}$, i.e. $$E^{~\underline{a}}_{{a}}\,u_i^{~\underline
b}\,\eta_{\underline{ab}}=0\,,$$ and complement the latter to an $SO(1,10)$ matrix (also called Lorentz vector harmonics) $$\label{lvh}
u_{\underline b}^{~\underline a}=(E_a^{~\underline
a},\,u_i^{~\underline a}),\qquad u_{\underline b}^{~\underline
a}\,u_{\underline d}^{~\underline
c}\,\eta_{\underline{ac}}=\eta_{\underline{bd}}\,, \qquad
u_{\underline b}^{~\underline a}\,u_{\underline d}^{~\underline
c}\,\eta^{\underline{bd}}=\eta^{\underline{ac}}\,.$$ Using the relations (\[5.2.1\])–(\[lvh\]) one can show that the fermionic equations (\[5.2.18\]) and (\[fe\]) are equivalent [@Howe:1997fb; @Sorokin:1999jx], with the projector matrix ${1\over 2}(1-\bar\Gamma)$ (\[5.2.19\]) having the following form in terms of the Lorentz spinor harmonics (\[5.2.04\]) and the self–dual tensor field $h_{abc}$ $$\label{pro}
{1\over 2}(1-\bar\Gamma)_{\underline\alpha\underline\beta}=
v_{\underline\alpha}^{~\beta p}(v_{\underline\beta,\,\beta p}
+C_{pq}\,h_{abc}\,\gamma^{abc}_{\beta\gamma}\,v^{~\gamma
q}_{\underline\beta})\,.$$
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[^1]: Other aspects of Dirac operators on branes have been considered e.g. in [@Claus:1997cq]-[@Gomis:2004pw].
[^2]: In our notation the underlined indices correspond to $D=11$ target superspace and not underlined ones correspond to the M5 brane worldvolume. The indices from the beginning of the Latin and Greek alphabet are vector and spinor tangent (super)space indices, while the indices from the middle of the Latin and Greek alphabet are that of local curved coordinates.
[^3]: The $\theta^2$ term was given in [@deWit:1998tk], the $\theta^3$ term was derived in [@Grisaru:2000ij] and the expression for $E^{\underline\beta}_{\underline m}$ up to the 5th order in $\theta$ was calculated in [@Tsimpis:2004gq] using a compact expression for the Wess–Zumino gauge, analogous to the one proposed for $D=4$ supergravity in [@Bandos:2002bx].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We calculate the full $ O(\alpha_s) $ radiative corrections to the three spin independent and five spin dependent structure functions that describe the angular decay distribution in the decay of a polarized top quark into a $ W $-boson (followed by the decay $ W^{+} \rightarrow l^{+} + \nu_l $ or by $ W^+ \rightarrow \bar q + q $) and a bottom quark. The angular decay distribution is described in cascade fashion, i.e. the decay $ t(\uparrow)
\rightarrow W^+ +X_b $ is analyzed in the top rest system while the subsequent decay $ W^+ \rightarrow l^+ + \nu_l $ (or $ W^+ \rightarrow \bar{q} + q $) is analyzed in the $ W $ rest frame. Since the structure function ratios depend on the ratio $ m_W/m_t $ we advocate the use of such angular decay measure ments for the determination of the top quark’s mass. Our results for the eight $ O(\alpha_s) $ integrated structure functions are presented in analy tical form keeping the mass of the bottom quark finite. In the limit $ m_b \rightarrow 0 $ the structure function expressions reduce to rather compact forms. We also present results on the $ m_b = 0 $ unpolarized and polarized $ O(\alpha_s) $ scalar structure functions relevant to the semi inclusive decay of a polarized top quark into a charged Higgs boson $ t(\uparrow) \rightarrow H^{+} + X_{b} $ in the Two Higgs Doublet Model when $ m_b = 0 $ .
---
MZ-TH/99-10\
hep-ph/0101322\
August 2001\
[**Complete angular analysis of polarized top**]{}\
[**decay at $ O(\alpha_s) $**]{}\
[M. Fischer, S. Groote, J.G. Körner and M.C. Mauser]{}\
Institut für Physik, Johannes-Gutenberg-Universität\
Staudinger Weg 7, D-55099 Mainz, Germany
**Introduction**
================
In the decay of an unpolarized or polarized top quark to the $ W $-gauge boson and a bottom quark the $ W^{+} $ is strongly polarized, or, phrased in a different language, the $ W^{+} $ has a nontrivial spin density matrix. Furthermore, the spin density matrix of the $ W $ can be tuned by changing the polarization of the top quark. The polarization of the $ W^{+} $ will reveal itself in the angular decay distribution of its subsequent decays $ W^{+}
\rightarrow l^{+} + \nu_{l} $ (or $ W^{+} \rightarrow \bar{q} + q $) [^1].
In the first stage one will aim to analyze the decay of unpolarized top quarks (or average over its polarization). The decay distribution of unpolarized top quark decay is governed by three structure functions which we shall refer to as $ H_U $ (“unpolarized-transverse”), $ H_L $ (“longitudinal”) and $ H_F $ (“forward-backward-asymmetric”). In fact, the CDF collaboration has already presented some results on the measurement of the longitudinal component of the $ W $ based on the limited RUN I data [@cdf]. The measurement has confirmed the expected dominance of the longitudinal mode. The error on this measurement is quite large ($ \approx 45 \, \% $) but is expected to be reduced significantly during RUN II at the TEVATRON to start in the spring of 2001. In RUN II one will produce $ (5 - 6) \times 10^3 $ top quark pairs per year and detector. This number will be boosted to $ 10^7 - 10^8 $ top quark pairs per year and detector at the LHC starting in 2006/2007. It is conceivable that the errors on the structure function measurements can be reduced to the $ 1-2 \, \% $ level in the next few years [@willenbrock]. If such an accuracy can, in fact, be achieved and, having in mind that the $ O(\alpha_s) $ corrections to the top decay rate amount to $ 8.5 \, \% $ [@c14; @c15; @c16; @c17; @c18; @ghinculov], it is quite evident that one needs to improve on the known theoretical Born level predictions for the above three structure functions by calculating their next-to-leading order radiative corrections.
At a later stage, when the data sample of polarized top quarks has become sufficiently large, one will be able to also analyze the decays of polarized top quarks. The top quark is very short-lived and therefore retains its full polarization content when it decays. Polarized top decay brings in five additional polarized structure functions which can be measured through an analysis of spin-momentum correlations between the polarization vector of the top quark and the momenta of its decay products.
Polarized top quarks will become available at hadron colliders through single top production which occurs at the $ 33 \, \% $ level of the top quark pair production rate [@c7]. Future $ e^{+} e^{-} $ colliders will also be copious sources of polarized top quark pairs [@c1; @c2; @c3; @c4; @c5; @c6]. For example, at the proposed TESLA collider one expects rates of $ (1 - 4) \times
10^5 $ top quark pairs per year. The polarization of these can be easily tuned through the availability of polarized beams (see e.g. [@fischer99]). Further, there is a high degree of correlation between the polarization of top and anti-top quarks produced in pairs either at $ e^{+} e^{-} $ colliders [@c8; @c9; @c10; @c11] or at hadron colliders [@c12] which can be probed through the joint decay distributions of the top and the anti-top quark.
In this paper we study momentum-momentum and spin-momentum correlations in the cascade decay process $ t \!\rightarrow\! W^+ \!+\! b $ followed by $ W^{+}
\!\rightarrow\! l^{+} \!+\! \nu_{l} $. The step-one decay $ t \!\rightarrow\!
W^{+} \!+\! b $ is analyzed in the $ t $-rest frame where we study the spin-momentum correlation between the spin of the top and the momentum of the $ W $. In step two we go to the rest frame of the $ W $ and analyze the correlation between the momentum of the lepton (or antiquark) and the initial momentum direction of the $ W $. In technical terms this means we analyze the double density matrix of the decaying top quark and the produced W-gauge boson. This must be contrasted with the [*center of mass*]{} analysis of polarized top decay where the spin-momentum correlations are all analyzed in the rest system of the top quark (for an $O(\alpha_s)$ analysis of this kind see [@cjkk]). Experimentally such a correlation measurement is easier, but from a theoretical point of view the cascade-type of analysis is advantageous because one can then better isolate the contribution of the longitudinal mode of the $ W $-gauge boson which is of relevance for the understanding of the electroweak symmetry breaking sector in the Standard Model. The results of the two analysis’ are of course related through a Lorentz boost along the $ W $ direction. However, the azimuthal correlations to be discussed later are not affected by such a Lorentz boost and are thus identical in both types of analysis.
The complete angular decay distribution is governed by altogether eight structure functions which we calculate analytically including their full $ O(\alpha_s) $ radiative corrections. One of the motivations for calculating the $ O(\alpha_s) $ radiative corrections is the fact that the radiative QCD corrections populate helicity configurations that are not accessible at the Born level. Take for example unpolarized top decay where, at the Born level, the $ W^{+} $ cannot be right handed, i.e. cannot have positive helicity, due to angular momentum conservation when $ m_{b} = 0 $. This implies that strictly forward $ l^{+} $ production does not occur at the Born level. However, when radiative corrections are taken into account, right-handed $W$’s do occur and strictly forward $ l^{+} $ production is allowed. As we shall see in Sec. 4 technically this means that the structure function combination $ (H_U + H_F)/2 $ vanishes at the Born term level but becomes nonzero at $ O(\alpha_s) $ [@FGKM]. We shall, however, see that the $ O(\alpha_s) $ population of the right-handed $ W $ is rather small [@FGKM]. The same statement holds true for the other structure function combinations that vanish at the Born term level.
In order to retain full control over the $ b $ mass dependence, and having also other applications in mind, we have kept a finite mass value for the $ b $ quark in our calculation. This improves on our earlier calculation of polarized top decay where the $ b $ quark mass was neglected and where we limited our attention to the six (diagonal) structure functions that govern the polar angle distribution in the cascade decay [@fischer99]. The additional two (non-diagonal) structure functions calculated in this paper describe the azimuthal correlation of the plane of the top quark’s polarization and the plane defined by the final leptons. In addition we determine the unpolarized and polarized scalar structure functions which are of relevance in the analysis of top decay into a bottom quark and a charged Higgs boson [@czarnecki]. We mention that our calculations have been done in the zero width approximation of the $ W $-boson. Finite width effects will be addressed in a forthcoming paper [@dgkm] (see also [@jk93]).
Most of the results in this paper are new. They have been checked against limiting cases and partial results obtained in other papers. We have checked our analytical $ O(\alpha_s) $ result for the total rate against the corresponding analytical rate result of Denner and Sack who also kept the $ b $ quark mass finite [@c14]. We find agreement. We took the zero $ b $-quark mass limit of the six diagonal structure functions and obtained agreement with our previous results in [@fischer99]. These had already been checked against the analytical results on the total rate obtained in [@c15; @c16; @c17; @c18] and on the longitudinal/transverse composition obtained in [@c19]. All six (mass zero) diagonal structure functions had also been checked against the corresponding numerical results given in [@c19; @c20; @c21]. The unpolarized scalar structure function has been checked against the results of [@czarnecki].
The central topic of this paper is the analysis of polarized top decay. We therefore mostly limit our attention to results valid in the limit $ m_{b}
\rightarrow 0 $ in the main part of our paper. This leads to enormous simplifications in the analytical rate formulas. The quality of the $ m_{b} =
0 $ approximation may be judged from the Born term rate which increases by $ 0.27 \, \% $ going from $ m_{b} = 4.8 \mbox{ GeV} $ to $ m_{b} = 0 $. The full $ m_{b} \neq 0 $ structure is given in Sec. 8 and the Appendices. Apart from retaining full control over $ m_b \ne 0 $ effects the finite mass results are needed e.g. in the theoretical analysis of semileptonic $ b \rightarrow c $ decays where the $ c $-quark mass can certainly not be neglected.
Our paper is structured as follows. In Sec. 2 we define a set of three spin independent and five spin dependent structure functions through the covariant expansion of the decay tensor resulting from the product of the two relevant current matrix elements. The eight invariant structure functions are related to eight helicity structure functions which form the angular coefficients of the angular decay distribution. In order to facilitate the calculation of the tree graph contributions we define a set of five covariant projection operators and a covariant representation of the spin vector of the top. These projectors can be used to covariantly project the requisite helicity structure functions from the hadron tensor. The advantage is that one thereby obtains the appropriate helicity structure functions and scalarizes the tensor integrands needed for the tree graph integration in one go. In Sec. 3 we derive the explicit form of the angular decay distribution in terms of the eight helicity structure functions for top decay. We also specify the changes in the angular decay distribution needed for antitop decay. Sec. 4 contains our Born term results. In Sec. 5 we list our results for the $ m_b = 0 $ one-loop contributions. In Sec. 6 we provide expressions for the $ O(\alpha_s) $ tree graph contributions and discuss technical details of how we have handled the necessary tree graph integrations. We mention that the infrared divergencies are regularized by a finite small gluon mass. In Sec. 7 we take the $ m_b \rightarrow 0 $ limit of the $ m_b \ne 0 $ results in Sec. 8 and present rather compact analytical $ O(\alpha_s) $ formulas for the various structure functions. Sec. 7 also contains our numerical results in the $ m_{b} = 0 $ approximation. Sec. 8 gives our analytical results on the tree graph integrations plus the one-loop contributions for $ m_b \ne 0 $. Sec. 9 provides a summary and our conclusions. In particular, we emphasize that angular measurements as advocated in this paper can be utilized to measure the mass of the top quark. In Appendix A we provide a complete list of $ m_b \ne 0 $ basis integrals that appear in the calculation of the tree graph contributions. This set of basis integrals should also be useful for other $ O(\alpha_s) $ or $ O(\alpha) $ radiative correction calculations. The requisite coefficient functions that multiply the basic integrals in the structure function expressions are listed in Appendix B. Appendix C, finally, contains the one-loop contribution in the $ m_b \ne 0 $ case.
**Invariant and helicity structure functions**
==============================================
The dynamics of the current-induced $ t \rightarrow b $ transition is embodied in the hadron tensor $ H^{\mu \nu} $ which is defined by
\[hadron-tensor1\] H\^(q\_0, q\^2=m\_W\^2, s\_t) & = & (2 )\^3 \_[X\_b]{} d\_f \^4(p\_t - q - p\_[X\_b]{})\
& & t(p\_t,s\_t) |J\^[+]{}| X\_b X\_b |J\^| t(p\_t,s\_t) ,
where $ d\Pi_f $ stands for the Lorentz-invariant phase space factor. In the Standard Model the weak current is given by $ J^\mu = \bar{q}_b \gamma^\mu P_L \bar{q}_t $ with $ P_L = \frac{1}{2} ( 1 - \gamma_5 ) $.
We are working in the narrow resonance approximation of the $ W $-boson and set $ q^2 = m_W^2 $ as indicated in the argument of the hadron tensor. Thus the hadron tensor is a function of the energy $ q_0 $ of the $ W $ alone. Since we are not summing over the top quark spin the hadron tensor also depends on the top spin $ s_t $ as indicated in Eq.(\[hadron-tensor1\]). The structure of the hadron tensor can be represented by a standard set of invariant structure functions defined by the expansion
\[tensor-expansion\] H\^ & = & ( - g\^ H\_1 + p\_t\^ p\_t\^ H\_2 - i \^ p\_[t,]{} q\_ H\_3 ) +\
& - & (q s\_t) ( - g\^ G\_1 + p\_t\^ p\_t\^ G\_2 - i \^ p\_[t,]{} q\_ G\_3 ) +\
& + & (s\_t\^ p\_t\^ + s\_t\^ p\_t\^ ) G\_6 + i \^ p\_[t ]{} s\_[t ]{} G\_8 + i \^ q\_ s\_[t ]{} G\_9 ,
where the $ H_i $ (i=1,2,3) and $ G_i $ (i=1,2,3,6,8,9) denote unpolarized and polarized structure functions, respectively.
In the expansion (\[tensor-expansion\]) we have kept only those structure functions that contribute in the zero lepton mass case. We have thus omitted covariants built from $ q^{\mu} $ and/or $ q^{\nu} $. We have also dropped contributions from invariants that are fed by [*T-odd*]{} or imaginary contributions which are both absent in the present case.
In the expansion (\[tensor-expansion\]) one has still overcounted by one term since there is a relationship between the three parity conserving (p.c.) spin dependent covariants appearing in (\[tensor-expansion\]) due to the identity of Schouten. The identity between the three covariants reads
$$ \label{schouten-identity}
q \!\cdot\! s_t \, \epsilon^{\mu \nu \rho \sigma} p_{t,\rho} q_{\sigma} -
q^2 \epsilon^{\mu \nu \rho \sigma} p_{t,\rho} s_{t \sigma} +
q \!\cdot\! p_t \, \epsilon^{\mu \nu \rho \sigma} q_{\rho} s_{t,\sigma} = 0$$
We shall, however, keep the overcounted set of nine invariant structure functions in (\[tensor-expansion\]) for reasons of computational convenience.
In this paper we shall only be concerned with two types of intermediate states in (\[hadron-tensor1\]), namely $ | X_{b} \rangle = | b \rangle $ (Born term and $ O(\alpha_s) $ one-loop contributions) and $ | X_{b} \rangle = | b + g
\rangle $ ($ O(\alpha_s) $ tree graph contribution). The Feynman diagrams contributing to the respective processes are drawn in Fig. 1.
The angular decay distribution that we are aiming for is given in terms of a set of angular decay coefficients which are linearly related to the set of unpolarized structure functions $ H_i $ and polarized structure functions $ G_i $ defined in Eq. (\[tensor-expansion\]). The relevant linear combinations are given by
$$\begin{aligned}
\label{lincombeg}
H_U & = & H_{++} + H_{--} = H_{11} + H_{22}, \\
H_L & = & H_{\mbox{oo}} = H_{33}, \\
H_F & = & H_{++} - H_{--} = i (H_{12} - H_{21}), \\
H_{U^P} & = & H_{++}(s_t^l) + H_{--}(s_t^l) =
H_{11}(s_t^l) + H_{22}(s_t^l), \\
H_{L^P} & = & H_{\mbox{oo}}(s_t^l) = H_{33}(s_t^l), \\
H_{F^P} & = & H_{++}(s_t^l) - H_{--}(s_t^l) =
i (H_{12}(s_t^l) - H_{21}(s_t^l)) , \\
H_{I^P} & = & \frac{1}{4} \Big( H_{+\mbox{o}}(s_t^{tr}) +
H_{\mbox{o}+}(s_t^{tr}) - H_{-\mbox{o}}(s_t^{tr}) -
H_{\mbox{o}-}(s_t^{tr}) \Big) \\ & = &
- \frac{1}{2 \sqrt{2}} (H_{13}(s_t^{tr}) + H_{31}(s_t^{tr})), \\
H_{A^P} & = & \frac{1}{4} \Big( H_{+\mbox{o}}(s_t^{tr}) +
H_{\mbox{o}+}(s_t^{tr}) + H_{-\mbox{o}}(s_t^{tr}) +
H_{\mbox{o}-}(s_t^{tr}) \Big) \\ & = &
\frac{i}{2\sqrt{2}} (H_{23}(s_t^{tr}) - H_{32}(s_t^{tr})),
\label{lincomend}
\end{aligned}$$
where $ H_{\lambda_W;\lambda_W'} = H_{\mu \nu} \epsilon^{\ast \mu}
(\lambda_W) \epsilon^{\nu}(\lambda_W') $ are the helicity projections of the polarized and unpolarized pieces of the structure functions $ H^{\mu \nu} $. The $ \epsilon^{\ast \mu}(\lambda_W) $ and $ \epsilon^{\nu}(\lambda_W) $ are the usual spherical components of the polarization vector of the $ W $ gauge boson. In the top quark rest system with $ q^{\mu} = (q_0;0,0,|\vec{q} \,|) $ and $| \vec{q} \,| = (q_0^2 - m_W^2)^{1/2} $ they read
\[helicity-projections\] \^(0) & = & (| |;0,0,q\_0),\
\^() & = & (0;1,-i,0).
In Eqs.(\[lincombeg\]–\[lincomend\]) we have also included the Cartesian components of the helicity structure functions in the $ W $-boson rest frame which are useful to have for some applications. For notational convenience we shall often refer to the set of helicity structure functions by their generic names. Thus we shall frequently use $ U $ for $ H_U $ and $ U^P $ for $ H_{U^P} $, etc..
The rest frame components of the longitudinal (“l”) and transverse (“tr”) polarization vector of the top are simply given by $ s_t^l = (0;0,0,1) $ and $ s_t^{tr} = (0;1,0,0) $. For the unpolarized helicity structure functions one sums over the the two diagonal spin configurations of the top while one takes the differences of these for the polarized helicity structure functions (in the $ z $-basis for $ s_t^l $ and in the $ x $-basis for $ s_t^{tr} $). When computing the polarized structure functions from the relevant Dirac trace expressions one thus has to replace $ ({p \hspace{-2mm} /}_t + m_t) $ in the unpolarized Dirac string by $ ({p \hspace{-2mm} /}_t + m_t)(1 + \gamma_5 {s \hspace{-2mm} /}_t) $. Note that the longitudinal component contributes only to the diagonal helicity structure functions $ U,L $ and $ F $ while the transverse component contributes only to the non-diagonal structure functions $ I $ and $ A $. The physics behind this will become clear when we write down the angular decay distribution in Sec. 3.
It turns out that it is rather convenient from the computational point of view to represent the helicity projections in (\[lincombeg\]–\[lincomend\]) (defined by the gauge boson polarization vectors and the top polarization vector) in covariant form. One has
H\_i\^ & = & H\_ \^\_i i = U,L,F,\
------------------------------------------------------------------------
H\_[i\^P]{} & = & H\_(s\_t\^l) \^\_i i = U,L,F,\
------------------------------------------------------------------------
H\_[i\^P]{} & = & H\_(s\_t\^[tr]{}) \^\_i i = I,A.
------------------------------------------------------------------------
The covariant projectors onto the diagonal density matrix elements are given by
\[kovprojanfang\] \^\_[L ]{} & = & (p\_t\^ - q\^ ) (p\_t\^ - q\^ ),\
\^\_[U+L]{} & = & - g\^ + ,\
\^\_[F ]{} & = & i \^ p\_[t,]{} q\_,
where $ \epsilon^{0123} = -1 $. We do not write out the projector for the unpolarized-transverse component $ U $ but note that it can be obtained from the combination $ {\mbox{I}\!\mbox{P}}^{\mu \nu}_{U+L} - {\mbox{I}\!\mbox{P}}^{\mu \nu}_{L} $.
The projectors onto the transverse-longitudinal non-diagonal density matrix elements are given by
\^\_I & = & + { \^(x) (p\_t\^ - q\^ ) + },\
\^\_A & = & - { i \^ \_(x) p\_[t,]{} q\_ (p\_t\^ - q\^ ) - }. \[kovprojende\]
They involve the the transverse polarization vector of the $ W $-gauge boson $ \epsilon_{\alpha}\!(x) \!=\! (0;1,0,0) $ pointing in the $x$-direction.
The covariant representation of the longitudinal component of the polarization vector of the top spin vector $ s_t^l $ is given by
$$ \label{polvektor1}
s_t^{l, \mu} = \frac{1}{| \vec{q} \,|}
\Big( q^{\mu} - \frac{p_t \!\cdot\! q}{m_t^2} p_t^{\mu} \Big),$$
whereas its transverse component $ s_t^{tr} $ reads
$$ \label{polvektor2}
s_t^{tr, \mu} = (0;1,0,0).$$
Note the inverse powers of $ | \vec{q} \,| = \sqrt{q_0^2 - m_W^2} $ that enter the $ L,T,F,I $ and $ A $ projectors and the longitudinal polarization vector. They come in for normalization reasons. These inverse powers of $ | \vec{q} \,| $ will make the necessary tree graph integrations to be dealt with in Sec. 6 and in the Appendices A and B somewhat more complicated than the total $ (U+L) $ rate integration which has a rather simple projector as Eq. (8.2) shows.
As mentioned in the Introduction, the covariant forms of the projection operators (\[kovprojanfang\]–\[kovprojende\]) and the polarization vectors (\[polvektor1\]) and (\[polvektor2\]) are quite convenient for the calculation of the $ O(\alpha_s) $ tree graph contributions to be dealt with in Sec.6 . The covariant projectors allow one to scalarize the tree graph tensor integrands and to project onto the requisite helicity structure functions in one go.
Although we shall mostly work in the helicity representation of the structure functions, it is sometimes convenient to have available the set of linear relations between the helicity and invariant structure functions. These can easily be worked out from the expansion (\[tensor-expansion\]), the projectors (\[kovprojanfang\]–\[kovprojende\]) and the polarization vectors (\[polvektor2\]). One has
\[system2anfang\] H\_[U]{} & = & 2 H\_1,\
------------------------------------------------------------------------
m\_W\^2 H\_L & = & m\_W\^2 H\_1 + | |\^2 m\_t\^2 H\_2,\
------------------------------------------------------------------------
H\_[F]{} & = & 2 | | m\_t H\_3,\
------------------------------------------------------------------------
H\_[U\^P]{} & = & 2 | | G\_1,\
------------------------------------------------------------------------
m\_W\^2 H\_[L\^P]{} & = & | | ( m\_W\^2 G\_1 + | |\^2 m\_t\^2 G\_2 - 2 q\_0 m\_t G\_6),\
------------------------------------------------------------------------
H\_[F\^P]{} & = & 2 ||\^2 m\_t G\_3 - 2 m\_t G\_8 - 2 q\_0 G\_9,\
------------------------------------------------------------------------
H\_[I\^P]{} & = & || G\_6,\
H\_[A\^P]{} & = & - G\_8 - m\_W G\_9. \[system2ende\]
Note that the three structure functions $ G_3 $, $ G_8 $ and $ G_9 $ always contribute in the two combinations $ (m_W^2 G_3 + G_8) $ and $ (q_0 m_1 G_3 -
G_9) $ proving again that there are only eight independent combinations of structure functions. If desired, Eqs.(\[system2anfang\]–\[system2ende\]) can be inverted such that the invariant structure functions can be expressed in terms of the helicity structure functions. The inversion has to be done in terms of the two above linear combinations of $ G_3 $, $ G_8 $ and $ G_9 $. Since our later results will always be presented in terms of the helicity structure functions, we shall not write down the inverse relations here.
**Angular decay distribution**
==============================
We are now in the position to write down the full angular decay distribution of polarized top decay into $ W^+ $ and $ b $ followed by the decay of the the $ W^{+} $ into $ (l^{+} + \nu_{l}) $. As noted before, the full angular decay distribution of the decay $ t(\uparrow) \rightarrow W^{+} (\rightarrow l^{+}
+ \nu_{l}) + X_{b} $, including polarization effects of the top quark, is completely determined by the three unpolarized and the five polarized helicity structure functions. Although the necessary manipulations to obtain the angular decay distribution involving Wigner’s $ D_{m m^{\prime}}^J(\theta,\phi) $- functions are standard (see e.g. [@c22]), it is quite instructive to reproduce the results here. To this end, it is useful to define helicity structure functions $ H_{\lambda_W \lambda^{\prime}_W}^{\lambda_t \,\,
\lambda^{\prime}_t} $ where the helicity label of the top quark is made explicit. Put in a different language the four-index object $ H_{\lambda_W \lambda^{\prime}_W}^{\lambda_t \,\, \lambda^{\prime}_t} $ is the unnormalized double density matrix of the top and the $ W $. The double density matrix is Hermitian, i.e. it satisfies
$$\Big( H_{\lambda_W \lambda^{\prime}_W}^
{\lambda_t \,\,\lambda^{\prime}_t} \Big)^{\ast} =
\Big( H_{\lambda^{\prime}_W \lambda_W}^
{\lambda^{\prime}_t \,\,\lambda_t} \Big).$$
As has been remarked on before the elements of the double density matrix are real in the present application. The double density matrix is therefore symmetric. The relation of the components of the double density matrix to the previously defined unpolarized and polarized helicity structure functions is given by
\[system3anfang\] H\_U & = & H\_[++]{}\^[++]{} + H\_[++]{}\^[–]{} + H\_[–]{}\^[++]{} + H\_[–]{}\^[–]{},\
H\_L & = & H\_\^[++]{} + H\_\^[–]{},\
H\_F & = & H\_[++]{}\^[++]{} + H\_[++]{}\^[–]{} - H\_[–]{}\^[++]{} - H\_[–]{}\^[–]{},\
H\_[U\^P]{} & = & H\_[++]{}\^[++]{} - H\_[++]{}\^[–]{} + H\_[–]{}\^[++]{} - H\_[–]{}\^[–]{},\
H\_[L\^P]{} & = & H\_\^[++]{} - H\_\^[–]{},\
H\_[F\^P]{} & = & H\_[++]{}\^[++]{} - H\_[++]{}\^[–]{} - H\_[–]{}\^[++]{} + H\_[–]{}\^[–]{},\
H\_[I\^P]{} & = & (H\_[+]{}\^[+-]{} + H\_[+]{}\^[-+]{} - H\_[-]{}\^[-+]{} - H\_[-]{}\^[+-]{}) = (H\_[+]{}\^[+-]{} - H\_[-]{}\^[-+]{}), \[HIP\]\
H\_[A\^P]{} & = & (H\_[+]{}\^[+-]{} + H\_[+]{}\^[-+]{} + H\_[-]{}\^[-+]{} + H\_[-]{}\^[+-]{}) = (H\_[+]{}\^[+-]{} + H\_[-]{}\^[-+]{}). \[HAP\]
For ease of notation we have used $ (\pm) $-labels for both the helicities of the top $ (\lambda_t = \pm 1/2) $ and the transverse helicities of the $ W $ gauge boson $ (\lambda_W = \pm 1) $. In the case of the non-diagonal structure functions $ H_{I^P} $ and $ H_{A^P} $ one can make use of the fact that the double density matrix is symmetric (for real coefficients !) to simplify the structure functions as indicated in the last two lines of Eqs. (\[HIP\]–\[HAP\]). From the fact that we are not observing the spin of the $ X_b $ system in our semi-inclusive measurement one has $ \lambda_{X_b} =
\lambda'_{X_b} $ leading to the constraint $ \lambda_W - \lambda^{\prime}_W =
\lambda_t - \lambda^{\prime}_t $. From this constraint it is immediately clear that the polarized structure functions $ U,L $ and $ F $ are associated with the longitudinal spin of the top and the structure functions $ I $ and $ A $ are associated with the transverse spin of the top.
The angular decay distribution can be obtained from the master formula
$$W(\theta_P,\theta,\phi) \propto
\sum\limits_{\lambda_W - \lambda^{\prime}_W = \lambda_t - \lambda^{\prime}_t}
e^{i (\lambda_W - \lambda^{\prime}_W) \phi} \,
d^1_{\lambda_W 1}(\theta) \, d^1_{\lambda^{\prime}_W 1}(\theta) \,
H_{\lambda_W \lambda^{\prime}_W}^{\lambda_t \: \lambda^{\prime}_t} \,
\rho_{\lambda_t \: \lambda^{\prime}_t} (\theta_P),
\label{masterformula}$$
where $ \rho_{\lambda_t \: \lambda^{\prime}_t} (\theta_P) $ is the density matrix of the top quark which reads
$$\rho_{\lambda_t \: \lambda^{\prime}_t} (\theta_P) =
\frac{1}{2} \pmatrix{
1 + P \cos \theta_P & P \sin \theta_P \cr
P \sin \theta_P & 1 - P \cos \theta_P }.$$
$ P $ is the magnitude of the polarization of the top quark. The sum in Eq. (\[masterformula\]) extends over all values of $ \lambda_W,
\lambda^{\prime}_W, \lambda_t $ and $ \lambda^{\prime}_t $ compatible with the constraint $ \lambda_W - \lambda^{\prime}_W = \lambda_t - \lambda^{\prime}_t
$. The second lower index in the small Wigner $ d(\theta) $-function $ d^1_{\lambda_W 1} $ is fixed at $ m = 1 $ for zero mass leptons because the total $ m $-quantum number of the lepton pair along the $ l^{+} $ direction is $ m = 1 $. Because there exist different conventions for Wigner’s $ d $-functions we explicate the requisite components that enter Eq. (\[masterformula\]): $ d^1_{11} = (1 + \cos\theta) / 2 $, $ d^1_{01} = \sin\theta / \sqrt{2} $ and $ d^1_{-11} = (1 - \cos \theta) / 2 $.
Including the appropriate normalization factor the four-fold decay distribution is given by
\[DiffRate\] & = & || { (H\_U + P \_p H\_[U\^P]{}) (1 + \^2 ) +\
& & + (H\_L + P \_p H\_[L\^P]{}) \^2 + (H\_F + P \_p H\_[F\^P]{})\
& & + P \_p H\_[I\^P]{} 2 + P \_p H\_[A\^P]{} }
We take the freedom to normalize the differential rate such that one obtains the total $ t \rightarrow b + W^{+} $ rate upon integration [*and not*]{} the total rate multiplied by the branching ratio of the respective $ W^{+} $ decay channel.
The polar angles $ \theta_P $ and $ \theta $, and the azimuthal angle $ \phi $ that arise in the full cascade-type description of the two-stage decay process $ t(\uparrow) \rightarrow W^{+} (\rightarrow l^{+} + \nu_{l}) + X_{b} $ are defined in Fig. 2. For better visibility we have oriented the lepton plane with a negative azimuthal angle relative to the hadron plane. For the hadronic decays of the $ W $ into a pair of light quarks one has to replace $ (l^{+},
\nu_{l}) $ by $ ( \bar{q}, q) $ in Fig. 2. We mention that we have checked the signs of the angular decay distribution Eq. (\[DiffRate\]) using covariant techniques.
As Eq. (\[DiffRate\]) shows the non-diagonal structure functions $ H_{I^P} $ and $ H_{A^P} $ are associated with azimuthal measurements. This necessitates the definition of a hadron plane which is only possible through the availability of the $ x $-component of the polarization vector of the top (see Fig. 2). This is the physical explanation of why the two structure functions $ H_{I^P} $ and $ H_{A^P} $ are functions only of the transverse component of the polarization vector of the top quark. For similar reasons the polarization dependent structure functions $ H_{U^P}, H_{L^P} $ and $ H_{F^P} $ depend only on the longitudinal component of the polarization vector.
Setting $P=0$ in Eq. (\[DiffRate\]) one obtains the decay distribution for unpolarized top decay. If desired, the transverse part of the unpolarized angular decay distribution can also be sorted in terms of decays into transverse-plus and transverse-minus $ W $-bosons given by the structure function combinations $ (U + F)/2 $ and $ (U - F)/2 $ which multiply the angular factors $(1 + \cos\theta)^2$ and $(1 - \cos\theta)^2$, resp., as done e.g. in [@FGKM].
If there were an imaginary part in the one-loop contribution one would have two additional contributions to the angular decay distribution proportional to $ \sin \phi $. This can be easily seen with the help of Eq. (\[masterformula\]). We concentrate on those terms in the angular decay distribution that are proportional to the off-diagonal terms $ \rho_{+-} $ in the density matrix of the top. The relevant terms read
H\_[+ ]{}\^[+-]{} e\^[+i ]{} + H\_[ +]{}\^[-+]{} e\^[-i ]{} = 2 ( (H\_[+ ]{}\^[+-]{}) - (H\_[+ ]{}\^[+-]{}) ),\
H\_[ -]{}\^[+-]{} e\^[+i ]{} + H\_[- ]{}\^[-+]{} e\^[-i ]{} = 2 ( (H\_[ -]{}\^[+-]{}) - (H\_[ -]{}\^[+-]{}) ).
The real contributions multiplying the angular factor $ \cos \phi $ have been included in the angular decay distribution (\[DiffRate\]) while the imaginary part contributions $ \mbox{Im}(H_{+ \mbox{o}}^{+-}) $ and $ \mbox{Im}
(H_{\mbox{o} -}^{+-}) $ multiplying $ \sin \phi $ do not appear in Eq. (\[DiffRate\]) since the $ O(\alpha_s) $ contributions calculated in this paper are purely real. The helicity structure functions $ \mbox{Im}
(H_{+ \mbox{o}}^{+-}) $ and $ \mbox{Im}(H_{\mbox{o} -}^{+-}) $ are conventionally called [*T-odd*]{} structure functions and are contributed to by the imaginary parts of loop contributions and/or by CP-violating contributions which, as has been emphasized before, are not present in this calculation.
Of interest is also the corresponding angular decay distribution for polarized anti-top decay $ \bar{t}(\uparrow) \rightarrow W^{-} (\rightarrow l^{-} +
\bar{\nu}_l) + X_{\bar{b}} $. The angular decay distribution is changed due to the fact that the total $ m $-quantum number of the lepton pair in the $ l^- $ direction is now $ m = - 1 $. The relevant components of the small Wigner $ d $-function are now $ d^1_{1-1} = (1 - \cos \theta) / 2 $, $ d^1_{0-1} = -\sin \theta / \sqrt{2} $ and $ d^1_{-1-1} = (1 + \cos \theta) / 2 $. This can be seen to result in a sign change for the angular factors multiplying the $ F $, $ F^P $ and $ A^P $ terms (and no sign change for the other terms). The structure functions of anti-top decay are related to those of top decay by $ CP $-invariance. The p.v. structure functions $ F, U^P, L^P $ and $ I^P $ will undergo a sign change whereas the p.c. structure functions $ U,L,F^P $ and $ A^P $ keep their signs. Overall this means that the signs of the unpolarized terms in Eq. (\[DiffRate\]) will not change their signs while the polarized terms will change signs when going from top decay to anti-top decay. To be quite explicit, if one wants to use the results of this paper to describe anti-top decay, the only required effective change is to change the signs of the terms multiplying the $ U^P,L^P,F^P,I^P $ and $ A^P $ structure functions in the angular decay distribution Eq. (\[DiffRate\]), using, however, the same structure functions as written down in this paper.
**Born term results**
=====================
The Born term tensor is calculated from the square of the Born term amplitude (see Fig. 2(a)) given by
$$ M^{\mu} = V_{tb}\, \frac{g}{ \sqrt{2}}
\bar{u}_{b} \gamma^{\mu} \frac{1}{2} (1 - \gamma_5) u_t.$$
We omit the coupling factor $ V_{tb}\, g /\sqrt{2} = 2 m_W V_{tb} (G_F/\sqrt{2})^{1/2} $ and write for the Born term tensor (the spin of the $b$ quark is summed)
$$ \label{born}
B^{\mu \nu} = \frac{1}{4} \mbox{Tr} ({p \hspace{-2mm} /}_b + m_b) \gamma^{\mu}
(1 - \gamma_5) ({p \hspace{-2mm} /}_t + m_t) (1 + \gamma_5 \slash{s}_t)
\gamma^{\nu} (1 - \gamma_5).$$
Since only even-numbered $ \gamma $-matrix strings survive between the two $ (1 - \gamma_5) $-factors in (\[born\]) one can compactly write
$$ \label{bornspur}
B^{\mu \nu} = 2 ( \bar{p}_t^{\nu} p_b^{\mu} + \bar{p}_t^{\mu} p_b^{\nu} -
g^{\mu \nu} \, \bar{p}_t \!\cdot\! p_b + i \epsilon^{\mu \nu \alpha \beta}
p_{b,\alpha} \bar{p}_{t,\beta}),$$
where
$$\bar{p}_t^{\mu} = p_t^{\mu} - m_t s_t^{\mu}.$$
It is not difficult to obtain the Born term helicity structure functions from (\[bornspur\]). This can be done in two ways. One can either read off the invariant structure functions according to the covariant expansion Eq. (\[tensor-expansion\]). The nonvanishing elements are given by $ B_{H_1} = m_t^2 (1-x^2+y^2) $, $ B_{H_2} = - 2 B_{H_3} = 4 $ for the unpolarized invariants and by $ B_{G_1} = B_{G_6} = B_{G_8} = - B_{G_9} = - 2 m_t $ for the polarized invariants (the notation is self-explanatory). These can then be converted to the helicity structure functions using the linear relations (\[system2anfang\]–\[system2ende\]). Or, one can directly compute the helicity structure functions from (\[bornspur\]) by using the covariant projectors defined in Sec.2 ([*cf.*]{} Eq.(\[kovprojanfang\]– \[kovprojende\])).
In order to find the relation of the Born term tensor $ B^{\mu \nu} $ to the hadron tensor $ H^{\mu \nu} $ defined in Sec. 2 one has to insert the appropriate one-particle $ b $-quark state into Eq. (\[hadron-tensor1\]) and then one has to do the requisite one-particle phase space integration. Technically this is done by rewriting the one-particle phase space as
$$ \label{1pps}
\int \!\! d\Pi_{b} =
\int \!\! \frac{d^3 \vec{p}_{b}}{2 \, E_b} =
\int \!\! d^4 p_b \, \delta(p_b^2 - m_b^2).$$
One can easily do the four-dimensional $ d^4 p_b $ integration in Eq.(\[hadron-tensor1\]) with the help of the four-dimensional $ \delta $-function $ \delta^4(p_t - q - p_b) $. This converts $ p_b^2 $ in the argument of the $\delta$-function in Eq.(\[1pps\]) into $ (p_t - q)^2 $. Rewriting the argument of the $ \delta $-function in terms of $ q_0 $ one finally arrives at
$$ \label{born+hadron-tensor}
H^{\mu \nu}(\mbox{Born}) = \frac{1}{4 m_t^2}
\delta(q_0 - \frac{m_t^2 + m_W^2 - m_b^2}{2 \, m_t}) B^{\mu \nu}.$$
We will present our results in table form where we use the scaled variables $ x = m_W/m_t $ and $ y = m_b/m_t $ as well as the abbreviation $ | \vec{q} \,|
= (m_t/2) \sqrt{ \lambda } $ with $ \lambda = \lambda (1,x^2,y^2) = 1 + x^4 +
y^4 - 2x^2 y^2 - 2x^2 - 2y^2 $. The first column in Table 1 contains the $ m_b
\neq 0 $, or equivalently, $ y \neq 0 $ results. In the second column we have set $ m_b = 0 $ $ (y = 0) $. In order to assess the quality of the $ m_b = 0 $ approximation for the various rate functions we have listed the percentage increments when going from the $ m_b \ne 0 $ case to the $ m_b = 0 $ case [*including*]{} the phase space factor $ | \vec{q} \,| $ that multiplies the helicity structure functions in the rate formula Eq. (\[DiffRate\]). In this comparison we have used $ m_b = 4.8 \mbox{ GeV} $ [@pivovarov] together with $m_t = 175$ GeV and $m_W=80.419$ GeV. The increment due to the phase space factor $ | \vec{q} \,| $ alone amounts to $ 0.15 \, \% $. Note that one may have overestimated the mass effect since a fixed pole mass, rather than a running mass which is smaller at the high scale of the top mass, is used. For example, taking one-loop running and the same bottom pole mass as above one has $ \bar{m}_b(m_t) = 1.79 $ GeV. The increment in the total rate going from $ \bar{m}_b(m_t) = 1.79 $ GeV to $ m_b = 0 $ would then only be $ 0.04 \, \% $ as compared to the $ 0.26 \, \% $ given in Table 1.
---------------------------------- ------------------------------------------------------------------------ --------------------------------------------------------------------------- -----------
Born $ m_b \neq 0 $ $ m_b = 0 $ increment
term
$ {\displaystyle}{B_{U+L}} $ $ {\displaystyle}{ m_t^2 \frac{1}{x^2} ((1 \!-\! y^2)^2 \!+\! x^2 $ {\displaystyle}{ m_t^2 \frac{1}{x^2} (1 \!-\! x^2)(1 \!+\! 2 x^2) +0.27 %
(1 \!-\! 2 x^2 \!+\! y^2))} $ \phantom{\Bigg(}} $
$ {\displaystyle}{B_{U^P+L^P}} $ $ {\displaystyle}{ m_t^2 \sqrt{\lambda} \frac{1}{x^2} (1 \!-\! $ {\displaystyle}{ m_t^2 \frac{1}{x^2} (1 \!-\! x^2)(1 \!-\! 2 x^2) +0.42 %
2 x^2 \!-\! y^2)} $ \phantom{\Bigg(}} $
$ {\displaystyle}{B_U} $ $ {\displaystyle}{2 m_t^2 (1 \!-\! x^2 \!+\! y^2)} $ $ {\displaystyle}{2 m_t^2 +0.05 %
(1 \!-\! x^2) \phantom{\Bigg(}} $
$ {\displaystyle}{B_{U^P}} $ $ {\displaystyle}{-2 m_t^2 \sqrt{\lambda}} $ $ {\displaystyle}{- 2 m_t^2 +0.29 %
(1 \!-\! x^2) \phantom{\Bigg(}} $
$ {\displaystyle}{B_L} $ $ {\displaystyle}{ m_t^2 \frac{1}{x^2} ((1 \!-\! y^2)^2 \!-\! x^2 $ {\displaystyle}{ m_t^2 \frac{1}{x^2} (1 \!-\! x^2) +0.36 %
(1 \!+\! y^2))} $ \phantom{\Bigg(}} $
$ {\displaystyle}{B_{L^P}} $ $ {\displaystyle}{ m_t^2 \sqrt{\lambda} \frac{1}{x^2} (1 \!-\! y^2)} $ $ {\displaystyle}{ m_t^2 \frac{1}{x^2} (1 \!-\! x^2) \phantom{\Bigg(}} $ +0.37 %
$ {\displaystyle}{B_F} $ $ {\displaystyle}{- 2 m_t^2 \sqrt{\lambda}} $ $ {\displaystyle}{- 2 m_t^2 (1 \!-\! +0.29 %
x^2) \phantom{\Bigg(}} $
$ {\displaystyle}{B_{F^P}} $ $ {\displaystyle}{2 m_t^2 (1 \!-\! x^2 \!+\! y^2)} $ $ {\displaystyle}{2 m_t^2 (1 \!-\! x^2) \phantom{\Bigg(}} $ +0.05 %
$ {\displaystyle}{B_S} $ $ {\displaystyle}{ m_t^2 \frac{1}{x^2} ((1 \!-\! y^2)^2 \!-\! x^2 $ {\displaystyle}{ m_t^2 \frac{1}{x^2} (1 \!-\! x^2) +0.36 %
(1 \!+\! y^2))} $ \phantom{\Bigg(}} $
$ {\displaystyle}{B_{S^P}} $ $ {\displaystyle}{ m_t^2 \sqrt{\lambda} \frac{1}{x^2} (1 \!-\! y^2)} $ $ {\displaystyle}{ m_t^2 \frac{1}{x^2} (1 \!-\! x^2) \phantom{\Bigg(}} $ +0.37 %
$ {\displaystyle}{B_{I^P}} $ $ {\displaystyle}{-\frac{1}{2} \sqrt{2} m_t^2 \sqrt{\lambda} $ {\displaystyle}{-\frac{1}{2} \sqrt{2} m_t^2 \frac{1}{x} (1 \!-\! x^2) +0.29 %
\frac{1}{x}} $ \phantom{\Bigg(}} $
$ {\displaystyle}{B_{A^P}} $ $ {\displaystyle}{\frac{1}{2} \sqrt{2} m_t^2 \frac{1}{x} (1 \!-\! $ {\displaystyle}{\frac{1}{2} \sqrt{2} m_t^2 \frac{1}{x} (1 \!-\! x^2) +0.24 %
x^2 \!-\! y^2)} $ \phantom{\Bigg(}} $
---------------------------------- ------------------------------------------------------------------------ --------------------------------------------------------------------------- -----------
: Born term helicity structure functions $ B_i $ ($ i = U \!+\! L $, $ U^P \!+\! L^P $, $ U $, $ U^P $, $ L $, $ L^P $, $ F $, $ F^P $, $ S $, $ S^P $, $ I^P $, $ A^P $ for $ m_b \ne 0 $ and $ m_b=0 $. Fourth column gives the percentage increment when going from $ m_b \ne 0 $ to $ m_b=0 $ including the phase space factor $ |\vec{q}| $.
In the $ m_b = 0 $ case listed in column 3 of Table 1 one observes the simple patterns $ B_U = - B_{U^P} = - B_F = B_{F^P} $, $ B_L = B_{L^P} $ and $ B_{I^P} = - B_{A^P} $. This pattern results from the fact that a massless $ b $-quark emerging from a $ (V-A) $ vertex is purely left-handed. Since from angular momentum conservation one has $ \lambda_t = \lambda_W - \lambda_b $ with $ \lambda_b=-1/2 $ one has the constraint $ \lambda_t - \lambda_W = 1/2 $. This implies that only the helicity configurations $ (\lambda_t = -1/2; \lambda_W =
-1) $ and $ (\lambda_t = +1/2; \lambda_W = 0) $ are non-vanishing. A quick look at the relations (\[system3anfang\]–\[HAP\]) allows one to readily verify the $ m_b = 0 $ pattern in Table 1. For $ m_b \ne 0 $ there is a leakage into right-handed bottom mesons resulting in a breaking of the above pattern as can be observed in the $ m_b \ne 0 $ column of Table 1. As noted in the Introduction these simple patterns are also not valid at $ O(\alpha_s) $ even for massless bottom mesons because of the additional gluon emission including an anomalous spin-flip contribution [@spinflip]. When the relevant $ m_b=0 $ Born term helicity structure functions from Table 1 are substituted in (\[DiffRate\]) we reproduce the angular decay distribution as written down in [@c8].
For completeness we have also included the two Born term scalar helicity structure functions $ B_S $ and $ B_{S^P} $ in Table 1. They are obtained by use of the scalar projector $ {\mbox{I}\!\mbox{P}}_S = q^{\mu} q^{\nu} / m_W^2 $. That they are identical to their longitudinal counterparts $ B_L $ and $ B_{L^P} $ even for $ m_b \neq 0 $ is a dynamical accident specific to the Born term level and does not hold true in general as e.g. evidenced by the $ O(\alpha_s) $ contributions to be discussed later on. These become equal to each other only in the limit $ m_t \rightarrow
\infty $ as will be discussed in Sec. 7. The $ m_b \neq 0 $ Born term equalities $ B_F = B_{U^P} $ and $ B_U = B_{F^P} $ can be seen to result from the fact that the double density matrix elements $ H_{++}^{--} $ and $ H_{--}^{++} $ vanish at the Born term level due to angular momentum conservation (see \[system3anfang\]).
In Fig.3 we present a lego plot of the two-fold ($ m_b=0 $) Born term angular decay distribution in $ \cos \theta $ and $ \cos\theta_P $ which results after taking the azimuthal average of Eq.(\[DiffRate\]). We have divided out the total Born term rate from the differential rate resulting in the hatted differential rate distribution as defined in Eq.(\[hatDiffRate\]). We have set $ P =1 $ in Fig.3. The lego plot shows that the $ \cos \theta $ and $ \cos\theta_P $ variation of the two-fold angular decay distribution around its average value of 0.25 is quite strong. This will facilitate the experimental measurement of the structure functions $\Gamma_U$, $\Gamma_L$, $\Gamma_F$, $\Gamma_{U^P}$, $\Gamma_{L^P}$ and $\Gamma_{F^P}$ .
Finally, for the sake of definiteness we list the Born term rate in terms of the Born term function $ B_{U+L} $. One has
$$\Gamma_0 = \frac{G_F \, m_W^2 \, |\vec{q}\,|}
{4 \sqrt{2} \, \pi \, m_t^2} \, |V_{tb}|^2 \, B_{U+L}.$$
**One-loop contribution**
=========================
The one-loop contributions to fermionic $(V-A)$ transitions have a long history. Since QED and QCD have the same structure at the one-loop level the history even dates back to QED times.
Our reference will be the work of Gounaris and Paschalis [@gounaris] (see also [@schilcher]) who used a gluon mass regulator to regularize the gluon IR singularity. The one-loop amplitudes are defined by the covariant expansion ($ J^V_\mu = \bar{q}_b \gamma_\mu q_t ,
J^A_\mu = \bar{q}_b \gamma_\mu \gamma_5 q_t $)
\[formfactor\] b(p\_b) | J\^V\_ | t(p\_t) & = & |[u]{}\_b(p\_b) { \_ F\_1\^V + p\_[t, ]{} F\_2\^V + p\_[b,]{} F\_3\^V } u\_t(p\_t),\
b(p\_b) |J\^A\_ | t(p\_t) & = & |[u]{}\_b(p\_b) { \_ F\_1\^A + p\_[t, ]{} F\_2\^A + p\_[b,]{} F\_3\^A } \_5 u\_t(p\_t).
In the Standard Model the appropriate current combination is given by $ J^V_\mu - J^A_\mu $.
We shall immediately take the limit $ m_b \rightarrow 0 $ of the one-loop expressions given in [@gounaris] (see also Appendix C)[^2]. Keeping only the finite terms and the relevant mass (M) ($ \ln y $ and $ \ln^2 y $) and infrared (IR) ($ \ln (\Lambda^2) $) singular logarithmic terms one obtains the rather simple result
F\_1\^V & = & F\_1\^A = 1 - C\_F ( 4 + (1 - x\^2) + ( ) +\
& & + 2 ( ) ( ) + 2 [\_2]{}(x\^2) ),\
F\_2\^V & = & - F\_2\^A = C\_F (+ 1 + (1 - x\^2) ),\
F\_3\^V & = & - F\_3\^A = C\_F (- 1 + (1 - x\^2) ),
where we have denoted the scaled gluon mass by $ \Lambda = m_g / m_t $. The dilog function $ \mbox{Li}_2(x) $ is defined by
$$\mbox{Li}_2(x) := - \int\limits_{0}^{x} \frac{\ln(1 - z)}{z} \, dz$$
Note that the one-loop contribution is purely real. This can be understood from an inspection of the one-loop Feynman diagram Fig. 1(b) which does not admit any nonvanishing physical two-particle cut. The fact that one has $ F^V_1 = F^A_1 $ and $ F^V_i = - F^A_i $ for $ i = 2,3 $ results from setting the $ b $-quark mass to zero. This can be seen by moving the chiral $ (1 -
\gamma_5) $ factor in the one-loop integrand numerator to the left. Because $ m_b $ is set to zero the Dirac numerator string will thus begin with $ \bar{u}_b (1 + \gamma_5) $ leading to the above pattern of relations between the loop amplitudes. We mention that the gluon mass regulator scheme can be converted to the dimensional reduction scheme by the replacement $ \log \Lambda^2 \rightarrow
1 / \epsilon - \gamma_E + \log 4 \pi \mu^2/q^2 $ where $ 2 \epsilon
= 4-N $, $ \gamma_E $ is the Euler-Mascharoni constant $ \gamma_E = 0.577
\ldots, $ and $ \mu $ is the QCD scale parameter.
**Tree graph contribution**
===========================
The tree graph contribution results from the square of the real gluon emission graphs shown in Fig.1c and 1d. Omitting again the weak coupling factor $ V_{tb} \, g / \sqrt{2} $ for the time being the corresponding hadron tensor is given by
\[Hadrontensor\] [H]{}\^ & = & - 4 \_s C\_F { - +\
& &
------------------------------------------------------------------------
+ +\
& &
------------------------------------------------------------------------
- (|[p]{}\_t p\_b) ( k\^ p\_b\^ + k\^ p\_b\^ - k p\_b g\^ - i \^ k\_ p\_[b,]{} ) + (p\_t p\_b) ( k\^ |[p]{}\_t\^ + k\^ |[p]{}\_t\^ - k |[p]{}\_t g\^ ) +\
& &
------------------------------------------------------------------------
- (k p\_b) ( p\_t\^ |[p]{}\_t\^ + p\_t\^ |[p]{}\_t\^ - p\_t |[p]{}\_t g\^ ) + (k p\_t) ( (p\_b + k)\^ |[p]{}\_t\^ + (p\_b + k)\^ |[p]{}\_t\^ + (p\_b + k) |[p]{}\_t g\^ ) +\
& &
------------------------------------------------------------------------
+ (k |[p]{}\_t) ( 2 p\_b\^ p\_b\^ - p\_b p\_b g\^ ) - i (\^ (k |[p]{}\_t) + \^ k\^ |[p]{}\_[t,]{} - \^ k\^ |[p]{}\_[t,]{} ) p\_[b,]{} p\_[t,]{} +\
& &
------------------------------------------------------------------------
+ i (\^ (p\_t |[p]{}\_t) + \^ p\^\_t |[p]{}\_[t,]{} - \^ p\^\_t |[p]{}\_[t,]{} ) k\_ p\_[b,]{} } + B\^ \_[SGF]{}
$$ \Delta_{SGF} := - 4 \pi \alpha_s \, C_F
\Big( \frac{m_b^2}{(k \!\cdot\! p_b)^2} + \frac{m_t^2}{(k \!\cdot\! p_t)^2} -
2 \frac{p_b \!\cdot\! p_t }{(k \!\cdot\! p_b)(k \!\cdot\! p_t)} \Big)$$
where $ k $ is the 4-momentum of the emitted gluon. $ \Delta_{SGF} $ is the IR-divergent [*soft gluon function*]{} and $ \bar{p}_t = p_t - m_t s_t $ as in Sec.4 [^3].
We have isolated the IR-singular part of the tree-graph contribution by splitting off a universal soft gluon factor which multiplies the lowest order Born term tensor $ B^{\mu \nu} $. This facilitates the treatment of the soft gluon singularity to be regularized by a (small) gluon mass $ m_g $. Since the soft gluon factor is universal in that it multiplies the lowest order Born contribution, the requisite soft gluon integration has to be done only once and is identical for all eight structure functions. The result for the integrated soft gluon function is given in Sec. 8. Integrating only the soft gluon function $ \Delta_{SGF} $ and neglecting the finite part in Eq.(\[Hadrontensor\]) amounts to what is called the soft gluon approximation. We emphasize that we always include the full tree-graph contribution (soft plus finite part) in our calculation. Also, we integrate over the full phase space of the gluon, and not only up to a given energy cut-off of the gluon.
We have deliberately used a calligraphic notation for the tree graph hadron tensor ${\cal H}^{\mu \nu}$ in (\[Hadrontensor\]) since ${\cal H}^{\mu \nu}$ is [*not*]{} the hadron tensor $ H^{\mu \nu} $ defined in Sec. 2. In fact, the mass dimension of ${\cal H}^{\mu \nu}$ differs from that of $H^{\mu \nu}$. To relate the two hadron tensors one has to do the appropiate phase space integration on the tree graph hadron tensor.
Next one makes use of the covariant projection operators and the covariant forms of the longitudinal and transverse polarization vectors defined in Sec. 2 to obtain the contributions to the three unpolarized and five polarized structure functions. Since we are aiming for a fully inclusive measurement regarding the $ X_b $ system the resulting expressions have to be integrated over the full two-dimensional phase space. As phase space variables we take the gluon energy $ k_0 $ and the W energy $ q_0 $ where the $ k_0 $ integration is done first. The phase space limits of the respective integrations are given by
$$\label{phasenraumgrenzen1}
k_{0,-} \le k_0 \le k_{0,+}$$
and $$\label{phasenraumgrenzen2}
m_{W} \le q_0 \le \frac{m_t^2 + m_W^2 - (m_b + m_g)^2}{2 m_t},$$ where $$\label{phasenraumgrenzen3}
k_{0,\pm} = \frac{(m_t - q_0)(M_{+}^2 - 2 q_0 m_t) \pm \sqrt{q_0^2 - m_W^2}
\sqrt{(M_{-}^2 - 2 q_0 m_t)^2 - 4 m_g^2 m_b^2}}{2 (m_t^2 + m_W^2 - 2 q_0 m_t)}$$ and $$\label{phasenraumgrenzen4}
M_{\pm}^2 := m_t^2 + m_W^2 - m_b^2 \pm m_g^2.$$
It is clear from Eq.(\[phasenraumgrenzen1\]-\[phasenraumgrenzen4\]) that the integration boundaries considerably simplify when the gluon mass is set to zero. In particular the second square root factor in the $ k_{0,\pm} $ boundary turns into a polynomial in $ q_0 $ which is an essential simplification for the second $ q_0 $ integration. This observation is at the core of our tree-level integration strategy exemplified by the partitioned form of Eq. (\[Hadrontensor\]). The soft gluon singularity has been isolated and brought into a simple form. The remaining part of the tree-graph contribution is IR finite and can be integrated without the gluon mass regulator.
The integration over the gluon energy $ k_0 $ ($ k_{0,-} \le k_0 \le k_{0,+} $) is simple and the results will not be presented here in explicit analytical form. Instead we present some representative results on the differential $ W $-boson energy distribution that result from the real gluon emission graphs Fig. 1c and 1d in graphical form in Figs. 4 and 5. Fig. 4 shows the $W$-boson energy distribution for the total rate $ d \Gamma_{U+L} / dq_0 $. The energy distribution rises sharply from the lower energy limit, where the $ W $-boson is produced at rest, then increases rapidly over the intermediate range of $ W $-boson energies and finally rises sharply again towards the end of the spectrum, where the soft gluon singularity is located. In Fig.5 we show the same distribution for the partial rate into positive helicity W-bosons $ d \Gamma_{+} / dq_0 $ $(\Gamma_+=\frac{1}{2}(\Gamma_U+\Gamma_F))$ for $ m_b=0 $ and for $ m_b \neq 0 $. As mentioned before there is no Born term contribution to $d \Gamma_{+} / dq_0 $ for $ m_b = 0 $ and thus $d \Gamma_{+} / dq_0 $ possesses no IR singularity in this limit. The absence of the IR singularity in the $ m_b = 0 $ case (dashed line) is quite apparent in Fig. 4. The distribution rises moderately fast from the lower end of the spectrum, then turns down over the intermediate range of energies and finally tends to zero at the end of the spectrum where the phase space closes. The $ m_b = 0 $ (dashed line) and $ m_b \neq 0 $ (full line) distributions lie on top of each other for most of the lower part of the spectrum. Starting at around 4.8 GeV below the upper phase space boundary the two distributions begin to diverge from each other. Whereas the $ m_b = 0 $ curve turns down and goes to zero at the end of the spectrum the $ m_b \neq 0 $ curve starts to rise again and, in fact, tends to infinity at the end of the spectrum due to its IR singular behaviour. Note the huge differences in scale of the $ d \Gamma_{U+L} / dq_0 $ and the $ d \Gamma_{+} / dq_0 $ distributions which will be reflected in big differences in the total $ \alpha_s $-corrections for the two respective rates.
The second integration over the energy of the $W$-boson is more difficult. Details can be found in Sec. 8 and in the Appendices. As it turns out the analytical $ m_b \neq 0 $ results are quite lengthy. We thus chose to present our $ m_b = 0 $ results first since they are sufficiently simple to be presented in compact form. They have been obtained by taking the $ m_b \rightarrow 0 $ limit of our $ m_b \neq 0 $ results written down in Sec. 8. For practical purposes the $ m_b = 0 $ results are sufficiently accurate for top decays since $ m_b \neq 0 $ effects are generally quite small. This is particularly true if a running $b$ quark mass at the top mass scale is used. Quantitative results on the $ \alpha_s $ $ m_b \neq 0 $ corrections are given at the end of Sec. 8 as well as in [@FGKM].
**Complete $ O(\alpha_s) $ results for $ m_b = 0 $**
====================================================
We are now in the position to put together our $ m_b = 0 $ results. We add together the Born term results from Sec. 4, the one-loop results from Sec. 5 and the $ m_b \rightarrow 0 $ limit of the integrated tree graph results according to Sec. 8. The mass and infrared singular log terms cancel among the $ O(\alpha_s) $ one-loop and tree-graph contributions as they must according to the Lee-Nauenberg theorem and one remains with a finite result. We choose to present our results in terms of scaled rate functions defined by $ \hat{\Gamma}_{i} := \Gamma_{i} / \Gamma_{0} $ ($i$ = $ U+L $, $ U^P +L^P $, $ U $, $ L $, $ F $, $ S $ , $ U^P $, $ L^P $, $ F^P $, $ S^P $, $ I^P $, $ A^P $) with $ \Gamma_{0} = \Gamma_{U+L} (\mbox{Born}) $ given by ($x=m_W/m_t$)
$$\Gamma_{0} = \Gamma_{U+L}(Born) =
\frac{G_F \, m_W^2 \, m_t}{8 \sqrt{2} \, \pi}
| V_{tb} |^2 \frac{(1 - x^2)^2 (1 + 2 x^2)}{x^2}.$$
The angular decay distribution reads
\[hatDiffRate\] & = & { (\_U + P \_p \_[U\^P]{}) (1 + \^2 ) +\
& & + (\_L + P \_p \_[L\^P]{}) \^2 +\
& & + (\_F + P \_p \_[F\^P]{}) +\
& & + \_[I\^P]{} P \_p 2 +\
& & + \_[A\^P]{} P \_p },
where $ P $ is the degree of polarization of the top quark. As mentioned before one recovers the angular decay distribution written down in [@c8] when substituting the $ m_b=0 $ Born term expressions from Table 1 in Eq.(\[hatDiffRate\]).
The various reduced rates $\hat {\Gamma}_i$ are given by
\_[U+L]{} & = & 1 + C\_F { -\
& & - (1 - x\^2) - (x) (1 - x\^2) - 4 (1 + x\^2)\
& & (1 - 2 x\^2) (x) - \_2(x\^2) },\
\_[(U+L)\^P]{} & = & + C\_F { - +\
& & + - (1 - x) - (1 + x) +\
& & - \_2(x) + \_2(-x) },
\_U & = & + C\_F { - (1 - x\^2)(19 + x\^2) + +\
& - & 2 (1 - x\^2) - 4 (5 + 7 x\^2 - 2 x\^4) (x) - 2 (1 - x)\^2\
& & (x) (1 - x) + (x) (1 + x) +\
& - & \_2(x) + \_2(-x) },\
\_[L]{} & = & + C\_F { -\
& & - (1 - x\^2) + 16 (1 + 2 x\^2) (x) - 2 (1 - x)\^2\
& & (1 - x) (x) - (x) (1 + x) +\
& - & \_2(x) - \_2(-x) },\
\_[F]{} & = & + C\_F { - 2 (1 - x)\^2(3 - 4 x) + +\
& + & (1 - x) + (1 + x) +\
& + & 8 (1 - x\^2)\^2 \_2(x) + 8 (1 + 3 x\^2 - x\^4) \_2(-x) },\
\_[S]{} & = & + C\_F { - +\
& + & (1 - x\^2) - 4 (1 - x\^2) (x) - (x) (1 - x\^2) +\
& - & \_2(x\^2) },\
\_[U\^P]{} & = & + C\_F { - -\
& & (2 + x\^2) + (1 - x) + (1 + x) +\
& + & 8 (1 - x\^2)\^2 \_2(x) - 8 ( 11 + 3 x\^2 + x\^4 ) \_2(-x) },\
\_[L\^P]{} & = & + C\_F { -(15 - 22 x + 105 x\^2 - 24 x\^3 + 4 x\^4)\
& & + - (1 - x) -\
& & (1 + x) - \_2(x) + \_2(-x) },\
\_[F\^P]{} & = & + C\_F { 2 (1 - x\^2) (4 + x\^2) - +\
& - & (1 - x\^2) - 4 (2 - 5 x\^2 - 2 x\^4) (x) - (x) (1 - x)\
& & + (x) (1 + x) +\
& - & \_2(x) + \_2(-x) },\
\_[S\^P]{} & = & + C\_F { - + +\
& + & (1 - x) + (1 + x) -\
& & \_2 (x) + \_2 (-x) },\
\_[I\^P]{} & = & + C\_F { -\
& & + (1 - x) +\
& & (1 + x) + \_2 (x) - \_2 (-x) },\
\_[A\^P]{} & = & + C\_F { -\
& & - (1 - x\^2) - (x) +\
& - & (x) (1 - x) - (x) (1 + x) +\
& - & \_2 (x) - \_2 (-x) }.
As mentioned in the Introduction the results for the total rate $ (U+L) $ agree with the analytical results given in [@c15; @c16; @c17; @c18] and in [@fischer99]. The six (mass zero) diagonal structure functions $ U,L,F $ and $ U^P,L^P,F^P $ have already been listed in [@fischer99]. They had been checked against the corresponding numerical results given in [@c19; @c20; @c21]. The results on the non-diagonal structure functions $ A^P $ and $ I^P $ are new. As concerns the unpolarized transverse structure functions explicit expressions for the two linear combinations $ T_{+} = \frac{1}{2} (U + F) $ and $ T_{-} = \frac{1}{2} (U - F) $ relevant for the interpretation of the CDF measurement [@cdf] have been given in [@FGKM].
We have also included $ O(\alpha_s) $ results on the unpolarized and polarized scalar structure functions $ \hat{\Gamma}_{S} $ and $ \hat{\Gamma}_{S^P} $. They determine the $ m_b = 0 $ unpolarized and polarized decay of the top quark into a charged Higgs ($ t \rightarrow b + H^+ $) as it occurs e.g. in the Two-Higgs-Doublet-Model ($ 2HDM $). This can be seen as follows. The scalar projection of the Standard Model (SM) left-chiral current structure $ \gamma^\mu P_L $ determines the coupling of the SM Goldstone boson, i.e. $ {q \hspace{-2mm} /}P_L \rightarrow (m_t P_R - m_b P_L) $. This would be the coupling structure of the charged Higgs in the $ 2HDM $ when the ratio of vacuum expectation values is taken to be one. It is then evident that, for $ m_b=0 $, the scalar structure functions $ \hat{\Gamma}_{S} $ and $ \hat{\Gamma}_{S^P} $ describe the decay $ t \rightarrow b + H^+$ in the $ 2HDM $ irrespective of the value of the ratio of vacuum expectation values. The unpolarized scalar structure function $ \hat{\Gamma}_{S} $ has been checked against the result of [@czarnecki]. The result on the polarized scalar structure function $ \hat{\Gamma}_{S^P} $ is new.
Before turning to the numerical evaluation of the various contributions we would like to discuss the large $ m_t $ limit of the various helicity structure functions. As expected from the statements of the Goldstone boson equivalence theorem the longitudinal and scalar contributions $ L, L^P, S $ and $ S^P $ dominate in this limit. In fact, setting $ x = 0 $ one finds
\_L & = & \_S = 1 + C\_F ( - \^2 ),\
\_[L\^P]{} & = & \_[S\^P]{} = 1 + C\_F ( - + \^2 ).
That $ \hat{\Gamma}_L = \hat{\Gamma}_S $ and $ \hat{\Gamma}_{L^P} =
\hat{\Gamma}_{S^P} $ for $ m_t \rightarrow \infty $ can be understood from the fact that the longitudinal and scalar polarisation vectors $ \epsilon^{\mu}(0) $ and $ \epsilon^{\mu}(S) $ become equal to each other in this limit since the longitudinal polarisation vector then simplifies to $ \epsilon^{\mu}(0) =
q^\mu / m_W + O (m_W/q_0) $. The same observation is also at the heart of the proof of the Goldstone boson equivalence theorem. As concerns the tree graph contribution the statement that $ \epsilon^{\mu}(0) = q^\mu / m_W + O(m_W/q_0) $ is certainly not true for all of three-body phase space as e.g. close to the phase space point where the $ W $-boson is at rest. The contribution from this phase space region to the three-body rate, however, becomes negligibly small when $ m_t \rightarrow \infty $.
We now turn to our numerical results. As numerical input values we take $
m_t = 175 \mbox{ GeV} $ and $ m_W = 80.419 \mbox{ GeV} $. The strong coupling constant is evolved from $ \alpha_s(M_Z) = 0.1175 $ to $ \alpha_s(m_t) =
0.1070 $ using two-loop running. The results are presented such that the reduced Born term rates are factored out from the reduced rates. This way of presenting the results allows one to quickly assess the size of the radiative corrections. One has
\_[U+L\^]{} & = & 1 - 0.0854,\
------------------------------------------------------------------------
\_[U\^]{} & = & 0.297 (1 - 0.0624),\
------------------------------------------------------------------------
\_[L\^]{} & = & 0.703 (1 - 0.0951),\
------------------------------------------------------------------------
\_[F\^]{} & = & - 0.297 (1 - 0.0687),\
------------------------------------------------------------------------
\_[(U+L)\^P]{} & = & 0.406 (1 - 0.1162),\
------------------------------------------------------------------------
\_[U\^P]{} & = & - 0.297 (1 - 0.0689),\
------------------------------------------------------------------------
\_[L\^P]{} & = & 0.703 (1 - 0.0962),\
------------------------------------------------------------------------
\_[F\^P]{} & = & 0.297 (1 - 0.0639),\
------------------------------------------------------------------------
\_[I\^P]{} & = & - 0.228 (1 - 0.0810),\
------------------------------------------------------------------------
\_[A\^P]{} & = & 0.228 (1 - 0.0820),\
------------------------------------------------------------------------
\_[S\^]{} & = & 0.703 (1 - 0.0895),\
------------------------------------------------------------------------
\_[S\^P]{} & = & 0.703 (1 - 0.0922).
The radiative corrections to the unpolarized and polarized rate functions are sizeable. They range from $ -6.2 \, \% $ for $ \hat{\Gamma}_U $ to $ -11.6 \, \% $ for $ \hat{\Gamma}_{(U+L)^P} $ compared to the rate correction of $ -8.5 \, \% $. The radiative corrections to the longitudinal and scalar contributions are largest. The radiative corrections all tend to go in the same direction. This is an indication that the bulk of the radiative corrections come from phase space regions close to the IR/M singular region where the radiative corrections are universal. When normalizing the rate functions to the total rate, as is appropriate for the definition of polarization observables, the size of the radiative corrections to the polarization observables is much reduced. For example, the $ O(\alpha_s) $ radiative corrections decrease the ratio $ \Gamma_L/\Gamma_{U+L} $ by $ 1.1 \, \% $ and increase the ratio $ \Gamma_U/\Gamma_{U+L} $ and the magnitude of the ratio $ \Gamma_F /\Gamma_{U+L} $ by $ 2.5 \, \% $ and $ 1.8 \, \% $, resp., relative to their Born term ratios. The relative ratio $ \Gamma_U/\Gamma_L $ is increased by $ 3.6 \, \% $. The values of the radiative corrections to the polarization observables are, however, large enough that they must be included in a meaningful comparison of future high precision data with the theoretical predictions of the Standard Model.
The combination $ (\hat{\Gamma}_U + \hat{\Gamma}_F)/2 $ determines the decay of an unpolarized top quark into a right-handed $ W $-boson. This combination vanishes at the Born term level for $ m_b=0 $ as Eqs. (39) and (41) show. Adding up the corresponding numerical values of the $ O(\alpha_s) $ contributions in Eq. (51) one finds that the right-handed $ W $-boson occurs only with $ 0.094 \, \% $ probability. The $ m_b \ne 0 $ effect in the Born term alone already amounts to $ 0.036 \, \% $ (see Table 1).
Altogether the $ O(\alpha_s) $ and the Born term $ m_b \neq 0 $ corrections to the transverse-plus rate occur only at the sub-percent level. It is safe to say that, if top quark decays reveal a violation of the Standard Model (SM) $(V-A)$ current structure that exceeds the 1% level, the violations must have a non-SM origin. In this context it is interesting to note that a possible $ (V+A) $ admixture to the SM $ t \rightarrow b $ current is already severely bounded indirectly to below 5% by existing data on $ b \rightarrow s + \gamma $ decays [@fy94; @cm94; @gp00].
The rate combination $ (\hat{\Gamma}_U + \hat{\Gamma}_F)/2 $ is in fact not the only combination that vanishes at the Born term level for $ m_b=0 $. Considering the fact that one must have $ \lambda_W - \lambda_t = -1/2 $ at the Born term level the only surviving Born term level rate expressions are $ \hat{\Gamma}_{--}^{--}$, $ \hat{\Gamma}_{\mbox{oo}}^{++} $ and $ \hat{\Gamma}_{-\mbox{o}}^{-+}$ as alluded to before in Sec. 4. The notation employed for the reduced rates follows the notation used in Eq. (\[system3anfang\]). The remaining rate expressions vanish at the Born term level but become populated at $ O(\alpha_s) $. They are
\[small-number\] \_[++]{}\^[++]{} & = & (\_[U]{} + \_[F]{} + \_[U\^P]{} + \_[F\^P]{}) = 0.000 833,\
\_\^[–]{} & = & (\_[L]{} - \_[L\^P]{}) = 0.000 389,\
\_[+]{}\^[+-]{} & = & (\_[I\^P]{} + \_[A\^P]{}) =- 0.000 236,\
\_[++]{}\^[–]{} & = & (\_[U]{} + \_[F]{} - \_[U\^P]{} - \_[F\^P]{}) = 0.000 093,\
\_[–]{}\^[++]{} & = & (\_[U]{} - \_[F]{} + \_[U\^P]{} - \_[F\^P]{}) = 0.000 120 .
As remarked on before the latter two reduced rates $ \hat{\Gamma}_{++}^{--} $ and $ \hat{\Gamma}_{--}^{++} $ vanish at the Born term level even for $ m_b \neq 0 $ since the net helicity of these transitions $ |\lambda_W - \lambda_t| = 3/2 $ exceeds that of the $ b $ quark $ |\lambda_b| = 1/2 $.
The four reduced rates $ \hat{\Gamma}_{++}^{++} $, $ \hat{\Gamma}_{\mbox{oo}}^{--} $, $ \hat{\Gamma}_{++}^{--} $ and $ \hat{\Gamma}_{--}^{++} $ are positive definite quantities since they result from squares of helicity amplitudes. Contrary to these $ \hat{\Gamma}_{+\mbox{o}}^{+-} $ is an interference contribution and thus can be negative as it in fact is. In Eq. (\[small-number\]) we have also included the numerical values for the above five structure function combinations resulting from the (tree graph) $ \alpha_s $ corrections. They are all very small at the sub per mille level.
In Sec.4 (Fig.3) we have shown a lego plot of the Born term two-fold angular decay distribution in $ \cos \theta $ and $ \cos \theta_P $. In order to be able to exhibit the size of the $ \alpha_s $ corrections we show in Fig.6 a contour plot of the same two-fold angular decay distribution with and without radiative corrections, again setting $ P = 1 $. The radiative corrections are not very large in the upper two quadrants and become largest in the lower left quadrant of the contour plot when both $ \cos \theta $ and $ \cos \theta_P $ tend to one.
Instead of analyzing the three-fold or two-fold angular decay distributions one can also consider single angle decay distributions. They are obtained by integrating over the two respective complementary decay angles. For the $ \cos \theta $ distribution one obtains
$$\frac{d \widehat{\Gamma}}{d \cos \theta} =
\frac{3}{8} (\widehat{\Gamma}_U +2 \widehat{\Gamma}_L)
(1 + \alpha_{\theta} \cos \theta + \beta_{\theta} \cos^2 \theta) ,$$
where
\_ & = & 2 ( = - = -0.349 ),\
\_ & = & ( = - = -0.651 ).
We have added the analytical and numerical Born term results for the asymmetry parameters in brackets using $ x^2=0.211 $. The $ O(\alpha_s) $ values for the asymmetry parameters are $ \alpha_\theta = -0.357 $ and $ \beta_\theta = -0.641 $, i.e. the $ \alpha_s $ corrections raise the magnitude of $ \alpha_\theta $ by $ 2.3 \,\% $ and lower the magnitude of $ \beta_\theta $ by $ 1.5 \,\% $. In Fig.7 we show the $ \cos \theta $ distribution both for the Born term case and the radiatively corrected case. There is a pronounced forward-backward asymmetry. In the forward direction the differential Born term rate drops to zero. As discussed before the $ O(\alpha_s)$ rate does not vanish in the forward direction due to real gluon emission. However, the radiative corrections are so small that the nonvanishing of the $ O(\alpha_s)$ rate in the forward direction cannot be discerned at the scale of the plot. In absolute terms the radiative corrections are largest for $ \cos \theta \approx 0 $ because of the large size of the radiative corrections to the longitudinal rate $ \hat{\Gamma}_{L} $. Note that $ \alpha_\theta $ is not the conventional forward-backward asymmetry parameter which is defined by
$$\alpha_{FB} = \frac
{d \Gamma (0 \ge \theta \ge \frac{\pi}{2}) -
d \Gamma (\frac{\pi}{2} \ge \theta \ge \pi)}
{d \Gamma (0 \ge \theta \ge \frac{\pi}{2}) +
d \Gamma (\frac{\pi}{2} \ge \theta \ge \pi)} =
\frac{3}{4} \frac{\widehat{\Gamma}_F}{\widehat{\Gamma}_{U+L}} \quad
\Bigg( = - \frac{3}{2} \frac{x^2}{1 + 2 x^2} = - 0.223 \Bigg).$$
The $ \alpha_s $ corrections raise $ \alpha_{FB} $ by $ 1.7\,\% $ in magnitude.
For the $ \cos \theta_P $ distribution one obtains
$$\frac{d \widehat{\Gamma}}{d \cos \theta_p} =
\frac{1}{2} (\widehat{\Gamma}_{U+L})
(1 + P \alpha_{\theta_p} \cos \theta_p),$$
where
$$\alpha_{\theta_p} = \frac{\widehat{\Gamma}_{(U+L)^P}}
{\widehat{\Gamma}_{U+L}} \qquad
\Bigg(= \frac{1 - 2 x^2}{1 + 2 x^2} = 0.406 \Bigg).$$
The $ \alpha_s $-corrections lower $ \alpha_{\theta_P} $ by $ 3.4\,\% $.
Finally, the $ \phi $ distribution reads
$$\frac{d \widehat{\Gamma}}{d \phi} =
\frac{1}{2 \pi} (1 +P \gamma_{\phi} \cos \phi),$$
where
$$\gamma_{\phi} = \frac{3 \pi^2}{8 \, \sqrt{2}}
\frac{\widehat{\Gamma}_{A^P}}{\widehat{\Gamma}_{U+L}} \qquad
\Bigg( = \frac{3 \pi^2}{16} \frac{x}{1 +2 x^2} = 0.597 \Bigg).$$
The $ \cos \phi $ dependent contribution from $ \hat{\Gamma}_{I_P} $ has dropped out because of having integrated over the full range of $ \cos \theta $. If desired the contribution of $ \hat{\Gamma}_{I_P} $ to the $ \phi $ distribution can be retained if one integrates only over half the range of $ \cos\theta $. The $ \alpha_s $-corrections raise $ \gamma_\phi $ by the small amount of $ 0.32\,\% $. In Fig.8 we show the $ \phi $ distribution both for the Born term case and the radiatively corrected case setting $ P=1 $.
**Complete $ O(\alpha_s) $ results for $ m_b \neq 0 $**
=======================================================
Differing from the presentation of our $ m_b =0 $ results in Sec. 7 we shall present our $ m_b \ne 0 $ results in a form where each of the separate contributions to the rate remains identified. In particular we do not explicitly cancel the IR terms coming from the one-loop and tree graph contributions. We thus write
\[complete\] \_[i]{}\^[QCD]{} & = & \_i() + { \_ \_[i, ]{} F\_ } + \_i() S() -\
& & { \_[n=-1,0]{} \_[(n), i]{} [R]{}\_[(n)]{} + \_[m,n]{} \_[(m,n), i]{} [R]{}\_[(m,n)]{} + \_[n=0,1]{} \_[(n), i]{} [S]{}\_[(n)]{} + \_[m,n]{} \_[(m,n), i]{} [S]{}\_[(m,n)]{} }
The first term in Eq. (\[complete\]) represents the Born term contribution which is given by
$$\label{complete1}
\Gamma_{i}(\mbox{Born}) =
\frac{G_F \, m_W^2 \, |V_{tb}|^2}
{8 \, \sqrt{2} \, \pi \, m_t}
\, \sqrt{\lambda} \, B_i$$
where the $ B_i $ are the Born term rates listed in Table 1. The Born term contribution $ \Gamma_{i}(\mbox{Born}) $ also appears as a factor in the third term where it multiplies the soft gluon factor $S(\Lambda)$. The index $i$ runs over the various structure function labels $ i = U+L $, $ U^P+L^P $, $ U $, $ U^P $, $ L $, $ L^P $, $ F $, $ F^P $, $ S $, $ S^P $, $ I^P $ and $ A^P $.
The second term in Eq. (\[complete\]) represents the one-loop contribution which is obtained by folding the one-loop amplitude in Appendix C with the Born term amplitude and then doing the appropriate projection onto the various structure functions. The appropriate coefficient functions $ \kappa_{i,\tau} $ are listed in Table 2. The coefficient functions $ \kappa_{i,\tau} $ multiply the $ \alpha_s $ one-loop amplitudes $ F_{\tau} = F_1^V, F_2^V, F_3^V,
F_1^A, F_2^A, F_3^A $ which are listed in Appendix C. We label the one-loop amplitudes consecutively by the index $ \tau = 1,\ldots ,6 $. Note that Table 2 contains only the vector current coefficient functions $ \kappa_{i,\tau} \, (\tau=1,2,3) $. The axial vector coefficient functions labelled by $ \tau = 4,5,6 $ can be easily obtained from the vector current coefficient functions by the substitution
$$\label{complete2}
\kappa_{F_1^A} = \kappa_{F_1^V} \Big|_{y \rightarrow -y}, \quad
\kappa_{F_2^A} = - \kappa_{F_2^V} \Big|_{y \rightarrow -y}, \quad
\kappa_{F_3^A} = - \kappa_{F_3^V} \Big|_{y \rightarrow -y} .$$
[|l|c|c|c|]{} i & $ \kappa_{F_1^V},i $ & $ \kappa_{F_2^V},i $ & $ \kappa_{F_3^V},i $\
$ U \!+\! L $ & $ {\scriptstyle}{\sqrt{\lambda} \, ((1 - y)^2 - x^2) ((1 + y)^2 + 2 x^2)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \sqrt{\lambda^3} \, (1 + y)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \sqrt{\lambda^3} \, (1 + y)} $\
$ U^P \!+\! L^P $ & $ {\scriptstyle}{\lambda \, (1 - 2 x^2 - y^2)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \lambda \, ((1 + y)^2 - x^2) (1 - y)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \lambda \, ((1 + y)^2 - x^2) (1 - y)} $\
$ U $ & $ {\scriptstyle}{2 \, \sqrt{\lambda} \, ((1 - y)^2 - x^2) \, x^2} $ & $ 0 $ & $ 0 $\
$ U^P $ & $ - 2 \, x^2 \, \lambda $ & $ 0 $ & $ 0 $\
$ L $ & $ {\scriptstyle}{\sqrt{\lambda} \, ((1 - y)^2 - x^2) (1 + y)^2} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \sqrt{\lambda^3} \, (1 + y)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \sqrt{\lambda^3} \, (1 + y)} $\
$ L^P $ & $ {\scriptstyle}{\lambda \, (1 - y^2)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \lambda \, ((1 + y)^2 - x^2)^2 (1 - y)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \lambda \, ((1 + y)^2 - x^2)^2 (1 - y)} $\
$ F $ & $ {\scriptstyle}{- 2 \, \lambda \, x^2} $ & $ 0 $ & $ 0 $\
$ F^P $ & $ {\scriptstyle}{2 \, \sqrt{\lambda} \, ((1 - y)^2 - x^2) \, x^2} $ & $ 0 $ & $ 0 $\
$ S $ & $ {\scriptstyle}{\sqrt{\lambda} \, ((1 + y)^2 - x^2) (1 - y)^2} $ & $ \begin{array}{c}
{\scriptstyle}{\frac{1}{2} \, m_t \, \sqrt{\lambda} \, ((1 + y)^2 - x^2) \times} \\
{\scriptstyle}{(1 + x^2 - y^2) (1 - y)}
\end{array} $ & $ \begin{array}{c}
{\scriptstyle}{\frac{1}{2} \, m_t \, \sqrt{\lambda} \, ((1 + y)^2 - x^2) \times} \\
{\scriptstyle}{(1 - x^2 - y^2) (1 - y)}
\end{array} $\
$ S^P $ & $ {\scriptstyle}{\lambda \, (1 - y^2)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \lambda \, (1 + x^2 - y^2) (1 + y)} $ & $ {\scriptstyle}{\frac{1}{2} \, m_t \, \lambda \, (1 - x^2 - y^2) (1 + y)} $\
$ I^P $ & $ {\scriptstyle}{- \frac{1}{\sqrt{2}} \, \lambda \, x} $ & $ {\scriptstyle}{- \frac{1}{4 \, \sqrt{2}} \, m_t \, \lambda \,
((1 + y)^2 - x^2) \, x} $ & $ {\scriptstyle}{- \frac{1}{4 \, \sqrt{2}} \, m_t \, \lambda \,
((1 + y)^2 - x^2) \, x} \rule[-2.5mm]{0mm}{3mm} $\
$ A^P $ & $ {\scriptstyle}{\frac{1}{\sqrt{2}} \, \sqrt{\lambda} \,
((1 - y)^2 - x^2) (1 + y) \, x} \rule[-2.5mm]{0mm}{3mm} $ & $ {\scriptstyle}{\frac{1}{4 \, \sqrt{2}} \, m_t \, \sqrt{\lambda^3} \, x} $ & $ {\scriptstyle}{\frac{1}{4 \, \sqrt{2}} \, m_t \, \sqrt{\lambda^3} \, x} $\
The third term in Eq. (\[complete\]) contains the result of integrating the soft gluon function $ \Delta_{SGF} $ in Eq. (\[Hadrontensor\]). The result depends on the (small) IR regularisation parameter $ \Lambda = m_g/m_t $ as indicated in the argument of the soft gluon factor $ S(\Lambda) $. The universal soft gluon factor $ S(\Lambda) $ is obtained by explicit integration and reads
\[complete3\] S() & = & - C\_F { (1 - x\^2 + y\^2) + 2 { ( ) - 2 } +\
& + & ( ) - 2 y\^2 (w\_1) },
where as in [@c14] we have used the abbreviations
$$ \label{w1wmu}
w_1 = \frac{x}{y} \cdot \frac{1 - x^2 + y^2 - \sqrt{\lambda}}
{1 + x^2 - y^2 + \sqrt{\lambda}}, \qquad
w_{\mu} = \frac{x}{y} \cdot \frac{1 - x^2 + y^2 - \sqrt{\lambda}}
{1 + x^2 - y^2 - \sqrt{\lambda}}.$$
In the limit $ y \rightarrow 0 $ one has
S() & = & - C\_F { (1 - x\^2) + x\^2 }.
In agreement with the Lee-Nauenberg theorem the logarithmic dependence on the IR regularisation parameter $ \Lambda $ can be seen to cancel between the loop and the soft gluon contributions for each of the ten structure functions.
The fourth term in Eq. (\[complete\]) finally contains the result of integrating the finite piece in the tree graph contribution Eq. (\[Hadrontensor\]), again after having done the appropriate projections. The result is given in terms of a set of standard integrals $ {\cal R}_{(n)} $, $ {\cal R}_{(m,n)} $, $ {\cal S}_{(n)} $ and $ {\cal S}_{(m,n)} $ which are listed in Appendix A. Appendix B gives the values of the coefficient functions $ \rho_{(n), \, i} $, $ \rho_{(m,n), \, i} $, $ \sigma_{(n), \, i} $ and $ \sigma_{(m,n), \, i} $ that multiply the standard set of integrals in the various helicity structure functions. In Table 3 we have listed the range of values of the parameters $ m $ and $ n $ that characterize the different types of tree graph integrals.
At this point it is perhaps appropiate to offer an excuse to the potential user of our $ m_b \ne 0 $ results that our results are presented in a multiply nested form to be collected from Eqs.(\[complete\], \[complete1\], \[complete2\], \[complete3\], \[w1wmu\]), Table 2 and Appendices A, B and C. Contrary to the $ m_b=0 $ results where a closed form representation was possible a presentation of unnested closed form expressions for $ m_b \ne 0 $ would require an extraordinary amount of space because of the presence of many different log and dilog functions and products thereof. Codes of the relevant expressions can be obtained from the authors on request.
When we have evaluated Eq. (\[complete\]) numerically the IR factors proportional to $\ln \Lambda $ in the one-loop and tree graph contributions were set to zero by hand. The numerical evaluation of the remaining part is quite stable numerically. In particular the limit $ m_b \rightarrow 0 $ is numerically quite smooth. This is demonstrated in Fig. 9 where we plot the bottom mass dependence of the total rate. Note that the $ O(\alpha_s) $ rate shows less dependence on the bottom mass than the Born term rate.
[|l|c|c|c|c|]{} i & $ \rho^{1}_{(n)},i $ & $ \rho^{0}_{(m,n)},i $ & $ \sigma^{1}_{(n)},i $ & $ \sigma^{0}_{(m,n)},i $
------------------------------------------------------------------------
\
$ U \!+\! L $ & $ - $ & $ (-2,-1)...(0,-1) $ & $ - $ & $ (0,0),(1,0) $\
$ U^P \!+\! L^P $ & $ -1,0 $ & $ (-2,0)...(1,0) $ & $ 0,1 $ & $ (0,0),(0,1)...(2,1) $\
$ U $ & $ - $ & $ (-2,1)...(2,1) $ & $ - $ & $ (0,2)...(3,2) $\
$ U^P $ & $ -1,0 $ & $ (-2,2)...(3,2) $ & $ 0,1 $ & $ (0,0),(0,3)...(4,3) $\
$ L $ & $ - $ & $ (-2,1)...(2,1) $ & $ - $ & $ (0,2)...(3,2) $\
$ L^P $ & $ -1,0 $ & $ (-2,2)...(3,2) $ & $ 0,1 $ & $ (0,0),(0,3)...(4,3) $\
$ F $ & $ -1,0 $ & $ (-2,0)...(1,0) $ & $ 0,1 $ & $ (0,0),(0,1)...(2,1) $\
$ F^P $ & $ - $ & $ (-2,1)...(2,1) $ & $ - $ & $ (0,2)...(3,2) $\
$ S $ & $ - $ & $ (-2,-1)...(0,-1) $ & $ - $ & $ (0,0),(1,0) $\
$ S^P $ & $ -1,0 $ & $ (-2,0)...(1,0) $ & $ 0,1 $ & $ (0,0),(0,1)...(2,1) $\
$ I^P $ & $ -1,0 $ & $ (-2,2)...(2,2) $ & $ 0,1 $ & $ (0,0),(0,3)...(3,3) $\
$ A^P $ & $ - $ & $ (-2,1)...(1,1) $ & $ - $ & $ (0,2)...(2,2) $\
The quality of the $ m_b = 0 $ approximation has been discussed before at the Born level. For example, at the Born term level the total rate is decreased by $ 0.27 \, \% $ when going from $ m_b = 0 $ to $ m_b = 4.8 \mbox{ GeV} $. Using the $ O(\alpha_s) $ $ m_b \ne 0 $ results from this section one finds that the $ m_b \ne 0 $ corrections to the total $ O(\alpha_s) $ rate reduce the rate by $ 0.16 \, \% $ compared to the Born term reduction of $ 0.27 \, \% $, i.e. the $ m_b \ne 0 $ corrections to the $ \alpha_s $-contribution alone tend to counteract the $ m_b \ne 0 $ effect in the Born term in the total rate (see also Fig. 9). The $ m_b \ne 0 $ corrections from the $ \alpha_s $-contributions alone are surprisingly large considering the fact that the factor multiplying the $ \alpha_s $-corrections $ C_F \, \alpha_s/(2 \pi) = 0.023 $ is a rather small number. This can be understood in part by noting that the $ \alpha_s $-contributions contain terms proportional to $ (m_b^2/m_W^2)
\ln(m_b^2/m_t^2) = -0.026 $ which is not a very small number. A further discussion of $ m_b \ne 0 $ effects for the $ \alpha_s $-contributions can be found in [@FGKM]. Noteworthy is a large 20% correction to the $O(\alpha_s)$ transverse-plus rate $ \hat{\Gamma}_+ $ due to $m_b$ effects [@FGKM]. That the bottom quark mass effect is so large in $ \hat{\Gamma}_+ $ can be appreciated in part by looking at the differential distribution in Fig. 5. We emphasize again that the mass effect may have been overestimated due to using a fixed pole mass, rather than a running mass which is smaller at the top mass scale.
**Summary and conclusion**
===========================
We have obtained analytical expressions for the $ O(\alpha_s) $ radiative corrections to the three unpolarized and five polarized structure functions that govern the decay of a polarized top quark. Although bottom quark mass effects are quite small in top quark decays we have retained the full bottom mass dependence in our calculation. In the limit $ m_b \rightarrow 0 $ the analytical results considerably simplify leading to compact expressions for the eight structure functions which are listed in the main text. The full mass dependence of our analytical results is written down in Sec. 8 and in the Appendices A and B. These finite mass results will prove useful for the theoretical description of $ b \rightarrow c $ bottom meson and bottom baryon decays (see e.g. [@fischer2000]).
For top quark decays the radiative corrections to the structure functions range from $ -6.2 \, \% $ to $ -11.6 \, \% $ where the radiative corrections to the unpolarized longitudinal structure functions $ \hat{\Gamma}_L $ and the polarized structure function $ \hat{\Gamma}_{(U+L)^P} $ are largest. These corrections are to be compared with the correction to the total rate which is $ -8.5 \, \% $. The radiative corrections to the structure functions all go in the same directions indicating that the bulk of the radiative corrections derive from contributions close to the IR/M region of phase space where the radiative corrections are universal. Nevertheless the span of values of the radiative corrections exceeds $ 5 \, \% $ and must be taken into account in a future comparison with precision experiments. The radiative corrections to rate combinations that vanish at the Born term level have been found to be rather small. In particular, the $\alpha_s$-correction to the normalized rate of an unpolarized top into positive helicity $W$-bosons amounts to only 0.1%. As discussed in Sec. 7, the minuteness of the $\alpha_s$-contribution to positive helicity $W$-bosons is of relevance when discussing a possible $(V+A)$-admixture to the Standard Model current.
We have also determined the $ O(\alpha_s) $ corrections to unpolarized and polarized $ q_1 \rightarrow q_2 $ scalar current transitions. For $ t
\rightarrow b $ transitions these scalar current transitions are relevant for top quark decays into a bottom quark and a charged Higgs as they occur in the two-Higgs doublet model. For $ b \rightarrow c $ transitions these transition matrix elements are needed e.g. for the description of the semi-inclusive decays of the $ B $-mesons and the $ \Lambda_b $ into spin-zero $ D_s $ mesons [@fischer2000; @aleksan].
In this paper we have only studied the first order QCD corrections to the structure functions in polarized top decays. For the total rate one obtains a correction of $ -8.5 \, \% $. Second order QCD corrections to the rate are expected to amount to $ -2.6 \, \% $ [@CHSS] while electroweak corrections are known to increase the rate by $ +1.7 \, \% $ [@c14; @EMMS]. For a high precision comparison of theory and experiment of the structure functions it would therefore be desirable to calculate the two-loop $ O(\alpha_s^2) $ and the electroweak one-loop corrections to the eight structure functions. While the two-loop QCD corrections to the structure functions are very difficult and are therefore not likely to be done in the next few years the calculation of the one-loop electroweak corrections to the eight structure functions is presently under way [@dgkm]. Finite width corrections will also have to be accounted for. They lower the total width by 1.56% [@dgkm; @jk93] and affect the different partial helicity rates by differing amounts [@dgkm].
We would like to conclude this paper with a speculative note concerning a possible top quark mass measurement from an angular decay analysis using the fact that the structure functions are top mass dependent. This suggestion is much in the spirit of the suggestion of Grunberg [*et al.*]{} who advocated a similar measurement of heavy quark masses in the context of $ e^{+} e^{-} $-annihilations [@grunberg]. Assume that the percentage measurement errors on a $ L/(U+L) $ and $ U/(U+L) $ measurement are $ \delta_L $ and $ \delta_U $, respectively. The percentage error on the mass measurement will be denoted by $ \delta $, i.e. we write $ m_t = \bar{m}_t(1 \pm \delta) $ where $ \bar{m}_t $ is some given central value of the top mass. From the dependence of the respective Born term ratios on the mass ratio $ x = m_W/m_t $ (assuming that the $ W $-mass is fixed) one finds that the percentage error on the top mass measurement is given by $ \delta = \delta_L (1+2 x_0^2)/(4x_0^2) $ and by $ \delta = \delta_U (1+2 x_0^2)/2 $, resp., where we write $ x^2 = x_0^2 ( 1 \mp 2 \delta) $ with $ x_0 = m_W / \bar{m}_t $. If we take $ m_t = 175 \mbox{ GeV} $ as central value ($ x_0^2 = 0.211 $) this would imply that an 1% error on the angular structure function measurement would allow one to determine the top quark mass with $ 1.7 \, \% $ and $ 0.7 \, \% $ accuracy, depending on whether the angular measurement was done on the longitudinal $ (L) $ or on the unpolarized-transverse $ (U) $ (or for that matter $ (F) $) mode. Since the radiative corrections change the ratios $ \Gamma_L/\Gamma_{U+L} $ and $ \Gamma_U/\Gamma_{U+L} $ by $ 1.1 \, \% $ and $ 2.4 \, \% $, respectively, it is clear that one has to use the full $ O(\alpha_s) $ results for the angular structure functions if such experimental accuracies can be reached. This is illustrated in Figs. 10 and 11 where we plot the top mass dependence of $ \Gamma_L/\Gamma_{U+L} $ and $ \Gamma_U/\Gamma_{U+L} $ for the Born term case and the $ O(\alpha_s) $ case for $ m_b = 0 $. Note that the $ O(\alpha_s) $ curves are horizontally displaced from the Born term curves by approximately 3 and 3.4 GeV, resp., meaning that one would make the correponding mistakes in the top mass determination from a measurement of the angular structure functions if the Born curves were used instead of the radiatively corrected ones. The present TEVATRON RUN I uncertainties on the top mass are around $ 4 \, \% $ which is anticipated to be improved to $ 1.7 \, \% $ during the initial stages of TEVATRON RUN II. It remains to be seen whether a mass determination based on angular measurements as proposed here can compete with the conventional method using invariant mass reconstruction.
[**Acknowledgements:**]{} M. Fischer and M.C. Mauser were partly supported by the DFG (Germany) through the Graduiertenkolleg “Teilchenphysik bei hohen und mittleren Energien” and its successor “Eichtheorien” at the University of Mainz. M.C. Mauser was also supported by the BMBF (Germany) under contract 05HT9UMB/4. S. Groote and J.G. Körner acknowledge partial support by the BMBF (Germany) under contract 06MZ865 and S.G. acknowledges support by the DFG. We would like to thank B. Lampe for initial participation in the project, H. Spiesberger for illuminating discussions and A. Arbuzov for checking some of the formula.
Integrals
=========
In this Appendix we catalogue the basic set of tree graph integrals that are needed in our $ m_b \neq 0 $ calculation and give their analytical results.
Basic integrals
---------------
In the first step of the tree graph integration one integrates over the gluon energy $ k_0 $. After having done the integration on the gluon energy it proves to be convenient to perform a shift in the $ W $-energy $ q_0 $ integration variable by introducing the variable $ z = 1 + x^2 - 2 q_0 / m_t $. One then encounters the following set of integrals
[R]{}\_[(m,n)]{} & := & \_[y\^2 + \^\_[2]{}]{}\^[(1 - x)\^2 - \^\_[1]{}]{} , \_[(n)]{} := \_[y\^2 + \^\_[2]{}]{}\^[(1 - x)\^2 - \^\_[1]{}]{} ,\
[S]{}\_[(m,n)]{} & := & \_[y\^2 + \^\_[2]{}]{}\^[(1 - x)\^2 - \^\_[1]{}]{} ( ),\
[S]{}\_[(n) ]{} & := & \_[y\^2 + \^\_[2]{}]{}\^[(1 - x)\^2 - \^\_[1]{}]{} ( ).
where $\lambda(1,x^2,z)= 1 + x^4 + z^2 - 2 x^2 z - 2 x^2 - 2 z $. The required range of values of the parameters $ m $ and $ n $ are listed in Table 3. The cut-off parameters $ \epsilon^{\prime}_{1} $ and $ \epsilon^{\prime}_{2} $ are needed to account for the spurious singularities which are artificially introduced by partial fractioning the integrands. The spurious singularities cancel as they must when all contributions to a particular helicity structure function are summed.
In order to get rid of the square roots the final substitution $ z =: 1 \!+\!
x^2 \!-\! x (r + 1)/r $ is introduced. The variable $ r $ has to be integrated in the interval $ [1 \!+\! \epsilon_1, \eta \!-\! \epsilon_2] $, where
$$\eta = (1 + x^2 - y^2 + \sqrt{\lambda})/2x
\qquad \mbox{and} \qquad \lambda = \lambda(1,x^2,y^2)$$
as before. The spurious cut-off parameters $ \epsilon_{1} $ and $ \epsilon_{2} $ replace the above cut-off parameters $ \epsilon^{\prime}_{1} $ and $ \epsilon^{\prime}_{2} $ and cancel in all final expressions.
In order to keep our results at a manageable length we introduce the following set of auxiliary functions
[L]{}\_1 & := & ( ), \_2 := ( ),\
[L]{}\_3 & := & ( ), \_4 := ( ),\
[L]{}\_5 & := & ( ), \_6 := ( ),\
[N]{}\_0 & := & [\_2]{}(x) + [\_2]{}( ) - 2 [\_2]{}(x), \_1 := [\_2]{}(x) - [\_2]{}( )\
[N]{}\_2 & := & - () (1+x) + ( ) ( ) +\
& - & [\_2]{}( ) + [\_2]{}( ) + [\_2]{}( ),\
[N]{}\_3 & := & - () (1-x) - ( ) ( ) +\
& - & [\_2]{}( - ) + [\_2]{}( ) + [\_2]{}( - ),
and
$$\beta_{+}(n) := (x-1)^n + (x+1)^n, \qquad
\beta_{-}(n) := (x-1)^n - (x+1)^n,$$
$$\beta(n) := \frac{(x-1)^n}{\eta-1} - \frac{(1+x)^n}{\eta+1}.$$
In the following we list our analytical results for the various types of integrals that are needed in our calculation.
Integrals of type R $ \!_{{\scriptstyle}(m,n)} $
------------------------------------------------
[R]{}\_[(-2,-1)]{} & = & + - ,\
[R]{}\_[(-1,-1)]{} & = & - \^[1/2]{} - (1+x\^2) + (1-x\^2) ,\
[R]{}\_[(0,-1) ]{} & = & (1+x\^2-y\^2) \^[1/2]{} - x\^2 ([L]{}\_2-[L]{}\_1),\
[R]{}\_[(1,-1) ]{} & = & - \^[3/2]{} + (1+x\^2) ( (1+x\^2-y\^2) \^[1/2]{} - x\^2 ([L]{}\_2-[L]{}\_1) ),\
[R]{}\_[(-2,0) ]{} & = & - , \_[(-1,0)]{}\^[0]{} =2 [L]{}\_5,\
[R]{}\_[(0,0) ]{} & = & (1-x)\^2 - y\^2, \_[(1,0)]{} = - , \_[(2,0)]{} = - ,\
[R]{}\_[(-2,1) ]{} & = & ( + ), \_[(-1,1)]{} = ,\
[R]{}\_[(0,1) ]{} & = & , \_[(1,1)]{} = - \^[1/2]{} + (1+x\^2) ,\
[R]{}\_[(2,1) ]{} & = & - y\^2 \^[1/2]{} - (1+x\^2) \^[1/2]{} + (1+4x\^2+x\^4) ,\
[R]{}\_[(3,1) ]{} & = & - \^[3/2]{} + (1+x\^2)(1+x\^2-y\^2) \^[1/2]{} - (3+x\^2)(1+3x\^2) \^[1/2]{} +\
& + & (1+x\^2)(1+8x\^2+x\^4) ,\
[R]{}\_[(-2, 2) ]{} & = & - + ( - ),\
[R]{}\_[[(-1, 2) ]{}]{} & = & - , \_[(0, 2)]{} = ([L]{}\_3-[L]{}\_4),\
[R]{}\_[(1, 2) ]{} & = & \_3 - \_4 - \_[-]{}(2) [L]{}\_5,\
[R]{}\_[(2, 2) ]{} & = & \_3 - \_4 - \_[-]{}(4) [L]{}\_5 + ( (1-x)\^2 - y\^2 ),\
[R]{}\_[(3, 2) ]{} & = & \_3 - \_4 - \_[-]{}(6) [L]{}\_5 + 3 ( (1-x)\^2 - y\^2 ) (1 + x\^2) - ,\
[R]{}\_[(4, 2) ]{} & = & \_3 - \_4 - \_[-]{}(8) [L]{}\_5 + ( (1-x\^2) - y\^2 )\
& & ( (1+x+x\^2-y\^2)\^2 + (6+17x\^2+6x\^4) )- 2(1+x\^2) .
Integrals of type R $ \!_{{\scriptstyle}(n)} $
----------------------------------------------
[R]{}\_[(-1)]{} & = & - \^[1/2]{} - (1+x\^2-y\^2) + \^[1/2]{} ( ),\
[R]{}\_[(0) ]{} & = & ( ) + ( ),
Integrals of type S $ \!_{{\scriptstyle}(m,n)} $
------------------------------------------------
[S]{}\_[(0,0)]{} & = & \^[1/2]{} - x\^2 ([L]{}\_2-[L]{}\_1) - y\^2 [L]{}\_1,\
[S]{}\_[(1,0)]{} & = & (1+5x\^2+y\^2) \^[1/2]{} - (2+x\^2) x\^2 - y\^4 ,\
[S]{}\_[(0,1)]{} & = & [N]{}\_0, \_[(1,1)]{} = (1+x\^2) [N]{}\_0 - \^[1/2]{} [L]{}\_1 + 2 (1-x\^2) [L]{}\_5 - ( (1-x)\^2-y\^2 ),\
[S]{}\_[(2,1)]{} & = & (1+4x\^2+x\^4) [N]{}\_0 - (3+3x\^2+y\^2) \^[1/2]{} [L]{}\_1 + 3(1-x\^4) [L]{}\_5 +\
& - & ( (1-x)\^2 - y\^2 ) ( (1-x)\^2+4+8x\^2+y\^2 ),\
[S]{}\_[(0,2)]{} & = & - ([N]{}\_2-[N]{}\_3),\
[S]{}\_[(1,2)]{} & = & - \_2 + \_3 + [N]{}\_1,\
[S]{}\_[(2,2)]{} & = & - \_2 + \_3 + 2(1+x\^2) [N]{}\_1 + \^[1/2]{} - x\^2 ([L]{}\_2-[L]{}\_1) - y\^2 [L]{}\_1,\
[S]{}\_[(3,2)]{} & = & - \_2 + \_3 + (3+x\^2)(1+3x\^2) [N]{}\_1 +\
& + & (9+13x\^2+y\^2) \^[1/2]{} - (6+5x\^2)x\^2 - y\^2 ( 4(1+x\^2)+y\^2 ) ,\
[S]{}\_[(0,3)]{} & = & { + \_3 - \_4 - \_1 + \_6 },\
[S]{}\_[(1,3)]{} & = & { 2(1-x) + (1-x) [L]{}\_3 - (1+x) [L]{}\_4 - \_1 + \_6 },\
[S]{}\_[(2,3)]{} & = & { 2(1-x)\^3 + (1-x)\^3 [L]{}\_3 - (1+x)\^3 [L]{}\_4 - \_1 + \_6 + 4x [N]{}\_0 },\
[S]{}\_[(3,3)]{} & = & { 2(1-x)\^5 + (1-x)\^5 [L]{}\_3 - (1+x)\^5 [L]{}\_4 - \_1 + \_6 +\
& + & 12 x (1 + x\^2) [N]{}\_0 + 8 x (1 - x\^2) [L]{}\_5 - 4x \^[1/2]{} [L]{}\_1 - 4x ( (1-x)\^2-y\^2) ) },\
[S]{}\_[(4,3)]{} & = & { 2(1-x)\^7 + (1-x)\^7 [L]{}\_3 - (1+x)\^7 [L]{}\_4 - \_1 + \_6 +\
& + & 24 x (1 + 3 x\^2 + x\^4) [N]{}\_0 + 28 x (1 - x\^4) [L]{}\_5 - 2x(7+7x\^2+y\^2) \^[1/2]{} [L]{}\_1 +\
& - & 2x ( (1-x)\^2-y\^2) ) (7+9x\^2) + x }
------------------------------------------------------------------------
Integrals of type S $ \!_{{\scriptstyle}(n)} $
----------------------------------------------
[S]{}\_[(0)]{} & = & - + [L]{}\_1 ( ) + ([L]{}\_2-[L]{}\_1) y +\
& + & [\_2]{}( x) - [\_2]{}( ) - 2 [\_2]{}( ),\
[S]{}\_[(1)]{} & = & { - + [L]{}\_1 ( ) + 2 [\_2]{}( - ) - 2 [\_2]{}( - ) },
Of all the many integrals listed in A.2–A.5 the total rate calculation done before in [@c14; @c15; @c16; @c17; @c18; @ghinculov] requires only the five basic integrals $ {\cal R}_{(-2,-1)} $, $ {\cal R}_{(-1,-1)} $, $ {\cal R}_{(0,-1) {\phantom{-}}} $, $ {\cal S}_{(0,0)} $ and $ {\cal S}_{(1,0)} $ compared to the 33 basic integrals that are needed for the full calculation. This may serve as a measure of the additional labour that is incurred when one calculates the complete set of structure functions as done in this paper.
Coefficient functions $ \rho_{(n)} $, $ \rho_{(m,n)} $, $ \sigma_{(n)} $ and $ \sigma_{(m,n)} $
===============================================================================================
In this Appendix we list the values of the various coefficient functions $ \rho_{(n),i} $, $ \rho_{(m,n),i} $, $ \sigma_{(n),i} $ and $ \sigma_{(m,n),i} $ ($ i = U \!+\! L $, $ U^P \!+\! L^P $, $ U $, $ U^P $, $ L $, $ L^P $, $ F $, $ F^P $, $ S $, $ S^P $, $ I^P $ and $ A^P $) that multiply the basic set of integrals listed in Appendix A as spelled out in the rate expression Eq. (\[complete\]). The coefficient functions involve polynomials in $ x^2 $ and $ y^2 $ which we sort by increasing powers of $ y^2 $. For reasons of conciseness we drop the suffix $ i $ denoting the particular type of structure function in the following listing. The contributions are collected in terms of powers of $ y^2 $.
Total rate $ i= U \!+\! L $
---------------------------
\_[(-2,-1)]{} & = & - ,\
\_[(-1,-1)]{} & = & ,\
\_[(0,-1) ]{} & = & - ,\
\_[(0,0) ]{} & = & - 2 ,\
\_[(1,0) ]{} & = & 2 ,
Polarized total rate $ i= U^P \!+\! L^P $
-----------------------------------------
\_[(-2,0)]{} & = & - ,\
\_[(-1,0)]{} & = & ,\
\_[(0,0) ]{} & = & ,\
\_[(1,0) ]{} & = & - ,\
\_[(-1) ]{} & = & 8 ,\
\_[( 0) ]{} & = & - 8 ,\
\_[(0,0) ]{} & = & - 4 ,\
\_[(0,1) ]{} & = & - 2 ,\
\_[(1,1) ]{} & = & 2 ,\
\_[(2,1) ]{} & = & 2 ,\
\_[(0) ]{} & = & - 4 ,\
\_[(1) ]{} & = & 4 ,
Longitudinal rate $ i = L $
---------------------------
\_[(-2,1)]{} & = & - ,\
\_[(-1,1)]{} & = & ,\
\_[(0,1) ]{} & = & - ,\
\_[(1,1) ]{} & = & ,\
\_[(2,1) ]{} & = & - 3 ,\
\_[(0,2) ]{} & = & - 2 ,\
\_[(1,2) ]{} & = & 2 ,\
\_[(2,2) ]{} & = & - 2 ,\
\_[(3,2) ]{} & = & 2 ,
Polarized longitudinal rate $ i = L^P $
---------------------------------------
\_[(-2,2)]{} & = & - ,\
\_[(-1,2)]{} & = & ,\
\_[(0,2) ]{} & = & 2 ,\
\_[(1,2) ]{} & = & - 2 ,\
\_[(2,2) ]{} & = & ,\
\_[(3,2) ]{} & = & - 7 ,\
\_[(-1) ]{} & = & 8 ,\
\_[(0) ]{} & = & - 8 ,\
\_[(0,0) ]{} & = & - 4 ,\
\_[(0,3) ]{} & = & - 2 +\
& + & 2 ,\
\_[(1,3) ]{} & = & 2 +\
& - & 2 ,\
\_[(2,3) ]{} & = & - 2 ,\
\_[(3,3) ]{} & = & 2 ,\
\_[(4,3) ]{} & = & 2 ,\
\_[(0) ]{} & = & - 4 ,\
\_[(1) ]{} & = & 4 ,
Unpolarized-transverse rate $ i = U $
-------------------------------------
\_[(-2,1)]{} & = & - 2 y\^2 (1 - x\^2)\^3,\
\_[(-1,1)]{} & = & 2 (1 - x\^2) ((1 - x\^2)\^2 - (1 - 5 x\^2) y\^2 - 2 y\^4),
------------------------------------------------------------------------
\
\_[(0,1) ]{} & = & 2 ((1 - 6 x\^2 + 5 x\^4) - 3 (5 - x\^3) y\^2 - 2 y\^4),
------------------------------------------------------------------------
\
\_[(1,1) ]{} & = & 2 ((17 - 5 x\^2) + y\^2),
------------------------------------------------------------------------
\
\_[(2,1) ]{} & = & 2,
------------------------------------------------------------------------
\
\_[(0,2) ]{} & = & 4 y\^2 ((5 - 4 x\^2 - x\^4) + 2 y\^2),
------------------------------------------------------------------------
\
\_[(1,2) ]{} & = & - 4 ((5 - 4 x\^2 - x\^4) - 2 (2 + x\^2) y\^2),
------------------------------------------------------------------------
\
\_[(2,2) ]{} & = & - 4 ((6 + 2 x\^2) + y\^2),
------------------------------------------------------------------------
\
\_[(3,2) ]{} & = & 4,
------------------------------------------------------------------------
Polarized unpolarized-transverse rate $ i = U^P $
-------------------------------------------------
\_[(-2,2)]{} & = & 2 y\^2 (1 - x\^2)\^4,\
\_[(-1,2)]{} & = & - 2 (1 - x\^2)\^2 ((1 - x\^2)\^2 + 2 (1 + 3 x\^2) y\^2 - 2 y\^4),
------------------------------------------------------------------------
\
\_[(0,2) ]{} & = & - 4 ((1-x\^2)\^2 (3+x\^2) - 2 (1 + 3 x\^4) y\^2 - 2 (5 - x\^2) y\^4),
------------------------------------------------------------------------
\
\_[(1,2) ]{} & = & 4 ((9 + 10 x\^2 + 5 x\^4) - (27 + 5 x\^2) y\^2 + y\^4),
------------------------------------------------------------------------
\
\_[(2,2) ]{} & = & 2 (10 (1 - x\^2) + 3 y\^2),
------------------------------------------------------------------------
\
\_[(3,2) ]{} & = & 6,
------------------------------------------------------------------------
\
\_[(-1) ]{} & = & - 16 ,
------------------------------------------------------------------------
\
\_[(0) ]{} & = & 16 ,
------------------------------------------------------------------------
\
\_[(0,0) ]{} & = & 8 ,
------------------------------------------------------------------------
\
\_[(0,3) ]{} & = & 4 (1 - x\^2) (4 x\^2 (1 - x\^2)\^2 + (1 + 4 x\^2 - 5 x\^4) y\^2 - 2 (4 - x\^2) y\^4),
------------------------------------------------------------------------
\
\_[(1,3) ]{} & = & 4 ((1 - x\^4)(3 - 11x\^2) + (9 - 22 x\^2 - 11 x\^4 ) y\^2 - 2 (1 - 2 x\^2) y\^4),
------------------------------------------------------------------------
\
\_[(2,3) ]{} & = & - 4 ((13 + 11 x\^4) - (15 + 7 x\^2) y\^2 + 2 y\^4),
------------------------------------------------------------------------
\
\_[(3,3) ]{} & = & - 4 ((1 - 5 x\^2) + y\^2),
------------------------------------------------------------------------
\
\_[(4,3) ]{} & = & - 4,
------------------------------------------------------------------------
\
\_[(0) ]{} & = & 8 (1 - x\^2 + y\^2),
------------------------------------------------------------------------
\
\_[(1) ]{} & = & - 8 (1 - x\^2 + y\^2)
------------------------------------------------------------------------
,
Scalar rate $ i = S $
---------------------
\_[(-2,-1)]{} & = & - ,\
\_[(-1,-1)]{} & = & ,\
\_[(0,-1) ]{} & = & - 3 ,\
\_[(0,0) ]{} & = & - 2 ,\
\_[(1,0) ]{} & = & 2 ,
Polarized scalar rate $ i = S^P $
---------------------------------
\_[(-2,0)]{} & = & - ,\
\_[(-1,0)]{} & = & ,\
\_[(0,0) ]{} & = & ,\
\_[(1,0) ]{} & = & - 7 ,\
\_[(-1) ]{} & = & 8 ,\
\_[(0) ]{} & = & - 8 ,\
\_[(0,0) ]{} & = & - 4 ,\
\_[(0,1) ]{} & = & - 2 ,\
\_[(1,1) ]{} & = & 2 ,\
\_[(2,1) ]{} & = & 2 ,\
\_[(0) ]{} & = & - 4 ,\
\_[(1) ]{} & = & 4 ,
Forward-backward-asymmetric rate $ i = F $
------------------------------------------
\_[(-2,0)]{} & = & - 2 y\^2 (1 - x\^2)\^2,\
\_[(-1,0)]{} & = & 2 ((1 - x\^2)\^2 + 4 x\^2 y\^2),
------------------------------------------------------------------------
\
\_[(0,0) ]{} & = & 2 (4 (2 + x\^2) - 7 y\^2),
------------------------------------------------------------------------
\
\_[(1,0) ]{} & = & - 2,
------------------------------------------------------------------------
\
\_[(-1) ]{} & = & 16 ,
------------------------------------------------------------------------
\
\_[(0) ]{} & = & - 16 ,
------------------------------------------------------------------------
\
\_[(0,0) ]{} & = & - 8 ,
------------------------------------------------------------------------
\
\_[(0,1) ]{} & = & - 4 (4 x\^2 (1 - x\^2) + (1 + 5 x\^2) y\^2 - 2 y\^4),
------------------------------------------------------------------------
\
\_[(1,1) ]{} & = & - 4 (3 (1 + x\^2) - y\^2),
------------------------------------------------------------------------
\
\_[(2,1) ]{} & = & 4,
------------------------------------------------------------------------
\
\_[(0) ]{} & = & - 8 ((1 - x\^2) + y\^2),
------------------------------------------------------------------------
\
\_[(1) ]{} & = & 8 ((1 - x\^2) + y\^2),
------------------------------------------------------------------------
Polarized forward-backward-asymmetric rate $ i=F^P $
----------------------------------------------------
\_[(-2,1)]{} & = & 2 y\^2 (1 - x\^2)\^3,\
\_[(-1,1)]{} & = & - 2 (1 - x\^2) ((1 - x\^2)\^2 - (1 - 5 x\^2) y\^2),
------------------------------------------------------------------------
\
\_[(0,1) ]{} & = & - 2 ((1 - 6 x\^2 + 5 x\^4) - (11 + x\^2) y\^2),
------------------------------------------------------------------------
\
\_[(1,1) ]{} & = & - 2 ((11 + x\^2) + 5 y\^2),
------------------------------------------------------------------------
\
\_[(2,1) ]{} & = & 10,
------------------------------------------------------------------------
\
\_[(0,2) ]{} & = & - 4 y\^2 (1 - x\^2) (5 + x\^2),
------------------------------------------------------------------------
\
\_[(1,2) ]{} & = & 4 ((1 - x\^2) (5 + x\^2) - 2 x\^2 y\^2),
------------------------------------------------------------------------
\
\_[(2,2) ]{} & = & 4 (2 x\^2 + y\^2),
------------------------------------------------------------------------
\
\_[(3,2) ]{} & = & - 4,
------------------------------------------------------------------------
Polarized longitudinal-transverse-interference rate $ i=I^P $
-------------------------------------------------------------
\_[(-2,2)]{} & = & ,\
\_[(-1,2)]{} & = & - ,\
\_[(0,2) ]{} & = & - ,\
\_[(1,2) ]{} & = & ,\
\_[(2,2) ]{} & = & ,\
\_[(-1) ]{} & = & - 4 ,\
\_[(0) ]{} & = & 4 ,\
\_[(0,0) ]{} & = & 2 ,\
\_[(0,3) ]{} & = & ,\
\_[(1,3) ]{} & = & ,\
\_[(2,3) ]{} & = & - ,\
\_[(3,3) ]{} & = & - ,\
\_[(0) ]{} & = & 2 ,\
\_[(1) ]{} & = & - 2 ,
Polarized parity-asymmetric rate $ i = A^P $
--------------------------------------------
\_[(-2,1)]{} & = & ,\
\_[(-1,1)]{} & = & - ,\
\_[(0,1) ]{} & = & ,\
\_[(1,1) ]{} & = & - ,\
\_[(0,2) ]{} & = & ,\
\_[(1,2) ]{} & = & - ,\
\_[(2,2) ]{} & = & ,
Loop integrals
==============
In this Appendix we list the $m_b \neq 0$ one-loop amplitude corrections to the process $ t \rightarrow b + W^{+} $. They are determined from the vertex correction Fig. 1b and the appropriate wave function renormalization constants $ Z_2 $. We present our results in terms of the three vector current amplitudes $ F_{i}^{V} $ $ (i=1,2,3) $ and the three axial vector current amplitudes $ F_{i}^{A} $ $ (i=1,2,3) $ defined in Eq. (\[formfactor\]) in Sec. 5. Using the abbreviations in Eq. (\[w1wmu\]) with $ q^2 = m_W^2 $, one has
F\_1\^V & = & 1 + C\_F { - - ( ) +\
& - & ( ) - 4 + (w\_1 w\_) ( - ) }\
F\_2\^V & = & C\_F { 2 - ( - ) ( ) +\
& - & ( - ) (w\_1 w\_) }\
F\_3\^V & = & F\_3\^V (m\_t,m\_b) = F\_2\^V (m\_b,m\_t).
As before the IR singularity is regularized by a small gluon mass $ m_g $. The axial vector amplitudes $ F_i^A $ can be obtained from the vector amplitudes by the replacement $ m_t \rightarrow - m_t $, i.e. one has $ F_i^A(m_t) = F_i^V(-m_t) $ $ (i=1,2,3) $. Our one-loop amplitudes are linearly related to the one-loop amplitudes given in [@gounaris]. The two sets of one-loop amplitudes agree with each other after correcting for a typo in [@gounaris] mentioned in Sec. 5.
(0,28)[a)]{} (0,28)[b)]{} (0,28)[c)]{} (0,28)[d)]{}
**Figure 1**
> Leading order Born term contribution (a) and $ O( \alpha_s ) $ contributions (b,c,d) to $ t \!\rightarrow\! b \!+\! W^{+} $.
**Figure 2**
> Definition of the polar angles $ \theta $ and $ \theta_P $, and the azimuthal angle $ \phi $. $\vec{P}$ is the polarization vector of the top quark.
(120,120) (112,030.7)[$ \mathbf{\theta} $]{} (047,014.0)[$ \mathbf{\theta} $]{}
**Figure 3**
Born term Lego plot of the two-fold angular decay distribution\
$ d \widehat{\Gamma} / d \! \cos \theta d \! \cos \theta_p $ with $ P = 1 $.
**Figure 4**
> Differential $ W $-boson energy distribution $ d\Gamma_{U+L}/dq_0 $ for the total rate resulting from $ O(\alpha_s) $ gluon emission ($ m_b = 4.8 \mbox{ GeV} $).
**Figure 5**
> Differential $ W $-boson energy distribution $d \Gamma_+/dq_0$ for the partial rate into positive helicity $ W $-bosons resulting from $ O(\alpha_s) $ gluon emission for $ m_b = 4.8 \mbox{ GeV} $ (solid line) and for $ m_b = 0 $ (dashed line).
**Figure 6**
> Contours of the decay distribution of a fully polarized ($ P = 1 $) top quark in the $ \cos \theta_p $ - $ \cos \theta $ plane for $ m_b = 0 $. The full lines are the distribution including the $ O(\alpha_s) $ corrections.
(120,120) (000,054.5)
**Figure 7**
> Charged lepton polar angular distribution in the $ W $ rest frame for $ m_b = 0 $ (Born term: full line; $ O(\alpha_s) $: dashed line). Also shown are average values of the decay distribution.
(120,120) (000,059)
**Figure 8**
> Azimuthal distribution of normalized rate for $ m_b = 0 $ (Born term: full line; $ O(\alpha_s) $: dashed line). Also shown are average values of the decay distribution.
**Figure 9**
Bottom mass dependence of the total rate $ \Gamma_{U + L} $\
(Born term: full line; $ O(\alpha_s) $: dashed line).
**Figure 10**
Top mass dependence of the rate ratio $ \Gamma_L / \Gamma_{U + L} $ for $ m_b = 0 $\
(Born term: full line; $ O(\alpha_s) $: dashed line).
**Figure 11**
Top mass dependence of the rate ratio $ \Gamma_U / \Gamma_{U + L} $ for $ m_b = 0 $\
(Born term: full line; $ O(\alpha_s) $: dashed line).
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[^1]: From this point on we shall drop explicit reference to the $ W^{+} \rightarrow
\bar{q} + q $ decay channel since it has the same angular decay distribution as $ W^{+} \rightarrow l^{+} + \nu_{l} $. In fact the branching fraction into the two hadronic channels $ (\bar{d}+u) $ and $ (\bar{s}+c) $ exceeds that of the sum of the three leptonic channels by a factor of approximately two because of the colour enhancement factor. Although not explicitly mentioned further on, the existence of the hadronic decay mode of the $ W^{+} $ is always implicitly assumed in the following.
[^2]: We have recalculated the one-loop results of Ref. [@gounaris] and have found an acknowledged typo in the scalar form factors $ F_3(Q^2) $ and $ H_3(Q^2) $ of [@gounaris]. The typo is corrected by replacing the factor $ (m_2 - m_1) / Q^2 $ in the last line of Eq.(A.8) of Ref. [@gounaris] by $ (m_2 - m_1) / (2 Q^2) $.
[^3]: Contrary to the Born term case the polarization of the top quark cannot be accounted for by replacing [*all*]{} $ p_t $ momenta by their barred counterparts $ \bar{p}_t $.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We study the repeated use of a *monotonic* recording medium—such as punched tape or photographic plate—where marks can be added at any time but never erased. (For practical purposes, also the electromagnetic “ether” falls into this class.) Our emphasis is on the case where the successive users act independently and selfishly, but not maliciously; typically, the “first user” would be a blind natural process tending to degrade the recording medium, and the “second user” a human trying to make the most of whatever capacity is left.
To what extent is a length of used tape “equivalent”—for information transmission purposes—to a shorter length of virgin tape? Can we characterize a piece of used tape by an appropriate “effective length” and forget all other details? We identify two equivalence principles. The *weak* principle is exact, but only holds for a sequence of infinitesimal usage increments. The *strong* principle holds for any amount of incremental usage, but is only approximate; nonetheless, it is quite accurate even in the worst case and is virtually exact over most of the range—becoming exact in the limit of heavily used tape.
The fact that strong equivalence does not hold *exactly*, but then it does *almost* exactly, comes as a bit of a surprise.
author:
- 'Tommaso Toffoli[^1]'
title: 'Thermodynamics of used punched tape: A weak and a strong equivalence principle'
---
Thermodynamics of write-once media, Equivalence principles for storage capacity of noisy medium
Bob—a poor computer science student—has found, rummaging through Alice’s dump, a large amount of used punched tape “in good conditions”. He doesn’t care for the data that is already on the tape: he would like to reuse the tape for storing his own data. He wants to be able to use a standard tape read/punch unit, which can sense holes in the tape and punch new ones but not remove holes that are already there. Since holes already made cannot be undone, the storage density Bob can expect to achieve is less than with virgin tape, and will depend on the actual conditions of the tape.
To what extent is a length of used tape “equivalent”—for information transmission purposes—to a shorter length of virgin tape? Are there any qualitative differences between tapes that have been used to different degrees, or can one characterize a piece of used tape simply by its “effective length” and forget all other details?
The theme we develop is complementary to that of Rivest and Shamir[@rivest-worm] (also cf. [@maier]). They stress the information-engineering aspects of reusing a tape generated by a cooperative partner in a pre-planned context. On the other hand, we are interested in a situation where the other party, while presumed non-malicious, volunteers no cooperation and pursues independent goals (if any goals can be made out); what we typically have in mind for “the other party” is *natural processes*.
The cumulative channel capacity of randomly-punched used tape was first investigated in [@wolf] (also see references therein), some of whose results we simplify and extend. References [@dolev] and [@heegard] discuss coding algorithms that dynamically adjust to “stuck-at-0” faults on the tape (cells that will not punch) sensed during punching, and “stuck-at-1” faults sensed during or before punching. A paper related to the present one in spirit if not in detailed substance is “Writing on dirty paper” by Costa[@costa], whose moral (“Do the best with what you have”) we make our own.
If you have no time at all, read just —a self-contained, intuitive debriefing.
Each position on the tape where a hole may appear is called a ; the two possible cell states are and . The instructions to the punch unit are and , with the following results on the tape $$\begin{array}{llll}
\hbox{\small\sc old state} &\hbox{\small\sc action}& &\hbox{\small\sc new state}\\\hline
\hbox{blank} & \hbox{spare} & \mapsto & \hbox{blank}\\
\hbox{hole} & \hbox{spare} & \mapsto & \hbox{hole}\\
\hbox{blank} & \hbox{punch} & \mapsto & \hbox{hole}\\
\hbox{hole} & \hbox{punch} & \mapsto & \hbox{hole}
\end{array}$$
A hole (or punch) distribution that factors into identical independent distributions for the individual cells—and is thus characterized by a single number, namely, the hole (or punch) density—will be called . We shall assume that on each round of usage or the tape starts with a canonical hole distribution of density $p$ and comes out with a uniform hole density $p'$; furthermore, we assume that the intervening punching process packs on the tape the *maximum* amount of new information compatible with startng density $p$ and target density $p'$. According to Shannon’s theorem, such maximum efficiency can asymptotically be achieved by means of sufficiently long block codes. From the above assumptions one can prove that both the punch distribution $q$ yielded by an optimal code and the resulting hole distribution $p'$ must be canonical as well. Thus, our usage assumptions imply that, starting from virgin tape—whose distribution is, of course, canonical with $p=0$—input, punch, and output distributions will be canonical at every successive stage. For this reason, in what follows all distributions will be tacitly understood to be canonical.
The result of applying a punch density $q$ to a hole density $p$ is a new hole density $$\eqlabel{pprime}
\pp=1-(1-p)(1-q).$$ A canonical punch distribution entails that, once the input hole density $p$ is known, there is no further advantage in knowing the position of the individual holes; in other words, overpunching can be carried out in a *data-blind* fashion.
Let’s examine a few distinguished cases.
[$\bullet$]{}
If $p=0$ the tape is blank—Bob can resell it as virgin tape.
If $p=1-p=1/2$, the tape has already been utilized by Alice at its maximum information capacity of one bit per cell. That would seem to leave Bob with no room for further information storage. But remember that he doesn’t care about the old information: punching new holes will destroy some of it but will encode some of his own! In fact (see below), with a punch density $q=3/5$, Bob can record on the tape as much as about .322 bits per cell.
If $p=1$, the tape carries no information for Alice—just as in the case $p=0$. However, now there is no way Bob can put any information on it. Alice wantonly spoiled the tape.
If $p$ is a probability, it will be convenient to write $\pbar$ for $1-p$. Thus, in , $$\pp=1-\pbar\qbar=\overline{\pbar\qbar},\quad \text{or}\quad
\qbar=\ppbar/\pbar.$$
We shall use natural logarithms throughout. It will be convenient to write $\LN x$ for $-\ln x$. The , defined as $$y = x\LN x,$$ will play an important role in the equivalence principles discussed here (see ). The , defined by $$H(p)=p\LN p+\pbar\LN\pbar,$$ is the average of the self-information function over the binary distribution $\{p,\pbar\}$.
Both self-information and binary entropy, as defined here, measure information in natural units or . Conversion of information quantities to binary units or is achieved by explicitly factoring out the constant $$\bit=\ln 2\approx.693;$$ thus, for example, the entropy of four equally probable messages is $\ln4=2\ln2 =2\ \bit$.
If $X$ and $Y$ are random variables, $P(x)$ will denote the probability that $X=x$, and $P(x.y)$ the probability that $X=x$ and $Y=y$. The between $X$ and $Y$ is defined as $$\{X;Y\} = \sum_{x,y}P(xy)\LN\frac{P(x)P(y)}{P(x.y)}.$$
For more background on information theory, see the excellent introduction by Abramson[@abramson].
Under the above assumptions (), used punched tape may be viewed as a communication channel affected by monotonic noise. In the channel diagram of , the input variable $X$ represents the instruction given to the punch unit while scanning a cell, and the output variable $Y$ represents the resulting cell state. An “error” occurs when a cell spared by the punch unit turns out already to contain a hole. The conditional probability $P(\hbox{hole}|\hbox{spare})$ associated with this transition equals the current hole density $p$.
From the joint and marginal distributions of $X$ and $Y$, namely, one obtains, for this channel operated at a punch density $q$, a mutual information The quantity $\mutu$ is the amount of *new* information that can be encoded on a tape having a hole density $p$ by punching it with a density $q$, resulting in a new hole density $\pp(p,q)=1-\pbar\qbar$.
*The relation expressed by equation —plotted in —completely characterizes the bulk properties of punched tape as a communication channel. The rest of this paper is devoted to extracting some of its implications.*
The $C$ of the channel is the maximum of $\mutu$ over all possible values of $q$ (or, equivalently, of $\pp$). By equating to zero the derivative of $\mutu$ with respect to $\pp$, $$\der\pp{\mutu} = \LN\frac\pp\ppbar+\frac{H(p)}\pbar=0,
% Der qM=\pbar\LN\frac{\pbar\,\qbar}{1-\pbar\,\qbar}+H(p),$$ one finds that this maximum occurs at where $\mutu$ attains the value as plotted in . In particular, for $p=1/2$, $$\qhat=\frac35,\ \pphat=\frac45,\ \hbox{and}\
C=\ln\frac54\approx.322\ \bit.$$
For sake of contrast with the current context of selfish, independent utilization of the tape by each successive party, in the following two subsections we shall briefly discuss the possibilities of cooperation and competition.
You shall receive an hundredfold
--------------------------------
In two successive selfish transmission stages starting from virgin tape, Alice got 1 bit’s worth of message out of each cell and Bob .322 bit, for a total of 1.322 bit. By collaborating, they could do much better[@rivest-worm; @maier]. In fact, if Alice and Bob worked in concert, with a very simple code they could each get two bits’ worth of message out of every three cells, for a total of $4/3\approx1.333$ bit/cell; with long block codes, they could get up to about 1.55 bit/cell. The advantages of collaboration show up even better when one can plan ahead a long series of transmission stages with a long length of tape: in this situation, the cumulative amount of message worth one can get out of an $n$-cell length of tape grows as $n\ln n$; therefore, the amount per unit length is unbounded!
Tape wars averted
-----------------
We have seen that, if the original tape was punched at a density $p$ by Alice, with an attendant rate $H(p)$ for her message, then Bob can achieve his channel capacity $C(p)$ as in by punching at density $\qhat(p)$ as in . In the process, Alice’s original message is, of course, degraded. In fact, if Alice tries to read back her message, she will find it contaminated by the same amount of one-way noise as if it had gone through the channel described by exchanging $q$ and $p$ in and table .
Suppose now that Alice, realizing that her tapes are going to be reused (or *concurrently* used—since, as we have seen, the two punching operations commute) by Bob, decides to encode her next batch of tapes so as to make her messages readable even after an anticipated punching by Bob at density $q$. According to and , as a preventive measure she will have to shift her punch density $p$ to a higher value than 1/2, thus achieving a lower rate but greater resistance to Bob’s tampering. When Bob realizes that, he will be forced to shift *his* punch density $q$ to a higher value—and so forth.
This is not a zero-sum game: as the arms race unfolds, each party will end up storing progressively less information on the tape. Will the race lead to the mutual destruction of information capacity? Fortunately, the curve of intersects the line $\qhat=p$ and there has a slope less than 1. Thus, the race converges to a stable point (with $q=p\approx 0.609$), where each party achieves an effective storage capacity of $\approx.240$ bit/cell.
The sum of the two capacities—and these are *coexisting* capacities, with both messages readable at the same time!—is about 0.48 bit/cell, to be compared with the 1 bit/cell Bob and Alice could have achieved by “space-sharing” the tape (e.g., one cell for Alice, one for Bob, and so forth). Thus, the attempt by the two parties to concurrently use the monotonic-write tape, performed in a selfish but rational way, results in an overall loss of storage capacity that is substantial but not crippling.
It must be noted that, even though at equilibrium they are in a symmetric situation, Alice and Bob cannot use the *very same* block code to encode their messages on the tape. To avoid interference the two codes must be practically uncorrelated or “mutually orthogonal”; this is always possible with long enough block codes.
We introduce the issue of *tape equivalence* by means of two dialogues. The $\ell$ of a piece of tape is the number of cells it contains. Because of the canonical distribution of both holes and punches (), the overall capacity of a tape of length $\ell$ is $\ell$ times the capacity of a single cell, and similarly for the mutual information.
\#1\#2
1em\#2\*, *\#2*:
Dialogue 1 {#dialogue-1 .unnumbered}
----------
[*Bob is now an old and stingy facilities officer at Caltech. He can no longer see the individual holes on the tape—his vision is blurred—and he wouldn’t any longer know how to start writing a block code. All he cares about is tape as a bulk commodity, and getting the most out of it. He is assisted by Sue, who physically handles the tape and knows how to devise appropriate block codes. Sue has standing instructions to recycle paper tape to the best of her capabilities and not to bother Bob with details.*]{}
Sue, we have to send a million-bit message to MIT. Get a piece of tape. I’ve got here a reel of tape with an overall capacity of one million bits. \[*She doesn’t tell Bob whether that’s a thousand feet of virgin tape, or perhaps ten thousand feet of heavily used tape.*\] Good! Here is the message. Don’t waste any capacity, and make sure you get the tape back from MIT so we can reuse it! By the way, what will be the capacity left on the tape after this message? I want to enter it as an asset in my inventory sheet. That will be 333,000 bits. So, using this tape at capacity will leave it “shrunk” to .333 of its previous capacity. Well, one third left is better than nothing! Sue, here is another message for MIT. Since it happens to be 333,000 bits long, let’s use the tape you got back from them. \[*We assume that, after decoding a message, MIT does not keep a record of the detailed hole pattern received. That might be used for improving transmission efficiency, but at substantial storage cost.*\] What will be the capacity left on the tape after this message? That will be about 119,000 bits. Hey, this time, it will only shrink to 119,000/333,333=.357 of its pre-transmission capacity \[*with an accusing tone*\] Are you sure you made the best use of my tape last time? Cool off, Bob! Every housewife knows that used tape shrinks less! In fact, really ripe tape only shrinks to $1/e\approx .368$ of its previous capacity upon each usage. And brand new tape? New tape is the worst! It will shrink to $\ln5/\ln2-2\approx.322$ of its previous capacity. Here’s the whole picture! ()
The two physical parameters of a piece of tape, namely, its length $\ell$ (in cells) and its current hole density $p$, completely characterize its “response”—in terms of amount of information transmitted and capacity left—upon each successive usage, including usages with a punch density $q<\qhat$ (where some of the capacity is saved for later) or $q>\qhat$ (where some capacity is wasted), according to equations , , and .
In particular, the capacity of a piece of tape of length $\ell$ is $\ell
C(p)$ (cf. ). This can be thought of—if we measure capacity in bits (see )—as the of the tape—i.e., the number of cells of virgin tape having the same overall capacity. Bob would have been delighted to find that two pieces of tape having the same reduced length are completely equivalent for information-transmission purposes. Such an equivalence principle would allow him to characterize a piece of tape by means of a single information-theoretical parameter—the reduced length—rather than the two physical parameters $\ell$ and $p$, and greatly simplify his inventory bookkeeping.
If such an equivalence held, then, as a specific consequence, the shrinkage coefficient of Dialogue 1, defined as $$s(p)=\frac{C(\pphat)}{C(p)},$$ would be independent of $p$. Unfortunately, as we have seen in the dialogue, this is only approximately true (). We’ll return to this problem, with better tools, in .
Dialogue 2 {#dialogue-2 .unnumbered}
----------
[*Sue is on vacation. Her temporary replacement, Willie, is being indoctrinated by Bob about the need to conserve tape. To test his coding capabilities, Bob chooses a spool of tape just like the one he gave Sue the first time.*]{}
Here is a length of used tape, Willie, and a million-bit message to be sent to MIT. Please transmit the message as efficiently as you can.
Is it urgent?
Not, really. Take your time, but do a good job!
Well, did you get the tape back from MIT?
Here it is!
What’s its capacity now?
580,000 bits, more or less.
What? It only shrank to .580 of its original length? How did you manage that?
You know, haste makes waste. So I first encoded only a small fraction of the message on the tape, using a very low punch density. MIT decoded that, wrote it down, and sent back the tape. Then I encoded on the same tape another increment of the message, sent it to MIT, and so on. The tape must have gone back and forth twenty times!
In the limit of an infinite number of infinitesimal increments, how much information could you transmit in this way?
Starting from virgin tape, about 2.37 bits/cell (precisely, $\frac{\pi^2}{\ln2}$).
That’s amazing!
And, of course, at any intermediate moment the transmission “mileage” already used plus that which is still left on the tape equals a constant—provided you always travel very slowly.
I got it! Your “mileage left” is the I was looking for. No matter how different they look physically, two pieces of tape (say, one short and fresh and the other long and stale) having the same effective length are equivalent for information transmission purposes.
Slow down, Bob! That is true only as long as you use them up *slowly*. By comparing Sue’s performance with mine, you realize that, when one tries to cram onto a tape a substantial fraction of its channel capacity at once, there are losses by “friction”, as it were. Well, one can tell the difference between fresh tape and well-worn tape by the fact that the former exhibits *just a little more friction* than the latter.
Let us explore in more detail what Bob discovered with Willie’s help.
Suppose that we start with virgin tape and record on it a small amount $d\info$ of information by punching it at a very low density. We ship the tape but ask the recipient to send it back to us after reading the message. We then record on this “slightly used” tape an additional small amount of information, further increasing the hole density. We continue in this way, sending one after the other a large number of messages each having a small information contents, until the tape is completely filled with holes. If at each stage the encoding is done optimally, what is the cumulative information $\int d\info$ of the messages we sent?
Assume that at a generic stage of this process we start with a hole density $p$ and increase it to $\pp=p+dp$ by issuing punch commands with a probability $dq$ per cell. The channel diagram is the same as , but with input and output probabilities as in .
The hole density increment is $dp=\pbar dq$, as the blanks, which appear with density $\pbar$, are turned into holes with probability $dq$, while the holes, with density $p$, remain unaffected. The mutual information of this infinitesimal punching operation, calculated from using $dq$ in place of $q$, is $$\begin{aligned}
d\info &=H(p+dp)-\frac{\overbar{p+dp}}\pbar H(p)\\
&= \left(\LN\frac p\pbar+\frac{H(p)}\pbar\right)dp
= \frac{\LN p}\pbar dp.
\end{aligned}$$ The indefinite integral of the integrand in the last expression is $$\int\frac{\LN p}{1-p}dp = -\mathrm{Li}_2(1-p),$$ where $\mathrm{Li}_2$ is the *dilogarithm* function. Thus, the of a tape of hole density $p$, i.e., the total amount of information that can be transmitted via it in successive small increments until all holes have been punched up, is as plotted in (compare with the qualitatively similar behavior of $C$, in ); for virgin tape ($p=0$), the effective capacity is $\mathrm{Li}_2(1)={\pi^2/6}$ (cf. [@wolf]). Note that, by , $$d\info=-dQ.$$ Since $Q$ is a *function of state* of the tape (i.e., it depends only on its state and not on the specific sequence of operation that led to that state), $d\info$ is an *exact* differential.
The above quantities are on a per-cell basis. Let us define the (cf. Dialog 2) of a piece of tape of length $\ell$ and hole density $p$ as $\lambda=\ell Q(p)$. If by a sequence of small incremental messages we transmit an amount of information $I$ per cell, and thus a total amount $S=I\ell$ for the entire piece of tape, the new effective length will be $\lambda'=\ell(Q-I)$. The corresponding shrinkage coefficient will be $$\frac{\lambda'}{\lambda}=1-\frac IQ=1-\frac S\lambda,$$ which is *independent* of the physical parameters $\ell$ and $p$ and depends only on the ratio between two information-theoretical quantities, i.e., the total amount $S$ of information transmitted and the effective length $\lambda$ of the tape. We shall call this the for monotonic-write media.
Let us now explore in more detail what Bob discovered with Sue’s help.
Whether we intend to utilize a piece of tape incrementally, as in Dialogue 2, or in discrete installements, as in Dialogue 1, the *effective length* $\lambda$ defined above provides a more natural measure of a tape’s information capacity than the *reduced length* introduced in Dialogue 1.
Armed with this measure, let us now turn our attention from the special case of the limit of an infinite sequence of infinitesimal messages to the general case of finite-size messages, where the weak principle is not applicable.
Our goal is to eliminate the physical parameters $p$ and $q$ between equations and , and thus write a relation directly between (a) the effective length $\lambda$ of a piece of tape before the transmission of a message, (b) the effective length $\lambda'$ after the transmission, and (c) the amount $S$ of information conveyed by the message. If such a relation exists, it may be assumed to be of the form $$f(\lambda,\lambda',S)=0$$ and, since we are assuming a canonical hole distribution before and after punching, it must satisfy the scaling property $$f(a\lambda,a\lambda',aS)=0\quad\hbox{for any}\ a.$$ Setting, as a special case, $a=1/\lambda'$, we obtain a relation between two variables $$g(\sigma,\mu)=f(\sigma,1,\mu)=0,$$ where $$\sigma=\frac{\lambda'}\lambda=\frac{Q(\pp)}{Q(p)}\quad \hbox{and}\quad
\mu=\frac S\lambda=\frac{\info(p,q)}{Q(p)}.$$ The variable $\mu$—which is the mutual information for a given stage of utilization of the tape—can be thought of as the information rate per unit of effective length of the tape, and $\sigma$ as the shrinkage coefficient attendant to that stage.
Since the variables $\sigma$ and $\mu$ depend on two parameters, $p$ and $q$, we cannot *a priori* expect to eliminate *both* parameters when solving for $\mu$ with respect to $\sigma$. However, for a given initial hole density $p$ treated as a fixed parameter, we can eliminate just $q$ and write $$\mu=\mu_p(\sigma).$$ The result of this elimination, performed numerically for different values of $p$, are shown in , which also shows the values of the eliminated parameter $q$ on the $\mu(\sigma)$ curves.
Paralleling the weak equivalence principle of the previous section—which states that tapes having the same effective capacity are indistinguishable at slow utilization rates—a would be one that is valid for any rate of utilization of the tape at any transmission stage, from an infinitesimal hole-density increment ($q$ close to 0) to gross overpunching ($q$ close to 1). I don’t know whether it is more surprising that, strictly speaking, punched tape does *not* obey a strong equivalence principle, or that, after all, it turns out to do so *to a very good approximation*. In fact, as is clear from , after eliminating $q$ between $\mu$ and $\sigma$ some dependence on $p$ remains, but this dependence is slight in any case and rapidly vanishes as $p$ approaches 1. Intuitively, the one-parameter family of curves of nearly collapses—when expressed in terms of a more natural set of variables—onto a *single curve* ().
The curves $\mu_p(\sigma)$ all have slope $-1$ at $\sigma=1$; this is an expression of the weak equivalence principle (i.e., for small $q$, the effective length decreases by an amount equal to the amount of information transmitted). They all have slope $\infty$ at $\sigma=0$, signifying that the waste of effective capacity increases precipitously when one punches at a density much greater than that needed for transmitting at channel capacity.
The worst-case departure of the $\mu_p$ curves from the limiting curve $\mu_1=\lim_{p\to1}\mu_p$ occurs near the maximum bulge of the curves, and is substantially the same as the departure of $s$ from its $1/e$ limit as plotted in . The curves $\mu_p$ are not likely to be expressible in closed form; however, as is easy to prove, the limiting curve $\mu_1$ is nothing but the familiar *self-information* function $\mu=\sigma\LN\sigma$. To the same approximation as the strong equivalence principle holds, *this function gives the information-transfer characteristics of punched tape (i.e., for any message, the capacity used by it, that wasted, and that left after the message) over the tape’s entire utilization range*.
Let us remark that the self-information function appears in the limit also in . In fact, one can show that $$\lim_{p\to1}\frac{\info(p,q)}{C(p)}=e\qbar\LN\qbar.$$
A piece of randomly punched tape is described by two physical parameters—its length $\ell$ and its hole density $p$. We have raised the question of whether the tape’s behavior as an information transmission commodity can be usefully characterized by a single information-theoretical parameter—its *effective length* $\lambda$. We have concluded that this is the case
[$\bullet$]{}
in the “quasi-static” limit of slow utilization rate (*weak equivalence principle*);
for any utilization rate (*strong equivalence principle*)
[$-$]{}
*exactly*, but only in the limit of already heavily used tape, and
*approximately*—but with good accuracy even in the worst case—over the whole range of previous and future uses of the tape.
This research was funded in part by NSF (9305227-DMS) and in part by ARPA through the Ultra Program (ONR N00014-93-1-0660) and the CAM-8 project (ONR N00014-94-1-0662). I am indebted to Peter Elias, Matteo Frigo, and Mark Smith for useful discussions, and to Ronald Rivest for some references.
[9]{}
, Norman, [*Information Theory and Coding*]{}, McGraw-Hill (1963). , Max, “Writing on dirty paper”, [*IEEE Trans.Info. Theory **IT-29***]{} (1983), 439–441. , Danny, David [Maier]{}, Harry [Marson]{}, and Jeffrey [Ullman]{}, “Correcting faults in write-once memory”, [*Proc.16th Anuual ACM Symp. on Theory of Computing*]{}, ACM (1984), 225–229. , Freeman, “Time without end: Physics and biology in an open universe”, [*Rev. Mod. Phys. **51***]{} (1979), 447–460. , Chris, and Abbas [El Gamal]{}, “On the capacity of computer memory with defects,” [*IEEE Trans. Info. Theory **IT-29***]{} (1983), 731–739. , David, “Using write-once memory for database storage”, [*1982 ACM Symposium on Principles of Database Systems*]{}, ACM (1982), 239–246. , Ronald, and Adi [Shamir]{}, “How to reuse a ‘Write-Once’ memory”, [*Information and Control **55***]{} (1982), 1–19. , Jack, Aaron [Wyner]{}, Jacob [Ziv]{}, and János [Körner]{}, “Coding for a Write-Once Memory”, [*AT&T Bell Lab.Tech. J. **63***]{} (1984), 1089–1112.
[Tommaso Toffoli]{} Tommaso Toffoli received a Doctorate in Physics from the University of Rome, Italy, in 1967, and a Ph.D. in Computer and Communication Science from the University of Michigan, Ann Arbor, in 1976. In 1977 he joined the MIT Laboratory for Computer Science, eventually becoming the leader of the Information Mechanics group. In 1995 he joined the faculty of the Boston University ECE Department. His main area of interest, namely Information Mechanics, deals with fundamental connections between physical and computational processes. He has developed and pioneered the use of cellular automata machines, as a way of efficiently studying a variety of synthetic dynamical systems that reflect basic constraints of physical law, such as locality, uniformity, and invertibility. Related areas of interest are: quantum computation; correspondence principles between microscopic laws and macroscopic behavior; quantitative measures of “computation capacity” of a system—as contrasted to “information capacity,” and connections between Lagrangian action and amount of computation. A new initiative, Personal Knowledge Structuring, aims at developing cultural tools that will help ordinary people turn the computer into a natural extension of their personal faculties.
[^1]: Tommaso Toffoli (), ECE Department, Boston University, 8 Saint Mary’s St., Boston, MA 02215.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
\
Los Alamos National Laboratory, Los Alamos, NM 87545 USA\
E-mail:
bibliography:
- 'PPG188.bib'
title: ' PHENIX results on system size dependence of J/$\psi$ nuclear modification in $p$, $d$, $^{3}$He+A collisions at $\sqrt{s_{NN}}$=200 GeV'
---
Introduction
============
Measurements of the bound states of charm quarks produced in collisions involving nuclei can provide information on a wide range of effects in both the early and late stages of the event. At the center of mass energies probed at the Relativistic Heavy ion Collider (RHIC), the production of primordial $c\bar{c}$ pairs usually proceeds through gluon fusion, making charm quark production sensitive to modifications of the gluon distribution in the nucleus [@RamonaOldShadowing]. After formation the pair must propagate through nuclear matter, where the quarks can experience energy loss [@CNM_Eloss] or multiple scattering which can affect the observed transverse momentum distributions of heavy flavor hadrons [@Cronin]. Such interactions can also disrupt the heavy quark precursor state before it has hadronized, resulting in quarkonia suppression. Even after exiting the nucleus and projecting onto a final state, the fully-formed quarkonia meson can be disrupted through interactions with co-moving hadrons [@Capella]. In collisions where quark-gluon plasma is formed, quarkonia suppression due to color screening is expected to be a large effect [@MatsuiSatz].
The PHENIX collaboration has previously analyzed data on quarkonia production in $d$+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV [@PPG109; @PPG125] to study these phenomena. In this small collision system, quarkonia suppression via color screening in a deconfined plasma is not expected to be the dominant effect. There have since been multiple calculations which can describe these results, by incorporating the previously described effects to varying degrees [@RamonaShadowing; @MaCGC; @Arleo2013; @Ferreiro_comovers]. The existing data are not sufficient to discriminate between these calculations and allow us to quantify the influence of each suppression mechanism. Therefore, additional measurements needed.
To this end, the PHENIX collaboration has measured $J/\psi$ production in $p$+Al, $p$+Au, and $^3$He+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. By changing the target nucleus (Al versus Au), the level of modification to the nucleon’s parton distribution function is expected to vary, with larger effects expected in the larger Au nucleus. The path length through the nucleus that the charm quarks travel also changes with nuclear size, so energy loss may be affected. This allows us to experimentally vary the initial state effects that charm quarks may be subject to. By changing the projectile between $p$, $d$, and $^3$He, we can experimentally vary the number of produced particles that co-move with the charmonium state outside the nucleus [@PPG221], which may affect the level of breakup that occurs in the late stages of the collision.
Measurement
===========
The PHENIX detector has two muon spectrometer arms [@MuonNIM], which cover the forward and backward rapidity intervals 1.2 $ < |y|< $ 2.2 and can measure $J/\psi$ through decays to $\mu^{+} \mu^{-}$ down to $p_{T}$ = 0. Muons produced in the collision travel through a hadron absorber and into a set of wire chambers that provide muon tracking. Further tracking and muon identification are provided by the MuID, which is layers of streamer tubes interleaved with additional steel absorber panels. To be considered in this analysis, muons must pass through all absorber material and be detected in the fourth layer of MuID tubes. The acceptance and efficiency of the detectors are found by running a full GEANT4 [@GEANT4] simulation of the PHENIX muon system.
The $^3$He+Au data set was recorded in 2014, and the $p$+Al and $p$+Au samples were recorded in 2015, all at the same center of mass energy per nucleon of 200 GeV. The two muon spectrometer arms allow data to be recorded in both the forward and backward rapidity ranges simultaneously. PYTHIA simulations show that in the forward (p,d, or$^3$He -going direction), $J/\psi$ production samples an $x$ range in the target nucleus of $x\approx$ 5$\times$10$^{-3}$, which is in the shadowing region of the gluon parton distribution function. In the backwards direction, the $x$ range sampled is in the anti-shadowing region near $x\approx$ 8$\times$10$^{-2}$.
Results and Discussion
======================
The $p_{T}$ dependence of the nuclear modification factor of $J/\psi$ produced in minimum bias $p$+Al collisions is shown in Fig. \[fig:RpAl\]. The error bars (boxes) represent the statistical (systematic) uncertainty on the points, and the global systematic due to the uncertainty on the $p+p$ denominator is shown as a black box around unity. The backward and forward rapidity results are shown in the left and right, respectively. In both cases, we see some evidence for a small amount of enhancement, although within uncertainties the nuclear modification factor is consistent with one. In this small system, there are no major effects observed on $J/\psi$ production.
![The nuclear modification factor of $J/\psi$ mesons produced at backward (left) and forward (right) rapidity in $p$+Al collisions at 200 GeV.[]{data-label="fig:RpAl"}](pAl_S_logo "fig:"){width="49.50000%"} ![The nuclear modification factor of $J/\psi$ mesons produced at backward (left) and forward (right) rapidity in $p$+Al collisions at 200 GeV.[]{data-label="fig:RpAl"}](pAl_N_logo "fig:"){width="49.50000%"}
This is in contrast to the larger $p$+Au and ${^3}$He+Au systems, where significant modification is observed at both forward and backward rapidity. At backward rapidity, in the Au-going direction (left panel of Fig. \[fig:RpAu\]), we see a suppression at low $p_{T}$ that disappears with increasing $p_{T}$. It is interesting to note that the modification is the same (within uncertainties) for both systems, despite the factor of $\sim$2 difference in the number of produced charged particles $dN_{ch}/d\eta$ between these system [@PPG221]. This may suggest that, over this range of $dN_{ch}/d\eta$, late stage breakup of $J/\psi$ is not sensitive to the co-moving particle density within these uncertainties.
At forward rapidity, Fig. \[fig:RpAu\], right panel, there is a similar structure observed. Here $J/\psi$ production samples the shadowing region of the gluon parton distribution function inside the nucleus. Again we see identical behavior for the $p$+Au and $^3$He+Au systems. Given that we are varying the projectile but not the nuclear target, this could suggest that initial state effects inside the nucleus such as energy loss or nuclear shadowing are the dominant effects on $J/\psi$ production in these small systems.
![The nuclear modification factor of $J/\psi$ mesons produced at backward (left) and forward (right) rapidity in $p$+Au and $^3$He+Au collisions at 200 GeV.[]{data-label="fig:RpAu"}](pAuHeAu_S_logo "fig:"){width="49.50000%"} ![The nuclear modification factor of $J/\psi$ mesons produced at backward (left) and forward (right) rapidity in $p$+Au and $^3$He+Au collisions at 200 GeV.[]{data-label="fig:RpAu"}](pAuHeAu_N_logo "fig:"){width="49.50000%"}
We now compare these new results in small systems to existing data on larger collision systems, using PHENIX data recorded in Cu+Cu, Cu+Au, Au+Au, and U+U collisions at RHIC [@PPG071; @PPG163; @PPG119; @PPG172]. The data are shown in the two panels of Fig. \[fig:RpAll\], where the same data is plotted in each rapidity range for symmetric systems. The $p_{T}$-integrated results from small systems are compared with these larger systems as a function of $N_{part}$. Within uncertainties, we observe a similar trend across all systems, with a large suppression developing with increasing system size. Taken together, these data probe a very large range of system size, temperature, and energy density.
![The nuclear modification factor of $J/\psi$ mesons produced at backward (left) and forward (right) rapidity in $p$+Al, $p$+Au, $^3$He+Au, Cu+Cu, Cu+Au, Au+Au, and U+U collisions at 200 GeV.[]{data-label="fig:RpAll"}](all_bkwd_prelim "fig:"){width="49.50000%"} ![The nuclear modification factor of $J/\psi$ mesons produced at backward (left) and forward (right) rapidity in $p$+Al, $p$+Au, $^3$He+Au, Cu+Cu, Cu+Au, Au+Au, and U+U collisions at 200 GeV.[]{data-label="fig:RpAll"}](all_fwd_prelim "fig:"){width="49.50000%"}
Summary
=======
Quarkonia measurements have been a cornerstone of the PHENIX physics program from the earliest days of RHIC. With these results, we have additional data that can constrain models of charmonium production in small systems, which allows us to quantify effects which are crucial for fully understanding the role color screening may play in the significant $J/\psi$ suppression observed in large systems. While PHENIX is no longer recording data, analysis of the large Au+Au data sets recorded in 2014 and 2016 is currently underway.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study magnon modes in the presence of a vortex in a circular easy–plane ferromagnet. The problem of vortex–magnon scattering is investigated for partial modes with different values of the azimuthal quantum number $m$ over a wide range of wave numbers. The analysis was done by combining analytical and numerical calculations in the continuum limit with numerical diagonalization of adequately large discrete systems. The general laws governing vortex–magnon interactions are established. We give simple physical explanations of the scattering results: the splitting of doublets for the modes with opposite signs of $m$, which takes place for the long wavelength limit, is an analogue of the Zeeman splitting in the effective magnetic field of the vortex. A singular behavior for the scattering amplitude, $\sigma_m\propto k$, takes place as $k$ diverges; it corresponds to the generalized Levinson theorem and can be explained by the singular behavior of the effective magnetic field at the origin.'
author:
- 'Denis D. Sheka'
- 'Ivan A. Yastremsky'
- 'Boris A. Ivanov'
- 'Gary M. Wysin'
- 'Franz G. Mertens'
title: |
Amplitudes for magnon scattering by vortices\
in two–dimensional weakly easy–plane ferromagnets
---
Introduction {#sec:introduction}
============
It is now firmly established that vortices play an important role in the condensed matter physics of two–dimensional (2D) systems with continuously degenerate ground states. In particular, the presence of vortices in 2D easy–plane (EP) magnets gives rise to the Berezinskiĭ–Kosterlitz–Thouless phase transition. [@Berezinsky72; @Kosterlitz73; @Kosterlitz74] Vortices play an essential role in the thermal and dynamical properties of 2D magnets, for a review see Ref. . A vortex signature in dynamical response functions can be observed experimentally; e.g. translational motion of vortices leads to a central peak in the dynamical correlation functions. [@Wiesler89] This peak had been predicted predicted both by a vortex gas theory and by combined Monte Carlo – spin dynamics simulations. [@Mertens89]
Recently there has been renewed attention to the problem of magnetic vortices for finite-size magnetic particles, especially their dynamics. It becomes very important in connection with novel composite magnetic materials such as magnetic dot arrays. [@Thurn00; @Cowburn00; @Shinjo00; @Sun00; @Demokritov01; @Erdin02; @Cowburn02] These magnetic dots are submicron–sized islands made from soft magnetic materials on a nonmagnetic substrate. They are important from a practical standpoint as high–density magnetic storage devices, [@Grimsditch98] and are interesting as fundamentally new objects in the basic physics of magnetism. The distribution of magnetization in such a dot is quite nontrivial: when the dot size is above a critical value, an inhomogeneous state with an out–of plane magnetic vortex occurs, which is stable due to competition between exchange and dipole interactions. [@Hubert98] It is expected that these nonuniform states will drastically change the dynamic and static properties of a dot in comparison with a uniformly magnetized magnetic disk.Recent experiments [@Novosad02a; @Park03] verify such properties; in particular, a mode with anomalously low frequency was detected,[@Park03] see also Refs. .
The general properties of vortex dynamics are intimately connected to the problem of vortex–magnon interactions. Usually this problem has been studied numerically for discrete models, mainly for circular samples cut from large lattice systems. [@Wysin94; @Wysin95; @Wysin96; @Wysin96a; @Ivanov98; @Wysin01; @Kovalev03a; @Zagorodny03] An analytical description of the problem in the framework of the continuum model has been proposed recently for different 2D magnets. [@Ivanov95g; @Ivanov98; @Ivanov99; @Sheka01; @Ivanov02] The most important effect of the vortex–magnon interaction is an excitation of certain magnon modes due to vortex motion and vice versa. Because the magnons in the EP ferromagnet have a gapless dispersion law, a possible Larmor dynamics of the vortex center is strongly coupled with a magnon cloud; [@Kovalev03a] therefore the corresponding motion has a non–Newtonian form. [@Ivanov98]
In this paper we consider the magnon modes which exist in a 2D Heisenberg EP ferromagnet in the presence of a vortex. We apply different, sometimes new, methods of analytical and numerical investigation, in order to extend the research of Ref. , presenting a wider range of results for the magnon scattering amplitude. In Sec. \[sec:model\] we demonstrate that the vortex acts on magnons in two ways. First of all, it provides for coupling between different directions of magnetization precession; the magnon modes are described by a “generalized” Schrödinger equation. Secondly, the problem naturally possesses an effective magnetic field, whose global properties are caused by the soliton topological charge. The scattering problem is treated both numerically and analytically for a wide range of wave numbers. The numerical study is carried out using two different approaches: solving the eigenvalue problem for the continuum limit (in the weak EP anisotropy limit), and extracting the scattering data from numerical diagonalization of discrete systems, see Sec. \[sec:scat-num\]. The analytical study of the scattering problem is developed (Sec. \[sec:scat-analyt\]) using both the long– and short–wavelength approximations. In contrast to the previous work [@Ivanov98] we describe analytically the splitting phenomenon of the doublets of magnon modes with opposite signs of the azimuthal quantum number, and give a physical picture of this effect: an effective magnetic field causes the Zeeman splitting of the magnon levels, see Sec. \[sec:k<<1\]. The singular behavior of the scattering amplitude is predicted in Sec. \[sec:k>>1\] for the short wavelength limit. This feature is caused by the specific singular effective magnetic field at the origin (vortex core); this study verifies the generalized version of the Levinson theorem, which we have established recently in Ref. for potentials with inverse square singularities.
Model and magnon modes {#sec:model}
======================
We consider the classical 2D Heisenberg ferromagnet (FM) with the Hamiltonian $$\label{eq:Hamiltonian}
\mathcal{H} = -J\sum_{\left({\bm{n}},{\bm{n}}'\right)}\left[
{\bm{S}}_{{\bm{n}}}\cdot{\bm{S}}_{{\bm{n}}'} -(1-\lambda){S}_{{\bm{n}}}^z
{S}_{{\bm{n}}'}^z \right],$$ where the spins ${\bm{S}}_{{\bm{n}}}$ are classical vectors on a square lattice with the lattice constant $a$. Here $\left({\bm{n}},{\bm{n}}'\right)$ denotes nearest–neighbor lattice sites, $J>0$ is the exchange integral, and $\lambda\in[0,1)$ describes easy–plane anisotropy.
We consider the continuum dynamics of the Heisenberg ferromagnet, which is adequate for model in the small–anisotropy case ($1-\lambda\ll1$). In a continuum limit the dynamics of the FM is described by Landau–Lifshitz equations for the normalized magnetization, $$\label{eq:m}
{\bm{m}}({\bm{r}},t) = \frac{{\bm{S}}({\bm{r}},t)}{S} =
\Bigl(\sin\theta\cos\phi;\sin\theta\sin\phi;\cos\theta\Bigr).$$ The equations of motion result from the Lagrangian $$\label{eq:Lagrangian}
L = \frac{S}{a^2}\int \text{d}^2x(1-\cos\theta)\frac{\partial\phi}{\partial t}
- E[\theta,\phi],$$ with the energy functional $$\label{eq:Energy}
E[\theta,\phi] = \frac{JS^2}{2}\!\!\int\!\! \text{d}^2x\! \left[\left(\nabla
\theta \right) ^2+\left( \nabla \phi \right) ^2\sin ^2\theta +
\frac{\cos^2\theta}{r_v^2}\right].$$ Here $r_v=\frac a2\sqrt{\lambda/(1-\lambda)}$ is the characteristic length scale (“magnetic length”). One should note that the strength of the weak easy-plane anisotropy ($\lambda\approx 1$), determines the magnetic length scale $r_v$; in this continuum analysis, results will depend on lengths scaled by $r_v$, and have no explicit dependence on the anisotropy strength.
The ground state of the magnet is continuously degenerate and isotropic within the easy plane (EP). The simplest elementary linear excitations of EP FMs that arise on the homogeneous background are the magnons belonging to a continuous spectrum. They have the form of elliptically polarized waves; if one chooses the homogenous spin distribution along the $x$–axis, then the spin wave takes the form: $$\label{eq:spin-wave}
m_x=1,\quad m_z+{i} m_y \cdot \frac{kr_v}{\sqrt{1+k^2r_v^2}}\propto \exp({i}
kx-{i} \omega t).$$ The dispersion law has the gapless form: $$\label{eq:dispersion_law} \omega({\bm{k}}) =
ck\sqrt{1+k^2r_v^2},$$ where $c=2aJS\sqrt{1-\lambda}$ is the characteristic magnon speed, ${\bm{k}}$ is the magnon wave vector, and $k=|{\bm{k}}|$ is its magnitude.
The simplest nonlinear excitation in the system is an out–of–plane (OP) vortex [@Gouvea89a; @Kosevich90] $$\label{eq:vortex}
\begin{split}
&\phi\equiv\phi_0=\varphi _0 + q \chi,\quad \theta=\theta_0(\rho),\\
&\theta_0(0)=\frac{1-p}{2}\pi,\quad \theta_0(\infty)=\pi/2,
\end{split}$$ where $\varphi_0$ is an arbitrary angle due to the EP symmetry, $\rho\equiv|{\bm{r}}|/r_v$ and $\chi$ are dimensionless polar coordinates in the plane of the magnet, the vorticity $q\in\mathbb{Z}$ plays the role of a $\pi_1$ topological charge, and the polarization $p=\pm1$ is connected with a $\pi_2$ topological charge (the Pontryagin index): $$\label{eq:Pontryagin-index}
Q=\frac{1}{4\pi}\int\mathcal{Q}\;\text{d}^2x,\qquad
\mathcal{Q}=\frac12\epsilon_{ij}\
{\bm{m}}\cdot\left(\partial_i{\bm{m}}\times\partial_j{\bm{m}}\right).$$ We term $\mathcal{Q}$ the gyrocoupling density, following @Thiele73; it has a sense as the density of the topological charge, see @Nicos91. For the vortex configuration the gyrocoupling density can be represented as $$\label{eq:Q}
\mathcal{Q}=\frac{q\sin\theta_0\cdot\theta_0^\prime}{\rho},$$ hence the Pontryagin index takes on half–integer or integer values $Q=qp/2$. Note that the presence of a nontrivial $\pi_2$–topological charge directly results in the gyrotropical dynamics of the vortex, which conserves the gyrovector ${\bm{G}}=Q\cdot2\pi\hslash S a^{-2}{\bm{e}}_z$.
The function $\theta_0$ is the solution of an ordinary differential equation, which can only be solved numerically. [@Kosevich90; @Bar'yakhtar93] Without an external magnetic field two oppositely polarized vortices are energetically equivalent; for definiteness we set $p=+1$.
To analyze magnons on the vortex background, we use a formalism and set of coordinates developed in Ref. , setting up the problem in terms of local Cartesian spin components. The unperturbed spins of the static vortex structure, ${\bm{m}}_0$, define *local* polar axes ${\bm{e}}_3$, different at every site, specifically, ${\bm{S}}_0({\bm{r}},t)=S{\bm{e}}_3$.
It is to be understood that these axes depend on the site chosen. The magnetic fluctuations occur perpendicular to these local axes, suggesting the definition of other axes, ${\bm{e}}_2$ being chosen along the direction of ${\bm{e}}_z
\times {\bm{e}}_3$, and ${\bm{e}}_1 = {\bm{e}}_2 \times {\bm{e}}_3$, to complete the mutually perpendicular set. One supposes that a dynamically fluctuating spin has small deviations along the ${\bm{e}}_1$ and ${\bm{e}}_2$ axes so that a spin is written as $$\label{eq:m1m2-def} {\bm{S}}({\bm{r}},t) = S\left({\bm{e}}_3 +m_1 {\bm{e}}_1+m_2{\bm{e}}_2\right).$$ The fields $m_1$ and $m_2$ have a simple physical significance, which can be seen if a given spin is supposed to have small deviations $\varphi$ and $\vartheta$ away from the vortex structure, determined by azimuthal and polar spherical angles, $\phi_0$ and $\theta_0$. We write $$\label{eq:var-def} \begin{split}
\frac{{\bm{S}}({\bm{r}},t)}{S}&= \cos(\theta_0+\vartheta){\bm{e}}_z +
\sin(\theta_0+\vartheta)\\ &\times \left[ \cos(\phi_0+\varphi)
{\bm{e}}_x+\sin(\phi_0+\varphi){\bm{e}}_y \right].
\end{split}$$ Linearizing in $\varphi$ and $\vartheta$, and using the definitions of $\{
{\bm{e}}_1, {\bm{e}}_2, {\bm{e}}_3 \}$, comparison of Eqs. and shows that $$\label{eq:m12-via-var} m_1=\vartheta, \quad m_2= \varphi \sin\theta_0.$$ Thus, the $m_1$ field measures spin rotations moving towards the polar (${\bm{e}}_z$) axis and the $m_2$ field measures spin rotations projected onto the $xy$–plane. In the absence of the vortex, we have $\theta_0=\pi/2$, $\phi_0=0$, and such oscillations correspond to the free magnons in the form .
The linearized equations for $m_1$ and $m_2$ can be described by a single complex–valued function $\psi({\bm{r}},t) = m_1 + {i} m_2$, which obeys the differential equation $$\label{eq:Gen-Schroedinger}
{i}\partial_\tau \psi=H\psi+W\psi^*,\qquad H=\left(-{i}{\bm{\nabla}}-{\bm{A}}\right)^2 + U,$$ with the “potentials”
\[eq:potentials\] $$\begin{aligned}
\label{eq:Gen-U}
U(\rho)&=&\frac12\sin^2\theta_0\left(1-\frac{q^2}{\rho^2}\right) -
\cos^2\theta_0 -
\frac{{\theta'_0}^2}{2},\\
\label{eq:Gen-W} W(\rho)&=&\frac12\sin^2\theta_0\left(1-\frac{q^2}{\rho^2}\right) +
\frac{{\theta'_0}^2}{2},\\
\label{eq:Gen-A} {\bm{A}}(\rho) &=& -\frac{q\cos\theta_0}{\rho}\cdot{\bm{e}}_\chi.\end{aligned}$$
Here we use the dimensionless coordinate variable $\rho=|{\bm{r}}|/r_v$, dimensionless time variable $\tau = t\cdot c/r_v$, and the operator ${\bm{\nabla}}=r_v\partial_{{\bm{r}}}$; prime denotes $\text{d}/\text{d}\rho$.
Let us note that the vector ${\bm{A}}$ acts in the Schrödinger–like operator $H$ in the same way as the vector–potential acts in the Hamiltonian of a charged particle. Then it is possible to conclude that there is an effective magnetic flux density $$\label{eq:B-eff}
{\bm{B}}={\bm{\nabla}}\times{\bm{A}}={\bm{e}}_z\cdot\frac{q\sin\theta_0\cdot
\theta'_0}{\rho}.$$ Note that the effective magnetic flux density can easily be rewritten through the gyrocoupling density as ${\bm{B}}=\mathcal{Q}\cdot{\bm{e}}_z$. Therefore the total flux is determined by the nontrivial $\pi_2$ topology of the vortex configuration. On first view when exploiting this analogy it is possible to look for the Aharonov–Bohm phenomenon for the scattering problem, because this magnetic flux density is localized in the region of the vortex core. However, one can see that the total magnetic flux $$\label{eq:flux}
\Phi=\int B_z \text{d}^2x = 4\pi Q = qp\Phi_0$$ is an integer multiple of the flux quantum $\Phi_0=2\pi$, so there is no Aharonov–Bohm scattering picture for the system.
A differential equation like is not a unique property of the vortex–magnon problem in the EP FM only. It appears for different kinds of anisotropy: it describes magnon modes on the soliton background in the easy–axis [@Sheka01] and isotropic magnets [@Ivanov99]. Note that for the specific case of an isotropic system with an exact analytical soliton solution of the Belavin–Polyakov type, the potential $W$ disappears, so the magnon modes satisfy the usual Schrödinger like equation ${i}\partial_\tau \psi = H\psi$, which describes, e.g., the quantum mechanical states for a charged particle in the axially symmetric potential $U(\rho)$ under the action of an external magnetic field with a vector potential ${\bm{A}}$.
For the anisotropic case, when $W\neq0$, the problem has important unusual properties, which are absent for the Belavin–Polyakov case. More generally, there appear properties which are forbidden for the usual quantum mechanics. In particular, an effective discrete Hamiltonian of the system is not necessarily Hermitian; in Refs. some constructive methods were elaborated to avoid these problems. Nevertheless we will discuss the features of Eq. in order to understand why the standard quantum mechanical intuition could fail.
The standard quantum mechanical equation ${i}\partial_\tau \psi = H\psi$ allows the conservation law $\partial_\tau|\psi|^2=-{\bm{\nabla}}\cdot{\bm{j}}$ for the current $$\label{eq:current}
{\bm{j}} = {i} \left(\psi{\bm{\nabla}}\psi^* - \psi^\star{\bm{\nabla}}\psi\right)
+ 2|\psi|^2{\bm{A}}.$$ The generalized Schrödinger–like equation with $W\neq0$ violates this conservation law, namely: $$\label{eq:div-j}
\partial_\tau|\psi|^2=-{\bm{\nabla}}\cdot{\bm{j}}-{i} W\left({\psi^*}^2 -
\psi^2\right).$$
Nonconservation of probability density has posed some problems in the passage from standard quantum mechanics to old pre–Feynmann quantum electrodynamics. The reason is that Eq. is formulated neither for a Hermitian, nor a linear operator; the last statement is due to the broken symmetry under the rescaling $\psi\to\lambda\psi$ with $\lambda\in\mathbb{C}$. There exists an analogy with relativistic theory: there can appear solutions with positive and negative energy in the passage from the Klein–Gordon to Dirac equation. In fact, our problem has the same origin. Let us reformulate the problem as an equation second order in time. One can calculate that the Klein–Gordon–like equation $$\label{eq:KG}
-\partial_{\tau\tau}\psi = \left(H^2-W^2\right)\psi$$ is valid far from the vortex center. What is important is that Eq. contains a Hermitian operator (similar arguments were used in Ref. ). Therefore, the eigenvalue problem (EVP) for $\omega^2$, not for $\omega$, is more appropriate for this system; the only problem is to separate solutions with positive and negative $\omega$. Note, that there appear $4^{\rm th}$ order operators with respect to the space coordinate, which causes the presence of master and slave functions in the solution, see below.
Such a problem, as well as a problem with nonconserved number of particles (probability amplitude), appears in the theory of a weakly nonideal Bose gas. It results, in fact, in the separation of positive and negative energy solutions under u–v Bogolyubov transformations. [@LandauIX]
Following this scheme we need to generalize the u–v transformation to the nonhomogeneous case.
We apply the partial–wave expansion, using the *ansatz* [@Ivanov98] $$\label{eq:Psi-via-u&v}
\begin{split}
\psi({\bm{r}},t)&=\sum_{\alpha} \left[u_\alpha(\rho)e^{{i}\Phi_\alpha} +
v_\alpha(\rho)e^{-{i}\Phi_\alpha}\right],\\
\Phi_\alpha(\chi,t) &= m\chi-\omega_\alpha t + \eta_m = m\chi-\varOmega_\alpha
\tau + \eta_m,
\end{split}$$ where $\alpha=(k,m)$ is a full set of eigenvalues, $m\in\mathbb{Z}$ being azimuthal quantum numbers, the $\eta_m$ are arbitrary phases, and $\varOmega=\omega r_v/c$ are dimensionless frequencies. This expansion leads to the following EVP for the radial eigenfunctions $u$ and $v$ (the index $\alpha$ will be omitted in the following): $$\label{eq:EVP-u&v} \textsf{H}\bm{\bigl|\varPsi\bigr>} =
\varOmega\bm{\bigl|\varPsi\bigr>}, \; \textsf{H}=
\begin{Vmatrix}H_+ & W \\ -W & -H_- \\ \end{Vmatrix},\;
\bm{\bigl|\varPsi\bigr>} =
\begin{Vmatrix} u \\ v \end{Vmatrix}.$$ Here $H_\pm = -\nabla_\rho^2+\mathcal{U}_0+1/2 \pm V$ is the 2D radial Schrödinger–like operator with the “potentials” $$\begin{aligned}
&\mathcal{U}_0(\rho)= U(\rho)+{\bm{A}}^2+\frac{m^2}{\rho^2}-\frac12 \nonumber \\
\label{eq:U} &= \frac{q^2+m^2}{\rho^2} -\frac{3q^2\sin^2\theta_0}{2\rho^2}
-\frac{3\cos^2\theta_0}{2} -
\frac{{\theta'_0}^2}{2},\\
\label{eq:V} &V(\rho) =-\frac{2m\left({\bm{A}}\cdot{\bm{e}}_\chi\right)}{\rho}
=\frac{2qm\cos\theta_0}{\rho^2},\end{aligned}$$ $\nabla_\rho^2=\text{d}^2/\text{d}\rho^2+(1/\rho)\text{d}/\text{d}\rho$ is the radial Laplace operator. In spite of the fact that the EVP is formulated for the Schrödinger operators $H_\pm$, this EVP is different in principle from the usual set of coupled Schrödinger equations, which is widely used, e.g. for the description of multichannel scattering. [@Newton82] The reason is that the matrix Hamiltonian $\textsf{H}$ is not Hermitian for the standard metric, for details see Ref. . To avoid this problem we introduce a corresponding bra–vector by the definition $$\label{eq:bra-vector}
\bm{\bigl<\varPsi\bigl|} = \bigl\| u\;; -v \bigr\|.$$ The Hilbert space for the ${\bm{\varPsi}}$–function has an indefinite metric, $$\label{eq:norm}
\bm{\bigl<\varPsi\bigr|\varPsi\bigr>} = (u|u)-(v|v),$$ where $(u|v)=\int_0^\infty u(\rho)v(\rho)\rho \text{d}\rho$ is the standard scalar product. By introducing such a Hermitian product, it is possible to define the standard energy functional, cp. Ref. : $$\label{eq:E-funct}
\mathcal{E}[u,v]=\bm{\bigl<\varPsi\bigl|\textsf{H}\bigr|\varPsi\bigr>} =
(u|H_+|u)+2(u|W|v)+(v|H_-|v).$$
Let us mention that Eq. is invariant under the conjugations $\varOmega\to-\varOmega$, $m\to-m$, and $u\leftrightarrow v$. In a classical theory we can choose either sign of the frequency; but in order to make contact with quantum theory, with a positive frequency and energy $\mathcal{E}_k=\hslash\omega_k$, we will discuss the case $\varOmega>0$ ($\omega>0$) only. Thus there are two different equations for the opposite signs of $m$. However, in the limiting case of the “zero modes” with $\varOmega=0$, the system again is invariant under conjugations $m\to-m$. For example, one of the zero modes, the so–called translational mode with $m=+1$, has the form $$\label{eq:zero-mode-m=+1}
u_{m=+1} = \frac{\sin\theta_0}{\rho} - \theta'_0,\qquad v_{m=+1} =
\frac{\sin\theta_0}{\rho} + \theta'_0,$$ which describes the position shift of the soliton. Because of the degeneration of the EVP at $\varOmega=0$, it leads to the existence of a zero mode with $m=-1$; the eigenfunction of this mode can be expressed just from under the conjugation $u\leftrightarrow v$. We use here notations for mode indices as in Refs. ; note that the mode with $m=+1$ corresponds in our notation to the mode with $m=-1$ in the notations of Refs. .
It should be stressed that the picture is quite different for the special case $W=0$, which corresponds to the isotropic magnet. [@Ivanov95g; @Ivanov99] Here we have two uncoupled equations for the functions $u$ and $v$. One of the equations (for the eigenfunction $v$) has the negative eigenvalue $-\varOmega$, from which it necessarily results that $v\equiv0$. In this special case the zero modes have the form $u_{+1}\propto \theta_0'$ and $v_{+1}=0$. Therefore the zero mode with $m=-1$ cannot be obtained by the simple conjugation. It explains the difference between the collective dynamics of the soliton in isotropic magnets, where it is enough to take into account only the mode with $m=+1$, and the EP FM, where translational modes with $m=-1$ and $m=+1$ must be taken into account. Nevertheless, the roles of the modes with $m=-1$ and $m=+1$ are not equal, for details see Sec. \[sec:Levinson\].
Scattering problem: numerical results {#sec:scat-num}
=====================================
Continuum approach {#sec:scat-cont}
------------------
We intend to describe the scattering of magnons by a vortex. However the EVP is not adjusted for the scattering problem, because it does not provide the asymptotic independence of the equations at infinity. To solve the problem it is convenient to make a unitary transformation of the eigenvector $\bm{\bigl|\varPsi\bigr>}$, $$\label{eq:tilde-Psi}
\bm{\bigl|\widetilde{\varPsi}\bigr>} = \textsf{A} \bm{\bigl|\varPsi\bigr>},
\quad \textsf{A}=
\begin{Vmatrix}
\cos\varepsilon & -\sin\varepsilon \\
\sin\varepsilon & \cos\varepsilon \\
\end{Vmatrix},
\; \bm{\bigl|\widetilde{\varPsi}\bigr>} = \begin{Vmatrix} \tilde{u} \\
\tilde{v}
\end{Vmatrix}.$$ The angle $\varepsilon$ of this unitary transformation is defined by the expression $$\label{eq:eps}
\tan2\varepsilon = \frac{1}{2\varOmega}.$$ Then we obtain the following partial differential equation for the function $\bm{\bigl|\widetilde{\varPsi}\bigr>} $:
\[eq:PDE4tilde-Psi+tilde-H\] $$\begin{aligned}
\label{eq:PDE4tilde-Psi}&\widetilde{\textsf{H}} \bm{\bigl|\widetilde{\varPsi}\bigr>} = \bm{\Lambda}
\bm{\bigl|\widetilde{\varPsi}\bigr>}, \; \widetilde{\textsf{H}} = \textsf{H}_0
+ \widetilde{\textsf{V}}, \;\bm{\Lambda}=
\text{diag}\left(\kappa^2;\varkappa^2\right),\\
\label{eq:H0} &\textsf{H}_0 =\text{diag}\left(\mathcal{H}_0;-\mathcal{H}_0\right),\quad
\mathcal{H}_0 = -\nabla_\rho^2+\mathcal{U}_0,\\
\label{eq:tilde-V} &\widetilde{\textsf{V}} = \left[V+\bm{g}\cdot\left(W-1/2\right)\right]
\textsf{A}^{-2},\end{aligned}$$
where $\bm{g}= \bigl\|\begin {smallmatrix}0&1\\-1&0\end{smallmatrix}\bigr\|$ is a metric spinor, the dimensionless wave number is $\kappa = kr_v$, and $\varkappa=\sqrt{\kappa^2+1}$.
First let us consider the magnon spectrum in the absence of a vortex (free fields). Without a vortex ($q=0$, $\theta_0=\pi/2$), Eqs. are uncoupled, which results in free magnons, $$\label{eq:modes_free}
\begin{split}
\tilde{u}_m(\rho) &\propto J_{|m|}(\kappa \rho) \underset{\kappa
\rho\gg1}{\sim} \sqrt{\frac{2}{\pi\kappa\rho}}\cdot\cos\left(\kappa \rho -
\frac{|m|\pi}{2} -\frac{\pi}{4}\right),\\
\tilde{v}_m(\rho) &=0,
\end{split}$$ where $J_m$ are Bessel functions. The free modes $\tilde{u}_m$ play the role of the partial cylinder waves of a plane spin wave $$\label{eq:plain_wave}
\exp\left({i}{\bm{k}}\cdot{\bm{r}}-{i}\omega t\right) = \sum_{m=-\infty}^\infty
{i}^mJ_m(\kappa \rho)e^{{i} m\chi-{i} \omega t}.$$
To describe magnon solutions in the presence of a vortex, one should note that far from the vortex center the potential $\widetilde{\textsf{V}}$ tends to zero, so the Eqs. become uncoupled, $$\label{eq:PDE-as}
\left( \nabla_\rho^2 + \kappa^2 \right) \tilde{u} =0,\; \left( \nabla_\rho^2 - \varkappa^2 \right) \tilde{v} =0,\; \rho\gg\max\left(1;\frac1\kappa;\frac1\varkappa\right)$$ with asymptotically independent solutions:
\[eq:u&v4x>>1\] $$\begin{aligned}
\tilde{u}_m(\rho) & \sim \frac{C_1}{\sqrt{\rho}}\cdot e^{{i}\kappa\rho} +
\frac{C_2}{\sqrt{\rho}}\cdot e^{-{i}\kappa\rho}\nonumber\\
\label{eq:u4x>>1} &\propto \frac{1}{\sqrt{\rho}}\cos\left(\kappa
\rho - \frac{|m|\pi}{2} -\frac{\pi}{4} + \delta_m \right),\\
\label{eq:v4x>>1} \tilde{v}_m(\rho) & \sim \frac{C_3}{\sqrt{\rho}}\cdot e^{\varkappa\rho} +
\frac{C_4}{\sqrt{\rho}}\cdot e^{-\varkappa\rho}.\end{aligned}$$
The scattering results in the quantity $\delta_m\equiv\delta_m(\kappa)$; it can be interpreted as the scattering phase shift, determining the intensity of the magnon scattering due to the presence of the vortex. Sometimes it is useful to introduce the scattering amplitude, $\sigma_m=-\tan\delta_m$. Using this notation, the oscillatory solution can be rewritten in the following form $$\label{eq:u-via-J&Y}
\tag{\ref{eq:u4x>>1}$'$} \tilde{u}_m(\rho) \propto J_{|m|}(\kappa \rho) + \sigma_m \cdot Y_{|m|}(\kappa
\rho).$$ where $Y_{|m|}$ are Neumann functions. Let us stress that the solution is valid only in the sense of the asymptotic form . As it follows from Eq. , the function $\tilde{v}$ has an exponential behavior, $$\label{eq:v-via-K&I}
\tag{\ref{eq:v4x>>1}$'$} \tilde{v}_m(\rho) \propto K_{|m|}(\varkappa \rho) + \gamma_m \cdot
I_{|m|}(\varkappa \rho)\propto \frac{e^{-\varkappa\rho}}{\sqrt{\rho}} +
\gamma_m\cdot \frac{e^{\varkappa\rho}}{\sqrt{\rho}},$$ where $K_{|m|}$ and $I_{|m|}$ are MacDonald and modified Bessel functions, respectively; at the same time $\tilde{u}$ yields oscillatory solutions. Naturally, the real modes have an oscillatory form here; we will use this fact below for the numerical analysis. It means that the function $\tilde{u}$ becomes a master function in the Eq. , while $\tilde{v}$ is a slave (note, that we choose $\varOmega>0$). This mirrors the difference between Eq. and a usual set of Schrödinger equations.
The scattering amplitude, or, equivalently, the phase shift, contains all information about the scattering processes. In particular, the general solution of the scattering problem for a plane wave can be expressed in the form, cp. Eq.
\[eq:G4plane-wave\] $$\label{eq:G4plane-wave1}
m_2-{i} m_1\cdot \frac{kr_v}{\sqrt{1+k^2r_v^2}} \propto
e^{{i}{\bm{k}}\cdot{\bm{r}}-{i}\omega t} + \mathcal{F}(\chi)\cdot
\frac{e^{{i}\kappa \rho-{i}\omega t}}{\sqrt{\rho}},$$ where the scattering function has the form [@Ivanov99] $$\label{eq:G4plane-wave2}
\mathcal{F}(\chi) = \frac{\exp(-{i}\pi/4)}{\sqrt{2\pi\kappa}}\cdot
\sum_{m=-\infty}^{\infty} \left(e^{2{i}\delta_m}-1\right)\cdot e^{{i} m\chi}.$$
The total scattering cross section is given by the expression $$\varrho = \int_0^{2\pi}\!|\mathcal{F}|^2\text{d}\chi =
\sum_{m=-\infty}^\infty\! \varrho_m\;,$$ where $\varrho_m = (4/k)\sin^2\delta_m$ are the partial scattering cross sections.
Let us switch to the numerical solution of the scattering problem in the continuum approach. The differential problem to be integrated consists of Eq. and asymptotic conditions at the center of the vortex and at infinity:
\[eq:PDE4num\] $$\begin{aligned}
\label{eq:PDE4num(1)} \widetilde{\textsf{H}} \bm{\bigl|\widetilde{\varPsi}\bigr>} &= \bm{\Lambda}
\bm{\bigl|\widetilde{\varPsi}\bigr>},\\
\label{eq:PDE4num(2)} \bm{\bigl|\widetilde{\varPsi}\bigr>} & \sim \textsf{A}\cdot
\begin{Vmatrix}\epsilon_m\cdot\rho^{|m+1|},\\ \rho^{|m-1|}\end{Vmatrix},\qquad \text{when}\;\rho\ll1\\
\nonumber \bm{\bigl|\widetilde{\varPsi}\bigr>} & \sim
\begin{Vmatrix}
J_{|m|}(\kappa\rho)+\sigma_m\cdot Y_{|m|}(\kappa\rho) \\ K_{|m|}(\varkappa\rho)
\end{Vmatrix},\\
\label{eq:PDE4num(3)} &\qquad\text{when}\;\rho\gg\max\left(1;\frac1\kappa; \frac1\varkappa\right).\end{aligned}$$
The presence of the matrix $\textsf{A}$ in the condition means that the functions $\tilde{u}$ and $\tilde{v}$ are not asymptotically independent even in the lowest approximation. In the next approximation there appears an additional “interaction” between $\tilde{u}$ and $\tilde{v}$, which is realized in the nonunit factor $\epsilon_m$; its value cannot be found through this asymptotic expansion.
We use the one–parameter shooting method, solving Eqs. , as described in Ref. . Choosing the shooting parameter $\epsilon_m$, we “kill” the growing exponent for the function $\tilde{v}_m$ in , where the coefficient $\gamma_m$ should be equal to zero; as a result we have obtained a well–pronounced exponential decay for $\tilde{v}_m\propto K_{|m|}(\varkappa \rho)$, and oscillating solutions for $\tilde{u}_m$. The scattering amplitude was found from these data by comparison with the asymptotes . The results are discussed in Sec. \[sec:scat-disc\].
Discrete approach {#sec:scat-discrete}
-----------------
In the discrete lattice approach, the small amplitude spin fluctuation modes in the presence of a vortex at the center of a finite circular system of radius $R$ are found. The spins occupy sites on a square lattice. We use the formalism and set of local coordinates as described in Sec. \[sec:model\] for the continuum model. Similar to the continuum expression , we describe the dynamically fluctuating spin on lattice site ${\bm{n}}$ as $$\label{eq:S-fluct} \tag{\ref{eq:m1m2-def}$'$} {\bm{S}}_{{\bm{n}}} = S\left({\bm{e}}_3 +m_1 {\bm{e}}_1+m_2{\bm{e}}_2\right),$$ where $m_1$ and $m_2$ measure spin rotations moving towards the polar axis and projected onto the $xy$–plane, respectively, see Eq. .
The spin dynamics equations of motion with an assumed $e^{-i\omega t}$ time dependence were linearized in $m_1$ and $m_2$, leading to an eigenvalue problem requiring numerical diagonalization. We assumed a Dirichlet boundary condition, $m_1=m_2=0$ at the edge of the system studied. For circular systems of radius $R$, we used a Gauss–Seidel relaxation scheme [@Wysin01] to calculate the frequencies and eigenfunctions of some of the lowest eigenmodes with a single vortex present at the system center. Before doing this, the vortex structure was relaxed to an accurate static structure using an energy minimization scheme. The diagonalization is partial; typically only the lowest 20 to 40 eigenstates were found, which substantially reduces the computing time needed, and relaxes constraints on the precision of the calculations. This limited diagonalization, however, gives only modes which have long wavelength spatial variations, which provides for a good comparison with continuum theory.
We considered different values of $\lambda$ close to $1$. Although the continuum limit would be better represented by using $\lambda$ very close to $1$, this could result in a vortex radius $r_v=\frac{a}{2}\sqrt{\lambda/(1-\lambda)}$ easily exceeding the system size that can be treated numerically. Therefore, data were calculated using $\lambda=0.99$, for which $r_v\approx 4.97a$. With this size of vortex length scale, discreteness effects due to the underlying lattice should be unimportant, and still, the vortex structure fits well within the confines of a system with a radius as small as $R\approx 10a$, so that finite size effects should also be negligible.
In general, a given mode has $e^{i m \chi}$ angular dependence on the azimuthal coordinate $\chi$, where $m$ is some integer azimuthal quantum number. In the continuum theory presented in Sec. \[sec:model\], $m$ is a good quantum number, due to rotational invariance. This symmetry is weakly broken on a lattice, but for long-wavelength and lower frequency modes, $m$ can be considered a good quantum number even on a lattice. (Generally, the calculated magnon wavefunctions were sometimes found to be composed of linear combinations of $+m$ and $-m$ components.) The numerically found modes were also characterized by a principle quantum number $n$, being the number of nodes in the wavefunction along the radial direction. For a mode of determined $m$ and $n$, the scattering amplitude $\sigma$ was found by a fitting procedure applied to the calculated eigenfunction for that mode (essentially, finding the ratio of outgoing and incoming waves), see Ref. for details.
In the continuum theory, scattering was analyzed as a function of wavevector $k$, or in terms of the dimensionless $kr_v$. For lattice calculations, the values of $k$ cannot be chosen freely, instead, they are determined by the actual system size. For a mode found to be oscillating at eigenfrequency $\omega$, the wavevector magnitude $k$ associated with the mode was found by supposing ${\bm{k}}=(k,0)$, and inverting the free magnon dispersion relation for the 2D EP FM on a lattice, $$\begin{split}
\omega_{{\bm{k}}}&=4JS\sqrt{(1-\gamma_{{\bm{k}}})(1-\lambda\gamma_{{\bm{k}}})}, \\
\gamma_{{\bm{k}}}&=\frac{1}{2}(\cos k_x + \cos k_y).
\end{split}$$ Therefore, a calculation of the modes for a single lattice size gives only specific values of $\kappa=kr_v$, one value corresponding to each mode. To get a wider and more continuous range of data for comparison with the continuum theory, calculations were carried out on lattices ranging in radius from $R=15a$ to $R=40a$. By plotting results as functions of $kr_v$, for fixed $m$ but from various $n$ and $R$, the data from the different system sizes superimposes smoothly, giving more slowly changing $kr_v$, which is more appropriate for comparison with the continuum limit.
Numerical results {#sec:scat-disc}
-----------------
![ \[fig:sigma-zero\] (Color online) Scattering data for different $m$ for small wave numbers, $kr_v<1.3$: from continuum theory (lines) and from discrete model numerical diagonalization (symbols) in circular square lattice systems of radii $R=15,
20, 25, 30, 35, 40$.](\myfig{1}){width="\columnwidth"}
![ \[fig:sigma\] (Color online) Scattering data for different $m$ for a wide region of wave numbers $k$: from continuum theory (lines) and from discrete model numerical diagonalization (symbols).](\myfig{2}){width="\columnwidth"}
Numerically, we have obtained the data of the vortex–magnon scattering by the two different approaches discussed above: solving the scattering problem using the shooting method for the continuum limit, and extracting the scattering data from numerical diagonalization of finite discrete systems. To be specific, data are presented for scattering from a vortex with unit vorticity, $q=+1$ and positive polarization, $p =+1$. One should note that results for vortex–magnon scattering for modes $m$ from other vortex types, as seen in Eq. , should be depend on the sign of $qpm$. The results are the following:
For all modes the scattering amplitude $\sigma_m(k)$ tends to zero as $k\to0$. In the long–wavelength limit the maximal scattering is related to the modes with $m=\pm1$. Except for the mode with $m=-1$, the scattering amplitude in the long–wavelength limit takes a negative value, see Fig. \[fig:sigma-zero\]. At extremely low values of wave number $\kappa\alt 0.01$, the scattering data contain sets of doublets for modes with opposite signs of $m$. In the long–wavelength limit the doublet splitting appears as a small correction, but the scattering picture changes when $k$ increases. For all modes $\sigma_m(k)$ diverges as $k\to\infty$: the scattering amplitude $\sigma_m(k\to\infty)\to
+\infty$ for all modes with $m\geq-1$, but $\sigma_m(k\to\infty)\to-\infty$ for $m<-1$, see Fig. \[fig:sigma\]. Naturally, there is no real divergence; it means that the physically observed phase shift does not tend to zero at infinity, but to a finite value $\delta_m(k\to\infty)\to\pm\pi/2$. The scattering data are presented in Figs. \[fig:sigma-zero\], \[fig:sigma\]. Comparison with the results of exact diagonalization on finite systems shows very good agreement between the two approaches.
Scattering problem: analytical description {#sec:scat-analyt}
==========================================
Scattering at long wavelength {#sec:k<<1}
-----------------------------
In order to analyze the scattering problem analytically in the long–wavelength limit, we start from the zero–frequency solutions, when $\varOmega=0$. First note that for the special cases $m=0,\pm1$ there exist so–called *half–bound states*. Recall that a zero–frequency solution of the Schrödinger–like equation is called a half bound state if its wave function is finite, but does not decay fast enough at infinity to be square integrable. We will refer to such modes as *half–local modes*. These modes correspond to the translational ($m=\pm1$) and rotational ($m=0$) symmetry of an infinite system, they have an exact analytical form: $$\label{eq:zero-mode-m<=1}
\tilde{u}_m^{(0)}=\frac{q\cdot\sin\theta_0}{\rho^{|m|}},\;
\tilde{v}_m^{(0)}=m\cdot\theta_0^\prime,\qquad m=0,\pm1.$$ Unlike the case of half–local modes with $m=0,\pm1$, all other zero–frequency solutions are nonlocal, and we are not able to construct exact expressions for them, but only the asymptotes for $\rho\gg1$: $$\label{eq:zero-mode-m>1}
\tilde{u}_m^{(0)}\propto \rho^{|m|},\qquad \tilde{v}_m^{(0)}\propto
\frac{e^{-\varkappa\rho}}{\sqrt{\rho}}.$$ Nevertheless, we will see that the knowledge of asymptotic solutions like will be enough to reconstruct the $\kappa$–dependence of the scattering amplitude. In order to solve the scattering problem in the long wavelength limit we apply a special perturbation theory, proposed in Ref. for the modes with $m=\pm1,0$, and extending it for all values of $m$. We construct the asymptotes of such a solution for a small but finite frequency by making the ansatz
\[eq:alpha&beta\] $$\begin{aligned}
\label{eq:alpha}
\tilde{v}(\rho) &=& \tilde{v}_0(\rho)\cdot\Bigl[1 + \kappa\alpha_1(\rho)
+\kappa^2\alpha_2(\rho)\Bigr],\\
\label{eq:beta} \tilde{u}(\rho) &=& \tilde{u}_0(\rho)\cdot\Bigl[1 +
\kappa\beta_1(\rho) + \kappa^2\beta_2(\rho)\Bigr],\end{aligned}$$
here $\alpha_1$, $\beta_1$ and $\alpha_2$, $\beta_2$ are first and second order corrections to the zeroth–solutions, respectively. Let us insert this ansatz into the set of Eqs. , multiply from the left with $\rho\cdot
\bm{\bigl<\widetilde{\varPsi}\bigr|}$ without integrating; then one obtains equations for the first and second order corrections:
$$\label{eq:prime-of-prime}
\begin{split}
&\Bigl[\rho\cdot\left(\alpha_k^\prime\cdot \tilde{v}_0^2 + \beta_k^\prime\cdot
\tilde{u}_0^2\right)\Bigr]^\prime = \Phi_k(\rho),\qquad k=1,2,\qquad
\Phi_1(\rho) = 2\rho\Bigl\{V\left( \tilde{u}_0^2- \tilde{v}_0^2\right) +
2(W-1/2)\tilde{u}_0\tilde{v}_0\Bigr\},\\
&\Phi_2(\rho) = \rho\biggl\{-\tilde{u}_0^2+\tilde{v}_0^2/2+ 2(W-1/2) \left(
\tilde{u}_0^2- \tilde{v}_0^2\right)+2V\left( \tilde{u}_0^2\beta_1 -
\tilde{v}_0^2\alpha_1\right) +
2\tilde{u}_0\tilde{v}_0\bigl[(W-1/2)(\alpha_1+\beta_1)-2V \Bigr]\biggr\}.
\end{split}$$
We are interested in the corrections $\beta_k$, which will give us a possibility to calculate the scattering amplitude. The formal solution of these equations can be written as $$\label{eq:beta-formal}
\begin{split}
\beta_k(\rho) &= \beta(0) + \int_0^\rho \frac{\alpha_k^\prime(\eta)
\tilde{v}_0^2(\eta) \text{d}\eta}{\tilde{u}_0^2(\eta)} + \int_0^\rho
\frac{\text{d}\eta}{\eta \tilde{u}_0^2(\eta)} \int_0^\eta
\Phi_k(\xi)\text{d}\xi.
\end{split}$$
Let us calculate the first order correction $\beta_1$. It is easy to see that the second RHS–term has an exponential decay as $\rho\to\infty$, while the third one has a slow algebraic decay only. Thus, far from the vortex core we have simply $$\label{eq:beta1}
\beta_1(\rho) \simeq \text{const} -\frac{\rho^{-2|m|}}{2|m|} \int_0^\infty
\Phi_1(\xi)\text{d}\xi,$$ valid in the region $\rho \gg 1$.
To calculate the second order correction, $\beta_2$, let us note that the last RHS–term of Eq. is divergent for $\rho\to\infty$, while the integral with $\alpha_2$ has an exponential decay, like the first order correction. To derive the divergent inner integral in Eq. we add and subtract the function $$\label{eq:Phi_2^0}
\Phi_2^{(0)}(\xi) = -\frac{\left[\xi^2\tilde{u}_0^2\right]^\prime}{2(|m|+1)} -
\frac{\left[\sin^2\theta_0\tilde{u}_0^2\right]^\prime}{2|m|}.$$ Then we arrive at an approximation for $\beta_2(\rho)$ in the important region $\rho\gg1$: $$\label{eq:beta2}
\begin{split}
\beta_2(\rho) &\simeq \text{const} -\frac{\rho^2}{4(|m|+1)} - \frac{\ln\rho}{2|m|}\\
&- \frac{\rho^{-2|m|}}{2|m|} \int_0^\infty
\left[\Phi_2(\xi)-\Phi_2^{(0)}(\xi)\right]\text{d}\xi.
\end{split}$$
Now we are in position to compare the magnon amplitude $\tilde{u}_m=\tilde{u}_0(1+\kappa\beta_1+\kappa^2\beta_2)$ with the scattering approach in order to extract the information about the scattering amplitude $\sigma_m(\kappa)$. To describe the scattering problem in the long–wavelength approximation we rewrite the differential problem for large distances $\kappa\rho\gg1$, only considering the terms with $\kappa^2$. In this scattering approach the oscillating function $\tilde{u}_m$ satisfies an equation $$\label{eq:eq4u-small-k}
\left(\nabla_\rho^2+\kappa^2-\frac{\nu^2}{\rho^2}\right)\tilde{u}_m=0,\qquad
\nu^2=m^2-\kappa^2.$$ The solution of this equation can be written as $$\label{eq:u-small-k}
\begin{split}
\tilde{u}_m(\rho)&\propto J_{|\nu|}(\kappa\rho)+\tilde{\sigma}_\nu(\kappa)Y_{|\nu|}(\kappa\rho)\\
&\propto\frac{1}{\sqrt{\rho}}\cos\left(\kappa\rho-\frac{|\nu|\pi}{2}-\frac{\pi}{4}
+\tilde{\delta}_m\right),
\end{split}$$ where the index of the Bessel and the Neumann function is noninteger. It results in a value of $\tilde{\delta}_m$ which differs from the real scattering phase shift $\delta_m$. Using asymptotic expansions and , the desired relation between the phase shift and $\tilde{\delta}_m$ can be written as $$\label{eq:delta-via-tilde-delta}
\delta_m(\kappa)=\tilde{\delta}_\nu(\kappa)+\frac{|m|-|\nu|}{2}\pi.$$ In the lowest order approximation in $\kappa$, the corresponding relation for the scattering amplitudes has the form: $$\label{eq:sigma-via-tilde-sigma}
\sigma_m(\kappa)=\tilde{\sigma}_\nu(\kappa) - \frac{\pi\kappa^2}{4|m|}.$$ To compare the scattering solution with the result of the perturbation theory we can expand the cylindrical functions in powers of the small quantity $|\nu|-|m|$ and represent through the cylindrical functions of integer order $|m|$, as done in Ref. . After that in the region $\kappa\rho\ll1$ we are able to use the asymptotes of the cylindrical functions at the origin; we arrive at the formula $$\label{eq:u-from-scat}
\begin{split}
\tilde{u}_m(\rho) &\simeq \rho^{|m|}\Biggl\{
1-\frac{\kappa^2\rho^2}{4(|m|+1)}-\frac{\kappa^2}{2|m|}\ln\left(\frac{\kappa\rho}{2}\right)
\\ &-\sigma_m\cdot\frac{(|m|!)^2}{\pi|m|}\left(\frac{2}{\kappa\rho}\right)^{2|m|}
\left[1 - \frac{\kappa^2}{4|m|}S_m\right]\Biggr\},\\
S_m &=\gamma+|m|\sum_{n=1}^{|m|-1}\frac{1}{n(|m|-n)},
\end{split}$$ where $\gamma$ is Euler’s constant.
Comparing this expression with the perturbation theory results \[see Eqs. , , \], in the region $1\ll\rho\ll1/\kappa$, where both are valid, we can restore the general dependence of the scattering amplitude in the long–wavelength approximation:
\[eq:sigma4k<<1+A+B\] $$\begin{aligned}
\label{eq:sigma4k<<1}
\sigma_m(k) &=-\mathcal{A}_m\left(\frac{\kappa}{2}\right)^{2|m|} +
m\mathcal{B}_m\left(\frac{\kappa}{2}\right)^{2|m|+1},\\
\label{eq:sigma4k<<1(A)}
\mathcal{A}_m&=\frac{2\pi|m|}{S_n(|m|!)^2}\int_0^\infty\left[\Phi_2(\xi)-
\Phi_2^{(0)}(\xi)\right]\text{d}\xi,\\
\label{eq:sigma4k<<1(B)}
\mathcal{B}_m&=-\frac{\pi}{m(|m|!)^2}\int_0^\infty\Phi_1(\xi)\text{d}\xi.\end{aligned}$$
Eq. solves the scattering problem except for factors $\mathcal{A}_m$ and $\mathcal{B}_m$. These factors can be found by the numerical integration of Eqs. and , using numerical data for $\tilde{u}_0$ and $\tilde{v}_0$. Thus, solving the equations for zero–modes once, we compute the whole dependence $\sigma_m(\kappa)$. Nevertheless, in order to discuss the analytical behavior let us note that for sufficiently large values of $|m|$, we can limit ourselves to the contribution of the term with $\tilde{u}_0^2$ in the function $\Phi_1$ and the term with $\tilde{u}_0^2\beta_1$ in the function $\Phi_2$, see Eq. . To calculate the integrals we need to have more information about the zero–modes. At small distances $\rho\ll1$ the isotropic (exchange) approximation works correctly, which leads to the following solutions:[@Ivanov95g] $$\tilde{u}_0\propto\rho^{|m|}\sin\theta_0,\quad\tilde{v}_0\propto
\rho^{|m|+1}\theta_0^\prime.$$ Such solutions have the correct asymptotic behavior at infinity and at the origin. Our numerical calculations justify the correctness of these assumptions for $m\gg1$; as a result we obtain analytical estimates for these factors: $$\label{eq:coef4k<<1}
\mathcal{A}_m \approx \frac{18(2|m|-1)!}{S_n((|m|-1)!)^2},\quad \mathcal{B}_m
\approx \frac{4\pi(2|m|-1)!}{(|m|!)^2}.$$ To compare the scattering results for different modes, we write explicitly the asymptotic expressions for all modes, taking into account , and asymptotes for half–local modes from Refs.
\[eq:sigma-as-all4k<<1\] $$\begin{aligned}
\label{eq:sigma-as-all4m=0} \sigma_{m=0}(\kappa)&\approx -\frac
\pi2\kappa^2\ln(1/\kappa),\\ \label{eq:sigma-as-all4m=pm1} \sigma_{m=\pm1}(\kappa) &\approx \mp\frac{\pi\kappa}{4},\\
\label{eq:sigma-as-all4|m|>1} \sigma_{m\neq0,\pm1}(\kappa) &\approx
-\mathcal{A}_m\left(\frac{\kappa}{2}\right)^{2|m|} +
m\mathcal{B}_m\left(\frac{\kappa}{2}\right)^{2|m|+1}. \end{aligned}$$
In the main approximation in $\kappa$ the scattering picture contains doublets for modes with opposite signs of $m$ for the modes with $|m|>1$. The splitting of the doublets (the last term in Eq. ) appears in the next order in $\kappa$. The splitting of the doublets for the magnon modes on the vortex background with given $m=\pm n$ was mentioned in the earliest papers on vortex–magnon scattering;[@Wysin95; @Wysin96] but it was explained, in fact, only for $m=\pm1$. [@Ivanov98] Our considerations on the basis of Eq. show that the splitting of the scattering data is the direct analogue of the Zeeman effect for electron states splitting in an external magnetic field. To follow this analogy one can rewrite the splitting constant $ \mathcal{B}_m$ in the form: $$\mathcal{B}_m\propto \int_0^\infty\! \text{d}\xi\;
\xi^{2|m|}\left({\bm{A}}(\xi)\cdot {\bm{e}}_\chi\right),$$ hence the splitting appears only in the effective magnetic field, which is described by the vector potential ${\bm{A}}$.
Using scattering results one can solve the scattering problem for a plane spin wave in the form . In the long–wavelength limit the maximum scattering is related to the translation modes with $m=\pm1$, which gives the scattering function in the form $$\label{eq:ScatF4k<<1-result}
\mathcal{F}(\chi) = \sqrt{\frac{\pi\kappa}{2}}e^{3{i} \pi/4}\sin\chi.$$ In this approximation the scattering is anisotropic, and the total scattering cross section is $\varrho = \pi^2\kappa/4$. To explain the origin of the anisotropic scattering, let us mention that the plane spin wave makes a spin flux, which influences the vortex as a whole, trying to move it by exciting translational modes. It is well–known that the vortex dynamics appears in the gradient of a magnetization field like the magnon flux. [@Nikiforov83] The dynamics of the vortex has a gyroscopical behavior (see for the review ): acting along the $x$–axis, the spin wave causes the translational motion of the vortex along the $y$–axis, which results in .
Scattering problem for short wavelength {#sec:k>>1}
---------------------------------------
![ \[fig:sigma-infty\] (Color online) Scattering data for different $m$ for the short–wavelength limit: from asymptotes of the continuum theory (lines) and from discrete model numerical diagonalization (symbols)](\myfig{3}){width="\columnwidth"}
For large $k$, in the main approximation to lowest order in $1/\kappa$, the scattering problem can be rewritten in the form:
\[eq:PDE4num-WKB\] $$\begin{aligned}
\label{eq:PDE4num-WKB(1)} \widetilde{\textsf{H}} \bm{\bigl|\widetilde{\varPsi}\bigr>} &= -\kappa^2
\bm{\bigl|\widetilde{\varPsi}\bigr>},\\
\label{eq:PDE4num-WKB(2)} \bm{\bigl|\widetilde{\varPsi}\bigr>} & \sim
\begin{Vmatrix}\epsilon_m\cdot\rho^{|m+1|},\\ \rho^{|m-1|}\end{Vmatrix},\qquad \text{when}\;\rho\ll1\\
\label{eq:PDE4num-WKB(3)} \bm{\bigl|\widetilde{\varPsi}\bigr>} & \sim
\begin{Vmatrix}
J_{|m|}(\kappa\rho)+\sigma_m\cdot Y_{|m|}(\kappa\rho) \\ K_{|m|}(\kappa\rho)
\end{Vmatrix},\qquad\text{when}\;\rho\gg1.\end{aligned}$$
We see that the functions $\tilde{u}$ and $\tilde{v}$ have independent asymptotes at the origin and at infinity . It means that the role of the “coupling potential” $W$ in the scattering problem is unimportant here. Therefore one can neglect the “coupling potential” and formulate the scattering problem for the master function $\tilde{u}$ only: $$\label{eq:Scroedinger4Um} \left[-\nabla_\rho^2+\mathcal{U}_m(\rho)\right] \tilde{u}_m = \kappa^2
\tilde{u}_m\;,$$ where the partial potential is $$\label{eq:Um} \mathcal{U}_m(\rho)= \mathcal{U}_0(\rho) + V(\rho) = U(\rho)-\frac12+\left[
\left({\bm{A}}(\rho)\cdot{\bm{e}}_\chi\right)-\frac{m}{\rho}\right]^2.$$
It is natural to suppose that the WKB–approximation is valid for this case. We use the WKB–method in the form proposed earlier for the description of the scattering for isotropic 2D magnets [@Ivanov99], and generalized after that for any singular potentials. [@Sheka02] We start from the effective 1D Schrödinger equation for the radial function $\tilde{u}_m(\rho) =
\psi_m(\rho)/\sqrt{\rho}$, which yields $$\label{eq:U-ef}
\begin{split}
&\left[-\frac{d^2}{d\rho^2}+\mathcal{U}_{\text{eff}}(\rho)\right] \psi_m =
\kappa^2
\psi_m\;,\\
&\mathcal{U}_{\text{eff}}(\rho)= \mathcal{U}_m(\rho)-\frac{1}{4\rho^2}\;.
\end{split}$$ The WKB–solution of Eq. , i.e. the 1D wave function $\psi_m^{WKB}$, leads to the following form of the partial wave $$\label{eq:WKB}
\tilde{u}_m^{WKB} = \frac{\psi_m^{WKB}}{\sqrt{\rho}} \propto
\frac{1}{\sqrt{\rho\cdot\mathcal{P}(\rho)}}\cos\left(\chi_0 +
\int_{\rho_0}^\rho \mathcal{P}(\rho')d\rho'\right),$$ where $\mathcal{P}=\sqrt{\kappa^2-\mathcal{U}_{\text{eff}}}$. Analysis shows that Eq. is valid for $\rho>a$, where $a$ is the turning point. The value of $a$ corresponds to the condition $\mathcal{P}(a)=0$, which results in $a\sim|m|/\kappa\ll 1$. We assume that the parameter $\rho_0$ satisfies the condition $a\ll \rho_0 \ll 1$.
On the other hand, at small distances $\rho\ll1$, the partial potential $\mathcal{U}_m$ has the asymptotic form $$\label{eq:U-as0}
\begin{split}
&\mathcal{U}_m \sim \frac{\nu^2}{\rho^2},\\ \nu&= m-\lim_{\rho\to0}\left[\rho\left({\bm{A}}(\rho)\cdot{\bm{e}}_\chi\right)
\right] = m+qp,
\end{split}$$ therefore one can construct asymptotically exact solutions (recall that we suppose $q=p=1$) $$\label{eq:psi-WKB0}
\tilde{u}_m \propto J_{|m+1|}(\kappa \rho),\qquad\text{when $\rho\ll 1$}.$$ For $\kappa\gg |m|$ there is a wide range of values of $\rho$, namely $$\label{eq:wide-range}
|m|/k\ll \rho\ll 1,$$ where we can use the asymptotic expression [@Ivanov99] for the Bessel function in the limit $k\rho\gg|m|$: $$\label{eq:psi-WKB4rho>>1}
\tilde{u}_m \propto \frac{1}{\sqrt{\rho}}\cos\Bigl(\kappa \rho -
\frac{|m+1|\pi}{2} -\frac\pi4+\frac{4|m+1|^2-1}{8\kappa \rho}\Bigr).$$ In the range of Eq. the solutions and coincide due to the overlap of the entire range of parameters, so one can derive the phase $\chi_0$ in the WKB–solution , $$\chi_0 = \kappa \rho_0 - \frac{|m+1|\pi}{2}
-\frac\pi4+\frac{4|m+1|^2-1}{8\kappa \rho_0}\;.$$ Therefore, we are able to calculate the short–wavelength asymptotic expression for the scattered wave phase shift by the asymptotic expansion of the WKB–solution : $$\label{eq:delta-via-P}
\begin{split}
\delta_m(\kappa) = \lim_{\rho\to\infty}&
\Biggl(\int_{\rho_0}^\rho\mathcal{P}(\rho')d\rho' +\chi_0 - \kappa \rho\\
&+\frac{|m|\pi}{2}+\frac{\pi}{4}-\frac{4m^2-1}{8\kappa \rho}\Biggr).
\end{split}$$ Under assumed conditions $k\rho\gg1$, the WKB–integral in can be calculated in the leading approximation in $1/k\rho$, $$\int_{\rho_0}^\rho\mathcal{P}(\rho')d\rho' \approx
\kappa(\rho-\rho_0)-\frac{1}{2\kappa} \int_{\rho_0}^\rho
\mathcal{U}_{\text{eff}}(\rho')d\rho'.$$ As a result, the scattering phase shift for large wave numbers, $\kappa\gg 1$, has the form $$\begin{split}
\delta_m(\kappa) &= \delta_m(\infty) - \frac{1}{2\kappa}\int_0^\infty
\Bigl[\mathcal{U}_m(\rho)- \frac{\nu^2}{\rho^2}\Bigr] d\rho,
\end{split}$$ with the limiting value $$\begin{aligned}
\label{eq:delta-infty}
\delta_m(\infty)&= -\frac\pi2\cdot\left(|\nu|-|m|\right) = -\frac\pi2\cdot
\operatorname{sgn}_+(m),\\
\nonumber \operatorname{sgn}_+(m) &=
\begin{cases}
1,& m\geq0,\\-1,&m<0.
\end{cases}\end{aligned}$$ Calculating the integral we obtain the phase shift in the form: $$\begin{aligned}
\label{eq:delta-via-D}
&&\delta_m(\kappa) = -\frac{\pi}{2}\cdot \operatorname{sgn}_+(m)
+\frac{\mathcal{D}_1+m\mathcal{D}_2}{\kappa}
\\
\nonumber &&\mathcal{D}_1=\frac14\int_0^\infty \left\{\frac{3\sin^2\theta_0}{\rho^2}
+3\cos^2\theta_0+\left(\theta_0^\prime\right)^2\right\}d\rho\approx 2.44,\\
\nonumber &&\mathcal{D}_2=\int_0^\infty \frac{1-\cos\theta_0}{\rho^2}d\rho\approx 1.38.\end{aligned}$$ The corresponding amplitude of the vortex–magnon scattering is $$\label{eq:sigma-via-C}
\sigma_m(\kappa) = \frac{\kappa}{\mathcal{D}_1+m\mathcal{D}_2}.$$ This linear divergence is well–pronounced in the numerical results, see Fig. \[fig:sigma-infty\]. To understand the origin of this divergence let us go back to the Eq. . One can see that the scattering phase shift at $k\to\infty$ does not vanish for potentials with inverse square singularity at the origin, with $\nu\neq m$, see Eq. . This is possible only in the magnetic field, which has singular behavior like $|{\bm{A}}| \sim1/\rho$.
Let us look for the consequences of this unusual behavior of the scattering, $\sigma_m\to\pm\infty$. We consider the scattering problem for a plane spin wave in the form . In the short–wavelength limit the WKB results for the phase shift are available. One can see that the scattering function tends to zero very quickly for large wave numbers, $\mathcal{F}(\chi) =
\mathcal{O}(\kappa^{-5/2})$, so there is no real divergence or singularity for a physically observable quantity such as the total scattering function $\mathcal{F}$ at large energies.
Levinson theorem {#sec:Levinson}
----------------
![ \[fig:delta\] (Color online) Scattering phase shifts for different $m$. Numerical results from the continuum theory.](\myfig{4a} "fig:"){width="3.5in"} ![ \[fig:delta\] (Color online) Scattering phase shifts for different $m$. Numerical results from the continuum theory.](\myfig{4b} "fig:"){width="3.5in"}
Now we can compare the scattering results in the long– and short–wavelength limits. The scattering is absent for the limit $k\to0$. However, the scattering amplitude has a linear divergence $\sigma\propto k$ for sufficiently large wave numbers, see Eq. . All these results were verified by the numerical calculations for continuum limit and for finite sized discrete lattice systems, see Figs. \[fig:sigma-zero\], \[fig:sigma\], \[fig:sigma-infty\]. According to our analytical calculations, see Eq. , the phase shift for the short–wavelength limit tends to the finite value $\delta_m(\infty) = -\operatorname{sgn}_+(m)\cdot \pi/2$. This result corresponds to the numerical data, see Fig. \[fig:delta\], except for the mode with $m=-1$, where the numerical data gives $\delta_{m=-1}(\infty)=-\pi/2$. However we need to note that the phase shift is determined with respect to $\pi$, in this sense values $\delta_m=\pi/2$ and $\delta_m=-\pi/2$ are identical. What is physically important is how $\delta_m(\kappa)$ changes from small to large $\kappa$. According to our numerical results, the total phase shift can be described by the formula $$\label{eq:Levinson}
\delta_m(0) - \delta_m(\infty) =
\begin{cases}
\frac\pi2\cdot\operatorname{sgn}_+(m), & m\neq-1, \\
\frac\pi2, & m=-1.
\end{cases}$$
It is well–known that the total phase shift is related to the number of bound states $N_m^{\text{b}}$ according to the Levinson theorem from the scattering problem for a spinless quantum mechanical particle without a magnetic field. This theorem was originally proved by Levinson for the 3D case, see the review . The two-dimensional version of the Levinson theorem reads [@Bolle86; @Lin97; @Dong98]
\[eq:Levinson-2D\] $$\label{eq:Levinson-2D(1)}
\delta_m(0) - \delta_m(\infty) = \pi\cdot N_m^{\text{b}}.$$ If there exist half–bound states (see notations in the Sec. \[sec:k<<1\]) for the $p$–wave ($m=1$), this is modified to [@Bolle86; @Dong98] $$\label{eq:Levinson-2D(2)}
\delta_1(0) - \delta_1(\infty) = \pi\cdot N_1^{\text{b}}+\pi.$$
For the 2D EP FM, the scattering picture is much more complicated. First, we have no standard Schrödinger equation, but the generalized one . This becomes apparent at most in the threshold behavior for the half–bound states, and the contribution of the half–bound states in the form may be not adequate, see below. Second, because of the role of the effective magnetic field, there appears an $m$–dependent potential: the symmetry $\delta_m(\kappa)=\delta_{-m}(\kappa)$ is broken, so it is not enough to take into account partial waves with $m\ge0$ only. As a result Levinson’s relation has a different form for the opposite signs of $m$.
Thus, except for the case of half–bound states one can hope that the Levinson theorem is adequate. However we see that the total phase shift contradicts the Levinson theorem in the form . The reason is that the partial potential $\mathcal{U}_m$ in the Schrödinger equation has an inverse square singularity at the origin, $\mathcal{U}_m\sim\nu^2/\rho^2$, where $\nu=m+qp$, see . Such a situation changes the statement of the Levinson theorem. As we have proved recently in Ref. , the generalized Levinson theorem for the Schrödinger–like equation for potentials with such singularities has form: $$\label{eq:Levinson-2D-gen}
\delta_m(0) - \delta_m(\infty) = \pi\cdot N_m^{\text{b}}
+\frac{\pi}{2}\left(|\nu|-|m| \right).$$ An additional $\pi$ can appear on the RHS of this equation, if the half–bound states exist for the $p$–wave ($|m|=1$), see . To explain the meaning of the extra term $(\pi/2)\cdot(|\nu|-|m|)$ in the generalized Levinson theorem , recall that in the partial wave method the scattering data are classified by the azimuthal quantum number $m$, which is the strength of the centrifugal potential. In the presence of a partial potential with an inverse square singularity at the origin such as $\mathcal{U}_m\sim \nu^2/\rho^2$, the effective singularity strength is shifted by the value $|\nu|-|m|$, which results in a change in the short–wavelength scattering phase shift by $(\pi/2)\cdot(|m|-|\nu|)$.
Let us compare the predictions of the generalized Levinson theorem , which is suitable for the Schrödinger–like equation, with our results for the vortex–magnon scattering problem in the 2D EP FM, which can be described by the generalized Schrödinger equation . In our case the singular potential is caused by the specific singular magnetic field at the origin, $|{\bm{A}}|\sim 1/\rho$, which results in $\nu=|m+1|$. The system has no bound states, $N_m^{\text{b}}=0$, therefore Eq. takes the form: $$\label{eq:Levinson-2D-gen1} \delta_m(0) - \delta_m(\infty) =
\frac\pi2\cdot\operatorname{sgn}_+m.$$ Our numerical results correspond to this formula for all modes with $m\neq-1$. The cause is the influence of the half–bound states. By comparison of and , one can adapt the generalized Levinson theorem for this case. It reads $$\label{eq:Levinson-final}
\begin{split}
&\delta_m(0) - \delta_m(\infty)\\ &=
\begin{cases}
\pi\cdot N_m^{\text{b}}+\frac{\pi}{2}\left(|\nu|-|m| \right), &
\text{when $m\neq-1$}\\
\pi\cdot N_m^{\text{b}}+\pi+\frac{\pi}{2}\left(|\nu|-|m| \right), & \text{when
$m=-1$}.
\end{cases}
\end{split}$$ Let us compare this result with Eq. . An extra $\pi$, which appears for the mode $m=-1$, is connected with the half–bound states, see Eq. . To explain the situation, let us stress again that our scattering problem is formulated not for the standard Schrödinger equation. However, the problem has a symmetry such that one eigenfunction becomes a master function, while the other is a slave. This makes it possible to use the main features of the standard quantum mechanical scattering theory. The appearance of the half–bound states is connected with the symmetry of the whole system, and both of the eigenfunctions are important. In the system there are three half–local modes, see Eq. . According to Eq. only one of the half–bound modes, namely, the mode with $m=-1$, gives an extra $\pi$ to the Levinson’s relation. More generally, this extra contribution corresponds to the half–bound mode with $m=-qp$, see Eq. . This result cannot be explained in the framework of the Levinson theorem for the standard Schrödinger equation, where both half–bound states with $m=+1$, and $m=-1$ should make contributions to the Levinson’s relation. The corresponding analogue of the Levinson theorem for the generalized Schrödinger equation takes into account the contribution of the half–bound state for only *one* value of $m$, namely, for $m=-qp$.
Conclusion {#sec:concl}
==========
We have presented a detailed study of vortex–magnon interactions in the 2D EP ferromagnet, having described this process by a “generalized” Schrödinger equation. The main features of the magnon scattering are connected with the special role of the effective magnetic field, which is created by the vortex. This effective field acts on magnons in the same way as a magnetic field influences an electron, leading to the appearance of the Lorentz force and the Zeeman splitting of the magnon states with opposite values of the azimuthal numbers $m$. The singular behavior of the effective magnetic field at the origin causes a divergence of the scattering amplitudes for all the partial waves; we have confirmed this study by a generalized version of the Levinson theorem for potentials with inverse square singularities.
Our investigations can be applied to the description of the internal dynamics of vortex state magnetic dots; the theory of the vortex–magnon scattering developed here could be a good guide for the study of the normal modes in vortex–state magnetic dots. It is clear that the EP FM cannot correspond quantitatively to the case of vortex–state magnetic dots, where the anisotropy is negligible and the static vortex structure is stabilized by magnetic–dipole interactions. We did not consider this type of interaction in this paper, as it is difficult to account for. Nevertheless, we believe that the main features of the problem studied above are generic. For example, an effective magnetic field exists due to the topological properties of the vortex only. Furthermore, we expect the appearance of modes with anomalously small frequencies, e.g., the mode of the translational oscillations of the vortex center. The nonzero frequency of this mode is caused by the interaction with the boundary only. Additionally, the splitting of the doublets for modes with opposite $m$ should appear due to the role of the effective magnetic field.
D.Sh. thanks the University of Bayreuth, where part of this work was performed, for kind hospitality and acknowledges support by the European Graduate School “Non–equilibrium phenomena and phase transitions in complex systems”, and the DLR Project No. UKR-02-011.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Linyi Li$^{\dagger}$ $^*$ Maurice Weber$^{\ddagger}$ $^*$ Xiaojun Xu$^{\dagger}$ [^1] Luka Rimanic$^\ddagger$ Tao Xie$^\mathsection$\
Ce Zhang$^\ddagger$ Bo Li$^\dagger$\
$^\dagger$ University of Illinois at Urbana-Champaign, USA {linyi2, xiaojun3, lbo}@illinois.edu\
$^\ddagger$ ETH Zurich, Switzerland {webermau, luka.rimanic, ce.zhang}@inf.ethz.ch\
$^\mathsection$ Peking University, China {taoxie}@pku.edu.cn
title: 'Provable Robust Learning Based on Transformation-Specific Smoothing'
---
[^1]: The first three authors contribute equally to this work.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The energy spectrum and primary composition of cosmic rays with energy between $3\times 10^{14}$ and $3\times10^{16}{\ensuremath{\,\mathrm{eV}}}$ have been studied using the CASA-BLANCA detector. CASA consisted of 957 surface scintillation stations; BLANCA consisted of 144 angle-integrating Cherenkov light detectors located at the same site. CASA measured the charged particle distribution of air showers, while BLANCA measured the lateral distribution of Cherenkov light. The data are interpreted using the predictions of the CORSIKA air shower simulation coupled with four different hadronic interaction codes.
The differential flux of cosmic rays measured by BLANCA exhibits a knee in the range of 2–3PeV with a width of approximately 0.5 decades in primary energy. The power law indices of the differential flux below and above the knee are $-2.72\pm0.02$ and $ -2.95\pm0.02$, respectively.
We present our data both as a mean depth of shower maximum and as a mean nuclear mass. A multi-component fit using four elemental species suggests the same composition trends exhibited by the mean quantities, and also indicates that QGSJET and VENUS are the preferred hadronic interaction models. We find that an initially mixed composition turns lighter between 1 and 3PeV, and then becomes heavier with increasing energy above 3PeV.
address:
- 'The Enrico Fermi Institute, University of Chicago, 5640 Ellis Avenue, Chicago, Illinois 60637-1433, USA'
- 'Dept. of Astronomy, Adler Planetarium and Astronomy Museum, Chicago, Illinois 60605, USA'
- 'High Energy Astrophysics Institute, Dept. of Physics, University of Utah, Salt Lake City, Utah 84112, USA'
author:
- 'J.W. Fowler'
- 'L.F. Fortson'
- 'C.C.H. Jui'
- 'D.B. Kieda'
- 'R.A. Ong'
- 'C.L. Pryke'
- 'P. Sommers'
title: A Measurement of the Cosmic Ray Spectrum and Composition at the Knee
---
PACS 95.85.R. Cosmic rays, Knee, Energy spectrum, Composition, Cherenkov.
Introduction {#sec.intro}
============
The all-particle energy spectrum of cosmic rays can be described by a steeply falling power law over many decades of energy. The smoothness of this drop in intensity with energy is broken by a change in index of the power law just above $10^{15}{\ensuremath{\,\mathrm{eV}}}$. While the origin of this break (referred to as the “knee”) is not yet fully understood, the prevailing theoretical models describe the knee as a result of the energy limit for particle acceleration in supernova shocks [@Cesarsky; @Drury94]. Further, these models predict that the composition of the primary cosmic rays should change from proton (or “light”-nuclei) dominated to iron (or “heavy”-nuclei) dominated as energy increases through the region of the knee. Measuring a composition trend from light to heavy would lend support to the supernova shock acceleration picture.
Determining the composition of cosmic rays with energies greater than $10^{15}{\ensuremath{\,\mathrm{eV}}}$ is a notoriously difficult problem. To detect primary cosmic rays above this energy directly by satellite experiments requires an unacceptable launch payload volume. Similarly, stratospheric balloon-borne experiments are limited by volume and flight time in their collection of primary particles. Thus, to investigate the composition of cosmic rays at the knee, we must rely on ground-based detection of air showers generated by the primary cosmic rays.
The Cherenkov light emission from the charged particle component of an air shower provides an integrated measurement of the longitudinal development [@Brennan58; @Chudakov60]. One approach is to sample the Cherenkov lateral distribution, the photon density as a function of distance from the air shower core. The Cherenkov intensity is proportional to the primary energy, while the slope of the lateral distribution is related to the depth of maximum shower development — and hence to the mass of the primary cosmic ray nucleus. Therefore measuring a large number of Cherenkov lateral distributions can provide information on how the composition changes with energy [@patterson_hillas]. Previous attempts to exploit this fact at the knee include [@Dawson89; @airobicc; @vulcan]. Optical photons suffer little absorption as they travel through the atmosphere. This means that the Cherenkov lateral distribution is much broader than that of charged particles. Additionally their numerical density is much higher. Thus it is possible to make high signal-to-noise measurements of Cherenkov lateral distributions using an array of detectors with smaller area and wider spacing than would be required for equivalent measurements of charged particles.
To obtain high quality Cherenkov lateral distribution data the Broad Lateral Non-imaging Cherenkov Array (BLANCA) was built at the Chicago Air Shower Array (CASA) installation in Dugway, Utah. Using CASA as the cosmic ray trigger, BLANCA operated on clear, moonless nights in 1997 and 1998. In the following analysis we use CASA to find the shower core position and arrival direction and BLANCA to make a precision measurement of the Cherenkov lateral distribution. This paper details the results obtained through these measurements on the energy spectrum and composition of cosmic rays in the energy range between $3\times 10^{14}$ and $3\times10^{16}{\ensuremath{\,\mathrm{eV}}}$.
The CASA-BLANCA Instrument {#sec.blanca}
==========================
The CASA-BLANCA instrument was located at the Dugway Proving Ground near Salt Lake City, Utah, USA, under a mean atmospheric overburden of $870{\ensuremath{\,\mathrm{g\,cm}}}^{-2}$. During BLANCA runs, CASA [@casa_nim] consisted of 957 scintillation counters which detected the charged particles in an air shower. The surface array covered approximately 0.2km$^2$. BLANCA [@blanca_instrument2] consisted of 144 angle-integrating detectors which recorded the lateral distribution of air shower Cherenkov light. The BLANCA detectors were not uniformly spaced but had an average separation of 35–40m. MIA, an array of buried muon detectors at the same site, was not used in this analysis. Figure \[fig.site\_map\] shows the site plan.
Each BLANCA detector contained a large Winston cone [@winston] which concentrated the light striking an 880cm$^2$ entrance aperture onto a photomultiplier tube. The concentrator had a nominal half-angle of 12.5$^\circ$ and truncated length of 60cm. The geometrical concentration ratio of this design was 19, but losses in the system reduced the effective concentration ratio to 15. Lab measurements and simulations show that the effective half-angle was actually $\sim 10^\circ$ because of a 6mm gap between the photomultiplier and the cone. The Winston cones were aligned vertically with $\sim 0.5^\circ$ accuracy. A two-output preamplifier increased the dynamic range of the detector. A typical BLANCA unit had a detection threshold of approximately one blue photon per cm$^{2}$.
The Cherenkov array did not have a separate trigger system, relying instead on triggers from CASA. Cherenkov data were recorded for all stations which exceeded a fixed threshold in coincidence with a surface array trigger. For showers in the BLANCA field of view, the CASA trigger threshold imposes a minimum energy of $\sim 100{\ensuremath{\,\mathrm{TeV}}}$ on the Cherenkov array. However, to eliminate composition bias inherent in the CASA trigger threshold, we use only showers with an energy of at least 200TeV as determined by BLANCA.
Data Collection and Calibration {#sec.data}
===============================
CASA-BLANCA operated on 90 moonless nights between January, 1997 and May, 1998. After removing periods of hazy or cloudy weather, approximately 460 hours of Cherenkov observations remain. Data cuts require events to have at least five good Cherenkov measurements from BLANCA and a reconstructed primary direction within [$9^\circ$]{} of zenith. Events are also cut if the core location reconstructed by CASA is outside the array or within 30m of the edge, because core location uncertainty increases towards the edge. The geometrical and temporal cuts result in an exposure to cosmic rays of [$1.83\times10^{10} {\ensuremath{\,\mathrm{m}}}^2 {\ensuremath{\,\mathrm{sr}}} {\ensuremath{\,\mathrm{s}}}$]{}.
For each night of data the BLANCA detectors are intercalibrated using the cosmic ray data itself to find their relative sensitivity to Cherenkov light. The intercalibration method depends on the circular symmetry of the Cherenkov light pool about the shower axis. Two detectors equidistant from a shower core receive, on average, equal Cherenkov photon densities, and any discrepancies can be attributed to different detector sensitivities. By averaging over suitable events in a run, we find the sensitivity ratio of each possible pair of BLANCA detectors. A maximum-likelihood method is then used to determine the set of 143 relative gains that best reproduces the pairwise ratios [@thesis]. The reliability of this method was verified with an *in situ* intercalibration using a portable stable blue LED flasher ($\lambda\sim 430{\ensuremath{\,\mathrm{nm}}}$). The relative detector sensitivities are distributed log-normally with typical RMS of 0.4 in natural log. The relative sensitivies are stable over time except for occasional changes in detector hardware.
The BLANCA absolute calibration also used the blue LED system as a reference source. The LED flasher was calibrated by using it to produce single photoelectrons in a photomultiplier with good charge resolution. Two BLANCA detectors were then calibrated in a dark box using the reference LED. The absolute calibration has a 20% systematic uncertainty, resulting mainly from the uncertain quantum efficiency of the reference photomultiplier. This absolute calibration error produces a similar uncertainty in the overall energy scale. The spectral response of the BLANCA photomultipliers was measured by the manufacturer for four representative tubes. As the relative response was similar for these four detectors, it was also assumed to be the same for all BLANCA detectors.
Air Shower Simulations {#sec.simulation}
======================
The CASA-BLANCA analysis compares the Cherenkov measurements with air showers simulated by the CORSIKA Monte Carlo version 5.621 [@corsika]. We used the EGS4 and GHEISHA codes for the electromagnetic and low-energy nuclear interactions. Nucleus-nucleus interactions at air shower energies are well beyond the reach of accelerator experiments. Therefore we are forced to rely on hadronic interaction models which attempt to extrapolate from the available data using different mixtures of theory and phenomenology. Several groups produce such models — in this paper we have used four: QGSJET [@qgsjet], VENUS [@venus], SIBYLL [@sibyll], and HDPM [@corsika]. BLANCA data are interpreted according to each model, indicating the systematic errors that depend on the choice of interaction model.
Simulated showers were produced for proton, helium, nitrogen, and iron primaries in equal numbers. Nitrogen was chosen to represent the entire CNO group. The shower libraries consist of 10,000 showers per primary species, per hadronic model. Primary energies are distributed uniformly in $\log(E)$ between $10^{14}$ and $10^{16.5} {\ensuremath{\,\mathrm{eV}}}$. Shower directions are uniform in solid angle to a maximum zenith angle of $12^\circ$. To produce such large simulated air shower libraries, it was necessary to employ the CORSIKA thinning option, which tracks only a representative sample of shower particles below a threshold energy. In all simulations, this threshold was $10^{-4}$ times the primary energy. Studies of simulated 1PeV showers showed that at this thinning level, the distributions of gross properties such as the depth of shower maximum and Cherenkov slope and intensity were indistinguishable from those of unthinned showers [@icrc_corsika].
Shower development and Cherenkov emission were simulated in Monte Carlo assuming the U. S. standard atmosphere. The CORSIKA program was modified to include a complete model of atmospheric scattering, both Rayleigh (molecular) and Mie (aerosol). Although the modified CORSIKA tracks scattered photons, few reach the ground within the BLANCA field of view, effectively making scattering an absorption process for Cherenkov photons. The scattering losses were generally similar in magnitude to the expected measurement errors and were correlated with the depth of shower maximum. Therefore, atmospheric scattering must not be ignored in analyzing the BLANCA data. On average, scattering reduces the Cherenkov intensity by $\sim 20\%$ and increases the inner slope by $\sim 7\%$. Scattering effects are smallest for late developing showers. Possible molecular absorption by oxygen and ozone was determined to be small compared with the systematic error in the absolute detector calibration and was consequently ignored.
The CORSIKA air showers were processed by a full BLANCA detector simulation which includes the measured wavelength dependence and angular response of the BLANCA detectors. Other simulated effects include detector alignment; unequal detector gains and saturation levels; night sky background light; photomultiplier response; and errors in the CASA core location and shower direction. The detector simulation produces “fake data” which is calibrated and fit like the real data. We have used this fake data to find the optimum transfer functions for converting Cherenkov measurements to air shower and primary cosmic ray parameters throughout this paper.
The Cosmic Ray Energy Spectrum {#sec.spectrum}
==============================
For each air shower event, raw BLANCA data are converted to photon densities, producing a Cherenkov lateral distribution. We fit this lateral distribution with an empirically motivated function which matches both the real and simulated data. The function is exponential in the range 30m–120m from the shower core and a power law from 120m–350m. It has three parameters: a normalization $C_{120}$, the exponential “inner slope” $s$, and the power law index $\beta$:
$$C(r) = \left\{
\begin{array}{ll}
C_{120}\ e^{s(120{\ensuremath{\,\mathrm{m}}}-r)}, & 30{\ensuremath{\,\mathrm{m}}}<r\leq 120{\ensuremath{\,\mathrm{m}}} \\
C_{120}\ (r/120{\ensuremath{\,\mathrm{m}}})^{-\beta}, & 120{\ensuremath{\,\mathrm{m}}}<r\leq
350{\ensuremath{\,\mathrm{m}}} \\
\end{array} \right.
\label{eq.ldf}$$
The energy of each air shower is derived using only the $C_{120}$ and $s$ parameters of the Cherenkov lateral distribution fit. The outer slope $\beta$ is not used, both because it correlates strongly with $s$ and because it is subject to larger measurement errors. The Monte Carlo fake data libraries (including detector simulation) are used to determine the relationship between measured quantities and energy. The energy depends primarily on $C_{120}$, the Cherenkov intensity 120m from the core. We fit the logarithm of the energy as a quadratic function of $\log C_{120}$ (the curvature is small, typically 0.005 decades$^{-1}$). In all hadronic models, $C_{120}$ grows approximately as $E^{1.07}$, because the fraction of primary energy directed into the electromagnetic component of the cascade increases with energy.
According to the Monte Carlo, the quadratic function used to estimate the energy works well for most showers but has a bias such that energies are slightly underestimated for showers with unusually large or small depths of maximum development. Therefore, a small correction is applied to the energy estimate. The correction depends on the Cherenkov slope $s$ and on the shower zenith angle. The magnitude of this energy correction is less than 10% for 85% of BLANCA showers.
Energies derived from the data in this manner have an error distribution which depends on the primary mass and energy. In general, the random errors on reconstructing a single shower’s energy are comparable to the systematic uncertainty due to the unknown composition. Assuming a mixed cosmic ray composition, the BLANCA energy resolution for a single air shower is approximately 12% for a 200TeV shower, falling to 8% for energies above 5PeV.
The differential all-particle cosmic ray flux measured by CASA-BLANCA is shown in Figure \[fig.spectrum\], scaled up by a factor of $(E/1{\ensuremath{\,\mathrm{GeV}}})^{2.75}$ to emphasize the structure. Table \[tab.spectrum\] lists the values of the observed spectrum. The energy spectrum is compared with that reported by several other groups. Although the CASA-BLANCA, DICE, and CASA-MIA experiments shared some instrumentation, their data sets and energy analysis methods are entirely independent. These three experiments show very good agreement in their spectrum determination. Most other results are consistent with CASA-BLANCA, given the 20% or larger energy systematic error typical of air shower measurements.
The spectrum shown in Figure \[fig.spectrum\] uses event energies derived from the Cherenkov predictions of CORSIKA with the QGSJET hadronic interaction model. The data can also be interpreted using the other available interaction models. The alternate energy estimates lead to spectra with no qualitatively different features. Instead, they amount only to a shift in energy scale of order 10%, less than the BLANCA instrumental energy scale uncertainty. HDPM and VENUS predict less Cherenkov light and hence assign higher energies than QGSJET does, while the SIBYLL simulations lead to lower assigned energies.
Figure \[fig.spectrum\] contains the knee region of the spectrum, near 3PeV. The CASA-BLANCA spectrum exhibits a smooth change rather than a sharp break here. However, measurements over a wider energy range show that the form of the cosmic ray spectrum is a power law well above and below 3PeV. Historically, many groups have found the knee to be quite sharp. In the spirit of the usual discussion of the knee, we have fit several similar functions to the data (Figure \[fig.knee\], top). The fits find simultaneously the position (energy) of the knee and the power law indices above and below the knee. A log-likelihood fit is performed in order to account for the Poisson statistics of discrete events in a binned energy distribution.
Of the trial functions, the smoothly changing power law fits the BLANCA data best: $$J(E) = J_k \left(\frac{E}{E_k}\right)^\alpha
\left[1+\left(\frac{E}{E_k}\right)^\frac{1}{w}\right]^{(\beta-\alpha)w}
\label{eq.knee}$$ $E_k$ is the energy at the center of the transition, *i.e.* the knee energy. For $E \ll E_k$, the function is a power law with index $\alpha$, while the spectral index becomes $\beta$ for $E \gg E_k$. Parameter $J_k$ sets the normalization at the knee. The fifth parameter, $w$, is the half-width in decades of the transition region. The lower panel of Figure \[fig.knee\] shows how the log-likelihood depends on the choice of $w$. The data favor a knee one-half decade wide ($w=0.25$). The best-fit knee energy is $2.0^{+0.4}_{-0.2}{\ensuremath{\,\mathrm{PeV}}}$, with power law indices of $\alpha=-2.72\pm0.02$ and $\beta=-2.95\pm0.02$. Figure \[fig.knee\_zoom\] shows the energy spectrum near the knee using fine bins 0.04 decades wide as well as the best fit curve.
Depth of Shower Maximum {#sec.xmax}
=======================
The depth of shower maximum ([$X_{max}$]{}) is an important characteristic of air shower development which, for given energy, is related to the mass of the primary particle. However, like all air shower parameters, the relationship depends on the choice of uncertain high energy hadronic interaction model parameters. Imaging experiments such as Fly’s Eye [@flys_eye], and to a lesser extent DICE [@dice], measure [$X_{max}$]{} rather directly. The importance of [$X_{max}$]{} for various other types of ground-based experiment has long been known [@PattersonHillas]; measured parameters can often be translated into the depth of shower maximum in a way which is rather independent of the hadronic model. Therefore [$X_{max}$]{} provides a useful middle ground on which experiments may publish and compare their results.
The mean [$X_{max}$]{} for a given primary type grows logarithmically with energy at an approximate elongation rate of [$80{\ensuremath{\,\mathrm{g\,cm}}}^{-2}$]{} per decade, although this value depends on the hadronic model used. The expected [$X_{max}$]{} is similar for two primaries of different mass if they have equal energy per nucleon.
To determine the optimum transfer function for converting Cherenkov lateral distributions into [$X_{max}$]{}, we study the same set of simulated showers used to derive the primary energy function. The fake data libraries use the four primary types and four hadronic interaction models processed through the BLANCA detector simulation. The simulated Cherenkov lateral distributions are fit to the function in Equation \[eq.ldf\], which is exponential with a slope $s$ from 30m to 120m from the core. The combined shower and detector simulations show that this inner slope is linearly related to [$X_{max}$]{} except for the deepest developing showers; an additional small quadratic term is required for slopes exceeding $s_\star=0.018{\ensuremath{\,\mathrm{m}}}^{-1}$.
The mean [$X_{max}$]{} is shown in Figure \[fig.xmax\] as a function of energy. Both quantities are derived from the CASA-BLANCA data using the CORSIKA/QGSJET Monte Carlo results. We indeed find that the results are very similar if any of the other hadronic models are used instead. Numerical results are given in Table \[tab.xmax\]. Figure \[fig.xmax\] also shows the mean [$X_{max}$]{} expected for pure samples of proton primaries and iron primaries. SIBYLL generally predicts deeper shower maximum than other models, while HDPM exhibits a steeper elongation rate than the others ($\sim{\ensuremath{90{\ensuremath{\,\mathrm{g\,cm}}}^{-2}}}$ per decade compared with 70–${\ensuremath{75{\ensuremath{\,\mathrm{g\,cm}}}^{-2}}}$ typical of the other models). The BLANCA results are clearly consistent with a mixed composition throughout the energy range, regardless of the preferred hadronic model. The data suggest that the composition becomes lighter approaching the knee and then becomes heavier at higher energies.
The shower depth estimated from Cherenkov observations is subject to a number of small systematic uncertainties. Random core errors lead to a systematic flattening of the Cherenkov inner slope, but the effect is only a very small bias toward deeper [$X_{max}$]{}. Photomultiplier saturation poses a potential problem at high energies. However, the nonlinearity has been characterized in laboratory studies of fourteen BLANCA detectors and the data corrected for its effects. The uncertainty on this correction leads to a systematic error in [$X_{max}$]{} which is only [$10{\ensuremath{\,\mathrm{g\,cm}}}^{-2}$]{} at the highest energies. A third error dominates at energies below 1PeV. This error arises from the limitations of the function which converts Cherenkov slope to an estimate of [$X_{max}$]{}. The function tends to overestimate the depth of showers at the extreme ends of the BLANCA energy range. It is difficult to overcome this weakness without introducing at the same time a much larger bias which depends on [$X_{max}$]{} itself. Instead, we take the error found in Monte Carlo studies as a systematic error on the measured [$X_{max}$]{}. The independent errors are added in quadrature to find the total systematic error (Figure \[fig.xmax\_systematic\]). Systematic errors are important only below 10PeV. At high energy, statistical errors are the more serious limitation on measuring mean [$X_{max}$]{}.
Mean Nuclear Mass {#sec.meanlna}
=================
The mean depth of shower maximum results presented above are essentially independent of a particular hadronic interaction model. They are therefore useful for comparison with other experiments and re-interpretation on the basis of future hadronic interaction models. However, a quantity which is of much more direct astrophysical interest is the mean nuclear mass of the primary cosmic rays.
We choose to derive mean primary mass directly from the Cherenkov lateral distribution slope $s$. It would make little difference if we were to do so via the [$X_{max}$]{} values discussed in the previous section. The important point is that while the transfer function from $s$ to [$X_{max}$]{} is rather independent of the hadronic interaction model, any interpretation in terms of absolute nuclear mass is not. This is clear from the disparity among the models of the mean proton and iron [$X_{max}$]{} values shown in Figure \[fig.xmax\].
At fixed primary energy, $s$ and [$X_{max}$]{}both depend linearly on the logarithm of nuclear mass $A$, as do most composition-sensitive air shower parameters. Following previous authors, we choose to work with the natural log, [$\ln(A)$]{}. Unlike $X_{max}$=$f_1(s)$, the transfer function $\ln(A)$=$f_2(s)$ depends on energy. Therefore we divide the Monte Carlo fake data into six bands of $C_{120}$ and perform a linear fit to [$\ln(A)$]{} versus $s$ in each. To interpret each real event the slope and intercept of the transfer function are interpolated between the appropriate bracketing bands.
This method has little systematic bias apart from the differences between hadronic interaction models. The mean reconstructed [$\ln(A)$]{}for a pure sample of each simulated primary species is accurate to to $\pm20\%$ over the BLANCA energy range. On the other hand, the random error is large on any single measurement of [$\ln(A)$]{}. Shower-by-shower estimates of primary mass cannot be made with any accuracy — the fluctuations inherent in the air shower process preclude them. Nevertheless, the mean value of [$\ln(A)$]{} provides a useful indicator of the cosmic ray mass composition.
The mean [$\ln(A)$]{} is shown as a function of primary energy in Figure \[fig.lna\]. The four sets of symbols show the BLANCA data interpreted using CORSIKA coupled with each of the hadronic codes. Numerical results are given in Table \[tab.lna\]. The dependence on interaction model is clear: the models set the overall mass scale differently, but they indicate the same mass variation with energy. The mean mass becomes lighter with increasing energy through the knee, then becomes heavier above $\sim 3{\ensuremath{\,\mathrm{PeV}}}$. The [$\ln(A)$]{} plot exhibits the same trends seen in the [$X_{max}$]{} results of the previous section.
A Multi-species Fit to the Cherenkov Data {#sec.multi}
=========================================
The techniques described above involve estimating [$X_{max}$]{} or [$\ln(A)$]{} for each shower and then taking the *average* over all showers in a given energy range. By considering only the average value we lose much of the available information. Instead the measured distribution of a composition sensitive parameter can be compared with those predicted for a number of simulated primary species, providing a more powerful technique to study cosmic ray composition.
Comparing measured and simulated distributions requires high statistics samples for both the real and Monte Carlo data sets. We separate the data into five logarithmic energy bins between $10^{14.5}$ and $10^{16.5}{\ensuremath{\,\mathrm{eV}}}$. This bin choice is a compromise between the need to include many showers in each range and the wish to examine trends on as fine an energy scale as possible. Within each range, we find the distribution of the Cherenkov inner slope ($s$) for the real data and for pure samples of each species in the Monte Carlo library (protons, He, N, and Fe). The simulated distributions of $s$ (with detector effects) are smoothed by a multiquadratic smoothing algorithm [@hbook]. To preserve information about their limited statistics, the data distributions are not smoothed.
The multi-species fit in each energy range finds the linear combination of the four simulated distributions which reproduces the data distribution best. Since each primary species has a characteristic shape of its $s$ distribution, this fit uses more information than simply the mean or even the width of $s$. We do not *a priori* require the fractional contribution of each primary type to lie in the physical range of 0–100%, nor is the sum of the fractions constrained to equal 1.0. In practice, however, the sum is always in the range $100\pm0.5\%$. The fits use a MINUIT-based log-likelihood maximization procedure [@minuit], which accounts properly for the Poisson probability distribution of data events in bins with low statistics.
As an example, Figure \[fig.multi\_example\] shows the multi-species fit in the energy range $10^{14.9}$–$10^{15.3}{\ensuremath{\,\mathrm{eV}}}$. The lower right panel (\#4) displays the full fit using all four available primary types. The other panels show the best fits that can be made when helium (\#2), nitrogen (\#3), or both (\#1) are omitted. The Monte Carlo predictions cannot match the data using protons and iron alone; the intermediate mass nitrogen species is also required. Panel \#2 shows that the best fit of p, N, and Fe to the data is very close but fails to match the shape at the peak and in the long tail to high values of $s$. The data strongly suggest that at least the four primary types considered here contribute to the cosmic rays just below the knee.
The fits demonstrated in Figure \[fig.multi\_example\] (lower right) are performed for all five energy ranges and using the predictions of all four high energy hadronic interaction models. Gauging the goodness of fit presents a problem. As stated above the multi species fit uses a log-likelihood maximization procedure; although this is an appropriate method for extracting abundance fractions from the binned data, the actual value of the likelihood is not useful [@minuit]. Therefore we calculate and use the familiar $\overline{\chi}^2$ quantity as a rough guide to the fit quality. In the lowest energy range, the number of data events is so large that no combination of the four cosmic ray species can reproduce the data adequately. This is probably the result of limitations of the shower and detector simulations, although it could also be due to the limited number of species considered. Conversely, the high energy ranges have too few events to constrain the abundances well. The results for all models are presented in Table \[tab.multi\_species\].
The results of the multi-species fit to the BLANCA Cherenkov slope data are shown in Figure \[fig.multi\_results\] for the QGSJET and VENUS models. The SIBYLL and HDPM models show similar trends but a heavier overall composition. These latter two models also give unphysical negative helium abundances in at least one energy bin, and systematically poorer fits. At 100TeV, data from the JACEE balloon direct measurements [@jacee97; @watson_rapp] are shown for comparison. The direct composition at 100TeV agrees well with the BLANCA data at 400TeV. The results of the multi-species fit also agree with the mean [$X_{max}$]{} and mean [$\ln(A)$]{} derived in the previous two sections. All three ways of interpreting the data indicate that the cosmic ray composition is lighter near 3PeV than it is at either 300TeV or 30PeV.
Conclusions {#sec.conclusions}
===========
The CASA-BLANCA experiment has studied cosmic rays in the energy range 0.3–30PeV. The primary energy and mass are found by measuring the Cherenkov lateral distribution for each air shower. In an effort to understand how results depend on the unknown physics of high energy nuclear interactions, we have interpreted the data using the CORSIKA air shower Monte Carlo program with four different hadronic interaction models: QGSJET, VENUS, SIBYLL, and HDPM.
The BLANCA energy spectrum agrees well with previous measurements and exhibits a smooth knee near 2–3PeV in primary energy. The model dependence of the energy scale is less than the absolute calibration uncertainty.
We find the transformation from measured Cherenkov lateral distribution slope to the depth of shower maximum [$X_{max}$]{} to be essentially model independent. In Figure \[fig.xmax\_all\] our results are compared to previous experiments over a wide energy range. The BLANCA data are well within the physically reasonable range bounded by the pure proton and iron curves; furthermore they are consistent at low energy with those expected from direct measurements and at high energy with the Fly’s Eye result [@flys_eye_result].
We have also interpreted our data as a mean nuclear mass. This is essentially equivalent to the [$X_{max}$]{} analysis but is a quantity of more direct astrophysical interest.
It has been a long held goal in the air shower field to choose an adequate hadronic interaction model *and* determine the nuclear composition of the primary cosmic rays simultaneously. With the advent of the powerful simulation tool provided by the CORSIKA group, and high collecting power arrays such as CASA-BLANCA, it seems that this ambition may be becoming a reality. A multi component fit of the type described in Section \[sec.multi\] is a much more efficient use of the available data than simply considering the mean value and spread of a quantity, and the experimental statistics are starting to justify this approach. The agreement between data and simulation in Figure \[fig.multi\_example\] is impressive.
On the basis of our data we favor the QGSJET and VENUS models and reject SIBYLL and HDPM. At the same time, both of the ways in which we have analyzed our data indicate that the cosmic ray composition is lighter near 3PeV than it is at either 300TeV or 30PeV. The trend towards heavier primary mass above 3PeV agrees with the canonical model of Galactic production and a rigidity-dependent time for escape, and is not consistent with acceleration at sites such as AGN, which require a pure proton composition well above the knee [@agn_protheroe].
The trend shown in our data to a lighter composition approaching the knee is puzzling but not without theoretical precedent. Swordy, arguing that there must be a minimum path length in the Galaxy even for the highest energy cosmic rays, predicts a light composition at the knee [@swordy_model].
We acknowledge the invaluable assistance of the CASA-MIA collaboration, as well as the University of Utah High-Resolution Fly’s Eye (HiRes) group and the command and staff of the U.S. Army Dugway Proving Ground. We thank D. Heck and the rest of the CORSIKA team for providing and maintaining their excellent program, and the authors of the hadronic interaction models to which it is linked. We thank C. Cassidy, J. Jacobs, J. Meyer, M. Pritchard, and K. Riley for helping with BLANCA’s construction and K. Anderson and C. Eberhardy for calibration work. We especially wish to thank M. Cassidy for his essential contributions as our technician. JF and CP acknowledge fellowships from the William Grainger Foundation and the Robert R. McCormick Foundation, respectively. This work was supported by the U.S. National Science Foundation. We would also like to thank S. Swordy for useful conversations.
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Data tables
===========
[cc]{} Energy range & Differential flux $J(E)$ in each bin\
$\log_{10}(E/1{\ensuremath{\,\mathrm{eV}}})$ & [(m$^{-2}$sr$^{-1}$s$^{-1}$GeV$^{-1}$)]{}\
[cll]{} Energy range & Mean $s$ & $ {\ensuremath{\langle X_{max} \rangle}}\pm{\ensuremath{\,\mathrm{stat.}}}\pm{\ensuremath{\,\mathrm{sys.}}}$\
$\log_{10}(E/{\ensuremath{\,\mathrm{eV}}})$ & $(10^{-3}{\ensuremath{\,\mathrm{m}}}^{-1})$ & $({\ensuremath{{\ensuremath{\,\mathrm{g\,cm}}}^{-2}}})$\
[cllll]{} Energy range &\
$\log_{10}(E/1{\ensuremath{\,\mathrm{eV}}})$ & QGSJET & VENUS & SIBYLL & HDPM\
[cccccr]{} Energy Range & & $\overline{\chi}^2$ of\
$\log_{10}(E/1{\ensuremath{\,\mathrm{eV}}})$ & p & He & N & Fe & Fit\
\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We introduce methods which allow observed galaxy clustering to be used together with observed luminosity or stellar mass functions to constrain the physics of galaxy formation. We show how the projected two-point correlation function of galaxies in a large semi-analytic simulation can be estimated to better than $\sim 10\%$ using only a very small subsample of the subhalo merger trees. This allows measured correlations to be used as constraints in a Monte Carlo Markov Chain exploration of the astrophysical and cosmological parameter space. An important part of our scheme is an analytic profile which captures the simulated satellite distribution extremely well out to several halo virial radii. This is essential to reproduce the correlation properties of the full simulation at intermediate separations. As a first application, we use low-redshift clustering and abundance measurements to constrain a recent version of the Munich semi-analytic model. The preferred values of most parameters are consistent with those found previously, with significantly improved constraints and somewhat shifted “best” values for parameters that primarily affect spatial distributions. Our methods allow multi-epoch data on galaxy clustering and abundance to be used as joint constraints on galaxy formation. This may lead to significant constraints on cosmological parameters even after marginalising over galaxy formation physics.'
author:
- |
Marcel P. van Daalen$^{1,2,3}$[^1], Bruno M. B. Henriques$^{1}$, Raul E. Angulo$^{4}$ and Simon D. M. White$^{1}$\
\
$^1$Max Planck Institute for Astrophysics, Karl-Schwarzschild Straße 1, 85741 Garching, Germany\
$^2$Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands\
$^3$Department of Astronomy, Theoretical Astrophysics Center, and Lawrence Berkeley National Laboratory,\
University of California, Berkeley, CA 94720, USA\
$^4$Centro de Estudios de Física del Cosmos de Aragón, Plaza San Juan 1, Planta-2, 44001, Teruel, Spain
bibliography:
- 'PhDbib.bib'
title: The galaxy correlation function as a constraint on galaxy formation physics
---
\[firstpage\]
galaxies: formation – cosmology: theory – cosmology: large-scale structure of Universe
Introduction
============
Galaxy formation is a complex process involving many astrophysical ingredients spanning a very wide range of scales. Because of this, any model of galaxy formation – be it hydrodynamical, analytical or semi-analytical in nature – has to rely on some set of observations in order to constrain the representation of physical processes that cannot be derived from first principles, or be simulated directly.
Hydrodynamical simulations can simulate baryonic processes directly on large scales while relying on sub-grid recipes to model relevant processes below the resolution limit. As such simulations are relatively expensive computationally, the values of the parameters in the sub-grid formulations usually have to be chosen by comparing a set of simulations run at lower resolution or in smaller volumes to some observational quantity, though these numerical settings themselves may impact which parameter values are appropriate. Still, as the available computational resources are ever growing, the number of processes which cannot be simulated directly is slowly decreasing [e.g. @Hopkins2014], and valiant efforts are currently being made to improve the accuracy of direct cosmological simulations (e.g. Illustris, @Vogelsberger2014, and EAGLE, @Schaye2015 [@Crain2015]).
Semi-analytic models (hereafter SAMs), on the other hand, necessarily require calibration of additional physical parameters to describe the baryonic processes which are not simulated directly on any scale. However, once the high-resolution collisionless simulations that they are based on have been run and stored, they can be carried out many times with different parameter values at relatively low computational cost. Coupled with a method to efficiently explore parameter space such as Monte Carlo Markov Chains (MCMC, a method introduced in SAMs by [@Kampakoglou2008] and [@Henriques2009]), this allows one to find the highest-likelihood set of parameters for any given model, based on a set of observational constraints.
Typically, SAMs use observational data sets of one-point functions, such as stellar mass or luminosity functions, as constraints for their model parameters (e.g. @Kauffmann1993 [@Baugh1996; @SomervillePrimack1998; @Kauffmann1999; @Cole2000; @Croton2006; @Bower2006; @Monaco2007; @Somerville2008; @Henriques2009; @Guo2011; @Henriques2013], see @Baugh2006 for a review on the general methodology). The resulting models of galaxy formation can then be tested against other observables (i.e. observables that are independent of those used as constraints) and be used to make predictions for these. A delicate balance must be maintained here: if the model has too many free parameters, prior regions that are too wide, or if there are too few (independent) observational constraints, degeneracies may occur (i.e. wide regions of high likelihood in parameter space, possibly with multiple peaks), while too little freedom or failing to include some relevant physical process may leave the model unable to match several observables at once.
SAMs constrained to match observed luminosity or stellar mass functions have typically had trouble matching the small-scale clustering of galaxies (e.g. @Kauffmann1999 [@Springel2005a; @Li2007; @Guo2011; @Kang2012]; but see e.g. @Kang2014 [@Campbell2015]). In order to determine the cause of this discrepancy, and to test whether the models retain enough freedom to match the observed clustering at all, it would be instructive to use clustering measurements as additional constraints while exploring parameter space. As galaxy clustering is determined by how galaxies with different properties populate haloes of different mass, it directly constrains galaxy formation, in a way that is complementary to, for example, the luminosity function.
However, this presents a problem: while one-point functions such as the stellar mass function can be quickly estimated with known uncertainty by running the model on a small sample of representative haloes, allowing large regions of parameter space to be rejected without having to run the model on the full dark matter simulation, the same cannot be done simply for two-point functions such as the correlation function. In principle, any observable that relies on spatial correlations between galaxies can only be calculated by running the model on the full simulation, which is computationally infeasible when thousands of parameter sets need to be explored. While running the SAM on a small sub-volume may allow one to measure small-scale correlations to some degree, cosmic variance will be an issue. Additionally, if one aims to compare to observations, where clustering is viewed in projection (unless line-of-sight velocities are used), one still has to account for large-scale correlations, even at small separations.
Here, we present an efficient method, based on the halo model, to estimate the projected correlation function, $w(r_\mathrm{p})$, to some known uncertainty from a small sample of haloes, and we apply it to constrain the version of the Munich semi-analytic model presented in @Guo2013 [, hereafter G13]. By measuring the properties of galaxies within individual haloes and making informed assumptions about the distribution of these haloes, we are able to circumvent the aforementioned problems, greatly reducing the CPU time needed to predict their two-point clustering.
This paper is organised as follows. In Section \[sec:methods\], we present our method for estimating $w(r_\mathrm{p})$ and briefly describe the semi-analytic model we apply it to. Next, in Section \[sec:results\], we show the results of using clustering as an additional constraint on parameter space, in addition to the oft-used $z=0$ stellar mass function. Finally, in Section \[sec:summary\] we present a summary of our work and discuss future improvements and applications.
Method {#sec:methods}
======
Estimating the correlation function {#subsec:estimator}
-----------------------------------
Our approach is slightly different to that of most previous works constructing a correlation function estimator based on the halo model, where the aim is typically to reproduce observations given some parametrised halo occupation distribution (HOD). Here, our goal is instead to reproduce the results of the semi-analytic model run on the full dark matter simulation to within some given accuracy, given the galaxy properties for a small sample of haloes. As we will show, we are able to reproduce the projected correlation function of the full simulated galaxy sample to within $\sim 10\%$, using the properties of semi-analytical galaxies occupying less than $0.04\%$ of the full halo sample ($0.14\%$ of the subhalo sample).
### The backbone of the model {#subsubsec:backbone}
Our starting point is the linear halo model, introduced independently by @Seljak2000, @MaFry2000 and @PeacockSmith2000. In what follows, we will adhere to the terminology of @CooraySheth2002. In the analytical halo model the power spectrum, $P(k)$, is written as the sum of two terms: $$P(k)=P^\mathrm{1h}(k)+P^\mathrm{2h}(k).
\label{eq:power}$$ Here $P^\mathrm{1h}(k)$ is the 1-halo term, describing the two-point clustering contribution of points within the same halo, and $P^\mathrm{2h}(k)$ is the 2-halo term, describing the contribution of points within separate haloes. For the clustering of matter, these are given by: $$\begin{aligned}
\nonumber
P_\mathrm{dm}^\mathrm{1h}(k) \!\!\!&=&\!\!\! \int n(M) \left(\frac{M}{\bar{\rho}}\right)^2 |u(k|M)|^2 \mathrm{d}M\\
P_\mathrm{dm}^\mathrm{2h}(k) \!\!\!&=&\!\! \int\!\!\int n(M_1) \left(\frac{M_1}{\bar{\rho}}\right) u(k|M_1) \times\\
\nonumber
& &\!\!\! n(M_2) \left(\frac{M_2}{\bar{\rho}}\right) u(k|M_2) P_\mathrm{hh}(k|M_1,M_2) \mathrm{d}M_1 \mathrm{d}M_2.\end{aligned}$$ Here $M=M_\mathrm{200mean}$ is the halo mass definition[^2] we will be using throughout, $n(M)$ is the halo mass function, $\bar{\rho}$ is the mean matter density of the Universe, $u(k|M)$ is the normalised Fourier transform of the density profile of a halo of mass $M$, and $P_\mathrm{hh}(k|M_1,M_2)$ is the halo-halo power contributed by two haloes of masses $M_1$ and $M_2$ on a Fourier scale $k$. We can rewrite the latter term assuming a linear scale-independent bias relation, $P_\mathrm{hh}(k|M_1,M_2)=b(M_1)b(M_2)P_\mathrm{lin}(k)$, where $b(M)$ is the halo bias and $P_\mathrm{lin}$ the linear theory matter power spectrum. We then obtain: $$P_\mathrm{dm}^\mathrm{2h}(k) = P_\mathrm{lin}(k)\left[\int n(M) b(M) \left(\frac{M}{\bar{\rho}}\right) u(k|M) \mathrm{d}M\right]^2\!\!\!\!.$$ From these expressions, one can straightforwardly derive a simple model for the galaxy power spectrum. Since we are interested in the clustering of galaxies instead of mass, we replace $M/\bar{\rho}$ by $\left<N_\mathrm{gal}|M\right>/\bar{n}_\mathrm{gal}$ and, since galaxies are discrete objects, $\left(M/\bar{\rho}\right)^2$ by $\left<N_\mathrm{gal}(N_\mathrm{gal}-1)|M\right>/\bar{n}_\mathrm{gal}^2$, leading to: $$\begin{aligned}
P_\mathrm{gal}^\mathrm{1h}(k) \!\!\!&=&\!\!\! \int n(M) \frac{\left<N_\mathrm{gal}(N_\mathrm{gal}-1)|M\right>}{\bar{n}_\mathrm{gal}^2} u_\mathrm{gal}(k|M)^p \mathrm{d}M\\
\nonumber
P_\mathrm{gal}^\mathrm{2h}(k) \!\!\!&=&\!\!\! P_\mathrm{lin}(k)\left[\int n(M) b(M) \frac{\left<N_\mathrm{gal}|M\right>}{\bar{n}_\mathrm{gal}} u_\mathrm{gal}(k|M) \mathrm{d}M\right]^2\!\!\!\!.\end{aligned}$$ Here the mean number density of galaxies is given by $n_\mathrm{gal}=\int n(M) \left<N_\mathrm{gal}|M\right>\mathrm{d}M$. Note that we have followed @CooraySheth2002 in replacing the normalised Fourier transform of the halo density profile, $u(k|M)$, by one describing the distribution of (satellite) galaxies, $u_\mathrm{gal}(k|M)$, and subsequently in changing the power-law index on this term in the 1-halo term by $p$. This is often done in the literature in order to be able to differentiate between contributions from central-satellite and satellite-satellite terms, with $p=1$ for the former and $p=2$ for the latter, based on the value of $\left<N_\mathrm{gal}(N_\mathrm{gal}-1)\right>$. $\left<N_\mathrm{gal}|M\right>$ – the most common form of the HOD – is often separated into contributions from centrals and satellites as well, with the former ($N_\mathrm{cen}$) being either $0$ or $1$, and the latter ($N_\mathrm{sat}$) being very well approximated by a (linear) power law [e.g. @GuzikSeljak2002; @Kravtsov2004; @Zehavi2005; @Tinker2005; @Zheng2005]. From this, approximate expressions for $\left<N_\mathrm{gal}(N_\mathrm{gal}-1)\right>$ in terms of $N_\mathrm{cen}$ and $N_\mathrm{sat}$ can be derived as well.
However, as our aim is to reproduce the results of the semi-analytic model, for which information on the HOD and the galaxy type is much more readily available than for observations, we can explicitly separate the contributions from central and satellite galaxies to the galaxy power spectrum without approximation. Keeping in mind that a halo will contain at most one central, meaning that $\left<N_\mathrm{cen}(N_\mathrm{cen}-1)|M\right>=0$, that $\left<N_\mathrm{cen}N_\mathrm{sat}|M\right>=\left<N_\mathrm{sat}N_\mathrm{cen}|M\right>$, and using that central galaxies reside in the centre of the halo and should therefore not be weighted by the profile, we derive: $$\begin{aligned}
\nonumber
P_\mathrm{gal}^\mathrm{1h}(k) \!\!\!\!\!\!&=&\!\!\!\!\!\! 2\!\!\int\!\! n(M) \frac{\left<N_\mathrm{cen}N_\mathrm{sat}|M\right>}{\bar{n}_\mathrm{gal}^2} \left[u_\mathrm{gal}(k|M) - W_{R}(k)\right] \mathrm{d}M\, +\\
\nonumber
& & \!\!\!\!\!\!\!\int\!\! n(M) \frac{\left<N_\mathrm{sat}(N_\mathrm{sat}\!\!-\!1)|M\right>}{\bar{n}_\mathrm{gal}^2} \!\left[u_\mathrm{gal}(k|M)^2 \!\!-\! W_{R}(k)^2\right]\! \mathrm{d}M\\
P_\mathrm{gal}^\mathrm{2h}(k) \!\!\!\!\!\!&=&\!\!\!\!\!\! P_\mathrm{lin}(k)\left[\int\!\! n(M) b(M) \frac{\left<N_\mathrm{cen}|M\right>}{\bar{n}_\mathrm{gal}} \mathrm{d}M\, +\right.\\
\nonumber
& & \left.\!\!\!\!\!\!\!\int\!\! n(M) b(M) \frac{\left<N_\mathrm{sat}|M\right>}{\bar{n}_\mathrm{gal}} u_\mathrm{gal}(k|M) \mathrm{d}M\right]^2\!\!\!\!.
\label{eq:haloterms}\end{aligned}$$ Note that we have followed @ValageasNishimichi2011 in adding a counterterm to the halo profiles in the 1-halo term, which ensures the 1-halo term goes to zero for $k \rightarrow 0$. Here $W_{R}(k)$ is the Fourier transform of a spherical top-hat of radius $R(M)=[3M/(4\pi\bar{\rho})]^{1/3}$, given by: $$W_{R}(k) = 3\left(\frac{\sin(kR)}{(kR)^3}-\frac{\cos(kR)}{(kR)^2}\right).$$
In our model, we take $P_\mathrm{lin}(k)$ to be the measured power spectrum of the initial conditions of the dark matter simulation. We calculate the halo mass function, $n(M)$, directly from the dark matter simulation too and spline-fit the results. Furthermore, we use the fit for the $M_\mathrm{200mean}$ halo bias function provided by @Tinker2010 for $b(M)$, and compute each of the four HOD terms directly from the SAM run on our halo subsample, spline-fitting these results as well.
### The galaxy distribution
The normalised Fourier transform of the galaxy distribution, $u_\mathrm{gal}(k|M)$, is often derived from the dark matter mass profile of the halo. This in turn is usually assumed to be equal to the @Navarro1997 [, NFW] profile, cut off at the virial radius $r_\mathrm{vir}=R_\mathrm{200mean}$, with some concentration-mass relation $c(M)$: $$\rho_\mathrm{NFW}(r)=\frac{\rho_0}{(r/r_\mathrm{s})(1+r/r_\mathrm{s})^2},$$ where $r_\mathrm{s}=r_\mathrm{vir}/c$ is the scale radius. The main advantage of using the one-parameter NFW profile is that this leads to an analytic expression for $u(k|M)$. However, many authors have shown that the @Einasto1965 profile provides a more accurate fit to the mean profile of haloes of a given mass, and to the distribution of dark matter substructure [e.g. @Navarro2004; @Merritt2005; @Merritt2006; @Gao2008; @Springel2008; @Stadel2009; @Navarro2010; @Reed2011; @DuttonMaccio2014]. The two-parameter Einasto density profile is given by: $$\rho_\mathrm{Ein}(r)=\rho_0 \exp\left\{-\frac{2}{\alpha}\left[\left(\frac{r}{r_\mathrm{s}}\right)^{\alpha}-1\right]\right\},$$ where the shape parameter $\alpha$ allows additional freedom in the slope of the profile. This function does not have an analytic Fourier transform, and an extra numerical integration step is therefore needed when replacing the NFW profile by an Einasto one. The larger degeneracies in fitting a two-parameter model also mean more data points are needed to obtain a reliable fit. Still, when the computational expense is acceptable and enough information on the measured profile is available, the increased accuracy will be worth the cost.
We find that the Einasto profile provides a very good fit to the distribution of satellite galaxies in the inner parts of haloes in our simulation. But even the Einasto profile over-predicts the number of galaxies at large radii, $r \ga 0.7r_\mathrm{vir}$. Furthermore, standard practice is to cut off the profile at the virial radius, while we find that $\sim 10\%$ of the satellite galaxies in our simulation are found at distances $1<r/r_\mathrm{vir}<3$. Note that these galaxies are not necessarily outside the virialised region, as haloes are typically not spherical objects. In addition, simulated haloes are truncated in a non-spherical manner at the boundary imposed by the Friends-of-Friends (FoF) group finder used to define them. Finally, subhaloes may travel outside the virial radius again after infall [“backsplash”, e.g. @Balogh2000; @Mamon2004; @Gill2005]. We therefore seek a profile with similar small-scale behaviour to the Einasto profile, while simultaneously fitting the galaxy population of simulated haloes out to $\sim 3 r_\mathrm{vir}$.
We find that the following functional form is capable of providing an excellent match to the galaxy distribution over the full range of scales we consider, and at any halo mass: $$n_\mathrm{sat}(r)=n_0 \left(\frac{r}{b}\right)^{a-3}\exp\left\{-\left(\frac{r}{b}\right)^c\right\}.
\label{eq:preprofile}$$ This fitting function has three parameters, $a$, $b$ and $c$. Note that the role of $b$ is similar to that of $r_\mathrm{s}$ in the Einasto profile. Both the Einasto and this new profile are near universal if defined in terms of $x \equiv r/r_\mathrm{vir}$. If we rewrite both profiles in terms of $x$ and integrate them to obtain $N(<r)$, the similarities and differences between the profiles are most easily appreciated. For the Einasto profile: $$N_\mathrm{Ein}(<r) = N_\mathrm{tot} \frac{\gamma\left[\frac{3}{\alpha},\frac{2}{\alpha}\left(\frac{x}{r_\mathrm{s}}\right)^\alpha\right]}{\gamma\left[\frac{3}{\alpha},\frac{2}{\alpha}\left(\frac{x_\mathrm{max}}{r_\mathrm{s}}\right)^\alpha\right]},$$ while for the profile given in equation : $$N_\mathrm{sat}(<r) = N_\mathrm{tot} \frac{\gamma\left[\frac{a}{c},\left(\frac{x}{b}\right)^c\right]}{\gamma\left[\frac{a}{c},\left(\frac{x_\mathrm{max}}{b}\right)^c\right]}.$$
![Galaxy number density profiles for all @Guo2011 halo members with stellar masses $10.27<\log_{10}(M_*/\mathrm{M}_{\sun})<10.77$, for five different halo mass bins (shown in different colours). The legend shows the mean logarithmic mass in each of the bins. Solid lines indicate the measured profiles, while dashed lines show our best fits (see equation ). The halo mass bins are dynamically chosen such that each contains roughly the same number of galaxies, and the fits are performed using 30 radial bins spaced equally in log-space between $\log_{10}x=-2.5$ and $\log_{10}x=0.5$. The three-parameter fit we use, given in equation , captures the measured satellite profile extremely well, even out to several times the virial radius, where the popular NFW and Einasto profiles often fail because of the edge imposed by halo definition in the simulation. For more information on how the fits are performed, see Appendix \[fittingapp\].[]{data-label="fig:profiles"}](plots/profile_53.ps){width="1.0\columnwidth"}
Here $\gamma(a,b)$ is the lower incomplete gamma function, and we have assumed the profiles cut off at some $x_\mathrm{max}$. The similarities in the two profiles are clear. The main difference is that the two parameters of the gamma function can be independently manipulated for our new profile, which effectively allows for a steeper profile at large $x$ and consequently a better match to the galaxy distribution around the virial radius. In practice, we fit a normalised number density profile $n_\mathrm{sat}(r)/\left<N_\mathrm{sat}\right>$ to the satellite distribution before numerically Fourier transforming this to obtain $u_\mathrm{gal}(k|M)$. For completeness, $n_\mathrm{sat}(r)/\left<N_\mathrm{sat}\right>$ is given by: $$\label{eq:profile}
\frac{n_\mathrm{sat}(r)}{\left<N_\mathrm{sat}\right>}=\frac{c}{4\pi b^3 r_\mathrm{vir}^3\gamma\left[\frac{a}{c},\left(\frac{x_\mathrm{max}}{b}\right)^c\right]} \left(\frac{x}{b}\right)^{a-3}\exp\left\{-\left(\frac{x}{b}\right)^c\right\}.$$ In our model we set $x_\mathrm{max}=5$. Even for small halo samples, the three parameters of the fit are sufficiently independent to ensure degeneracies are not a problem, i.e. fits with the new profile provide stable results even for a small number of sample points.
An example is given in Figure \[fig:profiles\], where we show the best-fit model for all halo members with stellar masses $10.27<\log_{10}(M_*/\mathrm{M}_{\sun})<10.77$ in the @Guo2011 semi-analytic model, for five different halo mass bins. The solid lines show the measured number density profiles, while the dashed lines show the best fits for the profile in equation . The radii of satellite galaxies in each halo mass bin are not binned when fitting, but instead used as direct inputs for a likelihood function which we maximise to find the best fit. This likelihood function is constructed assuming that the number of satellites found at each radius is a Poisson variable with mean given by the profile (see Appendix \[fittingapp\]).
After finding the best-fit profile parameters in each halo mass bin, we fit an Akima spline through each of the three parameters as a function of halo mass to obtain smooth functions that are stable to outliers. Not only does the profile given in equation fit the simulated satellite distribution extremely well for a large range in mass and radius, it is also parametrised in a way that yields very stable fits when only a few galaxies are available. This is important as we will only be using a very small set of haloes to inform our model, and the resulting clustering prediction is very sensitive to the satellite profile fits.
### Halo exclusion
The standard halo model does not account for halo exclusion, meaning that the distance between two haloes is allowed to be arbitrarily close to zero. As a consequence, the 2-halo term is overestimated on small scales. We have implemented halo exclusion following the approach of @Baldauf2013. They suggest a correction to the 2-halo term, such that $P_\mathrm{2h}'(k)=P_\mathrm{2h}(k)-P_\mathrm{corr}(k)$. If the sum of two halo radii (i.e. their minimum separation) is $R$, this correction term is given by $P_\mathrm{corr}(k)=V_\mathrm{excl}\left[P_\mathrm{2h}\ast W_{R}\right](k)+V_\mathrm{excl}W_{R}(k)$, where $[A \ast B](k)$ denotes a convolution integral between functions $A(k)$ and $B(k)$, and where $V_\mathrm{excl}=(4\pi/3)R^3$ is the effective excluded volume. For galaxies, the correction term can be split into separate contributions for central-central, central-satellite and satellite-satellite galaxy pairs, as: $$\begin{aligned}
\nonumber
P_\mathrm{corr}(k) \!\!\!\!\!\!&=&\!\!\!\!\!\! (1-f_\mathrm{sat})^2 P_\mathrm{corr,cc}(k)+\\
& & \!\!\!\!\!\!2 f_\mathrm{sat}(1-f_\mathrm{sat}) P_\mathrm{corr,cs}(k)+f_\mathrm{sat}^2 P_\mathrm{corr,ss}(k),\end{aligned}$$ where $f_\mathrm{sat}$ is the satellite fraction. We calculate averaged minimum separation radii $R_\mathrm{cc}$, $R_\mathrm{cs}$ and $R_\mathrm{ss}$ separately for each pairing and each stellar mass bin, by integrating over all halo masses. The final halo exclusion correction terms are then given by: $$\begin{aligned}
\nonumber
P_\mathrm{corr,cc}(k) \!\!\!\!\!\!&=&\!\!\!\!\!\! V_\mathrm{excl}\left[P_\mathrm{2h,cc}\ast W_{R_\mathrm{cc}}\right](k)+V_\mathrm{excl}W_{R_\mathrm{cc}}(k),\\
\nonumber
P_\mathrm{corr,cs}(k) \!\!\!\!\!\!&=&\!\!\!\!\!\! \left(u_\mathrm{gal}(k|M_\mathrm{h,sat})-W_{R_\mathrm{sat}}(k)\right)\times\\
\label{eq:haloexclusion}
& & \!\!\!\!\!\!\left(V_\mathrm{excl}\left[P_\mathrm{2h,cs}\ast W_{R_\mathrm{cs}}\right](k)+V_\mathrm{excl}W_{R_\mathrm{cs}}(k)\right),\\
\nonumber
P_\mathrm{corr,ss}(k) \!\!\!\!\!\!&=&\!\!\!\!\!\! \left(u_\mathrm{gal}(k|M_\mathrm{h,sat})^2-W_{R_\mathrm{sat}}(k)^2\right)\times\\
\nonumber
& & \!\!\!\!\!\!\left(V_\mathrm{excl}\left[P_\mathrm{2h,ss}\ast W_{R_\mathrm{ss}}\right](k)+V_\mathrm{excl}W_{R_\mathrm{ss}}(k)\right),\end{aligned}$$ where $M_\mathrm{h,sat}$ is the typical halo mass of the satellite population. Note that we have again taken into account the counterterms proposed by @ValageasNishimichi2011, with $R_\mathrm{sat}$ corresponding to the size of the Lagrangian region of the typical halo mass of the satellite population.
Implementing halo exclusion significantly improves our model predictions around the 1-halo and 2-halo transition scale ($r\sim 1\,\mathrm{Mpc}$) for our most massive stellar mass bin. For our fiducial model, we observed no noticeable effects on the projected correlation function for stellar masses $M_* \la 10^{11}{\,\mathrm{M}_{\sun}}$.
![The FoF halo mass function, showing the number of haloes available in the Millennium Simulation at $z=0$ (black) and the number randomly selected as a function of $M_\mathrm{200mean}$ in each subsample (red). The subsamples each comprise less than $0.04\%$ of the total halo sample, or $0.14\%$ of the total subhalo sample. The selection function was built iteratively by demanding that $\sim 99\%$ of the random samples it generates lead to projected correlation functions that are within $30\%$ of the full sample prediction. Low-mass haloes were favoured over high-mass haloes in order to suppress the size of the trees used in the SAM. Even so, the fraction of FoF groups needed to match the correlation function within some uncertainty at any stellar mass is higher for more massive haloes.[]{data-label="fig:selection"}](plots/selection_plot.ps){width="1.0\columnwidth"}
### Correction for non-sphericity and halo alignment
As is common, we have assumed a spherical distribution of satellite galaxies around each central. In reality, haloes and consequently their galaxy populations are triaxial. @vanDaalen2012 investigated the effect of assuming a spherical distribution on the two-point correlation function and galaxy power spectrum, and found that the effects can be quite large, with the true power being underestimated by $1\%$ around $k=0.2{\,h\,\mathrm{Mpc}^{-1}}$ to $10\%$ around $k=25{\,h\,\mathrm{Mpc}^{-1}}$, increasing even more towards smaller scales (see the right panel of their Figure 3). We have repeated their analysis and found that the functional shape of this underestimation of the power appears to be completely independent of the stellar mass of the galaxies. We therefore fit a function $e(k)$ through these results and use this to correct our halo model power spectra for the combined effects of non-sphericity and halo alignment. The final galaxy power spectrum that comes out of our model for a given set of galaxies is therefore: $$P_\mathrm{gal}=[P_\mathrm{gal}^\mathrm{1h}(k)+P_\mathrm{gal}^\mathrm{2h}(k)]/[1+e(k)],
\label{eq:totgalpow}$$ with $P_\mathrm{gal}^\mathrm{1h}(k)$ and $P_\mathrm{gal}^\mathrm{2h}(k)$ given by equation .
![The fractional difference between our model predictions of the projected galaxy correlation function and a direct calculation, for galaxies in the @Guo2011 semi-analytic model. Here we use the full simulated galaxy sample as an input to our model. Results are shown for six different stellar mass bins, indicated by lines of different colours, over the range where SDSS/DR7 data is available for each. The overall agreement is within $10\%$. The most massive stellar mass bin is most sensitive to the approximations made in the halo model, and has only a small number of satellites, making it difficult to get an accurate prediction for its (relatively) small-scale behaviour. However, this mass bin also has next to no impact on the constraints on the galaxy formation parameters, as it is only weakly sensitive to them. On the other hand, the four least massive stellar mass bins, which are the most sensitive to the parameters of the SAM, show excellent agreement between the real and predicted clustering functions.[]{data-label="fig:wguo"}](plots/projcompare_200mean.ps){width="1.0\columnwidth"}
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### Assembly bias
@Zentner2014 have shown that not including assembly bias in a clustering model leads to significant systematics. As @vanDaalen2012 showed, assembly bias is also reflected in the galaxies of the @Guo2011 model. However, assembly bias is generally only a significant systematic when the galaxies are split by a property other than mass, e.g. colour or age, which we do not do here. Most importantly, since our clustering estimator is based on the HOD and satellite profile of galaxies in the simulation, assembly bias is implicitly included in our results. Indeed, as we will show in §\[subsubsec:performance\], our estimator does not seem to contain any significant scale-independent systematics. We therefore do not include an explicit correction for assembly bias in our model.
### Converting to the projected correlation function
To obtain the projected correlation function from the galaxy power spectrum, we numerically perform two standard transformations. First, to obtain the 3D correlation function: $$\xi(r)=\frac{1}{2\pi^2}\int_0^\infty k^2 P(k)\frac{\sin kr}{kr}\,\mathrm{d}k,$$ and, finally, to obtain the projected galaxy correlation function: $$w(r_\mathrm{p})=2\int_0^{\pi_\mathrm{lim}} \!\!\xi\!\left(\!\sqrt{r_\mathrm{p}^2+\pi^2}\right)\mathrm{d}\pi=2\int_{r_\mathrm{p}}^{r_\mathrm{lim}} \!\!\frac{r\xi(r)}{\sqrt{r^2-r_\mathrm{p}^2}}\,\mathrm{d}r.$$ Here $r_\mathrm{p}$ and $\pi$ are the projected and line-of-sight separation, respectively, and $r_\mathrm{lim}=\sqrt{r_\mathrm{p}^2+\pi_\mathrm{lim}^2}$. Formally, the integration limit is $r_\mathrm{lim}=\infty$, but in order to directly compare our model $w(r_\mathrm{p})$ to that of observations we set $r_\mathrm{lim}=40{\,h^{-1}\,\mathrm{Mpc}}$, and convert all units from $\mathrm{Mpc/h}$ to $\mathrm{Mpc}$. As @vandenBosch2013 point out, this ignores the contribution of peculiar velocities beyond the integration limit, which may bias the projected correlation function on the largest scales probed. However, since the largest scales are the least interesting for our current investigation, we do not attempt to correct for this.
### Selection function {#subsubsec:selection}
Sample haloes are randomly selected following the selection function shown in Figure \[fig:selection\], a power law with a cut-off at a maximum of $200$ haloes per $0.1\,\mathrm{dex}$ in halo mass.
The selection function was initially constructed iteratively by demanding that the projected correlation functions resulting from $\ga 99\%$ of its random samples should agree with those from the full sample in each stellar mass bin to within $30\%$. Low-mass haloes were favoured over high-mass haloes by weighting each halo by the average number of subhaloes for its mass, in order to down-weight large merger trees. For the same reason, the constraint on the accuracy of the projected correlation function for galaxies with $M_*>10^{11.27}{\,\mathrm{M}_{\sun}}$ was relaxed, as it would require almost all of the highest-mass haloes to achieve $\sim 10\%$ accuracy consistently for the clustering of these rare (mostly central) galaxies.
After building several selection functions in this way, we found that on average they were well approximated by the combination of a constant value and a power law (rounded to integer values) shown as the red line in Figure \[fig:selection\]. The subsamples generated by this selection function each comprise less than $0.04\%$ of the total FoF halo sample, or $0.14\%$ of the total subhalo sample.
Parameter Description Units
-------------------------- -- -------------------------------------- -- ---------------------------------------------------------------------- -- --
$\alpha_{\rm{SF}}$ Star formation efficiency –
$\tilde{M}_{\rm{crit}}$ Star formation threshold $\mathrm{M}_{\sun}\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$
$\alpha_{\rm{SF,burst}}$ Star formation burst mode efficiency –
$\beta_{\rm{SF,burst}}$ Star formation burst mode slope –
$k_{\rm{AGN}}$ Radio feedback efficiency $h^{-1}\,\mathrm{M}_{\sun}\,\mathrm{yr}^{-1}$
$f_{\rm{BH}}$ Black hole growth efficiency –
$V_{\rm{BH}}$ Quasar growth scale $\mathrm{km}\,\mathrm{s}^{-1}$
$\epsilon$ SN mass-loading efficiency –
$V_{\rm{reheat}}$ Mass-loading scale $\mathrm{km}\,\mathrm{s}^{-1}$
$\beta_{1}$ Mass-loading slope –
$\eta$ SN ejection efficiency –
$V_{\rm{eject}}$ SN ejection scale $\mathrm{km}\,\mathrm{s}^{-1}$
$\beta_{2}$ SN ejection slope –
$\gamma$ Ejecta reincorporation scale factor –
$y$ Metal yield fraction –
$R_{\rm{merger}}$ Major-merger threshold ratio –
$\alpha_{\rm{friction}}$ Dynamical friction scale factor –
### Performance of the model {#subsubsec:performance}
We compare $w(r_\mathrm{p})$ predicted by applying our estimator to the full halo sample to that measured directly on the simulation for the @Guo2011 model in Figure \[fig:wguo\]. Here we show the relative difference between the two for six different bins in stellar mass, indicated as ranges in $\log_{10}(M_*/\mathrm{M}_{\sun})$. We only show the results over the range where we constrain $w(r_\mathrm{p})$ using observations. The model performs well, and deviations from the true correlation function are below $15\%$ except for the most massive galaxy bin. The magnitude of the mismatch tends to increase with stellar mass. The large-scale disagreement is caused by the model slightly under-predicting the power in the transition region between the 1-halo and 2-halo terms, while the small-scale offset is mostly due to the 1-halo term in the power spectrum being slightly overestimated around $k=1{\,h\,\mathrm{Mpc}^{-1}}$. However, overall the agreement is good, especially considering our relatively simple treatment of e.g. the halo bias (linear and scale-independent), and we leave further improvements – such as using a halo-halo power spectrum measured from the dark matter only simulation instead of a biased linear power spectrum – to future work. Note that the clustering predictions in the four lowest mass bins ($M_*<10^{10.27}{\,\mathrm{M}_{\sun}}$) are always accurate to better than $10\%$. This is important, as the clustering of these galaxies on sub-Mpc scales is sensitive to changes in the galaxy formation parameters, and therefore holds the most constraining power.
The true power of the model lies in its ability to reproduce the clustering prediction for the full sample from only a small subsample of FoF groups. In Figure \[fig:wsamples\] we compare the predictions for a large set of random subsamples selected according to the selection function shown in Figure \[fig:selection\] to the model prediction for the full sample. The dotted lines indicate offsets of $\pm 20\%$ for reference, and the colours indicate the same stellar mass bins as in Figure \[fig:wguo\]. For all but the highest mass bin, the model on average reproduces the clustering prediction of the full sample to the $1\%$ level, with a sample-to-sample scatter that is typically $<10\%$. This shows that the model is capable of reproducing the full sample estimate using only a small fraction of all the haloes.
While the model is sensitive to the number of high-mass haloes used, it is not sensitive to the low-mass end. Raising the low-mass ceiling of our selection function from $200$ to $1000$ haloes only slightly lowers the scatter for $r_\mathrm{p}<40\,\mathrm{kpc}$ while having no significant effect on larger scales, while lowering the ceiling to $40$ haloes increases the scatter on all scales by about $50\%$ while increasing the median offset only for $r_\mathrm{p}<40\,\mathrm{kpc}$, to on average $10\%$ on the smallest scales.
We find that the accuracy of the estimator is not sensitive to the galaxy formation parameters used, but is instead mainly determined by the particular haloes in the sample. Indeed, as we will see, our predictor works equally well for all sets of best-fit parameters we explore in §\[sec:results\]. This means that we could in principle construct an optimal (i.e. maximally representative) sample of haloes, given some selection function, and reasonably expect this sample to give highly accurate predictions throughout the SAM’s parameter space. Here, we have instead chosen the simpler approach of generating several random halo samples and using the one that lies closest to the medians shown in Figure \[fig:wsamples\].
The SAM and MCMC {#subsec:SAM}
----------------
As our estimator is able to quickly and accurately recover the projected correlation function from a very small subsample of haloes, this makes it ideally suited for constraining the parameter space of semi-analytic models using the projected correlation function. In this work we present a first application, where we constrain the model of G13, a recent version of the Munich semi-analytical code, using both the galaxy stellar mass function (SMF) and the projected galaxy correlation function. For this we utilise the same data sets as presented in G13. Since we will only utilise the Millennium Simulation, and not Millennium II, we only use constraints for stellar masses $M_*>10^9{\,h^{-1}\,\mathrm{M}_{\sun}}$.
{width="100.00000%"}
The G13 model includes $17$ parameters which together determine the outcome of galaxy formation. These are (see Table \[tab:params\]): the star formation efficiency ($\alpha_\mathrm{SF}$); the star formation criterion ($\tilde{M}_\mathrm{crit}$, or equivalently $\Sigma_\mathrm{crit}$); the star formation efficiency in the burst phase following a merger ($\alpha_\mathrm{SF,burst}$); the slope on the merger mass ratio determining the stellar mass formed in the burst ($\beta_\mathrm{SF,burst}$); the AGN radio mode efficiency ($k_\mathrm{AGN}$); the black hole growth efficiency ($f_\mathrm{BH}$); the typical halo virial velocity of the black hole growth process ($V_\mathrm{BH}$); three parameters governing the reheating and injection of cold disk gas into the hot halo phase by supernovae (SNe), namely the gas reheating efficiency ($\epsilon$), the reheating cut-off velocity ($V_\mathrm{reheat}$) and the slope of the reheating dependence on $V_\mathrm{vir}$ ($\beta_1$); three parameters governing the ejection of hot halo gas to an external reservoir, namely the gas ejection efficiency ($\eta$), the ejection cut-off velocity ($V_\mathrm{eject}$) and the slope of the ejection dependence on $V_\mathrm{vir}$ ($\beta_2$); a parameter controlling the gas return time from the external reservoir to the hot halo ($\gamma$); the yield fraction of metals returned to the gas phase by stars ($y$); the mass ratio separating major and minor merger events ($R_\mathrm{merger}$); and finally a parameter controlling the dynamical friction time scale of orphan galaxies, i.e. the time it takes for satellite galaxies of which the dark matter subhalo is disrupted (or at least no longer detected) to merge with the central galaxy ($\alpha_\mathrm{friction}$).
While in the original G13 paper some of these parameters were held fixed, here we allow all $17$ to vary in order to determine which of these are sensitive to the $w(r_\mathrm{p})$ constraints. We start our Monte Carlo Markov Chains (MCMCs) at the position in parameter space preferred by @Guo2011 and the WMAP1 version of G13, which was arrived at by requiring agreement with a variety of low-redshift observational data, primarily stellar mass and luminosity functions, but also gas fractions, gaseous and stellar metallicities, and central black hole masses, all as a function of stellar mass. Here we use MCMC techniques to find new sets of “best-fit” parameters (i.e. the parameters that result in the best agreement with the data) constrained by one or both of the low-redshift stellar mass function and the projected correlation function.
Since the error bars on the SDSS clustering data were derived from Poisson statistics alone, and so do not include cosmic variance, we artificially increase them for our fitting. Data points for the observed projected correlation function with uncertainties below $20\%$ had their error bar increased to this minimum value. The SMF was treated as having the same minimum uncertainty in order to avoid skewing our estimates. This is appropriate because we wish our MCMC procedures to exclude regions of parameter space where models substantially mismatch the data, rather than to attempt a statistically rigorous estimate of model parameters. As noted before, we do not use the clustering data below $M_*=10^{9.27}{\,\mathrm{M}_{\sun}}$, nor the stellar mass function data below $M_*<10^9{\,\mathrm{M}_{\sun}}$, when constraining the model, as the haloes hosting these galaxies are not well resolved in the Millennium Simulation which we are using as a basis for the SAM. When fitting to the SMF and clustering data simultaneously, we increase the relative weighting of the fit to the SMF by a factor of five to compensate for the fact that the clustering data is measured in five separate bins. This helps avoid sacrificing the excellent fit to the SMF in favour of matching the correlation function.[^3]
Note that while G13 used both WMAP1 and WMAP7 cosmologies, we here use the original WMAP1 cosmology only to avoid additional complications introduced by scaling to a different cosmology. In future work the results will be explored for more up-to-date cosmologies. As G13 showed, the change in cosmology does have some impact on the resulting correlation functions, although they are at least as sensitive to the SAM’s physical recipes. Besides updating the cosmology, the only change made from the WMAP1 @Guo2011 model to the newer G13 model is that the type 2 (orphan) satellite galaxy positions are now correctly updated in the code, meaning that their orbits now decay as intended and can therefore be disrupted earlier. This change was the reason for the improved agreement with clustering data in the WMAP1 version of G13 with respect to @Guo2011.
![The stellar mass functions of the models. The lines are as in Figure \[fig:corr\_constrained\], and here the estimates for the sample are in perfect agreement with the full calculation. Even when the SMF is not the only constraint, the model clearly has enough freedom to reproduce it to high precision, as the SMF+$w(r_\mathrm{p})$ results are within 1$\sigma$ of the data. The badly-matched $w(r_\mathrm{p})$-only model shows that reproducing the galaxy clustering does not guarantee reproducing the stellar mass function.[]{data-label="fig:smf_constrained"}](plots/plots_smf.ps){width="1.0\columnwidth"}
Results {#sec:results}
=======
Comparison with observations {#subsec:comparison}
----------------------------
The results of our MCMC chains for the projected correlation function are shown in Figure \[fig:corr\_constrained\], for six bins in stellar mass, as indicated in the panels. In each figure, we indicate the original results found by G13, where the galaxy formation parameters were set by hand without reference to clustering, as a green dotted line. The new results are shown in blue, orange and red; in blue, we show the correlation functions that follow from using the stellar mass function alone as a constraint (“SMF-only”), in orange we show the result of using the clustering data alone as a constraint (“$w(r_\mathrm{p})$-only”), while in red we show the results of constraining with both data sets simultaneously (“SMF+$w(r_\mathrm{p})$”). Note that the correlation functions for $w(r_\mathrm{p})$-only and SMF+$w(r_\mathrm{p})$ tend to coincide.
The dashed lines show the predictions made based on the sample of haloes used in the MCMC, as described in §\[subsec:estimator\]. The dotted lines show the true galaxy correlation function, as calculated directly from the full simulated galaxy catalogue for the same model parameters. The true values agree very well with the ones estimated from the sample (at the $\la 10\%$ level), as expected from the results of §\[subsubsec:performance\].
![Same as Figure \[fig:smf\_constrained\], but now showing the results for three sets of parameters instead of only the single best-fit. In each case we show the maximum likelihood model and two other models with only slightly lower likelihood but parameters spread over the allowed regions. Full and SMF+$w(r_\mathrm{p})$ results have been omitted for clarity. While the $w(r_\mathrm{p})$-only model in general prefers a lowered SMF around and above the knee, it does not necessarily increase the number of low-mass galaxies to match the clustering.[]{data-label="fig:smf_alt"}](plots/plots_smf_alt.ps){width="1.0\columnwidth"}
The resulting SMF+$w(r_\mathrm{p})$ correlation functions (red lines) provide a better fit to the data, bringing the small-scale clustering down considerably in comparison with the original G13 and SMF-only (blue lines) models. This effect is larger for low stellar masses, where the clustering discrepancy between the old model and the data was larger as well. The much improved match to observations indicates that the model retains enough freedom to match the clustering data. Note that the match to the projected galaxy correlation function for galaxies in the first mass bin is significantly improved as well, even though this data is not used to constrain the model. For the highest-mass galaxies, $11.27 < \log_{10}(M_*/\mathrm{M}_{\sun}) < 11.77$, all models perform equally well, as the clustering of these galaxies is relatively insensitive to the galaxy formation parameters.
In Figure \[fig:smf\_constrained\], we show how the models compare to the SMF data used to constrain them. The black points with error bars are derived by combining several observational data sets [see @Henriques2015]. The error bars show the uncertainties we used to calculate the likelihoods (see §\[subsec:SAM\]). The original G13 model, in which the parameters were set by hand, is again shown as a green dotted line, and matches the data well. When we use only the SMF as a constraint for the galaxy formation model, shown in blue, we obtain a marginally better fit to the data at low mass.
{width="100.00000%"}
When the projected galaxy correlation function is used as an additional constraint, shown in red, the agreement with the stellar mass function is almost as good. The simulated mass function for SMF+$w(r_\mathrm{p})$ shows slightly worse agreement with observations than SMF-only around the knee, where a decrease in clustering tends to necessitate a decrease in galaxy abundance, but the result is still within 1$\sigma$ of the data. The sample results (dashed lines) agree perfectly with the full catalogue ones (dotted lines) for all models, which is expected as the stellar mass function is a one-point function and requires a smaller halo sample to achieve the same accuracy as the correlation function.
It is clear that the SMF+$w(r_\mathrm{p})$ model is in much closer agreement with both the SMF and the clustering data *simultaneously* than both the original G13 and the SMF-only models, as the latter models are in (sometimes strong) disagreement with the clustering data for low-mass galaxies on small scales while the SMF+$w(r_\mathrm{p})$ model is generally in agreement with both the low-mass clustering data and the SMF within 1$\sigma$. This shows the merit of using clustering estimates as constraints while exploring parameter space. In §\[subsec:parameters\], we will show that adding the projected correlation functions as constraints not only markedly improves the constraints on almost all the SAM parameters, but that there is at least one model parameter which is *only* significantly constrained by including clustering data.
Looking at the results for $w(r_\mathrm{p})$-only (orange lines in Figures \[fig:corr\_constrained\] and \[fig:smf\_constrained\]), we see that its (predicted) projected correlation functions, like those for SMF+$w(r_\mathrm{p})$, are in almost perfect agreement with the data, while its stellar mass function is inconsistent with the observations at the many $\sigma$ level for both small and large stellar masses. It is clearly important to use both the number density and clustering data as constraints.
To show that reproducing the projected correlation functions does not dictate a certain SMF, we show in Figure \[fig:smf\_alt\] the stellar mass functions resulting from using two other $w(r_\mathrm{p})$-only models with only slightly lower likelihood but parameters spread over the allowed regions. One of these sets results in an SMF that happens to reproduce the observed SMF at the 1$\sigma$ level for $M_*<10^{10.3}{\,\mathrm{M}_{\sun}}$. We find that $w(r_\mathrm{p})$-only models that produce practically identical correlation functions (see Figure \[fig:corr\_alt\]) can show a wide range of number densities for low-mass galaxies, but will almost always underestimate the stellar mass function at high mass, especially at the knee. We explore the reason for this in the next section.
For good measure we also show results for three high-likelihood sets of parameters for SMF-only in both Figures \[fig:smf\_alt\] and \[fig:corr\_alt\]; virtually indistinguishable stellar mass functions can produce very different correlation functions, many of them highly incompatible with observations. Our results show that the clustering data, while powerful, should not be used as the only constraint for the model, and – to perhaps a larger degree – neither should the $z=0$ stellar mass function be the only observational constraint. We note that the latter point was also made by @Henriques2009, where large degeneracies were obtained when using only this constraint even when sampling only six parameters.
Because the observations we use here are not sufficient to constrain all model parameters, we do not consider the effect our best-fit parameters have on other observable quantities, but instead leave this to future work where additional constraints (such as high-redshift information) are adopted and/or some model parameters are held fixed.
Changes in parameters {#subsec:parameters}
---------------------
Even though we vary $17$ galaxy formation parameters, by far the largest role in bringing the clustering predictions in agreement with observations is played by only two of these: $\alpha_\mathrm{friction}$, which controls the time it takes for satellite galaxies to merge with the central once their dark matter subhalo has been disrupted, and $V_\mathrm{reheat}$, which indirectly controls the amount of cold ISM gas reheated to the hot halo by supernova feedback as a function of halo mass.
The way these parameters influence the clustering and stellar mass function predictions is as follows. When the clustering data are included as an additional constraint, the dynamical friction time scale of orphan galaxies decreases by about $25\%$ with respect to G13. This small but significant shift causes galaxies at small separation scales to merge with their centrals more quickly, flattening the galaxy distribution profile within the haloes and decreasing the amount of clustering on small scales, especially for low-mass satellites. This change in the galaxy distribution profiles from the G13 to the SMF+$w(r_\mathrm{p})$ model is shown in Figure \[fig:profilecompare\]. The halo mass bins are set to be the same for the two models to allow for an unbiased comparison. Note that the mass bins do change as a function of stellar mass in order to make sure each bin in halo mass is roughly equally populated. Although we only show the fits to the measured profiles here, we stress that each provides an excellent fit to the data, over the full range in scales shown. The change in slope of the profiles, caused mainly by the change in $\alpha_\mathrm{friction}$, is relatively small, meaning that the galaxy distributions are still consistent with SDSS data for rich clusters (see Figure 11 of G13).
The $25\%$ decrease in the friction time scale when using clustering as an additional constraint causes the number of type 2 galaxies of any mass at $z=0$ to decrease by a third; however, we note that this roughly one-to-one correspondence between $\alpha_\mathrm{friction}$ and the number of orphan galaxies is coincidental. In general, we find that the dynamical friction timescale can decrease by a factor of a few without lowering the number of satellites further, as the merging time scale for many of these galaxies is still long compared to the Hubble time. While the lower number density of orphans hardly affects the total number density of low-mass galaxies, the small measure of flattening experienced by the profile is enough to significantly decrease the small-scale clustering for these galaxies.
{width="100.00000%"}
The apparent sensitivity of low-mass clustering to a relatively small change in $\alpha_\mathrm{friction}$, which does not seem to be reflected in another observational quantity – at least not at current observational precision – means that using the clustering data as a constraint for semi-analytic models of galaxy formation may be one of the few ways in which the right merging time scale for orphan satellite galaxies can be determined. Whether SAMs use a parametrised dynamical friction time scale like the one employed here or another scheme to treat the merging of galaxies that no longer inhabit a (detectable) subhalo [e.g. @Campbell2015], there are always parameters involved that have historically been hard to constrain. Using the projected correlation functions together with an estimator as described in the current work offers a solution to this long-standing problem. Furthermore, the high sensitivity of the correlation functions to the details of the satellite distribution confirms the need for a highly accurate satellite profile if one aims to predict galaxy clustering.
{width="100.00000%"}\
{width="100.00000%"}\
{width="100.00000%"}
The parameter $V_\mathrm{reheat}$, on the other hand, shifts up by almost a factor of two with respect to G13 (from $70$ to $132{\,\mathrm{km}\,\mathrm{s}^{-1}}$), increasing the effectiveness of supernova feedback on ISM gas as a function of halo mass. This similarly changes the satellite profiles, but while the decrease in $\alpha_\mathrm{friction}$ causes many orphans to merge away faster, flattening the distribution at all masses, the change in $V_\mathrm{reheat}$ shifts the relative distribution of high- and low-mass satellites. The indirect result of a more effective ISM heating is that the galaxy distribution profiles for low-mass galaxies flattened, while those for satellites with $M_* \ga 10^{10.77}{\,\mathrm{M}_{\sun}}$ are slightly steepened. The latter, undesirable effect is mitigated through changes in other parameters, mainly by lowering the effectiveness of AGN feedback.
{width="100.00000%"}
The changes in $\alpha_\mathrm{friction}$ and $V_\mathrm{reheat}$ also impact the stellar mass function. First off, the decrease in the dynamical friction time scale causes the number density of galaxies above the knee ($M_* > 10^{10.5}{\,\mathrm{M}_{\sun}}$) to decrease. This counter-intuitive change in the SMF comes about because the cold gas in the merging satellites directly feeds the supermassive black holes in the centres of the central galaxies, increasing feedback from AGN and thereby the suppression of star formation. The increase in $V_\mathrm{reheat}$ has the same effect for $M_* > 10^{10.5}{\,\mathrm{M}_{\sun}}$, but increases the number density of galaxies below this mass scale. While this change in the SMF is quite intuitive, one might expect the overall clustering to increase/decrease with an increase/decrease in the number density of galaxies of the same mass, as the clustering scales with the HOD. While this does play a small role, it is the (normalised) galaxy distribution within each halo that is the real driving force behind the clustering predictions. Additionally, the effect of the HOD is partly mitigated by its normalization with $\bar{n}_\mathrm{gal}$, the mean number density of galaxies at some stellar mass. This also explains how a $w(r_\mathrm{p})$-only model that underestimates the SMF at any stellar mass could still lead to just as accurate clustering predictions as, say, SMF+$w(r_\mathrm{p})$.
Incidentally, these sometimes counter-intuitive changes in observable quantities caused by shifts in a single parameter, let alone the complicated interactions that can occur between different parameters changing at once, serve to demonstrate the importance of using a scheme like MCMC rather than attempting to set the values of a model – be it SAM or hydrodynamical – by hand.
The parameter changes in $\alpha_\mathrm{friction}$ and $V_\mathrm{reheat}$ with respect to the G13 parameters already produce clustering predictions that are very close to those of the SMF+$w(r_\mathrm{p})$ model. However, as they adversely affect the SMF, other parameter shifts are necessary to bring the SMF back into agreement with observation. We show the shift in all parameter values in Figure \[fig:params\]. We again indicate the results for all four models: the original G13 model (green dotted lines), the SMF-only model (blue lines), the $w(r_\mathrm{p})$-only model (orange lines), and the SMF+$w(r_\mathrm{p})$ model (red lines). Histograms indicate the (smoothed) Bayesian likelihood regions as derived from the full MCMC chains[^4], while the vertical dashed lines indicate the best-fit values.
Parameter G13 (WMAP1) SMF only SMF+w($r_\mathrm{p}$) w($r_\mathrm{p}$) only
------------------------------------------------------------------------------------------------- -- ------------- -- --------------------- -- ----------------------- -- ------------------------
$\tilde{M}_\mathrm{crit}\,[\mathrm{M}_{\sun}\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}]$ 0.38 $6.5\times 10^{-3}$ 2.3 $7.1\times 10^{-2}$
$V_\mathrm{reheat}\,[\mathrm{km}\,\mathrm{s}^{-1}]$ 70 2.39 132 169
$\beta_1$ 3.5 0.19 3.3 2.0
$V_\mathrm{eject}\,[\mathrm{km}\,\mathrm{s}^{-1}]$ 70 179 66.4 3.50
$\alpha_\mathrm{friction}$ 2.0 3.06 1.55 1.11
Without delving too much into details, this figure allows us to make several interesting observations. The best-fit values for the different models can be quite different from those of G13, even though the stellar mass and correlation functions it produces are not wildly different from those of our new models. As previously mentioned, the decrease in number density and the increase in clustering for high-mass galaxies in SMF+$w(r_\mathrm{p})$ caused by the shifts in $\alpha_\mathrm{friction}$ and $V_\mathrm{reheat}$ are mitigated by lowering the effectiveness of AGN feedback through a decrease in $k_\mathrm{AGN}$ and $f_\mathrm{BH}$. However, while in this case the best-fit parameters reflect these typical shifts, this is not the case for all high-likelihood parameter sets, i.e. the best-fit parameters do not necessarily fall near the peak of the likelihood region. For example, while the models that are constrained by the clustering data typically show the parameter shifts we just described, some high-likelihood parameter sets for the $w(r_\mathrm{p})$-only model (not shown here) leave $V_\mathrm{reheat}$ at its initial value and instead achieve a match to the clustering data by lower $\alpha_\mathrm{friction}$ only, and lowering it further than SMF+$w(r_\mathrm{p})$. Since it is more interesting – and indeed more Bayesian – to consider the typical solutions preferred by the model, rather than some best-fit set of parameters which may only perform marginally better than many others and do not inform us about the importance of its individual elements, we will only consider how the likelihood regions compare in what follows, both between the different models and to the G13 input values.
The likelihood regions for SMF-only are generally much wider than for the clustering-constrained models, which points to large degeneracies. This does not mean that the stellar mass function is unaffected by a shift in any of these parameters, but rather that a combination of shifts in other parameters can usually compensate for any undesirable changes caused. These degeneracies are already much less apparent when the clustering constraints are used, even if they are the only constraints. Indeed, by comparing the likelihood regions for $w(r_\mathrm{p})$-only and SMF+$w(r_\mathrm{p})$ (orange and red respectively), one can see that adding the stellar mass function as a constraint usually does not limit the parameter space traversed by the MCMC by much. As we used clustering data that was split into different stellar mass bins, this is not too surprising. It should be stressed, however, that modern semi-analytic models typically also take stellar mass or luminosity functions at higher redshift [e.g. @Henriques2013; @Henriques2015] as well other one-point functions such as colour information into account, which we have not done here. Regardless, it is clear that clustering can be a powerful and in some ways orthogonal addition to this list of constraints.
We further study the likelihood regions for a subset of the most interesting parameters in Figure \[fig:paramsubset\]. These are the five parameters for which the width of the likelihood region and/or the location of its peak relative to the G13 value changes most significantly when adding clustering constraints, meaning that the clustering constraints affect the limits on these parameters the most. It is clear from the relative widths of the likelihood regions that these parameters are more strongly constrained when the observed galaxy clustering is imposed. Besides showing wide likelihood regions, the SMF-only data often displayed multiple peaks, behaviour which is also reduced when clustering data is used. One notable exception is $\alpha_\mathrm{friction}$, although there the peaks do lie much closer together than for SMF-only.
The most essential parameters for lowering the small-scale clustering, $\alpha_\mathrm{friction}$ and $V_\mathrm{reheat}$, are also the ones that show the most interesting behaviour here. Not only are their distributions relatively narrow for SMF+$w(r_\mathrm{p})$ and $w(r_\mathrm{p})$-only, but their peaks are also clearly displaced from the input G13 values, favouring smaller dynamical friction time scales and an ISM reheating that is effective up to higher halo masses. While SMF-only also shows displays displaced peaks occasionally, there the G13 value is typically still in a region of high likelihood.
Finally, one interesting parameter to point out is $V_\mathrm{eject}$: this parameter is not noticeably more strongly constrained when clustering data is imposed, but does show a peak displacement from both SMF-only and G13, favouring a slightly lower effectiveness of supernovae ejecting material as a function of mass. This shift is mainly needed to counter the effect of the change in $V_\mathrm{reheat}$ on the abundance of low-mass galaxies, although other parameter combinations are able to serve a similar purpose (as evidenced by $V_\mathrm{eject}$’s still relatively wide likelihood region).
The best-fit parameter values for the five parameters shown in Figure \[fig:paramsubset\] are given in Table \[tab:paramvals\].
Summary {#sec:summary}
=======
We have developed a fast and accurate clustering estimator, capable of predicting the projected galaxy correlation function for a full simulated galaxy catalogue to within $\sim 10\%$ accuracy using only a very small subsample of haloes ($<0.1\%$ of the total sample). In this work, we have described our estimator and demonstrated its effectiveness for use in constraining parameter space for semi-analytic models of galaxy formation, using the @Guo2013 version of the Munich SAM as a test case. Central to the success of our estimator is a new, highly accurate satellite profile, presented in equation .
Our estimator determines the halo occupation distribution of galaxies in the subsample and fits a profile to the galaxy distribution within haloes as a function of halo mass, using these quantities in a halo model based approach to determine the galaxy clustering of the full sample. By being able to quickly predict the two-point galaxy correlation function for the first time while exploring parameter space, one can use clustering observations to limit the range allowed to the galaxy formation parameters of any SAM, adding constraints complementary to those of one-point functions typically used today, such as the stellar mass or luminosity function. As we have demonstrated, this substantially tightens constraints on parameters, and in some cases drives significant shifts in their preferred values. For suitable parameters, existing galaxy formation models nevertheless appear capable of reproducing well both clustering and abundance data for low-redshift galaxies. These results also imply that – at least on the scales considered here – the projected correlation function by itself may not be enough to constrain cosmology, as changes in galaxy formation physics can apparently compensate for using a set of cosmological parameters that significantly differ from current constraints.
For the G13 model tested here, the improved match to the correlation function is achieved mainly by significantly decreasing the time it takes for stripped (orphan) satellites galaxies to merge with their centrals (through a shift in $\alpha_\mathrm{friction}$), as well as increasing the effectiveness of SN feedback in heating the cold ISM gas as a function of halo mass (through $V_\mathrm{reheat}$). Both changes cause the galaxy distribution profiles within haloes to flatten, lowering the clustering on small scales. Other parameter shifts mainly serve to keep the changes in the SMF caused by the reduced time scales in check. The fact that the change in $\alpha_\mathrm{friction}$ has a stronger impact on the projected correlation function than on the measured satellite profiles implies that using clustering data as a constraint is likely the best way to find the right value for this type of parameter. This is an important result not just for semi-analytic models that use a dynamical friction time scale as employed in G13, but for any model that parametrises the merging of its orphan satellites.
While it is already accurate enough for our current application, several improvements could be made to the clustering estimator and/or its application. For example, the halo selection function can be calibrated to a higher accuracy than the $\sim 10\%$ accuracy we aimed for in the test case presented here, at the cost of a larger halo sample. This will improve the estimator’s performance for high-mass galaxies especially, which will likely be important when cosmological parameters are allowed to vary as well as galaxy formation parameters. The higher computational resources could, for example, be offset by varying fewer model parameters simultaneously. Furthermore, the inputs to the model that are not derived from the sample galaxy catalogue, such as the input matter/halo power spectrum, could be improved by e.g. using higher-order bias terms (such as those recently presented by @Lazeyras2016), or even a mass-dependent halo power spectrum measured from the base N-body simulation, which could be scaled with cosmology following @AnguloWhite2010b.
Relatively simple extensions of the methods set out in this paper will allow galaxy formation physics and cosmology to be constrained by abundances and clustering of galaxies at a variety of redshifts and separately for star-forming and passive systems. Galaxy-galaxy lensing could also be included among the constraints through a straightforward extension of our scheme. High-quality observational data are now (or soon will be) available in many of these areas, and it seems likely that requiring simultaneous and acceptable agreement with a single galaxy formation simulation will provide strong constraints not only on astrophysical but also on cosmological parameters. We will explore some of these topics in future work.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Joop Schaye for useful discussions and comments on an earlier version of the manuscript. MPvD also thanks Martin White for fruitful discussions on the Poisson distribution. The Millennium Simulation databases used in this paper and the web application providing online access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory. This work was supported in part by the Marie Curie Initial Training Network CosmoComp (PITN-GA-2009-238356), by Advanced Grant 246797 “GALFORMOD” from the European Research Council, and by the Theoretical Astrophysics Center at UCB. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grants ST/H008519/1 and ST/K00087X/1, STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure.
Fitting the satellite profile {#fittingapp}
=============================
The projected correlation function is very sensitive to small changes in the satellite profile. It is therefore important not only that the functional form of the profile provide a good match to the simulation, but also that the fitting procedure be as unbiased as possible. In our case, fitting bias is an issue because when binning the satellite distribution for a small halo sample in a limited range of halo masses, many radial bins may be empty. Discarding these bins (or equivalently, assigning them infinite error) will bias the profile high, while including them with some finite error may bias the profile low. Additionally, some information is lost when considering the number of satellites in different radial bins as independent measurements. We find that these seemingly small effects can bias the estimated projected correlation function by about $10\%$ for all scales below $1\,\mathrm{Mpc}$.
To avoid these biases, we follow the following procedure. We interpret the measured satellite profile as a realisation of a series of Poisson distributions, with radially-dependent means $\mu$ that scale with the profile given in equation . Specifically: $$\begin{aligned}
\nonumber
\mu(r,\mathbf{p}) \!\!\!\!\!&=&\!\!\!\!\! \frac{N_\mathrm{sat}}{N_\mathrm{h}}\frac{n_\mathrm{sat}(r,\mathbf{p})}{\left<N_\mathrm{sat}\right>}\,\mathrm{d}^3r\\
\!\!\!\!\!&=&\!\!\!\!\! n_\mathrm{sat}(r,\mathbf{p})\,\mathrm{d}^3r\\
\label{eq:means}
\nonumber
\!\!\!\!\!&=&\!\!\!\!\! \frac{c}{4\pi b^3 r_\mathrm{vir}^3\Gamma\left[\frac{a}{c}\right]} \left(\frac{x}{b}\right)^{a-3}\exp\left\{-\left(\frac{x}{b}\right)^c\right\}\left<N_\mathrm{sat}\right> \,\mathrm{d}^3r.\end{aligned}$$ Here we have explicitly shown the dependence of the profile on its parameters with $\mathbf{p}=\{a,b,c\}$. Note that we have slightly simplified the profile given in equation : for a sufficiently large maximum scaled radius $x_\mathrm{max}$, any reasonable set of parameters gives $\gamma\left[\frac{a}{c},\left(\frac{x_\mathrm{max}}{b}\right)^c\right] \approx \Gamma\left[\frac{a}{c}\right]$. For our chosen value of $x_\mathrm{max}=5$, we find that this approximation is well justified. We do however still only consider satellites for which $x=r/r_\mathrm{vir}<x_\mathrm{max}$.
Next, we consider infinitesimally small radial bins, such that the number of satellites in each bin, $N_i$, is either zero or one. The likelihood function is then given by the product of the Poisson distributions at each radius, which we convert to a log-likelihood (exploiting the binary nature of $N_i$): $$\begin{aligned}
\nonumber
\mathcal{L}(\mathbf{p}) \!\!\!\!\!&=&\!\!\!\!\! \prod_i \frac{\mu_i(\mathbf{p})^{N_i}\, e^{-\mu_i(\mathbf{p})}}{N_i!}\\
\Rightarrow \ln{\mathcal{L}(\mathbf{p})} \!\!\!\!\!&=&\!\!\!\!\! \sum_i \ln{\mu_i(\mathbf{p})} - \sum_i \mu_i(\mathbf{p})\\
\nonumber
\!\!\!\!\!&=&\!\!\!\!\! \sum_i \ln{n_\mathrm{sat}(r_i,\mathbf{p})} + \sum_i \ln{\mathrm{d}^3r_i} - \!\int\!\! n_\mathrm{sat}(r,\mathbf{p})\,\mathrm{d}^3r.
\label{eq:likelihood}\end{aligned}$$ Since the profile is by definition normalised, only the first term has any residual dependence on the parameters $\mathbf{p}$. Ignoring constants, we therefore seek to maximise: $$\sum_i \left[\ln{c}-3\ln{b}-\ln{\Gamma\left(\frac{a}{c}\right)}+(a-3)\ln{\frac{x_i}{b}}-\left(\frac{x_i}{b}\right)^c\right].
\label{eq:maxlikelihood}$$ The scaled radii of all satellites in the halo mass bin are thus directly fed into the likelihood function, without radial binning. To maximise this term we utilise the derivatives of the log-likelihood function with respect to the different parameters, which are presented here for completeness: $$\begin{aligned}
\nonumber
\frac{\partial\ln{\mathcal{L}(\mathbf{p})}}{\partial \ln{a}} \!\!\!\!&=&\!\!\!\! \sum_i \left[a\ln{\left(\frac{x_i}{b}\right)}-\frac{a}{c}\psi{\left(\frac{a}{c}\right)}\right],\\
\label{eq:likederivs}
\frac{\partial\ln{\mathcal{L}(\mathbf{p})}}{\partial \ln{b}} \!\!\!\!&=&\!\!\!\! \sum_i \left[c\left(\frac{x_i}{b}\right)^c-a\right],\\
\nonumber
\frac{\partial\ln{\mathcal{L}(\mathbf{p})}}{\partial \ln{c}} \!\!\!\!&=&\!\!\!\! \sum_i \left[1+\frac{a}{c}\psi{\left(\frac{a}{c}\right)}-c\ln{\left(\frac{x_i}{b}\right)}\left(\frac{x_i}{b}\right)^c\right].\end{aligned}$$ Here $\psi(x)$ is the digamma function, defined as the logarithmic derivative of $\Gamma(x)$.
Any set of parameters must satisfy $0\!<\!a\!<\!3$, $b\!>\!0$ and $\!c>\!0$. We have tested that this method does indeed yield unbiased profile fits.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: $M_\mathrm{200mean}$ is the mass within a spherical region with radius $R_\mathrm{200mean}$ and internal density $200\times \bar{\rho}=200\times \Omega_\mathrm{m}\rho_\mathrm{crit}$.
[^3]: We note that our results are not sensitive to the choice of the weight factor: we found that in our case using a factor of two or ten instead of the fiducial five yielded the exact same set of best-fit parameters.
[^4]: For each of the three models we ran $120$ parallel chains with $3000$ steps each.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'New experimental measurements are used to test model independent sum rules for charmed baryon masses. Sum rules for medium-strong mass differences are found to be reasonably well satisfied with increasing accuracy, and the new measurements permit an improved prediction of $2778\pm 9$ MeV for the mass of the $\Omega_c^{*0}$. But an isospin breaking sum rule for the $\Sigma_c$ mass splittings is still in significant disagreement posing a serious problem for the quark model of charmed baryons. Individual $\Sigma_c$ mass splittings are investigated, using the new CLEO measurement of the $\Xi_c^\prime$ mass splitting, but the accuracy is not yet sufficient for a good test.'
---
[**New experimental tests of sum rules\
for charmed baryon masses**]{}\
Jerrold Franklin\
[*Department of Physics, Temple University,\
Philadelphia, Pennsylvania 19122-6082*]{}\
[email protected]\
January 13, 1999
PACS numbers: 12.40.Yx., 14.20.-c, 14.40.-n
Model independent sum rules\[1-3\] were derived some time ago for heavy-quark baryon masses using fairly minimal assumptions within the quark model. The sum rules depend on standard quark model assumptions, and an additional assumption that the interaction energy of a pair of quarks in a particular spin state does not depend on which baryon the pair of quarks is in (“baryon independence"). This is a somewhat weaker assumption than full SU(3) symmetry of the wave function, which would require the same spatial wave function for each octet baryon, and each individual wave function to be SU(3) symmetrized. Instead, we use wave functions with no SU(3) symmetry, as described in Ref.[@sqm]. The wave functions can also be different for different quarks. For instance, a u-s pair in the $\Sigma^+$ hyperon can have a different spatial wave function than a u-d pair in the proton, but is assumed to have the same interaction energy as a u-s pair in the $\Xi^0$ hyperon.
In deriving the sum rules, no assumptions are made about the type of potential, and no internal symmetry beyond baryon independence is assumed. The sum rules allow any amount of symmetry breaking in the interactions and individual wave functions, but do rest on baryon independence for each quark-quark interaction energy. Several of the sum rules \[Eqs. (4), (5), and (6) below\] also rely on the assumption that there is no orbital angular momentum so that the three spin-$\frac{1}{2}$ quark spins add directly to spin-$\frac{1}{2}$ or spin-$\frac{3}{2}$. More detailed discussion of the derivation of the sum rules is given in Refs.and \[4\].
We have previously tested these sum rules in Refs. [@cb2] and [@cb3] using early measurements of heavy-quark baryon masses. Those tests showed reasonable agreement within fairly large experimental errors for two sum rules for medium-strong charmed baryon mass differences and for one sum rule for bottom baryon mass differences. But there was a relatively large, and worrisome, discrepancy for the isospin breaking mass differences between the $\Sigma_c$ charge states. Since those tests, there have been a number of new experiments\[6-11\] resulting in more accurate and more reliable values for some of the charmed baryon masses used in the sum rules. In this paper we look at the effect on the sum rules of these new experiments, especially the recent CLEO II measurement[@c98] of the $\Xi_c^{\prime +}$ and $\Xi_c^{\prime 0}$ masses.
The measured charmed baryon masses that will be used in the sum rules are listed in table I for the expected baryon assignments. The $\Xi_c^{+}$ baryon and the $\Xi_c^{\prime+}$ baryon are distinguished, in the quark model, by having different spin states for the u-s quark pair. The $\Xi_c^{+}$ is the spin-$\frac{1}{2}$ usc baryon having the u-s quarks in a spin zero state, and the $\Xi_c^{\prime+}$ has the u-s quarks in a spin one state. A similar distinction is made for the d-s quark pair in the $\Xi_c^{0}$ and $\Xi_c^{\prime0}$ charmed baryons. The numerical values in Table I are given in terms of appropriate mass differences when that corresponds to how the measurement was made. Where new experiments have given more accurate numbers since our previous test of the sum rules, a star has been put after the reference. Masses for light quark (u,d,s) baryons are all taken from the Review of Particle Physics[@pdg].
Baryon Mass (MeV) Reference
------------------- ------------------------------ -------------------------
$\Lambda_c^+$ $2284.9\pm .6$ [@pdg]
$\Sigma_c^{++}$ $\Lambda_c^+ +167.9\pm .2 $ [@pdg; @ait; @frab96]\*
$\Sigma_c^{0}$ $\Sigma_c^{++}-0.6\pm .2 $ [@pdg; @ait]\*
$\Sigma_c^{+}$ $\Sigma_c^0 +1.4\pm .6 $ [@crawf]
$\Sigma_c^{*++}$ $\Lambda_c^+ + 234.5\pm 1.4$ [@brand]\*
$\Sigma_c^{*0}$ $\Lambda_c^+ + 232.6\pm 1.3$ [@brand]\*
$\Xi_c^{+}$ 2465.6$\pm 1.4$ [@pdg; @edw]\*
$\Xi_c^{0}$ 2470.3$\pm 1.8$ [@pdg]
$\Xi_c^{\prime+}$ $\Xi_c^+ +107.8\pm 3.0$ [@c98]\*
$\Xi_c^{\prime0}$ $\Xi_c^0 +107.0\pm 2.9$ [@c98]\*
$\Xi_c^{*+}$ $\Xi_c^0 +174.3\pm 1.1$ [@gib]\*
$\Xi_c^{*0}$ $\Xi_c^+ +178.2\pm 1.1$ [@av95]
$\Omega_c^0$ 2704$\pm$4 [@pdg]
: Charmed baryon masses used in the sum rules.
The isospin breaking sum rule for the $\Sigma_c$ masses is[@cb2] $$\Sigma^+ +\Sigma^- -2\Sigma^0
= \Sigma^{*+}+\Sigma^{*-}-2\Sigma^{*0}
= \Sigma_c^{++}+\Sigma_c^0-2\Sigma_c^+,$$ -.09in $(1.7\pm .2)$$(2.6\pm 2.1)$$(-2.2\pm
1.2)$\
where we have written the experimental values in MeV below each equation. There is reasonable agreement for the $\Sigma-\Sigma^*$ sum rule, as well as for several other isospin breaking sum rules for light quark baryons[@cb; @sqm]. But the $\Sigma_c$ isospin splitting combination is significantly different from the other two combinations in Eq. (1). As noted in Ref. \[2\], this disagreement poses a serious problem because it is difficult to see how any reasonable quark model of charmed baryons could lead to the relatively large negative value for the $\Sigma_c$ combination in Eq. (1). A large number of specific quark model calculations[@theo] of charmed baryon masses generally satisfy the $\Sigma_c$ sum rule, and all predict large positive values for the $\Sigma_c$ mass combination in Eq. (1).
The experimental input that has been used for this combination of $\Sigma_c$ masses are the two separate mass difference measurements $$\begin{aligned}
\Sigma_c^{++}-\Sigma_c^0 & = & 0.6\pm .2\quad {\rm Ref.[5]}\\
\Sigma_c^{+}-\Sigma_c^0 & = & 1.4\pm .6\quad{\rm Ref.[12].}\end{aligned}$$ The $\Sigma_c^{++}-\Sigma_c^0$ mass difference results from four separate experiments that are reasonably consistent with one another, while there is only one experiment[@crawf] that has measured the $\Sigma_c^+ -\Sigma_c^0$ difference. There is no reason to question this experimental measurement of $\Sigma_c^+ -\Sigma_c^0$, and the result of Ref. \[12\] for $\Sigma_c^{++}
-\Sigma_c^0$ agrees well with the other experiments[@crawsr]. However, the extreme importance of the large discrepancy in the $\Sigma_c$ sum rule of Eq.(1) should make a new experimental measure of the mass difference $\Sigma_c^+
-\Sigma_c^0$ a high priority.
The new experimental measurement of the $\Xi_c^\prime$ masses[@c98] makes it possible, in principle, to test sum rules for separate mass differences of the $\Sigma_c$. These are $$\begin{aligned}
\Sigma_c^{++}-\Sigma_c^{0} & = & \Sigma^{*+}-\Sigma^{*-}
+2[(\Xi^{*-}-\Xi^{*0})+(\Xi_c^{\prime +}-\Xi_c^{\prime 0})] \\
(0.6\pm .2) & & \hspace{.9in} (-6.2 \pm 9.7)\nonumber\\
\Sigma_c^{+}-\Sigma_c^{0} & = & \Sigma^{*0}-\Sigma^{*-}
+(\Xi^{*-}-\Xi^{*0})+(\Xi_c^{\prime +}-\Xi_c^{\prime 0})\\
(1.4\pm .6) & &\hspace{.9in} (-4.2 \pm 4.9)\nonumber\end{aligned}$$ Unfortunately, the experimental errors on the $\Xi_c^\prime$ mass differences are still too large at this point to make an accurate comparison with the $\Sigma_c$ mass differences.
Although the discrepancy noted above for the $\Sigma_c$ mass differences puts any other quark model study of charmed baryons into question, we now look at sum rules for medium-strong mass differences, anticipating some eventual resolution (theoretical or experimental) of the difficulties posed by the $\Sigma_c$ mass splittings. A new measurement[@brand] of the masses of the $\Sigma^{*++}$ and $\Sigma^{*0}$ baryons makes possible a more accurate test of the sum rule[@cb3] $$\begin{aligned}
(\Sigma_c^{*+}-\Lambda_c^+)+\frac{1}{2}(\Sigma_c^+ -\Lambda_c^+)
& = & (\Sigma^{*0}-\Lambda^0)+\frac{1}{2}(\Sigma^0 -\Lambda^0).\\
(319\pm 2)\hspace{.55in} & &\hspace{.55in} (307)\nonumber\end{aligned}$$ We use the measured $\Sigma_c^{*++}$ mass for the $\Sigma_c^{*+}$ mass, but that difference is probably small. A corresponding sum rule[@cb3] for the b-quark baryons $\Sigma_b^{*0}$, $\Sigma_b^{0}$, $\Lambda_b^{0}$ has not changed, and is in good agreement .
In Ref.[@cb2] we used a sum rule to predict $2583\pm 3$ MeV for the $\Xi_c^{\prime +}$ mass. This mass has now been measured[@c98], and is listed in Table I. This permits a test of the sum rule, which we write here as
$$\begin{aligned}
\Sigma_c^{++}+\Omega_c^{0} -2\Xi_c^{\prime +}
& = & \Sigma^+ +\Omega^{-} -\Xi^0 -\Xi^{*0}\\
(10\pm 8)\hspace{.38in} & &\hspace{.44in} (15)\nonumber\end{aligned}$$
The two sum rules in Eqs. (6) and (7) are satisfied to about the same extent as light-quark baryon sum rules relating spin-$\frac{1}{2}$ baryon masses to spin-$\frac{3}{2}$baryon masses.[@cb; @sqm]
The new experimental measurements can be used to improve the accuracy of our previous prediction\[3\] of the as yet unmeasured $\Omega_c^{*0}$ mass $$\Omega_c^{*0}=\Omega_c^{0}+2(\Xi_c^{*+}-\Xi_c^{\prime+})
-(\Sigma_c^{*++}-\Sigma_c^{++})=2779\pm9,$$
In conclusion, we can say that increasingly accurate experimental mass determinations are making the model independent sum rules discussed here increasingly useful tests of the quark model for charmed baryons. We see that sum rules for medium-strong energy differences are satisfied at least as well for heavy-quark baryons as for light-quark baryons. However there remains a serious disagreement for the $\Sigma_c$ isospin breaking sum rule, which is violated by three standard deviations. Since sum rules in disagreement are of more concern than those which are satisfied, resolving the $\Sigma_c$ mass differences is of prime importance. Thus far no theoretical suggestion has been forthcoming.
[8]{} J. Franklin, Phys. Rev. [**D12**]{}, 2077 (1975). J. Franklin, Phys. Rev. [**D53**]{}, 564 (1996). J. Franklin, Phys. Rev. [**D55**]{}, 423 (1997). J. Franklin Phys. Rev. [**172**]{}, 1807 (1968). Review of Particle Properties, Phys. Rev. [**D50**]{}, 1173 (1994). E. M. Aitala [*et al.*]{} (E791), Phys. Lett. [**96B**]{}, 292 (1991). P. Frabetti [*et al.*]{} (E687), Phys. Lett. [**B365**]{}, 461 (1996). G. Brandenburg [*et al.*]{} (CLEO II), Phys. Rev. Lett. [**78**]{}, 2304 (1997). K. Edwards [*et al.*]{} (CLEO II), Phys. Lett. [**B373**]{}, 261 (1996). C. P. Jessup [*et al.*]{} “Observation of Two Narrow States Decaying Into $\Xi_c^{\prime +\gamma}$ and $\Xi_c^{\prime 0\gamma}$, CLNS 98/1581, CLEO 98-13, hep-ex/9810036 (unpublished). L. Gibbons[*et al.*]{}( CLEO II), Phys. Rev. Lett. [**75**]{}, 4364 (1996). Crawford [*et al.*]{} (CLEO II), Phys. Rev. Lett. [**71**]{}, 3259 (1993). P. Avery [*et al.*]{}( CLEO II), Phys. Rev. Lett. [**77**]{}, 810 (1995). C. Itoh [*et al.*]{}, Progr. Th. Phys. [**54**]{}, 908 (1975); K. Lane and S. Weinberg, Phys. Rev. Lett. [**37**]{}, 717 (1976); S. Ono, Phys. Rev. D [**15**]{}, 3492 (1977); N. G. Deshpande, [*et al.*]{}, Phys. Rev. D [**15**]{}, 1885 (1977); L-H. Chan, Phys. Rev. D [**15**]{}, 2478 (1977); [*ibid.*]{} [** 31**]{}, 204 (1985); D. B. Lichtenberg, Phys. Rev. D [**16**]{}, 231 (1977); C. S. Kalman and G. Jakimow, Lett. al Nuovo Cimento [**19**]{}, 403 (1997); A. C. D. Wright, Phys. Rev. D [**17**]{}, 3130 (1978); N. Isgur, Phys. Rev. D [**21**]{}, 779 (1980); J. M. Richard and P. Taxil, Z. Phys. C [**26**]{}, 421 (1984); W-Y. Wang and D. B. Lichtenberg, Phys. Rev. D [**35**]{}, 3526 (1987); S. Capstick, Phys. Rev. D [**36**]{}, 2800 (1987); S. Sinha, [*et al.*]{}, Phys. Lett. B [**218**]{}, 333 (1989); M. Genovese, [*et al.*]{}, Phys. Rev. D [**59**]{}, 014012 (1999). Reference [@crawf] is the only experiment to simultaneously measure both $\Sigma_c$ mass differences. For this single experiment, the $\Sigma_c$ sum rule combination gives $\Sigma_c^{++}+\Sigma_c^{0}-2\Sigma_c^{+}=-1.7\pm 1.0$.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We study pointwise convergence of entangled averages of the form $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f,$$ where $f\in L^2(X,\mu)$, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$, and the $T_i$ are ergodic measure preserving transformations on the standard probability space $(X,\mu)$. We show that under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, almost everywhere convergence holds for all $f\in L^2$. We also present results for the general $L^p$ case ($1\leq p<\infty$) and for polynomial powers, in addition to continuous versions concerning ergodic flows.'
address: 'MTA Alfréd Rényi Institute of Mathematics, P.O. Box 127, H-1364 Budapest, Hungary'
author:
- 'Dávid Kunszenti-Kovács'
title: Almost everywhere convergence of entangled ergodic averages
---
Introduction
============
Entangled ergodic averages go back to a paper by Accardi, Hashimoto and Obata [@AHO], where these were introduced motivated by questions from quantum stochastics. Given a map $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ and operators $A_i$ ($1\leq i\leq m-1$) and $T_i$ ($1\leq i\leq m$) on a Banach space $E$, the corresponding entangled averages are the multi-Cesàro means $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}}.$$ Originally, the question was about norm convergence of such averages, and this was further studied by Liebscher [@liebscher:1999], Fidaleo [@fidaleo:2007; @fidaleo:2010; @fidaleo:2014] and Eisner and the author [@EKK].\
In Eisner, K.-K. [@EKK2], attention was turned to pointwise almost everywhere convergence in the context of the $T_i$’s being Koopman operators on function spaces $E=L^p(X,\mu)$ ($1\leq p<\infty$), where $(X,\mu)$ is a standard probability space (i.e. a compact metrizable space with a Borel probability measure). The paper covered the one-parameter ($k=1$) case, and in the present paper, we push the ideas and methods presented there to study pointwise almost everywhere convergence of entangled ergodic averages in their full generality, allowing for multi-parameter entanglement.\
Note that in what follows, given a measure preserving transformation $S$ on a standard probability space $(X,\mu)$ and a measurable function $g$ on $X$, we shall write $Sg$ for the function $g\circ S$, i.e., we do not make a distinction between the transformation $S$ and the induced contractive operator on $L^q(X)$ ($1\leq q\leq\infty$).\
Our main result is as follows.
\[thm:main\] Let $m>1$ and $k$ be positive integers, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ a not necessarily surjective map, and $T_1,T_2,\ldots, T_m$ ergodic measure preserving transformations on a standard probability space $(X,\mu)$. Let $p\in[1,\infty)$, $E:=L^p(X,\mu)$ and let $E=E_{j,r}\oplus E_{j,s}$ be the Jacobs-Glicksberg-deLeeuw decomposition corresponding to $T_j$ $(1\leq j\leq m)$. Let further $A_j\in{\mathcal}{L}(E)$ $(1\leq j< m)$ be bounded operators. For a function $f\in E$ and an index $1\leq j\leq m-1$, write ${\mathscr}{A}_{j,f}:=\left\{A_jT_j^nf\left|\right.n\in{\mathbb}{N}\right\}$. Suppose that the following conditions hold:
- (Twisted compactness) For any function $f\in E$, index $1\leq j\leq m-1$ and $\varepsilon>0$, there exists a decomposition $E={\mathcal}{U}\oplus {\mathcal}{R}$ with $\dim {\mathcal}{U}<\infty$ such that $$P_{\mathcal}{R}{\mathscr}{A}_{j,f}
\subset B_\varepsilon(0,L^\infty(X,\mu)),$$ with $P_{\mathcal}{R}$ denoting the projection along ${\mathcal}{U}$ onto ${\mathcal}{R}$.
- (Joint ${\mathcal}{L}^\infty$-boundedness) There exists a constant $C>0$ such that we have $$\{A_jT^n_j|n\in{\mathbb}{N},1\leq j\leq m-1\}\subset B_C(0,{\mathcal}{L}(L^\infty(X,\mu)).$$
Then we have the following:
1. for each $f\in E_{1,s}$, $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f \right|\rightarrow 0$$ pointwise a.e.;
2. if $p=2$, then for each $f\in E_{1,r}$, $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f$$ converges pointwise a.e..
Note that it was proven in [@EKK2] that the Volterra operator $V$ on $L^2([0,1])$ defined through $$(Vf)(x):=\int_{0}^x f(z) {\mathrm}{d}z$$ as well as all of its powers can be decomposed into a finite sum of operators, each of which satisfy conditions (A1) and (A2) when paired with any Koopman operator. Hence the conclusions of Theorem \[thm:main\] apply whenever the operators $A_i$ are chosen to be powers of $V$.
Notations and tools {#sec:prelim}
===================
Before proceeding to the proof of our main result, we need to clarify some of the notions used, and introduce notations that will simplify our arguments.
Let ${\mathcal}{N}$ denote the set of all bounded sequences $\{a_n\}\subset \ell^\infty({\mathbb}{C})$ satisfying $$\lim_{N\to\infty}{\frac{1}{N}\sum_{n=1}^N}|a_n|=0.$$ By the Koopman-von Neumann lemma, see e.g. Petersen [@petersen:1983 p. 65], $(a_n)\in{\mathcal}{N}$ if and only if it lies in $\ell^\infty$ and converges to $0$ along a sequence of density $1$.\
Given a Banach space $E$ and an operator $T\in{\mathcal}{L}(E)$, the operator $T$ is said to have *relatively weakly compact orbits* if for each $f\in E$ the orbit set $\left\{ T^nf|n\in{\mathbb}{N}^+\right\}$ is relatively weakly closed in $E$. For any such operator, there exists a corresponding *Jacobs-Glicksberg-deLeeuw* decomposition of the form (cf. [@eisner-book Theorem II.4.8]) $$E=E_r \oplus E_s,$$ where $$\begin{aligned}
E_r&:=&\overline{\text{lin}}\{f\in E:\ Tf=\lambda f\mbox{ for some } |\lambda|=1\},\\
E_s&:=&\{f\in E:\ (\varphi( T^nf))\in{\mathcal}{N} \mbox{ for every } \varphi\in E'\}.\end{aligned}$$
Recall that power bounded operators on reflexive Banach spaces as well as positive contractions $T$ on $L^1(X,\mu)$ with $T\mathds{1}=\mathds{1}$ (cf. [@eisner-book Theorem II.4.8]) all have relatively weakly compact orbits. In particular, for all $1\leq p<\infty$, any Koopman operator on $E=L^p(X,\mu)$ has this property, and thus the corresponding decomposition above exists.
A sequence $(a_n)_{n=1}^\infty\subset {\mathbb}{C}$ is called a *good weight for the pointwise ergodic theorem* if for every measure preserving system $(X,\mu,T)$ and every $f\in L^1(X,\mu)$, the weighted ergodic averages $${\frac{1}{N}\sum_{n=1}^N}a_n T^nf$$ converge almost everywhere as $N\to\infty$.
Denote by ${\mathscr}{P}\subset \ell^\infty$ the set of almost periodic sequences, i.e., uniform limits of finite linear combinations of sequences of the form $(\lambda^n)$, $|\lambda|=1$. Such sequences play an important role in pointwise ergodic theorems. Indeed, every element in ${\mathscr}{P}$ is a good weight for the pointwise ergodic theorem. Also, it can easily be checked that the set ${\mathscr}{P}$ is closed under (elementwise) multiplication. For more details and the first part of the following example we refer to e.g. [@E].
\[ex:alm-per\]
1. Let $T$ have relatively weakly compact orbits on a Banach space $E$, $f\in E_r$ and $\varphi\in E'$. Then $(\varphi(T^n f))\in {\mathscr}{P}$.
2. Let $(a_n)_{n\in{\mathbb}{N}^+}\subset\ell^\infty$ such that for some $(q_n)_{n\in{\mathbb}{N}^+}\in\ell^1$ and $(\gamma_n)\subset {\mathbb}{C}$ with $|\gamma_n|=1$, $n\in{\mathbb}{N}$, $$a_n=
\sum_{k=0}^\infty \gamma_k^n\cdot q_k\quad \forall n\in{\mathbb}{N}.$$ Then $(a_n)\in {\mathscr}{P}$.
Proof of Theorem \[thm:main\] {#sec:general-case}
=============================
We shall proceed by successive splitting and reduction. For each operator $T_i$, starting from $T_2$, we split the functions it is applied to into several terms using condition (A1). Most of the obtained terms can be easily dealt with, but for the remaining ”difficult” terms, we move on to $T_{i+1}$, up to and including $T_m$.\
We first prove part (1), and then use this result to complete the proof for part (2). The details for the cases $m=2$ and $m=3$ will be worked out fully, and we shall then show how the proof extends to $m>3$.
Let $f\in E$ and $\epsilon>0$ be given. Then by assumption (A1) we have a finite-dimensional subspace ${\mathcal}{U}={\mathcal}{U}(f,\varepsilon/C^{m-1})\subset E$ and a decomposition $E={\mathcal}{U}\oplus {\mathcal}{R}$ such that $$P_{\mathcal}{R}{\mathscr}{A}_{1,f}\subset B_{\varepsilon/C^{m-1}}(0,L^\infty(X,\mu)).$$ Choose a maximal linearly independent set $g_1,\ldots,g_\ell$ in ${\mathcal}{U}$. We then for each $n\in{\mathbb}{N}^+$ have $$A_1T_1^{n}f=\lambda_{1,n}g_1+\ldots+\lambda_{\ell,n}g_\ell+r_n$$ for appropriate coefficients $\lambda_{j,n}\in{\mathbb}{C}$ and some remainder term $r_n\in {\mathcal}{R}$ with $\|r_n\|_\infty<\varepsilon/C^{m-2}$. There exist linear forms $\varphi_1,\ldots, \varphi_\ell\in E'$ such that $$\varphi_j(g_i)=\delta_{i,j}\quad \mbox{and}\quad \varphi_j|_{{\mathcal}{R}}=0\quad \mbox{ for every } i,j\in\{1,\ldots,\ell\}.$$ We then have $$\lambda_{j,n}= \varphi_j( A_1T_1^nf)= (A_1^*\varphi_j)(T_1^nf),$$ therefore $$\left|\lambda_{j,n}\right|\leq \|f\| \cdot\|A_1^*\| \max_{j\in\{1,\ldots,\ell\}}\| \varphi_j\|=:c$$ and, if $f\in E_{1,s}$, $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}\in{\mathscr}{N}$. Note that $c$ depends on $\varepsilon$.
With this splitting we have that $$\begin{aligned}
&&\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f\\
&=&\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} r_{n_{\alpha(1)}}\\
&&+\sum_{j=1}^\ell \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}g_j,\end{aligned}$$ and we shall investigate the multi-Cesàro convergence of each term separately.
Since $r_{n_{\alpha(1)}}\in L^\infty$, using condition (A2) and the fact that $T_m$ is an $L^\infty$-isometry we have for the first term the inequality $$\begin{aligned}
&& \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} r_{n_{\alpha(1)}}\right|(x)\\
&\leq& C^{m-2}
\frac{1}{N}\sum_{n_{\alpha(1)=1}}^N
\|r_{n_{\alpha(1)}}\|_\infty<\varepsilon\end{aligned}$$ for all $x$ from a set $R\subset X$ with $\mu(R)=1$.\
It remains to show that the second term also converges in the required sense.
We first turn our attention to part (1), and assume that $f\in E_{1,s}$. Recall that we then have for each $1\leq j\leq \ell$ that $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}\in{\mathscr}{N}$. Let us fix $1\leq j\leq \ell$, and consider the term $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}g_j.$$
Assume first that $m=2$. In this case we choose a function $\widetilde{g}_j\in L^\infty$ such that $\|g_j-\widetilde{g}_j\|_1\leq\|g_j-\widetilde{g}_j\|_p<\varepsilon/c\ell$. Then $$\begin{aligned}
&&
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}}\lambda_{j,n_{\alpha(1)}} g_j|\\
&\leq&
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}}\lambda_{j,n_{\alpha(1)}} (g_j-\widetilde{g}_j)|
+
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}}\lambda_{j,n_{\alpha(1)}} \widetilde{g}_j|.\end{aligned}$$
Since $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}\in{\mathscr}{N}$, the second term satisfies by (A2) $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}}\lambda_{j,n_{\alpha(1)}} \widetilde{g}_j|\leq \|\widetilde{g}_j\|_\infty\cdot\frac{1}{N}\sum_{n_{\alpha(1)}=1}^N |\lambda_{j,n_{\alpha(1)}}|\to 0$$ for all $x$ from a set $P_j\subset R$ with $\mu(P_j)=1$. Set $Q:=\cap_{j=1}^\ell P_j$.
It now remains to treat the first term, $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}}\lambda_{j,n_{\alpha(1)}} (g_j-\widetilde{g}_j)|.$$ Applying Birkhoff’s pointwise ergodic theorem to the operator $T_2$ and the function $|(g_j-\widetilde{g}_j)|$, there exists a set $S_{j,\varepsilon}\subset Q$ with $\mu(S_{j,\varepsilon})=1$ such that $$\lim_{N\to\infty}\frac{1}{N} \sum_{n_{\alpha(2)}=1}^N T_2^{n_{\alpha(2)}} |(g_j-\widetilde{g}_j)|=\|(g_j-\widetilde{g}_j)\|_1.$$ Since the sequence $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}$ is bounded in absolute value by $c$, we thus obtain $$\begin{aligned}
&&\overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}}\lambda_{j,n_{\alpha(1)}} (g_j-\widetilde{g}_j)|(x)\\
&\leq&
c\cdot\overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}}(g_j-\widetilde{g}_j)|(x)
=c\|(g_j-\widetilde{g}_j)\|_1<\varepsilon/\ell\end{aligned}$$ for each $x\in S_{j,\varepsilon}$.
Summing over all $1\leq j\leq \ell$, we have for each $x\in\cap_{j=1}^\ell S_{j,\varepsilon}=:S_\varepsilon$ that
$$\begin{aligned}
&&
\overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f|\\
&\leq&\overline{\lim_{N\to\infty}}\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |A_2T_2^{n_{\alpha(2)}} r_{n_{\alpha(1)}}|(x)\\
&&+\sum_{j=1}^\ell \overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}\widetilde{g}_j|(x)\\
&&+\sum_{j=1}^\ell \overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}(g_j-\widetilde{g}_j)|(x)
\\
&<&\varepsilon+0+\ell\cdot\varepsilon/\ell=2\varepsilon.\end{aligned}$$
But $\mu(S_\varepsilon)=1$, and hence we are done (with the case m=2 of part (1)).
Now assume $m>2$ and fix $1\leq j\leq \ell$. Again by assumption (A1), there exists a further finite dimensional subspace ${\mathcal}{U}_j={\mathcal}{U}(g_j,\varepsilon/\ell C^{m-3})\subset E$ and a corresponding decomposition $E={\mathcal}{U}_j\oplus{\mathcal}{R}_j$ such that $P_{{\mathcal}{R}_j}{\mathscr}{A}_{2,g_j}\subset B_{\varepsilon/\ell C^{m-3}}(0,L^\infty(X,\mu))$. Choose a maximal linearly independent set $g_{1,j},g_{2,j},\ldots,g_{\ell_j,j}$ in ${\mathcal}{U}_j$ and let\
$\varphi_{1,j},\ldots, \varphi_{\ell_j,j}\in E'$ have the property $$\varphi_{i,j}(g_{l,j})=\delta_{i,l}\quad \mbox{and}\quad \varphi_{i,j}|_{{\mathcal}{R}_j}=0\quad \mbox{ for every } i,l\in\{1,\ldots,\ell_j\}.$$ Then we may for each $n\in{\mathbb}{N}^+$ write $$A_2T_2^n g_j=\lambda_{1,j,n}g_{1,j}+\ldots+\lambda_{\ell_j,j,n}g_{\ell_j,j}+r_{j,n}$$ for appropriate coefficients $\lambda_{i,j,n}\in{\mathbb}{C}$ ($1\leq i\leq \ell_j$) and remainder term $r_{j,n}\in {\mathcal}{R}_j$ with $\|r_{j,n}\|_\infty<\varepsilon/\ell C^{m-3}$, and we have $$\lambda_{i,j,n}= \varphi_{i,j}( A_2T_2^n g_j)=(A_2^*\varphi_{i,j})(T_2^n g_j).$$ It follows that $$\left|\lambda_{i,j,n}\right|\leq \|g_j\|\cdot\|A_2^*\| \cdot\max_{i\in\{1,\ldots,\ell_j\}}\|\varphi_{i,j}\|=:c_j.$$
Since $\|r_{j,n_{\alpha(2)}}\|_\infty<\varepsilon/\ell C^{m-3}$, using condition (A2) and the fact that $T_m$ is an $L^\infty$-isometry we have for the contribution of the $r_{j,n}$ terms that $$\begin{aligned}
&& \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_3T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} r_{j,n_{\alpha(2)}}\right|(x)\\
&\leq& C^{m-3}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N}
\|\lambda_{j,n_{\alpha(1)}} r_{n_{\alpha(2)}}\|_\infty<\frac{\varepsilon}{\ell} \sum_{n=1}^N |\lambda_{j,n}|\end{aligned}$$ for all $x$ from some set $R_j\subset R$ with $\mu(R_j)=1$. But since $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}\in{\mathcal}{N}$, the Cesàro limit of $(|\lambda_{j,n}|)_{n\in{\mathbb}{N}}$ is equal to zero, so we in fact have $$\lim_{N\to\infty}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_3T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} r_{j,n_{\alpha(2)}}\right|(x)
=0$$ for all $x\in R_j$.
We now turn our attention to the term $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left| T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_3T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} \lambda_{i,j,n_{\alpha(2)}} g_{i,j}\right|.$$ If $m=3$, then as above for the case $m=2$, we split each $g_{i,j}$ ($1\leq i\leq \ell_j$) into $\widetilde{g}_{i,j}\in L^\infty$ and the remainder $g_{i,j}-\widetilde{g}_{i,j}$ such that $\|g_{i,j}-\widetilde{g}_{i,j}\|_1\leq \|g_{i,j}-\widetilde{g}_{i,j}\|_p\leq\varepsilon/cc_j\ell\ell_j$. Then $$\begin{aligned}
&& \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} \lambda_{i,j,n_{\alpha(2)}} g_{i,j}\right|(x)\\
&\leq&
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} \lambda_{i,j,n_{\alpha(2)}} \widetilde{g}_{i,j}\right|(x)\\
&&+ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} \lambda_{i,j,n_{\alpha(2)}} (g_{i,j}-\widetilde{g}_{i,j})\right|(x).\end{aligned}$$
Since $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}\in{\mathscr}{N}$, and $(|\lambda_{i,j,n}|)_{n\in{\mathbb}{N}^+}$ is bounded by $c_j$, the first term satisfies by (A2) $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} \lambda_{i,j,n_{\alpha(2)}} \widetilde{g}_{i,j}\right|(x)
\leq
c_j\|\widetilde{g}_{i,j}\|_\infty\cdot\frac{1}{N}\sum_{n_{\alpha(1)}=1}^N |\lambda_{j,n_{\alpha(1)}}|\to 0$$ for all $x$ from a set $P_{i,j}\subset R_j$ with $\mu(P_{i,j})=1$. Set $Q_j:=\cap_{i=1}^{\ell_j} P_{i,j}$.
Applying Birkhoff’s pointwise ergodic theorem this time to the operator $T_3$ and the function $|(g_{i,j}-\widetilde{g}_{i,j})|$, there exists a set $S_{i,j,\varepsilon}\subset Q_j$ with $\mu(S_{i,j,\varepsilon})=1$ such that $$\lim_{N\to\infty}\frac{1}{N} \sum_{n_{\alpha(3)}=1}^N T_3^{n_{\alpha(3)}} |(g_{i,j}-\widetilde{g}_{i,j})|=\|(g_{i,j}-\widetilde{g}_{i,j})\|_1.$$ Since the sequence $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}$ is bounded in absolute value by $c$ and $(|\lambda_{i,j,n}|)_{n\in{\mathbb}{N}^+}$ is bounded by $c_j$, we thus obtain $$\begin{aligned}
&&\overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} \lambda_{i,j,n_{\alpha(2)}} (g_{i,j}-\widetilde{g}_{i,j})\right|(x)\\
&\leq&
cc_j\cdot\overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_3^{n_{\alpha(3)}}(g_{i,j}-\widetilde{g}_{i,j})|(x)
=cc_j\|(g_j-\widetilde{g}_j)\|_1<\varepsilon/\ell\ell_j\end{aligned}$$ for each $x\in S_{i,j,\varepsilon}$.
Summing over all $1\leq i\leq \ell_j$ and then $1\leq j\leq\ell$ we therefore obtain $$\begin{aligned}
&&
\overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_3^{n_{\alpha(3)}}A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f|(x)\\
&\leq&\overline{\lim_{N\to\infty}}\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_3^{n_{\alpha(3)}}A_2T_2^{n_{\alpha(2)}} r_{n_{\alpha(1)}}|(x)\\
&&+\sum_{j=1}^\ell \overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} r_{j,n_{\alpha(2)}}|(x)\\
&&+\sum_{j=1}^\ell \sum_{i=1}^{\ell_j} \overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}}\lambda_{i,j,n_{\alpha(2)}} \widetilde{g}_{i,j}|(x)\\
&&+\sum_{j=1}^\ell \sum_{i=1}^{\ell_j} \overline{\lim_{N\to\infty}}
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} |T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}}\lambda_{i,j,n_{\alpha(2)}}(g_{i,j}-\widetilde{g}_{i,j})|(x)
\\
&<&\varepsilon+0+0+\sum_{j=1}^\ell \ell_j\cdot\varepsilon/\ell\ell_j=2\varepsilon.\end{aligned}$$ for all $x\in\cap_{j=1}^\ell\cap_{i=1}^{\ell_j} S_{i,j,\varepsilon}=:S_\varepsilon$, where $\mu(S_\varepsilon)=1$.\
This concludes the case $m=3$ of part (1). For general values of $m>3$, we follow the same procedure.\
Fix $\varepsilon>0$ and $f\in E_{1,s}$, and until and including reaching $A_{m-1}T_{m-1}^{n_{\alpha(m-1)}}$, we split the current functions $A_iT_i^ng_*$ according to property (A1) into a remainder term $r_{*,n}$ and a finite linear combination of functions $g_{b,*}$, with the new coefficients $\lambda_{b,*,n}$ forming a bounded sequence.\
Since each $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}$ lies in ${\mathcal}{N}$, all remainder terms beyond the $r_n$’s will have a zero contribution to the pointwise multi-Cesàro means (cf. the contribution of the $r_{j,n}$’s in the case $m=3$). The averaging out of the terms with $r_n$ will yield a contribution to the limes superior of at most $\varepsilon$.\
Once we have reached the stage $T_m^{n_{\alpha(m)}}g_{b,*}$, we split each $g_{b,*}$ further into a bounded function $\widetilde{g}_{b,*}$, and a term $(g_{b,*}-\widetilde{g}_{b,*})$ with very small $L^1$ norm.\
Again thanks to $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}\in{\mathcal}{N}$ and all other $\lambda_{*,n}$ sequences being bounded, the former terms will have a zero contribution, whilst the latter terms will, due to Birkhoff’s pointwise ergodic theorem, contribute to the limes superior with a total less than $\varepsilon$. Note that this is possible because in each step the number of functions we split into is only dependent on the current functions to be split and the value of $\varepsilon$ (cf. the details of the case $m=3$).
For part (2), assume $p=2$ and note that eigenfunctions in $E_{1,r}$ pertaining to different eigenvalues are always orthogonal. Let so $f\in E_{1,r}$ be fixed, and let $\left\{h_v\right\}_{v\in V}$ be an orthonormal basis in $E_{1,r}$ of eigenvectors pertaining to unimodular eigenvalues $\left\{\beta_v\right\}_{v\in V}$. Note that $V$ is a countable set. Then we can write $f=\sum_{v\in V} d_v h_v$, for some $\ell^2$-sequence $(d_v)$, and we have $$\begin{aligned}
\lambda_{j,n}
=\langle T_1^nf,A_1^*\varphi_j\rangle=\big\langle\sum_{v\in V}\beta_v^nd_v h_v,A_1^*\varphi_j\big\rangle=\sum_{v\in V}\beta_v^n \left(d_v\langle h_v,A_1^*\varphi_j\rangle\right),\end{aligned}$$ and so for each $1\leq j\leq \ell$ we have that $(\lambda_{j,n})_n\in{\mathscr}{P}$ since $\left(d_v\langle h_v,A_1^*\varphi_j\rangle\right)\in l^1$ by the Cauchy-Schwarz inequality.
For each $1\leq j\leq \ell$, we may split $g_j$ into a stable and a reversible part with respect to $T_2$, i.e. $g_j=g_j^s+g_j^r$ with $g_j^s\in E_{2,s}$ and $g_j^r\in E_{2,r}$. Then we have $$\begin{aligned}
&&\sum_{j=1}^\ell T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}g_j\\
&=&\sum_{j=1}^\ell T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}g_j^r\\
&&+\sum_{j=1}^\ell T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}g_j^s.\end{aligned}$$ To complete the proof of part (2), it remains to be shown that the multi-Cesàro means of the Left Hand Side converges pointwise for almost all $x\in R$. To that end, we shall show that each of the two sums on the Right Hand Side have this property.
Indeed, for the second sum, using part (1) applied to $(m-1)$ pairs $A_iT_i^n$, we obtain that $$\begin{aligned}
&&\left|
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \sum_{j=1}^\ell T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}g_j^s
\right|(x)\\
&\leq&
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \sum_{j=1}^{\ell}
\left|
T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}g_j^s
\right|(x)\\
&\leq&
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \sum_{j=1}^{\ell}
c \left|
T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} g_j^s
\right|(x)\\
&=&
c\cdot \sum_{j=1}^{\ell}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N}
\left|
T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} g_j^s
\right|(x)
\right)\to 0\end{aligned}$$ for all $x$ in a set $S\subset R$ with $\mu(S)=1$.
Now let us turn our attention to the first sum, involving the reversible parts $g_j^r$.\
In case $m=2$, this is in fact of the simple form $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \sum_{j=1}^\ell T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}}g_j^r (x)
=\sum_{j=1}^\ell \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \lambda_{j,n_{\alpha(1)}} T_2^{n_{\alpha(2)}} g_j^r (x).$$ Recall that the sequences $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}$ are almost periodic, hence on the one hand are good weights for the pointwise ergodic theorem, but also converge in the Cesàro sense. Using the former property in case $\alpha(1)=\alpha(2)$ and the latter otherwise, we therefore obtain that the above expression converges for almost all $x\in S$. This concludes the case $m=2$ for part (2).\
Now assume $m\geq 3$, and use property (A1) applied to $A_2,T_2$ to obtain the following. For each $1\leq j\leq \ell$ there exists a further finite dimensional subspace $${\mathcal}{U}_j={\mathcal}{U}(g_j^r,\varepsilon/cC^{m-2})\subset E$$ and a corresponding decomposition $E={\mathcal}{U}_j\oplus{\mathcal}{R}_j$ such that $$P_{{\mathcal}{R}_j}{\mathscr}{A}_{2,g_j^r}\subset B_{\varepsilon/cC^{m-2}}(0,L^\infty(X,\mu)).$$ Let $g_{1,j},g_{2,j},\ldots,g_{\ell_j,j}$ be an orthonormal basis in ${\mathcal}{U}_j$. Then we may for each $n\in{\mathbb}{N}^+$ write $$A_2T_2^n g_j^r=\lambda_{1,j,n}g_{1,j}+\ldots+\lambda_{\ell_j,j,n}g_{\ell_j,j}+r_{j,n}$$ for appropriate coefficients $\lambda_{i,j,n}\in{\mathbb}{C}$ ($1\leq i\leq \ell_j$) and remainder term $r_{j,n}\in {\mathcal}{R}_j$ with $\|r_{j,n}\|_\infty<\varepsilon/C^{a-2}$, and we have $$\lambda_{i,j,n}= \langle A_2T_2^n g_j^r,\varphi_{i,j}\rangle=\langle T_2^n g_j^r,A_2^*\varphi_{i,j}\rangle,$$ where each $\varphi_{i,j}$ is orthogonal to ${\mathcal}{R}_j$ and $\langle g_{l,j},\varphi_{i,j}\rangle=\delta_{l,i}$. (Note that in case ${\mathcal}{R}_j\perp{\mathcal}{U}_j$, we may choose $\varphi_{i,j}:=g_{i,j}$.) Thus $$\left|\lambda_{i,j,n}\right|\leq \|g_j^r\|_2\cdot\|A_2^*\|\max\{\|\varphi_{i,j}\|_2,\, i=1,\ldots,k_j\}=:c_j$$ and also $(\lambda_{i,j,n})_{n\in{\mathbb}{N}^+}\in{\mathscr}{P}$ by Example \[ex:alm-per\].
Therefore for each $1\leq j\leq \ell$ we have for almost every $x\in X$ $$\begin{aligned}
&\left|\overline{\lim}_{N\to\infty}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}} g_j^r
\right)(x)
\right.
\\
&
-
\left.
\underline{\lim}_{N\to\infty}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}} \lambda_{j,n_{\alpha(1)}} g_j^r
\right)(x)
\right|\\
&\leq
\left|\overline{\lim}_{N\to\infty}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_3T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} r_{j,n}
\right)(x)
\right.
\\
&
-
\left.
\underline{\lim}_{N\to\infty}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_3T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}} r_{j,n}
\right)(x)
\right|
\\
&+
\sum_{i=1}^{\ell_j}
\left|\overline{\lim}_{N\to\infty}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}\ldots A_3T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}}\lambda_{i,j,n_{\alpha(2)}} g_{i,j}
\right)(x)
\right.
\\
&
-
\left.
\underline{\lim}_{N\to\infty}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}\ldots A_3T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}}\lambda_{i,j,n_{\alpha(2)}} g_{i,j}
\right)(x)
\right|.\end{aligned}$$
Note that the first term on the Right Hand Side is bounded by $$2cC^{m-2}\|r_{j,n}\|_\infty<2\varepsilon$$ for all $x$ from a set $R_j\subset S$ with $\mu(R_j)=1$, and to complete our proof we need to handle the terms of the sum.\
Now if $m=3$, the sum simplifies to $$\begin{aligned}
\sum_{i=1}^{\ell_j}
&&\left|\overline{\lim}_{N\to\infty}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}}\lambda_{i,j,n_{\alpha(2)}} g_{i,j}
\right)(x)
\right.\\
&&-
\left.
\underline{\lim}_{N\to\infty}
\left(
\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_3^{n_{\alpha(3)}} \lambda_{j,n_{\alpha(1)}}\lambda_{i,j,n_{\alpha(2)}} g_{i,j}
\right)(x)
\right|\end{aligned}$$
The sequences $(\lambda_{i,j,n})_{n\in{\mathbb}{N}^+}$ and $(\lambda_{j,n})_{n\in{\mathbb}{N}^+}$ are almost periodic, hence on the one hand are good weights for the pointwise ergodic theorem, but also converge in the Cesàro sense, and the same is true for their product. In light of these properties, grouping them in a manner similar to what was done for the case $m=2$, we obtain that the above expression converges to $0$ for almost all $x\in R_j$. This concludes the case $m=3$ for part (2).
In case $m>3$, the above splitting procedure can be extended in a natural way. In each step, the functions $A_iT_i^ng_*$ are split using property (A1), the remainder terms $r_{*,n}$ giving rise to a total contribution of at most $2\varepsilon$ to the difference of the limsup and liminf. Given that the number of steps is $m-1$, this is a contribution over all steps of $\leq 2(m-1)\varepsilon$.\
Then each $g_{j,*}$ is further split into its stable and its reversible part with respect to $T_{i+1}$.\
The stable parts $g_{j,*}^s$ have a contribution of $0$ by part (1) of this theorem, proven above. The splitting is then repeated for the reversible parts $g_{j,*}^r$ until we reach $T_m$. Once at that stage, we group the coefficients $\lambda_{**,n_{\alpha(i)}}$ according to the value of $\alpha(i)$. Note that each sequence of $\lambda$’s is almost periodic, hence any product is as well. For each $b\neq\alpha(m)$ the corresponding product $\prod_{\alpha(i)=b}\lambda_{**,n_{\alpha(i)}}$ will in Cesàro average converge to some limit. For $b=\alpha(m)$ however, the corresponding product is the coefficient sequence of $T^{n_b}g_*$, and here we use the fact that almost periodic sequences are good weights for the pointwise ergodic theorem to obtain convergence (cf. the cases $m=2,3$).\
In total, the difference of the limsup and the liminf will not exceed $2(m-1)\varepsilon$ for all $x$ from a set of full measure, concluding the proof.
The pointwise limit is – if it exists – clearly the same as the stong limit, and takes the form given in [@EKK Thm. 3].
Weakly mixing transformations
=============================
In this section we wish to obtain a polynomial version of Theorem \[thm:main\], as well as trying to say something about the reversible part for the general $L^p$ case ($1\leq p<\infty$). To this end, we shall need stronger assumptions on the transformations $T_j$. Ergodicity will not suffice for our purpose, instead we shall require weak mixing.
A measure preserving transformation $T$ on a measure space $(X,\mu)$ is weakly mixing if $T \times T$ is an ergodic transformation on $(X \times X, \mu\times\mu)$.
The following lemma shows how this class of transformations becomes relevant for us.
\[le:weak\_mix\] Let $T$ be a weakly mixing measure preserving transformation on a standard probability space $(X,\mu)$, $1<p<\infty$, and $E=L^p(X,\mu)$. Then the reversible part $E_r$ of the Jacobs-Glicksberg-deLeeuw decomposition $E=E_r\oplus E_s$ corresponding to $T$ is one-dimensional and generated by the constant 1 function $\mathds{1}$.
The case $p=2$ follows from [@furst:book Thm. 4.30.], the general case from the arguments presented before that theorem ([@furst:book] pp. 96–97).
This spectral property of weakly mixing transformations simplifies the treatment of the reversible part. Also, as the next proposition shows, it guarantees that polynomial Cesàro means converge to an $L^\infty$ function.
\[prop:bourgain\] Let $T$ be an ergodic measure preserving transformation on a standard probability space $(X,\mu)$ and q(x) a polynomial with integer coefficients taking positive values on ${\mathbb}{N}^+$. Then for any $1<p<\infty$ and $f\in L^p(X,\mu)$ the limit $$\frac{1}{N} \sum_{n=1}^N T^{q(n)}f$$ exists almost surely. If in addition $T$ is weakly mixing, the limit is given by the constant function $\int_X f{\mathrm}{d}\mu$.
Our last ingredient is the following proposition, which allows us to establish properties of the coefficient sequences $(\lambda_{*,n})$ along polynomial indices. Recall that an operator is almost weakly stable if the stable part of the Jacobs-Glicksberg-deLeeuw decomposition is the whole space.
\[prop:uaws\_pol\] Let $T$ be an almost weakly stable contraction on a Hilbert space $H$. Then $T$ is almost weakly polynomial stable, i.e., for any $h\in H$ and non-constant polynomial $q$ with integer coefficients taking positive values on ${\mathbb}{N}^+$, the sequence $\{T^{q(j)}h\}_{j=1}^\infty$ is almost weakly stable.
Reformulating in the context needed in this paper we obtain the following.
\[cor:awps\] Let $T$ be a weakly mixing measure preserving transformation on the standard probability space $(X,\mu)$, $q$ a non-constant polynomial with integer coefficients taking positive values on ${\mathbb}{N}^+$ and $A$ an arbitrary operator on $L^2(X,\mu)$. Then for any $g,\varphi\in L^2(X,\mu)$ we have that the sequence $\langle AT^{q(n)}g,\varphi\rangle$ is bounded and lies in ${\mathcal}{N}$.
With these tools in hand, we can now state and prove almost everywhere pointwise convergence of entangled means on Hilbert spaces.
\[thm:main\_poly\] Let $m>1$ and $k$ be positive integers, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ a not necessarily surjective map, and $T_1,T_2,\ldots, T_m$ weakly mixing measure preserving transformations on a standard probability space $(X,\mu)$. Let $E:=L^2(X,\mu)$ and let $E=E_{j,r}\oplus E_{j,s}$ be the Jacobs-Glicksberg-deLeeuw decomposition corresponding to $T_j$ $(1\leq j\leq m)$. Let further $A_j\in{\mathcal}{L}(E)$ $(1\leq j< m)$ be bounded operators. Suppose that the the conditions (A1) and (A2) of Theorem \[thm:main\] hold.\
Further, let $q_1,q_2,\ldots, q_k$ be non-constant polynomials with integer coefficients taking positive values on ${\mathbb}{N}^+$. Then we have the following:
1. for each $f\in E_{1,s}$, $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} \left|T_m^{q_{\alpha(m)}(n_{\alpha(m)})}
\ldots A_2T_2^{q_{\alpha(2)}(n_{\alpha(2)})}A_1T_1^{q_{\alpha(1)}(n_{\alpha(1)})} f \right|\rightarrow 0$$ pointwise a.e.;
2. for each $f\in E$, the averages $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{q_{\alpha(m)}(n_{\alpha(m)})}
\ldots A_2T_2^{q_{\alpha(2)}(n_{\alpha(2)})}A_1T_1^{q_{\alpha(1)}(n_{\alpha(1)})} \mathds{1}$$ converge pointwise a.e. to the constant function $$\left(\langle f,\mathds{1}\rangle\prod_{i=1}^{m-1}\langle A_i\mathds{1},\mathds{1}\rangle\right)\cdot\mathds{1}.$$
We shall follow the proof of Theorem \[thm:main\], splitting functions in the exact same way, and then take a closer look at why the convergences still hold when averaging along polynomial subsequences.\
For part (1), the terms $r_n$ are still uniformly small in $L^\infty$, and the Cesàro averages remain small, regardless of the polynomial subsequences. Also, the coefficient sequences $\lambda_{*}$ are now subsequences of the ones in Theorem \[thm:main\], and so still bounded. In addition to this boundedness of coefficientss, the sequences $(\lambda_{j,n})_{n\in{\mathbb}{N}}$ being in ${\mathcal}{N}$ meant that the remainder terms $r_{*,n}$ arising from all futher decompositions average out to zero, and the same applies to the contribution of the bounded terms $\widetilde{g}_{b,*}$. In this polynomial setting this still holds by Corollary \[cor:awps\], and so we only have the terms with $g_{b,*}-\widetilde{g}_{b,*}$ left.\
For these, make use of the fact that all sequences $\lambda_*$ are bounded, and instead of Birkhoff’s pointwise ergodic theorem, apply Bourgain’s polynomial version, Proposition \[prop:bourgain\] in the weakly mixing case, to obtain that these terms also average out to a total contribution of at most $\varepsilon$ to the limsup.\
For part (2), it suffices to extend the proof of part (2) of Theorem \[thm:main\], and then investigate the limit. Here the key is that the reversible spaces $E_{i,r}$ are all one dimensional, spanned by $\mathds{1}$, with eigenvalue 1. Hence all coefficient sequences $\lambda_*$ arising from the reversible parts are constant 1, and as such average out to 1 even along polynomial subsequences. Therfore part (1) can be applied whenever a stable term $g_*^s$ comes into play, the remainder terms $r_*$ have a total contribution of at most $2\varepsilon$ in each splitting step, and we are left with the average of the terms that arise when at each split we choose the reversible part, with all coefficients $\lambda_*$ equal to 1. Pointwise convergence then follows from Proposition \[prop:bourgain\].\
When it comes to the limit, by the previous we have that the limit is determined by the purely reversible parts. Denoting by $$P_i:=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T_i^n$$ the mean ergodic projection onto $E_{i,r}={\mathrm}{Fix}(T_i)$ corresponding to $T_i$, this is equal to $$P_mA_{m-1}P_{m-1}\ldots A_1P_1f=\left(\langle f,\mathds{1}\rangle\prod_{i=1}^{m-1}\langle A_i\mathds{1},\mathds{1}\rangle\right)\cdot\mathds{1}.$$
In addition, weak mixing allows us to extend the convergence proven in Theorem \[thm:main\] to the reversible part for all $1\leq p<\infty$.
\[thm:main\_Lp\] Let $m>1$ and $k$ be positive integers, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ a not necessarily surjective map, and $T_1,T_2,\ldots, T_m$ weakly mixing measure preserving transformations on a standard probability space $(X,\mu)$. Let $p\in[1,\infty)$, $E:=L^p(X,\mu)$ and let $E=E_{j,r}\oplus E_{j,s}$ be the Jacobs-Glicksberg-deLeeuw decomposition corresponding to $T_j$ $(1\leq j\leq m)$. Let further $A_j\in{\mathcal}{L}(E)$ $(1\leq j< m)$ be bounded operators. Suppose that the conditions (A1) and (A2) of Theorem \[thm:main\] hold.\
Then we have for each $f\in E$ that the averages $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f$$ converge pointwise a.e. to the constant function $$\left(\langle f,\mathds{1}\rangle\prod_{i=1}^{m-1}\langle A_i\mathds{1},\mathds{1}\rangle\right)\cdot\mathds{1}.$$
We again follow in the steps of the proof of Theorem \[thm:main\]. Writing $f=\langle f,\mathds{1}\rangle\cdot\mathds{1}+(f-\langle f,\mathds{1}\rangle\cdot\mathds{1})$, we have by Lemma \[le:weak\_mix\] that the second term lies in $E_{1,s}$. By Theorem \[thm:main\] that part averages out to 0, so we only need to prove convergence for the function $\mathds{1}$. The key to part (2) of Theorem \[thm:main\] was showing that the coefficients $\lambda_{*,n}$ form an almost periodic sequence through the use of a basis of eigenfunctions in the reversible parts $E_{i,r}$. This would in general not be possible for $L^p$ spaces, as a non-orthogonal decomposition of a function into an infinite sum $\sum_{v\in V} d_vh_v$ would leave us with no good bound on the coefficients $d_v$. However now all our operators $T_i$ are weakly mixing, and hence the decomposition of a function $f\in E_{i,r}$ is the trivial $d\cdot\mathds{1}$ for the appropriate $d\in{\mathbb}{C}$, with the corresponding eigenvalue being 1. Consequently all coefficient sequences $\lambda_{*,n}$ arising from rotational parts are constant sequences, and a fortiori almost periodic. Therefore the proof presented for Theorem \[thm:main\] goes through in this case as well.\
Concerning the limit function itself, note that the proof actually yields (by part (1) of Theorem \[thm:main\]) that any stable part arising during the splitting averages to zero, and the remainder terms $r_*$ are uniformly small. Hence, again denoting by $P_i$ the mean ergodic projection onto $E_{i,r}={\mathrm}{Fix}(T_i)$ corresponding to $T_i$, the limit function is easily seen to be $$P_mA_{m-1}P_{m-1}\ldots A_1P_1f=\left(\langle f,\mathds{1}\rangle\prod_{i=1}^{m-1}\langle A_i\mathds{1},\mathds{1}\rangle\right)\cdot\mathds{1}.$$
The continuous case {#sec:ex}
===================
In this section, we finally turn our attention to a variant of the above results, where we replace the discrete action of the measure preserving operators with the continuous action of measure preserving flows. In other words, the semigroups $\{T_i^n|n\in{\mathbb}{N}^+\}$ are replaced by semigroups $\{T_i(t)|t\in[0,\infty)\}$.\
Note that both Birkhoff’s pointwise ergodic theorem, and the Jacobs-Glicksberg-deLeeuw decomposition possess a corresponding continuous version (for the latter, see e.g. [@eisner-book Theorem III.5.7]). Hence, with an analogous proof, we obtain the following continuous version of Theorem \[thm:main\].
\[thm:main-cont\] Let $m>1$ and $k$ be positive integers, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ a not necessarily surjective map and let $(T_1(t))_{t\geq 0}$,$\ldots $,$(T_m(t))_{t\geq 0}$ be ergodic measure preserving flows on a standard probability space $(X,\mu)$. Let $p\in[1,\infty)$, $E:=L^p(X,\mu)$ and let $E=E_{j,r}\oplus E_{j,s}$ be the Jacobs-Glicksberg-deLeeuw decomposition corresponding to $T_j(\cdot)$ $(1\leq j\leq m)$. Let further $A_j\in{\mathcal}{L}(E)$ $(1\leq j< m-1)$ be bounded operators. For a function $f\in E$ and an index $1\leq j\leq m-1$, write ${\mathscr}{A}_{j,f}:=\left\{A_jT_j(t)f\left|\right.t\in[0,\infty)\right\}$. Suppose that the following conditions hold:
- (Twisted compactness) For any function $f\in E$, index $1\leq j\leq m-1$ and $\varepsilon>0$, there exists a decomposition $E={\mathcal}{U}\oplus {\mathcal}{R}$ with $\dim {\mathcal}{U}<\infty$ such that $$P_{{\mathcal}{R}}{\mathscr}{A}_{j,f}
\subset B_\varepsilon(0,L^\infty(X,\mu)),$$ with $P_{{\mathcal}{R}}$ denoting the projection onto ${\mathcal}{R}$ along ${\mathcal}{U}$.
- (Joint ${\mathcal}{L}^\infty$-boundedness) There exists a constant $C>0$ such that we have $$\{A_jT_j(t)|\,t\in[0,\infty),1\leq j\leq m-1\}\subset B_C(0,{\mathcal}{L}(L^\infty(X,\mu)).$$
Then we have the following:
1. for each $f\in E_{0,s}$, $$\lim_{{\mathcal}{T}\to\infty}\frac{1}{{\mathcal}{T}^k}\int_{\left\{t_1,\ldots, t_k\right\}\in [0,{\mathcal}{T}]^k} \left|T_m(t_{\alpha(m)})
\ldots A_2T_2(t_{\alpha(2)})A_1T_1(t_{\alpha(1)}) f \right|\rightarrow 0$$ pointwise a.e.;
2. if $p=2$, then for each $f\in E_{1,r}$, $$\frac{1}{{\mathcal}{T}^k}\int_{\left\{ t_1,\ldots, t_k\right\}\in [0,{\mathcal}{T}]^k} T_m(t_{\alpha(m)})A_{m-1}T_{m-1}(t_{\alpha(m-1)})\ldots A_2T_2(t_{\alpha(2)})A_1T_1(t_{\alpha(1)}) f$$ converges pointwise a.e..
Since Lemma \[le:weak\_mix\] also extends to the time-continuous case, we have the following continuous version of Theorem \[thm:main\_Lp\], with a proof analogous to the original.
\[thm:main-cont\_Lp\] Let $m>1$ and $k$ be positive integers, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ a not necessarily surjective map and let $(T_1(t))_{t\geq 0}$,$\ldots $,$(T_m(t))_{t\geq 0}$ be weakly mixing measure preserving flows on a standard probability space $(X,\mu)$. Let $p\in[1,\infty)$, $E:=L^p(X,\mu)$ and let $E=E_{j,r}\oplus E_{j,s}$ be the Jacobs-Glicksberg-deLeeuw decomposition corresponding to $T_j(\cdot)$ $(1\leq j\leq m)$. Let further $A_j\in{\mathcal}{L}(E)$ $(1\leq j< m-1)$ be bounded operators. Suppose that the conditions (A1c) and (A2c) from Theorem \[thm:main-cont\_Lp\] hold.
Then for each $f\in E$, the multi-Cesàro averages $$\frac{1}{{\mathcal}{T}^k}\int_{\left\{ t_1,\ldots, t_k\right\}\in [0,{\mathcal}{T}]^k} T_m(t_{\alpha(m)})A_{m-1}T_{m-1}(t_{\alpha(m-1)})\ldots A_2T_2(t_{\alpha(2)})A_1T_1(t_{\alpha(1)}) f$$ converge pointwise a.e. to the constant function $$\left(\langle f,\mathds{1}\rangle\prod_{i=1}^{m-1}\langle A_i\mathds{1},\mathds{1}\rangle\right)\cdot\mathds{1}.$$
[10]{}
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|
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---
abstract: 'Hwang’s quasi-power theorem asserts that a sequence of random variables whose moment generating functions are approximately given by powers of some analytic function is asymptotically normally distributed. This theorem is generalised to higher dimensional random variables. To obtain this result, a higher dimensional analogue of the Berry–Esseen inequality is proved, generalising a two-dimensional version by Sadikova.'
address:
- 'Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Austria'
- 'Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Austria and Institute of Statistical Science, Academia Sinica, Taipei, Taiwan'
author:
- Clemens Heuberger
- Sara Kropf
bibliography:
- 'bib/cheub.bib'
title: 'Higher Dimensional Quasi-Power Theorem and Berry–Esseen Inequality'
---
Introduction
============
Asymptotic normality is a frequently occurring phenomenon in combinatorics, the classical central limit theorem being the very first example. The first step in the proof is the observation that the moment generating function of the sum of $n$ identically independently distributed random variables is the $n$-th power of the moment generating function of the distribution underlying the summands. As similar moment generating functions occur in many examples in combinatorics, a general theorem to prove asymptotic normality is desirable. Such a theorem was proved by Hwang [@Hwang:1998], usually called the “quasi-power theorem”.
Let $\{\Omega_n\}_{n\ge 1}$ be a sequence of integral random variables. Suppose that the moment generating function satisfies the asymptotic expression $$\label{eq:moments-1d}
M_n(s):={\mathbb{E}}(e^{\Omega_ns})=e^{W_n(s)}(1+O(\kappa_n^{-1})),$$ the $O$-term being uniform for $\abs{s}\le \tau$, $s\in{\mathbb{C}}$, $\tau>0$, where
1. $W_n(s)=u(s)\phi_{n}+v(s)$, with $u(s)$ and $v(s)$ analytic for $\abs{s}\le \tau$ and independent of $n$; and $u''(0)\neq 0$;
2. $\lim_{n\to\infty}\phi_{n}=\infty$;
3. $\lim_{n\to\infty}\kappa_n=\infty$.
Then the distribution of $\Omega_n$ is asymptotically normal, i.e., $$\sup_{x\in{\mathbb{R}}}\bigg\vert{\mathbb{P}}\bigg(\frac{\Omega_n- u'(0)\phi_{n}}{\sqrt{u''(0)\phi_{n}}} <
x\bigg)-
\Phi(x)\bigg\vert=O\bigg(\frac{1}{\sqrt{\phi_{n}}}+\frac{1}{\kappa_n}\bigg),$$ where $\Phi$ denotes the standard normal distribution $$\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\exp\Big(-\frac12
y^2\Big)\,dy.$$
See Hwang’s article [@Hwang:1998] as well as Flajolet-Sedgewick [@Flajolet-Sedgewick:ta:analy Sec. IX.5] for many applications of this theorem. A generalisation of the quasi-power theorem to dimension $2$ has been provided in [@Heuberger:2007:quasi-power]. It has been used in [@Heuberger-Prodinger:2006:analy-alter], [@Heuberger-Prodinger:2007:hammin-weigh], [@Eagle-Gao-Omar-Panario:2008:distr:short], [@Heuberger-Kropf-Wagner:2014:combin-charac] and [@Kropf:2015:varian-and]. In [@Drmota:2009:random Thm. 2.22], an $m$-dimensional version of the quasi-power theorem is stated without speed of convergence. Also in [@Bender-Richmond:1983:centr], such an $m$-dimensional theorem without speed of convergence is proved. There, several multidimensional applications are given, too.
In contrast to many results about the speed of convergence in classical probability theory (see, e.g., [@Gut:2005:probab]), the sequence of random variables is not assumed to be independent. The only assumption is that the moment generating function behaves asymptotically like a large power. This mirrors the fact that the moment generating function of the sum of independent, identically distributed random variables is exactly a large power. The advantage is that the asymptotic expression arises naturally in combinatorics by using techniques such as singularity analysis or saddle point approximation (see [@Flajolet-Sedgewick:ta:analy]).
The purpose of this article is to generalise the quasi-power theorem including the speed of convergence to arbitrary dimension $m$. We first state this main result in Theorem \[th:quasi-power-dD\] in this section. In Section \[sec:Berry–Esseen\], a new Berry–Esseen inequality (Theorem \[theorem:Berry-Esseen-dimension-m\]) is presented, which we use to prove the $m$-dimensional quasi-power theorem. In Section \[sec:exampl-mult-centr\], we give some applications of the multidimensional quasi-power theorem. The combinatorial idea behind the formulation of the Berry–Esseen inequality is discussed in Section \[sec:operator-Lambda\]. Our Berry–Esseen bound is proved in Section \[sec:proof-Berry–Esseen\]. The final Section \[sec:proof-quasi-power-theorem\] is then devoted to the proof of the quasi-power theorem.
We use the following conventions: vectors are denoted by boldface letters such as ${\mathbf{s}}$, their components are then denoted by regular letters with indices such as $s_j$. For a vector ${\mathbf{s}}$, $\|{\mathbf{s}}\|$ denotes the maximum norm $\max\{\abs{s_j}\}$. All implicit constants of $O$-terms may depend on the dimension $m$ as well as on $\tau$ which is introduced in Theorem \[th:quasi-power-dD\].
Our first main result is the following $m$-dimensional version of Hwang’s theorem.
\[th:quasi-power-dD\] Let $\{{\boldsymbol{\Omega}}_n\}_{n\ge 1}$ be a sequence of $m$-dimensional real random vectors. Suppose that the moment generating function satisfies the asymptotic expression $$\label{eq:moment-asymp}
M_n({\mathbf{s}}):={\mathbb{E}}(e^{\langle {\boldsymbol{\Omega}}_n,{\mathbf{s}}\rangle})=e^{W_n({\mathbf{s}})}(1+O(\kappa_n^{-1})),$$ the $O$-term being uniform for $\norm{{\mathbf{s}}}\le \tau$, ${\mathbf{s}}\in{\mathbb{C}}^m$, $\tau>0$, where
1. $W_n({\mathbf{s}})=u({\mathbf{s}})\phi_{n}+v({\mathbf{s}})$, with $u({\mathbf{s}})$ and $v({\mathbf{s}})$ analytic for $\norm{{\mathbf{s}}}\le \tau$ and independent of $n$; and the Hessian $H_u({\boldsymbol{0}})$ of $u$ at the origin is non-singular;
2. $\lim_{n\to\infty}\phi_{n}=\infty$;
3. $\lim_{n\to\infty}\kappa_n=\infty$.
Then, the distribution of ${\boldsymbol{\Omega}}_n$ is asymptotically normal with speed of convergence $O({\phi_n^{-1/2}})$, i.e., $$\label{eq:quasi-power-result}
\sup_{{\mathbf{x}}\in{\mathbb{R}}^{m}}\bigg\vert{\mathbb{P}}\bigg(\frac{{\boldsymbol{\Omega}}_n-\operatorname{grad}u ({\boldsymbol{0}})\phi_{n}}{\sqrt{\phi_{n}}} \le
{\mathbf{x}}\bigg)-
\Phi_{H_u({\boldsymbol{0}})}({\mathbf{x}})\bigg\vert=O\left(\frac{1}{\sqrt{\phi_{n}}}\right),$$ where $\Phi_{\Sigma}$ denotes the distribution function of the non-degenerate $m$-dimensional normal distribution with mean ${\boldsymbol{0}}$ and variance-covariance matrix $\Sigma$, i.e., $$\Phi_\Sigma({\mathbf{x}})=\frac{1}{(2\pi)^{m/2}\sqrt{\det \Sigma}}\int_{{\mathbf{y}}\le {\mathbf{x}}}\exp\Big(-\frac12
{\mathbf{y}}^\top \Sigma^{-1} {\mathbf{y}}\Big)\,d{\mathbf{y}},$$ where ${\mathbf{y}}\le {\mathbf{x}}$ means $y_\ell\le x_\ell$ for $1\le \ell\le m$.
If $H_{u}({\boldsymbol{0}})$ is singular, the random variables $$\frac{{\boldsymbol{\Omega}}_{n}-\operatorname{grad}u({\boldsymbol{0}})\phi_{n}}{\sqrt{\phi_{n}}}$$ converge in distribution to a degenerate normal distribution with mean ${\boldsymbol{0}}$ and variance-covariance matrix $H_{u}({\boldsymbol{0}})$.
Note that in the case of the singular $H_{u}({\boldsymbol{0}})$, a uniform speed of convergence cannot be guaranteed. To see this, consider the (constant) sequence of random variables $\Omega_{n}$ which takes values $\pm1$ each with probability $1/2$. Then the moment generating function is $(e^{t}+e^{-t})/2$, which is of the form with $\phi_{n}=n$, $u(s)=0$, $v(s)=\log (e^{t}+e^{-t})/2$ and $\kappa_{n}$ arbitrary. However, the distribution function of $\Omega_{n}/\sqrt{n}$ is given by $$\mathbb{P}\biggl(\frac{\Omega_{n}}{\sqrt{n}}\le x\biggr)=
\begin{cases}
0& \text{if }x<-1/\sqrt{n},\\
1/2& \text{if }-1/\sqrt{n}\le x<1/\sqrt{n},\\
1& \text{if }1/\sqrt{n}\le x,
\end{cases}$$ which does not converge uniformly.
In contrast to the original quasi-power theorem, the error term in our result does not contain the summand $O(1/\kappa_n)$. In fact, this summand could also be omitted in the original proof of the quasi-power theorem by using a better estimate for the error $E_n(s)=M_n(s)e^{-W_{n}(s)}-1$, cf. the proof of our Lemma \[lemma:characteristic-function-single-bound\].
The order of the error is optimal (without further assumptions on the random variables), as it is the case for the one-dimensional Berry-Esseen inequality. See, for example, the approximation of a binomial distribution by the normal distribution [@Petrov:2000:class-type § 1.2].
The proof of Theorem \[th:quasi-power-dD\] relies on an $m$-dimensional Berry–Esseen inequality (Theorem \[theorem:Berry-Esseen-dimension-m\]). It is a generalisation of Sadikova’s result [@Sadikova:1966:esseen; @Sadikova:1966:esseen:englisch] in dimension $2$. The main challenge is to provide a version which leads to bounded integrands around the origin, but still allows to use excellent bounds for the tails of the characteristic functions. To achieve this, linear combinations involving all partitions of the set $\{1,\ldots, m\}$ are used.
Note that there are several generalisations of the one-dimensional Berry–Esseen inequality [@Berry:1941:gauss; @Esseen:1945:fourier] to arbitrary dimension, see, e.g., Gamkrelidze [@Gamkrelidze:1977; @Gamkrelidze:1977:englisch] and Prakasa Rao [@Rao:2002:anoth-esseen]. However, using these results would lead to a less precise error term in , see the end of Section \[sec:Berry–Esseen\] for more details. For that reason we generalise Sadikova’s result, which was already successfully used by the first author in [@Heuberger:2007:quasi-power] to prove a $2$-dimensional quasi-power theorem. Also note that our theorem can deal with discrete random variables, too, in contrast to [@Roussas:2001:esseen], where density functions are considered.
For the sake of completeness, we also state the following result about the moments of ${\boldsymbol{\Omega}}_{n}$.
\[proposition:moments\] The cross-moments of ${\boldsymbol{\Omega}}_{n}$ satisfy $$\frac{1}{\prod_{\ell=1}^{m}k_{\ell}!}\mathbb E\bigg(\prod_{\ell=1}^{m}\Omega_{n,\ell}^{k_{\ell}}\bigg)=p_{{\mathbf{k}}}(\phi_{n})+O\big(\kappa_{n}^{-1}\phi_{n}^{k_{1}+\cdots+k_{m}}\big),$$ for $k_{\ell}$ nonnegative integers, where $p_{{\mathbf{k}}}$ is a polynomial of degree $\sum_{\ell=1}^{m}k_{\ell}$ defined by $$p_{{\mathbf{k}}}(X)=[s_{1}^{k_{1}}\cdots s_{m}^{k_{m}}]e^{u({\mathbf{s}})X+v({\mathbf{s}})}.$$
In particular, the mean and the variance-covariance matrix are $$\begin{aligned}
\mathbb E({\boldsymbol{\Omega}}_{n})&=\operatorname{grad}u({\boldsymbol{0}})\phi_{n}+\operatorname{grad}v({\boldsymbol{0}})+O(\kappa_{n}^{-1}),\\
\operatorname{Cov}({\boldsymbol{\Omega}}_{n})&=H_{u}({\boldsymbol{0}})\phi_{n}+H_{v}({\boldsymbol{0}})+O(\kappa_{n}^{-1}),\end{aligned}$$ respectively.
A Berry–Esseen Inequality {#sec:Berry--Esseen}
=========================
This section is devoted to a generalisation of Sadikova’s Berry–Esseen inequality [@Sadikova:1966:esseen; @Sadikova:1966:esseen:englisch] in dimension 2 to dimension $m$. Before stating the theorem, we introduce our notation.
Let $L=\{1,\ldots, m\}$. For $K\subseteq L$, we write ${\mathbf{s}}_K=(s_k)_{k\in K}$ for the projection of ${\mathbf{s}}\in{\mathbb{C}}^L$ to ${\mathbb{C}}^K$. For $J\subseteq K\subseteq L$, let $\chi_{J,K}\colon {\mathbb{C}}^{J}\to{\mathbb{C}}^{K}$, $(s_{j})_{j\in J}\mapsto
(s_{k}\iverson{k\in J})_{k\in K}$ be an injection from ${\mathbb{C}}^{J}$ into ${\mathbb{C}}^{K}$. Similarly, let $\psi_{J,K}\colon {\mathbb{C}}^{K}\to{\mathbb{C}}^{K}$, $(s_{k})_{k\in K}\mapsto (s_{k}\iverson{k\in J})_{k\in K}$ be the projection which sets all coordinates corresponding to $K\setminus J$ to $0$.
We denote the set of all partitions of $K$ by $\Pi_K$. We consider a partition as a set $\alpha=\{J_{1},\ldots,J_{k}\}$. Thus $\abs{\alpha}$ denotes the number of parts of the partition $\alpha$. Furthermore, $J\in\alpha$ means that $J$ is a part of the partition $\alpha$.
Now, we can define an operator which we later use to state our Berry–Esseen inequality. The motivation behind this definition is explained at the end of this section.
\[definition:Lambda-K\] Let $K\subseteq L$ and $h\colon {\mathbb{C}}^K\to {\mathbb{C}}$. We define the non-linear operator $$\Lambda_K(h):=\sum_{\alpha\in\Pi_K}\mu_\alpha \prod_{J\in
\alpha}h\circ \psi_{J, K}$$ where $$\mu_\alpha = (-1)^{\abs{\alpha}-1}(\abs{\alpha}-1)!\,.$$
We denote $\Lambda_{L}$ briefly by $\Lambda$.
For any random variable ${\mathbf{Z}}$, we denote its cumulative distribution function by $F_{\mathbf{Z}}$, its density function by $f_{\mathbf{Z}}$ (if it exists) and its characteristic function by $\varphi_{\mathbf{Z}}$.
With these definitions, we are able to state our second main result, an $m$-dimensional version of the Berry–Esseen inequality.
\[theorem:Berry-Esseen-dimension-m\] Let $m\ge 1$ and ${\mathbf{X}}$ and ${\mathbf{Y}}$ be $m$-dimensional random variables. Assume that $F_{\mathbf{Y}}$ is differentiable.
Let $$\begin{aligned}
A_j&=\sup_{{\mathbf{y}}\in{\mathbb{R}}^m}\frac{\partial F_{\mathbf{Y}}({\mathbf{y}})}{\partial y_j},\\
B_j&=\sum_{k=1}^{j} {\genfrac{\{}{\}}{0pt}{}{j}{k}} k!\ ,\\
C_1&=\sqrt[3]{\frac{32}{\pi\bigl(1-\bigl(\frac{3}{4}\bigr)^{1/m}\bigr)}},\\
C_2&=\frac{12}{\pi}\end{aligned}$$ for $1\le
j\le m$ where ${\genfrac{\{}{\}}{0pt}{}{j}{k}}$ denotes a Stirling partition number (Stirling number of the second kind).
Let $T>0$ be fixed. Then $$\label{eq:Berry-Esseen}
\begin{aligned}
\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{{\mathbf{X}}}({\mathbf{z}})-F_{{\mathbf{Y}}}({\mathbf{z}})}&\le
\frac{2}{(2\pi)^m} \int_{\norm{{\mathbf{t}}}\le
T}\abs[\bigg]{\frac{\Lambda(\varphi_{{\mathbf{X}}})({\mathbf{t}})-\Lambda(\varphi_{{\mathbf{Y}}})({\mathbf{t}})}{\prod_{\ell\in
L} t_\ell}}\,d{\mathbf{t}}\\
&\qquad+ 2\sum_{\emptyset\neq J\subsetneq
L}B_{m-\abs{J}}\sup_{{\mathbf{z}}_J\in{\mathbb{R}}^J}\abs[\big]{F_{{\mathbf{X}}_{J}}({\mathbf{z}}_J)-F_{{\mathbf{Y}}_{J}}({\mathbf{z}}_J)}
\\
&\qquad +\frac{2\sum_{j=1}^m A_j}{T}(C_1+C_2).
\end{aligned}$$ Existence of ${\mathbb{E}}({\mathbf{X}})$ and ${\mathbb{E}}({\mathbf{Y}})$ is sufficient for the finiteness of the integral in .
Let us give two remarks on the distribution functions occurring in this theorem: The distribution function $F_{\mathbf{Y}}$ is non-decreasing in every variable, thus $A_j>0$ for all $j$. Furthermore, our general notations imply that $F_{{\mathbf{X}}_J}$ is a marginal distribution of ${\mathbf{X}}$.
The numbers $B_j$ are known as “Fubini numbers” or “ordered Bell numbers”. They form the sequence [A000670](http://oeis.org/A000670) in [@OEIS:2015].
Recursive application of leads to the following corollary, where we no longer explicitly state the constants depending on the dimension.
\[corollary:Berry-Esseen\] Let $m\ge 1$ and ${\mathbf{X}}$ and ${\mathbf{Y}}$ be $m$-dimensional random variables. Assume that $F_{\mathbf{Y}}$ is differentiable and let $$A_j=\sup_{{\mathbf{y}}\in{\mathbb{R}}^m}\frac{\partial F_{\mathbf{Y}}({\mathbf{y}})}{\partial y_j}, \qquad 1\le
j\le m.$$
Then $$\begin{gathered}
\label{eq:Berry-Esseen-recursive}
\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{{\mathbf{X}}}({\mathbf{z}})-F_{{\mathbf{Y}}}({\mathbf{z}})}\\=
O\biggl(\sum_{\emptyset \neq K\subseteq L}\int_{\norm{{\mathbf{t}}_K}\le
T}\abs[\bigg]{\frac{\Lambda_K(\varphi_{{\mathbf{X}}}\circ\chi_{K, L})({\mathbf{t}}_K)-\Lambda_K(\varphi_{{\mathbf{Y}}}\circ\chi_{K, L})({\mathbf{t}}_K)}{\prod_{k\in
K} t_k}}\,d{\mathbf{t}}_K + \frac{\sum_{j=1}^m A_j}{T}\biggr)
\end{gathered}$$ where the $O$-constants only depend on the dimension $m$.
Existence of ${\mathbb{E}}({\mathbf{X}})$ and ${\mathbb{E}}({\mathbf{Y}})$ is sufficient for the finiteness of the integrals in .
In order to explain the choice of the operator $\Lambda$, we first state it in dimension $2$: $$\label{eq:Sadikova-simple}
\Lambda(h)(s_1, s_2) = h(s_1, s_2) - h(s_1, 0)h(0, s_2).$$ This coincides with Sadikova’s definition. This also shows that our operator is non-linear as, e.g., $\Lambda(s_{1}+s_{2})(s_{1},s_{2})\neq\Lambda(s_{1})(s_{1},s_{2})+\Lambda(s_{2})(s_{1},s_{2})$.
In Theorem \[theorem:Berry-Esseen-dimension-m\], we apply $\Lambda$ to characteristic functions; so we may restrict our attention to functions $h$ with $h({\boldsymbol{0}})=1$. From , we see that $\Lambda(h)(s_1, 0) = \Lambda(h)(0, s_2)=0$, so that $\Lambda(h)(s_1,
s_2)/(s_1s_2)$ is bounded around the origin. This is essential for the boundedness of the integral in Theorem \[theorem:Berry-Esseen-dimension-m\]. In general, this property will be guaranteed by our particular choice of coefficients. It is no coincidence that for $\alpha\in \Pi_L$, the coefficient $\mu_\alpha$ equals the value $\mu(\alpha, \{L\})$ of the Möbius function in the lattice of partitions: Weisner’s theorem (see Stanley [@Stanley:2012:enumer_1 Corollary 3.9.3]) is crucial in the proof that $\Lambda(h)({\mathbf{s}})/(s_1\ldots
s_m)$ is bounded around the origin (see the proof of Lemma \[lemma:Lambda-property\]).
The second property is that our proof of the quasi-power theorem needs estimates for the tails of the integral in Theorem \[theorem:Berry-Esseen-dimension-m\]. These estimates have to be exponentially small in every variable, which means that every variable has to occur in every summand. This is trivially fulfilled as every summand in the definition of $\Lambda$ is formulated in terms of a partition.
Note that Gamkrelidze [@Gamkrelidze:1977:englisch] (and also Prakasa Rao [@Rao:2002:anoth-esseen]) use a linear operator $L$ mapping $h$ to $$\label{eq:gamkrelidze}
(s_1, s_2) \mapsto h(s_1, s_2) - h(s_1, 0) - h(0, s_2).$$ When taking the difference of two characteristic functions, we may assume that $h(0, 0)=0$ so that the first crucial property as defined above still holds. However, the tails are no longer exponentially small in every variable: the last summand $h(0,s_{2})$ in is not exponentially small in $s_{1}$ because it is independent of $s_{1}$ and nonzero in general. However, the first two summands are exponentially small in $s_{1}$ by our assumption .
For that reason, using the Berry–Esseen inequality by Gamkrelidze [@Gamkrelidze:1977:englisch] to prove a quasi-power theorem leads to a less precise error term $O(\phi_{n}^{-1/2}\log^{m-1}\phi_n)$ in . It can be shown that the less precise error term necessarily appears when using Gamkrelidze’s result by considering the example of ${\boldsymbol{\Omega}}_n$ being the $2$-dimensional vector consisting of a normal distribution with mean $-1$ and variance $n$ and a normal distribution with mean $0$ and variance $n$. This is a consequence of the linearity of the operator $L$ in Gamkrelidze’s result.
Examples of Multidimensional Central Limit Theorems {#sec:exampl-mult-centr}
===================================================
In this section, we give two examples from combinatorics where we can apply Theorem \[th:quasi-power-dD\]. Asymptotic normality was already shown in earlier publications [@Drmota:1997:system-funct-equat; @Bender-Richmond:1983:centr], but we additionally provide an estimate for the speed of convergence.
Context-Free Languages
----------------------
Consider the following example of a context-free grammar $G$ with non-terminal symbols $S$ and $T$, terminal symbols $\{a,b,c\}$, starting symbol $S$ and the rules $$P=\{S\to aSbS,\, S\to bT,\, T\to bS,\, T\to cT,\, T\to a\}.$$ The corresponding context-free language $L(G)$ consists of all words which can be generated starting with $S$ using the rules in $P$ to replace all non-terminal symbols. For example, $abcabababba\in L(G)$ because it can be derived as $$S\to aSbS
\to abTbaSbS
\to abcTbabTbbT
\to abcabababba.$$
Let ${\mathbb{P}}({\boldsymbol{\Omega}}_{n}={\mathbf{x}})$ be the probability that a word of length $n$ in $L(G)$ consists of $x_{1}$ and $x_{2}$ terminal symbols $a$ and $b$, respectively. Thus there are $n-x_{1}-x_{2}$ terminal symbols $c$. For simplicity, this random variable is only $2$-dimensional. But it can be easily extended to higher dimensions.
Following Drmota [@Drmota:1997:system-funct-equat Sec. 3.2], we obtain that the moment generating function is $${\mathbb{E}}(e^{\langle
{\boldsymbol{\Omega}}_n,{\mathbf{s}}\rangle})=\frac{y_{n}(e^{{\mathbf{s}}})}{y_{n}({\boldsymbol{1}})}$$ with $y_{n}(\boldsymbol{z})$ defined in [@Drmota:1997:system-funct-equat]. Using [@Drmota:1997:system-funct-equat Equ. (4.9)], this moment generating function has an asymptotic expansion as in with $\phi_{n}=n$. Thus ${\boldsymbol{\Omega}}_{n}$ is asymptotically normally distributed after standardisation (as was shown in [@Drmota:1997:system-funct-equat]) and additionally the speed of convergence is $O(n^{-1/2})$.
Other context-free languages can be analysed in the same way, either by directly using the results in [@Drmota:1997:system-funct-equat] (if the underlying system is strongly connected) or by similar methods. This has applications, for example, in genetics (see [@Poznanovic-Heitsch:2014:asymp-rna]).
Dissections of Labelled Convex Polygons
---------------------------------------
Let $S_{1}{\mathbin{\mathaccent\cdot\cup}}\cdots{\mathbin{\mathaccent\cdot\cup}}S_{t+1}=\{3,4,\ldots\}$ be a partition. We dissect a labelled convex $n$-gon into smaller convex polygons by choosing some non-intersecting diagonals. Each small polygon should be a $k$-gon with $k\not\in S_{t+1}$. Define $a_{n}({\mathbf{r}})$ to be the number of dissections of an $n$-gon such that it consists of exactly $r_{i}$ small polygons whose number of vertices is in $S_{i}$, for $i=1$, …, $t$. For convenience, we use $a_{2}({\mathbf{r}})=[{\mathbf{r}}={\boldsymbol{0}}]$. Asymptotic normality was proved in [@Bender-Richmond:1983:centr Sec. 3], see also [@Bender:1974:asymp-method-enumer Ex. 7.1] for a one-dimensional version. We additionally provide an estimate for the speed of convergence.
Let $$f(z,{\mathbf{x}})=\sum_{\substack{n\geq2\\ {\mathbf{r}}\geq 0}}a_{n}({\mathbf{r}}){\mathbf{x}}^{{\mathbf{r}}}z^{n-1}.$$ Then choosing a $k$-gon with $k\in S_{1}{\mathbin{\mathaccent\cdot\cup}}\cdots{\mathbin{\mathaccent\cdot\cup}}S_{t}$ and gluing dissected polygons to $k-1$ of its sides translates into the equation $$f=z+\sum_{i=1}^{t}x_{i}\sum_{k\in S_{i}}f^{k-1}.$$ Following [@Bender:1974:asymp-method-enumer], this equation can be used to obtain an asymptotic expression for the moment generating function as in with $\phi_{n}=n$. The asymptotic normal distribution follows after suitable standardisation with speed of convergence $O(n^{-1/2})$.
Combinatorial Background of the Operator Λ {#sec:operator-Lambda}
==========================================
Before we start with the proof of Theorem \[theorem:Berry-Esseen-dimension-m\], we state and prove the property of our operator $\Lambda$ which motivates its Definition \[definition:Lambda-K\].
\[lemma:Lambda-property\] Let $K\subsetneq L$ and $h\colon {\mathbb{C}}^L\to {\mathbb{C}}$ with $h({\boldsymbol{0}})=1$. Then $$\Lambda(h)\circ \psi_{K, L}=0.$$
Before actually proving the lemma, we recall some of the theory about the Möbius function of a partially ordered set (poset), see also Stanley [@Stanley:2012:enumer_1 Section 3.7].
By the following definition, $\Pi_L$, the set of all partitions of $L$, is a poset: As usual, a partition $\alpha\in \Pi_L$ is said to be a refinement of a partition $\alpha'\in\Pi_L$ if $$\forall J\in\alpha\colon \exists J'\in\alpha'\colon J\subseteq J'.$$ In this case, we write $\alpha\le \alpha'$. This defines a partial order on $\Pi_L$.
The Möbius function on $\Pi_L$ is denoted by $\mu$: for $\alpha<
\alpha'$, we set $\mu(\alpha', \alpha')=1$ and $$\mu(\alpha, \alpha')=
-\sum_{\substack{\beta\in \Pi_L\\\alpha<\beta\le\alpha'}} \mu(\beta,
\alpha').$$ For $\alpha$, $\alpha'\in\Pi_L$, the infimum $\alpha\land \alpha'$ of $\alpha$ and $\alpha'$ is given by $$\{ J\cap J'\colon J\in\alpha, J'\in\alpha', J\cap J'\neq \emptyset\}.$$ In fact, $\Pi_L$ is a lattice (cf.Stanley [@Stanley:2012:enumer_1 Example 3.10.4]). The greatest element is $\{L\}$.
For $\alpha\in\Pi_L$, we have $$\mu(\alpha, \{L\})=(-1)^{\abs{\alpha}-1}(\abs{\alpha}-1)!=\mu_\alpha,$$ where $\abs{\alpha}$ denotes the number of parts of the partition, see Stanley [@Stanley:2012:enumer_1 (3.37)]. In particular, we may rewrite the definition of $\Lambda$ (Definition \[definition:Lambda-K\]) as $$\label{eq:Lambda_K_definition-2}
\Lambda(h):=\sum_{\alpha\in\Pi_L}\mu(\alpha, \{L\}) \prod_{J\in
\alpha}h\circ \psi_{J, L}.$$
For any $\gamma$, $\beta\in\Pi_L$ with $\gamma\le \beta< \{L\}$, Weisner’s theorem (see Stanley [@Stanley:2012:enumer_1 Corollary 3.9.3]) applied to the interval $[\gamma, \{L\}]$ asserts that $$\label{eq:Weisner}
\sum_{\substack{\alpha\in\Pi_L\\ \alpha\land \beta=\gamma}}\mu(\alpha, \{L\})=0.$$
We now turn to the actual proof of the lemma.
Consider the partition $\beta=\{K\} \cup
\{\{k\}\colon k\in L\setminus K\}$ of $L$, i.e., $\beta$ consists of $K$ as one part and a collection of singletons. As $K\neq L$, we have $\beta < \{L\}$.
By definition of $\psi$, we have $\psi_{J,L}\circ \psi_{K,L}=\psi_{J\cap K, L}$ for $J$, $K\subseteq L$. If $\alpha\in \Pi_L$, then $$\prod_{J\in \alpha} h\circ \psi_{J\cap K, L}=
\prod_{\substack{J\in
\alpha\land\beta\\
J\subseteq K}} h\circ \psi_{J, L}$$ because parts $J\in\alpha$ with $J\cap K=\emptyset$ contribute $h({\boldsymbol{0}})=1$. Therefore, collecting the sum according to $\alpha\land\beta$ yields $$\Lambda(h)\circ\psi_{K, L}:=\sum_{\alpha\in\Pi_L}\mu(\alpha, \{L\}) \prod_{J\in
\alpha}h\circ \psi_{J\cap K, L}=
\sum_{\gamma\in\Pi_L}\prod_{\substack{J\in
\gamma\\
J\subseteq K}} h\circ \psi_{J, L}
\sum_{\substack{\alpha\in\Pi_L\\\alpha\land\beta=\gamma}}\mu(\alpha, \{L\}).$$ As $\gamma\le\beta<\{L\}$, the inner sum vanishes by .
Proof of the Berry–Esseen Inequality {#sec:proof-Berry--Esseen}
====================================
This section is devoted to the proof of our Berry–Esseen inequality, Theorem \[theorem:Berry-Esseen-dimension-m\]. It is a generalisation of Sadikova’s proof.
We start with an auxiliary one-dimensional random variable.
\[lemma:P\] Let $P$ be the one-dimensional random variable with probability density function $$f_P(z)=\frac{3}{8\pi}\Bigl(\frac{\sin(z/4)}{z/4}\Bigr)^4.$$
Then its characteristic function is $$\label{eq:characteristic-function-P}
\varphi_P(t)=
\begin{cases}
1-6t^2+6\abs{t}^3&\text{if $0\le \abs{t}\le 1/2$},\\
2(1-\abs{t})^3&\text{if $1/2\le \abs{t}\le 1$},\\
0&\text{if $1\le \abs{t}$}
\end{cases}$$ and $$\begin{aligned}
{\mathbb{E}}(P^2)&=12,\notag\\
{\mathbb{E}}(\abs{P})&\le C_2.\label{eq:expectation-P-absolute-value}
\end{aligned}$$
Let $\lambda$ be the unique positive number such that $${\mathbb{P}}(P\le \lambda) = {\mathbb{P}}(P\ge -\lambda) = \Bigl(\frac{3}{4}\Bigr)^{1/m}.$$ Then $$\label{eq:lambda-estimate}
\lambda\le C_1.$$
The characteristic function is mentioned in [@Gnedenko-Kolmogorov:1954:limit Section 39]; it is computed by standard methods.
Differentiating $\varphi_P$ twice, we see that the second moment is $12$. To prove , we rewrite ${\mathbb{E}}(\abs{P})$ as $${\mathbb{E}}(\abs{P})
=\frac{12}{\pi} \int_0^1 \frac{\sin^4 z}{z^3}\, dz
+\frac{12}{\pi} \int_1^\infty \frac{\sin^4 z}{z^3}\, dz.$$ We use the estimates $\sin z\le z$ and $\abs{\sin z}\le 1$ on the intervals $[0, 1]$ and $[1, \infty)$, respectively. Thus $${\mathbb{E}}(\abs{P})\le \frac{12}{\pi}\Bigl(\frac12 + \frac12\Bigr)=\frac{12}{\pi}.$$
To obtain a bound for $\lambda$, we follow Gamkrelidze [@Gamkrelidze:1977:englisch]: we estimate the tail using $\abs{\sin^4(z)}\le 1$ and get $$1-\Bigl(\frac{3}{4}\Bigr)^{1/m} =\frac{3}{8\pi}\int_{\lambda}^\infty
\Bigl(\frac{\sin(z/4)}{z/4}\Bigr)^4\, dz\le
\frac{3}{2\pi}\int_{\lambda/4}^\infty\Bigl(\frac{1}{z}\Bigr)^4\,dz
=
\frac3{2\pi}\Bigl(-\frac13\Bigr)\frac{1}{z^3}\Bigr\rvert_{z=\lambda/4}^\infty
=\frac{32}{\pi\lambda^3}.$$ This results in .
In the next step, we consider tuples of random variables distributed as $P$. They will be used to ensure smoothness. We write ${\boldsymbol{1}}$ to denote a vector with all coordinates equal to $1$.
\[lemma:Q\] Let ${\mathbf{Q}}=(P_1/T, \ldots, P_m/T)$ be the $m$-dimensional random variable where the $P_j$ are independent random variables with the same distribution as $P$ in Lemma \[lemma:P\] and $T$ is the fixed constant defined in Theorem \[theorem:Berry-Esseen-dimension-m\].
Then ${\mathbf{Q}}$ has density function and characteristic function $$\begin{aligned}
f_{\mathbf{Q}}({\mathbf{z}})&=\prod_{j=1}^m Tf_P(Tz_j),\\
\varphi_{\mathbf{Q}}({\mathbf{t}})&=\prod_{j=1}^m\varphi_P\Bigl(\frac{t_j}{T}\Bigr),
\end{aligned}$$ respectively. The characteristic function vanishes outside $[-T, T]^m$.
Furthermore, $$\begin{aligned}
\int_{{\mathbf{z}}\in{\mathbb{R}}^m} \abs{z_j} f_{\mathbf{Q}}\Bigl({\mathbf{z}}+ \frac{\theta
\lambda}{T}{\boldsymbol{1}}\Bigr)\,d{\mathbf{z}}&\le \frac{C_2+\lambda}{T},\label{eq:Q-integral-1}\\
\int_{\theta {\mathbf{z}}\le 0} f_{\mathbf{Q}}\Bigl({\mathbf{z}}+ \frac{\theta
\lambda}{T} {\boldsymbol{1}}\Bigr)\,d{\mathbf{z}}=\frac{3}{4}\label{eq:Q-integral-2}
\end{aligned}$$ hold for $\theta\in\{\pm 1\}$ and $j\in\{1,\ldots, m\}$.
Because of independence, the distribution function and the characteristic function of ${\mathbf{Q}}$ is the product of the distribution functions and the characteristic functions of the $P_j/T$, respectively. Division by $T$ transforms the density and characteristic functions as claimed. As $\varphi_P(t)$ vanishes outside $[-1,
1]$ by , $\varphi_{\mathbf{Q}}({\mathbf{t}})$ vanishes outside $[-T, T]^m$.
By a simple translation, the integral on the left hand side of can be seen to be equal to $${\mathbb{E}}\Bigl(\abs[\Big]{Q_j-\frac{\theta\lambda}{T}}\Bigr).$$ Then is a simple consequence of $Q_j=P_j/T$, and the triangle inequality.
By the same translation and the definition of $\lambda$, the integral on the left hand side of is $${\mathbb{P}}\Bigl(\theta{\mathbf{Q}}\le \frac{\lambda}{T}{\boldsymbol{1}}\Bigr)=\prod_{j=1}^m {\mathbb{P}}(\theta
P_j\le \lambda)=\frac{3}{4}.$$
From now on, we let ${\mathbf{Q}}$ be as in Lemma \[lemma:Q\] and let ${\mathbf{Q}}$ be independent of ${\mathbf{X}}$ and independent of ${\mathbf{Y}}$. We first prove an inequality relating the difference between the distribution functions of ${\mathbf{X}}$ and ${\mathbf{Y}}$ to that of the distribution functions of ${\mathbf{X}}+{\mathbf{Q}}$ and ${\mathbf{Y}}+{\mathbf{Q}}$.
\[lemma:perturbation\]We have $$\label{eq:perturbation}
\begin{aligned}
\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{{\mathbf{X}}+{\mathbf{Q}}}({\mathbf{z}})-F_{{\mathbf{Y}}+{\mathbf{Q}}}({\mathbf{z}})}&\le
\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{\mathbf{X}}({\mathbf{z}})-F_{{\mathbf{Y}}}({\mathbf{z}})}\\
&\le
2\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{{\mathbf{X}}+{\mathbf{Q}}}({\mathbf{z}})-F_{{\mathbf{Y}}+{\mathbf{Q}}}({\mathbf{z}})}+\frac{2\sum_{j=1}^m
A_j}{T}(C_1+C_2).
\end{aligned}$$
Let $$\begin{aligned}
S&=\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{\mathbf{X}}({\mathbf{z}})-F_{{\mathbf{Y}}}({\mathbf{z}})}\\
S'&=\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{{\mathbf{X}}+{\mathbf{Q}}}({\mathbf{z}})-F_{{\mathbf{Y}}+{\mathbf{Q}}}({\mathbf{z}})}
\end{aligned}$$ and $\varepsilon>0$. We choose $\theta\in\{\pm 1\}$ such that $S=\sup_{z\in{\mathbb{R}}^m}\theta(F_{{\mathbf{X}}}({\mathbf{z}})-F_{{\mathbf{Y}}}({\mathbf{z}}))$.
There is a ${\mathbf{z}}_{\varepsilon}\in{\mathbb{R}}^m$ such that $$S-\varepsilon \le \theta(F_{{\mathbf{X}}}-F_{{\mathbf{Y}}})({\mathbf{z}}_\varepsilon).$$ Let ${\mathbf{w}}\in{\mathbb{R}}^n$ with $\theta{\mathbf{w}}\le {\boldsymbol{0}}$. By monotonicity of $F_{{\mathbf{X}}}$, we have $\theta F_{{\mathbf{X}}}({\mathbf{z}}_\varepsilon-{\mathbf{w}})\ge \theta
F_{{\mathbf{X}}}({\mathbf{z}}_\varepsilon)$. Thus $$\begin{aligned}
\theta(F_{{\mathbf{X}}}-F_{{\mathbf{Y}}})({\mathbf{z}}_\varepsilon-{\mathbf{w}})&\ge\theta
(F_{{\mathbf{X}}}-F_{{\mathbf{Y}}})({\mathbf{z}}_{\varepsilon})-\theta(F_{{\mathbf{Y}}}({\mathbf{z}}_\varepsilon-{\mathbf{w}})-F_{{\mathbf{Y}}}({\mathbf{z}}_\varepsilon))\\
&\ge S-\varepsilon - \sum_{j=1}^m A_j \abs{w_j}.
\end{aligned}$$ We multiply this inequality by $f_{{\mathbf{Q}}}\bigl({\mathbf{w}}+\frac{\theta\lambda}{T}{\boldsymbol{1}}\bigr)$ and integrate over all ${\mathbf{w}}\in{\mathbb{R}}^n$ with $\theta{\mathbf{w}}\le {\boldsymbol{0}}$. By and , we get $$\label{eq:integral-I_1}
I_1:=\int_{\theta{\mathbf{w}}\le{\boldsymbol{0}}}\theta(F_{{\mathbf{X}}}-F_{{\mathbf{Y}}})({\mathbf{z}}_\varepsilon-{\mathbf{w}})f_{{\mathbf{Q}}}\Bigl({\mathbf{w}}+\frac{\theta\lambda}{T}{\boldsymbol{1}}\Bigr)
\,d{\mathbf{w}}\ge \frac{3}{4}(S-\varepsilon)-\frac{C_2+\lambda}{T}\sum_{j=1}^m A_j.$$ Setting $$I_2:=\int_{\theta{\mathbf{w}}\nleq{\boldsymbol{0}}}\theta(F_{{\mathbf{X}}}-F_{{\mathbf{Y}}})({\mathbf{z}}_\varepsilon-{\mathbf{w}})f_{{\mathbf{Q}}}\Bigl({\mathbf{w}}+\frac{\theta\lambda}{T}{\boldsymbol{1}}\Bigr)
\,d{\mathbf{w}}$$ and using the estimate $\abs{\theta(F_{{\mathbf{X}}}-F_{{\mathbf{Y}}})({\mathbf{z}}_\varepsilon-{\mathbf{w}})}\le S$ yields $$\label{eq:integral-I_2}
\abs{I_2}\le S \int_{\theta{\mathbf{w}}\nleq{\boldsymbol{0}}}f_{{\mathbf{Q}}}\Bigl({\mathbf{w}}+\frac{\theta\lambda}{T}{\boldsymbol{1}}\Bigr)
\,d{\mathbf{w}}=\frac{S}{4}$$ by and the fact that $f_{{\mathbf{Q}}}$ is a probability density function.
Combining and yields $$\label{eq:I_1+I_2-bound}
\abs{I_1+I_2}\ge \abs{I_1}-\abs{I_2}\ge I_1-\abs{I_2}\ge \frac{S}{2}-\frac{C_2+\lambda}{T}\sum_{j=1}^m A_j-\frac{3\varepsilon}{4}.$$ As the sum of random variables corresponds to a convolution, we have $$\label{eq:convolution}
(F_{{\mathbf{X}}+{\mathbf{Q}}}-F_{{\mathbf{Y}}+{\mathbf{Q}}})({\mathbf{z}}) =
\int_{{\mathbb{R}}^m}(F_{{\mathbf{X}}}-F_{{\mathbf{Y}}})({\mathbf{z}}-{\mathbf{w}}) f_{{\mathbf{Q}}}({\mathbf{w}})\,d{\mathbf{w}}.$$ Replacing ${\mathbf{z}}$ and ${\mathbf{w}}$ by ${\mathbf{z}}_{\varepsilon}+\frac{\theta\lambda}{T}{\boldsymbol{1}}$ and ${\mathbf{w}}+\frac{\theta\lambda}{T}{\boldsymbol{1}}$, respectively, and using leads to $$S'\ge\abs[\Big]{(F_{{\mathbf{X}}+{\mathbf{Q}}}-F_{{\mathbf{Y}}+{\mathbf{Q}}})\Bigl({\mathbf{z}}_\varepsilon+\frac{\theta\lambda}{T}{\boldsymbol{1}}\Bigr)}=\abs{I_1+I_2}\ge\frac{S}{2}-\frac{C_2+\lambda}{T}\sum_{j=1}^m A_j-\frac{3\varepsilon}{4}$$ for all $\varepsilon>0$. Taking the limit for $\varepsilon\to 0$ and rearranging yields the right hand side of .
The left hand side of is an immediate consequence of .
We are now able to bound the difference of the distribution functions by their characteristic functions.
\[le:distribution-sum-bound\]We have $$\begin{gathered}
\label{eq:distribution-sum-bound}
\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs[\bigg]{\sum_{\alpha\in\Pi_L}\mu_\alpha\biggl(\prod_{J\in\alpha}
F_{{\mathbf{X}}_J+{\mathbf{Q}}_J}-\prod_{J\in\alpha}F_{{\mathbf{Y}}_J+{\mathbf{Q}}_J}\biggr)({\mathbf{z}})}\\\le
\frac1{(2\pi)^m}\int_{\norm{{\mathbf{t}}}\le T}\abs[\bigg]{\frac{\Lambda(\varphi_{{\mathbf{X}}})({\mathbf{t}})-\Lambda(\varphi_{{\mathbf{Y}}})({\mathbf{t}})}{\prod_{\ell\in
L} t_\ell}}\,d{\mathbf{t}}.
\end{gathered}$$
Let ${\mathbf{a}}$, ${\mathbf{z}}\in{\mathbb{R}}^m$ with ${\mathbf{a}}\le {\mathbf{z}}$.
The random variable ${\mathbf{X}}_J+{\mathbf{Q}}_J$ admits a density function, because ${\mathbf{Q}}_J$ admits a density function. In particular, ${\mathbf{X}}_J+{\mathbf{Q}}_J$ is a continuous random variable. By Lévy’s theorem (see, e.g., [@Ushakov:1999:selec-topic-charac-funct Thm. 1.8.4]), $${\mathbb{P}}({\mathbf{a}}_J \le {\mathbf{X}}_J+{\mathbf{Q}}_J\le
{\mathbf{z}}_J)=\frac1{(2\pi)^{\abs{J}}}\lim_{\substack{T_j\to\infty\\j\in
J}}\int_{\substack{-T_j\le t_j\le T_j\\ j\in J}} \varphi_{{\mathbf{X}}_J+{\mathbf{Q}}_J}({\mathbf{t}}_J)\prod_{j\in
J}\frac{e^{-it_jz_j}-e^{-it_ja_j}}{-it_j}\, d{\mathbf{t}}_J.$$ As $\varphi_{{\mathbf{X}}_J+{\mathbf{Q}}_J}({\mathbf{t}}_J)=\varphi_{{\mathbf{X}}_J}({\mathbf{t}}_J)\varphi_{{\mathbf{Q}}_J}({\mathbf{t}}_J)$ and $\varphi_{{\mathbf{Q}}_J}({\mathbf{t}}_J)$ vanishes outside $[-T, T]^J$ by Lemma \[lemma:Q\], we can replace the limit $T_j\to\infty$ by setting $T_j=T$, i.e., $${\mathbb{P}}({\mathbf{a}}_J \le {\mathbf{X}}_J+{\mathbf{Q}}_J\le
{\mathbf{z}}_J)=\frac{i^{\abs{J}}}{(2\pi)^{\abs{J}}}\int_{\norm{{\mathbf{t}}_J}\le T} \varphi_{{\mathbf{X}}_J}({\mathbf{t}}_J)\varphi_{{\mathbf{Q}}_J}({\mathbf{t}}_J)\prod_{j\in
J}\frac{e^{-it_jz_j}-e^{-it_ja_j}}{t_j}\, d{\mathbf{t}}_J.$$ Taking the product over all $J\in\alpha$ and summing over $\alpha\in\Pi_L$ yields $$\begin{gathered}
\label{eq:Levy-long-1}
\sum_{\alpha\in\Pi_L}\mu_\alpha\prod_{J\in\alpha}{\mathbb{P}}({\mathbf{a}}_J \le {\mathbf{X}}_J+{\mathbf{Q}}_J\le
{\mathbf{z}}_J) \\
= \frac{i^m}{(2\pi)^m}\int_{\norm{{\mathbf{t}}}\le T} \varphi_{{\mathbf{Q}}}({\mathbf{t}}) \prod_{\ell\in
L}\frac{e^{-it_\ell z_\ell}-e^{-it_\ell a_\ell}}{t_\ell}\sum_{\alpha\in
\Pi_L}\mu_\alpha\prod_{J\in\alpha}\varphi_{{\mathbf{X}}_J}({\mathbf{t}}_J) \, d{\mathbf{t}}\end{gathered}$$ where Fubini’s theorem and the fact that $\varphi_{\mathbf{Q}}({\mathbf{t}})=\prod_{J\in\alpha}\varphi_{{\mathbf{Q}}_J}({\mathbf{t}}_J)$ have been used. By definition of $\varphi_{{\mathbf{X}}}$, we have $\varphi_{{\mathbf{X}}_J}({\mathbf{t}}_J)=\varphi_{{\mathbf{X}}}(\psi_{J, L}({\mathbf{t}}))$. Therefore, we can use the definition of $\Lambda(\varphi_{\mathbf{X}})$ to rewrite to $$\sum_{\alpha\in\Pi_L}\mu_\alpha\prod_{J\in\alpha}{\mathbb{P}}({\mathbf{a}}_J \le {\mathbf{X}}_J+{\mathbf{Q}}_J\le
{\mathbf{z}}_J)
= \frac{i^m}{(2\pi)^m}\int_{\norm{{\mathbf{t}}}\le T} \frac{\Lambda(\varphi_{\mathbf{X}})({\mathbf{t}})}{\prod_{\ell\in L}t_\ell}\varphi_{{\mathbf{Q}}}({\mathbf{t}}) \prod_{\ell\in
L}(e^{-it_\ell z_\ell}-e^{-it_\ell
a_\ell}) \, d{\mathbf{t}}.$$ This equation remains valid when replacing ${\mathbf{X}}$ by ${\mathbf{Y}}$; taking the difference results in $$\begin{gathered}
\label{eq:Levy-long-X-Y}
\sum_{\alpha\in\Pi_L}\mu_\alpha\biggl(\prod_{J\in\alpha}{\mathbb{P}}({\mathbf{a}}_J \le {\mathbf{X}}_J+{\mathbf{Q}}_J\le
{\mathbf{z}}_J)-
\prod_{J\in\alpha}{\mathbb{P}}({\mathbf{a}}_J \le {\mathbf{Y}}_J+{\mathbf{Q}}_J\le
{\mathbf{z}}_J)\biggr)\\
= \frac{i^m}{(2\pi)^m}\int_{\norm{{\mathbf{t}}}\le T} \frac{\Lambda(\varphi_{\mathbf{X}})({\mathbf{t}})-\Lambda(\varphi_{\mathbf{Y}})({\mathbf{t}})}{\prod_{\ell\in L}t_\ell}\varphi_{{\mathbf{Q}}}({\mathbf{t}}) \prod_{\ell\in
L}(e^{-it_\ell z_\ell}-e^{-it_\ell
a_\ell}) \, d{\mathbf{t}}.
\end{gathered}$$ If the integral on the right hand side of is infinite, there is nothing to show. Thus we may assume that it is finite. This also implies that $$\frac{\Lambda(\varphi_{\mathbf{X}})({\mathbf{t}})-\Lambda(\varphi_{\mathbf{Y}})({\mathbf{t}})}{\prod_{\ell\in L}t_\ell}\varphi_{{\mathbf{Q}}}({\mathbf{t}})$$ is an integrable function on ${\mathbb{R}}^m$ (as it vanishes outside $[-T,
T]^m$). Then by the Riemann–Lebesgue lemma, we may take the limit $a_\ell\to-\infty$ for all $\ell\in L$ in to obtain $$\begin{gathered}
\sum_{\alpha\in\Pi_L}\mu_\alpha\biggl(\prod_{J\in\alpha}{\mathbb{P}}({\mathbf{X}}_J+{\mathbf{Q}}_J\le
{\mathbf{z}}_J)-
\prod_{J\in\alpha}{\mathbb{P}}({\mathbf{Y}}_J+{\mathbf{Q}}_J\le
{\mathbf{z}}_J)\biggr)\\
= \frac{i^m}{(2\pi)^m}\int_{\norm{{\mathbf{t}}}\le T} \frac{\Lambda(\varphi_{\mathbf{X}})({\mathbf{t}})-\Lambda(\varphi_{\mathbf{Y}})({\mathbf{t}})}{\prod_{\ell\in L}t_\ell}\varphi_{{\mathbf{Q}}}({\mathbf{t}}) e^{-i{\langle {\mathbf{t}}, {\mathbf{z}}\rangle}} \, d{\mathbf{t}}.
\end{gathered}$$ Taking absolute values and rewriting the left hand side in terms of marginal distribution functions yields .
We now bound the contribution of the lower dimensional distributions.
\[lemma:lower-dimension\] We have $$\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs[\bigg]{\sum_{\substack{\alpha\in\Pi_L\\\alpha\neq \{L\}}}\mu_\alpha\biggl(\prod_{J\in\alpha}
F_{{\mathbf{X}}_J+{\mathbf{Q}}_J}-\prod_{J\in\alpha}F_{{\mathbf{Y}}_J+{\mathbf{Q}}_J}\biggr)({\mathbf{z}})}
\le
\sum_{\emptyset\neq J\subsetneq
L}B_{m-\abs{J}}\sup_{{\mathbf{z}}\in{\mathbb{R}}^J}\abs[\big]{F_{{\mathbf{X}}_{J}}({\mathbf{z}})-F_{{\mathbf{Y}}_{J}}({\mathbf{z}})}
.$$
Let $\alpha=\{J_1,\ldots, J_r\}\in\Pi_L$. Then $$\begin{aligned}
\abs[\bigg]{\prod_{J\in\alpha} F_{{\mathbf{X}}_J+{\mathbf{Q}}_J}({\mathbf{z}}_J) - \prod_{J\in\alpha}
F_{{\mathbf{Y}}_J+{\mathbf{Q}}_J}({\mathbf{z}}_J)} &=
\biggl\lvert\sum_{k=1}^r \biggl(\prod_{j=1}^k F_{{\mathbf{X}}_{J_j}+{\mathbf{Q}}_{J_j}}({\mathbf{z}}_{J_j})\prod_{j={k+1}}^r
F_{{\mathbf{Y}}_{J_j}+{\mathbf{Q}}_{J_j}}({\mathbf{z}}_{J_j}) \\
&\qquad\qquad- \prod_{j=1}^{k-1} F_{{\mathbf{X}}_{J_j}+{\mathbf{Q}}_{J_j}}({\mathbf{z}}_{J_j})\prod_{j={k}}^r
F_{{\mathbf{Y}}_{J_j}+{\mathbf{Q}}_{J_j}}({\mathbf{z}}_{J_j})\biggr)\biggr\rvert\\
&=
\biggl\lvert\sum_{k=1}^r \prod_{j=1}^{k-1} F_{{\mathbf{X}}_{J_j}+{\mathbf{Q}}_{J_j}}({\mathbf{z}}_{J_j})\prod_{j={k+1}}^r
F_{{\mathbf{Y}}_{J_j}+{\mathbf{Q}}_{J_j}}({\mathbf{z}}_{J_j}) \\
&\qquad\qquad\times
\bigl(F_{{\mathbf{X}}_{J_k}+{\mathbf{Q}}_{J_k}}({\mathbf{z}}_{J_j})-F_{{\mathbf{Y}}_{J_k}+{\mathbf{Q}}_{J_k}}({\mathbf{z}}_{J_j})\bigr)\biggr\rvert\\
&\le \sum_{J\in\alpha}\abs[\big]{F_{{\mathbf{X}}_{J}+{\mathbf{Q}}_{J}}({\mathbf{z}}_J)-F_{{\mathbf{Y}}_{J}+{\mathbf{Q}}_{J}}({\mathbf{z}}_J)}
\end{aligned}$$ because the products over the distribution functions are bounded by $1$.
Therefore, $$\abs[\bigg]{\sum_{\substack{\alpha\in\Pi_L\\\alpha\neq \{L\}}}\mu_\alpha\biggl(\prod_{J\in\alpha}
F_{{\mathbf{X}}_J+{\mathbf{Q}}_J}-\prod_{J\in\alpha}F_{{\mathbf{Y}}_J+{\mathbf{Q}}_J}\biggr)({\mathbf{z}})}
\le
\sum_{\emptyset\neq J\subsetneq
L}\abs[\big]{F_{{\mathbf{X}}_{J}+{\mathbf{Q}}_{J}}({\mathbf{z}}_J)-F_{{\mathbf{Y}}_{J}+{\mathbf{Q}}_{J}}({\mathbf{z}}_J)}
\sum_{\substack{\alpha\in \Pi_{L}\\J\in\alpha}}\abs{\mu_\alpha}.$$ A partition $\alpha\in\Pi_L$ with $J\in \alpha$ can be uniquely written as $\alpha=\{J\}\cup \beta$ for a $\beta\in\Pi_{L\setminus J}$. Thus $$\sum_{\substack{\alpha\in
\Pi_{L}\\J\in\alpha}}\abs{\mu_\alpha}=\sum_{\beta\in\Pi_{L\setminus J}}
\abs{\beta}!=\sum_{k=1}^{m-\abs{J}}
{\genfrac{\{}{\}}{0pt}{}{m-\abs{J}}{k}}k!=B_{m-\abs{J}}$$ because there are ${\genfrac{\{}{\}}{0pt}{}{m-\abs{J}}{k}}$ partitions of $L\setminus J$ with $k$ parts. Using the left hand side of yields the assertion (more precisely, of a version of the left hand side of for marginal distributions).
Now, we can complete the proof of the theorem.
The estimate follows from Lemma \[lemma:perturbation\] (more precisely, the right hand side of ), Lemma \[le:distribution-sum-bound\] and Lemma \[lemma:lower-dimension\].
If the expectation of ${\mathbf{X}}$ exists, $\varphi_{{\mathbf{X}}}$ is differentiable. Therefore, $\Lambda(\varphi_{\mathbf{X}})$ is differentiable, too. By Lemma \[lemma:Lambda-property\], $\Lambda(\varphi_{\mathbf{X}})({\mathbf{t}})$ has a zero whenever one of the $t_\ell$, $\ell\in L$, vanishes. Thus $$\frac{\Lambda(\varphi_{\mathbf{X}})({\mathbf{t}})}{\prod_{\ell\in L}t_\ell}$$ is bounded around ${\boldsymbol{0}}$ and therefore bounded on $[-T, T]^m$. The same holds for ${\mathbf{Y}}$. Thus the integral on the right hand side of converges.
Proof of the Quasi-Power Theorem {#sec:proof-quasi-power-theorem}
================================
We may now prove the $m$-dimensional quasi-power theorem, Theorem \[th:quasi-power-dD\].
Let ${\boldsymbol{\mu}}_n=\phi_{n}\operatorname{grad}u({\boldsymbol{0}}) $ and $\Sigma=H_u({\boldsymbol{0}})$. We define the random vector ${\mathbf{X}}=\phi_{n}^{-1/2}({\boldsymbol{\Omega}}_n-{\boldsymbol{\mu}}_n)$. For simplicity, we ignore the dependence on $n$ in this and the following notations.
First, we establish bounds for the characteristic function of ${\mathbf{X}}$.
\[lemma:characteristic-function-single-bound\] For $\Sigma$ regular or singular, there exists an analytic function $V({\mathbf{s}})$ which is analytic for $\norm{{\mathbf{s}}}< \tau{\sqrt{\phi_{n}}}/2$ such that $$\varphi_{{\mathbf{X}}}({\mathbf{s}})=\exp\Bigl(-\frac12 {\mathbf{s}}^\top \Sigma {\mathbf{s}}+V({\mathbf{s}})\Bigr)$$ and $$\label{eq:V-bound}
V({\mathbf{s}})=O\Bigl(\frac{\norm{{\mathbf{s}}}^3+\norm{{\mathbf{s}}}}{{\sqrt{\phi_{n}}}}\Bigr)$$ hold for all ${\mathbf{s}}\in{\mathbb{C}}^K$ with $\norm{{\mathbf{s}}}< \tau{\sqrt{\phi_{n}}}/2$.
For $n\to\infty$, ${\mathbf{X}}$ converges in distribution to a normal distribution with mean ${\boldsymbol{0}}$ and variance-covariance matrix $\Sigma$. In particular, $\Sigma$ is positive (semi-)definite if it is regular (singular, respectively).
By replacing $u({\mathbf{s}})$ and $v({\mathbf{s}})$ by $u({\mathbf{s}})-u({\boldsymbol{0}})$ and $v({\mathbf{s}})-v({\boldsymbol{0}})$, respectively, we may assume that $u({\boldsymbol{0}})=v({\boldsymbol{0}})=0$. We define $E({\mathbf{s}})$ by the relation $M_n({\mathbf{s}})=e^{W_n({\mathbf{s}})}(1+E({\mathbf{s}}))$ and note that by assumption, $E({\mathbf{s}})=O(\kappa_n^{-1})$ uniformly for $\|{\mathbf{s}}\|\le\tau$. We note that this implies $E({\boldsymbol{0}})=0$.
By assumption, $M_n({\mathbf{s}})$ exists for $\norm{{\mathbf{s}}}\le \tau$. Therefore, it is continuous for these ${\mathbf{s}}$ and, by Morera’s theorem combined with applications of Fubini’s and Cauchy’s theorems, $M_n({\mathbf{s}})$ is analytic for $\norm{{\mathbf{s}}}\le\tau$. This also implies that $E({\mathbf{s}})$ is analytic for $\norm{{\mathbf{s}}}\le\tau$. By Cauchy’s formula, we have $$\frac{\partial E({\mathbf{s}})}{\partial s_j}=\frac1{2\pi
i}\oint_{\abs{\zeta_j}=\tau}\frac{E(s_1,\ldots, s_{j-1}, \zeta_j,
s_{j+1}, \ldots, s_d)}{(\zeta_j-s_j)^2}\,d\zeta_j=O\Bigl(\frac{1}{\kappa_n}\Bigr)$$ for $\norm{{\mathbf{s}}}<\tau/2$. Thus $$E({\mathbf{s}})=\int_{[0, {\mathbf{s}}]}{\langle \operatorname{grad}E({\mathbf{t}}), d{\mathbf{t}}\rangle}=O\Bigl(\frac{\norm{{\mathbf{s}}}}{\kappa_n}\Bigr)$$ for $\norm{s}<\tau/2$.
We calculate that $$\begin{aligned}
\varphi_{{\mathbf{X}}}({\mathbf{s}})&=M_n\big(i\phi_{n}^{-1/2}{\mathbf{s}}\big)\exp\big(-i\phi_{n}^{-1/2}{\langle {\boldsymbol{\mu}}_n, {\mathbf{s}}\rangle}\big)\\
&=\exp\Big(-\frac12 {\mathbf{s}}^\top \Sigma {\mathbf{s}}+ V({\mathbf{s}})\Big)
\end{aligned}$$ with $$V({\mathbf{s}})=u(i\phi_{n}^{-1/2}{\mathbf{s}})\phi_{n}+v(i\phi_{n}^{-1/2}{\mathbf{s}})-i\phi_{n}^{-1/2}{\langle {\boldsymbol{\mu}}_n, {\mathbf{s}}\rangle}
+ \frac12 {\mathbf{s}}^\top \Sigma {\mathbf{s}}+ \log(1+E(i\phi_{n}^{-1/2}{\mathbf{s}})).$$
Since $u({\boldsymbol{0}})=v({\boldsymbol{0}})=0$ and the first and second order terms of $u$ cancel out, we have $$\begin{aligned}
V({\mathbf{s}})=O\Bigl(\frac{\|{\mathbf{s}}\|^3+\|{\mathbf{s}}\|}{\sqrt{\phi_{n}}}\Bigr)
\end{aligned}$$ for $\|{\mathbf{s}}\| <\tau{\sqrt{\phi_{n}}}/2$.
Note that $$\lim_{n\to\infty} \varphi_{\mathbf{X}}({\mathbf{s}})=\exp\Big(-\frac12 {\mathbf{s}}^\top \Sigma {\mathbf{s}}\Big)$$ for ${\mathbf{s}}\in{\mathbb{C}}^m$, which implies that, in distribution, ${\mathbf{X}}$ converges to the normal distribution with mean zero and variance-covariance matrix $\Sigma$. Although we have to refine our estimates for applying Theorem \[theorem:Berry-Esseen-dimension-m\], we immediately conclude that $\Sigma$ is positive (semi-)definite depending on whether it is regular or not.
Let now $\Sigma$ be regular. By ${\mathbf{Y}}$ we denote a normally distributed random variable in ${\mathbb{R}}^m$ with mean ${\boldsymbol{0}}$ and variance-covariance matrix $\Sigma$. Its characteristic function is $$\varphi_{\mathbf{Y}}({\mathbf{s}})=\exp\Big(-\frac12 {\mathbf{s}}^\top \Sigma {\mathbf{s}}\Big).$$ The smallest eigenvalue of $\Sigma$ is denoted by $\sigma>0$.
We are now able to bound the functions occurring in the Berry–Esseen inequality.
\[lemma:characteristic-function-difference-bound\] There exists a $c<\tau/2$ such that $$\abs{\Lambda(\varphi_{\mathbf{X}})({\mathbf{s}})-\Lambda(\varphi_{\mathbf{Y}})({\mathbf{s}})}\le
\exp\Bigl(-\frac{\sigma}{4}\norm{{\mathbf{s}}}^2 + O(\norm{{\mathbf{s}}})\Bigr)O\Bigl(\frac{\norm{{\mathbf{s}}}^3+\norm{{\mathbf{s}}}}{{\sqrt{\phi_{n}}}}\Bigr)$$ holds for all ${\mathbf{s}}\in{\mathbb{C}}^L$ with $\norm{{\mathbf{s}}}\le c{\sqrt{\phi_{n}}}$ and $\norm{\Im
{\mathbf{s}}}\le 1$.
Let $\alpha\in\Pi_L$. Then by Lemma \[lemma:characteristic-function-single-bound\], we have $$\begin{gathered}
\label{eq:characteristic-function-multiple-bound-1}
\abs[\bigg]{\prod_{J\in \alpha} (\varphi_{\mathbf{X}}\circ\psi_{J,L})({\mathbf{s}}) - \prod_{J\in \alpha} (\varphi_{\mathbf{Y}}\circ\psi_{J,L})({\mathbf{s}})}\\
=\exp\biggl(-\frac12\Re\sum_{J\in \alpha} \psi_{J,L}({\mathbf{s}})^\top \Sigma
\psi_{J,L}({\mathbf{s}}) \biggr)
\abs[\bigg]{\exp\biggl(\sum_{J\in\alpha} V(\psi_{J,L}({\mathbf{s}}))\biggr)-1}.
\end{gathered}$$ For ${\mathbf{t}}\in{\mathbb{R}}^L$, we have ${\mathbf{t}}^\top \Sigma {\mathbf{t}}\ge \sigma {\mathbf{t}}^\top{\mathbf{t}}\ge
\sigma \norm{{\mathbf{t}}}^2$. For complex $w$, we have $|\exp(w)-1|\le
|w|\exp(|w|)$. Splitting ${\mathbf{s}}$ into its real and imaginary parts in the first summand and using these inequalities for the first and second factor of , respectively, yields $$\begin{gathered}
\abs[\bigg]{\prod_{J\in \alpha} (\varphi_{\mathbf{X}}\circ\psi_{J,L})({\mathbf{s}}) - \prod_{J\in \alpha} (\varphi_{\mathbf{Y}}\circ\psi_{J,L})({\mathbf{s}})}\\
\le \exp\Bigl(-\frac{\sigma}2\norm{{\mathbf{s}}}^2 +
O\Bigl(\norm{{\mathbf{s}}}+\frac{\norm{{\mathbf{s}}}^3+ \norm{{\mathbf{s}}}}{{\sqrt{\phi_{n}}}}\Bigr)\Bigr)
O\Bigl(\frac{\norm{{\mathbf{s}}}^3+ \norm{{\mathbf{s}}}}{{\sqrt{\phi_{n}}}}\Bigr)
\end{gathered}$$ by . For sufficiently small $c$, we obtain $$\abs[\bigg]{\prod_{J\in \alpha} (\varphi_{\mathbf{X}}\circ\psi_{J,L})({\mathbf{s}}) - \prod_{J\in \alpha} (\varphi_{\mathbf{Y}}\circ\psi_{J,L})({\mathbf{s}})}
\le \exp\Bigl(-\frac{\sigma}4\norm{{\mathbf{s}}}^2 +
O(\norm{{\mathbf{s}}})\Bigr)
O\Bigl(\frac{\norm{{\mathbf{s}}}^3+ \norm{{\mathbf{s}}}}{{\sqrt{\phi_{n}}}}\Bigr).$$ Multiplying by $\abs{\mu_\alpha}$ and summation over all $\alpha\in\Pi_L$ concludes the proof of the lemma.
The last ingredient to prove the quasi-power theorem is a bound of the integrals occurring in the Berry–Esseen inequality.
\[le:integral-bound\] Let $c$ be as in Lemma \[lemma:characteristic-function-difference-bound\]. Then $$\int_{\norm{{\mathbf{s}}}\le c{\sqrt{\phi_{n}}}}\abs[\Big]{\frac{\Lambda(\varphi_{{\mathbf{X}}})({\mathbf{s}})-\Lambda(\varphi_{\mathbf{Y}})({\mathbf{s}})}{\prod_{\ell\in L}s_\ell}}\, d{\mathbf{s}}= O\Bigl(\frac{1}{{\sqrt{\phi_{n}}}}\Bigr).$$
For simplicity, set $h=\Lambda(\varphi_{{\mathbf{X}}})-\Lambda(\varphi_{\mathbf{Y}})$. For a partition $\{J,K\}$ of $L$, set $${\mathcal{S}}(J, K)=\{ {\mathbf{s}}\in{\mathbb{R}}^L\colon \abs{s_j}\le 1\text{ for }j\in J,\ 1\le
\abs{s_k}\le c{\sqrt{\phi_{n}}}\text{ for }k\in K\}$$ and partition ${\mathbf{s}}$ into $({\mathbf{s}}_{J},{\mathbf{s}}_{K})$. We use the notation $$D^{J} = \frac{\partial^{\abs{J}}}{\partial z_{j_1}\cdots \partial z_{j_{\abs{J}}}}$$ when $J=\{j_1,\ldots, j_{\abs{J}}\}$. The product of the paths from $0$ to $s_j$ for $j\in J$ is denoted by $[{\boldsymbol{0}}, {\mathbf{s}}_{J}]$.
By Lemma \[lemma:Lambda-property\], we have $$\label{eq:integral-bound-integral}
h({\mathbf{s}})=\int_{[{\boldsymbol{0}},{\mathbf{s}}_{J}]}
D^{J}(h({\mathbf{z}}_{J}, {\mathbf{s}}_{K}))\,d{\mathbf{z}}_{J}.$$ By Cauchy’s integral formula, we have $$\label{eq:integral-bound-2}
D^{J}(h({\mathbf{z}}_{J}, {\mathbf{s}}_{K})) =
\frac1{(2\pi
i)^{\abs{J}}}\oint_{{\boldsymbol{\zeta}}_{J}}\frac{h({\boldsymbol{\zeta}}_{J},
{\mathbf{s}}_{K})}{\prod_{j\in J}(\zeta_j-z_j)^2}\,d{\boldsymbol{\zeta}}_{J}$$ where $\zeta_j$ is integrated over the circle of radius $1$ around $z_j$ for $j\in J$, thus $\norm{\Im {\boldsymbol{\zeta}}_{J}}\le 1$.
Using the estimate of Lemma \[lemma:characteristic-function-difference-bound\] yields $$\label{eq:integral-bound-bound}
\begin{aligned}
\abs{h({\boldsymbol{\zeta}}_{J}, {\mathbf{s}}_{K})}
&=\exp\Bigl(-\frac{\sigma}{4}\norm{({\boldsymbol{\zeta}}_{J},
{\mathbf{s}}_{K})}^2+O(\norm{({\boldsymbol{\zeta}}_{J}, {\mathbf{s}}_{K})})\Bigr)\\
&\qquad\times O\Bigl(\frac{\norm{({\boldsymbol{\zeta}}_{J},
{\mathbf{s}}_{K})}^3+\norm{({\boldsymbol{\zeta}}_{J}, {\mathbf{s}}_{K})}}{{\sqrt{\phi_{n}}}}\Bigr)\\
&=\exp\Bigl(-\frac{\sigma}4\norm{{\mathbf{s}}}^2+O(\norm{{\mathbf{s}}}+1)\Bigr) O\Bigl(\frac{\norm{{\mathbf{s}}}^3+1}{{\sqrt{\phi_{n}}}}\Bigr).
\end{aligned}$$
Combining , and leads to $$\begin{aligned}
&\int_{{\mathcal{S}}(J, K)}\abs[\Big]{\frac{h({\mathbf{s}})}{\prod_{\ell\in L}s_\ell}}\, d{\mathbf{s}}\\&\qquad=O\biggl(\frac1{{\sqrt{\phi_{n}}}}\int_{{\mathcal{S}}(J, K)}\frac{1}{\prod_{\ell\in
L}\abs{s_\ell}}\\
&\qquad\qquad\times\abs[\bigg]{\int_{[{\boldsymbol{0}}, {\mathbf{s}}_{J}]}\exp\Bigl(-\frac{\sigma}{4}\norm{{\mathbf{s}}}^2+ O(\norm{{\mathbf{s}}}+1)\Bigr)(\norm{{\mathbf{s}}}^3+1)
\, d{\mathbf{z}}_{J} }d{\mathbf{s}}\biggr).
\\&\qquad=O\biggl(\frac1{{\sqrt{\phi_{n}}}}\int_{{\mathcal{S}}(J, K)}\frac{1}{\prod_{\ell\in
L}\abs{s_\ell}}\exp\Bigl(-\frac{\sigma}{4}\norm{{\mathbf{s}}}^2+ O(\norm{{\mathbf{s}}}+1)\Bigr)(\norm{{\mathbf{s}}}^3+1)\\
&\qquad\qquad\times\abs[\bigg]{\int_{[{\boldsymbol{0}}, {\mathbf{s}}_{J}]}
\, d{\mathbf{z}}_{J} }d{\mathbf{s}}\biggr).
\end{aligned}$$ The inner integral results in $\abs{\prod_{j\in J}s_j}$. The factors $\abs{s_k}\geq1$ for $k\in K$ in the denominator can simply be omitted. If $K\neq \emptyset$, we still have to bound $$\begin{aligned}
&\int_{{\mathcal{S}}(J, K)}\exp\Bigl(-\frac{\sigma}{4}\norm{{\mathbf{s}}}^2+
O(\norm{{\mathbf{s}}}+1)\Bigr)(\norm{{\mathbf{s}}}^3+1)\,d{\mathbf{s}}\\
&\qquad=\sum_{k\in K} \int_{\substack{{\mathcal{S}}(J, K)\\\norm{{\mathbf{s}}}=\abs{s_k}}}
\exp\Bigl(-\frac{\sigma}{4}\norm{{\mathbf{s}}}^2+
O(\norm{{\mathbf{s}}}+1)\Bigr)(\norm{{\mathbf{s}}}^3+1)\,d{\mathbf{s}}\\
&\qquad=\sum_{k\in K} \int_{\substack{{\mathcal{S}}(J, K)\\\norm{{\mathbf{s}}}=\abs{s_k}}}
\exp\Bigl(-\frac{\sigma}{4}\abs{s_k}^2+
O(\abs{s_k}+1)\Bigr)(\abs{s_k}^3+1)\,d{\mathbf{s}}\\
&\qquad=\sum_{k\in K} \int_{1\le \abs{s_k}\le c{\sqrt{\phi_{n}}}}
\exp\Bigl(-\frac{\sigma}{4}\abs{s_k}^2+
O(\abs{s_k})\Bigr)(\abs{s_k}^3+1)\\
&\qquad\qquad\qquad\times\int_{{\boldsymbol{1}}\le\abs{{\mathbf{s}}_{K\setminus
\{k\}}}\le \abs{s_k}{\boldsymbol{1}}} \int_{\abs{{\mathbf{s}}_J}\le {\boldsymbol{1}}} \,d{\mathbf{s}}_J
d{\mathbf{s}}_{K\setminus \{k\}}ds_k\\
&\qquad=2^{\abs{L}-1}\sum_{k\in K} \int_{1\le \abs{s_k}\le c{\sqrt{\phi_{n}}}}
\exp\Bigl(-\frac{\sigma}{4}\abs{s_k}^2+
O(\abs{s_k})\Bigr)(\abs{s_k}^3+1)\abs{s_k}^{\abs{K}-1}\,ds_k
\end{aligned}$$ where the integration bounds are meant coordinate-wise. Then we use the fact that $$\int_{x\in{\mathbb{R}}} \exp\Bigl(-\frac{\sigma}{4} x^2\Bigr)\abs{x}^t\,dx$$ is finite for all constants $t\geq 0$. Thus, after completing the square in the argument of the exponential function, the integral over $s_k$ is bounded by a constant, i.e., $$\int_{{\mathcal{S}}(J, K)}\exp\Bigl(-\frac{\sigma}{4}\norm{{\mathbf{s}}}^2+
O(\norm{{\mathbf{s}}}+1)\Bigr)(\norm{{\mathbf{s}}}^3+1)\,d{\mathbf{s}}=O(1).$$ We conclude that $$\int_{{\mathcal{S}}(J, K)}\abs[\Big]{\frac{h({\mathbf{s}})}{\prod_{\ell\in L}s_\ell}}\, d{\mathbf{s}}=O\Bigl(\frac1{{\sqrt{\phi_{n}}}}\Bigr).$$ Summation over all partitions $\{J,K\}$ of $L$ completes the proof of the lemma.
We now collect all results to prove Theorem \[th:quasi-power-dD\].
We set $T=c{\sqrt{\phi_{n}}}$ with $c$ from Lemma \[lemma:characteristic-function-difference-bound\]. By Theorem \[theorem:Berry-Esseen-dimension-m\] and Lemma \[le:integral-bound\], we have $$\label{eq:quasi-power-one-step}
\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{{\mathbf{X}}}({\mathbf{z}})-F_{{\mathbf{Y}}}({\mathbf{z}})}=O\Bigl(\frac1{{\sqrt{\phi_{n}}}}\Bigr)+O\biggl(\sum_{\emptyset\neq
J\subsetneq L} \sup_{{\mathbf{z}}_J\in{\mathbb{R}}^J}\abs[\big]{F_{{\mathbf{X}}_{J}}({\mathbf{z}}_J)-F_{{\mathbf{Y}}_{J}}({\mathbf{z}}_J)}\biggr).$$ For $\emptyset\neq J\subsetneq L$, we have $\varphi_{{\mathbf{X}}_J}=\varphi_{{\mathbf{X}}}\circ \chi_{J, L}$. Therefore, all prerequisites for applying the quasi-power theorem on $({\boldsymbol{\Omega}}_n)_J$ are fulfilled. Therefore, we can apply recursively and finally obtain $$\sup_{{\mathbf{z}}\in{\mathbb{R}}^m}\abs{F_{{\mathbf{X}}}({\mathbf{z}})-F_{{\mathbf{Y}}}({\mathbf{z}})}=O\Bigl(\frac1{{\sqrt{\phi_{n}}}}\Bigr).$$
Note that it would also have been possible to apply Corollary \[corollary:Berry-Esseen\]; however, this would have required proving Lemmas \[lemma:characteristic-function-difference-bound\] and \[le:integral-bound\] for subsets $K$ of $L$, which would have required some notational overhead using $\chi_{K, L}$.
This follows by the same arguments as in [@Hwang:1998 Thm. 2].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using an instanton effective action formalism, we compute the critical coupling for the nonperturbative formation of a dynamical mass via instantons in non-Abelian gauge theories with $N_f$ massless fermions. Only continuous phase transitions are considered. For large values of $N_f$ the critical couplings are found to be much smaller than the equivalent critical couplings obtained from gauge exchange calculations in the ladder approximation.'
author:
- |
F. S. Roux\
[*Department of Physics, University of Toronto\
Toronto M5S1A7, CANADA* ]{}
title: Critical couplings for chiral symmetry breaking via instantons
---
0.5 cm 0.5 cm 0.2 cm 0 cm 0 cm -1 cm 17 cm 22 cm
introduction {#intro}
============
Nonperturbative gauge dynamics present many challenges that are yet to be fully understood. One of these is the phenomenon of chiral symmetry breaking. A theory with an $SU(N_c)$ gauge symmetry and $N_f$ fermions in the fundamental representation has a global chiral symmetry in the massless limit. Provided that $N_f < 5.5 N_c$ this theory is asymptotically free. Close to this threshold the coupling is bounded above due to an infrared fixed point in the beta function. As $N_f$ becomes smaller the fixed point value increases. Below a critical value for $N_f$ the coupling can grow strong enough to break the chiral symmetry down to the vector symmetry: $$SU(N_f)_R \times SU(N_f)_L \times U(1)_V \rightarrow SU(N_f)_V \times U(1)_V .
\label{v1}$$ As a result the fermions acquire a dynamical mass, which acts as an order parameter for the spontaneous breaking of the chiral symmetry. This much is well understood, but the respective parts played by gauge exchanges and instantons in this process still needs clarification.
It has been known for some time that gauge exchanges can generate a dynamical mass. To leading order (ladder approximation) in Landau gauge the critical coupling associated with gauge exchange dynamics is $$\alpha_c^{ge} = {2\pi N_c \over 3 (N_c^2-1)} .
\label{v2}$$ Assuming a two-loop beta function, one can show that the critical number of flavors, below which dynamical chiral symmetry breaking occurs, is [@r_atw] $$N_f^c = 4 N_c - {6 N_c \over 25 N_c^2-15} \approx 4 N_c .
\label{v3}$$
It is known that instanton [@r_bpst] dynamics can also break the chiral symmetry [@r_c; @r_cc]. Attempts to investigate this possibility have been hampered by the presence of an infrared divergence in the integral over the instanton size. In their recent investigation of the critical number of flavors for chiral symmetry breaking via instantons, Appelquist and Selipsky [@r_as] avoided the problem of infrared divergences by introducing the following two modifications of the Carlitz and Creamer gap equation [@r_cc]:
- They introduced a mass dependence for the fermion determinant that is not only valid for the small mass limit [@r_t] but also for the large mass limit [@r_ag]. This function simply makes a sharp transition from the small mass behavior to the large mass behavior. The effect is to introduce a fermion mass scale below which the fermions are integrated out.
- In Reference [@r_as] all couplings in the instanton expression are allowed to run according to the two-loop beta function. For those values of $N_f$ where the beta function has a fixed point the coupling remains fairly constant until it reaches the fermion mass scale. Fermions are integrated out below the fermion mass scale. This causes the beta function to revert back to normal asymptotic free running, which gives a suppression of the integrand in the infrared region because of the way it depends on the coupling.
The result of their investigation was that the critical number of flavors for instantons is about $4.77 N_c$. From this they concluded that the instanton contribution to chiral symmetry breaking is comparable to that of gauge exchanges.
In this paper we reinvestigate chiral symmetry breaking through instantons, but our investigation differs from that of Reference [@r_as] in two essential aspects:
- The recently proposed instanton effective action formalism [@r_rt] is used to determine the conditions under which the symmetric vacuum becomes unstable. This formalism reproduces the Carlitz and Creamer gap equation in the small mass limit for $SU(2)$. However, we are not interested in the shape of the mass function, but rather in the critical couplings and the critical number of flavors.
- We concentrate on continuous phase transitions, such as found for chiral symmetry breaking through gauge exchanges. The effective action formalism makes it clear that the instanton contribution responsible for this type of transition is not the one–instanton amplitude of Figure \[diagramme\]a but rather the instanton–anti-instanton amplitude with two mass insertions, shown in Figure \[diagramme\]b.
= 6 cm
The latter point needs clarification. The effective action can be interpreted as an effective potential. This makes it easier to determine when the stability of the symmetric vacuum is affected by the dynamics. For small couplings the global minimum of the effective potential is located at the origin, where the order parameter is zero. When the coupling is increased the global minimum remains at the origin until the coupling crosses the critical value where the phase transition occurs. At this point the global minimum can either jump discontinuously to a nonzero value of the order parameter, indicating a first order phase transition, or it can move continuously (though non-analytically) to a nonzero value of the order parameter, indicating a continuous phase transition. For a first order phase transition a local minimum must develop as the coupling is increased. This becomes deeper as the coupling is increased further until it is below the minimum at zero and so becomes the global minimum when the coupling crosses the critical value. A continuous phase transition appears when the global minimum ‘rolls’ out of the origin. This implies that the minimum at the origin must flatten out and become unstable. To investigate first order phase transitions one must be able to analyze the effective potential at arbitrarily large values of the order parameter. Unfortunately the effective potential is not reliable at large values of the order parameter because it is not possible to make a reliable truncation of the series of diagrams in the effective potential for large order parameters. For a continuous phase transition, on the other hand, the order parameter would be arbitrarily small close enough to the transition point. Hence, the order parameter can be used as an expansion parameter, allowing one to make reliable truncations of the series of diagrams. For this reason we restrict ourselves to the investigation of continuous phase transitions.
In a stability analysis for continuous phase transitions the one–instanton amplitude is irrelevant. The reason is as follows. The effective potential consists of terms with positive powers of the order parameter. Near the origin the nondynamical kinetic part of the potential is second order in the order parameter. Hence, to destabilize the potential at the origin the dynamical terms must be of second order or smaller. Terms with higher powers of the order parameter will become insignificant compared to the kinetic term when the order parameter approaches zero. Looking at the one–instanton diagram, shown in Figure \[diagramme\]a, one sees that this term always comes with as many powers of the order parameter as it takes to close off all the zeromode lines. For $N_f$ flavors there would be $N_f$ factors of the order parameter. This means that for $N_f > 2$ the one–instanton term cannot compete with the kinetic term near the origin. The leading contribution to the dynamic part of the potential that is second order in the order parameter is the instanton–anti-instanton term with two mass insertions shown in Figure \[diagramme\]b. Therefore only this term is considered in our calculation. It turns out that this term does not have an infrared problem in our stability analysis.
The main result of our stability analysis is presented in terms of an expression for arbitrary $N_f$ and $N_c$ from which one can compute the critical couplings for chiral symmetry breaking via instantons. We determine the critical numbers of flavors for $N_c = 3, 4, 5$ and $6$, assuming a two-loop beta function. Numerical values for critical couplings are provided in the fixed point regions of these values of $N_c$, up to the respective critical numbers of flavors. These results indicate that the critical couplings for the formation of 2-point functions via instantons are much smaller than those for gauge exchange dynamics when the number of flavors becomes large. The increasing strength of the instanton dynamics can be understood in terms of the increase in the number of fermion lines of the instanton vertices.
The instanton effective action formalism is briefly reviewed in Section \[s\_diag\]. We distinguish between the dynamical and nondynamical terms of the effective action. Section \[nondyn\] addresses the nondynamical terms and the dynamical term is considered in Section \[dyn\]. We present and discussed the results of this calculation in Section \[results\] and conclude with a short summary in Section \[summary\].
Instanton effective action formalism {#s_diag}
====================================
The 2-point effective action [@r_dm; @r_cjt] in the instanton formalism of Reference [@r_rt] is given by $$\Gamma[S] = {\rm Tr} \left\{ \ln \left( S^{-1} \right) \right\} + {\rm Tr}
\left\{ \left( S_m^{-1} - S^{-1} \right) S \right\} + W_{2PI}[S]
\label{v4}$$ where the fermion [*modal propagator*]{}, $S_m$, consists of different propagators for the different types of modes associated with an instanton background configuration. We parameterize the full fermion propagator $S$ by $$S(p) = {1 \over i p\!\!\!/ - \Sigma(p)} = {- i p\!\!\!/ - \Sigma(p) \over p^2 +
\Sigma^2(p)}
\label{v5}$$ where $\Sigma(p)$ is the dynamical mass function.
One can classify the terms of the effective action as either [*dynamical*]{} or [*nondynamical*]{}. Dynamical terms are those that vanish when the coupling is set to zero, while the nondynamical terms are those that remain. The first two one-loop kinetic terms in (\[v4\]) are nondynamical terms. They consist of one-loop diagrams that are only composed of the full fermion propagator $S$ and the fermion modal propagator, $S_m$.
The dynamical term $W_{2PI}[S]$ in (\[v4\]) is the sum of all 2PI vacuum diagrams constructed with Feynman rules for:
- the $2 N_f$-point instanton vertices, ${\cal V}_I$ and ${\cal V}_A$, given below;
- the full fermion propagator $S$, given in (\[v5\]); and
- an integral over the collective coordinates (sizes, positions and color orientations) of all instantons in the diagram.
The dynamics in this analysis are provided by the instantons and are represented by the instanton vertices: $${\cal V}_I = {\cal A} \prod_n^{N_f} \left( \delta_0 S_0^{-1} \psi^0_n \right) \left(
\overline{\psi}^0_n S_0^{-1} \overline{\delta}_0 \right) ,
\label{v7}$$ where $S_0^{-1}$ is the inverse bare propagator and the functional derivatives $\delta_0 (= \delta/ \delta \eta_0)$ and $\overline{\delta}_0 (= \delta/ \delta
\overline{\eta}_0)$ operate on the zeromode parts of the fermion sources, $\eta_0$ and $\overline{\eta}_0$. The expression for anti-instantons, ${\cal V}_A$, differs from the expression in (\[v7\]) only in that the zeromode functions, $\psi^0$ and $\overline{\psi}^0$, have the opposite helicities.
The coefficient of the instanton vertex is the integrand of the ’t Hooft instanton amplitude [@r_t]: $${\cal A} = \kappa \gamma(\alpha) \frac{1}{\rho^5} (\mu\rho)^{b_0+N_f} .
\label{v8}$$ Here $\rho$ is a scale parameter for the size of the instanton and $b_0 =
\frac{1}{3}(11 N_c - 2 N_f)$.
The dependence on the gauge coupling in (\[v8\]) is contained in $$\gamma \left( \alpha(\mu) \right) = \left( {2 \pi \over \alpha(\mu)} \right)^{2
N_c} \exp \left(- {2 \pi \over \alpha(\mu)} \right) .
\label{v9}$$ The factor $(\mu\rho)^{b_0}$ appearing in (\[v8\]) can be incorporated into the exponent part of $\gamma(\alpha)$. This then leads to one-loop running as a function of $\rho$ for the coupling in the exponent in $\gamma(\alpha)$. The instanton expression in (\[v8\]) can be computed to higher orders which would cause the coupling in the monomial part of $\gamma(\alpha)$ to run as a function of $\rho$ as well, but always at one order less than the coupling in the exponent. To all orders one would expect that both couplings run according to the exact beta function. Hence, assuming that the two-loop beta function is an accurate enough representation of the exact beta function, one can allow both couplings in $\gamma(\alpha)$ to run according to the two-loop beta function [@r_as]. This is what we shall assume so that we can make the replacement $\gamma(\alpha(\mu))
(\mu\rho)^{b_0} \rightarrow \gamma(\alpha(\rho))$.
The shape of $\gamma(\alpha)$ as a function of $\alpha$ is shown in Figure \[gamma\]. The instanton amplitude peaks at a value of $\alpha = \pi/N_c$.
= 8 cm
The prefactor in (\[v8\]) is $$\kappa = {2 \exp(5/6) \exp(B N_f-C N_c]) \over \pi^2 (N_c-1)! (N_c-2)!}
\label{v10}$$ where we follow [@r_as] and use the one-loop values: $B=0.2917$, $C=1.5114$ in MS-bar [@r_l].
Note that unlike the case in Reference [@r_rt] the integration over the color orientations has not yet been done in (\[v7\]). To alleviate the combinatorical calculations, this step is postponed until after the propagators are connected and unnecessary gauge transformations are cancelled. However, $\kappa$ in (\[v10\]) already contains the volume of the factor group, $SU(N_c)/G$, where $G$ is the subgroup that leaves the instantons invariant. All that remains of the integration over color orientations is a color averaging integral.
The expression of the fermion zeromode function in singular gauge that appears in (\[v7\]) is $$\psi^0(x) = {\sqrt{2} \rho\ x_{\mu} \gamma_{\mu} U^{\alpha} \over \pi \sqrt{\mu}
\sqrt{x^2} (\rho^2+x^2)^{3/2}} ,
\label{v11}$$ where $\mu$ is the renormalization scale[^1] and $x^2$ denotes an Euclidean dot-product: $x_{\mu} x_{\mu}$. The Euclidean Dirac matrices, $\gamma_{\mu}$, are here defined so that $\{ \gamma_{\mu}, \gamma_{\nu} \} = 2
\delta_{\mu\nu} \cdot {\bf 1}$, ${\rm tr} \{ \gamma_{\mu} \gamma_{\nu} \} = 4
\delta_{\mu\nu}$ and $\gamma_{\mu}^{\dag} = \gamma_{\mu}$. The $U^{\alpha}$ denotes a Dirac spinor with isospin index $\alpha$. (The isospin index is implicit in $\psi^0(x)$.) This spinor is normalized such that $\sum_{\alpha}
\overline{U}_{\alpha} U^{\alpha} = 1$ and it obeys the identity $\sum_{\alpha} U^{\alpha} \overline{U}_{\alpha} = \frac{1}{4} (1+\gamma_5)$.
We shall also later need the Fourier-transform of the zeromode function $$\Psi^0(p) = \int \psi^0(x) \exp(i x \cdot p)\ d^4 x = i {2 \pi \sqrt{2} \rho\ p_{\mu}
\gamma_{\mu} U^{\alpha} \over \sqrt{\mu} p^2} f\left( \frac{\rho \sqrt{p^2}}{2}
\right) ,
\label{v12}$$ where the function, $$f(x) = 2 x \left[ I_0(x) K_1(x) - I_1(x) K_0(x) \right] - 2 I_1(x) K_1(x) ,
\label{v13}$$ is defined in terms of modified Bessel functions [@r_AS].
The nondynamical terms {#nondyn}
======================
The trace over the inverse modal propagator in (\[v4\]) contains an integral over all the collective coordinates involved in the diagram. The general form of the result of this integration is the inverse bare propagator, $S_0^{-1} = i p\!\!\!/$, multiplied by a momentum-dependent scalar form factor, which we shall assume to be equal to 1.
In terms of (\[v5\]) the nondynamical terms in (\[v4\]), which we denote by $\Gamma_{kin}$, becomes $$\Gamma_{kin} = - \int \left( {4 \Sigma^2(p) \over p^2 + \Sigma^2(p)} - 2 \ln \left[
p^2 + \Sigma^2(p) \right] \right) {d^4p \over (2\pi)^4}.
\label{v14}$$
It is customary to introduce an infrared cutoff at the momentum scale where $p_0 =
\Sigma(p_0)$. Close to the phase transition point $\Sigma(p)$ is very small so that $p_0
\approx 0$. An expansion of $\Gamma_{kin}$ in term of $\Sigma(p)/p$ contains only even powers. We drop the terms that are independent of $\Sigma(p)$ because they do not affect the location of the minimum and we drop the terms that are higher than second order in $\Sigma(p)$ because they are suppressed. The result is $$\Gamma_{kin}^{(2)} = - {N_c N_f \over 4 \pi^2} \int_0^{\infty} \Sigma(p)^2 p\
dp.
\label{v15}$$
Now we make a redefinition of the integration variables and do a Fourier transformation. For this purpose we define $$p = \mu \exp(t) ~~~ {\rm and} ~~~ \Sigma(p) = \mu \exp(-t) \int_{-\infty}^{\infty}
\sigma(\omega) \exp(i\omega t)\ d \omega
\label{w1}$$ where $\mu$ is the renormalization scale. Now we have $$\begin{aligned}
\Gamma_{kin}^{(2)} & = & - {N_c N_f \mu^4 \over 4 \pi^2} \int_{-\infty}^{\infty}
\int_{-\infty}^{\infty} \sigma(\omega_1) \sigma(\omega_2) \int_{-\infty}^{\infty}
\exp(i\omega_1 t) \exp(i\omega_2 t)\ dt\ d \omega_1\ d \omega_2 \nonumber \\ & = &
- {N_c N_f \mu^4 \over 2 \pi} \int_{-\infty}^{\infty} \sigma(\omega) \sigma(-\omega)\
d \omega.
\label{w2}\end{aligned}$$
The dynamical term {#dyn}
==================
The instanton–anti-instanton term that is quadratic in the dynamical mass looks as follows: each mass insertion closes off two lines of an instanton vertex. The remaining lines are all connected between the two vertices. (See Figure \[diagramme\]b.) The resulting diagram is expressed as $$W_{2PI}^{(2)} = {\cal F} \kappa^2 \int \int \gamma(\alpha(\rho_1))\
\gamma(\alpha(\rho_2))\ (\mu^2\rho_1\rho_2)^{N_f} D(\rho_1) D(\rho_2)
G(\rho_1,\rho_2) \frac{1}{\rho_1^5} \frac{1}{\rho_2^5}\ d\rho_1\ d\rho_2
\label{v16}$$ where $\rho_1$ and $\rho_2$ denote the sizes of the two instantons; the combinatorial factor ${\cal F}$ simply counts the number of terms associated with this diagram; $D(\rho_1)$ and $D(\rho_2)$ are the parts of the expression due to the two mass loops and $G(\rho_1,\rho_2)$ is the part of the expression due to the remaining fermion loops.
The combinatorics and color averaging
-------------------------------------
To count the number of terms we start by considering all the possible ways that the propagators can close off the lines of the two instanton vertices to give a diagram with two mass loops. Then we perform the color averaging and add up all the different terms.
Recall that each line of an instanton vertex has a specific flavor associated with it. There is one incoming and one outgoing line for each flavor. Because the propagator is flavor diagonal it must connect lines of the same flavor. Hence, there are $N_f$ distinct ways to connect one of the two mass insertions, each leaving the other mass insertion and the remaining fermion lines with only one way to be connected.
Next we perform the averaging over the color orientations of the two respective instantons. The color orientations are represented by gauge transformations that operate on the zeromodes: $$\psi^0 \rightarrow g \psi^0 ~~~ {\rm and} ~~~ \overline{\psi}^0 \rightarrow
\overline{\psi}^0 g^{-1} .
\label{v17}$$ In the mass loops they cancel because there we have a $g$ and a $g^{-1}$ from the same instanton. The gauge transformations of the remaining lines can be combined into relative gauge transformations of one instanton with respect to the other: $g_1^{-1} g_2 = g_r$ or $g_2^{-1} g_1 = g_r^{-1}$. Thus the two color averaging integrals become one trivial ($=1$) and one nontrivial averaging integral. The nontrivial integral contains $(N_f-1)$ factors of $g_r$ and $g_r^{-1}$. The result of the averaging is a series of tensor structures each multiplied by a coefficient that depends on $N_f$ and $N_c$. General expressions for these coefficients are provided in Reference [@r_sam]. The tensor structures indicate how color indices of the same instanton are interconnected. This in turn determines how the fermion lines between the two instantons are closed off into fermion loops and thereby determines the form of the momentum integrals.
The evaluation of all these momentum integrals is too challenging to attempt. However, we expect that the sizes of the momentum integrals do not differ by more that an $O(1)$ factor and that the variations would average out. Therefore we set them all equal and estimate their typical size in Section \[glusse\]. Each tensor structure consists of $(N_f-1)!$ terms because of all the possible permutations of the $(N_f-1)$ pairs of fermion lines between the two instantons. The different tensor structures denote different [*relative*]{} permutations of one line with respect to the other in the $(N_f-1)$ pairs of fermion lines. All tensor structures that are associated with the same conjugacy class in the permutation group have the same coefficient.
Because we assume that the momentum integrals are the same, we only have to add all the coefficients of all the tensor structures multiplied by the number of terms associated with each tensor structure. It can be shown that for permutations of $(N_f-1)$ elements, $$\sum_{classes} \alpha_i N_i = {(N_c-1)! \over (N_c+N_f-2)!} ,
\label{v18}$$ where $N_i$ is the number of group elements in the $i$-th conjugacy class of the permutation group and $\alpha_i$ is the coefficient of the tensor structures in that class. To get the total count we have to multiply (\[v18\]) by $N_f$ for the number of ways to connect the mass loops and by $(N_f-1)!$ for the number of terms in each tensor structure. The resulting combinatorial factor is $${\cal F} = {N_f! (N_c-1)! \over (N_c+N_f-2)!} .
\label{v19}$$
The mass insertions
-------------------
Using (\[v5\]) and (\[v12\]) we obtain the following expressions for the mass loops $$\begin{aligned}
D(\rho) & = & \int \overline{\Psi}^0(p) S_0^{-1}(p) S(p) S_0^{-1}(p) \Psi^0(p)
{d^4p \over (2\pi)^4} \nonumber \\ & = & {\rho^2 \over \mu} \int
{\Sigma(p) \over p^2 + \Sigma^2(p)} f^2 \left( {\rho p \over 2} \right) p^3\ dp
\nonumber \\ & = & {\rho^2 \over \mu} \int \Sigma(p) f^2 \left( {\rho p \over 2}
\right) p\ dp + O \left( \Sigma^3(p) / p^3 \right)
\label{v20}\end{aligned}$$ where we again made use of the approximations introduced in Section \[nondyn\]. Next we again make a redefinition of the integration variables and do a Fourier transformation as in (\[w1\]). In addition we define $${\rho \mu \over 2} = \exp(\eta) ~~~ {\rm and} ~~~ f \left( \exp(a) \right) = f_0(a)
\label{v21}$$
As a result we have $$\begin{aligned}
D(\rho) & = & 4 \exp(2\eta) \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}
\sigma(\omega) \exp(i\omega t) f_0^2(t+\eta) \exp(t)\ dt\ d \omega \nonumber \\
& = & 4 \exp(\eta) \int_{-\infty}^{\infty} \sigma(\omega) \exp(-i\omega
\eta) F(\omega)\ d \omega
\label{v22}\end{aligned}$$ where $$F(\omega) = \int_{-\infty}^{\infty} \exp(i\omega a) f_0^2(a) \exp(a)\ da .
\label{v23}$$
The remaining fermion loops {#glusse}
---------------------------
The rest of the fermion lines between the two instanton vertices combine into $$G(\rho_1,\rho_2) = \prod_i^{2 N_f - 2} \left[ \int {d^4p_i \over (2\pi)^4}
\overline{\Psi}^0(p_i) S_0^{-1}(p_i) S(p_i) S_0^{-1}(p_i) \Psi^0(p_i) \right]
(2\pi)^4 \delta^4 \left( \sum_i p_i \right) .
\label{v24}$$ The fermion lines pair up into fermion loops due to the color contractions that appear after the averaging over color orientations. Considering one such loop, we find $$\begin{aligned}
I(\rho_1, \rho_2, p, q) & = & - \overline{\Psi}_a^0(p) S_0^{-1}(p) S(p) S_0^{-1}(p)
\Psi_b^0(p) \overline{\Psi}_b^0(q) S_0^{-1}(q) S(q) S_0^{-1}(q) \Psi_a^0(q)
\nonumber \\ & = & {2 (2\pi)^4 \over \mu^2} \rho_1^2 \rho_2^2 { p\cdot q \over p^2
q^2 } f \left( {\rho_1 p \over 2} \right) f \left( {\rho_2 p \over 2} \right) f
\left( {\rho_1 q \over 2} \right) f \left( {\rho_2 q \over 2} \right) + O \left(
\Sigma_0^2 / \mu^2 \right)
\label{v25}\end{aligned}$$ where in the first line we explicitly show the color indices on the zeromodes. By defining $$J(k,\rho_1,\rho_2) = \int I(\rho_1, \rho_2, p, p-k) {d^4p \over (2\pi)^4} ,
\label{v26}$$ one can write (\[v24\]) as $$G(\rho_1,\rho_2) = \prod_i^{N_f - 1} \left[ \int {d^4k_i \over (2\pi)^4} J(k_i,\rho_1,
\rho_2) \right] (2\pi)^4 \delta^4 \left( \sum_i k_i \right) .
\label{v27}$$ We shall not attempt to evaluate (\[v27\]) exactly, but instead estimate its size in terms of the number of fermion loops. It turns out that the $\rho$-integrals are dominated in the region where $\rho_1\approx\rho_2$. Therefore we set $\rho_1=\rho_2=\rho$ in the arguments of the $f$-functions. Then (\[v26\]) becomes $$J(k,\rho_1,\rho_2) = {8 \pi \rho_1^2 \rho_2^2 \over \mu^2} \int { p^2 - p\cdot k
\over p^2 (p-k)^2 } f^2 \left( {\rho p \over 2} \right) f^2 \left( {\rho
\sqrt{(p-k)^2} \over 2} \right) p^3 \sin^2(\theta)\ d\theta\ dp .
\label{v29}$$ For the region where $p>k$ we set $\sqrt{(p-k)^2} = p$ inside the $f$-function and for $p<k$ we set $\sqrt{(p-k)^2} = k$. Upon evaluating the angular integral we arrive at $$J(k,\rho_1,\rho_2) = {(2\pi)^2 \rho_1^2 \rho_2^2 \over \mu^2} \left[ \int_0^k {p^2
\over 2 k^2} f^2 \left( {\rho p \over 2} \right) f^2 \left( {\rho k \over 2} \right)
p\ dp + \int_k^{\infty} \left( 1 - {k^2 \over 2 p^2} \right) f^4 \left( {\rho p \over
2} \right) p\ dp \right] .
\label{v30}$$ If we assume that $\rho_1 > \rho_2$ then $\rho = \rho_1$, because the $f$-function with the largest $\rho$ dominates. Defining $$x = {p\rho_1 \over 2} ~~~ {\rm and} ~~~ y = {k\rho_1 \over 2}
\label{v31}$$ one can write (\[v30\]) as $$J(k,\rho_1,\rho_2) = {\pi^2 \rho_2^2 \over \mu^2} \left[ {8 f^2(y) \over y^2}
\int_0^y f^2(x)\ x^3\ dx + 16 \int_y^{\infty} f^4(x)\ x\ dx - 8 y^2 \int_y^{\infty}
\frac{1}{x} f^4(x)\ dx \right] \equiv {\pi^2 \rho_2^2 \over \mu^2} R(y) .
\label{v32}$$ The size of the area under $R(y)$ is $\approx \frac{1}{2}$. This function determines the momentum dependence under the remaining momentum integrals in (\[v27\]). To assess the size of the remaining integrals we note that (\[v27\]) has $(N_f-2)$ integrals over $k_i$. This will give $(N_f-3)$ angular integrals. (One can arrange it that each $k_i$ at most contracts with its two neighbors). Together with the definitions in (\[v31\]) we estimate the size of the remaining momentum integrals to be $$\prod_i^{N_f - 1} \left[ \int {d^4k_i \over (2\pi)^4} R \left( {k_i\rho_1 \over
2} \right) \right] (2\pi)^4 \delta^4 \left( \sum_i k_i \right) \sim \left( {1 \over
2} \right)^{N_f-1} \left[ {16 \over \rho_1^4} (2\pi)^{-2} \right]^{N_f-2} .
\label{v33}$$ Our estimate for the sizes of $G(\rho_1,\rho_2)$ then gives $$G(\rho_1,\rho_2) \sim {\rho_1^4 \pi^2 \over 8} \left[ {2 \rho_2^2 \over \mu^2
\rho_1^4} \right]^{N_f-1} .
\label{v35}$$
Results and discussion {#results}
======================
The integrand of (\[v16\]) is invariant under an interchange of $\rho_1$ and $\rho_2$. Therefore, one can split the $\rho_2$-integration into two regions, divided by the value of $\rho_1$. First, we consider only the part where $\rho_2 < \rho_1$. When we substitute (\[v35\]) into this part we find $$W_{\rho_2<\rho_1}^{(2)} = {2^{N_f} \pi^2\over 16} {\cal F} \kappa^2 \mu^2 \int \int
\gamma(\alpha(\rho_1))\ \gamma(\alpha(\rho_2))\ D(\rho_1) D(\rho_2) \rho_1^{3-3 N_f}
\rho_2^{3 N_f-7}\ d\rho_1\ d\rho_2 .
\label{v36}$$
One can see that the $\rho_2$-integral is dominated by the region near the $\rho_1$-boundary. For this reason one can set $\gamma(\alpha(\rho_2)) =
\gamma(\alpha(\rho_1))$. The integrand of the $\rho_1$-integral has its dominant region around some value $\rho_m$, close to $\mu$. Since we consider cases with large numbers of flavors, the coupling runs slowly. Therefore one can set $\gamma(\alpha(\rho_1)) = \gamma(\alpha(\rho_m)) = \gamma_0$.
Next we use (\[v21\]) to redefine the variables in (\[v36\]) and we substitute in (\[v22\]). The result is $$\begin{aligned}
W_{\rho_2<\rho_1}^{(2)} & = & \frac{1}{4} \mu^4 {\cal F} \kappa^2 \pi^2 2^{N_f}
\gamma_0^2 \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sigma(\omega_1)
\sigma(\omega_2) F(\omega_1) F(\omega_2) \nonumber \\ & & \times
\int_{-\infty}^{\infty} \int_{-\infty}^{\eta_1} \exp([5-3 N_f+i\omega_1] \eta_1)
\exp([3 N_f-5+i\omega_2] \eta_2)\ d\eta_2\ d\eta_1\ d \omega_1\ d\omega_2 \nonumber
\\ & = & \frac{1}{2} \mu^4 {\cal F} \kappa^2 \pi^3 2^{N_f} \gamma_0^2
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sigma(\omega_1) \sigma(\omega_2)
F(\omega_1) F(\omega_2) {\delta \left( \omega_1+\omega_2 \right) \over 3 N_f - 5 +
i \omega_2}\ d \omega_1\ d \omega_2
\label{v37}\end{aligned}$$
Now we add the part where $\rho_2 > \rho_1$. This is the same as (\[v37\]) but with $\omega_1$ and $\omega_2$ interchanged. Then we evaluate one of the $\omega$-integrals. The result is $$\begin{aligned}
W_{2PI}^{(2)} & = & W_{\rho_2<\rho_1}^{(2)} + W_{\rho_2>\rho_1}^{(2)} \nonumber \\ &
= & \mu^4 {\cal F} \kappa^2 \pi^3 2^{N_f} \gamma_0^2 \int_{-\infty}^{\infty}
\sigma(\omega) \sigma(-\omega) \left[ {(3 N_f - 5) F(\omega) F(-\omega) \over (3 N_f
- 5)^2 + \omega^2} \right]\ d \omega .
\label{v38}\end{aligned}$$ The maximum value of the part of the integrand in the brackets is at $\omega=0$ with $F(0)^2 \approx 0.3$. We notice that this term does not have an infrared divergence.
Comparing the result in (\[v38\]) with the nondynamical term in (\[w2\]) one can see that the potential becomes unstable at the origin when $${\cal F} \kappa^2 \pi^3 2^{N_f} \gamma_0^2 \left[ {F(0)^2 \over 3 N_f - 5} \right] >
{N_c N_f \over 2 \pi} .
\label{v41}$$ This can be expressed as $${\cal E} \gamma_0 > 1 ,
\label{v43}$$ where $${\cal E}(N_c,N_f) = {2 F(0) \exp(5/6) \exp(B N_f - C N_c) \over (N_c-2)!} \sqrt
{2^{N_f+1} (N_f-1)! \over (3 N_f-5) (N_f + N_c - 2)! N_c!}
\label{v44}$$ only depends on $N_c$ and $N_f$.
The critical coupling for instanton dynamics $\alpha_c^I$ is the one for which $$\gamma \left( \alpha(\rho_m) = \alpha_c^I \right) = \frac{1}{{\cal E}(N_c,N_f)} .
\label{v44a}$$ It necessarily has to be smaller than $\pi /N_c$. The requirement in (\[v44a\]), together with (\[v44\]), is the main result of our analysis. From it one can compute $\alpha_c^I$ for any value of $N_c$ and $N_f>4/3$. Numerical estimates of $\alpha_c^I$ for various values of $N_c$ and $N_f$ are provided in Figure \[alphac\].
Before we discuss these critical couplings we need to clarify the ranges of numbers of flavors for which these critical couplings are computed. These ranges are defined under the assumption that the running of the coupling is governed by the two-loop beta function – an assumption which becomes more accurate as the coupling becomes smaller. We only compute the critical couplings in the region of $N_f$ where the two-loop beta function has an infrared fixed point. The left-hand end of each curve in Figure \[alphac\] is therefore the point where a fixed point value appears, which moves down from infinity as a function of increasing $N_f$.
The right-hand ends of the curves in Figure \[alphac\] indicate the critical numbers of flavors $N_f^c$ above which chiral symmetry breaking does not occur. When there is a nontrivial infrared fixed point in the beta function the maximum coupling that can be reached is the fixed point coupling. Assuming a two-loop beta function the fixed point coupling is given by $$\alpha_* = -\frac{b}{c} = {- 4 \pi (11 N_c - 2 N_f) N_c \over 34 N_c^3 - 13 N_c^2
N_f + 3 N_f}
\label{v45}$$ where $b$ and $c$ are respectively the first and second coefficients in the beta function. The critical numbers of flavors $N_f^c$ is reached when the fixed point coupling given by (\[v45\]) falls below the critical coupling, obtained from solving (\[v44a\]). Therefore for values of $N_f$ above $N_f^c$ the coupling is unable to reach the critical value so that chiral symmetry breaking does not occur. The critical number of flavors, $N_f^c$ are $13.48, 17.95, 22.41$ and $26.87$ for $N_c=3, 4, 5,$ and $6$, respectively. This gives $N_f^c \approx 4.5 N_c$, compared to $4.77 N_c$, reported in Reference [@r_as].
= 10 cm
We computed critical couplings for $N_c=3, 4, 5$ and $6$ and for $N_f$ in the fixed point region up to $N_f^c$. The results are shown in Figure \[alphac\]. From these results we make the following observations:
- The critical coupling decreases as either $N_c$ or $N_f$ increases.
- The values of the critical couplings near $N_f^c$ are small compared to the equivalent critical coupling for gauge exchanges given in (\[v2\]). Compare, for example, the $N_c = 3$ values. We obtain $\alpha_c^I = 0.37$, while the equivalent gauge exchange value is $\alpha_c^{ge} = \pi/4 = 0.785$.
- Toward smaller values of $N_f$ the instanton critical couplings become more comparable to their gauge exchange counterparts.
On this basis we conclude that instanton dynamics are much stronger than gauge dynamics for large numbers of flavors.
summary
=======
We have used the instanton effective action formalism of Reference [@r_rt] to compute the critical couplings for the formation of 2-point functions. For this analysis we restricted ourselves to continuous phase transitions for chiral symmetry breaking, which implies that we only consider the instanton–anti-instanton amplitude with two mass insertions. We found that this amplitude does not suffer from an infrared divergence. Couplings are allowed to run according to the two-loop beta functions. The resulting critical couplings indicate that instanton dynamics are much stronger than gauge dynamics for large numbers of flavors.
Acknowledgements {#acknowledgements .unnumbered}
================
The author wants to express his appreciation for the valuable comments of Bob Holdom. He also wants to thank Craig Burrell for his help in the preparation of this manuscript.
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[^1]: The $\mu$ is included here to get a mass dimension of $\frac{3}{2}$ for the fermion zeromodes.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a generating function of permutation tableaux by the combinatorial interpretation of Corteel and Williams.
We prove bijectively two new combinatorial interpretations. The first one is in terms of weighted Motzkin paths called Laguerre histories and is obtained by refining a bijection of Foata and Zeilberger. Secondly we show that this partition function is the generating function of permutations with respect to right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by refining a bijection of Françon and Viennot.
Then we give a new formula for the partition function which generalizes the one of Blythe & al. It is proved in two combinatorial ways. The first proof is an enumeration of lattice paths which are known to be a solution of the Matrix Ansatz of Derrida & al. The second proof relies on a previous enumeration of rook placements, which appear in the combinatorial interpretation of a related normal ordering problem. We also obtain a closed formula for the moments of Al-Salam-Chihara polynomials.
author:
- |
Matthieu Josuat-Vergès[^1]\
Université Paris-sud and LRI,\
91405 Orsay CEDEX, FRANCE.\
`[email protected]`\
date: ' Mathematics Subject Classifications: 05A15, 05A19, 82B23, 60C05.'
title: 'Combinatorics of the three-parameter PASEP partition function'
---
Introduction
============
The PASEP partition function
----------------------------
The partially asymmetric simple exclusion process (also called PASEP) is a Markov chain describing the evolution of particles in $N$ sites arranged in a line, each site being either empty or occupied by one particle. Particles may enter the leftmost site at a rate $\alpha\geq0$, go out the rightmost site at a rate $\beta\geq0$, hop left at a rate $q\geq0$ and hop right at a rate $p>0$ when possible. By rescaling time it is always possible to assume that the latter parameter is 1 without loss of generality. It is possible to define either a continuous-time model or a discrete-time model, but they are equivalent in the sense that their stationary distributions are the same. In this article we only study some combinatorial properties of the partition function. For precisions, background about the model, and much more, we refer to [@BECE; @BCEPR; @CW1; @CW2; @DEHP; @TS]. We refer particularly to the long survey of Blythe and Evans [@BE] and all references therein to give evidence that this is a widely studied model. Indeed, it is quite rich and some important features are the various phase transitions, and spontaneous symmetry breaking for example, so that it is considered as a fundamental model of nonequilibrium statistical physics.
A method to obtain the stationary distribution and the partition function $Z_N$ of the model is the Matrix Ansatz of Derrida, Evans, Hakim and Pasquier [@DEHP]. We suppose that $D$ and $E$ are linear operators, ${\ensuremath{\langle W|}}$ is a vector, ${\ensuremath{|V\rangle}}$ is a linear form, such that: $$\label{ansatz}
DE-qED=D+E, \qquad {\ensuremath{\langle W|}} \alpha E = {\ensuremath{\langle W|}}, \qquad \beta D{\ensuremath{|V\rangle}} = {\ensuremath{|V\rangle}},\qquad {\ensuremath{\langle W|V \rangle}}=1,$$ then the non-normalized probability of each state can be obtained by taking the product ${\ensuremath{\langle W|}} t_1\dots t_N {\ensuremath{|V\rangle}}$ where $t_i$ is $D$ if the $i$th site is occupied and $E$ if it is empty. It follows that the normalization, or partition function, is given by ${\ensuremath{\langle W|}}(D+E)^N{\ensuremath{|V\rangle}}$. It is possible to introduce another variable $y$, which is not a parameter of the probabilistic model, but is a formal parameter such that the coefficient of $y^{k}$ in the partition function corresponds to the states with exactly $k$ particles (physically it could be called a [*fugacity*]{}). The partition function is then: $$\label{ZN}
Z_N = {\ensuremath{\langle W|}}(yD+E)^N{\ensuremath{|V\rangle}},$$ which we may take as a definition in the combinatorial point of view of this article (see Section \[Zfacts\] below for precisions). An interesting property is the symmetry: $$\label{Z_sym}
Z_N\big(\alpha,\beta,y,q\big) = y^N Z_N\big(\beta,\alpha,\tfrac1y,q\big),$$ which can be seen on the physical point of view by exchanging the empty sites with occupied sites. It can also be obtained from the Matrix Ansatz by using the transposed matrices $D^*$ and $E^*$ and the transposed vectors ${\ensuremath{\langle V|}}$ and ${\ensuremath{|W\rangle}}$, which satisfies a similar Matrix Ansatz with $\alpha$ and $\beta$ exchanged.
In section \[paths\], we will use an explicit solution of the Matrix Ansatz [@BECE; @BCEPR; @DEHP], and it will permit to make use of weighted lattice paths as in [@BCEPR].
Combinatorial interpretations
-----------------------------
Corteel and Williams showed in [@CW1; @CW2] that the stationary distribution of the PASEP (and consequently, the partition function) has a natural combinatorial interpretation in terms of [*permutation tableaux*]{} [@SW]. This can be done by showing that the two operators $D$ and $E$ of the Matrix Ansatz describe a recursive construction of these objects. They have in particular: $$\label{ZTP}
Z_N = \sum_{T\in PT_{N+1}}
\alpha^{-a(T)} \beta^{-b(T)+1} y^{r(T)-1} q^{w(T)},$$ where $PT_{N+1}$ is the set of permutation tableaux of size $N+1$, $a(T)$ is the number of 1s in the first row, $b(T)$ is the number of unrestricted rows, $r(T)$ is the number of rows, and $w(T)$ is the number of superfluous 1s. See Definition \[def\_PT\] below, and [@CW2 Theorem 3.1] for the original statement. Permutation tableaux are interesting because of their link with permutations, and it is possible to see $Z_N$ as a generating function of permutations. Indeed thanks to the Steingrímsson-Williams bijection [@SW], it is also known that [@CW2]: $$\label{depart}
Z_N = \sum_{\sigma\in\mathfrak{S}_{N+1}}
\alpha^{ -u(\sigma) } \beta^{ -v(\sigma) } y^{{{\rm wex}}(\sigma)-1} q^{{{\rm cr}}(\sigma)},$$ where we use the statistics in the following definition.
\[stat\_per\] Let $\sigma\in\mathfrak{S}_n$. Then:
- $u(\sigma)$ the number of [*special*]{} right-to-left minima, [*i.e.*]{} integers $j\in\{1,\dots,n\}$ such that $\sigma(j)=\hbox{min}_{j\leq i\leq n}\sigma(i)$ and $\sigma(j)<\sigma(1)$,
- $v(\sigma)$ is the number of [*special*]{} left-to-right maxima, [*i.e.*]{} integers $j\in\{1,\dots,n\}$ such that $\sigma(j)=\hbox{max}_{1\leq i\leq j}\sigma(i)$ and $\sigma(j)>\sigma(1)$,
- ${{\rm wex}}(\sigma)$ is the number of [*weak exceedances*]{} of $\sigma$, [*i.e.*]{} integers $j\in\{1,\dots,n\}$ such that $\sigma(j)\geq j$,
- and ${{\rm cr}}(\sigma)$ is the number of [*crossings*]{}, [*i.e.*]{} pairs $(i,j)\in\{1,\dots,n\}^2$ such that either $i<j\leq \sigma(i) <\sigma(j)$ or $\sigma(i)<\sigma(j)<i<j$.
It can already be seen that Stirling numbers and Eulerian numbers appear as special cases of $Z_N$. We will show that it is possible to follow the statistics in through the weighted Motzkin paths called [*Laguerre histories*]{} (see [@Co; @XGV] and Definition \[def\_hist\] below), thanks to the bijection of Foata and Zeilberger [@Co; @FZ; @MV]. But we need to study several subtle properties of the bijection to follow all four statistics. We obtain a combinatorial interpretation of $Z_N$ in terms of [*Laguerre histories*]{}, see Theorem \[histoires\] below. Even more, we will show that the four statistics in Laguerre histories can be followed through the bijection of Françon and Viennot [@Co; @FV]. Consequently we will obtain in Theorem \[main\] below a second new combinatorial interpretation: $$\label{main_int}
Z_N = \sum_{\sigma\in\mathfrak{S}_{N+1}}
\alpha^{-s(\sigma)+1} \beta^{-t(\sigma)+1} y^{{{\rm asc}}(\sigma)-1}q^{\hbox{\scriptsize {{\rm \hbox{31-2}}}}(\sigma)},$$ where we use the statistics in the next definition. This was already known in the case $\alpha=1$, see [@Co; @CN].
\[stat\_per2\] Let $\sigma\in\mathfrak{S}_n$. Then:
- $s(\sigma)$ is the number of right-to-left maxima of $\sigma$ and $t(\sigma)$ is the number of right-to-left minima of $\sigma$,
- ${{\rm asc}}(\sigma)$ is the number of [*ascents*]{}, [*i.e.*]{} integers $i$ such that either $i=n$ or $1\leq i \leq n-1$ and $\sigma(i)<\sigma(i+1)$,
- 31-2$(\sigma)$ is the number of generalized patterns 31-2 in $\sigma$, [*i.e.*]{} triples of integers $(i,i+1,j)$ such that $1\leq i<i+1<j\leq n$ and $\sigma(i+1)<\sigma(j)<\sigma(i)$.
Exact formula for the partition function
----------------------------------------
An exact formula for $Z_N$ was given by Blythe, Evans, Colaiori, Essler [@BECE Equation (57)] in the case where $y=1$. It was obtained from the eigenvalues and eigenvectors of the operator $D+E$ as defined in and below. This method gives an integral form for $Z_N$, which can be simplified so as to obtain a finite sum rather than an integral. Moreover this expression for $Z_N$ was used to obtain various properties of the large system size limit, such as phases diagrams and currents. Here we generalize this result since we also have the variable $y$, and the proofs are combinatorial. This is an important result since it is generally accepted that most interesting properties of a model can be derived from the partition function.
\[Z\_th\] Let ${\tilde\alpha}=(1-q)\frac1\alpha-1$ and ${\tilde\beta}=(1-q)\frac1\beta-1$. We have: $$\label{Z}
Z_N = \frac 1{(1-q)^N} \sum_{n=0}^N R_{N,n}(y,q) B_n( {\tilde\alpha}, {\tilde\beta}, y, q),$$ where $$R_{N,n}(y,q) = \sum_{i=0}^{\lfloor \frac{N-n}2 \rfloor } (-y)^i
q^{\binom{i+1}2} {\genfrac{[}{]}{0pt}{1}{n+i}{i}_q}
\sum_{j=0}^{N-n-2i}y^j\left( \tbinom Nj \tbinom N{n+2i+j}-
\tbinom N{j-1} \tbinom N{n+2i+j+1} \right)$$ and $$B_n({\tilde\alpha},{\tilde\beta},y,q) = \sum_{k=0}^n {\genfrac{[}{]}{0pt}{}{n}{k}_q} {\tilde\alpha}^k (y{\tilde\beta})^{n-k}.$$
In the case where $y=1$, one sum can be simplified by the Vandermonde identity $\sum_j \binom Nj \binom N{m-j}=\binom{2N}m$, and we recover the expression given in [@BECE Equation (54)] by Blythe & al: $$R_{N,n}(1,q) = \sum_{i=0}^{\lfloor \frac{N-n}2 \rfloor } (-1)^i
\left( \tbinom{2N}{N-n-2i} - \tbinom{2N}{N-n-2i-2} \right)
q^{\binom{i+1}2} {\genfrac{[}{]}{0pt}{1}{n+i}{i}_q}.
$$ In the case where $\alpha=\beta=1$, it was known [@CJPR; @MJV] that $(1-q)^{N+1}Z_N$ is equal to: $$\label{cas1}
\sum\limits_{k=0}^{N+1} (-1)^k
\Bigg( \sum\limits_{j=0}^{N+1-k} y^j\Big( \tbinom{N+1}{j}\tbinom{N+1}{j+k} -
\tbinom{N+1}{j-1}\tbinom{N+1}{j+k+1}\Big) \Bigg)
\left( \sum\limits_{i=0}^k y^iq^{i(k+1-i)} \right)$$ (see Remarks \[comp1\] and \[comp2\] for a comparison between this previous result and the new one in Theorem \[Z\_th\]). And in the case where $y=q=1$, from a recursive construction of permutation tableaux [@CN] or lattice paths combinatorics [@BCEPR] it is known that: $$Z_{N} = \prod_{i=0}^{N-1} \left(\frac1\alpha + \frac1\beta + i\right).$$
The first proof of is a purely combinatorial enumeration of some weighted Motzkin paths defined below in , appearing from explicit representations of the operators $D$ and $E$ of the Matrix Ansatz. It partially relies on results of [@CJPR; @MJV] through Proposition \[RNn\] below. In contrast, the second proof does not use a particular representation of the operators $D$ and $E$, but only on the combinatorics of the normal ordering process. It also relies on previous results of [@MJV] (through Proposition \[hats\] below), but we will sketch a self-contained proof.
This article is organized as follows. In Section \[Zfacts\] we recall known facts about the PASEP partition function $Z_N$, mainly to explain the Matrix Ansatz. In Section \[bij\] we prove the two new combinatorial interpretations of $Z_N$, starting from and using various properties of bijections of Foata and Zeilberger, Françon and Viennot. Sections \[paths\] and \[rooks\] respectively contain the the two proofs of the exact formula for $Z_N$ in Equation . In Section \[ALSC\] we show that the first proof of the exact formula for $Z_N$ can be adapted to give a formula for the moments of Al-Salam-Chihara polynomials. Finally in Section \[num\] we review the numerous classical integer sequences which appear as specializations or limit cases of $Z_N$.
Acknowledgement {#acknowledgement .unnumbered}
===============
I thank my advisor Sylvie Corteel for her advice, support, help and kindness. I thank Einar Steingrímsson, Lauren Williams and Jiang Zeng for their help.
Some known properties of the partition function $Z_N$ {#Zfacts}
=====================================================
As said in the introduction, the partition function $Z_N$ can be derived by taking the product ${\ensuremath{\langle W|}}(yD+E)^N{\ensuremath{|V\rangle}}$ provided the relations are satisfied. It may seem non-obvious that ${\ensuremath{\langle W|}}(yD+E)^N{\ensuremath{|V\rangle}}$ does not depend on a particular choice of the operators $D$ and $E$, and the existence of such operators $D$ and $E$ is not clear.
The fact that ${\ensuremath{\langle W|}}(yD+E)^N{\ensuremath{|V\rangle}}$ is well-defined without making $D$ and $E$ explicit, in a consequence of the existence of normal forms. More precisely, via the commutation relation $DE-qED=D+E$ we can derive polynomials $c^{(N)}_{i,j}$ in $y$ and $q$ with non-negative integer coefficients such that we have the [*normal form*]{}: $$(yD+E)^N = \sum_{i,j\geq 0} c^{(N)}_{i,j} E^i D^j$$ (this is a finite sum). See [@BHPSD] for other combinatorial interpretation of normal ordering problems. It turns out that the $c^{(N)}_{i,j}$ are uniquely defined if we require the previous equality to hold for any value of $\alpha$, $\beta$, $y$ and $q$, considered as indeterminates. Then the partition function is: $$Z_N(\alpha,\beta,y,q) = {\ensuremath{\langle W|}}(yD+E)^N{\ensuremath{|V\rangle}} =
\sum_{i,j\geq 0} c^{(N)}_{i,j} \alpha^{-i} \beta^{-j}.$$ Indeed, this expression is valid for any choice of ${\ensuremath{\langle W|}}$, ${\ensuremath{|V\rangle}}$, $D$ and $E$ since we only used the relations to obtain it. In particular $Z_N$ is a polynomial in $y$, $q$, $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ with non-negative coefficients. For convenience we also define: $$\bar Z_N\big(\alpha,\beta,y,q\big)= Z_N\big(\tfrac{1}{\alpha},\tfrac{1}{\beta},y,q\big).$$ For example the first values are: $$\bar Z_0=1, \qquad \bar Z_1=\alpha+y\beta, \qquad
\bar Z_2= \alpha^2 + y(\alpha + \beta + \alpha\beta + \alpha\beta q) + y^2\beta^2,$$ $$\begin{aligned}
\bar Z_3 &=& y^3\beta^3 + \left( \alpha\beta^2q+\alpha\beta^2 + \alpha + \alpha\beta +
\alpha\beta^2q ^2+\beta+\beta^2 q
+ 2\,a\beta q + 2\beta^{2} \right) y^2 \\
& & + \left( 2\alpha^2+\alpha^2q+\alpha+\beta\alpha^2q^{2} + \beta\alpha^2 + \beta\alpha^2q +
\alpha \beta + \beta + 2 \alpha \beta q \right) y+\alpha^3.\end{aligned}$$
Even if it is not needed to compute the first values of $Z_N$, it is useful to have explicit matrices $D$ and $E$ satisfying . The best we could hope is finite-dimensional matrices with non-negative entries, however this is known to be incompatible with the existence of phase transitions in the model (see section 2.3.3 in [@BE]). Let ${\tilde\alpha}=(1-q)\frac{1}{\alpha}-1$ and ${\tilde\beta}=(1-q)\frac{1}{\beta}-1$, a solution of the Matrix Ansatz is given by the following matrices $D=(D_{i,j})_{i,j\in\mathbb{N}}$ and $E=(E_{i,j})_{i,j\in\mathbb{N}}$ (see [@DEHP]): $$\label{defD}
(1-q)D_{i,i}= 1 + {\tilde\beta}q^i, \qquad (1-q)D_{i,i+1}= 1-{\tilde\alpha}{\tilde\beta}q^i,$$ $$\label{defE}
(1-q)E_{i,i}= 1 + {\tilde\alpha}q^i, \qquad (1-q)E_{i+1,i}= 1- q^{i+1},$$ all other coefficients being 0, and vectors: $${\ensuremath{\langle W|}}=(1,0,0,\dots), \qquad {\ensuremath{|V\rangle}} = (1,0,0,\dots)^*,$$ ([*i.e.*]{} ${\ensuremath{|V\rangle}}$ is the transpose of ${\ensuremath{\langle W|}}$). Even if infinite-dimensional, they have the nice property of being tridiagonal and this lead to a combinatorial interpretation of $Z_N$ in terms of lattice paths [@BCEPR]. Indeed, we can see $yD+E$ as a transfer matrix for walks in the non-negative integers, and obtain that $(1-q)^N Z_N$ is the sum of weights of Motzkin paths of length $N$ with weights: $$\label{Z_w}
\parbox{14.5cm}{
\begin{itemize}
\item $1-q^{h+1}$ for a step $\nearrow$ starting at height $h$,
\item $(1+y)+({\tilde\alpha}+y{\tilde\beta})q^h$ for a step $\rightarrow$ starting at height $h$,
\item $y(1-{\tilde\alpha}{\tilde\beta}q^{h-1})$ for a step $\searrow$ starting at height $h$.
\end{itemize}
}$$ We recall that a Motzkin path is similar to a Dyck path except that there may be horizontal steps, see Figures \[ex\_FZ\], \[fig\], \[ex\_FV\], \[decomp\_ex\] further. These weighted Motzkin paths are our starting point to prove Theorem \[Z\_th\] in Section \[paths\].
We have sketched how the Motzkin paths appear as a combinatorial interpretation of $Z_N$ starting from the Matrix Ansatz. However it is also possible to obtain a direct link between the PASEP and the lattice paths, independently of the results of Derrida & al. This was done by Brak & al in [@BCEPR], in the even more general context of the PASEP with five parameters.
Combinatorial interpretations of $Z_N$ {#bij}
======================================
In this section we prove the two new combinatorial interpretation of $Z_N$. Firstly we prove the one in terms of Laguerre histories (Theorem \[histoires\] below), by means of a bijection originally given by Foata and Zeilberger. Secondly we prove the one in terms in permutations (Theorem \[main\] below).
Permutation tableaux and Laguerre histories
-------------------------------------------
We recall here the definition of permutation tableaux and their statistics needed to state the previously known combinatorial interpretation .
\[def\_PT\] Let $\lambda$ be a Young diagram (in English notation), possibly with empty rows but with no empty column. A complete filling of $\lambda$ with 0’s and 1’s is a [*permutation tableau*]{} if:
- for any cell containing a 0, all cells above in the same column contain a 0, or all cells to the left in the same row contain a 0,
- there is at least a 1 in each column.
A cell containing a 0 is [*restricted*]{} if there is a 1 above. A row is [*restricted*]{} if it contains a restricted 0, and [*unrestricted*]{} otherwise. A cell containing a 1 is [*essential*]{} if it is the topmost 1 of its column, otherwise it is [*superfluous*]{}. The [*size*]{} of such a permutation tableaux is the number of rows of $\lambda$ plus its number of columns.
To prove our new combinatorial interpretations, we will give bijections linking the previously-known combinatorial interpretation , and the new ones. The main combinatorial object we use are the Laguerre histories, defined below.
\[def\_hist\] A [*Laguerre history*]{} of size $n$ is a weighted Motzkin path of $n$ steps such that:
- the weight of a step $\nearrow$ starting at height $h$ is $yq^i$ for some $i\in\left\{0,\dots,h\right\}$,
- the weight of a step $\rightarrow$ starting at height $h$ is either $yq^i$ for some $i\in\left\{0,\dots,h\right\}$ or $q^i$ for some $i\in\left\{0,\dots,h-1\right\}$,
- the weight of a step $\searrow$ starting at height $h$ is $q^i$ for some $i\in\left\{0,\dots,h-1\right\}$.
The [*total weight*]{} of the Laguerre history is the product of the weights of its steps. We call a [*type 1 step*]{}, any step having weight $yq^h$ where $h$ is its starting height. We call a [*type 2 step*]{}, any step having weight $q^{h-1}$ where $h$ is its starting height.
As shown by P. Flajolet [@Fla], the weighted Motzkin paths appear in various combinatorial contexts in connexion with some continued fractions called J-fractions. We also recall an important fact from combinatorial theory of orthogonal polynomials.
\[mu\_ortho\] If an orthogonal sequence $\{P_n\}_{n\in\mathbb{N}}$ is defined by the three-term recurrence relation $$xP_n(x) = P_{n+1}(x) + b_n P_n(x) + \lambda_n P_{n-1}(x),$$ then the moment generating function has the J-fraction representation $$\sum_{n=0}^\infty \mu_n t^n = \cfrac{1}{1 - b_0t -
\cfrac{\lambda_1 t^2}{1 - b_1t - \cfrac{\lambda_2t^2}{\ddots
}}},$$ equivalently the $n$th moment $\mu_n$ is the sum of weights of Motzkin paths of length $n$ where the weight of a step $\nearrow$ (respectively $\rightarrow$, $\searrow$) starting at height $h$ is $a_h$ (respectively $b_h$, $c_h$) provided $\lambda_n=a_{n-1}c_n$.
\[rem\_poly\] The sum of weights of Laguerre histories of length $n$ is the $n$th moment of some $q$-Laguerre polynomials (see [@KSZ08]), which are a special case of rescaled Al-Salam-Chihara polynomials. On the other hand $Z_N$ is the $N$th moment of shifted Al-Salam-Chihara polynomials (see Section \[ALSC\]). We will use the Laguerre histories to derive properties of $Z_N$, however they are related with two different orthogonal sequences.
The Foata-Zeilberger bijection
------------------------------
Foata and Zeilberger gave a bijection between permutations and Laguerre histories in [@FZ]. It has been extended by de Médicis and Viennot [@MV], and Corteel [@Co]. In particular, Corteel showed that through this bijection $\Psi_{FZ}$ we can follow the number weak exceedances and crossings [@Co]. The bijection $\Psi_{FZ}$ links permutations in $\mathfrak{S}_n$ and Laguerre histories of $n$ steps. The $i$th step of $\Psi_{FZ}(\sigma)$ is:
- a step $\nearrow$ if $i$ is a [*cycle valley*]{}, [*i.e.*]{} $\sigma^{-1}(i) > i < \sigma(i)$,
- a step $\searrow$ if $i$ is a [*cycle peak*]{}, [*i.e.*]{} $\sigma^{-1}(i) < i > \sigma(i)$,
- a step $\rightarrow$ in all other cases.
And the weight of the $i$th step in $\Psi_{FZ}(\sigma)$ is $y^\delta q^j$ with:
- $\delta=1$ if $i\leq\sigma(i)$ and 0 otherwise,
- $j=\#\{\; k \mid k < i \leq \sigma(k) < \sigma(i) \;\} $ if $i \leq \sigma(i)$,
- $j=\#\{\; k \mid \sigma(i) < \sigma(k) < i < k \; \} $ if $\sigma(i) < i$.
It follows that the total weight of $\Psi_{FZ}(\sigma)$ is $y^{{{\rm wex}}(\sigma)} q^{{{\rm cr}}(\sigma)}$. To see the statistics ${{\rm wex}}$ and ${{\rm cr}}$ in a permutation $\sigma$, it is practical to represent $\sigma$ by an arrow diagram. We draw $n$ points in a line, and draw an arrow from the $i$th point to the $\sigma(i)$th point for any $i$. This arrow is above the axis if $i\leq\sigma(i)$, below the axis otherwise. Then ${{\rm wex}}(\sigma)$ is the number of arrows above the axis, and ${{\rm cr}}(\sigma)$ is the number of proper intersection between arrows plus the number of chained arrows going to the right. See Figure \[ex\_FZ\] for an example with $\sigma=672581493$, so that ${{\rm wex}}(\sigma)=5$ and ${{\rm cr}}(\sigma)=7$.
(1,-2)(7,1) (1,0)(2,0)(3,0)(4,0)(5,0)(6,0)(7,0)(8,0)(9,0) (1,0)(3.5,1.6)(6,0) (2,0)(4.5,1.6)(7,0) (3,0)(2.5,-0.4)(2,0) (4,0)(4.5,0.5)(5,0) (5,0)(6.5,1)(8,0) (6,0)(3.5,-1.6)(1,0) (7,0)(5.5,-0.8)(4,0) (8,0)(8.5,0.4)(9,0) (9,0)(6,-1.6)(3,0)
(0,-0.5)(9,4) (0,0)(9,4) (0,0)(1,1)(2,2)(3,2)(4,3)(5,3)(6,2)(7,1)(8,1)(9,0) (0.5,1.1)[$yq^0$]{} (1.5,2.1)[$yq^1$]{} (2.5,2.6)[$q^0$]{} (3.5,3)[$yq^0$]{} (4.5,3.6)[$yq^3$]{} (5.7,3.2)[$q^2$]{} (6.7,2.2)[$q^0$]{} (7.7,1.6)[$yq^1$]{} (8.7,1.2)[$q^0$]{}
\[FZ\_lrmax\] Let $\sigma\in\mathfrak{S}_n$, and $1\leq i\leq n$. Then $i$ is a left-to-right maximum of $\sigma$ if and only if the $i$th step of $\Psi_{FZ}(\sigma)$ is a type 1 step (as in Definition \[def\_hist\]).
Let us call a $(\sigma,i)$-sequence a strictly increasing maximal sequence of integers $u_1,\dots,u_j$ such that $\sigma(u_k)=u_{k+1}$ for any $1\leq k\leq j-1$, and also such that $u_1 < i < u_j$. By maximality of the sequence, $u_1$ is a cycle valley and $u_j$ is a cycle peak. The number of such sequences is the difference between the number of cycle valleys and cycle peaks among $\{1,\dots,i-1\}$, so it is the starting height $h$ of the $i$th step in $\Psi_{FZ}(\sigma)$.
Any left-to-right maximum is a weak exceedance, so $i$ is a left-to-right maxima of $\sigma$ if and only if $i\leq\sigma(i)$ and there exists no $j$ such that $j < i \leq \sigma(i) < \sigma(j)$. This is also equivalent to the fact that $i \leq \sigma(i)$, and there exists no two consecutive elements $u_k$, $u_{k+1}$ of a $(\sigma,i)$-sequence such that $u_k < i \leq \sigma(i) < u_{k+1}$. This is also equivalent to the fact that $i\leq \sigma(i)$, and any $(\sigma,i)$-sequence contains two consecutive elements $u_k$, $u_{k+1}$ such that $u_k < i \leq u_{k+1} < \sigma(i)$.
By definition of the bijection $\Psi_{FZ}$ it is equivalent to the fact that the $i$th step of $\Psi_{FZ}(\sigma)$ has weight $yq^h$, [*i.e.*]{} the $i$th step is a type 1 step.
\[FZ\_rlmin\] Let $\sigma\in\mathfrak{S}_n$, and $1\leq i\leq n$. We suppose $i\neq\sigma(i)$. Then $i$ is a right-to-left minima of $\sigma$ if and only if the $i$th step of $\Psi_{FZ}(\sigma)$ is a type 2 step.
We have to pay attention to the fact that a right-to-left minimum can be a fixed point and we only characterize the non-fixed points here. This excepted, the proof is similar to the one of the previous lemma.
Before we can use the bijection $\Psi_{FZ}$ we need a slight modification of the known combinatorial interpretation , given in the following lemma.
\[vprim\] We have: $$\label{depart2}
\bar Z_N = \sum_{\sigma\in\mathfrak{S}_{N+1}}
\alpha^{ u'(\sigma) } \beta^{ v(\sigma) } y^{{{\rm wex}}(\sigma)-1} q^{{{\rm cr}}(\sigma)},$$ where $u'(\sigma)$ is the number of right-to-left minima $i$ of $\sigma$ satisfying $\sigma^{-1}(N+1) < i$.
This just means that in we can replace the statistic $u$ with $u'$, and this can be done via a simple bijection. For any $\sigma\in\mathfrak{S}_{N+1}$, let $\tilde\sigma$ be the reverse complement of $\sigma^{-1}$, [*i.e.*]{} $\sigma(i)=j$ if and only if $\tilde\sigma(N+2-j)=N+2-i$. It is routine to check that $u(\sigma)=u'(\tilde\sigma)$, ${{\rm wex}}(\sigma)={{\rm wex}}(\tilde\sigma)$, and $v(\sigma)=v(\tilde\sigma)$. Moreover, one can check that the arrow diagram of $\tilde\sigma$ is obtained from the one of $\sigma$ by a vertical symmetry and arrow reversal, so that ${{\rm cr}}(\sigma)={{\rm cr}}(\tilde\sigma)$. So and the bijection $\sigma\mapsto\tilde\sigma$ prove .
From Lemmas \[FZ\_lrmax\], \[FZ\_rlmin\], and \[vprim\] it possible to give a combinatorial interpretation of $\bar Z_N$ in terms of the Laguerre histories. We start from the statistics in $\mathfrak{S}_{N+1}$ described in Definition \[stat\_per\], then from and the properties of $\Psi_{FZ}$ we obtain the following theorem.
\[histoires\] The polynomial $y\bar Z_N$ is the generating function of Laguerre histories of $N+1$ steps, where:
- the parameters $y$ and $q$ are given by the total weight of the path,
- $\beta$ counts the type 1 steps, except the first one,
- $\alpha$ counts the type 2 steps which are to the right of any type 1 step.
Let $\sigma\in\mathfrak{S}_{N+1}$. The smallest left-to-right maximum of $\sigma$ is 1, and any other left-to-right maximum $i$ is such that $\sigma(1)<\sigma(i)$. So $1$ is the only left-to-right maximum which is not special. So by Lemma \[FZ\_lrmax\], $v(\sigma)$ is the number of type 1 steps in $\Psi_{FZ}(\sigma)$, minus 1.
Moreover, $\sigma^{-1}(N+1)$ is the largest left-to-right maximum of $\sigma$. Let $i$ be a right-to-left minimum of $\sigma$ such that $\sigma^{-1}(N+1)<i$. We have $i\neq\sigma(i)$, otherwise $\sigma$ would stabilize the interval $\{i+1,\dots,N+1\}$ and this would contradict $\sigma^{-1}(N+1)<i$. So we can apply Lemma \[FZ\_rlmin\], and it comes that $u'(\sigma)$ is the number of type 2 steps in $\Psi_{FZ}(\sigma)$, which are to the right of any type 1 step. So and the bijection $\Psi_{FZ}$ prove the theorem.
Before ending this subsection, we sketch how to recover a known result in the case $q=0$ from Theorem \[histoires\]. This was given in Section 3.2 of [@BGR] (see also Section 3.6 in [@BE]) and proved via generating functions. For any Dyck path $D$, let ${{\rm ret}}(D)$ be the number of returns to height 0, for example ${{\rm ret}}(\nearrow\searrow)=1$ and ${{\rm ret}}(\nearrow\searrow\nearrow\searrow)=2$, and the empty path $\cdot$ satisfies ${{\rm ret}}(\cdot)=0$. The result is the following.
When $y=1$ and $q=0$, the partition function is $Z_N=\sum (\frac1\beta)^{{{\rm ret}}(D_1)}(\frac1\alpha)^{{{\rm ret}}(D_2)}$ where the sum is over pairs of Dyck paths $(D_1,D_2)$ whose lengths sum to $2N$.
When $q=0$ we can remove any step with weight 0 in the Laguerre histories. When $y=1$, to distinguish the two kinds of horizontal steps we introduce another kind of paths. Let us call a [*bicolor*]{} Motzkin path, a Motzkin path with two kinds of horizontal steps ${\psset{unit=4.25mm}\begin{pspicture}(0.2,0)(1,0)
\psline[linestyle=dotted,dotsep=0.5mm,arrowsize=1.3mm,arrowlength=0.7,arrowinset=0.6]{->}(0.2,0.2)(1,0.2)
\end{pspicture}}$ and $\rightarrow$, and such that there is no ${\psset{unit=4.25mm}\begin{pspicture}(0.2,0)(1,0)
\psline[linestyle=dotted,dotsep=0.5mm,arrowsize=1.3mm,arrowlength=0.7,arrowinset=0.6]{->}(0.2,0.2)(1,0.2)
\end{pspicture}}$ at height 0. From Theorem \[histoires\], if $y=1$ and $q=0$ then $\beta\bar Z_N$ is the generating function of bicolor Motzkin paths $M$ of length $N+1$, where:
- there is a weight $\beta$ on each step $\nearrow$ or $\rightarrow$ starting at height 0,
- there is a weight $\alpha$ on each step $\searrow$ or ${\psset{unit=4.25mm}\begin{pspicture}(0.2,0)(1,0)
\psline[linestyle=dotted,dotsep=0.5mm,arrowsize=1.3mm,arrowlength=0.7,arrowinset=0.6]{->}(0.2,0.2)(1,0.2)
\end{pspicture}}$ starting at height 1 and being to the right of any step with a weight $\beta$.
There is a bijection between these bicolor Motzkin paths, and Dyck paths of length $2N+2$ (see de Médicis and Viennot [@MV]). To obtain the Dyck path $D$, each step $\nearrow$ in the bicolor Motzkin path $M$ is replaced with a sequence of two steps $\nearrow\nearrow$. Similarly, each step $\rightarrow$ is replaced with $\nearrow\searrow$, each step ${\psset{unit=4.25mm}\begin{pspicture}(0.2,0)(1,0)
\psline[linestyle=dotted,dotsep=0.5mm,arrowsize=1.3mm,arrowlength=0.7,arrowinset=0.6]{->}(0.2,0.2)(1,0.2)
\end{pspicture}}$ is replaced with $\searrow\nearrow$, each step $\searrow$ is replaced with $\searrow\searrow$. When some step $s\in\{\nearrow,\rightarrow,\searrow\}$ in $M$ has a weight $\beta$ or $\alpha$, and is transformed into steps $(s_1,s_2)\in\{\nearrow,\rightarrow,\searrow\}^2$ in $D$, we choose to put the weight $\beta$ or $\alpha$ on $s_1$. It appears that $D$ is a Dyck path of length $2N+2$ such that:
- there is a weight $\beta$ on each step $\nearrow$ starting at height 0,
- there is a weight $\alpha$ on each step $\searrow$ starting at height 2 and being to the right of any step with weight $\beta$.
Then $D$ can be factorized into $D_1\nearrow D_2\searrow$ where $D_1$ and $D_2$ are Dyck paths whose lengths sum to $2N$, and up to a factor $\beta$ it can be seen that $\beta$ (respectively $\alpha$) counts the returns to height 0 in $D_1$ (respectively $D_2$). More precisely the $\beta$s are on the steps $\nearrow$ starting at height 0 but there are as many of them as the number of returns to height 0. See Figure \[dycks\] for a an example.
(-1.6,0)(10,4) (0,0)(10,4) (0,0)(1,1) (1,1)(2,1) (2,1)(3,0)(4,0)(5,1)(6,1)(7,2)(8,1) (8,1)(9,1) (9,1)(10,0) (-1.7,1.6)[$M=$]{} (0.5,1.5)[$\beta$]{}(3.5,0.5)[$\beta$]{}(4.5,1.5)[$\beta$]{} (8.5,1.5)[$\alpha$]{}(9.5,1.5)[$\alpha$]{}
(20,5) (0,0)(20,5) (0,0)(1,1)(2,2)(3,1)(4,2)(5,1)(6,0)(7,1)(8,0)(9,1)(10,2)(11,3)(12,2)(13,3)(14,4)(15,3)(16,2)(17,1)(18,2)(19,1)(20,0) (-1.7,1.6)[$D=$]{} (0.5,1.5)[$\beta$]{}(6.5,1.5)[$\beta$]{}(8.5,1.5)[$\beta$]{} (16.5,2.5)[$\alpha$]{}(18.5,2.5)[$\alpha$]{}
\
(8,5) (0,0)(8,4) (0,0)(1,1)(2,2)(3,1)(4,2)(5,1)(6,0)(7,1)(8,0) (-1.8,1.6)[$D_1=$]{} (0.5,1.5)[$\beta$]{}(6.5,1.5)[$\beta$]{}
(10,5) (0,0)(10,4) (0,0)(1,1)(2,2)(3,1)(4,2)(5,3)(6,2)(7,1)(8,0)(9,1)(10,0) (-1.8,1.6)[$D_2=$]{} (7.5,1.5)[$\alpha$]{}(9.5,1.5)[$\alpha$]{}
The Françon-Viennot bijection
-----------------------------
This bijection was given in [@FV]. We use here the definition of this bijection given in [@Co]. The map $\Psi_{FV}$ is a bijection between permutations of size $n$ and Laguerre histories of $n$ steps. Let $\sigma\in\mathfrak{S}_n$, $j\in\{1,\dots,n\}$ and $k=\sigma(j)$. Then the $k$th step of $\Psi_{FV}(\sigma)$ is:
- a step $\nearrow$ if $k$ is a [*valley*]{}, [*i.e.*]{} $\sigma(j-1)>\sigma(j)<\sigma(j+1)$,
- a step $\searrow$ if $k$ is a [*peak*]{}, [*i.e.*]{} $\sigma(j-1)<\sigma(j)>\sigma(j+1)$,
- a step $\rightarrow$ if $k$ is a [*double ascent*]{}, [*i.e.*]{} $\sigma(j-1)<\sigma(j)<\sigma(j+1)$, or a [*double descent*]{}, [*i.e.*]{} $\sigma(j-1)>\sigma(j)>\sigma(j+1)$.
This is done with the convention that $\sigma(n+1)=n+1$, in particular $n$ is always an ascent of $\sigma\in\mathfrak{S}_n$. Moreover the weight of the $k$th step is $y^\delta q^i$ where $\delta=1$ if $j$ is an ascent and $0$ otherwise, and $i=$ 31-2$(\sigma,j)$. This number 31-2$(\sigma,j)$ is the number of patterns 31-2 such that $j$ correspond to the 2, [*i.e.*]{} integers $i$ such that $1<i+1<j$ and $\sigma(i+1)<\sigma(j)<\sigma(i)$. A consequence of the definition is that the total weight of $\Psi_{FV}(\sigma)$ is $y^{{{\rm asc}}(\sigma)}q^{\hbox{\scriptsize {{\rm \hbox{31-2}}}}(\sigma)}$. See Figures \[fig\] and \[ex\_FV\] for examples.
(0,0)(7,7) (0.3,1)[1]{} (0.3,2)[2]{} (0.3,3)[3]{} (0.3,4)[4]{} (0.3,5)[5]{} (0.3,6)[6]{} (0.3,7)[7]{} (1,0.2)[1]{} (2,0.2)[2]{} (3,0.2)[3]{} (4,0.2)[4]{} (5,0.2)[5]{} (6,0.2)[6]{} (7,0.2)[7]{} (1,1)(1,7)(1,1)(7,1) (2,1)(2,7)(1,2)(7,2) (3,1)(3,7)(1,3)(7,3) (4,1)(4,7)(1,4)(7,4) (5,1)(5,7)(1,5)(7,5) (6,1)(6,7)(1,6)(7,6) (7,1)(7,7)(1,7)(7,7) (1,4)(2,3)(3,7)(4,1)(5,2)(6,6)(7,5)
(0,-0.5)(7,3) (0,0)(7,3) (0,0)(1,1)(2,1)(3,2)(4,1)(5,2)(6,1)(7,0) (0.2,1.0)[$yq^0$]{} (1.3,1.5)[$yq^1$]{} (2.4,2.1)[$yq^0$]{} (3.5,2.2)[$q^0$]{} (4.5,2.2)[$yq^1$]{} (5.8,1.9)[$q^1$]{} (6.7,1)[$q^0$]{}
\[FV\_rlmin\] Let $\sigma\in\mathfrak{S}_n$ and $1\leq i\leq n$. Then $\sigma^{-1}(i)$ is a right-to-left minimum of $\sigma$ if and only if the $i$th step of $\Psi_{FV}(\sigma)$ is a type 1 step.
This could be done by combining the arguments of [@FV] and [@Co]. We sketch a proof introducing ideas that will be helpful for the next lemma.
We suppose that $j=\sigma^{-1}(i)$ is a right-to-left minimum. So $j$ is an ascent, and any $v$ such that $i>\sigma(v)$ is such that $v<j$. The integer 31-2$(\sigma,j)$ is the number of maximal sequence of consecutive integers $u,u+1,\dots,v$ such that $\sigma(u)>\sigma(u+1)>\dots>\sigma(v)$, and $\sigma(u) > i > \sigma(v)$. Indeed, any of these sequences $u,\dots,v$ is such that $v<j$ and so it is possible to find two consecutive elements $k,k+1$ in the sequence such that $\sigma(k+1)<\sigma(j)<\sigma(k)$, and these $k,k+1$ only belong to one sequence.
We call a $(\sigma,i)$-sequence a maximal sequence of consecutive integers $u,u+1,\dots,v$ such that $\sigma(u)>\sigma(u+1)>\dots>\sigma(v)$, and $\sigma(u)\geq i > \sigma(v)$. By maximality, $u$ is a peak and $v$ is a valley. The number of such sequences is the difference between the number of peaks and number of valleys among the elements of image smaller than $i$, so it is the starting height $h$ of the $i$th step in $\Psi_{FV}(\sigma)$.
So with this definition, we can check that $j=\sigma^{-1}(i)$ is a right-to-left minimum of $\sigma$ if and only if $j$ is an ascent and any $(\sigma,i)$-sequence $u,u+1,\dots,v$ is such that $v<j$. So this is equivalent to the fact that the $i$th step of $\Psi_{FV}(\sigma)$ is a type 1 step.
\[FV\_rlmax\] Let $\sigma\in\mathfrak{S}_n$, and $1\leq i\leq n$. We suppose $\sigma^{-1}(i)<n$. Then $\sigma^{-1}(i)$ is a right-to-left maximum of $\sigma$ if and only if
- the $i$th step of $\Psi_{FV}(\sigma)$ it is a type 2 step,
- any type 1 step is to the left of the $i$th step.
We keep the definition of $(\sigma,i)$-sequence as in the previous lemma. First we suppose that $\sigma^{-1}(i)$ is a right-to-left maximum strictly smaller than $n$, and we check that the two points are satisfied. If $\sigma^{-1}(j)$ is a right-to-left minimum, then $i>j$, so the second point is satisfied. A right-to-left maximum is a descent, so the $i$th step is $\rightarrow$ or $\searrow$ with weight $q^g$. We have to show $g=h-1$. Since $\sigma^{-1}(i)$ is a right-to-left maximum, there is no $(\sigma,i)$-sequence $u<\dots<v$ with $\sigma^{-1}(i)<u$. So there is one $(\sigma,i)$-sequence $u<\dots<v$ such that $u\leq \sigma^{-1}(i)<v$, and the $h-1$ other ones contains only integers strictly smaller than $\sigma^{-1}(i)$. So the $i$th step of $\Psi_{FV}(\sigma)$ has weight $q^{h-1}$.
Reciprocally, we suppose that the two points above are satisfied. There are $h-1$ $(\sigma,i)$-sequence containing integers strictly smaller than $\sigma^{-1}(i)$. Since $\sigma^{-1}(i)$ is a descent, the $h$th $(\sigma,i)$-sequence $u<\dots<v$ is such that $u\leq \sigma^{-1}(i)<v$. So there is no $(\sigma,i)$-sequence $u<\dots<v$ such that $\sigma^{-1}(i)<u$.
If we suppose that $i$ is not a right-to-left maximum, there would exist $k>i$ such that $\sigma^{-1}(k)>\sigma^{-1}(i)$. We take the minimal $k$ satisfying this property. Then the images of $\sigma^{-1}(k)+1,\dots,n$ are strictly greater than $k$, otherwise there would exist $\ell>\sigma^{-1}(k)$ such that $\sigma(\ell)>i>\sigma(\ell+1)$. But then $\sigma^{-1}(k)$ would be a right-to-left minimum and this would contradict the second point that we assumed to be satisfied.
In Theorem \[histoires\] we have seen that $\bar Z_N$ is a generating function of Laguerre histories, and the bijection $\Psi_{FV}$ together with the two lemmas above give our second new combinatorial interpretation of $\bar Z_N$.
\[main\] We have: $$\bar Z_N = \sum_{\sigma\in\mathfrak{S}_{N+1}}
\alpha^{s(\sigma)-1} \beta^{t(\sigma)-1} y^{{{\rm asc}}(\sigma)-1}q^{\hbox{\scriptsize {{\rm \hbox{31-2}}}}(\sigma)},$$ where we use the statistics in Definition \[stat\_per2\] above.
For example, in Figure \[ex\_FV\] we have a permutation $\sigma$ such that $$\alpha^{s(\sigma)-1} \beta^{t(\sigma)-1} y^{{{\rm asc}}(\sigma)-1}q^{\hbox{\scriptsize {{\rm \hbox{31-2}}}}(\sigma)}=
\alpha^2\beta^3y^5q^7.$$ Indeed $\Psi_{FV}(H)$ has total weight $y^5q^7$, has four type 1 steps and two type 2 steps to the right of the type 1 steps.
(0,0)(9,9) (0.3,1)[1]{} (0.3,2)[2]{}(0.3,3)[3]{}(0.3,4)[4]{}(0.3,5)[5]{}(0.3,6)[6]{} (0.3,7)[7]{}(0.3,8)[8]{}(0.3,9)[9]{}(1,0.2)[1]{}(2,0.2)[2]{} (3,0.2)[3]{}(4,0.2)[4]{}(5,0.2)[5]{}(6,0.2)[6]{}(7,0.2)[7]{} (8,0.2)[8]{}(9,0.2)[9]{} (1,1)(1,9)(1,1)(9,1) (2,1)(2,9)(1,2)(9,2) (3,1)(3,9)(1,3)(9,3) (4,1)(4,9)(1,4)(9,4) (5,1)(5,9)(1,5)(9,5) (6,1)(6,9)(1,6)(9,6) (7,1)(7,9)(1,7)(9,7) (8,1)(8,9)(1,8)(9,8) (9,1)(9,9)(1,9)(9,9) (1,8)(2,1)(3,2)(4,5)(5,6)(6,3)(7,9)(8,7)(9,4)
(0,-1)(9,4) (0,0)(9,4) (0,0)(1,1)(2,1)(3,2)(4,3)(5,3)(6,2)(7,2)(8,1)(9,0) (0.5,1.2)[$yq^0$]{} (1.5,1.6)[$yq^1$]{} (2.5,2.2)[$yq^1$]{} (3.5,3.2)[$yq^2$]{} (4.6,3.6)[$yq^1$]{} (5.5,3.2)[$q^1$]{} (6.5,2.6)[$q^1$]{} (7.7,2.1)[$q^0$]{} (8.7,1.1)[$q^0$]{}
We have mentioned in the introduction that the non-normalized probability of a particular state of the PASEP is a product ${\ensuremath{\langle W|}}t_1\dots t_N{\ensuremath{|V\rangle}}$. It is known [@CW1] that in the combinatorial interpretation , this state of the PASEP corresponds to permutation tableaux of a given shape. It is also known [@CW1] that in the combinatorial interpretation , this state of the PASEP corresponds to permutations with a given set of weak exceedances (namely, $i+1$ is a weak exceedance if and only if $t_i=D$, [*i.e.*]{} the $i$th site is occuppied). It is also possible to give such criterions for the new combinatorial interpretations of Theorems \[histoires\] and \[main\], by following the weak exceedances set through the bijections we have used. More precisely, in the first case the term ${\ensuremath{\langle W|}}t_1\dots t_N{\ensuremath{|V\rangle}}$ is the generating function of Laguerre histories $H$ such that $t_i=D$ if and only if the $(N+1-i)$th step in $H$ is either a step $\rightarrow$ with weight $yq^i$ or a step $\searrow$. In the second case, the term ${\ensuremath{\langle W|}}t_1\dots t_N{\ensuremath{|V\rangle}}$ is the generating function of permutations $\sigma$ such that $t_i=D$ if and only if $\sigma^{-1}(N+1-i)$ is a double ascent or a peak.
A first combinatorial derivation of $Z_N$ using lattice paths {#paths}
=============================================================
In this section, we give the first proof of Theorem \[Z\_th\].
We consider the set $\mathfrak{P}_N$ of weighted Motzkin paths of length $N$ such that:
- the weight of a step $\nearrow$ starting at height $h$ is $q^i-q^{i+1}$ for some $i\in\{0,\dots,h\}$,
- the weight of a step $\rightarrow$ starting at height $h$ is either $1+y$ or $({\tilde\alpha}+y{\tilde\beta})q^h$,
- the weight of a step $\searrow$ starting at height $h$ is either $y$ or $-y{\tilde\alpha}{\tilde\beta}q^{h-1}$.
The sum of weights of elements in $\mathfrak{P}_N$ is $(1-q)^N Z_N$ because the weights sum to the ones in . We stress that on the combinatorial point of view, it will be important to distinguish $(h+1)$ kinds of step $\nearrow$ starting at height $h$, instead of one kind of step $\nearrow$ with weight $1-q^{h+1}$.
We will show that each element of $\mathfrak{P}_N$ bijectively corresponds to a pair of weighted Motzkin paths. The first path (respectively, second path) belongs to a set whose generating function is $R_{N,n}(y,q)$ (respectively, $B_n({\tilde\alpha},{\tilde\beta},y,q)$) for some $n\in\{0,\dots,N\}$. Following this scheme, our first combinatorial proof of is a consequence of Propositions \[RNn\], \[Bn\], and \[decomp\] below.
The lattice paths for $R_{N,n}(y,q)$
------------------------------------
Let $\mathfrak{R}_{N,n}$ be the set of weighted Motzkin paths of length $N$ such that:
- the weight of a step $\nearrow$ starting at height $h$ is either $1$ or $-q^{h+1}$,
- the weight of a step $\rightarrow$ starting at height $h$ is either $1+y$ or $q^h$,
- the weight of a step $\searrow$ is $y$,
- there are exactly $n$ steps $\rightarrow$ weighted by a power of $q$.
In this subsection we prove the following:
\[RNn\] The sum of weights of elements in $\mathfrak{R}_{N,n}$ is $R_{N,n}(y,q)$.
This can be obtained with the methods used in [@CJPR; @MJV], and the result is a consequence of the Lemmas \[decomp2\], \[motzbi\] and \[core\] below.
\[decomp2\] There is a weight-preserving bijection between $\mathfrak{R}_{N,n}$, and the pairs $(P,C)$ such that for some $i\in\{0,\dots,\lfloor\frac{N-n}2\rfloor\}$,
- $P$ is a Motzkin prefix of length $N$ and final height $n+2i$, with a weight $1+y$ on every step $\rightarrow$, and a weight $y$ on every step $\searrow$,
- $C$ is a Motzkin path of length $n+2i$, such that $$\label{def_core}
\parbox{13cm}{
\begin{itemize}
\item the weight of a step $\nearrow$ starting at height $h$ is $1$ or $-q^h$,
\item the weight of a step $\rightarrow$ starting at height $h$ is $q^h$,
\item the weight of a step $\searrow$ is $1$,
\item there are exactly $n$ steps $\rightarrow$, and no steps $\nearrow\searrow$ both with weights 1.
\end{itemize}}$$
This is a direct adaptation of [@CJPR Lemma 1].
\[motzbi\] The generating function of Motzkin prefixes of length $N$ and final height $n+2i$, with a weight $1+y$ on every step $\rightarrow$, and a weight $y$ on every step $\searrow$, is $$\sum_{j=0}^{N-n-2i}y^j\left( \tbinom Nj \tbinom N{n+2i+j}-
\tbinom N{j-1} \tbinom N{n+2i+j+1} \right).$$
This was given in [@CJPR Proposition 4].
\[core\] The sum of weights of Motzkin paths of length $n+2i$ satisfying properties above is $(-1)^iq^{\binom{i+1}2}{\genfrac{[}{]}{0pt}{}{n+i}{i}_q}$.
A bijective proof was given in [@MJV Lemmas 3, 4].
Some precisions are in order. In [@CJPR] and [@MJV], we obtained the formula which is the special case $\alpha=\beta=1$ in $Z_N$, and is the $N$th moment of the $q$-Laguerre polynomials mentioned in Remark \[rem\_poly\]. Since $Z_N$ is also very closely related with these polynomials (see Section \[ALSC\]) it is not surprising that some steps are in common between these previous results and the present ones. See also Remark \[comp1\] below.
The lattice paths for $B_n({\tilde\alpha},{\tilde\beta},y,q)$
-------------------------------------------------------------
Let $\mathfrak{B}_n$ be the set of weighted Motzkin paths of length $n$ such that:
- the weight of a step $\nearrow$ starting at height $h$ is either $1$ or $-q^{h+1}$,
- the weight of a step $\rightarrow$ starting at height $h$ is $({\tilde\alpha}+y{\tilde\beta})q^h$,
- the weight of a step $\searrow$ starting at height $h$ is $-y{\tilde\alpha}{\tilde\beta}q^{h-1}$.
In this section we prove the following:
\[Bn\] The sum of weights of elements in $\mathfrak{B}_n$ is $B_n({\tilde\alpha},{\tilde\beta},y,q)$.
Let $\nu_n$ be the sum of weights of elements in $\mathfrak{B}_n$. It is homogeneous of degree $n$ in ${\tilde\alpha}$ and ${\tilde\beta}$ since each step $\rightarrow$ has degree 1 and each pair of steps $\nearrow$ and $\searrow$ has degree 2. By comparing the weights for paths in $\mathfrak{B}_n$, and the ones in , we see that $\nu_n$ is the term of $(1-q)^n Z_n$ with highest degree in ${\tilde\alpha}$ and ${\tilde\beta}$. Since ${\tilde\alpha}$ and $(1-q)\frac1\alpha$ (respectively, ${\tilde\beta}$ and $(1-q)\frac1\beta$) only differ by a constant, it remains only to show that the term of $\bar Z_n$ with highest degree in $\alpha$ and $\beta$ is $\sum_{k=0}^n{\genfrac{[}{]}{0pt}{}{n}{k}_q}\alpha^k(y\beta)^{n-k}$.
This follows from the combinatorial interpretation in Equation in terms of permutation tableaux (see Definition \[def\_PT\]). In the term of $\bar Z_n$ with highest degree in $\alpha$ and $\beta$, the coefficient of $\alpha^k\beta^{n-k}$ is obtained by counting permutations tableaux of size $n+1$, with $n-k+1$ unrestricted rows, $k$ 1s in the first row. Such permutation tableaux have $n-k+1$ rows, $k$ columns, and contain no 0. They are in bijection with the Young diagrams that fit in a $k\times(n-k)$ box and give a factor ${\genfrac{[}{]}{0pt}{}{n}{k}_q}$.
We can give a second proof in relation with orthogonal polynomials.
It is a consequence of properties of the Al-Salam-Carlitz orthogonal polynomials $U_k^{(a)}(x)$, defined by the recurrence [@ASCa; @KoSw98]: $$U_{k+1}^{(a)}(x) = xU_{k}^{(a)}(x) + (a+1)q^kU_{k}^{(a)}(x) + a(q^k-1) q^{k-1}U_{k-1}^{(a)}(x).$$ Indeed, from Proposition \[mu\_ortho\] the sum of weights of elements in $\mathfrak{B}_n$ is the $n$th moment of the orthogonal polynomial sequence $\{P_k(x) \}_{k\geq 0} $ defined by $$P_{k+1}(x) = xP_k(x) + ({\tilde\alpha}+y{\tilde\beta})q^kP_k + (q^k-1)y{\tilde\alpha}{\tilde\beta}q^{k-1}P_{k-1}.$$ We have $P_k(x)=(y{\tilde\beta})^k U_k^{(a)}( x (y{\tilde\beta})^{-1} )$ where $a={\tilde\alpha}(y{\tilde\beta})^{-1}$, and the $n$th moment of the sequence $\{U_k^{(a)}(x)\}_{k\geq 0}$ is $\sum_{j=0}^k{\genfrac{[}{]}{0pt}{}{k}{j}_q} a^j$ (see §5 in [@ASCa], or the article of D. Kim [@DK Section 3] for a combinatorial proof). Then we can derive the moments of $\{P_k(x)\}_{k\geq0}$, and this gives a second proof of Proposition \[Bn\].
Another possible proof would be to write the generating function $\sum_{n=0}^\infty \nu_n z^n$ as a continued fraction with the usual methods [@Fla], use a limit case of identity (19.2.11a) in [@CU] to relate this generating function with a basic hypergeometric series and then expand the series.
The decomposition of lattice paths
----------------------------------
Let $\mathfrak{R}^*_{N,n}$ be defined exactly as $\mathfrak{R}_{N,n}$, except that the possible weights of a step $\nearrow$ starting at height $h$ are $q^i-q^{i+1}$ with $i\in\{0,\dots,h\}$. The sum of weights of elements in $\mathfrak{R}^*_{N,n}$ is the same as with $\mathfrak{R}_{N,n}$, because the possible weights of a step $\nearrow$ starting at height $h$ sum to $1-q^{h+1}$. Similarly let $\mathfrak{B}^*_n$ be defined exactly as $\mathfrak{B}_n$, except that the possible weights of a step $\nearrow$ starting at height $h$ are $q^i-q^{i+1}$ with $i\in\{0,\dots,h\}$.
\[decomp\] There exists a weight-preserving bijection $\Phi$ between the disjoint union of $\mathfrak{R}^*_{N,n} \times \mathfrak{B}^*_n$ over $n\in\{0,\dots,N\}$, and $\mathfrak{P}_n$ (we understand that the weight of a pair is the product of the weights of each element).
To define the bijection, we start from a pair $(H_1,H_2)\in\mathfrak{R}^*_{N,n}
\times \mathfrak{B}^*_n$ for some $n\in\{0,\dots,N\}$ and build a path $\Phi(H_1,H_2)\in\mathfrak{P}_N$. Let $i\in\{1,\dots,N\}$.
- If the $i$th step of $H_1$ is a step $\rightarrow$ weighted by a power of $q$, say the $j$th one among the $n$ such steps, then:
- the $i$th step $\Phi(H_1,H_2)$ has the same direction as the $j$th step of $H_2$,
- its weight is the product of weights of the $i$th step of $H_1$ and the $j$th step of $H_2$.
- Otherwise the $i$th step of $\Phi(H_1,H_2)$ has the same direction and same weight as the $i$th step of $H_1$.
See Figure \[decomp\_ex\] for an example, where the thick steps correspond to the ones in the first of the two cases considered above. It is immediate that the total weight of $\Phi(H_1,H_2)$ is the product of the total weights of $H_1$ and $H_2$.
(-1.5,0)(11,4) (0,0)(11,3) (0,0)(1,1)(2,1)(3,2)(5,2)(6,3)(7,2)(8,1)(9,1)(10,0)(11,0) (0.5,1.3)
$1-q$
(1.5,1.4)
$q$
(2.5,2.6)
$q-q^2$
(3.5,2.5)
$q^2$
(4.5,2.5)
$q^2$
(5.5,3.3)
$1-q$
(6.5,3.1)
$y$
(7.5,2.1)
$y$
(8.5,1.4)
$q$
(9.5,1.1)
$y$
(10.5,0.4)
$q^0$
(-1.1,1.5)[$H_1=$]{}
(-1.5,0)(5,4) (0,0)(5,3) (0,0)(1,1)(2,1)(3,2)(4,1)(5,0) (0.5,1.3)
$1-q$
(1.5,2.4)
$({\tilde\alpha}+y{\tilde\beta})q$
(2.5,2.3)
$1-q$
(3.5,2.5)
$-y{\tilde\alpha}{\tilde\beta}q$
(4.5,1.5)
$-y{\tilde\alpha}{\tilde\beta}$
(-1.1,1.5)[$H_2=$]{}
(-2,0)(11,6.3) (0,0)(11,5) (0,0)(3,3)(4,3)(6,5)(11,0) (0.5,1.3)
$1-q$
(1.5,2.7)
$q-q^2$
(2.5,3.7)
$q-q^2$
(3.5,4.4)
$({\tilde\alpha}+y{\tilde\beta})q^3$
(4.5,4.7)
$q^2-q^3$
(5.5,5.3)
$1-q$
(6.5,5.2)
$y$
(7.5,4.2)
$y$
(8.5,3.5)
$-y{\tilde\alpha}{\tilde\beta}q^2$
(9.5,2.2)
$y$
(10.5,1.4)
$-y{\tilde\alpha}{\tilde\beta}$
(1,1)(2,2) (3,3)(4,3)(5,4) (8,3)(9,2) (10,1)(11,0) (-2.3,2.3)[$\Phi(H_1,H_2)=$]{}
The inverse bijection is not as simple. Let $H\in\mathfrak{P}_N$. The method consists in reading $H$ step by step from right to left, and building two paths $H_1$ and $H_2$ step by step so that at the end we obtain a pair $(H_1,H_2)\in\mathfrak{R}^*_{N,n} \times \mathfrak{B}^*_n$ for some $n\in\{0,\dots,N\}$. At each intermediate stage, we have built two Motzkin [*suffixes*]{}, [*i.e.*]{} some paths similar to Motzkin paths except that the starting height may be non-zero.
Let us fix some notation. Let $H^{(j)}$ be the Motzkin suffix obtained by taking the $j$ last steps of $H$. Suppose that we have already read the $j$ last steps of $H$, and built two Motzkin suffixes $H_1^{(j)}$ and $H_2^{(j)}$. We describe how to iteratively obtain $H_1^{(j+1)}$ and $H_2^{(j+1)}$. Note that $H_1^{(0)}$ and $H_2^{(0)}$ are empty paths. Let $h$, $h'$, and $h''$ be the respective starting heights of $H^{(j)}$, $H_1^{(j)}$ and $H_2^{(j)}$.
This iterative construction will have the following properties, as will be immediate from the definition below:
- $H_1^{(j)}$ has length $j$, and the length of $H_2^{(j)}$ is the number of steps $\rightarrow$ weighted by a power of $q$ in $H_1^{(j)}$.
- We have $h=h'+h''$.
- The map $\Phi$ as we described it can also be defined in the same way for Motzkin suffixes, and is such that $H^{(j)} = \Phi(H_1^{(j)},H_2^{(j)})$.
To obtain $H_1^{(j+1)}$ and $H_2^{(j+1)}$, we read the $(j+1)$th step in $H$ starting from the right, and add steps to the left of $H_1^{(j)}$ and $H_2^{(j)}$ according to the following rules:
------------------------------------------------------------ --------------------------- --------------------------------------------------------
step read in $H$ step added to $H_1^{(j)}$ step added to $H_2^{(j)}$
\[2mm\] $\searrow$ $-y{\tilde\alpha}{\tilde\beta}q^h$ $\rightarrow$ $q^{h'}$ $\searrow$ $-y{\tilde\alpha}{\tilde\beta}q^{h''}$
\[2mm\] $\searrow$ $y$ $\searrow$ $y$
\[2mm\] $\rightarrow$ $1+y$ $\rightarrow$ $1+y$
\[2mm\] $\rightarrow$ $({\tilde\alpha}+y{\tilde\beta})q^h$ $\rightarrow$ $q^{h'}$ $\rightarrow$ $({\tilde\alpha}+y{\tilde\beta})q^{h''}$
\[2mm\] $\nearrow$ $q^i-q^{i+1}$ $i<h'$ $\nearrow$ $q^i-q^{i+1}$
\[2mm\] $\nearrow$ $q^i-q^{i+1}$ $i\geq h'$ $\rightarrow$ $q^{h'}$ $\nearrow$ $q^{i-h'}-q^{i+1-h'}$
\[2mm\]
------------------------------------------------------------ --------------------------- --------------------------------------------------------
We can also iteratively check the following points.
- With this construction $H_1^{(j+1)}$ and $H_2^{(j+1)}$ are indeed Motzkin suffixes. This is because we add a step $\nearrow$ to $H_1^{(j)}$ only in the case where $i<h'$, hence $h'>0$. And we add a step $\nearrow$ to $H_2^{(j)}$ only in the case where $i\geq h'$, hence $h''>0$ (since $h=h'+h''>i$).
- The paths $H_1^{(j+1)}$ and $H_2^{(j+1)}$ are respectively suffixes of an element in $\mathfrak{R}_{N,n}$ and $\mathfrak{B}_n$ for some $n\in\{0,\dots,N\}$, [*i.e.*]{} the weights are valid.
- The set of rules we have given is the only possible one such that for any $j$ we have $H^{(j)} = \Phi(H_1^{(j)},H_2^{(j)})$.
It follows that $(H^{(N)}_1,H^{(N)}_2)\in\mathfrak{R}^*_{N,n}\times\mathfrak{B}^*_n$ for some $n\in\{0,\dots,N\}$, these paths are such that $\Phi(H^{(N)}_1,H^{(N)}_2)=H$, and it is the only pair of paths satisfying these properties. There are details to check, but we have a full description of $\Phi$ and of the inverse map $\Phi^{-1}$. See Figure \[inv\_ex\] for an example of the Motzkin suffixes we consider.
(-1.5,0)(5,4.5) (0,0)(5,4) (0,2)(1,3)(2,2)(3,2)(4,1)(5,0) (0.5,3.7)
$q-q^2$
(1.5,3.8)
$-y{\tilde\alpha}{\tilde\beta}q^2$
(2.5,3.6)
$({\tilde\alpha}+y{\tilde\beta})q^2$
(3.5,2.3)
$y$
(4.5,1.8)
$-y{\tilde\alpha}{\tilde\beta}$
(-1.5,1.5)[$H^{(j)}=$]{}
(-1.5,0)(5,4.5) (0,0)(5,4) (0,1)(1,1)(2,1)(3,1)(4,0)(5,0) (0.5,1.6)
$q^1$
(1.5,1.6)
$q^1$
(2.5,1.6)
$q^1$
(3.5,1.2)
$y$
(4.5,0.8)
$q^0$
(-1.5,1.5)[$H_1^{(j)}=$]{}
(-1.5,0)(4,4.5) (0,0)(4,4) (0,1)(1,2)(2,1)(3,1)(4,0) (0.5,2.5)
$1-q$
(1.5,2.6)
$-y{\tilde\alpha}{\tilde\beta}q$
(2.5,2.6)
$({\tilde\alpha}+y{\tilde\beta})q$
(3.5,1.5)
$-y{\tilde\alpha}{\tilde\beta}$
(-1.5,1.5)[$H_2^{(j)}=$]{}
Before ending this subsection we can mention another argument to show that $\mathfrak{P}_N$ and the disjoint union of $\mathfrak{R}^*_{N,n}\times\mathfrak{B}^*_n$ have the same cardinal. Thus we could just focus on the surjectivity of the map $\Phi$ and avoid making the inverse map explicit. The argument uses the notion of [*histories*]{} [@XGV] and their link with classical combinatorial objects, as we have seen in the previous section with Laguerre histories. As an unweighted set, $\mathfrak{P}_N$ is a set of colored Motzkin paths, with two possible colors on the steps $\rightarrow$ or $\searrow$, and $h+1$ possible colors for a step $\nearrow$ starting at height $h$. So $\mathfrak{P}_N$ is in bijection with colored involutions $I$ on the set $\{1,\dots,N\}$, such that there are two possible colors on each fixed point or each arch (orbit of size 2). So they are also in bijection with pairs $(I_1,I_2)$ such that for some $n\in\{0,\dots,N\}$:
- $I_1$ is an involution on $\{1,\dots,N\}$ with two possible colors on the fixed points (say, blue and red), and having exactly $n$ red fixed points,
- $I_2$ is an involution on $\{1,\dots,n\}$.
Using histories again, we see that the number of such pairs $(I_1,I_2)$ is the cardinal of $\mathfrak{R}^*_{N,n}\times\mathfrak{B}^*_n$.
Note that considering $\mathfrak{P}_N$ as an unweighted set is not equivalent to setting the various parameters to 1. For example the two possible colors for the horizontal steps correspond to the possible weights $1+y$ or $({\tilde\alpha}+y{\tilde\beta})q^i$. This bijection using colored involution is not weight-preserving but it might be possible to have a weight-preserving version of it for some adequate statistics on the colored involutions.
\[comp1\] The decomposition $\Phi$ is the key step in our first proof of Theorem \[Z\_th\]. This makes the proof quite different from the one in the case $\alpha=\beta=1$ [@CJPR], even though we have used results from [@CJPR] to prove an intermediate step (namely Proposition \[RNn\]). Actually it might be possible to have a direct adaptation of the case $\alpha=\beta=1$ [@CJPR] to prove Theorem \[Z\_th\], but it should give rise to many computational steps. In contrast our decomposition $\Phi$ explains the formula for $Z_N$ as a sum of products.
A second derivation of $Z_N$ using the Matrix Ansatz {#rooks}
====================================================
In this section we build on our previous work [@MJV] to give a second proof of . In this reference we define the operators $$\label{def_DE}
\hat D = \frac{q-1}q D + \frac 1q I \qquad \hbox{and}
\qquad \hat E = \frac{q-1}q E + \frac 1q I,$$ where $I$ is the identity. Some immediate consequences are $$\label{comrel}
\hat D \hat E - q \hat E \hat D = \frac{1-q}{q^2}, \qquad {\ensuremath{\langle W|}} \hat E =
-\frac {\tilde\alpha}q{\ensuremath{\langle W|}}, \qquad \hbox{and}\quad \hat D{\ensuremath{|V\rangle}} = -\frac {\tilde\beta}q {\ensuremath{|V\rangle}},$$ where ${\tilde\alpha}$ and ${\tilde\beta}$ are defined as in the previous sections. While the normal ordering problem for $D$ and $E$ leads to permutation tableaux, for $\hat D$ and $\hat E$ it leads to [*rook placements*]{} as was shown for example in [@AV]. The combinatorics of rook placements lead to the following proposition.
\[hats\] We have: $${\ensuremath{\langle W|}}(qy\hat D+q\hat E)^k{\ensuremath{|V\rangle}} =
\sum_{\substack{i+j \leq k \\ i+j \equiv k \hbox{\scriptsize \, mod }2 } }
{\genfrac{[}{]}{0pt}{}{i+j}{i}_q} (-{\tilde\alpha})^i (-y{\tilde\beta})^j M_{\frac{k-i-j}2 , k}$$ where $$M_{\ell,k} = y^{\ell} \sum_{u=0}^\ell (-1)^u q^{\binom{u+1}2} {\genfrac{[}{]}{0pt}{}{k-2\ell+u}{u}_q}
\left( \binom{k}{\ell-u} - \binom{k}{\ell-u-1} \right).$$
This is a consequence of results in [@MJV] (see Section 2, Corollary 1, Proposition 12). We also give here a self-contained recursive proof. We write the normal form of $(yq\hat D+q\hat E)^k$ as: $$\label{rec1}
(yq\hat D+q\hat E)^k = \sum_{i,j\geq0} d^{(k)}_{i,j} (q\hat E)^i(qy\hat D)^j.$$ From the commutation relation in we obtain: $$\label{rec2}
(qy\hat D)^j(q\hat E) = q^j(q\hat E)(qy\hat D)^j + y(1-q^j)(qy\hat D)^{j-1}.$$ If we multiply by $yq\hat D+q\hat E$ to the right, using we can get a recurrence relation for the coefficients $d^{(k)}_{i,j}$, which reads: $$d^{(k+1)}_{i,j} = d^{(k)}_{i,j-1} + q^j d^{(k)}_{i-1,j} + y(1-q^{j+1}) d^{(k)}_{i,j+1}.$$ The initial case is that $d^{(0)}_{i,j}$ is 1 if $(i,j)=(0,0)$ and 0 otherwise. It can be directly checked that the recurrence is solved by: $$d^{(k)}_{i,j} = {\genfrac{[}{]}{0pt}{}{i+j}{i}_q} M_{\frac{k-i-j}2,k}$$ where we understand that $M_{\frac{k-i-j}2,k}$ is $0$ when $k-i-j$ is not even. More precisely, if we let $e^{(k)}_{i,j} = {\genfrac{[}{]}{0pt}{}{i+j}{i}_q} M_{\frac{k-i-j}2,k}$ then we have: $$e^{(k)}_{i,j-1} + q^j e^{(k)}_{i-1,j} = {\genfrac{[}{]}{0pt}{}{i+j}{i}_q} M_{\frac{k-i-j+1}2,k},$$ and also $$y(1-q^{j+1}) e^{(k)}_{i,j+1} = y (1-q^{i+j+1}) {\genfrac{[}{]}{0pt}{}{i+j}{i}_q} M_{\frac{k-i-j-1}2,k}.$$ So to prove $d^{(k)}_{i,j}=e^{(k)}_{i,j}$ it remains only to check that $$M_{\frac{k-i-j+1}2,k} + y (1-q^{i+j+1}) M_{\frac{k-i-j-1}2,k} = M_{\frac{k-i-j+1}2,k+1}.$$ See for example [@MJV Proposition 12] (actually this recurrence already appeared more than fifty years ago in the work of Touchard, see [*loc. cit.*]{} for precisions).
Now we can give our second proof of Theorem \[Z\_th\].
From and we have that $(1-q)^N Z_N$ is equal to $${\ensuremath{\langle W|}}((1+y)I-qy\hat D - q \hat E )^N{\ensuremath{|V\rangle}} =
\sum_{k=0}^{N} \binom{N}{k}(1+y)^{N-k}(-1)^k {\ensuremath{\langle W|}}(qy\hat D + q\hat E)^k{\ensuremath{|V\rangle}}.$$ So, from Proposition \[hats\] we have: $$(1-q)^N Z_N = \sum_{k=0}^N
\sum_{\substack{i+j \leq k \\ i+j \equiv k \hbox{\scriptsize \, mod }2 } }
{\genfrac{[}{]}{0pt}{}{i+j}{i}_q} {\tilde\alpha}^i(y{\tilde\beta})^j\binom Nk (1+y)^{N-k}
M_{\frac{k-i-j}2,k}$$ (the $(-1)^k$ cancels with a $(-1)^{i+j}$). Setting $n=i+j$, we have: $$\begin{aligned}
(1-q)^N Z_N & = & \sum_{n=0}^N B_n({\tilde\alpha},{\tilde\beta},y,q)
\sum_{\substack{ n\leq k\leq N \\ k \equiv n \hbox{\scriptsize \, mod }2 } }
\binom Nk (1+y)^{N-k} M_{\frac{k-n}2,k}.\end{aligned}$$ So it remains only to show that the latter sum is $R_{N,n}(y,q)$. If we change the indices so that $k$ becomes $n+2k$, this sum is: $$\sum_{k=0}^{\lfloor \frac{N-n}2 \rfloor} \tbinom{N}{n+2k}(1+y)^{N-n-2k} y^{k}
\sum_{i=0}^k (-1)^i q^{\binom{i+1}2} {\genfrac{[}{]}{0pt}{}{n+i}{i}_q}
\left( \tbinom{n+2k}{k-i} - \tbinom{n+2k}{k-i-1} \right)$$ $$= \sum_{i=0}^{\lfloor \frac{N-n}2 \rfloor} (-y)^i q^{\binom{i+1}2} {\genfrac{[}{]}{0pt}{}{n+i}{i}_q}
\sum_{k=i}^{\lfloor \frac{N-n}{2} \rfloor } y^{k-i}\tbinom N{n+2k} (1+y)^{N-n-2k}
\left( \tbinom{n+2k}{k-i} - \tbinom{n+2k}{k-i-1} \right).$$ We can simplify the latter sum by Lemma \[idbinl\] below and obtain $R_{N,n}(y,q)$. This completes the proof.
\[idbinl\]For any $N,n,i\geq0$ we have: $$\label{idbin}
\begin{split}
\sum_{k=i}^{\lfloor \frac{N-n}{2} \rfloor } y^{k-i} \binom N{n+2k} (1+y)^{N-n-2k}
& \left( \tbinom{n+2k}{k-i} - \tbinom{n+2k}{k-i-1} \right) \\
& =
\sum_{j=0}^{N-n-2i}y^j\left( \tbinom Nj \tbinom N{n+2i+j}-
\tbinom N{j-1} \tbinom N{n+2i+j+1} \right).
\end{split}$$
As said in Lemma \[motzbi\], the right-hand side of is the number of Motzkin prefixes of length $N$, final height $n+2i$, and a weight $1+y$ on each step $\rightarrow$ and $y$ on each step $\searrow$. Similarly, $y^{k-i}(\tbinom{n+2k}{k-i} -
\tbinom{n+2k}{k-i-1})$ is the number of Dyck prefixes of length $n+2k$ and final height $n+2i$, with a weight $y$ on each step $\searrow$. From these two combinatorial interpretations it is straightforward to obtain a bijective proof of . Each Motzkin prefix is built from a shorter Dyck prefix with the same final height, by choosing where are the $N-n-2k$ steps $\rightarrow$.
\[comp2\] All the ideas in this proof were present in [@MJV] where we obtained the case $\alpha=\beta=1$. The particular case was actually more difficult to prove because several $q$-binomial and binomial simplifications were needed. In particular, it is natural to ask if the formula in for $Z_N|_{\alpha=\beta=1}$ can be recovered from the general expression in Theorem \[Z\_th\], and the (affirmative) answer is essentially given in [@MJV] (see also Subsection \[simpli\] below for a very similar simplification).
Moments of Al-Salam-Chihara polynomials {#ALSC}
=======================================
The link between the PASEP and Al-Salam-Chihara orthogonal polynomials $Q_n(x;a,b\mid q)$ was described in [@TS]. These polynomials, denoted by $Q_n(x)$ when we don’t need to specify the other parameters, are defined by the recurrence [@KoSw98]: $$\label{recASC}
2xQ_n(x) = Q_{n+1}(x) + (a+b)q^nQ_n(x) + (1-q^n)(1-abq^{n-1})Q_{n-1}(x)$$ together with $Q_{-1}(x)=0$ and $Q_0(x)=1$. They are the most general orthogonal sequence that is a convolution of two orthogonal sequences [@ASCh]. They are obtained from Askey-Wilson polynomials $p_n(x;a,b,c,d\mid q)$ by setting $c=d=0$ [@KoSw98].
Closed formulas for the moments
-------------------------------
Let $\tilde Q_n(x) = Q_n(\frac x2-1; {\tilde\alpha}, {\tilde\beta}\mid q)$, where ${\tilde\alpha}= (1-q)\frac1\alpha-1 $ and ${\tilde\beta}= (1-q)\frac1\beta-1$ as before. From now on we suppose that $a={\tilde\alpha}$ and $b={\tilde\beta}$ (note that $a$ and $b$ are generic if $\alpha$ and $\beta$ are). The recurrence for these shifted polynomials is: $$x\tilde Q_n(x) = \tilde Q_{n+1}(x) + (2+{\tilde\alpha}q^n+ {\tilde\beta}q^n)\tilde Q_n(x)
+ (1-q^n)(1-{\tilde\alpha}{\tilde\beta}q^{n-1})\tilde Q_{n-1}(x).$$ From Proposition \[mu\_ortho\], the $N$th moment of the orthogonal sequence $\{ \tilde Q_n(x) \}_{n\geq 0}$ is the specialization of $(1-q)^NZ_N$ at $y=1$. The $N$th moment $\mu_N$ of the Al-Salam-Chihara polynomials can now be obtained via the relation: $$\mu_N = \sum_{k=0}^N \binom Nk (-1)^{N-k} 2^{-k} (1-q)^k Z_k|_{y=1}.$$ Actually the methods of Section \[paths\] also give a direct proof of the following.
The $N$th moment of the Al-Salam-Chihara polynomials is: $$\label{asc_mom}
\begin{split}
\mu_N = \frac{1}{2^N}
\sum_{\substack{ 0\leq n \leq N \\ n\equiv N \hbox{\scriptsize mod }2 }}
\left( \sum_{j=0}^{\frac{N-n}2} (-1)^j q^{\binom{j+1}2} {\genfrac{[}{]}{0pt}{1}{ n+j }{j}_q}
\left( \binom{N}{\frac{N-n}2-j} - \binom{N}{\frac{N-n}2-j-1} \right)
\right) \\
\times \left( \sum_{k=0}^n {\genfrac{[}{]}{0pt}{}{n}{k}_q} a^k b^{n-k} \right).
\end{split}$$
The general idea is to adapt the proof of Theorem \[Z\_th\] in Section \[paths\]. Let $\mathfrak{P}'_N \subset \mathfrak{P}_N$ be the subset of paths which contain no step $\rightarrow$ with weight $1+y$. By Proposition \[mu\_ortho\], the sum of weights of elements in $\mathfrak{P}'_N$ specialized at $y=1$, gives the $N$th moment of the sequence $\{Q_n(\frac x2)\}_{n\geq 0}$. This can be seen by comparing the weights in the Motzkin paths and the recurrence . But the $N$th moment of this sequence is also $2^N\mu_N$.
From the definition of the bijection $\Phi$ in Section \[paths\], we see that $\Phi(H_1,H_2)$ has no step $\rightarrow$ with weight $1+y$ if and only if $H_1$ has the same property. So from Proposition \[decomp\] the bijection $\Phi^{-1}$ gives a weight-preserving bijection between $\mathfrak{P}'_N$ and the disjoint union of $\mathfrak{R}'_{N,n} \times \mathfrak{B}^*_n$ over $n\in\{0,\dots,N\}$, where $\mathfrak{R}'_{N,n} \subset \mathfrak{R}^*_{N,n}$ is the subset of paths which contain no horizontal step with weight $1+y$. Note that $\mathfrak{R}'_{N,n}$ is empty when $n$ and $N$ don’t have the same parity, because now $n$ has to be the number of steps $\rightarrow$ in a Motzkin path of length $N$. In particular we can restrict the sum over $n$ to the case $n\equiv N$ mod $2$.
At this point it remains only to adapt the proof of Proposition \[RNn\] to compute the sum of weights of elements in $\mathfrak{R}'_{N,n}$, and obtain the sum over $j$ in . As in the previous case we use Lemma \[decomp2\] and Lemma \[core\]. But in this case instead of Motzkin prefixes we get Dyck prefixes, so to conclude we need to know that $\tbinom{N}{(N-n)/2-j} - \tbinom{N}{(N-n)/2-j-1}$ is the number of Dyck prefixes of length $N$ and final height $n+2i$. The rest of the proof is similar.
We have to mention that there are analytical methods to obtain the moments $\mu_N$ of these polynomials. A nice formula for the Askey-Wilson moments was given by Stanton [@DS], as a consequence of joint results with Ismail [@IS equation (1.16)]. As a particular case they have the Al-Salam-Chihara moments: $$\label{asc_mom2}
\mu_N = \frac1{2^N} \sum_{k=0}^N (ab;q)_kq^k \sum_{j=0}^k \frac{ q^{-j^2} a^{-2j}
(q^{j} a + q^{-j}a^{-1})^N}{ (q,a^{-2}q^{-2j+1};q)_{j}(q,a^2q^{1+2j};q)_{k-j}},$$ where we use the $q$-Pochhammer symbol. The latter formula has no apparent symmetry in $a$ and $b$ and has denominators, but Stanton [@DS] gave evidence that can be simplified down to using binomial, $q$-binomial, and $q$-Vandermonde summation theorems. Moreover is equivalent to a formula for rescaled polynomials given in [@KSZ08] (Section 4, Theorem 1 and equation (29)).
Some particular cases of Al-Salam-Chihara moments {#simpli}
-------------------------------------------------
When $a=b=0$ in we immediately recover the known result for the continuous $q$-Hermite moments. This is 0 if $N$ is odd, and the Touchard-Riordan formula if $N$ is even. Other interesting cases are the $q$-secant numbers $E_{2n}(q)$ and $q$-tangent numbers $E_{2n+1}(q)$, defined in [@HZR] by continued fraction expansions of the ordinary generating functions: $$\sum_{n\geq0} E_{2n}(q)t^n=
\cfrac{1}{1-\cfrac{[1]_q^2t}{1-\cfrac{[2]_q^2t}{1-\cfrac{[3]_q^2t}{\ddots}}}}
\displaystyle
\quad\hbox{and}\quad
\sum_{n\geq0} E_{2n+1}(q)t^n=
\scriptstyle
\cfrac{1}{1-\cfrac{[1]_q[2]_qt}{1-\cfrac{[2]_q[3]_qt}{1-\cfrac{[3]_q[4]_qt}{\ddots}}}}.$$ The exponential generating function of the numbers $E_n(1)$ is the function $\mathrm{tan}(x)
+\mathrm{sec}(x)$. We have the combinatorial interpretation [@HZR; @MJV2]: $$E_n(q) = \sum_{\sigma\in\mathfrak{A}_n} q^{\hbox{\scriptsize{{\rm \hbox{31-2}}}}(\sigma)},$$ where $\mathfrak{A}_n\subset\mathfrak{S}_n$ is the set of alternating permutations, [*i.e.*]{} permutations $\sigma$ such that $\sigma(1)>\sigma(2)<\sigma(3)>\dots$. The continued fractions show that these numbers are particular cases of Al-Salam-Chihara moments: $$E_{2n}(q) =(\tfrac2{1-q})^{2n}\mu_{2n}|_{a=-b=i\sqrt{q}}, \qquad\hbox{and}\quad
E_{2n+1}(q)=(\tfrac2{1-q})^{2n}\mu_{2n}|_{a=-b=iq}$$ (where $i^2=-1$). From and a $q$-binomial identity it is possible to obtain the closed formulas for $E_{2n}(q)$ and $E_{2n+1}(q)$ that were given in [@MJV2], in a similar manner that can be simplified into when $\alpha=\beta=1$. Indeed, from we can rewrite: $$2^{2n}\mu_{2n} = \sum_{m=0}^n \big(\tbinom{2n}{n-m} - \tbinom{2n}{n-m-1}\big)
\sum_{j,k\geq0} (-1)^j q^{\binom{j+1}2} {\genfrac{[}{]}{0pt}{1}{2m-j}{j}_q}{\genfrac{[}{]}{0pt}{1}{2m-2j}{k}_q}
\left(\tfrac ba\right)^k a^{2m-2j}.$$ This latter sum over $j$ and $k$ is also $$\begin{aligned}
\sum_{j,k\geq0} (-1)^j q^{\binom{j+1}2}
{\genfrac{[}{]}{0pt}{1}{2m-j}{j+k}_q}{\genfrac{[}{]}{0pt}{1}{j+k}{j}_q} \left(\tfrac ba\right)^k a^{2m-2j}
= \sum_{\ell\geq j\geq0} (-1)^j q^{\binom{j+1}2}
{\genfrac{[}{]}{0pt}{1}{2m-j}{\ell}_q}{\genfrac{[}{]}{0pt}{1}{\ell}{j}_q} \left(\tfrac ba\right)^{\ell-j} a^{2m-2j}. \end{aligned}$$ The sum over $j$ can be simplified in the case $a=-b=i\sqrt q $, or $a=-b=iq$, using the $q$-binomial identities already used in [@MJV] (see Lemma 2): $$\sum_{j\geq0} (-1)^j q^{\binom j 2}{\genfrac{[}{]}{0pt}{}{2m-j}{\ell}_q}{\genfrac{[}{]}{0pt}{}{\ell}{j}_q} = q^{\ell(2m-\ell)},$$ and $$\sum_{j\geq0} (-1)^j q^{\binom {j-1} 2}{\genfrac{[}{]}{0pt}{}{2m-j}{\ell}_q}{\genfrac{[}{]}{0pt}{}{\ell}{j}_q} =
\tfrac{ q^{(\ell+1)(2m-\ell)} - q^{\ell(2m-\ell)} +q^{\ell(2m-\ell+1)} -
q^{(\ell+1)(2m-\ell+1)} }{ q^{2m-1}(1-q)}.$$ Omitting details, this gives a new proof of the Touchard-Riordan-like formulas [@MJV2]: $$\label{En}
E_{2n}(q) = \frac 1{(1-q)^{2n}} \sum_{m=0}^n \left( \tbinom{2n}{n-m}-
\tbinom{2n}{n-m-1} \right)
\sum_{\ell=0}^{2m} (-1)^{\ell+m} q^{\ell(2m-\ell)+m}$$ and $$\label{En2}
E_{2n+1}(q) = \frac 1{(1-q)^{2n+1}} \sum_{m=0}^n \left( \tbinom{2n+1}{n-m}-
\tbinom{2n+1}{n-m-1} \right) \sum_{\ell=0}^{2m+1} (-1)^{\ell+m} q^{\ell(2m+2-\ell)}.$$
Some classical integer sequences related to $\bar Z_N$ {#num}
======================================================
It should be clear from the interpretation given in that the polynomial $\bar Z_N$ contains quite a lot a of combinatorial information. When $\alpha=\beta=1$, the coefficients of $y^k$ in $\bar Z_n$ are the $q$-Eulerian numbers introduced by Williams [@LW]: $$\label{ZEhat}
\bar Z_N |_{\alpha=\beta=1} = \sum_{k=0}^N y^k \hat E_{k+1,N+1}(q),$$ where $\hat E_{k,n}(q)$ is defined in [@LW Section 6]. It was proved by Williams, that $\hat E_{k,n}(q)$ is equal to the Eulerian number $A_{n,k}$ when $q=1$, to the binomial coefficient $\binom{n-1}{k-1}$ when $q=-1$, and to the Narayana number $N_{n,k}=\frac1n\binom nk \binom n{k-1}$ when $q=0$. With the other parameters $\alpha$ and $\beta$, there are other interesting results.
Stirling numbers
----------------
Carlitz $q$-analog of the Stirling numbers of the second kind, denoted by $S_2[n,k]$, are defined when $1\leq k \leq n$ by the recurrence [@Car48]: $$\label{rec_stir}
S_2[n,k]=S_2[n-1,k-1] + [k]_q S_2[n-1,k], \quad S_2[n,k]=1 \hbox{ if } k=1 \hbox{ or } k=n.$$
If $\alpha=1$, the coefficient of $\beta^ky^{k}$ in $\bar Z_N$ is $S_2[N+1,k+1]$.
This follows from the interpretation in terms of permutation tableaux (see Definition \[def\_PT\]). Indeed, the coefficient of $\beta^ky^k$ in $\bar Z_N$ counts permutation tableaux of size $N+1$, with $k+1$ rows, and $k+1$ unrestricted rows. In a permutation tableau with no restricted row, each column contains a sequence of 0’s followed by a sequence of 1’s. Such permutation tableaux follow the recurrence where $n$ is the size and $k$ is the number of rows. Indeed, if the bottom row is of size 0 we can remove it and this gives the term $S_2[n-1,k-1]$. Otherwise the first column is of size $k$, this gives the term $[k]_q S_2[n-1,k]$ because the factor $[k]_q$ accounts for the possibilities of the first column, the factor $S_2[n-1,k]$ accounts for what remains after removing the first column.
The proof only relies on simple facts about permutation tableaux, and with no doubts it was previously noticed that $S_2[n,k]$ appears when we count permutation tableaux without restricted rows. Actually permutation tableaux with no restricted rows are equivalent to the 0-1 tableaux introduced by Leroux [@Ler90] as a combinatorial interpretation of $S_2[n,k]$.
From , it is possible to obtain a formula for $S_2[n,k]$. First, observe that the coefficient of $y^k$ in $\bar Z_N$ has degree $k$ in $\beta$. Hence, from the previous proposition: $$\label{stirZ}
\sum_{k=0}^{N} a^k S_2[N+1,k+1] = \lim_{y\to 0} \bar Z_N(1,\tfrac a y,y,q).$$ We have $R_{N,n}(0,q)=\binom Nn$. When $\alpha=1$ and $\beta=\frac ya$, we have ${\tilde\alpha}=-q$ and $y{\tilde\beta}=(1-q)a+y$. So from and it is straightforward to obtain: $$\label{carl1}
S_2[N+1,k+1] = \frac 1{(1-q)^{N-k}} \sum_{j=0}^{N-k} (-q)^j \binom{N}{k+j}{\genfrac{[}{]}{0pt}{}{k+j}{j}_q}.$$ Note that this differs from the expression originally given by Carlitz [@Car48]: $$\label{carl2}
S_2[N,k] = \frac 1{(1-q)^{N-k}} \sum_{j=0}^{N-k} (-1)^j \binom{N}{k+j}{\genfrac{[}{]}{0pt}{}{k+j}{j}_q},$$ but it is elementary to check that and are equivalent, using the two-term recurrence relations for binomial and $q$-binomial coefficients.
When $y=\alpha=1$, the coefficient of $\beta^k$ in $\bar Z_N$ is a $q$-analog of the Stirling number of the first kind $S_1(N+1,k+1)$. It is such that $q$ counts the number of patterns 31-2 in permutations of size $N+1$ and with $k+1$ right-to-left minima. Knowing the symmetry , we could also say that it is such that $q$ counts the number of patterns 31-2 in permutations of size $N+1$ and with $k+1$ right-to-left maxima. The combinatorial way to see the symmetry is the transposition of permutation tableaux [@CW3], so at the moment it is quite indirect to see that the two interpretations agree since we need all the bijections from Section \[bij\]. We have no knowledge of previous work concerning these $q$-Stirling numbers of the first kind.
Fine numbers
------------
The sequence of Fine numbers shares many properties with the Catalan numbers, we refer to [@DSh] for history and facts about them. We will show that a natural symmetric refinement of them appears as a specialization of $\bar Z_N$.
A [*peak*]{} of a Dyck path is a factor $\nearrow\searrow$, we denote by ${{\rm pk}}(P)$ the number of peaks of a path $P$. A Fine path is a Dyck path $D$ such that there is no factorization $D=D_1\nearrow\searrow D_2$ where $D_1$ and $D_2$ are Dyck paths. The Fine number $F_n$ is the number of Fine path of length $2n$, and more generally the polynomial $F_n(y)$ is $\sum y^{{{\rm pk}}(P)}$ where the sum is over Fine paths $P$ of length $2n$. These polynomials were considered in [@DSh] via their generating function.
An interesting property is that $F_n(y)$ is self-reciprocal, [*i.e.*]{} $F_n(y) = y^{n}F_n(\frac 1y)$ (a simple proof of this will appear below). This is reminiscent of the Dyck paths: the number of Dyck paths of length $2n$ with $k$ peaks is the Narayana number $N_{k,n}$ and we have $N_{k,n}=N_{n+1-k,n}$. The first values are: $$\begin{split}
F_1(y) = 0, \quad F_2(y) = y, \quad F_3(y) =y^2+y , \quad F_4(y) = y^3+4y^2+y,
\\
F_5(y) = y^4+8y^3+8y^2+y, \quad F_6(y) = y^5+13y^4+29y^3+13y^2+y.
\end{split}$$
When $\frac1\alpha=-y$, $q=0$, and $\beta=1$, we have $\bar Z_N= F_N(y)$.
In this case we have ${\tilde\beta}=0$, ${\tilde\alpha}=-1-y$. From the weights in the general case , we see that now $Z_N$ is the sum of weights of Motzkin paths such that:
- the weight of a step $\nearrow$ is 1, the weight of a step $\searrow$ is $y$,
- the weight of a step $\rightarrow$ is $1+y$, but there is no such step at height 0.
Let $H(t,y)=\sum_{N\geq 0} Z_Nt^N$. It is such that $H(t,y) = 1+yt^2G(t,y)H(t,y)$, where $G(t,y)$ counts the paths with the same weights but possibly with steps $\rightarrow$ at height 0. Let $L(t,y)=\sum t^{\ell(P)}y^{{{\rm pk}}(P)}$ where $\ell(P)$ is half the number of steps of $P$, and the sum is over all Dyck paths $P$. Some standards arguments show that $G(t,y)$ is linked with Narayana numbers in such a way that $L(t,y)=1+ytG(t,y)$. So we have $H(t,y) = 1+t\big(L(t,y)-1\big)H(t,y)$, which is precisely the functional equation given in [@DSh Section 7] for the generating function $\sum F_n(y)t^n$. This completes the proof.
When we substitute $y$ with $\frac1y$ in the Motzkin paths considered in the proof, we see that the weight of a step $\rightarrow$ is divided by $y$ and the weight of a step $\searrow$ is divided by $y^2$, so the total weight is divided by $y^n$ where $n$ is the length of the path. This proves the symmetry of the coefficients of $F_n(y)$.
Note that the symmetry of $Z_N$ obtained in this section is not a particular case of previously known symmetry . It may be a special case of another more general symmetry.
Concluding remarks
==================
We have used two kinds of weighted Motzkin paths to study $Z_N$. The first kind are the elements of $\mathfrak{P}_N$, [*i.e.*]{} the paths coming from the matrices $D$ and $E$ defined in and . They have the property that the weight of a step only depends on its direction and its height, so that there is a J-fraction expansion for the generating function $\sum_{N\geq0}Z_Nt^N$ with the four parameters $\alpha$, $\beta$, $y$ and $q$. The second kind of weighted Motzkin paths are the Laguerre histories, and their nice property is that they are linked bijectively with permutations. One might ask if there is a set of weighted Motzkin paths having both properties, but its existence is doubtful. Still it could be nice to have a direct simple proof that these two kinds of paths give the same quantity $Z_N$.
Our two new combinatorial interpretations in Theorems \[histoires\] and \[main\] complete the known combinatorial interpretations and , and this makes at least four of them. Although all is proved bijectively, there is not a direct bijection for any pair of combinatorial interpretations. In particular it would be nice to have a more direct bijection between permutation tableaux and permutations to link the right-hand sides of and , instead of composing four bijections (Steingrímsson-Williams, reverse complement of inverse, Foata-Zeilberger and Françon-Viennot). Permutation tableaux are mainly interesting because of their link with permutations, so in this regard it is desirable to have a direct bijection preserving the four parameters considered here.
We have given evidence that the lattice paths are good combinatorial objects to study the PASEP with three parameters. However, our combinatorial interpretation of $Z_N$ with the Laguerre histories relies on the previous one with permutation tableaux. To complete the lattice paths approach, it might be interesting to have a direct derivation of stationary probabilities in terms of Laguerre histories. For example in [@CW1], Corteel and Williams define a Markov chain on permutation tableaux which projects to the PASEP, similarly we could hope that there is an explicit simple description of such a Markov chain on Laguerre histories.
The three-parameter PASEP is now quite well understood since we have exact expressions for many interesting quantities. In a more general model, we allow particles to enter the rightmost site, and exit the leftmost site, so that there are five parameters. In this case the partition function is linked with the Askey-Wilson moments, in a similar manner that the three-parameter partition function is linked with Al-Salam-Chihara moments [@USW]. Recently, Corteel and Williams [@CW3] showed that there exist some staircase tableaux generalizing permutation tableaux, arising from this general model with five parameters. It is not clear whether a closed formula for the five-parameter partition function exists, and in the case it exists it might be unreasonably long. But knowing the results about the three-parameter partition function, we expect the five-parameter partition function to be quite full of combinatorial meaning.
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[^1]: Partially supported by the grant ANR08-JCJC-0011.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Models of debris disk morphology are often focused on the effects of a planet orbiting interior to or within the disk. Nonetheless, an exterior planetary-mass perturber can also excite eccentricities in a debris disk, via Laplace-Lagrange secular perturbations in the coplanar case or Kozai-Lidov perturbations for mutually inclined companions and disks. HD 106906 is an ideal example of such a a system, as it harbors a confirmed exterior 11 $\rm M_{\rm Jup}$ companion at a projected separation of 650 au outside a resolved, asymmetric disk. We use collisional and dynamical simulations to investigate the interactions between the disk and the companion, and to use the disk’s observed morphology to place constraints on the companion’s orbit. We conclude that the disk’s observed morphology is consistent with perturbations from the observed exterior companion. Generalizing this result, we suggest that exterior perturbers, as well as interior planets, should be considered when investigating the cause of observed asymmetries in a debris disk.'
author:
- 'Erika R. Nesvold, Smadar Naoz, Michael Fitzgerald'
title: 'HD 106906: A Case Study for External Perturbations of a Debris Disk'
---
Introduction {#sec:introduction}
============
Circumstellar debris disks are produced by the rocky and icy material leftover from the formation of the star and any planets in the system. To date, over 1700 debris disks have been detected via the excess infrared emission in their star’s spectral energy distribution [@Cotten2016], and over 40 have been resolved with optical or infrared imaging (http://circumstellardisks.org). The architecture of the underlying planetary system can leave a distinct imprint on the morphology of a debris disk [@Mouillet1997; @Wyatt1999a; @Matthews2014; @Nesvold2015; @Lee2016; @Nesvold2016].
Modeling debris disk morphology is often focused on the effects of a planetary-mass perturber orbiting interior to or within the disk [@Mouillet1997; @Chiang2009; @Pearce2015; @Nesvold2015a]. Nonetheless, exterior companions have been detected and inferred for several systems [@Rodriguez2012; @Bailey2014; @Mawet2015], and dynamical modeling suggests that an exterior perturber can also excite eccentricities of the particles in a debris disk, via Laplace-Lagrange secular perturbations in the near-coplanar case [@Thebault2012a] or Kozai-Lidov perturbations for mutually inclined companions and disks [@Nesvold2016], inducing asymmetries in the disk and triggering a collisional cascade. Collisions between the disk particles produce smaller dust grains whose thermal emission or scattered light can then be spatially resolved with infrared or optical imaging [@Wyatt2008].
HD 106906 is an ideal example of a system with an exterior perturber, as it harbors a confirmed exterior companion with a model-atmosphere-derived mass of 11 $\rm M_{\rm Jup}$ at a projected separation of $650$ AU outside a resolved disk [@Bailey2014]. Scattered-light imaging of the disk with the Gemini Planet Imager (GPI), the Hubble Space Telescope’s Advanced Camera for Surverys (HST/ACS), and SPHERE has revealed that the disk is a ring viewed nearly edge-on (inclination $\sim85^{\circ}$), with a inner region cleared of small dust grains [@Kalas2015; @Lagrange2016]. These observations noted four major features of the disk morphology and system geometry:
1. The position angle (PA) of the disk is oriented $\sim21^{\circ}$ counterclockwise from the position angle of the companion, which constrains the orbit of the companion relative to the disk.
2. The inner disk has little to no vertical extension. While @Kalas2015 tentatively suggested the presence of a “warp” in the disk’s vertical structure on the west side of the disk, this warp was not confirmed by @Lagrange2016. This lack of vertical extension indicates that the inclinations of the disk particles have not been excited.
3. The east side of the disk is brighter than the west side in GPI and SPHERE near-infrared images.
4. @Kalas2015 observed a faint extension on the west side of the disk out to nearly 500 au, but only diffuse emission on the east side .
These latter two features indicate that the disk may be an eccentric ring, which will exhibit a brightness asymmetry towards the pericenter side [“pericenter glow”, @Wyatt1999a] and a faint, extended tail towards the apocenter side [@Lee2016].
We modeled the HD 106906 system to demonstrate that the observed exterior companion can shape the disk into a flat, eccentric ring, and that all four of these morphological features can be reproduced without invoking the presence of a second companion. We also used the observed features and asymmetries of the HD 106906 disk to place constraints on the orbit of the observed companion. In Section \[sec:simulations\], we describe the collisional and dynamical simulations we performed of the parent bodies and dust grains in the HD 106906 disk. In Section \[sec:results\], we present the simulated brightness images produced by our simulations for comparison with observations. In \[sec:time\], we discuss the implication of these results and show how they can be used to constrain the orbit of HD 106906b. In Section \[sec:conclusions\], we summarize our conclusions and suggest opportunities for future work.
Simulations {#sec:simulations}
===========
Given that collisions between particles in a disk with sufficiently high optical depth ($L_{\rm IR}/L_{*} \approx1.4\times10^{-3}$ for HD 106906 [@Chen2011]) will both produce the small grains seen in observations and may affect the dynamics of the disk, we simulated the HD 106906 system using the Superparticle-Method Algorithm for Collisions in Kuiper belts and debris disks [SMACK, @Nesvold2013]. We then recorded the dust-producing encounters between parent bodies, simulated the orbits of the generated dust grains under the influence of radiative forces following the method of @Lee2016, then simulated the surface brightness of the dust using a Henyey-Greenstein scattering phase function [@Henyey1941].
SMACK Model
-----------
SMACK is based on the $N$-body integrator REBOUND [@Rein2012], but approximates each particle in the integrator as a collection of bodies with a range of sizes between 1 mm and 10 cm in diameter, traveling on the same orbit. This group of bodies is called a “superparticle” and is characterized by a size distribution, position, and velocity. The superparticles act as test particles in the integration, and orbit the star under the influence of perturbations by any planets in the simulation. Each superparticle is approximated as a sphere with some finite radius. When REBOUND detects that two superparticles are overlapping in space, SMACK statistically calculates the number of bodies within each superparticle that will collide and fragment, removes these bodies from their size distributions, and redistributes the fragments. SMACK also corrects the trajectories of the parent superparticles to conserve angular momentum and energy, compensating for the kinetic energy lost to fragmentation.
The parameters for the SMACK simulation of the HD 106906 system described in this work are listed in Table \[tab:init\]. The masses of the star and companion were 2.5 $\rm{M}_{\odot}$ and 11 $\rm{M_{Jup}}$, respectively. The initial semi-major axis range of $65-85$ au for the 10,000 superparticles in the simulated disk was chosen in anticipation that the disk would spread during the 15 Myr course of the simulation. The radial extent of the HD 106906 ring as observed in scattered-light imaging is $\sim50-100$ au [@Kalas2015; @Lagrange2016]. The orbital parameters of the companion were chosen such that the gravitational perturbations from the companion would excite the eccentricities of the disk particles without causing a vertical extension of the disk on the timescale of the system’s age, and such that the simulated companion’s orbit could reproduce the position of the observed companion.
Parameter Initial Disk Values HD 106906b
------------------------------------- --------------------- ------------
Semi-Major Axis (au) $65-85$ 700
Eccentricity $0-0.01$ 0.7
Inclination ($^{\circ}$) $0-0.29$ 8.5
Longitude of Nodes ($^{\circ}$) $0-360$ 90
Argument of Pericenter ($^{\circ}$) $0-360$ $-90$
Optical depth $1.4\times10^{-3}$ –
Density (g cm$^{-3}$) 1.0 –
: Initial conditions of the disk and companion for the simulation. \[tab:init\]
SMACK vs. Collisionless $N$-Body
--------------------------------
To measure the effects of collisions on the dynamics of the disk particles, we also performed a collisionless $N$-body simulation of the disk using the Wisdom-Holman integrator of REBOUND with collision detection and resolution turned off. The collisionless $N$-body simulation used the same companion and disk parameters as the SMACK simulation (Table \[tab:init\]), with 10,000 test particles to represent the disk. Figure \[fig:orbelcompare\] shows the time evolution of the simulated disk’s average eccentricity, inclination, longitude of nodes, and argument of pericenter. Although there are small variations, most notably in the average eccentricity, between the SMACK simulation and the collisionless $N$-body simulation, the maximum difference for each parameter is $\leq10\%$, indicating that fragmenting collisions have a minimal effect on the dynamics of this system.
![\[fig:orbelcompare\] Time evolution of the average eccentricity, inclination, longitude of nodes, and argument of pericenter for disk particles in the SMACK and collisionless $N$-body simulations described in this section. The dashed lines indicate the standard deviation of each orbital element for each simulation. The variation between the two simulations is $\leq10\%$ for each orbital element’s average](OrbElCompare_new.eps){width="\columnwidth"}
.
Dust Model {#sec:dust}
----------
To generate the simulated images of the dust grains in the HD 106906 system, we adapted the method of @Lee2016, which extended the dust orbit calculations of @Wyatt1999a to include estimates of the surface brightness. Our SMACK simulation output the locations of dust-producing collisions during the 15 Myr simulation, as well as the orbits of the parent bodies producing the dust. We selected the first $10^4$ dust production events occurring after time $t=5$ Myr. For each dust production event, we generated 10 dust orbits, each with a $\beta$ value randomly chosen from a power-law distribution with index 3/2 (where $\beta\approx F_{rad}/F_{grav}$ represents the ratio of the radiative and gravitational forces acting on a dust grain). The maximum possible value for the $\beta$ value was set by the parent body’s orbit: $$\beta_{\rm max} = \frac{1-e_{\rm p}}{2(1+e_{\rm p} \cos f_{\rm p})},$$ where $e_{\rm p}$ and $f_{\rm p}$ are the parent body’s eccentricity and true anomaly, respectively. The semi-major axis $a$, eccentricity $e$, and argument of pericenter $\omega$ of each dust orbit are given by the parent body’s orbit (specifically $a_{\rm p}$, $e_{\rm p}$, and $f_{\rm p}$) and the $\beta$ value assigned to the orbit: $$a = \frac{a_{\rm p}(1-e_{\rm p}^2)(1-\beta)}{1-e_{\rm p}^2-2\beta(1-e_{\rm p}\cos f_{\rm p})},$$ $$e = \frac{\sqrt{e_{\rm p}^2+2\beta e_{\rm p} \cos f_{\rm p} + \beta^2}}{1-\beta},$$ $$\omega = \omega_{\rm p} + \arctan\left(\frac{\beta \sin f_{\rm p}}{e_{\rm p}+\beta \cos f_{\rm p}}\right).$$ The inclination $i$ and longitude of nodes $\Omega$ of the dust orbit was set to be equal to the corresponding values for the parent body, $i_{\rm p}$ and $\Omega_{\rm p}$, respectively. For each dust orbit, we generated 10 dust grains, and assigned each a mean anomaly selected randomly from a uniform distribution between 0 and $360^{\circ}$. Thus, each dust production event produced 100 final dust grain locations.
After constructing the dust population from the SMACK results, we simulated the surface brightness of the dust using $\phi(g,\theta)/\beta^2 r^2$, where $\phi(g,\theta)$ is the Henyey-Greenstein scattering phase function with asymmetry parameter $g$, $\theta$ is the angle between the dust grain and the observer’s line-of-sight (with the vertex at the star), and $r$ is the distance between the dust grain and the star. Following @Lee2016, we used $g=0.5$.
Results {#sec:results}
=======
Figure \[fig:combined\] shows the simulated brightness of the dust produced by SMACK during after 5 Myr, scaled for comparison with Figure 1 of @Lagrange2016 and Figure 3 of @Kalas2015. The SMACK-produced dust population exhibits a brightness asymmetry at pericenter (Figure \[fig:combined\]a) as well as a faint extension on the apocenter side (Figure \[fig:combined\]b), although it does not reproduce the diffuse emission on the eastern side of the disk suggested by @Kalas2015.
![\[fig:combined\] Simulated surface brightness of the SMACK-simulated dust ring after 5 Myr of perturbations from a companion at semi-major axis $a_{\rm pl}=700$ au, eccentricity $e_{\rm pl}=0.7$, and inclination $i_{\rm pl} = 8.5^{\circ}$. The viewing inclination is $\sim5^{\circ}$ from edge-on and the pericenter side of the disk is towards the east. (a) The field of view and simulated coronagraphic mask were chosen for comparison with Figure 1 of the @Lagrange2016. Pericenter glow causes the east side of the disk to appear brighter. (b) The field of view and simulated coronagraphic mask were chosen for comparison with Figure 3 of @Kalas2015. The grey line in (b) indicates the orbit of the simulated companion, while the white dot (highlighted by the arrow) indicates the observed location of the companion.](Combined.png){width="\linewidth"}
The scattered-light brightness enhancement at pericenter is a signature of an eccentric ring [@Wyatt1999a; @Pan2016], indicating that the orbits of the disk particles in the simulation, initially assigned eccentricities $<0.01$ and random longitudes of node and arguments of pericenter, have become more eccentric and apsidally aligned due to their secular resonance with the distant companion [@Li2014 see Section \[sec:time\]]. The extended “tail” seen on the apocenter side of the disk is also a signature of an eccentric ring, in which high-eccentricity dust grains are produced near the ring’s pericenter on apsidally aligned orbits, and then observed as they travel to and from their distant apocenters [@Lee2016].
The relatively low inclination of the companion relative to the disk results in a flat, narrow ring, with no appreciable vertical extension after 5 Myr, and the observed relative inclination between the companion and the disk ($\sim21^{\circ}$) is reproduced by our choice of $5^{\circ}$ relative inclination and the $85^{\circ}$ viewing inclination. The observed exterior companion is therefore able to reproduce four of the observed morphological features of the disk with no requirement for a second companion.
The physical mechanisms described above allow us to place constraints on the orbit of the companion. The relative position angle of the companion and the disk’s line of nodes on the sky is related to the inclination of the companion’s orbit relative to the plane of the disk, but this relationship in complex and also depends on the longitude of nodes and argument of pericenter of the companion relative to the disk. In addition, these angles change with time, as the gravitational perturbations from the companion cause the orbits of the disk particles to precess coherently together (see Section \[sec:time\]). Instead, we can use the disk’s morphology to place constraints on the companion’s orbit. For example, the pericenter of the companion cannot be too close to the outer edge of the disk or the companion’s chaotic zone will truncate the disk. As a rough estimate, we can calculate the relationship between the radius of the inner edge of the companion’s chaotic zone, $r_{\rm z}$, and the companion’s pericenter distance, $r_{\rm pl}$, using the analytically derived classical chaotic zone relationship for circular orbits [@Wisdom1980], $(r_{\rm pl}-r_{\rm z})/r_{\rm pl} = 1.3\mu^{2/7}$, where $\mu$ is the companion-to-star mass ratio. If we set the inner edge of the companion’s chaotic zone to be the outer edge of the disk, $r_{\rm z}=100$ au, the minimum pericenter location for the companion is $r_{\rm pl}\approx138$ au. This constraint contains a degeneracy between the companion’s eccentricity and semi-major axis, $a_{\rm pl} (1-e_{\rm pl})\gtrsim138$ au.
There also exists an upper limit on the companion’s semi-major axis, as the secular timescale must be less than the age of the system for the companion’s secular perturbations to produce the observed asymmetries in the disk. If we constrain the secular timescale to be at least 10 Myr, the companion’s semi-major axis and eccentricity are constrained by $a_{\rm pl} \sqrt{1-e_{\rm pl}^2}\lesssim661$ au (see Section \[sec:time\]).
We can use the disk’s vertical extent to place an upper limit on the mutual inclination between the plane of the disk and the companion’s orbit. We ran three collisionless $N$-body simulations of the disk perturbed by a 11 $\rm M_{\rm Jup}$ companion with the same orbital parameters as in the SMACK simulation, but varying the mutual inclination between the companion and disk to $8.5^{\circ}$, $20^{\circ}$, or $30^{\circ}$. The $N$-body particles represented the parent bodies in the disk. We simulated the production of dust grains by generating and recording one dust orbit matching the location and velocity of each parent body every 10 years during each simulation. We then generated 100 dust grains from each dust orbit and produced simulated brightness images using the procedure described in Section \[sec:dust\]. Figure \[fig:verticalextent\] shows the simulated brightness of each disk, viewed $5^{\circ}$ from edge-on, at 10 Myr, as well as the location of the parent bodies in each disk. Perturbations from a higher companion inclination produce a larger vertical extent in the disk. Resolved images of the system show a flat disk, indicating that the companion’s inclination relative to the disk must be $i\lesssim20^{\circ}$. Our SMACK simulation demonstrates that the observed exterior companion can excite the necessary eccentricities in the ring within the age of the system without creating a significant vertical extension if the companion has a moderately large eccentricity ($e\approx0.7$) but a small inclination ($i\approx8.5^{\circ}$). Given these orbital parameters, the companion’s semi-major axis would need to be $\sim700$ AU to match its observed projected position.
![\[fig:verticalextent\] Left: Locations of the parent bodies in each disk at 10 Myr. Each disk is inclined $5^{\circ}$ from edge-on and perturbed by a companion with inclination $8.5^{\circ}$, $20^{\circ}$, or $30^{\circ}$. Right: Corresponding simulated surface brightness maps of each disk. The vertical extent of the disk increases with companion inclination, as well as with time.](VerticalExtent.png){width="\linewidth"}
Constraining the Companion’s Orbit {#sec:time}
==================================
At time $t=0$ yr in our SMACK simulation of the HD 106906 system, the disk particles have eccentricity $\leq0.01$ and longitudes of node and arguments of pericenter distributed randomly between $0-360^{\circ}$, forming a circular belt. By time $t=5$ Myr, gravitational perturbations from the companion have increased the average eccentricity of the particles to $\sim0.18$ (Fig. \[fig:alignment\]). Increasing the average eccentricity of the particles alone would only produce a broader circular disk, but the secular resonance formed with the companion also cause the particles’ longitudes of node and arguments of pericenter to converge. In other words, the orbits of the disk particles begin to apsidally align, producing a coherent eccentric ring. Fig. \[fig:alignment\] illustrates this with plots of the time evolution over 50 Myr of the longitudes of node, arguments of pericenter, eccentricities, and inclinations of ten randomly chosen disk particles in a collisionless $N$-body simulation with the same system parameters as the SMACK simulation described in Table \[tab:init\]. We used 10,000 particles to represent the disk, simulated with the Wisdom-Holman integrator in REBOUND with collisions turned off. The orbits of these ten randomly chosen particles become roughly apsidally aligned within $\sim4$ Myr. This behavior is consistent with the hierarchical nearly coplanar secular evolution, investigated in @Li2014, which showed that the resonance angle for low-inclination companions is the sum of the longitude of nodes ($\Omega$) and the argument of pericenter ($\omega$). The test particles, although they were initially assigned random values of $\Omega$ and $\omega$, are captured into resonance with the companion, which both pumps up their eccentricity and aligns their orbits, forming an eccentric disk.
![\[fig:alignment\] Time evolution of the argument of pericenter, longitude of nodes, eccentricity, and inclination of ten randomly chosen particles in a collisionless $N$-body simulation over 50 Myr. The dashed grey line in each plot indicates the standard deviation of the given orbital element for all the particles in the disk. The rough convergence of the argument of pericenter and longitude of nodes produces a coherent ring of particles. The eccentricities and inclinations of the particles oscillate with time.](AlignmentLongFull_new.eps){width="\columnwidth"}
The secular precession timescale of particle in the disk due an external planetary-mass companion with mass $M_{\rm pl}$ and eccentricity $e_{\rm pl}$ is defined as [@Naoz2016] $$\label{eq:tsec} t_{\rm sec}\sim \frac{16}{30\pi} \frac{M_{*}+M_{\rm pl}}{M_{\rm pl}}\frac{P_{\rm pl}^2}{P_{\rm d}}(1-e_{\rm pl}^2)^{3/2},$$ where $P_{\rm pl}$ is the companion’s period around the star (mass $M_{*}$) and $P_{\rm d}$ is the period of particles in the disk. The eccentric companion induces gravitational perturbations which result in the disk particles orbiting in a resonance, where the disk particles’ longitude of pericenter $\varpi=\omega+\Omega$ is the resonant angle [@Li2014].
The precession timescale can be used to place an upper limit on the companion’s semi-major axis. In order to perturb the entire disk (down to its inner edge at $\sim 50$ au) within the age of the system $t_{\rm age}\approx10$ Myr, the inner edge of the disk must have experienced at least one half-cycle of secular perturbation, so the secular timescale at 50 au must be $\frac{1}{2} t_{\rm sec}\lesssim t_{\rm age}$. Using a stellar mass of 2.5 ${\rm M}_{\odot}$ and a companion mass of 11 ${\rm M_{Jup}}$, and Equation \[eq:tsec\], this yields $$a_{\rm pl}\sqrt{1-e_{\rm pl}^2}\lesssim661~{\rm au}.$$
Summary and Conclusions {#sec:conclusions}
=======================
We have shown that the observed exterior companion in the HD 106906 system can shape the system’s debris disk into a flat, eccentric, dust-producing ring and reproduce its observed morphological features and asymmetries. Our SMACK simulations also allow us to place constraints on the orbit of the companion using the morphology of the disk.
While we have demonstrated that the perturbations from the observed, exterior companion can excite eccentricities in the HD 106906 ring, alternative mechanisms for eccentricity excitation also exist. For example, a second companion on an eccentric orbit interior to the debris ring could force an eccentricity on the ring. Future simulations investigating the plausibility of this scenario may be able to constrain the orbit of the outer companion based on stability requirements.
Constraining the orbit of HD 106906b could have implications for its formation scenario. Prior to the publication of resolved images of the disk, it was suggested (using $N$-body simulations) that the companion formed interior to the disk and was scattered onto a highly eccentric orbit [@Jilkova2015]. This study concluded that the disk can survive perturbations by a companion with an apocenter distance of 650 au and a pericenter distance interior to the disk if the companion’s inclination is $\gtrsim10^{\circ}$. However, this resulted in a significantly vertically perturbed disk by 10 Myr, regardless of the companion’s inclination. Our simulations indicate that a companion with an orbit completely exterior to the disk can reproduce the observed asymmetries without vertically extending the disk, supporting the scenario in which the companion formed *in situ*.
A more thorough exploration of the parameter space may be able to place further constraints on the companion’s orbit using the observed geometry of the system. The methodology we presented in this work can also be generalized to other debris disk observations to explore whether their observed asymmetries could be explained by the presence of an undetected distant exterior companion, and investigate how the morphology of these disks could constrain the orbit of their exterior perturbers. Other debris disk systems with exterior massive perturbers are likely not uncommon; surveys indicate that $\sim25\%$ of debris disks exist in binary or triple star systems [@Rodriguez2012]. Planetary-mass exterior companions like HD 106906b may be responsible for the asymmetries in observed debris disks such as HD 61005 [@Hines2007; @Esposito2016] and HD 15115 [@Rodigas2012; @Schneider2014; @MacGregor2015a], for example.
Numerical simulations were performed on the Memex High Performance Computing Cluster at the Carnegie Institution for Science. Erika Nesvold was supported by the Carnegie DTM Postdoctoral Fellowship. Smadar Naoz acknowledges partial support from a Sloan Foundation Fellowship. The authors wish to thank the anonymous referee for a prompt and helpful review.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'We propose a scheme for deterministic generation and long-term stabilization of entanglement between two electronic spin qubits confined in spatially separated quantum dots. Our approach relies on an electronic quantum bus, consisting either of quantum Hall edge channels or surface acoustic waves, that can mediate long-range coupling between localized spins over distances of tens of micrometers. Since the entanglement is actively stabilized by dissipative dynamics, our scheme is inherently robust against noise and imperfections.'
address:
- '$^{1}$Instituto de Ciencia de Materiales, CSIC, Sor Juana Ines de la Cruz, 3, Cantoblanco, 28049 Madrid, Spain'
- '$^{2}$Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany'
- '$^{3}$Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain'
- '$^{4}$Ikerbasque, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain'
author:
- 'M. Benito,$^{1,2,*}$ M. J. A. Schuetz,$^{2,*}$ J. I. Cirac,$^{2}$ G. Platero,$^{1}$ and G. Giedke$^{2,3,4}$'
title: 'Dissipative long-range entanglement generation between electronic spins'
---
[^1]
Introduction
============
The physical realization of a large-scale quantum information processing (QIP) architecture constitutes a fascinating problem at the interface between fundamental science and engineering [@HaAw08; @NielsenandChuang]. Further advances towards this goal hinge upon two major challenges: (i) control over the undesired influences of the environment which tend to corrupt genuine quantum properties such as entanglement, and (ii) long-range coupling between the logical qubits. The latter not only relaxes some serious architectural challenges [@ScBl14] but also allows for applications in quantum communication, distributed quantum computing, and some of the highest tolerances in error-correcting codes that are based on long-distance entanglement links [@NielsenandChuang; @knill05; @nickerson13].
In the solid state, electron spins confined in electrically defined semiconductor quantum dots (QDs) have emerged as a promising platform for QIP [@HKP+07; @Kloeffel2013]: Major building blocks such as initialization, single-shot readout, coherent control of single spins, and two-qubit gates between adjacent spins have been demonstrated successfully in proof-of-principle experiments. However, at present the integration of several qubits into a scalable architecture still remains a formidable challenge [@HKP+07; @Braakman2013; @Busl2013]. A large amount of wiring and control electronics needs to be accommodated on a very small scale, since interactions between QDs are very short-range, enabling QIP setups with nearest-neighbor interactions only. Therefore, a scalable design is likely to require *long-range* couplings over distances of several micrometers [@ScBl14; @ABD+13].
In this work, we propose a scheme for deterministic preparation of *steady-state* entanglement between *remote* qubits, defined by electron spins in spatially separated QDs. Our approach addresses the two challenges (i) and (ii) as described above within one unified framework: (i) By suitably engineering the continuous coupling of the system to its environment, our setup actively utilizes dissipation to create and stabilize quantum coherences, turning dissipation into the driving force behind the emergence of coherent quantum phenomena. This approach [@Plenio1999; @Braun2002; @Benatti2003; @Muschik2011] comes with potentially significant advantages over previous proposals [@Trifunovic2012; @Trifunovic2013; @Yang2015] which aim at a coherent coupling between remote spins, as dissipative methods are unaffected by timing and preparation errors and inherently robust against weak random perturbations, allowing us to stabilize entanglement for arbitrary times [@Krauter2011; @Sanchez2013; @Schuetz2013; @bohr15]. (ii) Our scheme directly builds upon recent experimental developments towards the realization of a solid-state electronic quantum bus, where flying electrons take over the role of photons in more conventional atomic, molecular, and optical based approaches in order to mediate long-range coupling between remote qubits. In particular, we consider quantum Hall edge (QHE) channels [@Komiyama1992; @Ji2003; @Stace2004; @Roulleau2008a; @Bocquillon2014; @Thalineau2014; @Yang2015] and surface acoustic waves (SAWs) [@wixforth89; @Barnes2000; @Stotz2005; @Hermelin2011; @McNeil2011; @SKG+13; @Bertrand] as exemplary candidate systems for the coherent transport of electron spins over long distances. Intuitively, the dissipative entanglement creation arises from a quantum interference effect in the *common* coupling of the localized spins $\mathbf{S}_{i} (i=1,2)$ to an adjacent electronic quantum channel, in which flying electrons continuously pass by the two localized spins. With any which-way information absent, first-order spin-flip processes between the localized spins and the flying ancilla spins occurring in the course of electron transport can happen either in the first *or* in the second node, which may lead to the formation of entanglement between the nodes, if two or more such processes with a unique common entangled steady-state dominate the dynamics [@TiVi12; @Schuetz2013]. This work is structured as follows. In Sec. \[sec:dissip\] we introduce two generic *dissipative* entanglement-generating dynamics, with a subsequent discussion on the robustness inherent to dissipative state preparation schemes. In Sec. \[sec:model\] we then propose and analyze two different physical setups, based on (i) QHE channels (see Sec. \[subsec:QHE\]) and (ii) SAW-induced moving quantum dots (see Sec. \[subsec:SAW\]), in order to approximately implement the paradigmatic schemes discussed in Sec. \[sec:dissip\]. In Sec. \[sec:results\] we turn to the central question of whether the steady-state entanglement found for the idealized dynamics can prevail in a realistic, noisy scenario. We discuss the dominant error sources, specify the experimental requirements, and provide a comprehensive comparison of the different setups. Finally, in Sec. \[sec:conclusions\] we draw conclusions and give an outlook on future directions of research.
Dissipative engineering {#sec:dissip}
=======================
Let us first consider two different generic dissipative entanglement-generating dynamics for the system’s density matrix (DM) $\rho$. A purely dissipative master equation (ME) with a unique entangled steady state is given by [@Muschik2011] $$\dot{\rho}=\alpha{\cal D}\left[\mu S_{1}^{+}+\nu S_{2}^{+}\right]\rho+\beta{\cal D}\left[\nu S_{1}^{-}+\mu S_{2}^{-}\right]\rho,\label{eq:goal1}$$ where $S_{i}^{\pm},\ i=1,2$ denote the (spin) raising and lowering operators for the two qubits and ${\cal D}\left[A\right]\rho=2A\rho A^{\dagger}-A^{\dagger}A\rho-\rho A^{\dagger}A$. For all rates $\alpha,\beta{>}0$, the dissipative evolution given in Eq. (\[eq:goal1\]) drives the system into the steady state $\left|\Psi_{\text{ss}}\right>=\mu \left| {\uparrow}{\downarrow} \right>-\nu \left|{\downarrow}{\uparrow}\right>$, which is unique and entangled for all $\mu,\nu{>}0,\ \mu{\neq}\nu$. While the entanglement is largest as $\mu\to\nu$, for equality the steady state is no longer unique (as is the case if one of the rates is zero). When there is more than one steady state, the long-time behavior depends on the initial state and may be strongly affected by small perturbations; for example, for $\beta=0$ (that is, for only one Lindblad term) in Eq. (\[eq:goal1\]). Still, a pure unique entangled steady state can be recovered by adding a suitable Hamiltonian term [@Stannigel2012], e.g., $$\dot{\rho} = -i\left[H,\rho\right]+\gamma{\cal D}\left[S_{1}^{+}+S_{2}^{+}\right]\rho,\label{eq:goal2}$$ where $H = 2\Omega (S_{1}^{x}+S_{2}^{x})-i\Delta (S_{2}^{-}S_{1}^{+}-S_{2}^{+}S_{1}^{-})$, with $S_{i}^{x}=\left(S_{i}^{+}+S_{i}^{-}\right)/2$. Here, the corresponding (unnormalized) steady state reads $\left|\Psi_{\text{ss}}\right>=\left|{\uparrow}{\uparrow}\right>+i\sqrt{2}\Omega/\Delta \left|{\cal S}\right>$, where $|{\cal S}\rangle=\left(\left|{\uparrow}{\downarrow}\right>-\left|{\downarrow}{\uparrow}\right>\right)/\sqrt{2}$ is the maximally entangled singlet state.
Our task in the following is then to find or engineer an environment for two physical spins $\mathbf{S}_i$ that leads to the effective dynamics described by Eqs. (\[eq:goal1\]) or (\[eq:goal2\]).
*Robustness.—*An important advantage of dissipative state preparation schemes is their robustness, i.e., that the relevant qualitative and quantitative features of the target state are preserved under perturbations ${\cal L}_1$ of the dynamics. It is a feature of the contractive dynamics generated by Lindblad-form Liouvillians that the schemes are inherently unaffected by transient, timing, and preparation errors; moreover, perturbations do not affect the steady-state eigenvalue, which remains 0. Standard perturbation theory (cf., e.g., [@Li2014; @Benatti2011]) shows that the changes to the steady state (and to the other eigenvalues) remain small (for a nondefective/nondegenerate ${\cal L}_0$) as long as $\alpha = \|{\cal L}_1\|$ (i.e., the strength of the perturbation) is small compared to the smallest (in modulus) nonzero eigenvalue of ${\cal L}_0$. This latter number is lower bounded by the “dissipative” or “spectral” gap of ${\cal L}_0$, determined by the eigenvalue of the Liouvillian with the largest real part different from zero, i.e., $\epsilon=-\max \left\{{\mathrm{Re}}(\lambda_{i}) \right\} $, where $\lambda_{i}$ are the nonzero eigenvalues of the Liouvillian.
The model {#sec:model}
=========
In what follows, we show how our general idea can be applied to two different exemplary physical setups, with the ultimate goal of approximately implementing the paradigmatic entanglement-generating dynamics given in Eqs. (\[eq:goal1\]) and (\[eq:goal2\]), using a fermionic environment. First, we investigate QHE states as this setup facilitates direct analogies to existing quantum optical schemes with photons [@Bocquillon2014]. Thereafter, we explore a setup based on electrically induced SAWs where the stroboscopic control over the effective interaction times between stationary and mobile electron spins [@Hermelin2011; @McNeil2011] results in larger amounts of entanglement. To treat each specific physical setup we employ two different input-output approaches tailored to the specific setups.
In all setups specified below, to controllably amplify the coupling between localized and flying electrons, we introduce auxiliary (ancilla) QDs that are tunnel-coupled to the QDs hosting the qubit electrons with spin $\mathbf{S}_{i} (i=1,2)$; by appropriate gating one can ensure that the system dots always stay occupied with a single electron each which opens up the possibility for storage of spin-spin entanglement between different (remote) quantum dots. An electron occupying the ancilla dot $j$ interacts locally with the system spin $\mathbf{S}_{i}$ via the Heisenberg exchange interaction [@Kloeffel2013] $$\begin{aligned}
H_{\text{IN}}^{i,j} & = & J_{i,j} \mathbf{S}_{i} \cdot \boldsymbol{\sigma}_{j}, \label{eq:heisenberg}\end{aligned}$$ where $\boldsymbol{\sigma}_{j} = \frac{1}{2} \sum_{\sigma,\sigma'}d_{j\sigma}^{\dagger} \boldsymbol{\tau}_{\sigma,\sigma'} d_{j\sigma'}$ refers to the spin-$1/2$ ancilla operator; here, $d_{j\sigma}^{\dagger}$ creates an electron with spin $\sigma={{\uparrow}},{{\downarrow}}$ in the ancilla dot $j$ and $ \boldsymbol{\tau}$ is the vector of Pauli matrices. The exchange coupling $J_{i,j}$ can be as large as several tens of $\mu\mathrm{eV}$ and controlled *in situ* by gating of the tunneling barrier between two nearby dots [@HKP+07; @Kloeffel2013].
The system is subject to an external magnetic field $\mathbf{B}$, taken along $\hat{z}$. In a suitable rotating frame the global homogeneous magnetic field drops out from the dynamics, and we are left with (small) inhomogeneous gradient fields, described by the Zeeman Hamiltonian $$\begin{aligned}
H_{\text{Z}} & = & \sum_{i}\delta_{i}S_{i}^{z}. \label{eq:ZeemanH}\end{aligned}$$ Here, the magnetic gradients $\delta_{i} {\lesssim} 2\mu\mathrm{eV}$ can be engineered via on-probe micro- [@PLOT+08] or nanomagnets [@Forster2015] and/or nuclear Overhauser fields [@Kloeffel2013].
. Scheme of the QHE-based setups. Two spatially separated qubits ($\mathbf{S}_{1},\mathbf{S}_{2}$) are coupled to auxiliary QDs, which are interconnected by a unidirectional QH edge channel. The upstream ancilla dot(s) are pumped selectively from a Fermi reservoir with a rate $\gamma_{\text{L}}$. While the first (purely dissipative) scheme requires two separate QHE channels, for the second scheme a single channel suffices (dashed box) together with local ESR driving fields of strength $\Omega_{i}$.](figure1-2){width="0.9\columnwidth"}
Transport via QHE states {#subsec:QHE}
------------------------
A two-dimensional electron gas (2DEG) in a large magnetic field supports QHE channels which have proven to provide an ideal test bed for electronic-optics-like experiments, since they allow for ballistic, one-dimensional, and chiral electron transport [@Bocquillon2014]; with backscattering drastically reduced due to chirality, in the QH regime the mean-free path of electrons is increased up to $\sim (0.1-1) \mathrm{mm}$ [@Komiyama1992; @Ji2003; @Stace2004]. Let us consider two nodes, consisting of just one system and one ancilla dot each, with the ancilla dots interconnected by such a chiral edge channel; compare the dashed box in Fig. \[fig:setup1\]. To describe the dynamical evolution of the system and ancilla degrees of freedom of this cascaded quantum system [@Stannigel2012; @Gardiner2004], we trace out the channel and employ the fermionic input-output formalism (see Appendix \[sec:cascaded-meq\]) [@Stace2004; @Gardiner2004]. We then arrive at the following Markovian ME for the reduced DM of system and ancilla dots, $$\begin{aligned}
\dot{\varrho} = -i\left[H_{\text{Z}}+H_{\text{IN}},\varrho\right]+{\cal L}_{\text{tr}}\varrho, \label{eq:MEtotal}\end{aligned}$$ where $H_{\text{Z}}$ accounts for Zeeman energies \[compare Eq. (\[eq:ZeemanH\])\], $H_{\text{IN}}$ describes *local* spin-spin interactions between system and auxiliary dots $$\begin{aligned}
H_{\text{IN}} = \sum_{\left<i,j\right>} H_{\text{IN}}^{i,j}, \label{eq:HIN}\end{aligned}$$ and ${\cal L}_{\text{tr}}\varrho=\sum_{\sigma} {\cal L}_{\text{tr},\sigma}\varrho$ describes electron transport. The latter reads explicitly $$\begin{aligned}
{\cal L}_{\text{tr},\sigma}\varrho & = & \frac{\gamma_{\text{L},\sigma}}{2}{\cal D}\left[d_{1\sigma}^{\dagger}\right]\varrho+\frac{\gamma}{2}{\cal D}\left[d_{1\sigma}+d_{2\sigma}\right]\varrho\nonumber \\
& + & {\frac{\gamma}{2}}\left[d_{1\sigma}^{\dagger}d_{2\sigma}-d_{2\sigma}^{\dagger}d_{1\sigma},\varrho\right]. \label{eq:cascaded}\end{aligned}$$ Here, the first term describes spin-selective pumping of the first ancilla dot, which could be achieved either via ferromagnetic leads or spin-filtering techniques [@Hanson2004]; in our dissipative setup, electron pumping (resulting in an effective electron source) is required in order to obtain a genuine nonequilibrium situation with continuous electron driving. The last two terms give the *nonlocal* incoherent and coherent contributions of the channel-mediated coupling between the ancilla dots, respectively. The theoretical treatment underlying Eq. (\[eq:cascaded\]) assumes weak coupling to the reservoir and a flat reservoir spectral density (Born-Markov approximation), an idealized dispersion-free channel, and the spin-resolved ancilla dot levels to be aligned within ${\lesssim} \gamma$ [@Stace2004]. Lastly, in accordance with the cascaded nature of the system, $\varrho$ in Eq. (\[eq:cascaded\]) accounts for a time delay between the nodes. For distances $\sim \mu\mathrm{m}$, however, one can neglect this time delay, since electron transport happens quasi-instantaneously on the relevant time scales (see Appendix \[sec:ancilla\] for an extended discussion).
For fast dissipation $(\gamma, \gamma_{\mathrm{L}} {\gg} J)$, the auxiliary dots settle into a quasisteady state ($\rho_{\text{a}}^{\text{ss}}$) on a time scale much shorter than the relevant system-dots dynamics. In this case, the system-bath coupling $H_{\text{IN}}$ can be treated perturbatively and one can adiabatically eliminate the ancilla coordinates yielding a coarse-grained equation of motion for the system spins ($\mathbf{S}_{1},\mathbf{S}_{2}$). The subsequent full calculation follows the general framework developed in [@Kessler2012] and is presented in detail in Appendix \[sec:Adiabatic-Elimination-and-effective-equation\]. The ensuing first-order contributions $\sim J$ result in effective, local magnetic fields for the system spins $\mathbf{S}_{i}$, which are oriented along the quantization axis $z$ and given by the mean value of the ancilla spins in the quasisteady state; i.e., $\left\langle \sigma_{i}^{z}\right\rangle _{\text{ss}}=\text{tr}_{\text{a}}\left\{ \sigma_{i}^{z}\rho_{\text{a}}^{\text{ss}}\right\} $ ($\mathrm{tr_{a}}[\dots]$ denotes the trace over the auxiliary degrees of freedom). As discussed in more detail below, via a suitable choice of local magnetic gradients $\delta_{i}$ in Eq. (\[eq:ZeemanH\]) these first-order terms can be chosen to vanish. To second order, nonlocal charge correlations inherent to the ancilla system are transferred to the system spins resulting in an effective master equation with one dominant nonlocal term. It reads $\Gamma_{+}^{\text{ff}}{\cal D}[{\bf v}_{\text{ff}}^{+}\cdot\left(S_{1}^{+},S_{2}^{+}\right)]\rho$, where $\rho=\mathrm{tr_{a}}[\varrho]$ and ${\bf v}_{\text{ff}}^{+}=\left(\cos\frac{\theta_{\text{ff}}}{2},\sin\frac{\theta_{\text{ff}}}{2}\right)$. Explicit expressions for $\theta_{\text{ff}}$ and $\Gamma_{+}^{\text{ff}}$ can be found in Appendix \[sec:Effective-master-equation\]. This nonlocal Lindblad term features two stationary states: $\left|\Psi_{\text{ss},1}\right>=\cos\frac{\theta_{\text{ff}}}{2} \left| \uparrow\downarrow \right>-\sin\frac{\theta_{\text{ff}}}{2} \left|\downarrow\uparrow\right>$ and a simple product state $\left|\Psi_{\text{ss},2}\right>= \left| \uparrow\uparrow\right>$. To destabilize the second (unentangled) stationary solution, we can either (i) add an extra channel or (ii) apply a coherent driving to the localized spins in order to (approximately) recover the dynamics stated in Eqs. (\[eq:goal1\]) and (\[eq:goal2\]), respectively. In this scenario (as opposed to the situation with just one nonlocal Lindblad term), the steady state is unique, which makes the scheme robust against initialization errors.
### Two channels and no driving
To mimic Eq. (\[eq:goal1\]), we consider a purely dissipative setting with two separate edge channels that are pumped spin-selectively by spin-up (spin-down) electrons only, respectively, interacting through different ancilla dots with the qubits; compare Fig. \[fig:setup1\]. Here, two separate channels are introduced in order to effectively obtain not only one, but two independent, nonlocal jump operators. The latter is needed to (approximately) emulate the paradigm master equation (1) with two independent jump operators, which (under the conditions specified in Sec. \[sec:dissip\]) ensures a unique steady state. The spin of the injected electron determines the type of nonlocal jump operator in the effective master equation for the system spins: Injecting a spin-up electron into the ancilla system will result in a collective flip $\mathcal{D}[\mu S_{1}^{+} + \nu S_{2}^{+}]\rho$, because the ancilla electron can only flip to spin-down (which comes with a spin-raising flip to the system spins), whereas injecting a spin-down electron into the ancilla system will lead to a collective flip of the form $\mathcal{D}[\nu S_{1}^{-} + \mu S_{2}^{-}]\rho$, because the ancilla electron can only flip to spin-up (which comes with a spin-lowering flip to the system spins). In this setting, the quantized levels in the ancilla dots help to suppress undesired, parasitic local processes where electrons are transferred from the lower (upper) to the upper (lower) edge channel by virtually occupying the system dot. For $J_{1}\equiv J_{1,1}=J_{2,4}$ and $J_{2}\equiv J_{2,2}=J_{1,3}$, the ensuing effective ME for the two qubits only reads $$\begin{aligned}
\dot{\rho} & = & +\Gamma_{+}^{\text{ff}}{\cal D}\left[{\bf v}_{\text{ff}}^{+}\cdot\left(S_{1}^{+},S_{2}^{+}\right)\right]\rho\label{eq:proposal1}\\
& & +\Gamma_{+}^{\text{ff}}{\cal D}\left[{\bf v}_{\text{ff}}^{+}\cdot\left(S_{2}^{-},S_{1}^{-}\right)\right]\rho+{\cal L}_{\text{n-id}}^{(1)}\rho. \nonumber \end{aligned}$$ Here, the external magnetic gradients have been chosen as $\delta_{1(2)}=\mp\left(J_{1}\langle\sigma_{1}^{z}\rangle_{ss}-J_{2}\langle\sigma_{2}^{z}\rangle_{ss}\right)$ (the index in parentheses refers to the lower sign) in order to cancel the first-order terms $\sim J$. Realistic numerical values for $\delta_{i}$ will be provided below. Explicit expressions for the mixing angle $\theta_{\text{ff}}$, the effective (second-order $\sim J^2$) rate $\Gamma_{+}^{\text{ff}}$ and the undesired terms ${\cal L}_{\text{n-id}}^{(1)}$ can be found in Appendix \[sec:Effective-master-equation\]. The ME given in Eq. (\[eq:proposal1\]) indeed features *nonlocal* transport-mediated jump terms of the same squeezing-type form as given in Eq. (\[eq:goal1\]), with $\mu\equiv\cos\frac{\theta_{\text{ff}}}{2}$ and $\nu\equiv\sin\frac{\theta_{\text{ff}}}{2}$; see inset in Fig. \[fig:result1\]a).
### One channel and driving
Next, we follow the same strategy to (approximately) recover Eq. (\[eq:goal2\]). To do so, we consider a potentially simpler setup, where a single channel suffices, but an additional (weak) resonant drive needs to be introduced; compare Fig. \[fig:setup1\]. As shown in detail in Appendix \[sec:Effective-master-equation\], again for $\gamma, \gamma_{\mathrm{L}} {\gg} J$, this system is described by $$\begin{aligned}
\dot{\rho} & = & -i\left[H_{\text{d}},\rho\right]-\Delta\left[S_{2}^{-}S_{1}^{+}-S_{1}^{-}S_{2}^{+},\rho\right]\label{eq:proposal2}\\
& + & \Gamma_{+}^{\text{ff}}{\cal D}[{\bf v}_{\text{ff}}^{+}\cdot(S_{1}^{+},S_{2}^{+})]\rho+{\cal L}_{\text{n-id}}^{(2)}\rho,\nonumber \end{aligned}$$ where $H_{\text{d}}=\sum_{i=1,2}2\Omega_{i}S_{i}^{x}$ describes electron-spin-resonance (ESR) driving of the spins in the rotating frame, and $\Delta$ is an effective, coherent spin-spin interaction mediated by the channel. Explicit expressions for $\theta_{\text{ff}}$, $\Gamma_{+}^{\text{ff}}$, $\Delta$ and ${\cal L}_{\text{n-id}}^{(2)}$ can be found in Appendix \[sec:Effective-master-equation\]. Here, the Zeeman energies have been chosen as $\delta_{i}=-J_{i}\langle\sigma_{i}^{z}\rangle_{ss}$. Again, realistic numerical values for $\delta_{i}$ will be provided below.
As evident from Eqs. (\[eq:proposal1\]) and (\[eq:proposal2\]) the *continuous* interaction of the two spin qubits with the entangled steady state of the ancilla electrons gives rise to more than just the desired Lindblad terms; cf. also Fig. \[fig:rates\].To address this limitation, we discuss below an alternative *stroboscopic* (that is, not continuous) setup which allows for better control of the system-ancilla interactions and therefore yields more ideal effective dynamics (as discussed in Sec. \[sec:dissip\]).
. Scheme of the SAW-based setups. Two spatially separated qubits ($\mathbf{S}_{1},\mathbf{S}_{2}$) are coupled to auxiliary QDs, which are interconnected by a depleted one-dimensional channel. Via mobile dots single electrons are continuously transferred between the two ancilla dots, where they interact successively with the system spins $\mathbf{S}_{i}$ for a controlled interaction time $\tau_{i}$. ](figure3-2){width="0.9\columnwidth"}
Transport via SAW moving dots {#subsec:SAW}
-----------------------------
To this end we replace the edge channels by mobile quantum dots based on SAWs. Here, we consider two ancilla QDs which are interconnected by a long depleted one-dimensional channel in a 2DEG; compare Fig. \[fig:setup2\]. Recently, it has been demonstrated experimentally that in such a setup SAWs can transfer reliably and on-demand single electrons from one dot to the other for distances of several micrometers [@Hermelin2011; @McNeil2011], with the potential to extend this to hundreds of micrometers [@Bertrand]. Our protocol then consists of a continuous train of mobile dots that interact successively with the two system spins $\mathbf{S}_{i}$ for a (electrostatically) controlled time $\tau_{i}$, very much like in a conveyor belt. Therefore, for a single ancilla electron the protocol comprises five steps: (i) load the first ancilla dot with electron spin $\sigma$, (ii) interact with system spin $\mathbf{S}_{1}$ via Heisenberg coupling (\[eq:heisenberg\]) for a time $\tau_{1}$, (iii) transfer the electron to the second ancilla dot (generically, $\mathbf{S}_{1}$ and the mobile electron are entangled by now), (iv) interact with system spin ${\mathbf{S}}_{2}$ via Heisenberg coupling (\[eq:heisenberg\]) for a time $\tau_{2}$, and (v) eject the electron from the second ancilla dot. The corresponding concatenated evolution for the two localized spins $\mathbf{S}_{i}(i=1,2)$ can be described by [@Christ2007] $$\begin{aligned}
\rho^{(n)} = \text{tr}_{\text{a}}[ e^{{\cal L}_{2,n}\tau_{2}}e^{{\cal L}_{1,n}\tau_{1}}(\rho^{(n-1)}\otimes|\sigma_{n-1}\rangle\langle\sigma_{n-1}|)], \label{eq:stroboscopicME}\end{aligned}$$ where $\rho^{(n)}$ defines the state after the $n-$th cycle of the protocol. Here, the trace is taken over the ancilla degrees of freedom and the Liouvillian ${\cal L}_{i,n}$ encodes both the interaction of the auxiliary electron with the main qubit $i=1,2$ via Eq. (\[eq:heisenberg\]) and Zeeman terms, Eq. (\[eq:ZeemanH\]). This model assumes perfect spin transfer which is approximately correct for distances much shorter than the characteristic dephasing length scale which we estimate as $\sim v_{s}T_{2}^{*} {\gtrsim} 100\mu\mathrm{m}$ for $v_{s}{\approx} 3\mu\mathrm{m}/\mathrm{ns}$ and $T_{2}^{*}{\approx} 100\mathrm{ns}$ [@McNeil2011].
Along the lines of our previous analysis, in what follows we present two SAW-based schemes: (i) a protocol with alternating spin directions and suitably synchronized exchange couplings and (ii) a spin-polarized protocol with a coherent driving. Both transport protocols will be shown to drive the localized spins to an entangled steady state, independently of the initial state.
### Alternating spin sequences
To recover the purely dissipative dynamics (\[eq:goal1\]), we assume alternating spin sequences (as could be realized by proper spin filtering on subnanosecond time scales [@Hanson2004]), together with appropriately synchronized interaction times $\tau_{i}$ or exchange couplings $J_{i}^{\sigma}\equiv J_{i,i}^{\sigma}$ (see Appendix \[sec:app-Effective-Stroboscopic-Evolution\] for a detailed derivation). This is necessary to achieve the desired asymmetry $\mu {\neq} \nu$. In the following, $\tau\equiv\tau_{1}=\tau_{2}$. Then, setting $\mu = J_{1}^{{\uparrow}}\tau=J_{2}^{{\downarrow}}\tau$, $\nu = J_{2}^{{\uparrow}}\tau=J_{1}^{{\downarrow}}\tau$, up to ${\cal O}(\tau^{3} J_{i}^{\sigma 3})$, the evolution of the DM simplifies to $$\begin{aligned}
\rho^{(n+1)} - \rho^{(n-1)} & = & \frac{1}{8} {\cal D}\left[\mu S_{1}^{+}+\nu S_{2}^{+}\right]\rho^{(n-1)}\nonumber\\
&+&\frac{1}{8} {\cal D}\left[\mu S_{2}^{-}+\nu S_{1}^{-}\right]\rho^{(n-1)},\label{eq:proposal3}\end{aligned}$$ Here, the inhomogeneous magnetic gradients have been chosen as $\delta_{1(2)}=\mp\frac{\mu-\nu}{8\tau}$, such that all first-order terms effectively vanish. Typical numerical values for $\delta_{i}$ will be provided below. Indeed, we recover *nonlocal* dissipators of the desired asymmetric (squeezing-type) form; compare Eq. (\[eq:goal1\]). Alternating sequences of spin-up and spin-down electrons (with suitably synchronized couplings) then yield approximately the desired entangling dynamics.
### Single spin-component and driving
Next, to emulate dynamics similar to Eq. (\[eq:goal2\]), we assume mobile dots with a single spin-filtered spin-component [@Hanson2004] and introduce an additional coherent external driving field. In this case, for asymmetric, but time-*independent* couplings ($\mu=J_{1}^{{\uparrow}}\tau$, $\nu=J_{2}^{{\uparrow}}\tau$), magnetic gradients $\delta_{i}=-J_{i}^{{\uparrow}}/4$ and weak driving $\Omega_{1,2}{\ll} J$, the evolution of the DM is approximately given by (see Appendix \[sec:app-Effective-Stroboscopic-Evolution\]) $$\begin{aligned}
\rho^{(n)} & = & \rho^{(n-1)}+\frac{\mu\nu}{8}\left[S_{1}^{-}S_{2}^{+}-S_{2}^{-}S_{1}^{+},\rho^{(n-1)}\right] \label{eq:proposal4} \\
& + & 1/8 {\cal D}\left[\mu S_{1}^{+}+\nu S_{2}^{+}\right]\rho^{(n-1)} -2i\tau\left[H_{\text{d}},\rho^{(n-1)}\right]. \nonumber\end{aligned}$$
Thus, we can realize the dynamics of Eqs. (\[eq:goal1\]) and (\[eq:goal2\]) with arbitrary accuracy by reducing the dwell times $\tau_i$.
Results and discussion {#sec:results}
======================
In the previous section, we have derived master-equation-based models for four different physical setups in total, two of them based on QHE channels and the remaining two based on SAW-induced moving quantum dots. In this section, we specify the experimental requirements and discuss in detail the results of our analysis, as quantified via the amount of entanglement that the different setups are able to generate between two remote spin qubits under realistic conditions. First, we discuss the QHE states based proposals, then the SAW-based proposals; we conclude the discussion with a comprehensive comparison of the different proposed setups.
![\[fig:result1\]Steady-state entanglement quantified via the $E_{F}$ for the two QHE-based proposals as a function of $\delta J$. (a) and (b) are based on Eq. (\[eq:proposal1\]) and Eq. (\[eq:proposal2\]), respectively. The solid lines refer to the ideal result, where the peak is reached for $\mu=\nu$ (see inset). The dashed lines also take into account the undesired terms, described by ${\cal L}_{\text{n-id}}^{(i)}$, while the dotted lines in addition account for nuclear dephasing (see text). Numerical parameters: $\gamma_{\text{L}}=\gamma=30\mu\text{eV}$, $J_{0}=3\mu\text{eV}$ and $\delta_{i} {\in}\left(-2,2\right)\mu\text{eV}$. In (b), for each value of $\delta J$, $\Omega_{i}$ has been optimized in the range $\Omega_{i}{\in}\left(0-50\right)\text{neV}$. ](figure2){width="1\columnwidth"}
QHE states {#sec:resultsQHE}
----------
Both Eqs. (\[eq:proposal1\]) and (\[eq:proposal2\]) potentially recover the ideal entanglement-generating dynamics given in Eq. (\[eq:goal1\]) and (\[eq:goal2\]), respectively, up to undesired terms absorbed into ${\cal L}_{\text{n-id}}^{(i)}$. We now turn to the central question of whether the entanglement inherent to the ideal dynamics can prevail in a realistic scenario. Due to the presence of the nonideal terms, even without further decoherence mechanisms, the steady state of Eqs. (\[eq:proposal1\]) and (\[eq:proposal2\]) is mixed. We confirm and quantify its entanglement using the entanglement of formation $E_F$ (see Appendix \[sec:EOF\]) [@Wootters1998]. As shown in Fig. \[fig:result1\], for a broad range of coupling parameters ($J_{1(2)}=J_0\mp \delta J$) the generation of steady-state entanglement persists in the two schemes even in the presence of the undesired terms ${\cal L}_{\text{n-id}}^{(i)}$. In order to obtain sizable steady-state entanglement (which arises from nonlocal second-order effects $\sim J^2$), the first-order contributions $\sim J$ have to be canceled via local magnetic fields as described by Eq. (\[eq:ZeemanH\]); compare our discussion in Sec. \[sec:model\]. For $\gamma_{\text{L}}=\gamma$ (as considered in the text), the Zeeman energies $\delta_i$ are typically of the order of (or smaller than) the Heisenberg coupling strengths $J_i$ (i.e., typically a few $\mu$eV); see Fig. \[fig:gradients\]. Using for example nanomagnets, gradients of this size can be readily achieved (e.g., in GaAs by local magnetic fields of a few 100mT) [@PLOT+08; @Forster2015].
![\[fig:gradients\]Value of the local magnetic fields $\delta_{1(2)}$ required to get (a) Eq. (\[eq:proposal1\]) and (b) Eq. (\[eq:proposal2\]), respectively, as a function of $\delta J$. Note that in (b) $\delta_{1,2}<0$ because we arbitrarily choose pumping with spin-up ancilla electrons. Correspondingly, for spin-down pumping the sign would be reversed. Numerical parameters: $\gamma_{\text{L}}=\gamma=30\mu\text{eV}$, $J_{0}=3\mu\text{eV}$.](SM-figure4){width="1\columnwidth"}
 Spectral gap of the dissipative dynamics (continuous red line) and dominating rate $\Gamma_{+}^{\text{ff}}$ (dotted black line) as a function of $\delta J$. (a) and (b) are based on Eq. (\[eq:proposal1\]) and Eq. (\[eq:proposal2\]), respectively. Numerical parameters: $\gamma_{\text{L}}=\gamma=30\mu\text{eV}$, $J_{0}=3\mu\text{eV}$ and $\delta_{i} {\in}\left(-2,2\right)\mu\text{eV}$. In (b), for each value of $\delta J$, $\Omega_{i}$ (green lines) have been optimized in the range $\Omega_{i}{\in}\left(0-50\right)\text{neV}$.](SM-figure6){width="1\columnwidth"}
Another important question is how long it approximately takes for the system to reach its steady state. This time scale is directly related to the spectral gap of the corresponding dissipative dynamics, which is shown in Fig. \[fig:app-gap\] for the two QHE-based proposals. The spectral gap is found to be proportional to $J_0^2/\gamma$, which can be increased for small values of $\gamma$, provided that the conditions for adiabatic elimination ($J_0\ll\gamma$) are still fulfilled. For the parameters $\gamma=30\mu$eV and $J_0=3\mu$eV (for which the adiabatic elimination of the fast degrees of freedom is perfectly valid), we then estimate $\epsilon\sim0.15\mu $eV and $\epsilon\sim0.03\mu $eV, respectively. Accordingly, the steady state is reached on a very fast time scale of roughly $\sim(5-25)$ns. Then, as discussed in Sec. \[sec:dissip\], any noise sources or imperfections that are slow compared to this very fast, zeroth-order time scale should not affect severely the qualitative and quantitative features of the steady state.
First, this is demonstrated explicitly for qubit dephasing due to nuclear spins in the (GaAs) host environment. As explained in more detail in Appendix \[sec:ap-Noise-sources\], the hyperfine interaction with the nuclei is modeled in terms of a random, slowly evolving effective magnetic field for the electron spins, yielding an extra Hamiltonian of the same form as Eq. (\[eq:ZeemanH\]), where the detuning parameters $\delta_{i}$ are sampled independently from a normal distribution with standard deviation $\sigma_{\text{nuc}}$ [@Kloeffel2013]. The resulting time-ensemble-averaged electron dephasing time $T_{2}^{*}=\sqrt{2}/\sigma_{\text{nuc}}$ has recently been extended up to $T_{2}^{*} {\approx} 3\mu\mathrm{s}$ [@Shulman2014]. As shown in Fig. \[fig:result1\], already for $T_2^*{\approx} 30$ns, the purely dissipative scheme is basically unaffected by nuclear noise.
Second, again because of the relatively large spectral gap $\epsilon$, perfect cancellation of the first-order terms $\sim J$ is not strictly required, provided that the residual (uncanceled) magnetic fields $\Delta_{i}$ are small compared to the gap; as shown in Appendix \[sec:ap-Noise-sources\], typically our scheme can tolerate residual gradients $\Delta_{i}$ of up to $\sim 0.1 \mu\text{eV}$ without severely affecting the generation of steady-state entanglement.
Lastly, in our analysis we have neglected several detrimental effects that may be encountered in an actual experiment, an approximation that we now justify: First, at sufficiently low temperatures $T{<}5\mathrm{K}$, dispersive effects and scattering out of the edge channel may be neglected for propagation distances ${\lesssim} 100\mu\mathrm{m}$ [@Stace2004]. Nevertheless, in Appendix \[sec:ap-Noise-sources\] we show that even a few percent of losses can be tolerated. Second, dephasing during propagation should be negligible for distances small compared to a characteristic coherence length scale $L_{\phi}$, which we estimate as $L_{\phi} = v_{d}T_{2}^{*} {\approx} (10^{2} - 10^{3}) \mu\mathrm{m}$ for a drift velocity $v_{d} {\approx} 10^4 \mathrm{m/s}$ and (due to motional narrowing) extended dephasing time $T_{2}^{*} {\approx} (10-100)\mathrm{ns}$ [@Stace2004; @McNeil2011; @SKG+13; @Stotz2005]. Then, in order to suppress errors due to nonresonant dot energies, these should be controlled with a precision ${\lesssim} 1 \mu \mathrm{eV}$ [@Stace2004]. Finally, based on QD experiments [@Hanson2004] where basically 100% bipolar spin-filter efficiency has been demonstrated, we have assumed perfect spin-selective driving. Still, with all these simplifications, the amount of steady-state entanglement that we obtain for a realistic scenario (with continuous ancilla-electron pumping) is modest $(E_{F} {\approx}0.2)$ as compared to the idealized cases discussed in Eqs. (\[eq:goal1\]) and (\[eq:goal2\]), respectively (even though it is still comparable to what has been predicted theoretically for two adjacent dots [@bohr15] and achieved experimentally for two atomic ensembles [@Krauter2011]). As shown below, one can largely circumvent this limitation by considering well-controlled stroboscopic interaction times between system and ancilla dots (as opposed to the arguably more simple continuous settings with largely fluctuating interaction times).
SAW moving dots {#sec:resultsSAW}
---------------
![\[fig:result2\]Steady-state entanglement quantified via the $E_{F}$ for the two SAW-based proposals as a function of $\delta J$, with $J_{1(2)}^{{\uparrow}}=J_{0}\mp\delta J$. (a) and (b) are based on Eq. (\[eq:proposal3\]) and Eq. (\[eq:proposal4\]), respectively. The solid lines refer to the ideal result, given by the lower order terms present in Eqs. (\[eq:proposal3\]) and (\[eq:proposal4\]), while the dashed lines correspond to the full evolution. The dotted lines also account for noise due to uncertainty in the dwell times and dephasing. Numerical parameters: $\sigma_{\text{\ensuremath{\tau}}}=5\%$, $J_{0}\tau{\approx}0.38$ and $T_{2}^{*}/\tau{\approx}300$. In (b), for each value of $\delta J$, $\Omega_{i}$ has been optimized in the range $\Omega_{i}\tau{\in}\left(0-1.5\right)\cdot 10^{-2}$. ](figure4){width="1\columnwidth"}
![\[fig:gradients-SAWs\]Value of the local magnetic fields $\delta_{1(2)}$ required to get (a) Eq. (\[eq:proposal3\]) and (b) Eq. (\[eq:proposal4\]), respectively, as a function of $\delta J$. Note that in (b) $\delta_{1,2}<0$ because we arbitrarily choose pumping with spin-up ancilla electrons. Correspondingly, for spin-down pumping the sign would be reversed. Numerical parameters: $J_{0}=2.5\mu\text{eV}$.](SM-figure7){width="1\columnwidth"}
The dynamical equations given in Eqs. (\[eq:proposal3\]) and (\[eq:proposal4\]) suggest that the system qubits will be driven to an entangled steady state regardless of the initial state (as long as $\tau J_{i}\ll1$). Our analytical results stated above have been confirmed by exact numerical simulations of Eq. (\[eq:stroboscopicME\]), where the ancilla degrees of freedom have not been eliminated. As demonstrated in Fig. \[fig:result2\], the generation of entanglement persists even in the presence of nuclear noise and residual time jitter. We include this noise source by choosing the interaction times $\tau_{i}$ randomly from a Gaussian distribution centered around the average $\tau$ with a standard deviation of $\sigma_{\tau}$ (see Appendix \[sec:ap-Noise-sources\] for a detailed analysis of noise sources). For sufficiently low time jitter and typical dephasing times $T_2^{*}=(30-300)\mathrm{ns}$, we find $E_{F} {\gtrsim} 0.4$, which extends up to $E_{F} {\gtrsim} 0.7$ for $T_2^{*}{\approx} 1\mu\mathrm{s}$. Typically, the steady state is reached after $\sim 10^3$ iterations, that is, within $\sim (0.1-1)\mu \mathrm{s}$ for $\tau {\approx} (0.1-1)\mathrm{ns}$. The local Zeeman energies required to effectively cancel the first-order terms are shown in Fig. \[fig:gradients-SAWs\]. However, we have also checked numerically that perfect cancellation of the first-order terms is not strictly required (for details see Appendix \[sec:ap-Noise-sources\]); accordingly, residual gradients of up to $\sim 0.03 \mu\text{eV}$ can be tolerated without severely affecting our results.
The ideal, analytical result given in Eq. (\[eq:proposal3\]) assumes the injection of alternating spin components of the form $\uparrow, \downarrow, \uparrow, \dots$. However, this condition can be relaxed to longer sequences of aligned ancilla spins, of the form $\uparrow, \uparrow, \dots, \downarrow, \downarrow,\dots,\uparrow,\uparrow,\dots$. This has been confirmed numerically in Fig. \[fig:app-strob-Passive-scheme-with-mobiledots\]. Accordingly, the switching times of the gates can be increased by about an order of magnitude without severely affecting the amount of steady-state entanglement.
. Steady-state entanglement quantified via the $E_{F}$ for the SAW-based proposal corresponding to Eq. (\[eq:proposal3\]) as a function of time ($t=2\text{n}\tau$) for two different initial states (continuous and dashed lines, respectively). Blue: Alternating spins. Orange: Alternating sequences of ten spins. Numerical parameters: $\delta J/J_{0}=0.28$ and $J_{0}\tau{\approx}0.38$ .](SM-figure8){width="1\columnwidth"}
Comparison of the Setups
------------------------
The presented proposals based on QHE states constitute continuous entangling generating setups in the sense that once the setup has been prepared there is no need to interact externally with the system before the entanglement measurement; moreover, they have been shown to drive the system to the steady state on very fast time scales (in a matter of few ns). However, this (arguably simple) continuous setting comes with the disadvantage of undesired terms in the master equations (\[eq:proposal1\]) and (\[eq:proposal2\]). As a consequence, even in the cleanest setup, we cannot go beyond a steady-state entanglement of $E_F\approx 0.2$ebits. As evidenced by our stroboscopic SAW-based scheme, this limitation can be overcome by suitably controlling the electron dwell times $\tau_i$ in the ancilla dots. In this way, the effective dynamics given in Eqs. (\[eq:proposal3\]) and (\[eq:proposal4\]) can be ensured to approach the ideal ones (by controlling the dwell times $\tau_i$). Therefore, in the limit $\tau_i\rightarrow0$ and without noise sources, we would recover the pure entangled steady states of Eqs.(\[eq:goal1\]) and (\[eq:goal2\]) and could approach *perfect* entanglement ($E_F = 1$). Here, we estimate an upper limit of $
E_F\approx 0.7$ when accounting for typical experimental parameters and imperfections. This better performance comes with the experimental challenge to transport many electrons via (for example) the SAW-created potentials reliably and with accurate (electrical) control of the electronic dwell times. Moreover, the proposal with alternating spin sequences comes with further requirements as the proper spin-filtering synchronized with the exchange couplings. However, based on recent progress demonstrated for single-electron transport experiments with SAW moving dots [@Hermelin2011; @McNeil2011; @Bertrand] and the robustness against errors (as we demonstrate here) a future, successful experimental realization of our scheme should be feasible.
. Upper and lower bounds of distillable entanglement in the steady-state quantified via the $E_{F}$ (orange) and $E_{D\to}$ (blue) for the two SAW-based proposals as a function of $\delta J$, with $J_{1(2)}^{{\uparrow}}=J_{0}\mp\delta J$. (a) and (b) show results based on Eq. (\[eq:proposal3\]) and Eq. (\[eq:proposal4\]), respectively. The solid lines correspond to the full evolution, while the dashed lines account for noise due to uncertainty in the dwell times and nuclear dephasing. Numerical parameters: $\sigma_{\text{\ensuremath{\tau}}}=5\%$, $J_{0}\tau{\approx}0.38$ and $T_{2}^{*}/\tau{\approx}300$. In (b), for each value of $\delta J$, $\Omega_{i}$ has been optimized in the range $\Omega_{i}\tau{\in}\left(0-1.5\right)\cdot 10^{-2}$.](ED2-saws-cut){width="1\columnwidth"}
Given the additional experimental challenges for an accurate control of the ancilla electron dwell times $\tau_{i}$ with synchronized (electrical) control of the Heisenberg coupling constants, one may wonder whether the increase in obtainable steady-state entanglement (in the stroboscopic SAW-based schemes) is worth the effort. This, of course, depends on the ultimate purpose of entanglement generation. When viewing entanglement production mainly as an experimental benchmark to demonstrate the capability to entangle, any entanglement measure (such as our canonical choice, the entanglement of formation $E_{\text{F}}$) would do; any state with nonzero $E_{F}$ can be shown (in principle) to be entangled either by measuring a suitable entanglement witness or by sufficiently precise state tomography. However, $E_{F}$ will not tell us, in general, how useful the state is for subsequent QIP tasks. Since most applications of entanglement require almost pure states, one of the most relevant uses of mixed-state entanglement is as an input to entanglement distillation protocols [@Bennett1996; @HHHH09]. Usefulness for such a task is measured by distillable entanglement [@BDSW96] $E_D(\rho)$, which quantifies how many pure Bell states can be obtained from many copies of $\rho$ by local operations and classical communication (per copy and in the limit of many copies). While $E_D(\rho)>0$ for all entangled states of two qubits, in general only upper and lower bounds are known. We use $E_{D\to}$, the entanglement that can be distilled using only one-way communication and which is given by [@DeWi03] $E_{D\to}(\rho)=\max\left\{ 0,S(\rho_1)-S(\rho),S(\rho_2)-S(\rho)
\right\}$, where $S$ is the von Neumann entropy and $\rho_i$ the reduced state at site $i=1,2$. Using this lower bound we find that the steady states in the *continuous* QHE-based protocols are too noisy to contain meaningful one-way distillable entanglement ($E_{D\to}(\rho_s)<0.01$), while the *stroboscopic* SAW-based schemes produce $0.1-0.2$ebits of $E_{D\to}$, cf. Fig. \[fig:ED\], showing that from a supply of $5n-10n$ such pairs we can distill $n$ high-fidelity Bell states which would, in turn, allow for, e.g., quantum teleportation or remote gate implementation. Similar considerations should apply for stroboscopic QHE-based settings with accurate control over the electron dwell times, as experimentally demonstrated for example in Ref. [@Bocquillon2014] .
Conclusions {#sec:conclusions}
===========
To conclude, we have presented a general scheme for the deterministic generation of entanglement between spins confined in spatially separated gate-defined QDs. We have detailed our ideas for two specific electron-based setups feasible with current state-of-the-art technology, for which the coherence length of the corresponding quantum channels should allow us to generate sizable entanglement ($E_F\approx 0.2-0.7$) over distances of up to $100\mu$m. While such noisy, modestly entangled two-qubit states can be used, e.g., for quantum teleportation, their main use lies in the fact that they can be distilled into highly entangled states by means of local operations on several copies [@Bennett1996; @Auer2015]. We have seen, in particular, that the stroboscopic schemes generate a sizable amount of distillable entanglement. Running our steady-state scheme on several spin qubits in parallel could provide deterministic inputs to such a distillation procedure. We have focused on GaAs-based systems, as these have been investigated most thoroughly in experiments, with the ambient nuclei posing one of the dominant sources of undesired noise. Two complementary strategies to address the role of nuclear spins in future studies would be (i) either to investigate nuclear-spin-free systems with $T_{2}^{*}{>}100\mu\mathrm{s}$ [@Hamaya2006; @Veldhorst2014] or (ii) to associate the Heisenberg coupling (\[eq:heisenberg\]) with the hyperfine interaction between ancilla electron spins and collective nuclear spin operators, with (possibly large) collective spin operators $\mathbf{I}_{i}(i=1,2)$ replacing the spin-$1/2$ system electron spins $\mathbf{S_{i}}$ considered in this work. By carefully choosing the spin-projection of the injected ancilla spins as well as the interaction times between electron and nuclear spins via the dwell times of the ancilla electrons in the QDs, one should be able to engineer a dissipative master equation of the form given in Eq. (\[eq:goal1\]), again with the replacement $\mathbf{S}_{i} \rightarrow \mathbf{I}_{i}$. Since nuclear spin ensembles typically comprise $10^{4}-10^{6}$ nuclei, this scheme could possibly generate large amounts of entanglement over mesoscopically large distances, provided that narrowed nuclear spin states with a width much smaller than the average polarization are prepared initially [@Schuetz2013].
*Acknowledgments.—*M.B. thanks the theory division of the Max Planck Institute of Quantum Optics for their hospitality. M.B. and M.J.A.S. would like to thank A. Gonzalez-Tudela for fruitful discussions. G.G. (M.B. and G.P.) acknowledges support by the Ministerio de Economia y Competitividad through Project No. FIS2014-55987-P (MAT2014-58241-P). M.J.A.S., G.G., and J.I.C. acknowledge support by the European Commission via project SIQS and by the Deutsche Forschungsgemeinschaft within the Cluster of Excellence NIM.
Entanglement of formation\[sec:EOF\]
====================================
The entanglement measure used in this work is the entanglement of formation ($E_F$) [@Wootters1998], defined as the minimum average entanglement of an ensemble of pure states that represents the mixed state $\rho$. It quantifies the necessary resources to create a given entangled state. For a mixed state $\rho$ of two qubits the concurrence is ${\cal C}=\text{max}\left\{ 0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\right\} $, where $\lambda_{i}$ are the square roots of the eigenvalues of the matrix $\rho A\rho^{*}A$ arranged in decreasing order, where $A$ is the antidiagonal matrix with elements $\left\{ -1,1,1,-1\right\} $. For two qubits it ranges from 0 (separable states) to 1 (maximally entangled states). The $E_F$ can be calculated from the concurrence as $$\begin{aligned}
E_F & = & -\frac{1+\sqrt{1-{\cal C}^{2}}}{2}\log_{2}\frac{1+\sqrt{1-{\cal C}^{2}}}{2}\nonumber \\
&& - \frac{1-\sqrt{1-{\cal C}^{2}}}{2}\log_{2}\frac{1-\sqrt{1-{\cal C}^{2}}}{2}\label{eq:EOF}\end{aligned}$$ and also ranges from $0$ to $1$.
Cascaded master equation for ancilla system {#sec:cascaded-meq}
===========================================
In Appendix \[sec:input-output\] we introduce the fermionic input-output formalism [@Gardiner2004] and apply it to “cascaded quantum systems”, which consist of quantum nodes connected through an ideal chiral reservoir. Then in Appendix \[sec:ancilla\] we employ the obtained cascaded master equation (ME) to model the ancilla quantum dots (QDs) connected via a quantum Hall edge (QHE) state as considered in the main text.
Fermionic input-output formalism {#sec:input-output}
--------------------------------
First of all, we address the interaction of a system with a Markovian reservoir of non-interacting fermions. The total Hamiltonian has the generic system Hamiltonian $H_{\text{S}}$, the bath Hamiltonian $$H_{B}=\int_{0}^{\infty}d\omega\omega f^{\dagger}(\omega)f(\omega)\ ,\label{eq:bath}$$ where $\omega$ is the bath energy and $f(\omega)$ are bath fermionic annihilation operators with anticommutation relations $\left[f(\omega),f(\omega')^{\dagger }\right]_+=\delta(\omega-\omega')$, and the interaction Hamiltonian $$H_{\text{SB}}=i\int_{0}^{\infty}d\omega\sqrt{\frac{\gamma}{2\pi}}\left\{ f^{\dagger}(\omega)d-d^{\dagger}f(\omega)\right\} \ ,\label{eq:interaction}$$ where $d$ is a fermionic annihilation operator acting on the system and the coupling to the reservoir is assumed to be independent of the frequency (Markov approximation). The Heisenberg equation of motion of the bath operators is $$\dot{f}(\omega)=-i\omega f(\omega)+\sqrt{\frac{\gamma}{2\pi}}d\ ,\label{eq:f(omega)1}$$ which can be formally integrated as $$f(\omega)=e^{-i\omega t}f(\omega,0)+\sqrt{\frac{\gamma}{2\pi}}\int_{0}^{t}dt'e^{-i\omega(t-t')}d(t')\ .\label{eq:f(omega)2}$$ Here $f\left(\omega,0\right)$ is the value of $f\left(\omega\right)$ at time $t=0$. A general system operator $a$ may commute or anticommute with the bath operators depending on its nature. We call it if even if it commutes with all bath operators and odd if not. The Heisenberg equation of motion is $$\begin{aligned}
\dot{a} & = & -\frac{i}{\hbar}\left[a,H_{\text{S}}\right]\label{eq:a1}\\
& + & \int_{0}^{\infty}d\omega\sqrt{\frac{\gamma}{2\pi}}\left\{ \mp f^{\dagger}(\omega)\left[a,d\right]_{\pm}-\left[a,d^{\dagger}\right]_{\pm}f(\omega)\right\} \ ,\nonumber \end{aligned}$$ where the top (bottom) signs apply for odd (even) $a$ operator and $\left[A,B\right]_{\pm}=AB\pm BA$. Inserting the expression (\[eq:f(omega)2\]) into Eq. (\[eq:a1\]) we derive the quantum Langevin equation $$\begin{aligned}
\dot{a} & = & -\frac{i}{\hbar}\left[a,H_{\text{S}}\right]\mp\left\{ \sqrt{\gamma}f_{\text{in}}^{\dagger}(t)+\frac{\gamma}{2}d^{\dagger}(t)\right\} \left[a,d\right]_{\pm}\nonumber \\
& & -\left[a,d^{\dagger}\right]_{\pm}\left\{ \sqrt{\gamma}f_{\text{in}}(t)+\frac{\gamma}{2}d(t)\right\} \ ,\label{eq:a2}\end{aligned}$$ where $$f_{\text{in}}(t)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega e^{-i\omega t}f(\omega,0)\label{eq:fin}$$ is called noise input field and is determined by the initial state of the bath. The noise output field, defined as the time-reversed evolution from the final time operator $f(\omega,t_{\text{f}})$, is related to it by $$f_{\text{out}}(t)-f_{\text{in}}(t)=\sqrt{\gamma}d(t)\ ,\label{eq:out-in}$$ an identity known as the input-output relation. Up to this point, no assumption has been made concerning the density operator of the bath. We will use the white-noise approximation which assumes the following correlation functions for the input field: $\left\langle f_{\text{in}}^{\dagger}\left(\omega\right)f_{\text{in}}\left(\omega'\right)\right\rangle =\bar{N}\delta\left(\omega-\omega'\right)$ and $\left\langle f_{\text{in}}\left(\omega\right)f_{\text{in}}^{\dagger}\left(\omega'\right)\right\rangle =\left(1-\bar{N}\right)\delta\left(\omega-\omega'\right)$. Here $\bar{N}$ is the Fermi distribution function of a thermal reservoir. Moreover we will assume a weak system-reservoir coupling in the sense that the correlation functions of the bath are not affected by the interaction.
The input-output formalism provides a powerful treatment for two or more subsystems sharing a common unidirectional reservoir [@H.J.Carmichael1993; @Gardiner1993; @Kolobov87], also known as cascaded quantum systems. Let us consider the case of two nodes coupled to the reservoir via Eq. (\[eq:interaction\]) with operators $d_{j}(j=1,2)$. Following the previous argument a system operator of subsystem $j$, $a_{j}$, follows the Eq. (\[eq:a2\]) with the change $d\rightarrow d_{j}$, $\gamma\rightarrow\gamma_{j}$ and $f_{\text{in}}\rightarrow f_{\text{in}}^{(j)}$. The fact that the reservoir is common and unidirectional implies a relation between the output of subsystem 1 and the input in 2. For a dispersion-free channel $f_{\text{in}}^{(2)}(t)=f_{\text{out}}^{(1)}(t-L/v)$, where $L$ is the distance between the two subsystems and $v$ the group velocity of the reservoir modes, i.e., all the output of the first subsystem is used later as the input into the second one, therefore we are able to write a generic equation for an odd (even) operator as [@Stace2004]
$$\begin{aligned}
\dot{a}(t) & = & -\frac{i}{\hbar}\left[a,H_{\text{S}}\right]\mp\left\{ \sqrt{\gamma_{1}}f_{\text{in}}^{\dagger(1)}(t)+\frac{\gamma_{1}}{2}d_{1}^{\dagger}(t)\right\} \left[a,d_{1}\right]_{\pm}-\left[a,d_{1}^{\dagger}\right]_{\pm}\left\{ \sqrt{\gamma_{1}}f_{\text{in}}^{(1)}(t)+\frac{\gamma_{1}}{2}d_{1}(t)\right\} \nonumber \\
& \mp & \left\{ \sqrt{\gamma_{2}}f_{\text{in}}^{(1)\dagger}(t-L/v)+\frac{\gamma_{2}}{2}d_{2}^{\dagger}(t)+\sqrt{\gamma_{1}\gamma_{2}}d_{1}^{\dagger}(t-L/v)\right\} \left[a,d_{2}\right]_{\pm}\nonumber \\
& - & \left[a,d_{2}^{\dagger}\right]_{\pm}\left\{ \sqrt{\gamma_{2}}f_{\text{in}}^{(1)}(t-L/v)+\frac{\gamma_{2}}{2}d_{2}(t)+\sqrt{\gamma_{1}\gamma_{2}}d_{1}(t-L/v)\right\} \ .\label{eq:a3}\end{aligned}$$
Since the coupling operators $d_{1,2}$ are fermionic annihilation (odd) operators, they (anti)commute with any (odd) even operator $a$ of the other system. Then it is clear from Eq. (\[eq:a3\]) that the time evolution of an operator of the second subsystem depends on the first one but not the other way around, which reflects the unidirectionality condition. Following [@Gardiner1993; @Cirac1997], for a dispersionless channel, the fixed time delay may be set to zero, i.e., one can choose $L/v=0^{+}$ without loss of generality. The previous equation can be easily rewritten as $$\begin{aligned}
\dot{a}(t) & = & -\frac{i}{\hbar}\left[a,H_{\text{S}}+\frac{i\sqrt{\gamma_{1}\gamma_{2}}}{2}\left(d_{1}^{\dagger}d_{2}-d_{2}^{\dagger}d_{1}\right)\right]\label{eq:a4}\\
& - & \left[a,d^{\dagger}\right]_{\pm}\left\{ \frac{d}{2}+f_{\text{in}}^{(1)}(t)\right\} \mp\left\{ \frac{d^{\dagger}}{2}+f_{\text{in}}^{\dagger(1)}(t)\right\} \left[a,d\right]_{\pm}\nonumber \end{aligned}$$ in terms of the nonlocal operator $d=\sqrt{\gamma_{1}}d_{1}+\sqrt{\gamma_{2}}d_{2}$. Once we have derived this quantum Langevin equation, we can find a ME for the partial density operator excluding the bath $\varrho$ by tracing out the bath degrees of freedom from the total density operator ${\cal W}$, $\varrho=\text{tr}_{\text{B}}\left\{ {\cal W}\right\} $. For this we make use of the relation $\text{tr}\left\{ \dot{a}(t){\cal W}\right\} =\text{tr}\left\{ a\dot{{\cal W}}(t)\right\} =\text{tr}_{\text{s}}\left\{ a\dot{\varrho}(t)\right\} $. Since any physical state is fully described by the expectation values of even observables (the odd ones have vanishing expectation value due to the parity superselection rule) we can restrict ourselves in Eq. (\[eq:a4\]) to the lower sign for all observables of interest and end up with the ME $$\begin{aligned}
\dot{\varrho} & = & -i\left[H_{\text{S}}+\frac{i\sqrt{\gamma_{1}\gamma_{2}}}{2}\left(d_{1}^{\dagger}d_{2}-d_{2}^{\dagger}d_{1}\right),\varrho\right]\nonumber \\
& + & \frac{1}{2}\left(1-\bar{N}\right){\cal D}\left[d\right]\varrho+\frac{1}{2}\bar{N}{\cal D}\left[d^{\dagger}\right]\varrho\ ,\label{eq:cascmeq}\end{aligned}$$ where ${\cal D}[A]\varrho=2A\varrho A^{\dagger}-A^{\dagger}A\varrho-\varrho A^{\dagger}A$ and $\bar{N}$ is the Fermi distribution function of the fermionic reservoir. This expression contains the nonlocal coherent and incoherent contributions of the coupling between subsystems mediated by the reservoir. For simplicity we have neglected the spin index in this derivation. Moreover, in the main text we work in a rotating frame such that the global homogeneous magnetic field drops out. If the ancilla dots energy levels are not aligned within $\gamma$, this would generate an undesired rotation of the nonlocal terms in Eq. (\[eq:cascmeq\]) [@Stace2004].
Ancilla quasisteady state {#sec:ancilla}
-------------------------
The dynamics of the ancilla QDs connected via a QHE state considered in the main text can be described by Eq. (\[eq:cascmeq\]). Note that we consider only the nearest resonant subband because the tunneling rates decrease exponentially with the distance from the dots [@Stace2004]. For simplicity, we restrict ourselves to the case $\gamma\equiv\gamma_1=\gamma_2$. Moreover, we consider the case of an empty channel $\bar{N}=0$, we need to account explicitly for spins and we add the contribution from the reservoir that pumps electrons into the first ancilla QD. Finally, if the spin-resolved levels of the two ancilla QDs are aligned, the system Hamiltonian term vanishes in a suitable rotating frame. Therefore the dynamics of the ancilla dots is described by the transport Liouville superoperator ${\cal L}_{\text{tr}}\varrho=\sum_{\sigma} {\cal L}_{\text{tr},\sigma}\varrho$ with $$\begin{aligned}
{\cal L}_{\text{tr},\sigma}\varrho & = & \frac{\gamma_{\text{L},\sigma}}{2}{\cal D}\left[d_{1\sigma}^{\dagger}\right]\varrho+\frac{\gamma}{2}{\cal D}\left[d_{1\sigma}+d_{2\sigma}\right]\varrho\nonumber \\
& + & \frac{\gamma}{2}\left[d_{1\sigma}^{\dagger}d_{2\sigma}-d_{2\sigma}^{\dagger}d_{1\sigma},\varrho\right]. \label{eq:cascaded-ap}\end{aligned}$$
For fast dissipation ($\gamma,\gamma_{\text{L}}\gg J$), the auxiliary dots settle into a quasisteady state ($\rho_{\text{a}}^{\text{ss}}$) on a time scale much shorter than the relevant system dots dynamics. We now compute and analyze this quasisteady state since it will play a central role for the system dots ME to be derived in Appendix \[sec:Effective-master-equation\]. If a single spin component is introduced, $\gamma_{\text{L},\downarrow}=0$ and $\gamma_{\text{L}}\equiv\gamma_{\text{L},\uparrow}$, the quasisteady state associated with Eq. (\[eq:cascaded-ap\]) is
$$\begin{aligned}
\rho_{\text{a}}^{\text{ss}} & = & \frac{1}{\left(\gamma_{\text{L}}+\gamma\right)\left(\gamma_{\text{L}}+2\gamma\right)^2} \left\{\gamma\left(2\gamma-\gamma_{\text{L}}\right)^2\left|0,0\right>\left<0,0\right|+\gamma_{\text{L}}\left(4\gamma^2+\gamma_{\text{L}}^2\right)\left|\uparrow,0\right>\left<\uparrow,0\right|+8\gamma_{\text{L}}\gamma^2\left|0,\uparrow\right>\left<0,\uparrow\right|\right.\nonumber \\
& - &\left.2\gamma\gamma_{\text{L}}\left(\gamma_{\text{L}}+2\gamma\right)\left( \left|\uparrow,0\right>\left<0,\uparrow\right|+\left|0,\uparrow\right>\left<\uparrow,0\right| \right)+4\gamma\gamma_{\text{L}}^2\left|\uparrow,\uparrow\right>\left<\uparrow,\uparrow\right|\right\} . \label{eq:steady-state}\end{aligned}$$
The average populations of the ancilla dots depend on the reservoir and channel rates as shown in Fig. \[fig:populations\].
![\[fig:populations\]Value of the diagonal elements of the ancilla steady state in Eq. (\[eq:steady-state\]) as a function of the ratio $\gamma_{\text{L}}/\gamma$. ](SM-figure1){width="1\columnwidth"}
For all $\gamma,\gamma_L\not=0$, the quasisteady state is entangled (due to the Markovian coupling to the common channel) and reaches an $E_F$ of $\sim 0.55$ at $\gamma_L=2\gamma$, at which point the steady state is a mixture of the two-electron state $\left| \uparrow,\uparrow\right>$ and the maximally entangled state $\left|\uparrow,0\right>-\left|0,\uparrow\right>$ that is a “dark state” for the collective coupling via the operator $(d_{1,\uparrow}+d_{2,\uparrow})$ in Eq. (\[eq:cascaded-ap\]). However, this entanglement comes in a form of limited usefulness as it involves a superposition of a single fermion in the first or in the second ancilla and due to fermionic superselection rules a single such state (while entangled [@Banuls2007; @2015arXiv151104450D]) cannot be distinguished from a separable state by local operations. Our scheme shows that this entanglement can still provide the quantum correlations necessary to produce a usable spin-qubit entanglement for the system spins, which are weakly coupled to this ancilla system.
In accordance with the cascaded nature of the system, $\varrho$ in Eq. (\[eq:cascaded-ap\]) takes into account a time delay between systems 1 and 2. If transport happens almost instantaneously even on the time-scale of the channel-ancilla coupling ($L/v\ll 1/\gamma$), the delay can be neglected and the quasisteady state in Eq. (\[eq:steady-state\]) can be understood as an equal-times state. However, this condition limits the length of the edge channels to $L<1\mu$m. For larger separations ($L/v\gg 1/\gamma$) we see that the first QD is driven into its steady state before the electrons that interact with it have time to reach the second QD. Hence we conclude that at any given time, QD1 and QD2 are not entangled; instead, QD1 is getting entangled with the bath (the electron modes in the channel connecting the two QDs). This notwithstanding, as the cascaded equation tells us, this system-bath entanglement is faithfully transported to QDs so that time-delayed measurements at the two dots show strong quantum correlations. If other quantum systems (such as the system spins in our setup) interact weakly with these two correlated ancillas they are exposed to an nonlocal master equation that can be effectively taken as an equal-time equation if $L/v$ is short compared to the time scale of the qubit dynamics, shown in Appendix \[sec:Adiabatic-Elimination-and-effective-equation\] to be on the order of $J_0^2/\gamma$. For realistic parameter values, we thus obtain a standard equal-time entangled steady state for channel lengths of up to a few tens of micrometers.
Adiabatic Elimination of the ancilla system \[sec:Adiabatic-Elimination-and-effective-equation\]
================================================================================================
Adiabatic Elimination \[sec:Adiabatic-Elimination\]
---------------------------------------------------
The adiabatic elimination is a useful method when one has a main system weakly coupled to an auxiliary system, which undergoes fast dynamics (given by a Liouvillian ${\cal L}_{0}$), since it allows us to determine the effective dynamics of the main system to (in principle) arbitrary order in the interaction [@Kessler2012]. Analogously to the Schrieffer-Wolff transformation for closed systems, it allows us to decouple the slow subspace, given by the steady state of the auxiliary system, i.e., ${\cal L}_{0}\rho_{\text{a}}^{\text{ss}}=0$ [@Note3], from the fast one. To this end, one defines the projector ${\cal P}$ by its action over the total density matrix (DM) ${\cal P}\varrho=\text{tr}_{\text{a}}\left\{ \varrho\right\} \otimes\rho_{\text{a}}^{\text{ss}}=\rho\otimes\rho_{\text{a}}^{\text{ss}}$, where we have introduced the reduced DM as the trace over the auxiliary system $\rho\equiv\text{tr}_{\text{a}}\left\{ \varrho\right\} $, and apply it to the total ME of the form $\dot{\varrho}=\left({\cal L}_{0}+{\cal V}\right)\varrho$, where ${\cal V}$ is the perturbative part. In this way we can obtain the subsequent orders of the effective Liouville operator expansion that governs the dynamics of the main system ($\dot{\rho}=\text{tr}_{\text{a}}\left\{ L_{\text{eff}}\varrho\right\} $) [@Kessler2012]. Defining the Laplace transform of ${\cal L}_{0}$ as ${\cal L}_{0}^{-1}=-\int_{0}^{\infty}d\tau e^{{\cal L}_{0}\tau}$, one can easily find $$\begin{aligned}
L_{\text{eff,1}} & = & {\cal P}{\cal V}{\cal P}\ ;\label{eq:Leff1}\\
L_{\text{eff,2}} & = & -{\cal P}{\cal V}{\cal Q}{\cal L}_{0}^{-1}{\cal Q}{\cal V}{\cal P}\ ;\label{eq:Leff2}\end{aligned}$$ where ${\cal Q}=1-{\cal P}$ is the projector into the fast subspace. The perturbation ${\cal V}$ contains the interaction between the main and auxiliary systems as well as a main-system Hamiltonian, i.e., in general $${\cal V}\varrho=-i\sum_{j=1}^{N}\left[A_{j}\otimes S_{j},\varrho\right]-i\sum_{j=1}^{N}a_{j}\left[S_{j},\varrho\right]\ .\label{eq:perturbation}$$ Here $A_{j}$ and $S_{j}$ are auxiliary and main-system operators, respectively, and $a_{j}\in {\rm I\!R}$. The first-order term of $\dot{\rho}$ is $$\text{tr}_{\text{a}}\left\{ L_{\text{eff,1}}\varrho\right\} =-i\sum_{j=1}^{N}\left[\left\langle A_{j}\right\rangle _{\text{ss}}S_{j},\rho\right]-i\sum_{j=1}^{N}a_{j}\left[S_{j},\rho\right]\ ,\label{eq:firstorder}$$ which means that to first order the main system experiences the effect of the mean values of the auxiliary-system operators in the quasisteady state, $\left\langle A_{j}\right\rangle _{\text{ss}}=\text{tr}_{\text{a}}\left\{ A_{j}\rho_{\text{a}}^{\text{ss}}\right\} $, plus the original main-system Hamiltonian. To second order, one can show $$\begin{aligned}
\text{tr}_{\text{a}}\left\{ L_{\text{eff,2}}\varrho\right\} & = & -\sum_{i,j}\text{tr}_{\text{a}}\left\{ \delta A_{i}{\cal L}_{0}^{-1}\delta A_{j}\rho_{\text{a}}^{\text{ss}}\right\} \left[S_{j}\rho,S_{i}\right]\nonumber \\
& - & \sum_{i,j}\text{tr}_{\text{a}}\left\{ \delta A_{i}{\cal L}_{0}^{-1}\rho_{\text{a}}^{\text{ss}}\delta A_{j}\right\} \left[S_{i},\rho S_{j}\right]\ ,\label{eq:trLeff2}\end{aligned}$$ where $\delta A_{j}$ are the fluctuations of the auxiliary-system operators: $\delta A_{j}=A_{j}-\left\langle A_{j}\right\rangle _{\text{ss}}$. Using the quantum regression theorem $$\begin{aligned}
\text{tr}_{\text{\text{a}}}\left\{ \delta A_{i}e^{{\cal L}_{0}\tau}\left[\delta A_{j}\rho_{\text{a}}^{ss}\right]\right\} & = & \left\langle \delta A_{i}(\tau)\delta A_{j}\right\rangle _{\text{ss}}\ ;\nonumber \\
\text{tr}_{\text{a}}\left\{ \delta A_{i}e^{{\cal L}_{0}\tau}\left[\rho_{\text{a}}^{ss}\delta A_{j}\right]\right\} & = & \left\langle \delta A_{j}\delta A_{i}(\tau)\right\rangle _{\text{ss}}\ ;\label{eq:QRT}\end{aligned}$$ and the relation $\left\langle \delta A_{j}\delta A_{i}(\tau)\right\rangle _{\text{ss}}^{*}=\left\langle \delta A_{i}^{\dagger}(\tau)\delta A_{j}^{\dagger}\right\rangle _{\text{ss}}$, Eq. (\[eq:Leff2\]) reads $$\begin{aligned}
\text{tr}_{\text{a}}\left\{ L_{\text{eff,2}}\varrho\right\} & = & \sum_{i,j}{\cal C}\left(A_{i},A_{j}\right)\left[S_{j}\rho,S_{i}\right] \nonumber \\
& + & \sum_{i,j} {\cal C}^*\left(A_{i}^{\dagger},A_{j}^{\dagger}\right)\left[S_{j}^{\dagger}\rho,S_{i}^{\dagger}\right]^{\dagger}\ ,\label{eq:trLeff2-2}\end{aligned}$$ where we introduce the correlation functions $$\begin{aligned}
{\cal C}\left(A_{i},A_{j}\right) & =&\text{tr}_{\text{a}}\left\{ \delta A_{i}{\cal L}_{0}^{-1}\delta A_{j}\rho_{\text{a}}^{\text{ss}}\right\} \ .\label{eq:Corr}\end{aligned}$$ In the specific case under consideration in the main text, ${\cal L}_{0}={\cal L}_{\text{tr}}$ and ${\cal V}\varrho=-i\left[H_{\text{Z}}+H_{\text{IN}},\varrho\right]$.\
\[sec:Effective-master-equation\]Effective master equation for the system spins
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In the following, we apply the method of adiabatic elimination developed in Appendix \[sec:Adiabatic-Elimination\] to the physical setup based on QHE states in order to eliminate the ancilla coordinates and obtain an effective ME for the system spins. An electron occupying the ancilla dot $j$ interacts locally with the system spin $\mathbf{S}_{i}$ via the Heisenberg exchange interaction [@Loss1998] $$\begin{aligned}
H_{\text{IN}}^{i,j} & = & J_{i,j} \mathbf{S}_{i} \cdot \boldsymbol{\sigma}_{j}, \label{eq:heisenberg-ap}\end{aligned}$$ where $\boldsymbol{\sigma}_{j} = \frac{1}{2} \sum_{\sigma,\sigma'}d_{j\sigma}^{\dagger} \boldsymbol{\tau} d_{j\sigma}$ refers to the spin-$1/2$ ancilla operator; here, $d_{j\sigma}^{\dagger}$ creates an electron with spin $\sigma=\uparrow,\downarrow$ in the ancilla dot $j$ and $ \boldsymbol{\tau}$ is the vector of Pauli matrices. The complete interaction Hamiltonian is then $H_{\text{IN}}=\sum_{\left<i,j\right>} H_{\text{IN}}^{i,j}$, which describes local spin-spin interactions between ancilla and system dots. According to Eq. (\[eq:perturbation\]), the generic ancilla operators $A_n$ are $\sigma_j^{\alpha}$, with $\alpha=x,y,z$ and $j=1,...4$, and the system operators $S_n$ are $J_{i,j}S_i^{\alpha}$, with $i=1,2$.
According to Eq. (\[eq:firstorder\]) the first-order contributions are given by the mean value of the magnetic field created by the ancilla electrons in the quasisteady state $\rho_{\text{a}}^{\text{ss}}$, i.e., $\left\langle \sigma_{i}^{z}\right\rangle _{\text{ss}}=\text{tr}_{\text{a}}\left\{ \sigma_{i}^{z}\rho_{\text{a}}^{\text{ss}}\right\} $, and the system Hamiltonian $$\begin{aligned}
\text{tr}_{\text{a}}\left\{ L_{\text{eff,1}}\varrho\right\} & = & -i\left[H_{\text{Z}}+\sum_{\left<i,j\right>}\left\langle \sigma_{j}^{z}\right\rangle _{\text{ss}}J_{i,j}S_{i}^{z},\rho\right]\ ;\label{eq:first-order-cancelation}\\
H_{\text{Z}} & = & \sum_{i}\delta_{i}S_{i}^{z}\ .\label{eq:Zeeman}\end{aligned}$$ The local constant fields in $H_\text{Z}$ can then be chosen such that they cancel Eq. (\[eq:first-order-cancelation\]) and will be on the order of the exchange coupling. Using Eq. (\[eq:trLeff2-2\]) we calculate the second-order contribution of the coupling to two ancilla dots connected via a unidirectional channel ($J_{i}\equiv J_{i,i}$). There is a term due to the parallel component of the Heisenberg interaction ($z-z$) $$\begin{aligned}
{\cal L}_{\text{zz}}\rho & = & \sum_{i=1}^{2}J_{i}^{2}{\cal C}\left(\sigma_{i}^{z},\sigma_{i}^{z}\right){\cal D}\left[S_{i}^{z}\right]\rho\label{eq:Lzz1}\\
& + & J_{1}J_{2}\left({\cal C}\left(\sigma_{2}^{z},\sigma_{1}^{z}\right)+{\cal C}\left(\sigma_{1}^{z},\sigma_{2}^{z}\right)\right)\left(\left[S_{1}^{z}\rho,S_{2}^{z}\right]+\left[S_{2}^{z},\rho S_{1}^{z}\right]\right)\nonumber \end{aligned}$$ and another one due to the perpendicular component ($flip-flop$) $$\begin{aligned}
{\cal L}_{\text{ff}}\rho & = & \sum_{i=1}^{2}{\cal C}\left(\sigma_{i}^{+},\sigma_{i}^{-}\right)\frac{J_{i}^{2}}{4}{\cal D}[S_{i}^{+}]\rho\label{eq:Lff1}\\
& + & {\cal C}\left(\sigma_{2}^{+},\sigma_{1}^{-}\right)\frac{J_{1}J_{2}}{4}\left\{ \left[S_{1}^{+}\rho,S_{2}^{-}\right]+\left[S_{2}^{+},\rho S_{1}^{-}\right]\right\} .\nonumber \end{aligned}$$ The correlation functions are defined in Eq. (\[eq:Corr\]). In Fig. \[fig:Ad-elim\], we represent schematically the second-order processes related to the operators $\sigma_{i}^{\pm}$. Note that the unidirectionality of the channel implies ${\cal C}\left(\sigma_{1}^{+},\sigma_{2}^{-}\right)=0$.
![\[fig:Ad-elim\]Schematic representation of the second-order correlation functions; compare Eq. (\[eq:Corr\]). The different components of $\rho_{\text{a}}^{\text{ss}}$ are coupled to the elements in rectangles via ancilla spin-flip operators $\sigma_{1,2}^{-}$. Then, the pseudoinverse of the transport Liouvillian ${\cal L}_{\text{tr}}$ couples them to the matrix elements shown in the bottom rectangles. Finally, a second application of the ancilla spin-flip operators couples the initial component to the components shown in the bottom ellipses. For simplicity, this example refers to the limiting case $\gamma_{\text{L}}\ll\gamma$,; in this regime one can restrict the discussion to the single-electron regime, where at most a single electron is found in the ancilla system (comprising the two ancilla dots) and the population of the state with one electron in each of the two auxiliary QD is negligibly small; moreover, double occupation of a single ancilla QD is disregarded due to strong Coulomb interaction effects. Note that this schematic representation refers to just two system QDs coupled to just two ancilla dots interconnected by a single channel. ](SM-figure2){width="1\columnwidth"}
For practical reasons, it is more adequate to express Eqs. (\[eq:Lzz1\]) and (\[eq:Lff1\]) by means of nonlocal terms. By simply diagonalizing the quadratic form we end up with
$${\cal L}_{\text{zz}}\rho = \Gamma_{+}^{\text{zz}}{\cal D}[\cos\frac{\theta_{\text{zz}}}{2}S_{1}^{z}+\sin\frac{\theta_{\text{zz}}}{2}S_{2}^{z}]\rho
+ \Gamma_{-}^{\text{zz}}{\cal D}[\sin\frac{\theta_{\text{zz}}}{2}S_{1}^{z}-\cos\frac{\theta_{\text{zz}}}{2}S_{2}^{z}]\rho\ \label{eq:Lzz}$$
and $${\cal L}_{\text{ff}}\rho = \Gamma_{+}^{\text{ff}}{\cal D}[\cos\frac{\theta_{\text{ff}}}{2}S_{1}^{+}+\sin\frac{\theta_{\text{ff}}}{2}S_{2}^{+}]\rho
+ \Gamma_{-}^{\text{ff}}{\cal D}[\sin\frac{\theta_{\text{ff}}}{2}S_{1}^{+}-\cos\frac{\theta_{\text{ff}}}{2}S_{2}^{+}]\rho
- \Delta\left[S_{2}^{-}S_{1}^{+}-S_{1}^{-}S_{2}^{+},\rho\right]\ .\label{eq:ff}$$ The rates in Eqs. (\[eq:Lzz\]) and (\[eq:ff\]) are all given in terms of the correlation functions as $$\begin{aligned}
\Gamma_{\pm}^{\text{zz}} & = & \frac{1}{2}\sum_{i=1}^{2}{\cal C}\left(\sigma_{i}^{z},\sigma_{i}^{z}\right)J_{i}^{2}\pm\frac{1}{2}\sqrt{\left[{\cal C}\left(\sigma_{1}^{z},\sigma_{1}^{z}\right)J_{1}^{2}-{\cal C}\left(\sigma_{2}^{z},\sigma_{2}^{z}\right)J_{2}^{2}\right]^{2}+\left[{\cal C}\left(\sigma_{1}^{z},\sigma_{2}^{z}\right)+{\cal C}\left(\sigma_{2}^{z},\sigma_{1}^{z}\right)\right]^{2}J_{1}^{2}J_{2}^{2}};\label{eq:ratezz}\\
\Gamma_{\pm}^{\text{ff}} & = & \frac{1}{8}\sum_{i=1}^{2}{\cal C}\left(\sigma_{i}^{+},\sigma_{i}^{-}\right)J_{i}^{2}\pm\frac{1}{8}\sqrt{\left[{\cal C}\left(\sigma_{1}^{+},\sigma_{1}^{-}\right)J_{1}^{2}-{\cal C}\left(\sigma_{2}^{+},\sigma_{2}^{-}\right)J_{2}^{2}\right]^{2}+{\cal C}\left(\sigma_{2}^{+},\sigma_{1}^{-}\right)^{2}J_{1}^{2}J_{2}^{2}};\label{eq:rateff}\end{aligned}$$
and the angles that define the nonlocal operators into the Lindblad dissipators as $$\begin{aligned}
\theta_{\text{zz}} & = & \arctan\frac{\left({\cal C}\left(\sigma_{1}^{z},\sigma_{2}^{z}\right)+{\cal C}\left(\sigma_{2}^{z},\sigma_{1}^{z}\right)\right)J_{1}J_{2}}{{\cal C}\left(\sigma_{1}^{z},\sigma_{1}^{z}\right)J_{1}^{2}-{\cal C}\left(\sigma_{2}^{z},\sigma_{2}^{z}\right)J_{2}^{2}}\ ;\label{eq:thetaz}\\
\theta_{\text{ff}} & = & \arctan\frac{{\cal C}\left(\sigma_{2}^{+},\sigma_{1}^{-}\right)J_{1}J_{2}}{{\cal C}\left(\sigma_{1}^{+},\sigma_{1}^{-}\right)J_{1}^{2}-{\cal C}\left(\sigma_{2}^{+},\sigma_{2}^{-}\right)J_{2}^{2}}\ .\label{eq:thetaff}\end{aligned}$$ Finally, the Hamiltonian term in Eq. (\[eq:ff\]) is an effective coherent spin interaction between the spatially separated spins mediated by the reservoir with strength $$\Delta=\frac{{\cal C}\left(\sigma_{2}^{+},\sigma_{1}^{-}\right)J_{1}J_{2}}{8}\ .\label{eq:Delta}$$
Following the intuition of spin-flip processes between the localized spins and the ancilla electrons, we expect that a nonlocal term may dominate over all other processes. In Fig. \[fig:rates\] a) the different rates contributing to Eqs. (\[eq:ratezz\]) and (\[eq:rateff\]) are shown as a function of the coupling strength difference $\delta J$, with $J_{1(2)}=J_{0}\mp\delta J$. Clearly, the rate $\Gamma_{+}^{\text{ff}}$ is found to dominate; however, other processes may not be neglected completely. Note that we have chosen the case of equal rates $\gamma_{\text{L}}=\gamma$ for simplicity because it is close to the optimum working point. For this particular case $$\begin{aligned}
\rho_{\text{a}}^{\text{ss}} & = & \frac{1}{18} \left\{
\left|0,0\right>\left<0,0\right|+5\left|\uparrow,0\right>\left<\uparrow,0\right|+8\left|0,\uparrow\right>\left<0,\uparrow\right|\right.\nonumber \\
& - &\left. 6\left( \left|\uparrow,0\right>\left<0,\uparrow\right|+\left|0,\uparrow\right>\left<\uparrow,0\right| \right)+4\left|\uparrow,\uparrow\right>\left<\uparrow,\uparrow\right|
\right\} . \label{eq:steady-state-2}\end{aligned}$$ Then the average fields are $\left\langle \sigma_{1}^{z}\right\rangle _{\text{ss}}=1/4$ and $\left\langle \sigma_{2}^{z}\right\rangle _{\text{ss}}=1/3$ and the correlation functions are ${\cal C}\left(\sigma_{1}^{+},\sigma_{1}^{-}\right)=1/(2\gamma)$, ${\cal C}\left(\sigma_{2}^{+},\sigma_{2}^{-}\right)=76/(63\gamma)$, ${\cal C}\left(\sigma_{2}^{+},\sigma_{1}^{-}\right)=22/(21\gamma)$, ${\cal C}\left(\sigma_{1}^{z},\sigma_{1}^{z}\right)=1/(32\gamma)$, ${\cal C}\left(\sigma_{2}^{z},\sigma_{2}^{z}\right)=1/(54\gamma)$, ${\cal C}\left(\sigma_{1}^{z},\sigma_{2}^{z}\right)=-1/(72\gamma)$ and ${\cal C}\left(\sigma_{2}^{z},\sigma_{1}^{z}\right)=1/(72\gamma)$.
 Rates of the effective ME for the system spins. Since the rate $\Gamma_{+}^{\text{ff}}$ dominates, we show in (b) the structure of the corresponding nonlocal operator $\cos\frac{\theta_{\text{ff}}}{2}S_{1}^{+}+\sin\frac{\theta_{\text{ff}}}{2}S_{2}^{+}$ as a function of $\delta J$, with $J_{1(2)}=J_{0}\mp\delta J$. ](SM-figure3){width="1\columnwidth"}
The dominating term $\Gamma_{+}^{\text{ff}}{\cal D}[\cos\frac{\theta_{\text{ff}}}{2}S_{1}^{+}+\sin\frac{\theta_{\text{ff}}}{2}S_{2}^{+}]\rho$ \[see the structure in Fig. \[fig:rates\] b)\] possesses two stationary states: $\left|\Psi_{\text{ss},1}\right>=\cos\frac{\theta_{\text{ff}}}{2} \left| \uparrow\downarrow \right>-\sin\frac{\theta_{\text{ff}}}{2} \left|\downarrow\uparrow\right>$ and $\left|\Psi_{\text{ss},2}\right>= \left| \uparrow\uparrow\right>$. To make it unique, we can (i) add an extra channel or (ii) apply a coherent driving to the localized spins.
### Two channels and no driving
We introduce an extra channel at the top with electrons flying in the opposite direction (from 4 to 3 in Fig. \[fig:setup1\]), opposite spin polarization and with the following symmetry in the exchange couplings: $J_{1}\equiv J_{1,1}=J_{2,4}$, $J_{2}\equiv J_{2,2}=J_{1,3}$. Summing up the first-order contributions from the two channels, the Zeeman energies (\[eq:ZeemanH\]) necessary to cancel the first-order term are (see Eq. \[eq:first-order-cancelation\]) $\delta_{1(2)}=\mp\left(J_{1}\langle\sigma_{1}^{z}\rangle_{ss}-J_{2}\langle\sigma_{2}^{z}\rangle_{ss}\right)$ (the index in parentheses refers to the lower sign), which in the case of equal rates become $\delta_{1(2)}=\pm\frac{J_{0}+7\delta J}{12}$.
For the second-order term of the adiabatic elimination we need to calculate the correlation functions ${\cal C}\left(\sigma_{i}^{+},\sigma_{j}^{-}\right)$ and ${\cal C}\left(\sigma_{i}^{z},\sigma_{j}^{z}\right)$; $i,j=1,...4$; in particular this includes cross-correlations between the two channels. As the ancilla dot 4 (3) is symmetric to 1 (2), the correlations into the same channel do not need to be computed again. Since the ancilla quasisteady state does not contain any cross-channel correlations, nonlocal, cross-channel correlators vanish (when one traces out the ancilla degrees of freedom). Then the new channel contributes mainly with the dissipator $\Gamma_{+}^{\text{ff}}{\cal D}\left[\cos\frac{\theta_{\text{ff}}}{2}S_{2}^{-}+\sin\frac{\theta_{\text{ff}}}{2}S_{1}^{-}\right]\rho$ (note the symmetry $S_{1}^{+(-)}\leftrightarrow S_{2}^{-(+)}$) and the effective ME for the system spins is $$\begin{aligned}
\dot{\rho} & = & +\Gamma_{+}^{\text{ff}}{\cal D}\left[{\bf v}_{\text{ff}}^{+}\cdot\left(S_{1}^{+},S_{2}^{+}\right)\right]\rho\label{eq:meqeff1}\\
& + & \Gamma_{+}^{\text{ff}}{\cal D}\left[{\bf v}_{\text{ff}}^{+}\cdot\left(S_{2}^{-},S_{1}^{-}\right)\right]\rho+{\cal L}_{\text{n-id}}^{(1)}\rho \ ,\nonumber \end{aligned}$$ where we have included all the non-dominating (nonideal) terms in $$\begin{aligned}
{\cal L}_{\text{n-id}}^{(1)}\rho & = & -2\Delta\left[S_{2}^{-}S_{1}^{+}-S_{1}^{-}S_{2}^{+},\rho\right]\label{eq:Lnoise1} \\
& + & \sum_{\text{\ensuremath{\sigma}=\ensuremath{\pm}}}\Gamma_{\sigma}^{\text{zz}}{\cal D}[{\bf v}_{\text{zz}}^{\text{\ensuremath{\sigma}}}\cdot\left(S_{1}^{z},S_{2}^{z}\right)]\rho\nonumber \\
& + & \sum_{\text{\ensuremath{\sigma}=\ensuremath{\pm}}}\Gamma_{\sigma}^{\text{zz}}{\cal D}[{\bf v}_{\text{zz}}^{\text{\ensuremath{\sigma}}}\cdot\left(S_{2}^{z},S_{1}^{z}\right)]\rho\nonumber \\
& + & \Gamma_{-}^{\text{ff}}{\cal D}[{\bf v}_{\text{ff}}^{\text{-}}\cdot\left(S_{1}^{+},S_{2}^{+}\right)]\rho+ \Gamma_{-}^{\text{ff}}{\cal D}[{\bf v}_{\text{ff}}^{\text{-}}\cdot\left(S_{2}^{-},S_{1}^{-}\right)]\rho\ .\nonumber\end{aligned}$$ Here ${\bf v}_{\text{a}}^{+}=\left(\cos\frac{\theta_{\text{a}}}{2},\sin\frac{\theta_{\text{a}}}{2}\right)$ and ${\bf v}_{\text{a}}^{-}=\left(\sin\frac{\theta_{\text{a}}}{2},-\cos\frac{\theta_{\text{a}}}{2}\right)$, for $\text{a}=\text{ff},\text{zz}$.
### One channel and driving
The second solution avoids the inclusion of a second channel and the extra ancilla QDs and consists of applying a weak coherent driving field in resonance with the Zeeman frequency, giving rise to the equation $$\begin{aligned}
\dot{\rho} & = & -i\left[H_{\text{d}},\rho\right]-\Delta\left[S_{2}^{-}S_{1}^{+}-S_{1}^{-}S_{2}^{+},\rho\right]\nonumber \\
& + & \Gamma_{+}^{\text{ff}}{\cal D}[{\bf v}_{\text{ff}}^{+}\cdot(S_{1}^{+},S_{2}^{+})]\rho+{\cal L}_{\text{n-id}}^{(2)}\rho\ ,\label{eq:meqeff2}\end{aligned}$$ with the nonideal part $$\begin{aligned}
{\cal L}_{\text{n-id}}^{(2)}\rho & = & \sum_{\text{\ensuremath{\sigma}=\ensuremath{\pm}}}\Gamma_{\sigma}^{\text{zz}}{\cal D}[{\bf v}_{\text{zz}}^{\text{\ensuremath{\sigma}}}\cdot\left(S_{1}^{z},S_{2}^{z}\right)]\rho\nonumber \\
& + & \Gamma_{-}^{\text{ff}}{\cal D}[{\bf v}_{\text{ff}}^{\text{-}}\cdot\left(S_{1}^{+},S_{2}^{+}\right)]\rho\ .\label{eq:Lnoise2}\end{aligned}$$ In this case, the Zeeman energies are $\delta_{i}=-J_{i}\langle\sigma_{i}^{z}\rangle_{ss}$.
. Steady-state entanglement between two remote qubits quantified via the $E_{F}$ for the two QHE-based proposals as a function of $\delta J$. The solid lines in (a) and (b) refer to Eq. (\[eq:meqeff1\]) and Eq. (\[eq:meqeff2\]), respectively, while the blue dots are calculated with the full ME including ancilla QDs in order to check the validity of our perturbative treatment. Numerical parameters: $\gamma_{\text{L}}=\gamma=30\mu\text{eV}$, $J_{0}=3\mu\text{eV}$ and $\delta_{i} {\in}\left(-2,2\right)\mu\text{eV}$. In (b), for each value of $\delta J$, $\Omega_{i}$ has been optimized in the range $\Omega_{i}{\in}\left(0-50\right)\text{neV}$. ](SM-figure5){width="1\columnwidth"}
Validity of adiabatic elimination\[sec:validity\]
-------------------------------------------------
In the main text, we discuss to what extent the entanglement of the localized spins inherent to the ideal dynamics persists despite the undesired terms absorbed into ${\cal L}_{\text{n-id}}^{(i)}$. These results are based on the previous adiabatic elimination of ancilla dots. To check the validity of our perturbative treatment, in Fig. \[fig:full-numerics\] we compare the entanglement in the steady state resulting from the full ME including ancilla QDs to the Eqs. (\[eq:meqeff1\]) and (\[eq:meqeff2\]), i.e., after adiabatic elimination. For the experimentally achievable parameters $\gamma=30\mu\text{eV}$ and $J_{0}=3\mu\text{eV}$ the agreement is very good, showing that the approximation is valid for physically achievable conditions and it is possible to work with the simplified effective ME for the system spins. Obviously, the approximation becomes less accurate for larger values of the coupling $J_{0}$ with respect to $\gamma$ (not shown).
Effective Stroboscopic Evolution\[sec:app-Effective-Stroboscopic-Evolution\]
============================================================================
In this appendix, we provide further details for the SAW-based setup explained in the main text. The protocol consists of a continuous train of mobile dots that interact successively with the two system spins. The concatenated evolution of the localized spins DM is described by $$\begin{aligned}
\rho^{(n)} & = & \text{tr}_{\text{a}}\left[ e^{{\cal L}_{2,n}\tau_{2}}e^{{\cal L}_{1,n}\tau_{1}}\left(\varrho^{(n-1)}\right)\right] \ ,\label{eq:rhon-1}\\
\varrho^{(n-1)} & = & \rho^{(n-1)}\otimes|\sigma_{n-1}\rangle\langle\sigma_{n-1}|\end{aligned}$$ where $\rho^{(n)}$ defines the state of the system after the $n$-th cycle of the protocol and the Liouvillian ${\cal L}_{i,n}$ encodes the interaction of the ancilla electron with the system spin $i$ and the Zeeman Hamiltonian (\[eq:Zeeman\]). Still, dephasing during transport could be included straightforwardly in this model by adding a corresponding super-operator in between the two interaction terms. For $J_i\tau_j,\delta_i\tau_j\ll1$, we can perform a short-time Taylor expansion $e^{{\cal L}_{i,n}\tau}=1+\tau{\cal L}_{i,n}+\frac{\tau^{2}}{2}{\cal L}_{i,n}^{2}+...$ to approximate $\rho^{(n)}$ to second order (let us employ for simplicity equal times $\tau\equiv\tau_{1}=\tau_{2}$)
$$\begin{aligned}
\rho^{(n)} & = & \text{tr}_{\text{a}}\left\{ \varrho^{(n-1)}-i\tau\left[H_{\text{Z}}+H_{\text{IN}}^{1,1},\varrho^{(n-1)}\right]-i\tau\left[H_{\text{Z}}+H_{\text{IN}}^{2,2},\varrho^{(n-1)}\right]\right\} \nonumber \\
& + & \text{tr}_{\text{a}}\left\{ \frac{\tau^{2}}{2}{\cal D}\left[H_{\text{Z}}+H_{\text{IN}}^{1,1}\right]\varrho^{(n-1)}+\frac{\tau^{2}}{2}{\cal D}\left[H_{\text{Z}}+H_{\text{IN}}^{2,2}\right]\varrho^{(n-1)}\right\} \label{eq:general-step-2-1}\\
& + & \tau^{2}\text{tr}_{\text{a}}\left\{ \left[H_{\text{Z}}+H_{\text{IN}}^{2,2},\varrho^{(n-1)}\left(H_{\text{Z}}+H_{\text{IN}}^{1,1}\right)\right]+\left[\left(H_{\text{Z}}+H_{\text{IN}}^{1,1}\right)\varrho^{(n-1)},H_{\text{Z}}+H_{\text{IN}}^{2,2}\right]\right\} +{\cal O}\left(\tau^{3}J^{3}\right)\ .\nonumber \end{aligned}$$
When the injected spin is $\left|\sigma_{n-1}\right\rangle =\left|\uparrow\right\rangle $, $$\begin{aligned}
\rho^{(n)} & = & \rho^{(n-1)}-2i\tau\left[\delta_{1}S_{1}^{z}+\delta_{2}S_{2}^{z},\rho^{(n-1)}\right]-\frac{i}{2}\tau\left[J_{1}^{\uparrow}S_{1}^{z}+J_{2}^{\uparrow}S_{2}^{z},\rho^{(n-1)}\right]\label{eq:up}\\
& + & \frac{\tau^{2}}{2}{\cal D}\left[\left(2\delta_{1}+\frac{J_{1}^{\uparrow}}{2}\right)S_{1}^{z}+\left(2\delta_{2}+\frac{J_{2}^{\uparrow}}{2}\right)S_{2}^{z}\right]\rho^{(n-1)}\nonumber \\
& + & \frac{1}{8}{\cal D}\left[\tau J_{1}^{\uparrow}S_{1}^{+}+\tau J_{2}^{\uparrow}S_{2}^{+}\right]\rho^{(n-1)}+\tau^{2}\frac{J_{1}^{\uparrow}J_{2}^{\uparrow}}{8}\left[S_{1}^{-}S_{2}^{+}-S_{2}^{-}S_{1}^{+},\rho^{(n-1)}\right]+{\cal O}\left(\tau^{3}J^{3}\right)\ \nonumber \end{aligned}$$ and if $\left|\sigma_{n}\right\rangle =\left|\downarrow\right\rangle$ the next step is given by $$\begin{aligned}
\rho^{(n+1)} & \simeq & \rho^{(n)}-2i\tau\left[\delta_{1}S_{1}^{z}+\delta_{2}S_{2}^{z},\rho^{(n)}\right]+\frac{i}{2}\tau\left[J_{1}^{\downarrow}S_{1}^{z}+J_{2}^{\downarrow}S_{2}^{z},\rho^{(n)}\right]\label{eq:down}\\
& + & \frac{\tau^{2}}{2}{\cal D}\left[\left(2\delta_{1}-\frac{J_{1}^{\downarrow}}{2}\right)S_{1}^{z}+\left(2\delta_{2}-\frac{J_{2}^{\downarrow}}{2}\right)S_{2}^{z}\right]\rho^{(n)}\nonumber \\
& + & \frac{1}{8}{\cal D}\left[\tau J_{1}^{\downarrow}S_{1}^{-}+\tau J_{2}^{\downarrow}S_{2}^{-}\right]\rho^{(n)}-\tau^{2}\frac{J_{1}^{\downarrow}J_{2}^{\downarrow}}{8}\left[S_{1}^{-}S_{2}^{+}-S_{2}^{-}S_{1}^{+},\rho^{(n)}\right]+{\cal O}\left(\tau^{3}J^{3}\right)\ .\nonumber\end{aligned}$$
Analogously to the two proposals of the QHE-based setup, we consider (i) a protocol with alternating spin directions and suitably synchronized exchange couplings and (ii) a spin-polarized protocol with a coherent driving. Both transport protocols drive the localized spins to an entangled state independent of the initial state.
### Alternating spin sequences
The concatenation of two steps with the injection of an opposite spin results in a first-order term that can be canceled by choosing the Zeeman energies as $\delta_{i}=-\frac{ J_{i}^{\uparrow}- J_{i}^{\downarrow}}{8}$. Setting in addition $\tau
J_{1}^{\uparrow} = \tau J_{2}^{\downarrow}\equiv \mu$ and $\tau J_{1}^{\downarrow} =
\tau J_{2}^{\uparrow}\equiv\nu$, this is simply a gradient of magnetic field between the two localized spins: $\delta_{1(2)}=\mp\frac{\delta J}{4}$, with $J_{1(2)}^{\uparrow}=J_0\mp\delta J$. Not only the first-order terms but also the dephasing second-order terms in Eqs. (\[eq:up\]) and (\[eq:down\]) cancel and it is readily seen that $$\begin{aligned}
\rho^{(n+1)} & = & \rho^{(n-1)}+\frac{1}{8}{\cal D}\left[\mu S_{1}^{+}+\nu S_{2}^{+}\right]\rho^{(n-1)}\label{eq:n+1-n-1-simple}\\
& + & \frac{1}{8}{\cal D}\left[\nu S_{1}^{-}+\mu S_{2}^{-}\right]\rho^{(n-1)}+{\cal O}\left(\tau^{3}J^{3}\right)\ ,\nonumber
\end{aligned}$$ whose second-order terms are the considered ideal dynamics in the main text because they have a unique pure entangled steady state.
### Single spin-component and driving
For the protocol with a single spin component the approximated stroboscopic evolution is given by Eq. (\[eq:up\]), therefore by choosing the magnetic fields with strengths $\delta_{i}=-J_{i}^{\uparrow}/4$, we cancel the first-order contribution. With the definitions $\mu=J_{1}^{\uparrow}\tau$ and $\nu=J_{2}^{\uparrow}\tau$ and applying a coherent driving $$H_{\text{d}}=\sum_{i=1,2}2\Omega_{i}S_{i}^{x}\label{eq:Hdriving}$$ such that $\Omega_{1,2}\ll J$, the stroboscopic evolution reads $$\begin{aligned}
\rho^{(n)} & = & \rho^{(n-1)}-2i\tau\left[H_{\text{d}},\rho^{(n-1)}\right]\label{eq:n-n-1-driving}\\
& + & \frac{\mu\nu}{8}\left[S_{1}^{-}S_{2}^{+}-S_{2}^{-}S_{1}^{+},\rho^{(n-1)}\right]\nonumber \\
& + & \frac{1}{8}{\cal D}\left[\mu S_{1}^{+}+\nu S_{2}^{+}\right]\rho^{(n-1)}+{\cal O}\left(\tau^{3}J^{3}\right)\ ,\nonumber \end{aligned}$$ which is also like the desired one up to second order.
Note that for a direct comparison of Eqs. (\[eq:n+1-n-1-simple\]) and (\[eq:n-n-1-driving\]) with a ME, one needs to assume infinitesimal interaction times, but we have confirmed that the schemes work for finite interaction times.
Noise Sources\[sec:ap-Noise-sources\]
=====================================
In this appendix we detail the different noise sources taken into account in the proposed setups. First of all, we account for qubit dephasing induced by nuclear spins in the (GaAs) host environment. Second, we consider electron losses due to imperfections in the transport mechanisms. Then, we analyze the effect of an imperfect cancellation of the first-order terms, i.e., the effect of some residual gradient. Finally, in the SAW-based proposal we account for imperfections due to uncertainties in the effective electron interaction times.
To account for dephasing due to the nuclear spins, we follow the standard treatment [@Chekhovich2013] and assume that the spins in the QDs experience non-Markovian noise. The fluctuations of the Overhauser field lead to a time-ensemble-averaged electron dephasing time $T_{2}^{*}$, that is related to the width of the nuclear field distribution $\sigma_{\text{nuc}}$ as $T_{2}^{*}=\sqrt{2}/\sigma_{\text{nuc}}$. In order to model this effect, we have to include the Hamiltonian [@Taylor2007; @Kloeffel2013; @Chekhovich2013] $$H_{\text{deph}}=\sum_{i=1,2}B_{i}^{\text{nuc}}S_{i}^{\text{z}}\label{eq:Ldeph}$$ with random parameters $B_{i}^{\text{nuc}}$ sampled independently from a normal distribution with standard deviation $\sigma_{\text{nuc}}$.
Before we proceed, we note that, due to the several time scales involved, our scheme should be very amenable to the inclusion of dynamical decoupling techniques, which allow for significantly extended electron coherence times, $T_2 \approx 10^{2}T_{2}^{*}$ [@Chekhovich2013; @Kloeffel2013; @Taylor2007].
Transport via QHE states {#sec:noiseQHE}
------------------------
The full MEs were derived in Appendix \[sec:Effective-master-equation\]. In Fig. \[fig:coop\] we plot the $E_F$ of the steady state for different values of the 9-like parameter $C=J_{0}^{2}/\gamma\sigma_{\text{nuc}}$, which compares desired $\sim J_0^2/\gamma$ to undesired $\sim \sigma_{\text{nuc}}\sim 1/T_{2}^{*}$ rates. As expected from the analysis of the spectral gap the purely dissipative proposal is typically found to be more robust. By choosing the values $\gamma=30\mu\text{eV}$ and $J_{0}=3\mu\text{eV}$ we can predict that a value of $\sigma_{\text{nuc}}=0.03\mu\text{eV}$, which corresponds to a cooperativity $C=10$, would be very good concerning the purely dissipative proposal. This standard deviation corresponds to a dephasing time $T_{2}^{*}\simeq30\text{ns}$, which is experimentally feasible and can be improved up to $3\mu\text{s}$ using nuclear-state-narrowing techniques [@Shulman2014; @Chekhovich2013].
![\[fig:coop\]Steady-state entanglement between two remote qubits quantified via the $E_{F}$ for the two QHE-based proposals as a function of the cooperativity $C=J_{0}^{2}/\gamma\sigma_{\text{nuc}}$. The solid (dotted) line results are based on Eq. (\[eq:meqeff1\]) and Eq. (\[eq:meqeff2\]), respectively. Numerical parameters: $\gamma_{\text{L}}=\gamma=30\mu\text{eV}$, $J_{0}=3\mu\text{eV}$, $\delta J/J_{0}=0.44$ ($\delta J/J_{0}=0.14$) for solid (dotted) line and $\delta_{i} {\in}\left(-2,2\right)\mu\text{eV}$. ](SM-figure9){width="1\columnwidth"}
To model the possible electron losses due to imperfections in the transport channel, we include a Lindblad operator with rate $\Gamma_l$ acting in the first ancilla QD, i.e., $\sum_{\sigma}\Gamma_l/2{\cal{D}}\left[d_1^{\sigma}\right]$ (also in $d_4^{\sigma}$ in the two-channels proposal). The result, shown in Fig. \[fig:losses-QHE\], predicts that we can afford a small percent of losses.
. Steady-state entanglement between two remote qubits quantified via the $E_{F}$ for the two QHE-based proposals as a function of $\delta J$. The solid lines in (a) and (b) refer to Eq. (\[eq:meqeff1\]) and Eq. (\[eq:meqeff2\]), respectively, while the dots are calculated with the full ME including ancilla QDs and different losses rates $\Gamma_{l}$. Numerical parameters: $\gamma_{\text{L}}=\gamma=30\mu\text{eV}$, $J_{0}=3\mu\text{eV}$ and $\delta_{i} {\in}\left(-2,2\right)\mu\text{eV}$. In (b), for each value of $\delta J$, $\Omega_{i}$ has been optimized in the range $\Omega_{i}{\in}\left(0-50\right)\text{neV}$. ](SM-figure10){width="1\columnwidth"}
Finally, we verify in Fig. \[fig:no-cancellation\] (a) that the perfect cancellation of the first-order terms is not necessary, provided that the residual gradients $\Delta_{i}$ are small compared to the gap.
![\[fig:no-cancellation\] Steady-state entanglement between two remote qubits quantified via the $E_F$ for two proposals as a function of $\delta J$. The solid lines in (a) and (b) refer to Eq. (\[eq:meqeff1\]) and Eq. (\[eq:n+1-n-1-simple\]), respectively, while the results in dashed and dotted lines account for different values of the residual gradient $\Delta_1$ ($\Delta_2=0$). Numerical parameters: (a) $\gamma_{\text{L}}=\gamma=30\mu\text{eV}$, $J_{0}=3\mu\text{eV}$. (b) $J_{0}=2.5\mu\text{eV}$, $\tau=0.1\text{ns}$.](sm-no-cancellation){width="1\columnwidth"}
Transport via SAW moving dots {#sec:noisesaws}
-----------------------------
The approximated Eqs. (\[eq:n+1-n-1-simple\]) and (\[eq:n-n-1-driving\]) suggest that the simulation of the full problem given in Eq. (\[eq:rhon-1\]) will drive the main qubits to an entangled steady state regardless of the initial state (as long as $\tau J_{i}\ll1$). However, in a realistic experimental situation, there will be also some noise sources. In the following, we account for: (i) dephasing due to the nuclear spins, (ii) imperfections due to the uncertainty in the dwell time $\tau$ (time jitter), (iii) electron losses due to imperfections in the transport mechanism and (iv) residual gradients. (i) As explained above, we include a dephasing Hamiltonian as in Eq. (\[eq:Ldeph\]) to model the non-Markovian noise due to the hyperfine interaction. We assume that the ancilla dots are refilled very quickly after every step and thus neglect the evolution in the short intermittent intervals when the ancilla dot is empty. (ii) In a realistic experimental situation, there will be also some noise associated with the uncertainty in the dwell times [@Bocquillon2013]. We include this noise source by choosing the times $\tau_{i}$ randomly from a Gaussian distribution centered around the average ($\tau$) with a standard deviation of $\sigma_{\tau}$. (iii) To model the losses we assume during the time simulation that with a certain probability an ancilla spin never interacts with the second localized spin. (iv) We estimate how large the imperfections in the magnetic gradients can be such that the entanglement generation is not severely affected.
![\[fig:app-noise-Passive-scheme-with-mobiledots\] (color online). Entanglement between two remote qubits quantified via the $E_F$ for the SAW-based proposal corresponding to Eq. (\[eq:n+1-n-1-simple\]) as a function of time ($t=2\text{n}\tau$) for two different initial states (solid and dashed lines, respectively) and $\delta J/J_{0}=0.28$, $J_{0}=2.5\mu\text{eV}$ and $\tau=0.1\text{ns}$. In both (a) and (b), the black curves depict the ideal case and the remaining curves show the effect of different kinds of noise (time jitter $\sigma_{\tau}$ and nuclear dephasing in (a); electron losses in (b) averaged over several random trajectories of the respective processes.](Fig17-final){width="1\columnwidth"}
In Fig. \[fig:app-noise-Passive-scheme-with-mobiledots\] we show the effect of the noise sources (i), (ii) and (iii) in the simulation in terms of $E_F$ of the state. The convergence is found after $\sim10^{3}$ iterations, which corresponds to the regime of $(0.1-1)\mu\text{s}$ for $\tau=(0.1-1)\text{ns}$. Note that if the product $J_{0}\tau$ is fixed, the results do not change, but the time to reach the steady state and consequently the undesired dephasing decrease with $\tau$. Once a small enough $\tau$ is fixed, the result improves as $J_0$ decreases but obviously the time grows and we need to find a compromise between the conditions $\tau J_0\ll 1$ and a time sufficiently short for the given nuclear dephasing time. In Fig. \[fig:no-cancellation\] b) we show the effect of (iv) in the entanglement generation scheme with alternating spins.
The short dephasing times considered within the main text force us to choose a quite large value of $\tau J_0=0.38$; therefore the amount of entanglement generated is bounded to $E_F\gtrsim0.4$. If the dephasing time reaches the maximal experimental reported value of $T_{2}^{*}=3\mu\text{s}$, the amount of steady-state entanglement increases up to $E_F\gtrsim0.7$, as shown in Fig. \[fig:optimistic-case\].
![\[fig:optimistic-case\]Steady-state entanglement between two remote qubits quantified via the $E_F$ for the two SAW-based proposals as a function of $\delta J$ ($J_{1(2)}^{\uparrow}=J_{0}\mp\delta J$). (a) and (b) show the results of Eq. (\[eq:n+1-n-1-simple\]) and Eq. (\[eq:n-n-1-driving\]), respectively. The solid lines refer to the ideal result, given by the lower order terms present in Eqs. (\[eq:n+1-n-1-simple\]) and (\[eq:n-n-1-driving\]), while the dashed lines correspond to the full evolution. The dotted lines also account for noise due to uncertainty in the dwell times and dephasing. Numerical parameters: $\sigma_{\text{\ensuremath{\tau}}}=5\%$, $J_{0}\tau{\approx}0.15$ and $T_{2}^{*}/\tau{\approx}30000$. In (b), for each value of $\delta J$, $\Omega_{i}$ has been optimized in the range $\Omega_{i}\tau\in\left(0-3\right)\cdot 10^{-3}$. ](SM-figure12){width="1\columnwidth"}
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[^1]: These authors have contributed equally to this work.
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---
abstract: 'We investigate the current-induced motion of ferromagnetic domain walls in presence of a Rashba spin-orbit interaction of the itinerant electrons. We show how a Rashba interaction can stabilize the domain wall motion, such that the Walker breakdown is shifted to larger current densities. The Rashba spin-orbit interaction creates a field-like contribution to the spin torque, which breaks the symmetry of the system and modifies the internal structure of the domain wall. Moreover, it can induce an additional switching of the chirality of the domain wall for sufficiently strong Rashba interactions. This allows one to choose the desired chirality by the choosing the direction of the applied spin-polarized current. Both the suppression of the Walker breakdown and the chirality switching affect the domain wall velocity significantly. This is even more pronounced for short current pulses, where an additional domain wall drift in either positive or negative direction appears after the pulse ends. By this, we can steer the final position of the domain wall. This mechanism may help to overcome the current limitations of the domain wall motion due to the Walker breakdown which occurs for rather low current densities in systems without a Rashba spin-orbit interaction.'
author:
- 'Martin Stier, Marcus Creutzburg, and Michael Thorwart'
title: Rashba induced chirality switching of domain walls and suppression of the Walker breakdown
---
Introduction
============
Magnetic memory devices are based on the presence of microscopic magnetic domains where the alignment, or more precisely the alternation of alignments, of the magnetization encodes the physical information. While in classical hard discs, these domains are directly switched by a magnetic field, it is energetically far more efficient to push them through a wire by an electrical current[@parkin2008magnetic; @malinowski2011current; @brataas2012current; @fert2013skyrmions]. The central feature of this current-induced domain wall (DW) motion can be rationalized within the picture of the standard $sd$-model of the localized electrons which form the local magnetic moments and the itinerant electrons injected into the $s$-band whose spins are polarized from outside in external contacts [@Zener; @berger1985]. Then, the local exchange interaction of the itinerant electron spins with the local magnetic moments generates a current-induced spin-transfer torque (STT). Two basic contributions of the STT are well known in materials without a breaking of the symmetry in the subsystem of the itinerant electrons[@zhang2004roles; @brataas2012current]. The adiabatic STT stems from the adiabatic alignment of the itinerant spins and the local magnetic moments, resulting in a DW motion due to the conservation of the total spin. Moreover, the nonadiabatic STT arises, which is also known as the $\beta$-term. It has its origin in a lag of the dynamics of the polarization of the itinerant electrons behind the dynamics of the local magnetic texture. This back-action of the local magnetic moments on the itinerant spins induces a relaxation dynamics for the latter during which an additional nonadiabatic current-induced STT is generated. Even though the non-adiabatic STT can lead to a considerable increase of the DW velocity with respect to the adiabatic motion, a very fast movement of the DW is limited by the Walker breakdown (WB) at a critical current density, which is accompanied by a precession of the magnetization[@hayashi2006direct] at the DW center. However, this precession can be suppressed in systems with a broken symmetry where a distinct direction of the magnetization or the electron spin is favored over the others. This may also imply a preference for a distinct chirality, or handedness, of the DW. Based on this observation, Miron *et al.*[@miron2011] have proposed to use the Rashba effect as a stabilizer of the DW chirality and the corresponding suppression of the WB and an increase of the DW velocity have been observed experimentally [@glathe2008experimental; @obata2008current]. The mechanism is similar to the action of a transverse magnetic field. In addition, other mechanisms to break the symmetry, such as the Dzyaloshinskii-Moriya interaction, are supposed to enhance the DW motion due to a preferred handedness[@brataas2013spintronics; @ryu2013chiral; @emori2013current]. Hence, by modifying the thickness of layers of distinct materials, the strength of the Dzyaloshinskii-Moriya interaction, and with it, the preferred chirality, can even be adjusted within certain limits[@chen2013tailoring].
The aim of this work is to reveal how a Rashba spin-orbit interaction of the itinerant electron spins acts on the chirality and the dynamics of a DW. We thereby consider the stabilization or destabilization of a distinct chirality in certain parameter regimes and their impact on the DW velocity. To illustrate the basic physical mechanism at work, we use a one-dimensional (1D) model which allows us to calculate the full STT including the Rashba-induced effective field in simple terms. By eventually solving the Landau-Lifshitz-Gilbert equation of motion of the DW and hereby calculating DW velocities for opposite chiralities, we identify several regimes of chirality-dependent DW motion. This actually includes a regime, where WB is suppressed and shifted to larger current densities, but also a current-dependent switching to the desired chirality. We show that the optimal chirality can be chosen by the direction of the applied current flow. Results are presented in a broad parameter range of the current density, the strength of the Rashba interaction and the lengths of the applied current pulse. Particularly for short current pulses, we find that the the average DW velocity may differ strongly from the steady current value.
Model
=====
We consider a 1D quantum wire in which a DW is formed by localized magnetic moments $M_s{\ensuremath{\mathbf{n}}}(x,t)$ described by a unit vector ${\ensuremath{\mathbf{n}}}(x,t)$ and its saturation magnetization $M_s$. The dynamics of these classical moments is well described by the Landau-Lifshitz-Gilbert equation of motion [@lakshmanan2011fascinating]. It describes the precessional motion of the moments which can be either induced by external global magnetic fields or local interactions. This precession is damped by the Gilbert damping term which lets the moments to actually align towards the (local or global) magnetic field. In our case the Landau-Lifshitz-Gilbert equation $${\ensuremath{\partial_t}}{\ensuremath{\mathbf{n}}}= -\gamma_0{\ensuremath{\mathbf{n}}}\times\mathbf H_{\rm{eff}} + \alpha {\ensuremath{\mathbf{n}}}\times{\ensuremath{\partial_t}}{\ensuremath{\mathbf{n}}}+\mathbf
T \label{eq::llg}\ ,$$ includes an effective field $\mathbf H_{\rm{eff}}$, the gyro-magnetic ratio $\gamma_0$, the Gilbert damping constant $\alpha$ and a spin torque $\mathbf T$ which is provided by external means and which is in the focus of the present work. In this work, spin transfer torque (STT) $\mathbf T$ has its origin in a spin polarized current which interacts with the magnetic moments via the exchange interaction.
The actual shape of the DW is created by an effective magnetic field which represents the interactions between the magnetic moments. We use the standard continuum form[@lakshmanan2011fascinating; @li2004] $$\mathbf H_{\rm{eff}}=-J_{\rm{IA}}{\ensuremath{\partial_x}}^2{\ensuremath{\mathbf{n}}}-K_{\parallel} n_{\parallel} \mathbf
e_{\parallel}+K_{\perp} n_{\perp} \mathbf e_{\perp}\ .$$ Here, the interaction (IA) strength $J_{\rm{IA}}=2A_{\rm ex}/M_s$ stems from the mutual exchange interaction among the spins with the strength $A_{\rm ex}$ and is a measure for the tendency of the magnetic moments to align parallel to each other. Moreover, the easy-axis anisotropy $K_{\parallel}$ and the hard axis anisotropy $K_{\perp}$ are the energies of the favorable and unfavorable directions for these moments, respectively. The explicit directions of these axes (e.g., the $x,y$ or $z$ direction) have to be defined according to the situation under consideration and different constellations arise. Even though the choice of the hard and the easy axis is hardly of any importance in highly symmetric systems, it becomes relevant if the symmetry is broken. Since this is the case for systems with a Rashba spin-orbit interaction in the focus here, we will consider different setups in this work.
A major point in this work is the calculation of $\mathbf T$ in Eq. (\[eq::llg\]) in presence of a Rashba spin-orbit interaction and a nonadiabatic relaxation channel for the itinerant electron spins. To do this, we make use of the standard $sd$-model [@Zener; @berger1985] of the electrons in the quantum wire and describe the localized magnetic moments ${\ensuremath{\mathbf{n}}}(x,t)$ as the spins of the localized electrons which typically live in $d$-like bands. They couple to the spins ${\ensuremath{\mathbf{s}}}(x,t)$ of the flowing or itinerant electrons in the 1D wire which typically live in $s$-like bands. Those are assumed to be non-interacting and are described by the kinetic Hamiltonian $H_{\rm{kin}}$. The two species couple via the exchange or the $sd$-interaction which gives rise to the Hamiltonian $H_{\rm{sd}}$. In addition, we add to this minimal 1D model the Rashba spin-orbit interaction $H_{\rm{Rashba}}$ for the itinerant electron spins in the spin polarized electron current which is imprinted at the ends of the quantum wire (see below). Since we are interested in nonadiabatic effects, we include a relaxational part $H_{\rm{relax}}$ for the itinerant electrons. In total, this yields the Hamiltonian $$H = H_{\rm{kin}} + H_{\rm{sd}} + H_{\rm{Rashba}} + H_{\rm{relax}}\
.\label{eq::totham}$$
 (a) Schematic view on the linearization of the original electron dispersion (dashed) to two branches of left/right moving particles (full lines) in the vicinity of the Fermi energy and the Fermi wave vector $\pm k_F$. This leads to left or right moving spin (polarized) currents which form a total spin current with a density $I_s$. Both, left and right moving electrons (blue), couple to the DW’s local moments (red) via the $sd$ interaction. (b/c) Schematic view of a Bloch(z) DW with hard $x$ anisotropy and positive chirality (b) and negative chirality (c).](figure1){width=".9\linewidth"}
We are only interested in 1D systems and it is very convenient to use the 1D Sugawara representation of the Hamiltonian of the spin sector. It is an appropriate description for 1D systems[@haldane1981luttinger; @gogolin2004bosonization] and provides a rather simple and straightforward way to calculate the STT [@thorwart2007; @stier2013]. In the Sugawara representation, we concentrate on the low energy sector of the electronic system. This means that we only focus on excitations in the vicinity of the Fermi level. As the dispersion is only varying slowly in this region we can linearize the dispersion which yields two chiral branches of the dispersion (cf. Fig. \[fig::scheme\]a). These branches can be associated to left and right moving electrons in the 1D wire. This yields the standard form of the kinetic part of the low energy Hamiltonian for both spin directions $$H_{\rm{kin}} = -i\hbar v\sum_{\sigma,p}\int dx c_{p\sigma}^{\dagger}(x) {\ensuremath{\partial_x}}c_{p\sigma}(x),$$ where $c_{p\sigma}^{(\dagger)}$ are the annihilators (creators) of electrons with spin $\sigma=\uparrow,\downarrow$ which are moving in the left or right direction ($p=L/R=-/+$) and the Fermi velocity $v$. To rewrite this in the Sugawara form, we define the spin density operators[@gogolin2004bosonization] $${\ensuremath{\mathbf{J}_p}}(x) = {\ensuremath{\frac{1}{2}}}:c_{p\sigma}^{\dagger}(x) \boldsymbol\sigma_{\sigma\sigma'}
c_{p\sigma'}(x):$$ with the Pauli matrices $\boldsymbol\sigma$ and the colons $:\dots :$ denoting normal ordering. The Hamiltonian now reads $$H_{\rm{kin}} = \hbar v \sum_p \int dx\ :{\ensuremath{\mathbf{J}_p}}\cdot{\ensuremath{\mathbf{J}_p}}: + H_{\rm{charge}}$$ with an irrelevant charge part. As we set-up the equation of motion for ${\ensuremath{\mathbf{J}_p}}$ below and ${\ensuremath{\left[{\ensuremath{\mathbf{J}_p}},H_{\rm{charge}}\right]_-}}=0$, the charge part does not contribute to the equation of motion. On the basis of the spin density operators for left and right moving particles, we find a simple definition of the total spin density $$\label{eq::spindens}
{\ensuremath{\mathbf{s}}}= {\ensuremath{\mathbf{J}}}_R+{\ensuremath{\mathbf{J}}}_R \, ,$$ and, more importantly, of the spin current density $${\ensuremath{\mathbf{J}}}= v({\ensuremath{\mathbf{J}}}_R-{\ensuremath{\mathbf{J}}}_L)\ ,\label{eq::spincurrdef}$$ which reduces to a vector instead of a tensor in 1D. All remaining parts of the total Hamiltonian (\[eq::totham\]) can also be expressed in terms of the ${\ensuremath{\mathbf{J}_p}}$. This is obvious for the $sd$ Hamiltonian which gives $$H_{\rm{sd}}={\ensuremath{\Delta_{\text{sd}}}}\int dx\ {\ensuremath{\mathbf{s}}}\cdot{\ensuremath{\mathbf{n}}}={\ensuremath{\Delta_{\text{sd}}}}\sum_p\int dx\ {\ensuremath{\mathbf{J}_p}}\cdot{\ensuremath{\mathbf{n}}}\ .$$ Regarding the Rashba Hamiltonian $H_{\rm{Rashba}}$, we can use further simplifications which arise in 1D wires. The conventional Rashba Hamiltonian $$H_{\rm{Rashba}} = \tilde\alpha_R(k_x\sigma_y - k_y\sigma_x)$$ reflects motion in 2D, while we may neglect the movement in one of the directions in 1D wires. Thus, we only keep $H_{\rm{Rashba}} = \tilde\alpha_R
k_x\sigma_y$ for our calculations and ignore some higher-order admixing of transverse states[@schulz2009low]. Additionally, in a low-energy model only wave vectors in the vicinity of the Fermi wave vector $k_F$ are relevant and we can replace $k_x\to k_F$ for the right moving and $k_x\to-k_F$ for the left moving particles. This yields the simplified 1D Rashba Hamiltonian in the Sugawara form $$H_{\rm{Rashba}} = {\ensuremath{\Delta_{\text{R}}}}\sum_p p\int dx\ {\ensuremath{\mathbf{J}_p}}\cdot \mathbf e_y,\quad
{\ensuremath{\Delta_{\text{R}}}}=2\tilde\alpha_R k_F.$$ For simplicity, we combine the two interactions to $$\begin{aligned}
H_{\rm{IA}} \equiv& H_{\rm{sd}}+H_{\rm{Rashba}}\nonumber\\
=& {\ensuremath{\Delta_{\text{sd}}}}\sum_p\int dx\ {\ensuremath{\mathbf{J}_p}}\cdot{\ensuremath{\mathbf m_p}}\label{eq::IA}\\
{\ensuremath{\mathbf m_p}}=& {\ensuremath{\mathbf{n}}}+p\alpha_R {\ensuremath{\mathbf e_{y}}}\label{eq::mp}\end{aligned}$$ and introduce a reduced Rashba interaction $\alpha_R = {\ensuremath{\Delta_{\text{R}}}}/{\ensuremath{\Delta_{\text{sd}}}}$. This also illustrates that the Rashba interaction provides an effective local magnetic field for the localized magnetic moments ${\ensuremath{\mathbf{n}}}(x,t)$ which depends on the chiral index $p=L/R$.
The last part of the total Hamiltonian (\[eq::totham\]) is the relaxation part, which we define here implicitly by the help of the commutator $$-\frac{i}{\hbar}
{\ensuremath{\left[{\ensuremath{\mathbf{J}_p}},H_{\rm{relax}}\right]_-}}=\frac{1}{\tau}({\ensuremath{\mathbf{J}_p}}-{\ensuremath{\mathbf{J}_p}}^{\rm{relax}})\ .$$ This form results from a standard relaxation time approximation with the relaxation time $\tau$ and can be derived from a microscopic system-bath Hamiltonian on the basis of a Bloch-Redfield-like approach. We refer to Ref. for further details.
Equation of motion for itinerant electrons
------------------------------------------
Next, we formulate the Heisenberg equation of motion (EOM) of ${\ensuremath{\mathbf{J}_p}}$ as $${\ensuremath{\partial_t}}{\ensuremath{\mathbf{J}_p}}=-\frac{i}{\hbar}{\ensuremath{\left[{\ensuremath{\mathbf{J}_p}},H\right]_-}}\ .$$ Its solution enters in Eq. (\[eq::spindens\]) and eventually yields the STT $$\mathbf T =
-\frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar}{\ensuremath{\mathbf{n}}}\times{\ensuremath{\mathbf{s}}}=-\frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar}{\ensuremath{\mathbf{n}}}\times\sum_p{\ensuremath{\mathbf{J}_p}}\
.\label{eq::stt}$$ Keeping in mind that the spin density operators in the low-energy description obey the Kac-Moody-Algebra[@gogolin2004bosonization] with the commutators $${\ensuremath{\left[J_p^{\mu}(x),J_{p'}^{\nu}(x')\right]_-}}=i[p{\ensuremath{\partial_x}}+\epsilon^{\mu\nu\lambda}J_p^{
\lambda}]\delta_{pp'}\delta(x-x'),$$ we find $$\begin{aligned}
({\ensuremath{\partial_t}}+vp{\ensuremath{\partial_x}}){\ensuremath{\mathbf{J}_p}}=&-\frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar}[{\ensuremath{\mathbf{J}_p}}\times{\ensuremath{\mathbf m_p}}+\beta({\ensuremath{\mathbf{J}_p}}-{\ensuremath{\mathbf{J}_p}}^{\rm{relax}})],
\label{eq::eom}\end{aligned}$$ where we have defined $\beta=\hbar/({\ensuremath{\Delta_{\text{sd}}}}\tau)$. The explicit form of the relaxation state is crucial for the resulting STT, since it not only changes the values of the equation but also influences the symmetry of the system. As shown by van der Bijl and Duine [@vanderbijl2012], this actually affects the existence of distinct STTs. To actually solve the EOM (\[eq::eom\]), we apply a gradient expansion scheme and express ${\ensuremath{\mathbf{J}_p}}$ in orders of derivatives of ${\ensuremath{\mathbf m_p}}$ as $${\ensuremath{\mathbf{J}_p}}= {\ensuremath{\mathbf{J}_p}}^{(0)}({\ensuremath{\mathbf m_p}}) + {\ensuremath{\mathbf{J}_p}}^{(1)}({\ensuremath{\partial_x}}{\ensuremath{\mathbf m_p}},{\ensuremath{\partial_t}}{\ensuremath{\mathbf m_p}})+\dots \
.\label{eq::gradient}$$ Notice that obviously $\partial_{x,t}{\ensuremath{\mathbf m_p}}= \partial_{x,t}{\ensuremath{\mathbf{n}}}$. The combination of the Ansatz (\[eq::gradient\]) and the EOM (\[eq::eom\]) allows for an arrangement by the orders of the derivatives on the respective left and right hand side of the equation as $$\begin{aligned}
0=&-{\ensuremath{\Delta_{\text{sd}}}}[{\ensuremath{\mathbf{J}_p}}^{(0)}\times{\ensuremath{\mathbf m_p}}-\beta({\ensuremath{\mathbf{J}_p}}^{(0)}-{\ensuremath{\mathbf{J}_p}}^{\rm{relax}})]\\
({\ensuremath{\partial_t}}+vp{\ensuremath{\partial_x}}){\ensuremath{\mathbf{J}_p}}^{(0)} =&-{\ensuremath{\Delta_{\text{sd}}}}[{\ensuremath{\mathbf{J}_p}}^{(1)}\times{\ensuremath{\mathbf m_p}}-\beta({\ensuremath{\mathbf{J}_p}}^{(1)}- 0 )]\\
\dots=&\dots\ .\end{aligned}$$ Every equation has the basic structure $$-{\ensuremath{\Delta_{\text{sd}}}}(\beta -{\ensuremath{\mathbf m_p}}\times){\ensuremath{\mathbf{J}_p}}^{(n)} = \mathbf X_p^{(n)}$$ with $\mathbf X_p^{(0)}=-{\ensuremath{\Delta_{\text{sd}}}}\beta{\ensuremath{\mathbf{J}_p}}^{\rm{relax}}$ and $\mathbf
X_p^{(1)}= ({\ensuremath{\partial_t}}+vp{\ensuremath{\partial_x}}){\ensuremath{\mathbf{J}_p}}^{(0)}$ and so on. The general solution reads[@thorwart2007] $${\ensuremath{\mathbf{J}_p}}^{(n)}=\frac{\beta^2\mathbf X_p-\beta\mathbf X_p^{(n)}\times{\ensuremath{\mathbf m_p}}+(\mathbf
X_p^{(n)}\cdot{\ensuremath{\mathbf m_p}}){\ensuremath{\mathbf m_p}}}{\beta(\beta^2+{\ensuremath{\mathbf m_p}}^2)}\ .\label{eq::gen_sol}$$ Starting from the zeroth order, we can now solve the equation successively to, in principle, arbitrary order. Since higher order terms become very involved, we restrict the calculation to zeroth and first order in this work.
Next, we have to address the relaxation state explicitly. In the literature, two approaches are discussed: the electron spin either relaxes towards the direction of the magnetization[@kim2012magnetization], such that ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$, or to the combined vector[@vanderbijl2012] ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf m_p}}$. To reveal the differences between these two approaches, we address below both.
\[sec::stt\]Spin torque
-----------------------
### Relaxation to ${\ensuremath{\mathbf{n}}}$
In the first approach, the relaxation occurs towards the magnetization ${\ensuremath{\mathbf{n}}}$ of the domain wall, such that $${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}=j_p {\ensuremath{\mathbf{n}}}$$ with some proportionality coefficients $j_{L/R}$. Considering small damping parameters $\beta^2\ll 1$, we find for the zeroth order $${\ensuremath{\mathbf{J}_p}}^{(0)}=j^{(0)}_p\frac{-\beta{\ensuremath{\mathbf{n}}}\times{\ensuremath{\mathbf m_p}}}{({\ensuremath{\mathbf{n}}}\cdot{\ensuremath{\mathbf m_p}})|{\ensuremath{\mathbf m_p}}|}+j_p^{(0)}
\frac{{\ensuremath{\mathbf m_p}}}{|{\ensuremath{\mathbf m_p}}|}\ .\label{eq::jp0_rel2n}$$ Here, we have introduced $j_p^{(0)}=j_p({\ensuremath{\mathbf{n}}}\cdot{\ensuremath{\mathbf m_p}})/|{\ensuremath{\mathbf m_p}}|$ to ensure that $|{\ensuremath{\mathbf{J}_p}}^{(0)}|\stackrel{\beta^2\ll 1}{=}j_p^ {(0)}$. By this, the spin current density far away from any magnetic texture, i.e., for $x\to \pm
\infty$ in the zeroth order follows from Eq. (\[eq::spincurrdef\]) as $$I_s\equiv v |{\ensuremath{\mathbf{J}}}^{(0)}_R-{\ensuremath{\mathbf{J}}}^{(0)}_L|\stackrel{\alpha_R^2,\beta^2\ll
1}{=}v(j_R^{(0)}-j_L^{(0)}) \, .\label{eq::is}$$ For an easier comparison with experimental data, this spin current density may be rewritten as $I_s=PI_c/(2eM_s)$, where $P$ is the spin polarity, $e$ the elementary charge, $M_s$ the saturated magnetic moment, and $I_c$ the charge current density of the imprinted spin polarized current. Using Eqs. (\[eq::stt\]) and (\[eq::is\]), we find the zeroth order contribution to the STT as $$\mathbf T^{(0)}=-\frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar
v}I_s\alpha_R\Big[{\ensuremath{\mathbf{n}}}\times{\ensuremath{\mathbf e_{y}}}-\beta{\ensuremath{\mathbf{n}}}\times({\ensuremath{\mathbf{n}}}\times{\ensuremath{\mathbf e_{y}}})\Big]+\mathcal
O(\alpha_R^3)\ .$$ This equation may be expressed in the form of a term with an effective magnetic field as it appears in the LLG (\[eq::llg\]), such that $$\mathbf T^{(0)} = -\gamma_0{\ensuremath{\mathbf{n}}}\times(\mathbf H_R^{(0)} + \mathbf
H_R^{\rm{anti}}) \, .$$ This form gives rise to the “Rashba field” $$\mathbf H_R^{(0)}= \frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar v\gamma_0}I_s\alpha_R{\ensuremath{\mathbf e_{y}}}\, ,$$ and to the “anti-damping field ” $$\mathbf H_R^{\rm{anti}}= -\frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar v\gamma_0}\beta
I_s\alpha_R{\ensuremath{\mathbf{n}}}\times{\ensuremath{\mathbf e_{y}}}\, .$$ From the zeroth order spin density (\[eq::jp0\_rel2n\]), we obtain the first order term which eventually yields the usual adiabatic and nonadiabatic contributions to the STT as $$\begin{aligned}
\mathbf T^{\rm{ad}} =& -I_s{\ensuremath{\partial_x}}{\ensuremath{\mathbf{n}}}+\mathcal O(\alpha_R^2)\\
\mathbf T^{\rm{non-ad}} =& \beta I_s{\ensuremath{\mathbf{n}}}\times{\ensuremath{\partial_x}}{\ensuremath{\mathbf{n}}}+\mathcal O(\alpha_R^2)\, ,\end{aligned}$$ as well as a first order contribution to the Rashba field as $$\mathbf H_R^{(1)}=
-\frac{I_s}{\gamma_0}\alpha_R^2\big[({\ensuremath{\mathbf{n}}}\times{\ensuremath{\partial_x}}{\ensuremath{\mathbf{n}}})\cdot{\ensuremath{\mathbf e_{y}}}\big]{\ensuremath{\mathbf e_{y}}}+\mathcal
O(\alpha_R^3)\ .$$ When the term $({\ensuremath{\mathbf{n}}}\times{\ensuremath{\partial_x}}{\ensuremath{\mathbf{n}}})\cdot{\ensuremath{\mathbf e_{y}}}$ is large, $\mathbf H_R^{(1)}$ may strongly affect the whole Rashba field, even though it is proportional to $\alpha_R^2$. However, for Bloch-like DWs this term appears to be small and we will not focus on it in this work.
Additional terms $\mathbf T_t\propto{\ensuremath{\partial_t}}{\ensuremath{\mathbf{n}}}$ also appear, which renormalize the Gilbert damping $\alpha$ in the LLG. As the origin of $\alpha$, and with it, the dependence on other parameters is not very well established in theory, we will neglect all torques $\propto{\ensuremath{\partial_t}}{\ensuremath{\mathbf{n}}}$ to obtain a constant model parameter $\alpha$ for all calculations shown below.
### Relaxation to ${\ensuremath{\mathbf m_p}}$
When the relaxation state is chosen as $${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}=j_p{\ensuremath{\mathbf m_p}}\, ,$$ the zeroth-order solution is $${\ensuremath{\mathbf{J}_p}}^{(0)}=j_p^{(0)}\frac{{\ensuremath{\mathbf m_p}}}{|{\ensuremath{\mathbf m_p}}|}\ ,$$ with $j_p^{(0)}=j_p|{\ensuremath{\mathbf m_p}}|$. Consequently, the anti-damping term is missing in the zeroth order contribution of the STT $$\mathbf T^{(0)}=-\gamma_0{\ensuremath{\mathbf{n}}}\times\mathbf H_R^{(0)},$$ while the Rashba field $$\mathbf H_R^{(0)} = \frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar v\gamma_0}I_s\alpha_R{\ensuremath{\mathbf e_{y}}}+\mathcal
O(\alpha_R^3)$$ still arises. The first-order contributions essentially remain of the same form,i.e., $$\begin{aligned}
\mathbf T^{\rm{ad}} =& -I_s{\ensuremath{\partial_x}}{\ensuremath{\mathbf{n}}}+\mathcal O(\alpha_R^2)\\
\mathbf T^{\rm{non-ad}} =& \beta I_s{\ensuremath{\mathbf{n}}}\times{\ensuremath{\partial_x}}{\ensuremath{\mathbf{n}}}+\mathcal O(\alpha_R^2) \,
. \end{aligned}$$ Only the first-order (nonadiabatic) Rashba field $$\mathbf H_R^{(1)}=
\beta\frac{I_s}{\gamma_0}\alpha_R^2({\ensuremath{\partial_x}}{\ensuremath{\mathbf{n}}}\cdot{\ensuremath{\mathbf e_{y}}}){\ensuremath{\mathbf e_{y}}}+\mathcal O(\alpha_R^3)\
.$$ is now proportional to ${\ensuremath{\partial_x}}{\ensuremath{\mathbf{n}}}\cdot{\ensuremath{\mathbf e_{y}}}$ and to $\beta\alpha_R^2$. As before, this term is only important for very steep DWs, which are not considered in this work here.
Domain wall chirality
---------------------
Bloch domain walls have additional degree of freedom which is the chirality. Chirality is defined as the clockwise ($C=+1$) or the counter-clockwise ($C=-1$) rotation of the magnetic moment in the according plane. For the system addressed below, an initial direction of the magnetic moment $n_{x(y)}>0$ at the DW center for the hard $y$ ($x$) axis means a negative chirality and vice versa (cf. Fig. \[fig::scheme\]). The chirality-dependent DW dynamics[@otalora2013breaking] has already been investigated previously for fixed chiralities[@linder2013chirality]. In contrast to that, we here allow the magnetization to dynamically tilt and also to eventually switch the chirality.
Results
=======
In 1D systems, we theoretically have the freedom to choose the directions of the easy and hard axes, the direction of the Rashba-induced field $\mathbf H_R$ and the chirality of the DW. We will consider systems which always have the easy axis in the $z$ direction, while the Rashba-induced field points in the $y$ direction. At the respective ends of the wire, we enforce $n_z(x\to\pm\infty)=\pm1$ as boundary conditions to solve the Landau-Lifshitz-Gilbert equation numerically. A crucial point of this work is the effect of the direction of the hard axis and the initial chirality of the DW on its dynamics. We will show results for four types of DWs: the hard axis in $x$ or $y$ direction and a positive or negative chirality $C=\pm 1$.
We choose model parameters which correspond to the estimated values[@miron2011] of Pt/Co/AlO$_x$. We have: $A_{ex}=10^{-11}\rm{J/m}$, $M_s=1090\rm{kA/m}$, $K_{\parallel}=0.92\rm{T}$, $K_{\perp}=0.03K_{\parallel}$, $2\alpha=\beta=0.12$. In addition, we set ${\ensuremath{\Delta_{\text{sd}}}}=0.5\rm{eV}$, $v_F=10^6\rm{m/s}$ and the polarity of the spin current $P=1$. The values of the Rashba interaction will be chosen around $\tilde\alpha_R=10^{-10}\rm{eVm}$ which corresponds to $\Delta_R=0.1\rm{eV}$ and $\alpha_R=\Delta_R/{\ensuremath{\Delta_{\text{sd}}}}=0.2$. The DW center ${x_{\rm{DW}}}$ is defined by the condition $n_z({x_{\rm{DW}}})=0$ and the DW velocity is calculated to be ${v_{\rm{DW}}}={\ensuremath{\partial_t}}{x_{\rm{DW}}}$.
Domain wall velocity
--------------------
{width="\linewidth"}
The DW velocity here depends on several quantities. First of all, it will be strongly determined by the size of the current density $I_c$ as well as by the strength of the Rashba spin-orbit interaction $\alpha_R$. Second, there are the topological features which influence the DW dynamics and which are determined by the sign of the initial chirality ($C=\pm 1$) and the direction of the hard axis (in $x$ or $y$ direction). Finally, we address the role of the different forms of the relaxation state (${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}},{\ensuremath{\mathbf m_p}}$) which effectively decides whether the anti-damping Rashba field $\mathbf H_R^{\rm{anti}}$ appears or not. We show the dependence of the DW velocity for all of these cases in this section.
In Fig. \[fig::pd\_rel2mp\], the long-time averaged DW velocity ${\ensuremath{\left\langlev_{\rm{DW}}\right\rangle}}=x(t_{\rm av})/t_{\rm av}$ , with $t_{\rm av}=100\rm{ns}$, is shown for the case of ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf m_p}}$, such that $\mathbf
H_R^{\rm{anti}}=0$. All four configurations ($C=\pm 1$, hard $x$ or $y$ axis) yield a qualitatively similar picture. As we have chosen $\beta=2\alpha$, we find a Walker breakdown which refers to the rather sharp velocity drop at small $\alpha_R$ in the vicinity of $I_c\approx0.3\times10^{12}\rm{A/m^2}$. The WB is accompanied by a precession of the magnetization at the DW center. For larger $\alpha_R$, the WB appears at larger current densities. Hence, the Rashba field $\mathbf H_R$ stabilizes the DW motion since it suppresses the precession of the DW as it acts as an additional effective anisotropy. The same observation has already been made previously[@miron2011; @linder2014wb]. To obtain a rough estimate of the critical current density $I_c^{(WB)}$, where the WB sets in, we introduce the total anisotropy field $K^* = K_{\perp}+H_R$. The critical current density is proportional to the total anisotropy field according to $$\begin{aligned}
I_c^{(WB)} =& \nu K^*\\
=& I_c^{(WB)}(\alpha_R = 0) \left(1 + \frac{H_R}{K_{\perp}}\right) \, .\end{aligned}$$ The remaining proportionality factor can be determined by the critical current density for a vanishing Rashba field. In our case, the Rashba field is given as $H_R\approx\frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar v_F \gamma_0}\alpha_RI_s=0.25\alpha_R
I_C[10^{12}\textrm{A/m}^2] \textrm{T}$. Thus, upon setting $I_c^{(WB)}(\alpha_R=0)=0.3\times10^{12}\rm{A/m^2}$, we find approximately that $$I_c^{(WB)} = \frac{0.3}{1-3\alpha_R} \, .$$ This yields to a complete suppression[@linder2014wb] of the WB for $\alpha_R\gtrsim 0.3$, which is reflected in Fig. \[fig::pd\_rel2mp\]. The differences between the four configurations shown in Fig. \[fig::pd\_rel2mp\] are discussed in more details in the next section.
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The second case when the relaxation state is ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$ yields an additional field-like (nonadiabatic) torque with the anti-damping field $\mathbf
H_R^{\rm{anti}}= -\frac{{\ensuremath{\Delta_{\text{sd}}}}}{\hbar v\gamma_0}\beta I_s\alpha_R{\ensuremath{\mathbf{n}}}\times{\ensuremath{\mathbf e_{y}}}$, which is perpendicular to $\mathbf H_R$. The most significant consequence of the anti-damping term is a possible movement of the DW against the current flow. This can be seen in Fig. \[fig::pd\_rel2n\] and has already been discussed in Ref. . Again, we devote the next section to a more detailed discussion of the differences between the four configurations.
Chirality switching
-------------------
![\[fig::v\_t\]DW velocities (left) and magnetization in $x$-direction at DW center (right) for three different current densities: (a) $I_c=0.125\times10^{12}$A/m$^2$, (b) $I_c=0.4\times10^{12}$A/m$^2$ and (c) $I_c=0.8\times10^{12}$A/m$^2$. The sign of the $n_x(x_{DW})$ indicates the (inverse) chirality of the DW. Three scenarios appear: (a) no switching of the chirality, (b) a single chirality switching for a positive initial chirality, and, (c) alternating chirality switching (and a Walker breakdown). Parameters used are $\alpha_R=0.1$, hard $y$ axis, and the relaxation to ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$.](figure4){width="\linewidth"}
Even though the DW velocities in Figures \[fig::pd\_rel2mp\] and \[fig::pd\_rel2n\] show a qualitatively similar behavior for all four configurations, several differences between them arise. They become more explicit when we investigate the time evolution of the DW velocity in more detail. Figure \[fig::v\_t\] shows the dynamical build-up of the DW velocity for the three parameter sets $\{\alpha_R,I_c\}$ indicated by the black arrows in Fig. \[fig::pd\_rel2n\]. In this case, the hard axis is in the $y$ direction and the results are shown for both initial chiralities $C=\pm 1$. Three scenarios can be identified: (a) For low current densities $I_c$, the two chiralities lead to different DW velocities for all times. (b) For intermediate $I_c$, the initially different velocities approach each other after some time. (c) Finally, for large current densities $I_c$, the two chiralities lead to equal but phase-shifted oscillating velocities with the same rather large average velocity. We also show in Fig. \[fig::v\_t\] the magnetization $n_x(x_{\rm{DW}})$ in the $x$ direction at the DW center. It is immediately clear what separates these three scenarios. In the case (a), the initial magnetizations of both chiralities remain unchanged over time and each chirality is conserved. In this case, neither the Rashba field $H_R\propto
\alpha_R I_c$ nor the non-adiabatic torque $T^{\rm{non-ad}}\propto\beta I_c$ are strong enough to overcome the field of the perpendicular anisotropy $K_{\perp}$. In the scenario (b), the Rashba field is stronger than $K_{\perp}$. In contrast to the field of the anisotropy $H_{\perp} = K_{\perp} n_{\perp}$, the Rashba field explicitly favors one direction of the magnetization $n_x(x_{\rm{DW}})\gtrless0$. For the case shown in Fig. \[fig::v\_t\] (b), a negative magnetization is preferred and an initially negative chirality is switched to a positive one after some time. Finally, for large current densities and scenario (c), the non-adiabatic torque may overcome both the perpendicular anisotropy field and the Rashba field. This yields, as usual, to a Walker breakdown. Here, both chiralities are alternatingly switched and the magnetization at the DW center precesses around the $z$ axis.
![\[fig::diffv\_rel2n\]Difference between the DW velocities ${\ensuremath{\left\langle\Delta v_{DW}\right\rangle}}={\ensuremath{\left\langlev_{DW}^{C=-1}\right\rangle}}-{\ensuremath{\left\langlev_{DW}^{C=+1}\right\rangle}}$ of both chiralities for (a) hard $y$ axis and (b) hard $x$ axis. White dashed lines separate the three scenarios shown in Fig. \[fig::v\_t\]. The relaxation occurs towards ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$.](figure5a "fig:"){width="\linewidth"} ![\[fig::diffv\_rel2n\]Difference between the DW velocities ${\ensuremath{\left\langle\Delta v_{DW}\right\rangle}}={\ensuremath{\left\langlev_{DW}^{C=-1}\right\rangle}}-{\ensuremath{\left\langlev_{DW}^{C=+1}\right\rangle}}$ of both chiralities for (a) hard $y$ axis and (b) hard $x$ axis. White dashed lines separate the three scenarios shown in Fig. \[fig::v\_t\]. The relaxation occurs towards ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$.](figure5b "fig:"){width="\linewidth"}
![\[fig::diffv\_rel2mp\]Same as in Fig. \[fig::diffv\_rel2n\], but the relaxation occurs towards ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf m_p}}$. ](figure6a "fig:"){width="\linewidth"} ![\[fig::diffv\_rel2mp\]Same as in Fig. \[fig::diffv\_rel2n\], but the relaxation occurs towards ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf m_p}}$. ](figure6b "fig:"){width="\linewidth"}
The conditions for the three scenarios are that in (a), there is no chirality switching if $H_R,T^{\rm{non-ad}}<K_{\perp}$, in (b) a *single* chirality switching to a preferred chirality occurs if $T^{\rm{non-ad}},K_{\perp}<H_R$, and in (c), a frequent alternating chirality switching of both initial chiralities and a WB arise, if $H_R,K_{\perp}<T^{\rm{non-ad}}$.
We may summarize these scenarios in a phase diagram. For this, we have calculated the difference ${\ensuremath{\left\langle\Delta
v_{\rm{DW}}\right\rangle}}={\ensuremath{\left\langlev_{DW}^{C=-1}\right\rangle}}-{\ensuremath{\left\langlev_{DW}^{C=+1}\right\rangle}}$ of the averaged velocities of both chirality classes. The results are shown in Figs. \[fig::diffv\_rel2n\] and \[fig::diffv\_rel2mp\]. For small current densities in scenario (a), large velocity differences arise and no chirality switching occurs. Both chiralities yield to permanently different velocities. Instead, for intermediate/high current densities, a single chirality switching occurs and the velocity differences are almost vanishing. Furthermore, at larger current densities but small $\alpha_R$, we find small oscillations of ${\ensuremath{\left\langle\Delta v_{\rm{DW}}\right\rangle}}$ which stem from the alternating chirality switchings in scenario (c). Even though we have performed a long-time average, the time interval $t\in[0,100\rm{ns}]$ over which the DW velocity is averaged, is not long enough to completely remove these oscillations. However, even though we could increase the time window of the average, we prefer to use these oscillations for an easier determination of the WB “phase”.
Finally, we note that in this work, we have focused on the case $\beta>\alpha$ which is a necessary condition for a WB to appear. For the case $\alpha=\beta$, the “phase” of the WB would vanish. In addition, in order to achieve a chirality switching, larger values of $\alpha_R$ or $I_c$ would be required. This is because the non-adiabatic torque assists the chirality switching in the same way as it already tends to switch the chirality frequently in the form of a WB.
To summarize this section, we have shown that it is in principle possible to control the chirality of a DW by the size of the applied current and that clearly separated phases arise which render this control achievable. This feature could be useful in technological applications in form of magnetic storage devices. Even though we have shown here only results where the switching to one preferred chirality is illustrated, it could be easily modified by the direction of the current flow. As $H_R\propto I_c$, a current applied in the opposite direction would change the sign of the Rashba field, which would then favor the other chirality.
Short current pulses
--------------------
 DW velocity vs time for (a) different current pulses with pulse lengths $t_p=\{4,8,12,16\}\rm{ ns}$ and the steady current (dashed line) at $I_c=0.8\times 10^{12}\rm{A/m^2}$ and (b) different current densities $I_c=\{0.4,0.8,1.6\} \times 10^{12}\rm{A/m^2}$ for $t_p=17\rm{ns}$. After the current pulse ends, the DW drifts some distance in either positive or negative direction depending on the state of the DW at the end of the pulse. Parameters: $\alpha_R=0.1$, hard $y$ axis, negative chirality $C=-1$.](figure7){width="\linewidth"}
Most of the results for the DW velocities in the previous section refer to averages over a intermediate-to-large time window. However, as shown in Fig. \[fig::v\_t\], relevant features arise on much shorter time scales, e.g., in the regime of a few nanoseconds[@stier2013; @thomas2006oscillatory; @meier2007direct]. For example a chirality switching as in Fig. \[fig::v\_t\] (b) only appears after some finite time. The long-time averaged velocity shows no difference, because for most of the time, both velocities match each other. However, an analysis in form of a shorter-time average could uncover the different velocities of the DWs of different initial chirality.
In this section, we address the DW dynamics on shorter time scales and use for this rather short current pulses, which basically includes short time averaging. It is known that several new features appear for short current pulses[@stier2013; @miron2011; @thomas2010dynamics]. First, possible oscillations, which would build up on longer times, will not average out for pulse lengths of the order of the oscillation period. This can readily be seen in Fig. \[fig::v\_vs\_t\_pulse\]. When a pulse ends, e.g., in the first half of an oscillation period, it can yield a drastically increased or decreased DW velocity, depending on the sign of the amplitude in the respective half-period. This should also lead to major chirality-dependent velocity differences since the oscillation amplitudes may differ for both chiralities \[cf. Fig. \[fig::v\_t\](c)\].
Second, the DW does not immediately stop after the end of the current pulse, but “drifts” a certain distance either in forward or backward direction[@stier2013; @miron2011; @thomas2010dynamics]. The direction of the drift is determined by the momentary state of the DW at the end of the pulse. As there are no spin torques any more acting after the pulse (neither any Rashba fields nor the conventional (non-)adiabatic torques are present), the DW strives to settle at its equilibrium position which is determined by the hard axis anisotropy. If the DW is tilted out of this position at the end of the pulse, it realigns in the fastest manner back to it and thereby moves some distance.
Not only the pulse length $t_p$ determines the state of the DW at the end of the pulse, but also the current density. For a constant $t_p$, an increasing $I_c$ leads to a smaller oscillation period as it is shown in Fig. \[fig::v\_vs\_t\_pulse\] (b). Thus the state at the end of the pulse, and with it the drift, is changed.
To see the effects of the short-time averaging and the drifting, we define two different averaged velocities. First, the averaged velocity during the pulse is ${\ensuremath{\left\langlev_{\rm{DW}}\right\rangle}}=x_{\rm{DW}}(t_p)/t_p$, and, second, an effective averaged velocity ${\ensuremath{\left\langlev^{\rm{eff}}_{\rm{DW}}\right\rangle}}=x_{\rm{DW}}(t\to\infty)/t_p$ is meaningful. The second quantity includes the drifting since we use the DW position $x_{\rm{DW}}(t\to\infty)$ after a long enough time. As it is difficult to decide at which time the DW actually stops, we also divide this position by the pulse length $t_p$. Thus, the effective velocity ${\ensuremath{\left\langlev^{\rm{eff}}_{\rm{DW}}\right\rangle}}$ is not a real time average, but an “effective” one which is compared to the pulse length.
 Average velocity over the current pulse length ${\ensuremath{\left\langlev_{DW}\right\rangle}}=x_p(t_p)/t_p$ and (b) effective velocity ${\ensuremath{\left\langlev_{DW}\right\rangle}}=x(t\to\infty)/t_p$ which includes the drifting of the DW after the pulse has ended, both for a hard $y$ axis and negative chirality $C=-1$. While the average velocity in (a) shows smooth oscillations at higher current densities due to the Walker breakdown, the drifting after the pulse leads to rather abrupt velocity changes. (c) Same as (b) but for positive chirality $C=+1$. Shown are the results for $\alpha_R=0.1$ and for the relaxation to ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$.](figure8a "fig:"){width="\linewidth"}  Average velocity over the current pulse length ${\ensuremath{\left\langlev_{DW}\right\rangle}}=x_p(t_p)/t_p$ and (b) effective velocity ${\ensuremath{\left\langlev_{DW}\right\rangle}}=x(t\to\infty)/t_p$ which includes the drifting of the DW after the pulse has ended, both for a hard $y$ axis and negative chirality $C=-1$. While the average velocity in (a) shows smooth oscillations at higher current densities due to the Walker breakdown, the drifting after the pulse leads to rather abrupt velocity changes. (c) Same as (b) but for positive chirality $C=+1$. Shown are the results for $\alpha_R=0.1$ and for the relaxation to ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$.](figure8b "fig:"){width="\linewidth"}  Average velocity over the current pulse length ${\ensuremath{\left\langlev_{DW}\right\rangle}}=x_p(t_p)/t_p$ and (b) effective velocity ${\ensuremath{\left\langlev_{DW}\right\rangle}}=x(t\to\infty)/t_p$ which includes the drifting of the DW after the pulse has ended, both for a hard $y$ axis and negative chirality $C=-1$. While the average velocity in (a) shows smooth oscillations at higher current densities due to the Walker breakdown, the drifting after the pulse leads to rather abrupt velocity changes. (c) Same as (b) but for positive chirality $C=+1$. Shown are the results for $\alpha_R=0.1$ and for the relaxation to ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$.](figure8c "fig:"){width="\linewidth"}
Figure \[fig::v\_vs\_pulse\] shows the DW velocities versus the current density $I_c$ and the pulse duration $t_p$. We find smooth oscillations of the average velocity ${\ensuremath{\left\langlev_{DW}\right\rangle}}$, i.e., without drifting. They occur because the oscillatory motion of the DW does not completely average out even for large current densities. For very short pulses, remarkably large velocities appear for a DW with negative chirality which can be traced back to the large velocities at the beginning of the motion for this chirality in systems with a hard $y$ axis \[cf. Fig. \[fig::v\_t\] (c)\]. When we include the drifting and consider the effective mean velocity ${\ensuremath{\left\langlev^{\rm{eff}}_{\rm{DW}}\right\rangle}}=x_{\rm{DW}}(t\to\infty)/t_p$, the changes in the velocity become more abrupt \[cf. Fig. \[fig::v\_vs\_pulse\] (b)\]. Then, the drifting partially compensates the oscillatory movement to some extent due its different moving directions. This lets the DW end up in distinct positions.
 Differences of effective velocities ${\ensuremath{\left\langle\Delta
v_{DW}\right\rangle}}=[x^{C=-1}(t\to\infty)-x^{C=+1}(t\to\infty)]/t_p$ for DWs with (a) a hard $y$ axis, and, (b) a hard $x$ axis. Particularly for short pulses, large differences occur. The blue color indicates positive and red negative values. Moreover, $\alpha_R=0.1$, and the relaxation occurs to ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$.](figure9a "fig:"){width="\linewidth"}  Differences of effective velocities ${\ensuremath{\left\langle\Delta
v_{DW}\right\rangle}}=[x^{C=-1}(t\to\infty)-x^{C=+1}(t\to\infty)]/t_p$ for DWs with (a) a hard $y$ axis, and, (b) a hard $x$ axis. Particularly for short pulses, large differences occur. The blue color indicates positive and red negative values. Moreover, $\alpha_R=0.1$, and the relaxation occurs to ${\ensuremath{\mathbf{J}_p}}^{\rm{relax}}\propto{\ensuremath{\mathbf{n}}}$.](figure9b "fig:"){width="\linewidth"}
In contrast to the rather large velocities of the DW with negative chirality, a positive chirality $C=+1$ leads to a strongly reduced DW velocity at very short pulses \[cf. Fig. \[fig::v\_vs\_pulse\] (c)\]. Thus, large velocity differences between the two cases of opposite initial chiralities at small $t_p$ are expected. Figure \[fig::diffv\_vs\_pulse\] indeed confirms this. In addition to the large absolute velocity differences at small $t_p$, these differences also vary very strongly and can actually change the sign and with that the motional direction of the DW.
These results illustrate the important role of the additional short-time effects for the DW dynamics, in particular in the presence of a Rashba spin-orbit interaction. Its impact on specific experiments may also be affected by additional phenomena not considered in this work, such as the presence of pinning centers[@van2013role], for instance.
Summary
=======
We have studied the influence of the direction of the hard axis and the DW chirality on the DW’s current induced dynamics in 1D Rashba wires. The spin transfer torque arises from a gradient expansion in the DW steepness and enters in the Landau-Lifshitz-Gilbert equation which we solve numerically. Two different relaxation states were used which either generate or suppress the Rashba anti-damping field-like torque $\mathbf H_R^{\rm{anti}}$. This is perpendicular to the regular Rashba-induced field-like torque $\mathbf H_R$, which arises in both cases.
As we focus on the case when the nonadiabaticity parameter $\beta > \alpha$, a Walker breakdown at sufficiently strong current densities $I_c$ arises. It is suppressed by the Rashba field $\mathbf H_R$, since it acts as an additional local anisotropy. For rather strong Rashba couplings $\alpha_R$, the WB is entirely suppressed. Even more interestingly, we identify a third “phase” in the $\{\alpha_R,I_c\}$ parameter space: For intermediate current densities and/or large $\alpha_R$, we find a single switching of the initial chirality to a preferred chirality, which can be chosen by the direction of the current flow.
Moreover, the effects of the chirality switching appear to be more pronounced at short times and we have considered short current pulses. As expected, we find a stronger influence of the momentary velocity state when a short-time averaging procedure is applied. Further phenomena such as a drifting of the DW after a short current pulse affect the short-time dynamics of a DW even more pronounced. This results partly in rather abrupt changes of the effective DW velocity and a completely different dependence on the current density as compared to the steady current arises. Then, even larger velocity differences between DWs of different chiralities result. This rich set of features shows that several possibilities arise for optimal parameter combinations in order to achieve a maximal DW velocity.
We acknowledge support from the DFG SFB 668 (project B16) and thank G. Meier for valuable discussions.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We provide a brief tutorial on the use of concentration inequalities as they apply to system identification of state-space parameters of linear time invariant systems, with a focus on the fully observed setting. We draw upon tools from the theories of large-deviations and self-normalized martingales, and provide both data-dependent and independent bounds on the learning rate.'
author:
- 'Nikolai Matni, Stephen Tu'
title: A Tutorial on Concentration Bounds for System Identification
---
Introduction {#sec:intro}
============
A key feature in modern reinforcement learning is the ability to provide high-probability guarantees on the finite-data/time behavior of an algorithm acting on a system. The enabling technical tools used in providing such guarantees are concentration of measure results, which should be interpreted as quantitative versions of the strong law of large numbers. This paper provides a brief introduction to such tools, as motivated by the identification of linear-time-invariant (LTI) systems.
In particular, we focus on the identifying the parameters $(A,B)$ of the LTI system $$x_{t+1} = Ax_t + Bu_t + w_t,
\label{eq:sys-intro}$$ assuming *perfect* state measurements. This is in some sense the simplest possible system identification problem, making it the perfect case study for such a tutorial. Our companion paper [@extended] shows how the results derived in this paper can then be integrated into self-tuning and adaptive control policies with finite-data guarantees. We also refer the reader to Section II of [@extended] for an in-depth and comprehensive literature review of classical and contemporary results in system identification. Finally, we note that most of the results we present below are not the sharpest available in the literature, but are rather chosen for the pedagogical value.
The paper is structured as follows: in Section \[sec:scalar\], we study the simplified setting when system is defined for a scalar state $x$, and data is drawn from independent experiments. Section \[sec:vector\] extends these ideas to the vector valued settings. In Section \[sec:single\] we study the performance of an estimator using all data from a single trajectory – this is significantly more challenging as all covariates are strongly correlated. Finally, in Section \[sec:data-dependent\], we provide data-dependent bounds that can be used in practical algorithms.
conclusion
==========
In this paper, we provided a brief introduction to tools useful for the finite-time analysis of system identification algorithms. We studied the full information setting, and showed how concentration of measure of sub-Gaussian and sub-exponential random variables are sufficient to analyze the independent trajectory estimator. We further showed that the analysis becomes much more challenging in the single-trajectory setting, but that tools from self-normalized martingale theory and small-ball probability are useful in this context. Finally, we provided computable data-dependent bounds that can be used in practical algorithms. In our companion paper [@extended], we show how these tools can be used to design and analyze self-tuning and adaptive control methods with finite-data guarantees. Although we focused on the full information setting, we note that many of the techniques described extend naturally to the partially observed setting [@oymak2018non; @sarkar2019finite; @tsiamis2019finite].
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'It is shown that quantum aspects of the energy-momentum tensor reveal dark matter and dark energy behavior for general relativity. Special meaning is assigned to the operator $X^2d^2/dX^2 +2Xd/dX$, for any field $X$ that is part of the energy-momentum tensor, and to the distinction between information on gravitational and non-gravitational effects. Changes in such information cause the Sun-Earth distance to increase by $11$ cm/yr.'
address: |
Kapteyn Astronomical Institute, University of Groningen\
P.O. Box 800, 9700 AV Groningen, The Netherlands\
[email protected]
author:
- 'M. Spaans'
title: 'Gravity and Information: Putting a Bit of Quantum into GR'
---
Quantum Aspects of the Energy-Momentum Tensor
=============================================
Einstein’s famous elevator/rocket thought experiment unambiguously connects energy-momentum to space-time curvature as the expression of gravity, and leads to the Einstein equation\[1\]. The a piori separation of space-time and energy-momentum was not to Einstein’s liking, despite the profound success of general relativity, and some deeper connection between the two may exist\[1,2,3,4,5,6,7,8\]. In this light, and as a follow-up to \[5\], quantum aspects are considered.
The energy momentum tensor is an observable quantity. Quantum-mechanically, this implies that it may be impacted by interaction with an observer. Indeed, the place, $s_1\equiv X\cdot$, and momentum, $s_2\equiv d/dX$, operators for some field $X$ in the energy-momentum tensor allow one to assign field values and the changes therein, but $s_1$ and $s_2$ do not commute. In fact, an observer that measures a field in the energy-momentum tensor (through $s_1$) also changes it (so acts with $s_2$). Through the Einstein equation this change propagates into space-time, which causes a secondary change (through $s_2$ again). It is this back-reaction that the observer should actually measure (through $s_1$ again). This leads to a “measurement” operator $M_X=s_1s_2s_2s_1=X^2d^2/dX^2 + 2Xd/dX$ that acts on any field in the energy-momentum tensor. The influence of the observer may seem small, [*but*]{} in reality any system observes itself unremittingly.
If implemented, $M_X$ would be a significant modification of Einstein gravity. Of course, one can argue that the measurement argument has no merit because the observer and observed parts of the gravitational system are assumed to be distinct entities in the first place. This very distinction, as an observable fact, implies that the action of the operator $M_X$ must be the identity. This is quite true. However, the distinction between observer and observed parts of the gravitational system can be made using [*only*]{} non-gravitational information. That is, irrespective of gravitational forces and, say, through exchange of information carried by light, the entanglement of an observer and a system part can be affected\[9\]. Consequently, $M_X$ does act if there is a difference between the non-gravitational and gravitational information that is necessary to describe the energy-momentum tensor. Therefore, one finds an operator $[M_X]^p$ with $p=0$ when a field $X$ is relevant to both gravitational and non-gravitational processes ($[M_X]^p$ is just the identity then) and $p=1$ when this is not the case.
Einstein gravity, for Newton’s constant $G_N$, is given by ${\bf G}_{\mu\nu}/8\pi G_N {\bf T}_{\mu\nu}=1$ in the usual notation. Typically, $T^{\mu\nu}=(\rho +P/c^2) U^\mu \times U^\nu -P g^{\mu\nu}$, with density $\rho$, pressure $P$ and four-velocity $U^\mu$. Semi-classically, one can just transform a classical field $X$ in $T^{\mu\nu}$ to $[M_X]^pX$. E.g., under $M_X$ one has eigenvalues 2 and 6 in $M_P P=2P$ and $M_U U^2=6U^2$, yielding $M_{P,U} T^{\alpha\beta}=(\rho +2P/c^2) 6U^\alpha \times U^\beta -2P g^{\alpha\beta}$. This specific $p=1$ case pertains to a collisionless fluid, since in the absence of collisions no non-gravitational information on pressure and velocity is needed, and only density is required for the non-gravitational aspects of the system to be tangible. For pure dust the pressure is zero and one finds a factor 6 increase in the strength of gravity. Interestingly, this is equivalent to the 5 times more dark matter than baryonic matter that is needed to describe the dynamics of (almost collisionless) stellar systems like galaxies\[5\]. For systems that are partially collisionless (e.g., due to stellar collisions or gas hydrodynamics), the discrete variable $p$ varies with position depending on the local gravitational and non-gravitational information needed. Interestingly, below a redshift of $1-2$ fewer galaxies merge and most stars are in place, and the universe as a whole is relatively quiet. So more recently than redshift $1-2$ processes that need mostly gravitational information occur less frequently, by a factor of more than a few looking at the mass accumulation history of galaxies, compared to those that require gravitational and non-gravitational information (like the evolution of stars and diffuse matter). This favors $p=0$ for the global matter distribution of the universe since a redshift of $1-2$, compared to $p=1$ for earlier times, and leads to an acceleration in cosmic expansion (dark energy behavior).
From Fields to Particles: the Sun-Earth System
==============================================
The operator $[M_X]^p$, when applied to individual particle excitations of an underlying field like density, should act exactly the same and weigh deviations in the required gravitational versus non-gravitational information of contributing particles. The Sun is obviously a collisionally dominated system and $[M_X]^p=1$. However, through particle interactions, the fusion cycle that powers the Sun causes our star to loose mass and $G_Nm(t)=G_Nm_0-G_N\delta m(t)+(f-1)G_N\delta m(t)$ for a starting mass $m_0$ in the classical limit. The $f$ term represents any extra gravitational information that is needed and follows from $T^{\mu\nu}$. For negligible pressure, $f=12$, i.e., 2 from $\rho$ times 6 from $U^2$ because now the non-gravitational information on [*changes*]{} due to the fusion cycle resides with individual particles only, independent of their mass density and four-velocity. In fact, one expects $f=36$ since 3 particles are removed for every produced $^4$He in the dominant fusion cycle of the Sun. This extra factor three is a pure particle effect not captured by the field description of $[M_X]^p$. In all, this translates into a correction of $34G_N\delta m(t)$ to $G_Nm_0$, which can be viewed as an information dependent change in the gravitational constant\[5\]. Therefore, the astronomical unit AU increases by $\sim 3\times 10^{-12}/yr$ due to the astronomical definition $G_Nm_\odot \equiv k^2 AU^3$\[10\], with $k$ Gauss’ constant. This yields $\sim 11$ cm/yr, close to what appears to be observed\[10\].
[0]{}
C.W. Misner, K.S. Thorne and J.A. Wheeler, [*Gravitation*]{} (W.H. Freeman and Company, New York, 1973) M. Spaans, [*Nuc. Phys. B*]{} [**492**]{} (1997) 526-542 M. Spaans, On Particle Mass Changes and GR: Space-time Topology Causes LHC Leakage, arXiv:0804.1688 M. Spaans, Comments on the Invariance of Physical Laws Under Particle Re-Arrangement, arXiv:0708.0215 M. Spaans, The Role of Information in Gravity, arXiv:0903.4315 M. Spaans, A Derivation of Einstein Gravity without the Axiom of Choice: Topology Hidden in GR, arXiv:0705.3902 M. Spaans, A Background Independent Description of Physical Processes, arXiv:gr-qc/0502004 M. Spaans, On the Topological Nature of Fundamental Interactions, arXiv:gr-qc/9901025 S. Dürr, T. Nonn and G. Rempe, [*Nature*]{} [**395**]{} (1998) 33-37 J.D. Anderson and M.M. Nieto, Astrometric Solar-System Anomalies, IAU Symp. 261, \# 7.02 and BAAS 41, 281; arXiv:0907.2469
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'T. Danilovich [^1]'
- 'S. Ramstedt'
- 'D. Gobrecht'
- 'L. Decin'
- 'E. De Beck'
- 'H. Olofsson'
bibliography:
- '33317.bib'
date: 'Received / Accepted '
subtitle: 'II: Abundances and distributions of CS and SiS'
title: 'Sulphur-bearing molecules in AGB stars'
---
[Sulphur has long been known to form different molecules depending on the chemical composition of its environment. More recently, the sulphur-bearing molecules SO and 2S have been shown to behave differently in oxygen-rich AGB circumstellar envelopes of different densities.]{} [By surveying a diverse sample of AGB stars for CS and SiS emission, we aim to determine in which environments these sulphur-bearing molecules most readily occur. We include sources with a range of mass-loss rates and carbon-rich, oxygen-rich, and mixed S-type chemistries. Where these molecules are detected, we aim to determine their CS and SiS abundances.]{} [We surveyed 20 AGB stars of different chemical types using the APEX telescope, and combined this with an IRAM 30 m and APEX survey of CS and SiS emission towards over 30 S-type stars. For those stars with detections, we performed radiative transfer modelling to determine abundances and abundance distributions.]{} [We detect CS towards all the surveyed carbon stars, some S-type stars, and the highest mass-loss rate oxygen-rich stars ($\dot{M}\geq 5\e{-6}\spy$). SiS is detected towards the highest mass-loss rate sources of all chemical types ($\dot{M}\geq 8\e{-7}\spy$). We find CS peak fractional abundances ranging from $\sim 4\e{-7}$ to $\sim2\e{-5}$ for the carbon stars, from $\sim 3\e{-8}$ to $\sim1\e{-7}$ for the oxygen-rich stars and from $\sim 1\e{-7}$ to $\sim8\e{-6}$ for the S-type stars. We find SiS peak fractional abundances ranging from $\sim 9\e{-6}$ to $\sim 2\e{-5}$ for the carbon stars, from $\sim 5\e{-7}$ to $\sim 2\e{-6}$ for the oxygen-rich stars, and from $\sim 2\e{-7}$ to $\sim 2\e{-6}$ for the S-type stars. ]{} [Overall, we find that wind density plays an important role in determining the chemical composition of AGB CSEs. It is seen that for oxygen-rich AGB stars both CS and SiS are detected only in the highest density circumstellar envelopes and their abundances are generally lower than for carbon-rich AGB stars by around an order of magnitude. For carbon-rich and S-type stars SiS was also only detected in the highest density circumstellar envelopes, while CS was detected consistently in all surveyed carbon stars and sporadically among the S-type stars.]{}
Introduction
============
After leaving the main sequence and passing through the red giant branch, low- to intermediate-mass stars become asymptotic giant branch (AGB) stars. These stars are characterised by intense mass loss, ejecting matter in a stellar wind which forms a circumstellar envelope (CSE) around the star [@Hofner2018]. These CSEs are known to be rich in different molecular species and are also a site of dust formation [@AGB]. The matter ejected in this manner contributes to the chemical enrichment of the interstellar medium (ISM) and the chemical evolution of galaxies [@Herwig2005].
The chemical characteristics of the CSE depend in large part on the chemical type of the AGB star, classified based on the photospheric carbon-to-oxygen ratio (C/O). Carbon-rich stars and oxygen-rich (M-type) stars have larger proportions of carbon and oxygen, respectively. S-type stars are believed to be intermediary transition objects with C/O $\sim 1$. The CSEs of carbon stars are known to contain a variety of carbon-bearing molecules [see for example @Olofsson1993a; @Cernicharo2000; @Gong2015], while the CSEs of oxygen-rich stars are typified by the presence of a variety of oxygen-bearing molecules [see for example @Velilla-Prieto2017]. S-type stars display a mixture of the more common oxygen- and carbon-bearing molecules [@Schoier2011; @Danilovich2014]. However, it is now known that high or low C/O do not preclude the formation of oxygen- or carbon-bearing molecules, respectively. For example, 2O and SiO have been detected and found to have unexpectedly high abundances towards carbon stars [@Schoier2006; @Lombaert2016], while HCN has been detected towards oxygen-rich stars [@Schoier2013].
Sulphur is a relatively abundant element which forms molecular bonds with both oxygen and carbon, among other species, and is hence found in a variety of molecules in the CSEs of AGB stars. For example, SO and 2 are commonly found in the CSEs of oxygen-rich AGB stars [@Danilovich2016], while CS has been found to be very abundant in carbon-rich AGB stars [@Olofsson1993a]. However, molecular abundances in AGB CSEs have been found to not only depend on the C/O of the CSE, but also on other factors, such as the density of the stellar wind, which is related to the mass-loss rate. For example, [@Danilovich2016] found different radial distributions of SO for low mass-loss rate AGB stars compared with higher mass-loss rate AGB stars, indicating that SO was formed at larger radii in the latter case. 2S, which contains neither carbon nor oxygen, is preferentially detected in higher mass-loss rate oxygen-rich stars [@Danilovich2017a] and, to date, has only been detected, weakly, towards one carbon-rich AGB star, CW Leo [@Cernicharo1987; @Omont1993; @Cernicharo2000]. Similar patterns of different molecular occurrences between high and low mass-loss have also been seen for other molecules such as SiO, which has been found to have higher abundances for lower mass-loss rate AGB stars [@Gonzalez-Delgado2003; @Schoier2006].
Previously, SiS was studied in a sample of oxygen-rich and carbon-rich AGB stars by [@Schoier2007], in which they generally find abundances of SiS in carbon-rich stars about an order of magnitude higher than in oxygen-rich stars. This strongly suggests that SiS is preferentially formed in carbon-rich CSEs. Although their initial models were based on a Gaussian abundance distribution profile, they find they needed to include a ‘core’ component with a higher abundance and a small radius to properly fit their SiS observations. [@Decin2010] found a similar result with a high-abundance inner component and a lower-abundance outer component when modelling SiS for the oxygen-rich star IK Tau.
[@Danilovich2015a] observed the SiS ($6\to5$) line concurrently with the CO ($1\to0$) line towards 29 AGB stars of various chemical types and mass-loss rates. They detected SiS towards 12 of the sample stars with the general trend being that SiS was only detected towards the higher mass-loss rate stars. Part of our goal in this work is to confirm whether such a mass-loss rate or density dependent trend exists for SiS. [@Lindqvist1988] searched for both CS and SiS (among other molecules) in a sample of 31 oxygen-rich stars. They detected CS ($2\to1$) in only four sources and SiS ($5\to4$) in only one source (TX Cam). [@Bujarrabal1994] searched for CS ($3\to2$) and ($5\to4$), and SiS ($5\to4$) in a diverse sample of evolved stars. They detected CS in six of the highest mass-loss rate oxygen-rich stars, all three S-type stars and the majority of their carbon-rich stars. They detect SiS in three nearby high mass-loss rate oxygen-rich stars, five carbon stars, and none of the S-type stars.
[@Olofsson1993a] surveyed a large sample of carbon-rich stars for several molecules and derived photospheric SiS abundances, and both circumstellar and photospheric CS abundances for many of them. In general, they found that the circumstellar abundances tended to be higher by factors of five to ten than the photospheric abundances. In some cases they found circumstellar CS abundances high enough to account for or exceed the total amount of sulphur expected to be present based on the solar sulphur abundance [@Asplund2009]. They suggest this may be due to under-predicting mass-loss rates, which affect the derived abundances, or the effect of a simplified radiative transfer analysis.
To properly constrain the occurrences of the most common sulphur-bearing molecules in AGB CSEs, we performed a survey of 20 AGB stars, which covered all three chemical types and a range of mass-loss rates [(low mass-loss rates: $\sim 10^{-8}$ to a few $10^{-7}\spy$, intermediate mass-loss rates: $\sim10^{-6}\spy$ and higher mass-loss rates: up to a few $10^{-5}\spy$). Our sample does not include the highest mass-loss rate OH/IR stars, which will be studied separately]{}. We focussed our search on rotational transitions of CS, SiS, SO, 2 and 2S and carried out the survey using the Atacama Pathfinder Experiment [APEX[^2], @Gusten2006]. The first results from this survey, for 2S, are presented in [@Danilovich2017a]. This survey was supplemented with a smaller survey of 9 M-type AGB stars with the Onsala 20 m telescope (OSO), focussing on 2, SO and SiS at low frequencies. Additionally, a survey of CS and SiS towards 33 S-type stars has also been included in this study, with observations gathered from the IRAM 30 m telescope and APEX. In this paper we focus on the detections of SiS and CS and perform radiative transfer analyses to determine the abundances and abundance distributions of these two molecules.
Sample and observations {#obs}
=======================
APEX sulphur survey
-------------------
We surveyed several rotational emission lines of sulphur-bearing species in a chemically diverse sample of 20 AGB stars, including seven M-type stars, five S-type stars and eight carbon stars and covering mass-loss rates from $\sim9\e{-8}\spy$ to $\sim2\e{-5}\spy$. The first results from this survey have already been presented in [@Danilovich2017a] for 2S, which also further describes the observing programme carried out [in March–April and August–December of 2016,]{} using the Swedish-ESO PI receiver [SEPIA Band 5, @Billade2012; @Belitsky2018] for APEX [@Gusten2006] and the Swedish Heterodyne Facility Instrument [SHeFI, @Belitsky2006; @Vassilev2008]. In this study we have focussed on the SiS and CS observations obtained during this survey. SO and 2 results will be presented in future papers in this series.
The full sample of stars for which SiS and CS were surveyed are listed in Table \[fullsample\] [along with the stars from the other observations discussed below]{}. The lines that were included in the survey are listed in Table \[trans\], as are other available lines from other telescopes that we used to constrain our models. Table \[SiSobs\] includes all the detected SiS lines and their integrated main beam intensities and Table \[SiSnondet\] lists the rms noise at $1~\kms$ for each observed SiS line, whether or not it was detected. The integrated main beam intensities for the CS lines are listed in Table \[CSobs\], as are the rms noise values for all observed lines.
------ ------------ --------- ------ --------------- -------------------- -----------------
Mol. Line Freq. Tel. $\theta$ $\eta_\mathrm{mb}$ $E_\mathrm{up}$
\[GHz\] \[$\arcsec$\] \[K\]
SiS $4\to3$ 72.618 OSO 45 0.55 9
$5\to4$ 90.772 IRAM 27 0.81 13
$6\to5$ 108.924 IRAM 21 0.78 18
$8\to7$ 145.227 IRAM 17 0.65 31
$9\to8$ 163.377 APEX 38 0.68 39
$10\to9$ 181.525 APEX 34 0.68 48
$11\to10$ 199.672 APEX 31 0.68 58
$12\to 11$ 217.818 APEX 29 0.75 68
$12\to 11$ 217.818 IRAM 11 0.63 68
$13\to 12$ 235.961 IRAM 10 0.59 79
$14\to 13$ 254.103 APEX 25 0.75 92
$16\to 15$ 290.381 APEX 22 0.75 119
$19\to 18$ 344.779 APEX 18 0.73 166
SiS $10\to 9$ 176.555 APEX 35 0.68 38
$11\to 10$ 194.205 APEX 32 0.68 47
CS $3\to2$ 146.969 IRAM 17 0.73 14
$4\to3$ 195.954 APEX 32 0.68 24
$5\to4$ 244.936 IRAM 10 0.59 35
$6\to5$ 293.912 APEX 21 0.75 49
$7\to6$ 342.883 APEX 18 0.73 66
CS $6\to5$ 289.382 APEX 22 0.75 49
------ ------------ --------- ------ --------------- -------------------- -----------------
: Observational parameters for the SiS and CS lines included in this study[]{data-label="trans"}
S-star survey
-------------
A total of 33 S-type stars were surveyed in CS and SiS emission using two telescopes. The stars from this sample are listed in the bottom part of Table \[fullsample\]. The CS ($3\to2$) and ($5\to4$) line emission was observed at the IRAM 30m telescope simultaneously with the HCN observations analysed in [@Schoier2013]. SiS (19$\to$18) was observed at the APEX 12m telescope using the SHeFI receiver in August to October, 2012. SiS (5$\to$4), (6$\to$5), (12$\to$11), and (13$\to$12) was observed at IRAM June 22–24, 2013.
The IRAM 30m telescope observations were performed in dual beamswitch mode using a beam throw of about 2, while at APEX position-switching was used with a reference position at +3. The pointing was checked regularly using strong CO and continuum sources and found to be consistent within $\approx$3 of the respective telescope pointing model.
At the telescope, the antenna temperature has been corrected for the attenuation of the atmosphere and spectra are first delivered in $T_{\rm{A}}^{\star}$-scale. They have been converted to $T_{\rm{mb}}$ scale using $T_{\rm{mb}}$=$T_{\rm{mb}}$/$\eta_{\rm{mb}}$, where $\eta_{\rm{mb}}$ is the main-beam efficiency. The adopted beam efficiencies and full-width at half-max beam sizes ($\theta_{\rm{mb}}$) are given in Table \[trans\]. The uncertainty in the absolute intensity scale is estimated to be about $\pm20$%.
The IRAM CS ($3\to2$) and ($5\to4$) observations are listed in Table \[csiramobs\], the APEX SiS ($19\to18$) observations are listed in Table \[sisapexobs\], and the IRAM SiS ($5 \to 4$), ($6 \to 5$), ($12 \to 11$), and ($13 \to 12$) observations are listed in Table \[sisiramobs\].
OSO 4 mm observations of M-type stars
-------------------------------------
A sample of eight northern M-type AGB stars, covering a range of mass-loss rates from $\sim9\e{-8}$ to $\sim3\e{-5}\spy$, were observed in the period 18–22 February 2016 as part of science verification for the new 4 mm HEMT amplifier receiver [@Belitsky2015] on the 20 m telescope at Onsala Space Observatory[^3] (OSO). The observations covered the SiS ($4\to 3$) emission line at 72.618 GHz for the eight sources listed in Table \[osoobs\], which also includes rms noise levels. The line was undetected in all sources except for , for which it was tentatively detected. We have not included BX Cam in our modelling since it is too northern to observe from APEX, and one tentatively detected line forms a dataset of insufficient quality to model well. However, we included the non-detected SiS ($4\to 3$) lines in our models of IK Tau and GX Mon, to place additional constraints on those models.
Supplementary observations
--------------------------
For the carbon star AI Vol we included two additional SiS lines, ($11\to10$) and ($10\to9$), which were observed with APEX/SEPIA Band 5 as part of an unbiased line survey whose results are yet to be published in full (De Beck et al, in prep). To better constrain our models, we included some previously published observations. These include IRAM 30 m observations of the SiS ($6\to5$) line at 108.924 GHz from [@Danilovich2015a]. We also included observations of IK Tau previously published by @Decin2010 and a few lines taken from the APEX archive. The full list of archival observations used in our study is found in Table \[supobs\].
Modelling
=========
Established parameters {#modparam}
----------------------
The bulk of our APEX sulphur survey and OSO samples was chosen from the stars with mass-loss rates determined through CO modelling by [@Danilovich2015a], [while most of the S star survey came from [@Ramstedt2009] and [@Ramstedt2014]]{}. We use the circumstellar model results from those studies as the basis for our SiS and CS modelling. For the stars not included in these studies, we used a variety of previously obtained mass-loss rates, as noted in Table \[fullsample\]. W Aql is included in our observing sample, however, the modelling results for this star presented here are based on line radiative transfer modelling of ALMA observations of CS and SiS by [@Brunner2018].
Some of the key stellar and circumstellar quantities for our sample — systemic velocity ($\upsilon_{\mathrm{LSR}}$), distance, mass-loss rate ($\dot{M}$), stellar effective temperature ($T_\mathrm{eff}$), and terminal expansion velocity ($\upsilon_\infty$) — are listed in Table \[fullsample\]. In one instance, IRC -10401, we recalculated the mass-loss rate based on newer data, which is explained in more detail in Sect. \[irc-10401\]. [A detailed discussion on the uncertainties in mass-loss modelling can be found in [@Ramstedt2008].]{}
[The referenced studies in Table \[fullsample\] also include models of the dust surrounding each star. Similar dust modelling methods are used in all the studies and the specific dust properties of each source can be found in its referenced study.]{}
Molecular data {#moldat}
--------------
For both SiS and CS we performed our radiative transfer analysis using molecular descriptions including rotational energy levels from $J=0$ to $J=40$ in the ground and first excited vibrational states. These energy levels are shown in Fig. \[ELD\] and are connected by 160 radiative transitions and 820 collisional transitions. For SiS, the energy levels and radiative transition parameters were all taken from the JPL spectroscopic database[^4] [@Pickett1998], while the collisional rates for SiS-2 are adopted from SiO-2 rates, which were themselves scaled and extrapolated from the SiO-He rates of @Dayou2006. For CS, the energy levels and radiative transition parameters were all taken from the Cologne Database for Molecular Spectroscopy [CDMS[^5], @Muller2005; @Endres2016]. The adopted collisional rates come from those of CO-2 computed by @Yang2010 and an assumed 2 ortho-to-para ratio of three.
![Rotational energy levels in the ground and first vibrationally excited states for SiS (*left*) and CS (*right*) included in our modelling. For both molecules the most energetic rotational level included is at $J=40$.[]{data-label="ELD"}](included_sis_levels_small.pdf "fig:"){height="8"} ![Rotational energy levels in the ground and first vibrationally excited states for SiS (*left*) and CS (*right*) included in our modelling. For both molecules the most energetic rotational level included is at $J=40$.[]{data-label="ELD"}](included_cs_levels.pdf "fig:"){height="8"}
Modelling procedure
-------------------
We performed our radiative transfer modelling using a one-dimensional accelerated lambda iteration method code (ALI), which is described in detail by [@Maercker2008] and [@Schoier2011], [and is based on the ALI scheme described by [@Rybicki1991]. ALI is able to deal with high optical depths]{} and has been used to model other S-bearing molecules such as SO, 2, and 2S [@Danilovich2016; @Danilovich2017a].
ALI is one-dimensional so we assumed a smooth, spherically symmetric wind with a constant mass-loss rate and velocity profile based on the stellar parameters listed in Table \[fullsample\] which are described in more detail in @Danilovich2015a. To fit our models to the observed data, we first assume a Gaussian molecular abundance distribution, $$f(r)= f_0 \exp\left(-\left(\frac{r}{R_e}\right)^2\right),$$ where $f_0$ is the peak central abundance and $R_e$ is the $e$-folding radius at which the the abundance has dropped to $f_0/e$. As we have no [a priori]{} constraints on the $e$-folding radius, we leave both $f_0$ and $R_e$ as free parameters in our modelling, to be adjusted to best fit the available data. This is only possible for the sources for which we have detected at least two different transitions with sufficiently distinct emitting regions. [Gaussian abundance profiles have been shown to be adequate fits for various molecules in the past, such as SiO [@Gonzalez-Delgado2003], 2O [@Maercker2016], and others [@Schoier2011; @Danilovich2014]. In the absence of more detailed information as to the radial distributions of CS and SiS (such as spatially resolved observations), we have chosen to use Gaussian abundance distribution profiles here based on these past results and due to the ease with which they can be adjusted to find the best fit for the data.]{}
To determine which models best fit the data, we minimised a $\chi^2$ statistic, which we define as $$\chi^2 = \sum^N_{i=1} \frac{\left(I_\mathrm{mod,i} - I_\mathrm{obs,i}\right)^2}{\sigma_i^2},$$ where $I$ is the integrated main beam line intensity, $\sigma$ is the uncertainty in the observed line intensities and $N$ is the number of lines being modelled. In general, we assumed an uncertainty of 20% in line intensity for our observed lines, except for those we identify as being tentatively detected, for which we assumed a 50% uncertainty. The uncertainties calculated for our model results are for a 90% confidence interval using this $\chi^2$ formulation. To better allow us to compare between stars for which differing numbers of observed lines might be available, we further defined a reduced-$\chi^2$ statistic: $\chi^2_\mathrm{red} = \chi^2/(N-p)$ where $p=2$ is the number of free parameters in our models (and hence for $N\leq3$ we leave $\chi_\mathrm{red}^2=\chi^2$). In the cases where only one line was detected for a particular source and molecule, we cannot calculate a $\chi^2$ value and our uncertainties are based on a 20% shift in model integrated intensity.
Modelling results
-----------------
We were able to successfully model the SiS and CS line emission in the CSEs of all the stars in our sample for which at least two lines per molecule were detected with only one exception. II Lup proved difficult to model using a spherically symmetric model with a smoothly accelerating wind and will be discussed in more detail in Appendix \[iilup\]. For the stars with only one detected line or with only two lines from adjacent SiS transitions detected, it was not possible to determine an $e$-folding radius. In these cases, we obtained the $e$-folding radius from a fit to our other results. This is discussed in more detail in Sect. \[analysis\].
The abundances, $f_0$, and $e$-folding radii, $R_e$, that we have derived are listed in Table \[results\]. A summary of the results is shown in Fig. \[starab\]. In Fig. \[abundancevsdensity\] we plot both SiS and CS abundances against wind density and in Fig. \[sisvscs\] we plot CS abundance against SiS abundance for the stars towards which both molecules were detected. The SiS results are discussed in more detail in Sect. \[sisresults\] and the CS results in Sect. \[csresults\]. Isotopologue modelling is discussed in Appendix \[isotopologues\].
{width="\textwidth"}
![Abundances of both SiS and CS plotted against stellar wind density, given by the mass-loss rate divided by the terminal expansion velocity.[]{data-label="abundancevsdensity"}](abundancesvsdensity.pdf){width="49.00000%"}
![Derived CS fractional abundances plotted against derived SiS fractional abundances.[]{data-label="sisvscs"}](sisvscsabundances.pdf){width="49.00000%"}
--------------------- ------------------------- --------------------- ----- ----------------------- ------------------ ----------------------------- --------------------- ----- ----------------------- -----------------
Star
$f_0$ $R_e$ $n$ $\chi^2_\mathrm{red}$ $N_\mathrm{SiS}$ $f_0$ $R_e$ $n$ $\chi^2_\mathrm{red}$ $N_\mathrm{CS}$
$[\e{-6}]$ \[$\e{15}$ cm\] \[cm$^{-2}$\] $[\e{-6}]$ \[$\e{15}$ cm\] \[cm$^{-2}$\]
*Carbon stars*
R Lep ... ... 0 ... ... $7.4_{-2.9}^{+3.6}$ $9.2_{-3.7}^{+7.1}$ 2 0.00 $1.2\e{17}$
V1259 Ori $9.0^{+2.8}_{-2.5}$ $13\pm3$ 5 3.4 $1.1\e{18}$ $1.3^{+?}_{-1.1}$ $\dagger$ $80_{-70}^{+?}$ \* 3 0.59 $1.7\e{18}$
AI Vol $24_{-8}^{+18}$ $6.2_{-1.1}^{+1.2}$ 7 3.2 $1.2\e{18}$ $18_{-10}^{+?}$ $\dagger$ $10^{+6}_{-4}$ 2 0.03 $8.9\e{17}$
X TrA ... ... 0 ... ... $11\pm7$ $4.0_{-1.5}^{+2.5}$ 2 0.01 $3.2\e{17}$
V821 Her $9.4\pm2.6$ $5.7\pm1.0$ 5 2.9 $2.8\e{17}$ $3.3_{-1.2}^{+1.6}$ $28_{-13}^{+36}$ 3 0.54 $9.9\e{16}$
U Hya ... ... 0 ... ... $0.42^{+0.09}_{-0.14}$ $4.8^{+4.7}_{-2.0}$ 3 7.1 $1.1\e{15}$
RV Aqr $11\pm6$ $2.9^{+0.5}_{-0.6}$ 3 0.71 $2.5\e{17}$ $7.0_{-3.4}^{+8.1}$ $22_{-11}^{+66}$ 2 0.00 $1.7\e{17}$
*Oxygen-rich stars*
IK Tau $1.7\pm0.4$ $5.3\pm1.0$ 7 1.1 $1.1\e{17}$ $0.11\pm0.04$ $23_{-12}^{+72}$ 2 0.02 $6.1\e{15}$
GX Mon $0.53\pm0.12$ $13\pm4$ 5 2.0 $3.5\e{16}$ $0.083^{+0.062}_{-0.065}$ $\gtrsim 200$ \* 2 0.36 $5.6\e{15}$
V1300 Aql $1.1\pm0.3$ $12\pm3$ 5 1.4 $7.6\e{16}$ $0.029\pm0.023$ $40_{-34}^{+?}$ \* 2 0.00 $2.5\e{15}$
*S-type stars*
R And ... ... 0 ... ... $0.15\pm0.5$ $18_{-8}^{+26}$ 2 0.01 $1.4\e{15}$
S Cas [$1.1^{+0.3}_{-0.2}$]{} $3.6$ \* 2 1.1 $2.4\e{16}$ $0.74_{-0.24}^{+0.26}$ $16^{+8}_{-6}$ 2 0.0 $1.7\e{16}$
IRC -10401 $0.60\pm0.12$ $3.0$ \* 1 ... $3.7\e{15}$ ... ... 0 ... ...
S Lyr ... ... 0 ... ... $8.2^{+3.7}_{-2.8}$ $19$ \* 1 ... $3.4\e{17}$
W Aql$^\ddagger$ $1.5\pm0.05$ $6.0$ ... ... $5.9\e{16}$ $1.2\pm0.05$ $7.0$ ... ... $4.7\e{16}$
EP Vul ... ... 0 ... ... $0.18\pm0.04$ $8.6$ \* 1 ... $1.1\e{15}$
$\chi$ Cyg $0.18\pm0.06$ $2.5$ \* 2 1.4 $1.5\e{15}$ $0.10\pm0.03$ $15^{+34}_{-7}$ 2 0.01 $5.7\e{14}$
RZ Sgr $0.28\pm0.06$ $7.2$ \* 1 ... $1.0\e{16}$ ... ... 0 ... ...
--------------------- ------------------------- --------------------- ----- ----------------------- ------------------ ----------------------------- --------------------- ----- ----------------------- -----------------
Analysis
========
SiS {#sisresults}
---
In the APEX survey, SiS was detected towards five out of the eight surveyed carbon stars and three out of seven M-type stars. In the S star survey, SiS was detected in five sources. [The low-energy SiS ($4\to3$) line was only tentatively detected for one of the stars observed with the OSO 20m telescope, although higher-energy transitions were detected for the two stars overlapping with the APEX sulphur survey (IK Tau and GX Mon).]{} In all cases, these were among the highest mass-loss rate sources in each category. The implications of this will be discussed further in Sect. \[disc\]. The SiS observations and model results for a representative carbon star, AI Vol, are plotted in Fig. \[SiSAIVolplots\] with the same for the remaining carbon stars plotted in [Figures \[SiSCplots-1\], \[SiSCplots-2\], and \[SiSCplots-3\]]{}. The SiS results for a representative oxygen-rich star, V1300 Aql, are plotted in Fig. \[SiSV1300Aqlplots\] with the same for the remaining oxygen-rich stars plotted in [Figures \[SiSMplots-1\] and \[SiSMplots-2\]]{}. In general we see higher abundances of SiS for carbon stars than for oxygen-rich or S-type stars. SiS abundances range from $\sim 9\e{-6}$ to $\sim 2\e{-5}$ for carbon stars, from $\sim3\e{-7}$ to $\sim2\e{-6}$ for oxygen rich stars, and from $\sim 2\e{-7}$ to $\sim 1\e{-6}$ for the S-type stars.
From the stars for which we were able to constrain the SiS $e$-folding radius, we found the following relation between $e$-folding radius and wind density [when weighting with the uncertainties listed in Table \[results\]]{} $$\label{resis}
\log_{10}(R_{e,\mathrm{SiS}}) = (21.3\pm0.2) + (0.84\pm0.03)\log_{10}\left(\frac{\dot{M}}{\upsilon_\infty}\right),$$ where $R_e$ is given in cm, $\dot{M}$ in $\spy$, $\upsilon_\infty$ in $\kms$, [and the errors are $1\sigma$ uncertainties]{}. This fit is represented by the dashed black line in the left panel of Fig. \[radvsdens\]. When modelling the S stars with only one SiS detection, or with only the SiS ($12\to11$) and ($13\to12$) lines detected, it was not possible to fit the $e$-folding radius with the available data. Hence, we derived $R_e$ for these sources using Eq. \[resis\] and then fitted the peak abundance, $f_0$, to the available data.
![Observations (black histograms) and model results (blue lines) for SiS towards AI Vol, a carbon star, plotted with respect to LSR velocity.[]{data-label="SiSAIVolplots"}](aivol-sis1.pdf "fig:"){width="49.00000%"} ![Observations (black histograms) and model results (blue lines) for SiS towards AI Vol, a carbon star, plotted with respect to LSR velocity.[]{data-label="SiSAIVolplots"}](aivol-sis0.pdf "fig:"){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for SiS towards V1300 Aql, an M-type star, plotted with respect to LSR velocity.[]{data-label="SiSV1300Aqlplots"}](v1300aql-sis1.pdf "fig:"){width="49.00000%"} ![Observations (black histograms) and model results (blue lines) for SiS towards V1300 Aql, an M-type star, plotted with respect to LSR velocity.[]{data-label="SiSV1300Aqlplots"}](spacer.pdf "fig:"){width="10.00000%"} ![Observations (black histograms) and model results (blue lines) for SiS towards V1300 Aql, an M-type star, plotted with respect to LSR velocity.[]{data-label="SiSV1300Aqlplots"}](v1300aql-sis0.pdf "fig:"){width="35.00000%"}
{width="49.00000%"} {width="49.00000%"}
CS {#csresults}
--
In the APEX survey, CS is detected in all of the surveyed carbon stars and in three out of the seven surveyed M-type stars. In the S star survey it is detected towards six sources.
For each source we have only two or three observed CS lines with which to constrain our models. However, in each case there was at least a pair of lines with transitions separated by $\Delta J=2$ which corresponded to an increase in upper energy level by a factor greater than two and different emitting regions within the CSE. This proved to be sufficient to constrain the $e$-folding radius for most of our sources. Some example CS results for AI Vol (carbon star), IK Tau (M-type), and $\chi$ Cyg (S-type) are plotted in Figures \[CSAIVolplots\], \[CSIKTauplots\], and \[CSSplots\], respectively. The remaining carbon-rich CS models are plotted in [Figures \[CSCplots-1\], \[CSCplots-2\], \[CSCplots-3\], \[CSCplots-4\], \[CSCplots-5\], and \[CSCplots-6\], while the remaining oxygen-rich CS models are plotted in Figures \[CSMplots-1\] and \[CSMplots-2\]]{}. CS abundances range from $\sim 4\e{-7}$ to $\sim 2\e{-5}$ for carbon stars, from only $\sim3\e{-8}$ to $\sim1\e{-7}$ for oxygen rich stars, and from $\sim1\e{-7}$ to $\sim8\e{-6}$ for the S-type stars.
We had some difficulty finding a conclusive $e$-folding radius for CS towards three of our sources: the carbon-rich V1259 Ori, and the oxygen-rich GX Mon and V1300 Aql. The main difficulty with the two oxygen-rich stars was the low signal-to-noise ratio for the CS lines, leading to ambiguity in fitting the models. The case of V1259 Ori is more complicated and is discussed in more detail in Sect. \[v1259ori\].
From the stars for which we were able to constrain the CS $e$-folding radius, we find the following relation between $e$-folding radius and wind density [when weighting with the uncertainties listed in Table \[results\]]{} $$\label{recs}
\log_{10}(R_{e,\mathrm{CS}}) = (18.9\pm0.2) + (0.40\pm0.03)\log_{10}\left(\frac{\dot{M}}{v_\infty}\right),$$ where $R_e$ is given in cm, $\dot{M}$ in $\spy$, $\upsilon_\infty$ in $\kms$, [and the errors are $1\sigma$ uncertainties]{}. This fit is represented by the dashed black line in the right panel of Fig. \[radvsdens\].
We also ran models for V1259 Ori, GX Mon, and V1300 Aql with the $R_e$ as obtained from Eq. \[recs\]. While we were able to find adequate models with this added restriction, the $\chi^2$ values of the new models were consistently higher than for the models listed in Table \[results\]. The new models also had systematically higher $f_0$ values by about 10–20%, but this increase does not change our overall conclusions. We generally refer to the original models, with $R_e$ as a free parameter, when we discuss results.
![Observations (black histograms) and model results (blue lines) for CS towards AI Vol, a carbon star, plotted with respect to LSR velocity.[]{data-label="CSAIVolplots"}](aivol-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards IK Tau, an M-type star, plotted with respect to LSR velocity.[]{data-label="CSIKTauplots"}](iktau-cs0.pdf){width="49.00000%"}
Discussion {#disc}
==========
Limitations of the modelling
----------------------------
As mentioned in Sect. \[csresults\], we did not have access to many CS lines, making the modelling uncertain. The accuracy of all our CS models would improve with access to higher-$J$ lines or interferometric data to better constrain the extents of the CS envelopes. When experimenting with high CS abundances, we found the issue that very high CS abundances ($\gtrsim 2\e{-5}$) lead to the model lines becoming fainter with increased abundance rather than brighter, as would ordinarily be expected, most likely due to high optical depths leading to saturation. This explains why, for some of the highest CS abundances, we were not able to find an uncertainty to the upper limit of the abundance in our models (see the results marked with a $\dagger$ in Table \[results\]). We did not run into similar problems with SiS [since, over a given energy range, SiS energy levels are more numerous, allowing the molecules to be spread over a larger number of states and hence reducing the optical depth of the lines. This is also why SiS lines are inherently less intense than CS lines [@Muller2005]]{}. Two stars for which we found somewhat anomalous results, AI Vol and II Lup, are discussed in more detail in Sections \[aivol\] and \[iilup\].
Trends seen in our results
--------------------------
The estimated SiS and CS fractional abundances are summarised in Fig. \[starab\], and the abundances shown as a function of CSE density in Fig. \[abundancevsdensity\]. SiS is only detected for the higher mass-loss rate objects for all three chemical types. Its abundance is $\approx$10$^{-5}$ for the carbon stars, and hence CS and SiS combined can account for [almost]{} all of the sulphur in these stars. For the oxygen-rich and S-type stars the average SiS abundance is about one order of magnitude lower. Based on our detection limits, we calculated an upper limit on the fractional SiS abundance for W Hya of $1\e{-6}$, for RR Aql of $3\e{-7}$, for R Lep of $6\e{-6}$ and [for S Lyr of $1.5\e{-5}$]{}. W Hya is the lowest mass-loss rate M-type star, RR Aql and R Lep are the highest mass-loss rate M-type and carbon stars, respectively, for which SiS was not detected, [and S Lyr is the highest mass-loss rate S-type star with a CS detection but not an SiS detection]{}. These upper limits do not rule out abundances comparable to those for similar stars for which SiS was detected. [We note, in particular, that the upper limit for SiS towards S Lyr is a similar amount (in dex) above the calculated CS abundance in Fig. \[abundancevsdensity\] as seen for the other S-type stars as well as the carbon stars with similar densities.]{}
The CS abundances are $\approx$10$^{-5}$ for the carbon stars independent of the CSE density (only the star with the lowest mass-loss rate is a significant exception to this). For the oxygen-rich stars CS was only detected for the higher mass-loss rate objects, and here the abundances are more than two orders of magnitude lower than for the carbon stars. The upper limits we calculated for W Hya, RR Aql, [and RZ Sgr]{} were $9\e{-8}$, $6\e{-8}$, [and $1.5\e{-7}$]{} respectively. [For the M-type stars this is]{} comparable to the abundances for the M-type stars with detected CS. [For RZ Sgr the CS upper limit is just below the SiS abundance, giving a similar difference between SiS and (upper limit) CS abundances as found for the other S-type stars.]{} For the S-type stars [as a whole,]{} there is a trend such that the CS abundances for the lower mass-loss rate stars are almost two orders of magnitude lower than for the carbon stars, which increase to values similar to that of carbon stars at the higher mass-loss rates. [However, RZ Sgr does not comply with this trend if the CS upper limit is taken into account. Similarly, a trend in SiS abundances could be seen with higher abundances correlated with higher densities if RZ Sgr were to be excluded. It is unclear from the available data, especially considering the small number of detections for CS and SiS towards S stars, whether RZ Sgr is an outlier due to inaccurate input parameters (such as mass-loss rate and/or distance) or whether it truly belies the apparent trends seen for the other S stars. Higher sensitivity observations of the S stars, providing a larger sample of detections, would help to confirm whether the trends we see here are real or a coincidental product of the sample. The potential anomaly of RZ Sgr aside,]{} the abundances of CS for the S-type stars fall between those of carbon and oxygen-rich stars.
In Fig. \[sisvscs\] we plot the modelled abundances of CS against those of SiS. The points appear to be grouped by chemical type with the carbon stars clustered in the top right, exhibiting the highest abundances of both molecular species. For the M-type and S-type stars there may be a correlation between the abundances of SiS and CS, possibly following slightly different trends. However, the small number of sources involved renders this only a tentative result.
While it might be expected that S-type stars with greater amounts of photospheric carbon — such as those classified as SC stars — would be more likely to produce circumstellar CS, there is no clear relationship in our results between spectral type and the detection of CS in the S-type stars. For example TT Cen and S Lyr are both SC stars, but CS was only detected in S Lyr, despite the expectation that (assuming a similar abundance) it should be easier to detect in TT Cen, which is a closer source and has a similar (even slightly higher) mass-loss rate.
[As can be seen in Fig. \[radvsdens\], we find generally smaller $e$-folding radii for SiS than for CS. This is most likely due to the fact that the binding energy of CS is higher than that of SiS [7.8 eV compared with 6.4 eV, @Herzberg1989; @Gail2013], making SiS more readily photodissociated by the interstellar radiation field.]{}
We also note the results of [@Gonzalez-Delgado2003], [@Schoier2006], and [@Ramstedt2009] who observed and modelled the abundances of SiO in oxygen-rich, carbon-rich and S-type AGB stars, respectively. Those studies find a trend of decreasing SiO abundance with increasing wind density, most clearly seen for the oxygen-rich and carbon-rich stars (since fewer high mass-loss rate S-type stars have been identified). [A similar trend was found by [@Massalkhi2018] for SiC$_2$ abundance decreasing with carbon star wind density.]{} Although we do not see a clear trend in SiS abundance with density (see Fig. \[abundancevsdensity\]), it is possible that the detection of SiS only in the highest mass-loss rate AGB stars is linked to the decreased abundances of SiO [and/or SiC$_2$]{} in the same stars. [Also, considering only the S-type stars and excluding RZ Sgr (see discussion above), there is a possible trend of increased SiS abundance with increased density, the opposite of the trend seen for SiO by [@Ramstedt2009].]{} For the carbon stars that have very high SiS abundances, accounting for roughly half the available S [for a solar abundance of S, taken from @Asplund2009], the SiS abundance can also account for a significant portion of the available Si (the solar abundance of which is approximately twice that of S). For some of the low mass-loss rate stars included in the [@Gonzalez-Delgado2003] and [@Schoier2006] studies, however, the abundances of SiO approaches the solar abundance of Si. For W Hya, a nearby low mass-loss rate star for which we did not detect SiS, [@Khouri2014a] found an SiO abundance high enough to account for almost all of the Si, while [@Danilovich2016] found that SO and 2 combined account for almost all of the S. [As Si, O and C are all known to play a part in dust formation, the depletion of these elements onto dust grains may play a part in the sulphur chemistry, especially if we consider that larger quantities of dust are generally associated with higher mass-loss rate AGB stars [@Justtanont1992]. In any case,]{} it seems from both earlier studies and from this work that the wind density plays an important role in determining the chemical composition of AGB CSEs.
Comparison with other observational studies
-------------------------------------------
[@Schoier2007] surveyed a sample of carbon- and oxygen-rich AGB stars and detected SiS towards eleven carbon stars and eight M-type AGB stars. They do not explicitly list any non-detections, but their presented SiS lines are mostly[^6] seen towards high mass-loss rate stars. In their radiative transfer modelling, [@Schoier2007] assumed the same photodissociation radius for SiS as for SiO. Therefore, they used the [@Gonzalez-Delgado2003] SiO empirical relation between mass-loss rate, wind velocity and photodissociation to find $e$-folding radii of SiS for their sample stars. They were unable to find good fits to the observed data using this assumption and found better fits by adding a central component with radius out to $1\e{15}$ cm and with a high SiS abundance of $2\e{-5}$. With this distribution SiS would account for most of the sulphur in the inner CSE. In contrast, we were able to find good fits with Gaussian SiS abundance distributions by leaving the $e$-folding radius as a free parameter. This approach did not require the inclusion of a central component of higher SiS abundance. For those sources which overlap with the [@Schoier2007] sample we found smaller $e$-folding radii than they did and fractional abundances larger than their Gaussian components but smaller than their inner components. [It is not surprising that leaving the $e$-folding radius as a free parameter gives a better fit to the observations than using the SiO $e$-folding radius does, since the dissociation energy of SiS is 6.4 eV, compared with 8.28 eV for SiO [@Gail2013]. These are sufficiently different that the extents of the corresponding molecular envelopes ought not to be identical.]{}
[@Decin2010] modelled several molecules, including SiS and CS, for the oxygen-rich AGB star IK Tau. They find an inner abundance for CS of $8\e{-8}$, relative to the 2 abundance, and use a non-Gaussian distribution based partly on the results of chemical modelling. Their CS result is [in good agreement with ours,]{} within a factor of $\sim1.4$, although we find a smaller extent for the CS envelope. This is mostly likely due to the addition of the lower-$J$ line CS ($4\to3$) in our study, compared with the use of only the ($7\to6$) and ($6\to5$) in the [@Decin2010] study. For SiS, [@Decin2010] found a similar core and extended plateau abundance distribution to that used by [@Schoier2007]. They found a high inner abundance of SiS of $1.1\e{-5}$, relative to 2, which drops to $8\e{-9}$ at $\sim1.5\e{15}$ cm and does not decrease again until $\sim3\e{16}$ cm. While this agrees well with [@Schoier2007], despite having been calculated using different methodology, it does not agree with our result for the same reasons discussed above. Our abundance of $1.7\e{-6}$ is intermediate to their two extremes and the extent of our SiS envelope is significantly smaller.
[@Olofsson1993a] surveyed a sample of about 40 carbon stars and detected the CS ($2\to1$) line in 11 of them. The only stars in both their sample and ours were U Hya, X TrA, and R Lep, for which they did indeed also detect CS. The CS abundances they calculated for all sources were upper limits, and our results fall well below these for U Hya and X TrA. For R Lep our calculated CS peak fractional abundance is in good agreement with the upper limit they found, although we find the $e$-folding radius to be larger by about a factor of two. In addition to the radio data, [@Olofsson1993a] also collected infrared photometry for about 60 stars and, based on LTE model atmospheres, calculated photospheric molecular abundances for a few species, including CS and SiS. For the three overlapping stars they found relatively high photospheric SiS abundances, while we did not detect SiS in their CSEs (in fact, they were the only three carbon stars for which we did not detect SiS). For CS, the photospheric abundance found for R Lep is more than two orders of magnitude smaller than our CSE abundance, for U Hya it is an order of magnitude larger than our CSE abundance, and for X TrA it is comparable with our CSE abundance, being only a factor of two larger. As [@Olofsson1993a] note, the photospheric CS abundance is very sensitive to temperature, and hence is likely to change significantly between different phases of pulsation.
In a search for both CS and SiS (among other molecules) in a sample of 31 oxygen-rich stars, [@Lindqvist1988] detected the CS ($2\to1$) line towards only four sources and the SiS ($5\to4$) line only towards TX Cam. They estimated CS abundances in the order of a few $10^{-7}$, with their IK Tau result a factor of about two higher than what we found for that source, and an SiS abundance of $\sim 1\e{-6}$, in agreement with our results. [@Bujarrabal1994] surveyed a sample of evolved stars (mostly AGB stars) using the IRAM 30 m telescope to observe several molecular species including CS and SiS. They estimated molecular abundances based on the integrated intensities of their observed lines and assuming a constant abundance within a given radius. They note that their estimates only hold for optically thin lines. Where their sample overlapped with ours, we found higher SiS abundances for IK Tau and V1300 Aql by factors of four and two, respectively, while our CS abundances were in agreement for IK Tau and $\chi$ Cyg, a factor of a half smaller for V1300 Aql, V1259 Ori, and W Aql, a factor of about three smaller for R And, and almost a factor of 6 higher for V821 Her. In the case of SiS in the oxygen-rich stars (no carbon star SiS observations overlapped with our sample and they did not detect SiS towards any S-type stars) and CS towards the carbon stars and W Aql, the discrepancies are most likely due to optical depth effects since our models indicate optically thick emission in these cases. For CS towards the oxygen-rich stars and in $\chi$ Cyg, which our models indicate to be optically thin, our results are in better agreement, with the discrepancy in the V1300 Aql and R And abundances most likely due to uncertainties caused by the weak emission in the case of both our observations and those of [@Bujarrabal1994]. It should also be noted that [@Olofsson1993a], [@Lindqvist1988] and [@Bujarrabal1994] all used similar and simple methods for estimating abundances.
Comparison with chemical models
-------------------------------
Since CS and SiS have long been known to occur in AGB CSEs, they are regularly included in chemical models of these stellar winds. Indeed, SiS and CS are commonly assumed to be parent species — molecules formed in the innermost regions, from which other species are subsequently formed — sometimes for CSEs of all chemical types. In this section we discuss some different existing chemical modelling results and compare them with our results.
### LTE models of the mid- and outer-CSE
[@Willacy1997] modelled the chemistry in the cooler outer regions of the CSE of an oxygen-rich AGB star. They used the characteristics of TX Cam ($\dot{M} = 3\e{-6}\spy$, $\upsilon_\infty=18~\kms$) as the basis for their models, although they also give some model results based on IK Tau and R Dor. They take SiS and 2S as parent species and model the CSE from $2\e{15}$ cm outwards. They predict a more extended SiS envelope than we find, by about an order of magnitude, but with a peak SiS abundance in reasonable agreement with ours: about twice the abundance we find for IK Tau. [The SiS column density they find for IK Tau, using a similar mass-loss rate and slightly higher expansion velocity than those used in our study ($\dot{M}=4.5\e{-6}\spy$, $\upsilon_\infty = 20~\kms$), is more than an order of magnitude lower than ours.]{} They also predict CS to be located in a shell around the star, with the peak in abundance falling at a few $10^{16}$ cm for TX Cam. The peak appears to be roughly in agreement with the $e$-folding radii we find for the oxygen-rich stars in our sample. For IK Tau they find a peak abundance for CS of $\sim3\e{-7}$, which is about three times what we calculate for our centrally peaked CS distribution. [Their column density for IK Tau is about two orders of magnitude lower than what we find.]{} The CS lines that we observe are formed in relatively cool regions, so they are not sensitive to a possible lower CS abundance in the inner regions.
[@Li2016] model a similar outer region of an oxygen-rich CSE, focussing on UV photochemistry, and using IK Tau as their fiducial model (with a similar mass-loss rate, $\dot{M}=4.5\e{-6}\spy$, and higher expansion velocity, $\upsilon_\infty = 24~\kms$, than used in the present study). Their parent species abundances are taken from observations, where available, and shock-induced non-LTE predictions otherwise. They include several S-bearing molecules in their list of parent species: SiS, CS, 2S, SO, 2, and HS. Their initial abundance of SiS, in particular, accounts for a significant portion of the sulphur budget and is about an order of magnitude higher than the peak abundance we found for IK Tau. Like [@Willacy1997], they find a more extended SiS envelope, which declines more slowly than our IK Tau model, resulting in their model being about an order of magnitude more extended than ours. Considering CS, however, their results are in reasonable agreement with ours, with a similar (centrally peaked) inner abundance of CS and a similarly large extent, although at very large extents their model deviates from our Gaussian assumption.
### Chemistry in a clumpy medium
[@Agundez2010] investigated the effects of clumpiness on AGB chemistry and, in particular, the penetration of UV photons. They included SiS and CS as parent species for the carbon-rich models and SiS, SO and 2S in their oxygen-rich models. They concluded that UV penetration in clumpy and low mass-loss rate CSEs (up to a few $10^{-7}\spy$) can trigger photochemistry in the warm inner regions of the CSEs, allowing the formation of CS (and HCN and NH) in oxygen-rich CSEs (as well as 2O and NH in carbon-rich CSEs). Their predictions of CS abundances in the range $10^{-8}-10^{-7}$ for all mass-loss rates are in agreement with the abundances we find for the higher mass-loss rate oxygen-rich stars and are not ruled out by our non-detections for the lower mass-loss rate stars.
[@Van-de-Sande2018] developed a chemical model incorporating a porosity formalism to treat the increased penetration of UV photons in a clumpy CSE, and considering the relative overdensity of the clumps. They run models for different clump parameters, including the density contrast and clump size, and find that larger deviations from a smooth (non-clumpy) outflow generally result in higher abundances of CS for a range of mass-loss rates ($\dot{M}=10^{-7}$, $10^{-6}$, and $10^{-5}\spy$) in the case of oxygen-rich CSEs. Although their derived radial abundance distribution profiles differ from the Gaussian profiles we use in this study, their peak CS abundances are in good agreement with the peak abundances that we found for our higher mass-loss rate oxygen-rich stars. [Their CS column densities for oxygen-rich AGB stars, which vary significantly with clumpiness, are in agreement for the most extreme clumpy models, and up to three orders of magnitude lower than our results, depending on the specific clumpiness of the outflow.]{}
### Our results from a chemical perspective
Regarding possible chemical mechanisms to explain our results, the only neutral-neutral formation rate for the SiS molecule is the radiative association Si + S $\to$ SiS reported by [@Andreazza2007], which has an activation barrier of only 66 K. The Arrhenius rate[^7] has a small pre-exponential factor and very weak temperature dependence. As a consequence, the process occurs efficiently at high densities and is negligible at low densities, in agreement with our SiS detection pattern. A temperature independent SiS photodissociation rate is reported by [@Prasad1980], [but based on the difference between the [@Prasad1980] rates for SiO and SiH photodissociation and those calculated more recently [e.g. by @Heays2017], it is likely to be off by at least an order of magnitude]{}. However, apart from these two reaction rates, the neutral SiS chemistry is poorly characterised. A more detailed description is required to construct an accurate chemical-kinetic network (i.e. interactions of SiS with SiO, silicates, other S-bearing compounds, etc).
To model the SiS abundance without specific rate prescriptions, we performed thermodynamic equilibrium (TE) calculations [@White1958; @Tsuji1973], using the same method as implemented by @Gobrecht2016 but extended to a larger sample space of temperatures and densities. The TE calculations are based on the minimisation of the total gas Gibbs free energy whose components are tabulated (NIST-JANAF Thermochemical Tables[^8]) for a given elemental mixture. For the conditions of an oxygen-rich CSE (C/O $= 0.75$) we find a strong density dependence of the SiS fractional abundance (see Table \[davidtable\]). In particular, at $T=1500$ K and densities of , which typically apply for 1–2 $R_*$, we find good agreement with the observed abundances in higher mass-loss rate sources. A similar argument could be applied to the carbon-rich sources.
$10^{14}$ $10^{13}$ $10^{12}$ $10^{11}$ $10^{10}$
------ ------------- ------------- -------------- -------------- --------------
2000 $3.4\e{-8}$ $2.1\e{-9}$ $3.4\e{-10}$ $2.7\e{-10}$ $2.5\e{-10}$
1500 $4.6\e{-6}$ $1.0\e{-6}$ $2.2\e{-7}$ $2.3\e{-8}$ $1.7\e{-9}$
1000 $1.2\e{-5}$ $1.2\e{-5}$ $1.2\e{-5}$ $1.2\e{-5}$ $1.1\e{-5}$
Regarding the presence of CS in oxygen-rich stars, its formation is thought to be greatly enhanced by shock chemistry [@Duari1999; @Cherchneff2006; @Gobrecht2016]. The extreme conditions in shocks free up C from CO, allowing CS to form even in carbon-deficient environments. We expect more extreme shock conditions in the higher mass-loss rate sources [@Mattsson2007], which can drive non-equilibrium reactions. Hence CS is more likely to form in these sources. In carbon stars, where there is abundant C, CS can form in thermal equilibrium and hence such extreme conditions are not required to form CS. This also explains the presence of CS in even the lowest mass-loss rate carbon stars. [The roughly constant abundance of CS in most of the carbon stars is most likely due to CS having a high binding energy and forming readily in the presence of abundant C.]{}
[Although sulphur does not readily condense onto dust grains, as evidenced by the lack of sulphur depletion found by studies of post-AGB stars [@Reyniers2007; @Waelkens1991], it is possible that dust-grain interactions may play a part in determining the abundances of CS and SiS. [@Gobrecht2016] performed a theoretical study on a inner winds of an oxygen-rich AGB star using an extensive chemical-kinetic network including the species CS, SiS, SO, 2, 2S, SH, and OCS. By comparing models with and without dust condensation, we find no significant difference in the abundances of CS and SiS. ]{}
Conclusions
===========
In this study we observed SiS and CS towards a large number of AGB stars. CS was detected towards all observed carbon stars, some S-type stars, and the highest mass-loss rate M-type stars. SiS was only detected towards the highest mass-loss rate sources for all chemical types.
We find higher abundances of both CS and SiS in carbon stars than S-type stars or M-type stars. More specifically, we found SiS abundances ranging from $\sim 9\e{-6}$ to $\sim 2\e{-5}$ for the carbon stars, from $\sim 5\e{-7}$ to $\sim 2\e{-6}$ for the oxygen-rich stars, and from $\sim 2\e{-7}$ to $\sim 2\e{-6}$ for the S-type stars. Our CS abundances ranged from $\sim 4\e{-7}$ to $\sim2\e{-5}$ for the carbon stars, from $\sim 3\e{-8}$ to $\sim1\e{-7}$ for the oxygen-rich stars and from $\sim 1\e{-7}$ to $\sim8\e{-6}$ for the S-type stars.
A correlation between CS abundance and CSE density for S-type stars is indicated by our results, [and may also be seen for SiS if one star is excluded from our sample. However,]{} no similar correlation can be seen for the carbon- or oxygen-rich stars, [although this could be partly due to the small number of sources.]{} Thermodynamic equilibrium calculations predict that SiS should form more readily in denser environments, in agreement with our observational results of only detecting SiS in such environments. Also, CS formation is thought to be strongly enhanced by shock chemistry, which would explain why it forms more readily in the higher mass-loss rate oxygen rich sources — where shocks are expected to be more extreme — than in the low mass-loss rate oxygen-rich sources. This also explains the trend for higher CS abundances with higher densities seen in the S-type stars. Carbon stars, with their plentiful C, do not require shocks to free up C and drive CS formation and can instead form CS in thermal equilibrium.
Based on observations made with APEX under programme IDs O-097.F-9318, O-098.F-9305, 077.F-9310 and 079.F-9325.
This work is based on observations carried out under project number 010-13 with the IRAM 30m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).
The Onsala 20 m telescope is operated by the Swedish National Facility for Radio Astronomy, Onsala Space Observatory at Chalmers University of Technology.
LD acknowledges support from the ERC consolidator grant 646758 AEROSOL and the FWO Research Project grant G024112N. EDB acknowledges financial support from the Swedish National Space Board. HO acknowledges financial support from the Swedish Research Council.
Observed lines and model plots
==============================
Observed lines
--------------
The integrated intensities of the SiS observations from the APEX sulphur survey are listed in Table \[SiSobs\] and the corresponding rms noise levels in mK are given in Table \[SiSnondet\]. The integrated intensities and rms noise levels in mK for CS observations from the APEX sulphur survey are given in Table \[CSobs\]. The rms noise levels for the stars in the S star survey are listed in Table \[csiramobs\] for the IRAM CS observations of ($5\to4$) and ($3\to2$), Table \[sisiramobs\] for the IRAM SiS observations of ($5\to4$), ($6\to5$), ($12\to11$), and ($13\to12$), and Table \[sisapexobs\] for the APEX SiS ($19\to18$) observations. For the stars with detected lines, integrated intensities and peak temperatures are also included. Observations of the SiS ($4\to3$) line at 72.618 GHz, carried out by the 4mm receiver at Onsala Space Observatory (OSO), are listed in Table \[osoobs\]. Previously published supplementary observations are listed in Table \[supobs\].
----------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
SiS
Star $9\to8$ $12\to11$ $14\to13$ $16\to15$ $19\to18$ $10\to9$ $11\to10 $
\[K $\kms$\] \[K $\kms$\] \[K $\kms$\] \[K $\kms$\] \[K $\kms$\] \[K $\kms$\] \[K $\kms$\]
R Lep x x ... x ... ... x
V1259 Ori 1.61 1.82 ... 2.03 1.78 x x
AI Vol 2.54 2.95 4.61 5.00 ... x 0.61$^{T}$
X TrA ... x ... x ... ... x
II Lup ... 5.22 ... 7.78 ... ... x
V821 Her 1.66 2.39 ... 3.64 3.27 x x
U Hya ... ... ... x x ... x
RV Aqr x 0.69 ... 1.09 ... ... x
R Hor x ... x ... ... ... ...
IK Tau ... 3.41 4.33 ... 5.61 x x
GX Mon 0.22$^T$ 1.03 0.85 ... 1.02 x x
W Hya x ... x ... ... x x
RR Aql x x x ... x x x
V1943 Sgr x ... x ... ... x x
V1300 Aql 1.18 1.90 2.23 ... 2.33 x x
T Cet x x ... x ... ... x
TT Cen x ... x x x x ...
RT Sco x ... ... x ... ... x
W Aql ... 0.95 ... ... ... x ...
RZ Sgr x ... x x x x x
----------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
----------- ---------- ----------- ----------- ----------- ----------- ---------- ------------
SiS
Star $9\to8$ $12\to11$ $14\to13$ $16\to15$ $19\to18$ $10\to9$ $11\to10 $
R Lep 9.6 16 ... 19 ... ... 8.9
V1259 Ori 19$^{D}$ 14$^D$ ... 13$^D$ 18$^D$ 31 15
AI Vol 21$^{D}$ 16$^D$ 20$^D$ 14$^D$ ... 39 22$^T$
X TrA ... 12 ... 14 ... ... 14
II Lup ... 9.9$^D$ ... 13$^D$ ... ... 20
V821 Her 21$^{D}$ 17$^D$ ... 18$^D$ 19$^D$ 29 16
U Hya ... ... ... 12 13 ... 9.0
RV Aqr 19 11$^D$ ... 19$^D$ ... ... 14
R Hor 12 ... 16 ... ... 19 13
IK Tau ... 15$^D$ 18$^D$ ... 18$^D$ 26 9.0
GX Mon 18$^{T}$ 11$^D$ 20$^D$ ... 21$^D$ 43 21
W Hya 16 ... 18 ... ... 40 20
RR Aql 21 13 19 ... 20 21 11
V1943 Sgr 16 ... 13 ... ... 15 13
V1300 Aql 14$^{D}$ 14$^D$ 19$^D$ ... 20$^D$ 20 13
T Cet 15 13 ... 11 ... ... 11
TT Cen 14 ... 16 13 17 50 ...
RT Sco 25 ... ... 11 ... ... 23
W Aql ... 14$^D$ ... ... ... 15 ...
RZ Sgr 13 ... 19 18 20 23 14
----------- ---------- ----------- ----------- ----------- ----------- ---------- ------------
-------------------- -------------- -------------- -------------- -------------- --------- --------- --------- ---------
CS CS
Star $4\to3$ $6\to5$ $7\to6$ $6\to5$ $4\to3$ $6\to5$ $7\to6$ $6\to5$
\[K $\kms$\] \[K $\kms$\] \[K $\kms$\] \[K $\kms$\] \[mK\] \[mK\] \[mK\] \[mK\]
[*Carbon stars*]{}
R Lep 2.30 4.14 ... x 8.3$^D$ 18$^D$ ... 18
V1259 Ori 4.15 5.20 4.28 x 15$^D$ 11$^D$ 19$^D$ 11
AI Vol 8.42 12.84 ... x 20$^D$ 15$^D$ ... 15
X TrA 0.81 1.58 ... x 14$^D$ 14$^D$ ... 14
II Lup 21.03 29.65 ... 0.78 19$^D$ 12$^D$ ... 12$^T$
V821 Her 6.01 10.83 9.58 x 15$^D$ 18$^D$ 21$^D$ 18
U Hya 0.17 0.19 0.42 x 8.8$^T$ 12$^T$ 13$^D$ 12
RV Aqr 4.61 6.70 ... x 14$^D$ 18$^D$ ... 18
[*M-type stars*]{}
R Hor x ... ... ... 12 ... ... ...
IK Tau 1.46 ... 2.57 ... 8.8$^D$ ... 29$^D$ ...
GX Mon 0.94 ... 0.71$^T$ ... 20$^D$ ... 20$^T$ ...
W Hya x ... ... ... 19 ... ... ...
RR Aql x ... x ... 11 ... 20 ...
V1943 Sgr x ... ... ... 12 ... ... ...
V1300 Aql 0.31$^T$ ... 0.48$^T$ ... 13$^T$ ... 19$^T$ ...
[*S-type stars*]{}
T Cet x x ... x 11 10 ... 10
TT Cen ... x x x ... 13 16 13
RT Sco x x ... x 21 9.8 ... 9.8
RZ Sgr x x x x 13 17 22 17
-------------------- -------------- -------------- -------------- -------------- --------- --------- --------- ---------
------------ ------------ ----------------- ----------------- -------------------
Source Transition rms $I_\mathrm{mb}$ $T_\mathrm{peak}$
(CS) \[mK, $T_A^*$\] \[K $\kms$\] \[K, $T_A^*$\]
$\chi$ Cyg ($5\to4$) 36.836 [4.93]{} 0.23
($3\to2$) 32.389 [2.55]{} 0.15
AA Cyg ($5\to4$) 11.639 ... ...
($3\to2$) 9.690 ... ...
AD Cyg ($5\to4$) 10.588 ... ...
($3\to2$) 9.795 ... ...
RX Lac ($5\to4$) 16.155 ... ...
($3\to2$) 15.322 ... ...
WY Cas ($5\to4$) 14.398 ... ...
($3\to2$) 11.886 ... ...
R And ($5\to4$) 13.773 1.4 ...
($3\to2$) 11.079 0.74 ...
W And ($5\to4$) 17.882 ... ...
($3\to2$) 13.771 ... ...
TV Dra ($5\to4$) 16.410 ... ...
($3\to2$) 12.780 ... ...
IRC-10401 ($5\to4$) 31.302 ... ...
($3\to2$) 20.778 ... ...
ST Sgr ($5\to4$) 19.392 ... ...
($3\to2$) 13.862 ... ...
W Aql ($5\to4$) 31.641 9.0 0.20
($3\to2$) 23.270 6.9 0.20
EP Vul ($5\to4$) 9.545 0.25 ...
($3\to2$) 9.205 ... ...
CSS2 41 ($5\to4$) 23.547 ... ...
($3\to2$) 16.491 ... ...
RZ Peg ($5\to4$) 10.691 ... ...
($3\to2$) 9.572 ... ...
V365 Cas ($5\to4$) 19.618 ... ...
($3\to2$) 14.244 ... ...
S Cas ($5\to4$) 20.352 6.9 0.15
($3\to2$) 14.621 2.9 0.07
T Cam ($5\to4$) 13.347 ... ...
($3\to2$) 11.082 ... ...
S Lyr ($5\to4$) 15.05 1.1 0.045
($3\to2$) 14.063 ... ...
ST Her ($5\to4$) 16.807 ... ...
($3\to2$) 13.845 ... ...
R Cyg ($5\to4$) 23.594 ... ...
($3\to2$) 19.983 ... ...
DK Vul ($5\to4$) 19.357 ... ...
($3\to2$) 13.983 ... ...
V386 Cep ($5\to4$) 18.576 ... ...
($3\to2$) 14.628 ... ...
AA Cam ($5\to4$) 14.88 ... ...
($3\to2$) 14.28 ... ...
R Lyn ($5\to4$) 7.522 $^a$ ... ...
($3\to2$) 6.538 $^b$ ... ...
Y Lyn ($5\to4$) 8.362 $^a$ ... ...
($3\to2$) 6.571 $^b$ ... ...
------------ ------------ ----------------- ----------------- -------------------
: IRAM 30m observations of CS ($5\to4$) and ($3\to2$) towards S type stars.[]{data-label="csiramobs"}
----------- ------------- ----------------- ----------------- -------------------
Source Transition rms $I_\mathrm{mb}$ $T_\mathrm{peak}$
(SiS) \[mK, $T_A^*$\] \[K $\kms$\] \[K, $T_A^*$\]
AFGL 2425 ($19\to18$) 8.266 ... ...
DY Gem ($19\to18$) 5.878 ... ...
EP Vul ($19\to18$) 11.777 ... ...
GI Lup ($19\to18$) 7.755 ... ...
IRC-10401 ($19\to18$) 6.579 0.11 0.012
R Gem ($19\to18$) 7.280 ... ...
RT Sco ($19\to18$) 9.301 ... ...
RZ Sgr ($19\to18$) 8.283 0.10 0.010
S Lyr ($19\to18$) 9.771 ... ...
ST Sco ($19\to18$) 6.227 ... ...
TT Cen ($19\to18$) 7.868 ... ...
W Aql ($19\to18$) 23.260 1.8 0.08
----------- ------------- ----------------- ----------------- -------------------
: APEX observations of SiS ($19\to18$) towards S type stars.[]{data-label="sisapexobs"}
------------ ------------- ----------------- ----------------- -------------------
Source Transition rms $I_\mathrm{mb}$ $T_\mathrm{peak}$
(SiS) \[mK, $T_A^*$\] \[K $\kms$\] \[K, $T_A^*$\]
DY Gem ($5\to4$) 5.973 ... ...
($6\to5$) 7.179 ... ...
($12\to11$) 13.759 ... ...
($13\to12$) 16.150 ... ...
R And ($5\to4$) 5.080 ... ...
($6\to5$) 6.476 ... ...
($12\to11$) 9.333 ... ...
($13\to12$) 14.126 ... ...
R Cyg ($5\to4$) 4.772 ... ...
($6\to5$) 6.023 ... ...
($12\to11$) 8.341 ... ...
($13\to12$) 12.810 ... ...
S Cas ($5\to4$) 5.436 ... ...
($6\to5$) 6.812 ... ...
($12\to11$) 11.203 0.54 0.013
($13\to12$) 16.152 1.1 0.026
W Aql ($5\to4$) 5.999 0.55 0.017
($6\to5$) 7.953 0.69 0.021
($12\to11$) 12.022 2.9 0.085
($13\to12$) 18.523 3.8 0.11
WY Cas ($5\to4$) 5.304 ... ...
($6\to5$) 6.528 ... ...
($12\to11$) 8.277 ... ...
($13\to12$) 12.678 ... ...
$\chi$ Cyg ($5\to4$) 5.471 ... ...
($6\to5$) 6.618 ... ...
($12\to11$) 9.220 0.73 0.036
($13\to12$) 13.837 1.2 0.030
------------ ------------- ----------------- ----------------- -------------------
: IRAM observations of SiS ($5\to4$), ($6\to5$), ($12\to11$), and ($13\to12$) towards S type stars.[]{data-label="sisiramobs"}
----------- --------------------------------- ----------------
Star $\int T_A^* \mathrm{d}\upsilon$ rms at $1\kms$
\[K $\kms$\] \[mK\]
IK Tau x 11
TX Cam x 7.2
NV Aur x 13
BX Cam 0.13$^T$ 7.0
GX Mon x 17
V1111 Oph x 10
T Cep x 12
R Cas x 8.1
----------- --------------------------------- ----------------
: OSO observations of SiS ($4\to3$) at 72.618 GHz[]{data-label="osoobs"}
[cccccc]{} Star & Transition & Telescope & $I_\mathrm{mb}$ & $\theta$& Ref.\
& & & \[K $\kms$\] & \[\]\
\
V1259 Ori & SiS ($6\to5$) & IRAM & 1.40 & 21 & 1\
AI Vol & SiS ($10\to9$) & APEX & 2.7 & 34 & 2\
& SiS ($11\to10$) & APEX & 3.0 & 31 & 2\
& SiS ($19\to18$) & APEX & 7.7 & 18 & 3\
& SiS ($10\to9$) & APEX & 0.76 & 35 & 2\
II Lup &CS ($7\to6$)& APEX & 20.3 & 18 &3\
&SiS ($19\to18$)& APEX & 3.5 & 18 &3\
V821 Her & SiS ($6\to5$) & IRAM & 1.33 & 21 & 1\
RV Aqr & SiS ($6\to5$) & IRAM & 0.39 & 21& 1\
\
IK Tau & SiS ($5\to4$) & OSO & 0.32 & 42 & 5\
& SiS ($6\to5$) & IRAM & 0.52 & 21 & 1\
& SiS ($8\to 7$) & IRAM & 7.5 & 17 & 4\
GX Mon & SiS ($6\to5$) & IRAM & 1.04 & 21 & 1\
V1300 Aql & SiS ($6\to5$) & IRAM & 1.53 & 21 & 1\
Plots
-----
SiS models and observations for the carbon stars are plotted in [Figures \[SiSCplots-1\], \[SiSCplots-2\], and \[SiSCplots-3\] with the same plotted for CS in Figures \[CSCplots-1\], \[CSCplots-2\], \[CSCplots-3\], \[CSCplots-4\], \[CSCplots-5\], and \[CSCplots-6\]]{}. The SiS model results and observations for the M-type stars are plotted in [Figures \[SiSMplots-1\] and \[SiSMplots-2\], with the same for CS plotted in \[CSMplots-1\] and \[CSMplots-2\]]{}. The CS model and observations for $\chi$ Cyg, an S-type star, are plotted in Fig. \[CSSplots\].
![Observations (black histograms) and model results (blue lines) for SiS towards the carbon star V1259 Ori, plotted with respect to LSR velocity.[]{data-label="SiSCplots-1"}](v1259ori-sis1.pdf "fig:"){width="49.00000%"} ![Observations (black histograms) and model results (blue lines) for SiS towards the carbon star V1259 Ori, plotted with respect to LSR velocity.[]{data-label="SiSCplots-1"}](v1259ori-sis0.pdf "fig:"){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for SiS towards the carbon star V821 Her, plotted with respect to LSR velocity.[]{data-label="SiSCplots-2"}](v821her-sis1.pdf "fig:"){width="49.00000%"} ![Observations (black histograms) and model results (blue lines) for SiS towards the carbon star V821 Her, plotted with respect to LSR velocity.[]{data-label="SiSCplots-2"}](v821her-sis0.pdf "fig:"){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for SiS towards the carbon star RV Aqr, plotted with respect to LSR velocity.[]{data-label="SiSCplots-3"}](rvaqr-sis0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for SiS towards the oxygen-rich star IK Tau, plotted with respect to LSR velocity.[]{data-label="SiSMplots-1"}](iktau-sis1.pdf "fig:"){width="49.00000%"} ![Observations (black histograms) and model results (blue lines) for SiS towards the oxygen-rich star IK Tau, plotted with respect to LSR velocity.[]{data-label="SiSMplots-1"}](iktau-sis0.pdf "fig:"){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for SiS towards the oxygen-rich star, GX Mon plotted with respect to LSR velocity.[]{data-label="SiSMplots-2"}](gxmon-sis1.pdf "fig:"){width="49.00000%"} ![Observations (black histograms) and model results (blue lines) for SiS towards the oxygen-rich star, GX Mon plotted with respect to LSR velocity.[]{data-label="SiSMplots-2"}](gxmon-sis0.pdf "fig:"){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards the carbon star V1259 Ori, plotted with respect to LSR velocity.[]{data-label="CSCplots-1"}](v1259ori-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards the carbon star V821 Her, plotted with respect to LSR velocity.[]{data-label="CSCplots-2"}](v821her-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards the carbon star R Lep, plotted with respect to LSR velocity.[]{data-label="CSCplots-3"}](rlep-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards the carbon star RV Aqr, plotted with respect to LSR velocity.[]{data-label="CSCplots-4"}](rvaqr-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards the carbon star U Hya, plotted with respect to LSR velocity.[]{data-label="CSCplots-5"}](uhya-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards the carbon star X TrA, plotted with respect to LSR velocity.[]{data-label="CSCplots-6"}](xtra-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards the oxygen-rich star GX Mon, plotted with respect to LSR velocity.[]{data-label="CSMplots-1"}](gxmon-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards the oxygen-rich star V1300 Aql, plotted with respect to LSR velocity.[]{data-label="CSMplots-2"}](v1300aql-cs0.pdf){width="49.00000%"}
![Observations (black histograms) and model results (blue lines) for CS towards $\chi$ Cyg, an S-type star, plotted with respect to LSR velocity.[]{data-label="CSSplots"}](chicyg-cs0.pdf){width="49.00000%"}
Further discussion of modelling
===============================
We encountered some modelling issues that only pertained to a few stars. For IRC -10401, we recalculated some key circumstellar parameters, including the mass-loss rate, based on an updated estimation of the period. For three carbon stars (V1259 Ori, AI Vol, and II Lup) we encountered various issues when we were modelling the molecular emission. We discuss these in detail below.
IRC -10401 {#irc-10401}
----------
For IRC -10401, we recalculated the mass-loss rate based on newly available data. When [@Ramstedt2009] first calculated the mass-loss rate of this S-type star, the period was not known and they assumed $L_* = 4000~\lsol$ to determine the distance based on dust radiative transfer modelling of the spectral energy distribution. However, now the period has been determined to be 480 days [@Kazarovets2016] and, based on the period luminosity relation of [@Whitelock2008], this gives $L_* = 7400~\lsol$, which in turn gives an updated distance of 585 pc (larger than the previous value of 430 pc). Using these updated parameters, we remodelled the same CO observations used by [@Ramstedt2009], using the same procedure, to find an updated higher mass-loss rate of $2\e{-6}\spy$ (compared with the earlier result of $3.5\e{-7}$). This updated result is what we base our SiS and CS models on in this work.
V1259 Ori
---------
For V1259 Ori we have clear CS detections with high signal-to-noise ratios that are nevertheless equally well fit by the model listed in Table \[results\] and by models with significantly larger $e$-folding radii, including $R_e > R_{1/2}(\mathrm{CO}) = 2.6\e{17}$ cm, the half abundance radius of CO. We do not expect CS to have a larger extent than CO, which is self-shielding and has a lower photodissocation rate than CS [@Heays2017]. This discrepancy in modelling CS is partly caused by the high optical depth of the observed CS lines, which reach optical depths of 20, 40, and 50 in the inner CSE for the ($4\to3$), ($6\to5$), and ($7\to6$) lines, and remain optically thick throughout most of the emitting region. Our model predicts that the ($1\to0$) line at 48.991 GHz would be optically thin throughout most of the envelope (with an optically thick peak in the inner CSE of only three) and is likely to allow us to properly constrain the CS envelope size. Alternatively, interferometric data that resolves the CS extent would also allow us to very precisely constrain the envelope size.
AI Vol {#aivol}
------
For AI Vol we found that the combined abundances of CS and SiS result in a total S abundance higher than expected based on the assumption that the solar S abundance is representative of the local environment within $\sim1$ kpc. Summing CS and SiS abundances, we find a total S abundance of $4.2\e{-5}$, more than one and a half times the solar abundance. However, the lower limit of $2.6\e{-5}$, based on our uncertainties, is equal to the solar abundance given by [@Asplund2009]. Since the CS model is based on only two observed lines and has errors on the fractional abundance in excess of 50% of the absolute value, it’s likely that a larger dataset with higher-$J$ observations would improve our model results. Similarly, although we have seven lines for the SiS model, the highest energy line is the SiS ($19\to18$) with an upper level energy of 166 K, which is not expected to be a good probe of the inner regions of the CSE. Hence higher-$J$ SiS observations would also improve our SiS model.
Another possible cause for our result of an apparent sulphur overabundance is the uncertainty in the mass-loss rate, which is based on several input parameters, each of which have their own uncertainty, as discussed in more detail in [@Danilovich2015a]. For example, uncertainty in the distance would affect the calculation of the mass-loss rate (as seen in Sect. \[irc-10401\] for ) and hence the abundances we derive in this work. The distance to AI Vol was originally taken from [@Woods2003] who note a typical uncertainty of up to a factor of two for their distances.
II Lup {#iilup}
------
{width="\textwidth"}
We were unable to satisfactorily model the CS and SiS emission towards II Lup. It was not possible to find a smoothly accelerating model with a constant mass-loss rate which agreed with the available observations for either CS or SiS. Some irregularities can be tentatively seen in Fig. \[iilupobs\] — and become clearer when attempting to fit models. These have also been seen in other molecular lines, such as CO, as discussed in more detail in [@Smith2014].
When attempting to model the data acquired in our APEX survey and the two lines — CS ($7\to6$) and SiS ($19\to18$) — retrieved from the APEX archive (see Table \[supobs\]) it was not possible to find a model which agreed with all three available lines for CS nor for SiS. We cannot constrain the molecular envelope size from the data, as we were able to do with most of the other observed stars, and radii calculated from Eqs. \[resis\] and \[recs\] do not give a satisfactory fit to the data. In the case of both molecules, a model which fits the two lower-$J$ lines well over-predicts the higher-$J$ line by a factor of approximately two. In the case of SiS, the ($19\to18$) is also narrower than the model prediction by $\sim 6~\kms$, even when taking an accelerating wind into account. These characteristics are suggestive of more complex conditions in the CSE than those taken into account by our models. To properly untangle the contributions from various components in the wind of II Lup, we require more observations of low- and high-energy emission from II Lup, preferably including some spatially resolved observations.
Isotopologues
=============
As noted in Tables \[SiSobs\] and \[CSobs\], in addition to the main isotopologues, our observations also [covered a line from each of]{} SiS, SiS, and CS. Of these, we only [tentatively]{} detected one SiS line, towards AI Vol (for which another SiS line was also available, see Table \[supobs\]), and one CS line towards II Lup.
To model the SiS emission towards AI Vol, we used a molecular data file constructed equivalently to the main isotopologue, with the same quantum-numbered energy levels and radiative transitions included. The same collisional rates were used as for the main isotopologue (see Sect. \[moldat\]). We adopted the $R_e$ found for the main isotopologue as a fixed parameter, and varied the peak abundance, $f_0$, to find a model that fit the isotopologue lines. The data and best-fitting model are plotted in Fig. \[Si34Splots\]. We find a peak SiS abundance of $(2.1^{+1.9}_{-1.6})\e{-6}$. Our result gives an SiS/SiS of $11.4 \pm 10.3$, smaller than the solar system value for S/S of 22.1 [@Asplund2009].
For II Lup we could not reliably model CS due to the problems we had modelling CS (discussed further in Sect. \[iilup\]). The observed isotopologue line is plotted in Fig. \[iilupobs\].
![Observations (black histograms) and model results (blue lines) for SiS towards AI Vol, plotted with respect to LSR velocity.[]{data-label="Si34Splots"}](aivol-si34s0.pdf){width="49.00000%"}
[^1]: Postdoctoral Fellow of the Fund for Scientific Research (FWO), Flanders, Belgium
[^2]: This publication is based on data acquired with the Atacama Pathfinder Experiment (APEX). APEX is a collaboration between the Max-Planck-Institut für Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory.
[^3]: The Onsala 20 m telescope is operated by the Swedish National Facility for Radio Astronomy, Onsala Space Observatory at Chalmers University of Technology.
[^4]: <https://spec.jpl.nasa.gov>
[^5]: <http://www.astro.uni-koeln.de/cdms>
[^6]: Indeed, the only star for which they detected SiS and which had a lower mass-loss rate ($\dot{M} = 5\e{-7}\spy$) than our lower-limit for SiS detections is the oxygen-rich R Cas. However, a more recent study of R Cas using higher-$J$ CO lines from *Herschel*/HIFI to determine the mass-loss rate, conducted by [@Maercker2016], found $\dot{M} = 8\e{-7}\spy$, equal to the lowest mass-loss rate star with an SiS detection in our sample.
[^7]: An Arrhenius reaction rate $k(T)$ is generally defined as\
$k(T) = A ({T}/{300~\mathrm{K}})^B \times \exp(- E_a/T)$\
where A is the pre-exponential factor, $B$ the temperature dependence and $E_a$ the activation barrier in K.
[^8]: <http://kinetics.nist.gov/janaf/>
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Francesco Mauro, Christian Moni Bidin, André-Nicolas Chené, Doug Geisler, Javier Alonso-García, Jura Borissova, Giovanni Carraro'
title: |
The [VVV-SkZ\_pipeline]{}: an automatic PSF-fitting photometric pipeline\
for the VVV survey
---
Introduction {#s_intro}
============
In the last decade, large-area surveys have revolutionized our understanding of the Galaxy and the Universe. Surveys like the Sloan Digital Sky Survey [@SDSS] and the Two Micron All Sky Survey (2MASS) [@2MASS] can be defined as two of “the most ambitious and influential surveys in the history of astronomy”[^1], for their contribution to astronomical knowledge. The importance of surveys in astronomical studies is underlined by the recent development of several survey-dedicated telescopes, like the 4-meter Visible and Infrared Survey Telescope for Astronomy (VISTA), the 2.5m VLT Survey Telescope, and the future 8.4m Large Synoptic Survey Telescope.
The “VISTA Variables in the Vía Láctea” (VVV) Survey [@Minniti10; @Saito10] is one of the six ESO Public Surveys operating on VISTA. VVV scans the Galactic bulge ($-10\leq l\leq+10$, $-10\leq b\leq +5$) and an adjacent part of the southern disk ($-65\leq l\leq-10$, $-2\leq b\leq +2$) in five near-IR bands ($ZYJHKs{}$) with the VIRCAM camera [@Emerson10], an array of sixteen non-buttable 2048$\times$2048 pixel detectors, with a pixel scale of $\sim 0\farcs 34/pix$. The VVV data are organized in contiguous $1{^\circ}\times1.5{^\circ}$ areas called “tiles”. Because of the large gaps among the CCDs, six offset exposures, called “pawprints”[^2], are required to survey a tile area covering each point at least twice. Each pawprint, on the other hand, is the combination of two exposures jittered by about 60 pixels in each axis. Multiple additional $2\times 4$s $K_s$-band images are obtained each year to find and monitor the variable stars in the VVV area. Observations are separated in observation blocks (OBs), that consist of exposures in $JHKs{}$, or $ZY$, or $2\times 4$s $Ks{}$ exposures of a single tile. VVV data are automatically processed at the Cambridge Astronomical Survey Unit (CASU)[^3] with the VIRCAM pipeline [@Irwin04], and the catalogs of aperture photometry (hereafter CASU catalogs) are produced and made publicly available through the ESO archive[^4] [@Saito12].
The VVV survey presents many situations in which the results of aperture photometry are clearly unsatisfactory, such as the very crowded regions of the Galactic center or stellar clusters, and the small-scale spatial variations of the background in the direction of star forming regions in the Galactic disk. In these cases, PSF-fitting photometry is mandatory to obtain optimal results. Moreover, aperture photometry does not return reliable results for saturated stars (${K_{s }}< 12$ for $2\times 4$s exposures). This prohibits the study of a variety of important objects (e.g. most of the red giant branch and RR Lyrae stars of bright globular clusters, or OB variables in open clusters). Fortunately, the wings of the PSF of these stars, unaffected by saturation, can be used to derive reliable PSF-fitting photometry.
Performing PSF-fitting photometry on multiple offsets in five bands requires a large amount of CPU time and also valuable interaction time to select optimal parameters. In addition, the image distortions [@Saito12] complicate the generation of stacked images to obtain a master list of stars when different chips or more than two offsets of the same chip are involved. The image distortions are up to about 10–15% across the wide field of view, and require a native WCS distortion model for the pawprints that varies radially as a fifth order polynomial.
In this paper, we present the [VVV-SkZ\_pipeline]{} (VSp), an automated pipeline designed to produce optimal PSF-fitting photometry with VVV data using the [DAOPHOT]{} suite [@DAOPHOT]. VSp is an adaptation of a previous procedure, developed by one of us (FM) to deal with data from a generic telescope. Its aims were to minimize the problem of the high degree of repetitive interaction required when [DAOPHOT]{} is used on a large amount of data, to produce accurate, precise and uniform photometry, and to leave to the user only the selection of the parameters. To achieve this result, algorithms were developed to manage a number of important tasks, like the rejection of the worst PSF stars. The software described here has already been applied in several papers with excellent results [e.g., @MoniBidin2011; @Majaess11; @Majaess12; @MauroAS; @Mauro2012; @MauroPhD; @BaumeAS; @BorissovaAS; @CheneAS; @Chene2012; @Chene2012b], where the reader can check the quality of the software products in different astrophysical contexts. The original pipeline has been applied to the “Southern Open Cluster Survey” (SOCS) project [@Kinemuchi2010; @MauroMSc].
In Section \[s\_general\] we give a general description of the [VVV-SkZ\_pipeline]{} and its main characteristics. In Section \[s\_det\] we discuss the pipeline in more detail. In Section \[s\_comp\] we show the advantages of VSp as a source of VVV photometry, discussing the comparison between the catalog produced by VSp, the 2MASS catalog (that was used for calibration) and the catalogs of other procedures available to obtain photometry from the VVV survey data.
General description {#s_general}
===================
The VSp performs PSF-fitting photometry through standard [DAOPHOT]{}<span style="font-variant:small-caps;">iv</span> and [ALLFRAME]{} routines [@ALLFRAME manuals and private communication], called by a series of Perl[^5] [@perl] scripts, that use programs written in C code with double precision when mathematical accuracy is required. The pipeline is composed of six parts, each one operated by a different script, whose input depends only on the output of the previous script. This design permits one to run the parts of the pipeline in series or separately, or to re-run the procedure from any point. The software is designed to perform photometric calibration using a generic catalog with standard $JH{K_{s }}$ magnitudes, but also with $Z$ and $Y$, if available. The astrometry is based on the World Coordinate System (WCS) present in the pawprints, determined by CASU and based on 2MASS astrometry. The pipeline depends on some external programs for specialized tasks (i.e., gnuplot for the plots, IRAF to extract images and header information; see Appendix \[ap\_progr\]). The pipeline is designed to work on the uncompressed pawprints, pre-reduced by CASU. In addition to the images, the required input consists of a list of frames, the configuration file of the pipeline, and a catalog of calibration stars with their equatorial coordinates and at least their standard $JH{K_{s }}$ magnitudes. The list of frames must contain the input file name, the number of the chip whose image has to be extracted, and the output file name (see details in Section \[ss\_prep\]). The default values in the option file, which governs the main parameters of the pipeline, have been extensively tested and should work well in most cases. However, the values of the parameters are tunable by the user. The only mandatory input information is related to the standard star catalog, since all the other information (such as seeing, filter name, etc) is obtained automatically from the image headers. The pipeline is characterized by a comprehensive procedure for the PSF calculation, including the possibility to give an optional list of integer coordinates of rectangular regions, whose stars are to be excluded from the PSF calculation. The user could desire to avoid stars in areas not suitable for the PSF calculation (i.e., variable background, highly saturated stars, highly crowded regions, defects of the image). This list must be given for each image that needs it. Additionally, any PSF star lying in the 60 pixel border area, covered by only one of the two jitters, is automatically rejected.
An important added value of the pipeline is the determination of the [DAOPHOT]{} parameter . Fixing it requires particular care, because its accurate determination permits good photometry even for partially saturated stars. There is no unique value suitable for all the VVV images, because the saturation level varies with chip, filter, and exposure time. Therefore, the pipeline reads the value to be used from an internal table, that provides the as a function of chip, filter, and OB type[^6]. The values were determined from an accurate statistical analysis of 48 pawprints. We found a strong correlation ($r_{xy}=0.97-0.99$) between the saturation level of different filter-OB pairs of the same chip, which helps simplify the internal table. Taking into account their variations (across the chip or between different exposures) we fixed the at 80% of the saturation level. To avoid problems with exposures whose sky value is particularly high (like $H$ exposures in the disk), we decided to use the following equation $$\label{eq:hgd}
\mathsf{high\, good\, datum}=sky+0.80(SatLvl-sky)$$ where $sky$ is the sky value, and $SatLvl$ the saturation level. The success of this determination is demonstrated by the comparison with external catalogs, like 2MASS (see Section \[ss\_comp2MASS\]). Of course, the user can still provide the pipeline with their own values of , and .
Once the pipeline has achieved a first measurement of the photometry, it stacks all the frames to create the master list of sources. This is a huge advantage over getting a list for every pawprint, since it provides much deeper photometry before source detection. The benefits of stacking different “stripes”[^7] are negligible, because of the very limited overlap. Hence, the pipeline by default combines only the images of the same “stripe” collected with the same chip (note that other choices can be selected in the option file, e.g. the stacking of all the frames of the same “stripe”, or separating them by offset). The combination of frames collected with the same chip, or with the central chips of the camera array, is generally not a problem. However, distortion across the wide field of the VIRCAM camera completely spoils the results when using widely-separated chips, even when [DAOMASTER]{} is used with cubic transformations, and combining frames from different chips is often problematic.
The final [ALLFRAME]{} output is astrometrized before matching the single output catalogs, to avoid the problems introduced by the field distortion. Consequently, their matching is simplified. The pipeline transforms the frame position in pixels into relative positions ($\cos{\delta}\Delta \alpha;\Delta \delta$) in arcseconds. The origin is centered on the point with the furthest south and west values, respectively in right ascension and declination, in order to have positive coordinates for all the stars. When the photometry in all the passbands are merged in the final catalog, the position in the equatorial system is calculated for each source. The relative coordinate system is also included in the final catalog (see Appendix \[ap\_Ctlg\]), in addition to the positions in equatorial coordinates.
Another important added value of the pipeline is the spurious-detection cleaning procedure. It is commonly known that saturated stars cause a non-negligible quantity of false star detections, that can adversely affect the results. A cleaning process, composed of two independent procedures, is included in the code for this reason. The first procedure is based on the trend with magnitude of the maxima of the distribution of photometric error $\sigma_m$, while the second procedure is based on both the typical loci of the spurious detections in the magnitude-error diagram, and their characteristic clustering around the saturated stars (see Section \[ss\_metrcalib\] for more details).
The source of the standard stars data is not constrained, but the procedure was tuned on 2MASS data and tested only with them. In particular, the coordinates of the observed stars are derived in the 2MASS astrometric system, hence a successful cross-identification with another input catalog is not guaranteed in the presence of an astrometric offset with respect to 2MASS. Feeding the pipeline with the complete list of standard stars in the field is recommended, as the code automatically selects the best stars, based on the magnitude, the quality of the catalog photometry, and the presence of contaminating nearby stars. When using 2MASS PSC as the reference catalog, the optimal range lies between the highly-saturated star regime (${K_{s }}=8-10$) and the 2MASS deviations at the fainter end [${K_{s }}=12-14$, see @MoniBidin2011], both varying between fields. It is therefore highly recommended to initially adopt the complete catalog with no pre-selection in magnitudes, then check the results and adjust the magnitude intervals, and rerun the final script. To facilitate this process, an astro-photometric comparison with the input standard catalog is executed at the end, the results are stored in a file, and a comprehensive set of graphs is generated to check the differences in both colors and magnitudes. The difference in position with the 2MASS catalog generally have a sample standard deviation less than 0.2 arcsec.
The final output catalog contains the calibrated magnitudes only in the $JH{K_{s }}$ bands. The 2MASS catalog does not provide $Y$ and $Z$ magnitudes, and there is a general lack of standard stars in these bands suitable for calibration in the survey area. We chose to leave these magnitudes in the instrumental system, since good calibration equations are not publicly available, or they depend on the spectral type. The equatorial position resolution in the final catalog is $10^{-6}$ degrees, which is the same resolution provided by the VizieR system[^8] and in other catalogs.
While the calibrated catalog is the final goal of the photometric work, the pipeline output also includes other auxiliary files, primarily aimed to check the quality of the results. In addition to the aforementioned astro-photometric comparison with the standard catalog, the plots of the calibration equations, and the plots related to the cleaning procedure, the code also produces a stellar density map in the format of a fits image with included WCS, and some additional useful graphics, e.g. a color-magnitude diagram $(J-{K_{s }}; {K_{s }})$, a reddening-free color-magnitude diagram $(c_3;{K_{s }})$ [see @Catelan11], and a color-color diagram $(H-{K_{s }}; J-H)$.
Salient details {#s_det}
===============
In this Section we give more relevant details of the main parts of the [VVV-SkZ\_pipeline]{}, and briefly describe how they operate. The fundamental steps of the reduction are: the creation of the input files, an accurate determination of the PSF, the creation of a complete master list of stars, the PSF-fitting photometry with [ALLFRAME]{}, and, finally, the astrometrization, calibration, and matching of all the data. We will use the convention that the script and program names will be in typewriter (e.g., `VVV-GetImgInfoHdr.pl`), the file names in italics (e.g., *[VVV-SkZ\_pipeline]{}.opt*), and the options in sans serif (e.g, ). Figure 1 shows a flow chart of how the program works.
![Flow chart of the pipeline procedure. The files the user provides are shown in an elliptical box, the scripts are in a rectangular box, while the produced files are in a rounded box. Open arrowheads indicate that the file will be used globally by the following scripts.[]{data-label="fig:flowchart"}](f1)
Preparing the Input Files {#ss_prep}
-------------------------
The [VVV-SkZ\_pipeline]{} requires three input files in addition to the decompressed pawprints: *login.cl* (the IRAF configuration file), *VVV-input* (the input file), and *[VVV-SkZ\_pipeline]{}.opt* (the configuration file).
The *login.cl* can be simply copied from the IRAF home directory without any change; the pipeline will set by itself the correct type of terminal, restoring the original file at the end.
The input file *VVV-input* must contain three entries per line (see Appendix \[ap\_input\]):
1. full name of the uncompressed pawprint (e.g., *v20100407\_00619\_st.fits*);
2. basename (name without extension) of the output image (e.g., M22-001);
3. space-separated list of chips that have to be extracted (e.g., or ).
The pipeline configuration file *[VVV-SkZ\_pipeline]{}.opt* (see Appendix \[ap\_opt\]) can be used to set a large quantity ($\sim 50$) of options, but all the parameters unspecified in the file will assume their default value. Hence, its minimum content is related to the catalog of standard stars: to specify the path of the catalog; to indicate the ordered list of passbands; to specify the column number of the right ascension (the declination must be in the following column) and of the first magnitude listed for the standard stars (the other magnitudes must be in the following columns, each followed by its error). Giving a name for the overall photometry with the option is preferable, otherwise the default name will be used.
### Extraction of images and information
The first step of the procedure is the extraction of the selected frames from the pawprints and of the required information from the headers. This operation is done by a Perl script that reads its input from the file *VVV-input*. The script automatically renames the output images. The file *VVV-imgdata* stores all the information which will be used later in the processing or can be useful to evaluate the data quality, such as seeing, airmass, sky level, ellipticity, version of the CASU pipeline used and the OB status.
PSF calculation {#ss_psfcal}
---------------
The PSF is calculated in five steps, increasing in complexity during the first three iterations: in the first step a purely analytic constant PSF model is used, then a constant look-up table of empirical corrections is added; in the following steps the look-up table is allowed to vary with position in the frame, quadratically in the last two iterations. In each step, the pipeline removes each selected PSF-building star that [DAOPHOT]{} marks as problematic. If the PSF calculation should fail, the pipeline changes the analytic function to find the problematic stars, removes them, and then returns to the chosen function. At the end of the first step, if the number of PSF stars is less than a given value (by default 50), the pipeline re-executes, increasing the initial number. By default, the VSp uses 400 stars as the initial number of the PSF-building stars (at the end of the first step only the 250 brightest stars are kept), Moffat3.5 as the analytic function, and 8 pixels as the PSF radius [$\sim 2\farcs 7$, where the maximum FWHM seeing allowed for the collection of VVV data is $0\farcs 8$, see @Minniti10]).
After each PSF calculation, [ALLSTAR]{} is run to perform a PSF-fitting photometry of all the stars. The procedure updates the global source list and the PSF-star list with their improved positions after each run of [ALLSTAR]{}. At the beginning of the first two reiterations, the procedure looks for missing stars in the star-subtracted image produced by [ALLSTAR]{}. Aperture photometry is subsequently performed on them, after subtracting neighbor stars. Stars to which this procedure is not applied are processed in the standard way. [ALLSTAR]{} is rerun to further improve the coordinates and the number of measurable stars. This information is used in the following aperture photometry steps, whose results are thus progressively improved. To reduce the contamination by nearby stars, the PSF is improved using an image where the stars in the proximity of the PSF-building stars are removed using SUBSTAR. Between the third and fourth step, the user is given the option to visually check the subtraction of the PSF stars and create a list of undetected stars. The last iteration simply refines the PSF, without adding stars or changing the complexity of the model PSF. The script gives a well-detailed output including number of stars used, rejected stars, analytic function used, and $\chi^2$ of the PSF provided by [DAOPHOT]{}.
Creation of the master list of stars and final PSF-fitting photometry {#ss_masterlist}
---------------------------------------------------------------------
The master list of stars is generated on a stacked image, created with Montage2. After a first run of [ALLFRAME]{} to improve the coordinates of the stars, the script generates a stacked image from the frames and their star-subtracted versions. The images in all the available bands are combined, while the combination of different VVV offsets depends on the option activated by the user, following the scheme outlined in Section \[s\_general\]. Using the stacked images and the previously generated input file *VVV-data4DpAlsMnt*, the script generates a master list of stars. This script uses [DAOPHOT]{} and [ALLSTAR]{} in five steps with a procedure similar to the one used for the PSF calculation, where the first step is executed on the stacked star-subtracted images. The default value of the threshold detection of sources is 4 times the background rms level for the stacked star-subtracted images, and 3 for the normal stacked images. This master list is then used by [ALLFRAME]{} to produce the final instrumental PSF-fitting photometry.
Final astrometrization, calibration and matching {#ss_metrcalib}
------------------------------------------------
The last script initially astrometrizes the [ALLFRAME]{} output using the WCS of the images, and applies the spurious detection cleaning procedure. The cleaning process is composed of two independent procedures. The first procedure is based on the trend of the maxima of the distribution of photometric error for each magnitude $\sigma_m$. We empirically found that the distribution of these points can be fitted by a function of the form $$\label{eq:err_m}
\log{\sigma_m}=a[(.4(m-m_0))^4+b]^{0.25}+c \;.$$ The star selection is done rejecting every source with an error greater than *n* times (where *n* is left as an option, with $n=3$ as the default value) the resulting $\sigma_m$ obtained by equation \[eq:err\_m\] (see Figure \[fig:spdetcln\], lower plot). The second procedure is based on both the typical loci of the spurious detections in the magnitude-error diagram, and their characteristic clustering around the saturated stars. The star selection is performed by an algorithm tuned to minimize the rejection of real stars. The cleaning process is configurable and reversible, and the positions, magnitude-error and magnitude-$\chi^2$ graphics of both the retained and rejected stars are plotted in an output file ( an example of the results is shown in Figure \[fig:spdetcln\]). Analyzing the photometry of several fields, the distribution of the ${K_{s }}$ magnitudes of the spurious detections is concentrated mainly around ${K_{s }}\approx 15-16$ with a dispersion of 1 magnitude. A similar peak is sometimes present at ${K_{s }}\approx 12-13$. Even in the other passbands, the distribution is concentrated similarly around magnitude $15-16$. The color distribution presents a peak, but varies with the field. In Figure \[fig:spdetcln\], the result of the spurious detection selection is shown. The false sources clearly tend to concentrate in magnitude. In the inserted plot, we show an example of how the procedure selects the majority of the spurious detections originated by saturated stars. Around the brightest saturated stars, the spurious detections are also arranged along rays radiating from the center, in a “cartwheel” distribution. As seen in Figure \[fig:spdetcln\], the sources lying in the 60 pixel border area have poorer photometry and this area, covered by only one of the two jitters, is a major source of spurious detections due to its lower signal-to-noise ratio. For this reason, any source in this 60-pixel frame is rejected, unless this option is disabled. The pipeline, in addition, rejects all the false detection located in the two empty $60\times 60$ pixel areas in the left-bottom and right-top corners of the images, generated by the jitter pattern. Similarly it rejects the false detections in the empty areas in chip 1, caused by masking of sensor defects[^9].
![Example of the results of the spurious detection cleaning procedure. We indicate in dark gray the retained sources, in black the rejected sources, in light gray the sources located within 60 pixels from the border. The main plot shows how the rejected sources are selected in the $(mag;\log{err})$ plane: the black solid line is the fit for the trend of the logarithm of the errors with the magnitude, the gray solid line is the sigma-clipping limit. Note how the empirical procedure manages to select more sources than sigma-clipping. The inserted plot shows an example of the positions of the sources in the relative coordinate system: the rejected ones mainly tend to clump around the position of saturated stars. The sources along the border present poorer photometry. []{data-label="fig:spdetcln"}](f5){width="\textwidth"}
After this cleaning process, the pipeline calculates the calibration parameters using the standard star catalog, calibrates the photometry, and matches all the data to produce a single output catalog for all the available bands and offsets. The calibration procedure considers only standard stars inside a magnitude interval given as input (by default or by the user). Some standard stars may have visual companions resolved in the VVV, but not in the standard catalog. If this companion is bright enough, the magnitude in the standard catalog is affected. The VSp rejects all standard stars with sources in the VVV catalog nearer than a tunable value, whose total light contamination exceeds a defined threshold. The default values are a distance of $2\farcs 2$ (tuned assuming 2MASS PSC as the standard star catalog, whose seeing[^10] varied between $2\farcs 5$ and $3\farcs 4$) and a contamination of 0.03 magnitudes. The least-squares fitting program assigns a weight-correction factor[^11] to the data to put less weight on the furthest points, instead of doing a sigma-clipping cut.
The calibration is performed twice. A first calibration is applied to the astrometrized [ALLFRAME]{} output for the classical correction for zero point and color term $$M_{STD}-m_{VSp}=a_1(J-{K_{s }})_{STD}+a_0\;.$$ Since the pipeline uses a local standard system, it does not correct for aperture or atmospheric extinction (both corrections are constants directly included in the $a_0$). The second calibration is applied to the output of [DAOMASTER]{} as a zero-point correction, after matching photometry of the same band and the same “stripe”. Eventually, the user is allowed to override the determination of $a_0$ and $a_1$ and of the zero-point corrections, imposing her/his preferred values.
The non-equivalence of the two VISTA and 2MASS photometric systems causes non-bijective linear transformation (i.e., stars with the same color could need different magnitude corrections). This degeneracy is difficult to disentangle because 2MASS PSC and VVV data overlap over a narrow interval of only 2-3 magnitudes. For this reason, CASU encourages one to calibrate onto the VISTA photometric system, using 2MASS magnitudes to define theoretical VISTA magnitudes, including correction for Galactic extinction. Of course, the user can directly choose the CASU catalogs of aperture photometry as input reference, thus calibrating directly into the VISTA system while retaining the advantage of the PSF-fitting procedure. Alternatively, the procedure followed by the VIRCAM pipeline run at CASU to produce the corresponding catalogs can be adopted to calibrate the VSp products onto the VISTA system, feeding the code with a 2MASS input catalog transformed as described in the CASU website. This flexibility of the VSp is important to guarantee a wide applicability of its products, because the use of the VISTA system, preferable from a theoretical point of view, is not desirable in all astrophysical cases. For example, many photometric indexes used to study the Galactic globular clusters [@Ferraro2006] are not defined in the VISTA system nor, to our knowledge, the transformation of the interstellar absorption among different photometric bands ($A_\lambda/A_V$).
M22: a test bed showing the advantages of the pipeline {#s_comp}
======================================================
In this section, we use the cluster M22 (NGC6656) as a test bed to compare the VSp products with alternative photometries and codes. This cluster was chosen for being a very crowded object in an uncrowded part of the survey. The test-bed is an area of $\approx 12\arcmin\times24\arcmin$ including the central part of the cluster M22 (see Figure \[fig:maps\]). The data comprise, for each of the three offsets covering the area, the three bands $JHK_\mathrm{s}$ taken in sequence. We first compare the VSp product with the 2MASS PSC (2M), which is a general reference for near-IR photometry. The aim is to show the quality of the anchoring of the pipeline photometry to the 2MASS system, and illustrate the brighter limit for good photometry of the VSp products. In addition, we compare the VSp catalog with two alternative photometric sources: the VVV CASU aperture-photometry catalog (the official catalog of the survey), both the tile and pawprint version (CasuT and CasuP, respectively); and a PSF-fitting photometry obtained with an updated version of [DoPhot]{} [DoP; @DoPhot1989; @Alonso2012]. The CASU catalogs are publicly available directly at the ESO archive, with no need to download the images. DoP was written “to be fast, highly automated, and flexible regarding the choice of PSF” [@DoPhot1993], therefore it is a valid solution to the problem of extracting PSF-fitting photometry from a large set of data.
The catalogs were matched with a quite elementary algorithm, which iteratively matches the sources in different catalogs based on both their coordinates and magnitudes simultaneously. The procedure worked satisfactorily since, while a few percent of unmatched stars is expected, $\approx 90\%$ of the sources were matched after the first iteration. The mean difference of astrometric positions of VSp sources with respect to 2MASS PSC is $-0\farcs 007$ in right ascension and $-0\farcs 001$ in declination, with sample standard deviation (sSD) of $0\farcs 187$ and $0\farcs 180$, respectively. The differences with the other catalogs are smaller than $0\farcs 05\simeq 0.15$ pixels ($0\farcs 001$ with the DoP catalog) with sSD $\lesssim 0\farcs 11\simeq 0.32$ pixels. The excellent agreement is not surprising, because all the procedures use the pawprint WCS for astrometrization, which was defined in the CASU pipeline in the 2MASS astrometric system. Additionally, the images come from the central chips, that observe the part of the field of view less affected by distortion, hence the merging of the different position measurements is less affected by systematic errors.
![Density map produced by the [VVV-SkZ\_pipeline]{} (upper figure) and a whole-field stacked image (lower figure) with the two subareas used to analyze the luminosity distribution in different condition of crowdedness: the “central area” (marked in white) and the “off-center area” (marked in black) are shown.[]{data-label="fig:maps"}](f2 "fig:"){width="\textwidth"} ![Density map produced by the [VVV-SkZ\_pipeline]{} (upper figure) and a whole-field stacked image (lower figure) with the two subareas used to analyze the luminosity distribution in different condition of crowdedness: the “central area” (marked in white) and the “off-center area” (marked in black) are shown.[]{data-label="fig:maps"}](f3 "fig:"){width="\textwidth"}
![Color-magnitude diagrams of the two subareas for the three catalogs based on pawprints: in the left panel the “central area” (c), and in the right panel the “off-center area” (o-c).[]{data-label="fig:cmdareas"}](f33){width="\textwidth"}
The luminosity distributions of the catalogs is one aspect that must be analyzed with care, because the different crowding in the field influences the source detection. For this reason, we studied the luminosity distribution in the whole field and in two subareas (see Figure \[fig:maps\]) of radius $2\farcm 5$. The first one was centered on M22, and characterized by strong crowding and saturated stars (hereafter the central area), while the second one, located $\sim 7'$ away, was an uncrowded area lacking saturated stars and, consequently, spurious detections (hereafter the off-center area). In Figure \[fig:cmdareas\], we also show the CMDs for the two selected areas for the three pawprint-based catalogs.
![Cumulative distribution of the ${K_{s }}$ magnitude for ${K_{s }}\geq 11$ for the entire field. The lower panel shows a zoom of the range ${K_{s }}=11-14$. In the in-plot figure the cumulative fraction distribution is showed.[]{data-label="fig:lumdistrC"}](f6 "fig:") ![Cumulative distribution of the ${K_{s }}$ magnitude for ${K_{s }}\geq 11$ for the entire field. The lower panel shows a zoom of the range ${K_{s }}=11-14$. In the in-plot figure the cumulative fraction distribution is showed.[]{data-label="fig:lumdistrC"}](f7 "fig:")
![Luminosity distribution in the ${K_{s }}$ band in the central and off-set areas (upper and lower panel, respectively).[]{data-label="fig:lumdistr25"}](f10 "fig:") ![Luminosity distribution in the ${K_{s }}$ band in the central and off-set areas (upper and lower panel, respectively).[]{data-label="fig:lumdistr25"}](f12 "fig:")
#### The luminosity distribution.
The global and fractional cumulative luminosity distributions of sources with magnitude ${K_{s }}\geq 11$ for the entire field are shown in Figure \[fig:lumdistrC\]. The VSp catalog goes deeper than the others, which are incomplete at ${K_{s }}\gtrsim 18$, even if the same threshold for source detection izxs set in the VSp and DoP procedures. The luminosity distributions of the off-center area (see Figure \[fig:lumdistr25\]) show how the four procedures detect the same quantity of stars in the range ${K_{s }}=11.75-16.25$, matching the luminosity distribution of the 2MASS catalog for ${K_{s }}=11.25-13.75$, when the crowding and the incidence of heavily saturated stars is low. The aperture photometry loses stars for ${K_{s }}\gtrsim 16.25$, and only the VSp luminosity distribution increases down to ${K_{s }}=18.75$. This trend is also clearly visible looking at the CMDs in the right panel of Figure \[fig:cmdareas\].
In the central area, instead, the limitations of aperture photometry with respect to PSF-fitting photometry emerge clearly (CasuP recovers only 40-65% of the sources with ${K_{s }}=12-15$ found by VSp and DoP, and CasuT only 10%), as seen in the CMDs in the left panel of Figure \[fig:cmdareas\]. According to the luminosity distributions, DoP apparently detects more stars with ${K_{s }}=14-16$. As discussed in Section \[s\_general\], we found that the spurious detections have their main peak in this magnitude range. The DoP procedure does not clean the spurious detections, and we suspect that this is the source of the increased quantity of apparent source detections. Further evidence is the “cartwheel” distribution of the DoP sources, not detected by VSp, and the presence of more stars with bluer and redder color in the DoP CMD in the left panel of Figure \[fig:cmdareas\].
In conclusion, the VSp catalog goes deeper. In addition, it then also returns more accurate photometric measurements, because the contamination from undetected faint sources is reduced. In addition, the VSp catalog is less affected by spurious detections, producing a cleaner CMD. The aperture photometry catalogs are limited with respect to source detection, especially in the most crowded areas.
![Frequency distribution of the color $(J-{K_{s }})$ for the four catalogs for stars in the range of ${K_{s }}=12.75-14.25$ (upper figure) and ${K_{s }}=15.75-16.75$ (lower figure). The FWHM of the distributions in color are indicated between parenthesis.[]{data-label="fig:distrJ-Ks"}](f14 "fig:") ![Frequency distribution of the color $(J-{K_{s }})$ for the four catalogs for stars in the range of ${K_{s }}=12.75-14.25$ (upper figure) and ${K_{s }}=15.75-16.75$ (lower figure). The FWHM of the distributions in color are indicated between parenthesis.[]{data-label="fig:distrJ-Ks"}](f16 "fig:")
#### The photometric errors.
The VSp catalog declares larger errors than the other catalogs for sources with ${K_{s }}<16$. However, this is not a good indication of the precision because some procedures might overestimate or underestimate the true photometric uncertainties. For this reason, we analyzed the $(J-{K_{s }})$ color distribution of stars in three magnitude ranges: ${K_{s }}=12.75-14.75$ (lower RGB stars, where the photometry is more accurate; upper panel of Figure \[fig:distrJ-Ks\]), ${K_{s }}=15.75-16.75$, and ${K_{s }}=16.75-17.25$ (MS stars, where the scatter due to photometric errors dominates, but the luminosity distributions are still comparable; lower panel of Figure \[fig:distrJ-Ks\]). The observed color distribution is determined by the convolution of the natural spread of the CMD with the photometric errors. The color spread of the VSp data has the lowest FWHM in all three magnitude ranges, hence they are the least affected by photometric errors. In the brightest range, the color width of the cluster RGB is $1.06-1.12$ times narrower in VSp data than in the other two pawprint-based catalogs, while the VSp catalog declares photometric errors $\sim 1.6$ times larger in color. The disagreement between a narrower dispersion and larger declared photometric errors is not present in the two faintest ranges.
The ratio $\eta=\sigma_{FWHM}/err_{max}$ between the FWHM-derived Gaussian standard deviation (StdDev) and the maximum of the $(J-{K_{s }})$ error distribution indicates the goodness of the photometric error estimate. For each color distribution in the three ranges, the FWHM was calculated and $\sigma_{FWHM}$ was obtained through the inversion of the equation $\mathrm{FWHM} = 2 \sqrt{2 \ln (2) } \; \sigma$. As stated above, this theoretical StdDev is the convolution of the photometric errors (which depend on the catalog) with the natural spread of the CMD (constant for all catalogs). Only the pawprint-based catalogs will be analyzed. In fact, the tile version of the CASU catalog missed most of the stars in the central part of M22 and it will be excluded from the following analysis. VSp data presents a nearly constant $\eta$, slightly increasing at fainter magnitudes (1.63;1.75;1.85 in the three magnitude ranges defined before), while it noticeably decreases for DoP and for CasuP data (2.71;2.08;1.64 and 2.96;1.63;1.27, respectively). Thus, $\eta$ is similar in the three catalogs in the faintest two ranges, while in the brighter range it is lower for the VSp data by a factor of 1.66-1.82. This is caused by the smaller color spread and the higher declared errors for ${K_{s }}<16$.
In conclusion, the VSp data present the smallest color spread at any magnitude, therefore the photometric errors are actually smaller. This improved precision is partially due to the higher detection rate of the faintest sources. Consequently, its larger declared errors could be an a overestimate, but the constancy with magnitude of the ratio $\eta$, compared to its rapid decrease in the other two catalogs, indicates that the errors are most probably underestimated in the latter.
#### The photometric differences among catalogs.
![Frequency distribution of the differences in ${K_{s }}$ (upper figure) and $(J-{K_{s }})$ (lower figure) among the catalogs for stars in the range of ${K_{s }}=13-14$ (the FWHM of each distribution is indicated in parenthesis).[]{data-label="fig:distrdiff13-14"}](f28 "fig:") ![Frequency distribution of the differences in ${K_{s }}$ (upper figure) and $(J-{K_{s }})$ (lower figure) among the catalogs for stars in the range of ${K_{s }}=13-14$ (the FWHM of each distribution is indicated in parenthesis).[]{data-label="fig:distrdiff13-14"}](f30 "fig:")
The frequency distributions of the photometric difference in ${K_{s }}$ and in $(J-{K_{s }})$ between the catalogs are plotted in Figures \[fig:distrdiff13-14\] for stars with magnitude ${K_{s }}=13-14$. This range is optimal for this comparison, because it is unaffected by saturation or large photometric errors. Contrary to the PSF catalogs, the CASU catalogs are calibrated on the VISTA photometric system. This difference in the photometric system affects mainly only the central position of the distributions of the photometric differences, but not the overall shape, since the considered interval in magnitude is small.
The FWHMs of the distributions are very similar both in magnitude and color. The ${K_{s }}$ differences between the VSp catalog and the other pawprint-based photometries show a plateau in the negative part down to $\sim-0.02\;mag$. A simple calculation reveals that this is explained by a contamination from sources with $\Delta m\simeq 4.3$ mag, i.e.${K_{s }}\simeq 17.3-18.3$, the magnitude range where the pipeline detects more sources than the other catalogs. This result underlines the importance of a higher detection rate of faint sources, because the contamination from nearby faint stars introduces errors in the estimated source luminosity. The distributions of the differences in $(J-{K_{s }})$ color are approximately symmetric, but have a narrower dispersion than what is predicted by the photometric errors alone. It is even narrower than the distribution of the differences in $J$ and ${K_{s }}$ magnitude. This behavior can be explained by a dispersion in ${K_{s }}$ due to photometric errors which is stochastic overall, but systematic for a single source, which is thus reduced when the color is calculated. Such errors can be generated by contamination from faint sources, variable from source to source, but systematic in all the passbands. The contamination from nearby undetected sources can worsen the photometric precision by hundredths of a magnitude, even in the brighter part of the photometric range.
![$(J-{K_{s }})$ color distribution for the four catalogs for stars in the range ${K_{s \,VSp}}=14.75-15.25$ (upper figure), ${K_{s \,VSp}}=15.25-15.75$ (central figure) and ${K_{s \,VSp}}=15.75-16.25$ (lower figure) []{data-label="fig:distrJ-Ks_pec"}](f32)
#### The detectability of features in the CMD.
Another aspect of the photometric accuracy is the feasibility of distinguishing real features of the CMD from the statistical noise. The $(J-{K_{s }})$ color distribution of the field sequences at $(J-{K_{s }})=0.75$ and 0.95 (see Figure \[fig:cmd\]) was analyzed at three different ranges of magnitude, ${K_{s }}=14.75-15.25$, ${K_{s }}=15.25-15.75$ and ${K_{s }}=15.5-16.25$ (see Figure \[fig:distrJ-Ks\_pec\]), using all four catalogs. We use the color distribution to evaluate the detectability since its analysis is quantifiable and objective, in contradistinction to an evaluation by eye of the CMD.
At ${K_{s }}\approx 15$ the field RGB is easily detectable in all the catalogs, but in the VSp data it stands more prominently above the minimum between it and cluster stars on its bluer side. At ${K_{s }}\approx 15.5$, it is mostly hidden by noise, but the VSp data present a narrower peak, disentangling it better from the spread of the [M22]{} stars; at ${K_{s }}\approx 16$ it is totally blended with the cluster main sequence. The secondary field sequence is generally undetectable in the CASU catalogs; at ${K_{s }}\approx 15$ it is sparsely populated (see Figure \[fig:cmd\]), and barely visible in the color distribution of VSp data. However, it is detectable at ${K_{s }}\approx 15.5$ as a peak $4.8\sigma$ higher than the background. The significance of the same peak is lower by about a factor of two ($2.1\sigma$) in DoP data. At ${K_{s }}\approx 16$ the peak is never significant, reaching only $1.7\sigma$ and $1.3\sigma$ above the background for the VSp and DoP catalogs, respectively.
Concurring with the analysis of photometric errors, the narrower spread in color of the VSp data help to spotlight less populated features present in the CMD, improving photometric analysis.
Comparison with 2MASS {#ss_comp2MASS}
---------------------
The 2MASS survey produced an all-sky homogeneous catalog, and it is therefore the reference data-set for the majority of IR photometric studies. In our case, the comparison with the [VVV-SkZ\_pipeline]{} catalog must take into account that 2MASS data were used as the standard source for the calibration.
![Upper figure: density map in logarithmic scale of the photometric differences in ${K_{s }}$ between 2MASS and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude (the dashed lines indicate the 1-sigma difference). Lower figure: density map in logarithmic scale of the photometric differences in $(J-{K_{s }})$ between 2MASS and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude.[]{data-label="fig:2mass"}](f20 "fig:") ![Upper figure: density map in logarithmic scale of the photometric differences in ${K_{s }}$ between 2MASS and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude (the dashed lines indicate the 1-sigma difference). Lower figure: density map in logarithmic scale of the photometric differences in $(J-{K_{s }})$ between 2MASS and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude.[]{data-label="fig:2mass"}](f21 "fig:")
As can be seen in Figure \[fig:2mass\], VSp catalog shows no significant offset from 2MASS PSC in the range ${K_{s }}\approx 9.5-14$. The partial saturation of stars brighter than ${K_{s }}\approx 12.0$ in the VVV survey does not cause any offset in the bright end of the VSp catalog. For these stars, digital saturation in the A/D converter occurs, but not physical saturation of the potential wells in the detectors, which would cause the migration of photo-electrons to nearby pixels. As a consequence, the [DAOPHOT]{} suite can fit the non-saturated wings of the PSF and recover the correct stellar magnitude, provided that an optimal value is chosen. This behavior at the bright end was found in all fields so far tested, and in some cases it extended to even brighter magnitudes. In the case of M22, the upper limits of the magnitude range where the VSp photometry is reliable are $J\approx 10.0$, $H\approx 10.0$, and $Ks\approx 9.5$. In $(J-Ks)$ no offset is observed in the full range $9.5<Ks<15$ (see Figure \[fig:2mass\]).
In the upper panel of Figure \[fig:2mass\], the trend of the error in the magnitude difference is also shown. This was calculated as the quadratic sum of the photometric errors of the two catalogs. The observed distribution of (${K_{s 2M}}-{K_{s VSp}}$) agrees well with the error at all magnitudes. This was, however, not found in all the fields tested. Depending on the stellar crowding, the photometry of some 2MASS stars can be contaminated by faint undetected sources, causing an overestimation in the resulting 2MASS luminosity. Under these circumstances, the distribution of the magnitude differences is asymmetric, with a predominance of negative values. Moreover, in crowded fields 2MASS also shows an underestimation of the luminosity of the fainter stars detected, as discussed by @MoniBidin2011, causing an asymmetry, with a predominance of positive values. As a consequence, the trend of (${K_{s 2M}}-{K_{s VSp}}$p) often turns upward in the faintest $1-1.5$ magnitudes of the 2MASS catalog, as also found by @Chene2012.
Comparison with CASU catalog {#ss_compCASU}
----------------------------
The CASU catalog is one of the official public products of the survey, and the astro-photometric catalog of a VVV field that can be obtained with the smallest effort. The CASU catalogs are offered in two versions: either a single-band catalog for each individual pawprint, or its equivalent for the stacked $1{^\circ}\times1\fdg 5$ tile. The catalogs are calibrated in the VISTA photometric system, as defined by 2MASS magnitudes and the theoretical transformation equations between the two systems (see Section \[ss\_metrcalib\]). We have compared our photometry with catalogs obtained from both pawprint (CasuP) and tile (CasuT) versions. They contain a similar quantity of stars, because the tile version detects more faint sources than the pawprint version, but it loses more stars in the more crowded part of the field (see upper plots in Figure \[fig:lumdistr25\]).
![Density map in logarithmic scale of the differences in ${K_{s }}$ magnitude between CASU (pawprint version in upper figure, tile version in lower figure) and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude (the dashed lines indicate the 1-sigma difference).[]{data-label="fig:casuPT"}](f24 "fig:") ![Density map in logarithmic scale of the differences in ${K_{s }}$ magnitude between CASU (pawprint version in upper figure, tile version in lower figure) and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude (the dashed lines indicate the 1-sigma difference).[]{data-label="fig:casuPT"}](f25 "fig:")
The photometric comparison is shown in Figure \[fig:casuPT\]. The CASU magnitudes clearly deviate in the bright-star regime, as expected for aperture photometry of saturated stars. This deviation starts at ${K_{s }}\approx 12$, $J\approx 12.7$ and $H\approx 12.3$ for both CasuP and CasuT catalogs. A similar behavior with respect to the 2MASS system for stars close to the saturation limit is also shown in Fig. 2 of @Gonzalez2011 and in the VMC survey paper by @Cioni2011. As expected, the catalogs have a small zero-point offset with respect to the VSp catalog, not being in the same photometric system. However, and quite unexpectedly, the two catalogs do not present the same exact offset, as can be seen in Figure \[fig:distrdiff13-14\]. The mean magnitude difference is $\Delta{K_{s }}=0.014mag$ for CasuP and $\Delta{K_{s }}=0.022mag$ for CasuT, similar to the value ${K_{s VVV}}-{K_{s 2M}}=0.028$ found by @Gonzalez2011 for tile b282. This means that the two CASU catalogs have a magnitude difference of $\Delta{K_{s }}=0.008$, explicable as a systematic error in their zero-point determination. The average difference in color is $\Delta(J-{K_{s }})=-0.058$mag for both catalogs, very similar to the offset found by @Gonzalez2011 between the 2MASS PSC and tile catalog for tile b282. Such magnitude differences are greater than what is predicted by the CASU transformation between 2MASS and VISTA systems.
Comparison with [DoPhot]{} {#ss_compDoP}
--------------------------
As a last evaluation of the photometric quality of the pipeline products, we compare the VSp catalog with an alternative PSF-fitting photometry code. Since the pipeline primarily aims to reduce the idle time and user interaction, the comparison catalog was based on an updated version of [DoPhot]{} [@DoPhot1989; @Alonso2012], a widely-used, fast, user-friendly and mostly automatic code. [DoPhot]{} needed 11 minutes of preparation, while the photometry was obtained in 24 minutes. The final step needed an additional 12 minutes. The whole photometric procedure took $\sim 47$ min on a MacBookPro with CPU 2.53GHz Intel Core2Duo. [DoPhot]{} was used with the threshold for source detection equal to the mean sky level plus three times the background signal-to-noise ratio.
![Upper figure: density map in logarithmic scale of the photometric differences in ${K_{s }}$ between DoP and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude (the dashed line indicates the 1-sigma difference). Lower figure: density map in logarithmic scale of the photometric differences in $(J-{K_{s }})$ between DoP and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude.[]{data-label="fig:DoP"}](f26 "fig:") ![Upper figure: density map in logarithmic scale of the photometric differences in ${K_{s }}$ between DoP and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude (the dashed line indicates the 1-sigma difference). Lower figure: density map in logarithmic scale of the photometric differences in $(J-{K_{s }})$ between DoP and VSp catalogs, as a function of VSp ${K_{s }}$ magnitude.[]{data-label="fig:DoP"}](f27 "fig:")
The match with the DoP catalog presented some problems, since sources with ${K_{s \,DoP}}=14.5-17$ were matched with sources brighter and fainter by $1-2$mag. A similar mismatch was present between DoP and CasuP catalogs. These sources are probably spurious detections, given the magnitude range (see former discussion) and the peculiarity in their position distribution, tending to cluster around the position of saturated stars, usually assuming a “cartwheel” distribution. We found that a $5\sigma$-clipping rejection in the matching procedure resolved the problem.
DoP was originally programmed to mask the saturated stars and their surroundings [@DoPhot1993]. This is a problem in many applications, both because many faint stars are lost in crowded regions, and because many important objects are found in the weakly saturated but still well photometered regime of VVV data, as discussed in Section \[s\_intro\]. As observed in Section \[ss\_comp2MASS\], the [VVV-SkZ\_pipeline]{} can correctly work up to about two-three magnitudes brighter than the saturation limit. We therefore removed the masking of bright stars in the DoP procedure [@Alonso2012]. This change, unfortunately, also causes the introduction of several spurious detections around saturated stars which are not removed in the procedure.
The photometric comparison (shown in Figure \[fig:DoP\]) shows that the DoP magnitudes deviate in the bright-star regime. Hence, even removing the masks, the magnitudes of the saturated stars are not recovered by DoP. The photometric offsets for fainter stars between the two PSF-fitting photometries are minimal: ${K_{s \,DoP}}-{K_{s \,VSp}}\approx -0.002$ and $(J-{K_{s }})_{DoP}-(J-{K_{s }})_{VSp}\approx 0.006$.
![Color-magnitude diagram of M22 overplotted with the photometrically reliable area allowed by the VSp catalog (within solid lines), and the photometric limits of other catalogs (within dotted line). VSp permits to use a magnitude range 50% larger.[]{data-label="fig:cmd"}](f4){width="\textwidth"}
[VVV-SkZ\_pipeline]{} performance {#ss_perf}
---------------------------------
Here, we discuss the typical processing time for an object. The code was run on an Intel(R) QuadCore(TM) i7 CPU 3.33GHz with normal (not high-performance) hard disks. The [DAOPHOT]{} suite was compiled in 32bit.
The PSFs of the nine images were calculated in $\sim 2.5$ hours, with a $\chi^2$ that varies from 0.013 to 0.024 (average value of 0.017), and using from 75 to 163 PSF-building stars for each image. The PSF-building stars have magnitudes $J\approx 12.8-13.4$, $H\approx 12.4-13.0$ and ${K_{s }}\approx 12.2-12.7$. The final cleaned VSp catalog of $\sim 242,000$ stars was obtained in about 10 hours. We reran the final script in order to select the optimal ranges of magnitude for the calibration. The final number of stars reduced per hour is $\approx 24,000$. Using two additional epochs in $K_\mathrm{s}$, the PSFs were calculated in $\sim 4$ hours, while the final cleaned catalog of $\sim 245,000$ stars was obtained in about 13 hours, with the number of stars reduced per hour of $\approx 19,000$. The average error of the color term of the calibration is $\approx 0.011$ (with a scatter of $0.005$).
Conclusions
===========
In this paper, we have presented the astro-photometric [VVV-SkZ\_pipeline]{} (VSp), a [DAOPHOT]{}+[ALLFRAME]{} procedure optimized to work on “VISTA Variables in the Vía Láctea” (VVV) ESO Public Survey data, a large-area survey covering the Galactic Bulge and southern disk. We demonstrated that, with very little effort, the user can obtain accurate results over a photometric range larger than provided by using other solutions such as the CASU catalogs or other automatic PSF-fitting photometry programs like [DoPhot]{} (DoP).
VSp is the only one of the four solutions analyzed in this paper that minimizes the effects of saturation; it provides a deep astro-photometric catalog, reliable more than 2 magnitudes brighter than the saturation limit (see Figures \[fig:2mass\] and \[fig:cmd\]). At variance with other solutions, VSp permits the study of interesting bright objects, and use techniques that rely on the brightest part of the CMD (like the determination of the metallicity with Calcium II Triplet equivalent width or the slope of the upper RGB) using VVV data. We showed that, when 2MASS PSC is used as a reference for calibration, VSp produces data well anchored to its photometric system, thanks to its larger photometric range. On the contrary, other solutions have to rely on stars with $J>12.7$ and ${K_{s }}>12$ (for bulge fields, fainter in the disk), where the 2MASS photometry presents deviations that can introduce non-negligible shifts. Systematic offsets are present also in the CASU catalogs, even if they are anchored to the VISTA system. In fact, we evidenced shifts in ${K_{s }}$ larger than those predicted by CASU transformations between 2MASS and VISTA system.
We showed that VSp yields more detections in crowded fields. The presence of spurious detections, which adds noise to the CMD and ruins the match with other photometries, is efficaciously handled, removing them. The VSp detects more faint sources, thus reducing their contamination in the photometry of the brighter sources. This contamination can worsen the photometric precision by some hundredths of a magnitude even in the brighter end of the VVV photometric range. These results allow one to produce a cleaner CMD, permitting a better analysis of the chosen target. Consequently, VSp data produce CMDs with narrower color dispersion, optimal to isolate and spot faint structures.
This pipeline permits the user to obtain a high-level PSF-fitting photometry with all the accuracy of the [DAOPHOT]{} suite, with the advantage of easy use and minimum idle time. In fact, the [VVV-SkZ\_pipeline]{} requires only the preparation of the data and possibly some additional iterations of the calibration procedure. In addition, VSp provides an adequate number of options to tune all the procedures according to the needs, and provides a set of output files to check in detail the result. The VSp pipeline will be distributed upon request, contacting the first author at
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully thank Peter B. Stetson for the [DAOPHOT]{} suite and his explanations. We thank Karen Kinemuchi for her help in the testing and checking of the reliability of the original pipeline and her suggestions for the paper. Dante Minniti, the PI of the VVV project, is warmly thanked for instigating this very important survey. We gratefully acknowledge support from the Chilean [*Centro de Astrofísica*]{} FONDAP No.15010003 and the Chilean Centro de Excelencia en Astrofísica y Tecnologías Afines (CATA) BASAL PFB-06/2007. CMB received support from The Milky Way Millennium Nucleus. ANC received support from Comite Mixto ESO-Gobierno de Chile and GEMINI-CONICYT No. 32110005. JAG was also supported by the Chilean Ministry for the Economy, Development, and Tourism’s Programa Iniciativa Científica Milenio through grant P07-021-F, awarded to The Milky Way Millennium Nucleus, by Proyecto Fondecyt Regular 1110326, by Proyecto Fondecyt Postdoctoral 3130552, and by Anillos ACT-86. JB is supported by FONDECYT No. 1120601 and by the Ministry for the Economy, Development, and Tourism’s Programa Inicativa Científica Milenio through grant P07-021-F, awarded to The Milky Way Millennium Nucleus. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
Alonso-García, J., Mateo, M., Sen, B., & al. 2012, , 143, 70 Baume, G., Feinstein, C., Borissova, J., et al. 2011, BAAA, 54, 191 Borissova, J., Clarke, J. R. A., Bonatto, C., et al. 2011, BAAA, 54, 191
Catelan, M., Minniti, D., Lucas, P. W., et al 2011, RR Lyrae Stars, Metal-Poor Stars, and the Galaxy, 145 Chené, A.-N., Clarke, J. R. A., Borissova, J., et al. 2011, BAAA, 54, 215 Chené, A.-N., Borissova, J., Clarke, J. R. A., et al. 2012, A&A, 545A, 54 Chené, A.-N., Borissova, J., Bonatto, C., et al. 2012b, ArXiv e-prints, arxiv:1211.2801 Cioni, M.-R. L., Clementini, G., Girardi, L., et al. 2011, A&A, 527, A116
Emerson, J., & Sutherland, W. 2010, The Messenger, 139, 2 Ferraro, F. R., Valenti, E., & Origlia, L. 2006, ApJ, 649, 243 Gonzalez, O. A., Rejkuba, M., Zoccali, M., Valenti, E., & Minniti, D. 2011, A&A, 534, A3 Hodgkin, S. T., Irwin, M. J., Hewett, P. C., & Warren, S. J. 2009, MNRAS, 394, 675 Irwin, M. J., Lewis, J., Hodgkin, S., et al. 2004, SPIE, 5493, 411 Kinemuchi, K., Sarajedini, A., Geisler, D., et al. 2010, BAAS, 215, 308.02
Majaess, D., Turner, D., Moni Bidin, C., et al. 2011, ApJ, 741, L27 Majaess, D., Turner, D., Moni Bidin, C., et al. 2012, A&A, 537, L4 Mateo, M.; Schechter, P. L. 1989, 1. ESO/ST-ECF Data Analysis Workshop, 31, 69 Mauro F., 2007, Master Thesis, Padova University Mauro, F., Moni Bidin, C., Geisler, D., 2011, BAAA, 54, 143 Mauro, F., Moni Bidin, C., Cohen, R., et al. 2012, ApJ, 761, L29 Mauro F., 2012, PhD Thesis, Universidad de Concepción Minniti, D., Lucas, P.W., Emerson, J.P., et al. 2010, New Astronomy, 15, 433 Moni Bidin, C., Mauro, F., Geisler, D., et al. 2011, A&A, 535, A33
Saito, R., Hempel, M., Alonso-García, J., et al. 2010, The Messenger, 141, 24 Saito, R., Hempel, M., Minniti, D., et al. 2012, A&A, 537, A107 Schechter, P. L., Mateo, M., & Saha, A 1993, PASP, 105, 1342 Skrutskie,M.F., Cutri,R.M., Stiening, R., et al. 2006, AJ, 131, 1163 Stetson, P. B. 1987, PASP, 199, 191 Stetson, P. B. 1994, PASP, 106, 250
Tody D., 1986, in Crawford D. L., ed., Proc. SPIE Vol. 627, Instrumentation in Astronomy VI. SPIE, Bellingham, p. 733 Wall, L. 1987, comp.sources.misc, v13i009:*Perl, a “replacement” for awk and sed* York, D. G., Adelman, J., Anderson, J. E., Jr., et al. 2000, AJ, 120, 1579 Zoccali, M., Renzini, A., Ortolani, S., et al. 2003, A&A, 399, 931
Mandatory External Programs {#ap_progr}
===========================
The pipeline calls some external programs, all freely distributed with no charge, that must be previously installed on the work machine:\
`DAOPHOT suite`. The pipeline uses the stand-alone version to obtain the photometry; the user has to ask it directly to P. B. Stetson and compile it.\
`Perl`. The pipeline is composed mainly by scripts written in Perl that operate as a front-end for the [DAOPHOT]{} suite. The interpreter is usually already installed in every Unix/Linux distribution (type *perl -v* on a command line to find out which version)[^12].\
`WCStools`. The pipeline uses part of the programs in this package (like sky2xy, xy2sky, cphdr) to manage the WCS transformations. It is downloadable from the Smithsonian Astrophysical Observatory site[^13].\
`NOAO IRAF`. The pipeline needs <span style="font-variant:small-caps;">iraf</span> [@iraf], v2.13 or following, mainly to extract the images from the pawprints and the header informations, and to create the density map in fits format; moreover the check of the PSF is operated with an interactive iraf task. It is downloadable from NOAO IRAF site[^14].\
`gnuplot`. The pipeline uses this program to generate the plots needed to check the outputs. It is usually included in software available for every Linux distribution, but also downloadable from its site[^15].\
`ds9`. This program is only needed if the user want to use the interactive iraf task to check the PSF calculation. It is downloadable from the Smithsonian Astrophysical Observatory site[^16].\
`system programs`. The pipeline uses several system programs (like date, pwd, tar, bzip2, wc, tail, cut, sort,...). They are usually already installed in every Unix/Linux distribution.\
\
`NASA HEASARC imcopy` is not used by the pipeline, but the user needs it to decompress the VVV data available at the Vista Science Archive[^17] (VSA). It is downloadable from the NASA[^18] or WFCAM Science Archive[^19] site.
*VVV-input* {#ap_input}
===========
Example of *VVV-input* input file. In this case the extraction of only one chip per pawprint was necessary.
v20100407_00619_st.fits M22-01 10
v20100407_00621_st.fits M22-02 11
v20100407_00623_st.fits M22-03 11
v20100407_00631_st.fits M22-04 10
v20100407_00633_st.fits M22-05 11
v20100407_00635_st.fits M22-06 11
v20100407_00643_st.fits M22-07 10
v20100407_00645_st.fits M22-08 11
v20100407_00647_st.fits M22-09 11
v20100825_00508_st.fits M22-10 10
v20100825_00510_st.fits M22-11 11
v20100825_00512_st.fits M22-12 11
v20100826_00420_st.fits M22-13 10
v20100826_00422_st.fits M22-14 11
v20100826_00424_st.fits M22-15 11
*[VVV-SkZ\_pipeline]{}.opt* {#ap_opt}
===========================
Example of *[VVV-SkZ\_pipeline]{}.opt* option file. Only the mandatory options are listed.
workname=M22
stdcat=../2MASSforVVV.dat
stdcat=../2MASSforb242.dat
filterstdcat=J,H,Ks
stdposcoomag=1,3
magcut=11.5:13.5,11.0:13,10.5:12.5
Format of the final catalog {#ap_Ctlg}
===========================
The catalogs have a three-line header. The first line gives the coordinates of the four corner of the field in the internal coordinate system. The number of the stars and of the passbands in the catalog are indicated in the second line. In the third line, the equatorial coordinates of the zero point of the internal coordinate system and the mean angle between the x axis of the images and the right ascension axis, both in radiant and degree units, are given.
The data for each star follow the header. They consist of:
Star ID
: in the form: number of tile (1-396) - number of stripe in the tile (1-8) - unique source ID in the stripe of 7 ciphers.
Equatorial coordinates
: of the star.
Internal coordinates
: in arcseconds from the zero point given in the header.
Magnitudes and associated errors
: for each passband in the order $JH{K_{s }}YZ$
Example of a final catalog.
# 1.449 598.415 1200.803 1.551 1498.941 603.678 306.326 1201.217
#nstars= 178092 nmag= 3
#ra0(J2000)= 278.8988800 dec0(J2000)= -24.0254100 angle -0.4601670 -26.3656280
24250000001 278.8993207 -23.8591836 1.449 598.415 18.6468 0.0758 17.8891 0.1218 ...
24250000002 278.9003358 -23.8595092 4.787 597.243 17.4458 0.0321 16.9051 0.0488 ...
24250000003 278.9005007 -23.8583633 5.329 601.368 17.3438 0.0219 17.0691 0.0561 ...
24250000004 278.9014003 -23.8555253 8.287 611.585 17.5538 0.0368 16.8541 0.0405 ...
24250000005 278.9015098 -23.8558219 8.647 610.517 17.4228 0.0372 16.6371 0.0394 ...
24250000006 278.9016773 -23.8583997 9.198 601.237 18.8868 0.1269 18.0951 0.1460 ...
24250000007 278.9019067 -23.8558033 9.952 610.584 19.0598 0.1164 18.6451 0.2118 ...
[...]
[^1]: <http://www.sdss.org/>
[^2]: <http://www.vista.ac.uk/>
[^3]: http://casu.ast.cam.ac.uk/
[^4]: <http://www.eso.org/sci/archive.html> <http://archive.eso.org/wdb/wdb/adp/phase3_main/form>
[^5]: www.perl.org
[^6]: The OBs are divided depending on the filters used (suffix j for the observations in $JHKs{}$, z for the observations in $ZY$ and v-$n$ for the additional epochs in $K_\mathrm{s}$), and whether they are bulge or disk tiles (prefix b and d). The $JHKs{}$ bulge OBs have a different exposure time than disk OBs and, even when bulge and disk OBs have the same exposure time, we find that it is better to consider them separately.
[^7]: A tile can be seen as the union of eight horizontal $\sim 2105$ pixel-wide “stripes” (combination of twelve frames), with only $\sim 150$ pixels of overlap, four fifths of which are covered by only one jitter for each stripe.
[^8]: <http://vizier.u-strasbg.fr/>
[^9]: <http://casu.ast.cam.ac.uk/surveys-projects/vista/technical/known-issues>
[^10]: <http://www.ipac.caltech.edu/2mass/releases/allsky/doc/sec4_4c.html>
[^11]: the algorithm is base on a series of five lectures presented at “V Escola Avancada de Astrofisica” by Peter B. Stetson <http://ned.ipac.caltech.edu/level5/Stetson/Stetson_contents.html> <http://www.cadc.hia.nrc.gc.ca/community/STETSON/homogeneous/Techniques/>
[^12]: <http://www.perl.org/get.html>
[^13]: <http://tdc-www.cfa.harvard.edu/wcstools/>
[^14]: <http://www.iraf.net>
[^15]: <http://www.gnuplot.info/>
[^16]: <http://hea-www.harvard.edu/RD/ds9/>
[^17]: <http://horus.roe.ac.uk/vsa/>
[^18]: <http://heasarc.gsfc.nasa.gov/docs/software/fitsio/cexamples.html#imcopy>
[^19]: <http://surveys.roe.ac.uk/wsa/qa.html#compress>
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Á. Skúladóttir'
- 'E. Tolstoy'
- 'S. Salvadori'
- 'V. Hill'
- 'M. Pettini'
- 'M. D. Shetrone'
- 'E. Starkenburg [^1]'
title: 'The first carbon-enhanced metal-poor star found in the Sculptor dwarf spheroidal[^2]'
---
R.A. Dec. $v_r$ (km s$^{-1}$) Obs. date Exp.time (min) Airmass (start) Airmass (end) Seeing
--------------------------------------- --------------------- --------------------- ------------ ---------------- ----------------- --------------- -------- --
$00^\text{h} 59^\text{m} 12^\text{s}$ $-33^\circ 46'21''$ 108.97 $\pm$ 0.32 2013-08-17 47.65 1.014 1.042 0.71
2013-09-01 47.65 1.129 1.261 1.04
2013-09-04 47.65 1.179 1.080 0.79
2013-09-04 47.65 1.068 1.023 0.85
Introduction
============
The chemical compositions of stellar photospheres provide detailed information about the interstellar medium (ISM) from which the stars were formed. Of particular interest are the very metal-poor (VMP) stars, $\text{[Fe/H]} \leq-2$ (where $\text{[Fe/H]}=\log_{10}(N_{\text{Fe}}/N_\text{H})_\star - \log_{10}(N_{\text{Fe}}/N_\text{H})_\odot$), in the Milky Way environment, which may still preserve imprints of the first generation of stars and the early chemical evolution of the Galaxy and its surroundings. Recent surveys have revealed that a significant fraction of these low-metallicity stars in the Milky Way halo are enhanced in carbon (\[C/Fe\] $\geq$ 0.7). The cumulative fraction of these carbon-enhanced metal-poor (CEMP) stars in the halo rises from $\sim$20% for $\text{[Fe/H]}\leq-2.5$ to $\sim$30% for $\text{[Fe/H]}\leq-3.0$ and up to 75% for $\text{[Fe/H]}\leq-4.0$ [@Lee2013]. Including the recently discovered carbon-enhanced, hyper iron-poor star with $\text{[Fe/H]}<-7.1$ [@Keller2014], the CEMP fraction at the lowest \[Fe/H\] becomes even higher.
Traditionally, CEMP stars are categorized by their heavy element abundance patterns. Some are enriched in heavy neutron-capture elements built by the slow-process (such as Ba) and are labeled CEMP-$s$ stars, and those that show significant abundances of heavy elements from the rapid-process (such as Eu) are referred to as CEMP-$r$ stars. Those that show enhancements of elements from both processes are labeled CEMP-$s/r$ stars. Finally, CEMP-no stars show no enhancements of the main $r$- or $s$-process elements.
The main $s$-process happens in low mass (M $\lesssim$ 4 M$_\odot$) asymptotic giant branch (AGB) stars, while the $r$-process requires a high energy, neutron-rich environment, so sites such as supernovae (e.g., @Travaglio2004) and neutron star mergers [@Tsujimoto2014] have been proposed. The lighter neutron-capture elements (such as Sr, Y and Zr) are created in the main $r$-process, but are overabundant at lower \[Fe/H\] compared to the heavier $r$-process elements, so an extra source, the weak $r$-process or the weak $s$-process, is needed to explain the observed abundances [@Travaglio2004; @Francois2007; @Frischknecht2012; @Cescutti2014].
In general, CEMP-$s$ and CEMP-$s/r$ stars have abundance patterns that suggest mass transfer from a companion in the AGB phase [@Lucatello2005]. Thus the C, N, O, and heavy elements for these stars do not reflect the ISM from which they were formed. The available data for CEMP-$s$ stars is consistent with $\sim$100% binary fraction and a maximum period of $\sim$20,000 days [@Starkenburg2014].
CEMP-no stars, however, are not especially associated with binaries. Though some of them do belong to binary systems, for many of them close binaries that favor mass transfer can be excluded (e.g., @Starkenburg2014). These stars are more frequent, and their carbon enhancement becomes more extreme at lower \[Fe/H\] (e.g., @Lee2013 [@Norris2013]), so it is difficult to explain the abundance pattern of such stars with mass transfer from an AGB-companion. Though other scenarios have been discussed, CEMP-no stars are generally believed to have formed out of carbon-enhanced gas clouds, enriched by low- and/or zero-metallicity stars. Thus, CEMP-no stars could provide direct information on the properties of the first generation of stars.
Among the proposed sources of C-enrichment in CEMP-no stars (see @Norris2013 for a detailed overview), two are of particular interest: (i) massive rapidly rotating zero-metallicity stars that produce large amounts of C, N, and O due to distinctive internal burning and mixing episodes [@Meynet2006]; (ii) faint SNe, associated with the first generations of stars, which experience mixing and fallback, ejecting large amounts of C but small amounts of Fe and the other heavier elements [@UmedaNomoto2003; @Iwamoto2005; @Tominaga2007].
Most of the observed CEMP stars have been found in the Galactic halo and a few have been found in the ultra faint (UF) galaxies around the Milky Way. Two stars with very high carbon values ($\text{[C/Fe]}>2$) have been found in the UF galaxies Bootes [@Lai2011] and Segue I [@Norris2013]. No star with $\text{[C/Fe]}>2$ has been found in the more luminous (L$_\text{tot}>10^5$ L$_\odot$), more distant dwarf spheroidal (dSph) galaxies, but some CEMP stars with lower carbon abundances have been observed, such as a star in Sextans dSph with $\text{[C/Fe]}\sim 1$ at $\text{[Fe/H]}\sim-3$ [@Honda2011]. In particular, no VMP star has been found in the Sculptor dSph with $\text{[C/Fe]}>0.1$ until now, despite extensive searches for low-metallicity stars (e.g., @Tafelmeyer2010 [@Starkenburg2013]).
Sculptor is a well-studied system with a magnitude of $M_V\approx-11.2$ and a distance of $86\pm 5$ kpc [@Pietrzynski2008]. It is at high Galactic latitude ($b=-83^\circ$) and has systemic velocity of $v_{\textsl{hel}}~=~+110.6~\pm~0.5$ km/s. The contamination by foreground Galactic stars is not significant, and most of it can be easily distinguished by velocity (e.g., @Battaglia2008b). The star formation history shows a peak in star formation $\sim$13 Gyr ago, with a slow decrease, so the galaxy is dominated by an old stellar population (>10 Gyr old), and has not formed any stars for the last $\sim$6 Gyr [@deBoer2012].
Large spectroscopic surveys of individual stars have been carried out in the central field of Sculptor. Abundances have been measured for Fe, Mg, Ca, Si, and Ti with intermediate-resolution (IR) spectroscopy [@Kirby2009] and high-resolution (HR) spectroscopy for $\sim$100 stars (Dwarf Abundances & Radial-velocities Team (DART) survey, @Tolstoy2009; Hill et al. in prep.). Because of the distance to Sculptor, in general only the brightest stars of the galaxy are available for HR spectroscopy. The HR sample is therefore mostly limited to the upper part of the red giant branch (RGB) ($0\lesssim$ $\log g$ $\lesssim2$). None of the large surveys of Sculptor have included measurements of carbon abundances, but several follow-up spectra of low-metallicity stars have been taken, many of them including C measurements [@Tafelmeyer2010; @Frebel2010; @Kirby2012; @Starkenburg2013].
In addition, there have been surveys of Carbon-stars and CH-stars in Sculptor (e.g., @Azzopardi1986), some of which that have been followed up with IR spectra (e.g., @Groenewegen2009). The carbon-enhancement of these stars is believed to come from internal processes or mass transfer, and does not reflect the ISM from which they were formed. Thus they will not be discussed further in this paper.
The star ET0097 is thus the most inherently carbon-rich star in the Sculptor dSph measured to date, with $\text{[Fe/H]}=-2.03\pm0.10$ and $\text{[C/Fe]}=0.51\pm0.10$. It was even more carbon-enhanced in the past, with $\text{[C/Fe]}\approx0.8$, making it the first CEMP observed in Sculptor. A detailed chemical analysis of this star is presented here, from an HR spectrum observed with the VLT/UVES telescope at the European Southern Observatory (ESO). In addition, carbon abundance estimates and upper limits for 85 other stars are presented, derived from CN molecular lines in the wavelength range 9100-9250 , from VLT/FLAMES spectra.
Observations and data reduction
===============================
From the DART survey [@Tolstoy2006], detailed abundance measurements are known for $\sim$100 stars, spread over a $25^\prime$ diameter field of view in the Sculptor dSph (Hill et al. in prep.; @Tolstoy2009). As a part of this project, ESO VLT/FLAMES/GIRAFFE HR spectroscopy was carried out over the wavelength range $\sim$9100-9300 , to measure S abundances in Sculptor (Skúladóttir et al. in prep.). In most of these spectra CN molecular lines were observed, with the exception of the most metal-poor stars ($\text{[Fe/H]}\lesssim-2.2$). This CN molecular band was exceptionally strong in the star ET0097 and to follow up that observation, an HR spectrum over a long wavelength range was taken for the star, using ESO VLT/UVES.
UVES is a dichroic HR optical spectrograph at the VLT [@Dekker2000], where the light beam from the telescope can be split into two arms, the Ultra Violet to the Blue arm and the Visual to the Red arm. The observations were taken in August and September of 2013, using a 1.2” slit, with a resolution of 40,000 in the blue, and 35,000 in the red. The observational details are listed in Table \[table:obs\].
The details regarding the observations and data reduction for the VLT/FLAMES data, along with the S measurements will be presented in an upcoming paper, Skúladóttir et al. in prep.
\[tab:SNR\]
Wavelength range () S/N
--------------------- -----
3770-4980 30
5760-7510 50
7660-9450 50
: Signal-to-noise ratios of the different parts of the final coadded spectrum.
Data reduction
--------------
The ESO VLT/UVES spectrum was reduced, extracted, wavelength calibrated, and sky-subtracted using the UVES pipeline provided by ESO [@Freudling2013]. The reduced spectra were corrected for telluric absorption using spectra of a blue horizontal branch star, taken the same nights as the observations. The spectra taken at different times all showed comparable counts, so they were combined using a median value of the four spectra. The usable wavelength range and their relative signal-to-noise (S/N) ratios are listed in Table \[tab:SNR\]. The S/N ratios were evaluated as the mean value over the standard deviation of the continuum in line-free regions.
Continuum normalization
-----------------------
In the red part of the spectrum ($\sim$5800-9400 ), the entire wavelength range was covered with CN molecular lines, most of them weak, but some stronger. To find proper continuum points for the spectrum, a synthetic spectrum was made, using rough estimates of the oxygen, carbon, and nitrogen abundances. An iterative comparison with the normalized observed spectrum was then used to find a better synthetic spectrum, which was used for a better determination of continuum points. This process was iterated until the result was stable.
A similar approach was used for the reddest part of the blue spectrum ($\sim$4500-5000 Å) that is covered in relatively weak CH and C$_2$ molecular lines. For the bluer part of the spectrum, true continuum points became rarer, and a continuum could only be estimated from points close to the continuum. At the bluest part of the spectrum, the B-X band of CN at 3888 Å$ $ is extremely strong, and wipes out all continuum points making normalization in the region very uncertain, so the bluest part of the spectrum ($\lesssim3900$ Å) could not be used in the abundance analysis.
Where enough continuum points were available, the spectrum was also renormalized around each line being measured by a constant factor for better accuracy, but this change in the height of the continuum was minimal, rarely more than 1-2%.
Colour mag. err E(V-X) $T_\textsl{eff}$ (K)
-------- -------- ------- -------- ----------------------
V 17.255 0.002
V-I 1.246 0.005 0.023 4382
V-J 2.125 0.004 0.041 4393
V-K 2.954 0.005 0.050 4378
: Photometry of ET0097.[]{data-label="table:photo"}
Stellar parameters
==================
The photometry for ET0097 comes from deep wide field imaging in the V and I bands [@deBoer2011] and the infrared photometry, bands J and K, come from VISTA survey observations, see Table \[table:photo\]. Although photometry is also available for the B band, it was not used, since in this carbon-rich star it is affected by the strong CH molecular band in the region.
The effective temperature of the star, $T_\textsl{eff}$, was determined from the photometry following the recipe from @RamirezMelendez2005 for giants with metallicity $\text{[Fe/H]}=-2.02$ and assuming a reddening correction, E(V-X), in the direction of the Sculptor dSph as evaluated by @Schlegel1998, see Table \[table:photo\]. The result, $T_\textsl{eff}~=~4383~\pm~35$ K, is 83 K higher than the temperature adopted for this star in Hill et al. in prep., but agrees within the error bars. Included in the estimated errors are the $\sigma(T_\textsl{eff})$ provided by @RamirezMelendez2005 for each color used, the photometric errors and the (minor) effect of the observational errors of \[Fe/H\].
The surface gravity for the star is obtained using the standard relation: $$\log g_{\star}=\log g_{\odot}+\log{\frac{\text{M}_{\star}}{\text{M}_{\odot}}}+ 4\log{ \frac{T_{\textsl{eff,}\star}}{T_{\textsl{eff,}\odot}} }+0.4(M_{\textsl{bol,}\star}-M_{\textsl{bol,}\odot})$$ which yields the surface gravity $\log g_\star = 0.75\pm0.13$.
The absolute bolometric magnitude for the star, $M_{\textsl{bol,}\star}=-3.01\pm0.08$, is calculated using a calibration for the V-band magnitude [@Alonso1999] and a distance modulus of $(m-M)_0~=~19.68\pm0.08$, from @Pietrzynski2008, which dominates the error on $M_{\textsl{bol,}\star}$, as the photometric errors are negligible. The mass of the star is assumed to be $\text{M}_\star~=~0.8\pm0.2 $ M$_\odot$. The solar values used are the following: $\log~g_\odot=4.44$, $T_{\textsl{eff,}\odot}~=~5790$ K and $M_{\textsl{bol,}\odot}~=~4.72$.
The microturbulence velocity, $v_t$, was spectroscopically derived by making sure the abundance measurements for Fe I do not show a trend with the equivalent width, $\log($EW$/\lambda)$. The result is $v_t~=~2.25~\pm~0.20$ km/s.
No significant slope was found between the iron abundances of individual lines and their relevant wavelength or their excitation potential, confirming the validity of the determined stellar parameters.
[l r r r r r l]{}
\
X$_i$ & $\lambda$ & $\chi_{ex}$ & $\log (gf)$ & $\log\epsilon(X_i)$ & $\delta_\textsl{noise,i}$ & Comment\
\
X$_i$ & $\lambda$ & $\chi_{ex}$ & $\log (gf)$ & $\log\epsilon(X_i)$ & $\delta_\textsl{noise,i}$ & Comment\
Li I &$ 6707.76 $&$ 0.000 $&$ -0.009 $&$ <0.17 $&$ $-$ $& blended\
Li I &$ 6707.91 $&$ 0.000 $&$ -0.309 $&$ $-$ $&$ $-$ $&\
O I &$ 6300.30 $&$ 0.000 $&$ -9.819 $&$ 7.30 $&$ 0.14 $&\
O I &$ 6363.78 $&$ 0.020 $&$ -10.303 $&$ 7.46 $&$ 0.30 $&\
Na I &$ 5889.95 $&$ 0.000 $&$ 0.117 $&$ 3.90 $&$ 0.26 $&\
Na I &$ 5895.92 $&$ 0.000 $&$ -0.184 $&$ 3.68 $&$ 0.20 $&\
Na I &$ 8183.26 $&$ 2.102 $&$ 0.230 $&$ 3.94 $&$ 0.28 $&\
Mg I &$ 4571.10 $&$ 0.000 $&$ -5.691 $&$ 6.02 $&$ $-$ $&\
Mg I &$ 4702.99 $&$ 4.346 $&$ -0.666 $&$ 5.80 $&$ $-$ $&\
Mg I &$ 5711.09 $&$ 4.346 $&$ -1.833 $&$ 5.96 $&$ $-$ $&\
Mg I &$ 8736.01 $&$ 5.946 $&$ -3.210 $&$ 6.04 $&$ $-$ $&\
Mg I &$ 8736.01 $&$ 5.946 $&$ -1.930 $&$ $-$ $&$ $-$ $&\
Mg I &$ 8736.02 $&$ 5.946 $&$ -3.300 $&$ $-$ $&$ $-$ $&\
Mg I &$ 8736.02 $&$ 5.946 $&$ -0.690 $&$ $-$ $&$ $-$ $&\
Mg I &$ 8736.02 $&$ 5.946 $&$ -1.970 $&$ $-$ $&$ $-$ $&\
Mg I &$ 8736.03 $&$ 5.946 $&$ -1.020 $&$ $-$ $&$ $-$ $&\
Mg I &$ 8806.76 $&$ 4.346 $&$ -0.137 $&$ 6.02 $&$ $-$ $&\
Al I &$ 3961.52 $&$ 0.014 $&$ -0.323 $&$ 3.62 $&$ 0.80 $& very blended\
Si I &$ 5708.40 $&$ 4.954 $&$ -1.470 $&$ 5.86 $&$ $-$ $&\
Si I &$ 5948.54 $&$ 5.082 $&$ -1.230 $&$ 5.80 $&$ $-$ $& blended\
Si I &$ 7034.90 $&$ 5.871 $&$ -0.880 $&$ 6.12 $&$ $-$ $&\
Si I &$ 7275.26 $&$ 6.206 $&$ -7.048 $&$ 5.90 $&$ $-$ $& blended\
Si I &$ 7275.26 $&$ 6.206 $&$ -8.389 $&$ $-$ $&$ $-$ $&\
Si I &$ 7275.30 $&$ 5.616 $&$ -0.847 $&$ $-$ $&$ $-$ $&\
Si I &$ 7409.08 $&$ 5.616 $&$ -0.880 $&$ 5.78 $&$ $-$ $&\
Si I &$ 7409.15 $&$ 5.964 $&$ -1.566 $&$ $-$ $&$ $-$ $&\
Si I &$ 7423.50 $&$ 5.619 $&$ -0.175 $&$ 5.42 $&$ $-$ $&\
Si I &$ 8752.01 $&$ 5.871 $&$ 0.079 $&$ 5.50 $&$ $-$ $&\
S I &$ 9212.86 $&$ 6.525 $&$ 0.420 $&$ 5.42 $&$ 0.32 $&\
S I &$ 9228.09 $&$ 6.525 $&$ 0.260 $&$ 5.46 $&$ 0.26 $&\
K I &$ 7664.91 $&$ 0.000 $&$ 0.130 $&$ 3.46 $&$ 0.10 $&\
K I &$ 7698.97 $&$ 0.000 $&$ -0.170 $&$ 3.44 $&$ 0.06 $&\
Ca I &$ 5857.45 $&$ 2.933 $&$ 0.240 $&$ 4.40 $&$ $-$ $&\
Ca I &$ 6102.44 $&$ 2.523 $&$ -2.805 $&$ 4.46 $&$ $-$ $&\
Ca I &$ 6102.72 $&$ 1.879 $&$ -0.793 $&$ $-$ $&$ $-$ $&\
Ca I &$ 6122.22 $&$ 1.886 $&$ -0.316 $&$ 4.52 $&$ $-$ $&\
Ca I &$ 6161.30 $&$ 2.523 $&$ -1.266 $&$ 4.50 $&$ $-$ $&\
Ca I &$ 6162.17 $&$ 1.899 $&$ -0.090 $&$ 4.50 $&$ $-$ $&\
Ca I &$ 6166.44 $&$ 2.521 $&$ -1.142 $&$ 4.56 $&$ $-$ $&\
Ca I &$ 6169.04 $&$ 2.523 $&$ -0.797 $&$ 4.60 $&$ $-$ $&\
Ca I &$ 6169.56 $&$ 2.526 $&$ -0.478 $&$ 4.48 $&$ $-$ $&\
Ca I &$ 6439.08 $&$ 2.526 $&$ 0.390 $&$ 4.44 $&$ $-$ $&\
Ca I &$ 6439.17 $&$ 5.490 $&$ -3.709 $&$ $-$ $&$ $-$ $&\
Ca I &$ 6439.24 $&$ 5.832 $&$ -4.094 $&$ $-$ $&$ $-$ $&\
Ca I &$ 6449.81 $&$ 2.521 $&$ -0.502 $&$ 4.48 $&$ $-$ $&\
Ca I &$ 6455.60 $&$ 2.523 $&$ -1.340 $&$ 4.56 $&$ $-$ $& blended\
Ca I &$ 6462.57 $&$ 2.523 $&$ 0.262 $&$ 4.34 $&$ $-$ $& blended\
Ca I &$ 6471.66 $&$ 2.526 $&$ -0.686 $&$ 4.46 $&$ $-$ $&\
Ca I &$ 6493.78 $&$ 2.521 $&$ -0.109 $&$ 4.48 $&$ $-$ $&\
Ca I &$ 6499.65 $&$ 2.523 $&$ -0.818 $&$ 4.50 $&$ $-$ $&\
Ca I &$ 6572.78 $&$ 0.000 $&$ -4.240 $&$ 4.44 $&$ $-$ $&\
Ca I &$ 6717.68 $&$ 2.709 $&$ -0.524 $&$ 4.60 $&$ $-$ $&\
Ca I &$ 6717.69 $&$ 5.883 $&$ -7.108 $&$ $-$ $&$ $-$ $&\
Ca I &$ 7148.15 $&$ 2.709 $&$ 0.137 $&$ 4.58 $&$ $-$ $&\
Ca I &$ 7202.20 $&$ 2.709 $&$ -0.262 $&$ 4.44 $&$ $-$ $&\
Ca I &$ 7202.56 $&$ 6.021 $&$ -4.660 $&$ $-$ $&$ $-$ $&\
Ca I &$ 7326.15 $&$ 2.933 $&$ -0.208 $&$ 4.42 $&$ $-$ $&\
Ca I &$ 7326.48 $&$ 6.023 $&$ -5.128 $&$ $-$ $&$ $-$ $&\
Ca I &$ 7326.48 $&$ 6.007 $&$ -4.670 $&$ $-$ $&$ $-$ $&\
Sc II &$ 4246.82 $&$ 0.315 $&$ 0.242 $&$ 1.14 $&$ $-$ $& blended\
Sc II &$ 4294.77 $&$ 0.605 $&$ -1.391 $&$ 0.90 $&$ $-$ $& very blended\
Sc I &$ 4320.62 $&$ 2.109 $&$ -1.920 $&$ 1.00 $&$ $-$ $& blended\
Sc II &$ 4320.73 $&$ 0.605 $&$ -0.252 $&$ $-$ $&$ $-$ $&\
Sc I &$ 4415.48 $&$ 3.083 $&$ -3.393 $&$ 1.24 $&$ $-$ $& blended\
Sc II &$ 4415.56 $&$ 0.595 $&$ -0.668 $&$ $-$ $&$ $-$ $&\
Sc I &$ 4431.23 $&$ 1.851 $&$ -6.387 $&$ 1.16 $&$ $-$ $&\
Sc II &$ 4431.35 $&$ 0.605 $&$ -1.969 $&$ $-$ $&$ $-$ $&\
Sc I &$ 4431.52 $&$ 3.083 $&$ -2.584 $&$ $-$ $&$ $-$ $&\
Sc II &$ 4670.41 $&$ 1.357 $&$ -0.576 $&$ 0.92 $&$ $-$ $& very blended\
Sc I &$ 4670.52 $&$ 3.172 $&$ -2.584 $&$ $-$ $&$ $-$ $&\
Sc II &$ 6245.64 $&$ 1.507 $&$ -1.030 $&$ 1.24 $&$ $-$ $& blended\
Sc I &$ 6279.57 $&$ 3.607 $&$ -1.673 $&$ 1.18 $&$ $-$ $&\
Sc II &$ 6279.75 $&$ 1.500 $&$ -1.265 $&$ $-$ $&$ $-$ $&\
Sc II &$ 6604.60 $&$ 1.357 $&$ -1.309 $&$ 1.16 $&$ $-$ $&\
Ti II &$ 4493.51 $&$ 1.080 $&$ -3.020 $&$ 3.12 $&$ $-$ $&\
Ti II &$ 4501.27 $&$ 1.116 $&$ -0.770 $&$ 2.78 $&$ $-$ $&\
Ti I &$ 4562.63 $&$ 0.021 $&$ -2.656 $&$ 3.00 $&$ $-$ $&\
Ti I &$ 4563.42 $&$ 2.427 $&$ -0.681 $&$ 2.92 $&$ $-$ $&\
Ti II &$ 4563.76 $&$ 1.221 $&$ -0.690 $&$ $-$ $&$ $-$ $&\
Ti II &$ 4568.31 $&$ 1.224 $&$ -2.940 $&$ 3.08 $&$ $-$ $&\
Ti II &$ 4583.41 $&$ 1.165 $&$ -2.920 $&$ 3.18 $&$ $-$ $&\
Ti II &$ 4609.26 $&$ 1.180 $&$ -3.430 $&$ 3.26 $&$ $-$ $&\
Ti I &$ 4609.34 $&$ 3.319 $&$ -2.072 $&$ $-$ $&$ $-$ $&\
Ti I &$ 4617.27 $&$ 1.749 $&$ 0.389 $&$ 2.96 $&$ $-$ $&\
Ti I &$ 4708.42 $&$ 3.199 $&$ -3.799 $&$ 3.18 $&$ $-$ $& blended\
Ti I &$ 4708.43 $&$ 1.873 $&$ -6.885 $&$ $-$ $&$ $-$ $&\
Ti II &$ 4708.66 $&$ 1.237 $&$ -2.340 $&$ $-$ $&$ $-$ $&\
Ti I &$ 4759.14 $&$ 2.778 $&$ -2.543 $&$ 3.12 $&$ $-$ $&\
Ti I &$ 4759.27 $&$ 2.256 $&$ 0.514 $&$ $-$ $&$ $-$ $&\
Ti II &$ 4764.52 $&$ 1.237 $&$ -2.950 $&$ 3.50 $&$ $-$ $&\
Ti I &$ 4792.25 $&$ 0.813 $&$ -3.468 $&$ 3.28 $&$ $-$ $& blended\
Ti II &$ 4792.43 $&$ 1.237 $&$ -3.328 $&$ $-$ $&$ $-$ $&\
Ti I &$ 4792.48 $&$ 2.334 $&$ -0.300 $&$ $-$ $&$ $-$ $&\
Ti II &$ 4805.08 $&$ 2.061 $&$ -0.960 $&$ 3.14 $&$ $-$ $&\
Ti I &$ 4805.42 $&$ 2.345 $&$ 0.150 $&$ $-$ $&$ $-$ $&\
Ti I &$ 4805.44 $&$ 3.062 $&$ -3.409 $&$ $-$ $&$ $-$ $&\
Ti I &$ 4840.87 $&$ 0.900 $&$ -0.509 $&$ 3.04 $&$ $-$ $&\
Ti II &$ 4865.61 $&$ 1.116 $&$ -2.790 $&$ 3.32 $&$ $-$ $&\
Ti I &$ 4865.78 $&$ 2.578 $&$ -0.398 $&$ $-$ $&$ $-$ $&\
Ti III &$ 4874.00 $&$ 18.252 $&$ 0.560 $&$ 3.12 $&$ $-$ $&\
Ti II &$ 4874.01 $&$ 3.095 $&$ -0.800 $&$ $-$ $&$ $-$ $&\
Ti I &$ 4885.08 $&$ 1.887 $&$ 0.358 $&$ 2.96 $&$ $-$ $& blended\
Ti I &$ 4885.20 $&$ 2.677 $&$ -1.681 $&$ $-$ $&$ $-$ $&\
Ti I &$ 4913.46 $&$ 2.506 $&$ -3.499 $&$ 3.18 $&$ $-$ $&\
Ti I &$ 4913.61 $&$ 1.873 $&$ 0.160 $&$ $-$ $&$ $-$ $&\
Ti I &$ 4981.73 $&$ 0.848 $&$ 0.504 $&$ 2.88 $&$ $-$ $&\
Ti I &$ 4981.90 $&$ 2.427 $&$ -3.637 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5866.23 $&$ 3.176 $&$ -3.522 $&$ 3.12 $&$ $-$ $&\
Ti I &$ 5866.37 $&$ 3.305 $&$ -0.186 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5866.45 $&$ 1.067 $&$ -0.840 $&$ $-$ $&$ $-$ $&\
Ti II &$ 5866.66 $&$ 8.089 $&$ -0.620 $&$ $-$ $&$ $-$ $&\
Ti II &$ 5899.03 $&$ 8.082 $&$ -2.325 $&$ 3.16 $&$ $-$ $&\
Ti I &$ 5899.29 $&$ 1.053 $&$ -1.154 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5899.50 $&$ 3.351 $&$ -2.307 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5921.92 $&$ 3.691 $&$ -2.186 $&$ 3.22 $&$ $-$ $&\
Ti II &$ 5921.94 $&$ 8.056 $&$ -0.024 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5922.11 $&$ 1.046 $&$ -1.466 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5922.14 $&$ 3.113 $&$ -1.602 $&$ $-$ $&$ $-$ $&\
Ti II &$ 5952.98 $&$ 8.071 $&$ -0.063 $&$ 3.00 $&$ $-$ $&\
Ti I &$ 5953.11 $&$ 3.090 $&$ -3.612 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5953.16 $&$ 1.887 $&$ -0.329 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5965.32 $&$ 3.409 $&$ -3.412 $&$ 3.14 $&$ $-$ $& blended\
Ti II &$ 5965.81 $&$ 8.089 $&$ -1.623 $&$ $-$ $&$ $-$ $&\
Ti I &$ 5965.83 $&$ 1.879 $&$ -0.409 $&$ $-$ $&$ $-$ $&\
Ti II &$ 5978.41 $&$ 8.093 $&$ -0.535 $&$ 3.14 $&$ $-$ $&\
Ti I &$ 5978.54 $&$ 1.873 $&$ -0.496 $&$ $-$ $&$ $-$ $&\
Ti II &$ 6126.00 $&$ 8.097 $&$ -1.650 $&$ 3.22 $&$ $-$ $&\
Ti I &$ 6126.22 $&$ 1.067 $&$ -1.425 $&$ $-$ $&$ $-$ $&\
Ti I &$ 6126.27 $&$ 3.154 $&$ -3.025 $&$ $-$ $&$ $-$ $&\
Ti I &$ 6257.80 $&$ 0.000 $&$ -4.297 $&$ 3.04 $&$ $-$ $&\
Ti I &$ 6258.10 $&$ 1.443 $&$ -0.355 $&$ $-$ $&$ $-$ $&\
Ti I &$ 6258.71 $&$ 1.460 $&$ -0.240 $&$ 3.14 $&$ $-$ $&\
Ti I &$ 6261.10 $&$ 1.430 $&$ -0.479 $&$ 3.16 $&$ $-$ $&\
Ti I &$ 6261.12 $&$ 3.128 $&$ -1.798 $&$ $-$ $&$ $-$ $&\
Ti I &$ 6261.28 $&$ 3.323 $&$ -3.656 $&$ $-$ $&$ $-$ $&\
Ti II &$ 6491.56 $&$ 2.061 $&$ -1.793 $&$ 2.98 $&$ $-$ $&\
Ti I &$ 6556.06 $&$ 1.460 $&$ -1.074 $&$ 3.38 $&$ $-$ $&\
Ti II &$ 6559.59 $&$ 2.048 $&$ -2.019 $&$ 3.04 $&$ $-$ $&\
Ti I &$ 7209.44 $&$ 1.460 $&$ -0.500 $&$ 3.14 $&$ $-$ $& blended\
Ti I &$ 7209.62 $&$ 3.424 $&$ -1.599 $&$ $-$ $&$ $-$ $&\
Ti I &$ 7244.85 $&$ 1.443 $&$ -0.810 $&$ 3.24 $&$ $-$ $& blended\
Ti I &$ 7251.71 $&$ 1.430 $&$ -0.770 $&$ 3.02 $&$ $-$ $& blended\
V II &$ 4002.80 $&$ 1.555 $&$ -5.601 $&$ 2.14 $&$ $-$ $&\
V II &$ 4002.94 $&$ 1.428 $&$ -1.447 $&$ $-$ $&$ $-$ $&\
V I &$ 4003.01 $&$ 2.332 $&$ -3.063 $&$ $-$ $&$ $-$ $&\
V I &$ 4003.12 $&$ 2.359 $&$ -4.625 $&$ $-$ $&$ $-$ $&\
V I &$ 4003.13 $&$ 2.505 $&$ -9.645 $&$ $-$ $&$ $-$ $&\
V I &$ 4003.17 $&$ 2.578 $&$ -1.161 $&$ $-$ $&$ $-$ $&\
V I &$ 4586.23 $&$ 2.578 $&$ -6.028 $&$ 1.88 $&$ $-$ $&\
V I &$ 4586.37 $&$ 0.040 $&$ -0.790 $&$ $-$ $&$ $-$ $&\
V I &$ 4586.44 $&$ 1.376 $&$ -6.348 $&$ $-$ $&$ $-$ $&\
V I &$ 4586.55 $&$ 2.505 $&$ -3.017 $&$ $-$ $&$ $-$ $&\
V I &$ 4827.46 $&$ 0.040 $&$ -1.478 $&$ 1.94 $&$ $-$ $&\
V II &$ 4827.47 $&$ 8.645 $&$ -3.405 $&$ $-$ $&$ $-$ $&\
V I &$ 4827.54 $&$ 2.040 $&$ -5.969 $&$ $-$ $&$ $-$ $&\
V I &$ 4827.69 $&$ 2.878 $&$ -3.382 $&$ $-$ $&$ $-$ $&\
V I &$ 4831.65 $&$ 0.017 $&$ -1.380 $&$ 1.96 $&$ $-$ $&\
V I &$ 4831.77 $&$ 1.955 $&$ -3.283 $&$ $-$ $&$ $-$ $&\
V I &$ 4864.73 $&$ 0.017 $&$ -0.960 $&$ 1.78 $&$ $-$ $&\
V I &$ 4864.83 $&$ 1.183 $&$ -1.240 $&$ $-$ $&$ $-$ $&\
V II &$ 4864.87 $&$ 6.857 $&$ -2.457 $&$ $-$ $&$ $-$ $&\
V II &$ 4875.22 $&$ 6.547 $&$ -4.285 $&$ 1.98 $&$ $-$ $&\
V I &$ 4875.42 $&$ 1.351 $&$ -4.161 $&$ $-$ $&$ $-$ $&\
V I &$ 4875.49 $&$ 0.040 $&$ -0.810 $&$ $-$ $&$ $-$ $&\
V II &$ 4875.50 $&$ 5.468 $&$ -1.533 $&$ $-$ $&$ $-$ $&\
V II &$ 4875.62 $&$ 4.005 $&$ -3.268 $&$ $-$ $&$ $-$ $&\
V I &$ 5703.58 $&$ 1.051 $&$ -0.211 $&$ 2.20 $&$ $-$ $&\
V II &$ 5703.65 $&$ 3.973 $&$ -4.727 $&$ $-$ $&$ $-$ $&\
V II &$ 5706.77 $&$ 6.901 $&$ -3.971 $&$ 2.18 $&$ $-$ $&\
V II &$ 5706.86 $&$ 9.031 $&$ -1.107 $&$ $-$ $&$ $-$ $&\
V I &$ 5706.98 $&$ 1.043 $&$ -0.454 $&$ $-$ $&$ $-$ $&\
V I &$ 6039.72 $&$ 1.064 $&$ -0.650 $&$ 2.10 $&$ $-$ $&\
V I &$ 6039.86 $&$ 3.517 $&$ -2.920 $&$ $-$ $&$ $-$ $&\
V I &$ 6039.86 $&$ 2.555 $&$ -4.421 $&$ $-$ $&$ $-$ $&\
V I &$ 6090.21 $&$ 1.081 $&$ -0.062 $&$ 1.98 $&$ $-$ $&\
V II &$ 6090.47 $&$ 3.799 $&$ -6.833 $&$ $-$ $&$ $-$ $&\
V I &$ 6090.54 $&$ 1.064 $&$ -2.600 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4545.94 $&$ 8.348 $&$ -2.289 $&$ 2.96 $&$ $-$ $&\
Cr I &$ 4545.95 $&$ 0.941 $&$ -1.370 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4546.02 $&$ 3.551 $&$ -3.528 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4546.03 $&$ 6.805 $&$ -3.416 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4588.20 $&$ 4.071 $&$ -0.845 $&$ 3.84 $&$ $-$ $&\
Cr I &$ 4588.24 $&$ 4.389 $&$ -3.887 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4588.40 $&$ 3.104 $&$ -4.542 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4591.13 $&$ 4.402 $&$ -3.455 $&$ 3.34 $&$ $-$ $&\
Cr I &$ 4591.39 $&$ 0.968 $&$ -1.740 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4591.42 $&$ 10.599 $&$ -4.794 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4591.46 $&$ 4.440 $&$ -1.925 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4591.48 $&$ 3.422 $&$ -1.888 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4591.60 $&$ 4.490 $&$ -3.992 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4616.12 $&$ 0.983 $&$ -1.190 $&$ 3.18 $&$ $-$ $&\
Cr II &$ 4616.24 $&$ 5.670 $&$ -2.346 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4625.92 $&$ 3.850 $&$ -0.310 $&$ 3.26 $&$ $-$ $&\
Cr II &$ 4625.94 $&$ 11.677 $&$ -0.760 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4626.02 $&$ 4.532 $&$ -0.960 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4626.09 $&$ 11.788 $&$ -4.123 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4626.17 $&$ 0.968 $&$ -1.320 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4634.00 $&$ 3.551 $&$ -1.808 $&$ 3.70 $&$ $-$ $& blended\
Cr II &$ 4634.07 $&$ 4.072 $&$ -1.236 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4651.13 $&$ 11.622 $&$ -1.756 $&$ 3.44 $&$ $-$ $&\
Cr II &$ 4651.23 $&$ 10.243 $&$ -5.278 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4651.28 $&$ 0.983 $&$ -1.460 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4651.40 $&$ 11.711 $&$ -3.420 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4652.04 $&$ 10.476 $&$ -1.599 $&$ 3.32 $&$ $-$ $&\
Cr I &$ 4652.16 $&$ 1.004 $&$ -1.030 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4652.27 $&$ 5.871 $&$ -4.565 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4755.95 $&$ 7.899 $&$ -5.302 $&$ 3.84 $&$ $-$ $&\
Cr I &$ 4756.05 $&$ 2.987 $&$ -2.912 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4756.11 $&$ 3.104 $&$ 0.090 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4756.16 $&$ 4.106 $&$ -3.511 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4756.31 $&$ 4.411 $&$ -4.185 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4829.18 $&$ 10.476 $&$ -2.906 $&$ 3.96 $&$ $-$ $&\
Cr II &$ 4829.22 $&$ 10.798 $&$ -2.554 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4829.22 $&$ 3.369 $&$ -2.423 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4829.31 $&$ 2.545 $&$ -1.604 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4829.37 $&$ 2.545 $&$ -0.810 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4848.06 $&$ 5.211 $&$ -1.253 $&$ 3.70 $&$ $-$ $&\
Cr II &$ 4848.23 $&$ 3.864 $&$ -1.280 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4876.40 $&$ 4.096 $&$ -2.973 $&$ 3.76 $&$ $-$ $&\
Cr II &$ 4876.40 $&$ 3.854 $&$ -1.580 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4876.47 $&$ 3.864 $&$ -2.093 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4876.67 $&$ 6.686 $&$ -2.966 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4876.69 $&$ 8.363 $&$ -6.334 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4922.16 $&$ 3.435 $&$ -3.337 $&$ 3.50 $&$ $-$ $&\
Cr II &$ 4922.22 $&$ 10.843 $&$ -2.464 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4922.27 $&$ 3.104 $&$ 0.270 $&$ $-$ $&$ $-$ $&\
Cr II &$ 4922.36 $&$ 10.454 $&$ -2.474 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4922.54 $&$ 3.011 $&$ -2.173 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4942.50 $&$ 0.941 $&$ -2.294 $&$ 3.58 $&$ $-$ $&\
Cr I &$ 4942.75 $&$ 3.449 $&$ -3.467 $&$ $-$ $&$ $-$ $&\
Cr I &$ 4964.93 $&$ 0.941 $&$ -2.527 $&$ 3.58 $&$ $-$ $&\
Cr II &$ 4965.00 $&$ 8.354 $&$ -5.841 $&$ $-$ $&$ $-$ $&\
Cr I &$ 6330.09 $&$ 0.941 $&$ -2.920 $&$ 3.56 $&$ $-$ $&\
Cr II &$ 6330.40 $&$ 11.144 $&$ -1.000 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4055.54 $&$ 2.143 $&$ -0.070 $&$ 3.06 $&$ $-$ $& blended\
Mn III &$ 4079.03 $&$ 11.095 $&$ -9.881 $&$ 2.88 $&$ $-$ $& very blended\
Mn III &$ 4079.18 $&$ 23.795 $&$ -0.432 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4079.22 $&$ 4.258 $&$ -0.161 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4079.24 $&$ 2.143 $&$ -0.530 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4079.41 $&$ 2.187 $&$ -0.420 $&$ $-$ $&$ $-$ $&\
Mn II &$ 4753.74 $&$ 6.528 $&$ -4.469 $&$ 3.30 $&$ $-$ $&\
Mn I &$ 4753.89 $&$ 5.214 $&$ -0.965 $&$ $-$ $&$ $-$ $&\
Mn II &$ 4754.03 $&$ 10.271 $&$ -3.379 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4754.04 $&$ 2.282 $&$ -0.086 $&$ $-$ $&$ $-$ $&\
Mn II &$ 4754.06 $&$ 6.139 $&$ -3.081 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4762.37 $&$ 2.888 $&$ 0.425 $&$ 2.94 $&$ $-$ $& blended\
Mn III &$ 4765.84 $&$ 14.459 $&$ -4.448 $&$ 3.22 $&$ $-$ $&\
Mn I &$ 4765.85 $&$ 2.941 $&$ -0.080 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4766.00 $&$ 4.425 $&$ -1.730 $&$ $-$ $&$ $-$ $&\
Mn III &$ 4766.29 $&$ 24.215 $&$ -1.865 $&$ 3.16 $&$ $-$ $&\
Mn I &$ 4766.42 $&$ 2.920 $&$ 0.100 $&$ $-$ $&$ $-$ $&\
Mn III &$ 4766.46 $&$ 21.564 $&$ -2.589 $&$ $-$ $&$ $-$ $&\
Mn II &$ 4766.52 $&$ 10.661 $&$ -4.115 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4766.66 $&$ 5.199 $&$ -0.560 $&$ $-$ $&$ $-$ $&\
Mn II &$ 4783.29 $&$ 10.283 $&$ -4.252 $&$ 3.32 $&$ $-$ $&\
Mn I &$ 4783.43 $&$ 2.298 $&$ 0.042 $&$ $-$ $&$ $-$ $&\
Mn I &$ 4823.52 $&$ 2.319 $&$ 0.144 $&$ 3.30 $&$ $-$ $& blended\
Mn II &$ 4823.65 $&$ 11.683 $&$ -2.969 $&$ $-$ $&$ $-$ $&\
Mn I &$ 6013.51 $&$ 3.072 $&$ -0.251 $&$ 3.00 $&$ $-$ $&\
Fe I &$ 5762.84 $&$ 4.301 $&$ -2.620 $&$ 5.48 $&$ $-$ $&\
Fe I &$ 5762.98 $&$ 4.191 $&$ -3.199 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5762.99 $&$ 4.209 $&$ -0.450 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5763.00 $&$ 4.191 $&$ -4.561 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5862.23 $&$ 3.640 $&$ -4.359 $&$ 5.30 $&$ $-$ $&\
Fe I &$ 5862.36 $&$ 4.549 $&$ -0.127 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5862.47 $&$ 5.334 $&$ -3.802 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5914.11 $&$ 4.608 $&$ -0.375 $&$ 5.34 $&$ $-$ $&\
Fe I &$ 5914.20 $&$ 4.608 $&$ -0.131 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5956.41 $&$ 5.352 $&$ -5.829 $&$ 5.68 $&$ $-$ $&\
Fe I &$ 5956.46 $&$ 4.283 $&$ -4.169 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5956.51 $&$ 5.352 $&$ -8.937 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5956.69 $&$ 0.859 $&$ -4.605 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5956.94 $&$ 4.580 $&$ -3.540 $&$ $-$ $&$ $-$ $&\
Fe I &$ 5976.69 $&$ 5.330 $&$ -5.182 $&$ 5.44 $&$ $-$ $&\
Fe I &$ 5976.78 $&$ 3.943 $&$ -1.243 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6002.77 $&$ 5.314 $&$ -3.755 $&$ 5.58 $&$ $-$ $&\
Fe I &$ 6002.79 $&$ 5.386 $&$ -4.886 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6003.01 $&$ 3.881 $&$ -1.120 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6003.02 $&$ 5.506 $&$ -7.385 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6003.11 $&$ 5.506 $&$ -7.472 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6003.11 $&$ 5.506 $&$ -8.111 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6003.17 $&$ 5.506 $&$ -7.907 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6055.76 $&$ 5.352 $&$ -7.003 $&$ 5.58 $&$ $-$ $&\
Fe I &$ 6055.78 $&$ 5.357 $&$ -7.928 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6055.89 $&$ 5.352 $&$ -7.777 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6055.94 $&$ 5.070 $&$ -2.322 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6055.94 $&$ 5.352 $&$ -5.833 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6055.95 $&$ 5.357 $&$ -6.627 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6055.95 $&$ 5.357 $&$ -8.619 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6056.01 $&$ 4.733 $&$ -0.460 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6056.09 $&$ 5.357 $&$ -7.290 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6056.09 $&$ 5.357 $&$ -7.768 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6056.25 $&$ 5.357 $&$ -7.202 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6065.48 $&$ 2.608 $&$ -1.530 $&$ 5.48 $&$ $-$ $&\
Fe I &$ 6065.49 $&$ 4.956 $&$ -3.471 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6082.46 $&$ 5.486 $&$ -8.673 $&$ 5.54 $&$ $-$ $&\
Fe I &$ 6082.54 $&$ 5.386 $&$ -5.272 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6082.58 $&$ 5.607 $&$ -9.656 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6082.58 $&$ 5.607 $&$ -3.403 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6082.61 $&$ 4.220 $&$ -3.746 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6082.71 $&$ 2.223 $&$ -3.573 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6082.85 $&$ 5.341 $&$ -3.667 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6082.89 $&$ 5.486 $&$ -8.142 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6082.89 $&$ 5.486 $&$ -9.294 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6136.25 $&$ 5.273 $&$ -3.556 $&$ 5.48 $&$ $-$ $&\
Fe I &$ 6136.62 $&$ 2.453 $&$ -1.400 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6136.99 $&$ 2.198 $&$ -2.950 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6137.28 $&$ 4.580 $&$ -2.160 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6137.47 $&$ 4.301 $&$ -3.741 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6137.50 $&$ 3.332 $&$ -2.514 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6137.69 $&$ 2.588 $&$ -1.403 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6151.62 $&$ 2.176 $&$ -3.299 $&$ 5.58 $&$ $-$ $&\
Fe I &$ 6151.69 $&$ 5.012 $&$ -3.761 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6173.01 $&$ 0.990 $&$ -7.794 $&$ 5.62 $&$ $-$ $&\
Fe I &$ 6173.03 $&$ 3.640 $&$ -4.961 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6173.15 $&$ 4.991 $&$ -3.920 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6173.29 $&$ 5.372 $&$ -5.600 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6173.33 $&$ 2.223 $&$ -2.880 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6173.34 $&$ 5.334 $&$ -4.314 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6173.49 $&$ 5.348 $&$ -7.269 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6173.64 $&$ 4.446 $&$ -3.400 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6180.20 $&$ 2.727 $&$ -2.586 $&$ 5.52 $&$ $-$ $&\
Fe I &$ 6180.29 $&$ 5.314 $&$ -4.453 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6191.56 $&$ 2.433 $&$ -1.417 $&$ 5.30 $&$ $-$ $&\
Fe I &$ 6191.57 $&$ 4.256 $&$ -5.483 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6191.77 $&$ 4.143 $&$ -5.123 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6200.27 $&$ 4.320 $&$ -3.931 $&$ 5.54 $&$ $-$ $&\
Fe I &$ 6200.31 $&$ 2.608 $&$ -2.437 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6200.51 $&$ 5.357 $&$ -9.018 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6213.27 $&$ 5.386 $&$ -4.197 $&$ 5.52 $&$ $-$ $&\
Fe I &$ 6213.43 $&$ 2.223 $&$ -2.482 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6218.94 $&$ 5.446 $&$ -8.948 $&$ 5.56 $&$ $-$ $&\
Fe I &$ 6219.14 $&$ 5.010 $&$ -2.270 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6219.28 $&$ 2.198 $&$ -2.433 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6219.31 $&$ 5.458 $&$ -5.082 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6219.53 $&$ 3.417 $&$ -4.100 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6219.58 $&$ 5.345 $&$ -4.691 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6230.48 $&$ 5.410 $&$ -5.491 $&$ 5.44 $&$ $-$ $&\
Fe I &$ 6230.72 $&$ 2.559 $&$ -1.281 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6252.38 $&$ 4.320 $&$ -7.012 $&$ 5.48 $&$ $-$ $&\
Fe I &$ 6252.56 $&$ 2.404 $&$ -1.687 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6252.78 $&$ 5.648 $&$ -8.652 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6254.26 $&$ 2.279 $&$ -2.443 $&$ 5.58 $&$ $-$ $&\
Fe I &$ 6301.50 $&$ 3.654 $&$ -0.718 $&$ 5.38 $&$ $-$ $&\
Fe I &$ 6301.77 $&$ 5.458 $&$ -2.889 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6317.76 $&$ 5.314 $&$ -5.205 $&$ 5.76 $&$ $-$ $&\
Fe I &$ 6318.02 $&$ 2.453 $&$ -2.261 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6318.04 $&$ 5.683 $&$ -9.049 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6318.05 $&$ 5.683 $&$ -5.610 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6318.05 $&$ 5.683 $&$ -8.040 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6318.35 $&$ 5.314 $&$ -5.989 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6322.69 $&$ 2.588 $&$ -2.426 $&$ 5.54 $&$ $-$ $&\
Fe I &$ 6322.74 $&$ 5.491 $&$ -5.869 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6335.33 $&$ 2.198 $&$ -2.177 $&$ 5.44 $&$ $-$ $&\
Fe I &$ 6344.02 $&$ 4.415 $&$ -3.572 $&$ 5.64 $&$ $-$ $&\
Fe I &$ 6344.07 $&$ 5.524 $&$ -8.598 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6344.08 $&$ 5.524 $&$ -6.712 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6344.15 $&$ 2.433 $&$ -2.923 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6344.28 $&$ 5.486 $&$ -8.925 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6344.28 $&$ 4.473 $&$ -6.227 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6344.37 $&$ 5.486 $&$ -9.075 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6344.37 $&$ 5.486 $&$ -8.548 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6358.43 $&$ 4.593 $&$ -3.620 $&$ 5.76 $&$ $-$ $&\
Fe I &$ 6358.51 $&$ 5.388 $&$ -4.570 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6358.63 $&$ 5.345 $&$ -5.414 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6358.63 $&$ 4.143 $&$ -1.657 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6358.65 $&$ 4.371 $&$ -3.448 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6358.70 $&$ 0.859 $&$ -4.468 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6399.53 $&$ 5.669 $&$ -9.756 $&$ 5.46 $&$ $-$ $&\
Fe I &$ 6399.61 $&$ 5.669 $&$ -8.100 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6399.61 $&$ 5.669 $&$ -9.605 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6399.65 $&$ 3.984 $&$ -3.287 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6399.69 $&$ 5.502 $&$ -5.292 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6399.77 $&$ 5.669 $&$ -9.938 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6399.79 $&$ 5.357 $&$ -6.843 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6399.85 $&$ 5.502 $&$ -6.988 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6400.00 $&$ 3.602 $&$ -0.290 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6400.06 $&$ 5.314 $&$ -4.911 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6400.13 $&$ 5.669 $&$ -7.575 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6400.32 $&$ 0.915 $&$ -4.318 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6400.34 $&$ 5.064 $&$ -3.635 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6400.37 $&$ 5.502 $&$ -7.493 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6400.37 $&$ 5.502 $&$ -8.377 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6411.65 $&$ 3.654 $&$ -0.595 $&$ 5.42 $&$ $-$ $&\
Fe I &$ 6411.99 $&$ 5.446 $&$ -5.559 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6421.21 $&$ 5.334 $&$ -3.994 $&$ 5.52 $&$ $-$ $&\
Fe I &$ 6421.35 $&$ 2.279 $&$ -2.027 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6421.57 $&$ 5.334 $&$ -5.293 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6494.98 $&$ 2.404 $&$ -1.273 $&$ 5.34 $&$ $-$ $&\
Fe I &$ 6545.85 $&$ 4.580 $&$ -3.977 $&$ 5.30 $&$ $-$ $&\
Fe I &$ 6545.99 $&$ 5.314 $&$ -3.547 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6546.20 $&$ 5.357 $&$ -7.253 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6546.24 $&$ 2.758 $&$ -1.536 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6546.47 $&$ 5.841 $&$ -7.040 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6546.50 $&$ 4.473 $&$ -6.658 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6592.61 $&$ 4.956 $&$ -1.262 $&$ 5.30 $&$ $-$ $&\
Fe I &$ 6592.91 $&$ 2.727 $&$ -1.473 $&$ $-$ $&$ $-$ $&\
Fe I &$ 6593.87 $&$ 2.433 $&$ -2.422 $&$ 5.50 $&$ $-$ $&\
Fe I &$ 6608.03 $&$ 2.279 $&$ -4.030 $&$ 5.52 $&$ $-$ $&\
Fe I &$ 7494.72 $&$ 1.557 $&$ -5.254 $&$ 5.26 $&$ $-$ $&\
Fe I &$ 7494.84 $&$ 5.458 $&$ -3.505 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7494.90 $&$ 4.985 $&$ -3.470 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7495.00 $&$ 3.695 $&$ -6.916 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7495.01 $&$ 5.720 $&$ -2.463 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7495.07 $&$ 4.220 $&$ 0.053 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7495.37 $&$ 5.693 $&$ -3.673 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7664.16 $&$ 4.835 $&$ -1.176 $&$ 5.42 $&$ $-$ $&\
Fe I &$ 7664.29 $&$ 2.990 $&$ -1.683 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7664.42 $&$ 5.849 $&$ -2.832 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7664.45 $&$ 4.733 $&$ -3.527 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7722.91 $&$ 5.539 $&$ -5.064 $&$ 5.78 $&$ $-$ $&\
Fe I &$ 7723.21 $&$ 2.279 $&$ -3.617 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7723.54 $&$ 5.793 $&$ -4.026 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7723.59 $&$ 5.539 $&$ -5.366 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7780.23 $&$ 5.519 $&$ -3.671 $&$ 5.24 $&$ $-$ $&\
Fe I &$ 7780.51 $&$ 5.683 $&$ -3.647 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7780.56 $&$ 4.473 $&$ 0.029 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7780.59 $&$ 5.693 $&$ -4.952 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7780.75 $&$ 5.930 $&$ -8.656 $&$ $-$ $&$ $-$ $&\
Fe I &$ 7831.86 $&$ 5.386 $&$ -4.697 $&$ 5.34 $&$ $-$ $&\
Fe I &$ 7832.20 $&$ 4.435 $&$ 0.111 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8046.05 $&$ 4.415 $&$ 0.031 $&$ 5.28 $&$ $-$ $&\
Fe I &$ 8387.77 $&$ 2.176 $&$ -1.493 $&$ 5.42 $&$ $-$ $&\
Fe I &$ 8387.96 $&$ 5.879 $&$ -7.512 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8388.07 $&$ 5.642 $&$ -3.588 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8515.11 $&$ 3.018 $&$ -2.073 $&$ 5.66 $&$ $-$ $&\
Fe I &$ 8621.60 $&$ 2.949 $&$ -2.321 $&$ 5.46 $&$ $-$ $&\
Fe I &$ 8621.89 $&$ 5.967 $&$ -7.684 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8674.31 $&$ 5.720 $&$ -6.667 $&$ 5.58 $&$ $-$ $&\
Fe I &$ 8674.36 $&$ 6.010 $&$ -7.879 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8674.54 $&$ 5.996 $&$ -8.466 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8674.58 $&$ 5.621 $&$ -6.068 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8674.58 $&$ 5.693 $&$ -4.781 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8674.75 $&$ 2.831 $&$ -1.800 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8675.19 $&$ 5.913 $&$ -5.414 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8763.97 $&$ 4.652 $&$ -0.146 $&$ 5.38 $&$ $-$ $&\
Fe I &$ 8975.11 $&$ 5.979 $&$ -5.119 $&$ 5.48 $&$ $-$ $&\
Fe I &$ 8975.16 $&$ 4.988 $&$ -2.087 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8975.27 $&$ 3.686 $&$ -7.036 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8975.31 $&$ 4.143 $&$ -4.759 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8975.40 $&$ 2.990 $&$ -2.233 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8975.42 $&$ 5.883 $&$ -5.666 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8975.69 $&$ 5.334 $&$ -6.162 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8975.80 $&$ 5.352 $&$ -8.779 $&$ $-$ $&$ $-$ $&\
Fe I &$ 8999.56 $&$ 2.831 $&$ -1.321 $&$ 5.40 $&$ $-$ $&\
Fe I &$ 9088.03 $&$ 5.502 $&$ -4.483 $&$ 5.40 $&$ $-$ $&\
Fe I &$ 9088.32 $&$ 2.845 $&$ -1.986 $&$ $-$ $&$ $-$ $&\
Fe I &$ 9088.35 $&$ 5.930 $&$ -5.925 $&$ $-$ $&$ $-$ $&\
Fe I &$ 9088.46 $&$ 6.010 $&$ -5.111 $&$ $-$ $&$ $-$ $&\
Fe I &$ 9088.50 $&$ 5.928 $&$ -3.861 $&$ $-$ $&$ $-$ $&\
Fe I &$ 9089.40 $&$ 2.949 $&$ -1.675 $&$ $-$ $&$ $-$ $&\
Fe I &$ 9089.50 $&$ 5.947 $&$ -4.040 $&$ $-$ $&$ $-$ $&\
Fe I &$ 9145.78 $&$ 6.252 $&$ -9.640 $&$ 5.22 $&$ $-$ $&\
Fe I &$ 9145.80 $&$ 5.693 $&$ -5.595 $&$ $-$ $&$ $-$ $&\
Fe I &$ 9146.13 $&$ 2.588 $&$ -2.804 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6247.14 $&$ 11.164 $&$ -4.093 $&$ 5.70 $&$ 0.14 $& blended\
Fe II &$ 6247.26 $&$ 11.237 $&$ -4.082 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6247.35 $&$ 6.209 $&$ -2.172 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6247.37 $&$ 10.909 $&$ -1.382 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6247.44 $&$ 12.276 $&$ -5.381 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6247.56 $&$ 3.892 $&$ -2.435 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6247.57 $&$ 5.956 $&$ -4.827 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6247.66 $&$ 11.591 $&$ -8.222 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6432.47 $&$ 10.448 $&$ -9.829 $&$ 5.72 $&$ 0.10 $& blended\
Fe II &$ 6432.68 $&$ 10.930 $&$ -1.236 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6432.68 $&$ 2.891 $&$ -3.687 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6456.38 $&$ 3.903 $&$ -2.185 $&$ 5.64 $&$ 0.12 $& blended\
Fe II &$ 6456.47 $&$ 11.547 $&$ -4.091 $&$ $-$ $&$ $-$ $&\
Fe II &$ 6516.08 $&$ 2.891 $&$ -3.432 $&$ 5.56 $&$ 0.08 $& blended\
Co I &$ 4233.71 $&$ 3.930 $&$ -3.106 $&$ 3.02 $&$ $-$ $&\
Co I &$ 4233.98 $&$ 0.000 $&$ -3.469 $&$ $-$ $&$ $-$ $&\
Co I &$ 4813.45 $&$ 2.871 $&$ -2.121 $&$ 3.28 $&$ $-$ $&\
Co I &$ 4813.47 $&$ 3.216 $&$ 0.050 $&$ $-$ $&$ $-$ $&\
Co 4 &$ 4813.71 $&$ 17.943 $&$ -6.686 $&$ $-$ $&$ $-$ $&\
Co II &$ 4840.10 $&$ 11.326 $&$ -2.988 $&$ 3.16 $&$ $-$ $&\
Co I &$ 4840.25 $&$ 3.170 $&$ 0.144 $&$ $-$ $&$ $-$ $&\
Co II &$ 4840.37 $&$ 3.459 $&$ -5.518 $&$ $-$ $&$ $-$ $&\
Co I &$ 6450.08 $&$ 2.137 $&$ -2.132 $&$ 2.96 $&$ $-$ $&\
Co I &$ 6450.25 $&$ 1.710 $&$ -1.698 $&$ $-$ $&$ $-$ $&\
Co I &$ 6872.39 $&$ 2.008 $&$ -1.589 $&$ 2.80 $&$ $-$ $&\
Co I &$ 7052.87 $&$ 1.956 $&$ -1.264 $&$ 2.58 $&$ $-$ $&\
Co I &$ 7084.98 $&$ 1.883 $&$ -1.018 $&$ 2.62 $&$ $-$ $&\
Ni I &$ 5711.88 $&$ 1.935 $&$ -2.270 $&$ 4.18 $&$ $-$ $& blended\
Ni I &$ 5715.07 $&$ 4.088 $&$ -0.352 $&$ 4.22 $&$ $-$ $& blended\
Ni II &$ 5748.25 $&$ 14.896 $&$ -0.858 $&$ 4.10 $&$ $-$ $&\
Ni I &$ 5748.35 $&$ 1.676 $&$ -3.260 $&$ $-$ $&$ $-$ $&\
Ni I &$ 5754.51 $&$ 3.941 $&$ -3.611 $&$ 4.38 $&$ $-$ $&\
Ni II &$ 5754.51 $&$ 14.733 $&$ -2.531 $&$ $-$ $&$ $-$ $&\
Ni I &$ 5754.65 $&$ 1.935 $&$ -2.330 $&$ $-$ $&$ $-$ $&\
Ni III &$ 5754.87 $&$ 22.952 $&$ -2.744 $&$ $-$ $&$ $-$ $&\
Ni II &$ 5846.77 $&$ 12.208 $&$ -3.583 $&$ 4.14 $&$ $-$ $&\
Ni I &$ 5846.99 $&$ 1.676 $&$ -3.210 $&$ $-$ $&$ $-$ $&\
Ni I &$ 5892.75 $&$ 4.154 $&$ -1.494 $&$ 4.32 $&$ $-$ $& blended\
Ni I &$ 5892.87 $&$ 1.986 $&$ -2.350 $&$ $-$ $&$ $-$ $&\
Ni II &$ 5892.94 $&$ 14.667 $&$ -2.139 $&$ $-$ $&$ $-$ $&\
Ni II &$ 5893.25 $&$ 15.008 $&$ -1.827 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6007.31 $&$ 1.676 $&$ -3.330 $&$ 4.14 $&$ $-$ $&\
Ni II &$ 6007.35 $&$ 13.080 $&$ -1.934 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6108.11 $&$ 1.676 $&$ -2.450 $&$ 4.06 $&$ $-$ $&\
Ni I &$ 6116.17 $&$ 4.089 $&$ -0.677 $&$ 4.26 $&$ $-$ $&\
Ni I &$ 6116.17 $&$ 4.266 $&$ -0.822 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6116.20 $&$ 15.026 $&$ -2.610 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6128.96 $&$ 1.676 $&$ -3.330 $&$ 4.18 $&$ $-$ $&\
Ni II &$ 6176.62 $&$ 15.008 $&$ -2.753 $&$ 4.16 $&$ $-$ $&\
Ni I &$ 6176.81 $&$ 4.088 $&$ -0.260 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6177.24 $&$ 1.826 $&$ -3.500 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6177.28 $&$ 11.969 $&$ -3.167 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6177.54 $&$ 4.236 $&$ -2.141 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6314.46 $&$ 12.904 $&$ 0.480 $&$ 3.62 $&$ $-$ $&\
Ni II &$ 6314.48 $&$ 14.903 $&$ -1.870 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6314.65 $&$ 1.935 $&$ -1.770 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6314.66 $&$ 4.154 $&$ -0.921 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6314.73 $&$ 14.906 $&$ -2.400 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6327.37 $&$ 12.457 $&$ -3.303 $&$ 4.14 $&$ $-$ $&\
Ni II &$ 6327.54 $&$ 14.837 $&$ -3.471 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6327.59 $&$ 1.676 $&$ -3.150 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6327.71 $&$ 14.685 $&$ -1.895 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6327.79 $&$ 15.022 $&$ -1.043 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6482.67 $&$ 14.912 $&$ -1.094 $&$ 3.94 $&$ $-$ $&\
Ni II &$ 6482.67 $&$ 14.912 $&$ -1.010 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6482.80 $&$ 1.935 $&$ -2.630 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6482.81 $&$ 14.744 $&$ -1.692 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6482.82 $&$ 14.744 $&$ -1.185 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6482.91 $&$ 14.912 $&$ -2.596 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6482.94 $&$ 14.912 $&$ -2.779 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6483.02 $&$ 14.912 $&$ -2.258 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6483.03 $&$ 14.912 $&$ -2.782 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6586.26 $&$ 14.739 $&$ -2.772 $&$ 4.10 $&$ $-$ $&\
Ni I &$ 6586.31 $&$ 1.951 $&$ -2.810 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6643.40 $&$ 14.731 $&$ -3.243 $&$ 4.26 $&$ $-$ $&\
Ni I &$ 6643.63 $&$ 1.676 $&$ -2.300 $&$ $-$ $&$ $-$ $&\
Ni II &$ 6767.54 $&$ 15.224 $&$ -2.851 $&$ 4.20 $&$ $-$ $&\
Ni I &$ 6767.77 $&$ 1.826 $&$ -2.170 $&$ $-$ $&$ $-$ $&\
Ni I &$ 6914.56 $&$ 1.951 $&$ -2.270 $&$ 4.12 $&$ $-$ $&\
Ni I &$ 7197.01 $&$ 1.935 $&$ -2.680 $&$ 4.28 $&$ $-$ $& blended\
Ni II &$ 7197.25 $&$ 14.334 $&$ -2.630 $&$ $-$ $&$ $-$ $&\
Ni I &$ 7197.39 $&$ 5.004 $&$ -3.128 $&$ $-$ $&$ $-$ $&\
Ni II &$ 7393.23 $&$ 15.059 $&$ -1.426 $&$ 4.70 $&$ $-$ $& blended\
Ni I &$ 7393.60 $&$ 3.606 $&$ -0.825 $&$ $-$ $&$ $-$ $&\
Ni II &$ 7408.96 $&$ 14.486 $&$ -3.296 $&$ 4.04 $&$ $-$ $& blended\
Ni II &$ 7409.01 $&$ 14.473 $&$ -2.127 $&$ $-$ $&$ $-$ $&\
Ni I &$ 7409.04 $&$ 5.497 $&$ -2.476 $&$ $-$ $&$ $-$ $&\
Ni I &$ 7409.25 $&$ 5.514 $&$ -1.790 $&$ $-$ $&$ $-$ $&\
Ni I &$ 7409.35 $&$ 3.796 $&$ -0.237 $&$ $-$ $&$ $-$ $&\
Ni I &$ 7414.50 $&$ 1.986 $&$ -2.570 $&$ 4.32 $&$ $-$ $& blended\
Cu I &$ 5782.13 $&$ 1.642 $&$ -1.720 $&$ 1.44 $&$ 0.26 $&\
Zn I &$ 4680.13 $&$ 4.006 $&$ -0.815 $&$ 2.82 $&$ 0.26 $& blended\
Zn I &$ 4722.15 $&$ 4.030 $&$ -0.338 $&$ 2.60 $&$ 0.26 $& blended\
Zn I &$ 4810.53 $&$ 4.078 $&$ -0.137 $&$ 2.64 $&$ 0.20 $&\
Ga I &$ 4172.04 $&$ 0.102 $&$ -0.270 $&$ 0.82 $&$ 0.44 $& very blended\
Sr II &$ 4077.71 $&$ 0.000 $&$ 0.167 $&$ 1.44 $&$ 0.34 $& very blended\
Sr I &$ 4607.33 $&$ 0.000 $&$ -0.570 $&$ 1.70 $&$ 0.16 $&\
Y II &$ 4682.32 $&$ 0.409 $&$ -1.510 $&$ 0.70 $&$ $-$ $& blended\
Y II &$ 4823.30 $&$ 0.992 $&$ -1.110 $&$ 0.56 $&$ $-$ $& blended\
Y II &$ 4854.86 $&$ 0.992 $&$ -0.380 $&$ 0.60 $&$ $-$ $&\
Y II &$ 4883.68 $&$ 1.084 $&$ 0.070 $&$ 0.50 $&$ $-$ $&\
Y I &$ 4900.08 $&$ 1.398 $&$ -0.360 $&$ 0.52 $&$ $-$ $& blended\
Y II &$ 4900.12 $&$ 1.033 $&$ -0.090 $&$ $-$ $&$ $-$ $&\
Y I &$ 4981.97 $&$ 1.983 $&$ -1.980 $&$ 0.44 $&$ $-$ $&\
Y II &$ 4982.13 $&$ 1.033 $&$ -1.290 $&$ $-$ $&$ $-$ $&\
Zr II &$ 3998.95 $&$ 0.559 $&$ -0.520 $&$ 1.08 $&$ $-$ $& blended\
Zr II &$ 4029.68 $&$ 0.713 $&$ -0.780 $&$ 0.98 $&$ $-$ $& blended\
Zr II &$ 4258.04 $&$ 0.559 $&$ -1.200 $&$ 0.74 $&$ $-$ $& blended\
Zr II &$ 4317.30 $&$ 0.713 $&$ -1.450 $&$ 0.92 $&$ $-$ $&\
Zr II &$ 4613.95 $&$ 0.972 $&$ -1.540 $&$ 0.98 $&$ $-$ $&\
Ba II &$ 4554.03 $&$ 0.000 $&$ 0.170 $&$ -0.34 $&$ $-$ $& blended\
Ba II &$ 4934.08 $&$ 0.000 $&$ -0.150 $&$ -0.32 $&$ $-$ $& blended\
Ba II &$ 5853.67 $&$ 0.604 $&$ -1.000 $&$ -0.48 $&$ $-$ $&\
Ba II &$ 6141.71 $&$ 0.704 $&$ -0.076 $&$ -0.44 $&$ $-$ $&\
Ba II &$ 6496.90 $&$ 0.604 $&$ -0.377 $&$ -0.14 $&$ $-$ $&\
La II &$ 3988.51 $&$ 0.403 $&$ 0.170 $&$ -1.34 $&$ $-$ $&\
La II &$ 3995.75 $&$ 0.173 $&$ -0.100 $&$ -1.00 $&$ $-$ $& blended\
La II &$ 4031.69 $&$ 0.321 $&$ -0.090 $&$ -0.96 $&$ $-$ $& very blended\
La II &$ 4086.71 $&$ 0.000 $&$ -0.070 $&$ -1.24 $&$ $-$ $& very blended\
La II &$ 4123.22 $&$ 0.321 $&$ 0.110 $&$ -0.54 $&$ $-$ $& very blended\
La II &$ 4333.75 $&$ 0.173 $&$ -0.060 $&$ -0.80 $&$ $-$ $& very blended\
La II &$ 4662.50 $&$ 0.000 $&$ -1.240 $&$ -1.12 $&$ $-$ $&\
La II &$ 4740.28 $&$ 0.126 $&$ -1.050 $&$ -1.06 $&$ $-$ $& blended\
La II &$ 4920.98 $&$ 0.126 $&$ -0.580 $&$ -1.46 $&$ $-$ $& blended\
La II &$ 4921.78 $&$ 0.244 $&$ -0.450 $&$ -1.26 $&$ $-$ $& blended\
Ce II &$ 4053.50 $&$ 0.000 $&$ -0.710 $&$ -0.62 $&$ $-$ $& very blended\
Ce II &$ 4053.53 $&$ 0.621 $&$ -2.740 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4053.57 $&$ 3.469 $&$ -2.910 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4053.59 $&$ 1.048 $&$ -1.490 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4418.78 $&$ 0.864 $&$ 0.280 $&$ -0.20 $&$ $-$ $&\
Ce II &$ 4483.89 $&$ 0.864 $&$ 0.150 $&$ -0.22 $&$ $-$ $& blended\
Ce III &$ 4484.06 $&$ 12.490 $&$ -1.860 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4486.88 $&$ 1.339 $&$ -1.200 $&$ -0.56 $&$ $-$ $&\
Ce II &$ 4486.91 $&$ 0.295 $&$ -0.260 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4487.00 $&$ 3.562 $&$ -1.100 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4487.12 $&$ 0.122 $&$ -3.350 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4487.17 $&$ 0.232 $&$ -2.540 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4539.42 $&$ 1.930 $&$ -0.810 $&$ -1.18 $&$ $-$ $& blended\
Ce II &$ 4539.50 $&$ 1.042 $&$ -3.320 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4539.58 $&$ 0.900 $&$ -1.060 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4539.61 $&$ 1.412 $&$ -0.910 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4539.75 $&$ 0.328 $&$ -0.020 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4539.85 $&$ 1.645 $&$ -2.050 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4562.28 $&$ 1.327 $&$ -2.120 $&$ -0.92 $&$ $-$ $&\
Ce II &$ 4562.36 $&$ 0.478 $&$ 0.230 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4572.28 $&$ 0.684 $&$ 0.290 $&$ -0.56 $&$ $-$ $& blended\
Ce II &$ 4628.16 $&$ 0.516 $&$ 0.200 $&$ -0.68 $&$ $-$ $&\
Ce II &$ 4628.19 $&$ 1.194 $&$ -3.280 $&$ $-$ $&$ $-$ $&\
Ce II &$ 4628.24 $&$ 1.366 $&$ -0.430 $&$ $-$ $&$ $-$ $&\
Pr II &$ 4408.82 $&$ 0.000 $&$ -0.278 $&$ <-1.30 $&$ $-$ $& blended\
Pr II &$ 4408.82 $&$ 0.000 $&$ -0.278 $&$ <-1.10 $&$ $-$ $& blended\
Nd II &$ 4004.00 $&$ 0.064 $&$ -0.570 $&$ -0.58 $&$ $-$ $& blended\
Nd II &$ 4021.33 $&$ 0.321 $&$ -0.100 $&$ -0.74 $&$ $-$ $&\
Nd II &$ 4351.28 $&$ 0.182 $&$ -0.610 $&$ -0.40 $&$ $-$ $& very blended\
Nd II &$ 4446.38 $&$ 0.205 $&$ -0.350 $&$ -0.84 $&$ $-$ $&\
Nd II &$ 4506.58 $&$ 0.064 $&$ -1.040 $&$ -0.88 $&$ $-$ $& blended\
Nd II &$ 4811.34 $&$ 0.064 $&$ -1.015 $&$ -0.38 $&$ $-$ $&\
Nd II &$ 4825.48 $&$ 0.182 $&$ -0.420 $&$ -0.62 $&$ $-$ $&\
Nd II &$ 4859.03 $&$ 0.321 $&$ -0.440 $&$ -0.62 $&$ $-$ $&\
Nd II &$ 4959.12 $&$ 0.064 $&$ -0.800 $&$ -0.44 $&$ $-$ $&\
Nd II &$ 6900.44 $&$ 0.000 $&$ -1.542 $&$ -0.60 $&$ $-$ $&\
Sm II &$ 4318.93 $&$ 0.277 $&$ -0.250 $&$ -1.02 $&$ 0.50 $& blended\
Sm II &$ 4434.32 $&$ 0.378 $&$ -0.070 $&$ -0.66 $&$ 0.32 $& blended\
Sm II &$ 4467.34 $&$ 0.659 $&$ 0.150 $&$ -1.04 $&$ 0.26 $& blended\
Eu II &$ 3907.11 $&$ 0.207 $&$ 0.170 $&$ -1.72 $&$ 0.38 $& very blended\
Eu II &$ 4129.72 $&$ 0.000 $&$ 0.220 $&$ -1.44 $&$ 0.38 $& very blended\
Eu II &$ 4205.04 $&$ 0.000 $&$ 0.210 $&$ -1.66 $&$ 0.66 $& very blended\
Eu II &$ 4435.58 $&$ 0.207 $&$ -0.110 $&$ -1.22 $&$ 0.40 $& very blended\
Eu II &$ 6645.06 $&$ 1.380 $&$ 0.120 $&$ <-1.22 $&$ $-$ $&\
Gd II &$ 4251.73 $&$ 0.382 $&$ -0.220 $&$ <-0.50 $&$ $-$ $&\
Tb II &$ 4752.53 $&$ 0.000 $&$ -0.550 $&$ <-1.38 $&$ $-$ $& blended\
Dy II &$ 3944.68 $&$ 0.000 $&$ 0.110 $&$ -0.86 $&$ 0.50 $& very blended\
Dy II &$ 3983.65 $&$ 0.538 $&$ -0.310 $&$ <-0.38 $&$ $-$ $& very blended\
Dy II &$ 4103.31 $&$ 0.103 $&$ -0.380 $&$ -1.00 $&$ 0.56 $& very blended\
Dy II &$ 4449.70 $&$ 0.000 $&$ -1.030 $&$ <-0.66 $&$ $-$ $& very blended\
Er II &$ 3896.23 $&$ 0.055 $&$ -0.241 $&$ <-0.60 $&$ $-$ $& very blended\
Pb I &$ 4057.81 $&$ 1.320 $&$ -0.170 $&$ <0.92 $&$ $-$ $& very blended\
Th II &$ 4250.34 $&$ 0.557 $&$ 0.000 $&$ <-1.20 $&$ $-$ $& blended\
\[table:linelist\]
[l r r r r r r]{} X & $\log \epsilon_\odot$ & $N_\textsl{X}$ & $\log \epsilon$ & \[X/Fe\] & $\delta_\textsl{noise}$ & $\delta_\textsl{total}$\
Fe I &$ 7.50 $&$ 49 $&$ 5.47 $&$ -2.03\tablefootmark{a} $&$ 0.02 $&$ 0.10 $\
Fe II &$ 7.50 $&$ 4 $&$ 5.64 $&$ -1.86\tablefootmark{a} $&$ 0.10 $&$ 0.14 $\
Li &$ 1.10 $&$ 1 $&$ <0.17 $&$ <1.10 $&$ $-$ $&$ $-$ $\
C &$ 8.52 $&$ $-$ $&$ 7.00 $&$ 0.51 $&$ 0.09 $&$ 0.10 $\
N &$ 7.92 $&$ $-$ $&$ 7.07 $&$ 1.18 $&$ 0.20 $&$ 0.20 $\
O &$ 8.83 $&$ 2 $&$ 7.33 $&$ 0.53 $&$ 0.18 $&$ 0.19 $\
Na &$ 6.33 $&$ 3 $&$ 3.81 $&$ -0.49 $&$ 0.24 $&$ 0.24 $\
Mg &$ 7.58 $&$ 5 $&$ 5.97 $&$ 0.42 $&$ 0.04 $&$ 0.06 $\
Al &$ 6.47 $&$ 1 $&$ 3.62 $&$ -0.82 $&$ 0.80 $&$ 0.80 $\
Si &$ 7.55 $&$ 7 $&$ 5.77 $&$ 0.25 $&$ 0.09 $&$ 0.11 $\
S &$ 7.33 $&$ 2 $&$ 5.44 $&$ 0.14 $&$ 0.29 $&$ 0.29 $\
K &$ 5.12 $&$ 2 $&$ 3.45 $&$ 0.36 $&$ 0.07 $&$ 0.08 $\
Ca &$ 6.36 $&$ 20 $&$ 4.49 $&$ 0.16 $&$ 0.02 $&$ 0.04 $\
Sc &$ 3.17 $&$ 9 $&$ 1.10 $&$ -0.04 $&$ 0.04 $&$ 0.06 $\
Ti &$ 5.02 $&$ 35 $&$ 3.12 $&$ 0.13 $&$ 0.02 $&$ 0.07 $\
V &$ 4.00 $&$ 10 $&$ 2.01 $&$ 0.04 $&$ 0.04 $&$ 0.06 $\
Cr &$ 5.67 $&$ 16 $&$ 3.53 $&$ -0.11 $&$ 0.07 $&$ 0.08 $\
Mn &$ 5.39 $&$ 9 $&$ 3.13 $&$ -0.23 $&$ 0.06 $&$ 0.07 $\
Co &$ 4.92 $&$ 7 $&$ 2.92 $&$ 0.03 $&$ 0.10 $&$ 0.11 $\
Ni &$ 6.25 $&$ 22 $&$ 4.18 $&$ -0.04 $&$ 0.04 $&$ 0.07 $\
Cu &$ 4.21 $&$ 1 $&$ 1.44 $&$ -0.74 $&$ 0.26 $&$ 0.26 $\
Zn &$ 4.60 $&$ 3 $&$ 2.68 $&$ 0.11 $&$ 0.23 $&$ 0.23 $\
Ga &$ 2.88 $&$ 1 $&$ 0.82 $&$ -0.03 $&$ 0.44 $&$ 0.44 $\
Sr &$ 2.97 $&$ 2 $&$ 1.65 $&$ 0.71 $&$ 0.20 $&$ 0.20 $\
Y &$ 2.24 $&$ 6 $&$ 0.55 $&$ 0.34 $&$ 0.04 $&$ 0.06 $\
Zr &$ 2.60 $&$ 5 $&$ 0.94 $&$ 0.37 $&$ 0.06 $&$ 0.07 $\
Ba &$ 2.13 $&$ 5 $&$ -0.34 $&$ -0.44 $&$ 0.06 $&$ 0.08 $\
La &$ 1.17 $&$ 10 $&$ -1.08 $&$ -0.22 $&$ 0.09 $&$ 0.09 $\
Ce &$ 1.58 $&$ 8 $&$ -0.62 $&$ -0.17 $&$ 0.12 $&$ 0.13 $\
Pr &$ 0.71 $&$ 1 $&$ <-1.10 $&$ <0.22 $&$ $-$ $&$ $-$ $\
Nd &$ 1.50 $&$ 10 $&$ -0.61 $&$ -0.08 $&$ 0.05 $&$ 0.07 $\
Sm &$ 1.01 $&$ 3 $&$ -0.91 $&$ 0.11 $&$ 0.32 $&$ 0.32 $\
Eu &$ 0.51 $&$ 4 $&$ -1.49 $&$ 0.03 $&$ 0.42 $&$ 0.42 $\
Gd &$ 1.12 $&$ 1 $&$ <-0.50 $&$ <0.41 $&$ $-$ $&$ $-$ $\
Tb &$ 0.35 $&$ 1 $&$ <-1.38 $&$ <0.30 $&$ $-$ $&$ $-$ $\
Dy &$ 1.14 $&$ 2 $&$ -0.92 $&$ -0.03 $&$ 0.53 $&$ 0.53 $\
Er &$ 0.93 $&$ 1 $&$ <-0.60 $&$ <0.50 $&$ $-$ $&$ $-$ $\
Pb &$ 1.95 $&$ 1 $&$ <0.92 $&$ <1.00 $&$ $-$ $&$ $-$ $\
Th &$ 0.09 $&$ 1 $&$ <-1.20 $&$ <0.74 $&$ $-$ $&$ $-$ $\
Abundance measurements
======================
All analysis was carried out using the spectral synthesis code TURBOSPEC developed by Bernand Plez [@AlvarezPlez1998; @Plez2012]. Because of the evolutionary stage of ET0097, its relatively low temperature and its high C and N abundances, most of the observed wavelength range is covered with CN and/or CH molecular lines. Synthetic spectra are therefore necessary to include the effects of these lines in the abundance evaluation and to distinguish between lines with different degrees of blending. The line list used was chosen to minimize the use of heavily blended lines, and when possible only lines with minimal or no blending were used (see Table \[table:linelist\]).
The stellar atmosphere models are adopted from MARCS [@Gustafsson2008] for stars with standard composition, 1D and assuming local thermodynamic equilibrium (LTE), interpolated to match the exact stellar parameters for ET0097. Parameters for the atomic lines are adopted from the DREAM data base [@Biemont1999], extracted via VALD (@Kupka1999 and references therein). The molecular parameters for C$_2$ come from M. Querci (private communication), and are described in @Querci1972. For CH molecular lines, we use the data from @Plez2007. The molecular parameters for CN are from T. Masseron (private communication), derived with similar methods and lab data as @Brooke2014 and @Sneden2014.
The results of the abundance analysis are listed in Table \[table:abundances\]. To maintain consistency with Hill et al. (in prep.), the solar values used are from @GrevesseSauval1998, and where data from the literature is included, it is shifted to match this scale.
Iron
----
Special care was taken to ensure that the measured Fe lines were not severely blended with CN, C$_2$, or CH molecular lines. Only Fe I lines that changed by less than 0.1 dex, when both the assumed carbon and nitrogen abundances were either increased or decreased by 0.2 dex, were used. Increasing both elements by this value had a strong effect on the CN line strength, so the real effect from the uncertainty on C and N abundance measurements should be much less than 0.1 dex on the selected Fe I lines and, in all cases, it is less than the statistical errors on the individual lines. In total, 49 Fe I lines could be used in the wavelength range 5600-9200 , see Table \[table:linelist\]. All Fe I lines in the bluer part of the spectrum were too blended to comply with the rather strict criteria for unblended lines.
For Fe II, no lines were found that fulfill the criteria, so the four least blended Fe II lines were used for the measurement.
\[!ht\] ![Measured oxygen lines, $\log \epsilon (\text{O})_{6300\text{\AA}}~=~7.30\pm0.14$ and $\log \epsilon (\text{O})_{6364\text{\AA}}~=~7.46\pm0.30$. Solid red lines show best fits, and dashed blue lines show upper and lower limits of error bars. Cases where \[O I\] lines have been removed from the linelist are shown with solid gray lines. The strength of CN molecular lines is very sensitive to oxygen abundance. Adopting a lower O abundance will make CN lines in the synthetic spectrum stronger, assuming fixed values of C and N. []{data-label="O"}](Oplot.eps "fig:"){width="\hsize"}
![Upper panel shows strong CN molecular band at 4215 (B-X system), while lower panel shows an example of the CN molecular band seen in the redder part of the spectrum. In both cases, solid red lines show the best fit of $\log \epsilon (\text{N})~=~7.07$, dashed blue lines show nitrogen values $\pm$0.20 from that fit, and solid gray lines what the spectrum would look like without any molecular CN lines. Note that the scale on the y-axis is not the same for both panels. []{data-label="CN"}](CNplot.eps){width="\hsize"}
Oxygen, carbon, and nitrogen
----------------------------
The oxygen abundance was derived from the forbidden \[O I\] lines at 6300 and 6364 Å, see Fig. \[O\]. Both lines were slightly blended with weak CN molecular lines, but changing the adopted C and N values by $\pm$0.20 around the best fit, changed the measured oxygen abundances only by $\sim$0.05 dex. The abundance measurements of the lines agree within the errorbars, $\log \epsilon (\text{O})_{6300\text{\AA}}~=~7.30~\pm~0.14$ and $\log \epsilon (\text{O})_{6364\text{\AA}}~=~7.46~\pm~0.30$. The final value of $\log \epsilon (\text{O})~=~7.33\pm0.18$ was adopted by weighting the two \[O I\] lines with their errors.
The \[O I\] lines of the triplet at 7774 were too weak to be accurately measured, considering the noise and slight blending, and they were therefore not included in the measurement.
The carbon abundance for ET0097 was determined by fitting three different molecular bands in regions of 20 Å. Parts of these bands are shown in Fig. \[CH\]. In some wavelength ranges, the molecular lines were saturated and in others very weak. Therefore, we only used regions where the $\chi^2$ of the fit was sensitive to the assumed C abundance. The CH G-band at $\sim$4300 , spreading throughout the range $\sim$4200-4400 Å, resulted in $\log \epsilon (\text{C})_{\text{CH:}4300\text{\AA}}=6.98$. The C$_2$ band around $\sim$4700 gave a result 0.20 dex lower, $\log \epsilon (\text{C})_{\text{C2:}4700\text{\AA}}=6.78$. A relatively weak CH band (A-X system) at $\sim$4850 Å$ $ was also used to measure carbon, yielding a value 0.25 dex higher than the stronger CH-band, $\log \epsilon (\text{C})_{\text{CH:}4850\text{\AA}}=7.23$. The final measured value of $\log \epsilon (\text{C})=7.00\pm0.10$ was adopted by averaging both CH bands and the C$_2$ band, weighting them with the size of the region available for the measurements.
The CN band at 4215 Å (B-X system) appeared clearly and without severe saturation (see Fig. \[CN\]). A big portion of the observed spectrum for ET0097 was covered with CN lines, due to the relatively low temperature of the star and the high C and N abundances. The nitrogen was therefore measured using both the band at 4215 and CN molecular lines in the wavelength range $\sim$6200-9400 (A-X system), giving the final result of $\log \epsilon (\text{N})=7.07\pm0.20$. Regions where the $\chi^2$ of the fit was insensitive to the nitrogen abundance were excluded. The nitrogen measurements, which cover 104 regions of 20 , show no trend with wavelength and are very consistent with a low scatter, $\sigma=0.05$. The measurements in the redder part of the spectrum and the CN band at 4215 are in perfect agreement, see Fig. \[CN\], indicating that the CN molecular parameters in the red are reliable. The uncertainty in the measurement from the CN band in the blue is higher than in the red, mostly due to the uncertain continuum determination. Using the larger wavelength range, however, the statistical errors for the nitrogen become negligible, and the real uncertainties come from errors in the carbon and oxygen measurements.
The B-X system of CN at 3888 was extremely strong in this star and wiped out all continuum points until the bluest end of the spectrum at $\sim$3770 . Thus, it was not included in the abundance measurement of nitrogen. However, as far as rough continuum estimation allowed, it was consistent with the result obtained using the other CN molecular bands.
Because CO locks away a sizable fraction of the C available in the star, both CH bands and the C$_2$ band are sensitive to the oxygen value, and the nitrogen abundance measured from the CN molecular bands is sensitive both to the oxygen and the carbon abundances. Therefore, the oxygen-carbon-nitrogen measurements were iterated several times to minimize the influence of the errors of one element on the other elements as much as possible.
![Example of the $^{13}$C features used for evaluating the $^{12}$C/$^{13}$C isotope ratio. Solid gray line shows the synthetic spectrum assuming $\log^{12}\text{C}/^{13}\text{C}=1.3$, value typical for unmixed giants [@Spite2006], solid red line shows $\log^{12}\text{C}/^{13}\text{C}=0.8$, and dashed blue lines show ratios $\pm$0.2 from that value. []{data-label="C13"}](C12plot.eps){width="\hsize"}
Indicators of mixing, $^{\textsl{12}}$C/$^{\textsl{13}}$C ratio and Li {#sec:mixing}
----------------------------------------------------------------------
When a star moves along the RGB, it undergoes a second episode of mixing (at the so-called RGB-bump) that lowers the carbon abundance at its surface, and increases the nitrogen. This upper RGB phase is reached at the luminosity $\log L_\star/L_\odot\sim2.2$, for stars in the metallicity range $-2.0~\lesssim~$\[Fe/H\]$~\lesssim-1.0$ [@Gratton2000] and when $\log L_\star/L_\odot\sim2.6$ for $\text{[Fe/H]}~\lesssim-2.5$ [@Spite2006]. Before an RGB star reaches this high luminosity, its C and N abundances do not change significantly during the star’s evolution [@Gratton2000].
Thus, a low C value and high N abundance are usually good indicators of this second episode of mixing. With the high luminosity of $\log L_\star/L_\odot=3.1\pm0.1$, ET0097 has a high nitrogen abundance, $\text{[N/Fe]}=1.18\pm0.20$, but also high $\text{[C/Fe]}=0.51\pm0.10$, so if it has undergone this mixing, it would be inherently more carbon-rich.
During mixing, the most fragile element, Li, is destroyed in the deeper layers of the star, so low Li abundances indicate that mixing has occured. For main sequence (MS) stars, the Li abundance generally lies along the so-called Spite plateau, $\log \epsilon(\text{Li})\approx2.2$. When a star enters the lower RGB phase, the Li at the surface is diluted with material coming from the deeper layers, and the Li abundance stabilizes at an average value of $\log\epsilon(\text{Li}) \approx 1$ along the lower RGB. However, when the second episode of mixing occurs, practically all the remaining Li is destroyed, yielding a very low value of $\log\epsilon(\text{Li}) \leq 0$ [@Gratton2000].
The lithium abundance for ET0097 was obtained by measuring the resonance doublet at 6707 , yielding a value of $\log \epsilon(\text{Li})=-0.12$, but due to the weakness of the line an upper limit of $\log \epsilon(\text{Li})<0.17$ was adopted, consistent with most of the Li being destroyed.
Another signature of deep mixing is a low isotope ratio, $\log ^{12}$C/$^{13}$C < 1.0 [@Spite2006]. To measure $^{12}$C/$^{13}$C, the total carbon abundance was kept constant and the $^{13}$C lines were fitted with synthetic spectra with different isotope ratios, see Fig. \[C13\]. The ratio was determined to be $\log^{12}\text{C}/^{13}\text{C}=0.77\pm0.03$.
The main signatures of mixing in a high luminosity RGB star are thus present in ET0097, with $\log L_\star/L_\odot=3.1$, $\text{[N/Fe]}=1.18$, $\log^{12}\text{C}/^{13}\text{C}=0.77$ and a best fit of $\log \epsilon(\text{Li})~=~-0.12$. We therefore conclude that this star has undergone mixing, and that its carbon abundance was higher at the earlier evolutionary stages.
By assuming most of the nitrogen present in the star was converted from carbon, we get an estimate of the original C abundance at the surface of the star: $\text{[C/Fe]}\approx0.8$. In the Galactic halo, mixing in RGB stars is observed to lower the surface C abundance on average by $\sim$0.5 [@Gratton2000; @Spite2005], with a scatter of $\sim$0.25 dex (@Spite2005, for $-3\leq\text{[Fe/H]}\leq-2$), consistent with the correction made here. An online tool to correct for mixing has been provided by @Placco2014, which gives $\text{[C/Fe]}\approx0.9$ as the initial abundance for ET0097, while here we adopt the more conservative value of $\text{[C/Fe]}\approx0.8$. The estimate for the star’s initial carbon enhancement thus falls under the classical definition of a CEMP ($\text{[C/Fe]}\geq0.7$) and the same conclusion is reached using the uncorrected value with the luminosity dependent definition of CEMP stars, from @Aoki2007.
Alpha elements
--------------
The main production sites of the alpha elements (made up of alpha particles) are normal core collapse Type II Supernovae. Early in the star formation history of any galaxy, Type II SN are believed to be the main contributors of metals, so the early ISM holds the imprint of their yields, and stars formed at earlier times typically show an enhancement in \[$\alpha$/Fe\] (e.g., $\text{[Mg/Fe]}\gtrsim0.3$). About 1-2 Gyr after the first SN II, Type Ia Supernova start to contribute (e.g., @deBoer2012), and so \[$\alpha$/Fe\] starts to decrease with increasing \[Fe/H\]. This so-called ‘knee’, happens around $\text{[Fe/H]}\sim-1.7$ in Sculptor [@Tolstoy2009].
In ET0097, the abundance measurements for Mg, Ca, and Ti were relatively straight forward, all having many lines that did not show any signs of blending from atomic or molecular lines, see Table \[table:linelist\]. On the other hand, all lines for Si were either weak and/or slightly blended. Though the blending was accounted for in the synthetic spectra, it added to the uncertainty of the measurements. However, there were many Si lines available in the observed wavelength range, and the final abundance was measured from the seven best lines. Some of these were weak and unblended, others slightly stronger but were blended. The number of lines and the reasonable scatter makes the result robust.
There were three S lines in the reddest part of the spectrum $\sim$9200 , which were affected by reasonably strong CN lines, see the lower panel of Fig. \[CN\]. Two S lines were relatively unblended, both of them yielding very similar results (agreeing within 0.05 dex). The line at 9237.5 was not used, however, it was consistent with the value obtained for the other two lines.
All alpha elements in ET0097 show an overabundance relative to iron, $\text{[$\alpha$/Fe]}>0$, similar to what is seen for other stars in Sculptor and in the Galactic halo at comparable iron abundances, $\text{[Fe/H]}\approx-2$.
Odd-Z elements
--------------
The abundances of Na, Al, and K were determined from resonance lines that are very sensitive to non-local thermodynamic effects (NLTE) effects. However, similar correction factors are expected for RGB stars with comparable metallicity, $T_\textsl{eff}$ and $\log g$, so when comparing abundances of ET0097 with similar stars in the Galactic halo or Sculptor, similar NLTE corrections can be applied. @Cayrel2004 adopted a correction of $-0.50$ dex for Na in giants, $+0.65$ dex for Al and $-0.35$ dex for K. Here, the same lines were used for these elements, so similar corrections can be made to the LTE values listed in Table \[table:abundances\]. More detailed NLTE calculations for stars with similar stellar parameters are presented in @Andrievsky2007 [@Andrievsky2008; @Andrievsky2010].
The Na abundance was determined from the D resonance lines at 5890 and 5896 , and a weaker line at 8183.3 . None of these lines showed any signs of blending and gave the weighted average of $\log \epsilon (\text{Na})=3.81\pm0.24$.
Two Al lines were visible in the spectrum, the resonance doublet at 3944 and 3961.5 . Both lines were heavily blended, in particular the line at 3944 , so it was not included in the measurement. The result remains rather uncertain, because of the blending and the difficult continuum evaluation in this region, $\log \epsilon (\text{Al})=3.62\pm0.80$, which is consistent with the line at 3944 .
The K abundance was determined from two strong lines at 7665 and 7699 . Both gave consistent results with a difference of only 0.02 dex. The adopted abundance is therefore $\log \epsilon (\text{K})=3.45\pm0.07$.
Both Sc and V had many unblended lines available, giving $\log \epsilon (\text{Sc})=1.10\pm0.04$ and $\log \epsilon (\text{V})=2.01\pm0.04$.
Iron-peak elements
------------------
In general, Fe-peak elements are believed to be created in supernova explosions. The elements Cr, Mn, Co, and Ni all had many available lines in the observed wavelength range, making it possible to discard those that were blended with molecular lines.
Only one rather weak line was available for Cu, at 5782 , in a region of the spectrum that was relatively free of molecular lines. The line barely showed any blending, giving $\log \epsilon (\text{Cu})=1.44\pm0.26$. Three lines were available for Zn. They only showed minor blending and agree well with each other, $\log \epsilon (\text{Zn})=2.68\pm0.23$.
Heavy elements
--------------
Abundances for three elements of the lighter $n$-capture elements were measured: Sr, Y and Zr. Two lines were observed for Sr, one very strong and blended Sr II line at 4078 Å, giving the result $\log \epsilon (\text{Sr})_{4078\text{\AA}}=1.44\pm0.34$, and a Sr I line free of blending at 4607 , $\log\epsilon(\text{Sr})_{4607\text{\AA}}=1.70\pm0.16$. The weighted average of the two yields $\text{[Sr/Fe]}=0.71\pm 0.20$. Yttrium had six lines in the wavelength range, all showing only minor blending and very little scatter, giving the final value $\text{[Y/Fe]}=0.34\pm0.06$. Five lines are used for the measurement of Zr, three of them slightly blended, giving the result $\text{[Zr/Fe]}=0.37\pm0.07$. The lighter neutron-capture elements in ET0097 therefore all show overabundance with respect to iron, $\text{[Sr,Y,Zr/Fe]}>0.3$.
The heavier $n$-capture elements Ba, La, Ce, and Nd, all had five or more lines available. Of those, La and Ce showed more scatter between lines, $\sigma\sim 0.30$ dex (compared to $\sigma\sim 0.15$ of Ba and Nd), which was to be expected since many of the measured lines for these elements were blended and/or weak. Three Sm lines were measured with a scatter between lines, $\sigma=0.23$, and the final result is $\text{[Sm/Fe]}=0.11\pm0.32$. Eu was difficult to measure in this star since the four lines available were all heavily blended. However, all lines agree reasonably well with each other, with a scatter between lines of $\sigma=0.23$, giving the rather low value $\text{[Eu/Fe]}=0.03\pm0.42$. No trace of the Eu line at 6645 was seen, but an upper limit of $\text{[Eu/Fe]}<0.30$ was determined, which is consistent with the four detected lines. The Dy abundance was derived from two weak and blended lines at 3945 Å$ $ and 4103 Å, $\text{[Dy/Fe]}=-0.03\pm0.53$. This is consistent with the best fits of two other weak lines at 3984 and 4450 . Only upper limits could be determined for those: $\text{[Dy/Fe]}_{3984\text{\AA}}<0.51$ and $\text{[Dy/Fe]}_{4450\text{\AA}}<0.23$. There were no detectable lines for the elements Pr, Gd, Tb, Er, and Pb, giving upper limits for these elements in the range $\text{[X/Fe]}\sim$0.2-1.0 dex, which excludes the possibility of extreme overabundances.
With low abundances for both the main $s$-process elements ($\text{[Ba/Fe]}<0$) and main $r$-process elements ($\text{[Eu/Fe]}<0.5$), ET0097 classifies as a CEMP-no star.
Error analysis
==============
To evaluate the statistical uncertainties in the abundance determination of a line, $\delta_{noise,i}$, the noise in line-free regions neighboring the line was measured. The error was then determined as when the $\chi^2$ of the fit became larger than that of the noise. Since the spectrum was dominated by molecular lines, line-free regions were not always available. Although the molecular bands were reasonably well fitted as a whole with the synthetic spectra, in some regions individual lines were not. In those cases, the typical deviation of the spectrum from the best synthetic fit was measured and included in the noise estimate.
The individual lines showed different degrees of blending, and for elements with fewer than five measured lines, this was accounted for by weighting the different measurements with their errors as follows:
$$\log \epsilon (X) = \frac{\sum\limits_{i=1}^{N_X} \log \epsilon (X)_i \cdot w_i}{\sum\limits_{i=1}^{N_X} w_i}$$
The sum runs over $N_X$ lines and the weights of individual lines were defined as: $$w_i=\frac{1}{\delta^2_\textsl{noise,i}}$$ where $\delta_\textsl{noise,i}$ is the statistical uncertainty of the abundance measurement of line $i$. For elements with five or more lines, this was not necessary and normal averages were used to determine the final abundances.\
The final error for elements with four or fewer measured lines was calculated as follows:
$$\delta_{\textsl{noise}} = \sqrt{\frac{N_X}{\sum_i w_i}}$$
For elements with five or more lines, the total error from the noise was defined from the scatter of the measurements:
$$\label{stat}
\delta_{\textsl{noise}} = \frac{\sigma}{\sqrt{N_X}}$$
Special care was taken in the evaluation of the errors on C and N abundances. These elements were measured over regions of 20 and the final value was determined from the average of all measured regions. Measurement errors were calculated using Eq. (\[stat\]). For C this gave an error of $\delta_C=0.05$ and $\delta_N=0.01$ for N.
However, the CH and C$_2$ lines are sensitive to O values, so the effect of the oxygen error on these abundances was measured and included, yielding a total statistical error of $\delta_{\textsl{noise,C}}=0.09$.
The CN molecular lines that were used to measure N are sensitive to both C and O abundances. Taking the effect of the uncertainties of these elements on the N measurements into account, the final error is $\delta_{\textsl{noise,N}}=0.20$.
The systematic errors coming from the uncertainties of the stellar parameters, $T_\textsl{eff}$, $\log g$ and $v_t$, were measured to be $\Delta\text{[Fe/H]}_\text{sp}=0.10$, and ranging from 0.02 to 0.06 dex for $\Delta$\[X/Fe\]$_\text{sp}$ (depending on the element). They were added quadratically to the $\delta_{\textsl{noise}}$ to obtain the adopted error, $$\label{eq:totalerr}
\delta_{\textsl{total}}(\text{[X/Fe]})=\sqrt{ \delta_{\textsl{noise}}(\text{X})^2+\delta_{\textsl{noise}}(\text{Fe})^2+ \Delta\text{[X/Fe]}_\text{sp}^2}$$
[l c c c r r r]{}
\
Star & RA & DEC &$\log L_\star/L_\odot$ & \[Fe/H\] & \[C/Fe\]$_\text{est}$ & \[C/Fe\]$_\text{lim}$\
\
Star & RA & DEC &$\log L_\star/L_\odot$ & \[Fe/H\] & \[C/Fe\]$_\text{est}$ & \[C/Fe\]$_\text{lim}$\
ET0024 & 1 00 34.04 & $-33$ 39 04.6 &$ 3.26 $&$ -1.24 $&$ -0.94 $&$ -0.74 $\
ET0026 & 1 00 12.76 & $-33$ 41 16.0 &$ 3.08 $&$ -1.80 $&$ -0.96 $&$ -0.74 $\
ET0027 & 1 00 15.37 & $-33$ 39 06.2 &$ 3.10 $&$ -1.50 $&$ -0.94 $&$ -0.70 $\
ET0028 & 1 00 17.77 & $-33$ 35 59.7 &$ 3.11 $&$ -1.22 $&$ -0.98 $&$ -0.78 $\
ET0031 & 1 00 07.57 & $-33$ 37 03.9 &$ 2.98 $&$ -1.68 $&$ -0.88 $&$ -0.58 $\
ET0033 & 1 00 20.29 & $-33$ 35 34.5 &$ 2.98 $&$ -1.77 $&$ -0.90 $&$ -0.60 $\
ET0043 & 1 00 13.95 & $-33$ 36 39.2 &$ 2.84 $&$ -1.24 $&$ -1.04 $&$ -0.88 $\
ET0048 & 0 59 55.63 & $-33$ 33 24.6 &$ 3.18 $&$ -1.90 $&$ -0.70 $&$ -0.24 $\
ET0051 & 0 59 46.41 & $-33$ 41 23.5 &$ 3.18 $&$ -0.92 $&$ -1.12 $&$ -1.08 $\
ET0054 & 0 59 56.60 & $-33$ 36 41.7 &$ 3.00 $&$ -1.81 $&$ -0.80 $&$ -0.44 $\
ET0057 & 0 59 54.21 & $-33$ 40 27.2 &$ 3.02 $&$ -1.33 $&$ -0.88 $&$ -0.62 $\
ET0059 & 0 59 38.11 & $-33$ 35 08.0 &$ 2.98 $&$ -1.53 $&$ -1.17 $&$ -1.13 $\
ET0060 & 0 59 37.74 & $-33$ 36 00.0 &$ 2.98 $&$ -1.56 $&$ -0.94 $&$ -0.72 $\
ET0062 & 0 59 47.21 & $-33$ 33 36.9 &$ 2.91 $&$ -2.27 $&$ <-0.50 $&$ <0.00 $\
ET0063 & 0 59 37.22 & $-33$ 37 10.5 &$ 2.99 $&$ -1.18 $&$ -0.96 $&$ -0.76 $\
ET0064 & 0 59 41.40 & $-33$ 38 47.0 &$ 2.96 $&$ -1.38 $&$ -0.88 $&$ -0.58 $\
ET0066 & 1 00 03.60 & $-33$ 39 27.1 &$ 2.94 $&$ -1.30 $&$ -1.00 $&$ -0.84 $\
ET0067 & 0 59 37.00 & $-33$ 30 28.4 &$ 2.90 $&$ -1.65 $&$ -0.96 $&$ -0.72 $\
ET0069 & 0 59 40.46 & $-33$ 35 53.8 &$ 2.86 $&$ -2.11 $&$ -0.82 $&$ -0.44 $\
ET0071 & 0 59 58.27 & $-33$ 41 08.7 &$ 2.92 $&$ -1.35 $&$ -0.98 $&$ -0.78 $\
ET0073 & 0 59 53.99 & $-33$ 37 42.1 &$ 2.82 $&$ -1.53 $&$ -0.90 $&$ -0.62 $\
ET0083 & 0 59 11.83 & $-33$ 41 25.3 &$ 2.97 $&$ -1.97 $&$ -0.76 $&$ -0.34 $\
ET0094 & 0 59 20.65 & $-33$ 48 56.6 &$ 3.18 $&$ -1.86 $&$ -0.82 $&$ -0.48 $\
ET0095 & 0 59 20.80 & $-33$ 44 04.8 &$ 3.09 $&$ -2.16 $&$ -0.80 $&$ -0.42 $\
ET0103 & 0 59 18.85 & $-33$ 42 17.3 &$ 2.95 $&$ -1.21 $&$ -1.04 $&$ -0.90 $\
ET0104 & 0 59 15.14 & $-33$ 42 54.6 &$ 2.90 $&$ -1.62 $&$ -0.82 $&$ -0.46 $\
ET0109 & 0 59 28.29 & $-33$ 42 07.2 &$ 3.24 $&$ -1.85 $&$ -0.76 $&$ -0.40 $\
ET0112 & 0 59 52.27 & $-33$ 44 54.8 &$ 3.11 $&$ -2.04 $&$ -0.72 $&$ -0.28 $\
ET0113 & 0 59 55.68 & $-33$ 46 40.1 &$ 3.08 $&$ -2.18 $&$ -0.74 $&$ -0.30 $\
ET0121 & 1 00 00.49 & $-33$ 49 35.8 &$ 2.94 $&$ -2.35 $&$ -0.83 $&$ -0.43 $\
ET0126 & 0 59 42.57 & $-33$ 42 18.1 &$ 2.99 $&$ -1.11 $&$ -0.96 $&$ -0.76 $\
ET0132 & 0 59 58.24 & $-33$ 45 50.8 &$ 2.88 $&$ -1.50 $&$ -0.88 $&$ -0.60 $\
ET0133 & 0 59 47.67 & $-33$ 47 29.5 &$ 2.88 $&$ -1.07 $&$ -1.04 $&$ -0.92 $\
ET0137 & 1 00 25.30 & $-33$ 50 50.8 &$ 3.27 $&$ -0.89 $&$ -0.98 $&$ -0.82 $\
ET0138 & 1 00 38.12 & $-33$ 48 16.9 &$ 3.12 $&$ -1.70 $&$ -0.92 $&$ -0.68 $\
ET0139 & 1 00 42.50 & $-33$ 44 23.5 &$ 3.18 $&$ -1.41 $&$ -0.96 $&$ -0.74 $\
ET0141 & 1 00 23.84 & $-33$ 42 17.4 &$ 3.08 $&$ -1.68 $&$ -0.82 $&$ -0.46 $\
ET0145 & 1 00 20.75 & $-33$ 47 11.1 &$ 3.00 $&$ -1.51 $&$ -1.26 $&$ -1.30 $\
ET0147 & 1 00 44.27 & $-33$ 49 18.8 &$ 3.03 $&$ -1.15 $&$ -1.24 $&$ -1.28 $\
ET0150 & 1 00 22.98 & $-33$ 43 02.2 &$ 3.05 $&$ -0.93 $&$ -1.18 $&$ -1.16 $\
ET0151 & 1 00 18.29 & $-33$ 42 12.2 &$ 2.98 $&$ -1.77 $&$ -0.86 $&$ -0.54 $\
ET0158 & 1 00 18.96 & $-33$ 45 14.8 &$ 2.85 $&$ -1.80 $&$ -0.96 $&$ -0.74 $\
ET0160 & 1 00 22.33 & $-33$ 50 24.0 &$ 2.90 $&$ -1.16 $&$ -1.00 $&$ -0.82 $\
ET0163 & 1 00 24.63 & $-33$ 44 28.9 &$ 2.82 $&$ -1.86 $&$ <-0.70 $&$ <-0.40 $\
ET0164 & 1 00 33.86 & $-33$ 44 54.4 &$ 2.82 $&$ -1.89 $&$ -1.08 $&$ -0.96 $\
ET0165 & 1 00 11.79 & $-33$ 42 16.9 &$ 2.88 $&$ -1.10 $&$ -0.92 $&$ -0.70 $\
ET0166 & 1 00 10.49 & $-33$ 49 36.9 &$ 2.83 $&$ -1.49 $&$ -0.90 $&$ -0.62 $\
ET0168 & 1 00 34.32 & $-33$ 49 52.9 &$ 2.83 $&$ -1.10 $&$ -1.10 $&$ -1.00 $\
ET0173 & 1 00 50.87 & $-33$ 45 05.2 &$ 3.23 $&$ -1.47 $&$ -0.84 $&$ -0.54 $\
ET0198 & 1 00 09.18 & $-33$ 36 09.4 &$ 2.78 $&$ -1.16 $&$ -1.11 $&$ -1.05 $\
ET0200 & 1 00 14.81 & $-33$ 36 49.9 &$ 2.77 $&$ -1.49 $&$ -0.96 $&$ -0.74 $\
ET0202 & 1 00 21.08 & $-33$ 33 46.4 &$ 2.72 $&$ -1.32 $&$ -1.04 $&$ -0.88 $\
ET0206 & 1 00 10.38 & $-33$ 41 05.0 &$ 2.72 $&$ -1.33 $&$ -0.98 $&$ -0.78 $\
ET0232 & 0 59 54.47 & $-33$ 37 53.4 &$ 2.80 $&$ -1.00 $&$ -1.19 $&$ -1.17 $\
ET0236 & 0 59 30.44 & $-33$ 36 05.0 &$ 2.74 $&$ -2.41 $&$ <-0.30 $&$ <0.40 $\
ET0237 & 0 59 50.78 & $-33$ 31 47.1 &$ 2.75 $&$ -1.61 $&$ -0.90 $&$ -0.62 $\
ET0238 & 0 59 57.60 & $-33$ 38 32.5 &$ 2.78 $&$ -1.57 $&$ -1.00 $&$ -0.80 $\
ET0239 & 0 59 30.49 & $-33$ 39 04.0 &$ 2.71 $&$ -2.26 $&$ <-0.40 $&$ <0.30 $\
ET0240 & 0 59 58.31 & $-33$ 34 40.4 &$ 2.77 $&$ -1.15 $&$ -1.02 $&$ -0.84 $\
ET0241 & 1 00 02.69 & $-33$ 30 25.3 &$ 2.79 $&$ -1.41 $&$ -0.94 $&$ -0.70 $\
ET0242 & 1 00 02.23 & $-33$ 40 21.1 &$ 2.86 $&$ -1.32 $&$ -0.94 $&$ -0.70 $\
ET0244 & 0 59 59.65 & $-33$ 39 31.9 &$ 2.73 $&$ -1.24 $&$ -1.04 $&$ -0.90 $\
ET0275 & 0 59 15.13 & $-33$ 39 43.8 &$ 2.70 $&$ -1.21 $&$ -1.08 $&$ -0.98 $\
ET0299 & 0 59 08.60 & $-33$ 42 29.4 &$ 2.70 $&$ -1.83 $&$ -0.66 $&$ -0.14 $\
ET0300 & 0 59 22.12 & $-33$ 49 03.7 &$ 2.75 $&$ -1.39 $&$ -0.98 $&$ -0.78 $\
ET0317 & 0 59 49.91 & $-33$ 44 05.0 &$ 2.81 $&$ -1.69 $&$ -0.96 $&$ -0.74 $\
ET0320 & 0 59 45.31 & $-33$ 43 53.8 &$ 2.76 $&$ -1.71 $&$ <-0.90 $&$ <-0.40 $\
ET0321 & 1 00 06.98 & $-33$ 47 09.7 &$ 2.78 $&$ -1.93 $&$ -1.10 $&$ -1.00 $\
ET0322 & 1 00 05.93 & $-33$ 45 56.5 &$ 2.74 $&$ -2.04 $&$ <-0.60 $&$ <0.00 $\
ET0327 & 0 59 37.56 & $-33$ 43 33.5 &$ 2.76 $&$ -1.32 $&$ -0.94 $&$ -0.68 $\
ET0330 & 1 00 04.16 & $-33$ 43 32.4 &$ 2.68 $&$ -2.00 $&$ -0.68 $&$ -0.18 $\
ET0339 & 0 59 44.90 & $-33$ 44 35.1 &$ 2.72 $&$ -1.08 $&$ -1.19 $&$ -1.17 $\
ET0342 & 0 59 35.02 & $-33$ 50 55.9 &$ 2.62 $&$ -1.35 $&$ -1.11 $&$ -1.03 $\
ET0350 & 0 59 41.95 & $-33$ 45 03.7 &$ 2.56 $&$ -1.90 $&$ <-0.50 $&$ <0.10 $\
ET0354 & 0 59 55.87 & $-33$ 45 43.7 &$ 2.56 $&$ -1.07 $&$ -1.15 $&$ -1.09 $\
ET0363 & 0 59 53.08 & $-33$ 43 58.5 &$ 2.52 $&$ -1.28 $&$ -1.06 $&$ -0.94 $\
ET0369 & 1 00 11.73 & $-33$ 44 50.4 &$ 2.80 $&$ -2.35 $&$ -0.81 $&$ -0.43 $\
ET0373 & 1 00 17.36 & $-33$ 43 59.6 &$ 2.74 $&$ -1.96 $&$ -0.88 $&$ -0.56 $\
ET0376 & 1 00 15.18 & $-33$ 43 11.0 &$ 2.78 $&$ -1.17 $&$ -0.96 $&$ -0.74 $\
ET0378 & 1 00 21.17 & $-33$ 46 01.3 &$ 2.77 $&$ -1.18 $&$ -0.96 $&$ -0.76 $\
ET0379 & 1 00 14.58 & $-33$ 47 11.6 &$ 2.72 $&$ -1.65 $&$ -1.11 $&$ -1.03 $\
ET0382 & 1 00 17.60 & $-33$ 46 55.2 &$ 2.72 $&$ -1.74 $&$ -1.04 $&$ -0.86 $\
ET0384 & 1 00 26.29 & $-33$ 44 45.7 &$ 2.71 $&$ -1.46 $&$ -1.10 $&$ -1.00 $\
ET0389 & 1 00 12.52 & $-33$ 43 01.3 &$ 2.68 $&$ -1.60 $&$ -0.98 $&$ -0.78 $\
ET0392 & 1 00 25.04 & $-33$ 42 28.1 &$ 2.67 $&$ -1.48 $&$ -0.86 $&$ -0.52 $\
\[table:CNdata\]
Results
=======
All element abundances for ET0097 are listed in Table \[table:abundances\].
Carbon in Sculptor
------------------
Carbon measurements have been previously attempted only for a limited number of stars in Sculptor, and the available measurements are shown in Fig. \[fig:cempL\]. The dashed line shows the definition of CEMP stars, as proposed by @Aoki2007, with a slope to account for mixing in RGB stars at higher luminosities, which decreases the carbon abundance and increases the nitrogen at the surface of the star [@Gratton2000; @Spite2006]. The high \[N/Fe\], low $\log \epsilon(\text{Li)}$, and low $^{12}\text{C}/^{13}\text{C}$ values show that ET0097 has undergone mixing, and was even more carbon-rich in the past, having $\text{[C/Fe]}\approx0.8$ (see Section \[sec:mixing\] for details).
With a measured $\text{[C/Fe]}=0.51\pm0.10$, ET0097 is the only known star in Sculptor that falls under the definition of a CEMP star, and it seems to be around $\sim$1.5 dex higher in carbon than other stars of similar luminosity.
The same stars are shown as a function of iron abundance in Fig. \[fig:cplot\]b. The most carbon-rich star in the sample, ET0097, is also the most iron-rich, while the fraction of carbon-rich stars increases with decreasing metallicity in the Galactic halo (e.g., @Lee2013 and references therein). Unlike the other Sculptor stars in Fig. \[fig:cempL\] and \[fig:cplot\]b, ET0097 was not chosen for closer observation based on low \[Fe/H\]. The UVES spectrum for ET0097 was taken after the C-enhancement was discovered from strong CN molecular lines around 9100-9250 . More than 80 other stars were also observed in this wavelength range (Skúladóttir et al. in prep.), and all but the most metal-poor stars have a clear detection of the molecular lines, but only ET0097 stands out, having exceptionally strong CN lines.
To use these CN lines to estimate the C abundances in this sample, some assumptions need to be made about the oxygen and nitrogen. All the stars in this sample are within the central $25^\prime$ diameter region of Sculptor and bright enough to ensure reasonable signal-to-noise at high spectral resolution. These high luminosity RGB stars ($\log L_\star/L_\odot>2.5$ for all stars, see Table \[table:CNdata\]) are expected to have undergone similar mixing to ET0097, decreasing the C at the surface and increasing the N abundance. Therefore, a typical value for mixed stars, $\text{[C/N]}=-1.2$ [@Spite2005], is adopted here for the sample. Some of the stars already have measured O abundances and show a simple trend with iron, see Fig. \[fig:cplot\]a. The same trend with \[Fe/H\] is therefore assumed for stars with unknown O abundances.
The C estimates of 85 stars, calculated with these assumptions, are also shown in Fig. \[fig:cplot\]b (and listed in Table \[table:CNdata\]). None of these stars show any sign of being carbon-enhanced, even if corrected for internal mixing, which has been observed to lower the \[C/Fe\] abundance on the surface of stars by $\sim$0.5 dex [@Gratton2000; @Spite2005].
A different approach can be applied. In the sample of @Spite2005, all mixed stars have $\text{[N/Fe]}>0.5$, so by assuming $\text{[N/Fe]}=0$, which is a very conservative lower limit for nitrogen in these stars, we are also able to obtain upper limits for \[C/Fe\] from the Sculptor CN measurements (assuming the same oxygen values as before), see Table \[table:CNdata\]. If there are any unmixed stars in our sample, then $\text{[N/Fe]}=0$ is a reasonable abundance estimate for these stars [@Spite2005]. Using these assumptions, all 85 stars have $\text{[C/Fe]}_\text{lim}\leq0.4$.
So with the exception of ET0097, which clearly stands out from the rest, no other star (mixed or unmixed) in this sample is likely to be inherently carbon-enhanced. In particular, by combining these estimates with the literature data, ET0097 is the only CEMP star from the sample of 22 stars in Sculptor with $\text{[Fe/H]}\leq-2$.
The general abundance pattern
-----------------------------
The abundance pattern of ET0097 is compared to what is seen in the Galactic halo and other stars in Sculptor, in Fig. \[fig:ele\]. Note that none of the abundances have been corrected for NLTE effects. For many elements this correction can be significant, in particular for Na, Al, and K, reaching up to $\sim$0.6 dex. However, the goal here is not to study the trends for these elements. Since both the Galactic halo and Sculptor samples consist of giant stars with similar $T_{\textsl{eff}}$, $\log g$ and \[Fe/H\] as ET0097, and have the same measured lines for these elements, any NLTE corrections are expected to be similar for all stars.
ET0097 is the only star in Sculptor in the metallicity range $-2.35\leq$\[Fe/H\]$\leq-1.75$ to have measured carbon and nitrogen, and, in fact, it has the only known nitrogen abundance in this galaxy. To estimate the C and N in stars of similar metallicity, we use the CN molecular lines in the region 9100-9250 , from VLT/FLAMES data, see the previous section for details.
Compared to those estimates, ET0097 seems to be enhanced both in C and N with respect to other stars in Sculptor of similar \[Fe/H\], where the difference in carbon is $\gtrsim$1 dex, and $\gtrsim$0.5 dex in nitrogen. Adopting a different \[C/N\] ratio could increase the N abundance estimate, bringing it closer to ET0097, but that would naturally decrease the C, making the difference there even bigger. Comparing ET0097 to carbon-normal, mixed RGB stars in the Galactic halo [@Spite2005], the nitrogen seems to be rather high, but it does not stand out significantly from the scatter. The C in this star is however clearly enhanced compared to similar stars in the Galactic halo. Finally, we note that the \[C+N/Fe\] in the Sculptor sample seems to be lower than what is observed in the Galactic halo.
In Fig. \[fig:ele\], it is clear that when other elements up to Zn are compared with the mean for Sculptor and the Galactic halo, ET0097 does not stand out significantly in any way, and falls within the scatter of stars with similar metallicity. However, ET0097 does show a different pattern in elements heavier than Zn. The lighter neutron-capture elements (sometimes called weak $r$-process elements), Sr, Y, and Zr, are enhanced compared to what is typical in Sculptor, while the heavier $n$-capture elements are depleted, or at the lower end of the trend (Ba). The Galactic halo shows a very large scatter of the $n$-capture elements in the metallicity range $-3.0\leq$\[Fe/H\]$\leq-2.0$, so both ET0097 and other stars in Sculptor fall within the scatter seen in these elements,with the exception of Sr and Y, which appear above the observed scatter.
Since ET0097 shows high abundances of light $n$-capture elements and low abundances of the heavier $n$-capture elements, naturally the abundance ratios \[Sr,Y,Zr/Ba\] are high, see Fig. \[fig:light\]. Here ET0097 is clearly different from the trend seen in the Galactic halo and in Sculptor. Similar abundance ratios are certainly seen in the halo (e.g., @Honda2004b [@Francois2007]), but typically at lower iron abundance ($\text{[Fe/H]}\lesssim-3$). This result is not limited to a comparison with Ba, ET0097 still stands out when any of the other heavier $n$-capture elements are used as a reference element. In fact, ET0097 shows the same relation to the heavier $n$-capture elements as seen in the Galactic halo, see Fig. \[fig:heavy\].
Origin of the abundance pattern
===============================
Alpha and iron-peak elements
----------------------------
In ET0097, alpha and Fe-peak elements from O to Zn show abundances comparable to what is seen both in the Galactic halo and in Sculptor for stars with similar \[Fe/H\], (see Fig. \[fig:ele\]). The most probable explanation is that the bulk of these elements comes from similar sources, such as low-metallicity Type II supernovae of 11-40 M$_\odot$ [@WoosleyWeaver1995], which are believed to be the main producers of these elements in the early universe, or possibly massive zero-metallicity SN of 10-100 M$_\odot$ [@HegerWoosley2010], which have been shown to predict similar abundances as seen in the Galactic halo, hence comparable with ET0097.
Carbon-enhancement
------------------
The origin of the carbon enhancement in CEMP-no stars is still debated and a variety of processes have been invoked (see, e.g., @Norris2013).
In some of these scenarios, the carbon enhancement is explained by mass transfer from a companion. Three different HR velocity measurements for ET0097 were obtained in Hill et al. (in prep.), Skúladóttir et al. (in prep.), and this work. A comparison was made between the two other studies for the 86 stars they had in common. ET0097 showed similar scatter in its velocity measurements compared to other stars in Sculptor. On average the difference between the samples was 1.5 km/s, and ET0097 had a difference of 1.4 km/s in the two measurements. Therefore, close binarity that favors mass transfer seems unlikely, though it cannot be completely excluded with the present data.
However, even if this star does have a companion or did at some point, the abundance pattern is not easily explained with binarity. Mass transfer from an AGB-companion as seen in CEMP-$s$ stars can be excluded because $\text{[Y/Ba]}<0$ is expected [@Travaglio2004], which is clearly not consistent with ET0097 (see Fig. \[fig:light\]). Mass transfer from rapidly rotating stars, which are known to produce a lot of C, is also a possible source for the carbon enhancement. But from these stars strong enhancements in N and O, comparable to the C enhancement, are also expected [@Meynet2006], and this is not consistent with the observations, which in particular show no enhancement in oxygen compared to other Sculptor and halo stars. This scenario can therefore be excluded.
Another possible scenario is that CEMP-no stars formed out of gas that has been enriched by faint supernovae with mixing and fallback which produce significant amounts of C but minimal Fe. Apart from the excess of carbon, faint SN show a general abundance pattern that is very different from normal Type II SN, with an excess of N and O. The ejecta from these stars are also predicted to have a very pronounced odd-even effect among iron-peak elements, showing up as very low abundance ratios, e.g., \[V/Fe\] and \[Mn/Fe\], which are not compatible with the results presented here.
However, by assuming ET0097 formed out of gas containing a mixture of yields from faint SN and normal SN, it is possible to explain the carbon enhancement in this star. By assuming that normal SN Type II enrich the gas up to $\text{[Fe/H]}\approx-2$, and that the gas was already pre-enriched with faint SN yields, as presented by @Iwamoto2005 to match the hyper metal-poor star HE0107-5240, we require that the fraction of faint SN to normal SN is such that the gas reaches $\text{[C/Fe]}=0.8$ (a reasonable assumption for the initial value for ET0097). Though these faint SN yields also show enhancements in N and O, they are considerably smaller than for C. Therefore, the addition of these elements in ET0097 from faint SN would be undetectable in this mixture of yields, falling well within the error of the measurements. The effects on the abundances of other elements in the star are even less pronounced. All the peculiarities of the faint SN yields are swamped by the SN Type II enrichment, leaving the high \[C/Fe\] value as the only evidence of its contribution. Therefore, it is indeed possible that the carbon enhancement in ET0097 is the result of (partial) enrichment with the products of faint SN. This is discussed in more detail in Section \[sec:PFS\].
The lighter $n$-capture elements
--------------------------------
The main $r$- and $s$-process are excluded as dominant sources of Sr, Y, and Zr in ET0097 because they are predicted to produce much higher abundance ratios of heavy to light neutron-capture elements than are consistent with the data. The enhancements of Sr, Y, and Zr, most probably come from the weak $r$-process (e.g., @ArconesMontes2011), which is predicted to produce significant amount of these elements, but minimal heavier $n$-capture elements ($Z\geq$56). Another possibility for the source of these elements is the weak $s$-process that occurs in fast-rotating massive zero-metallicity stars. Models of these stars have been able to reproduce the scatter of these elements observed in the Galactic halo [@Cescutti2014]. However, the data presented here are not sufficient to distinguish between the different possible scenarios, and models that predict excesses of Sr, Y, and Zr, without significant effects on other element abundances, are consistent with observations of ET0097.
In the Galactic halo, a few stars showing strong signatures of the weak $r$-process (or weak $s$-process) have been found and studied in detail. Two of those stars, HD 88609 ($\text{[Fe/H]}=-3.07$) and HD 122563 ($\text{[Fe/H]}=-2.77$) from @Honda2007 are compared to ET0097 in Fig. \[fig:hondasnedy\]. To ensure a useful comparison of the abundance pattern of the $n$-capture elements in these stars, they have all been normalized to the \[Y/H\] value of HD 88609. The absolute values of \[X/H\] for the $n$-capture elements are much higher in ET0097 than in the other, more metal-poor stars. In fact, when comparing to Fe, ET0097 is more enriched in Sr, Y, and Zr than the other two stars (\[Y/Fe\]$_{\text{ET0097}}=0.35$, \[Y/Fe\]$_{\text{HD~88609}}=-0.12$ and \[Y/Fe\]$_{\text{HD~122563}}=-0.37$). Also included in Fig. \[fig:hondasnedy\] (with the same normalization) is the $n$-capture rich star CS 22892-052 from @Sneden2003, which is believed to show a pure signature of the main $r$-process.
The relative abundance pattern in the $n$-capture elements of ET0097 is comparable to the two Honda stars (see Fig. \[fig:hondasnedy\]), making it very likely that these stars were polluted by similar processes. The only exception is Pr, which seems to be much lower in ET0097 than in the others.[^3] Apart from this one element, the abundance patterns of the three stars are comparable. The main $r$-process rich star, CS 22892-052, shows a completely different pattern, and so it is clear that the origin of its $n$-capture elements is different from the other stars. ET0097 and the two Honda stars show similar signatures of the weak $r$-process, possibly with some contamination from the main $r$-process.
Finally, it should be noted that the weak $s$-process in fast-rotating, zero-metallicity massive stars has been proposed to simultaneously enrich gas with C, N O and the lighter neutron-capture elements [@Chiappini2006; @Frischknecht2012; @Cescutti2014]. However, these models predict an enhancement of O comparable with the C enhancement, which is not observed in ET0097. It should also be noted that neither of the Honda stars are enhanced in carbon, \[C/Fe\]$\lesssim-0.40$ for both stars [@Honda2004b], which is consistent with the idea that the enhancements of carbon and the light $n$-capture elements come from two different processes. This is also supported by the top panel of Fig. \[fig:light\], where the CEMP-no stars do not show any obvious trend of \[Sr/Ba\] with \[Fe/H\], different from the normal population, and the same is true for \[Sr/Fe\] with \[Fe/H\] (See, e.g., @Cescutti2014, their Fig. 1).
Possible formation scenario {#sec:PFS}
===========================
To explain the abundance pattern seen in ET0097, the formation scenario must include a plausible explanation for carbon and lighter neutron-capture enhancements at such high $\text{[Fe/H]}=-2.03\pm0.10$. Usually these are seen at lower values, $\text{[Fe/H]}\lesssim-3$, in the Galactic halo. The star ET0097 is also depleted in the heavier $n$-capture elements compared to other stars of similar iron abundance in Sculptor, see Fig. \[fig:ele\].
These peculiarities of ET0097 therefore seem to indicate that it was not formed from the same material as most other stars observed in Sculptor. One possible scenario is that this star was formed in one of the progenitor (mini)-halos of Sculptor that formed at high-redshift and initially evolved independently (e.g., @SalvadoriFerrara2009), also sometimes called inhomogeneous mixing.
Although the C-enhancement in CEMP-no stars is usually associated with faint SNe, which have relatively high C yields compared to their Fe-peak elements production (e.g., @Iwamoto2005), pollution by faint SNe alone cannot enrich a gas cloud up to a metallicity of $\text{[Fe/H]}\approx-2$ [@Salvadori2012; @CookeMadau2014], and the general abundance pattern of the yields of such stars is very different from ET0097. However, as discussed in the previous section, it is possible that ET0097 was formed out of material that contained a mixture of yields from faint SNe and normal core collapse SNe.
It has been shown that massive zero-metallicity SN are able to pollute small self-enriched systems up to high $\text{[Fe/H]}\approx-2$ [@Salvadori2007; @Karlsson2008]. So one possibility is that ET0097 was formed in a mini-halo that had only been enriched by the first generation of stars, a population containing both zero-metallicity core collapse SN and faint SN. If that is the case, we should be able to estimate the required relative contributions of faint and “normal” primordial supernovae to simultaneously account for the observed C and Fe abundances and the lighter $n$-capture elements.
To test this idea, we assume a very simple scenario where this primordial population forms in a single burst of mass $M_\star$, with a Salpeter IMF in the mass range 10-100 M$_\odot$. A fraction $F_{F}$ of the star-forming gas goes into faint SN (as described by @Iwamoto2005 for HE0107-5240), and a fraction $F_{N}=1-F_{F}$ goes into normal zero-metallicity SN, (as described by @HegerWoosley2010 with standard mixing and energy $E_{SN}=1.2 \cdot 10^{51}$ erg)[^4].
The iron abundance of a gas cloud with mass $M_g$, and a mass of iron $M_\text{Fe}$ can be approximated by: $$\label{eq:FeH}
\text{[Fe/H]}=\log\left(\frac{ M_\text{Fe}}{M_g}\right)-\log\left(\frac{M_\text{Fe}}{M_\text{H}} \right)_\odot$$ Similarly, we get an expression for the mass of carbon in the gas, $M_\text{C}$,
$$\label{eq:CFe}
\text{[C/Fe]}=\log\left(\frac{ M_\text{C}}{M_\text{Fe}}\right)-\log\left(\frac{M_\text{C}}{M_\text{Fe}} \right)_\odot$$
By assuming solar abundances from @GrevesseSauval1998 the mass of Fe and C needed to enrich the gas up to \[Fe/H\]$=-2.03$ and \[C/Fe\]=0.8 is $$\begin{aligned}
M_\text{Fe}&=&10^{-4.78}\cdot M_g \label{eq:Mfe}\\
M_\text{C}&=&14.2 \cdot M_\text{Fe} \label{eq:McMfe}
$$
The total amount of iron and carbon produced in a star forming episode of total mass $M_\star$, is therefore $$\begin{aligned}
M_\text{Fe}&=&\mathcal{Y}_{N}(\text{Fe}) F_{N} M_\star + \mathcal{Y}_{F}(\text{Fe})F_{F} M_\star \label{eq:MFe2}\\M_\text{C}&=&\mathcal{Y}_{N}(\text{C}) F_{N} M_\star + \mathcal{Y}_{F}(\text{C}) F_{F} M_\star \label{eq:MC2}\end{aligned}$$ where the yields $\mathcal{Y}$ of Fe and C, are the masses of these elements (in M$_\odot$) that are produced and released into the environment by each M$_\odot$ of gas transformed into stars. The Fe yields in faint SNe are negligible, $\mathcal{Y}_{F}(\text{Fe})\ll \mathcal{Y}_{N}(\text{Fe})=2.0\cdot 10^{-3}$, and the carbon yields of the faint and normal SNe are respectively: $\mathcal{Y}_{F}(\text{C})=4.2\cdot 10^{-3}$ [@Iwamoto2005], and $\mathcal{Y}_{N}(\text{C})=1.0\cdot 10^{-2}$ [@HegerWoosley2010]. Thus Eq. (\[eq:Mfe\])-(\[eq:MC2\]) give $F_{N}=0.19$ and $F_{F}=0.81$. By defining the star formation efficiency, $f_\star$, as the fraction of gas turned into stars, the total mass of the primordial population is $$M_\star=f_\star M_g$$ Combining this with Eq. (\[eq:Mfe\]) and (\[eq:MFe2\]), gives $f_\star$=0.044. This is an upper limit, since more realistic calculations should take into account that stars with different masses do not explode all at once and some of the gas is ejected by SNe of massive stars before the lower mass stars start to contribute, leaving less gas to enrich.
This scenario is also able to explain the overabundance of the lighter $n$-capture elements (Sr, Y, Zr), seen in ET0097, if certain requirements are fulfilled. @ArconesMontes2011 describe the weak $r$-process in neutrino-driven winds from core collapse supernovae of progenitor mass 10 M$_\odot \leq M \leq$ 25 M$_\odot$. Using their yields and going through similar calculations to those above, the weak process has to occur in $\sim$10% of the total stellar mass formed as a normal SN in the mass range 10-25 M$_\odot$, to account for the Sr, Y, and Zr abundances observed in ET0097.
@HegerWoosley2010 have shown that their yields are consistent with the general abundance pattern seen in low-metallicity halo stars (such as in @Cayrel2004) for other elements up to Zn, and it is therefore reasonable to conclude that they are also consistent with ET0097.
With this very simple calculation, we show that ET0097 is compatible with having been enriched with the first stellar generation. We are able to explain both the high iron abundance, and the overabundances of carbon and the lighter $n$-capture elements.
Another (similar) possibility is that this star was formed in a (mini)-halo, where the first stellar generation was dominated by faint SN, and then normal (nonzero-metallicity) Type II SNe [@WoosleyWeaver1995], which have comparable C and Fe yields to the zero-metallicity case, enriched the gas up to $\text{[Fe/H]}\approx-2$.
In both of these scenarios, it is necessary that the (mini)-halo is large enough to retain some of its gas for the next generation(s) of stars, and that the stellar population of zero-metallicity stars is dominated by faint SN. This is consistent with @deBennassuti2014 who show that a stellar population of zero-metallicity stars dominated by faint SN is able to produce the CEMP fraction observed in the halo.
The CEMP-no fraction in Sculptor
================================
In total, 22 stars in Sculptor with $\text{[Fe/H]}\leq-2$ have C measurements or upper limits (@Frebel2010 [@Tafelmeyer2010; @Kirby2012; @Starkenburg2013]; and this work). Only one of them falls into the category of a CEMP-no star, making the fraction $4.5^{+10.5}_{-3.8}$% (errors are derived using @Gehrels1986).
In the Galactic halo, the proportion of CEMP-no stars (when the RGB stars have been corrected for internal mixing and CEMP-$s$ and CEMP-$s/r$ stars have been excluded from the sample) is $20\pm2$% (compilation of data and correction for mixing comes from @Placco2014[^5], with Poisson errors derived from @Gehrels1986). If this fraction was the same in Sculptor, $p=0.20$, then from a sample of $N=22$ stars, the expected number of CEMP-no stars with $\text{[C/Fe]}\geq0.7$ is $R_{exp}=Np\pm\sqrt{N(1-p)p}=4.4\pm 1.9$. However, only one is found. In the lower metallicity range, $\text{[Fe/H]}\leq-3$, the fraction of CEMP stars in the Galactic halo is $p=0.43^{+0.06}_{-0.05}$ [@Placco2014]. In Sculptor, $N=8$ stars in this range have measured carbon or upper limits, so the expected number of stars with $\text{[C/Fe]}\geq 0.7$ (once they have been corrected for mixing), is $R_{exp}=3.4\pm 1.4$, but none are found. The fraction of CEMP-no stars in the currently observed Sculptor sample is thus lower than in the Galactic halo, and the difference is statistically significant.
This ratio can be affected by the fact that no stars with $\text{[Fe/H]}\leq-4.0$ have been found so far in Sculptor, while those stars predominantly fall into the CEMP-no category in the halo. Looking at the range $-2.5\leq\text{[Fe/H]}\leq-2$, the fraction of CEMP-no stars in Sculptor is $8^{+19}_{-7}$%, which is consistent with that found in the halo, $5^{+3}_{-2}\%$ (data coming from @Placco2014, with Poisson errors derived from @Gehrels1986). The expected number of CEMP-no stars in this range, should the Sculptor fraction be the same as in the halo, is $R_{exp}=0.6^{+0.8}_{-0.6}$ stars, which is consistent with the one star found. Only one observed (C-normal) star in Sculptor falls in the range $-3<\text{[Fe/H]}<-2.5$, so little can be said about the CEMP-no fraction there. However, for the lowest metallicity range, $-4\leq\text{[Fe/H]}\leq-3$, no CEMP-no star is found in Sculptor out of a sample of eight stars, giving the Poisson upper limit of the fraction, $23\%$. In the same metallicity range in the halo, the fraction is $39^{+6}_{-5}\%$. The expected number of CEMP-no stars in the Sculptor sample, should the fraction be the same as in the halo is $R_{exp}=3.1\pm1.4$ stars, while none are found. Although still poorly constrained due to low number statistics, the CEMP-no fraction in Sculptor is therefore consistent with the Galactic halo at higher metallicities ($-2.5\leq\text{[Fe/H]}\leq-2$), while it appears to be different at the lowest metallicity end, $-4\leq\text{[Fe/H]}\leq-3$ (see also: @Starkenburg2013).
It remains puzzling that no CEMP-no stars are found at lower metallicities in Sculptor, while ET0097 has a rather high metallicity for such objects, $\text{[Fe/H]}=-2.03\pm0.10$. If the CEMP-no stars indeed show imprints of the very first stars, they would be expected to be more common at the lowest metallicities, also in dwarf spheroidal galaxies such as Sculptor. This apparent discrepancy is not easily explained, and it cannot be excluded that CEMP-no stars are a mixed population with different formation scenarios (an overview is given in @Norris2013). However, we want to emphasize that the CEMP-no fraction in Sculptor is still very poorly constrained, with an upper limit of $\sim$25% both at the high and the low-metallicity end, and not very constraining lower limits ($<2\%$). Although the CEMP-no fraction at the low-metallicity end in Sculptor seems to be different from the halo, a similar trend is still possible where the fraction increases with lower \[Fe/H\], and should be expected. The effect of the environment on the CEMP-no fraction is still not well understood, and will be explored in greater detail in Salvadori et al. in prep.
Conclusions
===========
After unusually strong CN molecular lines were discovered in the star ET0097, a follow-up spectrum at high-resolution and over a long wavelength range was taken with the ESO/VLT/UVES spectrograph. Detailed abundance analysis shows that with \[C/Fe\]=0.51$\pm$0.10, ET0097 is the most carbon-rich VMP star known in Sculptor. Having a luminosity of $\log L_\star/L_\odot=3.1$, this star is expected to have undergone mixing, lowering the carbon at the surface of the star and increasing the nitrogen. This is confirmed by the high N of the star, \[N/Fe\]=1.18$\pm$0.20, the low isotope ratio $\log^{12}$C/$^{13}$C=0.77$\pm$0.03, and the low Li abundance $\log \epsilon(\text{Li})<0.17$. The original C abundance of ET0097 is therefore estimated to be $\text{[C/Fe]}\approx0.8$, making this the only known CEMP in Sculptor.
The star shows normal abundances for all alpha and Fe-peak elements from O to Zn, consistent with what is seen both in the Galactic halo and Sculptor for giants of similar metallicities. Compared to other stars in Sculptor, ET0097 is enhanced in the lighter $n$-capture elements (Sr, Y, Zr), and shows low abundances of the heavier $n$-capture elements, making this a CEMP-no star. The abundance ratios \[Sr,Y,Zr/Ba\] are high, especially in relation to the Fe abundance of the star. The abundance pattern of the $n$-capture elements is comparable to what is seen in stars that are believed to have been polluted by the weak $r$-process.
One possible scenario to explain the peculiar abundance pattern of ET0097 is that the star formed in one of the low-mass progenitor halos of Sculptor, which had only been enriched by a primordial population consisting of a mixture of faint SN ($\sim$80%) and zero-metallicity core collapse SN ($\sim$20%). Another possibility is that this star was formed in a halo where the first stellar generation consisted of faint SN only, and then normal Type II SNe polluted the gas up to $\text{[Fe/H]}\approx-2$.
In addition to the abundance analysis for ET0097, ESO/VLT/FLAMES data in the wavelength range 9100-9250 was used to determine estimates and upper limits for carbon in 85 Sculptor stars, including 11 stars with $\text{[Fe/H]}\leq-2$. No other star in the sample was found to be carbon-enhanced.
In the Galactic halo, @Placco2014 have carefully determined the fraction of CEMP stars in a sample of $\sim$500 stars with $\text{[Fe/H]}\leq-2$. The RGB stars in their sample have been corrected for internal mixing, and CEMP-$s$ and CEMP-$s/r$ stars have been excluded. Using these very detailed results, we are able to compare the CEMP-no fraction observed in Sculptor to that seen in the halo. The fraction of CEMP-no to C-normal stars in the entire sample of observed Sculptor stars with $\text{[Fe/H]}\leq-2$, is $4.5^{+10.5}_{-3.8}$%. This is lower than that seen in the Galactic halo, $20\pm2\%$ [@Placco2014], and the difference is statistically significant.
If we explore this further in different metallicity bins, then for $-2.5\leq\text{[Fe/H]}\leq-2$, the observed CEMP-no fraction in Sculptor is $8^{+19}_{-7}$%, consistent with that found in the halo, $5^{+3}_{-2}\%$. At the lowest metallicity end in Sculptor, $-4\leq\text{[Fe/H]}\leq-3$, the CEMP-no fraction is $0^{+23}$%, which is statistically significantly lower then that observed in the Galactic halo, $39^{+6}_{-5}\%$. The carbon measurements in Sculptor are still few, so the CEMP-no fraction is poorly constrained, but, at least at the lowest metallicity end, the CEMP-no fraction in Sculptor seems to be fundamentally different from what is seen in the Galactic halo.
We thank ESO for granting us Directors Discretionary time to allow us to rapidly follow up this very interesting star. The authors are indebted to the International Space Science Institute (ISSI), Bern, Switzerland, for supporting and funding the international team “First stars in dwarf galaxies”. Á. S. thanks Anna Frebel for useful advice and insightful suggestions. S. S. acknowledges support from the Netherlands Organisation for Scientific Research (NWO), VENI grant 639.041.233. E. S. gratefully acknowledges the Canadian Institute for Advanced Research (CIFAR) Global Scholar Academy.
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[^1]: CIFAR Global Scholar
[^2]: Based on observations made with ESO telescopes at the La Silla Paranal observatory under program IDs 291.B-5036 (director’s discretionary time) and 089.B-0304(B)
[^3]: The Pr lines at 4179.4 and 4189.5 , used by @Honda2007, are severely blended with CH molecular lines in ET0097, and therefore cannot be used for the abundance determination, but are consistent with no lines being observed. Instead the upper limit for Pr was determined from lines at 4408.8 and 4496.5 , both giving similar results.
[^4]: In the normal SN case, the yields for carbon are taken from Table 8 in @HegerWoosley2010, and the iron taken from $^{56}$Ni yields in their Table 6 for the same mass values. Both elements are integrated in the same way over a Salpeter IMF.
[^5]: The @Placco2014 sample is based on the most recent version of the SAGA database [@Suda2008] and the compilation of literature data by @Frebel2010 and data published since then. For individual references see @Placco2014.
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|
---
abstract: 'Generating photorealistic images of human subjects in any unseen pose have crucial applications in generating a complete appearance model of the subject. However, from a computer vision perspective, this task becomes significantly challenging due to the inability of modelling the data distribution conditioned on pose. Existing works use a complicated pose transformation model with various additional features such as foreground segmentation, human body parsing etc. to achieve robustness that leads to computational overhead. In this work, we propose a simple yet effective pose transformation GAN by utilizing the Residual Learning method without any additional feature learning to generate a given human image in any arbitrary pose. Using effective data augmentation techniques and cleverly tuning the model, we achieve robustness in terms of illumination, occlusion, distortion and scale. We present a detailed study, both qualitative and quantitative, to demonstrate the superiority of our model over the existing methods on two large datasets.'
author:
- Arnab Karmakar
- Deepak Mishra
bibliography:
- 'mybibliography.bib'
title: A Robust Pose Transformational GAN for Pose Guided Person Image Synthesis
---
Introduction
============
Given an image of a person, a pose transformation model aims to reconstruct the person’s appearance in another pose. While for humans it is very easy to imagine how a person would appear in a different body pose, it has been a difficult problem in computer vision to generate photorealistic images conditioned only on pose; given a single 2D image of the human subject. The idea of pose transformation can help construct a viewpoint invariant representation. This has several interesting applications in 3D reconstruction, movie making, motion prediction or human computer interaction etc.
The task of pose transformation given a single image and a desired pose, is achieved by any machine learning model in basically two steps: (1) learning the significant visual features of the person-of-interest along with the background from the given image, and (2) imposing the desired pose on the person-of-interest, generate a photorealistic image while preserving the previously learned features. Generative Adversarial Networks (GAN) [@goodfellow-gan] have been widely popular in this field due to its sharp image generation capability. While the majority of successful pose transformation models use different variation of GANs as their primary component, they give little importance to efficient data augmentation and utilization of inherent CNN features to achieve robustness. Recent developments in this field have been targeted to develop complex deep neural network models with the use of multiple external features such as human body parsing [@soft-gated], semantic segmentation [@soft-gated][@balakrishnan], spatial transformation [@balakrishnan][@deformable-gans] etc. Although this is helpful in some scenarios, there is accuracy issues and computational overhead due to each intermediate step that affects the final result.
In this work, we aim to develop an improved end-to-end model for pose transformation given only the input image and the desired pose, and without any other external features. We make use of the Residual learning strategy [@resnet] in our GAN architecture by incorporating a number of residual blocks. We achieve robustness in terms of occlusion, scale, illumination and distortion by using efficient data augmentation techniques and utilizing inherent CNN features. Our results in two large datasets, a low-resolution person re-identification dataset Market-1501 [@market-1501] and high-resolution fashion dataset DeepFashion [@deepfashion] have been demonstrated. Our contributions are two folds: First, we develop an improved pose transformation model to synthesize photorealistic images of a person in any desired pose, given a single instance of the person’s image, and without any external features. Second, we achieve robustness in terms of occlusion, scale and illumination by efficient data augmentation techniques and utilizing inherent CNN features.
Related work
============
There has been a lot of research in the field of generative image modelling using deep learning techniques. One line of work follow the idea of Variational Autoencoders (VAE) [@kingma-vae] which uses the reparameterization trick to maximize the lower bound of data likelihood [@mypaper]. VAEs have been popular for its image interpolation capability, but the generated images lack sharpness and high frequency details. GAN [@goodfellow-gan] models make use of adversarial training for generating images from random noise. Most works in pose guided person image generation make use of GANs because of its capability to produce fine details.
Amongst the large number of successful GAN architectures, many were developed upon the DCGAN [@dcgan] model that combines Convolutional Neural Network (CNN) with GANs. Pix2pix [@pix2pix] proposed a conditional adversarial network (CGAN) for image-to-image translation by learning the mapping from condition image to target image. Yan et al. [@skeleton-aided] explored this idea for pose conditioned video generation, where the human images are generated based on skeleton poses. GANs with different variations of U-Net [@unet] have been extensively used for pose guided image generation. The PG$^2$ [@pg2] model proposes a 2-step process with a U-Net-like network to generate an initial coarse image of the person conditioned on the target pose and then refines the result based upon the difference map. Balakrishnan et al. [@balakrishnan] uses separate foreground and background synthesis using a spatial transformer network and U-Net based generator. Ma et al. [@disentangled-pig] uses pose sampling using a GAN coupled with an encoder-decoder model. Dong et al. [@soft-gated] produces state-of-the-art results in pose driven human image generation and uses human body parsing as an additional attribute for Warping-GAN rendering. These additional attribute learning generates an overhead in computational capability and affects the final results. Other significant works for pose transfer in the field of person re-identification [@pn-gan] [@fd-gan] mostly deals with low resolution images and a complex training procedure. In this work, we propose a simplified end-to-end model for pose transformation without using additional feature learning at any stage.
Methodology
===========
![Proposed architecture of the pose transformational GAN (pt-GAN). The idea is to transform the given person image to the desired pose. The additional classification branch of the Discrminator helps the Generator’s learning to produce realistic images.[]{data-label="fig:gan_arch"}](figs/gan_full.pdf){width="\linewidth"}
Pose Estimation
---------------
The image generation is conditioned on an input image and the target pose represented by a pose vector. In order to get the encoded pose vector, we use off-the-shelf pose detection algorithm OpenPose [@openpose], which is trained without using either of the datasets deployed in this work. Given an input person image $I_i$, the pose estimation network OpenPose produces a pose vector $P_i$, which localizes and detects 25 anatomical key-points.
Generator
---------
The Image generator ($G_P$) aims at producing the same person’s images under different poses. Particularly, given an input person image $I_i$ and a desired pose $P_j$, the generator aims to synthesize a new person image $I_{P_j}$, which contains the same identity but with a different pose defined by $P_j$. The image vector obtained using a pretrained ResNet-50 [@resnet] model (ImageNet [@imagenet]), and the pose vector are concatenated and fed to the generator. The architecture is depicted in Figure \[fig:generator\].
The generator consists of multiple Convolution and Transposed Convolution layers. The key element of the proposed Generator is the residual blocks. Each residual block performs downsampling using convolution followed by upsampling using transposed convolution and then re-using the input by addition (Figure \[fig:both\](b)). The motivation is to take advantage of Residual Learning ($y = F(x)+x$) that can be used to pass invariable information (e.g. clothing color, texture, background) from the bottom layers to the higher layers and change the variable information (pose) to synthesize more realistic images, achieving pose transformation at the same time.
![Architecture of the Generator Network. The Generator network consists of 9 residual blocks, which helps the GAN to preserve low level features (clothing, texture), while transforming high level features (pose) of the subject.[]{data-label="fig:generator"}](figs/generator.png){width="0.9\linewidth"}
Discriminator
-------------
In our implementation, the Discriminator ($D_P$) predicts the class label for the image along with the binary classification of determing whether the image is real or generated. Studies [@discrim] show that incorporating classification loss in discriminator along with the real/fake loss, in turn increases the generator’s capability to produce sharp images with high details. The Discriminator consists of stacked Conv-ReLU-Pool layer and the final fully connected layer has been modified to incorporate both binary loss and classification loss (Figure \[fig:both\](a)).
![**(a)** Architecture of the discriminator of pt-GAN. A classification task is added with the real/fake prediction. This simultaneously helps the Generator to produce more realistic images. **(b)** Architecture of the Residual Blocks used in the Generator. The Residual Learning strategy preserves low level features (color, texture) and learns high level features (pose) simultaneously.[]{data-label="fig:both"}](figs/new_both.png){width="\linewidth"}
Data Augmentation
-----------------
1. **Image Interpolation**: The input images have been resized to $256\times 256$ before passing through ResNet. Market-1501 images ($128\times 64$) are resized to $256\times 128$, and zero-padded to make $256\times 256$. The images in DeepFashion are of the desired dimension by default.
2. **Random Erasing [@randomerasing]**: Random erasing is helpful in achieving robustness against occlusion. A random patch of the input image is given random values while the reconstruction is expected to be perfect. Thus, the GAN learns to reconstruct (and remove) the occluded regions in the generated images.
3. **Random Crop**: The input image is randomly cropped and upscaled to the input dimension ($256\times 256$) to augment the cases where the human detection is inaccurate or only the partial body is visible.
4. **Jitter**: We use random jitter in terms of brightness, Contrast, Hue and Saturation (random jitter to each channel) to augment the effects of illumination variations.
5. **Random Horizontal Flip**: Inspecting the dataset, it is seen that most human subjects has both left-right profile images. Hence flipping the image left-right is a good choice for image augmentation.
6. **Random distortion**: We have incorporated random distortion with a grid size of $10$, to compensate the distortion in the generated image as well as enforce our model to learn important features of the input image even in the presence of non-idealities.
![The data augmentation techniques used in this work: (a) Original Image, (b) Random Erasing, (c) Random Crop, (d) Random Distortion, (e)-(g) Random Jitter: (e) Brightness, (f) Contrast, (g) Saturation; (h) Random Flip[]{data-label="fig:data_augmentation"}](figs/data_augmentation.png){width="0.7\linewidth"}
The CNN by itself enforces scale invariance through max-pooling and convolution layers. Thereafter we claim to have achieved invariance from distortion, occlusion, illumination and scale. A demonstration of the data augmentation techniques is shown in Figure \[fig:data\_augmentation\].
Experiments
===========
Datasets
--------
### DeepFashion:
The DeepFashion (In-shop Clothes Retrieval Benchmark) dataset [@deepfashion] consists of 52,712 in-shop clothes images, and 200,000 cross-pose/scale pairs. The images are of 256$\times$256 resolution. We follow the standard split adopted by [@pg2] to construct the training set of 146,680 pairs each composed of two images of the same person but different poses.
### Market-1501:
We also show our results on the re-identification dataset Market-1501 [@market-1501] containing 32,668 images of 1,501 persons. The images vary highly in pose, illumination, viewpoint and background in this dataset, which makes the person image generation task more challenging. Images have size 128$\times$64. Again, we follow [@pg2] to construct the training set of 439,420 pairs, each composed of two images of the same person but different poses.
Implementation and Training
---------------------------
For image descriptor generation, We have used a pretrained ResNet-50 network whose weights were not updated during the training of the generator and discrimiator. The input image and the target image are of the same class with different poses. The reconstruction loss (MSE) is incorporated with the negative discriminator loss to update the Generator. In our implementation, we have used 9 Residual blocks sequentially in the generator architecture. The discriminator is trained on the combined loss (binary crossentropy and categorical crossentropy).
The architecture of the proposed model is described in detail in section \[methodology\]. For training the Generator as well as Discriminator we have used Adam optimizer with $\beta_1=0.5$ and $\beta_2=0.999$. The initial learning rate was set to 0.0002 with a decay factor 10 at every 20 epoch. A batch size of 32 is taken as standard.
Results and Discussion
======================
Qualitative Results
-------------------
We demonstrate a series of results in high resolution fashion dataset DeepFashion [@deepfashion] as well as a low resolution re-identification dataset Market-1501 [@market-1501]. In both the datasets, by visual inspection, we can say that our model performs good reconstruction and is able to learn invariable information like the colour and texture of clothing, characteristics of make/female attributes such as hair and face while successfully performing image transformation into the desired pose. The results on DeepFashion is better due to good details and simple background, whereas the low resolution affects the quality of the generated images in Market-1501. The results are demonstrated in Figure \[fig:main\_result\].
![Qualitative results on DeepFashion and Market-1501 datasets. The proposed model is able to reproduce good details, and also learn invariable information like the colour and texture of clothing, characteristics of make/female attributes such as hair and face while successfully perform image transformation into the desired pose.[]{data-label="fig:main_result"}](figs/main_result_gan_2.png){width="0.98\linewidth"}
Quantitative Results
--------------------
We use two popular measures of GAN performance, namely Structural Similarity (SSIM) [@ssim] and Inception Score (IS) [@is] for verifying the performance of our model. We compare our work with the already existing methods based on SSIM and IS scores on both DeepFashion and Market-1501 datasets in Table \[tab:result-table\]. Our model achieves the best IS score in Market-1501 dataset while achieving second best results in SSIM score in both the datasets. However, the deviation from the state-of-the-art is $\sim 1.5\%$ in these cases which can be overcome through rigorous testing and hyperparameter tuning. We also inspect the improvement incorporated by data augmentation as seen in Table \[tab:result-table\]. The proposed augmentation methods give an average improvement of $\sim 9\%$. This essentially strengthens our argument that a significant boost in performance can be gained by exploring effective training schemes, without changing the model parameters or loss function.
[ C[4cm]{} C[1.2cm]{} C[1.2cm]{} C[1.2cm]{} C[1.2cm]{} ]{}\
& &\
\
Model & SSIM & IS & SSIM & IS\
\
pix2pix [@pix2pix] & 0.692 & 3.249 & 0.183 & 2.678\
PG2 [@pg2] & 0.762 & 3.090 & 0.253 & 3.460\
DSCF [@deformable-gans] & 0.761 & & 0.290 & 3.185\
BodyROI7 [@disentangled-pig] & 0.614 & 3.228 & 0.099 &\
Dong et al. [@soft-gated] & & & & 3.409\
\
Ours w/o augmentation & 0.713 & 3.006 & 0.268 & 3.425\
Ours (full) & & 3.238 & &\
Failure Cases
-------------
![Failure Cases in our pt-GAN model. If the input contains fine details (text, stripes) or the target pose is incomplete then the reconstruction is poor. The external attribute (handbag) learning is of limited success.[]{data-label="fig:fail_result"}](figs/fail_result_gan.png){width="0.98\linewidth"}
We analyse some of our failure cases in both the datasets to understand the shortcoming of our model. As seen in Figure \[fig:fail\_result\], the text in clothing as well as very fine patterns of clothing (stripes, dots) are not modelled properly. The external attribute features (e.g. the handbag in Figure \[fig:fail\_result\]) are not learned properly as it is difficult to map external attributes to the output image when conditioned only on pose. The accuracy is also dependent on the completeness of the target pose. Finally, there is some limitation in cases where a rare complex pose is presented which has scarce training data. In Market-1501, the reconstruction of faces is not very good due to poor resolution.
Further Analysis
----------------
Along with the quantitative and qualitative results, we demonstrate a special case to show the improvement caused by data augmentation methods. As seen in Figure \[fig:occlusion\_inv\], the occlusion in the input image is partially carried forward when the data augmentation methods are not used. With data augmentation the generated image is better in quality and the artifacts generated in the edges are less.
![Occlusion invariance using the proposed model. The occlusion is partially carried forward when data augmentation methods are not used. With data augmentation, the resultant image is free from the artifacts.[]{data-label="fig:occlusion_inv"}](figs/result_data_augmentation.png){width="0.6\linewidth"}
Conclusion
==========
In this work, we proposed an improved end-to-end pose transformation model to synthesize photorealistic images of a given person in any desired pose without any external feature learning. We make use of the residual learning strategy with effective data augmentation techniques to achieve robustness in terms of occlusion, scale, illumination and distortion. For future work, we plan to achieve better results by utilising feature transport from the source image and conditioning the discriminator on both source image and target pose, alongwith using a perceptual (content) loss for reconstruction.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Tunka-Rex is the radio extension of Tunka-133 located in Siberia close to Lake Baikal. The latter is a photomultiplier array registering air-Cherenkov light from air showers induced by cosmic-ray particles with initial energies of approximately $10^{16}$ to $10^{18}$ eV. Tunka-Rex extends this detector with 25 antennas spread over an area of 1 km$^2$. It is triggered externally by Tunka-133, and detects the radio emission of the same air showers. The combination of an air-Cherenkov and a radio detector provides a facility for hybrid measurements and cross-calibration between the two techniques. The main goal of Tunka-Rex is to determine the precision of the reconstruction of air-shower parameters using the radio detection technique. It started operation in autumn 2012. We present the overall concept of Tunka-Rex, the current status of the array and first analysis results.'
address:
- 'Institut für Kernphysik, Karlsruhe Institute of Technology (KIT), Germany'
- 'Institute of Applied Physics ISU, Irkutsk, Russia'
- 'Institut für Prozessdatenverarbeitung und Elektronik, Karlsruhe Institute of Technology (KIT), Germany'
- 'Skobeltsyn Institute of Nuclear Physics MSU, Moscow, Russia'
- 'Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia'
- 'DESY, Zeuthen, Germany'
author:
- 'D. Kostunin'
- 'N.M. Budnev'
- 'O.A. Gress'
- 'A. Haungs'
- 'R. Hiller'
- 'T. Huege'
- 'Y. Kazarina'
- 'M. Kleifges'
- 'A. Konstantinov'
- 'E.N. Konstantinov'
- 'E.E. Korosteleva'
- 'O. Krömer'
- 'L.A. Kuzmichev'
- 'R.R. Mirgazov'
- 'L. Pankov'
- 'V.V. Prosin'
- 'G.I. Rubtsov'
- 'C. Rühle'
- 'F.G. Schröder'
- 'E. Svetnitsky'
- 'R. Wischnewski'
- 'A. Zagorodnikov'
title: 'Tunka-Rex: Status and Results of the First Measurements'
---
Tunka-Rex , Tunka-133 ,Tunka ,ultra-high energy cosmic rays ,extensive air showers ,radio detection
Introduction
============
Since the measurements of cosmic rays have reached energies of the GZK [@Greisen:1966jv; @Zatsepin:1966jv], the main challenge for the physics of ultra-high cosmic rays is to increase the statistics and the measurement quality close to the breakdown of the cosmic ray flux at approximate 60 EeV. To obtain a sufficient statistics we need to build economically reasonable large-area detectors with high duty cycle. The radio detection could be one of the perspective techniques for future investigations of ultra-high energy cosmic rays.
Radio emission from extensive air showers was theoretically predicted [@Askaryan1961; @Kahn1966; @Castagnoli1969; @Hough1973] and first detected [@Jelley; @Vernov; @Allan1971] about 50 years ago. The radio detection techniques became popular in the last decade again, because standard detection methods have reached technological and economical limits: measurements by surface particle detectors depend on models, whose accuracy is limited at high energies due to extrapolation; optical fluorescence and air-Cherenkov detectors are limited by their duty cycle due to duration of dark nights and weather. Thus, a number of modern experiments [@FalckeNature2005; @CODALEMA; @AERA; @LOFAR] aims at obtaining the main properties of extensive air showers, such as arrival direction, energy, shower maximum and primary particle[^1] using the radio detection technique. These experiments proved that the radio emission can be detected from air showers with energies above $10^{17}$ eV, with an angular resolution for the arrival direction better than $1^{\circ}$ [@SchroederLOPES_ARENA2012].
The open question is the precision of the reconstruction for primary energy and shower maximum. Up to now, the experiments have given only upper limits for these quantities ($20\,\%$ for the energy and about $100\,$g/cm^2^ for $X_\mathrm{max}$) [@GlaserAERA_ARENA2012; @RebaiCODALEMAenergy2012; @ApelLOPES_MTD2012; @PalmieriLOPES_ICRC2013]. In a very recent report from the experiment LOFAR, it was estimated that a precision of the $X_{\mathrm{max}}$ reconstruction can reach up to $20$ g/cm$^2$ [@LOFAR_ICRC2013]. This precision would be comparable with the fluorescence technique. The current challenge is to reach a competitive precision with an economic radio array which can be scaled to very large areas. The main goal of Tunka-Rex, the radio extension of the Tunka observatory for air showers, is to answer this question, i.e., to determine the precision for the reconstruction of the energy and the atmospheric depth of the shower maximum based on the cross-calibration with a air-Cherenkov detector. For this purpose, Tunka-Rex is built within the Tunka-133 photomultiplier (PMT) array. The latter is measuring the air-Cherenkov light of air showers in the energy range between $10^{16}$ and $10^{18}\,$eV [@TunkaRICAP2013; @TunkaICRC2013]. Data of both detectors are recorded by a shared data-acquisition system, and the radio antennas are triggered by the photomultiplier measurements. This setup automatically provides hybrid measurements of the radio and the air-Cherenkov signal, and consequently allows us to perform a cross-calibration of both techniques. In particular, we can test the sensitivity of the Tunka-Rex radio measurements for the energy and for $X_\mathrm{max}$ by comparing them to the measurements of the established air-Cherenkov array.
Setup and hardware properties
=============================
Tunka-Rex currently consists of 20 antennas attached by cables to the cluster centers of the Tunka-133 photomultiplier array (Fig. \[fig\_map\]), which is organized in 25 clusters formed by 7 PMTs each.
![Map of Tunka. One Tunka-Rex antenna, consisting of two SALLAs, is attached to each cluster of seven non-imaging photo-multipliers (PMTs). In the season from October 2012 to April 2013, 18 out of 20 Tunka-Rex antennas have been operating. In autumn 2013, five additional antennas will be installed to complete the array of 25 antennas in total.[]{data-label="fig_map"}](fig1.pdf){width="1.0\linewidth"}
The spacing between the antennas in the inner clusters is approximately $200\,$m, covering an area of roughly $1\,$km^2^. At each antenna position there are two orthogonally aligned SALLAs (short aperiodic loaded loop antenna) [@AERAantennaPaper2012] with 120 cm diameter (Fig. \[fig-SALLA\]).
![A Tunka-Rex antenna in front of a Tunka-133 cluster center box and its central PMT.[]{data-label="fig-SALLA"}](fig2.pdf){width="1.0\linewidth"}
Unlike most radio experiments the antennas in Tunka-Rex are not aligned along the north-south and east-west axis, but rotated by 45$^{\circ}$, like in LOFAR [@LOFAR]. Since the radio signal from cosmic ray air showers is predominantly east-west polarized [@polarization], this should result in more antennas with signal in both channels but also less events with signal in at least one channel. The SALLA has been chosen as antenna not only for economic reasons, but also because its properties depend only little on environmental conditions, particularly, the influence of the ground on the antenna gain and, thus, the measurement accuracy, is suppressed by a load attached to the bottom part of the antenna. The signal chain is continued by a low noise amplifier (LNA) placed in the isolated metal box, connected directly to the top of the SALLA; a buried coaxial cable connecting the antenna to the main amplifier and a filter at each cluster center of Tunka-133 hosting a flash ADC board for the digital data acquisition (Fig. \[chain\]).
![Signal chain of a Tunka-Rex antenna with the corresponding transformations.[]{data-label="chain"}](fig3.pdf){width="1.0\linewidth"}
![The mean background spectrum in one night of operation, as measured with Tunka-Rex station 15. Background interferences are under active investigation, more details see Ref. [@HillerTunkaRex_ICRC2013][]{data-label="fig-spectrum"}](fig4.pdf){width="1.0\linewidth"}
Tunka-Rex is triggered by the photomultipliers and records the radio signal from the air showers between $30\,$ and $80\,$MHz, where the signal outside of this band is suppressed with an analog filter (Fig. \[fig-spectrum\]). First, this improves the signal-to-noise ratio of the air-shower radio signals; second, this ensures that we measure entirely in the first Nyquist domain and can fully reconstruct the signal in this frequency band. Each of the clusters features its own local DAQ independently (for an air-shower event a coincidence in at least 3 PMTs is required). There the signal from both, the antennas and the PMTs, is digitized and transmitted to the central DAQ via optical fibers where it is stored on disk (see Fig. \[fig\_daq\]). Then, during the offline analysis, all independent entries from clusters are merged into hybrid events.
Based on the known Tunka-133 and Tunka-Rex hardware properties, particularly, on the differences of cable lengths, we can estimate the window for the radio signal (see Fig. \[fig\_signal\]) $$t = \frac{N_{FADC}}{2}T_{BW} + \frac{L_R - L_C}{\varepsilon c} - \frac{d}{\sqrt{2}c} -\tau \pm \Delta t\,,$$ where $N_{FADC} = 1024$ is the number of FADC records in the trace, $T_{BW} = 5$ ns is the binwidth, $L_R \approx 30$ m, $L_C \approx 90$ m are the cable lengths to radio antennas and PMTs containers respectively, $c \approx 3\cdot 10^8$ m/s the speed of light, $\varepsilon c \approx (2/3)c$ is the signal velocity in the coaxial cable, $d \approx 80$ m is the typical distance between each cluster center and the surrounding PMTs, $\tau \approx 50$ ns is the PMT signal width and $\Delta t \approx d/(\sqrt{2}c) = 200$ ns is the bound for the radio signal window. The first term in this expression just gives the center of the trace (approximate position of the signal from last PMT), the second term is the delay due to the difference of cable lengths, the third term are possible delays due to geometry of the detector (for a typical zenith angle of 45$^\circ$), and the fourth term describes possible delay between shower arrival and triggering (passing amplitude threshold for PMT). Finally we have chosen $\Delta t$ to take into account all possible geometries (i.e. for the range of zenith angles from $0^\circ$ to $70^\circ$) of air showers. Thus, our estimation for the time window of the radio signal is $2000\pm200$ ns.
For more details on the Tunka-Rex hardware and the systematic uncertainties on signal reconstruction, see Ref. [@HillerTunkaRex_ICRC2013].
![Scheme of the data acquisition of Tunka-133/Tunka-Rex. Each cluster contains a local clock. The time for each single cluster event is calculated as cluster time plus the delay of the optical fiber. We assume that signals from the same air-shower are within a 7000 ns time window corresponding to the time needed by the shower front to cover the entire air-Cherenkov detector.[]{data-label="fig_daq"}](fig5.pdf){width="0.85\linewidth"}
Event selection and data analysis
=================================
![Trace of the radio signal from air-showers before correcting for the antenna pattern. We take the first 500 ns (left shaded area) for the noise level estimate and estimate the bounds for the signal window (central shaded area) from cable lengths, hardware delays and detector geometry.[]{data-label="fig_signal"}](fig6.pdf){width="1.0\linewidth"}
Tunka-Rex started operation on 8 October 2012. Since then operates within the Tunka-133 trigger, i.e. in dark moonless nights with good weather excluding the summer months from May to September.
By design, the maximum zenith angle for each PMT illumination is 50$^\circ$ (PMTs are placed inside isolated metal barrels whose top is covered by plexiglas). The zenith angle for triggering can be extended up to 70$^\circ$ due to indirect detection of light reflected from the inner surfaces of barrels. Thus, all radio events are divided in two groups based on geometry:
- “vertical” events: zenith angle $\theta < 50^\circ$ with reconstructed geometry, energy, shower maximum available from the air-Cherenkov detector. These events are good candidates for the cross-calibration. The main disadvantage of them is the low statistics and small number of antennas per event due to the steepness of the lateral distribution of the radio signal.
- “horizontal” (inclined) events: zenith angle $\theta \ge 50^\circ$ with reliable reconstruction of the shower direction from the air-Cherenkov detector. For these events, the other shower parameters cannot be reconstructed from the air-Cherenkov measurements. Thus, if it is possible to reconstruct the shower parameters (energy, $X_{\mathbf{max}}$) from the radio measurements, this could increase the total statistics of usable events at Tunka.
For the exposure and flux estimation from the radio detector we still have to study the background in more details and make performance simulations.
![Zenith angle distribution of radio detected events. The maximum efficiency is reached at the Tunka-133 reconstruction threshold of $\theta \approx 50^\circ$. The statistics at smaller angles is mainly suppressed by the steeply falling lateral distribution of the radio signal, the statistics at larger angles is suppressed by trigger detection capabilities.[]{data-label="fig_angle"}](fig7.pdf){width="0.70\linewidth"}
{width="0.85\linewidth"}
For a first analysis, we used only high quality events which have a clear radio signal (signal$^2$ / noise$^2$ $> 6$) in at least three antennas. Based on the detector specifications (opening angle for the PMT, typical distance between radio stations) one can assume that the maximum efficiency for hybrid events could be reached near the Tunka-133 reconstruction threshold $\theta \approx 50^\circ$ (Fig. \[fig\_angle\]). Due to these reasons, only a small fraction of the air-Cherenkov events have also a clear radio signal (see Fig. \[fig\_exampleEvent\] for an example). Moreover we demand that the direction reconstructed from the arrival times of the radio signal agrees within $5^\circ$ with the direction obtained from the photomultiplier array. This cut excludes most of the background events, which by chance pass the signal-to-noise cut. In future, we plan to develop improved quality criteria based on the radio signal alone, to distinguish real from false events. For analysis of the radio measurements, we use the radio extension of the Offline software framework developed by the Pierre Auger Collaboration [@AugerOffline2007; @RadioOffline2011]. It features a correction of the measured radio signal for the properties of the used hardware and a reconstruction of the electric field-strength vector at each antenna position. Since the absolute calibration of the antennas is still under evaluation we use a simulated pattern for the SALLA.
![Arrival directions of the Tunka-Rex events passing the quality cuts. Due to the geomagnetic effect, the radio signal is expected to be on average stronger for events coming from North, which explains the asymmetry in the detection efficiency: 89 of the 131 events are in the northern half of the plot, and 42 in the southern half.[]{data-label="fig_angularDistribution"}](fig9.pdf){width="0.8\linewidth"}
![Tunka events with energies above $10^{17}\,$eV detected by the PMT array, and those events passing the quality cuts for the radio measurements. The efficiency increases with energy $E$, the sine of the geomagnetic angle $\sin \alpha$, and the zenith angle $\theta$.[]{data-label="fig_efficiency"}](fig10.pdf){width="\linewidth"}
First Results
=============
Until now, we found 49 events with a zenith angle $\theta \le 50^\circ$, and 82 events with $\theta > 50^\circ$ in an effective measurement time of 392 hours (Table \[tab\_EventStatistics\]). Generally, the radio efficiency increases not only with large geomagnetic angles $\alpha$, i.e. the angle between the shower axis and the geomagnetic field[^2] (Figs. \[fig\_angularDistribution\] and \[fig\_efficiency\]), but also with larger zenith angles (Fig. \[fig\_angle\]). In addition, we expect that the event rate will increase when we complete the array to 25 antennas this autumn, and optimize our algorithms for digital background suppression.
-------------------- ----------- ----------------------- ---------------------
effective
measurement period time $\theta \le 50^\circ$ $\theta > 50^\circ$
06 - 23 Nov 2012 56 h 9 11
04 - 23 Dec 2012 65 h 8 12
03 - 21 Jan 2013 114 h 14 23
01 - 17 Feb 2013 87 h 12 22
01 - 17 Mar 2013 70 h 6 14
Total sum 392 h 49 82
-------------------- ----------- ----------------------- ---------------------
: Statistics of Tunka-Rex events passing the quality cuts in dependence of the zenith angle $\theta$, excluding the period from 08 to 24 Oct 2012 used for commissioning of the detector. The effective measurement time and counting rate is limited by the PMT array, i.e. light (moon) and weather conditions.[]{data-label="tab_EventStatistics"}
To test the expected sensitivity of the Tunka-Rex measurements to air shower parameters, in particular to the energy, we reconstructed the lateral distribution of the radio signal for the 49 events with $\theta \le 50^\circ$. In a first approach, we used the shower geometry provided by the denser air-Cherenkov array to calculate the distance from each antenna to the shower axis, and then plotted the maximum absolute value of the electric field-strength vector as function of this axis distance. To estimate the uncertainties of the amplitude measurements and to correct the measured amplitudes for a bias due to background, we used formulas developed for the east-west aligned antennas of LOPES [@SchroederLOPESnoise_ARENA2010], and then fitted an exponential function (Fig. \[fig\_exampleLDF\] and \[fig\_ldf2\])[^3].
![Lateral distribution of the radio amplitude (i.e maximum field strength) for the example event shown in Fig. \[fig\_exampleEvent\]. Light grey are antennas with signal-noise ratio below threshold (correspond to grey crosses in Fig. \[fig\_exampleEvent\]).[]{data-label="fig_exampleLDF"}](fig11.pdf){width="\linewidth"}
Consistent with several historic and modern experiments [@Allan1971; @SchroederLOPES_ARENA2012; @GlaserAERA_ARENA2012; @RebaiCODALEMAenergy2012], the amplitude parameter of the lateral distribution is correlated with the primary energy (Fig. \[fig\_energy\]). However, the analysis is still preliminary, e.g., because of the status of the calibration and because the impact of the background at the Tunka site has to be studied in more detail. Moreover, we expect that the slope of the lateral distribution is sensitive to the position of the shower maximum [@HuegeUlrichEngel2008; @deVries2010], which we will analyze in near future by comparing Tunka-Rex measurements to the $X_\mathrm{max}$ reconstruction of the PMT array Tunka-133. Future work will be dedicated to find an optimal reconstruction algorithm for the energy and $X_\mathrm{max}$, and to test the achievable precision by comparison to the air-Cherenkov measurements. In addition to using the lateral distribution, $X_\mathrm{max}$ might also be obtained form the radio measurements via the shape of the radio wavefront [@Lafebre2010; @SchroederWavefrontICRC2011], or the slope of the frequency spectrum [@GrebeAERA_ARENA2012].
![Typical values of fitted parameters $\varepsilon_{100}$ (amplitude at 100 m), $\eta$ (slope parameter) in comparison with shower parameters obtained from the air-Cherenkov detector.[]{data-label="fig_ldf2"}](fig12.pdf){width="\linewidth"}
![Correlation between the radio field strength at 100 m normalized by sine of geomagnetic angle and the energy reconstructed with the air-Cherenkov measurements. The given values for the field strength are based on a simple exponential fit function and preliminary calibration.[]{data-label="fig_energy"}](fig13.pdf){width="\linewidth"}
Conclusion
==========
As a result of the first weeks of operation, Tunka-Rex registered more than hundred events with significant radio signal from extensive air showers with energies above $10^{17}\,$eV in combination with the Tunka air-Cherenkov array. This shows that the Tunka observatory is able to provide hybrid measurements which is the pre-requisite to perform a cross-calibration between the air-Cherenkov and the radio signal. Our measurements are compatible with the picture that the radio emission originates dominantly from the geomagnetic deflection of the charged particles in the air shower. In future, we plan to optimize the reconstruction techniques, and to compare our measurements to simulations and other experiments. A detailed background study might help to improve the signal-to-noise ratio and to increase the precision of the reconstruction of the shower maximum. Moreover, we plan to trigger Tunka-Rex also by the planned scintillator extension of Tunka [@TunkaICRC2013], and thus can measure also during day to increase the duty cycle by an order of magnitude. Finally, we will test a joint operation with Tunka-HiSCORE [@HiSCORE_RICAP2013; @HiSCORE_ICRC2013] by deploying additional antennas. By this, we can also study to which extent a denser array of radio detectors can increase the detection efficiency and precision for the energy and $X_{\mathbf{max}}$ reconstruction.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge the support of the Russian Federation\
Ministry of Education and Science (G/C 14.518.11.7046,\
14.B25.31.0010, 14.14.B37.21.0785, 14.B37.21.1294), the Russian Foundation for Basic Research (Grants 11-02-00409, 13-02-00214, 13-02-12095, 13-02-10001), the Helmholtz association (grant HRJRG-303).
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J. Maller, these proceedings.
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S. Buitink, (LOFAR Collaboration) Proc. 33rd ICRC, paper 0579, Rio de Janeiro, Brazil (2013)
V. Prosin, these proceedings. N. Budnev, for the Tunka Collaboration, Proc. 33rd ICRC, paper 0418, Rio de Janeiro, Brazil (2013)
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[^1]: The chemical composition of cosmic rays (i.e, the primary particles) can be reconstructed only by indirect methods from air-shower measurements, for example, by combining study of primary energy and shower maximum, which can be obtained by the optical detectors.
[^2]: The field strength $\varepsilon$ mainly depends on the vector product\
$|\mathbf A \times \mathbf B| = |\mathbf A| \cdot |\mathbf B| \sin\alpha$, where $\mathbf A$ is the shower axis and $\mathbf B$ the geomagnetic field, $\alpha = \angle(\mathbf A, \mathbf B)$ is the geomagnetic angle.
[^3]: The exponential fit function has chosen according to the pioneer LOPES and CODALEMA experiments. First, this simple approximation was sufficient for the precision reached on these experiments, second, two parameters in that fit function could be easily connected to shower parameters. By the ongoing experiments and simulations it could be shown that the lateral distribution is more complicated and contains an azimuthal asymmetry due to the interference of the geomagnetic and the Askaryan effect.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
\
(On behalf of the Belle collaboration)\
Laboratoire de Physique des Hautes Énergies,\
École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland\
E-mail:
bibliography:
- 'bib.bib'
title: '$\mathbf{{{B_s^0}}}$ Decays at Belle'
---
Introduction {#introduction .unnumbered}
============
The Belle experiment [@NIMA_479_117], located at the interaction point of the KEKB asymmetric-energy $e^+e^-$ collider [@NIMA_499_1], was designed for the study of $B$ mesons[^1] produced in $e^+e^-$ annihilation at a center-of-mass (CM) energy corresponding to the mass of the ${\Upsilon(4S)}$ resonance ($\sqrt s\approx10.58~{{\hbox{GeV}}}$). After having recorded an unprecedented sample of $\sim800$ millions of $B\bar B$ pairs, the Belle collaboration started to record collisions at higher energies, opening the possibility to study other particles, like the ${{B_s^0}}$ meson. Up to now, ${{L_{\textrm{int}}}}=23.6~{\,{\rm fb}^{-1}}$ of data have been analyzed at the energy of the ${\Upsilon(5S)}$ resonance ($\sqrt s\approx10.87~{{\hbox{GeV}}}$).
The ${\Upsilon(5S)}$ resonance is above the ${{B_s^0}}\bar{{B_s^0}}$ threshold and it was naturally expected that the ${{B_s^0}}$ meson could be studied with ${\Upsilon(5S)}$ data as well as the $B$ mesons are with ${\Upsilon(4S)}$ data. The large potential of such ${\Upsilon(5S)}$ data was quickly confirmed [@PRL_98_052001; @PRD_76_012002] with the 2005 engineering run representing 1.86 ${\,{\rm fb}^{-1}}$. The main advantage with respect to the hadronic colliders is the possibility of measurements of absolute branching fractions. However, the abundance of ${{B_s^0}}$ mesons in ${\Upsilon(5S)}$ hadronic events has to be precisely determined. Above the $e^+e^-\to u\bar u, d\bar d, s\bar s, c\bar c$ continuum events, the $e^+e^-\to b\bar b$ process can produce different kinds of final states: seven with a pair of non-strange $B$ mesons ($B^{\ast}\bar B^{\ast}$, $B^{\ast}\bar B$, $B\bar B$, $B^{\ast}\bar B^{\ast}\pi$, $B^{\ast}\bar B\pi$, $B\bar B\pi$ and $B\bar B\pi\pi$), three with a pair of ${{B_s^0}}$ mesons (${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$, ${{B_s^{\ast}}}\bar{{B_s^0}}$ and ${{B_s^0}}\bar{{B_s^0}}$), and final states involving a bottomonium resonance below the $B\bar B$ threshold [@PRL_100_112001]. The $B^{\ast}$ and ${{B_s^{\ast}}}$ mesons always decay by emission of a photon. The total $e^+e^-\to b\bar b$ cross section at the ${\Upsilon(5S)}$ energy was measured to be $\sigma_{b\bar b}=(302\pm14)$ pb [@PRL_98_052001; @PRD_75_012002] and the fraction of ${{B_s^0}}$ events to be $f_s=\sigma(e^+e^-\to{{{{B_s^{(\ast)}}}{\bar B_s}^{(\ast)}}})/\sigma_{b\bar b}=(19.3\pm2.9)$ % [@PLB_667_1]. The dominant ${{B_s^0}}$ production mode, $b\bar b\to{{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$, represents $f_{{{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}}=\fss$ of the $b\bar b\to{{{{B_s^{(\ast)}}}{\bar B_s}^{(\ast)}}}$ events [@PRL_102_021801].
For all the exclusive modes presented here, the ${{B_s^0}}$ candidates are fully reconstructed from the final-state particles. From the reconstructed four-momentum in the CM, $(E_{{{B_s^0}}}^{\ast},p_{{{B_s^0}}}^{\ast})$, two variables are formed: the energy difference ${{\Delta E}}=E_{{{B_s^0}}}^{\ast}-\sqrt s/2$ and the beam-constrained mass ${{M_{\textrm{bc}}}}=\sqrt{s/4-p_{{{B_s^0}}}^{\ast2}}$. The signal coming from the dominant $e^+e^-\to{{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ production mode is extracted from a two-dimensional fit performed on the distribution of these two variables. The corresponding branching fraction is then extracted using the total efficiency (including sub-decay branching fractions) determined with Monte-Carlo simulations, $\sum\varepsilon{{\mathcal B}}$, and the number of ${{B_s^0}}$ mesons produced via the $e^+e^-\to{{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ process, $N_{{{B_s^0}}}=2\times{{L_{\textrm{int}}}}\times\sigma_{b\bar b}\times f_s\times f_{{{B_s^{\ast}}}\bar{{B_s^{\ast}}}}=(2.48\pm0.41)\times10^6$.
Observation of three new CKM-favored $\mathbf{{{B_s^0}}}$ decays
================================================================
Following our recent measurement of ${{{{B_s^0}}\to D_s^-\pi^+}}$ [@PRL_102_021801], we present for the first time an extension of this analysis which includes decays with photons in the final state and report the first observations of the decays ${{{{B_s^0}}\to D_s^{\ast-}\pi^+}}$, ${{{{B_s^0}}\to D_s^-\rho^+}}$ and ${{{{B_s^0}}\to D_s^{\ast-}\rho^+}}$. The leading process of these three modes is a $b\to c\bar ud$ tree-level transition of order $\lambda^2$ with a spectator $s$ quark. The ${{D_s^-}}$ mesons are reconstructed via three modes : ${{D_s^-}}\to\phi(\to K^+K^-)\pi^-$, ${{D_s^-}}\to K^{\ast0}(\to K^+\pi^-)K^-$ and ${{D_s^-}}\to{{K_S^0}}(\to\pi^+\pi^-)K^-$. Based on the ratio of the second and the zeroth Fox-Wolfram moments [@PRL_41_1581], $R_2$, the $e^+e^-\to u\bar u,d\bar d,s\bar s,c\bar c$ continuum events are efficiently rejected by taking advantage of the difference between their event geometry (jet like, high $R_2$) and the signal event shape (spherical, low $R_2$). The ${{{{B_s^0}}\to D_s^{\ast-}\pi^+}}$ (${{{{B_s^0}}\to D_s^-\rho^+}}$ and ${{{{B_s^0}}\to D_s^{\ast-}\rho^+}}$) candidates with $R_2$ smaller than 0.5 (0.35) are kept for further analysis. A best candidate selection, based on the intermediate particle reconstructed masses, is then implemented to keep only one ${{B_s^0}}$ candidate per event. The ${{M_{\textrm{bc}}}}$ and ${{\Delta E}}$ distribution of the selected candidates for the three modes are shown in Fig. \[fig:dominant\], where the various components of the fit are described.
![\[fig:dominant\] Histograms with the fit results. The left (right) plots shows the ${{M_{\textrm{bc}}}}$ (${{\Delta E}}$) projections for the candidates with ${{\Delta E}}$ (${{M_{\textrm{bc}}}}$) in the ${{B_s^{\ast}}}\bar{{B_s^{\ast}}}$ signal region, The red-dashed (green-dotted, black-dotted, solid-blue) lines represent the signal (peaking ${{B_s^0}}$ background, continuum, total) fitted PDF. (a): ${{{{B_s^0}}\to D_s^{\ast-}\pi^+}}$ candidates. ${{{{B_s^0}}\to D_s^-\pi^+}}$ (${{{{B_s^0}}\to D_s^-\rho^+}}$) events can be distinguished on the right (left) of the signal peak in the ${{\Delta E}}$ plot. (b): ${{B_s^0}}\to D_s^-\rho^+$ candidates. The peaking background comes from ${{B_s^0}}\to D_s^{\ast-}\rho^+$ events. (c): ${{B_s^0}}\to D_s^{\ast-}\rho^+$ candidates.](./proj_dominant.pdf){width="\linewidth"}
The numerical results are summarized in Table \[tab:dom\]. The systematic error coming from $f_s$ is separated from the other source of uncertainties. The ${{{{B_s^0}}\to D_s^{\ast-}\rho^+}}$ mode is the decay of the spin-less ${{B_s^0}}$ to two spin-1 particles. The proportion of longitudinal polarization, $f_L$, is not known, and the efficiencies depend significantly on this parameter. A 13% relative error, denoted “pol.”, is added to the branching fraction uncertainty to take this into account.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Mode $N_{\rm sig}^{{{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}}$ $\sum\varepsilon{{\mathcal B}}$ ($10^{-3}$) $S$ Branching fraction
-------------------------------------- ----------------------------------------------------- --------------------------------------------- -------------- --------------------------------------------------------------------------------
${{{{B_s^0}}\to D_s^{\ast-}\pi^+}}$ $53.4^{+10.3}_{-9.4}$ $9.1\pm0.6$ $8.4\sigma$ ${(
2.4{}^{+0.5}_{-0.4}({\rm stat.})\pm0.3({\rm syst.})\pm0.4(f_s))\times10^{-3}}$
${{{{B_s^0}}\to D_s^-\rho^+}}$ $92.2^{+14.2}_{-13.2}$ $4.4\pm0.3$ $10.6\sigma$ ${(
8.5{}^{+1.3}_{-1.2}({\rm stat.})\pm1.1({\rm syst.})\pm1.3(f_s))\times10^{-3}}$
${{{{B_s^0}}\to D_s^{\ast-}\rho^+}}$ $86.6^{+15.1}_{-14.0}$ $2.7\pm0.2$ $8.7\sigma$ ${(13.0{}^{+2.3}_{-2.1}({\rm stat.})
\pm1.7({\rm syst.})\pm1.7({\rm pol.})\pm1.9(f_s))\times10^{-3}}$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: \[tab:dom\]Signal yields for the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ production process with statistical error, $N_{\rm sig}$, total efficiencies, $\sum\varepsilon{{\mathcal B}}$, statistical significances, $S$, and measured branching fractions for the ${{{{B_s^0}}\to D_s^{\ast-}\pi^+}}$, ${{{{B_s^0}}\to D_s^-\rho^+}}$ and ${{{{B_s^0}}\to D_s^{\ast-}\rho^+}}$ modes.
These first observations confirm the large predominance of the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ production mode as no significant excess is seen in the other two signal regions. Within uncertainties, no deviation from the $SU(3)$-related $B^0$ decays is seen.
First observation of the $\mathbf{CP}$-eigenstate decay $\mathbf{{{{{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta}}}$
===========================================================================================================
${{B_s^0}}$ decays to $CP$ eigenstates are important for $CP$-violation parameter measurements [@PRD_63_114015] and preliminary results about the first observation of ${{{{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta}}$ [@hepex_0905_2959v1] are reported. The ${{J\!\!/\!\!\psi}}$ candidates are formed with oppositely-charged electron or muon pairs, while $\eta$ candidates are reconstructed via the $\eta\to\gamma\gamma$ and $\eta\to\pi^+\pi^-\pi^0$ modes. A mass (mass and vertex) constrained fit is then applied to the $\eta$ (${{J\!\!/\!\!\psi}}$) candidates. If more than one candidate per event satisfies all the selection criteria, the one with the smallest fit residual is selected. The main background is the continuum, which is reduced by requiring $R_2<0.4$. The combined ${{M_{\textrm{bc}}}}$ and ${{\Delta E}}$ distributions are presented in Fig. \[fig:jpsieta\]. The ${{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta,\,\eta\to\gamma\gamma$ and ${{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta,\,\eta\to\pi^+\pi^-\pi^0$ candidates are fitted separately. Table \[tab:jspieta\] presents the numerical results.
![\[fig:jpsieta\]Projection of the ${{{{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta}}$ candidates (points with error bars) and the fitted PDF (solid line) in the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ signal region. The sub-modes $\eta\to\gamma\gamma$ and $\eta\to\pi^+\pi^-\pi^0$, which are fitted separately, are summed in these plots. The blue-dotted line represents the continuum component of the PDF. The small peak in the ${{M_{\textrm{bc}}}}$ plot is the ${{B_s^{\ast}}}\bar{{B_s^0}}$ contribution, as the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ signal range for ${{\Delta E}}$ overlaps the one for ${{B_s^{\ast}}}\bar{{B_s^0}}$ signal. ](./jpsieta_deltae.pdf){height="3.8cm"}
![\[fig:jpsieta\]Projection of the ${{{{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta}}$ candidates (points with error bars) and the fitted PDF (solid line) in the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ signal region. The sub-modes $\eta\to\gamma\gamma$ and $\eta\to\pi^+\pi^-\pi^0$, which are fitted separately, are summed in these plots. The blue-dotted line represents the continuum component of the PDF. The small peak in the ${{M_{\textrm{bc}}}}$ plot is the ${{B_s^{\ast}}}\bar{{B_s^0}}$ contribution, as the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ signal range for ${{\Delta E}}$ overlaps the one for ${{B_s^{\ast}}}\bar{{B_s^0}}$ signal. ](./jpsieta_mbc.pdf){height="3.8cm"}
${{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta,\,\eta\to\gamma\gamma$ ${{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta,\,\eta\to\pi^+\pi^-\pi^0$
----------------------------------------------------------------------- -------------------------------------------------------------- -----------------------------------------------------------------
${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ Signal Yield $12.1\pm3.8$ $5.6\pm2.5$
Stat. Significance 5.9$\sigma$ 4.0$\sigma$
${{\mathcal B}}({{{{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta}})$ (combined)
: \[tab:jspieta\]${{{{B_s^0}}\to{{J\!\!/\!\!\psi}}\,\eta}}$ results: yields, significances and branching fraction.
Observation of $\mathbf{{{{{B_s^0}}\to K^+K^-}}}$ and searches for $\mathbf{{{{{B_s^0}}\to\pi^+\pi^-}}}$, $\mathbf{{{{{B_s^0}}\to K^+\pi^-}}}$ and $\mathbf{{{B_s^0}}\to{{K_S^0}}{{K_S^0}}}$
============================================================================================================================================================================================
Finally, we present our results for the ${{{{B_s^0}}\to K^+K^-}}$, ${{{{B_s^0}}\to K^+\pi^-}}$, ${{{{B_s^0}}\to\pi^+\pi^-}}$ and ${{B_s^0}}\to{{K_S^0}}{{K_S^0}}$ charmless decays. The ${{{{B_s^0}}\to K^+K^-}}$ mode is particularly interesting because it can be used for the determination of the CKM angle $\gamma$ [@PLB_459_306] and may be sensitive to New Physics [@PRD_70_031502]. The charged pion and kaon candidates are selected using charged tracks and identified with energy deposition, momentum and time-of-flight measurements. The ${{K_S^0}}$ candidates are reconstructed via the ${{K_S^0}}\to\pi^+\pi^-$ decay, by selecting two oppositely-charged tracks matching various geometrical requirements [@phd_ffang]. A likelihood based on a Fisher discriminant using 16 modified Fox-Wolfram moments [@PRL_91_261801] is implemented to reduce the continuum, which is the main source of background.
We do observe a 5.8$\sigma$ excess of $24\pm6$ events in the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ region for the ${{{{B_s^0}}\to K^+K^-}}$ mode (Fig. \[fig:kk\]). The branching fraction ${{\mathcal B}}({{{{B_s^0}}\to K^+K^-}})={{(3.8{}^{+1.0}_{-0.9}({\rm stat.})\pm0.7({\rm syst.}))\times 10^{-5}}}$ is derived. On the other hand, no significant signal is seen for the other modes. Including the systematics uncertainties, we set the following upper limits at 90% confidence level: ${{\mathcal B}}({{{{B_s^0}}\to\pi^+\pi^-}})<{{1.2\times10^{-5}}}$, ${{\mathcal B}}({{{{B_s^0}}\to K^+\pi^-}})<{{2.6\times10^{-5}}}$ and, assuming ${{\mathcal B}}({{B_s^0}}\to{{K_S^0}}{{K_S^0}})={{\mathcal B}}({{B_s^0}}\to{{K_L^0}}{{K_L^0}})$, ${{\mathcal B}}({{{{B_s^0}}\to K^0\bar K^0}})<{{6.6\times 10^{-5}}}$. The later is the first limit set for the ${{{{B_s^0}}\to K^0\bar K^0}}$ mode. All the other values are compatible with the CDF results [@PRL_97_211802; @PRL_103_031801].
![\[fig:kk\]Projections of the ${{{{B_s^0}}\to K^+K^-}}$ candidates (points with error bars) and the fitted PDF (solid blue line) in the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ signal region. The solid-red (dotted-grey) line represents the signal (continuum) component of the PDF.](./bskk_mbc.pdf){height="4cm"}
![\[fig:kk\]Projections of the ${{{{B_s^0}}\to K^+K^-}}$ candidates (points with error bars) and the fitted PDF (solid blue line) in the ${{B_s^{\ast}}}{{{\bar B_s}^{\ast}}}$ signal region. The solid-red (dotted-grey) line represents the signal (continuum) component of the PDF.](./bskk_deltae.pdf){height="4cm"}
Conclusion {#conclusion .unnumbered}
==========
All these studies demonstrate the great potential of the Belle data set recorded at ${\Upsilon(5S)}$ energy. The sensitivity obtained for several ${{B_s^0}}$ modes allows many interesting studies. The branching fraction determinations of several important modes is ongoing and eight new measurements have been reported here. So far, the full Belle sample has reached 100 ${\,{\rm fb}^{-1}}$, and the KEKB collider may continue delivering collisions at the ${\Upsilon(5S)}$ energy during the fall 2009. Of course, many more interesting results are expected with the full Belle ${\Upsilon(5S)}$ data set.
[^1]: The notation “$B$” refers either to a $B^0$ or a $B^+$. Moreover, charge-conjugated states are implied everywhere.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The majority of stars are thought to form in clusters. Cluster formation in dense clumps of molecular clouds is strongly influenced, perhaps controlled, by supersonic turbulence. We have previously shown that the turbulence in regions of active cluster formation is quickly transformed by the forming stars through protostellar outflows, and that the outflow-driven protostellar turbulence is the environment in which most of the cluster members form. Here, we take initial steps in quantifying the global properties of the protostellar turbulence through 3D MHD simulations. We find that collimated outflows are more efficient in driving turbulence than spherical outflows that carry the same amounts of momentum. This is because collimated outflows can propagate farther away from their sources, effectively increasing the turbulence driving length; turbulence driven on a larger scale is known to decay more slowly. Gravity plays an important role in shaping the turbulence, generating infall motions in the cluster forming region that more or less balance the outward motions driven by outflows. The resulting quasi-equilibrium state is maintained through a slow rate of star formation, with a fraction of the total mass converted into stars per free fall time as low as a few percent. Magnetic fields are dynamically important even in magnetically supercritical clumps, provided that their initial strengths are not far below the critical value for static cloud support. They contain an energy comparable to the turbulent energy of the bulk cloud material, and can significantly reduce the rate of star formation. We find that the mass weighted probability distribution function (PDF) of the volume density of the protostellar turbulence is often, although not always, approximately lognormal. The PDFs of the column density deviate more strongly from lognormal distributions. There is a prominent break in the velocity power spectrum of the protostellar turbulence, which may provide a way to distinguish it from other types of turbulence.'
author:
- 'Fumitaka Nakamura and Zhi-Yun Li'
title: Protostellar Turbulence Driven by Collimated Outflows
---
Introduction {#intro}
============
We investigate the role of protostellar outflows in cluster formation. The formation of stellar clusters is important to study for at least two reasons. First, the majority of stars are thought to form in clusters (Lada & Lada 2003; Allen et al. 2006). This follows from the observations that most stars are born in giant molecular clouds (GMCs) and that in nearby GMCs such as the Orion molecular clouds where systematic surveys are available the majority of young stellar objects are found in clusters. Just as importantly, it is in clusters that most, if not all, massive stars are produced. Understanding cluster formation is likely a prerequisite to understanding massive star formation.
Outflows play an important role in star formation in general and cluster formation in particular. Individually, they may drive the accretion in the inner disk (Königl & Pudritz 2000), brake the stellar rotation (Shu et al. 2000), and perhaps help defining the most fundamental quantity of a star—the mass (Nakano et al. 1995; Matzner & McKee 2000; Shu et al. 2004). Collectively, they have the potential to replenish the energy and momentum dissipated in a star-forming cloud. This possibility was first examined in detail by Norman & Silk (1980). They envisioned the star-forming clouds to be constantly stirred up by the winds of optically revealed T Tauri stars. The idea was strengthened by the discovery of molecular outflows (McKee 1989), which point to even more powerful outflows from the stellar vicinity during the embedded, [*protostellar*]{} phase of star formation (Lada 1985; Bontemps et al. 1996). Shu et al. (1999) estimated the momentum output from protostellar outflows based on the Galactic star formation rate, and concluded that it is sufficient to sustain in all molecular gas a level of turbulence at $\sim 1 - 2$ km/s, not far from the line widths actually observed in typical GMCs. If the majority of stars are formed in localized parsec-scale dense clumps that occupy a small fraction of the GMC volume (Lada et al. 1991), their ability to influence the dynamics of the bulk of the GMC material will probably be reduced; other means of turbulence maintenance may be needed in regions of relatively little star formation, as concluded by Walawender et al. (2005) in the case of the Perseus molecular cloud. The concentration of star formation should, however, make the outflows more important in the spatially limited, but arguably the most interesting, regions of a GMC—the regions of cluster formation, where the majority of stars are thought to form.
We have made a start in simulating cluster formation in magnetized parsec-scale dense clumps including outflows (Li & Nakamura 2006). We find that, in agreement with previous work (Mac Low et al. 1998; Stone et al. 1998; Padoan & Nordlund 1999), the initial turbulence that the clump inherits from its formation process decays away quickly. It is replaced by the motions driven by the protostellar outflows associated with star formation. It is in this protostellar outflow-driven turbulence (“protostellar turbulence” for short hereafter) that the majority of the cluster members are produced. In this picture, quantifying the protostellar turbulence becomes a pressing issue. The current investigation is a step in this direction.
An important issue that we seek to address is how fast stars form in a cluster. The star formation rate (SFR) can be constrained by observations. For example, Lada et al. (1996) derived an average SFR of ${\dot M}_*\approx 4.5\times 10^{-5}$ M$_\odot$ yr$^{-1}$ for the nearby embedded cluster associated with the reflection nebula NGC 1333 in the Perseus molecular cloud, based on an estimate of the total stellar mass ($\sim 45$ M$_\odot$) and the duration of star formation ($\sim 10^6$ years; see also Aspin 2003). It is to be compared with the limiting rate expected in the case of unimpeded star formation (where most of the cloud mass is converted into stars in one free-fall time) $${\dot M}_{\rm ff}= {M\over {\bar t}_{\rm ff}} = \left({32 G\over 3\pi}
\right)^{1/2} \left({M\over L}\right)^{3/2} = 3.91 \times 10^{-3}
\left({M\over 10^3 M_\odot}\right)^{3/2} \left({1 {\rm pc} \over
L}\right)^{3/2} M_\odot {\rm yr}^{-1},
\label{sfr_ff}$$ where $M$ and $L$ are the mass and size of the region, and ${\bar
t}_{\rm ff}$ is the free-fall time at the average density ${\bar
\rho}=M/L^3$. In the case of NGC 1333, Warin et al. (1996) carried out a detailed mapping of the molecular gas in $^{13}$CO and C$^{18}$O, and estimated a mass of 2900 M$_\odot$ from $^{13}$CO and 950 M$_\odot$ from C$^{18}$O in an area of 650 arcmin$^2$, corresponding to a size $L=2.60$ pc for an adopted distance of 350 pc. The two masses yield ${\dot M}_{\rm ff}=4.62\times 10^{-3}$ and $8.66\times 10^{-4}$ M$_\odot$ yr$^{-1}$, respectively. For a smaller core region of 140 arcmin$^2$ in area (or $L=1.20$ pc), a mass of 450 M$_\odot$ is estimated from C$^{18}$O, yielding ${\dot M}_{\rm ff}=8.93\times 10^{-4}$ M$_\odot$ yr$^{-1}$. Note that the values of SFR from C$^{18}$O are comparable on the two scales; the decrease in mass for the core is nearly offset by a decrease in size. These three values of ${\dot M}_{\rm ff}$ are some 20-100 times larger than the actual SFR inferred by Lada et al. (1996). Unless most stars in the NGC 1333 cluster form within a period of time much shorter than $10^6$ years (which is not supported by the presence of relatively evolved Class II YSOs and the detailed analyses of Lada et al. \[1996\] and Aspin \[2003\], although a mini-burst of star formation involving a fraction of the stars, particularly Class 0 objects, appears to be ongoing in this deeply embedded cluster \[Bally et al. 1996\]), the SFR must be reduced by at least an order of magnitude below the free-fall value in order to match observations. This conclusion is strengthened by the recent HCO$^+$ and N$_2$H$^+$ observations of Walsh et al. (2006), which yield a cluster-wide mass accretion rate of order $10^{-4}$ M$_\odot$ yr$^{-1}$. The rate is comparable to the star formation rate ${\dot M}_*$ and much lower than the free-fall rate ${\dot M}_{\rm ff}$.
The ratio of the actual to limiting star formation rates, ${\dot
M}_*/{\dot M}_{\rm ff}$, has a simple physical meaning: the fraction of the cloud mass turned into stars in one free-fall time. It is the dimensionless star formation rate ${\rm SFR}_{\rm ff}$ defined in Krumholz & McKee (2005) and Krumholz & Tan (2006). The latter authors estimated values of a few percent for ${\rm SFR}_{\rm ff}={\dot M}_*/{\dot
M}_{\rm ff}$ in three types of objects of different characteristic densities, including giant molecular clouds, infrared dark clouds, and Galactic and extragalactic dense gas traced by HCN. The values, while uncertain, are compatible with the range that we estimated above ($\sim 1\%-5\%$) for NGC 1333. The small values for SFR$_{\rm ff}$ imply that star formation does not proceed at the limiting free-fall rate in molecular clouds in general (Zuckerman & Evans 1974) and localized cluster-forming regions in particular. It must be retarded somehow.
Cluster formation may be retarded by magnetic fields. The magnetic field is particularly well observed in OMC1—the nearest region of ongoing massive star formation. It has a hourglass shape (produced perhaps by gravitational contraction; Schleuning 1998; Houde et al. 2004), a line-of-sight field strength of $360\pm 80\ \mu$G from CN Zeeman measurements (Crutcher 1999). If we adopt an inclination angle of the field line of $\vartheta=65^\circ$ to the line of sight determined by Houde et al. (2004), the total field strength would become $\sim 850 \mu$G, which would imply a global magnetic flux-to-mass ratio $\sim 2.6$ times below the critical value for cloud support by a static magnetic field. The ratio was estimated based on the column density measured along the line of sight, which should be higher than the column density along the minor axis of the (elongated) clump. Correcting for this projection effect would bring the intrinsic flux-to-mass ratio closer to the critical value. Crutcher (2005) reviewed the available observational data on other regions of cluster/massive star formation and concluded that their magnetic fields, when detected, appear to have strengths not far from the critical values as well. We are thus motivated to include in our simulations moderately strong magnetic fields, with strengths up to half of the critical value. We do not consider magnetically subcritical clouds, where star formation is enabled by ambipolar diffusion, a process not treated in the present paper.
Supersonic turbulence can also slow down (global) star formation. A major concern is that the turbulence tends to decay quickly, even in the presence of a strong magnetic field (e.g., Mac Low et al. 1998; Stone et al. 1998; Padoan & Nordlund 1999). Unless the cloud is short-lived, the observed supersonic turbulence must be replenished. A common practice is to drive the turbulence in Fourier space. The extent to which such a treatment is adequate for the turbulence observed in molecular clouds remains to be ascertained (e.g., Elmegreen & Scalo 2004). In particular, it is unclear whether the treatment is applicable to the turbulence in regions of active cluster formation, which we have argued previously has a unique origin: it is driven from within by protostellar outflows accompanying star formation (Li & Nakamura 2006; see also Cunningham et al. 2006 and Matzner 2007). The goal of the paper is to quantify this outflow-driven turbulence and the rate of star formation in it through numerical simulation.
The rest of the paper is organized as follows. In § 2, we describe the formulation of the physical model, including the simulation setup and numerical code used. The results of the simulations are presented in § 3 and § 4. We discuss the results and their implications in § 5, and summarize our main conclusions in § 6.
Model Formulation {#setup}
=================
Initial and Boundary Conditions
-------------------------------
We consider a self-gravitating cloud in a cubic box of length $L$ on each side. Standard periodic conditions are imposed at the boundaries. As in Li & Nakamura (2006), we choose an initial density distribution that is centrally condensed. The distribution is motivated by the fact that embedded clusters, such as NGC 1333 (Ridge et al. 2003) and the Serpens core (Olmi & Testi 2002), are often surrounded by envelopes with density decreasing away from the cluster. Compared with the oft-used uniform density distribution, it has the advantage of better isolating the central region of active cluster formation gravitationally from its mirror images introduced by the periodic boundary condition. The adopted functional form for the density profile is $$\rho (r) = \left\{
\begin{array}{ll}
\frac{\displaystyle \rho_0}{\displaystyle 1+(r/r_0)^2}, & r \le r_e=L/2\\
\frac{\displaystyle \rho_0}{\displaystyle 1+(r_e/r_0)^2}, & r > r_e=L/2
\end{array} \right.
\label{density}$$ where $\rho_0$ is the peak density at the center and $r$ the spherical radius. The two characteristic radii, $r_0$ and $r_e$, measure the sizes of the central plateau region and the cloud as a whole. We adopt $r_0=r_e/3$, corresponding to a central-to-edge density contrast of 10. The density in the region between the cubic simulation box and the sphere of radius $r_e$ is held constant.
To ensure that the cloud contains many Jeans masses for fragmentation into a cluster, we choose a size for the simulation box that is 9 times the Jeans length at the cloud center $$L_J= \left({\pi c_s^2/G \rho_0}\right)^{1/2}.$$ The isothermal sound speed $c_s=2.66\times 10^4 (T/20 K)^{1/2}$ cm/s, where $T$ is the cloud temperature, and the central density $\rho_0 =
4.68\times 10^{-24} n_{H_2,0}$ g cm$^{-3}$, with $n_{H_2,0}$ being the central number density of molecular hydrogen, assuming 1 He for every 10 H atoms. We adopt a fiducial value $L=1.5$ pc, typical of the dimensions of cluster-forming clumps. It corresponds to a Jeans length $L_J=L/9=0.17$ pc. Scaling the box size $L$ by 1.5 pc, we obtain a central number density, $$n_{H_2,0}=2.69 \times 10^4 \left(\frac{T}{20 K}\right)
\left(\frac{1.5{\rm pc}}{L}\right)^2 {\rm cm}^{-3},$$ and a total cloud mass, $$M_{\rm tot}=9.39\times 10^2 M_\odot \left(\frac{T}{20 K}\right)
\left(\frac{L}{1.5{\rm pc}}\right)$$ inside the computation domain. The mass is also typical of the nearby cluster-forming clumps for the fiducial parameters (e.g., Ridge et al. 2003).
From the Jeans length and sound speed, we define a gravitational collapse time $$t_g= {L_J \over c_s} = 6.12 \times 10^5 \left({L\over 1.5{\rm pc}}
\right) \left({ 20 K\over T}\right)^{1/2} ({\rm years}).
\label{gravtime}$$ It is longer than the free fall time at the cloud center $$t_{\rm ff}=\left({3\pi \over 32 G \rho_0}\right)^{1/2} = 1.88 \times
10^5 \left({L\over 1.5{\rm pc}} \right) \left({ 20
K\over T}\right)^{1/2} ({\rm years}),
\label{freefalltime}$$ by a factor of 3.27. The time $t_g$ is $26\%$ longer than the free fall time ${\bar t}_{\rm ff}$ at the average cloud density ${\bar \rho}=0.15
\rho_0$.
We impose a uniform magnetic field along the $x$ axis at the beginning of simulation. The field strength is specified by the parameter $\alpha$, the ratio of magnetic to thermal pressure at the cloud center, through $$B_0=4.73 \times 10^{-5} \alpha^{1/2} \left({T\over 20 K}\right)
\left({1.5{\rm pc}\over L}\right) ({\rm Gauss}).
\label{fieldstrength}$$ In units of the critical value $2\pi G^{1/2}$ (Nakano & Nakamura 1978), the flux-to-mass ratio in the central flux tube is $$\Gamma_0 = { 2^{1/2} \alpha^{1/2} \over 3\pi\; \tan^{-1}(r_e/r_0)}
=0.12 \alpha^{1/2}.
\label{centralgamma}$$ The dimensionless flux-to-mass ratio for the cloud as a whole is ${\bar \Gamma}=0.33 \alpha^{1/2}$, which is nearly three times larger than the central value. The envelope is more strongly magnetized than the central region.
In addition to the magnetic field, we include in our simulation a modest level of rotation. Although slow rotation is generally difficult to measure in a strongly turbulent cluster-forming environment, it has been claimed in some cases, such as the Serpens core (Olmi & Testi 2002). The inferred level of rotation is typically not high enough to prevent the collapse of the cloud as a whole. It could become more important on smaller scales. We assume that the clump rotates slowly around the $z$ axis (perpendicular to the initial magnetic field lines), with a profile $$V_{\rm rot} = \left\{
\begin{array}{ll}
\frac{\displaystyle 3 (\varpi/r_0) c_s} {\displaystyle
1+(\varpi/r_0)^2}, & \ \ \ \varpi < r_e \\ 0, & \ \ \ \varpi
\geq r_e \end{array}\right.
\label{rotation}$$ where $\varpi$ is the distance from the rotation axis that passes through the center of the simulation box. The rotation speed peaks at a cylindrical radius $\varpi=r_0=L/6$, with a maximum value of $1.5 c_s$. It is set to zero at the simulation boundaries to satisfy the periodic condition. This distribution of rotation speed is used in all simulations.
We stir the cloud at the beginning of the simulation with a turbulent velocity field of power spectrum $v_k^2 \propto k^{-3}$ and a mass weighted rms Mach number ${\cal M}=10$. Following the standard practice (e.g., Ostriker et al. 2001), random realization of the power spectrum is done in Fourier space. The turbulence is allowed to decay freely, except for the feedback from protostellar outflows. We have experimented with other choices for the initial turbulence, including different random realizations of the same power spectrum and different power-laws for the spectrum (e.g., $v_k^2\propto k^{-1}, k^{-2}$, and $k^{-4}$), and obtained qualitatively similar results. In particular, the fully developed protostellar turbulence is insensitive to the variations in the initial turbulence.
Prescriptions for Stellar Mass and Outflow {#prescription}
------------------------------------------
How the mass of a star is determined is a long-standing, unresolved problem. Low-mass stars are observed to form in dense cores of molecular clouds. There is evidence that only a fraction of the core material eventually ends up inside stars. For example, Onishi et al. (2002) derived an average virial mass of about 5$M_\odot$ for the H$^{13}$CO$^+$ cores in the Taurus molecular clouds. This mass is an order of magnitude higher than that of a typical star formed in the region, which is only $\sim 0.5~M_\odot$ (Kenyon & Hartmann 1995). Dense cores in more crowded cluster-forming regions are more difficult to resolve observationally. They may convert a higher fraction of their mass into stars (Motte et al. 1998), although this is uncertain. In regions far from massive stars, the masses of low-mass stars are probably determined by the competition between protostellar infall and outflow, neither of which is fully understood. In our treatment, we will leave the stellar mass as a quantity to be prescribed.
To specify the stellar mass, we adopt the following recipe for mass extraction. When the density in a cell crosses a threshold $\rho
_{\rm th}= 100 \rho_0$, we define around it a “supercell” that includes all cells in direct contact with the central cell, either through a surface, a line or a point. The supercell is thus a cubic region having 3 cells on each side and 27 cells in total. From each of the 27 cells, we extract 20% of the mass and put it in a Lagrangian particle located at the supercell center; the particle represents a formed “star.” The percentage of the mass extracted is chosen to yield a stellar mass $M_*\approx
0.5~M_\odot$, typical of low mass stars. We have examined the physical conditions of the gas in a large number of supercells, and found the gas to be self-gravitating and well on the way to star formation. At formation, we assume that the star moves with the velocity of its host cell. Its subsequent evolution through the cluster potential, which includes contributions from both gas and stars[^1], is followed numerically.
Following Matzner & McKee (2000), we assume that each star injects into the ambient medium a momentum that is proportional to the stellar mass $M_*$. We denote the proportionality constant by $P_*$, and normalize it by $100$ km/s. In the Appendix, we find a likely range for the scaled outflow parameter $f=P_*/100$ km/s between $\sim 0.1$ and $\sim 1$. We will adopt $f=0.5$ as our fiducial value, corresponding to a round number $P_*=50$ km/s, which is close to the value 40 km/s adopted by Matzner & McKee (2000) in their semi-analytic model of cluster formation. For comparison, we will also discuss in some depth a simulation with weaker outflows specified by $P_*=25$ km/s.
We assume that the outflow momentum $M_* P_*$ ejected by a star of mass $M_*$ is shared instantaneously by the material left in the supercell immediately after the stellar mass extraction. This prescription is similar to the one used by Allen & Shu (2000) in their simulation of outflow-driven motions in critically magnetized, sheet-like GMCs. In our initial study (Li & Nakamura 2006), we let the momentum-carrying material in the supercell move radially away from the center, with the same speed in all directions. However, there is ample observational evidence and strong theoretical arguments for the protostellar outflows being collimated in general (Bachiller & Tafalla 1999; Shu et al. 1995). Here, we go beyond the simplest prescription, and adopt a two-component structure for the outflow, with each component serving a distinct purpose: a collimated, “jet” component that facilitates the energy and momentum transport to large distances, surrounded by a spherical component that reverses the local infall after a star is formed. As in Allen & Shu (2000), we pick the direction of the magnetic field in the central cell as the jet axis. Included in the jet component are all cells of the supercell whose centers lie within 30$^\circ$ of the axis; the remaining cells are assigned to the spherical component. The jet carries a fraction $\eta$ of the total outflow momentum $M_* P_*$; the remaining fraction is carried by the spherical component. The material in each component is driven away from the center radially, with a speed given by the momentum divided by the mass in that component.
The above prescriptions for stellar mass and outflow, while idealized, serve as a reasonable first approximation in our view. Possible future refinements are described in § \[improvement\].
Numerical Code
--------------
The MHD equations that govern the cloud evolution are solved using a 3D MHD code based on an upwind TVD scheme. To ensure that the magnetic field is divergence free, we replace the magnetic field $\mbox{\boldmath
$B$}$ by $\mbox{\boldmath $B$}^{\rm new} = \mbox{\boldmath $B$}-\nabla
\Phi_B$ (where $\nabla^2 \Phi_B = \nabla\cdot\mbox{\boldmath $B$}$) after each time step. The code does not contain any artificial viscosity, which allows the shocks to be captured sharply. We have tested the code against standard shock tube problems and the problem of point explosion. The tests show that our code is second-order accurate in both space and time, and that it suppresses a numerical instability that appears just behind a shock front for non-TVD code like the Lax-Wendroff code \[see e.g., Fig. 7 of Nakamura et al. (1999) and Fig. 7 of Nakamura (2000)\]. The 3D code is an extension of the well-tested 2D codes that we have used in several previous studies (e.g., Nakamura et al. 1999 and Li & Nakamura 2002).
We solve the Poisson equation for gravitational potential using fast Fourier transform, taking advantage of the periodic boundary conditions imposed. As is the standard practice, we set the $k = 0$ component of the gravitational potential to zero in Fourier space. This is equivalent to using $\rho-\bar{\rho}$ (where $\bar{\rho}$ is the mean density of the cloud), rather than $\rho$, as the source term in the Poisson equation. Formed stars are treated separately from the gas, as Lagrangian particles. Their equations of motion are solved using a symplectic method.
In the simulations to be presented in this paper, we adopt a uniform grid of $128^3$. This resolution is relatively low compared to some recent simulations of MHD turbulence in molecular clouds (e.g., Li et al. 2004). However, our simulations include a new ingredient—protostellar outflows—and are followed to a time significantly later than the previous simulations. Both the outflow and longer evolution exacerbate the time step problem associated with large Alfvén speeds in rarefied regions. Restricting ourselves to a relatively modest resolution allows us to carry out an exploration of parameters essential to the problem. Furthermore, our grid does contain more than 2 million cells, which enable us to resolve the global structure of the outflow-driven turbulence reasonably well. We are running a couple of higher resolution simulations with $256^3$. Preliminary analyses show that our main results do not depend sensitively on resolution.
Protostellar Outflow-driven Turbulence: the Standard Model {#outflowsection}
==========================================================
A number of parameters are needed to fully specify our simulations. In this paper, we will concentrate on three that most directly affect the properties of the protostellar outflow-driven turbulence. These include the parameters $f$ and $\eta$, which characterize the strength and collimation of the outflow respectively, and the ratio of magnetic-to-thermal energy at the cloud center $\alpha$, which specifies the degree of cloud magnetization. Other parameters, such as those specifying the initial mass distribution and turbulent velocity field, are kept fixed to facilitate model comparison. In this section, we focus on a “standard” model with $\alpha=2.5$, $f=0.5$ and $\eta=0.75$ (Model S0 in Table 1). The choice $\alpha=2.5$ yields a dimensionless flux-to-mass ratio of 0.19 (in units of the critical value $2\pi
G^{1/2}$) in the central flux tube and 0.52 for the cluster-forming clump as a whole. The clump is therefore magnetically supercritical, although only moderately so for the bulk of the cloud material. The choice $f=0.5$ corresponds to a relatively strong outflow, and $\eta=0.75$ means that the outflow momentum is dominated by the relatively narrow “jet” component. The combination of model parameters is chosen to yield a relatively low rate of star formation. This simulation will serve as the standard against which other simulations, listed in Table 1 and to be discussed in the next section, are compared.
Star Formation Rate
-------------------
One of the most important quantities of cluster formation is the star formation efficiency (SFE hereafter), defined as the ratio of the mass of all stars to the total mass of stars and gas. It determines, among other things, whether a cluster would remain gravitationally bound upon gas removal. In Fig. \[SFEstandard\], we plot the SFE of the standard model as a function of time, in units of the gravitational time $t_g$, which is $26\%$ longer than the average free-fall time ${\bar t}
_{\rm ff}$. The first several stars form in quick succession around $0.4~t_g$, followed by a period of relative quiescence. Episodes of similar mini-bursts of star formation are also evident at later times. By the last time shown in Fig. \[SFEstandard\] ($t=2~t_g$ or $1.22 [L/1.5{\rm pc}][20K/T]^{1/2}$ Myrs), a total of 114 stars have formed, with an accumulative SFE of $\sim 6\%$. The average mass of a star is $\sim 0.5 \ M_\odot (T/20 K) (L/1.5{\rm pc})$, typical of low-mass stars, as mentioned earlier. From the fact that within a time interval $\sim 1.6~t_g$ (or $\sim
2~{\bar t}_{\rm ff}$) since the formation of the first star $\sim
6\%$ of the gas has been converted into stars, we obtain a star formation efficiency per free-fall time ${\rm SFR}_{\rm ff}\approx 3\%$. It is in the range of star formation rate inferred for the embedded cluster NGC 1333 and other types of objects (see discussion in § \[intro\]). At this rate, it takes some $33$ free-fall times to completely deplete the gas in the cluster forming region through star formation[^2]. To understand why the star formation is slowed down to such a remarkable extent, we examine in some depth the internal structure and dynamics of the cluster-forming clump, starting from global quantities such as the total energy and momentum. The effects of the magnetic field are discussed separately in the next section (§ \[magfield\]).
Evolution of Scalar Momentum and Gravitational Energy
-----------------------------------------------------
The momentum is more useful to keep track of than the kinetic energy in our standard simulation. This is because the kinetic energy can easily be dominated by the fast moving outflows, especially the jet components. Impulsive injection and rapid dissipation of the outflow energy generate large variations in the total kinetic energy, even though the kinetic energy of the bulk cloud material away from the active outflows remains relatively constant. The total (scalar) momentum is not expected to vary as strongly, since the outflows interact with the ambient medium in a momentum-conserving fashion. In Fig. \[momentum\], we show the evolution of the total (scalar) momentum divided by the total mass of the gas. The ratio is the specific momentum or simply the mass-weighted turbulent speed, which we will denote by $v_{\rm turb}$. It provides a better measure of the turbulent speed of the bulk material than the oft-used mass-weighted rms velocity; the latter is more dominated by active outflows. As expected, the specific momentum decreases rapidly at the beginning of the simulation, as a result of the momentum-canceling (head-on) collisions that develop in the highly compressive initial turbulent velocity field. The decline is stopped by the formation of stars, whose outflows appear to have kept the turbulent speed $v_{\rm turb}$ close to $\sim5\ c_s$ at late times. The near constancy of $v_{\rm
turb}$ indicates that the turbulent gas has reached a quasi-equilibrium state.
The specific momentum of the turbulent gas is to be compared with the specific momentum injected into the gas by outflows. The latter is given by $v_{\rm inj}= M_{*,{\rm tot}} P_*/M_{\rm gas}$, where $M_{*,{\rm tot}}$ and $M_{\rm gas}$ are the total masses of the stars and gas, respectively, and $P_*$ is the outflow momentum per unit stellar mass. If only a small fraction of the gas is converted into stars, as in the standard simulation, we have $v_{\rm inj}\approx {\rm SFE}\times P_*$, where SFE is the star formation efficiency. For the adopted $P_*=50$ km/s, we have $v_{\rm inj} \approx $3 km/s at the end of the simulation ($2\ t_g$), when ${\rm SFE} \approx 6\%$. It is about twice the mass-weighted turbulent speed $v_{\rm turb}\approx 1.5$ km/s at the same time. In Fig. \[momentum\], we show the speed $v_{\rm inj}$ as a function of time along with the turbulent speed $v_{\rm turb}$. From the slope of the $v_{\rm inj}$ curve, we estimate the amount of specific momentum injected into the cloud by outflows per gravitational time $t_g$ to be $\sim 7\ c_s$. At this rate, it takes $\sim0.7\ t_g$ (or $\sim0.9\ {\bar
t}_{\rm ff}$) to replenish the equilibrium specific momentum $v_{\rm turb}\approx 5\ c_s$. In a steady state, this should also be the momentum dissipation time.
Another indication that a quasi-equilibrium state is reached at late times comes from the evolution of the absolute value of the gravitational energy[^3] per unit mass, $E_g$, shown in Fig. \[gravenergy\]. The initial increase in $E_g$ is caused by turbulence dissipation, which leads to cloud contraction, starting from an initial mass distribution that is already somewhat centrally condensed. The contraction is slowed down, and eventually arrested by the outflows associated with star formation. The near constancy of the gravitational energy at late times implies that the system is neither collapsing nor expanding rapidly as a whole. It signals that a quasi-equilibrium state has been reached. In what follows, we shall examine in some detail the structure and dynamics of the quasi-equilibrium state, focusing on a representative time $1.5~t_g$, when 80 stars have formed and the outflow-driven protostellar turbulence appears fully developed.
Quasi-Equilibrium State {#decay}
-----------------------
Global quantities such as the total scalar momentum and gravitational energy do not capture the full complexity of the gas dynamics. This complexity is illustrated in Fig. \[velvector\], where we show a color map of the density distribution, with velocity vectors and contours of the gravitational potential superposed. Naively, one might expect the gas to slide down the gravitational potential well towards the bottom more or less freely, producing an infall-dominated velocity field that leads to rapid star formation at the limiting free-fall rate given by equation (1). Infall motion does appear in some regions, but not in others: many patches have more or less coherent motions in directions around or away from the bottom of the potential well. The patchy appearance of the velocity field is a general feature of the protostellar turbulence that we also see at other times and in other simulations. The patches are often separated by regions of enhanced density, which are probably created by converging flows. An important question is: are the infall and outflow motions more or less balanced globally so as to keep the self-gravitating system close to a dynamical equilibrium?
To address the above question quantitatively, we show in Fig. \[velPDF\] the mass weighted probability distribution function (PDF) for the radial component of the velocity, $v_r$, towards or away from the location of minimum gravitational potential. The PDF peaks around $v_r\approx -1~c_s$, and a simple integration yields that $61\%$ of the total mass has a negative radial velocity and is thus infalling. The outflowing gas moves somewhat faster on average, however. If we define an average infall (outflow) speed from the total inward (outward) momentum divided by the total infalling (outflowing) mass, we obtain a value of $3.99\ c_s$ for outflow and $-2.40\ c_s$ for infall. The faster outflow speed is consistent with the fact that the PDF shows a stronger wing towards positive $v_r$. The [*net*]{} radial velocity, defined as the [*net*]{} total radial momentum divided by the total mass, is only $0.08\ c_s$, much smaller than either the average infall or outflow speed. It is only a small fraction of the sound speed, indicating the outflow and infall motions are nearly balanced globally, leaving the system in a rough, dynamical equilibrium in the radial direction. The balance is generally true at other times as well, although the net radial speed can go up to several tenths of the sound speed.
Mass Distribution
-----------------
It is well known that supersonically turbulent media are clumpy, with a wide range of densities. In Fig. \[SFRff\](a), we plot the mass weighted PDF for the volume density at the representative time. The distribution appears roughly lognormal, peaking around $\sim 0.6~\rho_0$ (or $\sim 4$ times the average density), broadly consistent with previous simulations (e.g., Ostriker et al. 2001), despite the fact that in our simulation the supersonic turbulence is created and maintained in a potential well of considerable depth and gravity plays a more dominant role. There is some deviation at the high density end, where the PDF is dominated by dense cores and filaments, some of which are self-gravitating and are on the verge of collapse to form a new generation of stars. In Fig. \[SFRff\](b), we plot the fraction of the total mass that resides in regions above a given density. At low densities, the mass fraction asymptotes to a value slightly below unity, because a few percent of the mass has already been converted into stars at the time under consideration. It decreases quickly towards the high density end. Only $12\%$ of the mass resides in regions denser than $5~\rho_0$, and this fraction drops to $5.8\%$ above $10~\rho_0$. The rapid decrease in the amount of mass available for star formation towards the high density end is a key ingredient in the scenario of turbulence regulated star formation (e.g., Elmegreen & Scalo 2004; Krumholz & McKee 2005).
To gauge the effect of the mass distribution on the star formation rate, we plot in Fig. \[SFRff\](b) the SFR in the limit that all mass above a given density $\rho$ is converted into stars in one free fall time at the average density of that mass. The limiting rate is normalized by the global free-fall rate defined in equation (1), which in our simulation has a value of $${\dot M}_{\rm ff} = 4.32\times 10^2\ {c_s^3\over G},$$ much larger than the classical value of $0.975 c_s^3/G$ for the inside-out collapse of a singular isothermal sphere (Shu 1977). The distribution of the free-fall rate has a broad peak around $0.4~\rho_0$. At lower densities, it increases with density, as a result of mass clumping into moderately overdense regions which, by itself, worsens the problem of rapid star formation. At densities above the peak, the mass fraction drops off with density faster than inverse square root of the density, leading to a decline of the free fall rate. For comparison, we note that the actual rate of star formation in the simulation is $\sim 0.03 {\dot M}_{\rm ff}$, or ${\dot M}_* \approx 13\
c_s^3/G$.
We next consider the distribution of the column density, which is more accessible to direct measurement than the volume density. In Fig. \[coldenPDF\], we plot the column density PDFs at the representative time along the three axes. The PDFs along the $y$- and $z$-axes are broader than that along the $x$-axis. The broader distributions are caused by mass settling along the field lines; when viewed perpendicular to the field lines, the column density is enhanced in the plane of mass concentration and decreased away from it. The PDFs deviate significantly from lognormal distributions, especially towards the high density end. The deviation is larger than those found by, for example, Ostriker et al. (2001) for decaying turbulence. One difference is that our turbulence is driven, indeed in a specific way—by collimated outflows. Perhaps more importantly, our prescription of star formation enables us to run the simulation longer, which allows more time for the cloud material to condense gravitationally into cores and filaments. In addition, the global gravitational potential well is deeper in our simulation. The cluster-wide gravity tends to concentrate dense cores and filaments towards the bottom of the potential well, increasing their chance for overlap, especially when viewed along the plane of mass concentration perpendicular to the large scale magnetic field. The overlap skews the PDF towards the high column density end.
To examine the spatial distribution of the mass more quantitatively, we plot in Fig. \[Dprofile\] the average density as a function of radius at the representative time. The averaging is done in concentric shells centered on the location of minimum gravitational potential. We exclude in the log-log plot the central part of the cloud where the number of grid cells is small and the shell radius is ill-defined. There is a clear trend for the averaged density outside the central (excluded) region to drop off with radius, in an approximately power-law fashion. A similar drop-off is found at the two other times (1.0 and 2.0 $t_g$) shown in Fig. \[Dprofile\]. A power-law $\rho\propto
r^{-1.5}$ is also plotted for comparison. Although the power-law provides a fair (although not unique) description of the averaged density distribution, it should be kept in mind that the medium is very clumpy. The clumpiness is illustrated vividly in Fig. \[velvector\], where the volume density on a representative slice through the computational domain is displayed.
Velocity Power Spectrum
-----------------------
The power spectrum of a turbulence in the inertial range (between the energy input and dissipation scales) can often be approximated by a power law, $E_k= k^2
v_k^2 \propto k^{-n}$, where $k$ is the wavenumber. The power index $n$ holds clues to the nature of the turbulence. For example, the incompressible Kolmogorov turbulence has $n=5/3$. For the shock-dominated Burgers turbulence, the index is $n=2$. Some simulations of driven turbulence obtained a power index $n\approx
1.74$ for isothermal gas (Boldyrev et al. 2002), which is closer to the Kolmogorov than Burgers value. The exact value of the power index depends, however, on model parameters, particularly the degree of cloud magnetization (Vestuto et al. 2003). In our standard simulation, the power spectra deviate strongly from a single power-law. The deviation is illustrated in Fig. \[spectra\], where the spectra at three times (1.0, 1.5 and 2.0 $t_g$) are plotted. They all appear to have a break around $k_b \approx
0.6\; (2 \pi/L_J)$. Below $k_b$, the spectra are rather flat. They steepen to approximately $k^{-2.5}$ above the break. The shape of the spectra indicates that the bulk of the power resides near the break rather than at the smallest wavenumber.
The break $k_b$ in the power spectrum corresponds to a characteristic length scale $L_b=2\pi k_b^{-1} \approx 1.7 L_J$, which is about 1/5 of the size of the simulation box. The scale $L_b$ can plausibly be identified with the typical outflow length $L_f$. A crude estimate of $L_f$ comes from the distance that the (conical) jet component of the outflow travels before it is slowed down to the ambient turbulent speed ($v_{\rm turb} \approx 5 c_s$). Assuming a constant ambient density at the average value ($0.15 \rho_0$), we obtain $L_f \approx
2.8 L_J$, which is about 65% larger than $L_b$. However, the stars are formed preferentially in dense regions near the bottom of the potential well. The higher-than-average ambient density should lower the estimated outflow length somewhat, bringing it to a closer agreement with the characteristic break length. The outflow may be further shortened by magnetic tension if it propagates perpendicular to the large scale magnetic field. These considerations lead us to believe that the break in the power spectrum is produced by the momentum injection from protostellar outflows, although the strong inhomogeneity in the mass distribution makes it difficult to predict the length of any individual outflow accurately. Some outflows may be trapped close to where they are produced, while others may cross the entire cloud through largely empty regions. Matzner (2007) independently argued for the existence of a break in the power spectrum using a similar reasoning.
Breaks are also seen in the power spectra of the turbulence driven in Fourier space. For example, Vestuto et al. (2003) drove their turbulence with a power peaking at a scale that is 1/8 of the box size. They found a break near the driving wavenumber. In the inertia range above the break, they found that the power index $n$ decreases with increasing magnetic field strength. Our standard simulation has a field strength comparable to their model B, which has an estimated power index $n=2.0$. This index is smaller than that in our simulation. A potential cause for the difference is that their model B has a resolution higher than ours by a factor of 2, and Vestuto et al. have shown that a higher resolution tends to yield a smaller index. However, our preliminary analysis of a higher resolution ($256^3$) simulation indicates that in the inertial range the power index is essentially the same as that of our current simulation. The difference may instead result from the fact that self-gravity is included in our simulations but not in theirs and, perhaps more importantly, that our turbulence is driven by discrete, highly anisotropic outflows rather than isotropically in Fourier space. Anisotropy due to the collimation of individual outflows and the correlation of outflow directions is an important feature that distinguishes our turbulence simulations from the others and this feature is usually not taken into account in Kolmogorov-type dimensional analyses; the effects of anisotropic driving remain to be fully explored. The difference in turbulence driving is also reflected in the flattening of our power spectra at the highest wavenumbers (see Fig. \[spectra\]). The flattening is most likely produced by the spherical component of the outflow, which supplies energy on small scales.
The relatively flat power spectrum at small wavenumbers below the break $k_b$ is more interesting. Some of the power may come from inverse cascade. However, the inverse cascade in the simulations of Vestuto et al. (2003) produced an $E_k$ below the break wavenumber that decreases quickly towards small $k$ roughly as $E_k \propto k^2$. In contrast, the $E_k$ in our simulation generally remains flat or continues to increase slowly towards small $k$. We believe that most of the power below the break $k_b$ is supplied directly by collimated outflows rather than through inverse cascade. Movies of the standard simulation show clearly that many outflows can break out from the dense cores surrounding their driving sources and propagate to large distances, sometimes across the entire simulation box, injecting energy and momentum on the largest possible scale. Since the outflow propagation is well resolved in our simulations, we believe that the general conclusion that the power spectrum $E_k$ flattens at small wavenumbers is robust, although the details may depend on the outflow treatment and may be affected by the periodic boundary condition.
The power spectrum of the protostellar turbulence can be decomposed into a solenoidal and compressible component. We find that the solenoidal component always dominates the compressible component. This is in qualitative agreement with previous simulations. Quantitatively, the solenoidal component is larger by a factor of $\sim 10$ at wavenumbers above the break $k_b$. Below the break, the factor is somewhat lower. Overall, the ratio of the two components is comparable to those found by Boldyrev et al. (2002) and Vestuto et al. (2003), who drove their turbulence by a purely solenoidal velocity field. The similarity indicates that the driving of protostellar turbulence by collimated outflows is probably mostly solenoidal, as one may expect from the large velocity shear between the fast outflow and the ambient medium.
Stellar Component {#stars}
-----------------
The simple prescription adopted in our model for star formation makes detailed comparison between the predicted stellar properties and observations premature at this stage of model development. Nevertheless, there are a few general features of the model stars that are worth noting. These include the spatial distribution of the stars relative to the gas, and their velocity dispersion.
We illustrate the stellar distribution in Fig. \[starmap\], where the positions of the 80 stars at the representative time $1.5~t_g$ are projected onto the $x$-$y$ plane. Also plotted in the figure are contours of the column density along the $z$-axis, which delineate the gas distribution. The densest regions appear to form an “S-shaped” ridge. The elongation is caused mainly by mass settling along the large-scale magnetic field lines, which run more or less horizontally in the plot. There are two separate concentrations of dense gas, with the main one near the center of the column density map, and the secondary one to the lower right. The majority of stars are clustered around the main gas concentration. Specifically, slightly more than half of all stars (41 out of 80) are located within a radius $0.75~L_J$ (or 0.125 pc for the fiducial parameters) of the center. In this localized region, the SFE is $15.7\%$, much higher than that of the clump as a whole ($4.2\%$). The less massive concentration of dense gas is forming a smaller group of stars.
The degree of stellar clustering is expected to depend on age. To show the age dependence, we divide the stars into two groups: those born before and after $1~t_g$. They have 34 and 46 members respectively. The two groups of stars are represented by two different symbols and colors in Fig. \[starmap\]. It is clear that the younger stars are more closely associated with the dense gas at the current time and the older stars are more spread out, as expected. Part of the reason for the older stars to be more widely dispersed is that they have had more time to spread out. Another part is that the older stars are bound less tightly to the cluster-forming region as a whole. Indeed, 12 of the 34 older stars (or 35%) have positive total (kinetic minus gravitational) energies, and are thus formally unbound to the cluster at the time under consideration. The unbound fraction goes down to 8.7% (or 4 out of 46) for the younger stars. The difference in the unbound fraction comes primarily from the tendency for the older stars to locate higher up in the potential well. As a result, they are able to move around more freely. If the gas in the cluster-forming region that gravitationally binds the majority of the stars together were to be removed suddenly, the stellar cluster would dissolve quickly.
An important quantity that can in principle be directly measurable is the velocity dispersion of the stars. In Fig. \[dispersion\], we plot the velocity dispersion $\sigma_*$, defined as the rms value of the stellar velocities relative to the mean[^4], as a function of time for the standard model. Except for a brief initial period (when the number of stars is still small), the dispersion $\sigma_*$ ranges from $\sim 4$ to $\sim 6$ $c_s$. It is comparable to the mass-weighted average turbulent speed $v_{\rm turb}$ of the gas, which is also shown in the figure for comparison. Even though $\sigma_*$ and $v_{\rm turb}$ are defined somewhat differently, the fact that they are comparable is significant: both speeds reflect the depth of the global gravitational potential well. In particular, there is a significant increase in both $\sigma_*$ and $v_{\rm turb}$ starting around $1~t_g$. Their increase appears to track the increase in the absolute value of the gravitational energy around the same time (see Fig. \[gravenergy\]).
Before leaving the section on the standard model, we comment briefly on the dense cores, a number of which show up prominently in Fig. \[starmap\] (see also Fig. \[velvector\]). These cores are the “crown jewels” of the protostellar turbulence—the basic units of individual star formation. They tend to cluster near the bottom of the potential well. Whether they are created mostly in-situ near the bottom is unclear; some of them could be produced higher up in the potential well and later “sink” to the bottom. In any case, the spatial concentration of dense cores increases the chance for core-core interaction, particularly coalescence, which can build up the core mass. One the other hand, the mass of a core may be limited by gravitational collapse and disruption through outflows associated with star formation. The resulting core mass spectrum may hold the key to the determination of IMF (Motte et al. 1998). We will postpone a detailed investigation of these core-related topics (including the role of outflow triggering) to future, higher resolution studies.
Variations on the Standard Model
================================
Spherical versus Collimated Outflows
------------------------------------
Protostellar outflows are collimated, particularly during the earliest, Class 0 phase of (low-mass) star formation. It is during this phase that the bulk of the mass of a star is assembled. If the outflows are driven by the gravitational energy release associated with mass accretion, then the bulk of their energy and momentum is expected to be injected into the ambient medium during this phase. This expectation is consistent with the observations of Bontemps et al. (1996), which showed that molecular outflows—the ambient material set into motion by the outflow-ambient interaction—are most powerful for the Class 0 sources; they tend to become weaker and broader with time (Bachiller & Tafalla 1999). It is the momentum carried in the powerful molecular outflows in the early stages of star formation that we seek to capture with our outflow prescription (see the Appendix); the optical jets observed at later times (Bally & Reipurth 2001) and possibly the more tenuous wide-angle winds that are predicted to surround the jets in the magnetocentrifugal wind theory (Shu et al. 1995) are not included explicitly in our current model; their inclusion would only make the feedback from star formation stronger.
We illustrate the effects of outflow collimation using three variants of the standard model. These variants include a spherical model that does not have a jet component at all (Model E1 in Table 1), and two models with weaker jet components (Model E2 with a jet momentum fraction $\eta=0.25$, and Model E3 with $\eta=0.50$). Other quantities, including the dimensionless parameter $f$ for outflow strength, are kept the same as in the standard model. In Fig. \[SFEf05\], we plot the time evolution of the SFE for all four models. The SFEs remain relatively close together until about $1.0~t_g$. Between $1.0$ and $2.0~t_g$, there is a clear trend that the SFE decreases as the strength of the jet component increases relative to the spherical component. In particular, the SFE increases from $\sim 0.02$ to $\sim 0.20$ in the spherical model and to only $\sim 0.06$ in the standard jet model. The difference between the two in the total mass of the stars formed during this time interval is a factor of $\sim 4.5$. We conclude that collimated outflows are more efficient in suppressing star formation in the protostellar turbulence than the spherical outflow carrying the same amount of momentum.
The above difference in SFE can be understood in terms of where the outflow momentum is deposited. Collimated outflows can propagate well outside the dense regions near the bottom of the potential well, where most stars form. They deposit a large fraction of their momenta in the outer envelope, where most mass resides. Spherical outflows, on the other hand, are more easily trapped. They tend to drive turbulent motions on a smaller scale, which are dissipated more quickly. The higher rate of turbulence dissipation in the spherical outflow model is compensated by a faster turbulence replenishment, through a higher rate of star formation. To fuel the higher star formation rate, more material must be concentrated into dense regions that can collapse to form stars. The mass concentration is reflected in the PDFs of the volume and column densities, both of which tend to be strongly skewed towards the high value end compared with lognormal distributions. The column density PDF is particularly noteworthy. In addition to a main peak similar to that shown in Fig. \[coldenPDF\] for the standard jet case, it often displays a broad shoulder or even a second peak to the right of the main peak, as illustrated in Fig. \[coldenPDF\_sph\]. Such strong deviations from a lognormal distribution may be detectable in regions of active cluster formation when the outflows are trapped relatively close to their driving sources.
Effects of Outflow Strength {#magstrength}
---------------------------
The exact value of the outflow parameter $f$ is uncertain. In the Appendix, we have estimated a plausible range of $P_*=
10-100$ km/s for the outflow momentum per solar mass of stellar material, corresponding to $f=0.1-1$. To illustrate the effects of outflow strength, we have rerun the standard simulation but with a lower ($f=0.25$, Model F1) and higher ($f=0.75$, Model F2) outflow strength. The SFEs of these simulations are plotted in the upper panel of Fig. \[SFEf\]. There is a clear trend for the SFE to increase with decreasing outflow strength. Indeed, the SFE is nearly inversely proportional to the outflow strength parameter $f$, so that their product is roughly the same for all three cases, as shown in the lower panel of the figure. Apparently, the total amount of momentum injected into the cloud is insensitive to the strength of individual outflow: the reduction in the momentum supplied per outflow is more or less compensated by the increase in the number of stars (and thus outflows). In contrast, the total amount of kinetic energy injected increases roughly linearly with the outflow strength. The difference is consistent with the expectation that the momentum of protostellar outflow is more directly relevant for turbulence replenishment than the energy; the latter dissipates more readily.
Model F1 with $f=0.25$ is particularly interesting. It demonstrates that even relatively weak outflows are capable of driving a robust protostellar turbulence in which the SFR is reduced well below the limiting free-fall rate. In the reminder of the subsection, we will discuss this model in some detail, and contrast it with the standard model, starting with the energy evolution. As shown in Fig. \[energyf025\], the total kinetic energy drops quickly, before being pumped up by the outflows associated with the (bursty) star formation. Compared with the standard model, the kinetic energy here is more comparable with the gravitational energy, because it is less dominated by the (weaker) active outflows. The absolute value of the gravitational energy is somewhat higher than that in the standard model, indicating that the bulk of the cloud material is more tightly bound. It increases initially, as a result of gravitational settling of the mass in the cluster-forming region towards the bottom of the potential well due to turbulence dissipation. The increase is stopped around $1~t_g$ when enough motions are generated by outflows to arrest further contraction. There is some undulation at later times, which is also evident in the standard model (see Fig. \[gravenergy\]). The existence of such mild oscillations in the gravitational energy may not be too surprising, given the bursty nature of the star formation. The amplitude of the oscillation is small, again indicating that the cluster-forming system is hovering close to an equilibrium.
We have examined the quasi-equilibrium state of the weaker outflow model at the representative time $1.5~t_g$, and found it similar to that of the standard model discussed in § \[outflowsection\]. In particular, the turbulent velocity field contains many distinct patches of more or less coherent motions. The average infall and outflow speeds are $-1.86$ and $2.57\ c_s$, respectively. The net average radial speed is only $1.09\times 10^{-2}\ c_s$, much smaller than the average infall and outflow speeds and the sound speed. The infall and outflow momenta nearly balance each other, with the slower infall speed compensated by a larger amount of infalling mass, as in the standard model. The mass weighted PDF of the volume density can again be fitted reasonably well with a lognormal distribution. The PDFs of the column density deviate more strongly from lognormal, especially along directions perpendicular to the initial magnetic field direction. A prominent break is present in the velocity power spectrum, as in the standard case. The similarities lead us to conclude that the gross properties of the protostellar turbulence are insensitive to the outflow strength.
Effects of Magnetic Field {#magfield}
-------------------------
We illustrate the effects of the magnetic field using three models of different field strengths. They are Models M1, M2, and S0 in Table 1, specified by $\alpha=10^{-6}$, $0.5$ and $2.5$, respectively. For the standard model ($\alpha=2.5$), the dimensionless mass-to-flux ratio ${\bar \Gamma}=0.52$ for the clump as a whole, as mentioned earlier. This ratio drops to $0.23$ and $3.3\times 10^{-4}$ for $\alpha=0.5$ and $10^{-6}$. In all three cases, the ratio is substantially less than unity, indicating that the global magnetic field is not strong enough to suppress star formation altogether. Stars are indeed formed in all three cases, as shown in Fig. \[mag\], where the SFEs of the models are plotted. There is a clear trend for the SFE to decrease with increasing field strength, as one might expect. Specifically, the magnetic field in the standard model ($\alpha=2.5$) has reduced the rate of star formation by a factor of $\sim 2.2$ compared to the negligible field case ($\alpha=10^{-6}$). Even the relatively weak field in the $\alpha=0.5$ model appears to have a significant effect on the rate of star formation, reducing it by a factor of $\sim 1.6$.
A simple, albeit crude, way to gauge the dynamical importance of a magnetic field is to compare its energy to the kinetic energy of the gas. In our case, the comparison is complicated by the fact that the kinetic energy is often dominated by active outflows. For example, in the standard simulation with $\alpha=2.5$, $81\%$ of the energy is carried by $8.6\%$ of the mass that moves faster than $10\ c_s$ at the representative time $1.5~t_g$. The remaining energy is carried by the bulk, more slowly moving material. It has a value of $8.80
\times 10^2$ in units of $\rho_0 c_s^2 L_J^3$, corresponding to a mass-weighted rms speed of $4.55\ c_s$, comparable to the specific (scalar) momentum (see Fig. \[momentum\]). This portion of kinetic energy is smaller than the total magnetic energy at the same time ($2.63\times 10^3\
\rho_0 c_s^2 L_J^3$). However, the magnetic energy is dominated by the background uniform field, which accounts for $1.82\times 10^3\
\rho_0 c_s^2 L_J^3$. The remaining $8.06\times 10^2\ \rho_0 c_s^2
L_J^3$, carried by the distorted magnetic field, is remarkably close to the kinetic energy carried by the bulk of the turbulent material away from the active outflows ($8.80 \times 10^2\ \rho_0 c_s^2
L_J^3$). The similarity indicates that an energy equipartition is reached between the distorted magnetic field and the turbulent motions of the bulk material for the standard model, which has a moderately strong magnetic field to begin with.
The weaker field case of $\alpha=0.5$ (Model M2) is more intriguing. At the beginning of the simulation, the magnetic energy is only $3.65\times 10^2\ \rho_0 c_s^2 L_J^3$, well below the kinetic energy. By the representative time $1.5~t_g$, it has nearly quadrupled to $1.38\times 10^3\ \rho_0 c_s^2 L_J^3$. Most of this energy is stored in the distorted magnetic field, which accounts for $1.02\times 10^3
\rho_0 c_s^2 L_J^3$. The distorted field energy is higher than its counterpart in the standard model by $27\%$, despite the fact that the standard model is more strongly magnetized to begin with. This energy is again close to the kinetic energy carried by the bulk of the cloud material that moves at a speed below $10\ c_s$ ($1.11\times 10^3\ \rho_0 c_s^2
L_J^3$). Apparently, the magnetic energy has been amplified to an equipartition level in this initially weaker field case. Part of the amplification comes from the concentration of mass towards the bottom of the potential well, which drags the field lines into pinched configurations. The pinched field lines are evident in Fig. \[3Dfield\], which shows the field structure and isodensity surfaces in 3D. More dramatic amplification comes from the stretching of field lines by fast moving outflows, which creates large magnetic distortions that are relatively long lived. Further amplification comes from the turbulent motions of the (slower) bulk material. These processes remain to be quantified.
The amplification factor is even larger for the weakest field model of $\alpha=10^{-6}$. The initial magnetic energy is $7.29\times
10^{-4}\ \rho_0 c_s^2 L_J^3$. It increases to $0.46\ \rho_0 c_s^2
L_J^3$ at $1.5~t_g$, by a factor of $630$. Despite the large enhancement factor, the magnetic energy is still orders of magnitude below the kinetic energy, indicating that the magnetic field is dynamically unimportant in this extreme case. We conclude that magnetic fields are dynamically important in protostellar turbulence as long as their strengths are not much below the critical value to begin with. The role of magnetic fields is discussed further towards the end of § \[nature\] below.
Discussion
==========
The Nature of Protostellar Turbulence {#nature}
-------------------------------------
A salient feature of protostellar turbulence is the simultaneous existence of infall and outflow motions. Fluid parcels are pushed up the gravitational potential well by outflows. Once slowed down, they are pulled back towards the bottom of the potential well by gravity, setting up a vigorous circulation of material between the dense central region where most stars form and the outer envelope, where most of the mass resides. The circulation is in a way reminiscent of convection, although the fluid motions are highly supersonic, unlike the conventional buoyancy-driven convection. In this picture, the gravity plays a role as important as the outflows. It drives infall motions that close the global circulation, enabling the system to reach a quasi-equilibrium state, in which the infall and outflow motions are globally balanced. Simultaneous infall and outflow are observed in a number of embedded clusters, including NGC 1333 (Walsh et al. 2006; Knee & Sandell 2000), the Serpens cloud core (Williams & Myers 2000; Olmi & Testi 2002; Davis et al. 1999), and NGC 2264 (Peretto et al. 2006; Williams & Garland 2002; Wolf-Chase et al. 2003). Such observations underpin our notion of cluster formation in outflow-driven protostellar turbulence.
There are several lines of evidence for a quasi-equilibrium state in all of our simulations. First, the global infall and outflow momenta nearly cancel, yielding a net mass-weighted velocity in the radial direction much smaller than the average infall or outflow velocity overall. There are, however, regions where one dominates the other and the equilibrium is locally upset. Second, both the specific momentum and gravitational energy approach, and oscillate with a small amplitude around, a constant value at late times, indicating that an equilibrium has been reached. The total kinetic energy is not as good an indicator of equilibrium, since it is typically dominated by fast moving transients. The kinetic energy for the regions away from active outflows is, however, comparable to the gravitational energy, suggesting a rough virial equilibrium for the bulk material. In addition, the spherically averaged density distribution as a function of radius appears to settle into a power-law of roughly constant power index in the envelope. The density PDFs and velocity power spectra also maintain similar shapes at late times. All these lines of evidence point to the existence of a quasi-equilibrium state with fully developed protostellar turbulence; it is in such an environment that the majority of the cluster members form.
Stars form at a relatively slow rate. In our standard model, only $\sim 3\%$ of the gas is converted into stars per free fall time at the average density. This corresponds to a remarkably long gas depletion time of $\sim 33$ free fall times. The exact value of the star formation rate depends on several factors, including the strength and degree of collimation of the outflow, as well as the initial magnetic field strength (and perhaps topology, which is not explored here). Although there is considerable uncertainty in estimating each of these factors, we believe that the general conclusion that outflows can maintain the turbulence in a cluster forming clump is robust. At a fundamental level, there is enough momentum in the outflows to replenish the momentum dissipated in the turbulence for a reasonable star formation efficiency of order $10\%$, as stressed by McKee (1989) and Shu et al. (1999), among others, as long as the momentum dissipation time is not much shorter than the free fall time. Our detailed simulations allowed us to estimate the dissipation time self-consistently. For the set of models with relatively strongly magnetic fields ($\alpha
=2.5$) and collimated outflows (Models S0, M1 and M2), the average dissipation time, as measured by the ratio of the equilibrium (scalar) momentum and the rate of momentum input, is close to the free-fall time ${\bar t}_{\rm ff}$ at the average density, more or less independent of the outflow strength. The dissipation time increases with the strength of the magnetic field and the degree of outflow collimation.
The velocity field of protostellar turbulence is dominated by the solenoidal (or shear) component. Such a velocity field is perhaps to be expected, given that there is a large shear between the collimated outflows and the ambient medium, and that any fast compressive motions that may have also been generated by the interaction are readily dissipated in shocks. Indeed, the large ratios of the kinetic energies in the solenoidal and compressive components obtained in our simulations are similar to those obtained in Boldyrev et al. (2002) and Vestuto et al. (2003), where the turbulence is driven isotropically in Fourier space using a purely solenoidal velocity field. Our protostellar turbulence is a special case of driven turbulence: it is driven by discrete, fast-moving, collimated outflows (see also Mac Low 2000), that come from the protostars formed preferentially near the bottom of the potential well. For turbulence driven in Fourier space, the dissipation rate depends on the scale of driving (e.g., Mac Low 1999), with those driven on small scales decaying faster than those driven on large scales. A well recognized problem with driving in Fourier space is that it affects the gas in the entire computation box simultaneously (e.g., Elmegreen & Scalo 2004). The more physical outflow-driving acts sequentially on a range of scales, from near the protostars to large distances, and anisotropically, both on the local scale of individual star formation and on the global scale of cluster-forming clump; the latter comes about because the outflow orientations are not completely random in the presence of a strong magnetic field. The amounts of energy and momentum deposited on different scales depend on both the intrinsic properties of the outflows (such as the flow speed, degree of collimation, duration, etc), and the (generally anisotropic) distributions of mass and magnetic field in both the dense cores that surround the outflow-driving protostars and the general turbulent background. Despite these differences, our finding that the turbulence driven by collimated outflows decays more slowly than that driven by spherical outflows is qualitatively consistent with the previous result, since collimation effectively increases the scale of driving.
Magnetic fields can influence the protostellar turbulence in several ways. If strong enough, they can flatten the mass distribution and affect, perhaps even control, the directions of outflow ejection and propogation. In our simulations, we find that the energy stored in the distorted magnetic field is comparable to the kinetic energy of the bulk cloud material away from active outflows, as long as the field strength is not far below the critical value to begin with; relatively weak fields are amplified by gas motions to an equipartition value. An implication is that the turbulent motions for the bulk of the gas is roughly Alfv[é]{}nic, and the magnetic field is dynamically important. Indeed, the field may be crucial in transmitting the outflow energy and momentum to regions not directly impacted by outflows, through large amplitude Alfv[é]{}n waves. It may also prevent dense fragments from moving freely in the clump potential, since they need to drag along the ambient material linked to them through magnetic field lines (Elmegreen 2006). These effects remain to be fully quantified.
Approximate equipartition between the magnetic and turbulent kinetic energies is inferred in regions with Zeeman measurements of the magnetic field strength (e.g., Myers & Goodman 1988; Crutcher 1999). For the energy in the measured (ordered) magnetic field to be comparable to the turbulent kinetic energy, the flux-to-mass ratio probably needs to lie within a factor of a few of the critical value (as in the standard simulation). Such a ratio may be obtained naturally if the cluster-forming clumps are created quickly out of a more diffuse, magnetically subcritical medium through strong shocks, driven perhaps by HII regions or supernova explosions. The strong C-shock may reduce the flux-to-mass ratio in the shocked layer below the critical value, inducing a relatively rapid subsequent evolution (collapse and star formation) that freezes the flux-to-mass ratio at a value somewhat below the critical value. In this picture of externally triggered clump formation out of a (perhaps moderately) magnetically subcritical cloud (to be quantified elsewhere), the magnetic energy would be automatically comparable to the turbulent kinetic energy, and be a significant factor in regulating star formation in clusters.
To summarize, the protostellar turbulence in active regions of cluster formation is outflow-driven, gravity-assisted, and magnetically mediated.
Connection to Previous Work and Observations
--------------------------------------------
The present investigation is a first step towards a quantitative theory of cluster formation in outflow-driven protostellar turbulence. Our emphasis has been on the global properties of the turbulence. Supersonic turbulence in molecular clouds has been subject to intensive numerical studies in recent years (as reviewed, e.g., by Elmegreen & Scalo 2004). A longstanding problem is that it decays away quickly, which prompted many workers to drive the turbulence in Fourier space. The most important distinction of our simulations is that we drive the turbulence in physical space, using outflows that are ubiquitously observed in star formation.
There have been a few previous studies of turbulent motions driven by outflows. Allen & Shu (2000) examined the effects of outflows on the dynamics of critically magnetized, sheet-like GMCs. The same 2D geometry is adopted in our previous investigation of star formation in turbulent, magnetically subcritical clouds that includes both ambipolar diffusion and outflows (Nakamura & Li 2005). These 2D calculations are extended in this paper to three dimensions (but without ambipolar diffusion). The only other 3D study is that of Mac Low (2000). He conducted a preliminary investigation of continuous driving of turbulence in a non-self-gravitating cloud by outflows ejected from a number of randomly distributed locations that are fixed in space and time. We went beyond this study by including self-gravity and connecting the outflows to the discrete events of star formation, which occur preferentially near the bottom of the potential well.
The protostellar outflow has a unique feature that can be potentially observable. It has a velocity power spectrum characterized by a prominent break. At small wavenumbers, the spectrum remains relatively flat. It steepens to approximately $E_k\propto
k^{-2.5}$ above the break. The broken power-law spectrum implies that the bulk of the energy resides near the break, rather than at the smallest wavenumber. Given that the turbulence is driven by a collection of outflows of varying lengths, there is no prior reason why the power should be dominated by the largest scale of the system. In particular, the Larson’s (1981) linewidth-size relation is unlikely to be strictly applicable [*inside*]{} parsec-scale clumps of active star formation, as recognized independently by Matzner (2007). High resolution observations of nearby embedded clusters that resolve the gas kinematics on different scales can be used to test this proposition (e.g., Caselli & Myers 1995; Saito et al. 2006).
Another way to probe the nature of the turbulence is through probability distribution functions (PDFs). Although the PDFs of the volume density in protostellar turbulence can often, but not always, be fitted reasonably well by lognormal distributions, the PDFs of the column density show considerable deviations, especially towards the high value end and in regions of rapid star formation. The strong deviation is not necessarily a unique signature of the protostellar turbulence per se. More likely, it is a reflection of the crucial role that the gravity plays in our simulations. The gravity allows dense cores to form and collapse individually, and to cluster near the bottom of the potential well collectively. The ability to follow the cloud evolution to a later stage of gravitational evolution distinguishes our simulations from other grid-based simulations of cluster formation in turbulent clouds (e.g., Heitsch et al. 2001; Ostriker et al. 2001; Li et al. 2004; Tilley & Pudritz 2004; Vazquez-Semadeni et al. 2005; for examples of SPH simulation of cluster formation, see Klessen et al. 1998 and Bate et al. 2003). Such simulations are typically terminated before the runaway collapse of the densest core, before the primordial turbulence is transformed into a protostellar turbulence through the outflows associated with star formation. We are able to go well beyond the collapse of the first core and follow the formation of multiple generations of stars and their feedback into the cluster-forming environment, albeit using simple prescriptions for the subgrid physics. We hope to refine these prescriptions in the future.
Limitations and Future Refinements {#improvement}
----------------------------------
The most restrictive prescription is perhaps the mass of individual stars. We have on purpose kept the stellar mass close to a single value, about $0.5~M_\odot$ for the fiducial choices of parameters, which is comparable to the typical mass of low mass stars. The prescription, however, decoupled the properties of the dense cores and the stars that they produce. This is obviously an oversimplification. A better prescription is perhaps to assume a constant accretion time for all stars. Some motivation for this prescription comes from the analysis of Myers & Fuller (1993), who inferred a spread in mass accretion time for stars between 0.3 and 30 M$_\odot$ much smaller than the spread in stellar mass. Ultimately, high resolution calculations, using perhaps adaptive mesh refinement (AMR), are required to follow the process of mass accretion in detail to determine the stellar mass self-consistently. Even with AMR, there are uncertainties of physical origin that cannot be eliminated with higher resolution: the effects of jets and winds from young stellar objects (as well as radiation for massive stars), which can interfere with the process of mass accretion and limit the stellar mass (Nakano et al. 1995; Matzner & McKee 2000; Shu et al. 2004).
In our simulation, the momentum of the primary outflow from close to a star is assumed to be imparted instantaneously to a small region immediately next to the star. While we believe that this simple prescription captures the essence of the outflow feedback, a more realistic treatment would be to include the primary protostellar wind in the simulation for a finite duration. A potential difficulty with such a treatment is that the high wind speed may reduce the time step to the point of making the global simulation prohibitive. Another possibility is to carry out separate calculations of core collapse and outflow propagation (e.g., Shang et al. 2006), and find general rules that can be used in the global calculations. In any case, a detailed treatment of outflow-ambient interaction is needed for not only the determination of the mass of an individual star in a given core, but also the outflow-shaped environment for the cluster formation. The theoretical treatment can benefit greatly from systematic observations that quantify the dependence of outflow properties on the stellar mass and star-forming environments, if any. There is a pressing need for such observations.
Another area of future improvement will be in spatial resolution. Higher resolution is desirable for at least two reasons. First, the protostellar turbulence appears to be dominated by shearing rather than compressible motions. The shear may excite small scale turbulence that may be washed out by numerical diffusion on a relatively coarse grid, although the magnetic field may suppress the development of such a turbulence. More importantly, higher resolution will allow us to study the properties of dense cores with greater confidence. Dense cores in some cluster forming regions appear to have mass spectra similar to the IMF. Understanding this important observation will be a focus of our future higher resolution simulations.
Lastly, we reiterate that the standard periodic boundary condition is adopted in our simulations. It has the undesirable effect of preventing energy and momentum, carried by either outflowing material or MHD waves, from leaving the simulated region. This effect may be compensated to some extent by the energy and momentum fed into the region from the ambient environment. A consequence of this restriction is that we are unable to follow cloud disruption by outflows, which may terminate the formation of relatively poor clusters that do not contain massive stars.
Summary
=======
We have carried out numerical simulations of cluster formation in protostellar outflow-driven turbulence. Our emphasis was on the global properties of the gas and the star formation rate. The main results are summarized below.
1\. Protostellar outflows of strength in the observationally inferred range can readily replenish the supersonic turbulence in clumps of active cluster formation against dissipation, and keep the clumps close to a dynamical equilibrium. The protostellar turbulence is characterized by the coexistence of infall driven by gravity and outflow motions. These motions are roughly balanced so as to yield a net mass flux towards the bottom of the gravitational potential well much smaller than that expected in a global free-fall collapse.
2\. The protostellar turbulence is maintained by outflows associated with star formation at rates as low as a few percent per free fall time. Collimated outflows are more efficient in driving the turbulence than spherical outflows carrying the same amount of momentum. Collimation enables an outflow to propagate farther away from its source, effectively increasing the scale for energy and momentum injection. This result is in agreement with the previous finding that turbulence driven on a larger scale decays more slowly. Although protostellar outflows tend to retard global star formation, they can directly trigger star formation in localized regions through shock compression.
3\. There is a prominent break in the velocity power spectrum of the protostellar turbulence. Below the break wavenumber, the spectrum is relatively flat. It steepens at higher wavenumbers. The break may provide a handle to distinguish the outflow-driven protostellar turbulence from other types of turbulence. In particular, it is unlikely that the Larson’s linewidth-size holds on the scales of relatively flat power spectrum. Another general feature is that the turbulent velocity field is dominated by the solenoidal component, produced perhaps by the strong shear between the outflow and the ambient medium. The high degree of anisotropy in the turbulence driving is an important feature of the protostellar turbulence that warrants further investigation.
4\. We find that the PDFs of the volume density in the cluster-forming clump can be approximated reasonably well by lognormal distributions in general. The PDFs of the column density, on the other hand, often show large deviations from lognormal, especially when the ordered magnetic field is strong enough to flatten the mass distribution along the field lines, and in regions of rapid star formation. Dense cores and filaments, which cluster near the bottom of the potential well, tend to skew the column density PDFs towards the high value end. The mass distribution in the cluster forming region is very clumpy. Nevertheless, the spherically averaged density typically increases with decreasing radius in an approximately power-law fashion in the envelope, with $\rho\propto r^{-1.5}$ providing a reasonable fit to the profile in many cases.
5\. Magnetic fields are dynamically important for cluster formation even in magnetically supercritical clumps, as long as the initial field strength is not much below the critical value to begin with. Moderately strong fields can significantly reduce the rate of star formation. Even a relatively weak magnetic field can be amplified to an equipartition level by the outflow-driven turbulent motions. The magnetic field is expected to influence the directions of outflow ejection and propogation and the transmission of outflow energy and momentum to the ambient medium, although these effects remain to be fully quantified.
6\. We find that the stellar velocity dispersion is comparable to the turbulent speed of the gas. More detailed predictions on the stellar properties await a refined treatment of the subgrid physics of individual star formation.
This work is supported in part by a Grant-in-Aid for Scientific Research of Japan (1540117), a Grant for Promotion of Niigata University Research Projects, and NSF and NASA grants (AST-0307368 and NAG5-12102). We thank Chris Matzner for sharing some results prior to publication and the referee for comments that improved the presentation of the paper. The numerical calculations were carried out mainly on SX6 at Niigata University, on SX8 at YITP in Kyoto University, and on SR8000 at Center for Computational Science in University of Tsukuba.
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APPENDIX: Protostellar Outflow Momentum {#outflow}
=======================================
In this Appendix, we estimate a plausible range of the outflow momentum from several observations. Most, perhaps all, low-mass stars go through a phase of strong outflow during formation. The origin of the outflow is unclear. The leading scenario is that they are driven by rotating magnetic fields from close to the central object (e.g., Königl & Pudritz 2000; Shu et al. 2000). These winds interact with the environments that surround the forming stars, feeding energy and momentum back into the star-forming cloud. The best evidence for the wind-ambient interaction comes from bipolar molecular outflows (Lada 1985), which are thought to be the cloud material that has been set into motion by the interaction. Since the interaction is likely momentum (rather than energy) conserving, a useful quantity for characterizing the strength of the feedback is the total (time-integrated) momentum of the wind divided by the stellar mass, denoted by $P_*$ (Matzner & McKee 2000). Its value can be constrained from both observations and theoretical considerations.
The best observational constraint on the wind momentum per unit stellar mass $P_*$ comes from the famous source of bipolar molecular outflow, L1551 IRS5. The CO outflow is the first to be discovered (Snell et al. 1980), and remains arguably the best studied source (Stojimirovic et al. 2006). Just as importantly, the mass of its central object has been determined dynamically, based on radio observations of the orbital motions of the central binary system (Rodriguez et al. 2003); this mass is typically not available for other strong outflow sources, which are deeply embedded in general.
Stojimirovic et al. (2006) carried out a thorough study of the structure and kinematics of the outflow, and obtained a flow momentum between $20.5 - 26.5\ M_\odot$ km s$^{-1}$ (depending on the excitation temperature adopted), corrected for the effects of CO optical depth but not inclination. The inclination angle $i$ of the outflow to the plane of the sky is somewhat uncertain. Fridlund & Liseau (1994) found values ranging from $\sim 24^{\circ}$ to $\sim 43^{\circ}$ from the radial velocities and proper motions of HH knots that bisect the blue outflow lobe. Consistent with this range is the inclination angle ($i\sim 30^\circ$) for the symmetry axis of the system inferred from the modeling of IR images in scattered light (Lucas & Roche 1996) and continuum images of circumstellar disks at 7 mm (Rodriguez et al. 1998). If we adopt the same range for the molecular outflow as well, the inclination-corrected momentum would increase to $\sim 30.1 - 65.2 M_\odot$ km s$^{-1}$. This is likely a lower limit to the total momentum carried by the primary wind, because it does not include the contribution from the slowest part of the outflow (within 1.5 km/s of the systematic velocity) that is difficult to disentangle from the ambient cloud and there is evidence that the wind may have punched through part of the visible boundary of the L1551 cloud. Using the total stellar mass of $\sim 1.2$ M$_\odot$ for the binary estimated by Rodriguez et al. (2003), we find $P_* \gsim 25 - 54$ km s$^{-1}$ for this particular source. L1551 IRS5 is considered prototypical of moderately collimated, “classical” molecular outflows (Bachiller & Tafalla 1999). There is another class of highly collimated outflows, driven mostly by Class 0 objects. One of the best studied outflows in this class is L1157, driven by a low luminosity protostar of $\sim 11 L_\odot$ (Umemoto et al. 1992). Bachiller et al. (2001) obtained a total flow momentum of $4.71 M_\odot$ km/s, after correcting for the effects of CO optical depth but not inclination. A large inclination correction factor is needed, since the outflow appears to lie close to the plane of the sky. If one adopts the value $i\sim
9^\circ$ deduced by Gueth et al. (1996), the momentum would increase by a factor of $6.4$, to $\sim 30 M_\odot$ km/s. The mass of the central object is uncertain. For a rough estimate, we assume that most of the bolometric luminosity $L_{\rm bol}$ comes from the accretion luminosity $$L_{\rm acc} = {G {\dot M_*} M_* \over R_*}
\label{accretionL}$$ and that the mass accretion rate ${\dot M_*}$ is given by $M_*/
t_{\rm dyn}$, where $M_*$ is the stellar mass and $t_{\rm dyn}$ is the dynamical time for the outflow, which is measured to be $\sim 3\times 10^4$ years for L1157 (Bachiller et al. 2001). Under these assumptions, we find $$M_* \approx \left({L_{\rm bol} R_* t_{\rm dyn} \over G}\right)^{1/2}
=0.17 \left({L_{\rm bol}\over 10 L_\odot}\right)^{1/2} \left({
R_*\over 3 R_\odot}\right)^{1/2} \left({t_{\rm dyn}\over 3\times
10^4 {\rm yrs} }\right)^{1/2} M_\odot,
\label{stellarM}$$ where the radius $R_*$ of the (low-mass) protostar should be within a factor of 2 of the scaling $3 R_\odot$ (Stahler 1988). Froebrich et al. (2003) estimated a somewhat lower stellar mass ($0.1 M_
\odot$) for L1157, adopting a more elaborate, time dependent model for mass accretion and a lower $L_{\rm bol}=7.6\pm 0.8 L_\odot$. These crude estimates, if not too far off the true value, would point to a large value for the wind momentum per unit stellar mass $P_*$ of more than $10^2$ km/s for this particular case.
The above values of $P_*$ estimated for the best studied cases may not be representative, however. They are likely biased toward high values because of observational selection effects; weaker outflows are more difficult to study in detail. To account for the possibility of a range of values for $P_*$, we introduce a dimensionless parameter $$f= \left({P_* \over 100\ {\rm km/s} }\right) = \left({V_{\rm w}
\over 10^2\ {\rm km/s}}\right) \left({M_{\rm w}\over M_*}\right).
\label{windpara}$$ It is the product of the speed of the primary wind $V_{\rm w}$ in units of $10^2$ km/s and the fraction of the stellar mass that is ejected in the wind. For revealed T Tauri stars where the wind speed can be measured directly, $V_{\rm w}$ is typically a few hundred km/s. In the X-wind theory (Shu et al. 2000), the most natural value for the ratio $M_{\rm w}/M_*$ is $\sim 1/3$. It is therefore conceivable that $f$ may be as high as unity. Some support for a relatively high value of $f$ comes from the unpublished work of Cabrit & Shepherd (quoted in Richer et al. 2000), which indicates that $(M_{\rm w}
V_{\rm w}) / (M_* V_{\rm K}) \sim 0.3$ (where $V_{\rm K}=[GM_*/R_*]
^{1/2}$ is the Keplerian speed at the stellar surface) over a wide range of stellar luminosity. For a typical young star of mass $M_* \approx 0.5 M_\odot$ and radius $R_*
\approx 3 R_\odot$, we have $P_* \sim 0.3\ V_{\rm K}\approx 53$ km/s (or $f\approx 0.5$. On the other hand, the wind during the deeply embedded phase (when most of the driving of molecular outflows occurs) may be somewhat slower, both because of a smaller stellar mass (and thus a shallower gravitational potential well) and because of a likely higher wind mass loading which, in the magnetocentrifugal wind theory, would lead to a slower outflow (e.g., Anderson et al. 2005). Also, there is evidence that the ratio of mass loss rate in the wind and accretion rate onto the star ${\dot M}_{\rm w}/{\dot M}_* \approx 0.1$ during both the revealed (Calvet 1997) and the embedded phase (Bontemps et al. 1996). These estimates, although fairly uncertain, point to values for $f$ as low as $\sim 0.1$.
[llllll]{} S0 & 2.5 & 0.5 & 0.75 & standard model, with jet component\
E1 & 2.5 & 0.5 & n/a & spherical outflow, no jet component\
E2 & 2.5 & 0.5 & 0.25 & significantly weaker jet component\
E3 & 2.5 & 0.5 & 0.50 & somewhat weaker jet component\
F1 & 2.5 & 0.25 & 0.75 & weaker outflow\
F2 & 2.5 & 0.75 & 0.75 & stronger outflow\
M1 & $10^{-6}$ & 0.50 & 0.75 & extremely weak magnetic field\
M2 & 0.5 & 0.5 & 0.75 & weaker magnetic field\
[^1]: In computing the contribution of a formed star (or Lagrangian particle) to the gravitational potential, we spread its mass evenly in its host cell to avoid singularity.
[^2]: It does not mean that the clump will live for a time as long as the depletion time. The clump will likely be dispersed either internally or externally long before the conversion of gas to stars is completed.
[^3]: We remind the reader that the $k=0$ component of the gravitational potential is set to zero in Fourier space to accommodate the periodic boundary condition. This treatment reduces the absolute value of the gravitational energy in the model compared to the case where the cloud exists in isolation.
[^4]: In computing $\sigma_*$, we exclude the two stars that move at speeds of 81.6 and 135 $c_s$, respectively, much faster than the other stars. Such fast moving stars are produced, on rare occasions, when the density in a fast moving outflow is pushed above the threshold for mass extraction. They may disappear with a refined outflow treatment.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present an approximation scheme for support vector machine models that use an RBF kernel. A second-order Maclaurin series approximation is used for exponentials of inner products between support vectors and test instances. The approximation is applicable to all kernel methods featuring sums of kernel evaluations and makes no assumptions regarding data normalization. The prediction speed of approximated models no longer relates to the amount of support vectors but is quadratic in terms of the number of input dimensions. If the number of input dimensions is small compared to the amount of support vectors, the approximated model is significantly faster in prediction and has a smaller memory footprint. An optimized `C++` implementation was made to assess the gain in prediction speed in a set of practical tests. We additionally provide a method to verify the approximation accuracy, prior to training models or during run-time, to ensure the loss in accuracy remains acceptable and within known bounds.'
author:
- 'Marc Claesen[^1]\'
- 'Frank De Smet[^2]\'
- 'Johan A.K. Suykens\'
- Bart De Moor
bibliography:
- 'bibliography.bib'
title: 'Fast Prediction with SVM Models Containing RBF Kernels[^3]'
---
Introduction
============
Kernel methods form a popular class of machine learning techniques for various tasks. An important feature offered by kernel methods is the ability to model complex data through the use of the kernel trick [@scholkopf2002learning]. The kernel trick allows the use of linear algorithms to implicitly operate in a transformed feature space, resulting in an efficient method to construct models which are nonlinear in input space. In practice, despite the computationally attractive kernel trick, the prediction complexity of models using nonlinear kernels may prohibit their use in favor of faster, though less accurate, linear methods.
We present an approach to reduce the computational cost of evaluating predictive nonlinear models based on RBF kernels. This is valuable in situations where model evaluations must be performed in a limited time span. Several applications in the computer vision domain feature such requirements, including object detection [@cao:inria-00325810; @10.1109/TPAMI.2012.62] and image denoising [@mika1999kernel; @yang2004two]. The widely used Radial Basis Function (RBF) kernel is known to perform well on a large variety of problems. It effectively maps the data onto an infinite-dimensional feature space. The RBF kernel function $\kappa(\cdot,\cdot)$ is defined as follows, with kernel parameter $\gamma$: $$\kappa(\mathbf{x}_i,\mathbf{x}_j)=e^{-\gamma\|\mathbf{x}_i-\mathbf{x}_j\|^2}. \label{eq:rbf}$$
Support vector machines (SVMs) are a prominent class of kernel methods for classification and regression problems [@burges1998tutorial]. The decision functions of SVMs take a similar form for various types of SVM models, including classifiers, regressors and least squares formulations [@Cortes:1995:SN:218919.218929; @LSSVM]. For lexical convenience, we will use common SVM terminology in this text though the technique applies to all kernel methods.
The run-time complexity of kernel methods using an RBF kernel is $O(n_{SV}\times d)$ where $n_{SV}$ is the number of support vectors and $d$ is the input dimensionality. When run-time complexity is crucial and the number of support vectors is large, users are often forced to use linear methods which have $O(d)$ prediction complexity at the cost of reduced accuracy [@10.1109/TPAMI.2012.62]. We suggest a method which can significantly lower the run-time complexity of models with RBF kernels for many learning tasks.
In our approach, the decision function of SVM models that use an RBF kernel is approximated via the second-order Maclaurin series approximation of the exponential function. This approach was first proposed by Cao et al. [@cao:inria-00325810]. We extend their work by using fewer assumptions, providing a conservative bound on the approximation error for a given data set and reporting results of an extensive empirical analysis. Using this approximation, prediction speed can be increased significantly when the number of dimensions $d$ is low compared to the number of support vectors $n_{\rm SV}$ in a model. The proposed approximation is applicable to all models using an RBF kernel in popular SVM packages like LIBSVM [@CC01a], SHOGUN [@SonRaeHenWidBehZieBonBinGehFra10] and LS-SVMlab [@lssvmlabguide].
We will derive the proposed approximation in the context of SVMs but its use easily extends to other kernel methods. Particularly, the approximation is applicable to all kernel methods that exploit the representer theorem [@scholkopf2001generalized]. This includes methods such as Gaussian processes [@rasmussen2006gaussian], RBF networks [@poggio1990networks], kernel clustering [@Filippone:2008:SKS:1284917.1285173], kernel PCA [@scholkopf1998nonlinear; @suykens2003support] and kernel discriminant analysis [@788121].
Related Work
============
A large variety of methods exist to increase prediction speed. Three main classes of approaches can be identified: (i) pruning support vectors from models, (ii) approximating the feature space by a low-dimensional input space and (ii) approximating the decision function of a given model directly. Our proposed approach belongs to the latter class.
Reducing Model Size by Pruning Support Vectors
----------------------------------------------
Pruning support vectors linearly increases prediction speed because the run-time complexity of models with RBF kernels is proportional to the amount of support vectors. Pruning methods have been devised for SVM [@scholkopf1998fast; @liang2010effective] and least squares SVM formulations [@suykens2002weighted; @hoegaerts2004comparison].
Feature Space Approximations
----------------------------
Rahimi and Recht proposed using standard linear methods after explicitly mapping the input data to a randomized low-dimensional feature space, which is designed such that the inner products therein approximate the inner products in feature space [@rahimi2007random]. This approach results in linear prediction complexity, as the resulting model is linear in the randomized input space. This is a general technique applicable to a large variety of kernel functions. For the RBF kernel, our specialized approach approximates each kernel evaluation to within $\epsilon = 0.03$ at complexity $O(d^2)$ when adhering to the proposed bounds. The complexity of random Fourier features is much higher than $O(d^2)$ for low-dimensional input spaces, where the RBF kernel is most useful [@rahimi2007random; @cotter2011explicit].
Direct Decision Function Approximations
---------------------------------------
Approaches that focus on approximating the decision function directly typically involve some form of approximation of the kernel function. Such approximations need not retain the structure and interpretation of the original model, provided that the decision function does not change significantly. Kernel approximations may leave out the interpretation of support vectors completely by reordering computations [@herbster2001learning], or by aggregating support vectors into more efficient structures [@cao:inria-00325810]. Neural networks have also been used to approximate the SVM decision function directly [@Kang20144989], in which case prediction speed depends on the chosen architecture.
A second-order approximation of the exponential function for RBF kernels was first introduced by Cao et al. [@cao:inria-00325810]. The basic concept of our paper resembles their work. In terms of training complexity, this approximation was analyzed in [@cotter2011explicit]. Here we focus exclusively on prediction speed. Cao et al. [@cao:inria-00325810] make two assumptions regarding normalization in deriving the approximations that may needlessly constrain their applicability. These assumptions are:
1. Feature vectors are normalized to unit length, to simplify $\kappa$ to $\kappa(\mathbf{x}_i,\mathbf{x}_j)=e^{-2\gamma}e^{2\gamma\mathbf{x}_i^T\mathbf{x}_j}$.
2. Feature values must always be positive such that $0 \leq \mathbf{x}_i^T\mathbf{z} \leq 1$ holds.
We will perform a more general derivation that requires none of these assumptions. Our derivation is agnostic to data normalization and we provide a conservative bound to assess the validity of the approximation during prediction (Eq. ). Additionally, we derive the full approximation in matrix-form using the gradient and Hessian of the approximated part of the decision function. This allows the use of highly optimized linear algebra libraries in implementations of our work. Our benchmarks demonstrate that the use of such libraries yields a significant speed-up. Finally, we freely provide our implementation to facilitate comparison with competing approaches.
Second-Order Maclaurin Approximation
====================================
Predicting with SVMs involves computing a linear combination of inner products in feature space between the test instance $\mathbf{z} \in \mathbb{R}^{d}$ and all support vectors. In subsequent equations, $\mathbf{X} \in \mathbb{R}^{d \times {n_{\rm SV}}}$ represents a matrix of ${n_{\rm SV}}$ support vectors. We will denote the $i$-th support vector by $\mathbf{x}_i$ (the $i$-th column of $\mathbf{X}$). Via the representer theorem [@scholkopf2001generalized], the decision values are a linear combination of kernel evaluations between the test instance and all support vectors: $$f(\cdot) : \mathbb{R}^d \rightarrow \mathbb{R} : f(\mathbf{z}) = \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i \kappa(\mathbf{x}_i,\mathbf{z}) \label{eq:generalcompact} + b,$$ where $b$ is a bias term, $\alpha$ contains the support values, $\mathbf{y}$ contains the training labels and $\kappa(\cdot,\cdot)$ is the kernel function. Expanding the RBF kernel function in Eq. yields: $$\begin{aligned}
f(\mathbf{z}) &= \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i e^{-\gamma\|\mathbf{x}_i-\mathbf{z}\|^2} + b \nonumber \\
&= \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i e^{-\gamma \|\mathbf{x}_i\|^2}e^{-\gamma \|\mathbf{z}\|^2} \underbrace{e^{2\gamma \mathbf{x}_i^T\mathbf{z}}} + b. \label{eq:general}\end{aligned}$$
The exponentials of inner products between support vectors and the test instance – underbraced in Equation – induce prediction complexity $O({n_{\rm SV}}\times d)$. Large models with many support vectors are slow in prediction, because each SV necessitates computing the exponential of an inner product in $d$ dimensions for every test instance $\mathbf{z}$. We use a second-order Maclaurin series approximation for these exponentials of inner products as described by [@cao:inria-00325810] (see the appendix for details on the Maclaurin series), which enables us to bypass the explicit computation of inner products.
The exponential per test instance $e^{-\gamma \|\mathbf{z}\|^2}$ can be computed exactly in $O(d)$. Before approximating the factors $e^{2\gamma \mathbf{x}_i^T\mathbf{z}}$, we reorder Equation by moving the factor $e^{-\gamma \|\mathbf{z}\|^2}$ in front of the summation: $$\begin{aligned}
f(\mathbf{z})&=e^{-\gamma \|\mathbf{z}\|^2} \big( \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i e^{-\gamma \|\mathbf{x}_i\|^2}e^{2\gamma \mathbf{x}_i^T\mathbf{z}} \big)+b, \nonumber \\
&=e^{-\gamma \|\mathbf{z}\|^2} g(\mathbf{z})+b, \label{eq:generalreordered}\end{aligned}$$ with: $$g(\cdot) : \mathbb{R}^d \rightarrow \mathbb{R} : g(\mathbf{z}) = {\sum_{i=1}^{n_{\rm SV}} \alpha_i y_i e^{-\gamma \| \mathbf{x}_i \|^2} e^{2\gamma \mathbf{x}_i^T\mathbf{z}}}. \label{eq:g}$$
The exponentials of inner products can be replaced by the following approximation, based on the second-order Maclaurin series of the exponential function (see the appendix): $$e^{2\gamma \mathbf{x}_i^T\mathbf{z}}\approx1+2\gamma \mathbf{x}_i^T\mathbf{z}+2\gamma^2(\mathbf{x}_i^T\mathbf{z})^2. \label{eq:approxexp}$$
Approximating the exponentials $e^{2\gamma \mathbf{x}_i^T\mathbf{z}}$ in $g(\mathbf{z})$ via Equation yields: $$\begin{aligned}
\hat{g}(\mathbf{z}) &= \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i e^{-\gamma \|\mathbf{x}_i\|^2}\big(1+2 \gamma \mathbf{x}_i^T\mathbf{z}+2 \gamma^2(\mathbf{x}_i^T\mathbf{z})^2\big) , \nonumber \\
&= c+\mathbf{v}^T\mathbf{z}+\mathbf{z}^T\mathbf{M}\mathbf{z}, \label{eq:ghat}\end{aligned}$$ with: $$\begin{aligned}
c \in \mathbb{R} &= g(\mathbf{0}^{d}) = \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i e^{-\gamma \|\mathbf{x}_i\|^2}, \\
\mathbf{v} \in \mathbb{R}^{d} \rightarrow v_j &= \nabla g(\mathbf{z}) \\ &= 2 \gamma \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i e^{-\gamma \|\mathbf{x}_i\|^2} X_{j,i}, \\
\mathbf{v} &= \mathbf{X}\mathbf{w}, \\
\mathbf{M} \in \mathbb{R}^{d\times d} \rightarrow M_{j,k} &= \frac{\partial^2 g(\mathbf{z})}{\partial z_j \partial z_k} \\
&= 2 \gamma^2 \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i e^{-\gamma \|\mathbf{x}_i\|^2} X_{j,i} X_{k,i},\\
\mathbf{M} &= \mathbf{X}\mathbf{D}\mathbf{X}^T.\end{aligned}$$
The vector $\mathbf{v}$ and matrix $\mathbf{M}$ represent the gradient and Hessian of $g$, respectively. Here $\mathbf{w} \in \mathbb{R}^{{n_{\rm SV}}}$ is a weighting vector: $w_i=2 \gamma \alpha_i y_i e^{-\gamma \|\mathbf{x}_i\|^2}$ and $\mathbf{D} \in \mathbb{R}^{{n_{\rm SV}}\times {n_{\rm SV}}}$ is a diagonal scaling matrix: $D_{i,i}=2 \gamma^2 \alpha_i y_i e^{-\gamma \|\mathbf{x}_i\|^2}$ and $D_{i,j}=0$ if $i\neq j$. Finally, the approximated decision function $\hat{f}(\mathbf{z})$ is obtained by using $\hat{g}(\mathbf{z})$ in Eq. : $$\hat{f}(\mathbf{z}) = e^{-\gamma \|\mathbf{z}\|^2} \hat{g}(\mathbf{z})+b = e^{-\gamma \|\mathbf{z}\|^2} \big( c+\mathbf{v}^T\mathbf{z}+\mathbf{z}^T\mathbf{M}\mathbf{z} \big) + b. \label{eq:approxcompact}$$
The parameters $c$, $\mathbf{v}$, $\mathbf{M}$ and $b$ are independent of test points and need only be computed once. The complexity of a single prediction becomes $O(d^2)$ – due to $\mathbf{z}^T\mathbf{Mz}$ – instead of $O({n_{\rm SV}}\times d)$ for an exact RBF kernel.
The model size and prediction complexity of the proposed approximation is independent of the amount of support vectors in the exact model. This is especially interesting for least squares SVM formulations, which are generally not sparse in terms of support vectors [@LSSVM]. The RBF approximation loses its appeal when the number of input dimensions grows very large. For problems with high input dimensionality, the feature mapping induced by an RBF kernel often yields little improvement over using the linear kernel anyway [@hsu2003practical].
Approximation Accuracy {#acc}
----------------------
The relative error of the second-order Maclaurin series approximation of the exponential function is less than $3.05\%$ for exponents in the interval $[-0.5,0.5]$ (see Eq. in Appendix \[app:maclaurin\]). Adhering to this interval guarantees that the relative error of any given term in the linear combination of $\hat{g}(\mathbf{z})$ is below $3.05\%$, compared to $g(\mathbf{z})$ (Eqs. and , respectively). This translates into the following bound for our approximation: $$|2\gamma \mathbf{x}_i^T \mathbf{z}| < \frac{1}{2}, \quad \forall i. \label{eq:bound1}$$
The inner product can be avoided via the Cauchy-Schwarz inequality: $$|\mathbf{x}_i^T\mathbf{z}| \leq \|\mathbf{x}_i\|\|\mathbf{z}\|, \quad \forall i. \label{eq:cauchy}$$
Combining Eqs. and yields a way to assess the validity of the approximation in terms of the support vector $\mathbf{x}_M$ with maximal norm ($\forall i: \|\mathbf{x}_M \| \geq \| \mathbf{x}_i \|$): $$\|\mathbf{x}_M\|^2\|\mathbf{z}\|^2 < \frac{1}{16\gamma^2}. \label{eq:bound}$$ Storing $\|\mathbf{x}_M\|^2$ in the approximated model enables checking adherence to the bound in Eq. during prediction, based on the squared norm of the test instance $\|\mathbf{z}\|^2$. Observe that this bound can be verified during prediction at no extra cost because $\|\mathbf{z}\|^2$ must be computed anyway (see Eq. ). Our tools can additionally report an upper bound for $\gamma$ for a given data set prior to training a model. In this case, the upper bound is obtained based on the maximum norm over all instances. The obtained upper bound for $\gamma$ may be slightly overconservative, because the data instance with maximum norm will not necessarily become a support vector.
Relation to Degree-2 Polynomial Kernel
--------------------------------------
The RBF approximation yields a quadratic form which can be related to a degree-2 polynomial kernel. We use the following general form for the degree-2 polynomial kernel: $$\kappa(\mathbf{x}_i, \mathbf{x}_j) = \big(\gamma \mathbf{x}_i^T \mathbf{x}_j + \beta\big)^2.$$ Note that $\gamma$ has a similar effect in the degree-2 polynomial kernel as in the RBF kernel (though not identical). To relate the second-order approximation of the RBF kernel with a degree-2 polynomial kernel we must expand the polynomial kernel in a similar fashion as in Equation . Note that this expansion is exact for the polynomial kernel instead of an approximation as it is for the RBF kernel.
$$\begin{aligned}
\textbf{approximated RBF} &\quad\longleftrightarrow\quad \textbf{exact degree-2 polynomial} \nonumber \\
{\color{blue}e^{-\gamma \|\mathbf{z}\|^2}} \big(c+\mathbf{w}^T\mathbf{X}\mathbf{z}+\mathbf{z}^T\mathbf{XDX}^T\mathbf{z}\big)+b
&\quad\longleftrightarrow\quad
c+\mathbf{w}^T\mathbf{X}\mathbf{z}+\mathbf{z}^T\mathbf{XDX}^T\mathbf{z} + b \label{eq:rbf-2d-decfun} \\
c = \sum_{i=1}^{{n_{\rm SV}}}\alpha_i y_i {\color{blue}e^{-\gamma \|\mathbf{x}_i\|^2}}
&\quad\longleftrightarrow\quad
c = \beta^2 \sum_{i=1}^{n_{SV}} \alpha_i y_i \label{eq:rbf-2d-c} \\
w_i = 2 \gamma \alpha_i y_i {\color{blue}e^{-\gamma \|\mathbf{x}_i\|^2}}
&\quad\longleftrightarrow\quad
w_i = 2 \beta \gamma \alpha_i y_i \label{eq:rbf-2d-w} \\
D_{i,i} = {\color{blue}2} \gamma^2 \alpha_i y_i {\color{blue}e^{-\gamma \|\mathbf{x}_i\|^2}}
&\quad\longleftrightarrow\quad
D_{i,i} = \gamma^2 \alpha_i y_i \label{eq:rbf-2d-d}\end{aligned}$$
Equations to contrast an approximated RBF model with an exact model with degree-2 polynomial kernel. Fixing $\beta$ at $1$ facilitates the comparison which exposes two key differences between both models: (i) the nonlinearity $e^{-\gamma \|\mathbf{z}\|^2}$ in the approximated RBF model in Equation and (ii) a higher relative weight on second-order terms in the RBF approximation in Equation . The other exponential factors in terms of the support vectors in Eqs. - act as scaling factors, which can be incorporated in the $\alpha$ values of the model with polynomial kernel to obtain an equivalent effect, e.g. $\alpha_i^{(2D)} = \alpha_i^{(RBF)} e^{-\gamma \|\mathbf{x}_i\|^2}$.
The extra scaling in Eq. adds flexibility to approximated RBF models compared to exact models with a polynomial kernel. The scaling causes the relative impact of the bias term $b$ in the model on the overall decision to vary per test instance $\mathbf{z}$. Adhering to the approximation bound defined in Equation limits this scaling effect to the interval $(e^{-0.25}, 1]$, assuming $\|\mathbf{x}_M\| \geq \|\mathbf{z}\|,\ \forall \mathbf{z}$.
Implementation
--------------
In order to benchmark the approximation against exact evaluations, we have made a `C++` implementation to approximate LIBSVM models and predict with the approximated model.[^4] The implementation features a set of configurations to do the main computations. The configurations differ in the use of linear algebra libraries and vector instructions. Different configurations have consequences in two aspects: (i) approximating an SVM model and (ii) predicting with the approximated model.
#### Approximation Speed
The key determinant of approximation speed is matrix math. Approximation time is dominated by the computation of $\mathbf{M}=\mathbf{X}\mathbf{D}\mathbf{X}^T$, which involves large matrices if $d$ and ${n_{\rm SV}}$ are large. The following implementations have been made:
1. `LOOPS`: uses simple loops to implement matrix math (default).
2. `BLAS`: uses the Basic Linear Algebra Subprograms (BLAS) for matrix math [@Blackford01anupdated]. The BLAS are usually available by default on modern Linux installations (in `libblas`). This default version is typically not heavily optimized.
3. `ATLAS`: uses the Automatically Tuned Linear Algebra Software (ATLAS) routines for matrix math [@Whaley00automatedempirical]. ATLAS provides highly optimized versions of the BLAS for the platform on which it is installed. The performance of ATLAS is comparable to vendor-specific linear algebra libraries such as Intel’s Math Kernel Library [@eddelbuettel2010benchmarking].
#### Prediction Speed
The main factor in prediction speed for approximated models is evaluating $\mathbf{z}^T\mathbf{M}\mathbf{z}$ where $\mathbf{M}$ is a symmetric $d\times d$ matrix. This simple operation can exploit Single Instruction Multiple Data (SIMD) instruction sets if the platform supports them. The use of vector instructions can be enabled via compiler flags. We observed no significant gains in prediction speed when using the BLAS or ATLAS.
Results and discussion {#classification}
======================
To illustrate the speed and accuracy of the approximation, we used it for a set of classification problems. The exact models were always obtained using LIBSVM [@CC01a]. We investigated the amount of labels that differ between exact and approximated models as well as speed gains. The accuracies are listed in Table \[tab:accuracies\]. We report the accuracy of the exact model and the percentage of labels which differ between the exact model and the approximation (note that not all differences are misclassifications). Table \[tab:timings\] reports the results of our speed measurements. Before discussing these results, we briefly summarize the data sets we used.
Data Sets
---------
To facilitate verification of our results, we used data sets that are freely available in LIBSVM format at the website of the LIBSVM authors.[^5] We used all the data sets as they are made available, without extra normalization or preprocessing. We used the following classification data sets:
- `a9a`: the Adult data set, predict who has a salary over $\$50.000$, based on various information [@Platt:1999:FTS:299094.299105]. This data set contains two classes, $d=123$ features (most are binary dummy variables) and $32,561$/$16,281$ training/testing instances.
- `mnist`: handwritten digit recognition [@lecun1998gradient]. This data set contains 10 classes – we classified class $1$ versus others, $d=780$ features and $60,000$/$10,000$ training/testing instances.
- `ijcnn1`: used for the IJCNN 2001 neural network competition [@prokhorov2001ijcnn]. There are 2 classes, $d=22$ features and $49,990$/$91,701$ training/testing instances.
- `sensit`: SensIT Vehicle (combined), vehicle classification [@duarte2004vehicle]. This data set contains 3 classes – we classified class $3$ versus others, $d=100$ features and $78,823$/$19,705$ training/testing instances.
- `epsilon`: used in the Pascal Large Scale Learning Challenge.[^6] This data set contains 2 classes, $d=2,000$ features and $400,000$/$100,000$ training/testing instances. To reduce training time, we switched the training and test set.
Accuracy
--------
The accuracies we obtained in our benchmarks are listed in Table \[tab:accuracies\]. This table contains the key parameters per data set: number of dimensions $d$ and the maximum value that should be used for $\gamma$ in order to guarantee validity of the approximation ($\gamma_{MAX}$). Here $\gamma_{MAX}$ is obtained via Eq. after data normalization. The last column shows that only a very minor number of labels are predicted differently by the exact and approximated models.
Some of the listed results do not adhere to the bound, e.g. $\gamma > \gamma_{MAX}$. We used these parameters to illustrate that even though the accuracy of some terms in the linear combination may be inaccurate (e.g. relative error larger than $3\%$), the overall accuracy may still remain very good. In practice, we always recommend to adhere to the bound which guarantees high accuracy. Ignoring this bound is equivalent to abandoning all guarantees regarding approximation accuracy, because it is impossible to assess the approximation error which increases exponentially (shown in Figure \[fig:maclaurinrelerr\] in the appendix).
When the bound was satisfied, the fraction of erroneous labels was consistently less than $1\%$ (`a9a`, `mnist` and `ijcnn1`). In the last experiment for `a9a` we used a value for $\gamma$ that is over five times larger than $\gamma_{MAX}$ and still get only $3.5\%$ of erroneous labels. These results demonstrate that the approximation is very acceptable in terms of accuracy.
The experiments on `sensit` and `epsilon` illustrate that a large number of dimensions $d$ safeguards the validity of the approximation to some extent, even when $\gamma$ becomes too large. The fraction $\gamma/\gamma_{MAX}$ is larger for `epsilon` than it is for `sensit` but due to the higher number of dimensions in `epsilon`, the fraction of erroneous labels remains lower ($0.53\%$ for `epsilon` versus $0.95\%$ for `sensit`). This occurs because the Cauchy-Schwarz inequality (Equation ) is a worst-case upper bound for the inner product. When $d$ grows large, it is less likely for $|\mathbf{x}_i^T\mathbf{z}|$ to approach $\|\mathbf{x}_i\| \|\mathbf{z}\|$. In other words, the bound we use – based on the Cauchy-Schwarz inequality – is more conservative for larger input dimensionalities.
data set $d$ $\gamma_{MAX}$ $\gamma$ $n_{\rm test}$ ${n_{\rm SV}}$ $acc$ (%) $diff$ (%)
------------- -------- ---------------- ----------- ---------------- ---------------- ----------- ------------ --
adult (a9a) $122$ $0.018$ $0.01$ $16,281$ $11,834$ $84.8$ $0.2$
adult (a9a) $122$ $0.018$ $0.02$ $16,281$ $11,674$ $84.9$ $1.3$
adult (a9a) $122$ $0.018$ $0.10$ $16,281$ $11,901$ $85.0$ $3.5$
mnist $780$ $10^{-3}$ $10^{-4}$ $10,000$ $2,174$ $99.3$ $0.08$
ijcnn1 $22$ $0.064$ $0.05$ $91,701$ $4,044$ $97.7$ $0.46$
sensit $100$ $0.0025$ $0.003$ $19,705$ $25,722$ $86.5$ $0.95$
epsilon $2000$ $0.25$ $0.35$ $400,000$ $36,988$ $89.2$ $0.53$
Speed Measurements
------------------
Timings were performed on a desktop running Debian Wheezy. We used the default BLAS that are prebundled with Debian, which appear to be somewhat optimized, but not as much as ATLAS. We ran benchmarks on an Intel i5-3550K, which supports the Advanced Vector Extensions (AVX) instruction set for SIMD operations.
Table \[tab:timings\] contains timing results of prediction speed between exact models and their approximations. The speed increase for the approximation is evident: ranging from $7$ to $137$ times when the time to approximate is disregarded, or $4.4$ to $69$ times when it is accounted for. We can see that the speed increase also holds for a large number of dimensions ($2000$ for the `epsilon` data set). The model for `mnist` contains few SVs compared to the number of dimensions, which causes a smaller speed increase in favor of the approximated model.
data set approach math $t_{approx}$ (s) SIMD $t_{pred}$ (s) ratio 1 ratio 2
----------- ---------- ------------- ------------------ -------------- --------------------------- --------- ---------
`a9a` exact $/$ $/$ $/$ $\mathbf{13.75\pm 0.060}$ $1$ $1$
approx [`BLAS`]{} $0.05\pm 0.002$ $\times$ $0.160\pm 0.002$ $86$ $65$
[`LOOPS`]{} $0.56\pm 0.021$ $\checkmark$ $0.146\pm 0.003$ $94$ $19$
optimal [`BLAS`]{} $0.05\pm 0.002$ $\checkmark$ $0.146\pm 0.003$ $94$ $70$
`mnist` exact $/$ $/$ $/$ $\mathbf{12.81\pm 0.016}$ $1$ $1$
approx [`BLAS`]{} $0.036\pm 0.001$ $\times$ $1.757\pm 0.008$ $7.3$ $7.1$
[`LOOPS`]{} $1.480\pm 0.005$ $\checkmark$ $1.405\pm 0.006$ $9.1$ $4.4$
optimal [`BLAS`]{} $0.036\pm 0.001$ $\checkmark$ $1.405\pm 0.006$ $9.1$ $8.9$
`ijcnn1` exact $/$ $/$ $/$ $\mathbf{15.87\pm 0.012}$ $1$ $1$
approx [`BLAS`]{} $0.010\pm 0.000$ $\times$ $0.679\pm 0.012$ $23$ $23$
[`LOOPS`]{} $0.010\pm 0.000$ $\checkmark$ $0.667\pm 0.016$ $24$ $23$
optimal [`BLAS`]{} $0.010\pm 0.000$ $\checkmark$ $0.667\pm 0.016$ $24$ $23$
`sensit` exact $/$ $/$ $/$ $\mathbf{79.62\pm 0.127}$ $1$ $1$
approx [`BLAS`]{} $0.670\pm 0.000$ $\times$ $0.590\pm 0.000$ $134$ $63$
[`LOOPS`]{} $1.437\pm 0.036$ $\checkmark$ $0.581\pm 0.012$ $137$ $39$
optimal [`ATLAS`]{} $0.565\pm 0.005$ $\checkmark$ $0.581\pm 0.012$ $137$ $69$
`epsilon` exact $/$ $/$ $/$ $\mathbf{622.1\pm 0.165}$ $1$ $1$
`*` approx [`BLAS`]{} $1.161\pm 0.003$ $\times$ $10.78\pm 0.110$ $58$ $52$
[`LOOPS`]{} $43.98\pm 0.495$ $\checkmark$ $9.68\pm 0.03$ $64$ $12$
optimal [`ATLAS`]{} $0.442\pm 0.029$ $\checkmark$ $9.68\pm 0.03$ $64$ $61$
In terms of approximation speed, the impact of specialized linear algebra libraries is apparant as shown in columns 3 and 4 of Table \[tab:timings\]. ATLAS consistently outperforms BLAS and both are orders of magnitude faster than the naive implementation, particularly when the matrix $\mathbf{X}$ gets large (over $100\times$ faster for `epsilon`, where $\mathbf{X}$ is $2.000\times 36.988$).
The impact of vector instructions is clear, with gains of up to $25\%$ in prediction speed when they are used (cfr. `mnist` results). Note that most of the time is spent on file IO for these benchmarks, which may give a pessimistic misinterpretation of the speed increase of vector instructions.
A competing method approximates the decision function using artificial neural networks (ANN) with a single hidden layer [@Kang20144989]. In this approach, prediction complexity is $O(n_{HN}\times d)$ where $n_{HN}$ is the number of hidden nodes in the network (typically $n_{HN} < n_{SV}$). [@Kang20144989] report prediction speedups of a factor $5$ to $28$ on models with few support vectors (which enables using few hidden nodes in the approximating ANN). The empirical speedup of using our quadratic approximation ranges from a factor $9$ to $137$ for models with many support vectors. When the number of support vectors grows, the decision boundary becomes more complex and will require more hidden units to be approximated effectively, which reduces the appeal of using ANNs. In contrast, the complexity of our approach is not influenced by the number of support vectors.
Applications
============
The most straightforward applications of the proposed approximation are those which require fast prediction. This includes many computer vision applications such as object detection, which require a large amount of predictions, potentially in real-time [@cao:inria-00325810; @10.1109/TPAMI.2012.62]. Complementary to featuring faster prediction, the approximated kernel models are often smaller than exact models. The approximated models consist of three scalars ($b$, $c$ and $\gamma$), a dense vector ($\mathbf{v} \in \mathbb{R}^{d}$) and a dense, symmetric matrix ($\mathbf{M} \in \mathbb{R}^{d\times d}$). When the number of dimensions is small compared to the number of SVs, these approximated models are significantly smaller than their exact counterparts. We included Table \[tab:modelsize\] to illustrate this property: it shows the model sizes per classification data set. In our experiments the approximated models are smaller for all data sets except one. The biggest compression ratio we obtained was $290$ times (for the `sensit` data set). If we would approximate least squares SVM models, the compression ratios would be even larger due to the larger amount of SVs in least squares SVM models [@LSSVM].
data set $d$ ${n_{\rm SV}}$ exact approx ratio
----------- --------- ---------------- ---------- ---------- --------
`a9a` $122$ $11,834$ $830$ KB $111$ KB $7.5$
`mnist` $780$ $2,174$ $3.2$ MB $3.7$ MB $0.86$
`ijcnn1` $22$ $4,044$ $628$ KB $4.2$ KB $150$
`sensit` $100$ $25,722$ $32$ MB $113$ KB $290$
`epsilon` $2,000$ $36,988$ $1.1$ GB $42$ MB $27$
: Comparison of model sizes (both types are stored in text format).[]{data-label="tab:modelsize"}
Finally, a subtle side effect of our method is the fact that training data is obfuscated in approximated models. Data obfuscation is a technique used to hide sensitive data [@1366117]. Training data may be proprietary and/or contain sensitive information that cannot be exposed in contexts such as biomedical research [@murphy2002as]. In standard SVM models, the support vectors are exact instances of the training set. This renders SVM models unusable when data dissemination is an issue. The approximated models consist of complicated combinations of the support vectors (and typically $d \ll {n_{\rm SV}}$), which makes it very challenging to reverse-engineer parts of the original data from the model. The approximation can be considered a surrogate one-way function of the support vectors [@DBLP:journals/corr/cs-CR-0012023]. As such, these approximations may allow SVMs to be used in situations where they are currently not considered [@1366117].
Conclusion {#conclusion .unnumbered}
==========
We have derived an approximation for SVM models with RBF kernels, based on the second-order Maclaurin series approximation of the exponential function. The applicability of the approximation is not limited to SVMs: it can be used in a wide variety of kernel methods. The proposed approximation has been shown to yield significant gains in prediction speed.
Our benchmarks have shown that a minor loss in accuracy can result in very large gains in prediction speed. We have listed some example applications for such approximations. In addition to applications in which low run-time complexity is desirable, applications that require compact or data-hiding models benefit from our approach.
Our work generalizes the approximation proposed by [@cao:inria-00325810]. The derivation we performed made no implicit assumptions regarding data normalization. An easily verifiable bound was established which can be used to guarantee that the relative error of individual terms in the approximation remains low.
A competing method to approximate SVM models with an RBF kernel uses neural networks [@Kang20144989]. The advantages of our approach are (i) known bounds on the approximation error, (ii) faster to approximate an exact model (linear combination of SVs versus training a neural network) and (iii) faster in prediction when the number of dimensions is low. An advantage of the neural network approximation is that it can always be used, in contrast to our quadratic approximation whose validity depends on the data and choice of $\gamma$ as explained in Section \[acc\].
Approximation of exponential function {#app:maclaurin}
=====================================
The Maclaurin series for the exponential function and its second-order approximation are: $$\begin{aligned}
e^x &= \sum_{k=0}^\infty \frac{1}{k!} x^k, \nonumber \\
&\approx 1+x+\frac{1}{2}x^2. \label{eq:maclaurin}\end{aligned}$$
Figure \[fig:maclaurinrelerr\] illustrates the absolute relative error of the second-order Maclaurin series approximation. The relative error is smaller than $\pm 3\%$ when the absolute value of the exponent $x$ is small enough, e.g. $|x|<0.5$: $$|x| < \frac{1}{2} \quad \Rightarrow \quad |\frac{e^x-1-x-0.5x^2}{e^x}| < 0.0305. \label{eq:maclaurinbound}$$
This can be used to verify the validity of the approximation.
![Absolute relative error of the second-order Maclaurin series approximation: $y(x) = \big|\big(e^x-(1+x+0.5x^2)\big)/e^x\big|$.[]{data-label="fig:maclaurinrelerr"}](maclaurin_err.eps){width="\linewidth"}
[^1]: [KU Leuven, dept. of Electrical Engineering ESAT – STADIUS KU Leuven – iMinds dept. of Medical IT Kasteelpark Arenberg 10, box 2446; 3001 Leuven, Belgium]{}
[^2]: KU Leuven, dept. of Public Health and Primary Care, Environment and Health; Kapucijnenvoer 35 blok d, box 7001; 3000 Leuven, Belgium
[^3]: Research supported by: Research Council KUL: ProMeta, GOA MaNet, KUL PFV/10/016 SymBioSys, START 1, OT 09/052 Biomarker, several PhD/postdoc $\&$ fellow grants; Flemish Government: IOF: IOF/HB/10/039 Logic Insulin, FWO: PhD/postdoc grants, projects: G.0871.12N research community MLDM; G.0733.09; G.0824.09; IWT: PhD Grants; TBM-IOTA3, TBM-Logic Insulin; FOD: Cancer plans; Hercules Stichting: Hercules III PacBio RS; EU-RTD: ERNSI; FP7-HEALTH CHeartED; COST: Action BM1104, Action BM1006: NGS Data analysis network; ERC AdG A-DATADRIVE-B.
[^4]: Our implementation is available at <https://github.com/claesenm/approxsvm>.
[^5]: Available at <http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/>.
[^6]: Available at <http://largescale.ml.tu-berlin.de/instructions/>.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We carry out a coarse-grained molecular dynamics simulation of phospholipid vesicles with transmembrane proteins. We measure the mean and Gaussian curvatures of our protein-embedded vesicles and quantitatively show how protein clusters change the shapes of their host vesicles. The effects of depletion force and vesiculation on protein clustering are also investigated. By increasing the protein concentration, clusters are fragmented to smaller bundles, which are then redistributed to form more symmetric structures corresponding to lower bending energies. Big clusters and highly aspherical vesicles cannot be formed when the fraction of protein to lipid molecules is large.'
author:
- Amir Houshang Bahrami
- Mir Abbas Jalali
title: Vesicle deformations by clusters of transmembrane proteins
---
Introduction
============
Biological membranes are found in various complex shapes [@Lip95; @Sei97] which are closely related to membrane functions such as biconcave shape of erythrocytes or corkscrew shape of spirochetes. Shape variety depends mainly on protein concentration, operation of external forces on membranes in cellular environments [@Zim06], membrane movement, fusion and budding processes, variations in lipid composition, and vesicle trafficking [@Mc05]. Membrane local curvature is representative for its shape, and proteins are believed to play an important role in membrane conformation. Proteins behave as both generating [@Zim06; @Mc05] and sensing [@Vog06] elements for the membrane curvature [@Mc05; @Phi09]. While protein aggregation induces shape transformation, membrane curvature may also generate feedback on protein aggregation and yield attractive [@Phi09; @Rey07] or repulsive [@Phi09; @Kim98] curvature-mediated interactions between them. It has also been shown that bending rigidity and membrane thickness affect protein functioning [@And07; @Jen04].
Proteins affect membrane curvature through various ways such as scaffold and local curvature mechanisms, and by their integration (as transmembrane proteins) with the membrane [@Zim06; @Mc05; @Phi09; @Koz10]. However, the experimental measurements of variations in the membrane curvature are not easy. The role of proteins on the membrane deformations has been studied using continuum elastic modeling [@Phi09; @Kim98; @Wei98], particle-based [@Cha08] and mesoscopic [@Ven05] simulations, and hybrid elastic-discrete particle models [@Naj09]. In all these studies, lipid bilayers represent a liquid environment with freely diffusing lipid chains that dissolve topological deformations. The aggregation of somewhat rigid proteins remarkably reduces the diffusion of lipid chains and causes shape variations.
In this study, we use a coarse-grained model [@Ven06] and generalize the method of Markvoort et al. [@Mar05] to generate phospholipid vesicles from initially rectangular and flat bilayers with different concentrations of transmembrane proteins. We discuss our simulation method in section \[sec:simulation-procedure\] and model the vesicle surface using spherical harmonics. In section \[sec:curvatures\], we present a method for computing the mean and Gaussian curvatures at the locations of hydrophilic heads of lipid chains and proteins. The local curvature information together with the sizes of protein clusters help us to investigate the deformations of host vesicles in section \[sec:examples\]. We also study the size distribution, formation and fragmentation of protein clusters. We summarize our fundamental results in section \[sec:conclusions\].
Model Description {#sec:simulation-procedure}
=================
Mesoscopic models have been widely used to study the physics of membranes [@Ven06]. Lipids can spontaneously aggregate and form various membranes [@Goe98; @Goe99]. When lipid chains are assembled in the form of a closed 3D surface and trap water molecules, a vesicle is generated. Entropy is the main driving mechanism of this process [@Mar05]. To construct vesicles, we insert initially flat rectangular lipid bilayers (which may contain proteins) inside a box of water molecules. Such a configuration is unstable because the tails of boundary lipids are repelled by water molecules and the bilayer is compressed by in-plane forces. Consequently, the bilayer buckles and closes itself to acquire a minimum potential energy state. Vesicles formed through this bilayer $\rightarrow$ vesicle transition process, with the progenitor bilayer being surrounded by solvent particles (without touching simulation boundaries), have a more relaxed pressure distribution. Furthermore, using bilayer $\rightarrow$ vesicle transition process we obtain vesicles of different sizes in a more controllable process.
Our simulation box has dimensions of $L_x\times L_y\times L_z$ with the $x$-axis being normal to the initial bilayer mid plane. We use periodic boundary conditions, and choose a sufficiently large box so that the bilayer does not reach to the boundaries before vesiculation. The number of protein and lipid molecules are denoted by $N_p$ and $N_l$, respectively. The particles that constitute the elements of our setups either are water particles (type 1), or have hydrophobic (type 2) and hydrophilic (type 3) natures. We follow reference \[19\], and model each lipid chain by one hydrophilic head and four hydrophobic tail particles (Fig. \[fig:fig1\][*a*]{}).
All particles interact through Lennard-Jones (LJ) potential [@Allen87] with a cut off radius $r_c = 2.5 \sigma$.
We use the integer subscripts $i$ and $j$ for particle types, and index a particle in a molecule by the subscript $s$. For the pairs $(i,j)=(1,2)$ and (2,3) the interaction potential is $V_{ij} = 4\epsilon_{ij}(\sigma_{ij}/r)^{9}$, and for other pairs we use $U_{ij} = 4\epsilon_{ij}[(\sigma_{ij}/r)^{12}-(\sigma_{ij}/r)^{6}]$ where $\sigma_{ij},\epsilon_{ij}$=1 $(i,j=1,2,3)$ [@Goe98]. A harmonic bond potential of the form $U^{\rm bond}_{s,s+1} = K_{b}(\left|r_{s,s+1}\right|-\sigma)^{2}$ is applied between neighboring particles inside a lipid chain, and we have set $K_{b}=5000 \epsilon_{s,s+1} \sigma^{-2}$ where $\epsilon_{s,s+1}=1$. With these assumptions, less than 10% of bond lengthes fluctuate more than 2% around $\sigma$ \[19\]. In some case studies, to include bending rigidity in our lipid and protein chains, we use the potential $$\begin{aligned}
U^{\rm bend}_{s-1,s,s+1} &=&
K_{\rm bend} \left (\! 1 \!-\! \frac{{\textit{\textbf{r}} }_{s-1,s}.{\textit{\textbf{r}} }_{s,s+1}}
{\left|{\textit{\textbf{r}} }_{s-1,s}\right|\left|{\textit{\textbf{r}} }_{s,s+1}\right|} \right ), \\
&=& K_{\rm bend}(1-\cos\phi_s),\end{aligned}$$ where $K_{\rm bend}$ is the bending coefficient of non-flexible chains [@Goe98]. $\phi_s$ defines the bending angle between adjacent bonds in a single chain. Protein molecules (Fig. \[fig:fig1\][*b*]{}) are composed of seven strands in a hexagonal arrangement [@Ven05]. A particle in each strand interacts with all of its neighbors, within the same protein, through the potentials $U^{\rm bond}_{s,s+1}$ and $U^{\rm bend}_{s-1,s,s+1}$ defined above. In this way, lipid-protein interactions are the same as lipid-lipid interactions.
In this study, we work with simple proteins whose lengths are adjusted in a way that the hydrophobic mismatch effect is minimum, and such that a single protein does not affect the membrane conformation considerably. Shape variations that we report, are thus caused [*only*]{} by clustering. Both flexible and rigid chains can be used in lipid and protein molecules. The experiments of section \[sec:examples\] show that adding bending rigidity does not induce remarkable qualitative or quantitative changes in the protein tilting angle or in the aggregation products. In fact, the strong harmonic bond potentials assumed between neighboring strands (in a protein molecule) provide enough bending rigidity for our relatively short protein molecules.
We implement equilibrium molecular dynamics simulation of an NVT ensemble [@Allen87] with velocity Verlet algorithm for integration in the time domain. We use the integration time step $\delta t = 0.005 t_{0}$ with $$\begin{aligned}
t_{0} = \sqrt{\frac{m \sigma^2}{48 k_{\rm B} T}} = \frac{1}{\sqrt{64.8}},
~ k_{\rm B}=1,~ T=1.35,\end{aligned}$$ and set the number density of particles[@Goe98] to $\rho = 2/3$. All Particles have equal masses of $m=1$, and all lengths are scaled by $\sigma$. The parameters used in our models produce correct physical properties of bilayers, including diffusion coefficients, density profiles and mechanical properties like surface tension and stress distribution [@Goe98; @Mar05]. After vesicle formation, the position vectors of particles are measured with respect to the vesicle center, and the lipid or protein heads exposed to water molecules outside the vesicle are tagged as surface particles. We assign an integer number $n$ to each surface particle, and denote the total number of surface particles by $N$. It is remarked that there is not a meaningful correlation between the physical location of each particle and its number $n$. The identifier $n$ is used only for statistical purposes. Furthermore, a single number $n$ is assigned to each surface particle, i.e., there are, respectively, one and seven particle identifiers corresponding to each lipid chain and protein molecule.
Local mean and Gaussian curvatures {#sec:curvatures}
==================================
To measure the local curvature of vesicles, with and without proteins, we express the radial distance of surface particles from the vesicle center in terms of spherical harmonics [@Arf85] as $$\begin{aligned}
r(\theta,\phi)=\sum_{l=0}^{l_{\rm max}}\sum_{m=0}^{l} \left [ a^m_l A^m_l(\phi,\theta)
+ b^m_l B^m_l(\phi,\theta) \right ],
\label{eq:position-vector}\end{aligned}$$ where $$\begin{aligned}
A^m_l &=& {\rm Re} \left [ Y^m_l \right ],~
B^m_l={\rm Im} \left [ Y^m_l \right ], \\
Y^m_l &=& (-1)^m e^{{\rm i}m\phi}
\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)!}} P^m_l(\cos\theta).\end{aligned}$$ Here $P^m_l$ are associated Legendre functions and ${\rm i}=\sqrt{-1}$. We find the coordinates of surface particles $(r_n,\phi_n,\theta_n)$, define $$\begin{aligned}
{\textit{\textbf{R}} }= \left [
\begin{array}{cccc}
r_{1}(\phi_{1},\theta_1) ~ & r_{2}(\phi_{2},\theta_2) ~ & \ldots ~ &
r_{N}(\phi_N,\theta_{N} )
\end{array}
\right ]^{\rm T},
\label{eq:r-vector}\end{aligned}$$ and collect all constants coefficients $a^m_l$ and $b^m_l$ in single column vectors ${\textit{\textbf{a}} }$ and ${\textit{\textbf{b}} }$, respectively. The superscript T means transpose. We also define the matrices ${\textbf{A} }$ and ${\textbf{B} }$ whose elements are $A^m_l(\phi_n,\theta_n)$ and $B^m_l(\phi_n,\theta_n)$, respectively. The discrete form of equation (\[eq:position-vector\]) thus becomes $${\textit{\textbf{R}} }={\textbf{Y} }\cdot {\textit{\textbf{x}} },~~
{\textbf{Y} }= \left [
\begin{array}{cc}
{\textbf{A} }& {\textbf{B} }\end{array}
\right ], ~~ {\textit{\textbf{x}} }=\left \{
\begin{array}{c}
{\textit{\textbf{a}} }\\
{\textit{\textbf{b}} }\end{array}
\right \},
\label{eq:reduced-matrix-equation}$$ which in practice, has more equations than unknowns.
We calculate ${\textit{\textbf{x}} }$ using the singular value decomposition of ${\textbf{Y} }$, and obtain the position of any surface particle from (\[eq:position-vector\]) and $$\begin{aligned}
{\textit{\textbf{r}} }= r\left [ \sin (\theta)\cos (\phi) \hat {\textit{\textbf{i}} }+
\sin (\theta)\sin (\phi) \hat {\textit{\textbf{j}} }+
\cos (\theta) \hat {\textit{\textbf{k}} }\right ],
\label{eq:define-r-vs-phi-theta}\end{aligned}$$ where $(\hat {\textit{\textbf{i}} },\hat {\textit{\textbf{j}} },\hat {\textit{\textbf{k}} })$ are the unit vectors in Cartesian coordinates. Let us define ${\textit{\textbf{r}} }_{\phi}$ and ${\textit{\textbf{r}} }_{\theta}$ as the first order, and ${\textit{\textbf{r}} }_{\phi \phi}$, ${\textit{\textbf{r}} }_{\phi \theta}$, and ${\textit{\textbf{r}} }_{\theta \theta}$ as the second order partial derivatives of ${\textit{\textbf{r}} }$ in (\[eq:define-r-vs-phi-theta\]) with respect to $\phi$ and $\theta$. The coefficients of the first fundamental form of the surface are thus determined as $$\begin{aligned}
E={\textit{\textbf{r}} }_{\phi} \cdot {\textit{\textbf{r}} }_{\phi},~~
F={\textit{\textbf{r}} }_{\phi} \cdot {\textit{\textbf{r}} }_{\theta},~~
G={\textit{\textbf{r}} }_{\theta} \cdot {\textit{\textbf{r}} }_{\theta}.\end{aligned}$$ Defining the unit vector normal to the vesicle surface as $\hat {\textit{\textbf{n}} }= ({\textit{\textbf{r}} }_{\phi} \times {\textit{\textbf{r}} }_{\theta})/|{\textit{\textbf{r}} }_{\phi}\times {\textit{\textbf{r}} }_{\theta}|$, one finds the coefficients of the second fundamental form: $$\begin{aligned}
L={\textit{\textbf{r}} }_{\phi\phi} \cdot \hat {\textit{\textbf{n}} },~~
M={\textit{\textbf{r}} }_{\phi\theta} \cdot \hat {\textit{\textbf{n}} },~~
N={\textit{\textbf{r}} }_{\theta\theta} \cdot \hat {\textit{\textbf{n}} }.\end{aligned}$$ The mean curvature $H$ and the Gaussian curvature $K$ at the location of each surface particle can thus be computed using [@Saf94] $$\begin{aligned}
H=\frac{LN-M^2}{EG-F^2},~~K=\frac{EN-2FM+GL}{2(EG-F^2)}.
\label{eq:K-and-H}\end{aligned}$$ The principal curvatures ($C_{1},C_{2}$) are related to $H$ and $K$ through $H=(C_{1}+C_{2})/2$ and $K=C_{1} C_{2}$ [@Saf94]. In our numerical experiments we have truncated the series (\[eq:position-vector\]) at $l_{\rm max}=5$. Including $l_{\rm max}>5$ terms had $\approx 3\%$ improvement in fractional errors. To perform a global shape classification of vesicles, we compute the average curvatures $$\begin{aligned}
\left \{
\begin{array}{l}
\bar H \\
\bar K
\end{array}
\right \}
&=& \frac{1}{N} \sum_{n=1}^{N}
\left \{
\begin{array}{l}
H(n) \\
K(n)
\end{array}
\right \},
\label{eq:averaged-curvatures}\end{aligned}$$ and their corresponding standard deviations $\tilde H$ and $\tilde K$ for particles living on the surface of model vesicles.
\
Simulation Results {#sec:examples}
==================
We first consider the case of lipid and protein chains with no bending rigidity. Our first model, which is called $V_1$, is a vesicle without proteins and it is composed of $N_l=2300$ similar lipid chains. The initial flat bilayer is bent inside the water (solvent) particles and gradually forms an almost spherical vesicle shown in the left panels of Fig. \[fig:fig2\]. To measure the sphericity of this vesicle quantitatively, we use a spherical harmonics expansion with $l_{\rm max}=5$, and fit a 3D surface to the surface particles. The mean local curvature $H(n)$ is computed from (\[eq:K-and-H\]) and used in (\[eq:averaged-curvatures\]) to find $\bar H=0.0564$ and $\tilde H=0.0068$. The small value of $\tilde H$ compared to $\bar H$ confirms the spherical nature of vesicle $V_1$.
Protein-embedded vesicles {#subsec:protein-embedded-vesicles}
-------------------------
We replace some lipid chains of our initially flat bilayer by protein molecules while keeping the area of generating bilayer almost constant. We randomly distribute proteins in the bilayer sheet, but observe that they form small clusters after the bending of membrane and during vesicle formation. The vesiculation process takes from $\approx 5\times 10^{5} \delta t$ for model $V_{1}$ to $\approx 2\times 10^{6}\delta t$ for protein-embedded vesicles. That is because proteins (or clusters of proteins) resist against buckling by decreasing the fluidity of the progenitor membrane. After vesicle formation, we have waited for about $5\times 10^{5}$ time steps to ensure that vesicles have reached to an equilibrium condition so that the number and area of protein clusters remain constant.
[![Top views of our flat bilayers $B_3$ (left panel) and $B_4$ (right panel) with transmembrane proteins. The bilayers $B_3$ and $B_4$ have the same number of proteins and areas of the vesicles $V_3$ and $V_4$, respectively.[]{data-label="fig:fig3"}](fig3a.eps "fig:"){width="20.00000%"}]{} [![Top views of our flat bilayers $B_3$ (left panel) and $B_4$ (right panel) with transmembrane proteins. The bilayers $B_3$ and $B_4$ have the same number of proteins and areas of the vesicles $V_3$ and $V_4$, respectively.[]{data-label="fig:fig3"}](fig3b.eps "fig:"){width="20.00000%"}]{}
We have constructed three vesicles with initial bilayers of approximately equal surface areas and different protein concentrations. We name these vesicles $V_2$, $V_3$ and $V_4$, which have been formed inside $N_{w}\approx 161000$ water molecules. The numbers of lipid and protein molecules in our models have been given in Table \[tab:table1\]. Fig. \[fig:fig2\] demonstrates three dimensional views of these vesicles and their cross sections with the largest diameter so that the majority of clusters are visualized. We have also shown some water particles inside and outside vesicles. It is seen that larger protein clusters have induced lower curvatures in their neighborhood, and consequently, prominent deformations in their host vesicles.
$V_1$ $V_2$ $V_3$ $V_4$
------------ -------- -------- -------- --------
$N_l$ 2300 2120 1940 1760
$N_p$ 0 20 40 60
$\bar H_p$ - 0.0460 0.0381 0.0418
$\bar H_l$ 0.0564 0.0549 0.0582 0.0567
$\bar H$ 0.0564 0.0541 0.0544 0.0527
$\tilde H$ 0.0068 0.0135 0.0155 0.0158
$\bar K$ 0.0030 0.0029 0.0029 0.0027
$\tilde K$ 0.0014 0.0015 0.0015 0.0015
$\bar p$ - 1.66 5 4.61
$\bar p_b$ - 1.53 2.66 3.52
$Q_c$ 0 3 5 8
$Q_f$ 0 9 3 5
$p(1)$ - 6 14 13
$p(2)$ - 3 11 11
$p(3)$ - 2 5 11
$p(4)$ - - 5 6
$p(5)$ - - 2 6
$p(6)$ - - - 4
$p(7)$ - - - 2
$p(8)$ - - - 2
: \[tab:table1\]The protein and lipid content, and averaged curvatures of the simulated vesicles $V_{1}$–$V_{4}$. The quantity $\bar p_b$ has been computed for the bilayers $B_2$–$B_4$.
[{width="45.00000%"}]{} [{width="45.00000%"}]{}\
\
\
We also investigate several protein-embedded flat bilayers where the aggregation of proteins is caused mainly by the depletion force. Comparing the sizes and population of clusters formed in bilayers and vesicles helps us to better understand the roles of entropy- and curvature-driven aggregation of proteins during vesiculation. We simulate three flat bilayers, which extend to the sides of the simulation box. The surface areas of these bilayers and the population of their proteins match those of the vesicles $V_2$–$V_4$. We label these bilayers as $B_2$, $B_3$ and $B_4$. They remain in a flat equilibrium state because of periodic boundary conditions that keep them in touch with the sides of the simulation box. Fig. \[fig:fig3\] displays the snapshots of $B_3$ and $B_4$.
We split the molecules of the outer layer of vesicles into two groups of lipids and proteins, and denote by $H_l(n)$ and $H_p(n)$ the mean curvatures at the locations of lipid and protein heads, respectively. We track only initially tagged particles living on the outer surface of vesicle (because the probability of flip-flop motions is low) and compute the local curvature having their coordinates $(r_n,\phi_n,\theta_n)$ for $n=1,2,\ldots,N$. We have plotted $H_{\mu}$ and $K_{\mu}$ ($\mu\equiv l,p$) in Fig. \[fig:fig4\] for models $V_1$–$V_4$. The curvatures at different surface points have been marked by different symbols and colors. The scattered plots provide a quantitative insight to the effect of proteins on the vesicle shape. It is seen that the fluctuations of $H$ are higher in models with proteins. As we mentioned before, there is no correlation between the location of particles and their corresponding identifier $n$. Therefore, in Fig. \[fig:fig4\], points with close values of $n$ are not close physically. The oscillatory behavior of the graphs could change by renumbering the surface points but the major minima always coincide with protein clusters, which flatten their host vesicles locally.
We define $\bar H_l$ and $\bar H_p$ as the averages of mean curvatures (taken over the particles of the same kind) corresponding to lipid and protein heads, respectively. The average mean curvature $\bar H$ for all surface particles (protein and lipid heads) and its corresponding standard deviation $\tilde H$ have been given in Table \[tab:table1\] together with $\bar H_l$ and $\bar H_p$. Lower values of $\bar H$ for models $V_{2}$–$V_{4}$ confirm that transmembrane proteins reduce the averaged mean curvature through creating low-curvature clusters. During vesiculation, protein molecules aggregate and form clusters. The number of clusters and the population of proteins in each cluster are determined by (i) the random arrangement of proteins in the generating bilayer membrane (ii) the interplay between depletion force and curvature induced interactions (iii) system temperature and the concentration of protein molecules. If protein molecules are scattered in a vesicle, the membrane will maintain its sphericity. However, if the same number of proteins build a single cluster, the highest variation in vesicle shape will be observed. In our simulations we have not seen these two extreme cases. Several clusters with different populations of proteins are usually formed in our vesicles.
Curvature-mediated clustering {#subsec:curvature-mediated-clustering}
-----------------------------
Let us denote the number of scattered free proteins of a model by $Q_f$, and the number of its clusters by $Q_c$. Moreover, we indicate by $p(m)$ the number of proteins in the $m$th cluster. For models $V_2$, $V_3$ and $V_4$, we have reported the values of $Q_f$, $Q_c$ and $p(m)$ in Table \[tab:table1\]. It is seen that the biggest cluster have been formed in vesicle $V_{3}$ and not in $V_{4}$, which has more proteins. Consequently, $\bar H_p$ is lower in vesicle $V_{3}$ than $V_{4}$. The standard deviation $\tilde{H}$ is $\approx 10\%$ of $\bar H$ for the near-spherical vesicle $V_{1}$, but it increases remarkably for protein-embedded vesicles. As clusters grow, the curvature associated with the ensemble of lipids increases, and the vesicle becomes more aspherical. This can be understood from the larger value of $\bar H_l$ in vesicle $V_{3}$. A reverse phenomenon is also possible: if during the vesicle formation the curvature decreases in certain regions, proteins will migrate there and form low-curvature clusters. To show the correlation between the number of proteins in clusters and $\bar H$, we have computed the quantity $$\begin{aligned}
\bar p = \frac {1}{Q_f+Q_c} \left [ Q_f + \sum_{m=1}^{Q_c} p(m) \right ],
\label{eq:define-bar-p}\end{aligned}$$ and given its magnitude in Table \[tab:table1\]. We also define $\bar p_b$ using a formula similar to (\[eq:define-bar-p\]), but for our model bilayers $B_i$ that correspond to vesicles $V_i$ ($i=2,3,4$). The computed values of $\bar p_b$ are given in Table \[tab:table1\].
Since bilayers remain flat during simulation, the effect of membrane curvature on the aggregation process is ignorable and $\bar p_b$ indicates the contribution of the depletion force to cluster formation. Comparing the values of $\bar p$ and $\bar p_b$ clearly shows that curvature induced interactions, during vesiculation, facilitate the cluster growth and we get $\bar p > \bar p_b$. Moreover, $\bar p_b$ is a monotonic function of $N_p$ while $\bar p$ is not. The reason is the lack of an effective fragmentation mechanism in flat bilayers: larger protein concentrations always lead to bigger clusters. In vesicles, however, the tendency to form a structure with minimum bending energy leads to the fragmentation of big clusters at the turning (maximum) point of the function $\bar p(N_p)$.
$V_4$ $V_5$ $V_6$
---------- ------- ------- -------
$\bar p$ 4.61 4.61 4.61
$Q_c$ 8 9 9
$Q_f$ 5 4 4
$p(1)$ 13 14 14
$p(2)$ 11 11 9
$p(3)$ 11 8 8
$p(4)$ 6 7 7
$p(5)$ 6 6 6
$p(6)$ 4 3 5
$p(7)$ 2 3 3
$p(8)$ 2 2 2
$p(9)$ - 2 2
: \[tab:table2\]The number and sizes of protein clusters for vesicles $V_{4}$–$V_{6}$ with $N_p=60$ and different initial random arrangements of proteins.
For all surface points including lipid and protein heads, we have also calculated (see Table \[tab:table1\]) the average Gaussian curvature $\bar K$ and its standard deviation $\tilde{K}$ using (\[eq:K-and-H\]) and (\[eq:averaged-curvatures\]). Right panels of Fig. \[fig:fig4\] show the distribution of $K$ for lipid and protein heads. Based on the Gauss-Bonnet theorem, any compact manifold ${\cal M}$ without boundary, is topologically equivalent to a sphere and the surface integral $\int_{{\cal M}} K ~{\rm d}A$ of Gaussian curvature will be invariant. Since our vesicles become aspherical through the shape transformations induced by protein clustering, Gauss-Bonnet theorem applies and all models $V_1$–$V_4$ must be topologically equivalent. Given that the head groups of proteins and lipids are identical in our simulations and the areas of generating bilayers are almost the same, the surface integral will be approximately equal to $A \bar K$. Data in Table \[tab:table1\] show that both $\bar K$ and $\tilde K$ remain invariant (from one vesicle to another) within a reasonable error threshold. This confirms the self-consistency of our models and the results obtained from spherical harmonic expansions.
Convergence tests {#subsec:convergence}
-----------------
We have continued our simulations until the size and number of clusters become constant in a relaxed equilibrium state. To assure that simulated vesicles are in equilibrium, we have carried out various experiments. In the first experiment we generated two vesicles from the same progenitor bilayer of $V_4$, but using two different sets of randomly distributed proteins. We have given the properties of new vesicles $V_5$ and $V_6$ in Table \[tab:table2\]. Although the vesicles $V_4$, $V_5$ and $V_6$ start from different initial conditions, there are minor differences between the properties of their clusters. Notably, they posses the same shape indicator $\bar p$=$4.61$.
To demonstrate that we usually reach a physical equilibrium and not a kinetically trapped state, we designed a second experiment and produced a vesicle from an initial bilayer where $N_p$=$20$ proteins (similar to vesicle $V_2$) had formed an initial big cluster. In the resulting vesicle $V_7$, the proteins of the initial single cluster are dissociated into three smaller separate clusters as shown in Fig. \[fig:fig5\]. This result is, again, consistent with the general features of $V_2$, which had been obtained from a completely different initial condition. It is worth noting that we have observed a transient interplay between clustering and fragmentation well before reaching the equilibrium state.
[![Three dimensional views of dissociated clusters in vesicle $V_7$ with $N_p=20$ protein molecules initially located in a single big cluster.[]{data-label="fig:fig5"}](fig5.eps "fig:"){width="20.00000%"}]{}
We have repeated our simulations with $\rho=0.8$ and with non-flexible lipid and protein chains. Using a bending stiffness $K_{\rm bend}=5$ in lipids and $K_{\rm bend}=80$ in protein strands, the bilayer $\rightarrow$ vesicle transition is slowed down but we do not observe considerable change, either quantitative or qualitative, in the clustering phenomenon and the shape transformation of vesicles: the numbers and sizes of final clusters and the shape parameter $\bar p$ are similar in all models. By making stiffer molecular chains and increasing the density, lipid diffusion is decreased, which in turn, yields a longer relaxation time.
Conclusions {#sec:conclusions}
===========
We have studied the phenomena of protein clustering and membrane shape transformation during membrane vesiculation and afterwards. Comparing relaxed vesicles and bilayers shows that protein clustering during vesiculation occurs due to both entropy-driven depletion force and the curvature-mediated interactions. The latter effect enhances the generation of larger protein clusters and determines bilayer’s bending rigidity. Once the vesicle is formed, protein clusters locally flatten their host vesicles and increase the bending energy as $\bar H$ and $\bar H_l$ increase. The system, however, cannot tolerate the increase in the bending energy for protein concentrations beyond a critical value. By increasing the protein concentration, bigger protein clusters are not formed in our simulations, or they break apart. We anticipate a uniform distribution of fragmented clusters, like the shape of a soccer ball. Our observations show that low protein concentrations do not lead to efficient cluster formation.
This work was partially supported by the research vice-presidency at Sharif University of Technology. We thank the referees for their constructive comments, which substantially improved the paper.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Projected entangled pair states (PEPS) provide a natural ansatz for the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We develop a framework to describe on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, in terms of virtual symmetries of the local tensors expressed as a set of matrix product operators (MPOs) labeled by distinct group elements. These MPOs describe the possibly anomalous symmetry of the edge theory, whose local degrees of freedom are concretely identified in a PEPS. A classification of SPT phases is obtained by studying the obstructions to continuously deforming one set of MPOs into another, recovering the results derived for fixed-point models \[X. Chen *et al.*, Phys. Rev. B 87, 155114 (2013)\] [@GuWen]. Our formalism accommodates perturbations away from fixed point models, opening the possibility of studying phase transitions between different SPT phases. We also demonstrate that applying the recently developed quantum state gauging procedure to a SPT PEPS yields a PEPS with topological order determined by the initial symmetry MPOs. The MPO framework thus unifies the different approaches to classifying SPT phases, via fixed-points models, boundary anomalies, or gauging the symmetry, into the single problem of classifying inequivalent sets of matrix product operator symmetries that are defined purely in terms of a PEPS.'
author:
- 'Dominic J.'
- Nick
- Michael
- 'Mehmet B.'
- Jutho
- Frank
bibliography:
- 'SPT.bib'
title: 'Matrix product operators for symmetry-protected topological phases: Gauging and edge theories'
---
Introduction {#intro}
=============
The phase diagrams of quantum many-body systems become much richer when global [symmetries]{} are imposed. It has become clear of late that in the presence of a global symmetry there exist distinct phases which cannot be distinguished via local order parameters. These phases are referred to as *symmetry-protected topological* (SPT) phases [@GuWen]. In contrast to topologically ordered systems [@Wen90], all SPT phases become trivial if the symmetry is allowed to be explicitly broken. While this implies that SPT ground states possess only short-range entanglement[,]{} they cannot be adiabatically connected to a product state without breaking the symmetry. Furthermore they exhibit interesting edge properties when defined on a finite system with nontrivial boundary.
In recent years there has been a growing interplay between the theory of quantum many-body systems and quantum information. This has led to the development of tensor network ansatz for the ground states of local, gapped Hamiltonians [@Fannes92; @VerstraeteCirac06; @VerstraeteMurgCirac08; @GarciaVerstraeteWolfCirac08; @bridgeman2016hand]. Tensor network methods have proven particularly useful in understanding the emergence of topological phenomena in quantum many-body ground states. In one dimension, Matrix Product States were used to completely classify SPT phases via the second cohomology group of their symmetry group [@SchuchGarciaCirac11; @1Done; @1Dtwo]. In two dimensions, Projected-Entangled Pair States (PEPS) have been used to characterize systems with intrinsic topological order [@Ginjectivity; @transfermatrix; @Buerschaper14; @MPOpaper; @chiral4], symmetry-protected topological order [@Chen] and chiral topological insulators [@chiral1; @chiral2; @chiral3].
[The first goal of this work is to present a general framework for the description of on-site symmetries within the PEPS formalism. The framework includes symmetry actions on states with topological order and thus provides a natural setting for the study of symmetry-enriched topological phases [@turaev2000homotopy; @kirillov2004g; @drinfeld2010braided; @etingof2009fusion; @bombin2010topological; @hung2013quantized; @mesaros2013classification; @barkeshli2014symmetry; @tarantino2015symmetry; @teo2015theory] with PEPS [@michael]. We then restrict to PEPS without topological order and provide a complete characterization of bosonic SPT order by formulating sufficient conditions to be satisfied by the individual PEPS tensors. Previously some powerful results for renormalization group (RG) fixed-point states with SPT order were presented by Chen *et al.* [@Chen; @GuWen]]{}, the present work extends these results to systems with a finite correlation length. Furthermore, application of the quantum state gauging procedure of Ref.[@Gaugingpaper] within the framework presented here illuminates the correspondence between SPT phases and certain topologically ordered phases [in the language of PEPS, providing a complementary description to the Hamiltonian gauging construction of Levin and Gu [@LevinGu]]{}. This naturally brings together the classification of SPT phases via fixed-point models, gauging and anomalous boundary symmetries into a single unified approach that focuses only on MPOs which are properties of the ground states alone.
To achieve these goals we have developed tools to deal with orientation dependent MPO tensors. These tools allow us to calculate the symmetry action on monodromy defected and symmetry twisted states and also modular transformations, pre- and post- gauging, in a local way that is governed by a single tensor.
[We first outline the general formalism for characterizing gapped phases in PEPS using matrix product operators (MPOs) in Section \[formalism\]. Section \[globalsymmetry\] presents a set of local conditions that lead to a large class of PEPS with global symmetries which fit within the general formalism. Next, in Section \[sptpeps\], we identify a class of short-range entangled PEPS and discuss how SPT order manifests itself in these models via their anomalous edge physics. Section \[gaugingsptpeps\] explains how gauging a SPT PEPS with a discrete symmetry group yields a long-range entangled PEPS with topological order. In Section \[six\] we study symmetry twists and monodromy defects of SPT PEPS. These concepts are then illustrated with a family of examples that fall within the framework of SPT PEPS in Section \[exfpspt\]. We show explicitly that gauging these states yields ground states of the twisted quantum double models [@pasquier; @tqd], which are the Hamiltonian formulations of Dijkgraaf-Witten discrete gauge theories [@DijkgraafWitten; @Dewilde].]{}
The appendices are organized into sections that review relevant background and others that provide technical details of results which are used throughout the paper. We first review the relevant properties of MPO-injective PEPS in Appendix \[a\], provide an argument that a MPO-injective PEPS with a single block projection MPO is the unique ground state of its parent Hamiltonian in Appendix \[b\] and review the definition of the third cohomology of a single block MPO group representation in Appendix \[c\]. In Appendix \[newapp1\] we present results concerning possible orientation dependencies of MPO group representations. In Appendix \[newapp2\] we discuss different crossing tensors, their composition and the effect of modular transformations. Appendix \[d\] contains a brief review of the quantum state gauging formalism and a proof that a gauged SPT PEPS is MPO-injective [@MPOpaper]. In Appendix \[e\] we present an extension of the quantum state gauging procedure of Ref.[@Gaugingpaper] to arbitrary flat $\mathsf{G}$-connections and use it to prove that the gauging procedure is gap preserving for arbitrary topologies and to furthermore construct the full topological ground space of a gauged SPT model. In Appendix \[g\] we develop a description of symmetry twisted states, topological ground states and monodromy defected states in terms of MPOs and calculate their transformation under the residual symmetry group. Finally in Appendix \[gaugingham\] we demonstrate that the quantum state gauging procedure for finite groups is equivalent to the standard minimal coupling scheme for gauging Hamiltonians.
Characterizing topological phases with matrix product operators {#formalism}
================================================================
In this section we present a general framework for the classification of gapped phases with PEPS in terms of universal and discrete labels that arise directly from tensor network states. These discrete labels emerge from the set of MPO symmetries of the PEPS tensors and should remain invariant under continuous deformation of the MPOs.
A 2D PEPS can be defined on any directed graph ${\ensuremath{\Lambda}}$ (most commonly a regular lattice) embedded in an oriented 2D manifold ${\mathcal{M}}$ given a tensor $${\ensuremath{A}}_v:=\sum_{i_v=1}^d\sum_{\{i_e\}=1}^D ({\ensuremath{A}}_v)_{{\{i_e\}}}^{i_v}\ket{i_v}\bigotimes_{e\in E_v}{\ensuremath{(i_e|}}$$ for every vertex $v\in {\ensuremath{\Lambda}}$, where $E_v$ is the set of edges with $v$ as an endpoint, see Fig.\[e1\]. Here $i_v$ is the physical index running over a basis for the Hilbert space of a single site $\mathbb{C}^d$ and each $i_e$ is a virtual index of dimension $D$ along an edge $e$ adjacent to $v$ in the graph ${\ensuremath{\Lambda}}$.
$$\begin{aligned}
a) \vcenter{\hbox{
\includegraphics[height=0.25\linewidth]{Figures/e1}}} \hspace{1cm} b)
\vcenter{\hbox{
\includegraphics[height=0.25\linewidth]{Figures/e2}}} \end{aligned}$$
For any simply connected region ${\mathcal{R}}\subset{\mathcal{M}}$ whose boundary $\partial {\mathcal{R}}$ forms a contractible closed path in the dual graph ${\ensuremath{\Lambda}}^*$ we define the PEPS map $${\ensuremath{A}}_{\mathcal{R}}:(\mathbb{C}^D)^{\otimes |\partial {\mathcal{R}}|_e}\rightarrow (\mathbb{C}^d)^{\otimes |{\mathcal{R}}|_v},$$ from $|\partial {\mathcal{R}}|_e$ virtual indices on the edges that cross $\partial{\mathcal{R}}$ to $|{\mathcal{R}}|_v$ physical indices on the vertices in ${\mathcal{R}}$, by taking the set of tensors $\{ A_v \, |\, v\in{\mathcal{R}}\}$ and contracting each pair of indices that are assigned to an edge within ${\mathcal{R}}$, to yield $${\ensuremath{A}}_{\mathcal{R}}:=\sum_{\{i_v\}_{v\in {\mathcal{R}}}}\sum_{\{i_e\}_{e\in \overline{{\mathcal{R}}}}}\bigotimes_{v\in{\mathcal{R}}}(A_v)^{i_v}_{{\{i_e\}}_{e\in E_v}}\bigotimes_{v\in{\mathcal{R}}}\ket{i_v}\bigotimes_{e\in\partial{\mathcal{R}}}{\ensuremath{(i_e|}}$$ where $\overline{{\mathcal{R}}}:={\mathcal{R}}\cup\partial{\mathcal{R}}$, see Fig.\[e2a\].
![The PEPS map ${\ensuremath{A}}_{\mathcal{R}}$ from virtual indices on edges in $\partial {\mathcal{R}}$ to physical indices on vertices in ${\mathcal{R}}$.[]{data-label="e2a"}](Figures/e34){width="0.33\linewidth"}
Universal properties of the phase of matter containing the PEPS wave function are manifest in the local symmetries of ${\ensuremath{A}}_{\mathcal{R}}$. The specific symmetries we consider are of the form $U^{\otimes|{\mathcal{R}}|_v}{\ensuremath{A}}_{\mathcal{R}}={\ensuremath{A}}_{\mathcal{R}}O^{\partial {\mathcal{R}}}$, where $U$ is an on-site unitary corresponding to a physical symmetry that is respected in our classification of phases. Since physical symmetries necessarily form a group under multiplication, we henceforth use the notation $U(g),\ g\in\mathsf{G}$ (we do not consider non on-site symmetries such as lattice symmetries [@jiang2015symmetric]). $O^{\partial {\mathcal{R}}}$ is a MPO acting on the virtual space associated to the edges crossing ${\partial {\mathcal{R}}}$. In general, $$O^{\partial {\mathcal{R}}}=\sum_{\{i_n\},\{i_n'\}=1}^D {\ensuremath{\text{Tr}[B_{\sigma_{i}}^{i_1,i_1'}\cdots B_{\sigma_N}^{i_{N},i_{N}'}]}} \ket{i_1\dots i_{N}}\bra{i_1'\dots i_{N}'}$$ where the edges crossing ${\partial {\mathcal{R}}}$ are ordered 1 to $N:=|{\partial {\mathcal{R}}}|_e$, by fixing an arbitrary base point and following the orientation of ${\partial {\mathcal{R}}}$ (specifically the orientation induced by ${\mathcal{M}}$). Each $(B_{\sigma_n}^{i,i'})_{a,b}$ is a $\chi\times\chi$ matrix, see Fig.\[e1\], which can depend on the handedness $\sigma_n=\pm$ of the crossing of $\partial{\mathcal{R}}$ and edge $n$ ($+$ for right, $-$ for left).
Any truly topological symmetries should persist under arbitrary deformations of the region ${\mathcal{R}}$, hence the relevant task is to find a complete set $\mathcal{S}_g$ of linearly independent single block [@MPSrepresentations] MPOs $O_{\alpha}^{\partial {\mathcal{R}}}(g) $ for every symmetry transformation $U(g)$ such that for every region ${\mathcal{R}}$ (satisfying the conditions outlined above) we have $$\label{singlelayer}
U(g)^{\otimes|{\mathcal{R}}|_v}{\ensuremath{A}}_{\mathcal{R}}={\ensuremath{A}}_{\mathcal{R}}O^{\partial {\mathcal{R}}}_{\alpha}(g)$$ see Fig.\[e2\].
$$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.33\linewidth]{Figures/e3}}} \quad = \quad
\vcenter{\hbox{
\includegraphics[width=0.33\linewidth]{Figures/e4}}} \end{aligned}$$
There is an important subtlety in finding inequivalent MPOs that satisfy Eq. since two linearly independent solutions $O^{\partial {\mathcal{R}}}_{1}(g),\, O^{\partial {\mathcal{R}}}_{2}(g)$ may coincide on the support of ${\ensuremath{A}}_{\mathcal{R}}$. This occurs precisely when they differ by an operator supported on the kernel of ${\ensuremath{A}}_{\mathcal{R}}$. To remove this redundancy one must first find the set of all single block MPO symmetries $\mathcal{S}_1$ for $U(1)=\openone$. Assuming these MPOs are complete in the following sense $\sum_{\alpha}O_{\alpha}^{\partial{\mathcal{R}}}(1)={\ensuremath{A}}_{\mathcal{R}}^+{\ensuremath{A}}_{\mathcal{R}}$, where ${\ensuremath{A}}_{\mathcal{R}}^+$ is a distinguished generalized inverse of ${\ensuremath{A}}_{\mathcal{R}}$, any MPO $\hat O^{\partial {\mathcal{R}}}$ can be projected onto the support of ${\ensuremath{A}}_{\mathcal{R}}$ to yield another MPO ${\ensuremath{A}}_{\mathcal{R}}^+{\ensuremath{A}}_{\mathcal{R}}\hat O^{\partial {\mathcal{R}}}$ with a (multiplicative) constant increase in the bond dimension. Hence the set of inequivalent single blocked MPO symmetries $\mathcal{S}_g:=\{O^{\partial {\mathcal{R}}}_{\alpha}(g)\}_\alpha$ can be found by taking all linearly independent MPOs satisfying Eq., projecting them onto the support subspace ${\ensuremath{A}}_{\mathcal{R}}^+{\ensuremath{A}}_{\mathcal{R}}$ and collecting the linearly independent single block MPOs that result.
Eq. implies that $\mathcal{S}:=\bigcup_g \mathcal{S}_g$ has a $\mathsf{G}$-graded algebra structure. This algebra structure and the number of elements in $\mathcal{S}$ must be independent of $\mathcal{R}$. Note the MPO matrices $B_{\sigma_e,\alpha}^{ij}(g)$ also do not depend on $\mathcal{R}$ hence for every region the MPO $O^{\partial {\mathcal{R}}}_{\alpha}(g)$ is constructed from the same local tensors. The symmetry relations of Eq., the graded algebra structure of $\mathcal{S}$ and any discrete labels of the MPO representation of this graded algebra provide universal labels of a quantum phase, independent of the details of the local tensors ${\ensuremath{A}}_v$.
A discrete set of labels that fully specify a symmetry-enriched topological phases of matter can be derived from $\mathcal{S}$, the MPO representation of $\mathsf{G}$, in a purely local fashion and these labels remain invariant under continuous physical perturbations.
This set of labels can be calculated by following a similar approach to Ref.[@nick], and they should describe the emergent symmetry defects and their $\mathsf{G}$-graded fusion and $\mathsf{G}$-crossed braiding properties. Note this data subsumes the underlying anyon theory and the possibly fractional symmetry transformation of the defects.
Intrinsic topological order is defined without reference to any symmetry and thus corresponds to the $\mathsf{G}=\{1\}$ case, in which PEPS are classified by $\mathcal{S}_1$. Injective PEPS [@GarciaVerstraeteWolfCirac08] always posses trivial topological order and have $\mathcal{S}_1=\{\openone^{\otimes|\partial{\mathcal{R}}|}\}$ whereas all known topological ordered PEPS [@Ginjectivity; @Buerschaper14; @transfermatrix; @MPOpaper; @chiral4] satisfy Eq. with a nontrivial $\mathcal{S}_1$. This was formalized in the framework of MPO-injectivity in Ref.[@MPOpaper], which was shown to capture all Levin-Wen string-net models (the Hamiltonian version of Turaev-Viro state sum invariants [@turaev1992state]). In Ref.[@MPOpaper] the independence of the MPO tensors from the region ${\mathcal{R}}$ was guaranteed by the intuitive *pulling through* property and the more technical *generalized* and *extended inverse* properties, all of which were purely local conditions.
By taking a global symmetry $\mathsf{G}$ into account, a finer classification is achieved in terms of $\mathcal{S}$ where $|\mathcal{S}_g|>0$ $\forall g\in G$. This classification contains symmetry-protected phases for $|\mathcal{S}_1|=1$ and symmetry-enriched topological phases for $|\mathcal{S}_1|>1$. In the next section we demonstrate how solutions of Eq. can be obtained for nontrivial elements $g\in\mathsf{G}$ in a similar fashion to Ref.[@MPOpaper].
Global symmetry in PEPS {#globalsymmetry}
========================
In this section we present a set of local conditions that lead to a general class of solutions to Eq..
Consider a PEPS on a trivalent directed graph ${\ensuremath{\Lambda}}$ embedded in an oriented manifold ${\mathcal{M}}$, built from four index tensors $A$ which we interpret as linear maps from the virtual to physical indices $A:(\mathbb{C}^D)^{\otimes 3}\rightarrow\mathbb{C}^d$. Firstly, we require that the tensors $A$ satisfy the axioms of MPO-injectivity [@MPOpaper], a framework describing general gapped phases without symmetry. Thus (potentially after some blocking of lattice sites, which we assume has already been carried out) the projection $P:=A^+A$ onto the subspace within which the tensor $A$ is injective can be written as a matrix product operator $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.14\linewidth]{Figures/e5}}} \label{n36}
\ \ =
\vcenter{\hbox{
\includegraphics[width=0.25\linewidth]{Figures/e6}}}\end{aligned}$$ here the MPO tensors are denoted as black squares and satisfy the axioms listed in [@MPOpaper], see Appendix \[a\] for a brief review. These axioms ensure that the same MPO is obtained for any larger region, independent of the order in which the generalized inverses are applied, and furthermore that this closed MPO is a projector independent of its length.
We now describe purely local sufficient conditions for a PEPS to be invariant under the on-site action $U(g)$ of a global symmetry group $\mathsf{G}$. Hereto, we introduce another set of closed MPOs $\{V^{\partial {\mathcal{R}}}(g)\, | \, g \in \mathsf{G}\}$ which inherit an orientation from $\partial {\mathcal{R}}$. These MPOs are composed of four index tensors that depend on a group element $g$. The tensors are depicted by filled circles in the following diagrams and are defined by conditions and $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.18\linewidth]{Figures/e7}}} \label{n23}
\ = \vcenter{\hbox{\raisebox{0.25cm}{
\includegraphics[width=0.17\linewidth]{Figures/e8}}}}\end{aligned}$$ where $U(g)$ is a unitary representation of $\mathsf{G}$, and $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.18\linewidth]{Figures/e9}}} \label{n25}
\ = \vcenter{\hbox{
\includegraphics[width=0.18\linewidth]{Figures/e10}}}.\end{aligned}$$ Note Eq. with the directions reversed is implied by the above conditions. The orientation of the MPO tensors is significant as pulling the MPO through a PEPS tensor in a right handed fashion, as in Eq., induces an action $U(g)$ on the physical index while pulling through in a left handed fashion results in a physical action $U^\dagger(g)$, this follows directly from Eq. since $U$ is a unitary representation.
With these two properties, it is clear that the ground space of a MPO-injective PEPS constructed from the tensor $A$ on any closed system of arbitrary size is invariant under the global symmetry action $U(g)^{\otimes N}$. Hence such MPO-injective PEPS that are unique ground states must be eigenvectors of the global symmetry. For the special case of injective PEPS [@GarciaVerstraeteWolfCirac08] the MPO $P$ is simply the identity $P=\openone$ (i.e. a MPO with bond dimension $1$), the symmetry MPOs $V(g)$ can always be factorized into a tensor product of local gauge transformations [@canonicalPEPS] and the ground state is unique.
From Eqs. and it immediately follows that the PEPS tensors are intertwiners, i.e. $U(g)A = AV(g)$, where $V(g)$ denotes a closed MPO acting on the three virtual indices. Without loss of generality, and in accordance with the general framework of Section \[formalism\], we impose that the MPOs $V(g)$ act within the support space of $A$ such that $PV(g)=V(g)$, i.e. $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.29\linewidth]{Figures/e11}}} \label{n39}
\ = \vcenter{\hbox{
\includegraphics[width=0.25\linewidth]{Figures/e12}}}\end{aligned}$$ and in particular $V(1)=P$, for $1$ the identity group element. Hence the MPOs $V(g)$ form a representation of $\mathsf{G}$ since we have $AV(g_1g_2) = U(g_1g_2)A = U(g_1)U(g_2)A = AV(g_1)V(g_2)$, and thus $PV(g_1g_2) = PV(g_1)V(g_2)$ where $P:=A^+A$, see Eq. (note $PV(g)P={\ensuremath{A}}^+U(g){\ensuremath{A}}P=PV(g)$). A similar argument shows that the symmetry MPO $V_\text{rev}(g)$ along the path with reversed orientation (inducing reversed arrows) equals $V(g^{-1})$ since ${\ensuremath{A}}V_{\text{rev}}(g)=U^\dagger(g){\ensuremath{A}}=U(g^{-1}){\ensuremath{A}}={\ensuremath{A}}V(g^{-1})$ which implies $ PV_{\text{rev}}(g)=PV(g^{-1})$. The above two arguments extend to arbitrary contractible regions ${\mathcal{R}}$ and boundary MPOs $P_{\partial {\mathcal{R}}},\,V^{\partial{\mathcal{R}}}(g)$.
If we do not project the boundary symmetries onto the support subspace of $A$ there are many equivalent choices for the symmetry action on the boundary. In particular, there might be choices for which the action is factorizable into a tensor product (see e.g. Ref.[@Chen]), even if the support projector is not. However, the resulting boundary actions will generically not form a representation of the relevant symmetry group $\mathsf{G}$. The procedure we have outlined of projecting these actions onto the injectivity subspace provides an unambiguous recipe to identify the relevant set of boundary operators that form a MPO representation of the physical symmetry group $\mathsf{G}$. For the particular case of renormalization group fixed-point models, our recipe matches the results of Ref.[@Chen], as illustrated in Section \[exfpspt\].
With these properties it is clear that the class of symmetric PEPS satisfying Eqs. and constitute a special case of the general framework described in Section \[formalism\]. Let $V^{\partial\mathcal{R}}(g)$ denote the MPO corresponding to group element $g$ acting on the boundary of region $\mathcal{R}$ then we have $$U(g)^{\otimes|{\mathcal{R}}|_v}{\ensuremath{A}}_{\mathcal{R}}={\ensuremath{A}}_{\mathcal{R}}V^{\partial\mathcal{R}}(g)
\, .$$ Note in the general case we may need to decompose $V^{\partial \mathcal{R}}(g)$ into a sum of single block MPOs to be consistent with Section \[formalism\].
This general class of solutions show that the formalism of Section \[formalism\] accommodates the description of both symmetries and topological order, and furthermore nontrivial actions of symmetries on states with topological order. Hence the formalism is well suited to describe symmetry-enriched topological phases within the PEPS framework. We plan to pursue this direction explicitly in future work[@michael].
Symmetry-protected topological PEPS {#sptpeps}
====================================
Having discussed the general framework for gapped phases and global symmetries in PEPS, we now focus on the subclass corresponding to states with symmetry-protected topological order. In the first subsection we identify the characteristic properties of short-range entangled SPT PEPS. We proceed in the second subsection with an analysis of the edge properties of non-trivial SPT PEPS.
Identifying SPT PEPS
---------------------
First we must identify the relevant set of PEPS that accurately capture the short-range entanglement property characteristic of SPT phases. As shown in Ref.[@MPOpaper] and argued in the previous sections, MPO-injective PEPS can describe topological phases with long-range entanglement. To single out the short-range entangled PEPS that are candidates to describe SPT states we require that the projection MPO $P$ has a single block when brought into its canonical form. Let $B_P^{ij}$ denote the MPO matrix with external indices $i$ and $j$, the single block property is equivalent to the transfer matrix $\mathbb{E}_P: = \sum_{ij}B_P^{ij}\otimes \bar{B}_P^{ij}$ having a unique eigenvalue of largest magnitude with a corresponding unique eigenvector of full rank. For RG fixed-point PEPS, which are injective on the support subspace of $P$, we argue that the single block property implies the topological entanglement entropy [@KitaevPreskill; @levin:topological-entropy] is zero.
For a RG fixed-point (zero correlation length) MPO-injective PEPS with a single blocked projector MPO $P$, the topological entanglement entropy of the PEPS is zero
Note the rank of the reduced density matrix $\rho_{\mathcal{R}}$ on a finite homotopically trivial region ${\mathcal{R}}$ of a MPO-injective PEPS on a sphere equals the rank of the projection MPO surrounding that region, i.e. $\mathrm{rank}({\rho_{\mathcal{R}}})=\mathrm{rank}(P_{\partial {\mathcal{R}}})$ [@MPOpaper]. Since the MPO $P$ is a projection, we have $\mathrm{rank}(P_{\partial {\mathcal{R}}}) = \mathrm{tr}(P_{\partial {\mathcal{R}}}) = \mathrm{tr}(P_{\partial {\mathcal{R}}}^2) = \mathrm{tr}(\mathbb{E}_P^L)$, where $L=|\partial{\mathcal{R}}|_e$ is the number of virtual bonds crossing the boundary of the region ${\mathcal{R}}$ under consideration. We then use the uniqueness of the largest eigenvalue $\lambda_{\text{max}}$ of $\mathbb{E}_P$ to conclude that, for large regions, the rank of the reduced density matrix scales as $\lambda_{\text{max}}^L$. This implies that the zero Rényi entropy has no topological correction and for RG fixed-points this furthermore implies that the topological entanglement entropy is zero [@topologicalrenyi]. We expect this property to hold throughout the gapped phase containing the fixed-point.
A further crucial property of a SPT phase without symmetry breaking is the existence of a unique ground state on any closed manifold. For a PEPS to be a unique ground state its transfer matrix must have a unique fixed-point. This excludes both symmetry-breaking and topological degeneracy [[@boundarypaper; @transfermatrix]]{}. By taking a PEPS sufficiently close to its isometric form [@Ginjectivity; @Buerschaper14; @MPOpaper] we avoid the symmetry-breaking case (and assure the gap condition [@SchuchGarciaCirac11]). Furthermore, in Appendix \[b\] we present an argument showing that MPO-injective PEPS with single block projection MPOs do not lead to topological degeneracy.
We have argued above that SPT PEPS should be MPO-injective on the support subspace of a single blocked projection MPO. In the language of Section \[formalism\] this implies $|\mathcal{S}_1|=1$ for SPT PEPS. We now show that in this case the symmetry MPOs are also single blocked.
For any MPO-injective PEPS with a single blocked projection MPO, all symmetry MPOs of that PEPS can be chosen to be single blocked.
Assume $V(g)$ contains multiple blocks when brought into canonical form $V(g)=\sum_i V_i(g)$, then we have $PV(g)=\sum_i V_{\pi(i)}(g)$ in canonical form (for some permutation $\pi$) since $V(g)=PV(g)$ for all lengths. This follows from the fact that a pair of MPOs which are equal for all lengths exhibit the same blocks when brought into canonical form [@dpg]. Furthermore $\pi=1$ since $V_{{i}}(g)=P V_{\pi^{-1}({{i}})}(g)=P^2 V_{\pi^{-1}({{i}})}(g)=V_{\pi({{i}})}(g)$.
We have $$P=V(g^{-1})V(g)=\sum_{{{i}}} V(g^{-1})V_{{{i}}}(g)$$ and since this equality holds for all lengths and $P$ has a single block, there can be only one block on the right hand side after bringing it into canonical form [@dpg]. Hence one term in the sum gives rise to a $P$ block along with zero blocks in the canonical form and the others give rise only to zero blocks. Writing this out we have $$P=V(g^{-1})V_{{{i}}}(g)$$ multiplying by $V(g)$ from the left and making use of the invariance under $P$ implies $$V(g)=V_{{{i}}}(g)$$ which has a single block (after throwing away the trivial zero blocks).
The arguments in this subsection show that the subclass of symmetric, MPO-injective PEPS satisfying Eqs. and which accurately describe SPT phases are precisely those with a single blocked projection MPO, provided they are taken sufficiently close to an isometric form to discount the possibility of a phase transition.
Hence the framework of Section \[formalism\] yields a classification of SPT phases in terms of the discrete labels of the (necessarily single blocked) MPO group representation $V(g)$ of the physical symmetry group $\mathsf{G}$ which include the group structure and the third cohomology class $[\alpha] \in H^3(\mathsf{G},\mathsf{U(1)})$ [@Chen] (see Appendix \[c\] for a review).
Edge properties
----------------
We now focus on how the MPO symmetries affect the edge physics of a SPT PEPS and discuss how this can be used to diagnose nontrivial SPT order.
A short-range entangled PEPS with MPO symmetries $V(g)$ that satisfy Eqs. and has non-trivial SPT order if the third cohomology class $[\alpha]$ of the MPO representation is non-trivial. The existence of this non-trivial SPT order can be inferred by analyzing the edge physics when such a PEPS is defined on a finite lattice ${\mathcal{R}}$ with a physical edge (boundary) $\partial {\mathcal{R}}$. In this case the PEPS has open (uncontracted) virtual indices along the physical boundary and all virtual boundary conditions give rise to exact ground states of the canonical PEPS (bulk) parent Hamiltonian $H_{\text{PEPS}}$ (note boundary conditions orthogonal to the support of $P_{\partial {\mathcal{R}}}$ yield zero). Hence the ground space degeneracy scales exponentially with the length of the boundary, which is a generic property of any PEPS (bulk) parent Hamiltonian. The physically relevant question is whether the Hamiltonian can be perturbed by additional local terms $H_{\text{pert}}=\sum_v H_v$, which are invariant under $\mathsf{G}$, to gap out these edge modes and give rise to a unique symmetric ground state.
In Ref.[@boundarypaper] an isometry $\mathcal{W}$ was derived that maps any operator $O$ acting on the physical indices of the PEPS to an effective operator acting on the virtual indices of the boundary $O\mapsto {\ensuremath{\mathcal{W}^{\ensuremath{A}}_{\mathcal{R}}[O]}}$. [Let ${\ensuremath{A}}_{\mathcal{R}}=WH$ be a polar decomposition of ${\ensuremath{A}}_{\mathcal{R}}$, where $W$ is an isometry from the virtual to physical level $(\mathbb{C}^D)^{\otimes |\partial {\mathcal{R}}|_e}\rightarrow (\mathbb{C}^d)^{\otimes |{\mathcal{R}}|_v}$. This induces the following isometry ${\ensuremath{\mathcal{W}^{\ensuremath{A}}_{\mathcal{R}}[O]}}:=W^\dagger O W$ that maps bulk operators to the boundary in an orthogonality preserving way. Note there is some freedom in choosing $W$ precisely when $P_{\partial {\mathcal{R}}}$ is nontrivial, in this case we make the choice that best preserves locality. Regardless of our choice of $W$ we always have $P_{\partial {\mathcal{R}}}{\ensuremath{\mathcal{W}^{\ensuremath{A}}_{\mathcal{R}}[O]}}P_{\partial {\mathcal{R}}}=H^+{\ensuremath{A}}_{\mathcal{R}}^\dagger O {\ensuremath{A}}_{\mathcal{R}}H^+$, where $H^+$ is defined to be the pseudoinverse of $H$, see Fig.\[e3\].]{}
$$\begin{aligned}
\vcenter{\hbox{\includegraphics[trim={0cm 9cm 26cm 0cm},clip,width=0.3\linewidth]{Figures/e14a}}}\end{aligned}$$
Away from an RG fixed-point, however, it has not been proven that this isometry preserves locality. To this point we venture the following conjecture, [which was numerically illustrated for a particular non-topological PEPS in Ref.[@PEPSedges]]{},
\[conj1\] The boundary isometry of any PEPS with exponentially decaying correlations maps a local operator $O_v$ acting on the physical indices near the boundary to a (quasi-) local operator $\tilde O_e^v:={\ensuremath{\mathcal{W}^{\ensuremath{A}}_{\mathcal{R}}[O_v]}}$ acting on the virtual degrees of freedom along the boundary.
From properties and it is clear that acting with $U(g)$ on every physical site is equivalent to acting with the MPO $V^{\partial {\mathcal{R}}}(g)$ on the virtual boundary indices of the PEPS, [hence a $\mathsf{G}$-symmetric local perturbation $H_v$ to the Hamiltonian at the physical level $H_{\text{PEPS}}$ is mapped to an effective (quasi-) local Hamiltonian term on the virtual boundary $\tilde{H}^v_e$ that is invariant under $V^{\partial {\mathcal{R}}}(g)$. The full symmetric edge Hamiltonian is given by $$\begin{aligned}
\tilde{H}_{\text{edge}}&=P_{\partial {\mathcal{R}}}\, {\ensuremath{\mathcal{W}^{\ensuremath{A}}_{\mathcal{R}}[H_{\text{pert}} ]}}\, P_{\partial {\mathcal{R}}}
\nonumber \\
&=V^{\partial {\mathcal{R}}}(1)\, \left(\sum\limits_{e\in\partial{\mathcal{R}}}\sum\limits_{v\mapsto e} \tilde{H}^v_e \right)\, V^{\partial {\mathcal{R}}}(1)\end{aligned}$$ where $v\mapsto e$ denotes that the bulk perturbation centered on site $v$ becomes a (quasi-) local boundary term centered on virtual bond $e$.\
Ground states of the perturbed physical Hamiltonian $H_{\text{bulk}}=H_{\text{PEPS}}+H_{\text{pert}}$ are given by contracting the virtual boundary indices of the ground state PEPS network with ground states of the effective edge Hamiltonian, i.e. $\ket{\Psi_0^{\text{bulk}}}={\ensuremath{A}}_{\mathcal{R}}{\ensuremath{|\psi_0^{\text{edge}})}}$. If the edge Hamiltonian $\tilde{H}_\text{edge}$ is gapped and does not exhibit spontaneous symmetry breaking then its ground state ${\ensuremath{|\psi_0^{\text{edge}})}}$ is well approximated by an injective MPS that is invariant under $V^{\partial {\mathcal{R}}}(g)$.]{} However it was shown by Chen *et al.* that this results in a contradiction, since an injective MPS cannot be invariant under the action of [a single blocked MPO group representation $V(g)$]{} with non-trivial third cohomology [@Chen].
[Consequently, the effective edge Hamiltonian $\tilde{H}_\text{edge}$ either exhibits spontaneous symmetry breaking, in which case the MPS is not injective, or must be gapless, in which case its ground state cannot be well approximated by a MPS. In the former case, the physical state ${\ensuremath{A}}_{\mathcal{R}}{\ensuremath{|\psi_{0,i}^\text{edge})}}$ obtained by contracting the virtual boundary indices of the PEPS network with ${\ensuremath{|\psi_{0,i}^\text{edge})}}$, one of the symmetry breaking ground states of $\tilde{H}_\text{edge}$, also exhibits symmetry breaking and hence does not qualify as a symmetric state. The latter case, on the other hand, implies that a local symmetric perturbation to the physical Hamiltonian is unable to gap out the gapless edge modes, which is one of the hallmarks of non-trivial SPT order. ]{}
[ Here we have again relied on a form of Conjecture \[conj1\], specifically that a PEPS with exponentially decaying correlations has a gapped transfer matrix, which implies that the gapless modes on the virtual boundary of the PEPS network are approximately identified, via the PEPS map ${\ensuremath{A}}_{\mathcal{R}}$, with physical degrees of freedom that are an order of the correlation length from the boundary.]{} Note this explicit identification of the gapless edge mode degrees of freedom is a major strength of the PEPS framework [@PEPSedges]. Our conjecture is consistent with the intuition that as a SPT PEPS is tuned to criticality the gap of the transfer matrix shrinks and the edge modes extend further into the bulk, and is also supported by the results of Ref.[@SPTphasetransition] concerning phase transitions between symmetry-protected and trivial phases.
In this section we have identified a subclass of symmetric PEPS with short-range entanglement that are MPO-injective with respect to a single blocked projection MPO. This led to a classification of SPT phases within the framework of Section \[formalism\] in terms of the third cohomology class of the MPO symmetry representation. Finally we described the influence of the possibly anomalous MPO symmetry action on the boundary physics of the PEPS. In the next section we explore an alternative approach to classifying SPT phases with PEPS via gauging.
Gauging SPT PEPS {#gaugingsptpeps}
=================
In this section we discuss how gauging a SPT PEPS yields a long range entangled PEPS whose topological order is determined by the symmetry MPOs. We then proceed to show that the gauging procedure preserves the energy gap of a symmetric Hamiltonian. Our approach explicitly identifies how the symmetry MPOs that determine the boundary theory of a SPT model are mapped to topological MPOs that describe the anyons of a topological theory [@nick].
Gauging SPT PEPS to topologically ordered PEPS
-----------------------------------------------
We first outline the application of the gauging procedure from Ref.[@Gaugingpaper] to SPT PEPS and the effect this has upon the MPO symmetries.
Conditions and ensure that the SPT PEPS described in Section \[sptpeps\] are invariant under the global action $U(g)^{\otimes|{\mathcal{M}}|_v}$ of a symmetry group $\mathsf{G}$, hence the quantum state gauging procedure of Ref.[@Gaugingpaper] is applicable. [It was shown in Ref.[@Gaugingpaper] that the virtual boundary action of the physical symmetry in an injective PEPS becomes a purely virtual topological symmetry of the gauged tensors, with a trivial physical action. More precisely, it was shown that the gauging procedure transforms an injective PEPS, with virtual bonds in $\mathbb{C}^D$ and a virtual symmetry representation that factorizes as $V^{\partial\mathcal{R}}(g)=v(g)^{\otimes L}$ (with $v(g):\mathbb{C}^D\to\mathbb{C}^D$), into a $\mathsf{G}$-injective PEPS, with virtual bonds in $\mathbb{C}^D\otimes \mathbb{C}[\mathsf{G}]$, that is injective on the support subspace of the projector $\sum_{g\in \mathsf{G}} \left[ v(g) \otimes R(g) \right]^{\otimes L}$. Here, $L := |\partial {\mathcal{R}}|_e$ is the number of virtual bonds crossing the boundary of the region $\mathcal{R}$ under consideration and $R(g)\ket{h}:=\ket{hg^{-1}}$ denotes the right regular representation of $\mathsf{G}$ on the new component $\mathbb{C}[\mathsf{G}]$ of the virtual bonds.]{} Let us recast this in the framework of Section \[formalism\]. The ungauged symmetric injective PEPS map satisfies $${\ensuremath{A}}_{\mathcal{R}}V^{\partial\mathcal{R}}(g)=U(g)^{\otimes|{\mathcal{R}}|_v}{\ensuremath{A}}_{\mathcal{R}}$$ for any region ${\mathcal{R}}\subset{\mathcal{M}}$ and $g\in\mathsf{G}$. Now let $O^{\partial\mathcal{R}}(g) := \left[ v(g) \otimes R(g)\right ]^{\otimes L}$, then the gauged PEPS map ${\ensuremath{A}}^\mathsf{g}_{\mathcal{R}}$ for any region ${\mathcal{R}}$ satisfies $${\ensuremath{A}}_{\mathcal{R}}^\mathsf{g} O^{\partial\mathcal{R}}(g) ={\ensuremath{A}}_{\mathcal{R}}^\mathsf{g}$$ for all $g\in\mathsf{G}$, which implies that the gauged PEPS ${\ensuremath{A}}^\mathsf{g}$ is in the same phase as a quantum double model constructed form $\mathsf{G}$, provided it is sufficiently close to a fixed-point to ensure there is no symmetry breaking [@qdouble; @Ginjectivity].
The result of Ref.[@Gaugingpaper] can be extended to the general case outlined in Section \[sptpeps\] and Appendix \[b\] where the PEPS map ${\ensuremath{A}}_{\mathcal{R}}$ in region $\mathcal{R}$ has a non-factorizable MPO representation of the symmetry on the virtual level, given by $V^{\partial\mathcal{R}}(g):(\mathbb{C}^D)^{\otimes L} \to (\mathbb{C}^D)^{\otimes L}$, and is only injective on the support subspace of the projection MPO $P_{\partial \mathcal{R}}=V^{\partial \mathcal{R}}(1)$ which is required to be single blocked. Hence we have $$\begin{aligned}
{\ensuremath{A}}_{\mathcal{R}}P_{\partial \mathcal{R}}&= {\ensuremath{A}}_{\mathcal{R}}\\
{\ensuremath{A}}_{\mathcal{R}}V^{\partial\mathcal{R}}(g)&=U(g)^{\otimes|{\mathcal{R}}|_v}{\ensuremath{A}}_{\mathcal{R}}\end{aligned}$$ for all $g\in\mathsf{G}$; note we have explicitly separated the $g=1$ case for emphasis. In the language of Section \[formalism\] we have $\mathcal{S}_g=\{ V^{\partial {\mathcal{R}}}(g)\},\ \forall g\in\mathsf{G}$.
The gauged PEPS $A^\mathsf{g}$ obtained by applying the procedure of Ref.[@Gaugingpaper] to $A$ has virtual bonds in $\mathbb{C}^D\otimes \mathbb{C}[\mathsf{G}]$ and satisfies the axioms of MPO-injectivity [@MPOpaper], but is now injective on the support subspace of the projection MPO $P^\mathsf{g}_{\partial \mathcal{R}} := \frac{1}{|\mathsf{G}|}\sum_{g \in \mathsf{G}}O^{\partial\mathcal{R}}(g)$, where $ O^{\partial\mathcal{R}}(g):= V^{\partial\mathcal{R}}(g) \otimes R(g)^{\otimes L}$, see Appendix \[d\] for a detailed proof. Writing these conditions out, we have $${\ensuremath{A}}^\mathsf{g}_{\mathcal{R}}O^{\partial\mathcal{R}}(g) = {\ensuremath{A}}^\mathsf{g}_{\mathcal{R}}$$ for all $g\in\mathsf{G}$, which implies ${\ensuremath{A}}^\mathsf{g}_{\mathcal{R}}P^\mathsf{g}_{\partial \mathcal{R}}={\ensuremath{A}}^\mathsf{g}_{\mathcal{R}}$. Note every MPO $O^{\partial\mathcal{R}}(g)$ is one of the original MPO symmetries $V^{\partial\mathcal{R}}(g)$ tensored with a tensor product representation on the new component $\mathbb{C}[\mathsf{G}]$ of the virtual space that was introduced by gauging. The MPO representation of $P^\mathsf{g}_{\partial\mathcal{R}}$ thus has a canonical form with multiple blocks labeled by $g\in\mathsf{G}$ that correspond to the single block MPOs $O^{\partial\mathcal{R}}(g)$. Hence for the gauged PEPS $\mathcal{S}_1=\{ O^{\partial {\mathcal{R}}}(g) \,|\, g\in\mathsf{G}\}$. Importantly, tensoring with a local action $R(g)$ on the additional virtual space $\mathbb{C}[\mathsf{G}]$ does not change the bond dimension nor the third cohomology class of the MPO representation.
The topological order of the gauged SPT PEPS is a twisted Dijkgraaf-Witten model (provided it is sufficiently close to a fixed-point to ensure there is no symmetry breaking) which is shown explicitly in Section \[gaugingfixptspt\]. We emphasize that up to the trivial operators $R(g)^{\otimes L}$ the same MPOs determine both the gapless edge modes of the SPT phase and, as argued in [@Buerschaper14; @MPOpaper], the topological order of the gauged model. This realizes the gauging map from SPT models with a finite symmetry group to models with intrinsic topological order, explored at the level of Hamiltonians by Levin and Gu [@LevinGu], explicitly on the level of states. In Appendix \[gaugingham\] we apply the gauging procedure of Ref.[@Gaugingpaper] to families of SPT Hamiltonians with an arbitrary finite symmetry group, which yields an unambiguous gauging map to families of topologically ordered Hamiltonians.
We note that the PEPS gauging procedure can equally well be applied to gauge any normal subgroup $\mathsf{N}\unlhd\mathsf{G}$ of the physical symmetry group $\mathsf{G}$. This gives rise to states with symmetry-enriched topological order, where the topological component corresponds to a gauge theory with gauge group $\mathsf{N}$ and the global symmetry is given by the quotient group $\mathsf{G}/\mathsf{N}$; we plan to investigate this direction further in future work [@michael].
Gauging preserves the gap {#gptg}
--------------------------
We now show that the gauging procedure of Ref.[@Gaugingpaper] preserves the energy gap of a symmetric Hamiltonian, which implies by contrapositive that two SPT PEPS are in different phases when the corresponding gauged PEPS lie in distinct topological phases.
Let $H_\mathsf{m}$ denote a local gapped symmetric ‘matter’ Hamiltonian, which captures the particular case of parent Hamiltonians for SPT PEPS. The Hamiltonian is a sum of local terms $H_\mathsf{m} := \sum_v h_v$, where each $h_v$ acts on a finite region within a constant distance of vertex $v$. Without loss of generality we take the Hamiltonian to satisfy $[h_v,U(g)^{\otimes|{\mathcal{M}}|_v}]=0,\,\forall g\in\mathsf{G}$ and shift the lowest eigenvalue of $H_\mathsf{m}$ to 0. The gap to the first excited energy level is denoted by $\Delta_m>0$. We now apply the gauging procedure of Ref.[@Gaugingpaper] to obtain the gauged matter Hamiltonian defined by $H^{\ensuremath{\mathcal{G}}}_\mathsf{m}:=\sum_v{\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_v}[ h_v]$, for ${\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_v}$ given in Eq.. This Hamiltonian is also local since each ${\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_v}$ is locality preserving.
The gauging procedure introduces gauge fields on the links of the PEPS network and the full Hamiltonian of the gauged system contains local flux constraint terms $H_\mathcal{B}:=\sum_p (\openone-\mathcal{B}_p)$ acting on these gauge fields by adding an energy penalty when the flux through a plaquette $p$ is not the identity group element. Each local term $\mathcal{B}_p$ is a Hermitian projector acting on the edges around plaquette $p$ which has eigenvalue 1 on any gauge field configuration ($\mathsf{G}$-connection) that satisfies the flux constraint and 0 otherwise, see Eq.. Furthermore $\mathcal{B}_p$ is diagonal in the group basis on the edges, hence $[\mathcal{B}_p,\mathcal{B}_{p'}]=0$.
The full Hamiltonian may also contain a sum of local commuting projections onto the gauge invariant subspace $H_P:=\sum_v (\openone-P_v)$, see Eq., this corresponds to a model with an effective low energy gauge theory rather than a strict gauge theory. Hence the full Hamiltonian on the gauge and matter system is given by the following sum $$H_\text{full}=H^{\ensuremath{\mathcal{G}}}_\mathsf{m}+\Delta_\mathcal{B} H_\mathcal{B}+\Delta_P H_P$$ where $\Delta_\mathcal{B},\Delta_P\geq 0$. Note a strictly gauge invariant theory is recovered in the limit $\Delta_P\rightarrow \infty$. It is easy to verify that the components of the full Hamiltonian commute, i.e. $[H^{\ensuremath{\mathcal{G}}}_\mathsf{m},H_\mathcal{B}]=[H^{\ensuremath{\mathcal{G}}}_\mathsf{m},H_P]=[H_\mathcal{B},H_P]=0$, and hence are simultaneously diagonalizable. Furthermore, $H_\mathcal{B}$ and $H_P$ each have lowest eigenvalue $0$ and gap $1$.
Assuming $\Delta_P$ is sufficiently large, the low energy subspace of $H_\text{full}$ lies within the ground space of $H_P$ and hence is spanned by states of the form $P[\, \ket{\lambda}_{{\ensuremath{\Lambda}}_v}\otimes\ket{\phi}_{{\ensuremath{\Lambda}}_e}]$, with $P = \prod_{v\in\Lambda}P_v$, for a basis $\ket{\lambda}$ of the matter (vertex) degrees of freedom (we will consider the eigenbasis of $H_\mathsf{m}$) and a basis $\ket{\phi}$ of the gauge (edge) degrees of freedom (we will consider the group element basis).
Similarly, assuming $\Delta_\mathcal{B}$ is sufficiently large, the low energy subspace of $H_\text{full}$ lies within the ground space of $H_\mathcal{B}$ which is spanned by states whose gauge fields form a flat $\mathsf{G}$-connection on the edge degrees of freedom. Since we additionally have $[\mathcal{B}_p,P]=0$ the common ground space of $H_\mathcal{B}$ and $H_P$ is spanned by states of the form $P[\, \ket{\lambda}_{{\ensuremath{\Lambda}}_v}\otimes\ket{\phi_\text{flat}}_{{\ensuremath{\Lambda}}_e}]$, for a basis $\ket{\phi_\text{flat}}$ of the flat $\mathsf{G}$-connections on the edge degrees of freedom (note these are product states).
$\mathsf{G}$-connections form equivalence classes under the local gauge operations ${\ensuremath{a}}_v^g:=\bigotimes_{e\in E_v^+} R_e(g)\bigotimes_{e\in E_v^-} L_e(g)$ (see appendix \[d\] for a more detailed definition of $a^g_v$). On a 1-homotopy trivial manifold (no noncontractible loops) there is only 1 such equivalence class given by all connections of the form $\ket{\phi_\text{flat}}=\prod_i a_{v_i}^{g_i}\ket{1}_{{\ensuremath{\Lambda}}_e}$, where $\ket{1}_{{\ensuremath{\Lambda}}_e}:=\ket{1}^{\otimes |{\ensuremath{\Lambda}}_e|}$.
\[prop3\] For a 1-homotopy trivial manifold, the states $G\ket{\lambda}$ (for a basis $\ket{\lambda}$) span the common ground space of both $H_\mathcal{B}$ and $H_P$, where $G$ is the quantum state gauging map defined in Eq..
Since $P_v=\int\mathrm{d} g\, U_v(g) \otimes {\ensuremath{a}}_v^g$ one can easily see $P_v {\ensuremath{a}}_v^g = P_v U^\dagger_v(g)$ and hence for any state in the intersection of the ground spaces of $H_\mathcal{B}$ and $H_P$ we have $$\begin{aligned}
P[\, \ket{\psi}_{{\ensuremath{\Lambda}}_v}\otimes\ket{\phi_\text{flat}}_{{\ensuremath{\Lambda}}_e}]&=P[\, \ket{\psi}_{{\ensuremath{\Lambda}}_v}\otimes\prod_i a_{v_i}^{g_i}\ket{1}_{{\ensuremath{\Lambda}}_e}]
\nonumber \\
&=
P[\, [\prod_i U_{v_i}(g_i)]^\dagger\ket{\psi}_{{\ensuremath{\Lambda}}_v}\otimes\ket{1}_{{\ensuremath{\Lambda}}_e}]
\nonumber \\
&= G [\prod_i U_{v_i}(g_i)]^\dagger\ket{\psi}_{{\ensuremath{\Lambda}}_v}\end{aligned}$$ where we have started from our above characterization of the common ground space.
We now proceed to show that any eigenstate of $H_\mathsf{m}$ is mapped to an eigenstate of $H^{\ensuremath{\mathcal{G}}}_\mathsf{m}$ by the quantum state gauging map $G$. See appendix \[d\] for the details about the operator and state gauging maps $\mathcal{G}$ and $G$ as constructed in [@Gaugingpaper].
\[p4\] The identity ${\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}[O]G = GO$ holds for any symmetric operator $O$.
Suppose $O$ acts on the sites $v\in{\ensuremath{\Gamma}}\subset{\ensuremath{\Lambda}}$ where ${\ensuremath{\Gamma}}$ is a subgraph of the full lattice which contains all the edges between its vertices, then we have $$\begin{aligned}
&{\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}[O]G = \int\prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} h_v\bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(h_v) O \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v^\dagger(h_v)
\nonumber \\
&\bigotimes_{e\in{\ensuremath{\Gamma}}} \ket{h_{v_e^-}h_{v_e^+}^{-1}}\bra{h_{v_e^-}h_{v_e^+}^{-1}} \int\prod_{v\in{\ensuremath{\Lambda}}} \mathrm{d} g_v\bigotimes_{v\in{\ensuremath{\Lambda}}} U_v(g_v)
\bigotimes_{e\in{\ensuremath{\Lambda}}} \ket{g_{v_e^-}g_{v_e^+}^{-1}}
\nonumber \\
&\phantom{{\ensuremath{\mathcal{G}}}[O]G}=\int\prod_{v\in{\ensuremath{\Lambda}}} \mathrm{d} g_v \prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} h_v \bigotimes_{v\in{\ensuremath{\Lambda}}} U_v(g_v) \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v^{-1} h_v) \,
\nonumber \\
& O\, \bigotimes_{v\in{\ensuremath{\Gamma}}} U^{\dagger}_v(g_v^{-1}h_v)
\prod_{e\in{\ensuremath{\Gamma}}} \delta_{(g_{v_e^-}^{-1}h_{v_e^-}),\, (g_{v_e^+}^{-1}h_{v_e^+})}
\bigotimes_{e\in{\ensuremath{\Lambda}}} \ket{g_{v_e^-}g_{v_e^+}^{-1}}
\nonumber \\
&\phantom{{\ensuremath{\mathcal{G}}}[O]G}=G\, O\end{aligned}$$ where edge $e$ runs from vertex $v_e^+$ to $v_e^-$. The last equality follows since the $\delta$ condition forces $(g_v^{-1}h_v)$ to be equal for all $v\in{\ensuremath{\Gamma}}$ (assuming ${\ensuremath{\Gamma}}$ is connected) and the operator $O$ is symmetric under the group action $[ O, \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g)] = 0$.
This implies that any eigenstate $\ket{\psi_\lambda}$ of $H_\mathsf{m}$ with eigenvalue $\lambda$ gives rise to an eigenstate $G\ket{\psi_\lambda}$ of $H^{\ensuremath{\mathcal{G}}}_\mathsf{m}$ with the same eigenvalue. Note we have assumed that $G\ket{\psi_\lambda}\neq0$, which is the case when the representation under which $\ket{\psi_\lambda}$ transforms contains the trivial representation. This always holds for a unique ground state (possibly after redefining the matrices of the group representation by multiplicative phases $U(g)\mapsto e^{i\theta(g)}U(g)$).
If $H_\mathsf{m}$ has a unique ground state $\ket{\lambda_0}$ the ground state of the full Hamiltonian is given by $G\ket{\lambda_0}$ (since $H^{\ensuremath{\mathcal{G}}}_\mathsf{m}\geq0$ for $H_\mathsf{m}\geq 0$) and its gap satisfies $\Delta_\text{full}\geq\min(\Delta_\mathsf{m},\Delta_\mathcal{B},\Delta_P)$.
Hence if two local SPT Hamiltonians are connected by a gapped, continuous and symmetric path of local Hamiltonians then the gauged models are also connected by a gapped and continuous path of local Hamiltonians.
In Appendix \[e\] we extend this proof to SPT Hamiltonians on topologically nontrivial manifolds where the gauging procedure leads to a topological degeneracy of the ground space. Orthogonal topological ground states are obtained by gauging distinct symmetry twisted SPT states, which are the subject of the next section.
Symmetry twists and monodromy defects {#six}
======================================
In this section we argue that symmetry twists and monodromy defects have a natural description in the tensor network formalism in terms of symmetry MPOs that correspond to anyons in the gauged model. We harness this description to calculate the effect that modular transformations have upon symmetry twisted and topological ground states via their effect on a four index *crossing tensor*. Similarly we calculate the projective transformation of a monodromy defect by composing two *crossing tensors*. Our approach explicitly identifies how the symmetry MPOs that describe defects of a SPT model become topological MPOs that describe the anyons of a topological model [@nick].
Symmetry twists in SPT PEPS
---------------------------
We first describe the construction of a symmetry twisted SPT PEPS in terms of the original SPT PEPS, symmetry MPOs and a crossing tensor. We then calculate the transformation of this state under the residual symmetry group.
For a flat gauge field configuration there is a well defined procedure for applying a corresponding symmetry twist to a local symmetric Hamiltonian, given by conjugating each local term by a certain product of on-site symmetries (see Appendix \[e\]). On a trivial topology such a symmetry twist can be applied directly to a symmetric state by acting with a certain product of on-site symmetries. For example a symmetry twist on an infinite plane, specified by a pair of commuting group elements $(x,y)\in\mathsf{G}\times \mathsf{G}$ and oriented horizontal and vertical paths $p_x,p_y$ in the dual lattice, acts on a state $\ket{\psi}$ in the following way $$\begin{aligned}
\ket{\psi}^{(x,y)}:= \bigotimes_{v\in \mathcal{U}} U_v(x) \bigotimes_{v\in\mathcal{R}} U_v(y) \ket{\psi}\end{aligned}$$ where ${\mathcal{R}}$ is the half plane to the right of $p_y$, $\mathcal{U}$ the half plane above $p_x$, see Fig.\[e4\]. Note $x$ and $y$ must commute for the relevant gauge field configuration to be flat. One can also understand why they must commute by first applying the $x$-twist which reduces the symmetry group to $\mathsf{C}(x)$ (the centralizer of $x$) and hence it only makes sense to implement a second twist for $y\in\mathsf{C}(x)$. With this definition applying a symmetry twist to an eigenstate of a symmetric Hamiltonian (on a trivial topology) yields an eigenstate of the symmetry twisted Hamiltonian with the same eigenvalue.
$$\begin{aligned}
a) \vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/e15}}} \quad b)
\vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/e16}}} \end{aligned}$$
The framework of SPT PEPS provides a natural prescription for the application of a symmetry twist directly to a PEPS on any topology, given by acting with symmetry MPOs on the virtual level of the PEPS. In the above example, assuming $\ket{\psi}$ is a SPT PEPS with local tensor $A$ and symmetry MPOs $V(g)$, Eq. implies that the symmetry twisted state $\ket{\psi}^{(x,y)}$ is given by acting on the virtual level of the PEPS $\ket{\psi}$ with the MPOs $V^{{\ensuremath{p}}_x}(x),V^{{\ensuremath{p}}_y}(y)$ (with inner indices contracted with the four index *crossing tensor* $Q_{x,y}$ where ${\ensuremath{p}}_x,{\ensuremath{p}}_y$ intersect) see Fig.\[e5\].
![$(x,y)$ symmetry twisted PEPS, for infinite or periodic boundary conditions.[]{data-label="e5"}](Figures/e17){width="0.43\linewidth"}
The *crossing tensor* $Q_{x,y}$ is defined in terms of the local reduction tensor of the MPO representation $X(x,y)$ (see Eqs.(\[n27\],\[na21a\])) $$\begin{aligned}
\label{cxy}
Q_{x,y}:&=X(x,y) X^{+}(y,x)
\\
&= {\ensuremath{W_{R}^{x}(y)}}
\nonumber
\\
\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/e18}}}
\ :&=
\vcenter{\hbox{
\includegraphics[width=0.25\linewidth]{Figures/e19}}}
\nonumber\end{aligned}$$ where $X^+(y,x)$ is the pseudoinverse of $X(y,x)$. Eq. and the *zipper* condition for $X(x,y)$ imply that the $Q_{x,y}$ tensor contracted with MPOs $V^{{\ensuremath{p}}_x}(x),V^{{\ensuremath{p}}_y}(y)$ can be moved through the PEPS on the virtual level by applying appropriate on-site symmetries to the physical level.
This prescription extends straightforwardly to an arbitrary topology (see Appendix \[e\]) as we now demonstrate with the example of a symmetry twist on a torus for a pair of commuting group elements $(x,y)$ along distinct noncontractible cycles $p_x,p_y$. The symmetry twisted SPT PEPS $\ket{\psi}^{(x,y)}$ is again given by applying the MPOs $V^{{\ensuremath{p}}_x}(x),V^{{\ensuremath{p}}_y}(y)$ (with inner indices contracted with the crossing tensor $Q_{x,y}$) to the virtual level of the untwisted PEPS $\ket{\psi}$. Importantly this prescription fulfills the condition that applying a symmetry twist to a PEPS groundstate of a symmetric frustration free Hamiltonian yields a groundstate of the symmetry twisted Hamiltonian due to Eq.. We note that similar tensor network techniques allow a construction of symmetry twists for time reversal symmetry [@Gaugingtime].
A symmetry twisted state with conjugated group elements $(x^g,y^g)$ is related, up to a phase, to the symmetry twisted state with group elements $(x,y)$ via a global symmetry action as follows $\theta^{x,y}_g \ket{\psi}^{(x^g,y^g)}=U(g)^{\otimes |{\mathcal{M}}|_v}\ket{\psi}^{(x,y)}$. Similarly a symmetry twisted state for a local deformation of the paths $(p_x,p_y)\mapsto(\tilde p_x,\tilde p_y)$ is related to the symmetry twisted state for $(p_x,p_y)$ by a product of on-site symmetries corresponding to the deformation via Eq.. Hence the number of distinct classes of symmetry twisted states on a torus, under local operations, is given by the number of conjugacy classes of commuting pairs of group elements, which equals the number of irreducible representation of the quantum double ${D}(\mathsf{G})$ [@Ginjectivity].
It is apparent that a symmetry twisted state $\ket{\psi}^{(x,y)}$ forms a 1D representation under the physical action of the residual symmetry group $\mathsf{C}(x,y)$, where $\mathsf{C}(S)$ denotes the centralizer of a subset $S\subseteq G$. Assuming that the untwisted ground state $\ket{\psi}$ is symmetric under $\mathsf{G}$ (which can always be achieved after rephasing the physical representation) the symmetry twisted states may still form nontrivial 1D representations of their respective residual symmetry groups, this fact becomes important when counting the ground space dimension of the gauged model. Calculating these 1D representations explicitly within the PEPS framework yields the result $\theta^{x,y}_g={\ensuremath{{\alpha}^{(x,y)}(g)}}$ the second slant product of the 3-cocycle $\alpha$ that arose from the MPO group representation (see Appendix \[newapp2\], Eq.\[na27\]). Hence an $(x,y)$ symmetry twisted state is symmetric under $\mathsf{C}(x,y)$ iff ${\ensuremath{{\alpha}^{(x,y)}}}\equiv 1$, in which case $y$ is called ${\ensuremath{{\alpha}^{(x)}}}$-regular. If this property is satisfied by a given $y\in\mathsf{C}(x)$ it is also holds for all conjugates of $y$. Furthermore the number of ${\ensuremath{{\alpha}^{(x)}}}$-regular conjugacy classes is known to be equal to the number of irreducible projective representations with 2-cocycle ${\ensuremath{{\alpha}^{(x)}}}$ [@DijkgraafWitten].
Gauging the symmetry twisted SPT PEPS
-------------------------------------
We now outline how the application of an appropriate gauging procedure to a symmetry twisted SPT PEPS yields a topological ground state.
There is a twisted version of the gauging procedure of Ref.[@Gaugingpaper] for each flat gauge field configuration which maps a symmetric Hamiltonian with the corresponding symmetry twist to a gauged Hamiltonian, the same one as obtained by applying the untwisted gauging procedure to the untwisted symmetric Hamiltonian (see Appendix \[e\] for more detail). For a fixed representative $(x,y)$ the twisted gauging operator $G_{x,y}$ is given by contracting the tensor product operators $R(x)^{\otimes |{\ensuremath{p}}_x|},R(y)^{\otimes |{\ensuremath{p}}_y|}$ with the virtual level of the original gauging operator $G$. The twisted versions of the state gauging map are orthogonal for distinct symmetry twists in general and furthermore the fixed representatives satisfy ${G_{x',y'}^\dagger G_{x,y}=\delta_{[x',y'],[x,y]}\int \mathrm{d}g \, U(g)^{\otimes |{\mathcal{M}}|_v} \delta_{g\in\mathsf{C}(x,y)}}$ (see Appendix \[e\].3 for a proof of this). Hence each conjugacy class of symmetry twisted states that are symmetric under the residual symmetry group is mapped to an orthogonal ground state, while those that form a nontrivial 1D representation are mapped to 0. Consequently the dimension of the ground space for the gauged model is given by the number of irreducible representations of the twisted quantum double $D^\alpha(\mathsf{G})$ which can not be larger than the ground space dimension of a gauged trivial SPT model with the same symmetry group.
Given a SPT PEPS ground state $\ket{\psi}$, the orthogonal ground states of the gauged model can be constructed by applying the gauging tensor network operator and acting with the SPT symmetry MPO and a product of on-site symmetry actions $[V(g) \otimes R(g)^{\otimes L}]$ along noncontractible cycles on the virtual level of the gauged tensor network $G\ket{\psi}$. For a fixed representative $(x,y)$ of a symmetric class of symmetry twists the corresponding gauged ground state is given by contracting the MPOs $[V^{{\ensuremath{p}}_x}(x) \otimes R(x)^{\otimes |{\ensuremath{p}}_x|}],[V^{{\ensuremath{p}}_y} (y) \otimes R(y)^{\otimes |{\ensuremath{p}}_y|}]$ (with the crossing tensor $Q_{x,y}$ at the intersection point ${\ensuremath{p}}_x\cap\, {\ensuremath{p}}_y$ [@MPOpaper]) with the virtual level of the gauged PEPS $G\ket{\psi}$.
Modular transformations
-----------------------
We calculate the effect of modular transformation on symmetry twisted and topological ground states via their effect on a set of four index crossing tensors.
Symmetry twisted ground states have been used to identify non trivial SPT order via the matrix elements of modular transformations taken with respect to them [@symmetrytwist; @huang2015detecting]. We have calculated the SPT $\tilde S$ & $\tilde T$ matrices, corresponding to a $\frac{\pi}{2}$ rotation and a Dehn twist respectively, using our framework to find (see Eq.) $$\begin{aligned}
\bra{x',y'} \tilde S \ket{x,y} &= {\ensuremath{{\alpha}^{(y)}(x^{-1},x)}}^{-1} \braket{x',y'|y,x^{-1}}
\\
\bra{x',y'} \tilde T \ket{x,y} &= \alpha(x,y,x) \braket{x',y'|x,xy}\end{aligned}$$ where we have used the abbreviation $\ket{x,y}:=\ket{\psi}^{(x,y)}$, $\alpha^{(y)}$ is the slant product of $\alpha$ (see Appendix \[newapp2\], Eq.\[slant\]) and note $y\in\mathsf{C}(x)$. The gauging procedure elucidates the precise correspondence between these matrix elements and the $S$ & $T$-matrix of the gauged theory [@vishwanath; @MPOpaper; @moradi] which we have also calculated within the ground space (again see Eq.) $$\begin{aligned}
S &= \sum_{xy=yx} {\ensuremath{{\alpha}^{(y)}(x^{-1},x)}}^{-1} \ket{[y,x^{-1}]}\bra{[x,y]} \\ T&= \sum_{ xy=yx} \alpha(x,y,x) \ket{[x,xy]}\bra{[x,y]} \end{aligned}$$ where $\ket{[x,y]}:=G_{x,y}\ket{\psi}^{(x,y)}$ denotes a ground state of the gauged model. Note in our framework we consider a fixed but arbitrary choice of representative for each conjugacy class, rather than group averaging over them.
We have explicitly verified that $S$ & $T$ generate a linear representation of the modular group in agreement with known results for lattice gauge theories (See Subsection \[appmodular\]).
Projective symmetry transformation of monodromy defects
--------------------------------------------------------
Here we describe an explicit construction of the projective representation that acts upon a monodromy defect. We calculate the 2-cocycle of this projective representation by considering the composition of pairs of crossing tensors.
Monodromy defects can be understood as symmetry twists along paths with open end points and have proven useful for the identification of SPT phases [@Wen1; @Zaletel]. The prescription for applying symmetry twists to SPT PEPS extends naturally to a construction of a pair of monodromy defects at the ends of a path ${\ensuremath{p}}_g$, for $g\in\mathsf{G}$. This is given by applying a symmetry MPO $V^{{\ensuremath{p}}_g}(g)$ to the virtual level of the PEPS with an open inner index at either end of the path, which may be contracted with defect tensors replacing the PEPS tensors at each of the defects, see Fig.\[e7\]. A defect tensor must lie in the support subspace of the projector ${\ensuremath{{\ensuremath{\mathcal{V}}}}}^g(1)$ acting on its virtual indices, see Eq.. This may leave some freedom in choosing the tensor which correspond to internal degrees of freedom of the defect, see Ref.[@nick] for further details. Applying the twisted gauging procedure for the corresponding gauge field configuration (which is flat except near the defect points) explicitly maps the symmetry twisted PEPS to a PEPS that describes a pair of anyon excitations in the gauged theory, see Appendix \[g\] and Refs.[@Ginjectivity; @nick].
We now study a pair of monodromy defects on a twice punctured sphere topology, with a defect in each puncture, see Fig.\[e7\]. This captures the case of a symmetry twist $g$ applied to a path $p_g$ along a cylinder, from one boundary to the other, and also the case of a pair of monodromy defects on a sphere, where each puncture is formed by removing a PEPS tensor and replacing it with a tensor that describes the defect.
$$\begin{aligned}
a) \vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/a1}}} \quad b)
\vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/a2}}} \end{aligned}$$
Treating a symmetry twisted SPT PEPS on a cylinder (of fixed radius) as a one dimensional system, it is clear that the bulk is invariant under the residual symmetry group $\mathsf{C}(g)$ since the symmetry twisted SPT PEPS on a torus formed by closing the cylinder (such that $p_g$ becomes a noncontractible cycle) is symmetric. In this case the PEPS can be interpreted as a MPS and standard results in this setting imply that the global symmetry $U(h)^{\otimes|{\mathcal{M}}|_v}$ is intertwined by the PEPS to a tensor product of projective symmetry representations on the left and right virtual boundaries ${\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_L(h)\otimes {\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_R(h)$.
The projective boundary action ${\ensuremath{{\ensuremath{\mathcal{V}}}}}_R^g(h)$ of the symmetry can be explicitly constructed within the SPT PEPS framework. We find that it is given by a symmetry MPO acting on the PEPS virtual bonds entering the puncture, with its inner indices at the intersection of $p_g$ and the boundary of the puncture contracted with the tensor ${\ensuremath{Y_{R}^{g}(h)}}$ (see Eq.) that acts on the inner index of the symmetry twist MPO $V^{{\ensuremath{p}}_g}(g)$ entering the puncture. $$\label{e8}
{\ensuremath{{\ensuremath{\mathcal{V}}}}}_R^g(h)=\vcenter{\hbox{
\includegraphics[width=0.21\linewidth]{Figures/newfig68}}}$$
The multiplication of physical symmetries induces a composition rule for the ${\ensuremath{Y_{R}^{g}(\cdot)}}$ tensors, see Appendix \[g\] for details. Explicit calculation of these products yields the 2-cocycle factor set $\omega^g$ of the projective boundary representation ${\ensuremath{{\ensuremath{\mathcal{V}}}}}_R^g(k){\ensuremath{{\ensuremath{\mathcal{V}}}}}_R^g(h)=\omega^g(k,h){\ensuremath{{\ensuremath{\mathcal{V}}}}}_R^g(kh)$ in terms of the 3-cocycle $\alpha$ of the MPO symmetry representation $\omega^g(k,h)\sim \frac{\alpha(g,k,h)\alpha(k,h,g)}{\alpha(k,g,h)}$. This is consistent with the results of Ref.[@Zaletel]. Note that altering $\alpha$ by a 3-coboundary induces a 2-coboundary change to the 2-cocycle $\omega^g$, which hence forms a robust label of the SPT phase. The projective symmetry action is closely related to the braiding of anyons in the gauged theory.
Example: fixed-point SPT states {#exfpspt}
================================
Inspired by the illuminating examples in Refs.[@GuWen] and [@Chen] we now present a family of SPT PEPS with symmetry group $\mathsf{G}$ and 3-cocycle $\alpha$ satisfying Eqs. and , and explicitly demonstrate that gauging these states [@Gaugingpaper] yields MPO-injective PEPS that are the ground states of twisted quantum double Hamiltonians [@tqd; @DijkgraafWitten].
Fixed-point SPT PEPS
---------------------
We describe our construction of fixed-point SPT PEPS and calculate the MPOs induced by the symmetry action on a site. We explicitly give the fusion tensors for these MPOs and verify that they satisfy the *zipper* condition before determining the 3-cocycle of the MPO representation.
Our short-range entangled PEPS are defined on any trivalent lattice embedded in an oriented 2-manifold (dual to a triangular graph). They realize states equivalent to a standard SPT fixed-point construction on the triangular graph [@GuWen; @else2014classifying]. To this end we specify an ordering on the vertices of the triangular graph which induces an orientation of each edge, pointing from larger to smaller vertex. With this information we assign the following PEPS tensor $A_\triangle:\mathbb{C}(\mathsf{G})^{\otimes 6}\rightarrow\mathbb{C}(\mathsf{G})^{\otimes 3}$ to each vertex of the trivalent lattice $$\begin{aligned}
\label{n3}
{\ensuremath{A}}_\triangle :=
\int \prod_{v\in\triangle} \mathrm{d} g_v \ &\tilde{\alpha}_\triangle \bigotimes_{v\in \triangle} \ket{g_v}_{\triangle,v}
\nonumber \\
&\bigotimes_{e \in \triangle}( g_{v_e^-}|_{\triangle,e,v_e^-}( g_{v_e^+}|_{\triangle,e,v_e^+}\end{aligned}$$ where edge $e$ is oriented from $v_e^+$ to $v_e^-$ (hence $v_e^- < v_e^+$) in the triangular graph. The phase $\tilde{\alpha}_\triangle$ is defined on a vertex of the trivalent PEPS lattice dual to plaquette $\triangle$ of the triangular lattice, whose vertices appear in the order $v,\ v',\ v''$ following the orientation of the 2-manifold (note the choice of starting vertex is irrelevant), by a 3-cocycle $\alpha$ as follows $\tilde{\alpha}_{\triangle}:=\alpha^{\sigma_\pi}(g_1g_2^{-1},g_2g_3^{-1},g_3)$. Where $(g_1,g_2,g_3):=\pi (g_v,g_{v'},g_{v''})$ with $\pi$ the permutation that sorts the group elements into ascending vertex order and $\sigma_\pi=\pm 1$ is the parity of the permutation (equivalently the orientation of $\triangle$ relative to the 2-manifold). In the following example the tensor ${\ensuremath{A}}_\triangle$, possessing six virtual and three physical indices, has non zero entries given by $$\vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/d24}}} \label{d1} = \alpha(g_1g_2^{-1},g_2g_3^{-1},g_3)\, .$$ Note the tensor diagrams in this section use the convention that physical vertex indices are written within the body of the tensor. Moreover we only depict the virtual and physical index combinations that give rise to non-zero values of the tensor.
The global symmetry of the PEPS on a closed manifold is ensured by the following transformation property of each local tensor $$\begin{aligned}
\label{tensorsym}
R(h)^{\otimes 3} {\ensuremath{A}}_\triangle = {\ensuremath{A}}_\triangle \bigotimes_{e\in\triangle} [ Z^{\sigma_{\triangle,e}}_{e} (h) R(h)^{\otimes 2} ] \, ,\end{aligned}$$ where $ Z_{e} (h) := \int \mathrm{d} g_{v_e^-} \mathrm{d} g_{v_e^+} \alpha (g_{v_e^-}g_{v_e^+}^{-1},g_{v_e^+},h) \vert g_{v_e^-},g_{v_e^+}) \allowbreak (g_{v_e^-},g_{v_e^+}\vert$, and $\sigma_{\triangle,e}=\pm 1$ is $+1$ if $e$ is directed along the clockwise orientation of $\partial\triangle$, and $-1$ otherwise. With this definition one can check that Eq. is equivalent to the cocycle condition . Note the boundary actions on the shared edge of two neighboring tensors ${\ensuremath{A}}_\triangle, {\ensuremath{A}}_{\triangle'}$, induced by group multiplication on the physical sites $\triangle,\triangle'$, cancel out since $\sigma_{\triangle,e}=-\sigma_{\triangle',e}$ from which it follows that the full PEPS (on a closed manifold) is invariant under the group action applied to all physical indices. In our example the symmetry property is[^1] $$\vcenter{\hbox{
\includegraphics[width=0.414\linewidth]{Figures/d35}}} =\hspace{-0.2cm}\vcenter{\hbox{
\includegraphics[width=0.505\linewidth]{Figures/d36}}} \label{d3}$$ Where the left side of the equality depicts the physical symmetry acting on a single tensor, and the right side depicts the virtual representation of the symmetry.
Note that a tensor product of the virtual symmetry matrices $[ Z^{\sigma_{\triangle,e}}_{e} (h) R(h)^{\otimes 2} ]$ in general do not constitute a representation of $\mathsf{G}$. A representation of $\mathsf{G}$ on the virtual level, $V(g)$, is obtained by projecting these matrices onto the subspace on which the PEPS tensor is injective. By doing so we construct MPOs that cannot be factorized as a tensor product. For the current fixed-point example we project $[ Z^{\sigma_{\triangle,e}}_{e} (h) R(h)^{\otimes 2} ]$ onto the subspace of virtual boundary indices corresponding to non-zero values of ${\ensuremath{A}}_\triangle$, Eq.. This yields a MPO $V(h)$ constructed from the following tensors $$\label{mpotensor}
\vcenter{\hbox{
\includegraphics[width=0.3\linewidth]{Figures/d25}}} = \alpha (g_1g_2^{-1},g_2,h)$$ note that for fixed $h$ these MPOs possess a single block. We introduce the isometry $X(h_1,h_2)$ $$\vcenter{\hbox{
\includegraphics[width=0.25\linewidth]{Figures/d26}}} \label{d5} = \alpha(g,h_1,h_2)\, ,$$ to describe the multiplication of two MPO tensors. With this isometry we have the following relation $$\vcenter{\hbox{
\includegraphics[width=0.515\linewidth]{Figures/d29}}} =\vcenter{\hbox{
\includegraphics[width=0.33\linewidth]{Figures/d30}}} \label{d9}$$ where the left most tensor of Eq. is $X^\dagger(h_1,h_2)$ and we have made use of the 3-cocycle condition . This implies that the MPOs $V(h)$ with fixed inner indices indeed form a representation of $\mathsf{G}$. Note the stronger *zipper* condition $$\vcenter{\hbox{
\includegraphics[width=0.52\linewidth]{Figures/d28}}} =\vcenter{\hbox{
\includegraphics[width=0.28\linewidth]{Figures/d27}}} \label{e9}$$ also holds for this MPO representation.
From Eq. it is clear that the PEPS tensors ${\ensuremath{A}}_\triangle$, Eq., together with the MPOs $V(h)$, defined by Eq., have SPT order described by the framework of Section \[sptpeps\]. We now calculate the third cohomology class of the MPOs to determine which SPT phase the model belongs to. For this we see that $X$ obeys the following associativity condition $$\begin{gathered}
\label{n19}
X(h_1h_2,h_3) [X(h_1,h_2)\otimes \mathds{1}_{h_3}] = \\ \alpha^{-1}(h_1,h_2,h_3) \, X(h_1,h_2h_3) [\mathds{1}_{h_1}\otimes X(h_2,h_3)]\, ,\end{gathered}$$ which is again the 3-cocycle condition Eq.. From Eq. we thus conclude that the short-range entangled states described by the tensors of Eq. lie in a symmetry-protected topological phase labeled by the cohomology class $[\alpha^{-1}]\in H^3(G,U(1))$, see Appendix \[a\].
One may be surprised to notice that one layer of strictly local unitaries (equivalent to the local unitary circuit $D_\alpha$ ) acting on the vertices of the PEPS built from the tensors in Eq. can remove the 3-cocycles, thus mapping it to a trivial product state. Superficially this seems to contradict the fact that SPT states cannot be connected to the trivial product state by low-depth local unitary circuits that preserve the symmetry. However, this is not the case as this definition requires every individual gate of the circuit to preserve the symmetry [@chen2010local], which is not true for the circuit just described.
Gauging the fixed-point SPT PEPS {#gaugingfixptspt}
---------------------------------
We now apply the quantum state gauging procedure of Ref.[@Gaugingpaper] to gauge the global symmetry of the fixed-point SPT PEPS defined in the previous subsection. For this we construct a gauging tensor network operator (matching that of Ref.[@Gaugingpaper] on the dual triangular graph) that couples gauge degrees of freedom to a given matter state. We proceed by applying a local unitary circuit to disentangle the gauge constraints and explicitly demonstrate that the resulting tensor describes the ground state of a twisted Dijkgraaf-Witten gauge theory.
The gauging map is defined by the following local tensors $G^\triangle:\mathbb{C}(\mathsf{G})^{\otimes 6}\otimes \mathbb{C}(\mathsf{G})^{\otimes 3} \rightarrow\mathbb{C}(\mathsf{G})^{\otimes 6}$ $$\begin{aligned}
G^\triangle:=\int \prod_{v\in\triangle} \mathrm{d} h_v \bigotimes_{v\in\triangle} &R_{\triangle,v}(h_v) \bigotimes_{e\in\triangle}[\, \ket{h_{v_e^-}h_{v_e^+}^{-1}}_{\triangle,e}
\nonumber \\
&\otimes( h_{v_e^+}|_{\triangle,e,v_e^+} ( h_{v_e^-}|_{\triangle,e,v_e^-}\, ] ,\end{aligned}$$ note $G^\triangle$ introduces gauge degrees of freedom on the edges. For our example the gauging tensor is $$\vcenter{\hbox{
\includegraphics[width=0.45\linewidth]{Figures/d31}}} \label{d10}$$ We can apply the gauging tensors locally to the SPT PEPS to form tensors for a gauge and matter PEPS $$\begin{aligned}
&\bar{{\ensuremath{A}}}_\triangle:=\int \prod_{v\in\triangle} \mathrm{d} h_v \mathrm{d} g_v\, \tilde{\alpha}_\triangle \bigotimes_{v\in\triangle} \ket{g_vh_v^{-1}}_{\triangle,v}
\nonumber \\
&\bigotimes_{e\in\triangle} \ket{h_{v_e^-}h_{v_e^+}^{-1}}_{\triangle,e}
( g_{v_e^+},h_{v_e^+}|_{\triangle,e,v_e^+}( g_{v_e^-}, h_{v_e^-}|_{\triangle,e,v_e^-}\end{aligned}$$ in our example these are $$\vcenter{\hbox{
\includegraphics[width=0.49\linewidth]{Figures/d32}}} = \alpha(g_1g_2^{-1},g_2g_3^{-1},g_3)\, .
\label{d11}$$ The gauged PEPS $\ket{\psi_\mathsf{g}}$, built from the tensors $\bar{\ensuremath{A}}_\triangle$, satisfies local gauge constraints $\tilde{P}_v\ket{\psi_\mathsf{g}} = \ket{\psi_\mathsf{g}}$ for every vertex $v$, where $$\begin{aligned}
\tilde{P}_v:=\int \mathrm{d} g_v \bigotimes_{\triangle \ni v} [ R_{\triangle,v}(h) \bigotimes_{e\in E_v^+} R_{\triangle,e}(g_v) \bigotimes_{e\in E_v^-} L_{\triangle,e}(g_v) ] \nonumber\end{aligned}$$ The gauge and matter tensor $\bar{{\ensuremath{A}}}_\triangle$ is MPO-injective with respect to a purely virtual symmetry inherited from the symmetry transformation of the SPT tensor ${\ensuremath{A}}_\triangle$ and it also intertwines a physical symmetry to a virtual symmetry due to the transformation of the gauging tensors $$\begin{aligned}
\bar{{\ensuremath{A}}}_\triangle \bigotimes_{e\in\triangle} [ Z^{\sigma_{\triangle,e}}_{e} (h) R(h)^{\otimes 2} ] \otimes R(h)^{\otimes 2} =\bar{{\ensuremath{A}}}_\triangle
\\
\bigotimes_{v\in\triangle} R_{\triangle,v}(h) \bigotimes_{e\in\triangle} R_{\triangle,e}(h) L_{\triangle,e}(h) \bar{{\ensuremath{A}}}_\triangle
\nonumber \\
= \bar{{\ensuremath{A}}}_\triangle \bigotimes_{e\in\triangle} \openone^{\otimes 2} \otimes L(h)^{\otimes 2}\end{aligned}$$ the latter symmetry reflects the invariance of the full PEPS under the gauge constraints $\tilde{P}_v$.
We next apply a local unitary circuit $\tilde{C}_{\ensuremath{\Lambda}}$ to explicitly map the gauge and matter model to a twisted quantum double ground state on the gauge degrees of freedom alone. This circuit is given by the tensor product of the following local unitary on each site $$\begin{aligned}
\tilde{C}_\triangle := \int \prod_{v\in\triangle} \mathrm{d} g_v \bigotimes_{v\in\triangle} \ket{g_v}\bra{g_v}_v \bigotimes_{e\in\triangle} L_e(g_{v_e^-}) R_e(g_{v_e^+})\, , \nonumber\end{aligned}$$ which maps the gauge constraints to local rank one projectors on the matter degrees of freedom at each vertex $\tilde{C}_{\ensuremath{\Lambda}}\tilde{P}_v \tilde{C}_{\ensuremath{\Lambda}}= \int \mathrm{d} g_v \bigotimes_{\triangle \ni v} R_{\triangle,v}(h)$, fixing the state of the matter to be $\int \mathrm{d} g_v \bigotimes_{\triangle \ni v} \ket{g_v}_{\triangle,v}$. From this we infer that the circuit $\tilde{C}_{\ensuremath{\Lambda}}$ disentangles the gauge from the matter degrees of freedom. To see this explicitly we apply the circuit locally to each PEPS tensor, along with a unitary change of basis on the virtual level (leaving the physical state invariant) to form the tensor $\bar{\bar{{\ensuremath{A}}}}_\triangle$ which is defined as follows $$\begin{aligned}
&
\bar{\bar{{\ensuremath{A}}}}_\triangle:= \tilde{C}_\triangle \bar{{\ensuremath{A}}}_\triangle \bigotimes_{e\in\triangle} U_{\triangle,e,v_e^+}\otimes U_{\triangle,e,v_e^-}
\nonumber \\
& \phantom{\bar{\bar{{\ensuremath{A}}}}_\triangle}\
= \int \prod_{v\in\triangle} \mathrm{d} k_v \mathrm{d} g_v \,\tilde{\alpha}_\triangle \bigotimes_{v\in\triangle} \ket{k_v}_{\triangle,v}
\bigotimes_{e\in\triangle} [\, \ket{g_{v_e^-}g_{v_e^+}^{-1}}_{\triangle,e}
\nonumber \\
& \phantom{\bigotimes_{e\in\triangle} \ket{g_{v_e^-}g_{v_e^+}^{-1}} }
\otimes ( g_{v_e^+},k_{v_e^+}|_{\triangle,e,v_e^+}( g_{v_e^-}, k_{v_e^-}|_{\triangle,e,v_e^-} \, ]\end{aligned}$$ where $U:=\int \mathrm{d} g \ket{g}\bra{g} \otimes S L^\dagger(g)$, with $S\ket{g}:=\ket{g^{-1}}$, satisfies $(g,h| U= (g,gh^{-1}| $. For our example this tensor is given by $$\vcenter{\hbox{
\includegraphics[width=0.47\linewidth]{Figures/d33}}} = \alpha(g_1g_2^{-1},g_2g_3^{-1},g_3)
\label{d12}$$ This disentangled PEPS tensor $\bar{\bar{{\ensuremath{A}}}}_\triangle$ is now MPO-injective on the support subspace of the projection MPO given by a normalized sum of the symmetry MPOs from the SPT PEPS. Moreover the intertwining condition maps the physical vertex symmetry to a trivial action on the virtual space $$\begin{aligned}
\bar{\bar{{\ensuremath{A}}}}_\triangle \bigotimes_{e\in\triangle}[ Z^{\sigma_{\triangle,e}}_{e} (h) R(h)^{\otimes 2} ] \otimes \openone ^{\otimes 2} = \bar{\bar{{\ensuremath{A}}}}_\triangle
\\
\bigotimes_{v\in\triangle} R_{\triangle,v}(h) \bar{\bar{{\ensuremath{A}}}}_\triangle = \bar{\bar{{\ensuremath{A}}}}_\triangle \bigotimes_{e\in\triangle} \openone^{\otimes 2} \otimes R(h)^{\otimes 2}\, .\end{aligned}$$ From this we see that $\bar{\bar{A}}_\triangle$ separates into a trivial local component on the matter degrees of freedom yielding the state $\bigotimes_v \int\mathrm{d} g_v \bigotimes_{\triangle \ni v} \ket{g_v}_{\triangle,v}$, and the following tensors on the gauge degrees of freedom $$\begin{aligned}
\label{n12}
\int \prod_{v\in\triangle} \mathrm{d} g_v \tilde{\alpha}_\triangle
\bigotimes_{e\in\triangle} \ket{g_{v_e^-}g_{v_e^+}^{-1}}
( g_{v_e^+}|_{\triangle,e,v_e^+}( g_{v_e^-}|_{\triangle,e,v_e^-} .\end{aligned}$$ These tensors define a PEPS on the gauge degrees of freedom that is a ground state of a 2D twisted quantum double with 3-cocycle $\alpha$. Note this PEPS matches the standard representation of the ground state on the subspace obtained by mapping $\bigotimes_{\triangle\ni v}\ket{g}_{\triangle,v}\mapsto \ket{g}_v$ and $\bigotimes_{\triangle\ni e}\ket{g}_{\triangle,e}\mapsto \ket{g}_e$. For our example this tensor is $$\vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/d34}}} = \alpha(g_1g_2^{-1},g_2g_3^{-1},g_3)
\label{d13}$$ note in the Abelian case the tensors in Eq. reduce to the string-net tensors [@stringnet1; @stringnet2] after a suitable mapping between 3-cocycles and $F$-symbols [@ZNstringnet] (in the non-Abelian case one has to change to the basis of irreducible representations to make the identification).
Perturbations away from fixed-points
------------------------------------
The examples presented thus far in this section are all fixed-point states under a real space blocking renormalization group flow and have zero correlation length. This corresponds to the PEPS tensor that builds the state being of MPO-isometric type [@MPOpaper]. More generally one could add an arbitrary perturbation that lies within the MPO-injectivity subspace (this can be constructed by applying the MPO projector to an arbitrary perturbation) to the MPO-isometric PEPS tensor to find a new MPO-injective PEPS that will generically have a finite correlation length. For a sufficiently small symmetric perturbation the resulting MPO-injective PEPS will lie in the same phase of matter as the fixed-point MPO-isometric PEPS [@cirac2013robustness; @Buerschaper14].
The simplest explicit perturbations away from fixed-point tensors are given by local filtering operations on the physical indices. For a given MPO-injective PEPS tensor ${\ensuremath{A}}$ local filtering by a projector $P$ generates a family of MPO-injective deformations $ \{P(\lambda)A \ |\ \lambda\in [0,1)\}$ where ${P(\lambda):=(1-\lambda)\openone + \lambda P}$. For topological PEPS $P$ can be an arbitrary projector on the physical index, while for SPT PEPS it must commute with the on-site symmetry action. This path of deformations can move from one phase of matter to another, for instance if we let $P=\ket{0}\bra{0}$ the deformation can induce an anyon condensation transition if ${\ensuremath{A}}$ describes a topologically ordered ground state [@Gaugingpaper; @shadows; @marien2016condensation]. In the SPT case with on-site group action $R(g)$ one can consider ${P=\ket{\tilde e}\bra{\tilde e}}$, the projector onto the trivial representation, where ${\ket{\tilde e}=\frac{1}{|\mathsf{G}|} \sum\limits _{g\in \mathsf{G}} \ket{g} }$ to find a symmetric interpolation to the trivial phase. A framework to understand these transitions in terms of symmetry breaking of the virtual symmetry is described in Refs. [@shadows; @marien2016condensation].
Conclusions
============
[We have presented a unified picture for the characterization of all gapped phases, possibly with respect to certain physical symmetries, within the framework of PEPS in terms of virtual MPO symmetries. To achieve this we developed a characterization of global symmetry in the framework of MPO-injective PEPS [@Buerschaper14; @MPOpaper]. In contrast to the injective case [@canonicalPEPS], where the symmetry representation on the virtual indices factorizes into a tensor product, a MPO-injective PEPS tensors can have a virtual symmetry representation given by unfactorizable MPOs. We subsequently identified the short-range entangled PEPS to be those having a single block in the projection MPO onto the injectivity subspace. If the accompanying single block MPO virtual symmetry representation has a non-trivial third cohomology class it gives rise to unconventional edge properties and thus to symmetry-protected topological PEPS.]{} Our identification of the virtual entanglement structure of PEPS with SPT order opens new routes to study transitions between SPT phases by utilizing methods that have been developed to study anyon condensation transitions of topological phases [@shadows; @transfermatrix].
[We demonstrated that applying the quantum state gauging procedure [@Gaugingpaper] to a SPT PEPS transforms its MPO representation of $\mathsf{G}$ into a purely virtual symmetry of the gauged tensors. This implies that the resulting gauge-invariant PEPS also satisfies the axioms of MPO-injectivity, but with a projection MPO onto the injectivity subspace with a block structure labeled by the group elements $g\in\mathsf{G}$. This block structure of the projection MPO, together with the third cohomology class label, characterizes the phases of the twisted quantum double models which are known to have intrinsic topological order.]{} It was shown in Ref.[@MPOpaper] that the projection MPO determines all the topological properties of the gauged PEPS. This relation explains the mechanism behind the braiding statistics approach to SPT phases [@LevinGu] at the level of the corresponding quantum states. It furthermore reveals that both the gauging and boundary theory approaches to classifying SPT phases are recast in the PEPS framework as the classification of a common set of MPOs. We have illustrated these concepts for a family of RG fixed-point states, containing a representative for all two-dimensional bosonic SPT phases with a finite on-site symmetry group.
To prove these results we developed new tools to deal with orientation dependent MPO tensors and used them to calculate the symmetry action on monodromy defected and symmetry twisted states and also modular transformations, before and after gauging, in terms of a single tensor.
The general formalism presented in this paper describes both local physical symmetries and topological order of PEPS with virtual MPO symmetries. Furthermore, it captures the general action of a symmetry on a PEPS with topological order and hence yields a natural framework for the study of symmetry-enriched topological phases (SET). The quantum state gauging procedure can be adapted to gauge only a normal subgroup of the global symmetry group of a SPT PEPS, which allows one to explicitly construct families of SET PEPS. An open question is how the corresponding MPOs encode the discrete, universal labels of the SET phase and how to extract them. We further expect that a better understanding of excitations in MPO-injective PEPS [@nick] will yield insights into the physical properties of SET phases such as symmetry fractionalization. We plan to study these matters in future work [@michael].
In this work we only explicitly consider finite on-site unitary symmetry actions. It is an interesting and relevant question to generalize this to time-reversal and continuous Lie group symmetries as well as lattice translation and point group symmetries. Progress has been made on incorporating these types of symmetries into PEPS in Ref. [@jiang2015symmetric]. In particular since time-reversal can be realized as a local action on the PEPS tensors [@Gaugingtime] a similar approach to that used here should apply, with some extra care necessary due to the possible action of time reversal on the symmetry MPOs.
Another question which presents itself is how to generalize the constructions presented in this paper to fermionic systems. Partial progress has been made in the direction of applying the same principles to the formalism of fermionic PEPS [@fpeps]. This has led to a (partial) classification of fermionic SPT phases [@williamson2016fermionic; @bultinck2016fermionic] based on supercohomology [@supercohomology] and the existence of Majorana-type defects [@cheng2015towards]. The quantum state gauging procedure works equally well for fermionic systems, but the gauge degrees of freedom are always bosonic. It would thus be interesting to see how fermionic SPT order can be probed in this way.
Our identification of SPT PEPS in 2D as being injective with respect to an injective MPO hints at a hierarchical definition of SPT PEPS in arbitrary dimension with an injective tensor network object associated to each codimension. This appears to recover the cohomological classification of bosonic SPT states in arbitrary dimensions by a generalization of the argument from [@Chen]. We plan to explore this further in future work.\
\
*Acknowledgments -* We acknowledge helpful discussions with N. Schuch. This work was supported by EU grant SIQS and ERC grant QUERG, the Odysseus grant from the Research Foundation Flanders (FWO) and the Austrian FWF SFB grants FoQuS and ViCoM. M.M. and J.H. further acknowledge the support from the Research Foundation Flanders (FWO).
Axioms for MPO-injectivity {#a}
===========================
This section reviews [the axioms of MPO-injectivity as presented]{} in Ref.[@MPOpaper].
We interpret the tensors $A$ of a MPO-injective PEPS as linear maps from the virtual to the physical space and apply a distinguished generalized inverse $A^+$, which gives rise to a projector that can be written as a MPO: $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.14\linewidth]{Figures/a30}}} \label{n41}
\ = \vcenter{\hbox{
\includegraphics[width=0.25\linewidth]{Figures/a31}}}\end{aligned}$$ We further require this MPO to satisfy the *pulling through* property shown in Eq.. $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.18\linewidth]{Figures/a32}}} \label{n42}
\ = \vcenter{\hbox{
\includegraphics[width=0.18\linewidth]{Figures/a33}}}\end{aligned}$$ The same property should also hold where the MPO gets pulled from three virtual indices to one or vice versa. This makes the presence of this MPO locally undetectable in the PEPS. Using the pulling through property, it is easy to check that the requirement for the MPO to be a projector is equivalent to the property shown in Eq. $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.13\linewidth]{Figures/a34}}} \label{n43}
\ = \vcenter{\hbox{
\includegraphics[width=0.13\linewidth]{Figures/a35}}}\end{aligned}$$ We also need a technical requirement such that the properties of the PEPS grow in a controlled way with the number of sites. For example, we want two concatenated tensors to be injective on the support subspace of the projection MPO surrounding these two tensors. For this we need that there exists a tensor $E$, depicted in (\[n44\]), $$\begin{aligned}
E:= \label{n44}
\vcenter{\hbox{
\includegraphics[width=0.15\linewidth]{Figures/a36}}}\end{aligned}$$ such that we have the *extended inverse* property (\[n45\]). $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.25\linewidth]{Figures/a37}}} \label{n45}
\ = \vcenter{\hbox{\raisebox{1.9 cm}{
\includegraphics[width=0.25\linewidth]{Figures/a38}}}}\end{aligned}$$ The extended inverse property allows one to prove many useful things such as the intersection property or an explicit expression for the ground state manifold on a torus [@MPOpaper]. It turns out that under very reasonable assumptions about the projection MPO the extended inverse condition is automatically satisfied [@nick].
Uniqueness of SPT PEPS ground state {#b}
===================================
In this appendix we demonstrate that the parent Hamiltonian of a MPO-injective PEPS with a single block projection MPO has a unique ground state on the torus (i.e. no topological degeneracy). A similar argument holds for higher genus surfaces.\
For a Hermitian projection MPO there is no need to keep track of a direction on the internal leg of the MPO, we also ignore the explicit directions on the edges of the PEPS as they are irrelevant to our arguments. We require the following condition (stronger than Eq.)
We assume the projection MPO has been brought into a form satisfying the *zipper* condition, i.e. there are no off diagonal blocks in the product of two MPO tensors after it has been brought into canonical form, equivalently $$\begin{aligned}
\label{xinv}
\vcenter{\hbox{
\includegraphics[width=0.16\linewidth]{Figures/newfig1}}}=\vcenter{\hbox{
\includegraphics[height=0.16\linewidth]{Figures/newfig2}}}\end{aligned}$$ where $X$ is the reduction tensor for multiplication of copies of the MPO which forms a single block representation of the trivial group. This is true of the MPOs arising from fixed-point models. For this representation we have the following version of Eq. $$\begin{aligned}
\label{amove}
\vcenter{\hbox{
\includegraphics[height=0.15\linewidth]{Figures/newfig3}}}=
\alpha
\vcenter{\hbox{
\includegraphics[height=0.15\linewidth]{Figures/newfig4}}}.\end{aligned}$$ We now rewrite this equality in a more suggestive fashion $$\begin{aligned}
\label{amove}
\vcenter{\hbox{
\includegraphics[width=0.16\linewidth]{Figures/fig6}}}
\ = \, \alpha \vcenter{\hbox{
\includegraphics[width=0.16\linewidth,angle=90]{Figures/fig6}}} .\end{aligned}$$ In the above, and throughout the remainder of this appendix, we ignore explicit direction dependence as it does not affect the arguments made.\
In the framework of MPO-injectivity different ground states of the PEPS parent Hamiltonian on the torus are spanned by tensor networks closed with different $Q$ tensor solutions (see Ref.[@MPOpaper]) connected to MPOs on the virtual level along the inequivalent noncontractible loops of the torus $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/fig7}}}.\end{aligned}$$ From the physical level one only has access to the $Q$ tensor projected onto the support subspace of a MPO loop along the closure of the system. $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/fig8}}}\end{aligned}$$ Note this closure gives rise to the same ground state as the closed loop is a symmetry of the closed MPO-injective tensor network. Using condition repeatedly (within the closed tensor network) leads to the following crossing tensor $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/fig9}}}\end{aligned}$$ which again gives rise to the same ground state. Following several more applications of Eqs. & we arrive at $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/fig10}}}\end{aligned}$$ Note the overall phase of the ground state is irrelevant. Since the $Q$ tensor can be placed anywhere in the tensor network we have that the following matrix $$\begin{aligned}
M_Q:=\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/fig11}}}\end{aligned}$$ commutes through the virtual level of the single block (injective) projection MPO and hence must be proportional to the identity $M_Q=1$. Plugging this in we have the crossing tensor $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/fig12}}}\end{aligned}$$ which, by Eq., yields the same state as the following $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/fig13}}}\end{aligned}$$ and with several applications of Eqs. & one can verify that this is equivalent to $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.2\linewidth]{Figures/fig14}}}\end{aligned}$$ which is easily seen to be a symmetry of a closed MPO-injective tensor network which hence yields the trivial ground state. To summarize we have seen that any $Q$ tensor solution gives rise to the unique ground state obtained by closing the tensor network without any MPOs on the virtual level.
Third cohomology class of a single block MPO group representation {#c}
==================================================================
In this appendix we recount the definition of the third cohomology class of an injective MPO representation of a finite group $\mathsf{G}$, as first introduced in Ref.[@Chen]. For details about group cohomology theory in the context of SPT order we refer the reader to Ref.[@GuWen].
In a MPO representation of $\mathsf{G}$, multiplying a pair of MPOs labeled by the group elements $g_0$ and $g_1$ is equal to the MPO labeled by $g_0g_1$ for every length. Since the MPOs are injective we again know there exists a gauge transformation on the virtual indices of the MPO that brings both representations into the same canonical form [@MPSrepresentations]. This implies that there exists an operator (the reduction tensor) $X(g_0,g_1):(\mathbb{C}^{{\ensuremath{\chi}}})^{\otimes 2}\rightarrow \mathbb{C}^{ {\ensuremath{\chi}}}$ such that Eq. holds. $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.35\linewidth]{Figures/n56}}} \label{n27}
= \vcenter{\hbox{
\includegraphics[height=0.16\linewidth]{Figures/n57}}}\end{aligned}$$ note $X(g_0,g_1)$ is only defined up to multiplication by a complex phase $\beta(g_0,g_1)$. If we now multiply three MPOs labeled by $g_0$, $g_1$ and $g_2$ there are two ways to reduce the multiplied MPOs to the MPO labeled by $g_0g_1g_2$. When only acting on the right virtual indices these two reductions are equivalent up to a nonzero complex number labeled by $g_0$, $g_1$ and $g_2$. This is shown in Eq.. $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[height=0.33\linewidth]{Figures/n58}}} \label{n29}
= \alpha(g_0,g_1,g_2)\vcenter{\hbox{
\includegraphics[height=0.33\linewidth]{Figures/n59}}}\end{aligned}$$ By multiplying four MPOs one sees that $\alpha$ has to satisfy certain consistency conditions as the two different paths achieving the same reduction, shown in Eq., should give rise to the same complex number. $$\begin{aligned}
\begin{split}
\vcenter{\hbox{
\includegraphics[height=0.35\linewidth]{Figures/n60}}} \label{n31}
\ \rightarrow \vcenter{\hbox{
\includegraphics[height=0.35\linewidth]{Figures/n61}}}
\ \rightarrow \vcenter{\hbox{
\includegraphics[height=0.35\linewidth]{Figures/n62}}}
\\
\downarrow \hspace{5cm} \downarrow\hspace{1cm}
\\
\vcenter{\hbox{
\includegraphics[height=0.35\linewidth]{Figures/n64}}}
\hspace{0.8cm} \rightarrow \hspace{0.8cm}
\vcenter{\hbox{
\includegraphics[height=0.35\linewidth]{Figures/n63}}}
\hspace{0.6cm}
\end{split}\end{aligned}$$ Using Eq. one can easily verify that the consistency conditions are $$\frac{\alpha(g_0,g_1,g_2)\alpha(g_0,g_1g_2,g_3)\alpha(g_1,g_2,g_3)}{\alpha(g_0g_1,g_2,g_3)\alpha(g_0,g_1,g_2g_3)} = 1 \label{cocycle}$$ which are exactly the 3-cocycle conditions and hence $\alpha$ is a 3-cocycle. As mentioned above $X(g_0,g_1)$ is only defined up to a complex number $\beta(g_0,g_1)$. This freedom can change the 3-cocycle defined in Eq. by $$\alpha'(g_0,g_1,g_2) = \alpha(g_0,g_1,g_2)\frac{\beta(g_1,g_2)\beta(g_0,g_1g_2)}{\beta(g_0,g_1)\beta(g_0g_1,g_2)}$$ thus we see that $\alpha$ is only defined up to a 3-coboundary. For this reason the single block MPO group representation is endowed with the label $[\alpha]$ from the third cohomology group $H^3(\mathsf{G},\mathbb{C})$. Using the fact that $H^d(\mathsf{G},\mathbb{R}) = \mathbb{Z}_1$ [@GuWen] (and that $\mathbb{R}$ as an additive group is isomorphic to $\mathbb{R}^+$ as a multiplicative group), we thus obtain that the third cohomology class of the MPO representation $[\alpha]$ is an element of $H^3(\mathsf{G},\mathsf{U(1)})$.
Orientation dependencies of MPO group representations {#newapp1}
=====================================================
In this appendix we go beyond previous treatments of MPO group representations to consider subtleties that arise due to possible orientation dependencies of the tensors. We find a gauge transformation that reverses the orientation of MPO tensors, and use it to define the Frobenius-Schur indicator. We then find several *pivotal* phases and relate them to the 3-cocycle of the MPO group representation.
Orientation reversing gauge transformation
------------------------------------------
To describe the most general bosonic SPT phases one must use lattices with oriented edges, the internal index of the MPO also carries an orientation which leads to the definition of a pair of possibly distinct MPO tensors which depend on the handedness of the crossing upon which they sit $$\begin{aligned}
&B_+(g)=\vcenter{\hbox{
\includegraphics[height=0.16\linewidth]{Figures/newfig6}}}
\, ,
&B_-(g)=\vcenter{\hbox{
\includegraphics[height=0.16\linewidth]{Figures/newfig7}}} \label{na1}\end{aligned}$$ As shown in Section \[globalsymmetry\] reversing the orientation of the internal MPO index corresponds to inverting the group element which the MPO represents, i.e. $V_{\text{rev}}(g)=V(g^{-1})$. Since this holds for any injective group MPO of arbitrary length standard results from the theory of MPS imply that the local tensors are related by an invertible gauge transformation which we denote $Z_g$ $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[height=0.16\linewidth]{Figures/newfig16}}}&=\vcenter{\hbox{
\includegraphics[height=0.16\linewidth]{Figures/newfig18}}}
\\
\vcenter{\hbox{
\includegraphics[height=0.16\linewidth]{Figures/newfig17}}}
&=\vcenter{\hbox{
\includegraphics[height=0.16\linewidth]{Figures/newfig19}}} \label{na2}\end{aligned}$$ where we use the following graphical notation for $Z_g$ and related matrices $$\begin{aligned}
Z_g&=\raisebox{-0.15cm}{\hbox{\includegraphics[width=0.16\linewidth]{Figures/newfig8}}} \, ,
&Z_g^T&=\raisebox{-0.15cm}{{\hbox{\includegraphics[width=0.16\linewidth]{Figures/newfig10}}}}
\\
Z_g^{-1}&=\raisebox{-0.15cm}{\hbox{\includegraphics[width=0.16\linewidth]{Figures/newfig9}}} \, ,
&(Z_g^{-1})^T&=\raisebox{-0.15cm}{\hbox{\includegraphics[width=0.16\linewidth]{Figures/newfig11}}}\end{aligned}$$ which satisfy the relations $$\begin{aligned}
\raisebox{-0.1cm}{\hbox{\includegraphics[height=0.08\linewidth]{Figures/newfig12}}}&=\raisebox{-0.0cm}{\hbox{\includegraphics[height=0.07\linewidth]{Figures/newfig14}}}
\\
\raisebox{-0.1cm}{\hbox{\includegraphics[height=0.08\linewidth]{Figures/newfig13}}}&=\raisebox{-0.0cm}{\hbox{\includegraphics[height=0.06\linewidth]{Figures/newfig15}}}
\label{na3}\end{aligned}$$ note while it seems *a priori* that the gauge transformations in Eq. could be independent, the fact that the equation $V_{\text{rev}}(g)=V(g^{-1})$ holds for arbitrary orientations of the PEPS bonds implies that they can be chosen to be the same.
Applying the gauge transformation twice we arrive at the equality $$\begin{aligned}
\vcenter{\hbox{\includegraphics[height=0.16\linewidth]{Figures/newfig16}}}=\vcenter{\hbox{\includegraphics[height=0.16\linewidth]{Figures/newfig20}}} \label{na4}\end{aligned}$$ which implies $Z_g(Z_{g^{-1}}^{-1})^T=\chi_g \openone$ for some $\chi_g\in U(1)$ since the MPO is injective. Hence $Z_g=\chi_g Z_{g^{-1}}^{T}$ i.e. $$\begin{aligned}
\raisebox{-0.15cm}{\hbox{\includegraphics[height=0.08\linewidth]{Figures/newfig8}}}
=\chi_g \, \raisebox{-0.15cm}{\hbox{\includegraphics[height=0.08\linewidth]{Figures/newfig21}}} \label{na5}\end{aligned}$$ where $\chi_g$ is analogous to the Frobenius–Schur indicator and can be seen to satisfy $\chi_g=\chi_{g^{-1}}^{-1}$. Note $\chi_g$ can be absorbed by redefinition of $Z_g$ whenever $g\neq g^{-1}$, but we will not do so at this point.
Pivotal phases
--------------
Since the multiplication of the injective MPOs forms a representation of $\mathsf{G}$ we have a local reduction as in Eq.. Again since this holds for arbitrary orientations of the PEPS bonds the reduction matrix $X(g_0,g_1)$ is the same for left and right handed MPOs. From here on we will work with a stronger restriction on the form of the MPOs such that the following *zipper* condition holds $$\begin{aligned}
\raisebox{-.61cm}{\hbox{\includegraphics[height=0.2\linewidth]{Figures/newfig23}}}=\vcenter{\hbox{\includegraphics[width=0.16\linewidth]{Figures/newfig22}}} \label{a3}\end{aligned}$$ this is equivalent to there being no off diagonal blocks in the product of two MPO tensors after it has been brought into canonical form, and is true for MPOs that arise from fixed-point models.
Let us now derive a relation between ${\openone_{g}\otimes (Z_{h}^{-1})^T }\ X^{+}(g,h)$ and $X(gh,h^{-1})$ in terms of a *one-line pivotal phase* which we then proceed to calculate in terms of the three cocycle $\alpha$ of the MPO group representation. Consider $$\begin{aligned}
\vcenter{\hbox{\includegraphics[height=0.27\linewidth]{Figures/newfig24}}}\ \
&=
\hspace{1cm}\vcenter{\hbox{\includegraphics[width=0.16\linewidth]{Figures/newfig25}}}
\nonumber \\
&=\hspace{.53cm}\vcenter{\hbox{\includegraphics[height=0.37\linewidth]{Figures/newfig26}}}
\nonumber \\
&=\hspace{.35cm}\vcenter{\hbox{\includegraphics[height=0.36\linewidth]{Figures/newfig27}}}
\nonumber \\
&=\vcenter{\hbox{\includegraphics[height=0.35\linewidth]{Figures/newfig28}}}
\label{na7}\end{aligned}$$ which yields the desired equality $$\begin{aligned}
\raisebox{-.68cm}{\hbox{\includegraphics[height=0.25\linewidth]{Figures/newfig29}}}
= \gamma(gh,h^{-1}) \
\raisebox{-.6cm}{\hbox{\includegraphics[height=0.2\linewidth]{Figures/newfig30}}}
\label{na8}\end{aligned}$$ where $\gamma(gh,h^{-1})$ is some yet to be determined one-line pivotal phase. We now separate $\gamma(gh,h^{-1})$ into a product of a phase specified by the cocycle $\alpha$ and another phase $b(g,h)$ which we show to be trivial. Multiplying Eq. by $X^{-1}(g_0g_1,g_1^{-1})$ yields $$\begin{aligned}
\gamma(gh,h^{-1}) \, \raisebox{.01cm}{\hbox{\includegraphics[height=0.05\linewidth]{Figures/newfig31}}}
\ = \raisebox{-0.6cm}{\hbox{\includegraphics[height=0.25\linewidth]{Figures/newfig32}}}
\nonumber \\
=\alpha^{-1}(g,h,h^{-1}) \, \raisebox{-1.02cm}{\hbox{\includegraphics[height=0.25\linewidth]{Figures/newfig33}}}
\nonumber \\
=\alpha^{-1}(g,h,h^{-1}) \, b(g,h) \, \raisebox{.0cm}{\hbox{\includegraphics[height=0.05\linewidth]{Figures/newfig31}}}
\label{na9}\end{aligned}$$ Now considering $$\begin{aligned}
\vcenter{\hbox{\includegraphics[height=0.3\linewidth]{Figures/newfig34}}}
= b(g,h) \, \raisebox{.01cm}{\hbox{\includegraphics[height=0.05\linewidth]{Figures/newfig35}}}
\label{na10}\end{aligned}$$ after an application of Eq. to the left most reductions tensors we see that $b(g,h)=b(xg,h),\, \forall x$ and hence $b$ has no dependence on the first input and can be absorbed into the definition of $Z_h$. Similar reasoning yields another useful equality $$\begin{aligned}
\vcenter{\hbox{\includegraphics[height=0.25\linewidth]{Figures/newfig36}}}
= \alpha(g^{-1},g,h) \
\raisebox{-.6cm}{\hbox{\includegraphics[height=0.2\linewidth]{Figures/newfig37}}}
\label{na11}\end{aligned}$$ In summary we have calculated the one-line pivotal phases $$\begin{aligned}
\gamma(gh,h^{-1})&=\alpha^{-1}(g,h,h^{-1}) \nonumber \\
\gamma'(gh,h^{-1})&=\alpha(g^{-1},g,h) \label{na12}\end{aligned}$$
We now proceed to define a *pivotal* phase relating the following different reductions of the same left handed MPO tensors $$\begin{aligned}
\vcenter{\hbox{\includegraphics[width=0.16\linewidth]{Figures/newfig38}}}
&= \raisebox{-.63cm}{\hbox{\includegraphics[height=0.2\linewidth]{Figures/newfig39}}}
\label{na13}
\\
&=
\raisebox{-.63cm}{\hbox{\includegraphics[height=0.2\linewidth]{Figures/newfig40}}}
\nonumber\end{aligned}$$ Hence $$\begin{aligned}
\raisebox{-.7cm}{\hbox{\includegraphics[height=0.21\linewidth]{Figures/newfig41}}}
= \beta(g,h) \
\raisebox{-.55cm}{\hbox{\includegraphics[height=0.2\linewidth]{Figures/newfig42}}}
\label{na15}\end{aligned}$$ for some pivotal phase $\beta(g,h) \in U(1)$. By making use of Eqs.(\[na8\],\[na11\],\[na12\]) we calculate $\beta$ directly to find $$\begin{aligned}
\beta(g,h)=\varepsilon(g)\varepsilon(h) \tilde \beta(g,h) \label{na16}\end{aligned}$$ where $$\begin{aligned}
\varepsilon(g)&:=\chi_g\, \alpha(g,g^{-1},g) \\
\tilde \beta(g,h)&:=\frac{\alpha(h,g,g^{-1})}{\alpha(hg,g^{-1},h^{-1})}\end{aligned}$$ we proceed to show that $\varepsilon \cong 1$ and hence $\beta \cong \tilde \beta$.
Evaluating $\beta$ in two different ways as follows $$\begin{aligned}
\beta(g,h) \
\raisebox{-.16cm}{\hbox{\includegraphics[height=0.1\linewidth]{Figures/newfig45}}}
=\raisebox{-.7cm}{\hbox{\includegraphics[height=0.21\linewidth]{Figures/newfig43}}}
\nonumber \\
= {\chi_g \chi_h} \
\raisebox{-.7cm}{\hbox{\includegraphics[height=0.21\linewidth]{Figures/newfig44}}}
\nonumber \\
= \frac{\chi_g \chi_h \beta(h^{-1},g^{-1})}{\chi_{gh}} \
\raisebox{-.16cm}{\hbox{\includegraphics[height=0.1\linewidth]{Figures/newfig45}}}
\label{na17}\end{aligned}$$ leads to the relation on $\varepsilon$ $$\begin{aligned}
\varepsilon(k)\varepsilon(h)\varepsilon(hk)=1 \label{na18}\end{aligned}$$ after several applications of the 3-cocycle condition for $\alpha$.
Using Eq. we find $$\begin{aligned}
\raisebox{-.88cm}{\hbox{\includegraphics[height=0.3\linewidth]{Figures/newfig46}}}
&= \frac{\alpha(g^{-1},h^{-1},h^{-1})}{\alpha(k,h,g)} \nonumber \\
&\times \raisebox{-1.34cm}{\hbox{\includegraphics[height=0.3\linewidth]{Figures/newfig47}}}
\label{na19}\end{aligned}$$ applying Eq. twice to both sides yields the further constraint on $\beta$ $$\label{na20}
d\beta (a,b,c):=\frac{\beta(a,b)\beta(ab,c)}{ \beta(b,c) \beta(a,bc)} =\frac{\alpha(a,b,c)}{\alpha(c^{-1},b^{-1},a^{-1})}$$ hence $\alpha$ forms a potential obstruction to $\beta$ being a 2-cocycle. Note that $\tilde \beta$ also satisfies Eq. as a consequence of the 3-cocycle condition for $\alpha$ and hence the function $\theta(a,b):=\varepsilon(a)\varepsilon(b)$ satisfies the 2-cocycle condition $d\theta(a,b,c) = 1$. This 2-cocycle condition, together with Eq., implies that $\varepsilon(a)=\varepsilon(c),\,\forall a,c\in \mathsf{G}$ and since $\varepsilon(1)=1$ consequently $\varepsilon \equiv 1$ is the constant function. This of course implies $\beta\equiv \tilde \beta$ which is the desired result $$\begin{aligned}
\label{betafin}
\beta(g,h)=\frac{\alpha(h,g,g^{-1})}{\alpha(hg,g^{-1},h^{-1})}.\end{aligned}$$
Crossing tensors {#newapp2}
================
In this Appendix we define four crossing tensors and demonstrate that they are related by phases involving only the 3-cocycle of the MPO representation. We proceed to define a composition operation on the crossing tensors and calculate the resulting crossing tensor. Building upon this result we determine the transformation of a crossing tensor under the global symmetry. Finally we calculate the effect of modular transformations on the crossing tensors.
Definitions
-----------
We now introduce several different forms for the crossing tensor (see Eq.) that are related by phases which play an important role in our calculations $$\begin{aligned}
{\ensuremath{W_{R}^{g}(h)}}:&=X(g,h) X^{+}(h,g) = \ \vcenter{\hbox{\includegraphics[width=0.24\linewidth]{Figures/newfig48}}} \label{na21a}
\\
{\ensuremath{W_{L}^{g}(h)}}:&= X(h,g) X^{+}(g,h) = \ \raisebox{-.62cm}{\hbox{\includegraphics[width=0.24\linewidth]{Figures/newfig49}}} \label{na21b}
\\
{\ensuremath{Y_{R}^{g}(h)}}:&= X^{+}(gh,h^{-1}) [X^{+}(h,g)\otimes Z_h^{-1}]
\nonumber \\
&= \ \raisebox{-.94cm}{\hbox{\includegraphics[width=0.24\linewidth]{Figures/newfig50}}} \label{na21c}
\\
{\ensuremath{Y_{L}^{g}(h)}}:&= [X(h,g)\otimes Z_h] X(gh,h^{-1})
\nonumber \\
&=\ \raisebox{-.94cm}{\hbox{\includegraphics[width=0.24\linewidth]{Figures/newfig51}}}
\, \sim \, \raisebox{-.6cm}{\hbox{\includegraphics[width=0.24\linewidth]{Figures/newfig52}}}
\label{na21d}\end{aligned}$$ note $h\in\mathsf{C}(g)$ and each tensor above is treated as a representative of an equivalence class of all crossing tensors that give rise to equal PEPS. Using Eqs.(\[n29\],\[na12\]) one finds ${\ensuremath{Y_{L}^{g}(h)}}=\alpha(g,h,h^{-1}){\ensuremath{W_{L}^{g}(h)}}$, ${\ensuremath{W_{L}^{g}(h)}}= {\ensuremath{{\alpha}^{(g)}(h,h^{-1})}}^{-1} {\ensuremath{W_{R}^{g}(h^{-1})}}$, and ${\ensuremath{W_{R}^{g}(h)}}=\alpha(g,h,h^{-1}) {\ensuremath{Y_{R}^{g}(h)}}$, i.e. $$\begin{aligned}
\begin{CD}
{\ensuremath{W_{L}^{g}(h)}} @> {\ensuremath{{\alpha}^{(g)}(h,h^{-1})}} >> {\ensuremath{W_{R}^{g}(h^{-1})}}\\
@V \alpha(g,h,h^{-1}) VV @AA \alpha(g,h^{-1},h) A\\
{\ensuremath{Y_{L}^{g}(h)}} @>> \omega^g(h,h^{-1}) > {\ensuremath{Y_{R}^{g}(h^{-1})}}
\end{CD}
\label{na22}\end{aligned}$$ where $$\begin{aligned}
\label{slant}
{\ensuremath{{\alpha}^{(g)}(k,h)}}:=\alpha(g,k,h) \alpha(k,h,g) \alpha^{-1}(k,g,h)\end{aligned}$$ is the slant product of $\alpha$ (which is a 2-cocycle) and $$\begin{aligned}
\label{omegag}
\omega^g(k,h):={\ensuremath{{\alpha}^{(g)}(k,h)}}\frac{\alpha(g,kh,(kh)^{-1})}{\alpha(g,k,k^{-1})\alpha(g,h,h^{-1})}\end{aligned}$$ is an equivalent 2-cocycle, i.e. $[\omega^g]=[{\ensuremath{{\alpha}^{(g)}}}]$. One can easily verify that changing $\alpha$ by a 3-coboundary alters ${\ensuremath{{\alpha}^{(g)}}}$ by a 2-coboundary and hence the cohomology class $[\alpha]$ is mapped to $[{\ensuremath{{\alpha}^{(g)}}}]$ by the slant product.
Composition rule
----------------
There is a natural composition operation on the ${\ensuremath{Y_{R}^{g}(h)}}$ tensors induced by the action of a global symmetry $U(k)^{\otimes |{\mathcal{M}}|_v},\, k\in\mathsf{C}(g,h),$ upon a symmetry twisted ground state as follows $$\begin{aligned}
{\ensuremath{Y_{R}^{g}(k)}} \times {\ensuremath{Y_{R}^{g}(h)}} := \, \raisebox{-1.6cm}{\hbox{\includegraphics[width=0.35\linewidth]{Figures/newfig53}}} \label{na23}\end{aligned}$$ which includes a reduction of the tensors by $X(k,h)$ and $X^+(k,h)$, note this product is associative but not commutative. The ${\ensuremath{Y_{R}^{g}(h)}}$ tensors in fact form a projective representation under this composition rule since $$\begin{aligned}
\begin{CD}
\raisebox{-1.6cm}{\hbox{\includegraphics[width=0.35\linewidth]{Figures/newfig53}}} @>\alpha(k,gh,h^{-1})^{-1}>> \raisebox{-1.6cm}{\hbox{\includegraphics[width=0.35\linewidth]{Figures/newfig54}}}\\
@. @V \alpha(gkh,h^{-1},k^{-1}) VV\\
\raisebox{-2.15cm}{\hbox{\includegraphics[height=0.23\linewidth,angle=90]{Figures/newfig56}}} @< \alpha(k,h,g)^{-1}<< \raisebox{-2.2cm}{\hbox{\includegraphics[width=0.28\linewidth]{Figures/newfig55}}} \\
@VV \beta(h,k) V @. \\
\raisebox{-.82cm}{\hbox{\includegraphics[width=0.21\linewidth]{Figures/newfig58}}}
\end{CD}\nonumber
\end{aligned}$$ which yields $$\begin{aligned}
\label{na25}
{\ensuremath{Y_{R}^{g}(k)}} \times {\ensuremath{Y_{R}^{g}(h)}}=& \frac{\alpha(k,gh,h^{-1})\alpha(k,h,g)}{\alpha(gkh,h^{-1},k^{-1}) \beta(h,k)}
{\ensuremath{Y_{R}^{g}(kh)}}
\nonumber \\
=& {\ensuremath{{\alpha}^{(g)}(k,h)}} \frac{\alpha(g,kh,h^{-1}k^{-1})}{\alpha(g,k,k^{-1}) \alpha(g,h,h^{-1})}
\nonumber \\
&\frac{\alpha(k,h,h^{-1}k^{-1})}{\alpha(h,h^{-1},k^{-1})\beta(h,k)}
{\ensuremath{Y_{R}^{g}(kh)}}
\nonumber \\
=& \omega^g(k,h) {\ensuremath{Y_{R}^{g}(kh)}}\end{aligned}$$ after several applications of the 3-cocycle condition for $\alpha$, see Eq..
Symmetry action
---------------
We are now in a position to calculate the effect of applying a global symmetry $k\in\mathsf{C}(g,h)$ to an $(x,y)$ symmetry twisted SPT PEPS on a torus as follows $$\begin{aligned}
\label{na26}
\vcenter{\hbox{\includegraphics[width=0.38\linewidth]{Figures/newfig63}}}
\, \sim \,
\vcenter{\hbox{\includegraphics[width=0.38\linewidth]{Figures/newfig64}}}
\\
\sim \, \vcenter{\hbox{\includegraphics[width=0.38\linewidth]{Figures/newfig65}}}
\\
\sim \, \vcenter{\hbox{\includegraphics[width=0.38\linewidth]{Figures/newfig66}}}
\\
= {\ensuremath{Y_{R}^{g}(k)}} \times {\ensuremath{W_{R}^{g}(h)}} \times {\ensuremath{Y_{L}^{g}(k)}}
\\
= \frac{\alpha(g,h,h^{-1}) }{\omega^g(k,k^{-1})} \ {\ensuremath{Y_{R}^{g}(k)}} \times {\ensuremath{Y_{R}^{g}(h)}} \times {\ensuremath{Y_{R}^{g}(k^{-1})}}
\\
= \frac{\alpha(g,h,h^{-1}) \omega^g (k,h)}{\omega^g(k,k^{-1})} \ {\ensuremath{Y_{R}^{g}(k h)}} \times {\ensuremath{Y_{R}^{g}(k^{-1})}}
\\
= \frac{\omega^g (k,h) \omega^g (kh,k^{-1}) \alpha(g,h,h^{-1})}{\omega^g(k,k^{-1})} \ {\ensuremath{Y_{R}^{g}(h)}}
\\
= \frac{\omega^g (k,h)}{\omega^g (h,k) } \ {\ensuremath{W_{R}^{g}(h)}} \end{aligned}$$ where we have made use of the 3-cocycle condition on $\alpha$ and the relations from Eq.. Hence we have found the group action $\pi_k[\cdot]$ induced on the crossing tensor by the physical symmetry to be $$\begin{aligned}
\pi_k [{\ensuremath{W_{R}^{g}(h)}} ] &= (\omega^g)^{(h)} (k)^{-1} \, {\ensuremath{W_{R}^{g}(h)}}
\nonumber
\\
&= {\ensuremath{{\alpha}^{(g,h)}(k)}}^{-1} \, {\ensuremath{W_{R}^{g}(h)}}
\label{na27}\end{aligned}$$ where $(\omega^g)^{(h)}$ is the slant product of $\omega^g$ (it is easy to see this equals the coeficient in Eq.) and hence a 1D representation of $\mathsf{C}(g,h)$ which equals the twice slant product of alpha, i.e. $(\omega^g)^{(h)}={\ensuremath{{\alpha}^{(g,h)}}}$ (since the slant product maps cohomology classes to cohomology classes). Now by the orthogonality of characters we have that the projector $\Pi_{g,h}[\cdot]:=\sum\limits_{k\in\mathsf{C}(g,h)} \pi_k[\cdot]$ maps a nonzero ${\ensuremath{W_{R}^{g}(h)}}$ to zero iff ${\ensuremath{{\alpha}^{(g,h)}}}$ is nontrivial i.e. $$\begin{aligned}
\label{na27b}
\Pi_{g,h}[ {\ensuremath{W_{R}^{g}(h)}} ]\neq 0 \iff {\ensuremath{{\alpha}^{(g,h)}}}\equiv 1 .\end{aligned}$$
Modular transformations {#appmodular}
-----------------------
In this section we will calculate the effects of the $S$ and $T$ transformations ($\frac{\pi}{2}$ rotation and Dehn twist respectively) on the crossing tensor ${\ensuremath{W_{R}^{g}(h)}}$ which is relevant for both symmetry twisted and topological ground states. We use the following left handed convention $$\begin{aligned}
\label{na28}
\begin{CD}
\vcenter{\hbox{\includegraphics[height=.16\linewidth]{Figures/newfig59}}} @>S>> \vcenter{\hbox{\includegraphics[width=0.23\linewidth]{Figures/newfig60}}}
\\
\vcenter{\hbox{\includegraphics[height=.16\linewidth]{Figures/newfig59}}} @> T>> \vcenter{\hbox{\includegraphics[height=0.16\linewidth]{Figures/newfig61}}}
\end{CD}
\\
\sim \
\vcenter{\hbox{\includegraphics[width=0.22\linewidth]{Figures/newfig62}}}. \nonumber\end{aligned}$$ Using Eqs.(\[cocycle\],\[na8\],\[na11\],\[na12\],\[na22\]) and the 3-cocycle condition on $\alpha$ we find $$\begin{aligned}
\label{na29}
S[ {\ensuremath{W_{R}^{g}(h)}} ]&= {\ensuremath{{\alpha}^{(h)}(g^{-1},g)}}^{-1} {\ensuremath{W_{R}^{h}(g^{-1})}}
\\
T[ {\ensuremath{W_{R}^{g}(h)}} ]&= \alpha(g,h,g) {\ensuremath{W_{R}^{g}(gh)}} \end{aligned}$$ with these formulas we have explicitly verified that the action of $S$ and $T$ generate a linear representation of the modular group, i.e. they satisfy the relations $$S^4=\openone,\ (ST)^3=S^2.$$ It was sufficient to simply consider the multiplication of these generators since the gauge theories we deal with are doubled topological orders and consequently have zero modular central charge. We do not reproduce the tedious calculation here.
Gauging SPT PEPS yields topological PEPS {#d}
=========================================
In this appendix we recount the definition of the quantum state gauging procedure of Ref.[@Gaugingpaper] and generalize their proof to show that gauging a SPT PEPS results in a MPO-injective PEPS with a projection MPO that has multiple blocks in its canonical form, labeled by the group elements.
Quantum state gauging procedure
-------------------------------
Let us first recount the definition of the global projector onto the gauge invariant subspace. This is defined on a directed graph ${\ensuremath{\Lambda}}$ in which the vertices are enumerated and the edges are directed from larger to smaller vertex. To each vertex $v\in{\ensuremath{\Lambda}}$ we associate a Hilbert space ${\ensuremath{\mathbb{H}}}_v$ together with a representation $U_v(g)$ of the group $\mathsf{G}$ and to each edge $e\in{\ensuremath{\Lambda}}$ we associate a Hilbert space isomorphic to the group algebra ${\ensuremath{\mathbb{H}}}_e\cong \mathbb{C}[\mathsf{G}]$. We define the matter Hilbert space ${\ensuremath{\mathbb{H}}}_\mathsf{m}:=\bigotimes_{v\in{\ensuremath{\Lambda}}}{\ensuremath{\mathbb{H}}}_v$ and the gauge Hilbert space ${\ensuremath{\mathbb{H}}}_\mathsf{g}:=\bigotimes_{e\in{\ensuremath{\Lambda}}}{\ensuremath{\mathbb{H}}}_e$ which together form the full Hilbert space ${\ensuremath{\mathbb{H}}}_\mathsf{g,m}:={\ensuremath{\mathbb{H}}}_\mathsf{g}\otimes{\ensuremath{\mathbb{H}}}_\mathsf{m}$. The states in ${\ensuremath{\mathbb{H}}}_\mathsf{g,m}$ that are relevant for the gauge theory satisfy a local gauge invariance condition at each vertex. Specifically, they lie in the simultaneous $+1$ eigenspace of the following projection operators $$P_v:=\int \mathrm{d} g_v U_v(g_v) \bigotimes_{e\in E_v^+} R_e(g_v) \bigotimes_{e\in E_v^-} L_e(g_v)\label{constraint}$$ where $E_v^+$ ($E_v^-$) is the set of adjacent edges directed away from (towards) vertex $v$. $R(g),L(g)$ are the right and left regular representations, respectively. The projector onto the gauge invariant subspace is given by $P_{\ensuremath{\Lambda}}:=\prod_v P_v$ and the analogous projector ${\ensuremath{\mathcal{P}}}_{\ensuremath{\Gamma}}$ for any operator $O$ supported on a subgraph ${\ensuremath{\Gamma}}\subset {\ensuremath{\Lambda}}$ (which contains the bounding vertices of all its edges) is defined to be $$\begin{aligned}
{\ensuremath{\mathcal{P}}}_{\ensuremath{\Gamma}}[O] := &\int \prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} g_v [\bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v) \bigotimes_{e\in{\ensuremath{\Gamma}}} L_e(g_{v_e^-})R_e(g_{v_e^+})]
\nonumber \\
& \, \times O\ [\bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v) \bigotimes_{e\in{\ensuremath{\Gamma}}} L_e(g_{v_e^-})R_e(g_{v_e^+})]^\dagger \label{operator}\end{aligned}$$ where edge $e$ points from $v_e^+$ to $v_e^-$.
We proceed to describe a gauging procedure for models defined purely on the matter degrees of freedom ${\ensuremath{\mathbb{H}}}_\mathsf{m}$. To apply $P_{\ensuremath{\Lambda}}$ and ${\ensuremath{\mathcal{P}}}_{\ensuremath{\Gamma}}$ we first require a procedure to embed states and operators from ${\ensuremath{\mathbb{H}}}_\mathsf{m}$ into ${\ensuremath{\mathbb{H}}}_\mathsf{g,m}$. For this we define the gauging map for matter states $\ket{\psi}\in {\ensuremath{\mathbb{H}}}_\mathsf{m}$ by $${\ensuremath{G}}\ket{\psi} := P[\, \ket{\psi} \bigotimes_{e}\ket{1}_e] \, ,\label{gaugingmap}$$ and for matter operators $O \in \mathbb{L}({\ensuremath{\mathbb{H}}}_\mathsf{m})$ acting on a subgraph ${\ensuremath{\Gamma}}\subseteq {\ensuremath{\Lambda}}$ (containing all edges between its vertices) by $${\ensuremath{{\ensuremath{\mathcal{G}}}}}_{\ensuremath{\Gamma}}[O]:={\ensuremath{\mathcal{P}}}_{\ensuremath{\Gamma}}[O\bigotimes_{e\in {\ensuremath{\Gamma}}} \ket{1}\bra{1}_e]\,. \label{gaugingoperator}$$
Gauging SPT PEPS {#gauging-spt-peps}
----------------
In this section we show that a gauged SPT PEPS satisfies the axioms of MPO-injectivity.
Consider a region ${\mathcal{R}}$ of a SPT PEPS $\ket{\psi}\in{\ensuremath{\mathbb{H}}}_\mathsf{m}$ built from local tensor ${\ensuremath{A}}$. The PEPS map ${\ensuremath{A}}_{\mathcal{R}}$ on this region satisfies ${\ensuremath{A}}_{\mathcal{R}}^+{\ensuremath{A}}_{\mathcal{R}}=P_{\partial{\mathcal{R}}}$ and hence is injective on the support subspace of a single block projection MPO $P_{\partial{\mathcal{R}}}=V^{\partial {\mathcal{R}}}(1)$ given by $\text{supp}(P_{\partial{\mathcal{R}}})\subseteq(\mathbb{V}_e)^{\otimes L}$ where $\mathbb{V}_e$ denotes the Hilbert space of a virtual index and $L:=|\partial{\mathcal{R}}|_e$.
For the gauged PEPS $G\ket{\psi}\in{\ensuremath{\mathbb{H}}}_{\mathsf{g,m}}$, the region ${\mathcal{R}}$ is defined to include only those edges between vertices within ${\mathcal{R}}$, i.e. excluding the edges $e\in\partial{\mathcal{R}}$. Note our proof is easily adapted to the case where the edge degrees of freedom are ‘doubled’ and absorbed into the neighboring vertex degrees of freedom, as in Section \[exfpspt\].
The gauged PEPS map on region ${\mathcal{R}}$, ${\ensuremath{A}}_{\mathcal{R}}^\mathsf{g}:(\mathbb{V}_e \otimes \mathbb{C}[\mathsf{G}])^{\otimes L}\rightarrow \mathbb{H}_v^{\otimes |{\mathcal{R}}|_v}\otimes \mathbb{H}_e^{\otimes |{\mathcal{R}}|_e}$, naturally decomposes into the original PEPS map and a gauging tensor network operator multiplying the physical degrees of freedom ${\ensuremath{A}}_{\mathcal{R}}^\mathsf{g}=G_{\mathcal{R}}{\ensuremath{A}}_{\mathcal{R}}$ where $$\begin{aligned}
G_{\mathcal{R}}:=\int\prod_{v\in{\mathcal{R}}}\mathrm{d} g_v \bigotimes_{v\in{\mathcal{R}}} U_v(g_v) \bigotimes_{e\in{\mathcal{R}}} \ket{g_{v_e^-}g_{v_e^+}^{-1}}_e \bigotimes_{e\in\partial {\mathcal{R}}} {\ensuremath{(g_{v_e^\pm}|}}_e\end{aligned}$$ where $v_e^{\pm}\in{\mathcal{R}}$ denotes the unique vertex in ${\mathcal{R}}$ adjacent to the edge $e\in\partial {\mathcal{R}}$.
A generalized inverse of the gauged PEPS is given by $(A_{\mathcal{R}}^\mathsf{g})^+={\ensuremath{A}}_{\mathcal{R}}^+G_{\mathcal{R}}^\dagger$ which satisfies $({\ensuremath{A}}_{\mathcal{R}}^\mathsf{g})^+{\ensuremath{A}}_{\mathcal{R}}^\mathsf{g}=\frac{1}{|G|} \sum\limits_{g\in\mathsf{G}} V^{\partial {\mathcal{R}}}(g)\otimes R(g)^{\otimes L}$. Furthermore, the gauged PEPS is MPO-injective with respect to the projection MPO $\frac{1}{|G|}\sum\limits_{g\in\mathsf{G}} V^{\partial {\mathcal{R}}}(g)\otimes R(g)^{\otimes L}$ which is a sum of single block injective MPOs labeled by $g\in\mathsf{G}$.
Firstly we have $$\begin{aligned}
G_{\mathcal{R}}^\dagger G_{\mathcal{R}}=&\int\prod_{v\in{\mathcal{R}}}\mathrm{d} h_v\mathrm{d} g_v \bigotimes_{v\in{\mathcal{R}}} U_v(h_v^{-1}g_v) \nonumber \\
&\bigotimes_{e\in{\mathcal{R}}} \braket{h_{v_e^-}h_{v_e^+}^{-1}|g_{v_e^-}g_{v_e^+}^{-1}} \bigotimes_{e\in\partial {\mathcal{R}}} {\ensuremath{|h_{v_e^\pm})}}{\ensuremath{(g_{v_e^\pm}|}}_e
\nonumber
\\
=& \int \mathrm{d} g \bigotimes_{v\in{\mathcal{R}}} U_v(g) \bigotimes_{e\in\partial{\mathcal{R}}} R_e(g)\end{aligned}$$ since the delta conditions $\braket{h_{v_e^-}h_{v_e^+}^{-1}|g_{v_e^-}g_{v_e^+}^{-1}}$ force $h_{v_e^-}^{-1}g_{v_e^-}=h_{v_e^+}^{-1}g_{v_e^+}$ and hence $h_v^{-1}g_v=:g$ is constant across all $v\in{\mathcal{R}}$, assuming ${\mathcal{R}}$ is connected. Hence $$\begin{aligned}
{\ensuremath{A}}_{\mathcal{R}}^+ G_{\mathcal{R}}^\dagger G_{\mathcal{R}}{\ensuremath{A}}_{\mathcal{R}}= P_{\partial{\mathcal{R}}} \int \mathrm{d} g \ V^{\partial {\mathcal{R}}}(g) \bigotimes_{e\in\partial{\mathcal{R}}} R_e(g)\end{aligned}$$ since $U(g)^{\otimes |{\mathcal{R}}|_v}{\ensuremath{A}}_{\mathcal{R}}={\ensuremath{A}}_{\mathcal{R}}V^{\partial {\mathcal{R}}}(g)$ for a SPT PEPS (see Section \[globalsymmetry\]) then the result follows as ${P_{\partial {\mathcal{R}}} V^{\partial {\mathcal{R}}} (g) =V^{\partial {\mathcal{R}}} (g)}$.
Let us now address the remaining conditions for MPO-injectivity. Most importantly the pulling through condition is easily seen to hold by Eq. and since $P_v U^\dagger_v(g) = P_v \bigotimes_{e\in E_v^+} R_e(g) \bigotimes_{e\in E_v^-} L_e(g)$, see Appendix \[e\], Proposition \[prop11\] for more detail. The trivial loops condition for the MPO $V^{\partial {\mathcal{R}}}(g)\otimes R(g)^{\otimes L}$ follows directly from the trivial loops condition for $V^{\partial {\mathcal{R}}}(g)$ and the convention that $R(g)$ is inverted depending on the orientation of the crossing of the MPO loop with the virtual bond edge of the PEPS graph, see Eqs.,. Finally, as discussed at the end of Appendix \[a\] the extended inverse condition is automatically satisfied when the projection MPO has a canonical form with injective blocks [@nick], which is the case for the MPO $V^{\partial {\mathcal{R}}}(g)\otimes R(g)^{\otimes L}$.
Generalizing the gauging procedure to arbitrary flat $\mathsf{G}$-connections {#e}
=============================================================================
In this section we outline a generalization of the gauging procedure defined in Ref.[@Gaugingpaper] to arbitrary flat $\mathsf{G}$-connections. For equivalent $\mathsf{G}$-connections the gauging maps are related by local operations while for inequivalent $\mathsf{G}$-connections, which are necessary to construct the full ground space of a gauged model on a nontrivial manifold, the gauging maps are topologically distinct. The gauging maps for nontrivial flat $\mathsf{G}$-connections take inequivalent symmetry twisted states of the initial SPT models to orthogonal ground states of the topologically ordered gauged models.
Elementary definitions
----------------------
A $\mathsf{G}$-connection $\phi$ on a directed graph ${\ensuremath{\Lambda}}$, embedded in an oriented 2-manifold ${\mathcal{M}}$, is given by specifying a group element ${\ensuremath{\phi}}_e\in\mathsf{G}$ for each edge $e\in{\ensuremath{\Lambda}}$. $$\begin{aligned}
{\ensuremath{\phi}}:{\ensuremath{\Lambda}}_e &\rightarrow\mathsf{G}
\\
e&\mapsto{\ensuremath{\phi}}_e \end{aligned}$$
where $\Lambda_e$ is the set of edges in $\Lambda$ and $\phi$ can be thought of as a labeling $\{ {\ensuremath{\phi}}_e \}$ of the edges in ${\ensuremath{\Lambda}}$ by group elements ${\ensuremath{\phi}}_e\in\mathsf{G}$. We view these connections as basis states $\ket{{\ensuremath{\phi}}}:=\bigotimes_e \ket{{\ensuremath{\phi}}_e}_e \in \mathbb{C}[\mathsf{G}]^{\otimes |{\ensuremath{\Lambda}}_e|}$.\
Each $\mathsf{G}$-connection $\phi$ defines a notion of transport along any oriented path (with origin and end point specified) ${\ensuremath{p}}\in{\ensuremath{\Lambda}}$ on the edges of the graph, the transport is specified by the group element $${\ensuremath{\phi}}_{\ensuremath{p}}:=\prod_{i =|{\ensuremath{p}}|_e}^{1} {\ensuremath{\phi}}_{e_i}^{\sigma_i}={\ensuremath{\phi}}_{e_{|p|}}^{\sigma_{|p|}} \cdots {\ensuremath{\phi}}_{e_1}^{\sigma_1}$$ where the edges $e_i\in {\ensuremath{p}}$ are ordered as they occur following ${\ensuremath{p}}$ along its orientation, and $\sigma_i$ is 1 if the orientation of $e_i$ matches that of ${\ensuremath{p}}$ and $-1$ if it does not, see Fig.\[e10\]. Note for paths ${\ensuremath{p}}^1$, from $v_0$ to $v_1$, and ${\ensuremath{p}}^2$, from $v_1$ to $v_2$, we have the following relation ${\ensuremath{\phi}}_{{\ensuremath{p}}^2}{\ensuremath{\phi}}_{{\ensuremath{p}}^1}={\ensuremath{\phi}}_{{\ensuremath{p}}^{12}}$, where ${\ensuremath{p}}^{12}:={\ensuremath{p}}^1\cup{\ensuremath{p}}^2$ is given by composing paths 1 and 2.
A pair of $\mathsf{G}$-connections $\phi,\varphi$ are considered equivalent if they are related by a sequence of local gauge transformations from the set $$\begin{aligned}
\label{gceq}
&\{{\ensuremath{a}}_v^g:=\bigotimes_{e\in E_v^+} R_e(g)\bigotimes_{e\in E_v^-} L_e(g) \, |\, \forall g\in\mathsf{G},v\in{\ensuremath{\Lambda}}\}
\\
&\text{i.e.}\quad\quad \phi\sim\varphi \iff \ket{\phi}=\prod_i {\ensuremath{a}}_{v_i}^{g_i}\ket{\varphi} . \nonumber\end{aligned}$$ One can easily verify that this constitutes an equivalence relation. Importantly, this equivalence relation preserves the conjugacy class of the $\mathsf{G}$-holonomy ${\ensuremath{\phi}}_{\ensuremath{p}}$ of any closed path ${\ensuremath{p}}\in{\ensuremath{\Lambda}}$ with a fixed base point.
An important class of connections are the flat $\mathsf{G}$-connections which are defined to have trivial holonomy along any contractible path.
A $\mathsf{G}$-connection $\phi$ is flat iff ${\ensuremath{\phi}}_{\ensuremath{p}}=1$ for any closed path ${\ensuremath{p}}\in{\ensuremath{\Lambda}}$ that is contractible in the underlying manifold ${\mathcal{M}}$.
This definition immediately implies that ${\ensuremath{\phi}}_{\ensuremath{p}}={\ensuremath{\phi}}_{{\ensuremath{p}}'}$ for any pair of homotopic oriented paths ${\ensuremath{p}},{\ensuremath{p}}'$ with matching endpoints. It is easy to see that a $\mathsf{G}$-connection is flat if and only if it satisfies the local condition ${\ensuremath{\phi}}_{\partial {\ensuremath{q}}}=1$ for every plaquette ${\ensuremath{q}}$ of the graph ${\ensuremath{\Lambda}}\subset{\mathcal{M}}$, where $\partial {\ensuremath{q}}\subset{\ensuremath{\Lambda}}$ is the boundary of ${\ensuremath{q}}$ with the orientation inherited from ${\mathcal{M}}$. Moreover, one can easily verify that flatness is preserved under the equivalence relation and hence the flat $\mathsf{G}$-connections form equivalence classes under this relation. Note there can be multiple flat equivalence classes since it is possible for a flat $\mathsf{G}$-connection to have a nontrivial holonomy ${\ensuremath{\phi}}_{\ensuremath{p}}\neq1$ along a noncontractible loop ${\ensuremath{p}}\in{\ensuremath{\Lambda}}\subset {\mathcal{M}}$.
One can easily show that any contractible region ${\ensuremath{\Gamma}}\subseteq {\ensuremath{\Lambda}}\subset{\mathcal{M}}$ (formed by a set of vertices and the edges between them) of a flat $\mathsf{G}$-connection $\ket{{\ensuremath{\phi}}}$ can be ‘cleaned’ by a sequence of operations $\prod_i a_{v_i}^{g_i}$, where each $v_i\in{\ensuremath{\Gamma}}$, such that the resulting equivalent connection $\ket{{\ensuremath{\phi}}'}:=\prod_i a_{v_i}^{g_i}\ket{{\ensuremath{\phi}}}$ satisfies ${\ensuremath{\phi}}'_e=1,\forall e\in{\ensuremath{\Gamma}}$.
$$\begin{aligned}
a) \vcenter{\hbox{
\includegraphics[width=0.22\linewidth]{Figures/e21}}} \hspace{.5cm} b)\,
\vcenter{\hbox{\includegraphics[width=0.6\linewidth]{Figures/newfig5}}} \end{aligned}$$
\
Utilizing the cleaning procedure leads one to the following conclusion
\[conl\] The equivalence class $[{\ensuremath{\phi}}]$ of a flat $\mathsf{G}$-connection ${\ensuremath{\phi}}$ on an oriented 2-manifold (w.l.o.g. a genus-$n$ torus or $n$-torus) ${\mathcal{M}}$ is labeled uniquely by the conjugacy class of n pairs of group elements that commute with their neighbors, i.e. $\left\{ [ (x_1,y_1),\dots,(x_n,y_n) ]\, | \,\exists x_i,y_i\in\mathsf{G},x_iy_i=y_ix_i,\right.$ $\left.y_ix_{i+1}=x_{i+1}y_i\right\}$, the set of such labels is henceforth referred to as $\mathcal{I}_{\mathcal{M}}$.
The argument proceeds as follows: any $\mathsf{G}$-connection can be ‘cleaned’ onto the set of edges that cross any of the $2n$ closed paths $\{(p^i_x,p^i_y)\}$ in the dual graph ${\ensuremath{\Lambda}}^*$ (where each $(p_x^i,p_y^i)$ and $(p_y^i,p_x^{i+1})$ pair intersect once) that span the inequivalent noncontractible loops of the $n$-torus, see Fig.\[e10\]. Now by the flatness condition the group elements along any loop must be the same (assuming w.l.o.g. the edges on that loop have the same orientation) and the group elements $(x_i,y_i)$ and $(y_i,x_{i+1})$ of each pair of intersecting loops must commute. Furthermore, equivalence under the application of $\bigotimes_{v\in{\ensuremath{\Lambda}}} a_v^g,\,\forall g\in\mathsf{G}$ implies that every set of labels in the same conjugacy class are equivalent.
Note there is a uniquely defined set of group elements $$\begin{aligned}
\label{glcon}
\left\{ (x_1,y_1),\dots,(x_n,y_n)\, | \,x_i,y_i\in\mathsf{G},x_iy_i=y_ix_i,\right. \nonumber \\
\left. y_ix_{i+1}=x_{i+1}y_i\right\}\end{aligned}$$ for each flat $\mathsf{G}$-connection ${\ensuremath{\phi}}$ which are specified by the $\mathsf{G}$-holonomies $x_i:={\ensuremath{\phi}}_{\tilde p^i_x},\, y_i:={\ensuremath{\phi}}_{\tilde p^i_y}$ of pairs of paths $(\tilde p^i_y,\tilde p^i_x)$ in the graph ${\ensuremath{\Lambda}}$, where $\tilde p^i_x$ is defined to be a path that intersects $p^i_x$ once and all other paths $p^k_y,p^j_x,\, j\neq i$, zero times ($\tilde p^i_y$ is defined similarly). Moreover, the conjugacy class $[(x_1,y_1),\dots,(x_n,y_n)]:=\{(x^h_1,y^h_1),\dots,(x^h_n,y^h_n) \, |\, \forall h\in\mathsf{G}\}$ labels the equivalence class $[{\ensuremath{\phi}}]$ of the $\mathsf{G}$-connection ${\ensuremath{\phi}}$, where $x^h:=hxh^{-1}$.
For a fixed representative $\gamma=\{(x_i,y_i)\}$ of conjugacy class $[\gamma]\in\mathcal{I}_{\mathcal{M}}$ and choice of paths $\{(p^i_x,p^i_y)\}$ spanning the inequivalent noncontractible cycles of the $n$-torus, we construct a particularly simple representative flat $\mathsf{G}$-connection as follows
\[flaty\] The simple representative flat $\mathsf{G}$-connection ${\ensuremath{\phi}}^\gamma$ is defined by setting ${\ensuremath{\phi}}_e^\gamma:=x_i^{\sigma^i_e}$ if ${\ensuremath{p}}_x^i$ crosses $e$ and ${\ensuremath{\phi}}_e^\gamma:=y_i^{\sigma^i_e}$ if ${\ensuremath{p}}_y^i$ crosses $e$, where $\sigma^i_e$ is +1 if the crossing is right handed and -1 if it is left handed, and otherwise ${\ensuremath{\phi}}_e^\gamma:=1$ for edges that are not crossed by either ${\ensuremath{p}}_x^i,{\ensuremath{p}}_y^i$.
Note an arbitrary flat connection $\ket{{\ensuremath{\phi}}}$ is related to some $\ket{{\ensuremath{\phi}}^\gamma}$ by a sequence of local operations $\ket{{\ensuremath{\phi}}}=\prod_i a_{v_i}^{g_i} \ket{{\ensuremath{\phi}}^\gamma}$. In particular, the representative connection $\ket{\tilde{{\ensuremath{\phi}}}^\gamma}$ corresponding to a deformation of the paths $(p^i_x,p^i_y)\mapsto(\tilde p^i_x,\tilde p^i_y)$ that does not introduce additional intersections (a planar isotopy) is related to $\ket{{\ensuremath{\phi}}^\gamma}$ by a sequence of local operations $\prod_i a_{v_i}^{g_i}$ that implements the deformation.
![ A representative flat $\mathsf{G}$ connection labeled by $(x,y)$.[]{data-label="e11"}](Figures/e15){width="0.4\linewidth"}
Twisting and gauging operators and states
-----------------------------------------
For any local operator $O$ acting on the matter degrees of freedom in a contractible region ${\ensuremath{\Gamma}}\subseteq{\ensuremath{\Lambda}}$ there is a well defined notion of twisting $O$ by a flat $\mathsf{G}$-connection ${\ensuremath{\phi}}$. Fixing a base vertex $v_0\in{\ensuremath{\Gamma}}$ the twisted operator is given by $$\begin{aligned}
O^{\ensuremath{\phi}}:=\int\mathrm{d}g\bigotimes_{v\in{\ensuremath{\Gamma}}}U_v( {\ensuremath{\phi}}_{{\ensuremath{p}}_v} g)\, O \bigotimes_{v\in{\ensuremath{\Gamma}}}U_v^\dagger( {\ensuremath{\phi}}_{{\ensuremath{p}}_v} g)\end{aligned}$$ where ${\ensuremath{p}}_v$ is any path from $v_0$ to $v$ within ${\ensuremath{\Gamma}}$ (the choice does not matter since the connection is flat and ${\ensuremath{\Gamma}}$ is contractible). The choice of distinguished base vertex $v_0$ is irrelevant since a change $v_0\mapsto v_0'$ can be compensated by shifting $g\mapsto {\ensuremath{\phi}}_{{\ensuremath{p}}'}^{-1}g$, where ${\ensuremath{p}}'$ is a path from $v_0'$ to $v_0$, which has no effect since $g$ is summed over. Note this definition of $O^{\ensuremath{\phi}}$ first projects $O$ onto the space of symmetric operators, hence the sum over $g$ is unnecessary if $O$ is already symmetric. One can verify that $O^{\ensuremath{\phi}}$ commutes with the following twisted symmetry $ \bigotimes_{v\in{\ensuremath{\Gamma}}}U_v( g^{{\ensuremath{\phi}}_{{\ensuremath{p}}_v}}),\,\forall g\in\mathsf{G}$, where $g^h=hgh^{-1}$, independent of the choice of base point $v_0$ and paths ${\ensuremath{p}}_v\in{\ensuremath{\Gamma}}$ from $v_0$ to $v$.
The twisted state gauging map $G_{\ensuremath{\phi}}$, for a flat $\mathsf{G}$-connection ${\ensuremath{\phi}}$, is defined by the following action $$\begin{aligned}
&G_{\ensuremath{\phi}}\ket{\psi}:=P[\, \ket{\psi} \otimes \ket{{\ensuremath{\phi}}} ]
\nonumber \\
&\phantom{G_{\ensuremath{\phi}}}
=\int \prod_{v\in{\ensuremath{\Lambda}}} \mathrm{d} g_v [\, \bigotimes_{v\in{\ensuremath{\Lambda}}}U_v(g_v)] \ket{\psi} \bigotimes_{e\in{\ensuremath{\Lambda}}} \ket{g_{v_e^-}{\ensuremath{\phi}}_eg_{v_e^+}^{-1}}_e\end{aligned}$$ where $\ket{\psi}\in{\ensuremath{\mathbb{H}}}_\mathsf{m}$ is a state of the matter degrees of freedom. One can verify that ${G_{\ensuremath{\phi}}^\dagger G_{\ensuremath{\phi}}=\int \mathrm{d} g \bigotimes_{v\in{\ensuremath{\Lambda}}}U_v(g^{{\ensuremath{\phi}}_{{\ensuremath{p}}_v}}) \prod_i \delta_{gx_i,x_ig}\delta_{gy_i,y_ig}}$ is the projection onto the symmetric subspace of the twisted symmetry, where $(x_i,y_i)$ are the pairs of commuting group elements that label ${\ensuremath{\phi}}$, see Eq.. The $\delta$ conditions arise since the state overlaps force the conjugation of $g$ by the transport group elements ${\ensuremath{\phi}}_{{\ensuremath{p}}_v},{\ensuremath{\phi}}_{{\ensuremath{p}}_v'}$ to agree for non homotopic paths ${\ensuremath{p}}_v,{\ensuremath{p}}_v'$ from $v_0$ to $v$. These $\delta$ conditions also ensure the choice of fixed base point $v_0$ is irrelevant.\
The twisted operator gauging map ${\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}^{\ensuremath{\phi}}$ is defined similarly $$\begin{aligned}
{\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}^{\ensuremath{\phi}}[O]:=\int\prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} g_v \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v)\, O \bigotimes_{v\in{\ensuremath{\Gamma}}} U^\dagger_v(g_v)
\nonumber \\
\bigotimes_{e\in{\ensuremath{\Gamma}}} \ket{g_{v_e^-}{\ensuremath{\phi}}_e g_{v_e^+}^{-1}}\bra{g_{v_e^-}{\ensuremath{\phi}}_e g_{v_e^+}^{-1}}\end{aligned}$$ where $O$ is an operator that acts on the matter degrees of freedom on sites $v\in{\ensuremath{\Gamma}}\subseteq{\ensuremath{\Lambda}}$, and ${\ensuremath{\Gamma}}$ is defined to include all the edges between its vertices. ${\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}^{\ensuremath{\phi}}$ is invertible on the space of ${\ensuremath{\phi}}$-twisted symmetric local operators $O^{\ensuremath{\phi}}$ in the following sense $$\begin{aligned}
&\text{Tr}_{e\in{\ensuremath{\Gamma}}}[{{\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}^{\ensuremath{\phi}}[O^{\ensuremath{\phi}}]\bigotimes_{e\in{\ensuremath{\Gamma}}}\ket{{\ensuremath{\phi}}_e}\bra{{\ensuremath{\phi}}_e}_e}]=
\int\prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} g_v \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v)
\nonumber \\
&\phantom{= \int\prod_{v\in{\ensuremath{\Gamma}}}\mathrm{d} g_v \bigotimes_{v\in{\ensuremath{\Gamma}}}}
\times O^{\ensuremath{\phi}}\bigotimes_{v\in{\ensuremath{\Gamma}}} U^\dagger_v(g_v) \prod_{e\in{\ensuremath{\Gamma}}} \delta_{g_{v_e^-}{\ensuremath{\phi}}_e g_{v_e^+}^{-1},{\ensuremath{\phi}}_e}
\nonumber \\
&\phantom{\int\prod_{v\in{\ensuremath{\Gamma}}}}= \int \mathrm{d} g_{v_0} \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_{v_0}^{{\ensuremath{\phi}}_{{\ensuremath{p}}_v}})
\, O^{\ensuremath{\phi}}\bigotimes_{v\in{\ensuremath{\Gamma}}} U^\dagger_v(g_{v_0}^{{\ensuremath{\phi}}_{{\ensuremath{p}}_v}})
\nonumber\\
&\phantom{\int\prod_{v\in{\ensuremath{\Gamma}}}}=O^{\ensuremath{\phi}}\end{aligned}$$ where the final equality follows from the twisted symmetry of $O^{\ensuremath{\phi}}$ and the second equality follows since the $\delta$ conditions force $g_{v_e^-}=g_{v_e^+}^{{\ensuremath{\phi}}_e}$ which implies, after fixing a base point $v_0\in{\ensuremath{\Lambda}}$, that $g_v=g_{v_0}^{{\ensuremath{\phi}}_{{\ensuremath{p}}_v}}$ for any path ${\ensuremath{p}}_v$ from $v_0$ to $v$ within ${\ensuremath{\Gamma}}$ which is assumed to be contractible in the underlying manifold ${\mathcal{M}}$.
For the twisted gauging procedure we also have a version of Proposition \[p4\], which states the useful equality ${\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}[O]G=G O$ for symmetric $O$. In the twisted case it must be modified in the following way
\[prop7\] The identity ${\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}^{\ensuremath{\phi}}[O^{\ensuremath{\phi}}]G_{\ensuremath{\phi}}=G_{\ensuremath{\phi}}O^{\ensuremath{\phi}}$ holds for any symmetric operator $O$.
We now proceed to show this $$\begin{aligned}
\label{oggo}
&{\ensuremath{\mathcal{G}}}_{\ensuremath{\Gamma}}^{\ensuremath{\phi}}[O^{\ensuremath{\phi}}]G_{\ensuremath{\phi}}= \int\prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} h_v\bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(h_v) O^{\ensuremath{\phi}}\bigotimes_{v\in{\ensuremath{\Gamma}}} U_v^\dagger(h_v)
\nonumber \\
&\phantom{{\ensuremath{\mathcal{G}}}_{\ensuremath{\phi}}}
\bigotimes_{e\in{\ensuremath{\Gamma}}} \ket{h_{v_e^-}{\ensuremath{\phi}}_eh_{v_e^+}^{-1}}\bra{h_{v_e^-}{\ensuremath{\phi}}_eh_{v_e^+}^{-1}} \int\prod_{v\in{\ensuremath{\Lambda}}} \mathrm{d} g_v\bigotimes_{v\in{\ensuremath{\Lambda}}} U_v(g_v)
\nonumber \\
&\phantom{{\ensuremath{\mathcal{G}}}_{\ensuremath{\phi}}[O^{\ensuremath{\phi}}]}
\bigotimes_{e\in{\ensuremath{\Lambda}}} \ket{g_{v_e^-}{\ensuremath{\phi}}_eg_{v_e^+}^{-1}}
\nonumber \\
&
=\int\prod_{v\in{\ensuremath{\Lambda}}} \mathrm{d} g_v \prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} h_v \bigotimes_{v\in{\ensuremath{\Lambda}}} U_v(g_v)
\bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v^{-1}h_v) \, O^{\ensuremath{\phi}}\nonumber \\
& \bigotimes_{v\in{\ensuremath{\Gamma}}} U^\dagger_v(g_v^{-1}h_v)
\prod_{e\in{\ensuremath{\Gamma}}} \delta_{(g_{v_e^-}^{-1}h_{v_e^-}),\, (g_{v_e^+}^{-1}h_{v_e^+})^{{\ensuremath{\phi}}_e}}
\bigotimes_{e\in{\ensuremath{\Lambda}}} \ket{g_{v_e^-}{\ensuremath{\phi}}_e g_{v_e^+}^{-1}}
\nonumber \\
&
= G_{\ensuremath{\phi}}O^{\ensuremath{\phi}}\end{aligned}$$ the last equality follows since the $\delta$ condition forces $g_v^{-1}h_v=(g_{v_0}^{-1}h_{v_0})^{{\ensuremath{\phi}}_{{\ensuremath{p}}_v}}$ (for a fixed choice of vertex $v_0$ and path ${\ensuremath{p}}_v\in{\ensuremath{\Gamma}}$ from $v_0$ to $v$ which has no effect on the outcome) implying $\bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v^{-1}h_v) = \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(\, (g_{v_0}^{-1}h_{v_0})^{{\ensuremath{\phi}}_{{\ensuremath{p}}_v}})$ which is precisely a twisted symmetry that commutes with $O^{\ensuremath{\phi}}$ to yield the desired result.
For a symmetric local Hamiltonian that has been twisted by a flat $\mathsf{G}$-connection ${\ensuremath{\phi}}$, $H_\mathsf{m}^{\ensuremath{\phi}}=\sum_v h_v^{\ensuremath{\phi}}$, we define the twisted gauged Hamiltonian $ (H_\mathsf{m}^{\ensuremath{\phi}})^{{\ensuremath{\mathcal{G}}}^{\ensuremath{\phi}}} :=\sum_v{\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_v}^{\ensuremath{\phi}}[h_v^{\ensuremath{\phi}}]$ in a locality preserving way similar to the untwisted case. With this definition we pose the following proposition
\[gceg\] For all flat $\mathsf{G}$-connections ${\ensuremath{\phi}}$ we have $(H_\mathsf{m}^{\ensuremath{\phi}})^{{\ensuremath{\mathcal{G}}}^{\ensuremath{\phi}}}=H_\mathsf{m}^{{\ensuremath{\mathcal{G}}}}$.
To prove this it suffices to consider a generic local term $h_v^{\ensuremath{\phi}}$ acting on the subgraph ${\ensuremath{\Gamma}}_v$ $$\begin{aligned}
\label{ghv}
{\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_v}^{\ensuremath{\phi}}[h_v^{\ensuremath{\phi}}]
&=\int\prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} g_v \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v) U_v( {\ensuremath{\phi}}_{{\ensuremath{p}}_v})\, h_v \bigotimes_{v\in{\ensuremath{\Gamma}}}U_v^{\dagger}( {\ensuremath{\phi}}_{{\ensuremath{p}}_v})
\nonumber \\
& \phantom{=\int\prod_{v\in{\ensuremath{\Gamma}}}}
U^\dagger_v(g_v) \bigotimes_{e\in{\ensuremath{\Gamma}}} \ket{g_{v_e^-}{\ensuremath{\phi}}_e g_{v_e^+}^{-1}}\bra{g_{v_e^-}{\ensuremath{\phi}}_e g_{v_e^+}^{-1}}
\nonumber \\
&= \int\prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} g_v \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(g_v {\ensuremath{\phi}}_{{\ensuremath{p}}_v})\, h_v \bigotimes_{v\in{\ensuremath{\Gamma}}}U_v^\dagger( g_v{\ensuremath{\phi}}_{{\ensuremath{p}}_v})
\nonumber \\
&\phantom{=\int\ }
\bigotimes_{e\in{\ensuremath{\Gamma}}} \ket{g_{v_e^-}{\ensuremath{\phi}}_{{\ensuremath{p}}_{v_e^-}}{\ensuremath{\phi}}_{{\ensuremath{p}}_{v_e^+}}^{-1} g_{v_e^+}^{-1}}\bra{g_{v_e^-}{\ensuremath{\phi}}_{{\ensuremath{p}}_{v_e^-}}{\ensuremath{\phi}}_{{\ensuremath{p}}_{v_e^+}}^{-1} g_{v_e^+}^{-1}}
\nonumber \\
&= \int\prod_{v\in{\ensuremath{\Gamma}}} \mathrm{d} \tilde{g}_v \bigotimes_{v\in{\ensuremath{\Gamma}}} U_v(\tilde{g}_v )\, h_v \bigotimes_{v\in{\ensuremath{\Gamma}}}U_v^\dagger( \tilde{g}_v)
\nonumber \\
&\phantom{=\int\prod_{v\in{\ensuremath{\Gamma}}}}
\bigotimes_{e\in{\ensuremath{\Gamma}}} \ket{\tilde{g}_{v_e^-}\tilde{g}_{v_e^+}^{-1}}\bra{\tilde{g}_{v_e^-}\tilde{g}_{v_e^+}^{-1}}
\nonumber \\
&= {\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_v}[h_v]\end{aligned}$$ for the first equality we use the symmetry of $h_v$, for the second we use the fact ${\ensuremath{\phi}}_e={\ensuremath{\phi}}_{{\ensuremath{p}}_{v_e^-}}{\ensuremath{\phi}}_{{\ensuremath{p}}_{v_e^+}}^{-1}$, note the choice of base point $v_0$ and paths ${\ensuremath{p}}_v$ from $v_0$ to $v$ in ${\ensuremath{\Gamma}}$ have no effect since $h_v$ is symmetric and ${\ensuremath{\Gamma}}$ is contractible, for the third we use the invariance of the Haar measure under the change of group variables $g_v\mapsto\tilde{g}_v:=g_v{\ensuremath{\phi}}_{{\ensuremath{p}}_v}$.
Gauging preserves the gap and leads to a topological degeneracy
---------------------------------------------------------------
We are now in a position to prove that gauging a SPT Hamiltonian defined on an arbitrary oriented 2-manifold ${\mathcal{M}}$ preserves the energy gap, generalizing the proof presented in Section \[gptg\].
The full gauged Hamiltonian is given by ${H_\text{full}:=H_\mathsf{m}^{{\ensuremath{\mathcal{G}}}}+\Delta_\mathcal{B} H_\mathcal{B}+\Delta_P H_P}$, see Section \[gptg\] for a discussion of each term in the Hamiltonian. Note by Proposition \[gceg\] the same full Hamiltonian $H_\text{full}$ is achieved by gauging any ${\ensuremath{\phi}}$-twist of a given SPT Hamiltonian.
As argued in Section \[gptg\], for $\Delta_\mathcal{B},\, \Delta_P$ sufficiently large, the low energy subspace of $H_\text{full}$ lies within the common ground space of $H_\mathcal{B}$ and $H_P$. This subspace is spanned by the states $P[\, \ket{\lambda}_\mathsf{m}\otimes \ket{{\ensuremath{\phi}}}_\mathsf{g}]=G_{\ensuremath{\phi}}\ket{\lambda}$, where the matter states $\ket{\lambda}$ form a basis of $\mathbb{H}_\mathsf{m}$, and the gauge states $\ket{{\ensuremath{\phi}}}$ span the flat $\mathsf{G}$-connections. This leads to a generalization of Proposition \[prop3\] to arbitrary 2-manifolds
\[prop9\] For an oriented 2-manifold ${\mathcal{M}}$ the set of states $\{ G_{{\ensuremath{\phi}}^\gamma}\ket{\lambda} \}$, for $\{\ket{\lambda}\}$ a basis of $\mathbb{H}_\mathsf{m}$ and a fixed choice of representatives $\gamma\in[\gamma]\in\mathcal{I}_{\mathcal{M}}$, span the common ground space of $H_\mathcal{B}$ and $H_P$.
Firstly, by Proposition \[conl\], an arbitrary flat connection $\ket{{\ensuremath{\phi}}}$ is related to $\ket{{\ensuremath{\phi}}^\gamma},\, \exists[\gamma]\in\mathcal{I}_{\mathcal{M}}$ by a sequence of local operations $\ket{{\ensuremath{\phi}}}=\prod_i a_{v_i}^{g_i} \ket{{\ensuremath{\phi}}^\gamma}$. Since $P_v=\int\mathrm{d} g U_v(g) \otimes {\ensuremath{a}}_v^g$ one can easily see $P_v {\ensuremath{a}}_v^g = P_v U^\dagger_v(g)$ and hence we have $$\begin{aligned}
G_{{\ensuremath{\phi}}}\ket{\psi}_\mathsf{m} &=P[\ket{\psi}_\mathsf{m}\otimes\prod_i a_{v_i}^{g_i}\ket{{\ensuremath{\phi}}^\gamma}_\mathsf{g}]
\nonumber \\
&=
P[\, [\prod_i U_{v_i}(g_i)]^\dagger\ket{\psi}_\mathsf{m}\otimes\ket{{\ensuremath{\phi}}^\gamma}_\mathsf{g}]
\nonumber \\
&= G_{{\ensuremath{\phi}}^\gamma} [\prod_i U_{v_i}(g_i)]^\dagger\ket{\psi}_\mathsf{m}\, .\end{aligned}$$ Therefore the common ground space of $H_\mathcal{B}$ and $H_P$ is spanned by the states $\{G_{{\ensuremath{\phi}}^\gamma}\ket{\lambda}\}_{(\lambda,\gamma)}$ for a basis $\{\ket{\lambda}\}_\lambda$ of $\mathbb{H}_\mathsf{m}$ and a representative $\gamma$ of each conjugacy class $[\gamma]\in\mathcal{I}_{\mathcal{M}}$.
We now bring together the definitions and propositions laid out thus far to show the following
[Gauging a gapped SPT Hamiltonian on an arbitrary oriented 2-manifold ${\mathcal{M}}$ yields a gapped local Hamiltonian with a topology dependent ground space degeneracy]{}.
Let $\ket{\lambda^\gamma}$ denote an eigenstate of the twisted SPT Hamiltonian $H_\mathsf{m}^{{\ensuremath{\phi}}^\gamma}$ with eigenvalue $\lambda$. From Propositions \[prop7\] & \[gceg\] it follows that gauging an eigenstate of a ${\ensuremath{\phi}}$-twisted SPT Hamiltonian yields an eigenstate of the gauged Hamiltonian, so we have $H_\mathsf{m}^{{\ensuremath{\mathcal{G}}}} G_{{\ensuremath{\phi}}^\gamma}\ket{\lambda^\gamma}=\lambda G_{{\ensuremath{\phi}}^\gamma}\ket{\lambda^\gamma}$.
If $H_\mathsf{m}$ has a unique ground state $\ket{\lambda_0}$ Proposition \[prop9\] implies the ground space of the full Hamiltonian $H_\text{full}$ is spanned by the states $\{G_{{\ensuremath{\phi}}^\gamma}\ket{\lambda^\gamma_0}\}_\gamma$ and its gap satisfies ${\Delta_\text{full}\geq\min(\Delta_m,\Delta_\mathcal{B},\Delta_P)}$
In the above we have assumed that $G_{{\ensuremath{\phi}}^\gamma}\ket{\lambda^\gamma_0}\neq0$, for some $\gamma$. Note $G\ket{\lambda_0}\neq 0$ always holds for a unique ground state $\ket{\lambda_0}$ of a symmetric Hamiltonian (possibly after rephasing the matrices of the physical group representation which is assumed to have occurred).
We now proceed to show that the ground space degeneracy is equal to the number of distinct equivalence classes of symmetry twists which are invariant under the residual physical symmetry. This relies on the assumption that the distinct symmetry twisted SPT Hamiltonians $H_\mathsf{m}^{{\ensuremath{\phi}}^\gamma}$ each have a nonzero unique ground state $\ket{\lambda^\gamma_0}$ with the same energy $\lambda_0$. We show this to be the case, when the original frustration free SPT Hamiltonian $H_\mathsf{m}$ has a SPT PEPS ground state, by explicitly constructing tensor network representations of the twisted ground states, see Definition \[stp\].
\[overlapprop\] The overlap matrix of the gauged ground states ${\ensuremath{M_{[\gamma'],[\gamma]}}}:=\bra{\lambda^{\gamma'}_0}G_{{\ensuremath{\phi}}^{\gamma'}}^\dagger G_{{\ensuremath{\phi}}^\gamma}\ket{\lambda^\gamma_0}$ is diagonal, where $\gamma,\gamma'$ are drawn from a fixed set of representatives for the conjugacy classes in $\mathcal{I}_{\mathcal{M}}$. Furthermore, ${\ensuremath{M_{[\gamma'],[\gamma]}}}$ is invariant under a change of representatives and ${\ensuremath{M_{[\gamma],[\gamma]}}}=0$ iff $\ket{\lambda^\gamma_0}$ transforms as a nontrivial representation of the physical symmetry action of $\mathsf{C}(\gamma)$.
The operators $G^\dagger_\varphi G_{\ensuremath{\phi}}$ that appear in the overlaps of the gauged twisted ground states imply that they are orthogonal. To see this consider the following $$\begin{aligned}
&G^\dagger_\varphi G_{\ensuremath{\phi}}=\int \prod_{v\in{\ensuremath{\Lambda}}} \mathrm{d} k_v \mathrm{d} g_v \bigotimes_{v\in{\ensuremath{\Lambda}}}U_v(k_v^{-1}g_v)
\nonumber\\
\label{GG}
&\phantom{G^\dagger_\varphi G_{\ensuremath{\phi}}=\int \prod_{v\in{\ensuremath{\Lambda}}}}
\prod_{e\in{\ensuremath{\Lambda}}} \braket{k_{v_e^-}\varphi_e k_{v_e^+}^{-1}|g_{v_e^-}{\ensuremath{\phi}}_eg_{v_e^+}^{-1}}
\\
&=\int \mathrm{d} g_{v_0} \bigotimes_{v\in{\ensuremath{\Lambda}}}U_v(\varphi_{p_v} g_{v_0} {\ensuremath{\phi}}_{p_v}^{-1}) \prod_i \delta_{x_i'g_{v_0},g_{v_0}x_i} \delta_{y_i'g_{v_0},g_{v_0}y_i}
\nonumber\end{aligned}$$ where we have fixed an arbitrary base vertex $v_0$, $p_v$ is any path from $v_0$ to $v$, and $\{(x_i',y_i')\}_i,\{(x_i,y_i)\}_i$ label the connections $\varphi,{\ensuremath{\phi}}$ respectively. The delta conditions arise since the overlaps in Eq. force the transported group element $\varphi_{p_v}g_{v_0} {\ensuremath{\phi}}_{p_v}^{-1} $ to agree for any choice of path $p_v$ (which may be homotopically distinct). This implies that $G^\dagger_\varphi G_{\ensuremath{\phi}}=0$ whenever the labels $\{(x_i',y_i')\}_i,\{(x_i,y_i)\}_i$ fall into distinct equivalence classes of $\mathcal{I}_{\mathcal{M}}$.\
For the particular case of the simple representative $\mathsf{G}$-connections ${\ensuremath{\phi}}^\gamma$ we have $$\begin{aligned}
G_{{\ensuremath{\phi}}^{\gamma'}}^\dagger G_{{\ensuremath{\phi}}^{\gamma}}
= \delta_{[\gamma'],[\gamma]} \int \mathrm{d} g \bigotimes_{v\in{\ensuremath{\Lambda}}}U_v(g) \prod_i \delta_{x_i'g,gx_i } \delta_{y_i'g,gy_i } \end{aligned}$$ for equivalence classes $[\gamma'],[\gamma]\in\mathcal{I}_{\mathcal{M}}$. Furthermore, if $\gamma'\sim\gamma$ then there exists a group element $h\in\mathsf{G}$ such that $(x_i',y_i')=(x_i^g,y_i^g),\, \forall i\, \iff g\in h\, \mathsf{C}(\gamma) $, a left coset of the centralizer of $\gamma=\{x_i,y_i\}_i$. In this case $$G_{{\ensuremath{\phi}}^{\gamma'}}^\dagger G_{{\ensuremath{\phi}}^{\gamma}}=\int \mathrm{d} g \bigotimes_{v\in{\ensuremath{\Lambda}}}U_v(g)\, \delta_{g\in h \mathsf{C}(\gamma) }$$ and ${H_\mathsf{m}^{{\ensuremath{\phi}}^{\gamma'}}=U(g)^{\otimes |{\ensuremath{\Lambda}}|_v} H_\mathsf{m}^{{\ensuremath{\phi}}^\gamma} U^\dagger (g)^{\otimes |{\ensuremath{\Lambda}}|_v} }$ for any ${g\in h\, \mathsf{C}(\gamma) }$, which implies ${\theta_{g}^\gamma \ket{\lambda^{\gamma'}_0}=U(g)^{\otimes |{\ensuremath{\Lambda}}|_v} \ket{\lambda^\gamma_0} }$ for some phase $\theta_{g}^\gamma\in \mathsf{U(1)}$. Hence $${\bra{\lambda^\gamma_0}G_{{\ensuremath{\phi}}^\gamma}^\dagger G_{{\ensuremath{\phi}}^{\gamma}}\ket{\lambda^\gamma_0}=\theta_h^{\gamma} \bra{\lambda^\gamma_0}G_{{\ensuremath{\phi}}^\gamma}^\dagger G_{{\ensuremath{\phi}}^{\gamma'}}\ket{\lambda^{\gamma'}_0}\iff [\gamma]=[\gamma']}.$$ Moreover since $\ket{\lambda^\gamma_0} $ is the unique groundstate of a $\mathsf{C}(\gamma)$-symmetric Hamiltonian $\theta_{(\cdot)}^\gamma$ is a 1D representation of $\mathsf{C}(\gamma)$. By the orthogonality of characters we have ${G_{{\ensuremath{\phi}}^\gamma}^\dagger G_{{\ensuremath{\phi}}^{\gamma}}\ket{\lambda^\gamma_0}\neq 0\iff\theta_{(\cdot)}^\gamma\equiv 1}$. Note $\theta_{(\cdot)}^\gamma\equiv 1$ is in fact a property of a conjugacy class as it does not depend on the choice of representative $\gamma$.
Consequently the choice of representative symmetry twist $\gamma\in[\gamma]\in\mathcal{I}_{\mathcal{M}}$ does not matter as all lead to the same gauged state $\ket{\lambda_0,[\gamma]}:=G_{{\ensuremath{\phi}}^{\gamma}}\ket{\lambda^\gamma_0}$. Hence the overlap matrix of the gauged twisted SPT groundstates is given by $$\begin{aligned}
{\ensuremath{M_{[\gamma'],[\gamma]}}}&=\braket{\lambda_0,[\gamma']|\lambda_0,[\gamma]}
\nonumber \\
&= \delta_{[\gamma'],[\gamma]} \, \delta_{\theta_{(\cdot)}^\gamma, 1} \frac{|\mathsf{C}(\gamma)|}{|G|} \braket{\lambda^\gamma_0|\lambda^\gamma_0}\end{aligned}$$ and the set of states ${\{\ket{\lambda_0,[\gamma]}|\, {[\gamma]}\in\mathcal{I}_{\mathcal{M}},\,\theta^\gamma_{(\cdot)}\equiv 1\}}$ form an orthogonal basis for the ground space of the full gauged Hamiltonian $H_\text{full}$.
Symmetry twists & monodromy defects {#g}
====================================
In this appendix we describe a general and unambiguous procedure for applying symmetry twists to SPT PEPS using virtual symmetry MPOs. We furthermore demonstrate that the gauging procedure maps the symmetry MPOs to freely deformable topological MPOs on the virtual level and hence the gauged symmetry twisted PEPS are locally indistinguishable while remaining globally orthogonal, implying that they exhibit topological order. We move on to discuss how the same MPOs can be arranged along open paths to describe monodromy defects in SPT PEPS and anyons in the gauged PEPS. Moreover, we explicitly calculate the projective transformation of individual monodromy defects under the residual symmetry group using tensor network techniques.
Symmetry twisted states
-----------------------
In this section we discuss the ground states of symmetry twisted Hamiltonians in more detail and show that the PEPS framework naturally accommodates a simple construction of these states.
On a trivial topology a symmetry twist can be applied directly to a state by acting on some region of the lattice with the physical symmetry. For example on an infinite square lattice in the 2D plane a symmetry twist $(x,y)$ along an oriented horizontal and vertical path $p_x,p_y$, in the dual lattice, acts on a state $\ket{\psi}$ via $$\begin{aligned}
\ket{\psi}^\phi:&=\int\mathrm{d}g\bigotimes_{v\in{\ensuremath{\Gamma}}}U_v( {\ensuremath{\phi}}_{{\ensuremath{p}}_v} g) \ket{\psi}
\\
&= \bigotimes_{v\in \mathcal{U}} U_v(x) \bigotimes_{v\in\mathcal{R}} U_v(y) \int\mathrm{d}g\bigotimes_{v\in{\ensuremath{\Gamma}}}U_v(g)\ket{\psi}\end{aligned}$$ where ${\ensuremath{\phi}}$ is the simple representative connection with label $(x,y)$ on paths $p_x,p_y$, see Definition \[flaty\], and ${\mathcal{R}}$ is the half plane to the right of $p_y$, $\mathcal{U}$ the half plane above $p_x$, see Fig.\[e4\]. Note this definition implicitly projects $\ket{\psi}$ onto the trivial representation and we have $O^{\ensuremath{\phi}}\ket{\psi}^{\ensuremath{\phi}}= (O\ket{\psi})^{\ensuremath{\phi}}$ for symmetric operators $O$. Hence twisting an eigenstate $\ket{\lambda}$ of a SPT Hamiltonian $H_\mathsf{m}$ yields an eigenstate $\ket{\lambda}^{\ensuremath{\phi}}$ of the twisted Hamiltonian $H_\mathsf{m}^{\ensuremath{\phi}}$ with the same eigenvalue. Note $x$ and $y$ must commute for ${\ensuremath{\phi}}$ to be a flat connection, equivalently if one thinks of first applying the $x$ twist to a symmetric Hamiltonian, then the resulting operator will only be symmetric under the centralizer subgroup of $x$, $\mathsf{C}(x)\leq\mathsf{G}$, and hence it only makes sense to apply a second twist for an element $y\in \mathsf{C}(x)$.
The effect of such a symmetry twist on a SPT PEPS $\ket{\psi}$ is particularly simple, it can be achieved by adding the virtual symmetry MPOs $V^{{\ensuremath{p}}_x}(x)$ and $V^{{\ensuremath{p}}_y}(y)$ (with inner indices contracted with the four index crossing tensor $Q_{x,y}={\ensuremath{W_{R}^{x}(y)}}$ (\[cxy\],\[na21a\]) where ${\ensuremath{p}}_x,{\ensuremath{p}}_y$ intersect, see Fig.\[e5\]) to the virtual level of the PEPS. Let us denote the resulting tensor network state $\ket{\psi^{(x,y)}}$, then by Eq. we have $\ket{\psi^{(x,y)}}=\ket{\psi}^{\ensuremath{\phi}}$.
For nontrivial topologies the symmetry twist on a state $\ket{\psi}^{\phi^\gamma}$ is not well defined in terms of a physical symmetry action since two homotopically inequivalent paths ${\ensuremath{p}}_v,{\ensuremath{p}}_v'$ can give rise to distinct transport elements $\phi_{{\ensuremath{p}}_v}\neq\phi_{{\ensuremath{p}}_v'}$. Note this problem does not arise when symmetry twisting local operators, such as the terms in a local Hamiltonian, since each operator acts within a contractible region. The PEPS formalism yields a simple resolution to this problem since the process of applying a symmetry twist ${\ensuremath{\phi}}^\gamma$ on the virtual level of a PEPS $\ket{\psi^\gamma}$ remains well defined, see Definition \[stp\] and Fig.\[e12\].
![ An $(x,y)$ symmetry twisted PEPS on a torus.[]{data-label="e12"}](Figures/e27a){width="0.5\linewidth"}
The general scenario is as follows; we have a local gapped frustration free SPT Hamiltonian $H_\mathsf{m}$ defined on an oriented 2-manifold ${\mathcal{M}}$ with a SPT PEPS $\ket{\lambda_0}$ as its unique ground state (note SPT PEPS parent Hamiltonians satisfy these conditions) and we want to apply a symmetry twist along paths ${\ensuremath{p}}_x^i,{\ensuremath{p}}_y^i$ in the dual graph labeled by $\gamma=\{(x_i,y_i)\}_i$.
\[stp\] For a SPT PEPS $\ket{\psi}$ and a symmetry twist $\gamma$, specified by a set of pairwise intersecting paths in the dual graph $\{{\ensuremath{p}}_x^i,{\ensuremath{p}}_y^i\}_i$ and pairwise commuting group elements $\{(x_i,y_i)\}_i$ in $\mathsf{G}$, the symmetry twisted PEPS $\ket{\psi^\gamma}$ is constructed by taking the tensor network for $\ket{\psi}$ with open virtual indices on edges that cross $\{{\ensuremath{p}}_x^i,{\ensuremath{p}}_y^i\}_i$ and contracting these virtual indices with the MPOs $\{V^{{\ensuremath{p}}_x^i}(x_i),V^{{\ensuremath{p}}_y^i}(y_i)\}_i$. Moreover, at the intersection of the paths ${\ensuremath{p}}_x^i\cap{\ensuremath{p}}_y^i$ the internal indices of the MPOs $V^{{\ensuremath{p}}_x^i}(x_i),V^{{\ensuremath{p}}_y^i}(y_i)$ are contracted with four index crossing tensors $Q_{x_i,y_i}={\ensuremath{W_{R}^{x_i}(y_i)}}$, defined in Eqs.(\[cxy\],\[na21a\]) and similarly with $Q_{y_{i-1},x_i}={\ensuremath{W_{R}^{y_{i-1}}(x_i)}}$ at the intersections ${\ensuremath{p}}_y^{i-1}\cap{\ensuremath{p}}_x^{i}$. This is depicted in Fig. \[e12\].
It follows from Eq. and the *zipper* condition for $X(x_i,y_i)$ that the symmetry twisted ground state SPT PEPS $\ket{\lambda_0^{\gamma}}$ is the ground state of the twisted SPT Hamiltonian $H_\mathsf{m}^{{\ensuremath{\phi}}^\gamma}$. More generally for any SPT PEPS $\ket{\psi}$ that is an eigenstate of each local term in $H_\mathsf{m}$, Eq. implies that $\ket{\psi^{\gamma}}$ is an eigenstate of $H_\mathsf{m}^{{\ensuremath{\phi}}^\gamma}$ with the same eigenvalue (thereby justifying the notation). Note the twisted SPT PEPS $\ket{\psi^{\gamma}}$ for different choices of representative $\gamma$ from the same conjugacy class $[\gamma]\in\mathcal{I}_{\mathcal{M}}$ are all related by the action of some global symmetry, which again follows from Eqs., and Proposition \[conl\].
\[prop10\] A $\gamma$-twisted SPT PEPS $\ket{\psi^\gamma}$ transforms as the following 1D representation $${\theta_{(\cdot)}^\gamma={\ensuremath{{\alpha}^{(x_0,y_0)}(\cdot)}}^{-1}\prod\limits_{i=1} [{\ensuremath{{\alpha}^{(y_{i-1},x_i)}(\cdot)}}{\ensuremath{{\alpha}^{(x_i,y_{i})}(\cdot)}}]^{-1}}$$ under the physical action of the residual symmetry group $\mathsf{C}(\gamma)$.
The physical action of the symmetry ${U(k)^{\otimes |{\mathcal{M}}|_v}}$ induces a local action $\pi_k$ on each crossing tensor ${\{{\ensuremath{W_{R}^{x_i}(y_i)}},{\ensuremath{W_{R}^{y_{i-1}}(x_i)}}\}_i}$ and by Eq. we find the combined action to be ${{\ensuremath{{\alpha}^{(x_0,y_0)}(\cdot)}}^{-1}\prod\limits_{i=1} [{\ensuremath{{\alpha}^{(y_{i-1},x_i)}(\cdot)}}{\ensuremath{{\alpha}^{(x_i,y_{i})}(\cdot)}}]^{-1}}$ as claimed.
Topological ground states
-------------------------
We now show that the twisted gauging procedure maps the virtual symmetry MPO to a freely deformable topological MPO on the virtual level.
\[prop11\] Applying the twisted gauging map $G_{{\ensuremath{\phi}}^\gamma}$ to a nonzero twisted SPT PEPS $\ket{\psi^\gamma}$ yields the MPO-injective PEPS $G\ket{\psi}$ with a set of freely deformable MPOs joined by crossing tensors, specified by $[\gamma]$, acting on the virtual level. The gauged state is zero iff $\ket{\psi^\gamma}$ transforms nontrivially under the residual symmetry group $\mathsf{C}(\gamma)$, this property depends only on $[\gamma]$ and $[\alpha]$.
We will first show that the tensor network $G_{{\ensuremath{\phi}}^{\gamma}}\ket{\psi^{\gamma}}$ is given by contracting the MPOs $[V^{{\ensuremath{p}}_x^i}(x_i)\bigotimes_{e\in{\ensuremath{p}}_x^i} R_e(x_i)],[V^{{\ensuremath{p}}_y^i}(y_i)\bigotimes_{e\in{\ensuremath{p}}_y^i} R_e(y_i)]$ (contracted with the crossing tensor $Q_{x_i,y_i}={\ensuremath{W_{R}^{x_i}(y_i)}}$ at $p_x^i\cap p_y^i$) with the virtual indices of $G\ket{\psi}$ on edges that cross the paths $\{{\ensuremath{p}}_x^i,{\ensuremath{p}}_y^i\}$.
In general $G_\phi$ is a projected entangled pair operator (PEPO) with vertex tensors $G_\phi^v=\int \mathrm{d} g \, U_v(g) \bigotimes_{e\in E_v} {\ensuremath{(g|}}=G^v$ and edge tensors $G_\phi^e=\int \mathrm{d} g_{v_e^+} \mathrm{d} g_{v_e^-} L_e(g_{v_e^-})R_e(g_{v_e^+})\ket{\phi_e}\otimes{\ensuremath{( g_{v_e^+}|}}{\ensuremath{( g_{v_e^-}|}}$ [@Gaugingpaper]. Furthermore the edge tensors satisfy $G_\phi^e=G_1^e (R(\phi_e)\otimes\openone)=G_1^e (\openone\otimes R^\dagger(\phi_e)\, )$. Hence the PEPO $G_\phi$ is given by the untwisted gauging map $G$ with the tensor product operators $\{\bigotimes_{e\in{\ensuremath{p}}_x^i} R_e(x_i),\bigotimes_{e\in{\ensuremath{p}}_y^i} R_e(y_i)\}$ applied to the virtual indices that cross $\{{\ensuremath{p}}_x^i,{\ensuremath{p}}_y^i\}$.
Eqs. and together with $P_v {\ensuremath{a}}_v^g = P_v U^\dagger_v(g)$ imply $G_{{\ensuremath{\phi}}^{\gamma}}\ket{\psi^{\gamma}}=G_{{\ensuremath{\phi}}^{\tilde\gamma}}\ket{\psi^{\tilde\gamma}}$ for any deformation $\tilde\gamma=\{\tilde{\ensuremath{p}}_x^i,\tilde{\ensuremath{p}}_y^i\}$ of the paths $\gamma=\{{\ensuremath{p}}_x^i,{\ensuremath{p}}_y^i\}$ that does not introduce additional intersections (a planar isotopy). This furthermore implies that the MPOs $[V^{{\ensuremath{p}}}(g)\bigotimes_{e\in{\ensuremath{p}}} R_e(g)]$ satisfy the pulling through condition of Ref.[@MPOpaper] for any path $p$. Consequently, the MPO $\frac{1}{|G|}\sum_g[V^{{\ensuremath{p}}}(g)\bigotimes_{e\in{\ensuremath{p}}} R_e(g)]$, that was shown to be the projection onto the injectivity subspace of the gauged PEPS in Appendix \[d\], also satisfies the pulling through condition.
By Proposition \[overlapprop\] the gauged SPT PEPS $G_{{\ensuremath{\phi}}^\gamma}\ket{\psi^\gamma}$ is zero iff $\theta^\gamma_{(\cdot)}$ is nontrivial, which is a property of the conjugacy class $[\gamma]$. Now by Proposition \[prop10\] and the fact that the slant product maps cohomology classes to cohomology classes we have the stated result.
Hence *the nonzero gauged symmetry twisted PEPS ground states $\ket{\lambda_0,[\gamma]}:=G_{{\ensuremath{\phi}}^{\gamma}}\ket{\lambda^\gamma_0}$ are topologically ordered* since the tensors $Q_{x_i,y_i}$ that determine the ground state are locally undetectable, which follows from the pulling through condition satisfied by the topological MPOs and Eq., while for $[\gamma]\neq[\gamma']$ the states are globally orthogonal $\braket{\lambda_0,[\gamma]|\lambda_0,[\gamma']}=0$, as shown above.
[ In fact there is a slight subtlety, as while the reduced density matrices for all $[\gamma],[\gamma']\in\mathcal{I}_{\mathcal{M}}$ are supported on the same subspace $\rho_{\mathcal{R}}^{\lambda_0,[\gamma]},\rho_{\mathcal{R}}^{\lambda_0,[\gamma']}\in \text{Im}({\ensuremath{A}}_{\mathcal{R}}\otimes {\ensuremath{A}}^\dagger_{\mathcal{R}})$ for any contractible region ${\mathcal{R}}$, they are not necessarily equal [@Ginjectivity] (or even exponentially close in the size of the region). One might also fret over the possibility that the state exhibits spontaneous symmetry breaking.\
However neither of these complications can occur for the gauged symmetry twisted SPT PEPS, since an exact isometric fixed-point SPT PEPS does not exhibit symmetry breaking and is gauged to a topologically ordered fixed-point state which also does not exhibit symmetry breaking (see Section \[exfpspt\]). Furthermore the gauging map is gap preserving, hence gauging any SPT PEPS in the same phase as an SPT fixed-point maps it to a topological PEPS in the same phase as the gauged topological fixed-point PEPS. ]{}
Monodromy defects in SPT PEPS
-----------------------------
Monodromy defects can be created in a SPT theory by applying a symmetry twist along an open ended path in the dual graph ${\ensuremath{p}}_g$ from plaquette $q_0$ to $q_1$, specified by a $\mathsf{G}$-connection ${\ensuremath{\phi}}^{{\ensuremath{p}}_g}$, where ${\ensuremath{\phi}}^{{\ensuremath{p}}_g}_e=1$, for $ e\notin {\ensuremath{p}}_g$ and ${\ensuremath{\phi}}^{{\ensuremath{p}}_g}_e=g^{\sigma_e}$ for $e\in {{\ensuremath{p}}_g}$ ($\sigma_e$ is +1 if ${\ensuremath{p}}_{g}$ crosses $e$ in a right handed fashion and -1 for left handed crossings) hence ${\ensuremath{\phi}}^{{\ensuremath{p}}_g}$ is flat on every plaquette except $q_0,q_1$, the end points of ${{\ensuremath{p}}_g}$, see Fig.\[e13\]. The defect states can be realized as ground states of some twisted Hamiltonians $H_\mathsf{m}^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}=\sum_{q\in{\ensuremath{\Lambda}}\backslash{\partial {{\ensuremath{p}}_g}}} h_q^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}+h'_{q_0}+h'_{q_1}$ where the choice of the end terms $h'_{q_0},h'_{q_1}$ is somewhat arbitrary. These monodromy defects can be introduced into a SPT PEPS $\ket{\psi}$ following the framework set up for symmetry twists.
A monodromoy defect specified by ${\ensuremath{p}}_g$ in a SPT PEPS $\ket{\psi}$ is described by a set of tensor network states parametrized by a pair of tensors $B_0,B_1$ where $B_0:(\mathbb{C}^D)^{\otimes |E_{v_0}|}\otimes\mathbb{C}^{\ensuremath{\chi}}\rightarrow\mathbb{C}^d$ is a local tensor associated to a vertex $v_0\in\partial q_0$ with a set of indices matching those of the tensor $A_{v_0}$, and an extra virtual index of the same bond dimension ${\ensuremath{\chi}}$ as the internal index of the MPO ($B_1$ is defined similarly).\
The monodromy defected tensor network states $\ket{\psi^{{\ensuremath{p}}_g},B_0,B_1}$ are constructed from the SPT PEPS $\ket{\psi}$ by replacing the PEPS tensors ${\ensuremath{A}}_{v_0},{\ensuremath{A}}_{v_1}$ with $B_0,B_1$ and contracting the extra virtual indices thus introduced with the open end indices of the MPO $V^{{\ensuremath{p}}_g}(g)$ which acts on the virtual indices of the PEPS that cross ${\ensuremath{p}}_g$. This is depicted in Fig. \[a1\] b).
$$\begin{aligned}
\vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/e24}}} \hspace{.7cm} \end{aligned}$$
This provides an ansatz [@nick] for symmetry twists by choosing appropriate boundary tensors $B_0,B_1$ to close the free internal MPO indices at $q_0,q_1$, the possibility of different boundary conditions corresponds to the ambiguity in the local Hamiltonian terms $h'_{q_0},h'_{q_1}$, see Figs.\[e13\],\[a1\]. Eq. implies that the defect state ansatz $\ket{\psi^{{\ensuremath{p}}_g},B_0,B_1}$ is in the ground space of the sum of Hamiltonian terms away from the end points of ${\ensuremath{p}}_g$, $\sum_{q\in{\ensuremath{\Lambda}}\backslash{\partial {{\ensuremath{p}}_g}}} h_q^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}$, for any choice of tensors $B_0,B_1$.
Since the connection ${\ensuremath{\phi}}^{{\ensuremath{p}}_g}$ is flat everywhere but $q_0,q_1,$ the gauging map can be applied, in the usual way, to operators that are supported away from these plaquettes. Hence the twisted gauged defect Hamiltonian is $(H_\mathsf{m}^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}})^{{\ensuremath{\mathcal{G}}}^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}}:=\sum_{q\in{\ensuremath{\Lambda}}\backslash{\partial {{\ensuremath{p}}_g}}} {\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_q}^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}[h_q^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}]+h''_{q_0}+h''_{q_1}$ where again there is an ambiguity in the choice of end terms $h''_{q_0},h''_{q_1}$. The SPT PEPS with monodromy defect ${\ensuremath{p}}_g$ can be gauged via the standard gauging procedure for the $\mathsf{G}$-connection ${\ensuremath{\phi}}^{{\ensuremath{p}}_g}$ to yield the tensor network $G_{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}\ket{\psi^{{{\ensuremath{p}}_g}},B_0,B_1}$. Similar to the case of symmetry twists on closed paths, the gauged defected SPT PEPS $G_{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}\ket{\psi^{{{\ensuremath{p}}_g}},B_0,B_1}$ is constructed from the untwisted gauged SPT PEPS $G\ket{\psi}$ by removing the tensors $G^{v_0}A_{v_0},G^{v_1}A_{v_1}$ and replacing them with the pair of tensors $G^{v_0}B_{0},G^{v_1}B_{1}$ connected by a virtual MPO $[V^{{{\ensuremath{p}}_g}}(g)\bigotimes_{e\in p_g}R_e(g)]$ acting on the virtual indices of the PEPS that cross ${\ensuremath{p}}_g$. Note the dimension of the inner indices of this MPO match the extra indices of $G^{v_0}B_{0},G^{v_1}B_{1}$ since the newly introduced component of the MPO $\bigotimes_{e\in p_g}R_e(g)$ has trivial inner indices. To achieve a more general ansatz one may want to replace $G^{v_0}B_0,G^{v_1}B_1$ by arbitrary tensors $\tilde B_0,\tilde B_1:(\mathbb{C}^D\otimes \mathbb{C}[\mathsf{G}])^{\otimes |E_{v}|}\otimes\mathbb{C}^{\ensuremath{\chi}}\rightarrow\mathbb{C}^d$.
As shown above, the MPO $[V^{{{\ensuremath{p}}_g}}(g)\bigotimes_{e\in p_g}R_e(g)]$ satisfies the pulling through condition of Ref.[@MPOpaper] and hence $G_{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}\ket{\psi^{{{\ensuremath{p}}_g}},B_0,B_1}=G_{{\ensuremath{\phi}}^{{\ensuremath{p}}'_g}}\ket{\psi^{{{\ensuremath{p}}'_g}},B_0,B_1}$ for ${\ensuremath{p}}_g'$ an arbitrary, end point preserving, deformation of ${\ensuremath{p}}_g$. By Eq. we have ${\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_{q}}^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}[h_q^{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}]= {\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_{q}}[h_q]$ and hence the gauged defected SPT PEPS $G_{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}\ket{\psi^{{{\ensuremath{p}}_g}},B_0,B_1}$, for all $B_0,B_1$, is in the ground space of the sum of gauged Hamiltonian terms away from the end points $\sum_{q\in{\ensuremath{\Lambda}}\backslash{\partial {{\ensuremath{p}}_g}}} {\ensuremath{\mathcal{G}}}_{{\ensuremath{\Gamma}}_{q}}[h_q]$. Consequently $G_{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}\ket{\psi^{{{\ensuremath{p}}_g}},B_0,B_1}$ must represent a superposition of anyon pairs, localized to the plaquettes $q_0,q_1$, on top of the vacuum (ground state). Furthermore the freedom in choosing $B_0,B_1$ leads to a fully general anyon ansatz within the framework of MPO-injective PEPS [@nick].
Projective symmetry transformation of monodromy defects
-------------------------------------------------------
We proceed to show that the internal degrees of freedom of a monodromy defect ${\ensuremath{p}}_g$ transform under a projective representation of the residual global symmetry group $\mathsf{C}(g)$ via a generalization of the mechanism for virtual symmetry actions in MPS [@SchuchGarciaCirac11; @1Done; @1Dtwo].
We consider a SPT PEPS on an oriented manifold ${\mathcal{M}}$ with a twice punctured sphere topology and a symmetry twist ${\ensuremath{p}}_g$ running from one puncture ${\ensuremath{{\Pi}}}_0$ to the other ${\ensuremath{{\Pi}}}_1$. This captures both the case of a symmetry twisted SPT model defined on a cylinder (when the virtual bonds that enter the punctures are left open), and the case of a pair of monodromy defects on a sphere (when the punctures are formed by removing a pair of PEPS tensors ${\ensuremath{A}}_{v_0},{\ensuremath{A}}_{v_1}$ and contracting the virtual indices thus opened with $B_0,B_1$), see Fig. \[a1\].
The bulk of the symmetry twisted state is invariant under the physical on-site representation $U(h)^{\otimes |{\mathcal{M}}|_v}$ of $\mathsf{C}(g)\leq\mathsf{G}$, but this may have some action on the virtual indices that enter the punctures. Treating the SPT PEPS on a cylinder of fixed radius as a one dimensional symmetric MPS implies, by well established arguments [@SchuchGarciaCirac11; @1Done], that the action of the symmetry on the virtual boundaries ${\ensuremath{{\ensuremath{\mathcal{V}}}}}_0^g(h)\otimes {\ensuremath{{\ensuremath{\mathcal{V}}}}}_1^g(h)$ forms a representation, while each individual boundary action ${\ensuremath{{\ensuremath{\mathcal{V}}}}}_0^g(h), {\ensuremath{{\ensuremath{\mathcal{V}}}}}_1^g(h)$ is free to form a projective representation.
$$\begin{aligned}
a) \vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/a1}}} \quad b)
\vcenter{\hbox{
\includegraphics[width=0.4\linewidth]{Figures/a2}}} \end{aligned}$$
Assuming the symmetry MPOs satisfy the $\emph{zipper}$ condition one can directly calculate the effect that a physical symmetry action $U(h)^{\otimes |{\mathcal{M}}|_v},\, h\in \mathsf{C}(g)$ has on the virtual boundary, simultaneously demonstrating the symmetry invariance of the bulk. $$\begin{aligned}
\vcenter{\hbox{
\includegraphics[height=0.21\linewidth]{Figures/newfig69}}}
\ = \vcenter{\hbox{
\includegraphics[height=0.21\linewidth]{Figures/newfig70}}}
\\ \label{a4}
= \vcenter{\hbox{
\includegraphics[height=0.205\linewidth]{Figures/newfig71}}}\end{aligned}$$ Hence the symmetry action ${\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_1(h)$ (see Eq.) on the boundary of a single puncture ${\ensuremath{{\Pi}}}_1$ is given by the MPO $V^{\partial {\ensuremath{{\Pi}}}_1^{-}}(h)$, acting on the virtual indices along $\partial {\ensuremath{{\Pi}}}_1$, contracted with the crossing tensor ${\ensuremath{Y_{R}^{g}(h)}}$ (see Eq.) acting on the inner MPO index of the symmetry twist $V^{p_g}(g)$ that enters the puncture. Similarly ${\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_0(h)$, acting on the boundary of the other puncture ${\ensuremath{{\Pi}}}_0$, is given by contracting the MPO $V^{\partial {\ensuremath{{\Pi}}}_0^{-}}(h)$ with the crossing tensor ${\ensuremath{Y_{L}^{g}(h)}}$.
There is a natural composition operation on the crossing tensors ${\ensuremath{Y_{R}^{g}(\cdot)}}$ that is induced by applying a product of global symmetries $U(k)^{\otimes |{\mathcal{M}}|_v}U(h)^{\otimes |{\mathcal{M}}|_v}$ and utilizing the reduction of Eq. twice and then zipping the MPOs $V^{\partial {\ensuremath{{\Pi}}}_1^{-}}(k)V^{\partial {\ensuremath{{\Pi}}}_1^{-}}(h)=X(k,h)V^{\partial {\ensuremath{{\Pi}}}_1^{-}}(kh)X^+(k,h)$ by Eq.. This is nothing but the product ${{\ensuremath{Y_{R}^{g}(k)}}\times {\ensuremath{Y_{R}^{g}(h)}}}$ that was previously defined in Eq..
Since the physical action $U(h)^{\otimes |{\mathcal{M}}|_v}$ forms a representation of the symmetry group $\mathsf{C}(g)$ the simultaneous virtual action on both boundaries ${\ensuremath{{\Pi}}}_0,{\ensuremath{{\Pi}}}_1$ together ${\ensuremath{{\ensuremath{\mathcal{V}}}}}_0^g(h)\otimes {\ensuremath{{\ensuremath{\mathcal{V}}}}}_1^g(h)$ must also form a representation. However, there is a multiplicative freedom in the multiplication rule of the representation on a single boundary $${\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_1(k){\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_1(h)=\omega^g_1(k,h){\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_1(kh)$$ (and similarly for ${\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_0(h)$), under the constraint $\omega^g_0(k,h)\omega^g_1(k,h)=1$, allowing the possibility of projective representations.
Using the result of Eq., ${{\ensuremath{Y_{R}^{g}(k)}}\times {\ensuremath{Y_{R}^{g}(h)}}=\omega^g(k,h) {\ensuremath{Y_{R}^{g}(kh)}}}$, we can pin down the 2-cocycle $\omega^g_1$ explicitly in terms of the 3-cocycle $\alpha$ of the injective MPO representation $V^{\partial {\ensuremath{{\Pi}}}_0^{-}}(\cdot)$ as follows $\omega^g_1(k,h)=\omega^g(k,h)$ (see Eq. for definition of $\omega^g$). Hence the cohomology class of the projective representation ${\ensuremath{{\ensuremath{\mathcal{V}}}}}^g_1(\cdot)$ is given by $$[\omega^g_1(k,h)]=[{\ensuremath{{\alpha}^{(g)}(k,h)}}].$$
It was shown above that a gauged SPT PEPS with a pair of defects $G_{{\ensuremath{\phi}}^{{\ensuremath{p}}_g}}\ket{\psi^{{\ensuremath{p}}_g},B_0,B_1}$ describes a superposition of anyon pairs in the resulting topological theory. The projective transformation of the monodromy defects is intimately related to the braiding of the resulting anyons, which can be inferred from the following process, depicted in Fig. \[a14\]. First consider an isolated anyon formed by creating a pair of anyons and then moving the other arbitrarily far away. Next create a second pair and move them to encircle the isolated anyon, at this point one should fuse these anyons, but the full description of such fusion requires a systematic anyon ansatz which is beyond the scope of the current paper (see Ref.[@nick]). Instead we drag the pair arbitrarily far away as demonstrated in Fig. \[a14\] and use the fact that this can be rewritten as some local action on the internal degrees of freedom of the isolated anyon, plus another locally undetectable action that can be moved arbitrarily far away.
$$\begin{aligned}
&\vcenter{\hbox{
\includegraphics[width=0.35\linewidth]{Figures/a24}}} \ \rightarrow
\vcenter{\hbox{
\includegraphics[width=0.35\linewidth]{Figures/a25}}}
\\
&\hspace{5.2cm}\downarrow
\\
&\vcenter{\hbox{
\includegraphics[width=0.35\linewidth]{Figures/newfig72}}} \ =\,
\vcenter{\hbox{
\includegraphics[width=0.35\linewidth]{Figures/a26}}} \end{aligned}$$
Gauging symmetric Hamiltonians and ground states {#gaugingham}
================================================
In this appendix we apply the gauging procedure developed in Ref.[@Gaugingpaper] to families of trivial and SPT Hamiltonians with symmetric perturbations and find that they are mapped to perturbed quantum double and twisted quantum double models respectively. We then go on to describe gauging the (unperturbed) fixed-point ground states.
Gauging the Hamiltonian
-----------------------
First we apply the gauging procedure to a symmetric Hamiltonian defined on the matter degrees of freedom, each with Hilbert space ${\ensuremath{\mathbb{H}}}_v\cong \mathbb{C}[\mathsf{G}]$ and symmetry action $U_v(g)=R_v(g)$, associated to the vertices of a directed graph ${\ensuremath{\Lambda}}$ embedded in a closed oriented 2-manifold ${\mathcal{M}}$. The Hamiltonian is given by $$H_\mathsf{m}=\alpha \sum_{v\in{\ensuremath{\Lambda}}} h_v^{0} + \sum_{m\in\mathsf{G}} \beta_m \sum_{e\in{\ensuremath{\Lambda}}} \mathcal{E}_e^m$$ the vertex terms are $ h_v^{0}:=\int \mathrm{d} \hat{g}_v \mathrm{d} g_v \ket{\hat{g}_v}\bra{g_v}$ while the edge interaction terms are $\mathcal{E}_e^m:=\int \mathrm{d} g_{v_e^-} \mathrm{d} g_{v_e^+}\, \delta_{g_{v_e^-}g_{v_e^+}^{-1},m}\, | g_{v_e^-} \rangle \langle g_{v_e^-} | \otimes | g_{v_e^+} \rangle \langle g_{v_e^+}| $. Each term in this Hamiltonian is symmetric under the group action $\bigotimes_v R_v(g)$. For $\alpha,\beta_m<0$ and $|\alpha|\gg |\beta_m|$ this Hamiltonian describes a symmetric phase with trivial SPT order, while for $|\beta_m| \gg |\alpha|$ the Hamiltonian describes different symmetry broken phases.
We construct the gauge and matter Hamiltonian $H_\mathsf{g,m}$ by first gauging the local terms $h_v^0$, which leaves them invariant ${\ensuremath{{\ensuremath{\mathcal{G}}}}}_{v}[h_v^0]=h_v^0$. Next we gauge the interaction terms $\mathcal{E}_e^m$ with the gauging map on $\bar{e}$ (the closure of edge $e$) $$\begin{aligned}
&{\ensuremath{{\ensuremath{\mathcal{G}}}}}_{\bar{e}}[\mathcal{E}_e^m]=\int \mathrm{d} g_{v_e^-} \mathrm{d} g_{v_e^+} \mathrm{d} h_{v_e^-} \mathrm{d} h_{v_e^+}\ \delta_{g_{v_e^-}g_{v_e^+}^{-1},m} \, | g_{v_e^-}h^{-1}_{v_e^-} \rangle
\nonumber \\
&\langle g_{v_e^-}h^{-1}_{v_e^-} | \otimes | h_{v_e^-}h^{-1}_{v_e^+} \rangle \langle h_{v_e^-}h^{-1}_{v_e^+} |_e
\otimes | g_{v_e^+}h^{-1}_{v_e^+} \rangle \langle g_{v_e^+}h^{-1}_{v_e^+} | .
\nonumber\end{aligned}$$
Finally we consider additional local gauge invariant Hamiltonian terms acting purely on the gauge degrees of freedom: symmetric local fields $$\mathcal{F}^{{\ensuremath{c}}}_e:=\int \mathrm{d} \hat{g}_e \mathrm{d} g_e \, \delta_{\hat{g}_eg_e^{-1}\in {\ensuremath{c}}} \, \ket{\hat{g}_e}\bra{g_e}$$ where ${\ensuremath{c}}\in {\ensuremath{\mathfrak{C}(\mathsf{G})}}$ are conjugacy classes of $\mathsf{G}$, and plaquette flux-constraints $$\label{bp}
\mathcal{B}_p^m:=\int \prod_{e\in \partial p} \mathrm{d} g_e\ \delta_{g^{\sigma^{e_1}}_{e_1}\cdots g^{\sigma^{e_{|\partial p|}}}_{e_{|\partial p|}},m} \, \bigotimes_{e\in \partial p} | g_e \rangle\langle g_e |\, ,$$ where each plaquette $p$ has a fixed orientation induced by the 2-manifold ${\mathcal{M}}$, and the group elements $\{ g_{e_1},\dots,g_{e_{|\partial p|}}\}$ are ordered as the edges are visited starting from the smallest vertex label and moving against the orientation of $\partial p$, then $\sigma^{e_i}=\pm1$ is $+1$ if the edge $e_i$ points in the same direction as the orientation of $p$ and ${-}1$ otherwise. Finally we require that the group elements lie in the center of the group $m\in \mathsf{C}(\mathsf{G})$ which renders the choice of vertex from which we begin our traversal of $\partial p$ irrelevant.
The full gauge and matter Hamiltonian is thus given by $$\begin{aligned}
H_\mathsf{g,m}=\alpha \sum_{v\in{\ensuremath{\Lambda}}} & h_v^0 + \sum_{m\in\mathsf{G}} \beta_m \sum_{e\in{\ensuremath{\Lambda}}} {\ensuremath{{\ensuremath{\mathcal{G}}}}}_{\bar{e}}[\mathcal{E}_e^m] + \sum_{{\ensuremath{c}}\in{\ensuremath{\mathfrak{C}(\mathsf{G})}}} \gamma_{\ensuremath{c}}\sum_{e\in{\ensuremath{\Lambda}}} \mathcal{F}_e^{\ensuremath{c}}\nonumber \\
&+ \sum_{m\in \mathsf{C}(\mathsf{G})} \varepsilon_m \sum_{p\in{\ensuremath{\Lambda}}} \mathcal{B}_p^m\, .\end{aligned}$$ Note that each term commutes with all local gauge constraints $\{P_v\}$, see Eq., and the physics takes place within this gauge invariant subspace.
Disentangling the constraints
-----------------------------
To see more clearly that this gauge theory is equivalent to an unconstrained quantum double model we will apply a local disentangling circuit to reveal a clear tensor product structure, allowing us to ‘spend’ the gauge constraints to remove the matter degrees of freedom.
We define the disentangling circuit to be the product of local unitaries $C_{\ensuremath{\Lambda}}:=\prod_v C_v$, where $C_v:=\int \mathrm{d} g_v \ket{g_v} \bra{g_v}_v \prod_{e\in E_v^+} R_e(g) \prod_{e\in E_v^-} L_e(g)$. Note the order in the product is irrelevant since $[C_v,C_{v'}]=0$. This circuit induces the following transformation on the gauge projectors: $C_{\ensuremath{\Lambda}}P_v C_{\ensuremath{\Lambda}}^\dagger=\int \mathrm{d} g R_v(g)$, hence any state $\ket{\psi}$ in the gauge invariant subspace (simultaneous $+1$ eigenspace of all $P_v$) is disentangled into a tensor product of symmetric states on all matter degrees of freedom with an unconstrained state $\ket{\psi'}\in {\ensuremath{\mathbb{H}}}_\mathsf{g}$ on the gauge degrees of freedom $C_{\ensuremath{\Lambda}}\ket{\psi} = \ket{\psi'} \bigotimes_v \int \mathrm{d} g_v \ket{g_v}$.
Now we apply the disentangling circuit to the Hamiltonian $H_\mathsf{g,m}$. First note the pure gauge terms $\mathcal{F}_e^{\ensuremath{c}}$ and $\mathcal{B}_p^m$ are invariant under conjugation by $C_v$. The vertex terms are mapped to $C_v h_v^0 C_v^\dagger = \int \mathrm{d} g_v R_v(g_v) \bigotimes_{e\in E_v^+} R_e(g_v) \bigotimes_{e\in E_v^-} L_e(g_v)$. Since the disentangled vertex degrees of freedom are invariant under $R_v(g_v)$ we see that this Hamiltonian term acts as $\int \mathrm{d} g_v \bigotimes_{e\in E_v^+} R_e(g_v) \bigotimes_{e\in E_v^-} L_e(g_v)$ on the relevant gauge degrees of freedom. We recognize this as the vertex term of a quantum double model. Finally we examine the transformation of the interaction terms $C_v {\ensuremath{{\ensuremath{\mathcal{G}}}}}_{\bar{e}}[\mathcal{E}_e^m] C_v^\dagger = \frac{1}{|\mathsf{G}|}\ket{m}\bra{ m}_e$, which yield local fields on the gauge degrees of freedom that induce string tension. Hence we see that the gauge plus matter Hamiltonian after disentangling becomes a local Hamiltonian $H_\mathsf{g}:=C_v H_\mathsf{g,m} C_v^\dagger$ acting purely on the gauge degrees of freedom $$\begin{aligned}
& H_\mathsf{g} = \alpha \sum_{v} \int \mathrm{d} g_v \bigotimes_{e\in E_v^+} R_e(g_v) \bigotimes_{e\in E_v^-} L_e(g_v) + \sum_{m\in\mathsf{G}} \frac{\beta_m }{|\mathsf{G}|}
\nonumber \\
&\times \sum_{e\in{\ensuremath{\Lambda}}} \ket{m}\bra{ m}_e + \sum_{{\ensuremath{c}}\in{\ensuremath{\mathfrak{C}(\mathsf{G})}}}\gamma_{\ensuremath{c}}\sum_{e\in{\ensuremath{\Lambda}}} \mathcal{F}_e^{\ensuremath{c}}+ \sum_{m\in \mathsf{C}(\mathsf{G})} \varepsilon_m \sum_{p\in{\ensuremath{\Lambda}}} \mathcal{B}_p^m\end{aligned}$$ which describes a quantum double model with string tension and flux perturbations. Note that a spontaneous symmetry breaking phase transition in the ungauged model is mapped to a string tension induced anyon condensation transition by the gauging procedure.
Gauging nontrivial SPT Hamiltonians
-----------------------------------
The gauging procedure extends to nontrivial SPT Hamiltonians which are defined on triangular graphs embedded in closed oriented 2-manifolds ${\mathcal{M}}$. The only modification required is to replace the trivial vertex terms $h_v^0$ by nontrivial terms $h_v^\alpha$ which are defined by $$\label{hvalpha1}
\int \mathrm{d} \hat{g}_v \mathrm{d} g_v \prod_{v' \in L(v)}\mathrm{d} g_{v'} \prod_{\triangle \in S(v)} \alpha_{\triangle} \, \ket{\hat{g}_v}\bra{g_v} \bigotimes_{v'\in L(v)} \ket{g_{v'}}\bra{g_{v'}}$$ where $S(v)$ is the star of $v$, $L(v)$ is the link of $v$ and $\alpha_\triangle\in\mathsf{U}(1)$ for plaquette $\triangle$, whose vertices are given counterclockwise (relative to the orientation of the 2-manifold) by $v,\ v',\ v'',$ is defined by the 3-cocycle $\alpha_{\triangle}:=\alpha^{\sigma_\pi}(g_1g_2^{-1},g_2g_3^{-1},g_3g_4^{-1})$ where $(g_1,g_2,g_3,g_4):=\pi (\hat{g}_v,g_v,g_{v'},g_{v''})$ for $\pi$ the permutation that sorts the group elements into ascending vertex label order (with the convention that $\hat{g}_v$ immediately precedes $g_v$) and $\sigma_\pi=\pm 1$ is the parity of the permutation. The terms $h_v^{\alpha}$ are clearly symmetric under global right group multiplication and are seen to be Hermitian since conjugation inverts the phase factor $\alpha_\triangle$ and interchanges the role of $\hat{g}_v$ and $g_v$ which inverts the parity of $\pi$ thereby compensating the conjugation of $\alpha_\triangle$.
We apply the gauging map on the region $\bar{S}(v)$ (the closure of the star of $v$) to $h_v^\alpha$ $$\begin{aligned}
{\ensuremath{{\ensuremath{\mathcal{G}}}}}_{\bar{S}(v)}[h_v^\alpha]
=& \int \mathrm{d} \hat{g}_v \mathrm{d} g_v \mathrm{d} h_v \prod_{v' \in L(v)}\mathrm{d} g_{v'} \mathrm{d} h_{v'} \prod_{\triangle \in S(v)} \alpha_{\triangle}
\nonumber \\
\phantom{{\ensuremath{{\ensuremath{\mathcal{G}}}}}_{\bar{S}(v)}[h_v^\alpha] =}
&\ket{\hat{g}_v h_v^{-1}} \bra{g_v h_v^{-1}} \bigotimes_{v'\in L(v)} \ket{g_{v'}h_{v'}^{-1}}\bra{g_{v'}h_{v'}^{-1}}
\nonumber \\
\phantom{{\ensuremath{{\ensuremath{\mathcal{G}}}}}_{\bar{S}(v)}[h_v^\alpha] =}
&\bigotimes_{e\in \bar{S}(v)} | {h_{v_e^-}h^{-1}_{v_e^+}} \rangle \langle {h_{v_e^-}h^{-1}_{v_e^+}} |_e\end{aligned}$$ followed by the disentangling circuit $C_{\ensuremath{\Lambda}}{\ensuremath{{\ensuremath{\mathcal{G}}}}}_{\bar{S}(v)}[h_v^\alpha] C_{\ensuremath{\Lambda}}^\dagger$ which yields $$\begin{aligned}
& \int \mathrm{d} \hat{g}_v \mathrm{d} g_v \mathrm{d} h_v \prod_{v' \in L(v)}\mathrm{d} g_{v'} \mathrm{d} h_{v'} \prod_{\triangle \in S(v)} \alpha_{\triangle} \, \ket{\hat{g}_v h_v^{-1}}\bra{g_v h_v^{-1}}
\nonumber \\
& \bigotimes_{v'\in L(v)} \ket{g_{v'}h_{v'}^{-1}}\bra{g_{v'}h_{v'}^{-1}} \bigotimes_{e\in E_v^+} | {g_{v_e^-}\hat{g}^{-1}_{v}} \rangle \langle {g_{v_e^-}g^{-1}_{v}} |
\nonumber \\
& \ \bigotimes_{e\in E_v^-} | {\hat{g}_{v}g^{-1}_{v_e^+}} \rangle \langle {g_{v}g^{-1}_{v_e^+}} |
\bigotimes_{e\in L(v)} | {g_{v_e^-}g^{-1}_{v_e^+}} \rangle \langle {g_{v_e^-}g^{-1}_{v_e^+}}| .\end{aligned}$$ Note, importantly, the phase functions $\alpha_\triangle$ now depend only on the gauge degrees of freedom. Finally in Eq. we rewrite the pure gauge Hamiltonian terms without reference to the matter degrees of freedom, which become irrelevant as the matter degrees of freedom in any gauge invariant state are fixed to be in the symmetric state $\int \mathrm{d} g_v \ket{g_v}_v$ by the disentangling circuit $$\begin{aligned}
& \int \mathrm{d} \hat{g}_v \mathrm{d} g_v \prod_{v' \in L(v)}\mathrm{d} g_{v'} \prod_{\triangle \in S(v)} \alpha_{\triangle} \bigotimes_{e\in E_v^+} | {g_{v_e^-}\hat{g}^{-1}_{v}} \rangle \langle {g_{v_e^-}g^{-1}_{v}} |
\nonumber \\
& \phantom{=\int} \bigotimes_{e\in E_v^-} | {\hat{g}_{v}g^{-1}_{v_e^+}} \rangle \langle {g_{v}g^{-1}_{v_e^+}} |
\bigotimes_{e\in L(v)} | {g_{v_e^-}g^{-1}_{v_e^+}} \rangle \langle {g_{v_e^-}g^{-1}_{v_e^+}}|
\label{appehv}\end{aligned}$$ This can be recognized as the vertex term of a 2D twisted quantum double model (the lattice hamiltonian version of a twisted Dijkgraaf Witten theory for the group $\mathsf{G}$ and cocycle $\alpha$).
Gauging SPT groundstates
------------------------
In this section we apply the gauging procedure directly to the ground states of the nontrivial SPT Hamiltonian that was defined in Eq.. These ground states are constructed using the following local circuit [@else2014classifying] $$\label{dalpha}
D_\alpha:=\int \prod_{v\in{\ensuremath{\Lambda}}}\mathrm{d} g_v \prod_{\triangle\in {\ensuremath{\Lambda}}} \tilde{\alpha}_{\triangle} \bigotimes_{v\in{\ensuremath{\Lambda}}} \ket{g_v}\bra{g_v}$$ where $\tilde{\alpha}_\triangle\in\mathsf{U}(1)$ is a function of the degrees of freedom on plaquette $\triangle$, whose vertices are given counterclockwise, relative to the orientation of the 2-manifold ${\mathcal{M}}$, by $v,\ v',\ v''$ (note the choice of starting vertex is irrelevant) and is defined by a 3-cocycle as follows $\tilde \alpha_{\triangle}:=\alpha^{\sigma_\pi}(g_1g_2^{-1},g_2g_3^{-1},g_3)$ where $(g_1,g_2,g_3):=\pi (g_v,g_{v'},g_{v''})$ with $\pi$ the permutation that sorts the group elements into ascending vertex label order and $\sigma_\pi=\pm 1$ is the parity of the permutation (equivalently the orientation of $\triangle$ embedded within the 2-manifold ${\mathcal{M}}$). Note $D_\alpha$ is easily expressed as a product of commuting 3-local gates.
To define SPT fixed-point states we start with the trivial state ${\ensuremath{\ket{\text{SPT}(0)}}}:=\bigotimes_v \int \mathrm{d} g_v \ket{g_v}_v$, which is easily seen to be symmetric under global right group multiplication. One can also check that $D_\alpha$ is symmetric under conjugation by global right group multiplication by utilizing the 3-cocycle condition satisfied by each $\tilde{\alpha}_\triangle$. With this we define nontrivial SPT fixed-point states ${\ensuremath{\ket{\text{SPT}(\alpha)}}}:=D_\alpha {\ensuremath{\ket{\text{SPT}(0)}}}$, which are symmetric by construction. To see that ${\ensuremath{\ket{\text{SPT}(\alpha)}}}$ is the ground state of the SPT Hamiltonian $\sum_v h_v^{\alpha}$ we note $h_v^\alpha=D_\alpha h_v^0 D_\alpha^\dagger$ which again is proved using the 3-cocycle condition.
We will now gauge the SPT fixed-point states by applying the state gauging map to $D_\alpha$, since the input variables of the circuit carry the same information as the virtual indices of the fixed-point SPT PEPS we hope this makes the correspondence between the two pictures more clear $$\begin{aligned}
{\ensuremath{G}}D_\alpha = \int \prod_{v\in{\ensuremath{\Lambda}}}\mathrm{d} g_v \mathrm{d} h_v \prod_{\triangle\in {\ensuremath{\Lambda}}} \tilde{\alpha}_{\triangle} \, \bigotimes_{e\in {\ensuremath{\Lambda}}} | h_{v_e^-}h^{-1}_{v_e^+} \rangle
\nonumber \\
\bigotimes_{v\in{\ensuremath{\Lambda}}} \ket{g_v h_v^{-1}}\bra{g_v} .\end{aligned}$$ Under the local disentangling circuit this transforms to $$\begin{aligned}
C_{\ensuremath{\Lambda}}{\ensuremath{G}}D_\alpha =\int \prod_{v\in{\ensuremath{\Lambda}}}\mathrm{d} g_v \prod_{\triangle\in {\ensuremath{\Lambda}}} \tilde{\alpha}_{\triangle} \, \bigotimes_{e\in{\ensuremath{\Lambda}}} & | g_{v_e^-}g^{-1}_{v_e^+} \rangle \bigotimes_{v\in{\ensuremath{\Lambda}}}\bra{g_v}
\nonumber \\ \label{gaugeda}
&\bigotimes_{v\in{\ensuremath{\Lambda}}} \int \mathrm{d} k_v \ket{k_v} ,\end{aligned}$$ where it is clear that the matter degrees of freedom have been disentangled into symmetric states and the cocycles $\tilde{\alpha}_{\triangle}$ depend on both the group variables on the edges and the inputs on the vertices (which correspond to the PEPS virtual degrees of freedom).
The explicit connection to the fixed-point SPT PEPS is made by replacing the basis at each vertex $\ket{g_v}_v$ by an analogous basis of the diagonal subspace of variables at each plaquette surrounding the vertex $\bigotimes_{\triangle \in S(v)} \ket{g_v}_{\triangle,v}$. This construction lends itself directly to a PEPS description where a tensor is assigned to each plaquette of the original graph (i.e. the PEPS is constructed on the dual graph). This in turn is why we must apply a seemingly modified version of the gauging operator of Ref.[@Gaugingpaper] to gauge the PEPS correctly and we note that on the subspace where redundant variables are identified the modified PEPS gauging operator becomes identical to the standard gauging operator.
[^1]: Note the following subtlety, our tensor diagrams depict the coefficients of the map $A_\triangle$ and hence the group action $R(h)$ on the physical kets is equivalent to $R(h^{-1})$ on the coefficients, i.e. $R(h) \int \mathrm{d} g f(g) \ket{g}=\int \mathrm{d} g f(gh) \ket{g}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The long-time behavior of the survival probability for unstable multilevel systems that follows the power-decay law is studied based on the $N$-level Friedrichs model, and is shown to depend on the initial population in unstable states. A special initial state maximizing the asymptote of the survival probability at long times is found and examined by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field.'
author:
- Manabu Miyamoto
title: |
Initial state maximizing the nonexponentially decaying survival probability\
for unstable multilevel systems
---
One of the crucial characters of unstable systems is the famous exponential-decay law. Observations of the law were made for many quantum systems, and its theoretical description also proved to be attributed to the poles on the second Riemann sheet of the complex energy plane [@Nakazato(1996)]. However, the deviation from the exponential decay law was also predicted both for short times and for long times [@Khalfin(1957)]. Indeed, despite an apparent difficulty [@Greenland(1988)], a nonexponential decay law at short times was successfully observed [@Wilkinson(1997)]. On the other hand, the long-time deviation has still not been detected, even though expected for all systems coupled with the continuum of the lower-bounded energy spectrum. The main reasons behind the matter could be ascribed to too small survival probability, that is, the component of the initial state remaining in the state at long times.
The unstable systems are described by the Friedrichs model [@Friedrichs(1948); @Exner(1985)], which enables us to investigate the time evolution involving such processes as the spontaneous emission of photons from the atoms [@Facchi(1998); @Antoniou(2001)] and the photodetachment of electrons from the negative ions [@Rzazewski(1982); @Haan(1984); @Antoniou(2001); @Nakazato(2003)]. In the former often only the first excited level is counted, while other higher ones are neglected, and in the latter the negative ion is assumed to have only one electron bound state. These single-level approximations (SLA) could be verified as long as the lowest level is quite separate from the higher ones. However, the multilevel treatment of the model gives us another advantage: the choice of coherently superposed initial-states extending over various levels. In fact, it can yield a variety of temporal behavior that is never found in the SLA [@Frishman(2001); @Antoniou(2003); @Antoniou(2004); @Miyamoto(2003)]. Such multilevel effects on temporal behavior are still not well studied except for Refs. [@Frishman(2001); @Antoniou(2003); @Antoniou(2004); @Miyamoto(2003); @Davies(1974)], and much less examined with respect to nonexponential decay at long times.
In the present article, we consider the long-time behavior of the survival probability $S(t)$ by examining the $N$-level Friedrichs model. In particular, restricting ourselves to the weak coupling case, we clarify how the asymptote of $S(t)$ depends on the initial states. By choosing the initial state localized at the lowest level, we look at the SLA from a multilevel treatment. Then, the result in the $N$-level model turns out to agree with that in the SLA in the weak coupling regime. Furthermore, among the various initial states, we can find a special one that maximizes the asymptote of $S(t)$ at long times. Initial states that eliminate the first term of the asymptotic expansion of $S(t)$ are also obtained. For clarity of discussion, we assume all form factors to vanish at zero energy. However, the existence of such special initial states is proved to be quite general and independent of other details of the form factors.
The $N$-level Friedrichs model describes the couplings between the discrete spectrum and the continuous spectrum. The Hamiltonian of the model is defined by $$H=H_0 +\lambda V ,
\label{eqn:2.10}$$ where $H_0$ denotes the free Hamiltonian $$H_0 = \sum_{n=1}^{N} \omega_{ n} | n \rangle \langle n |
+ \int_0^{\infty} d \omega \ \omega | \omega \rangle \langle \omega |,
\label{eqn:2.20}$$ and $\lambda V$ the interaction Hamiltonian $$V =
\sum_{n=1}^{N}
\int_0^{\infty} d \omega \
\left[ v_n^* (\omega ) | \omega \rangle \langle n | +
v_n (\omega ) | n \rangle \langle \omega | \right],
\label{eqn:2.50}$$ with the coupling constant $\lambda$. The eigenvalues $\omega_{n} $ of $H_0$ were supposed not to be degenerate, i.e., $\omega_n < \omega_{n^{\prime}}$ for $n < n^{\prime}$. Both $| n \rangle $ and $| \omega \rangle $ are the bound and scattering eigenstates of $H_0$, respectively, and satisfy the orthonormality condition: $\langle n | n^{\prime} \rangle = \delta_{n n^{\prime}}$, $\langle \omega | \omega^{\prime} \rangle =\delta (\omega - \omega^{\prime})$, and $\langle n | \omega \rangle = 0 $, where $\delta_{n n^{\prime}}$ is Kronecker’s delta and $\delta (\omega - \omega^{\prime})$ is Dirac’s delta function. They also compose the complete orthonormal system with the resolution of identity. In Eq. (\[eqn:2.50\]), $v_n (\omega )$ denotes the form factor characterizing the transition between $| n \rangle$ and $| \omega \rangle$. In the latter discussion, we will simplify the model with the assumption that the form factor $v_n (\omega )$ is an analytic function in a complex domain including the cut $(0, \infty )$, and behaves like $$v_n (\omega) =
\left\{
\begin{array}{ll}
q_n \omega^{p_n} & ( \omega \rightarrow +0 ) \\
s_n \omega^{-r_n} & ( \omega \rightarrow \infty )
\end{array}
\right. ,
\label{eqn:6.40}$$ where $p_n$ and $r_n$ are the positive constants, while $q_n$ and $s_n$ are appropriate ones. The small-energy condition ensures that the integral $\int_{0}^{\infty} d \omega
v_n (\omega ) v_{n^{\prime }}^* (\omega )/\omega$ is definite. The large-energy condition ensures that $\int_{0}^{\infty} d \omega
v_n (\omega ) v_{n^{\prime }}^* (\omega )/(z -\omega) $ is definite for all complex numbers $z \notin [0, \infty )$. Both of the conditions are satisfied by several systems involving the spontaneous emission process of photons [@Facchi(1998); @Antoniou(2001)] and the photodetachment process of electrons [@Rzazewski(1982); @Haan(1984); @Antoniou(2001); @Nakazato(2003)]. Note that this small-energy condition excludes the photoionization processes associated with the Coulomb interaction [@Wigner(1948)]; however, the formulation developed below could be applied to those cases.
The initial state $| \psi \rangle $ of our interest is an arbitrary superposition of the unstable states $| n \rangle $, $$| \psi \rangle = \sum_{n=1}^{N} c_n | n \rangle ,
\label{eqn:6.10}$$ where $c_n$’s are complex numbers satisfying the normalization condition $\sum_{n=1}^{N} |c_n |^2 =1$. Then, the survival probability $S(t)$ of the initial state $| \psi \rangle$, that is, the probability of finding the initial state in the state at a later time $t$, is defined by $S(t)= |A(t) |^2 $. The $A(t)$ denotes the survival amplitude of $| \psi \rangle $, i.e., $A(t) =\langle \psi | e^{- i tH} | \psi \rangle $. In general, the Hamiltonian (\[eqn:2.10\]) has the possibility of possessing not only the scattering eigenstates $| \psi_{\omega}^{(\pm )} \rangle $, but also the bound eigenstates [@bound-states]. We shall here restrict ourselves to studying the decaying part of $A(t)$, and merely call it the survival amplitude with the same symbol as $$A(t)=\int_{0}^{\infty }
d \omega e^{- i t\omega }
|\langle \psi_{\omega }^{(\pm)} | \psi \rangle |^2 .
\label{eqn:6.20}$$ In order to estimate the long time behavior of $A(t)$, let us evaluate the scattering eigenstates $| \psi_{\omega }^{(\pm )} \rangle$ by solving the Lippmann-Schwinger equation, i.e., $
| \psi_{\omega }^{(\pm )} \rangle
= | \omega \rangle + (\omega \pm i 0 -H_0)^{-1} \lambda V
| \psi_{\omega }^{(\pm )} \rangle
$. In the case of our Hamiltonian, this equation can be solved in the form, $
%\begin{equation}
| \psi_{\omega }^{(\pm)} \rangle
%&=&
=
| \omega \rangle +
\sum_{n=1}^{N} F_n^{(\pm)} (\omega )
%\nonumber \\
%&&
%\times
\left[
| n \rangle +
\int_0^{\infty } d \omega^{\prime }
\frac{\lambda v_n^* (\omega^{\prime} )}{\omega - \omega^{\prime} \pm i 0}
| \omega^{\prime} \rangle
\right] ,
%\label{eqn:3.45}
%\end{equation}
$ from which the integrand of $A(t)$ reads, $$\langle \psi_{\omega }^{(\pm)} | \psi \rangle
=\sum_{n=1}^{N} F_n^{(\pm) *} (\omega ) c_n .
\label{eqn:220}$$ The $F_n^{(\pm)} (\omega )$ is determined by an algebraic equation $$\sum_{n^{\prime} =1}^{N} G^{-1}_{n n^{\prime }} (\omega \pm i0)
F_{n^{\prime}}^{(\pm)} ( \omega ) =-\lambda v_n (\omega ) ,
\label{eqn:3.50}$$ where $$G^{-1}_{n n^{\prime }} (z ) \equiv
(\omega_{n} -z )\delta_{n n^{\prime }}
+ \lambda^2 s_{n n^{\prime }} (z ) ,
\label{eqn:4.20}$$ which is the $(n, n^{\prime })$-th component of the $N \times N$ matrix $G^{-1} (z )$, and $s_{n n^{\prime}} (z)$ is defined by $$s_{n n^{\prime}} (z) \equiv
\int_{0}^{\infty} d \omega^{\prime }
\frac{ v_n (\omega^{\prime } ) v_{n^{\prime}}^* (\omega^{\prime} ) }
{z - \omega^{\prime }} ,
\label{eqn:3.60}$$ for all $z=re^{ i \varphi }$ ($r>0$, $0< \varphi < 2\pi$). Under the large-energy condition of Eq. (\[eqn:6.40\]), $s_{n n^{\prime }} (z )$ is guaranteed to be analytic in the whole complex plane except the cut $[0, \infty )$. For the later convenience, $G^{-1} (z)$ is defined as an inverse of $G(z)$, where $G(z)$ is assumed to be regular. Note that $G(z)$ is nothing more than the reduced (or partial) resolvent $G_{n n^{\prime }}(z) = \langle n |(H -z)^{-1} | n^{\prime } \rangle$. One can confirm this fact by following the discussion in section 3.2 of Ref. [@Exner(1985)]. Since the behavior of $A(t)$ at long times is characterized by that of $F_n^{(\pm)} (\omega)$ in Eq. (\[eqn:220\]) at small energies, we need to estimate the small-energy behavior of $G(z)$. Note that under the condition (\[eqn:6.40\]) we have $$\begin{aligned}
G^{-1}_{n n^{\prime }} (\omega \pm i0)
&=&
(\omega_{n} -\omega )\delta_{n n^{\prime }}
\nonumber \\
&&
+ \lambda^2
\bigl[
I_{n n^{\prime }} (\omega ) \mp i\pi v_n (\omega ) v_{n^{\prime }}^* (\omega )
\bigr]
\nonumber \\
&=&
\omega_n \delta_{n n^{\prime }} + \lambda^2 I_{n n^{\prime }} (0) +o(1) ,
\label{eqn:110}\end{aligned}$$ as $\omega \rightarrow +0$, where $s_{n n^{\prime }} (\omega \pm i0) =
I_{n n^{\prime }} (\omega )
\mp i\pi v_n (\omega ) v_{n^{\prime }}^* (\omega )$ and $
%\begin{equation}
I_{n n^{\prime}} (\omega )
\equiv
P \int_0^{\infty} d\omega^{\prime}
\frac{\varphi_n (\omega^{\prime} )
\varphi_{n^{\prime }}^* (\omega^{\prime} )}
{\omega -\omega^{\prime} }
%\label{eqn:260}
%\end{equation}
$, where $P$ denotes the principle value of the integral. The existence of $I_{n n^{\prime}} (0)$ may be just guaranteed by the small-energy condition of Eq. (\[eqn:6.40\]). Supposing that $G_{n n^{\prime }} $ is of the form $$G_{n n^{\prime }} (\omega \pm i0) =g_{n n^{\prime }}
+o(1),
%+\tilde{g}_{n n^{\prime }} (\omega )
\label{eqn:120}$$ as $\omega \rightarrow +0$, one obtains that $$\begin{aligned}
\delta_{n n^{\prime }}
&=&
\sum_{m=1}^{N}
G_{n m} G^{-1}_{m n^{\prime }}
\nonumber \\
&=&
\sum_{m=1}^{N}
g_{n m}
\bigl[
\omega_{m} \delta_{m n^{\prime }} + \lambda^2 I_{m n^{\prime }} (0)
\bigr]
+o(1),
\label{eqn:130}\end{aligned}$$ which leads to $$g_{n n^{\prime }}
=\frac{1}{\omega_{n^{\prime }} }
\left[
\delta_{n n^{\prime }} -
\lambda^2 \sum_{m=1}^{N} g_{n m} I_{m n^{\prime }} (0)
\right] .
\label{eqn:140}$$ We solve this equation by assuming that $g_{n n^{\prime }} $ can be expanded for small $\lambda$ as $$g_{n n^{\prime }}
=
\sum_{j=0}^{\infty }
g_{n n^{\prime }}^{(j)} \lambda^{2j} .
\label{eqn:150}$$ By substituting Eq. (\[eqn:150\]) into (\[eqn:140\]), it follows that $$g_{n n^{\prime }}^{(0)}
=
\delta_{n n^{\prime }} /\omega_{n^{\prime }}
,~~~
g_{n n^{\prime }}^{(1)}
=
-I_{n n^{\prime }} (0)/\omega_n \omega_{n^{\prime }} ,
\label{eqn:160}$$ and for $j \ge 1$ $$g_{n n^{\prime }}^{(j)}
=
-\frac{1}{\omega_{n^{\prime }}}
\sum_{m=1}^{N} g_{n m}^{(j-1)} I_{m n^{\prime }} (0) ,
\label{eqn:170}$$ where we have assumed that all $\omega_n$ does not vanish. Note that $g_{n n^{\prime }}^{(0)} $ and $g_{n n^{\prime }}^{(1)} $ derived here accord with at least those for solvable cases, where $G(z)$ is explicitly obtained [@Antoniou(2004); @Davies(1974)]. We can then obtain $$F_n^{(\pm)} (\omega )
=
-\lambda f_n \omega^p
+o(\omega^p ) ,
\label{eqn:180}$$ with $$f_n \equiv
\frac{\tilde{q}_n }{\omega_n}
-\lambda^2
\sum_{n^{\prime} =1}^{N}
\frac{I_{n n^{\prime }} (0) \tilde{q}_{n^{\prime }} }
{\omega_n \omega_{n^{\prime }} } +O(\lambda^4) ,
\label{eqn:240}$$ where $$\tilde{q}_n =
\left\{
\begin{array}{ll}
q_n & ( p_n =p ) \\
0 & ( p_n \neq p )
\end{array}
\right. ,
\label{eqn:230}$$ where $p =\min \{ p_n \}$. With use of the $\tilde{q}_n$ instead of $q_n$, we extracted only the dominant part of $F_n^{(\pm)} (\omega )$ at small $\omega$.
The long time behavior of $A (t)$ can be simply obtained by applying to Eq. (\[eqn:6.20\]) the asymptotic method for the Fourier integral [@Copson]. As mentioned before, the long time behavior is determined by the small-energy behavior of its integrand. By inserting Eq. (\[eqn:180\]) into (\[eqn:220\]), the integrand of $A (t)$ turns out to behave asymptotically $$|\langle \psi_{\omega }^{(\pm)} | \psi \rangle |^2
=
\lambda^2 \left| \sum_{n=1}^{N} f_n^* c_n \right|^2
\omega^{2p} +o(\omega^{2p}) ,
\label{eqn:6.100}$$ as $\omega \rightarrow +0$. Applying the asymptotic formula for Fourier integrals, we obtain from Eq. (\[eqn:6.100\]) the asymptotic behavior of Eq. (\[eqn:6.20\]) reading, $$A (t) =
\lambda^2 \frac{\Gamma (2p+1)}{(it)^{2p+1}}
\left| \sum_{n=1}^{N} f_n^* c_n \right|^2
+o(t^{-2p-1}) ,
\label{eqn:6.110}$$ as $t \rightarrow \infty$, where $ i ^{d+1-p } =e^{ i (d+1-p )\pi /2}$, and $\Gamma(z+1)=\int_{0}^{\infty} d x x^{z} e^{-x}$. We can clearly perceive $A(t) \sim t^{-2p-1} $, the power decay law.
Using the above result, let us first consider the higher-level effects on the long-time behavior that starts from the localized initial state at the lowest level. This study is directed to an examination of the SLA. For such an initial state, i.e., $c_n = \delta_{n 1}$, Eq. (\[eqn:6.110\]) becomes $$A (t) =
\lambda^2 \frac{\Gamma (2p+1)}{(it)^{2p+1}}
\frac{|q_1 |^2 }{\omega_1^2} [1+ O(\lambda^2 ) ]
+o(t^{-2p-1}) ,
\label{eqn:8.13}$$ where we supposed that $\tilde{q}_1 \neq 0$. Since there are no factors related to the higher levels in Eq. (\[eqn:8.13\]), it implies that the long-time asymptotic behavior of $A (t)$ could agree with that in the SLA for a sufficiently small $\lambda$.
On the other hand, we can find a special superposition of unstable states $| n \rangle$ that maximizes the asymptote of $A(t)$ at long times. It is worth noting that its dependence on the initial states only appears in Eq. (\[eqn:6.110\]) through the factor $\sum_{n=1}^{N} f_n^* c_n$, which can be rewritten by an inner product as $$\sum_{n=1}^{N} f_n^* c_n
=
\langle \chi | \psi \rangle ,
\label{eqn:7.60}$$ where we have introduced an auxiliary vector defined by $$| \chi \rangle
\equiv
\sum_{n=1}^{N} f_n | n \rangle .
\label{eqn:7.20}$$ Thus, resorting to the Schwarz inequality, we see that the maximum of the factor (\[eqn:7.60\]) is just attained by if and only if $| \psi \rangle \propto | \chi \rangle $, i.e., $$c_n = c f_n
/ \|\chi \| ,
%\Biggl/ \left[ \sum_{k=1}^{N} |f_n |^2 \right]^{1/2} ,
\label{eqn:7.70}$$ where $c$ is an arbitrary complex number with $|c|=1$. Therefore, preparing the initial state $|\psi \rangle$ according to the above weights (\[eqn:7.70\]), we can maximize the asymptote of $A(t)$ at long times. Substituting Eq. (\[eqn:7.70\]) into Eq. (\[eqn:6.110\]), one obtains that $$\begin{aligned}
A (t)
&=&
\lambda^2 \frac{\Gamma (2p+1) %\sum_{n=1}^{N} |f_n |^2
}{(it)^{2p+1}}
\| \chi \|^2
+o(t^{-2p-1})
\label{eqn:8.25} \\
&\simeq&
\lambda^2 \frac{\Gamma (2p+1) %+O(\lambda^2 )
}{(it)^{2p+1}}
\sum_{n=1}^{N} \left| \frac{\tilde{q}_n }{\omega_n } \right|^2
%+o(t^{-2p-1})
.
\label{eqn:260b}\end{aligned}$$ It should be remarked that the initial state extended over unstable states $| n \rangle$ has the possibility of increasing the intensity of $A(t)$ more than a localized one would. This possibility may be interpreted as follows. Let us consider the spontaneous emission process for an atom interacting with the electromagnetic field, where $|n\rangle$ is identified with the $(n+1)$-th excited state of the atom with the vacuum state of the field and $|\omega \rangle$ is the ground state with the one-photon state. In this process, an initially excited atom makes a transition to the ground state with emitting a photon, while the atom that fell into the ground state can be reexcited by absorbing a photon. In the latter process, there are various candidates for the excited state. Repopulation of each excited level can make the intensity of $A(t)$ grow, providing that the initial state possesses those excited levels. However, if the initial state only consists of a specific excited state, the other excited states composing the state at a later time $t$ are discarded without any contribution to $A(t)$ [@comment]. This is the reason why the decay of the $A(t)$ for extended states can be relaxed more than that for localized states.
Note that the above argument also suggests the possibility of finding another kinds of initial states that are coherently superposed to eliminate the factor (\[eqn:7.60\]). This is indeed achieved by the initial states that are orthogonal to $| \chi \rangle $, $$\langle \chi | \psi \rangle
%=\sum_{n=1}^{N} f_n^* c_n
=0 .
\label{eqn:7.80}$$ In this case, the first term in the rhs of Eq. (\[eqn:6.110\]) becomes zero. This fact means that $A(t)$ for such an orthogonal state asymptotically decays faster than $t^{-2p-1}$.
The maximizing initial state seems to be desirable for an experimental verification of the power-decay law. Let us now discuss the value of $|A(t)|^2$ for such an initial state at long times. In particular, we shall evaluate this value at the time $t_{ep}$ of the transition from the exponential to the power decay law. We have to know the exact values of both $p_n$ and $q_n$ for many $n$’s; however, this requirement is satisfied by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field (see also Refs. [@Facchi(1998); @Antoniou(2001)]). This time, $|n\rangle$ is interpreted as the $(n+1)p$-state of the atom with the vacuum state of the field, and $|\omega \rangle$ as the $1s$-state with the one-photon state. It then follows that $p_n =1/2$ for all $n$ [@Seke(1994)], and $|q_n|$ is determined through the relation $\gamma_n = 2\pi \lambda^2 |v_n (\omega_n )|^2 +O(\lambda^2)
\simeq 2\pi \lambda^2 |q_n|^2 |\omega_n|$, where the last estimation is confirmed in the dipole approximation. $\gamma_n$ is the decay rate of the $(n+1)p$-state, which is estimated as $\gamma_n %= 2\pi \lambda^2 |v_n (\omega_n )|^2
\simeq
8.0 \times 10^{9} \times 2^8 (n+1)n^{2n} / 9(n+2)^{2n+4}
$ $\mathrm{s}^{-1}$ [@Bethe]. From these facts it follows that $$\lambda^2 \biggl| \frac{q_n}{\omega_n} \biggr|^2
\simeq
\frac{8.0 \times 10^{9} \times 6 (n+1)^7 n^{2n} }
{\pi \Omega^3 (n+2)^{2n+4} [(n+1)^2 -1]^3 }
~\mathrm{s}^2 ,
\label{eqn:270}$$ where $\omega_{ n} =\frac{4}{3}\Omega [1-(n+1)^{-2}]$ with $\Omega = 1.55 \times 10^{16}$ $\mathrm{s}^{-1}$, and we also choose $\lambda=6.43\times 10^{-9}$. Thus, we see that $|q_n|^2 / |\omega_n|^2 \sim n^{-3}$ for a large $n$.
----------------------------------------------------------------------------------------------------------------------------
Number of levels $N$ 1 10 50
------------------------------------------------------- ---------------------- ---------------------- ----------------------
$\bigl| \frac{\omega_1}{q_1} \bigr|^2 1.00 1.28 1.29
\sum_{n=1}^{N} \bigl| \frac{q_n}{\omega_n} \bigr|^2 $
$t_N ~(\mathrm{s})$ $1.60\times 10^{-9}$ $3.18\times 10^{-7}$ $3.18\times 10^{-5}$
$t_{ep}~(\mathrm{s})$ $2.00\times 10^{-7}$ $4.23\times 10^{-5}$ $4.59\times 10^{-3}$
----------------------------------------------------------------------------------------------------------------------------
: \[tab:levels\] The level-number dependence of $\sum_{n=1}^{N} |q_n /\omega_n |^{2}$, the decay time $t_N$ of the $(N+1)p$-state, and the transition time $t_{ep}$ from the exponential to the power decay law.
In Table \[tab:levels\], the numerical values of $%|\omega_1 /q_1|^{2}
\sum_{n=1}^{N} |q_n /\omega_n |^{2}$ ($\simeq \| \chi \|^2 $), the decay time $t_N$ ($=1/\gamma_N$) of the $(N+1)p$-state, and the time $t_{ep}$ are listed for the three cases of the level numbers $N=1$, $10$, and $50$. Here, we define $t_{ep}$ as the maximum time which equates the square modulus of the asymptote (\[eqn:260b\]) to that of the following $A(t)$ at intermediate times [@Antoniou(2003); @Davies(1974)], $$A(t) \simeq %\langle \chi |\chi \rangle^{-1}
\sum_{n=1}^{N} |c_n|^2 e^{-it\omega_n -t\gamma_n /2} ,
\label{eqn:280}$$ where $c_n$ is chosen as Eq. (\[eqn:7.70\]). It is worth noting that when $t \gg t_N$, $|A(t)|^2$ can approximate $|c_N |^4 e^{-t\gamma_N}$ because the decay time $t_n$ lengthens with $n$ in this case. We see from Table \[tab:levels\] that $t_{ep}$ is much longer than $t_N$, so that $t_{ep}$ is roughly estimated as the root of the equation, $$|c_N |^4 e^{-t\gamma_N} =
\frac{\lambda^4 }{t^4}
\Biggl| \sum_{n=1}^{N} \biggl|\frac{q_n }{\omega_n} \biggr|^2 \Biggr|^2 .
\label{eqn:290}$$ On the other hand, the factor $\sum_{n=1}^{N} |q_n /\omega_n |^{2}$ is essentially unchanged with $N$, whereas $t_{ep}$ rapidly increases with $N$ (see Table \[tab:levels\]). These facts and Eq. (\[eqn:290\]) imply that $A(t_{ep} )$ rather decreases as $N$ increases [@A(tep)]. Hence, we should unfortunately conclude that the maximizing initial state does not provide any help for an observation of the power decay law for the spontaneous emission from a hydrogen atom.
In summary, we have considered the long-time behavior of the unstable multilevel systems and estimated the asymptotic behavior of the survival amplitude $A(t)$ for an arbitrary initial state in the long-time region where $A(t)$ obeys a power decay law. We have then found two special initial states. One of them asymptotically maximizes $A(t)$ at long times, and the other eliminates the first term of the asymptotic expansion of $A(t)$. The latter fact may imply that the exponent of the power decay of $A(t)$ is determined by not only the small-energy behavior of the form factors but also the initial population in unstable states. Such relations between the initial states and the power decay law were studied with respect to the long-time behavior of wave packets, both for the free-particle system [@Miyamoto(free)] and for finite-range potential systems [@Miyamoto(potential)]. In the case of the experimental verification of the power decay laws, the existence of the maximizing initial states seems preferable. This expectation is probably misplaced for the spontaneous emission process of a hydrogen atom, however, a possibility still could remain for the systems allowing the photodetachment or the photoionization process. We then should require of them the property that both $\gamma_n$ and $|q_n /\omega_n |$ do not decrease as $n$ increases. More important states for this aim are those states which maximize $A(t)$ at the transition time from the exponential to the power decay law. The relation between such a maximizing state and the discussed one is still unclear. It will be addressed in a future issue.
The author would like to thank Professor I. Ohba and Professor H. Nakazato for useful comments, and Dr. J. Harada for helpful discussions. He would also like to thank the Yukawa Institute for Theoretical Physics at Kyoto University, where this work was initiated during the YITP-03-16, Quantum Mechanics and Chaos: From Fundamental Problems through Nanosciences. This work is partly supported by a Grant for The 21st Century COE Program at Waseda University from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
[99]{}
For a review, see, for example, H. Nakazato, M. Namiki, and S. Pascazio, Int. J. Mod. Phys. B [**10**]{}, 247 (1996).
L. A. Khalfin, Zh. Eksp. Theor. Fiz. [**33**]{}, 1371 (1957) \[Sov. Phys. JETP [**6**]{}, 1053 (1958)\].
P. T. Greenland, Nature (London) [**335**]{}, 298 (1988).
S. R. Wilkinson, [*et al*]{}, Nature (London) [**387**]{}, 575 (1997).
K. O. Friedrichs, Commun. Pure Appl. Math. [**1**]{}, 361 (1948).
P. Exner, (Reidel, Doredrecht, 1985).
P. Facchi and S. Pascazio, Phys. Lett. A [**241**]{}, 139 (1998).
I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, Phys. Rev. A [**63**]{}, 062110 (2001).
K. Rz[a]{}[. z]{}ewski, M. Lewenstein, and J. H. Eberly, J. Phys. B [**15**]{}, L661 (1982).
S. L. Haan and J. Cooper, J. Phys. B [**17**]{}, 3481 (1984).
H. Nakazato, in “Fundamental Aspects of Quantum Physics”, edited by L. Accardi and S. Tasaki (World Scientific, New Jersey, 2003).
E. Frishman and M. Shapiro, Phys. Rev. Lett. [**87**]{}, 253001 (2001); Phys. Rev. A [**68**]{}, 032717 (2003).
I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, Int. J. Theor. Phys. [**42**]{}, 2403 (2003).
I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, quant-ph/0402210 (2004).
M. Miyamoto, in Proceedings of the Workshop on [*Quantum Mechanics and Chaos: From Fundamental Problems through Nanosciences*]{} (2003), to be appeared in Bussei Kenkyu (Kyoto) (in Japanese).
E. B. Davies, J. Math. Phys. [**15**]{}, 2036 (1974).
E. P. Wigner, Phys. Rev. [**73**]{}, 1002 (1948).
This is the case for the strong-coupling interactions (see, e.g., [@Nakazato(2003)]), and for the degenerated multilevel systems [@Antoniou(2004)].
E. T. Copson, (Cambridge Univ. Press, Cambridge, 1965), Chap. 3.
Note that this interpretation is not necessarily applied to all of the time region. The reason is that such repopulation processes are naively expected to occur under the dominance of the terms of $O(\lambda^2)$ for our interaction Hamiltonian $\lambda V$. This does not contradict the behavior of $A(t)$ at long times [@Antoniou(2003)].
J. Seke, Physica A [**203**]{}, 269 (1994).
H. A. Bethe and E. E. Salpeter, (Springer-Verlag, Berlin, 1957), Sect. 63.
For instance, we obtain, $|A(t_{ep})|^2 \simeq 4.41 \times 10^{-55}$ for $N=1$, an extremely small value.
K. Unnikrishnan, Am. J. Phys. [**65**]{}, 526 (1997); [**66**]{}, 632 (1998); F. Lillo and R. N. Mantegna, Phys. Rev. Lett. [**84**]{}, 1061 (2000); [**84**]{}, 4516 (2000); J. A. Damborenea, I. L. Egusquiza, and J. G. Muga, Am. J. Phys. [**70**]{}, 738 (2002); M. Miyamoto, J. Phys. A [**35**]{}, 7159 (2002); Phys. Rev. A [**68**]{}, 022702 (2003).
M. Miyamoto, Phys. Rev. A [**69**]{}, 042704 (2004).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This template helps you to create a properly formatted LaTeX manuscript.'
address:
- 'Radarweg 29, Amsterdam'
- '1600 John F Kennedy Boulevard, Philadelphia'
- '360 Park Avenue South, New York'
author:
- Elsevier
- Elsevier Inc
- Global Customer Service
bibliography:
- 'mybibfile.bib'
title: Elsevier LaTeX template
---
`elsarticle.cls`,LaTeX,Elsevier ,template 00-01,99-00
The Elsevier article class
==========================
#### Installation
If the document class *elsarticle* is not available on your computer, you can download and install the system package *texlive-publishers* (Linux) or install the LaTeX package *elsarticle* using the package manager of your TeX installation, which is typically TeX Live or MikTeX.
#### Usage
Once the package is properly installed, you can use the document class *elsarticle* to create a manuscript. Please make sure that your manuscript follows the guidelines in the Guide for Authors of the relevant journal. It is not necessary to typeset your manuscript in exactly the same way as an article, unless you are submitting to a camera-ready copy (CRC) journal.
#### Functionality
The Elsevier article class is based on the standard article class and supports almost all of the functionality of that class. In addition, it features commands and options to format the
- document style
- baselineskip
- front matter
- keywords and MSC codes
- theorems, definitions and proofs
- lables of enumerations
- citation style and labeling.
Front matter
============
The author names and affiliations could be formatted in two ways:
(1) Group the authors per affiliation.
(2) Use footnotes to indicate the affiliations.
See the front matter of this document for examples. You are recommended to conform your choice to the journal you are submitting to.
Bibliography styles
===================
There are various bibliography styles available. You can select the style of your choice in the preamble of this document. These styles are Elsevier styles based on standard styles like Harvard and Vancouver. Please use BibTeX to generate your bibliography and include DOIs whenever available.
Here are two sample references: [@Feynman1963118; @Dirac1953888].
References {#references .unnumbered}
==========
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This guide is for authors who are preparing papers for *Monthly Notices of the Royal Astronomical Society* using the document preparation system and the [mn2e]{} class file.'
date: Released 2002 Xxxxx XX
title: |
Monthly Notices of the Royal Astronomical Society:\
style guide for authors
---
\[firstpage\]
– class files: `mn2e.cls` – sample text – user guide.
Introduction
============
The standard format for papers submitted to *Monthly Notices* is . The layout design for *Monthly Notices* has been implemented as a class file. The [mn2e]{} class file is based on the [mn]{} style file, which in turn is based on `article` style as discussed in the LaTeXmanual [@la]. Commands that differ from the standard LaTeXinterface, or that are provided in addition to the standard interface, are explained in this guide. This guide is not a substitute for the LaTeX manual itself. We also refer authors to @kd and @kn. Authors planning to submit their papers in LaTeX are advised to use `mn2e.cls` as early as possible in the creation of their files.
The mn2e document class
-----------------------
The use of LaTeX document classes allows a simple change of class to transform the appearance of your document. The [mn2e]{} class file preserves the standard LaTeX interface such that any document that can be produced using the standard LaTeX `article` class can also be produced with the [mn2e]{} class file. However, the measure (or width of text) is narrower than the default for `article`, and even narrower than for the `A4` style, therefore line breaks will change and long equations may need re-setting.
When your article is printed in the journal, it is typeset in 9/11 pt Times Roman. As most authors do not have this font, it is likely that the make-up will change with the change of font. For this reason, we ask you to ignore details such as slightly long lines, page stretching, or figures falling out of synchronization, because these details can be dealt with at a later stage.
General style issues
--------------------
For general style issues, authors are referred to the ‘Instructions for Authors’ on the journal web page.[^1] Authors who are interested in the details of style are referred to @bu and @ch. The language of the journal is British English and spelling should conform to this.
Use should be made of symbolic references (`\ref`) in order to protect against late changes of order, etc.
Submission of articles to the journal
--------------------------------------
Authors should refer to the journal web page for instructions on how to submit a new manuscript for refereeing, and on how to supply the files for an accepted paper to Blackwell Publishing. Abbreviated instructions are also available on the inside back cover of the journal. Note that the standard procedure is to supply a PDF file at submission stage and TeX/LaTeX file(s) of the text plus EPS files of the figures at acceptance stage. If for some reason you cannot do this, please notify the RAS as soon as possible.
The correct *Monthly Notices* House Style should be used – again, details are given in the Instructions for Authors on the journal web pageand in the Style Guide published in the 1993 January 1 issue (, 260, 1). Ensure that any author-defined macros are gathered together in the file, just before the `\begin{document}` command.
Using the mn2e class file
=========================
If the file `mn2e.cls` is not already in the appropriate system directory for LaTeX files, either arrange for it to be put there or copy it to your working directory. The [mn2e]{} document class is implemented as a complete class, [*not*]{} a document style option. In order to use the MN document class, replace `article` by `mn2e` in the `\documentclass` command at the beginning of your document:
\documentclass{article}
is replaced by
\documentclass{mn2e}
In general, the following standard document style options should [*not*]{} be used with the [mn2e]{} class file:
1. , , – unavailable;
2. [twoside]{} (no associated style file) – [ twoside]{} is the default;
3. [fleqn]{}, [leqno]{}, [titlepage]{} – should not be used (`fleqn` is already incorporated into the MN style);
4. [twocolumn]{} – is not necessary as it is the default style.
The [mn2e]{} class file has been designed to operate with the standard version of `lfonts.tex` that is distributed as part of LaTeX. If you have access to the source file for this guide, `mn2eguide.tex`, and to the specimen article, `mn2esample.tex`, attempt to typeset both of these. If you find font problems you might investigate whether a non-standard version of `lfonts.tex` has been installed in your system.
Authors using LaTeX wishing to create PDF files with smooth fonts are advised to read Adobe FaxYI Document Number 131303 by Kendall Whitehouse. Type 1 PostScript versions of the Computer Modern fonts are now freely available and are normally installed with new TeX/LaTeX software. Information is available from Y&Y.[^2] Alternatively, you may wish to use Times font when creating your PDF file, e.g.\
\documentclass[useAMS]{mn2e}
\usepackage{times}
Additional document style options
---------------------------------
The following additional style options are available with the [mn2e]{} class file:
[onecolumn]{} – to be used [*only*]{} when two-column output is unable to accommodate long equations;
[landscape]{} – for producing wide figures and tables which need to be included in landscape format (i.e. sideways) rather than portrait (i.e. upright). This option is described below.
[doublespacing]{} – this will double-space your article by setting `\baselinestretch` to 2.
[referee]{} – 12/20pt text size, single column, measure 16.45 cm, left margin 2.75 cm on A4 page.
[galley]{} – no running heads, no attempt to align the bottom of columns.
`useAMS` – this enables the production of upright Greek characters $\upi$, $\umu$ and $\upartial$.
`usedcolumn` – this uses the package file `dcolumn.sty` to define two new types of column alignment for use in tables.
`usenatbib` – this uses Patrick Daly’s `natbib.sty` package for cross-referencing.
`usegraphicx` – this enables the use of the [graphicx]{} package for inclusion of figures. Note that the standard LaTeX graphics package `graphicx.sty` is required in order to use the `usegraphicx` option.
Please place any additional command definitions at the very start of the LaTeX file, before the `\begin{document}`. Author-defined macros should be kept to a minimum. Please do not customize the MNRAS macros or class file, or redefine macros that are already in the class file, and please do not include additional definitions unless they are actually used in the paper.
Landscape pages
---------------
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1. Use the `table*` or `figure*` environments in your document to create the space for your table or figure on the appropriate page of your document. Include an empty caption in this environment to ensure the correct numbering of subsequent tables and figures. For instance, the following code prints a page with the running head, a message half way down and the figure number towards the bottom. If you are including a plate, the running headline is different, and you need to key in the three lines that are marked with `% **`, with an appropriate headline.
% ** \clearpage
% ** \thispagestyle{plate}
% ** \plate{Opposite p.~812, MNRAS, {\bf 261}}
\begin{figure*}
\vbox to220mm{\vfil Landscape figure to go here.
\caption{}
\vfil}
\label{landfig}
\end{figure*}
2. Create a separate document with the [mn2e]{} document class but also with the `landscape` document style option, and include the `\pagestyle` command, as follows:
\documentclass[landscape]{mn2e}
\pagestyle{empty}
3. Include your complete tables and illustrations (or space for these) with captions using the `table*` and `figure*` environments.
4. Before each float environment, use the `\setcounter` command to ensure the correct numbering of the caption. For example,
\setcounter{table}{0}
\begin{table*}
\begin{minipage}{115mm}
\caption{The Largest Optical Telescopes.}
\label{tab1}
\begin{tabular}{@{}llllcll}
:
\end{tabular}
\end{minipage}
\end{table*}
The corresponding example for a figure would be:
\clearpage
\setcounter{figure}{12}
\begin{figure*}
\vspace{144mm}
\caption{Chart for a cold plasma.}
\label{fig13}
\end{figure*}
Additional facilities
=====================
In addition to all the standard LaTeX design elements, the [mn2e]{} class file includes the following features.
1. Extended commands for specifying a short version of the title and author(s) for the running headlines.
2. A `keywords` environment and a `\nokeywords` command.
3. Use of the `description` environment for unnumbered lists.
4. A `\contcaption` command to produce captions for continued figures or tables.
In general, once you have used the additional `mn2e.cls` facilities in your document, do not process it with a standard LaTeX class file.
Titles and author’s name
------------------------
In the MN style, the title of the article and the author’s name (or authors’ names) are used both at the beginning of the article for the main title and throughout the article as running headlines at the top of every page. The title is used on odd-numbered pages (rectos) and the author’s name appears on even-numbered pages (versos). Although the main heading can run to several lines of text, the running headline must be a single line ($\le 45$ characters). Moreover, new line commands (e.g. `\\`) are not acceptable in a running headline. To enable you to specify an alternative short title and an alternative short author’s name, the standard `\title` and `\author` commands have been extended to take an optional argument to be used as the running headline. The running headlines for this guide were produced using the following code:
\title[Monthly Notices: \LaTeXe\ guide for authors]
{Monthly Notices of the Royal Astronomical
Society: \\ \LaTeXe\ style guide for authors}
and
\author[A. Woollatt et al.]
{A.~Woollatt,$^1$\thanks{Affiliated to ICRA.}
M.~Reed,$^1$ R.~Mulvey,$^1$ K.~Matthews,$^1$
D.~Starling,$^1$ Y.~Yu,$^1$
\newauthor % starts a new line in the
% author environment
A.~Richardson,$^1$
P.~Smith,$^2$\thanks{Production Editor.}
N. Thompson$^2$\footnotemark[2]
and G. Hutton$^2$\footnotemark[2] \\
$^1$Cambridge University Press, Shaftesbury
Road, Cambridge CB2 2BS\\
$^2$Blackwell Publishing,
23 Ainslie Place, Edinburgh EH3 6AJ}
The `\thanks` note produces a footnote to the title or author. The footnote can be repeated using the `\footnotemark[]` command.
Key words and abstracts
-----------------------
At the beginning of your article, the title should be generated in the usual way using the `\maketitle` command. Immediately following the title you should include an abstract followed by a list of key words. The abstract should be enclosed within an `abstract` environment, followed immediately by the key words enclosed in a `keywords` environment. For example, the titles for this guide were produced by the following source:
\maketitle
\begin{abstract}
This guide is for authors who are preparing
papers for \textit{Monthly Notices of the
Royal Astronomical Society} using the \LaTeXe\
document preparation system and the {\tt mn2e}
class file.
\end{abstract}
\begin{keywords}
\LaTeXe\ -- class files: \verb"mn2e.cls"\ --
sample text -- user guide.
\end{keywords}
\section{Introduction}
:
The heading ‘[**Key words**]{}’ is included automatically and the key words are followed by vertical space. If, for any reason, there are no key words, you should insert the `\nokeywords` command immediately after the end of the `abstract` environment. This ensures that the vertical space after the abstract and/or title is correct and that any `thanks` acknowledgments are correctly included at the bottom of the first column. For example,
\maketitle
\begin{abstract}
:
\end{abstract}
\nokeywords
\section{Introduction}
:
The key words list is common to MNRAS, ApJ and A&A, and is available from the journal web page.
Lists
-----
The [mn2e]{} class file provides numbered lists using the `enumerate` environment and unnumbered lists using the `description` environment with an empty label. Bulleted lists are not part of the journal style and the `itemize` environment should not be used.
The enumerated list numbers each list item with roman numerals:
1. first item
2. second item
3. third item
Alternative numbering styles can be achieved by inserting a redefinition of the number labelling command after the `\begin{enumerate}`. For example, the list
1. first item
2. second item
3. etc…
was produced by:
\begin{enumerate}
\renewcommand{\theenumi}{(\arabic{enumi})}
\item first item
:
\end{enumerate}
Unnumbered lists are provided using the `description` environment. For example,
First unnumbered item which has no label and is indented from the left margin.
Second unnumbered item.
Third unnumbered item.
was produced by:
\begin{description}
\item First unnumbered item...
\item Second unnumbered item.
\item Third unnumbered item.
\end{description}
Captions for continued figures and tables {#contfigtab}
-----------------------------------------
The `\contcaption` command may be used to produce a caption with the same number as the previous caption (for the corresponding type of float). For instance, if a very large table does not fit on one page, it must be split into two floats; the second float should use the `\contcaption` command:
\begin{table}
\contcaption{}
\begin{tabular}{@{}lccll}
:
\end{tabular}
\end{table}
Some guidelines for using\
standard facilities
==========================
The following notes may help you achieve the best effects with the [mn2e]{} class file.
Sections
--------
provides four levels of section headings and they are all defined in the [mn2e]{} class file:
`\section`
`\subsection`
`\subsubsection`
`\paragraph`
Section numbers are given for section, subsection, subsubsection and paragraph headings. Section headings are automatically converted to upper case; if you need any other style, see the example in Section \[headings\].
Illustrations (or figures)
--------------------------
The [mn2e]{} class file will cope with most positioning of your illustrations and you should not normally use the optional positional qualifiers on the `figure` environment that would override these decisions. See the instructions for authors on the journal web page for details regarding submission of artwork. Figure captions should be *below* the figure itself, therefore the `\caption` command should appear after the figure or space left for an illustration. For example, Fig. \[sample-figure\] is produced using the following commands:
\begin{figure}
%\includegraphics[width=84mm]{fig1.ps}
%% to include a figure, or
\vspace{3.5cm}
%% to leave a blank space
\caption{An example figure.}
\label{sample-figure}
\end{figure}
Tables
------
The [mn2e]{} class file will cope with most positioning of your tables and you should not normally use the optional positional qualifiers on the `table` environment which would override these decisions. Table captions should be at the top, therefore the `\caption` command should appear *above* the body of the table.
The `tabular` environment can be used to produce tables with single horizontal rules, which are allowed, if desired, at the head and foot and under the header only. This environment has been modified for the [mn2e]{} class in the following ways:
1. additional vertical space is inserted on either side of a rule;
2. vertical lines are not produced.
Commands to redefine quantities such as `\arraystretch` should be omitted in general. For example, Table \[symbols\] is produced using the following commands. Note that `\rmn` will produce a roman character in math mode. There are also `\bld` and `\itl`, which produce bold face and text italic in math mode.
Class $\gamma _1$ $\gamma _2$ $\langle \gamma \rangle$ $G$ $f$ $\theta _{\rmn{c}}$
--------- ------------- ------------- -------------------------- -------- --------------------- ---------------------
BL Lacs 5 36 7 $-4.0$ $1.0\times 10^{-2}$ 10$\degr$
FSRQs 5 40 11 $-2.3$ $0.5\times 10^{-2}$ 14$\degr$
: Radio-band beaming model parameters for FSRQs and BL Lacs.[]{data-label="symbols"}
is the slope of the Lorentz factor distribution, i.e. $n(\gamma)\propto \gamma ^G$, extending between $\gamma _1$ and $\gamma_2$, with mean value $\langle \gamma \rangle$, [*f*]{} is the ratio between the intrinsic jet luminosity and the extended, unbeamed luminosity, while $\theta_{\rmn{c}}$ is the critical angle separating the beamed class from the parent population.
\begin{table}
\caption{Radio-band beaming model parameters
for FSRQs and BL Lacs.}
\label{symbols}
\begin{tabular}{@{}lcccccc}
\hline
Class & $\gamma _1$ & $\gamma _2$
& $\langle \gamma \rangle$
& $G$ & $f$ & $\theta _{\rmn{c}}$ \\
\hline
BL Lacs &5 & 36 & 7 & $-4.0$
& $1.0\times 10^{-2}$ & 10$\degr$ \\
FSRQs & 5 & 40 & 11 & $-2.3$
& $0.5\times 10^{-2}$ & 14$\degr$ \\
\hline
\end{tabular}
\medskip
{\em G} is the slope of the Lorentz factor
:
class from the parent population.
\end{table}
If you have a table that is to extend over two columns, you need to use `table*` in a minipage environment, i.e. you can say
\begin{table*}
\begin{minipage}{126mm}
\caption{Caption which will wrap round to the
width of the minipage environment.}
\begin{tabular}{%
:
\end{tabular}
\end{minipage}
\end{table*}
The width of the minipage should more or less be the width of your table, so you can only guess on a value on the first pass. The value will have to be adjusted when your article is typeset in Times, so do not worry about making it the exact size.
Running headlines
-----------------
As described above, the title of the article and the author’s name (or authors’ names) are used as running headlines at the top of every page. The headline on left-hand pages can list up to three names; for more than three use et al. The `\pagestyle` and `\thispagestyle` commands should [*not*]{} be used. Similarly, the commands `\markright` and `\markboth` should not be necessary.
Typesetting mathematics {#TMth}
-----------------------
### Displayed mathematics
The [mn2e]{} class file will set displayed mathematics flush with the left margin, provided that you use the standard of open and closed square brackets as delimiters. The equation $$\sum_{i=1}^p \lambda_i = \rmn{trace}(\mathbfss{S})$$ was typeset using the [mn2e]{} class file with the commands
\[
\sum_{i=1}^p \lambda_i = \rmn{trace}(\mathbfss{S})
\]
Note the difference between the positioning of this equation and of the following centred equation, $$\alpha_{j+1} > \bar{\alpha}+ks_{\alpha}$$ which was wrongly typeset using double dollars as follows:
$$ \alpha_{j+1} > \bar{\alpha}+ks_{\alpha} $$
Please do not use double dollars.
### Bold math italic / bold symbols
To get bold math italic you should use `\bmath`, e.g.
\[
d(\bmath{s_{t_u}}) = \langle [RM(\bmath{X_y}
+ \bmath{s_t}) - RM(\bmath{x_y})]^2 \rangle
\]
to produce: $$d(\bmath{s_{t_u}}) = \langle [RM(\bmath{X_y}
+ \bmath{s_t}) - RM(\bmath{x_y})]^2 \rangle$$ Working this way, scriptstyle and scriptscriptstyle sizes will take care of themselves.
### Bold Greek {#boldgreek}
Bold lowercase Greek characters can now be obtained by prefixing the normal (unbold) symbol name with a ‘b’, e.g. `\bgamma` gives $\bgamma$. This rule does not apply to bold `\eta`, as this would lead to a name clash with `\beta`. Instead use `\boldeta` for bold eta. Note that there is no `\omicron` (so there is no `\bomicron`), just use ‘o’ in math mode for omicron ($o$) and ‘`\bmath{o}`’ for bold omicron ($\bmath{o}$). ‘`\bmath{}`’ can also be used for other Greek characters.
For bold uppercase Greek, prefix the unbold character name with `\mathbf`, e.g. `\mathbf\Gamma` gives $\mathbf\Gamma$. Upper and lowercase Greek characters are available in all typesizes.
You can then use these definitions in math mode, as you would normal Greek characters:
\[
\balpha_{\bmu} = \mathbf{\Theta} \alpha.
\]
will produce $$\balpha_{\bmu} = \mathbf{\Theta} \alpha.$$
### Upright Greek characters {#upgreek}
You can obtain upright Greek characters if you have access to the American Maths Society Euler fonts (version 2.0), but you may not have these. In this case, you will have to use the normal math italic symbols and the typesetter will substitute the corresponding upright characters. You will make this easier if you can use the macros `\upi`, `\umu` and `\upartial` etc. in your text to indicate the need for upright characters, together with the [useAMS]{} global option: (`\documentclass[useAMS]{mn2e}`). Characters $\upi$, $\umu$ and $\upartial$ will appear upright only on systems that have the Euler roman fonts (`eurm`*xx*); characters $\leq$ and $\geq$ appear slanted only on systems that have the AMS series A fonts (`msam`*xx*). On systems that do not have these fonts, the standard forms of the characters appear in the printout; however, they should be correct in the final typeset paper if the correct LaTeX commands have been used.
### Special symbols {#SVsymbols}
The macros for the special symbols in Tables \[mathmode\] and \[anymode\] have been taken from the Springer Verlag ‘Astronomy and Astrophysics’ design, with their permission. They are directly compatible and use the same macro names. These symbols will work in all text sizes, but are only guaranteed to work in text and displaystyles. Some of the symbols will not get any smaller when they are used in sub- or superscripts, and will therefore be displayed at the wrong size. Do not worry about this as the typesetter will be able to sort this out. Authors should take particular note of the symbols `\la`, `\ga`, `\fdg` and `\sun`.
Input Explanation Output Input Explanation Output
------------------- ------------------- ----------- ---------- ---------------------- ----------
`\la` less or approx $\la$ `\ga` greater or approx $\ga$
\[2pt\] `\getsto` gets over to $\getsto$ `\cor` corresponds to $\cor$
\[2pt\] `\lid` less or equal $\lid$ `\gid` greater or equal $\gid$
\[2pt\] `\sol` similar over less $\sol$ `\sog` similar over greater $\sog$
\[2pt\] `\lse` less over simeq $\lse$ `\gse` greater over simeq $\gse$
\[2pt\] `\grole` greater over less $\grole$ `\leogr` less over greater $\leogr$
\[2pt\] `\loa` less over approx $\loa$ `\goa` greater over approx $\goa$
Input Explanation Output Input Explanation Output
--------------------- ----------------------- -------- ----------- -------------------- ---------
`\sun` sun symbol $\sun$ `\degr` degree $\degr$
\[2pt\] `\diameter` diameter `\sq` square
\[2pt\] `\fd` fraction of day `\fh` fraction of hour
\[2pt\] `\fm` fraction of minute `\fs` fraction of second
\[2pt\] `\fdg` fraction of degree `\fp` fraction of period
\[2pt\] `\farcs` fraction of arcsecond `\farcm` fraction of arcmin
\[2pt\] `\arcsec` arcsecond `\arcmin` arcminute
Bibliography
------------
References to published literature should be quoted in text by author and date: e.g. Draine (1978) or (Begelman, Blandford & Rees 1984). Where more than one reference is cited having the same author(s) and date, the letters a,b,c, … should follow the date; e.g. Smith (1988a), Smith (1988b), etc. When a three-author paper is cited, you should list all three authors at the first citation, and thereafter use ‘et al.’.
### Use of [natbib]{}
If the `usenatbib` global option is specified, Patrick Daly’s `natbib.sty` package will be used for for cross-referencing. If the `usenatbib` option is specified, citations in the text should be in one of the following forms (or one of the additional forms documented within `natbib.sty` itself).
- `\citet{`*key*`}` produces text citations, e.g. Jones et al. (1990),
- `\citep{`*key*`}` produces citations in parentheses, e.g. (Jones et al. 1990),
- `\citealt{`*key*`}` produces citations with no parentheses, e.g. Jones et al. 1990.
For three-author papers, a full author list can be forced by putting a `*` just before the `{`. To add notes within the citation, use the form `\citep[`*pre\_reference\_text*`][`*post\_reference\_text*`]{`*key*`}` (note that either of *pre\_reference\_text* and *post\_reference\_text* can be blank).
Items in the reference list must be of the form\
`\bibitem[\protect\citeauthoryear{`*author\_names*`}`\
`{`*year*`}]{`*key*`}` Text of reference ...\
for one-, two- and multi-author papers, or\
`\bibitem[\protect\citeauthoryear{`*three\_author\_names*`}`\
`{`*first\_author\_etal*`}{`*year*`}]{`*key*`}` Text of reference ...\
for three-author papers.
Note that Patrick Daly’s package `natbib.sty` is required in order to use the `usenatbib` option.
We recommend that authors use `natbib.sty` as their standard cross-referencing package, because of the flexibility in citation style that it provides.
### The list of references
The following listing shows some references prepared in the style of the journal; the code produces the references at the end of this guide. The following rules apply for the ordering of your references:
1. if an author has written several papers, some with other authors, the rule is that the single-author papers precede the two-author papers, which, in turn, precede the multi-author papers;
2. within the two-author paper citations, the order is determined by the second author’s surname, regardless of date;
3. within the multi-author paper citations, the order is chronological, regardless of authors’ surnames.
<!-- -->
\begin{thebibliography}{}
\bibitem[\protect\citeauthoryear{Butcher}{1992}]{bu}
Butcher J., 1992, Copy-editing: The Cambridge
Handbook, 3rd edn. Cambridge Univ. Press,
Cambridge
\bibitem[\protect\citeauthoryear{The Chicago Manual}%
{1982}]{ch} The Chicago Manual of Style, 1982.
Univ. Chicago Press, Chicago
\bibitem[\protect\citeauthoryear{Blanco}{1991}]{bl}
Blanco P., 1991, PhD thesis, Edinburgh
University
\bibitem[\protect\citeauthoryear{Brown \& Jones}%
{1989}]{bj} Brown A. B., Jones C. D., 1989,
in Robinson E. F., Smith G. H., eds,
Proc. IAU Symp. 345, Black Dwarfs.
Kluwer, Dordrecht, p. 210
\bibitem[\protect\citeauthoryear{Edelson}{1987}]{ed}
Edelson R. A., 1987, ApJ, 313, 651
\bibitem[\protect\citeauthoryear{Knuth}{1998}]{kn}
Knuth D. E., 1998, The \TeX book. Addison-Wesley,
Reading, MA
\bibitem[\protect\citeauthoryear{Kopka \& Daly}%
{1999}]{kd} Kopka H., Daly P. W., 1999, A Guide
to \LaTeX, 3rd edn. Addison-Wesley, Harlow
\bibitem[\protect\citeauthoryear{Lamport}{1986}]{la}
Lamport L., 1986, \LaTeX: A Document
Preparation System. Addison--Wesley, New York
\bibitem[\protect\citeauthoryear{Mirabel \& Sanders}%
{1989}]{ms} Mirabel I. F., Sanders D. B., 1989,
ApJ, 340, L53
\bibitem[\protect\citeauthoryear{Misner et al.}%
{1973}]{mtw} Misner C. W., Thorne K. S.,
Wheeler J. A., 1973, Gravitation.
Freeman, San Francisco
\bibitem[\protect\citeauthoryear{Sopp \& Alexander}%
{1991}]{sa} Sopp H. M., Alexander P., 1991,
MNRAS, 251, 112
\bibitem[\protect\citeauthoryear{Stella \& Campana}%
{1991}]{sc} Stella L., Campana S., 1991, in
Treves A., Perola G. C., Stella L., eds,
Iron Line Diagnostic in X-ray Sources.
Springer--Verlag, Berlin, p. 230
\end{thebibliography}
Each entry takes the form
\bibitem[\protect\citeauthoryear{Author(s)}%
{Date}]{tag} Bibliography entry
where `Author(s)` should be the author names as they are cited in the text, `Date` is the date to be cited in the text, and `tag` is the tag that is to be used as an argument for the vatious `\cite` commands. `Bibliography entry` should be the material that is to appear in the bibliography, suitably formatted.
Please, wherever possible, supply the formatted .bbl file of your reference list rather than the ‘raw’ .bib file(s).
Appendices
----------
The appendices in this guide were generated by typing:
\appendix
\section{For authors}
:
\section{For editors}
You only need to type `\appendix` once. Thereafter, every `\section` command will generate a new appendix which will be numbered A, B, etc. Figures and tables that appear in the Appendices should be called Fig. A1, Table A1, etc.
Example of section heading with\
S[**MALL**]{} C[**APS**]{}, , , and bold\
Greek such as $\bmu^{\bkappa}$ {#headings}
=========================================
There are at least two ways of achieving this section head. The first involves the use of `\boldmath`. You could say:
\section[]{Example of section heading with\\*
S{\sevensize\bf MALL} C{\sevensize\bf APS},
\lowercase{lowercase}, \textbfit{italic},
and bold\\* Greek such as
\mbox{\boldmath{$\mu^{\kappa}$}}}
Many implementations of LaTeX do not support `\boldmath` at 9pt, so you may need to use the bold Greek characters as described in Section \[boldgreek\], and typeset the section head as follows:
\section[]{Example of section heading with\\*
S{\sevensize\bf MALL} C{\sevensize\bf APS},
\lowercase{lowercase}, \textbfit{italic},
and bold\\* Greek such as
$\bmu^{\bkappa}$}
Was produced with:
\section[]{Example of section heading with\\*
S{\sevensize MALL} C{\sevensize APS},
\lowercase{lowercase}, \textbfit{italic},
and bold\\* Greek such as
$\bmu^{\bkappa}$}
Butcher J., 1992, Copy-editing: The Cambridge Handbook, 3rd edn. Cambridge Univ. Press, Cambridge The Chicago Manual of Style, 1982. Univ. Chicago Press, Chicago Blanco P., 1991, PhD thesis, Edinburgh University Brown A. B., Jones C. D., 1989, in Robinson E. F., Smith G. H., eds, Proc. IAU Symp. 345, Black Dwarfs. Kluwer, Dordrecht, p. 210 Edelson R. A., 1987, ApJ, 313, 651 Knuth D. E., 1998, The TeXbook. Addison-Wesley, Reading, MA Kopka H., Daly P. W., 1999, A Guide to LaTeX, 3rd edn. Addison-Wesley, Harlow Lamport L., 1986, LaTeX: A Document Preparation System. Addison–Wesley, New York Mirabel I. F., Sanders D. B., 1989, ApJ, 340, L53 Misner C. W., Thorne K. S., Wheeler J. A., 1973, Gravitation. Freeman, San Francisco Sopp H. M., Alexander P., 1991, MNRAS, 251, 112 Stella L., Campana S., 1991, in Treves A., Perola G. C., Stella L., eds, Iron Line Diagnostic in X-ray Sources. Springer–Verlag, Berlin, p. 230
For authors
===========
Table \[authors\] is a list of design macros which are unique to the Monthly Notices class and style files. The list displays each macro’s name and description.
For detailed guidelines on style, authors are referred to the journal web page. Adherence to correct style from the start will obviously save time and effort later on, in terms of fewer proof corrections. The notes given on the journal web page relate to common style errors found in Monthly Notices manuscripts, and are [*not*]{} intended to be exhaustive. Please see the editorials in issues 257/2 and 260/1, as well as any recent issue of the journal, for more details.
----------------------------------------------------- -------------------------------------------------------------------
`\title[optional short title]{long title}` short title used in running head
`\author[optional short author(s)]{long author(s)}` short author(s) used in running head
`\newauthor` starts a new line in the author environment
`\begin{abstract}...\end{abstract}` for abstract on titlepage
`\begin{keywords}...\end{keywords}` for keywords on titlepage
`\nokeywords` if there are no keywords on titlepage
`\begin{figure*}...\end{figure*}` for a double spanning figure in two-column mode
`\begin{table*}...\end{table*}` for a double spanning table in two-column mode
`\plate{Opposite p.~812, MNRAS, {\bf 261}}` used with `\thispagestyle{plate}` for plate pages
`\contcaption{}` for continuation figure and table captions
`\bmath{math text}` Bold math italic / symbols.
`\textbfit{text}`, `\mathbfit{text}` Bold text italic (defined in the preamble of `mnsample.tex`).
`\textbfss{text}`, `\mathbfss{text}` Bold text sans serif (defined in the preamble of `mnsample.tex`).
----------------------------------------------------- -------------------------------------------------------------------
For editors
===========
The additional features shown in Table \[editors\] may be used for production purposes. The most commonly used of these is `\bsp`, which produces the ‘This paper $\ldots$’ statement. This should be placed at the end of the document.
------------------------ -------------------------------------------------------------------
`\pagerange{000--000}` for catchline, note use of en-rule
`\pagerange{L00--L00}` for letters option, used in catchline
`\volume{000}` volume number, for catchline
`\pubyear{0000}` publication year, for catchline
`\journal` replace the whole catchline at one go
`[doublespacing]` documentstyle option for doublespacing
`[galley]` documentstyle option for running to galley
`[landscape]` documentstyle option for landscape illustrations
`[letters]` documentstyle option, for short communications (adds L to folios)
`[onecolumn]` documentstyle option for one-column
`[referee]` documentstyle option for 12/20pt, single col, 39pc measure
`\bsp` typesets the final phrase ‘This paper has been typeset from a\
& TeX/LaTeX file prepared by the author.’
------------------------ -------------------------------------------------------------------
\#1[ @font @ @ plus .1pt 40004000 ‘=1000]{} =
Troubleshooting
===============
Authors may from time to time encounter problems with the preparation of their papers in TeX/LaTeX. The appropriate action to take will depend on the nature of the problem – the following is intended to act as a guide.
1. If a problem is with TeX/LaTeX itself, rather than with the actual macros, please refer to the appropriate handbooks for initial advice [@kd; @kn]. If the solution cannot be found after discussion with colleagues, and you suspect that the problem lies with the macros, then please contact the RAS Journal Production team at Blackwell Publishing, 23 Ainslie Place, Edinburgh EH3 6AJ, UK \[Tel: +44 (0)131 226 7232; Fax: +44 (0)131 226 3803\]. The Blackwell Publishing office may also be reached using the journal e-mail address ([email protected]). Please provide precise details of the problem (what you were trying to do – ideally, include examples of source code as well – and what exactly happened; what error message was received).
2. Problems with page make-up, particularly in the two-column mode (e.g. large spaces between paragraphs, or under headings or figures; uneven columns; figures/tables appearing out of order). Please do [*not*]{} attempt to remedy these yourself using ‘hard’ page make-up commands – the typesetters will sort out such problems during the typesetting process. (You may, if you wish, draw attention to particular problems when submitting the final version of your paper.)
3. If a required font is not available at your site, allow TeXto substitute the font and report the problem on your FTP submission form.
4. If you choose to use `\boldmath`, you may find that boldmath has not been defined locally for use with a particular size of font. If this is the case, you will get a message that reads something like:
LaTeX Warning: No \boldmath typeface in this size,
using \unboldmath on input line 44.
If you get this message, you are advised to use the alternative described in this guide for attaining bold face math italic characters, i.e. `\bmath{...}`.
\[here\]
Fixes for coding problems
-------------------------
The new versions of the class file and macros have been designed to minimize the need for user-defined macros to create special symbols. Authors are urged, wherever possible, to use the following coding rather than create their own. This will minimize the danger of author-defined macros being accidentally ‘overridden’ when the paper is typeset in 9/11 pt Times Roman (see Section \[TMth\], ‘Typesetting mathematics’, in the LaTeXauthor guide).
1. Fonts in sections and paper titles. The following are examples of styles that sometimes prove difficult to code.
### P {#p .unnumbered}
is produced by:
\title[A survey of {\rm IRAS} galaxies at
$\delta > 50\degr$]
{A survey of \textit{IRAS} galaxies at
$\bmath{\delta > 50\degr}$}
is produced by:
\title[Observations of compact H\,{\normalsize
\it II} regions]
{Observations of compact H\,{\Large\bf II}
regions}
### S {#s .unnumbered}
is produced by:
\section[]{The \textit{IRAS} data for
$\bmath{\delta > 50\degr}$
is produced by:
\section[]{H\,{\sevensize\bf II} galaxies at
$\bmath{\lowercase{z} > 1.6}$}
### S {#s-1 .unnumbered}
is produced by:
\subsection[]{The \textit{IRAS} data for
$\bmath{\delta > 50\degr}$: galaxies
at $\bmath{z > 1.5}$}
is produced by:
\subsection[]{Observations of compact
H\,{\sevensize\bf II} regions}
is produced by:
\subsubsection[]{A survey of radio galaxies for
$\delta > 50\degr$}
is produced by:
\subsubsection[]{Determination of $T_{\rm eff}$ in
compact H\,{\sevensize\it II} regions}
1. Small capitals and other unusual fonts in table and figure captions:
is produced by:
\caption{Profiles of the H$\alpha$ and
N\,{\sc iii} lines observed.}
2. Multiple author lists (to get the correct vertical spacing and wraparound on the title page of a multiple-author paper).
is produced by:
\title[The variation in the strength of low-$l$
solar p modes: 1981--2]%
{The variation in the strength of
low-$\bmath{l}$ solar
p modes: 1981--2}
\author[Y. Elsworth et al.]
{Y. Elsworth, R. Howe, G.R. Isaac, \newauthor
C.~P. McLeod, B.~A. Miller, R. New, \newauthor
C.~C. Speake and S.~J. Wheeler}
3. Ionized species (as used in the examples above). The correct style calls for the use of small capitals and a thin space after the symbol for the element: e.g. for , use the code `\mbox{H\,{\sc ii}}`. The use of the `\mbox` will stop the H and the [ii]{} being separated.
4. Lower case greek pi ($\pi$), mu ($\mu$) and partial ($\partial$). In certain circumstances, the *Monthly Notices* style calls for these to be roman \[when pi is used to denote the constant 3.1415$\ldots$, mu is used to denote ‘micro’ in a unit (e.g.$\umu$m, $\umu$Jy), and partial is a differential symbol\]. See Section \[upgreek\] for instructions.
5. Decimal degrees, arcmin, arcsec, hours, minutes and seconds. The symbol needs to be placed vertically above the decimal point. For example, the sentence
> The observations were made along position angle 1205, starting from the central coordinates $\rmn{RA}(1950)=19^{\rmn{h}}~22^{\rmn{m}}~18\fs2$, $\rmn{Dec.}~(1950)=45\degr~18\arcmin~36\farcs 4$
uses the following coding:
The observations were made along position angle
120\fdg 5, starting from the central coordinates
$\rmn{RA}(1950)=19^{\rmn{h}} 22^{\rmn{m}} 18\fs2$,
$\rmn{Dec.}~(1950)=45\degr 18\arcmin 36\farcs 4$
6. The correct coding for the prime symbol is `\arcmin`, and that for is `\arcsec`; see the two tables on special symbols. Note that these symbols should *only* be used for coordinates. The words ‘arcmin’ and ‘arcsec’ should be used for units (e.g. ‘4.5-arscec resolution’).
7. N-rules, hyphens and minus signs (see the instructions for authors on the journal web page for correct usage). To create the correct symbols in the sentence
> The high-resolution observations were made along a line at an angle of $-15\degr$ (east from north) from the axis of the jet, which runs north–south
you would use the following code:
The high-resolution observations were made along
a line at an angle of $-15\degr$ (east from north)
from the axis of the jet, which runs north--south
8. Vectors and matrices should be bold italic and bold sans serif respectively. To create the correct fonts for the vector $\bmath{x}$ and the matrix , you should use `$\bmath{x}$` and `\textbfss{P}` respectively; `\mathbfss` is for use in math mode. Bold face text italic can be obtained by using `\textbfit{..}` and `\mathbfit{..}` for math mode.
9. Bold italic superscripts and subscripts. To get these to come out in the correct font and the right size, you need to use `\bmath`. You can create the output $\bmath{k_x}$ by typing `$\bmath{k_x}$`. Try to avoid using LaTeX commands to determine script sizes that are already defined in the style file. For example, macros such as
\newcommand{\th}{^\mbox{\tiny th}}
are generating extra work;
\newcommand{\th}{^{th}}
will do, and will get the size of the superscript right whether in main text, tables or captions (the use of `\tiny` over-rides the style file). Also, the `\mbox` is not necessary, as TeX will not split a superscript/subscript from its variable at a line break.
10. Calligraphic letters (uppercase only). Normal uppercase calligraphic can be produced with `\mathcal` as normal (in math mode). Bold calligraphic can be produced with `\bmath`. e.g. `$\bmath{\mathcal A}$` gives $\bmath{\mathcal
A}$.
11. Automatic scaling of brackets. The codes `\left` and `\right` should be used to scale brackets automatically to fit the equation being set. For example, to get $$v = x \left( \frac{N+2}{N} \right)$$ use the code
\[
v = x \left( \frac{N+2}{N} \right)
\]
12. Roman font in equations. It is often necessary to make some symbols roman in an equation (e.g. units, subscripts). For example, to get the following output: $$\sigma \simeq (r/13~h^{-1}~\rmn{Mpc})^{-0.9},
\qquad \omega = \frac{N-N_{\rmn{s}}}{N_{\rmn{R}}},$$ you should use:
\[
\sigma \simeq (r/13~h^{-1}~\rmn{Mpc})^{-0.9},
\qquad \omega=\frac{N-N_{\rmn{s}}}{N_{\rmn{R}}},
\]
13. Continuation figure and table captions. See Section \[contfigtab\].
Springer-Verlag macro names
---------------------------
These have been incorporated from the Astronomy & Astrophysics LaTeXstyle file, to aid in the creation of various commonly used astronomical symbols. Please see Section \[SVsymbols\] for details.
\[lastpage\]
[^1]: http://www.blackwell-science.com/mnr/
[^2]: http://www.yandy.com/resources.htm
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Till Bargheer
- Song He
- Tristan McLoughlin
- 'Till Bargheer, Song He, and Tristan McLoughlin'
bibliography:
- 'CS.bib'
title: 'New Relations for Three-Dimensional Supersymmetric Scattering Amplitudes'
---
=
[myabstract[We provide evidence for a duality between color and kinematics in three-dimensional supersymmetric Chern–Simons matter theories. We show that the six-point amplitude in the maximally supersymmetric, ${\cal N}=8$, theory can be arranged so that the kinematic factors satisfy the fundamental identity of three-algebras. We further show that the four- and six-point ${\cal N}=8$ amplitudes can be “squared" into the amplitudes of ${\cal N}=16$ three-dimensional supergravity, thus providing evidence for a hidden three-algebra structure in the dynamics of the supergravity.]{}]{}
[keywords[Chern-Simons theory, supergravity, scattering amplitudes, three-algebra, gauge theory, supersymmetry, BCJ relations, numerators, color-kinematic duality, squaring relations]{}]{}
UUITP-06/12
AEI-2012-019
[affiliation[Department of Physics and Astronomy, Uppsala University, SE-751 08 Uppsala, Sweden]{}]{}
[affiliation[Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Potsdam, Germany]{}]{}
****
<span style="font-variant:small-caps;"></span>
*${}^a$Department of Physics and Astronomy, Uppsala University, SE-751 08 Uppsala, Sweden\
${}^b$Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Potsdam, Germany*
`[email protected], {song.he,tmclough}@aei.mpg.de`
**Abstract**
Introduction
============
Scattering amplitudes have provided a rich vein of insight into the hidden structures underlying our theories of gauge and gravitational interactions. One particularly suggestive result is the color-kinematics duality discovered by Bern, Carrasco, and Johansson (BCJ) [@Bern:2008qj]. At tree-level, color dressed scattering amplitudes in Yang–Mills (YM) theories can, quite generally, be written as a sum over cubic graphs \[eq:colordecomp\] [A]{}\_n= g\^[n-2]{} \_
[l]{} i
, where the $c_i$’s are color structures made from the usual Lie algebra structure constants, and the $n_i$’s are kinematic factors from which we have removed products of inverse propagators $p^2_{\ell_i}$ associated to internal lines of the respective cubic diagram. BCJ proposed that there exists a representation of the amplitude such that for any set of color structures related by a Jacobi identity, there is a corresponding relation between their numerator factors, i.e. \[eq:Jacobi\] c\_1 + c\_2 + c\_3 = 0 n\_1 + n\_2 + n\_3 = 0 . This duality implies non-trivial relations between different tree-level color-ordered subamplitudes, so-called BCJ relations [@Bern:2008qj].
Moreover, as is well known, via the Kawai–Lewellen–Tye (KLT) relations [@Kawai:1985xq; @Berends:1988zp], such Yang–Mills amplitudes can be used to express tree-level scattering in related gravity theories. BCJ [@Bern:2008qj; @Bern:2010ue] proposed that it is possible to express gravity amplitudes in terms of the gauge theory data by simply replacing the color factors by another copy of the kinematic numerators and summing over the same cubic diagrams.
In this note, we provide evidence for a non-trivial analogue of the color-kinematics duality in supersymmetric Chern–Simons matter (SCS) theories and for a corresponding “double-copy" construction leading to the $ E_{8(8)}$ symmetric, three-dimensional ${\cal N}=16$ supergravity.
N=8 SCS scattering amplitudes
=============================
The maximally supersymmetric Chern–Simons theory, the Bagger–Lambert–Gustavsson(BLG) theory, constructed in [@Bagger:2006sk; @Gustavsson:2007vu; @Bagger:2007jr], is the unique three-dimensional gauge theory with $OSp(8|4)$ superconformal symmetry. The on-shell, physical states comprise eight scalars, $X^I$, in the $\bf{8}_v$ of $SO(8)$ and eight fermions, $\Psi^{\dot I}$, in the $\bf{8}_c$. An important feature of the original construction was the appearance of three-algebras. Briefly, a three-algebra is a vector space, $T^a$, $a=1,
\dots, N$, with a trilinear product $$\begin{aligned}
[T^a, T^b, T^c]=f^{abc}{}_d T^d\,,\end{aligned}$$ where the structure constants $f^{abc}{}_d$ satisfy the fundamental three-algebra identity, \[eq:threealg\] f\^[efg]{}\_d f\^[abc]{}\_g=f\^[efa]{}\_g f\^[bcg]{}\_d+f\^[efb]{}\_g f\^[cag]{}\_d+f\^[efc]{}\_g f\^[abg]{}\_d . Moreover there is a trace form, $h^{ab}={\rm Tr}(T^a T^b)$, which can be used to raise and lower indices. The structure constants with all indices raised are completely anti-symmetric, $f^{abcd}=f^{[abcd]}$. All on-shell fields are three-algebra valued fundamental fields, e.g.$X^I=\sum_{a=1}^{N}(X^I)^a\,T^a$. The only known finite-dimensional example is where the three-algebra is four-dimensional, while the structure constants are proportional to the invariant four-index tensor $f^{a_1 a_2a_3 a_4}\propto {\epsilon}^{a_1 a_2a_3 a_4}$.
As we are interested in scattering amplitudes, it is convenient to make use of the spinor-helicity formalism, whereby three-momenta are expressed as the product of two-component real spinors:[^1] $p^{\alpha\beta}=\lambda^\alpha\lambda^\beta$ where $\alpha, \beta=1, 2$. The on-shell fields can be grouped into a single superfield,[^2] by introducing four Gra[ß]{}mann parameters $\gamma^i$, $i=1,\dots, 4$. This construction breaks manifest $SO(8)$ R-symmetry by rewriting the $\bf{8}_v$ scalars as $X^I=\{\bar{X}, X^{[ij]},X\}$ and similarly for the fermions, $\Psi^{\dot A}=\{\psi_i, \bar{\psi}^i\}$, so that the on-shell superfield is \_[BLG]{}&=&|[X]{}+\^[i]{}[ ]{}\_[i]{}+\_[[i]{}[j]{}[k]{}l]{} \^[i]{}\^[j]{}X\^[\[[k]{}l\]]{}\
& &+\_[[i]{}[j]{}[k]{}l]{}\^[i]{}\^[j]{}\^[k]{}[|]{}\^l+\_[[i]{}[j]{}[k]{}l]{}\^[i]{}\^[j]{}\^[k]{}\^l X . \[eqn:superfields\] $OSp(8|4)$ invariant four-point scattering amplitudes, \[eq:4ptCS\] [A]{}\_4= f\^[a\_1a\_2a\_3a\_4]{} , have previously been constructed [@Huang:2010rn]. In this formula, the delta-functions impose conservation of momenta, $P^{\alpha\beta}=\sum_{j=1}^4 p^{\alpha\beta}_j$, and supermomenta, $Q^{\alpha i }=\sum_{j=1}^4 \lambda^{\alpha}_j \gamma_j^i$, while the kinematic invariants are defined as $\langle j k\rangle={\epsilon}_{\alpha\beta}\lambda^\alpha_j\lambda^\beta_k$. The overall form of the amplitude is fixed by the superconformal symmetries, while the normalization, dependence on the Chern–Simons coupling $k$, and color structure, are fixed by explicit Feynman diagram calculation of any component amplitude.
Quite generally we can write an $n$-point amplitude in the BLG theory in the form , but with the $c_i$ corresponding to three-algebra color structures.[^3] The sum is now over diagrams with quartic vertices, and the color structures are found by associating to each vertex a factor $f^{abcd}$, and to each internal line a metric $h_{ab}$. For example, Fig. \[fig:quarticdiag\] corresponds to $c_{(123)(456)}:=f^{a_1a_2a_3 b}h_{bc}f^{c a_4a_5a_6 }$.
![Quartic diagrams[]{data-label="fig:quarticdiag"}](FigColorTree.mps){width="1.5cm" height="1.8cm"}
${{\settowidth{\apb@width}{\includegraphics[width=1.5cm, height=1.3cm]{FigColorTree1.mps}}\parbox[c]{\apb@width}{\includegraphics[width=1.5cm, height=1.3cm]{FigColorTree1.mps}}}}={{\settowidth{\apb@width}{\includegraphics[width=1.5cm,height=1.3cm]{FigColorTree2.mps}}\parbox[c]{\apb@width}{\includegraphics[width=1.5cm,height=1.3cm]{FigColorTree2.mps}}}}+
{{\settowidth{\apb@width}{\includegraphics[width=1.5cm,height=1.3cm]{FigColorTree3.mps}}\parbox[c]{\apb@width}{\includegraphics[width=1.5cm,height=1.3cm]{FigColorTree3.mps}}}}+
{{\settowidth{\apb@width}{\includegraphics[width=1.5cm,height=1.3cm]{FigColorTree4.mps}}\parbox[c]{\apb@width}{\includegraphics[width=1.5cm,height=1.3cm]{FigColorTree4.mps}}}}$
![Six point quartic diagram.[]{data-label="fig:quarticdiag"}](FigColorTree.mps)
${{\settowidth{\apb@width}{\includegraphics[]{FigColorTree1.mps}}\parbox[c]{\apb@width}{\includegraphics[]{FigColorTree1.mps}}}}={{\settowidth{\apb@width}{\includegraphics[]{FigColorTree2.mps}}\parbox[c]{\apb@width}{\includegraphics[]{FigColorTree2.mps}}}}+
{{\settowidth{\apb@width}{\includegraphics[]{FigColorTree3.mps}}\parbox[c]{\apb@width}{\includegraphics[]{FigColorTree3.mps}}}}+
{{\settowidth{\apb@width}{\includegraphics[]{FigColorTree4.mps}}\parbox[c]{\apb@width}{\includegraphics[]{FigColorTree4.mps}}}}$
A key feature is that due to the fundamental identities not all of the color structures are independent. Namely, given $c_s=\dots f^{efg}{}_d f^{abc}{}_g\dots $, $c_t=\dots f^{efa}{}_g f^{bcg}{}_d\dots $, $c_u=\dots f^{efb}{}_g f^{cag}{}_d\dots $ and $c_v=\dots
f^{efc}{}_g f^{abg}{}_d\dots$, where the “$\dots$" denote factors common to all diagrams, then $c_s=c_t+c_u+c_v$. Our first proposal is that corresponding numerators $n_s$, $n_t$, $n_u$ and $n_v$ can always be found such that (see Fig. \[fig:funiden\]) \[eq:funiden\] c\_s=c\_t+c\_u+c\_v n\_s=n\_t+n\_u+n\_v .
We do not have a general proof for these relations, instead we will provide evidence for their existence by considering the first non-trivial case, i.e. six points.
At six points, all color structures consist of the contractions of two tensors as in Fig. \[fig:quarticdiag\]. Accounting for the anti-symmetry of $f^{abcd}$, there are ten distinct color structures $c_i$, labeled by partitions of the six color labels into groups of three, e.g. $c_{1}=c_{(123)(456)}$.[^4] At six points there are five independent three-algebra relations between different color structures. Our claim is that there is a choice of numerators such that they satisfy the same three-algebra fundamental identities, however the numerators are not uniquely defined and finding explicit forms is not straightforward. Instead, we will show the existence of such numerators, and give a recipe for calculating them, by considering the color-ordered subamplitudes of ${\cal N}=6$ Aharony–Bergman–Jafferis–Maldacena (ABJM) theory [@Aharony:2008ug].
New relations for color-ordered subamplitudes
=============================================
As is well known, the BLG theory can be rewritten [@VanRaamsdonk:2008ft] as a special case ($N=2$) of the $SU(N)\times SU(N)$ ${\cal
N}=6$ Chern–Simons theories with bi-fundamental matter, that is ABJM-theories. The ABJM on-shell fields can be grouped into two superfields, $\hat \Phi^{\scriptscriptstyle A}_{\scriptscriptstyle
\bar A}$, transforming as $(N,\bar N)$, and $\hat {\bar
\Phi}^{\scriptscriptstyle\bar B}_{\scriptscriptstyle B}$, transforming as $(\bar N, N)$ [@Bargheer:2010hn]. This formalism is manifestly $U(3)$ symmetric, making use of three Gra[ß]{}mann parameters $\gamma^{\hat i}$, $\hat i=1,2,3$. For $N=2$ the conjugate representations are equivalent and the two superfields can be combined: $\Phi_{\rm BLG}={\hat \Phi}+\gamma^4 \hat{\bar \Phi}$.
Scattering amplitudes in BLG theory can be found from those of ABJM by identifying the appropriate fields and color structures. ABJM scattering amplitudes can however be decomposed into color-ordered subamplitudes. Each color-ordered subamplitude will contribute to several kinematical coefficients of the BLG color structures $c_i$. We claim that every color-ordered ABJM subamplitude can be written as a certain combination of numerators $n_i$ with propagators, in such a way that the corresponding BLG amplitudes take the form , with the numerators satisfying the three-algebra identities . This implies non-trivial relations among the color-ordered ABJM subamplitudes, and thus is a slightly stronger claim than the proposition that the BLG amplitudes decompose as with satisfied. In the following, we provide evidence for this proposal by examining the six-point amplitudes.
Four-point amplitudes in ABJM[^5] were considered in [@Agarwal:2008pu], the six-point color-ordered subamplitude for ABJM were first calculated in [@Bargheer:2010hn], see also [@Gang:2010gy]. As a representative component amplitude, we consider the six-point amplitude involving a single flavor of complex scalar $\phi(p)^{A}_{\bar A}$ and its conjugate $\bar \phi(p)^{\bar B}_{ B}$, A\_[6]{}= A(1,2,3,4,5,6)\^[[|B\_2]{}]{}\_[|A\_1]{} \^[A\_2]{}\_[B\_2]{} \^[|B\_4]{}\_[|A\_3]{} \^[A\_5]{}\_[ B\_4]{}\^[|B\_6]{}\_[|A\_5]{} \^[A\_1]{}\_[ B\_6]{}+… , \[eq:abjm6phi\] with the ellipses denoting other color orderings.
At six points, we propose that the color-ordered ABJM subamplitudes take the form
$$\begin{aligned}
A(i,j,k,p,q,r)=\frac{n_{(ijk)(pqr)}}{p_{ijk}^2}+\frac{n_{(qri)(jkp)}}{p_{qri}^2}+\frac{n_{(rij)(kpq)} }{p_{kpq}^2}~.\end{aligned}$$
There are six independent subamplitudes, all others are related to those by cyclic double-shifts and by inversions, e.g.$A(3,4,5,6,1,2)=A(1,2,3,4,5,6)$, $A(1,2,5,6,3,4)=A(1,4,3,6,5,2)$. $$\begin{aligned}
A(3,4,5,6,1,2)&=A(1,2,3,4,5,6)~,{\nonumber}\\
A(1,2,5,6,3,4)&=A(1,4,3,6,5,2)~.{\nonumber}\end{aligned}$$ If, as we claim, it is possible to choose numerators satisfying , we can solve for five of the numerators, for example by setting $$\begin{aligned}
n_9=-n_1+n_{10}+n_4~, & &~~~n_8=-n_2+n_3+n_4\\
n_7=n_1-n_3-n_4~, & &~~~n_6=n_{10}-n_2+n_4\\
&&\kern-55pt n_5 =-n_1+n_{10}+n_3+n_4~.\end{aligned}$$ We can now solve for four further numerators $n_2$, $n_3$, $n_4$ and $n_{10}$ in terms of known expressions [@Bargheer:2010hn; @Gang:2010gy] for $A(1,2,3,4,5,6)$, $A(1,2,3,6,5,4)$, $A(1,2,5,4,3,6)$, and $A(1,4,3,6,5,2)$. We thus derive identities for $A(1,4,5,2,3,6)$ and $A(1,4,3,2,5,6)$ in terms of these subamplitudes and the undetermined numerator $n_1$. The expressions for the numerators and correspondingly the identities are rather complicated, however it is straightforward to numerically check, by choosing explicit numerical values for external momenta, that they are in fact satisfied. Importantly, the undetermined kinematical factor, $n_1$, does not appear in any of these relations and so corresponds to a generalized gauge freedom analogous to that found in the YM case [@Bern:2008qj].
E8(8) supergravity theory
=========================
The three-dimensional ${\cal N}=16$ supergravity with $E_{8(8)}$ symmetry ($E_8$-theory), originally constructed by Marcus and Schwarz [@Marcus:1983hb], consists of 128 scalar bosons and 128 fermions which are in inequivalent real spinor representations of $SO(16)$, the maximal compact subgroup of $E_{8(8)}$. An immediate consequence of this, as explained in [@Marcus:1983hb], is that non-trivial scattering amplitudes must have an even number of external particles, as products of odd numbers of spinors cannot form a singlet. Consequently the S-matrix is naively different than the dimensional reduction of the four-dimensional ${\cal N}=8$ supergravity with $E_{7(7)}$ symmetry ($E_7$-theory). However, as is long known, e.g. [@Breitenlohner:1987dg], on-shell the two theories are related by performing a duality transformation, after dimensional reduction, of all the vector fields into scalars, which then combine with the scalars from dimensional reduction, including those originally in the $E_{7(7)}/SU(8)$ coset of the ${\cal N}=8$ supergravity, to become those of the $E_{8(8)}/SO(16)$ coset.
The $E_{8(8)}$ algebra comprises 120 compact $SO(16)$ generators $X^{IJ}$, $I,J=1,\dots,16$, and 128 non-compact generators $Y^A$, $A=1,\dots,
128$.
It is convenient to fix the unitary-gauge, whereby a generic group element is written as $g=e^{\varphi^A Y^A}$ with $\varphi^A$ the physical scalars. The $E_{8(8)}/SO(16)$-coset action is constructed from the algebra-valued current $P_\mu =\frac{1}{2}\left(e^{-\varphi}\partial
e^{\varphi}-e^{\varphi}\partial e^{-\varphi}\right)$. The bosonic action is [@Marcus:1983hb], $$\begin{aligned}
{\cal L}_{\rm bos}=\frac{1}{4\kappa^2} \sqrt{-g} R-\frac{1}{4\kappa^2} \sqrt{-g} g^{\mu\nu} P_\mu^A P_\nu^A,\end{aligned}$$ where the first term is the usual gravity action. Using this action (the fermionic terms are also known), with appropriate gauge fixing, one can straightforwardly calculate scattering amplitudes using Feynman diagrams. At four-points such amplitudes for four scalars receive contributions from graviton exchange and from contact interactions that arise upon expanding the coset term to quartic order in fields, ${\cal L}_{\varphi^4}\sim (\varphi
\Gamma^{IJ}\partial_\mu \varphi)( \varphi \Gamma^{IJ}\partial_\mu \varphi)$. In the simplest case we can consider the scattering of four scalars all carrying the same coset index, e.g. all fields being $\varphi^1$, in which case there is no contribution from contact terms. Combining all graviton exchange diagrams we find M\_4=( ++) . It is not difficult to calculate other component amplitudes, however we can make use of the supersymmetry to determine the full four-point superamplitude.
For the $E_8$-theory we can define an on-shell superfield by using eight Gra[ß]{}mann parameters $\eta^I$, $I=1,\dots, 8$ which breaks the $SO(16)$ R-symmetry to $U(8)$. Splitting the 128 scalars $\varphi^A$ into the fields $\{\xi, {\bar \xi}, \xi_{IJ},{\bar \xi}^{IJ},\xi_{IJKL}\}$ with, for example $\xi=\tfrac{1}{2}(\varphi^1+i \varphi^2)$, and similarly for the fermionic fields, we can write the superfield[^6] = +\^I \_I+\^I\^J \_[IJ]{}+…+\^8[|]{} .
By using super-Poincaré symmetry and matching to the component amplitude, the four-point superamplitude is \[eq:E8fourpt\] [M]{}\_4 = . Here, the 16-dimensional fermionic delta-function is given by the product of two eight-dimensional fermionic delta-functions, $\delta^{(16)}(Q)\sim
\delta^{(8)}(Q^1)\delta^{(8)}( Q^2)$, such as appeared in . Stripping off the overall normalization and momentum delta-function we see that this is the “square" of . This then suggests an analogue of the KLT relation [@Kawai:1985xq; @Berends:1988zp] between ${\cal N}=4$ supersymmetric Yang–Mills (SYM) and the $E_7$ supergravity theory to one between ${\cal N}=8$ BLG and the $E_8$-theory. As zeroth order checks, we note that the spectra of the $E_8$-theory and that of the BLG theory squared match; furthermore in both cases all non-trivial amplitudes have even numbers of legs. Of course the direct dimensional reduction of ${\cal N}=4$ SYM and the $E_7$-theory amplitudes to three dimensions are related by the usual KLT relations, and for fields which are unchanged by the duality transformation, in particular the scalars originating in the $E_{7(7)}/SU(8)$ coset, the three-dimensional scattering amplitudes are just those of the four-dimensional theory evaluated on three-dimensional kinematics. However, after the duality transformation to the $E_8$-theory, this ceases to be the case for all amplitudes; as a simple example there is no $E_8$-theory three-point amplitude corresponding to the square of the three-dimensional SYM three-point amplitude. As ${\cal N}=8$ BLG theory can be found from supersymmetric three-dimensional Yang–Mills [@Mukhi:2008ux] via a “Higgsing"-procedure reminiscent of the duality transformation, it is perhaps not surprising that it should be thus related to the $E_8$ supergravity theory.
Three-dimensional gravity as the square of Chern–Simons
=======================================================
Given the suggestion that the BLG amplitudes can be written in terms of numerators satisfying the three-algebra color-kinematics duality, it is natural to ask if the gravity theory amplitudes can be written as a “double-copy" as in [@Bern:2008qj], \[eq:gravsquare\] M\_n=i ()\^[n-2]{}\_
[l]{} i
, where the $n_i$’s are the numerators appearing in the BLG amplitude and the sum is over the same $n$-point quartic diagrams. This relation obviously holds at four points for the superamplitudes, and at six points we can perform an explicit check by making use of the numerators calculated from the six-point color-ordered ABJM subamplitudes for specific components. For example, the pure scalar ABJM amplitude $\hat A_{6\phi}$ can be used to calculate the numerators for the $A_6(X_1 {\bar X}_2X_3 {\bar X}_4X_5 {\bar X}_6)$ in the BLG theory, which potentially squares into the $M_6(\xi_1 {\bar \xi}_2 \xi_3 {\bar \xi}_4\xi_5 {\bar \xi}_6)$ gravity amplitude. That this is indeed the case can in principle be shown by comparing with the result of a direct Feynman diagram calculation. Equivalently, but significantly more efficiently, one can take this complex scalar to have originated in the $E_{7(7)}/SU(8)$ coset, so that the squared amplitude can be compared with the dimensional reduction of the six-scalar $E_7$ supergravity amplitude. The latter can be found from a scalar component of ${\cal N}=4$ SYM Next-to-Maximally-Helicity-Violated (NMHV) amplitude, conveniently written as a sum of the so-called R-invariants, multiplied by an MHV pre-factor [@Drummond:2008vq], \^[NMHV]{}\_6=[A]{}\^[MHV]{}\_6\_[3s+1 < t 5]{}R\_[6st]{} and by making use of the KLT relations [@Kawai:1985xq; @Berends:1988zp]. It is then straightforward to check, again by choosing a range of numerical values for external momenta, that the resulting pure scalar amplitude in fact agrees with the squared BLG amplitude .
For higher-point amplitudes it would be possible to prove, along the lines of [@Bern:2010yg], that holds if there were Britto–Cachazo–Feng–Witten (BCFW) recursion relations [@Britto:2004ap; @Britto:2005fq] for the $E_8$-theory. Recursion relations for ABJM theories, and thus BLG theories, have been proven in [@Gang:2010gy]. The key step is proving that the superamplitude falls off sufficiently fast for large deformations of the momenta under a complex non-linear shift: $ \hat {\cal A}(\{\lambda_1(z), \lambda_l(z)\}) \sim{\cal
O}\left(1/z\right) $ as $z\rightarrow \infty$ with $\lambda_1(z)=\tfrac{z+z^{-1}}{2} \lambda_1-\tfrac{z-z^{-1}}{2i } \lambda_l$, $\lambda_l(z)=\tfrac{z-z^{-1}}{2i} \lambda_1+\tfrac{z+z^{-1}}{2 } \lambda_l$ and similar shifts for the Gra[ß]{}mann parameters. The proof of a sufficient fall-off for $E_8$ superamplitudes does not currently exist. However, it is possible to naively apply the method of [@Gang:2010gy] and use the four-point amplitude to construct a candidate six-point superamplitude in $E_8$ supergravity. We find that the relevant scalar component, $M_6(\xi_1 {\bar \xi}_2 \xi_3 {\bar \xi}_4\xi_5 {\bar \xi}_6)$, of this superamplitude agrees with the amplitude calculated by squaring the numerators . This shows that at least to six points the BCFW recursion relations of [@Gang:2010gy] hold for the $E_8$-theory.
Outlook
=======
In order to confirm the proposed “double-copy" relations for the $E_8$-theory, it would be very useful to prove in general the BCFW relations for the three-dimensional supergravity. Relatedly, numerator identities for YM and squaring relations for gravity have been conjectured to extend to all-loop diagrams [@Bern:2010ue], and it would be interesting to check whether similar relations hold for the three-dimensional Chern–Simons and gravity theories beyond tree-level. If this does indeed hold it would demonstrate the existence of a hidden three-algebra structure in three-dimensional gravity. This is interesting as a non-trivial model for similar structures in four dimensional gravity, particularly with regard to knotty issues of quantum gravity, and as an important intermediary step to two-dimensions where gravity is known to posses infinite dimensional symmetries [@Geroch:1972yt; @Julia:1980gr].
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank N. Beisert, G. Bossard, F. Loebbert, R. Monteiro, H. Nicolai, D. O’Connell, and R. Roiban for helpful discussions. We would further like to thank the organizers of the Nordita program on Exact Results in Gauge-String Dualities, where parts of this work were presented, for their hospitality. Finally, TMcL would like to thank CPTh, Ecole Polytechnique for their hospitality during the completion of this work.
[^1]: In fact, only particles with positive energy correspond to real spinors. For negative energies, $\lambda$ is taken to be purely imaginary.
[^2]: Our construction closely parallels the oscillator construction of the $OSp(8|4)$ algebra [@Gunaydin:1985tc] and so is only $U(4|2)$ covariant, corresponding to the Jordan decomposition of $OSp(8|4)$ with respect to a $U(1)\in
U(4|2)$. An equivalent on-shell superfield formulation of BLG was constructed in [@Huang:2010rn].
[^3]: Note that the gauge field is non-dynamical and thus only fundamental matter fields appear as external states.
[^4]: To be explicit we will also choose $c_2=c_{(156)(234)}$, $c_3=c_{(612)(345)}$, $c_4=c_{(125)(436)}$, $c_5=c_{(136)(245)}$, $c_6=c_{(145)(236)}$, $c_7=c_{(124)(356)}$, $c_8=c_{(143)(256)}$, $c_9=c_{(146)(235)}$, $c_{10}=c_{(135)(246)}$. We use the same notation for labeling the numerators.
[^5]: Actually of a one parameter family of mass deformed theories.
[^6]: This superfield is very similar to that of the $E_7$-theory and indeed making the formal identification $\xi=h$, $\bar \xi=\bar h$, $\xi_{IJ}=B_{IJ}$, $\bar \xi^{IJ}={\bar B}^{IJ}$, $ \xi_{IJKL}=D_{IJKL}$ to the fields of the $E_7$-theory this becomes more apparent.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Shujaat Khan
- Jaeyoung Huh
- Jong Chul Ye
title: |
— Supplemental Document —\
Deep Learning-based Universal Beamformer for Ultrasound Imaging[^1]
---
Experimental Results {#sec:results}
====================
### Focused mode imaging results on phantom dataset
Fig \[fig:results\_view\_focused\] shows the reconstruction results on phantom dataset. We also compared the CNR, GCNR, PSNR, and SSIM distributions of reconstructed B-mode images obtained from $159$ *phantom* test frames. Table \[tbl:results\_vSTATS\_phantom\] showed that the proposed deep beamformer consistently outperformed the standard DAS beamformer for all subsampling schemes and ratios.
### Comparison with deep RF interpolation [@yoon2018efficient]
In [@yoon2018efficient], deep learning approach was designed for interpolating missing RF data, which are later used as input for standard beamformer (BF). On the other hand, the proposed method is an end-to-end CNN-based beamforming pipeline, without requiring additional BF. Consequently, our approach is much simpler and can be easily incorporated to replace the standard beamforming pipeline.
To verify that the proposed DeepBF still outperforms the deep RF interpolation [@yoon2018efficient], we also compared the CNR, GCNR, PSNR and SSIM distributions of reconstructed B-mode images obtained from $360$ *in-vivo* test frames. Fig. \[fig:results\_STATS\_Efficient\] show distribution of aforementioned statistics on 4$\times$ sub-sampled *in-vivo* dataset. From results it can be easily seen that the proposed deep beamformer consistently outperformed the standard DAS beamformer and the Deep RF Interpolation [@yoon2018efficient]. Specifically, note that on the in-vivo test dataset, the proposed network also outperform the Deep RF Interpolation [@yoon2018efficient], by $0.07$ and $0.016$ units in CNR and GCNR, respectively. Whereas, in PSNR and SSIM the proposed method achieved $1.40$ dB, $0.05$ units improvement respectively. Fig \[fig:results\_view\_Efficient\], shows the reconstruction results on 4$\times$ sub-sampled *in-vivo* data using conventional DAS, Deep RF Interpolation [@yoon2018efficient] and the proposed DeepBF. In short, the novelty of this work is the end-to-end deep learning to replace the standard BF, which was never considered in [@yoon2018efficient].
![Quantitative comparison using invivo data on 4$\times$ subsampling scheme: ([first column]{}) CNR value distribution, ([second column]{}) GCNR value distribution, ([third column]{}) PSNR value distribution, ([fourth column]{}) SSIM value distribution. []{data-label="fig:results_STATS_Efficient"}](invivo_sc0rx4_cnr_multi.png "fig:"){width="4cm"}![Quantitative comparison using invivo data on 4$\times$ subsampling scheme: ([first column]{}) CNR value distribution, ([second column]{}) GCNR value distribution, ([third column]{}) PSNR value distribution, ([fourth column]{}) SSIM value distribution. []{data-label="fig:results_STATS_Efficient"}](invivo_sc0rx4_gcnr_multi.png "fig:"){width="4cm"}![Quantitative comparison using invivo data on 4$\times$ subsampling scheme: ([first column]{}) CNR value distribution, ([second column]{}) GCNR value distribution, ([third column]{}) PSNR value distribution, ([fourth column]{}) SSIM value distribution. []{data-label="fig:results_STATS_Efficient"}](invivo_sc0rx4_psnr_multi.png "fig:"){width="4cm"}![Quantitative comparison using invivo data on 4$\times$ subsampling scheme: ([first column]{}) CNR value distribution, ([second column]{}) GCNR value distribution, ([third column]{}) PSNR value distribution, ([fourth column]{}) SSIM value distribution. []{data-label="fig:results_STATS_Efficient"}](invivo_sc0rx4_ssim_multi.png "fig:"){width="4cm"}
[^1]: This work is supported by National Research Foundation of Korea, Grant Number: NRF-2016R1A2B3008104.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We are interested in the comparison of transcript boundaries from cells which originated in different environments. The goal is to assess whether this phenomenon, called alternative splicing, is used to modify the transcription of the genome in response to stress factors. We address this question by comparing the change-points locations in the individual segmentation of each profile, which correspond to the RNA-Seq data for a gene in one growth condition. This requires the ability to evaluate the uncertainty of the change-point positions, and the work of [@rigaill_exact_2011] provides an appropriate framework in such case. Building on their approach, we propose two methods for the comparison of change-points, and illustrate our results on a dataset from the yeast specie. We show that the UTR boundaries are subject to alternative splicing, while the intron boundaries are conserved in all profiles. Our approach is implemented in an R package called EBS which is available on the CRAN.'
address:
- |
AgroParisTech, UMR 518, and\
16 rue Claude Bernard,\
75005 Paris, France.\
- |
INRA, UMR 518,\
16 rue Claude Bernard,\
75005 Paris, France.\
author:
-
-
bibliography:
- 'Biblio.bib'
title: 'Comparing change-point locations of independent profiles with application to gene annotation'
---
,
Introduction
============
Segmentation problems arise in a large range of domains such as economy, biology or meteorology, to name a few. Many methods have been developed and proposed in the literature in the last decades to detect change-points in the distribution of the signal along one single series. Yet, more and more applications require the analysis of several series at a time to better understand a complex underlying phenomenon. Such situations refer for example to the analysis of the genomic profiles of a cohort of patients [@picard2011joint], of meteorological series observed in different locations [@ehsanzadeh2011simultaneous] or of sets astronomical series of photons abundance [@dobigeon2007joint].
When dealing with multiple series, two approaches can be typically considered. The first consists in the [*simultaneous*]{} segmentation of all series, looking for changes that are common to all of them. This approach amounts to the segmentation of one single multivariate series but might permit the detection of change-points in series with too low a signal to allow their analysis independently. The second approach consists in the [*joint*]{} segmentation of all the series, each having its specific number and location of changes. This allows to account for dependence between the series without imposing that the changes occur simultaneously.
We are interested here in a third kind of statistical problem, which is the comparison of change-point locations in several series that have been segmented separately. To our knowledge, this problem has not yet been fully addressed.
Indeed, comparing change-point is connected to the evaluation of the uncertainty of the change-point positions. An important point is that the standard likelihood-based inference is very intricate, since the required regularity conditions for the change-point parameters are not satisfied [@Fed75]. Most methods to obtain change-point confidence intervals are based on their limit distribution estimators [@Fed75; @BaP03] or the asymptotic use of a likelihood-ratio statistic [@Mug03]. Bootstrap techniques have also been proposed (see [@HuK08] and references therein). Comparison studies of some of these methods can be found in [@bib_climate] for climate applications or in [@ToL03] for ecology. Recently, [@rigaill_exact_2011] proposed a Bayesian framework to derive the posterior distributions of various quantities of interest – including change-point locations – in the context of exponential family distributions with conjugate prior.
[As for the comparison of change-points, the most common approaches rely on classification comparison techniques such as the Rand Index [@rand1971objective]; and aim at assessing the performances of segmentation methods on single datasets, by comparing their outputs between themselves or using the truth as reference. The notion of change-point location difference as a quantity of interest has, to our knowledge, never been considered. ]{}
Our work is a generalization of [@rigaill_exact_2011] to the comparison of change point location. It is motivated by a biological problem detailed in the next paragraph.
#### Differential splicing in yeast
Differential splicing is one of the mechanism that living cells use to modify the transcription of their genome in response to some change in their environment, such as a stress. More precisely, differential splicing refers to the ability for the cell to choose between versions (called isoforms) of a given gene by changing the boundaries of the regions to be transcribed.
New sequencing technologies, including RNA-Seq experiments, give access to a measure of the transcription at the nucleotide resolution. The signal provided by RNA-Seq consists in a count (corresponding to a number of reads) associated to each nucleotide along the genome. This count is proportional to the transcription level of the nucleotide. This technology therefore allows to locate precisely the boundaries of the transcribed regions, to possibly revise the known annotation of the genomes and to study the variation of these boundaries across conditions.
We are interested here in an RNA-Seq experiment made on a given specie, yeast, grown under several conditions. The biological question to be addressed is ’Does yeast use differential splicing of a given gene as a response to a change in its environment?’.
#### Contribution
In this paper we develop a Bayesian approach to compare the change-point location of independent series corresponding to the same gene [under several conditions]{}. We suppose that we have information on the structure of this gene (such as the number of introns) so that the number of segments of each segmentation is assumed to be known. In Section \[model\], we recall the Bayesian segmentation model introduced in [@rigaill_exact_2011] and its adaptation to our framework. In Section \[sec:cred\] we derive the posterior distribution of the shift between the change-point locations in two independent profiles, while in Section \[method\] we [introduce the calculation of the posterior probability for change-points to share the same location in different series. The]{} performances are assessed in Section \[simuls\] via a simulation study designed to mimic real RNA-Seq data. We finally apply the proposed methodology to study the existence of differential splicing in yeast in Section \[appli\]. Our approach is implemented in an R package `EBS` which is available on the CRAN repository.
All the results we provide are given conditional on the number of segments in each profiles. Indeed comparing the location of, say, the second change-points in each series implicitly refers to a total number of change-points in each of them. Yet, most of the results we provide can be marginalized over the number of segments.
Model for one series {#model}
====================
In this section we introduce the general Bayesian framework for the segmentation of one series and recall preceding results on the posterior distribution of change-points.
Bayesian framework for one series
---------------------------------
The general segmentation problem consists in partitioning a signal of $n$ data-points $\{y_t\}_{t \in [\![1, n]\!]}$ into $K$ segments. The model is defined as follows: the observed data $\{y_t\}_{t=1,\ldots,n}$ are supposed to be a realization of an independent random process $Y=\{Y_t\}_{t=1,\ldots,n}$. This process is drawn from a probability distribution $\mathcal{G}$ which depends on a set of parameters among which one parameter $\theta$ is assumed to be affected by $K-1$ abrupt changes, called change-points and denoted $\tau_k$ ($1 \leq k \leq K-1$). A partition $m$ is defined as a set of change-points: $m = (\tau_0,\tau_1,\dots,\tau_{K})$ with conventions $\tau_0=1$ and $\tau_{K}=n+1$ and a segment $J$ is said to belong to $m$ if $J = [\![\tau_{k-1};\tau_{k}[\![$ for some $k$.
The Bayesian model is fully specified with the following distributions:
- the prior distribution of the number of segments $P(K)$;
- the conditional distribution of partition $m$ given $K$: $P(m|K)$;
- the parameters $\theta_J$ for each segment $J$ are supposed to be independent with same distribution $P(\theta_J)$;
- the observed data $Y = (Y_t)$ data are independent conditional on $m$ and $(\theta_J)$ with distribution depending on the segment: $$(Y_t | m, J\in m, \theta_J, t\in J) \sim \mathcal{G}(\theta_J,\phi)$$ where $\phi$ is some parameter that is constant across the segments that will be supposed to be known.
Exact calculation of posterior distributions {#distributions}
--------------------------------------------
@rigaill_exact_2011 show that if distribution $\mathcal{G}$ possesses conjugate priors for $\theta_J$, and if the model satisfies the factorability assumption, that is, if $$\begin{aligned}
\label{Eq:PYm}
P(Y,m) & = & C \prod_{J \in m} a_J P(Y_J|J), \nonumber \\
\text{where} \qquad
P(Y_J|J) & = & \int P(Y_J|\theta_J)P(\theta_J)d\theta_J, \end{aligned}$$ quantities such that $P(Y,K)$, posterior change-point location distributions or the posterior entropy can be computed exactly and in a quadratic time. Examples of satisfying distributions are
- the Gaussian heteroscedastic: $$\mathcal{G}(\theta_J,\phi) =\mathcal{N}(\mu_J,\sigma^2_J) \; \text{with} \; \theta_J=(\mu_J, \sigma^2_J), \ \phi=\emptyset ,$$
- the Gaussian homoscedastic with known variance $\sigma^2$: $$\mathcal{G}(\theta_J,\phi) =\mathcal{N}(\mu_J,\sigma^2)\; \text{with} \; \theta_J=\mu_J, \ \phi=\sigma^2,$$
- the Poisson: $$\mathcal{G}(\theta_J,\phi) =\mathcal{P}(\lambda_J)\; \text{with} \; \theta_J=\lambda_J, \ \phi=\emptyset,$$
- or the negative binomial homoscedastic with known dispersion $\phi$: $$\mathcal{G}(\theta_J,\phi) =\mathcal{NB}(p_J,\phi)\; \text{with} \; \theta_J=p_J, \ \phi=\phi.$$
Note that the Gaussian homoscedastic does not satisfy the factoriability assumption if $\sigma$ is unknown, and that the negative binomial heteroscedastic does not belong to the exponential family and does not have a conjugate prior on $\phi$.\
The factorability assumption also induces some constraint on the distribution of the segmentation $P(m|K)$. In this paper, we will limit ourselves to the uniform prior: $$P(m|K) = \mathcal U\left(\mathcal M_{K}^{1,n+1}\right)$$ where ${{\mathcal M}}_{K}^{1,n+1}$ stands for the set of all possible partitions of $[\![1,n+1[\![$ into $K$ non-empty segments.
Posterior distribution of the shift {#sec:cred}
===================================
The framework described above allows to compute a set of quantities of interest in an exact manner. In this paper, we are mostly interested in the location of change-points. We first remind how posterior distributions can be computed and then propose a first exact comparison strategy.
Posterior distribution of the change-points
-------------------------------------------
The key ingredient for most of the calculations is the $(n+1) \times (n+1)$ matrix $A$ that contains the probabilities of all segments: $$\begin{aligned}
\label{eq:matrixA}
\forall 1\leq i < j \leq n+1, \qquad [A]_{i,j}= P(Y_{[\![i,j[\![}|[\![i,j[\![)\end{aligned}$$ where $P(Y_J|J)$ is given in .
The posterior distribution of change-points can be deduced from this matrix in a quadratic time with the following proposition:
\[Prop:PostTauk\] Denoting $p_k(t;Y; K) = P(\tau_{k}=t|Y,K)$ the posterior distribution of the ${k}$[th]{} change-point, we have $$p_k(t;Y;K) = \dfrac{\left[(A)^{k}\right]_{1,t}\left[(A)^{K-k}\right]_{t,n+1}}{\left[(A)^{K}\right]_{1,n+1}}.$$
[[*Proof.*]{}]{} We have $$p_k(t;Y;K) =\frac{\sum_{m\in \mathcal{B}_{K,k}(t)}p(Y|m)p(m|K)}{P(Y|K)}$$ where $\mathcal{B}_{K,k}(t)$ is the set of partitions of $\{1,\dots,n\}$ in $K$ segments with $k$th change-point at location $t$. [Note that $\mathcal{B}_{K,k}(t)=\mathcal{M}_k^{1,t}\otimes\mathcal{M}_{K-k}^{t,n+1}$ (i.e. all $m \in \mathcal{B}_{K,k}(t) $ can be decomposed uniquely as $m=m_1\cup m_2$ with $m_1 \in \mathcal{M}_k^{1,t}$ and $m_2 \in \mathcal{M}_{K-k}^{t,n+1}$ and reciprocally).]{} Then using the factoriability assumption, we can write $$p_k(t;Y;K) =\dfrac{\sum_{m_1\in \mathcal{M}_k^{1,t}}p(Y|m_1)\sum_{m_2\in \mathcal{M}_{K-k}^{t,n+1}}p(Y|m_2)\; p(m|K)}{\sum_{m \in {{\mathcal M}}_{K}^{1,n+1}}p(Y|m)\; p(m|K)}$$ [$\square$\
]{}
Comparison of two series
------------------------
We now propose a first procedure to compare the location of two change-points in two independent series. Consider two independent series $Y^1$ and $Y^2$ with same length $n$ and respective number of segments $K^1$ and $K^2$. The aim is to compare the locations of the $k_1$th change-point from of series $Y^1$ (denoted $\tau_{k_{1}}^{1}$) with the $k_2$th change-point of series $Y^2$ (denoted $\tau_{k_{2}}^{2}$). The posterior distribution of the difference between the location of the two change-points can be derived with the following Proposition.
\[Prop:PostDelta\] Denoting $\delta_{k_1, k_2}(d; K^1, K^2) = P(\Delta=d|Y^1, Y^2, K^1, K^2)$ the posterior distribution of the difference $\Delta = \tau_{k_{1}}^{1}-\tau_{k_{2}}^2$, we have $$\delta_{k_1, k_2}(d; K^1, K^2) =\sum_t p_{k_1}(t;Y^1; K^1) p_{k_2}(t-d ;Y^2; K^2).$$
[[*Proof.*]{}]{}This simply results from the convolution between the two posterior distributions $p_{k_1}$ and $ p_{k_2}$. [$\square$\
]{}
The posterior distribution of the shift can therefore be computed exactly and in a quadratic time. The non-difference between the two change-point locations $\tau_{k_{1}}^{1}$ and $\tau_{k_{2}}^{2}$ can then be assessed, looking at the position of 0 with respect to the posterior distribution $\delta$.
Comparison of change point locations {#method}
====================================
We now consider the comparison of change-point locations between more than 2 series. In this case, the convolution methods described above does not apply anymore so we propose a comparison based on the exact computation of the posterior probability for the change-points under study to have the same location.
Model for $I$ series
--------------------
We now consider $I$ independent series $Y^\ell$ (with $1\leq \ell \leq I$) with same length $n$. We denote $m^\ell$, their respective partitions and $K^\ell$ their respective number of segments. We further denote $\tau_k^\ell$ the $k$[th]{} change-point in $Y^\ell$ so $m^\ell = (\tau_0^\ell, \tau_1^\ell, \dots, \tau_{K^\ell}^\ell)$. Similarly, $\theta_J^\ell$ denotes the parameter for the series $\ell$ within segment $J$ provided that $J \in m^\ell$ and $\phi^\ell$ the constant parameter of series $\ell$. In the following, the set of profiles will be referred to as ${{\bf Y}}$ and respectively for the vector of segment numbers (${{\mathbf K}}$), the set of all partitions (${{\bf m}}$) and the set of all parameters (${\text{\mathversion{bold}{$\theta$}}}$).
In the perspective of change-point comparison, we introduce the following event: $${{E}_0}= \{\tau_{k_1}^{1}=\dots=\tau_{k_I}^{I}\}.$$ We further denote ${{E}_1}$ its complementary and define the binary random variable $${{E}}= {\mathbb{I}}\{{{E}_1}\} = 1 - {\mathbb{I}}\{{{E}_0}\}.$$ The complete hierarchical model is displayed in Figure \[Fig:GraphModel\] and is defined as follows:
- The random variable ${{E}}$ is drawn conditionally on ${{\mathbf K}}$ as a Bernoulli $\mathcal B(1- p_0({{\mathbf K}}))$ where $p_0({{\mathbf K}}) = P({{E}_0}| {{\mathbf K}})$;
- The parameters ${\text{\mathversion{bold}{$\theta$}}}$ are drawn independently according to $P({\text{\mathversion{bold}{$\theta$}}}| {{\mathbf K}})$;
- The partitions are drawn conditionally on ${{E}}$ according to $P({{\bf m}}| {{\mathbf K}}, {{E}})$;
- The observations are generated according to the conditional distribution $P({{\bf Y}}| {{\bf m}}, {\text{\mathversion{bold}{$\theta$}}})$.
More specifically, denoting ${{\mathcal M}}_{{{\mathbf K}}}^{1,n+1} = \bigotimes_\ell {{\mathcal M}}_{K^\ell}^{1,n+1}$, the partitions are assumed to be uniformly distributed, conditional on $E$, that is $$P({{\bf m}}| {{\mathbf K}}, {{E}_0}) = {{\mathcal U}}({{\mathcal M}}_{{{\mathbf K}}}^{1,n+1} \cap {{E}_0}),
\qquad
P({{\bf m}}| {{\mathbf K}}, {{E}_1}) = {{\mathcal U}}({{\mathcal M}}_{{{\mathbf K}}}^{1,n+1} \cap {{E}_1}).$$
![[**Graphical model.**]{} Hierarchical model for the comparison of $I$ series.[]{data-label="Fig:GraphModel"}](GraphModel.png){width=".2\textwidth"}
Posterior probability for the existence of a common change-point
----------------------------------------------------------------
We propose to assess the existence of a common change-point location between the $I$ profiles based on the posterior probability of this event, namely $P({{E}_0}| {{\bf Y}}, {{\mathbf K}}).$
\[Prop:PosteriorE0\] The posterior probability of ${{E}_0}$ can be computed in $O(Kn^2)$ as $$\begin{aligned}
P({{E}_0}| {{\bf Y}}, {{\mathbf K}}) & = & \frac{p_0({{\mathbf K}})}{q_0({{\mathbf K}})} Q({{\bf Y}}, {{E}_0}| {{\mathbf K}})\; . \\
&& \left[ \frac{1 - p_0({{\mathbf K}})}{1 - q_0({{\mathbf K}})} Q({{\bf Y}}| {{\mathbf K}}) + \frac{p_0({{\mathbf K}}) - q_0({{\mathbf K}})}{q_0({{\mathbf K}})[1 - q_0({{\mathbf K}})]} Q({{\bf Y}}, {{E}_0}| {{\mathbf K}}) \right]^{-1}
\end{aligned}$$ where $$\begin{aligned}
Q({{\bf Y}}| {{\mathbf K}}) & = & \prod_\ell \left[(A_\ell)^{K_\ell}\right]_{1, n+1}, \\
Q({{\bf Y}}, {{E}_0}| {{\mathbf K}}) & = & \sum_t \prod_\ell \left[(A_\ell)^{k_\ell}\right]_{1, t} \left[(A_\ell)^{K_\ell-k_\ell}\right]_{t+1, n+1},\\
\text{and } q_0({{\mathbf K}}) = Q({{E}_0}|{{\mathbf K}}) &=& \sum_t \prod_\ell \begin{small} {\left(\begin{array}{c} t-2 \\ k_\ell-1 \end{array} \right)} {\left(\begin{array}{c} n-t \\ K_\ell-k_\ell-1 \end{array} \right)} \left/ {\left(\begin{array}{c} n-1 \\ K_\ell-1 \end{array} \right)} \right. \end{small} .
\end{aligned}$$ and $A_\ell$ stands for the matrix $A$ as defined in (\[eq:matrixA\]), corresponding to series $\ell$.
[[*Proof.*]{}]{}We consider the surrogate model where the partition ${{\bf m}}$ is drawn uniformly and independently from ${{E}}$, namely $Q({{\bf m}}| {{\mathbf K}}) = {{\mathcal U}}({{\mathcal M}}_{{{\mathbf K}}}^{1,n+1})$ (note that this corresponds to choosing $p_0({{\mathbf K}}) = q_0({{\mathbf K}})$). All probability distributions under this model are denoted by $Q$ along the proof. The formulas for probabilities $Q({{\bf Y}}| {{\mathbf K}})$ and $Q({{\bf Y}}, {{E}_0}| {{\mathbf K}})$ derive from [@rigaill_exact_2011]. It then suffices to apply the probability change as $$P({{\bf Y}}, {{E}_0}| {{\mathbf K}}) = \frac{p_0({{\mathbf K}})}{q_0({{\mathbf K}})} Q({{\bf Y}}, {{E}_0}| {{\mathbf K}}),
\quad
P({{\bf Y}}, {{E}_1}|{{\mathbf K}}) = \frac{1 - p_0({{\mathbf K}})}{1 - q_0({{\mathbf K}})} Q({{\bf Y}}, {{E}_1}|{{\mathbf K}}).$$ The result then follows from the decomposition of $P({{\bf Y}}| {{\mathbf K}})$ as $P({{\bf Y}}, {{E}_0}|{{\mathbf K}}) + P({{\bf Y}}, {{E}_1}|{{\mathbf K}})$ and the same for $Q({{\bf Y}}| {{\mathbf K}})$. [$\square$\
]{}
The Bayes factor is sometimes preferred for model comparison; it can be computed exactly in a similar way:
The Bayes factor can be computed in $O(Kn^2)$ as $$\begin{aligned}
\frac{P({{\bf Y}}| {{E}_0}, {{\mathbf K}})}{P({{\bf Y}}|{{E}_1}, {{\mathbf K}})} & = & \frac{1 - q_0({{\mathbf K}})}{q_0({{\mathbf K}})} \; \frac{Q({{\bf Y}}, {{E}_0}| {{\mathbf K}})}{Q({{\bf Y}}| {{\mathbf K}}) - Q({{\bf Y}}, {{E}_0}| {{\mathbf K}})}
\end{aligned}$$ using the same notations as in Proposition \[Prop:PosteriorE0\].
[[*Proof.*]{}]{}The proof follows this of Proposition \[Prop:PosteriorE0\]. [$\square$\
]{}
Simulation study {#simuls}
================
Simulation design {#description}
-----------------
We designed a simulation study to identify the influence of various parameters on the performances of our approach. The design is illustrated in Figure \[plansimu\]: we compared $3$ independent profiles with $7$ segments, with all odd (respectively even) segments sharing the same distribution. The first two profiles have identical segmentation $m$ given by $m=(1,101,201,301,401,501,601,701)$ and the change-point locations of the third one are progressively shifted apart as $\tau_{k}^3=\tau_{k}^1+2^{k-1}$, for each $1\leq k \leq 6$. We shall denote $d_k=\tau_k^3-\tau_k^1$ and drop the index $k$ when there is no ambiguity on it.
Our purpose is to mimic data obtained by RNA-Seq experiments, so that the parameters for the negative binomial distribution were chosen to fit typical real-data. Considering the model where odd segments are sampled with distribution ${{\mathcal NB}}(p_0,\phi)$, and even with ${{\mathcal NB}}(p_1,\phi)$, we chose two different values of $p_0$, $0.8$ and $0.5$, and for each of them, we made $p_1$ vary so that the odd-ratio $s := p_1 /(1-p_1) / [p_0 / (1-p_0)]$ is $4$, $8$ and $16$. Finally, we used different values of $\phi$ as detailed in Table \[paramvalue\] in order to explore a wide range of possible dispersions while keeping a signal/noise ratio not too high. Note that the higher $\phi$, the less overdispersed the signal. From our experience, the configuration of parameter combinations with $p_0=0.5$ is the more typical of observed values for RNA-Seq data.
-------- ------------ -------- --------------
$p_1$ $\phi$ $p_1$ $\phi$
$0.5$ $5$ $0.2$ $0.08^{1/8}$
$0.33$ $\sqrt{5}$ $0.1$ $0.08^{1/4}$
$0.2$ $0.8$ $0.05$ $0.08^{1/2}$
$0.64$ $0.08$
-------- ------------ -------- --------------
: [**Values of parameters used in the simulation study**]{}[]{data-label="paramvalue"}
[Provided that the ratio $\lambda=\phi(1-p)/p$ remains constant, the negative binomial distribution with dispersion parameter $\phi$ going to infinity converges to the Poisson distribution $\mathcal{P}(\lambda)$. We propose an identical simulation study based on the Poisson distribution for the comparison with non-dispersed datasets. Specifically, we used for $\lambda_0$ the values $1.25$ and $0.73$ so that the odd-ratios $s=4;8;16$ corresponded to the respective values $\lambda_1=5;10;20$ and $2.92; 5.83;11.7$]{} In practice there is little chance that the overdispersion is known. We propose to estimate this parameter from the data and use the obtained value in the analysis. The results presented here used the estimator inspired from [@jonhson_kotz]: starting from sliding window of size $15$, we compute the method of moments estimator of $\phi$, using the formula $\phi = {{E}}^2(X)/(V(X)-{{E}}(X))$, and retain the median over all windows. When this median is negative (which is likely to happen in datasets with many zeros), we double the size of the window. In practice however, results are very similar when using maximum likelihood or quasi-maximum likelihood estimators on sliding windows.
Results
-------
We compute the posterior probability $P({{E}_0}| {{\bf Y}}, {{\mathbf K}})$ for each simulation and each value of $d$. Figures \[Pois\] to \[BN2\] in Appendix \[abacus\] represent the boxplots of this probability for each configuration. For sake of visibility, the outliers were not drawn in those figures. Note that in each figure, the first boxplot corresponds to $d=0$ and thus to model ${{E}_0}$, while $d\neq 0$ for left boxplots so that the true model is ${{E}_1}$. These plots can be understood as abacus for the detection power of the proposed approach. For example, the perfect scenario corresponds to $s=16$ in the Poisson case of Figure \[Pois\].
As expected, these results show that the lower the value of $\phi$ (the Poisson distribution is interpreted here as $\phi = +\infty$), the most difficult the decision becomes. The trend is identical for decreasing values of the odd-ratio $s$ and decreasing values of $d$. In the most difficult scenario of very high dispersion compared to signal value, the method fails to provide satisfying decisions whatever the level of odd-ratio or distance between change-points. However, in most configurations, the method is adequate as soon as $d\geq 16$.
An important question is the impact of the estimation of the dispersion parameter. Interestingly, in the simulation study with $p_0=0.8$, our estimator tended to under-estimate $\phi$ (and thus over-estimate the dispersion) while it was the contrary in the simulation study with $p_0=0.5$. This affects the performance of the decision rule, which behaves better when $\phi$ is higher. For instance, Figure \[phi\] shows, for $s=16$ and $d=16$, that knowing the true value of $\phi$ improves the results when $p_0=0.8$ but worsens them when $p_0=0.5$.
![[**Impact of estimating the dispersion parameter.**]{} Boxplot of the posterior probability of ${{E}_0}$ for $s=16$ and $d=16$ when estimating the value of $\phi$ (left boxplot of each subdivision) or when using the known value (right boxplot of each subdivision).[]{data-label="phi"}](ClR13-Fig3){width="12cm"}
Comparison of transcribed regions in yeast {#appli}
==========================================
#### Experimental design.
We now go back to our first motivation a consider a study from the Sherlock lab in Stanford [@Risso_norma]. In their experiment, they grew a yeast strain, *Saccharomyce Cerevisiae*, in three different environments: ypd, which is the traditional (rich) media for yeast, delft, a similar but poorer media, and glycerol. [In the last decade many studies (see for instance [@polyA1; @Tian-polyA]) have showed that a large proportion of genes have more than one polyadenylation sites, thus can express multiple transcripts with different 3’ UTR sizes. Similarly, the 5’ capping process is dependent on environment conditions [@5cap], and the 5’ UTR size may vary according to stress factors. We may therefore expect that the yeast cells grown in different conditions (they ferment in the first two media, while they respire in glycerol) will produce transcripts of unequal sizes. On the contrary, the intron-exon boundaries are not expected to differ between conditions]{}
#### Change-point location.
We applied our procedure to gene YAL013W which has two exons. The RNA-Seq series were segmented into $5$ segments to allow one segment per transcribed region separated by segments of non-coding regions. Figure \[real-data\] illustrates the posterior distribution of each change-point in each profile.
![[**Posterior distribution of change-point location.**]{} Segmentation in $5$segments of gene YAL013W in three different media: ypd (top), delft (middle) and glycerol (bottom). Black dots represent the number of reads starting at each position of the genome (left scale) while blue curves are the posterior distribution of the change-point location (right scale).[]{data-label="real-data"}](compyeastresult.pdf){width="12cm"}
#### Credibility intervals on the shift.
For each of the first to the fourth change-point, we [computed the posterior distribution of the difference between change-point locations for each pairs of conditions]{}. [For the biological reasons stated above, we expect to observe more differences for the first and last change-points than for the other two, which can be used as a verification of the decision rule.]{} Figure \[credibility\] [provides]{} the [posterior]{} distribution of [these differences]{}, as well as the $95$% credibility intervals.
#### Posterior probability of common change-point.
[We then computed the probability that the change-point is the same across several series, taking $p_0 = 1/2$.]{} Table \[res:BF\] provides, for the simultaneous comparison of the three conditions and for each pair of conditions, the value of the posterior probability of ${{E}_0}$ at each change-point ($\tau_1^\ell$ is associated with the $5'$ UTR, $\tau_2^\ell$ to the 5’ intron boundary, $\tau_3^\ell$ to the $3'$ intron boundary and $\tau_4^\ell$ to the $3'$ UTR). Reassuringly, in most cases the change-point location is identical when corresponding to intron boundaries. [On the contrary, UTR boundaries seem to differ from one condition to another.]{}
![[**Distribution of change-point location and $95$% credibility intervals.** ]{} For each of the two by two comparison (top: ypd-delft; middle: ypd-glycerol; bottom delft-glycerol), posterior distribution of the change-point difference for each of the first to the fourth change-point.[]{data-label="credibility"}](cred-yeast.pdf){width="13cm"}
---------------- ------------------------- ------------------------ ------------------------ ------------------------
$\qquad \tau_1 \qquad $ $\qquad \tau_2 \qquad$ $\qquad \tau_3 \qquad$ $\qquad \tau_4 \qquad$
all media $10^{-3}$ $0.99$ $0.99$ $6\;10^{-3}$
ypd-delft $0.32$ $0.30$ $0.99$ $10^{-5}$
ypd-glycerol $4\;10^{-4}$ $0.99$ $0.99$ $6\;10^{-3}$
delft-glycerol $5\;10^{-2}$ $0.60$ $0.99$ $0.99$
---------------- ------------------------- ------------------------ ------------------------ ------------------------
: [Posterior probability of a common change point across conditions for gene YAL013W]{}[]{data-label="res:BF"}
#### Differential splicing in yeast.
We finally applied our comparison procedure to a set of $50$ genes from the yeast genome which all possess two exons and which were expressed in all three conditions at the time of the experiment. The left figure of Figure \[50genes\] shows the distribution of the posterior probability of ${{E}_0}$ for the simultaneous comparison of the three conditions when $p_0({{\mathbf K}})=1/2$. Once again the results strengthens the expectation that intron boundaries should not vary between conditions while more difference is observed for the UTRs. A closer look at the five genes for which we have evidence of either the second or third change-point difference reveals that one of the two exons was not expressed in the Glycerol medium. Moreover, a discussion with Dr Sherlock suggests that about $10$% of the genes should be liable to differential splicing. We therefore performed the analysis over again removing the $5$ outliers and setting $p_0=0.9$ for $\tau_1$ and $\tau_4$ and $p_0=0.99$ for the other two. Results are illustrated in the right figure of Figure \[50genes\]. For these new prior values, we observe that $9$ genes have a $3'$ UTR length which varies, and $16$ for the $5'$ UTR.
![[**Distribution of $P({{E}_0}| {{\bf Y}}, {{\mathbf K}})$ for a set of $50$ genes with two values of $p_0$.**]{} We set $p_0=1/2$ in the left figure, and $p_0=0.9$ for $\tau_1$ and $\tau_4$, $p_0=0.99$ for the intron boundaries in the right figure.[]{data-label="50genes"}](ClR13-Fig6){width="13cm"}
Conclusion
==========
We have proposed two exact approaches for the comparison of change-point location. The first is based on the posterior distribution of the shift in two profiles, while the second is adapted to the comparison of multiple profiles and studies the posterior probability of having a common change-point. These procedures, when applied to RNA-Seq datasets, confirm the expectation that transcription starting and ending sites may vary between growth conditions while the localization of introns remains the same.
While we have illustrated these procedures with count datasets, they can be adapted to all distributions from the exponential family verifying the factoriability assumption as described in Section \[distributions\]. They are in fact implemented in an R package `EBS` for the negative binomial, Poisson, Gaussian heteroscedastic and Gaussian homoscedastic with known variance parameter. This package is available on the CRAN repository at <http://cran.r-project.org/web/packages/EBS/index.html>.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors deeply thank Sandrine Dudoit, Marie-Pierre Etienne, Emilie Lebarbier Eric Parent and Gavin Sherlock for helpful conversations and comments on this works.
Appendix section {#abacus}
================
![[**Boxplot of posterior probabilities of ${{E}_0}$ for Poisson.**]{} Plotted as $d$ increases in simulation studies for the Poisson distribution with $\lambda_0=0.73$ (Top) and $\lambda_0=2.92$ (Bottom) and for each value of $s$ (in columns).[]{data-label="Pois"}](ClR13-Fig7){width="12cm"}
![[**Boxplot of posterior probabilities of ${{E}_0}$ for negative Binomial, with $p_0=0.8$.**]{} Plotted as $d$ increases in simulation studies for the negative binomial distribution with $p_0=0.8$ and for each value of $s$ (in columns) and each value of $\phi$ (in rows) as detailed in the left side of Table \[paramvalue\]. The overdispersion is estimated as detailed in Section \[description\].[]{data-label="BN1"}](ClR13-Fig8){width="12cm"}
![[**Boxplot of posterior probabilities of ${{E}_0}$ for negative Binomial, with $p_0=0.5$.**]{} Plotted as $d$ increases in simulation studies for the negative binomial distribution with $p_0=0.5$ and for each value of $s$ (in columns) and each value of $\phi$ (in rows) as detailed in the right side of Table \[paramvalue\]. The overdispersion is estimated as detailed in Section \[description\].[]{data-label="BN2"}](ClR13-Fig9){width="12cm"}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We briefly review the spin structure of the nucleon and show that it is best thought in the light-front formulation. We discuss in particular the longitudinal and transverse spin sum rules, the proper definition of canonical orbital angular momentum and the spin-orbit correlation.'
author:
- Cédric Lorcé
date: 'Received: date / Accepted: date'
title: Spin structure of the nucleon on the light front
---
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
Introduction
============
One of the major challenges in hadronic physics is to understand how the nucleon spin arises from the spin and orbital motion of its constituents. Unlike atomic systems, relativistic and non-perturbative effects are essential to understand this spin structure.
The decomposition of the nucleon spin is not unique and is therefore sometimes considered as unphysical. Already at the classical level there exist two definitions (canonical and kinetic) of orbital angular momentum (OAM). While there are often no practical differences between these two definitions, the situation changes in presence of gauge fields, triggering longstanding debates about which definition has to be considered as the fundamental, primary or “physical” one. In the context of the nucleon spin decomposition, it has recently been recognized that both definitions are equally interesting, as they are in principle both measurable and reflect complementary aspects of the intricate bound system.
Since measurable quantities are necessarily gauge invariant, recent theoretical works demonstrated that canonical quantities can be made gauge invariant by sacrifying locality or manifest Lorentz covariance. This can be done in infinitely many ways, but only few variants have a clear relation with actual experimental observables. One of the crucial questions now is how to connect the two definitions of OAM to experimental observables.
Another delicate question is the meaningful separation of spin and OAM. The light-front formalism, with its preferred direction, turns out to provide the most consistent and intuitive picture. Over the last two decades, so many relations and sum rules have been proposed in the literature that they created some sort of confusion. One of the current tasks is to clarify the validity and scope of these relations and sum rules.
In this proceeding, we sketch a portrait of the present situation and present some recent developments. In section \[sec2\], we briefly introduce the two families of nucleon spin decompositions. In section \[sec3\], we discuss various spin sum rules and relations. In section \[sec4\], we define the quark spin-orbit correlation and show how it can be expressed in terms of parton distributions. Finally, we conclude with section \[sec5\]. For the interested reader, more detailed discussions can be found in the recent reviews [@Leader:2013jra; @Wakamatsu:2014zza].
Canonical and kinetic spin decompositions {#sec2}
=========================================
![The nucleon spin decompositions. See text for more details.[]{data-label="fig:1"}](Decompositions_nucleon.eps){width="8cm"}
There are essentially two families of nucleon spin decomposition: kinetic and canonical. In the kinetic family, one has the Belinfante, Ji and gauge-invariant kinetic (gik) decompositions, depending on how many pieces the total angular momentum is split into, see Fig. \[fig:1\]. In the Belinfante decomposition, the total angular momentum is simply decomposed into quark and gluon contributions $${\vec{J}}={\vec{J}}^q_\text{Bel}+{\vec{J}}^G_\text{Bel}.$$ Ji decomposed the quark angular momentum into spin and OAM contributions [@Ji:1996ek] $${\vec{J}}={\vec{S}}^q+{\vec{L}}^q_\text{Ji}+{\vec{J}}^G_\text{Ji}\qquad\text{with} \qquad{\vec{J}}^G_\text{Ji}={\vec{J}}^G_\text{Bel}.$$ Finally, decomposing further the gluon angular momentum into spin and OAM gives the gauge-invariant kinetic decomposition discussed by Wakamatsu [@Wakamatsu:2010qj; @Wakamatsu:2010cb] $$\label{gik}
{\vec{J}}={\vec{S}}^q+{\vec{L}}^q_\text{gik}+{\vec{S}}^G+{\vec{L}}^G_\text{gik}\qquad\text{with} \qquad{\vec{L}}^q_\text{gik}={\vec{L}}^q_\text{Ji}.$$ The gauge-invariant canonical (gic) decomposition obtained by Chen *et al.* [@Chen:2008ag; @Chen:2009mr] $$\label{gic}
{\vec{J}}={\vec{S}}^q+{\vec{L}}^q_\text{gic}+{\vec{S}}^G+{\vec{L}}^G_\text{gic}$$ can be understood as a gauge-invariant version (or extension) of the Jaffe-Manohar decomposition [@Jaffe:1989jz], and differs from the gauge-invariant kinetic decomposition in how the total OAM is split into quark and gluon contributions $${\vec{L}}={\vec{L}}^q_\text{gik}+{\vec{L}}^G_\text{gik}={\vec{L}}^q_\text{gic}+{\vec{L}}^G_\text{gic}.$$ A nice physical interpretation of the difference has been proposed in Ref. [@Burkardt:2012sd].
The operators associated with the various contributions are $$\begin{aligned}
{\vec{S}}^q&=\int{\mathrm{d}}^3x\,\psi^\dag\tfrac{1}{2}{\vec{\Sigma}}\psi, \qquad&{\vec{S}}^G&=\int{\mathrm{d}}^3x\,{\vec{E}}^a\times{\vec{A}}^a_{\text{phys}},\\
{\vec{L}}^q_\text{gic}&=\int{\mathrm{d}}^3x\,\psi^\dag({\vec{x}}\times i{\vec{D}}_{\text{pure}})\psi,\qquad&{\vec{L}}^G_\text{gic}&=-\int{\mathrm{d}}^3x\,E^{ai}({\vec{x}}\times{\vec{\mathcal D}}^{ab}_{\text{pure}}) A^{bi}_{\text{phys}},\\
{\vec{L}}^q_\text{gik}&=\int{\mathrm{d}}^3x\,\psi^\dag({\vec{x}}\times i{\vec{D}})\psi,\qquad&{\vec{L}}^G_\text{gik}&={\vec{L}}^G_\text{gic}-\int{\mathrm{d}}^3x\,({\vec{\mathcal D}}\cdot{\vec{E}})^a\,{\vec{x}}\times{\vec{A}}^a_{\text{phys}},
\end{aligned}$$ where the gauge field has been decomposed into two parts ${\vec{A}}={\vec{A}}^{\text{pure}}+{\vec{A}}^{\text{phys}}$, and the pure-gauge covariant derivatives are given by ${\vec{D}}_{\text{pure}}=-{\vec{\nabla}}-ig{\vec{A}}_{\text{pure}}$ and ${\vec{\mathcal D}}_{\text{pure}}=-{\vec{\nabla}}-ig[{\vec{A}}_{\text{pure}},\quad]$.
The complete gauge-invariant decompositions and seem to be in contradiction with textbook claims that it is not possible to write down gauge-invariant expressions for gluon spin and OAM contributions. This is actually not the case since textbooks refer only to local expressions, whereas ${\vec{A}}^{\text{pure}}$ and ${\vec{A}}^{\text{phys}}$ are non-local expressions of the gauge field [@Lorce:2012rr; @Lorce:2012ce]. The pure-gauge part ${\vec{A}}^{\text{pure}}$ plays essentially the role of a background field [@Lorce:2013gxa; @Lorce:2013bja]. Background dependence implies that the split ${\vec{A}}={\vec{A}}^{\text{pure}}+{\vec{A}}^{\text{phys}}$ is accompanied by a new freedom $$\label{Stueckelberg}
{\vec{A}}^{\text{pure}}\mapsto{\vec{A}}^{\text{pure}}+{\vec{B}},\qquad{\vec{A}}^{\text{phys}}\mapsto{\vec{A}}^{\text{phys}}-{\vec{B}},$$ referred to as Stueckelberg symmetry [@Lorce:2012rr; @Stoilov:2010pv]. The crucial point is that it is the actual experimental conditions that determine the form of the background field to be used [@Lorce:2012rr; @Wakamatsu:2014toa].
Spin sum rules and relations {#sec3}
============================
Using the Belinfante-Rosenfeld energy-momentum tensor, Ji obtained the remarkable result that the quark and gluon total kinetic angular momentum can be expressed in terms of twist-2 generalized parton distributions (GPDs) [@Ji:1996ek] $$\label{Jirel}
\langle J^{q,G}\rangle=\tfrac{1}{2}\int{\mathrm{d}}x\,x[H_{q,G}(x,0,0)+E_{q,G}(x,0,0)].$$ This relation holds for the longitudinal component $J_L={\vec{J}}\cdot{\vec{P}}/|{\vec{P}}|$ where ${\vec{P}}$ is the nucleon momentum [@Leader:2012md]. By rotational symmetry, it holds also for the transverse component, but only in the nucleon rest frame. Considering the transverse component of the Pauli-Lubanski vector does not prevent frame dependence of the separate quark and gluon contributions [@Leader:2013jra; @Leader:2012ar; @Hatta:2012jm; @Harindranath:2013goa].
Subtracting from Eq. the longitudinal quark spin contribution given by the isoscalar axial-vector form factor (FF) in the $\overline{MS}$ scheme $$\langle S^q\rangle=\tfrac{1}{2}\,G^q_A(0),$$ one gets the longitudinal quark kinetic OAM $$\label{OAMeq}
\langle L^q_\text{gik}\rangle=\tfrac{1}{2}\int{\mathrm{d}}x\,x[H_{q,G}(x,0,0)+E_{q,G}(x,0,0)]-\tfrac{1}{2}\,G^q_A(0).$$ The same quantity is also directly related to a twist-3 GPD [@Penttinen:2000dg; @Kiptily:2002nx; @Hatta:2012cs] $$\langle L^q_\text{gik}\rangle=-\int{\mathrm{d}}x\,xG^q_2(x,0,0).$$
Using the light-front formalism which is particularly suitable for the parton model picture, the most intuitive expression for OAM is as a phase-space integral [@Lorce:2011kd; @Lorce:2011ni] $$\label{OAMWigner}
\langle L^q(\mathcal W)\rangle=\int{\mathrm{d}}x\,{\mathrm{d}}^2k_\perp\,{\mathrm{d}}^2b_\perp\,(\vec b_\perp\times\vec k_\perp)_z\,\rho^{[\gamma^+]q}_{++}(x,\vec k_\perp,\vec b_\perp;\mathcal W),$$ where the relativistic phase-space or Wigner distribution $\rho^{[\gamma^+]q}_{++}(x,\vec k_\perp,\vec b_\perp;\mathcal W)$ can be interpreted as giving the quasi-probability to find an unpolarized quark with momentum $(xP^+,{\vec{k}}_\perp)$ and transverse position ${\vec{b}}_\perp$ inside a longitudinally polarized nucleon. Note that the Euclidean subgroup of the light-front formalism plays here a crucial role in providing a well-defined transverse center of the nucleon [@Soper:1976jc; @Burkardt:2000za; @Burkardt:2005hp]. The phase-space distributions are related by Fourier transform to the generalized transverse-momentum dependent distributions (GTMDs) [@Meissner:2009ww; @Lorce:2011dv; @Lorce:2013pza], leading to the simple relation [@Lorce:2011kd; @Hatta:2011ku; @Kanazawa:2014nha] $$\langle L^q(\mathcal W)\rangle=-\int{\mathrm{d}}x\,{\mathrm{d}}^2k_\perp\,\tfrac{{\vec{k}}^2_\perp}{M^2}\,F^q_{14}(x,0,{\vec{k}}_\perp,{\vec{0}}_\perp;\mathcal W).$$ Depending on the shape of the Wilson line $\mathcal W$, one obtains either kinetic $\langle L^q_\text{gik}\rangle=\langle L^q(\mathcal W_\text{straight})\rangle$ or canonical $\langle L^q_\text{gic}\rangle=\langle L^q(\mathcal W_\text{staple})\rangle$ OAM [@Burkardt:2012sd; @Lorce:2012ce; @Ji:2012sj]. Unfortunately, it is not known so far how to extract quark GTMDs from actual experiments. They are however in principle calculable on the lattice [@Ji:2013dva].
Some quark model calculations suggested that the canonical OAM might also be expressed in terms of a transverse-momentum dependent distributions (TMDs) $$\langle L_\text{gic}^q\rangle=-\int{\mathrm{d}}x\,{\mathrm{d}}^2k_\perp\,\tfrac{{\vec{k}}_\perp^2}{2M^2}\,h_{1T}^{\perp q}(x,{\vec{k}}^2_\perp),$$ but this relation does not hold in general [@Lorce:2011kn] just like other relations among the TMDs [@Lorce:2011zta].
Spin-orbit correlation {#sec4}
======================
The so-called quark OAM contribution to the nucleon spin corresponds to the correlation between the quark OAM and the nucleon spin. Another interesting quantity is the correlation between the quark spin and OAM which is given by the following operators $$\begin{aligned}
C^q_\text{gic}&=\int{\mathrm{d}}^3x\,\psi^\dag\gamma_5({\vec{x}}_\perp\times i{\vec{D}}_{{\text{pure}},\perp})_z\psi,\\
C^q_\text{gik}&=\int{\mathrm{d}}^3x\,\psi^\dag\gamma_5({\vec{x}}_\perp\times i{\vec{D}}_\perp)_z\psi.
\end{aligned}$$ These operators are very similar to the longitudinal quark OAM operators and represent, respectively, the canonical and kinetic versions of the quark spin-orbit correlation [@Lorce:2011kd; @Lorce:2014mxa].
Following a similar approach to Ref. [@Ji:1996ek], we derived an expression which relates measurable parton distributions to the kinetic version of the quark spin-orbit correlation [@Lorce:2014mxa] $$\label{SOtwist2}
\langle C^q_\text{gik}\rangle=\tfrac{1}{2}\int{\mathrm{d}}x\,x\tilde H_q(x,0,0)-\tfrac{1}{2}\,[F^q_1(0)-\tfrac{m_q}{2M_N}\,H^q_1(0)],$$ Beside its resemblance with Eq. , the remarkable fact about this expression is that it provides a physical interest in the second moment of the helicity distribution. We also found a corresponding twist-3 GPD relation $$\label{SOtwist3}
\langle C^q_\text{gik}\rangle=-\int{\mathrm{d}}x\,x[\tilde G^q_2(x,0,0)+2\tilde G^q_4(x,0,0)].$$
Like the quark OAM, the most intuitive expression for the quark spin-orbit correlation is as a phase-space integral [@Lorce:2011kd; @Lorce:2014mxa] $$\label{OAMWigner}
\langle C^q(\mathcal W)\rangle=\int{\mathrm{d}}x\,{\mathrm{d}}^2k_\perp\,{\mathrm{d}}^2b_\perp\,(\vec b_\perp\times\vec k_\perp)_z\,\rho^{[\gamma^+\gamma_5]q}_{++}(x,\vec k_\perp,\vec b_\perp;\mathcal W),$$ where the phase-space distribution $\rho^{[\gamma^+\gamma_5]q}_{++}(x,\vec k_\perp,\vec b_\perp;\mathcal W)$ can be interpreted as giving the difference between the quasi-probabilistic distributions of quarks with polarization aligned and antialigned with respect to the longitudinal direction. The simple relation in terms of GTMDs is [@Lorce:2011kd; @Kanazawa:2014nha; @Lorce:2014mxa] $$\langle C^q(\mathcal W)\rangle=\int{\mathrm{d}}x\,{\mathrm{d}}^2k_\perp\,\tfrac{{\vec{k}}^2_\perp}{M^2}\,G^q_{11}(x,0,{\vec{k}}_\perp,{\vec{0}}_\perp;\mathcal W).$$ Once again, depending on the shape of $\mathcal W$, one obtains either kinetic $\langle C^q_\text{gik}\rangle=\langle C^q(\mathcal W_\text{straight})\rangle$ or canonical $\langle C^q_\text{gic}\rangle=\langle C^q(\mathcal W_\text{staple})\rangle$ spin-orbit correlation [@Lorce:2014mxa].
Interestingly, since $F^u_1(0)=2$ and $F^d_1(0)=1$ and since the tensor FF $H^q_1(0)$ can safely be neglected because of the mass ratio $m_{u,d}/4M_N\sim 10^{-3}$, the essential input we need is the second moment of the quark helicity distribution $$\int_{-1}^1{\mathrm{d}}x\,x\tilde H_q(x,0,0)=\int_0^1{\mathrm{d}}x\,x[\Delta q(x)-\Delta\overline q(x)].$$ Contrary to the lowest moment $\int_{-1}^1{\mathrm{d}}x\,\tilde H_q(x,0,0)=\int_0^1{\mathrm{d}}x\,[\Delta q(x)+\Delta\overline q(x)]$, this second moment cannot be extracted from deep-inelastic scattering (DIS) polarized data without extra assumptions about the polarized sea-quark distributions. However, by combining inclusive and semi-inclusive deep-inelastic scattering, separate quark and antiquark contributions can be extracted [@Leader:2010rb] $$\int_{-1}^1{\mathrm{d}}x\,x\tilde H_u(x,0,0)\approx 0.19,\qquad \int_{-1}^1{\mathrm{d}}x\,x\tilde H_d(x,0,0)\approx -0.06,$$ at the scale $\mu^2=1$ GeV$^2$, leading to the values $C^u_z\approx -0.9$ and $C^d_z\approx -0.53$. These values seem consistent with recent Lattice calculations by the LHPC collaboration [@Bratt:2010jn], see table \[Modelresults\].
\[Modelresults\]
[ccccc]{} Model [@Lorce:2011dv]&$\int^1_{-1}{\mathrm{d}}x\,\tilde H_u(x,0,0)$&$\int^1_{-1}{\mathrm{d}}x\,\tilde H_d(x,0,0)$&$\int^1_{-1}{\mathrm{d}}x\,x\tilde H_u(x,0,0)$&$\int^1_{-1}{\mathrm{d}}x\,x\tilde H_d(x,0,0)$\
NQM&$4/3$&$-1/3$&$4/9$&$-1/9$\
LFCQM&$0.995$&$-0.249$&$0.345$&$-0.086$\
LF$\chi$QSM&$1.148$&$-0.287$&$0.392$&$-0.098$\
LSS [@Leader:2010rb]&$0.82$&$-0.45$&$\approx 0.19$&$\approx -0.06$\
Lattice [@Bratt:2010jn]&$0.82(7)$&$-0.41(7)$&$\approx 0.20$&$\approx -0.05$\
Noting that the second moment of the quark helicity distribution is a valence-like quantity with tamed low-$x$ region, one may expect phenomenological quark model predictions to be more accurate than for the lowest moment. In table \[Modelresults\] we provide the first two moments of the up and down quark helicity distributions obtained within the naive quark model (NQM), the light-front constituent quark model (LFCQM) [@Boffi:2002yy; @Boffi:2003yj; @Pasquini:2005dk; @Pasquini:2006iv; @Pasquini:2008ax] and the light-front chiral quark-soliton model (LF$\chi$QSM) [@Lorce:2006nq; @Lorce:2007as; @Lorce:2007fa] at the scale $\mu^2\sim 0.26$ GeV$^2$. From these estimates, we expect a negative quark spin-orbit $C^q_z$ for both quark flavors ($C^u_z\approx -0.8$ and $C^d_z\approx -0.55$), meaning that the quark spin and kinetic OAM are expected to be, in average, antiparallel. On the contrary, the canonical version of the quark spin-orbit correlation turns out to be positive in the models [@Lorce:2011kd], displaying the importance of the quark-gluon interaction.
Conclusion {#sec5}
==========
There are essentially two types or families of nucleon spin decompositions: the canonical one and the kinetic one. It has recently been recognized that both are in principle measurable. The crucial piece which is currently missing is the contribution coming from the quark and gluon orbital angular momentum. Many relations and sum rules have been proposed, but few turned out to be of practical significance. The light-front formalism is the best suited for describing and interpreting the high-energy scattering experiments involving nucleons. It is therefore hardly surprizing that it gives the proper formulation for decomposing in a meaningful way the nucleon spin. Finally, the quark spin-orbit correlation is a new independent quantity characterizing the nucleon spin structure. Like the quark orbital angular momentum, this information can be extracted from measurable parton distribution.
I benefited a lot from many discussions and collaborations with E. Leader, B. Pasquini and M. Wakamatsu. This work was supported by the Belgian Fund F.R.S.-FNRS *via* the contract of Chargé de Recherches.
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---
author:
- Adam Coogan
- Logan Morrison
- Stefano Profumo
bibliography:
- 'hazma.bib'
title: 'Hazma: A Python Toolkit for Studying Indirect Detection of Sub-GeV Dark Matter'
---
LM and SP are partly supported by the U.S. Department of Energy grant number de-sc0010107. We acknowledge with gratitude early collaboration on this project with Francesco D’Eramo. Francesco D’Eramo was the first to put forward the idea of using chiral perturbation theory in the context of accurately calculating the annihilation and decay products of MeV dark matter. He was also responsible for several of the early calculations, part of the writing of the text, and for overall educating us all on numerous topics in chiral perturbation theory. We thank Michael Peskin for important input and feedback, and Regina Caputo for discussions about MeV gamma-ray observatories. [We thank Graham White as well as Nicholas Rodd for pointing out typos in an earlier version of the manuscript.]{} Finally, we thank Eulogio Oset and Jose Antonio Oller for clarifying some points about unitarized chiral perturbation theory.
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\
[**Even and Odd Symplectic and Kählerian Structures on Projective Superspaces** ]{}\
[O. M. Khudaverdian]{}\
[*Yerevan State University* ]{}\
[*Yerevan, Armenia* ]{}\
[A. P. Nersessian[^1]]{}\
[*Laboratory of Theoretical Physics,*]{}\
[*Joint Institute for Nuclear Recearch*]{}\
[*Dubna, Head Post Office, P.O.Box 79, 101 000 Moscow, Russia*]{}\
[**Abstract**]{}
Supergeneralization of $\DC P(N)$ provided by even and odd Kählerian structures from Hamiltonian reduction are construct.Operator $ \Delta$ which used in Batalin– Vilkovisky quantization formalism and mechanics which are bi-Hamiltonian under corresponding even and odd Poisson brackets are considered.
[*Submitted to J. Math. Phys.*]{}
0
0
0
Introduction
=============
On the supermanifolds it is possible to define not only even, but also odd symplectic structures \[1\]. Phase space structure corresponded to even symplectic structure and odd one.
For example on the superspace $E^{2N,M}$ with coordinates $z^A = (x^1,...,x^{2N}, \theta^1,..., \theta^{M})$ one can consider an even symplectic structure with corresponding even canonical Poisson bracket: $$\{f,g\}_0
=
\sum_{i=1}^N \left(
\frac{\partial f}{\partial x^i}
\frac{\partial g}{\partial x^{i+N} }
-
\frac{\partial f}{\partial x^{i+N}}
\frac{\partial g}{\partial x^i}
\right)
+
\sum_{\alpha =1}^{M}
\epsilon _\alpha \frac{\partial^{R} f}{\partial \theta^ \alpha}
\frac{\partial^{L} g }{\partial \theta^{\alpha}} ,\;\;\epsilon_{\alpha} =\pm
1 .$$ On $E^{N.N}$ one can consider an odd one with corresponding canonical odd Poisson bracket (Buttin bracket, antibracket): $$\{f,g\}_1 =
\sum_{i=1}^{N}\left(
\frac{\partial f}{\partial x^i}
\frac{\partial^{L} g}{\partial\theta^i}
+
\frac{\partial^{R} f}{\partial \theta^i}
\frac{\partial g}{\partial x^i}
\right)$$ In the \[2, 3\] Batalin and Vilkovisky used odd bracket for formulating Lagranian BRST quantization formalism (BV -formalizm). Its provides a possibility to give covariant and the most elegant formulation of the conditions on all the ghosts . BV-formalism is an effective method for quantization of gauge theories with open Lie algebra. An attempt to consider it as a framework of background independent open-string field theory was made \[4, 5\].
On other hand the possiblity importance of the odd bracket in twistorial program and supersymmetric mechanics was emphasised \[6-9\]. The problem of reformulation of supersymmetric mechanics in terms of odd bracket, using the supercharge as a new Hamiltonian and the attempts to quantize it were performed in \[8-10\].
There is no doubt that odd bracket needs to be geometrically investigated.
It is possible to formulate Hamiltonian mechanics in term of odd brackets as well as in term of even one \[11\]. Arbitrary even nondegenerate bracket can be reduced (locally) to canonical form (1.1), and arbitrary odd one – to canonical form (1.2) \[12\]. But in the general case even and odd brackets cannot be simultaneously reduced to form (1.1) and (1.2). The structure of the supergroup of transformations which preserve both brackets, depends on their mutual position. Anyway this supergroup is finite-dimensional and it is the different grading of the brackets that leads to this fact \[13\].
There are nontrivial geometrical objects depending on second derivatives which are invariant under transformations preserving odd bracket and the volume form connected with even bracket. It is the “operator $\Delta $” \[13\] which used in BV -formalism \[2, 3\] and the semidensity constructed in \[14\]. These objects have no analogs in a classical case.
These results strongly indicate that nontrivial geometry arises on the supermanifolds which are provided by Poisson brackets of different grading. Geometrical properties of superspaces provided by Poisson brackets of different gradings were investigated in \[13, 15-17\]. Superspaces, provided simultaneously by even and odd canonical one was investigated in \[15\]. It was shown in \[16\] that exists a large class of supermanifolds (the supermanifolds, assotiated with tangent bundles of Kählerian manifolds) on which one can defined simultaneously even add odd symplectic (and Kählerian ) structures. These structures turn out to be lifting of the corresponding structures on the underlying manifolds. Therefore their properties have to be expressible in terms of classical geometrical objects. They are good models for revealing geometrical properties of two -bracket supermanifolds.
But there don’t support by nontrivial examples where even and odd symplectic structures have natural geometrical origin.\
In this work we construct the example of such supermanifold as reduced phase superspaces of the superoscillator. (The dynamics of the superoscillator in the superspace $E^{2N,2N}$ can be described either in terms of the canonical even bracket (1.1) or in terms of the canonical odd one (1.2). In the second case the role of the Hamiltonian is played by one of its supercharges \[15\].)
In the [*Section 2*]{} we demonstrate reduction procedure on the simple examples constructing phase superspaces reduced by the Hamiltonian of the superoscillator. This procedure performed in terms of even and odd structures leeds to the two different supermanifolds .
In the [*Section 3*]{} we perform the reduction procedure by Hamiltonian of the superoscillator and by its supercharges. In terms of both symplectic structures we come to the same supermanifold which naturally inherits even and odd structures of the initial superspace. Canonical complex structure on the initial superspace $E^{2N,2N}$ provides this supermanifold with the complex structure and with the even and odd Kählerian structures corresponding to them.
It occurs that this supermanifold is associated to the tangent space of the underlying manifold - complex projective space.
This supermanifold obtained by reduction procedure can be naturally included in the family of the supermanifolds ( which are associated with tangent bundles of arbitraty Kählerian manifold) with even and odd Kählerian structures lifted from the Kählerian structure of the underlying manifold, which was investigated in \[16\].
In the [*Section 4*]{} we investigate the bi-Hamiltonian mechanics (i.e. the even vector fields preserving both symplectic structures) and “operator $ \Delta$ ” on the constructed supermanifold and discuss theirs connection with the geometrical objects on underlying manifold.
In the [*Appendix 1*]{} for the general case we briefly mention the method of Hamiltonian reduction in the terms convenient for our purposes.
In the [*Appendix 2*]{} we recall the connection between the supermanifolds and the linear bundles to the extent necessary for our purposes. In this Appendix we suggest a natural lifting in the general case of the reduction procedure from the manifolds to their corresponding supermanifolds with odd symplectic structures.
For rigorous definitions and conventions in supermathematics used here we refer too \[1\].
The preliminary results of this article were published in \[17\].
0
Examples of Kählerian Supermanifolds and Hamiltonian Reduction
================================================================
We mostly consider symplectic structures (odd or even one) as the part of corresponding Kählerian structures. In the same way as in the bosonic case \[18\] complex supermanifold is provided by even (odd) Kählerian structure if symplectic structure is defined by real closed nondegenerated even (odd) two-form $\Omega^{\kappa}$ which in local complex coordinates $z^A$, is given by the following expression $$\Omega^{\kappa}=i(-1)^{p(A)(p(B)+\kappa+1)}g^\kappa_{A {\bar B}}
dz^A \wedge d{\bar z}^B,$$ where $$g^\kappa_{A {\bar B}} =
(-1)^{(p(A)+\kappa+1)(p(B)+\kappa+1)+\kappa +1}
\overline {g^\kappa _{B {\bar A}}},\quad p(g^\kappa _{A\bar B})=p_A
+p_B+\kappa$$ Here and further index $\kappa =0(1)$ denote even(odd) case.
Then there exists a local real even (odd) function $K^\kappa(z,{\bar z})$ (Kählerian potential), such that $$g^\kappa_{A {\bar B}} =
\frac{\partial ^L}{\partial z^A}
\frac{\partial ^R}{\partial {\bar z}^B}
K^\kappa (z,{\bar z})$$ (As well as in usual case \[18\] the potential $K$ is defined with precision define up to arbitrary analytic and antianalytic functions.)
To even (odd) form $\Omega^{\kappa}$ there corresponds the even (odd) Poisson bracket $$\{ f,g\}_\kappa
=
i\left(
\frac{\partial ^R f}{\partial \bar z^A}
g^{{\bar A}B}_\kappa
\frac{\partial ^L g}{\partial z^B}
-
(-1)^{(p(A)+\kappa)(p(B)+\kappa)}
\frac{\partial ^R f}{\partial z^A}
g^{{\bar A}B}_\kappa
\frac{\partial ^L g }{\partial \bar z^B}
\right),$$ where $$g^{{\bar A}B}_\kappa g_{B{\bar C}}^\kappa=\delta^{\bar A}_{\bar C}
\;\;,\;\;\;\; \overline{g^{{\bar A}B}_\kappa}
= (-1)^{(p(A)+\kappa)(p(B)+\kappa)}g^{{\bar B}A}_\kappa .$$ Its satisfied to conditions of reality and “antisimmetricity” $$\overline{\{ f, g\}_\kappa }=\{\bar f ,\bar g \}_\kappa ,\;\;\;\{ f, g
\}_\kappa = -
(-1)^{(p(f)+\kappa)(p(g)+\kappa)}\{ g, f \}_\kappa ,$$ and Jacobi identities : $$( -1)^{(p(f)+\kappa)(p(h)+\kappa)}\{ f,\{ g, h \}_{\kappa}\}_\kappa +{\rm
{(cicl. perm.)}} = 0$$ On the complex superspace $\DC ^{N+1,N+1}$ with complex coordinates $z =(z^n,\eta ^n ),$ $n=0,1,...,N$ canonical symplectic structure $$\Omega^0 = i(dz^n \wedge {\bar dz}^n -i d\eta^n \wedge d{\bar \eta}^ n)$$ with corresponding even Poisson bracket $$\{ f,g\}_0 = i\left(\frac{\partial f}{\partial z^n}
\frac{\partial g}{\partial {\bar z}^n} -
\frac{\partial f}{\partial {\bar z}^n}
\frac{\partial g}{\partial z^n}\right) +
\frac{\partial ^R f}{\partial \eta^n}
\frac{\partial ^L g}{\partial {\bar \eta}^n} +
\frac{\partial ^R f}{\partial {\bar \eta}^n}
\frac{\partial ^L g }{\partial \eta^n}$$ defines even Kählerian structure, and canonical odd symplectic structure $$\Omega^1 = dz^n \wedge d{\bar \eta}^n + d{\bar z}^n \wedge d\eta^n$$ with corresronding odd bracket $$\{ f,g\}_1 =\frac{\partial f}{\partial z^n}
\frac{\partial^L g}{\partial {\bar \eta}^n} +
\frac{\partial f}{\partial {\bar z}^n}
\frac{\partial^L g}{\partial \eta^n} -
\frac{\partial ^R f}{\partial {\bar \eta}^n}
\frac{\partial g}{\partial z^n} -
\frac{\partial ^R f}{\partial \eta^n}
\frac{\partial g}{\partial {\bar z}^n}$$ defines odd Kählerian structure. One can obtaines more nontrivial examples by Hamiltonian reduction.
It is well known that for the harmonic oscillator in $(N+1)_{\DC}$–dimensional phase space using the energy integral for decreasing by one the complex degrees of freedom we go to $N$-dimensional complex projective space and Kählerian metric corresponding to reduced symplectic structure on it coincides with canonical one \[19\]. The straightforward generalization of this procedure on supercase gives us the following example.
Let $$H= z^n {\bar z}^n -i \eta^n {\bar \eta}^n$$ be the Hamiltonian of the superoscillator in the complex phase superspace $\DC^{N+1,N+1}$ with even Poisson bracket (2.6). $ H$ defines Hamiltonian action of group $U(1)$ on $\DC^{N+1,N+1}$ via motion equations $${\dot f}=\{ H,f \}_0\quad,\quad z \rightarrow e^{it}z$$ As well as in the ordinary case the $(N.N+1)$ - dimensional complex projective superspace $\DC P(N.N+1)$ (the manifold of $(1.0)$ - dimension complex subspaces in the $\DC ^{(N+1.N+1)}$ ) is obtained as the factorization of the $(2N+1.2N+2)_{\DR}$ -dimensional level supersurface $$H=h$$ by Hamiltonian action (2.9) of the group $U(1)$. One can choose as the local coordinates of the supermanifold $\DC P(N,N+1)$ in the map $z^m \neq 0$ the functions $w^A_{(m)}=(w^a_{(m)}, \eta ^k _{(m)}),
a \neq m $, where $$w^a_{(m)}=\frac{z^a}{z^m}\quad,\quad
\theta^k _{(m)}=\frac{\eta ^k}{z^m}$$ restricted on the supersurface (2.10). The transition functions for these coordinates from the map $z^n\neq 0$ to the map $z^m\neq 0$ are $$w^a_{(m)} =\frac{w^a_{(n)}}{w^m _{(n)}}, \;\;
\quad \theta^k _{(m)} =
\frac{\theta^k _{(n)}}{w^m _{(n)}}, \quad {\rm where}\quad w^m_{(n)}=
(w^a_{(n)}, w^n_{(n)}=1).$$ These coordinates are invariant under $U(1)$ group action: $$\{ w^a_{(m)} , H \}_0=
\{ \theta^k _{(m)} , H \}_0= 0.$$ So the inherited Poisson bracket on $\DC P(N,N+1)$ is naturally defined by the relation $$\{f,g\}_0^{\rm red}=
\{f,g\}_0\mid_{H=h},$$ where $f,g$ are functions depending on the coordinates $w^A_{(m)}, {\bar w^A_{(m)}}$ (see for details Appendix 1 or \[19\]).
The calculations give us $$\begin{aligned}
\{ w^A_{(m)},w^{ B}_{(m)}\}_0^{\rm red}& =&
\{{\bar w}^{ A}_{(m)},{\bar w}^{ B}_{(m)}\}_0^{\rm red} = 0,\nonumber \\
\{w^A_{(m)},{\bar w}^{ B}_{(m)}\}_0^{\rm red}&=&
(-1)^{p_A p_B +1}\{{\bar w}^B_{(m)},{w}^{A }_{(m)}\}_0^{\rm red} = \\
&= &(i)^{p_A p_B +1} \frac{1+(-i)^{p_C} w^C_{(m)} {\bar w}^{ C}_{(m)}}{h}
(\delta^{AB}+(-i)^{p_A p_B} w^A_{(m)} {\bar w}^{B}_{(m)}) .\nonumber\end{aligned}$$ From (2.12) one obtain that the coordinates $w^A_{(m)}$ provide $\DC P(N.N+1)$ by complex structure and to Poisson bracket (2.13) correspond Kählerian structure with potential: $$K_{(m)}= h\log (1+(-i)^{p_C} w^C_{(m)} {\bar w}^{ C}_{(m)}) .$$
Let us consider now the reduction of the odd Poisson bracket (2.7) on the $\DC^{N+1,N+1}$ by Hamiltonian action of $U(1)$ group. It is easy to check that it defined by an odd Hamiltonian $$Q_2=i(z^k {\bar \eta}^{ k} -
{\bar z}^{ k} \eta^{ k}),$$ ( which is supercharge of previous one), because it is easy to check that for arbitrary function $f$: $${\dot f}=\{H,f\}_0=\{Q_2,f\}_1$$ where $\{\quad,\quad\}_1$ is odd Poisson bracket (2.7). Performingng the reduction as above we obtain the supermanifold $M_{\DR}^{2N+1.2N+1}$ of real dimension $(2N+1,2N+1)$ which evidently can not has (even) complex structure. We define an odd symplectic structure on it similary to even case : the $U(1)$ -invariant functions $(w^a, \theta^a, {\bar w^a}, {\bar \theta^a}, H_0, Q_1)$ where $w^a, \theta^a,$ are defined by (2.11) , $$\begin{aligned}
H_0 &=& z^k {\bar z}^{ k}, \\
Q_1 &=& z^k {\bar \eta}^{ k} +
{\bar z}^{ k} \eta^{ k} ,\end{aligned}$$ restricted on the level supersurface $$Q_2=q_2$$ can be seen as local coordinates of $M_{\DR}^{2N+1.2N+1}$. In these coordinates the odd Poisson bracket is defined by following basic relations $$\{w^a_{(m)},{\bar \theta }^b_{(m)}\}_1^{\rm red}=
\frac{i(1+ w^c_{(m)} {\bar w}^c_{(m)})}{ H_0}\delta^{ab} ,\quad
\{\theta^a_{(m)},H_0 \}_1^{\rm red}=-w^a_{(m)} , \quad
\{Q_1,H_0 \}_1^{\rm red}=H_0 .$$ We see that the same transformations group $U(1)$ of the complex superspace $\DC^{(N+1,N+1)}$ which Hamiltonian action in both cases is defined by (2.9) reduces this superspace to rather different symplectic supermanifolds.
In the following section by Hamiltonian reduction we construct a complex supermanifold which can be considered as a reduction of both of them and which have naturally defined even and odd Kählerian structures.
0
Supergeneralization of $\DC P(N)$ with Even and Odd Kählerian Structures
=========================================================================
In this section we do Hamiltonian reduction of initial $\DC^{N+1,N+1}$ with canonical even structure (2.6) by one generalization of $U(1)$ and the reduction of $\DC^{N+1,N+1}$ with canonical odd structure (2.7) by another generalization of $U(1)$. The complex supermanifolds obtained in both cases appear to be the same (up to diffeomorfism) and can be considered as “intersection” of supermanifolds considered above. This supermanifold provided by even and odd Kählerian structures turns to be associated to the tangent bundle of complex projective space $\DC P(N)$.
Reduction by Even Bracket
-------------------------
Now let us consider at first the reduction of even structure (2.6) on the superspace $\DC^{N+1,N+1}$ by Hamiltonian $H$ and its supercharges $Q_1$ and $Q_2$ (which defined by (2.8), (2.14), (2.16)). They form the superalgebra $$\{Q_r,Q_s\}_0 =2 \delta_{rs}H,\quad \{Q_r ,H \}_0 =\{H ,H\}_0 =0
,\quad r,s=1,2.$$ This superalgebra defines the Hamiltonian action of $(1.2)$– dimensional group of transformations of the $\DC^{N+1,N+1}$ . To every even element $\tilde H =\alpha H + \beta Q_1 +\gamma Q_2$ (where $\alpha$ is even and $\beta$ and $\gamma$ is odd constants) of this superalgebra corresponds one-parametric transformation $z \rightarrow {\tilde z}(t,z)$ via motions equations ${\dot z}=\{\tilde H,z\}_0$. The group of these transformations is the supergeneralization of the $U(1)$ group transformations (2.9). We denote it by $U^s(1)$. Lets define in $\DC^{(N+1,N+1)}$ the level supersurface $M_{h,q_{1},q_{2}}$ by equations $$H=h, \;\;\;\; Q_1=q_1, \;\;\;\; Q_2=q_2.$$ Reduced phase superspace is the factorization of $M_{h,q_{1},q_{2}}$ by the action of $U(1)$ subgroup of $U^s(1)$, because transformations corresponding to $Q_1$ and to $Q_2$ do not preserve (3.2). For pulling down Poisson bracket (2.6) on it we have to choose convenient local coordinates which are $U^s(1)$ -invariant functions on $\DC^{(N+1,N+1)}$ restricted on $M_{h,q_{1},q_{2}}$ (see for details Appendix 1) These coordinates are following $$\begin{aligned}
\sigma^a_{(m)}&=&-i\{w^a_{(m)},Q_+\}=
\theta^a_{(m)}-\theta^m_{(m)} w^a_{(m)}, \\
x^a_{(m)}&=& w^a_{(m)}+i\frac{Q_-}{H}\sigma^a_{(m)},\end{aligned}$$ where $w^a_{(m)}, \theta^a_{(m)}, \theta^m_{(m)}$ are defined by (2.11) and $$Q_{\pm} =\frac{Q_1 \pm iQ_2}{2}.$$ These coordinates provide reduced superspace by complex structure (see Subsection 3.3).
If $f$ and $g$ are $U^s(1)$–invariant functions then $\{f,g\}$ is $U^s(1)$–invariant function too, so from (3.1), (3.3) using Jacoby identity (2.5) one can obtain that their Poisson brackets depend only on $x^a$, ${\bar x^a}$, $ \sigma^a$ $ {\bar \sigma^a}$, and $H$ . The inherited Poisson bracket as well as in previous section is defined by the relation $$\{f,g\}_0^{\rm red}=
\{f,g\}_0\mid_{H=h,Q_{1,2}=q_{1,2}},$$ where $f,g$ are $U^s(1)$ -invariant functions, $\{\quad,\quad\}_0$ is the canonical even bracket (2.6) on $\DC^{N+1,N+1}$. Substituing (3.3), (3.4) in this relation and taking into account (2.13), (3.1), (3.2) and $U^s(1)$–invariance one obtain by straightforward calculations $$\begin{aligned}
\{x^A,x^B\}_0^{\rm red}&=&\{{\bar x}^A,{\bar x}^B\}_0^{\rm red}= 0 ,\quad {\rm
where}\quad x^A =(x^a ,\sigma^a) \nonumber \\
\{x^a,{\bar x^b}\}^{\rm red}_0&=&
i\frac{A}{h}(\delta^{ab}+x^a{\bar x}^b) -
\frac{\sigma^a{\bar \sigma}^b}{h}, \nonumber \\
\{ x^a,\bar\sigma^b \}_0^{\rm red}&=&
i\frac{A}{h}\left( x^a{\bar \sigma}^b +
\mu (\delta^{ab}+x^a{\bar x}^b )\right) \\
\{\sigma^a,{\bar \sigma^b}\}^{\rm red}_0 &=&
\frac{A}{h}\left( (1+i\mu{\bar \mu})\delta^{ab}+x^a{\bar x}^b +
i(\sigma^a +\mu x^a)({\bar \sigma}^b+{\bar \mu}{\bar x}^b \right),\nonumber
\end{aligned}$$ (other relations are obtaned from (3.5) taking into account (2.4)) where $$A= 1+x^a{\bar x}^a -i\sigma^a {\bar \sigma}^a +
\frac{i\sigma^a {\bar x^a} {\bar \sigma}^b x^b}{1+x^c{\bar x}^c }, \;\;\;\;
\mu= \frac{{\bar x}^a \sigma^a}{1+x^b{\bar x}^b } .$$
One can show that to odd structure (3.5) corresponds Kählerian structure with potential $$K= h\log\left(1+x^a \bar x^a - i\sigma^a\bar\sigma^a+\frac{i\sigma^a{\bar
x^a}
\bar \sigma^b x^b}{1+x^c \bar x^c} \right)$$
Reduction by Odd Bracket
------------------------
In the same way we consider the reduction of the $\DC^{(N+1.N+1)}$ with odd structure (2.7) by another supergeneralization $U^{\tilde s}(1)$ of the group $U(1)$ generated by $Q_2$ and $H_0$ (defined by (2.14), (2.15))(as it was mentioned above $Q_2$ defines $U(1)$ group action (2.9) in terms of odd bracket). This group is abelian : $$\{H_0,Q_2\}_1= \{H_0,H_0\}_1=\{Q_2,Q_2\}_1=0$$ so reduced phase superspace have real dimension $(2N.2N)$. The functions $w^a_{(m)}, \sigma^a_{(m)}$, defined by (2.11) and (3.3) commute with $Q_2$ and $H_0$ so their restriction on levels supermanifold $$Q_2=q_2,\quad H_0=h_0$$ are the appropriate local coordinates for pulling down odd Poisson bracket on a reduced superspace. The inherited odd Poisson bracket is defined in the same way as (3.5): $$\{f,g\}_1^{\rm red}=
\{f,g\}_{1}\mid_{H_0=h_0,Q_2=q_2},$$ where $f,g$ are $U^{\tilde s}(1)$–invariant functions, $\{\quad,\quad\}_1$ is the canonical odd bracket (2.7) on $\DC^{N+1,N+1}$ . The calculations give $$\begin{aligned}
\{w^A,w^B\}^{\rm red}_1&=&\{{\bar w}^A,{\bar w}^B\}^{\rm red}_1=0 , \quad{\rm
where} \quad w^A=(w^a,\sigma^a)\nonumber\\
\{w^a,{\bar w^b}\}^{\rm red}_1 &=& 0, \nonumber \\
\{w^a,{\bar \sigma}^b\}_1^{\rm red} &=&
\frac{1+w^c{\bar w}^c}{h_0}(\delta^{ab}+w^a{\bar w}^b) , \\
\{\sigma^a,{\bar \sigma^b}\}_1^{\rm red}&=&
\frac{1+w^c{\bar w}^c}{h_0}(\sigma^a{\bar w}^b-w^a{\bar \sigma}^b)
+\nonumber \\
&+& \left(\frac{\sigma^c{\bar w}^c-w^c{\bar \sigma}^c}{h_0}+
iq_2(1+w^c{\bar w}^c)\right)(\delta^{ab}+w^a{\bar w}^b) \nonumber.\end{aligned}$$ (other relations are obtaned from (3.7) taking into account (2.4)). Corresponding odd Kählerian structure (the local coordinates $(w^a_{(m)},\sigma^a_{(m)})$ provide reduced superspace by complex structure (see Subsection 3.3)). is given by potential $$K_1=
h_ 0\frac
{i(w^a {\bar \sigma^a}-{\bar w^a}\sigma^a)}{1+w^b{\bar w^b}}
+
q_2 \log (1+w^a{\bar w^a}) .$$
Investigation of the Global Properties
---------------------------------------
We obtain two reduced superspaces one with coordinates $x^a, \sigma^a$ and even Kählerian structure with potential (3.6), another with coordinates $w^a, \sigma^a$ and odd Kählerian structure with potential (3.8) . Now we show that they coincide up to diffeomorphism and clarify their global structure. It is not useless for these purposes to investigate the transitions functions from map to map for coordinates $w^a_{(m)}, \sigma^a_{(m)}$ and $x^a_{(m)}, \sigma^a_{(m)}$.
The coordinates $\sigma^a_{(m)}$ transform like differentiatiales of the $w^a_{(m)}$ according their definition (3.3). $$\begin{aligned}
w^a_{(n)}\rightarrow w^a _{(m)}
& =&\frac{ w^a _{(n)}}{w^m _{(n)}} , \\
\sigma^a _{(n)}\rightarrow \sigma^a _{(m)}
&=&\frac{\sigma^a_{(n)}w^m _{(n)} - w^a _{(n)}\sigma^m_{(n)}}
{(w^m _{(n)})^2} , \quad k=0,..., N. \nonumber\end{aligned}$$ where $(w^n _{(n)} =1 ,\;
\sigma^n_{(n)}=0 )$.
From (3.4) and (3.9) it is easy to see that the coordinates $(x^a_{(m)}, \sigma^a_{(m)})$ transform like $(w^a_{(m)}, \sigma^a_{(m)})$: $$\begin{aligned}
x^a_{(n)}\rightarrow x^a_{(m)}&=&\frac{x^a _{(n)}}{ x^m _{(n)}},\\
\sigma^a_{(n)}\rightarrow \sigma^a_{(m)}
& =& \frac{\sigma^a_{(n)}x^m _{(n)} - x^a _{(n)}\sigma^m_{(n)}}
{(x^m _{(n)})^2} \quad (x^n_{(n)}=1,\sigma^n_{(n)}=0) \nonumber\end{aligned}$$ As seen, this supermanifolds have global complex structures.
It allows us to consider these two reduced superspaces as the same because one can identify $(w^a_{(m)}, \sigma^a_{(m)})$ with $(x^a_{(m)}, \sigma^a_{(m)})$. The correspondence $(x^a_{(m)},\sigma^a_{(m)})\rightarrow (w^a_{(m)},\sigma^a_{(m)})$ preserving under the transformations (3.9), (3.10) sets up isomorphism from the functions defining on the reduced superspace with even structure (3.5) on the functions defining on the reduced superspace with odd structure (3.7). The obtained phase superspace we denote by $ S\DC P(N) $.\
Now let us summarize our results . The phase superspace $\DC P(N.N+1)$ which was constructed in the Section 2 as the reduction of $\DC^{N+1.N+1}$ with even canonic structure by the Hamiltonian of superoscillator (without using its supercharges) and now constructed $S\DC P(N)$ have the same underlying manifold - $N$-dimensional complex projective space $\DC P(N)$ . The Kählerian structure which corresponded to (2.13) on the $\DC P(N.N+1)$ as well as the even Kählerian structure with (3.5) for $S\DC P(N)$ pull down to the standard Kählerian structure of underlying complex projective space. $S\DC P(N)$ can be considered as the further reduction of $\DC P(N.N+1)$ by the supercharges. In contrary to $\DC P(N)$ it have naturally defined odd Kählerian structure with potential (3.8) and can be considered as further reduction of $M_{\DR}^{2N+1.2N+1}$ by $H_0$ too $$\DC^{(N+1.N+1)} { \buildrel H\over \longrightarrow}\DC P(N.N+1)
{\buildrel {Q_1,Q_2}\over \longrightarrow} S\DC P(N)
\quad{\rm ({even \quad reduction})}$$ $$\DC^{(N+1.N+1)}{\buildrel {Q_2}\over
\longrightarrow}M^{2N+1.2N+1}_{\DR}
{\buildrel {H_0}\over \longrightarrow}S\DC P(N)
\quad{\rm {(odd \quad reduction})}.$$ Moreover from the equations (3.9), (3.10) it is easy to see that $S\DC P(N)$ with local coordinates $x^a_{(m)}, \sigma^a_{(m)}$ is associated to the $T\DC P(N)$ - tangent bundle of the underlying manifold $\DC P(N)$ because the even coordinates from map to map transform through themself only and odd coordinates transform as differentials of even ones \[1\] (see also Appendix 2).\
From this point of view it becomes natural the following property of the odd symplectic structure (3.7). One can show that in the coordinates $${\tilde \sigma}_a = g_{a\bar b}\bar{\sigma^b} ,$$ where $g_{ab}$ is the Kählerian metric of the underlyind projective space , the odd symplectic structure turns out to be canonical one if $Q_2 =0$ (for general case, if $Q_2 \neq 0 $ see Appendix 2). $${\tilde \Omega}^1 = dw^a \wedge d{\tilde \sigma}_a + {d\bar w^a}\wedge
{d\bar {\tilde \sigma}_a}$$ Indeed in the coordinates $(w^a ,\sigma_a)$ $S\DC P(N)$ is associated to $T^* \DC P(N)$ - cotangent bundle of $\DC P(N)$, which have naturally defined canonical symplectic structure \[19\].\
It has been mentioned in Introduction that these constructions have general meaning. Indeed for every Kählerian manifold $M$ with local complex coordinates $w^a$ one can consider the complex supermanifold ${SM}$ ($dim_{\DC} SM = (dim_{\DC}M. dim_{\DC}M)$) with local coordinates $w^a, \sigma^a$ which is associated to $TM$. Then the local functions $$\begin{aligned}
K_0(w,{\bar w},\sigma,{\bar \sigma})&=& K(w,{\bar w})+
F(ig_{a{\bar b}}(w,{\bar w})\sigma^a{\bar \sigma}^b), \quad p( K_0 )=0\\
K_1(w,{\bar w},\sigma,{\bar \sigma})&=&
\epsilon\frac{\partial K(w,{\bar w})}{\partial w^a}\sigma^a+
{\bar \epsilon}\frac{\partial K(w,{\bar w})}
{\partial {\bar w^a}}{\bar \sigma^a}+\alpha K(w,\bar w),\quad p( K_1 )=1
\end{aligned}$$ ( where $K(w,\bar w)$ is the Kählerian potential of $M$ , $g_{a\bar b}$- corresponding Riemannian metric, $F(r)$-arbitrary scalar function such that $F'(0) \neq 0$, $\epsilon$ is even complex constant an $\alpha$ is real odd one) can be considered as the potentials which correctly define global even and odd Kählerian structures on $SM$ \[16\].
In the case $M=\DC P(N)$ we obtain immediately the structures constructed above putting in (3.11), (3.12) $ K(w,\bar w)=\log (1+w^a \bar w^a
),
F(r)=\log (1-r), \epsilon=i, \alpha=q_2$. 0
Operator $\Delta$ and bi-Hamiltonian Mechanics
===============================================
Now we want to discuss the properties of some supergeometrical constructions which can be defined in natural way on the supermanifolds provided by even and odd symplectic structures studying them on the supermanifold constructed above.
The supermanifolds which are associated in some coordinates to tangent bundle (see Appendix 2) can be considered as “gauge fixing” objects for the studying the supergeometrical constructions which in this case have to reduce to the well-known geometrical objects. So this constructions can be considered as the generalization on supercase of the corresponding geometrical objects.
From this point of view it is interesting to look at the explicit expressions for the “operator $\Delta$” and the bi-Hamiltonian mechanics on the $S\DC P(N)$ provided by odd and even brackets (3.6), (3.9) (similar expressions for the supermanifolds provided by two Kählerian structures with potentials (3.11), (3.12) see in \[16\]).
Operator $ \Delta$ on $S\DC P(N)$
-----------------------------------
On the supermanifold ${\cal M}^{m.m}$ with coordinates $z^A=(x^1,\ldots,x^{m},\theta^1,\ldots,\theta^m)$ which is provided by odd symplectic structure with Poisson bracket $\{\quad,\quad\}_1$ and the volume form $dv=\rho(x, \theta)d^{m}x d^{m}\theta$ one can invariantly define the odd differential operator of the second order, so called “operator $ \Delta$” which is invariant under the transformations preserving the symplectic structure and the volume form \[2, 11\]. Its action on the function $f(x,\theta)$ is the divergence of the Hamiltonian vector field ${\bf D}_{f} =\{f,z^A \}_1\frac{\partial^L}{\partial z^A}$ with the volume form $dv$: $$\Delta f =div ^{\rho} {\bf D}_{f} \equiv \frac{{\cal L}_{{\bf D}_f} dv}{dv},$$ where ${\cal L}_{{\bf D}f}$ — Lie derivative along ${{\bf D}_f}$ \[1\]. In coordinate form $$\Delta f=\frac{1}{\rho}(-1)^{p(A)}
\frac{\partial^L}{\partial z^A}\left(\rho\{z^A,f\}_1\right)$$ The “operator $ \Delta$” have no analogs with even symplectic structures — the oddness of the Poisson bracket $\{\quad,\quad\}_1$ which force that operator (4.2) to have dependence of second derivatives.
If the density $\rho=1$ and $\{\quad,\quad\}_1$ has the canonical form (1.2) then $\Delta$ is in the canonical form $$\Delta ^{\rm can} f =2\frac{\partial ^2 f}{\partial x^i \partial \theta ^i}
,$$ which is well-known from BV-formalism \[2-4\].
It is easy to obtain from (4.1) using Jacobi identities, Leibnitz rules and the transformation law of integral density $\rho(z)$ that generelized operator $\Delta$ (4.2) is connected with corresponding odd bracket by the same expressions as canonical operator $\Delta^{\rm can}$ (4.3) connected with canonical odd bracket (1.2) \[3, 4\]: $$\begin{aligned}
\Delta \{f,g \}_1& =&\{f,\Delta g \}_1 +(-1)^{p(g)+1}\{\Delta f ,g \}_1
\nonumber \\
(-1)^{p(g)}\{f,g \}_1 &=&\frac{1}{2}\left ( \Delta(fg) - f\Delta g
-(-1)^{p(g)}(\Delta f)g \right )\nonumber \\
\Delta'f &=& \Delta f +\{\log{{\cal J}} ,f \}_1 ,\nonumber\end{aligned}$$ where ${\cal J}$-Jacobian of canonical transformation of odd bracket, $\Delta'$- operator $\Delta$ in new coordinates. However the nilpotency condition $$\Delta^2=0$$ are violated for arbitrary $\rho (x,\theta)$.
For example , if symplectic structure is canonical, (4.4) hold if $\rho (x,\theta)$ satisfy to the equation $$\Delta \rho =0$$ which is master equation of BV-formalism for the action $S=\log \rho$ . Then $\Delta$ corresponding to operator of BRST transformation \[2-4\]. It is interesting to study the connection between the condition (4.4) and the possibility to reduce (4.2) to (4.3) by the suitable transformation of the coordinates.
If the supermanifold ${\cal M}$ provided by even symplectic structure $\Omega^0$ also here one can put into (4.2) the density $\rho$ , which is invariant under canonical transformations of $\Omega^0 $ \[19, 20\]: $$\rho(z)=\sqrt{{\rm Ber} \Omega_{AB}}.$$ Let ${\cal M} = S\DC P(N)$ provided by odd Poisson bracket (3.7) ( with $q_2
=0$) and even one (3.5).The invariant (under canonical transformations of (3.5)) density $\rho$ on it has the form $$\rho(w, \bar w, \theta, \bar \theta) =(1-r)^2 \equiv (1-ig_{a\bar b}\theta^a
\bar{\theta^b})^2$$ where $$g_{a\bar b}=\frac{1}{1+w^c \bar w^c}\left (\delta_{a\bar b} -
\frac{\bar w^a w^b}{1+w^c \bar w^c} \right)$$ –Kählerian metric of $\DC P(N)$ ($r$ corresponds to cohomologies on $\DC P(N)$) . The operator $\Delta$ on $S\DC P(N)$ with this density takes the folowing form $$\Delta f = \frac{1}{\rho}
\left(\nabla^a \frac{\partial^L}{\partial\theta^a} +
{\overline {\nabla^a}}\frac{\partial^L}{\partial{\bar \theta}^a}\right)(\rho
f),$$ where $$\nabla_a=\frac{\partial}{\partial w^a} -
\Gamma^c_{ab}\theta^b\frac{\partial^L}{\partial\theta^c},\quad
{\overline {\nabla^a}}=g^{{\bar a}b}\nabla_b ,$$ $\Gamma^c_{ab}=g^{\bar d c}g_{a \bar d ,b} \equiv
-\frac{\bar w^a \delta^c _b +\bar w^b \delta^c_a}{1+w^d \bar w^d}$ – the Christoffel symbols of the Kählerian metric (4.7) on $\DC P(N)$. Nilpotency condition (4.4) is satisfied obviously . The operator (4.8) corresponds to the operator of covariant divergency $\delta = \ast d \ast$ on $\DC P(N)$.
Since ${\cal M}=SM$ with Kählerian potentials (3.11), (3.12) ($\epsilon =i, \alpha = 0$) operator $\Delta$ is also defined by the expression (4.8) \[16\], where $\Gamma^c_{ab}$– the Christoffel symbols of the Kählerian metric on underlieng manifold $M$, $$\rho= \frac{{\rm det}(\delta^a_b+iF^{\prime}(r)
R^a_{bc{\bar d}}\theta^c\theta^d)}
{F^{\prime}(r)^{N-1}(F^{\prime}(r) +F^{\prime\prime}(r)r)}$$ where $ R^a_{bc\bar d}=(\Gamma^a_{bc})_{,\bar d}$ is the curvature tensor on $M$, $r=ig_{a\bar b}$. (We see that $\rho$ depends on Chern classes of the underlying Kählerian manifold.It is interesting to compare (4.9) with the general formulas for characteristic classes on the supermanifolds \[20\].) To operator $\Delta$ on $SM$ is corresponds the covariant divergence on the underlying Kählerian manifold $M$.
Bi-Hamiltonian Mechanics on $S\DC P(N)$
----------------------------------------
Here we deliver explicit formulae for the even vector fields preserving even and odd Poisson brackets (bi-Hamiltonian mechanics) (3.5), (3.7) on ${S\DC P(N)}$. In other words we have to find the pairs of the functions $(H,Q)$ ($p(H)=0$, $p(Q)=1$) on $S\DC P(N)$ such that for arbitraty function $f:$ $$\{H,f\}_0 = \{Q,f\}_1,$$ where $\{\quad,\quad\}_0$ ($\{\quad,\quad\}_1$) defines by (3.5), (3.7). To every pair $(H,Q)$ the solution of (4.10) corresponds vector field $${\bf D}_{H,Q} =\{H,z^A \}_0\frac{\partial^L}{\partial z^A}=\{Q,z^A
\}_1\frac{\partial^L}{\partial z^A}$$ These fields form a finite-dimensional Lie algebra \[13\] and they are defined by Killing vectors of the underlying manifold $M$ \[16\]. The solutions of the (4.10) is following: $$\begin{aligned}
H&=&H_0 -
\frac{i}{1-r}\frac{\partial^2 H_0}{\partial w^a \partial {\bar w}^b}
\theta^a{\bar \theta}^b, \nonumber\\
Q&=&i\left(\frac{\partial H_0}{\partial w^a}\theta^a -
\frac{\partial H_0}{\partial {\bar w}^a}{\bar \theta}^a\right),\nonumber\end{aligned}$$ where $$H_0=\frac{h_{a{\bar b}}w^a {\bar w}^b - {\rm tr} h + h_a w^a +
{\overline {h_a w^a}}} {1+w^c{\bar w}^c},$$ $h_{a{\bar b}}$ are arbitrary Hermitian matrices and $h_a$ – arbitrary complex numbers. Corresponding vector field $${\bf D}_{H, Q} = V^a (w)\frac{\partial}{\partial w^a}+ V^a_c(w)\theta ^c
\frac{\partial}{\partial \theta ^a} ,$$ where $$V^{a}(w)=ig^{\bar b a}\frac{\partial H_0(w,\bar w)}{\partial \bar w^b}$$ is the Killing vector of $\DC P(N)$. Since ${\bf D}_{H, Q}$ defined by (4.11) is holomorphic and Hamiltonian for the both brackets, it is the Killing vector for both Kählerian structures.
Bi-Hamiltonian mechanics on superanifold $SM$ with symplectic structures, defining by (3.11), (3.12) have a similar form (4.11), where $V^a$ is Killing vector of underlying Kählerian manifold $M$ \[16\].
Acknowledgments
===============
We would like to thank A.V. Karabegov for useful discussions , R.L. Mkrtchian and V.I. Ogievetsky for interest to this work.
1. On Procedure of Hamiltonian Reduction
=========================================
In this Appendix we retell the main algebraic notions of Hamitonian reduction mechanism using language which is maximally adapted for our purposes and can be evidently generalized on supercase.(The detailed considerations see in \[19\]). Let $M$ be symplectic space with symplectic structure $\Omega$ and $\Gamma(M)$ be an algebra of functions on $M$ . Poisson bracket $\{\quad,\quad\}$ corresponding to $\Omega$ defined by the following relation $\{f,g\} = \Omega({\bf D}_f,{\bf D}_g)$ where ${\bf D}_f$ is the vector field corresponding to $f$ via the equation $$\Omega({\bf D}_f, {\bf V})=df({\bf V})$$ for arbitrary vector field ${\bf V}$.
Let $C$ be an subalgebra in $\Gamma(M)$ which is closed under $\{\quad,\quad\}$ . The algebra of functions $\Gamma(M)$ has two algebraic operations – usual multiplicative structure and Lie algebra multiplication provided by Poisson bracket $\{\quad,\quad\}$. Further if it is not pointed we suppose the first operation only.)
Let the functions $F_1,\ldots,F_k$ be generators of $C$. In this case $\{F_i,F_j\}=c_{ij}^k F_k$ where $c_{ij}^k$ are constants, the functions $\{F_i\}$ generate Hamiltonian action of the group $G$ (corresponding to Lie algebra with structure constants $c_{ij}^k$) on the M . To every function $F_i$ corresponds $G$ group infinitesimal transformation via the vector field ${\bf D}_{F_i}.$ and ${\bf D}_{\{F_i,F_j\}}=[{\bf D}_{F_i},{\bf D}_{F_i}]$.
To subalgebra $C$ corresponds the reduction procedure from $M$ to symplectic manifold $M^{red}$.
Let $M_p$ be the level manifold in $M$ defined by $$F_i=p_i$$ and $G_p$ – its isotropy group: $G_p=\{g\in G: gp_ig^{-1}=p_i\}$. Then $M^{\rm red}=M_p /G_{p_i}$ and $\Omega$ is pulling down on the $\Omega^{\rm red}$ on $M^{\rm red}$ defining its symplectic structure.
For supercase it is more convenient to describe $M^{\rm red}$ and $\Omega^{\rm red}$ correspondingly in the terms of $\Gamma^{\rm red}$ – algebra of the functions on it and $\{\quad,\quad\}^{\rm red}$ – Poisson bracket corresponding to $\Omega^{\rm red}$. (The generators of $\Gamma^{\rm red}$ are the coordinates of $M^{\rm red}$.) Let $B(M)$ be an subalgebra of the functions which is “orthogonal" to subalgebra $C$ by Poisson bracket $\{\quad,\quad\}$. $$B=\{\Gamma\ni f: \{f,g\}=0 \quad\forall g\in C\}$$ in other words $f\in B$ iff $\{f,F_i\}=0$.
Because of Jacoby identity $B$ is Lie algebra too: $$f,g\in B\Rightarrow \{f,g\}\in B,$$ so $B$ is the subalgera of $G$ — invariant functions of $\Gamma$.
To level manifold $M_p$ corresponds the ideal ${\cal J}$ in the algebra $\Gamma$ generating by the functions $F_i - p_i$ $${\cal J}=\{\Gamma \ni f: f=\sum \alpha_i(F_i - p_i)
\quad{\rm where}\quad \alpha_i\in \Gamma\}$$ $B\cap{\cal J}$ is the ideal in B too, so on can consider subalgebra $$\Gamma^{\rm red}=B/B\cap {\cal J}.$$ It is the algebra of functions on reducted space $M^{\rm red}$. The Poisson bracket $\{\quad,\quad\}^{\rm red}$ on $ \Gamma^{\rm red}$ is defined in the following way. For any $[f],[g]\in B$, where $[f]$ is the equivalence class of the function $f\in B$ in $ \Gamma^{\rm red}$ $$\{[f],[g]\}^{\rm red} = [\{f,g\}].$$ To check the correctness of this definition we note that if $f,g\in B$ then $\{f,g\}\in B$ too . If $f\rightarrow {\tilde f}=f+h$ where $h= \sum {\alpha_i(F_i - p_i)}\in B\cap {\cal J}$ then $\{h,g\}\in B$ and $$\{h,g\}=\{\sum {\alpha_i (F_i - p_i)}, g\}=
\sum \{\alpha_i, g\}(F_i -p_i) \in {\cal J}$$ because $\{F_i-p_i,g\}=0$ . So $\{{\tilde f},g\}-\{f,g\}
\in B\cap {\cal J}$.
The reduction procedure leads to the fact that if dynamical system on $M$ is described by Hamiltonian $H$ which is $G$ - invariant ($H\in B$) and at $t=0$ the conditions $F_i = p_i$ hold then
i\) these conditions preserve in a time,
ii\) $[f^t]=[f]^t$,
where $h^t$ we denote the evolution of the function $h$ in the time $t$ via motion equations ${\dot h}=\{H,h\}$ $[{\dot h}]=\{[H],[h]\}^{\rm red}$ .
As example we retell in these terms the reduction procedure performed in the Subsection 3.1.
We consider as $C$ the algebra of functions on the $\DC^{(N+1.N+1)}$ which explicitly depend on the functions $H,Q_1,Q_2$ playing the role of generators $F_i$. The “orthogonal” subalgebra $B$ of $U^s(1)$ - invariant functions is the algebra of functions explicitly depending on $x^a$, ${\bar x^a}$, $ \sigma^a$ $ {\bar \sigma^a}$, and $H$ functions. The functions $f(H-h)+g(Q_1-q_1)+r(Q_2-q_2)$ where $f,g,r$ are arbitrary functions consist the ideal ${\cal J}$ . $B\cap {\cal J}$ - the $U^s(1)$ - invariant part of this ideal consists on the functions depending only on $H$. So the generators of the algebra $\Gamma^{red}=B/B\cap {\cal J}$ are $[x^a]$, $[{\bar x^a}]$, $[ \sigma^a]$ $ [{\bar \sigma^a}]$ and the functions (coordinates) $x^a$, ${\bar x^a}$, $ \sigma^a$ $ {\bar \sigma^a}$ are their corresponding representatives.
2. Supermanifolds and Linear Bundles
====================================
In this Appendix we briefly mention the connection between supermanifolds and corresponding linear bundles to the extent necessary for our purposes. (See in details in \[1\].)
Let $TM$ be the tangent bundle to the manifold $M$. $x^a_{(m)}$ are the local coordinates on the $M$ in $m$-th map and the $(x^a_{(m)}, v^a_{(m)})$ are the corresponding local coordinates on $TM$ ( $v^a_{(m)}$ are coordinates of tangent space in the basic $\frac{\partial}{\partial x^a_{(m)}}$). From map to map $$x^a_{(k)} \rightarrow x^a_{(m)} =x^a_{(m)} (x^a_{(k)}),\quad
v^a_{(k)} \rightarrow v^a_{(m)} =
\frac{\partial x^a_{(m)}}{\partial x^b_{(k)}}v^b_{(k)}.
\eqno (A2.1)$$ Considering for every map the superalgebra generating by $(x^a_{(m)}, \theta^a_{(m)})$ where $x^a_{(m)}$ are even and $\theta^a_{(m)}$ are odd, transforming from map to map like $(x^a_{(m)}, v^a_{(m)})$ in the (A2.1) ($v\leftrightarrow \theta$) we go to supermanifold ${\cal M}$ which is associated to $TM$ in the coordinates $(x^a_{(m)}, \theta^a_{(m)})$ For the coordinates $(x^a_{(m)}, \theta^a_{(m)})$ on the ${\cal M}$ the more general class of transformations is admittable : $$x^a\rightarrow {\tilde x}^a(x^a,\theta^a) \quad
\theta^a\rightarrow {\tilde \theta}^a(x^a,\theta^a)$$ which do not correspond to (A2.1). In particularly if $\theta^a \to {\tilde \theta}_a= g_{ab}\theta^b,$ where $g_{ab}$ is some Riemanian metric on $M$ then the supermanifold ${\cal M}$ in the coordinates $(x^a,{\tilde \theta}_a )$ is associated to the cotangent bundle $T^*M$ of $M$.
On the supermanifolds which can be associated in some coordinates to tangent or cotangent bundle the superstructures evidently are reduced to the standard geometrical objects.
For example on the supermanifold ${\cal M}$ considered here the canonical odd (Buttin) bracket $\{\quad,\quad\}_1$ (defined by basic relations $\{x^i , {\tilde \theta}_j\}_1=\delta^i_j )$ is corresponding to the Schouten bracket $[\quad,\quad]$ of the polyvector fields on $M$: To polyvector field ${\bf T}= T^{j_1,\ldots,j_k}(x)$ on $M$ corresponds the function $\rho {\bf T} = T(x,\theta)=
T^{j_1,\ldots,j_k}(x)\theta_{j_1}\ldots\theta_{j_k}$ on the ${\cal M}$ such that $$\{\rho{\bf T},\rho{\bf U}\}_1= \rho[{\bf T},{\bf U}] .$$ Similarly operator $D=\theta^a\frac{\partial}{\partial x^a}$ on the ${\cal M}$ corresponds to the exterior differentiation operator on $T^*M$ and operator $\Delta $ to the divergence \[1 \].
On one hand these type supermanifolds can be served as the good tests for studying superstructures on other hand we can use them as condensed language for constructed the geometrical structures in superterms. We deliver one example which is strightly connected with the considerations in the Subsection 3.2.
The reduction procedure performed in Section 3 was indeed the prolongation of the $M{\buildrel {\rm {reduction}}\over \rightarrow}M^{\rm red}$ to the $TM{\buildrel {\rm {reduction}}\over \rightarrow}TM^{\rm red}$ in the case $M = \DC ^{N+1}$, $M^{\rm red} = \DC P(N).$
Now for the odd structure reduction we show that in the general case. Let $M$ be the symplectic manifold with symplectic structure defined by Poisson bracket $\{\quad,\quad\}$ and the functions $I_r.$ generate Hamiltonian action of the Lie group $G$ on it: $$\{I_r,I_s\}=c^t_{rs}I_m$$ where $c^t_{rs}$ are the structure constants of the Lie algebra ${\cal G}$ of $G$. Let $M^{\rm red}$ be the manifold obtained by reduction: $M^{\rm red} = M_p /G_p$ where $M_p =\{x\in M: I_r(x)=p_r\}$ is the level manifold and $G_p$ - is its isotropy group.
Let $x^i$ be the local coordinates on $M$ and $y^a$ - the local ones on $M^{\rm red}$ in which the reduction was performed : $\{y^a(x), I_r(x)\}=0$. Then $$w^{ab} =\{y^a, y^b\}^{\rm red} = \{y^a(x), y^b(x)\}\mid_{I_r (x)=p_r}
\eqno(A2.2)$$ defines the reduced Poisson bracket (and symplectic structure) on $M^{\rm red}$.
If ${\cal M}$ is supermanifold associated to $T^{*}M$ in the local coordinates $(x^i,\theta_i)$ and Poisson bracket $\{\quad,\quad\}_1$ defines the odd canonical structure on it then it is easy to see that the functions $$Q_r=\{I_r(x),x^i\}\theta_i=\{I_r,F\},\quad {\rm where}\quad
F=\frac{1}{2}\{x^i,x^j\}\theta_i\theta_j$$ define the same Hamiltonian action of the group $G$ on the $M$ in the terms of odd bracket: for arbitrary function $f(x)$ on $M$ $$\{f(x),I_r(x)\}=\{f(x),Q_r(x,\theta )\}_1.$$ Moreover the functions $(Q_r, I_r)$ define the Hamiltonian action of the supergeneralization of the group $G$ on the ${\cal M}$ in the terms of odd bracket: $$\{Q_r, Q_s\}_1=c^t_{rs}Q_t, \quad \{Q_r, I_s\}_1=c^t_{rs}I_t,
\quad \{I_r, I_s\}_1=0 .$$
One can show that the functions $y^A =(y^a, \eta^a =\{y^a(x), x^i\}\theta_i
=\{y^a (x), F(x, \theta)\}\_1 )$ play the role of local coordinates on $( dim M^{\rm red}. dim M^{\rm
red})$–dimensional reduced supermanifold ${\cal M}^{\rm red}$ : $$\{y^A(x,\theta ),I_r \}_1 =\{y^A (x,\theta ),Q_r \}_1 =0,$$ and in this coordinates ${\cal M^{\rm red}}$ associated to $TM^{\rm red}$. The functions $y^A =(y^a, \eta^a)$ one can used for reduction of odd bracket $\{\quad ,\quad \}_1$ on ${\cal M}^{\rm red}$: $$\{y^a,y^b\}^{\rm red}_1 =0,\quad
\{y^a,\eta^b\}^{\rm red}_1 =w^{ab},$$ $$\{\eta^a,\eta^b\}^{\rm red}_1 = \frac{\partial w^{ab}}{\partial
y^c}\eta^c
+\frac{\partial w^{ab}}{\partial p^r} q^r$$ where $w^{ab}(y,p)$ is given by (A2.2) and $I_r=p_r,\;\;
Q_r=q_r$ define the level supermanifold in ${\cal M}$.
One can construct local coordinates $(y^a,{\tilde \eta}_a)$ such that in these coordinates ${\cal M}^{\rm red}$ is associated to $ T^{*}M^{\rm red}$ : $${\tilde \eta}_a = w_{ab}\eta^b -\frac{\partial A_a}{\partial p^k} q^k,$$ where $$w_{bc}w^{ca}=\delta^a_b \;\;\; {\rm and} \;\;\;
\frac{\partial A_a}{\partial y^b}-
\frac{\partial A_b}{\partial y^a} =w_{ab}.$$ In this coordinates reduced symplectic structure coincides with canonical one : $$\{y^a,y^b\}^{\rm red}_1=0,\quad \{\tilde \eta^a ,\tilde \eta^b\}^{\rm
red}_1 =0 ,\\
\{y^a,{\tilde \eta}^b\}^{\rm red}_1=\delta^a_b\quad .$$
[33]{}
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[^1]: E-MAIL:[email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Due to the evolving nature of power systems and the complicated coupling relationship of power devices, it has been a great challenge to identify the contingencies that could trigger cascading blackouts of power systems. This paper provides an effective approach to identifying the initial contingency in power transmission networks, which are equipped with flexible alternating current transmission system (FACTS) devices, high-voltage direct current (HVDC) links and protective relays. Essentially, the problem of contingency identification is formulated in the framework of nonlinear programming, which can be solved by the Jacobian-Free Newton-Krylov (JFNK) method to circumvent Jacobian matrix and reduce the computational cost. Notably, the proposed identification approach is also applied to complicated cascading failure models of power systems. Finally, numerical simulations are carried out to validate the proposed identification approach on IEEE $118$ Bus Systems. The proposed approach succeeds in reconciling the rigorous optimization formulation with the practical modeling of cascading blackouts.'
author:
- 'Chao Zhai, Hehong Zhang, Gaoxi Xiao and Tso-Chien Pan [^1]'
title: Contingency Identification of Cascading Failures in Power Transmission Networks
---
Keywords: Cascading failures, contingency identification, power transmission networks, nonlinear programming
Introduction
============
The past decades have witnessed several large blackouts in the world such as India Blackout (2012), US-Canada Blackout (2003), Italy Blackout (2003) and Southern Brazil Blackout (1999) to name just a few, which have left millions of residents without power supply and caused huge financial losses [@mcl09]. In such catastrophe events, the initial contingencies ($e.g.$ extreme weather, terrorist attack and operator error) play a crucial role in triggering the cascading outage of power systems. It is reported that the mal-operation of a protection relay is the key “trigger" of the final line outage sequence in most blackouts [@beck05]. For instance, conventional relays may lead to unselective tripping under high load conditions, which could initiate the chain reaction of branch outages under certain conditions (e.g., a wrong relay operation of Sammis-Star line in the 2003 US-Canada Blackout [@beck05]). The reliability and resilience of power grids are closely related to the proactive elimination of disruptive initial contingencies. Thus, it is vital to identify the initial contingency that causes the most severe blackouts and work out remedial schemes against cascading blackouts in advance.
In practice, electrical power devices such as FACTS devices, HVDC links and protective relays serve as the major protective barrier against cascading blackouts. To be specific, FACTS devices significantly contribute to the stability improvement of power systems, while HVDC links behave like a “firewall" to prevent the propagation of cascading outages. Actually, the FACTS devices have been widely installed in power transmission networks to improve the capability of power transmission, controllability of power flow, damping of power oscillation and post-contingency stability. As a series FACTS device, the thyristor-controlled series capacitor (TCSC) allows fast and continuous adjustments of branch impedance in order to control the power flow and improve the transient stability [@jov05]. In addition, the HVDC links assist in preventing cascades propagation and restoring the power flow after faults. For example, Québec power system in Canada survived the cascades in the 2003 US-Canada Blackout due to its DC interconnection to the US power systems [@beck05]. As the most common protection device, protective relays of power system react passively to the system oscillation and promptly remove the overloading elements without affecting the normal operation of the rest of the system. Meanwhile it allows for time delay of abnormal oscillations to neglect the trivial disturbances and avoid the overreaction to the transient state changes [@jia16]. It is necessary to take into account the protection mechanism of the above power devices for the practical cascading dynamics of power systems.
Owing to simplicity, efficiency and scalability in the simulation, the DC power flow model has been widely adopted to investigate cascading failures of power systems [@alm15; @yan15]. It is demonstrated that the DC power flow model is able to assess the vulnerability of power grids and reveal informative details of cascading failure process, including the size, contributing factors and the duration of cascading failures [@yan15]. Additionally, the model predictive control can be applied to mitigate the cascading effect of severe line-overload disturbances in power systems [@alm15]. Actually, the DC power flow model is usually regarded as a good substitute for the AC based model in high voltage power transmission networks [@zhai17a; @stot09]. As a result, the DC power flow equation is employed in this work to compute the transmission power on branches of power transmission networks.
So far, cascading blackouts of power systems have been investigated through two distinct routes. Specifically, some researchers aim at the strict mathematical formulation for the exploration of vulnerable elements in power systems regardless of the transient response and protection mechanisms [@alm15; @tae16], while others focus on the practical physical process and accurate modeling of cascading blackouts [@jia16; @yan15]. While the former may fail to reflect the real physical characteristic of cascading failures, the latter is in lack of a rigorous theoretical framework. This work attempts to fill the gap between the practical modeling of cascading blackouts and the strict mathematical formulation by properly decoupling the optimization problem and cascading dynamics of power grids. The main contributions of this paper are listed as follows
1. Propose the cascading dynamics of power transmission networks equipped with FACTS devices, HVDC links and protective relays.
2. Formulate the problem of contingency identification with nonlinear programming and solve it via the efficient numerical method.
3. Validate the proposed approach on the large-scale power transmission networks using different protection schemes.
The outline of this paper is organized as follows. Section \[sec:prob\] presents the cascading dynamics of power transmission networks. Section \[sec:opt\] provides the optimization formulation and theoretical results on the contingency identification, followed by numerical methods in Section \[sec:num\]. Next, the identification approach is validated in Section \[sec:sim\]. Finally, we conclude the paper and discuss future work in Section \[sec:con\].
Cascading Dynamics {#sec:prob}
==================
This section aims to characterize the cascading evolution of power transmission networks subject to the initial contingency and system stresses. Figure \[cascade\] presents the cascading process of power systems after the initial contingency is added on the system. First of all, the FACTS devices take effect to adjust the branch admittance and balance the power flow for relieving the stress of power networks. If the stress is not eliminated, protective relays will be activated to serve the overloading branches on the condition that the timer of circuit breakers runs out of the preset time. Under certain circumstances, the outage of overloading branches may result in severer stress of power networks and ends up with having cascading blackouts of power systems. To describe the cascading process, we introduce the concept of cascading step. Essentially, a cascading step is defined as one topological change ($e.g.$, one branch outage) of power networks due to contingencies, human interferences or the branch overloads. The models of the DC power flow, FACTS devices, HVDC links and protective relays are presented in sequence.
\[0.07\][![\[cascade\] Cascading failure process of power transmission networks.](pro.jpeg "fig:")]{}
DC Power Flow Model
-------------------
For high-voltage power transmission networks, the DC power flow equation is well qualified to describe the quantitative relationship of injected bus power, branch susceptance and voltage angle as follows $$\label{dc_pfe}
P_b=A^Tdiag(B)A\theta$$ where $A$ denotes the branch-bus incidence matrix [@stag68] and $\theta$ refers to the vector of voltage angles. $P_b$ represents the vector of injected power on each bus. Additionally, $B=(B_{1},B_{2},...,B_{n})$ is the susceptance vector for branches, and each element $B_{i}$ is given by $$B_{i}=-\frac{1}{X_{C,i}+X_i}, \quad i\in I_n=\{1,2,...,n\}$$ where $X_{C,i}$ denotes the reactance of TCSC equipped on Branch $i$, and $X_i$ represents the original reactance of Branch $i$. The DC power flow equation (\[dc\_pfe\]) can be solved as $$\theta=\left(A^Tdiag(B)A\right)^{-1^*}P_b$$ where the operator $-1^{*}$ denotes the operation of matrix inverse, which is defined in [@cz17]. Then the vector of transmission power on each branch can be computed by $$\label{p_e}
P_e=diag(B)A\left(A^Tdiag(B)A\right)^{-1^*}P_b$$ Notably, the generator bus connected to the largest generating station is selected as the slack bus, and thus the power variation of slack bus accounts for a small percentage of its generating capacity.
FACTS Devices
-------------
\[0.07\][![\[tcsc\_con\] Control diagram of TCSC on branches.](TCSC.jpeg "fig:")]{}
FACTS devices can greatly enhance the stability and transmission capability of power systems. As an effective FACTS device, TCSC has been widely installed to control the branch impedance and relieve system stresses. The dynamics of TCSC is described by a first order dynamical model [@pas95] $$\label{tcsc}
T_{C,i}\frac{d{X}_{C,i}}{dt}=-X_{C,i}+X_{ref,i}+u_i, \quad X_{\min,i}\leq X_{C,i}\leq X_{\max,i}\quad i\in I_n$$ where $X_{ref,i}$ refers to its reference reactance of Branch $i$ for the steady power flow. $X_{\min,i}$ and $X_{\max,i}$ are the lower and upper bounds of the branch reactance $X_{C,i}$ respectively and $u_i$ represents the supplementary control input, which is designed to stabilize the disturbed power system [@son00]. For simplicity, PID controller is adopted to regulate the power flow on each branch $$\label{pid}
u_i(t)=K_P\cdot e_i(t)+K_I\cdot\int_{0}^{t}e_i(\tau)d\tau+K_D\cdot\frac{de_i(t)}{dt}$$ where $K_P$, $K_I$ and $K_D$ are tunable coefficients, and the error $e_i(t)$ is given by $$e_i(t)=\left\{
\begin{array}{ll}
P^{ref}_{e,i}-|P_{e,i}(t)|, & \hbox{$|P_{e,i}(t)|\geq P^{ref}_{e,i}$;} \\
0, & \hbox{otherwise.}
\end{array}
\right.$$ Here, $P^{ref}_{e,i}$ and $P_{e,i}(t)$ denote the reference transmission power and the actual transmission power of Branch $i$, respectively. Note that TCSC fails to function when the transmission line is severed. Figure \[tcsc\_con\] presents the diagram about the operation of TCSC via PID controller to reach the reference transmission power. First of all, we compute the error $e_i(t)$ between the actual power $P_{e,i}(t)$ and the reference power $P^{ref}_{e,i}$. Next, the PID controller produces the control input $u_i(t)$ based on $e_i(t)$, which regulates the reactance of TCSC on Branch $i$. Finally, the actual power $P_{e,i}(t)$ will converge to the reference power $P^{ref}_{e,i}$ as time goes.
HVDC Links
----------
\[0.075\][![\[hvdc\]Schematic diagram of monopolar HVDC link and its equivalent circuit.](HVDC.jpeg "fig:")]{}
In practice, the HVDC link works as a protective barrier to prevent the propagation of cascading outages, and it is normally composed of a transformer, a rectifier, a DC line and an inverter (see Fig. \[hvdc\]). Actually, the rectifier terminal can be regarded as a bus with real power consumption $P_{r}$, and the inverter terminal can be treated as a bus with real power generation $P_{i}$. The direct current from the rectifier to the inverter is computed as follows [@kun94] $$I_d=\frac{3\sqrt{3}(\cos\alpha-\cos\gamma)}{\pi(R_{cr}+R_L-R_{ci})},$$ where $\alpha\in[\pi/30,\pi/2]$ denotes the ignition delay angle of the rectifier, and $\gamma\in[\pi/12,\pi/9]$ represents the extinction advance angle of the inverter. $R_{cr}$ and $R_{ci}$ refer to the equivalent communicating resistances for the rectifier and inverter, respectively. Additionally, $R_L$ denotes the resistance of the DC transmission line. Thus the power consumption at the rectifier terminal is $$\label{pr}
P_r=\frac{3\sqrt{3}}{\pi}I_d\cos\alpha-R_{cr}I^2_d,$$ and at the inverter terminal is $$\label{pi}
P_i=\frac{3\sqrt{3}}{\pi}I_d\cos\gamma-R_{ci}I^2_d=P_r-R_LI^2_d.$$ Notably, $P_r$ and $P_i$ keep unchanged when $\alpha$ and $\gamma$ are fixed.
Protective Relay
----------------
The protective relays are indispensable components in power systems protection and control. When the transmission power exceeds the given threshold of the branch, the timer of circuit breaker starts to count down from the preset time [@jia16]. Once the timer runs out of the preset time, the transmission line is severed by circuit breakers and its branch admittance becomes zero. Specifically, the vector of branch susceptance at the $k$-th cascading step is given by $$\label{relay}
B^k=G(P^{k-1}_{e},\sigma)\circ F(B^{k-1})$$ where the operator $\circ$ denotes the Hadamard product, and $\sigma=(\sigma_1,\sigma_2,...,\sigma_n)$ represents the threshold vector of transmission power on each branch. $F(B^{k-1})$ provides the vector of branch susceptance at the $(k-1)$-th cascading step, and it updates constantly due to the dynamics of FACTS devices. Additionally, the vector function $G(P^{k-1}_{e},\sigma)$ is used to characterize the branch outage as follows $$G(P^{k-1}_{e},\sigma)=\left(
\begin{array}{c}
g(P^{k-1}_{e,1},\sigma_1) \\
g(P^{k-1}_{e,2},\sigma_2) \\
. \\
g(P^{k-1}_{e,n},\sigma_n) \\
\end{array}
\right)\in R^n$$ And each element of $G(P^{k-1}_{e},\sigma)$ is a step function as follows $$g(P^{k-1}_{e,i},\sigma_i)=\left\{
\begin{array}{ll}
0, & \hbox{$|P^{k-1}_{e,i}|>\sigma_i$ and $t_c>T$;} \\
1, & \hbox{otherwise.}
\end{array}
\right.$$ where $T$ is the preset time of the timer in protective relays, and $t_c$ denotes the counting time of the timer. Intuitively, the branch outage occurs when its transmission power is larger than the threshold and meanwhile its timer runs out.
The evolution time of cascading failure is introduced to allow for the time factor of cascading blackouts. Essentially, the time interval between two consecutive cascading steps basically depends on the preset time of the timer in protective relays [@jia16]. Thus, the evolution time of cascading failure is roughly estimated by $t=kT$ at the $k$-th cascading step.
Optimization Formulation {#sec:opt}
========================
Since cascading blackouts result in the severe damage of power transmission, we focus on the power transmission at the end of cascading outages and thus design the cost function as follows $$\label{cost}
J(\delta, B^m)=\frac{1}{2}\|P^m_{e}(\delta)\|^2$$ where $P^m_{e}(\delta)$ denotes the vector of transmission power on each branch at the $m$-th cascading step, and $\delta\in[\underline{\delta},\bar{\delta}]$ characterizes the admittance change of the selected branch caused by the initial contingency. Specifically, the vector of transmission power $P^m_{e}(\delta)$ after the contingency can be computed by $$P^m_{e}(\delta)=diag(B^m)A(A^T diag(B^m)A)^{-1^*}P_b$$ with $$B^k=G(P^{k-1}_{e},\sigma)\circ F(B^{k-1}), \quad k\in I_m=\{1,2,...,m\}.$$ As mentioned before, $F(B^{k-1})$ characterizes the dynamical adjustment of FACTS devices at the $k$-th cascading step with $B^{1}=B^{0}+\delta$. Notably, $P_b$ refers to the vector of injected power on buses after the rectifier and inverter terminals of HVDC links are treated as the loads and generators, respectively. Therefore, the problem of identifying initial contingencies in power transmission networks is formulated as $$\label{formulation}
\begin{split}
&~~~~~~\min_{\delta} J(\delta,B^m) \\
&s.~t.~\underline{\delta}\leq\delta\leq\bar{\delta} \\
&~~~~~~B^k=G(P^{k-1}_{e},\sigma)\circ F(B^{k-1}),~k\in I_m \\
&~~~~~~P^k_{e}(\delta)=diag(B^k)A(A^T diag(B^k)A)^{-1^*}P_b \\
\end{split}$$ where the objective function $J(\delta,Y_p^m)$ is defined in equation (\[cost\]). Then it follows from the KKT conditions that we obtain necessary conditions for optimal solutions to Optimization Problem (\[formulation\]) as follows [@man94].
The optimal solution $\delta^{*}$ to Optimization Problem (\[formulation\]) with the multipliers $\mu_1$ and $\mu_2$ satisfies the KKT conditions $$\label{kkt}
\begin{split}
&P^m_{e}(\delta^*)^T\left(\frac{\partial P^m_{e}}{\partial\delta}|_{\delta^*}\right)+\mu_1-\mu_2=0 \\
&\delta^*-\bar{\delta}+x^2_1=0 \\
&\delta^*-\underline{\delta}-x^2_2=0 \\
&\mu_1(\delta^*-\bar{\delta})=0 \\
&\mu_2(\delta^*-\underline{\delta})=0 \\
&\mu_1-y_1^2=0 \\
&\mu_2-y_2^2=0 \\
\end{split}$$ where $x_i$ and $y_i$, $i\in I_2$ are the unknown variables.
The KKT conditions for Optimization Problem (\[formulation\]) are composed of four components: stationary, primal feasibility, dual feasibility and complementary slackness. Specifically, stationary condition allows us to obtain $$\frac{\partial J(\delta,Y^m_p)}{\partial\delta}|_{\delta^*}+\mu_1-\mu_2=0,$$ which is equivalent to $$P^m_{e}(\delta^*)^T\left(\frac{\partial P^m_{e}}{\partial\delta}|_{\delta^*}\right)+\mu_1-\mu_2=0$$ using equation (\[cost\]). Additionally, the primal feasibility leads to $\underline{\delta}\leq\delta^*\leq\bar{\delta}$, which can be converted into equality constraints $$\delta^*-\bar{\delta}+x_1^2=0, \quad \delta^*-\underline{\delta}-x_2^2=0$$ with the unknown variables $x_1,x_2 \in R$. Moreover, the dual feasibility corresponds to $\mu_1,\mu_2\geq0$, which can be replaced by $$\mu_1-y_1^2=0,\quad \mu_2-y_2^2=0$$ with the unknown variables $y_1,y_2 \in R$. Finally, the complementary slackness gives $$\mu_1(\delta^*-\bar{\delta})=0, \quad \mu_2(\delta^*-\underline{\delta})=0$$ This completes the proof.
To reduce the computation burden, the partial derivative of $P^m_{e}$ with respective to $\delta$ can be approximated by $$\label{app}
\frac{\partial P^m_{e}}{\partial\delta}|_{\delta^*}\approx\frac{P^m_{e}(\delta^*+\epsilon)-P^m_{e}(\delta^*)}{\epsilon}$$ with the sufficiently small $\epsilon$.
Numerical Method {#sec:num}
================
To avoid the computation of partial derivatives and reduce computation costs, the Jacobian Free Newton Krylov (JFNK) Method is employed to solve the system of nonlinear algebraic equations without forming the Jacobian matrix. Essentially, the JFNK methods are synergistic combinations of Newton methods for solving nonlinear equations and Krylov subspace methods for solving linear equations [@kno04]. To facilitate the analysis, we rewrite Equation (\[kkt\]) in matrix form $$\label{mequation}
\mathcal{F}(\mathbf{z})=\mathbf{0}$$ where $\mathbf{z}=(\delta^*,\mu_1,\mu_2,x_1,x_2,y_1,y_2)^T\in R^7$ denotes the unknown vector, and $\mathbf{0}\in R^7$ refers to a zero vector. To obtain the iterative formula for solving (\[mequation\]), we compute Taylor series of $\mathcal{F}(\mathbf{z})$ at $\mathbf{z}^{s+1}$ as follows $$\label{taylor}
\mathcal{F}(\mathbf{z}^{s+1})=\mathcal{F}(\mathbf{z}^{s})+\mathfrak{J}(\mathbf{z}^s)(\mathbf{z}^{s+1}-\mathbf{z}^{s})+O(\delta\mathbf{z}^{s})$$ with $\delta\mathbf{z}^{s}=\mathbf{z}^{s+1}-\mathbf{z}^{s}$. By neglecting the high-order term $O(\delta\mathbf{z}^{s})$ and setting $\mathcal{F}(\mathbf{z}^{s+1})=\mathbf{0}$, we obtain $$\label{lequ}
\mathfrak{J}(\mathbf{z}^s)\cdot\delta\mathbf{z}^{s}=-\mathcal{F}(\mathbf{z}^{s}),\quad \quad s\in Z^{+}$$ where $\mathfrak{J}(\mathbf{z}^s)$ represents the Jacobian matrix and $s$ denotes the iteration index. Thus, solutions to Equation (\[mequation\]) can be approximated by implementing Newton iterations $$\mathbf{z}^{s+1}=\mathbf{z}^{s}+\delta\mathbf{z}^{s}$$ where $\delta\mathbf{z}^{s}$ is obtained by Krylov methods. First of all, the Krylov subspace is constructed as follows $$\label{kspace}
K_i=\mathrm{span}\left(\mathbf{r}^s,~\mathfrak{J}(\mathbf{z}^s)\mathbf{r}^s,~\mathfrak{J}(\mathbf{z}^s)^2\mathbf{r}^s,...,~\mathfrak{J}(\mathbf{z}^s)^{i-1}\mathbf{r}^s\right)$$ with $$\mathbf{r}^s=-\mathcal{F}(\mathbf{z}^{s})-\mathfrak{J}(\mathbf{z}^s)\cdot\delta\mathbf{z}_0^{s},$$ where $\delta\mathbf{z}_0^{s}$ is the initial guess for the Newton correction and is typically zero [@kno04]. Actually, the optimal solution to $\delta\mathbf{z}^{s}$ is the linear combination of elements in Krylov subspace $K_i$. $$\label{gmres}
\delta \mathbf{z}^s=\delta \mathbf{z}_0^s+\sum_{j=1}^{i-1}\beta_j\cdot \mathfrak{J}(\mathbf{z}^s)^j\mathbf{r}^s$$ where $\beta_j$, $j\in\{1,2,...,i-1\}$ is obtained by minimizing $\|\mathfrak{J}(\mathbf{z}^s)\delta z^s+\mathcal{F}(\mathbf{z}^s)\|_2$ with Generalized Minimal RESidual (GMRES) method [@saad86]. In particular, matrix-vector products in (\[gmres\]) can be approximated by $$\label{jv}
\mathfrak{J}(\mathbf{z}^s)\mathbf{r}^s\approx\frac{\mathcal{F}(\mathbf{z}^s+\xi\mathbf{r}^s)-\mathcal{F}(\mathbf{z}^s)}{\xi}$$ where $\xi$ is a sufficiently small value [@saad90]. In this way, we avoid the computation of Jacobian matrix via matrix-vector products in (\[jv\]) while solving Equation (\[mequation\]).
Table \[jfnk\] summarizes the JFNK method for solving Equation (\[mequation\]). First of all, we set the initial step $s=0$, the initial tolerance $\epsilon_0$ and the minimum tolerance $\epsilon_{\min}$ for evaluating the termination condition of loop iterations. Then the residual $\mathbf{r}^s$ is calculated in each iteration, which allows us to construct the Krylov subspace $K_i$. For elements in $K_i$, the matrix-vector products are approximated by Equation (\[jv\]) without forming the Jacobian. Next, the gradient $\delta\mathbf{z}^{s}$ for Newton iterations is obtained via GMRES method. Finally, we update the tolerance $\epsilon_s$ and step number $s$ after implementing the Newton iteration for $\mathbf{z}^s$. And a new iteration loop is launched if the termination condition $\epsilon_{s}\leq\epsilon_{\min}$ does not hold.
------------------------------------------------------------------------------------------------------------------------------------ -- --
1: Set $s=0$, $\epsilon_{0}$ and $\epsilon_{\min}$ satisfying $\epsilon_{0}>\epsilon_{\min}$
2: **while** ($\epsilon_{s}>\epsilon_{\min}$)
3: Calculate the residual $\mathbf{r}^s=-\mathcal{F}(\mathbf{z}^{s})-\mathfrak{J}(\mathbf{z}^s)\cdot\delta\mathbf{z}_0^{s}$
4: Construct the Krylov subspace $K_i$ in (\[kspace\])
5: Approximate matrix-vector products in (\[gmres\]) using (\[jv\])
6: Compute $\beta_j$ in (\[gmres\]) with GMRES method
7: Compute $\delta\mathbf{z}^{s}$ with (\[gmres\])
8: Update $\mathbf{z}^{s+1}=\mathbf{z}^{s}+\delta\mathbf{z}^{s}$
9: Update $\epsilon_{s+1}=\|\delta\mathbf{z}^{s}\|/\|\mathbf{z}^s\|$
10: Update $s=s+1$
11: **end while**
------------------------------------------------------------------------------------------------------------------------------------ -- --
: \[jfnk\] JFNK Method.
---------------------------------------------------- -- --
1: Select the disturbed branch
2: Set $l_{\max}$, $l=0$ and $\delta=0$
3: **while** ($l<l_{\max}$)
4: Compute $\delta^*$ with the JFNK method
5: **if** ($J(\delta^*,B^m)<J(\delta,B^m)$)
6: $\delta=\delta^*$
7: **end if**
8: Update $l=l+1$
9: **end while**
---------------------------------------------------- -- --
: \[cia\] Contingency Identification Algorithm.
Table \[cia\] presents the explicit process of implementing Contingency Identification Algorithm (CIA). First of all, we select a branch in power transmission networks to add the disturbance with the initial value $\delta=0$, and the maximum iterative step $l_{\max}$ is specified with the initial iterative step $l=0$. Then we compute the optimal disturbance $\delta^*$ with the JFNK method. The disturbance value $\delta^*$ in (\[kkt\]) is saved if it leads to the worse cascading blackout ($i.e.$, $J(\delta^*,B^m)<J(\delta,B^m)$). The above algorithm does not terminate until the iterative step $l$ is larger than or equal to $l_{\max}$.
Essentially, the proposed approach to identifying initial disturbances is universal, which also applies to power distribution systems using the AC power flow model and more complicated protective mechanisms. The main difference lies in the computation of transmission power at the final cascading step, $i.e.$, $P_e^m$ in Equation (\[kkt\]).
Simulation and Validation {#sec:sim}
=========================
In this section, we implement the proposed CIA in Table \[cia\] to search for the disruptive disturbances on selected branches of IEEE 118 Bus System [@zim11]. The numerical results on disruptive disturbances are validated by disturbing the selected branch with the computed magnitude of disturbance. Per-unit system is adopted with the base value of $100$ MVA in numerical simulations, and the power flow threshold for each branch is specified in Table \[thre\]. The power flow on each branch is close to the saturation, although it does not exceed their respective threshold. In this way, the power system is vulnerable to initial contingencies, and thus is likely to suffer from cascading blackouts.
**Power Flow Threshold** **Branch ID**
---------------------------------------------------------------------------------------- -- --
7 32
6 18 31
5 7 8 9
4 1 12 13 14 21 33 36 37 96
3 11 15 41 51 141
2 3 5 6 10 17 19 20 22 23 25 26 27 28 29 30
2 34 39 42 43 54 62 90 93 94 97 98 99 104 105
106 107 108 126 127 137 139 163 178 179 183
1 all other branches
: \[thre\] Thresholds of the transmission power on each branch.
Figure \[Step1\] shows the normal state of IEEE 118 Bus System, which includes 53 generator buses, 64 load buses, 1 reference bus (Bus $69$) and 186 branches. Branch $8$ (red link connecting Bus $5$ to Bus $8$) is randomly selected as the disturbed element of power networks. And the HVDC links are denoted by blue lines including Branch 4 connecting Bus $3$ to Bus $5$, Branch 16 connecting Bus $11$ to Bus $13$ and Branch 38 connecting Bus $26$ to Bus $30$. The maximum iterative step $l_{\max}$ is equal to $10$ in the CIA. Other parameters are given as follows: $\epsilon=10^{-2}$ in Equation (\[app\]), $\epsilon_{\min}=10^{-8}$ in the JFNK method, $\underline{\delta}=0$, $\bar{\delta}=37.45$ and $m=12$. For simplicity, we specify the same values for the parameters of three HVDC links as follows: $R_{ci}=R_{cr}=R_L=0.1$, $\alpha=\pi/15$ and $\gamma=\pi/4$. Regarding the FACTS devices, we set $X_{min,i}=0$, $X_{max,i}=10$ and $X_{ref,i}=0$ for the TCSC dynamics, and $K_P=4$, $K_I=3$ and $K_D=2$ for its PID controller. Additionally, the reference transmission power $P^{ref}_{e,i}$ is equal to the threshold of transmission power $\sigma_i$. We consider two preset values of the timer in protective relays, $i.e.$, $T=0.5$s and $T=1$s. Contingency Identification Algorithm is carried out to search for the disturbance that results in the worst-case cascading failures of power systems. For the IEEE 118 Bus System without the FACTS devices, the computed magnitude of disturbance on Branch $8$ is $37.45$, which exactly leads to the outage of Branch 8. For the power system with the FACTS devices and the preset time of the timer $T=0.5$s, the computed disturbance magnitude is $36.77$, while it is $35.98$ for $T=1$s.
Next, we validate the proposed approach by adding the computed disturbances on Branch 8 of IEEE 118 Bus Systems. Specifically, Fig. \[nofacts\] demonstrates the final state of IEEE 118 Bus System with no FACTS devices and $T=1$s. The cascading process terminates with 95 outage branches and the cost function value of $53.28$ after 16 seconds, and the system collapses with 42 islands in the end. These 42 islands include $24$ isolated buses and $18$ subnetworks encircled by the dashed lines. In contrast, Figure \[facts05\] presents the final configuration of IEEE 118 Bus Systems with the protection of the FACTS devices and $T=0.5$s. The cascading process ends up with 40 outage branches and the cost function value of $102.56$ after 10 seconds, and the power system is separated into $17$ islands, which include $6$ subnetworks and $11$ isolated buses. Figure \[facts1\] gives the final state of power systems with the FACTS devices and $T=1$s. It is observed that the power network is eventually split into 3 islands (Bus 14, Bus 16 and a subnetwork composed of all other buses) with only $6$ outage branches and the cost function of $153.69$. Note that the initial disturbances from CIA fail to cause the outage of Branch $8$ in the end for both $T=0.5$s and $T=1$s. The above simulation results demonstrate the advantage of the FACTS devices in preventing the propagation of cascading outages. The larger preset time of timer enables the FACTS devices to sufficiently adjust the branch impedance in response to the overload stress. And thus the less severe damages are caused by the contingency for the larger preset time of timer.
Figure \[comp\] presents the time evolution of outage branches in IEEE 118 Bus System as a result of disturbing Branch $8$ in three different scenarios. The cyan squares denote the number of outage branches with no FACTS devices and $T=1$s, while the green and blue ones refer to the numbers of outage branches with the FACTS devices for $T=0.5$s and $T=1$s, respectively. The computed disturbances are added to change the admittance of Branch $8$ at $t=0$s. With no FACTS devices, the cascading outage of branches propagates quickly from $t=2$s to $t=10$s and terminates at $t=16$s. When the FACTS devices are adopted and the preset time of timer is $T=0.5$s, the cascading failure starts at $t=2$s and speeds up till $t=8$s and stops at $t=10$s. For $T=1$s, the cascading outage propagates slowly due to the larger preset time of timer and comes to an end with only $6$ outage branches at $t=8$s. Together with protective relays and HVDC links, the FACTS devices succeed in protecting power systems against blackouts by adjusting the branch impedance in real time. More precisely, the number of outage branches decreases by $57.9\%$ with FACTS devices and $T=0.5$s and decreases by $93.7\%$ with FACTS devices and $T=1$s.
Conclusions {#sec:con}
===========
In this paper, we investigated the problem of identifying the initial contingencies that lead to cascading blackout of power transmission networks equipped with FACTS devices, HVDC links and protective relays. An optimization formulation was proposed to identify the contingencies in the framework of nonlinear programming, and an efficient numerical method was presented to solve the optimization problem. Numerical simulations were carried out on IEEE 118 Bus Systems to validate the proposed approach. Significantly, the contingency identification algorithm allows us to detect some nontrivial disturbances that lead to the severe cascading failure of power transmission networks, other than severing the branch. It is demonstrated that the coordination of FACTS devices and protective relays greatly enhances the capability of power grids against blackouts. In the next step, we will investigate the contingency identification problem for the AC-based power distribution systems with transient process.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is partially supported by the Future Resilience System Project at the Singapore-ETH Centre (SEC), which is funded by the National Research Foundation of Singapore (NRF) under its Campus for Research Excellence and Technological Enterprise (CREATE) program. It is also supported by Ministry of Education of Singapore under Contract MOE2016-T2-1-119.
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[^1]: Chao Zhai, Hehong Zhang, Gaoxi Xiao and Tso-Chien Pan are with Institute of Catastrophe Risk Management, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798. They are also with Future Resilient Systems, Singapore-ETH Centre, 1 Create Way, CREATE Tower, Singapore 138602. Chao Zhai, Hehong Zhang and Gaoxi Xiao are also with School of Electrical and Electronic Engineering, Nanyang Technological University. Corresponding author: Gaoxi Xiao. Email: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In a Hilbert space setting $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation $$\ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t),$$ where $\nabla\Phi$ is the gradient of a convex continuously differentiable function $\Phi: \mathcal H \to \mathbb R$, $\alpha$ is a positive parameter, and $g: [t_0, + \infty[ \rightarrow \mathcal H$ is a “small” perturbation term. In this damped inertial system, the viscous damping coefficient $\frac{\alpha}{t}$ vanishes asymptotically, but not too rapidly. For $\alpha \geq 3$, and $\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty$, just assuming that ${{\rm argmin}\kern 0.12em}\Phi \neq \emptyset$, we show that any trajectory of the above system satisfies the fast convergence property $$\begin{aligned}
\Phi(x(t))- \min_{\mathcal H}\Phi \leq \frac{C}{t^2}.\end{aligned}$$ For $\alpha > 3$, we show that any trajectory converges weakly to a minimizer of $\Phi$, and we show the strong convergence property in various practical situations. This complements the results obtained by Su-Boyd- Candès, and Attouch-Peypouquet-Redont, in the unperturbed case $g=0$. The parallel study of the time discretized version of this system provides new insight on the effect of errors, or perturbations on Nesterov’s type algorithms. We obtain fast convergence of the values, and convergence of the trajectories for a perturbed version of the variant of FISTA recently considered by Chambolle-Dossal, and Su-Boyd-Candès.
address:
- 'Institut de Mathématiques et Modélisation de Montpellier, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France'
- 'Laboratoire IBN AL-BANNA de Mathématiques et applications (LIBMA), Cadi Ayyad university, Faculty of Sciences Semlalia, Mathematics, 40000 Marrakech, Morroco'
author:
- Hedy Attouch
- Zaki Chbani
date: 'July 6, 2015'
title: 'FAST INERTIAL DYNAMICS AND FISTA ALGORITHMS IN CONVEX OPTIMIZATION. PERTURBATION ASPECTS.'
---
[^1]
Introduction
============
Throughout the paper, $\mathcal H$ is a real Hilbert space which is endowed with the scalar product $\langle \cdot,\cdot\rangle$, with $\|x\|^2= \langle x,x\rangle $ for any $x\in \mathcal H$. Let $\Phi : \mathcal H \rightarrow \mathbb R$ be a convex differentiable function, whose gradient $\nabla \Phi$ is Lipschitz continuous on bounded sets. We suppose that $S={{\rm argmin}\kern 0.12em}\Phi$ is nonempty. Let us give $\alpha$ a positive parameter. We are going to study the asymptotic behaviour (as $t \to + \infty$) of the trajectories of the second-order differential equation $$\label{edo01}
\mbox{(AVD)}_{\alpha,g} \quad \quad \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t)$$ and consider similar questions for the corresponding algorithms. Let us give some $t_0 >0$. The second-member $g: [t_0, + \infty[ \rightarrow \mathcal H$ is a perturbation term (integrable source term), such that $g(t)$ is small for large $t$. Precisely, in our main result, Theorem \[fastconv-thm\], assuming that $\alpha \geq 3$, and $\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty$, we show that any trajectory of (\[edo01\]) satisfies the fast convergence property $$\begin{aligned}
\label{basic-fast}
\Phi(x(t))- \min_{\mathcal H}\Phi \leq \frac{C}{t^2}.\end{aligned}$$ This extends the fast convergence of the values obtained by Su, Boyd and Candès in [@SBC] in the unperturbed case $g=0$. In Theorem \[Thm-weak-conv\], when $\alpha > 3$, we show that any trajectory of (\[edo01\]) converges weakly to a minimizer of $\Phi$, which extends the convergence result obtained by Attouch, Peypouquet, and Redont in [@APR1] in the case $g=0$.
This inertial system involves a viscous damping which is attached to the term $\frac{\alpha}{t} \dot{x}(t)$. It is an isotropic linear damping with a viscous parameter $\frac{\alpha}{t}$ which vanishes asymptotically, but not too rapidly. The asymptotic behaviour of the inertial gradient-like system $$\label{edo2}
\mbox{(AVD)} \quad \quad \ddot{x}(t) + a(t) \dot{x}(t) + \nabla \Phi (x(t)) = 0,$$ with Asymptotic Vanishing Damping ((AVD) for short), has been studied by Cabot, Engler and Gaddat in [@CEG1]-[@CEG2]. As a main result, they proved that, under moderate decrease of $a(\cdot)$ to zero, i.e., $a(t) \to 0$ as $t \to +\infty$ with $\int_0^{\infty} a(t) dt = + \infty$, then for any trajectory $x(\cdot)$ of (\[edo2\]) $$\label{edo3}
\Phi (x(t)) \to \min_{\mathcal H}\Phi.$$ As a striking property, for the specific choice $a(t)= \frac{\alpha}{t}$, with $\alpha \geq 3$ , for example when considering $$\label{edo4}
\ddot{x}(t) + \frac{3}{t} \dot{x}(t) + \nabla \Phi (x(t)) = 0,$$ it has been proved by Su, Boyd, and Candès in [@SBC] that the fast convergence property of the values (\[basic-fast\]) is satisfied by the trajectories of (\[edo4\]). In the same article [@SBC], the authors show that (\[edo4\]) can be seen as a continuous version of the fast convergent method of Nesterov, see [@Nest1]-[@Nest2]-[@Nest3]-[@Nest4]. For the continuous dynamic, a related study concerning the case $a(t)= \frac{1}{t^{\theta}}$, $0<\theta <1$ has been developed by Jendoubi and May in [@JM], with roughly speaking $\mathcal O (\frac{1}{t^{1+ \theta}})$ convergence. The analysis developped in [@JM] does not contain the case $a(t)= \frac{\alpha}{t}$, where the introduction of an additional scaling, due to the coefficient $\alpha$, requires a specific analysis. That’s our main concern in this paper.
Our results provide new insight on the effect of perturbations or errors in the associated algorithms. They provide a guideline for the study of the preservation, under small perturbations, of the fast convergence property of the corresponding Nesterov type algorithms. Specifically we consider a perturbed version of the variant of FISTA recentely considered by Chambolle and Dossal [@CD], and Su, Boyd and Candès [@SBC]. We obtain fast convergence of the values in the case $\alpha \geq 3$, and convergence of the trajectories in the case $\alpha > 3$. Convergence of the trajectories in the case $\alpha =3$, which corresponds to Nesterov algorithm, is still an open question.
Fast Convergence of the values
==============================
Let $\Phi : \mathcal H \rightarrow \mathbb R$ be a convex function, whose gradient $\nabla \Phi$ is Lipschitz continuous on bounded sets. Let $t_0 >0$, $\alpha >0$, and $g: [t_0, + \infty[ \rightarrow \mathcal H$ such that $\displaystyle{\int_{t_0}^{+\infty} \|g(t)\| dt < + \infty}$. We consider the second-order differential equation $$\label{energy-00}
\mbox{{\rm(AVD)}}_{\alpha, g} \quad \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t).\quad\quad$$ From Cauchy-Lipschitz theorem, for any Cauchy data $x(t_0) =x_0 \in \mathcal H, \ \dot{x}(t_0)= x_1 \in \mathcal H $ we immediately infer the existence and uniqueness of a local solution to (\[energy-00\]). The global existence follows from the energy estimate proved in Proposition \[energy-thm-1\], in the next paragraph. Throughout this paper we will use the following Gronwall-Bellman lemma, see [@Bre1 Lemme A.5] for a proof.
\[GB-lemma\] Let $m\in L^1 (t_0,T; \mathbb R)$ such that $m \geq 0$ a.e. on $]t_0,T[$ and let $c$ be a nonnegative constant. Suppose that $w$ is a continuous function from $[t_0,T]$ into $\mathbb R$ that satisfies, for all $t\in[t_0,T]$ $$\frac{1}{2}w^2 (t) \leq
\frac{1}{2} c^2 + \int_{t_0}^t m(\tau) w(\tau) d\tau .$$ Then, for all $t\in[t_0,T]$ $$|w(t| \leq c + \int_{t_0}^t m(\tau) d\tau .$$
Energy estimates
----------------
The following estimates are obtained by considering the global energy of the system, and showing that it is a strict Lyapunov function.
\[energy-thm-1\] Suppose $\alpha >0 $, and $\displaystyle{\int_{t_0}^{+\infty} \|g(t)\| dt < + \infty}$ . Then, for any orbit $x: [t_0, +\infty[ \rightarrow \mathcal H$ of $ \mbox{{\rm(AVD)}}_{\alpha,g}$ $$\begin{aligned}
& \sup_t \| \dot{x}(t) \| < + \infty , \label{et1} \\
& \int_{t_0}^{+\infty} \frac{1}{t} \| \dot{x}(t) \|^2 dt < + \infty . \label{et2} \end{aligned}$$ Precisely, for any $t\geq t_0$ $$\label{energy-001}
\| \dot{x}(t) \| \leq \| \dot{x}(t_0) \| + \sqrt{2}\left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) + \int_{t_0}^{\infty} \| g(\tau) \| d\tau ,$$ $$\label{energy-002}
\int_{t_0}^{\infty} \frac{\alpha}{\tau} \| \dot{x}(\tau) \|^2 d\tau \leq \frac{1}{2} \| \dot{x}(t_0) \|^2 + \left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) +
\left( \| \dot{x}(t_0) \| + \sqrt{2}\left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) + \|g\|_{L^1 (t_0, \infty)} \right) \|g\|_{L^1 (t_0, \infty)} .$$
Let us give some $T >t_0$. For $t_0 \leq t \leq T$, let us define the energy function $$\label{energy-01}
W_T(t):= \frac{1}{2} \| \dot{x}(t) \|^2 + \left( \Phi(x(t))- \min_{\mathcal H}\Phi \right) +
\int_t^T \langle \dot{x}(\tau) ,
g(\tau) \rangle d\tau .$$ Because of $\dot{x}$ continuous, and $g$ integrable, the energy function $W_T$ is well defined. After time derivation of $W_T$, and by using $ \mbox{{\rm(AVD)}}_{\alpha,g}$, we obtain $$\begin{aligned}
\dot{W_T} (t)&:= \langle \dot{x}(t), \ddot{x}(t) + \nabla \Phi (x(t)) - g(t) \rangle \\
&= \langle \dot{x}(t), -\frac{\alpha}{t} \dot{x}(t) \rangle ,\end{aligned}$$ that is $$\begin{aligned}
\label{energy-02}
\dot{W_T} (t) + \frac{\alpha}{t} \| \dot{x}(t) \|^2 \leq 0.\end{aligned}$$ Hence $W_T(\cdot)$ is a decreasing function. In particular, $ W_T(t) \leq W_T(t_0)$, i.e., $$\frac{1}{2} \| \dot{x}(t) \|^2 + \left( \Phi(x(t))- \min_{\mathcal H}\Phi \right) +
\int_t^T \langle \dot{x}(\tau) ,
g(\tau) \rangle d\tau
\leq \frac{1}{2} \| \dot{x}(t_0) \|^2 + \left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) +
\int_{t_0}^T \langle \dot{x}(\tau) ,
g(\tau) \rangle d\tau .$$ As a consequence, $$\frac{1}{2} \| \dot{x}(t) \|^2 \leq \frac{1}{2} \| \dot{x}(t_0) \|^2 + \left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) + \int_{t_0}^t \|\dot{x}(\tau)\| \| g(\tau) \| d\tau.$$ Applying Gronwall-Bellman lemma \[GB-lemma\], we obtain $$\begin{aligned}
\| \dot{x}(t) \| &\leq \left( \| \dot{x}(t_0) \|^2 + 2\left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) \right)^{\frac{1}{2}} + \int_{t_0}^t \| g(\tau) \| d\tau \\
& \leq \| \dot{x}(t_0) \| + \sqrt{2}\left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) + \int_{t_0}^t \| g(\tau) \| d\tau .\end{aligned}$$ This being true for arbitrary $T>t_0$, and $t_0 \leq t \leq T$, we deduce that $$\label{energy-03}
\| \dot{x}(t) \| \leq \| \dot{x}(t_0) \| + \sqrt{2}\left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) + \int_{t_0}^{\infty} \| g(\tau) \| d\tau ,$$ which gives (\[et1\]) and (\[energy-001\]). As a consequence, the function $W$ (corresponding to $T= +\infty$) $$\label{energy-04}
W(t):= \frac{1}{2} \| \dot{x}(t) \|^2 + \left( \Phi(x(t))- \min_{\mathcal H}\Phi \right) +
\int_t^{\infty} \langle \dot{x}(\tau) ,
g(\tau) \rangle d\tau ,$$ is well defined, and is minorized by $$\label{energy-05}
-\| \dot{x} \|_{ L^{\infty} (t_0, + \infty)} \int_{t_0}^{\infty} \| g(\tau) \| d\tau .$$ By (\[energy-02\]) we have $$\begin{aligned}
\label{energy-06}
\dot{W} (t) + \frac{\alpha}{t} \| \dot{x}(t) \|^2 \leq 0.\end{aligned}$$ Integrating (\[energy-06\]) from $t_0$ to $t$, and using (\[energy-03\]), (\[energy-05\]), we obtain $$\begin{aligned}
\int_{t_@}^{\infty} \frac{\alpha}{\tau} \| \dot{x}(\tau) \|^2 d\tau & \leq \frac{1}{2} \| \dot{x}(t_0) \|^2 + \left( \Phi(x(t_0))- \min_{\mathcal H}\Phi \right) + \| \dot{x} \|_{ L^{\infty} (t_0, + \infty)} \int_{t_0}^{\infty} \| g(\tau) \| d\tau < + \infty\\
& \leq \frac{1}{2} \| \dot{x}(t_0) \|^2 + \left( \Phi(x(t_0))- \min_{\mathcal H}\Phi \right) +
\left( \| \dot{x}(t_0) \| + \sqrt{2}\left( \Phi(x_0)- \min_{\mathcal H}\Phi \right) + \|g\|_{L^1 (t_0, +\infty)} \right) \|g\|_{L^1 (t_0, +\infty)} ,\end{aligned}$$ which gives (\[et2\]) and (\[energy-002\]).
The main result
---------------
Let us state our main result.
\[fastconv-thm\] Suppose that $\alpha \geq3$, and $\displaystyle{\int_{t_0}^{+\infty} \tau \|g(\tau)\| d\tau < + \infty}$. Then, for any orbit $x: [t_0, +\infty[ \rightarrow \mathcal H$ of $ \mbox{{\rm(AVD)}}_{\alpha,g}$, we have the following fast convergence of the values: $$\Phi(x(t))- \min_{\mathcal H}\Phi = \mathcal O \left( \frac{1}{t^2}\right) .$$ Precisely $$\begin{aligned}
\label{Liap-001}
\frac{2}{\alpha-1}t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi ) \leq C+
2 \left(\left( \frac{C}{\alpha-1}\right)^{ \frac{1}{2} } + \frac{1}{\alpha-1}
\int_{t_{0}}^{\infty} \tau \| g(\tau) \| d\tau \right) \int_{t_{0}}^{\infty} \tau \|g(\tau) \| d\tau ,\end{aligned}$$ with $$C= \frac{2}{\alpha-1}t^2 (\Phi(x_0)- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x_0 - x^{*} + \frac{t_0}{\alpha-1} \dot{x}(t_0) \|^2 .$$ Moreover $$\begin{aligned}
\label{Liap-001-b}
\sup_{t \geq t_0}\| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \| \leq \left( \frac{C}{\alpha-1}\right)^{ \frac{1}{2} } + \frac{1}{\alpha-1}
\int_{t_{0}}^{\infty} \tau \| g(\tau) \| d\tau <+\infty .\end{aligned}$$
The proof is an adaptation to our setting (with an integrable source term $g$) of the argument developed by Su-Boyd-Candès in [@SBC]. Let us give some $T >t_0$, and $x^{*} \in S= \mbox{\rm{argmin}} \Phi$. For $t_0 \leq t \leq T$, let us define the energy function $$\label{basic-Liap}
\mathcal E_{\alpha,g, T} (t):= \frac{2}{\alpha-1}t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \|^2 +
2\int_t^{T} \tau \langle x(\tau) - x^{*} + \frac{\tau}{\alpha-1} \dot{x}(\tau), g(\tau) \rangle d\tau.$$ Let us show that $$\dot{\mathcal E}_{\alpha,g,T} (t) + 2 \displaystyle{\frac{\alpha-3}{\alpha-1}} t (\Phi(x(t))- \min_{\mathcal H}\Phi ) \leq 0 .$$ Derivation of $\mathcal E_{\alpha, g,T} (\cdot)$ gives $$\begin{aligned}
\dot{\mathcal E}_{\alpha,g,T} (t)&:= \frac{4}{\alpha-1}t (\Phi(x(t))- \inf_{\mathcal H}\Phi ) +
\frac{2}{\alpha-1}t^2 \langle \nabla \Phi (x(t)), \dot{x}(t) \rangle \\
&+ 2(\alpha-1) \langle x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) , \dot{x}(t) +
\frac{1}{\alpha-1} \dot{x}(t) +\frac{t}{\alpha-1} \ddot{x}(t)
\rangle - 2t\langle x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) , g(t) \rangle \\
&= \frac{4}{\alpha-1}t (\Phi(x(t))- \inf_{\mathcal H}\Phi ) +
\frac{2}{\alpha-1}t^2 \langle \nabla \Phi (x(t)), \dot{x}(t) \rangle \\
&+ 2(\alpha-1) \langle x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) ,
\frac{t}{\alpha-1} \Big( \frac{\alpha}{t}\dot{x}(t) + \ddot{x}(t) -g(t)\Big) \rangle. \end{aligned}$$ Then use $\mbox{{\rm(AVD)}}_{\alpha, g}$ in this last expression to obtain $$\begin{aligned}
\dot{\mathcal E}_{\alpha,g,T} (t)=& \frac{4}{\alpha-1}t (\Phi(x(t))- \inf_{\mathcal H}\Phi ) + \frac{2}{\alpha-1}t^2 \langle \nabla \Phi (x(t)), \dot{x}(t) \rangle \\
&- 2 t\langle x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t), \nabla \Phi (x(t)) \rangle\\
=& \frac{4}{\alpha-1}t (\Phi(x(t))- \inf_{\mathcal H}\Phi ) - 2 t \langle x(t) - x^{*} , \nabla \Phi (x(t)) \rangle.\label{rfastoche1}\end{aligned}$$ By convexity of $\Phi$ $$\Phi (x^{*}) \geq \Phi (x(t))+ \langle x^{*} - x(t) , \nabla \Phi (x(t)) \rangle.$$ Replacing in (\[rfastoche1\]) we obtain $$\dot{\mathcal E}_{\alpha,g,T} (t) + \left(2- \frac{4}{\alpha-1}\right) t (\Phi(x(t))- \inf_{\mathcal H}\Phi) \leq 0.$$ Equivalently $$\label{basic-energy-Liap}
\dot{\mathcal E}_{\alpha,g,T} (t) + 2 \frac{\alpha-3}{\alpha-1} t (\Phi(x(t))- \inf_{\mathcal H}\Phi ) \leq 0 .$$ As a consequence, for $\alpha \geq 3$, the function ${\mathcal E}_{\alpha,g}$ is nonincreasing. In particular, ${\mathcal E}_{\alpha,g} (t) \leq {\mathcal E}_{\alpha,g} (t_0)$, which gives $$\begin{aligned}
&\frac{2}{\alpha-1}t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \|^2 +
2\int_t^{T} \tau \langle x(\tau) - x^{*} + \frac{\tau}{\alpha-1} \dot{x}(\tau), g(\tau) \rangle d\tau \\
&\leq \frac{2}{\alpha-1}{t_0}^2 (\Phi(x_0)- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x_0 - x^{*} + \frac{t_0}{\alpha-1} \dot{x}(t_0) \|^2 +
2\int_{t_{0}}^{T} \tau \langle x(\tau) - x^{*} + \frac{\tau}{\alpha-1} \dot{x}(\tau), g(\tau) \rangle d\tau .\end{aligned}$$ Equivalently $$\begin{aligned}
\label{energy-08}
\frac{2}{\alpha-1}t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \|^2 \leq C+
2\int_{t_{0}}^{t} \tau \langle x(\tau) - x^{*} + \frac{\tau}{\alpha-1} \dot{x}(\tau), g(\tau) \rangle d\tau ,\end{aligned}$$ with $$C= \frac{2}{\alpha-1}{t_0}^2 (\Phi(x_0)- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x_0 - x^{*} + \frac{t_0}{\alpha-1} \dot{x}(t_0) \|^2 .$$ From (\[energy-08\]) we infer $$\begin{aligned}
\label{energy-09}
\frac{1}{2} \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \|^2 \leq \frac{C}{2(\alpha-1)}+ \frac{1}{\alpha-1}
\int_{t_{0}}^{t} \| x(\tau) - x^{*} + \frac{\tau}{\alpha-1} \dot{x}(\tau)\| \|\tau g(\tau) \| d\tau .\end{aligned}$$ Applying once more Gronwall-Bellman lemma \[GB-lemma\], we obtain $$\begin{aligned}
\label{energy-1}
\| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \| \leq \left( \frac{C}{\alpha-1}\right)^{ \frac{1}{2} } + \frac{1}{\alpha-1}
\int_{t_{0}}^{t} \tau \| g(\tau) \| d\tau .\end{aligned}$$ Since $\displaystyle{\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty}$, it follows that $$\begin{aligned}
\label{energy-10}
\sup_t \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \| \leq \left( \frac{C}{\alpha-1}\right)^{ \frac{1}{2} } + \frac{1}{\alpha-1}
\int_{t_{0}}^{\infty} \tau \| g(\tau) \| d\tau <+\infty .\end{aligned}$$ Returning to (\[energy-08\]), we conclude that $$\begin{aligned}
\label{energy-11}
\frac{2}{\alpha-1}t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi ) \leq C+
2 \left(\left( \frac{C}{\alpha-1}\right)^{ \frac{1}{2} } + \frac{1}{\alpha-1}
\int_{t_{0}}^{\infty} \tau \| g(\tau) \| d\tau \right) \int_{t_{0}}^{\infty} \tau \|g(\tau) \| d\tau .\end{aligned}$$
As a consequence the energy function $$\label{basic-Liap-c}
\mathcal E_{\alpha,g} (t):= \frac{2}{\alpha-1}t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \|^2 +
2\int_t^{+\infty} \tau \langle x(\tau) - x^{*} + \frac{\tau}{\alpha-1} \dot{x}(\tau), g(\tau) \rangle d\tau.$$ is well defined, and is a Lyapunov function for the dynamical system $ \mbox{{\rm(AVD)}}_{\alpha,g}$.
Convergence of trajectories
===========================
In the case $\alpha >3$, provided that the second member $g(t)$ is sufficiently small for large $t$, we are going to show the convergence of the trajectories of the system
$$\mbox{(AVD)}_{\alpha,g} \quad \quad \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t).$$
Main statement, and preliminary results
---------------------------------------
The following convergence result is an extension to the perturbed case (with a source term $g$) of the convergence result obtained by Attouch-Peypouquet-Redont in [@APR1].
\[Thm-weak-conv\] Let $\Phi : \mathcal H \rightarrow \mathbb R$ a convex continuously differentiable function such that $S={{\rm argmin}\kern 0.12em}\Phi$ is nonempty. Suppose that $\alpha >3$ and $\displaystyle{\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty}$. Let $t_0 >0$, and $x: [t_0, +\infty[ \rightarrow \mathcal H$ be a classical solution of [$ \mbox{(AVD)}_{\alpha,g}$]{}. Then, the following convergence properties hold:
a\) (weak convergence) There exists some $x^{*} \in {{\rm argmin}\kern 0.12em}\Phi$ such that $$\label{conv-basic}
x(t) \rightharpoonup x^{*} \ \mbox{weakly as} \ t \rightarrow + \infty.$$
b\) (fast convergence) There exists a positive constant $C$ such that $$\label{rfast1}
\Phi(x(t))- \min_{\mathcal H}\Phi \leq \frac{C}{t^2}$$ $$\label{energy2}
\int_{t_0}^{\infty} t \left( \Phi(x(t)) - \inf_{\mathcal H}\Phi \right) dt < + \infty.$$ c) (energy estimate) $$\begin{aligned}
\label{energy1}
&\int_{t_0}^{\infty} t\| \dot{x}(t) \|^2 dt < + \infty\\
&\| \dot{x}(t) \| \leq \frac{C}{t} \label{conv1}\end{aligned}$$ and hence $$\label{conv2}
\lim_{t\to\infty} \| \dot{x}(t) \| = 0.$$
In order to analyze the convergence properties of the trajectories of system (\[edo01\]), we will use the Opial’s lemma [@Op] that we recall in its continuous form; see also [@Bruck], who initiated the use of this argument to analyze the asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces.
\[Opial\] Let $S$ be a non empty subset of $\mathcal H$ and $x:[0,+\infty[\to \mathcal H$ a map. Assume that $$\begin{aligned}
(i) & & \mbox{for every }z\in S,\>\lim_{t\to+\infty}\|x(t)-z\|\mbox{ exists};\\
(ii) & & \mbox{every weak sequential cluster point of the map }x\mbox{ belongs to }S.\end{aligned}$$ Then $$w-\lim_{t\to+\infty}x(t)=x_{\infty}\ \ \mbox{ exists, for some element }x_{\infty}\in S.$$
We also need the following result concerning the integration of a first-order nonautonomous differential inequation, see [@APR1].
\[basic-edo\] Suppose that $\delta >0$, and let $w: [\delta, +\infty[ \rightarrow \mathbb R$ be a continuously differentiable function that satisfies the following differential inequality $$\label{basic-edo1}
\dot{w}(t) + \frac{\alpha}{t} w(t) \leq k(t),$$ for some $\alpha > 1$, and some nonnegative function $k: [\delta, +\infty[ \to \mathbb R$ such that $t \mapsto tk(t) \in L^1 (\delta, +\infty)$. Then $$\label{basic-edo2}
w^{+} \in L^1 (\delta, +\infty).$$
Proof of the convergence results
--------------------------------
**Step 1.** Let us return to the decrease property (\[basic-energy-Liap\]) which is satisfied by the Lyapunov function ${\mathcal E}_{\alpha,g}$: $$\dot{\mathcal E}_{\alpha,g} (t) + 2 \frac{\alpha-3}{\alpha-1} t (\Phi(x(t))- \inf_{\mathcal H}\Phi) \leq 0.$$ By integration of this inequality, we obtain $${\mathcal E}_{\alpha,g} (t) +
2 \frac{\alpha-3}{\alpha-1} \int_{t_0}^t \tau(\Phi(x(\tau))- \inf_{\mathcal H}\Phi) d\tau \leq {\mathcal E}_{\alpha,g} (t_0).$$ By definition of ${\mathcal E}_{\alpha,g}$, and neglecting its nonnegative terms, we infer $$2\int_t^{+\infty} \tau \langle x(\tau) - x^{*} + \frac{\tau}{\alpha-1} \dot{x}(\tau), g(\tau) \rangle d\tau +
2 \frac{\alpha-3}{\alpha-1} \int_{t_0}^t \tau(\Phi(x(\tau))- \inf_{\mathcal H}\Phi) d\tau \leq {\mathcal E}_{\alpha,g} (t_0).$$ Hence $$2 \frac{\alpha-3}{\alpha-1} \int_{t_0}^t \tau(\Phi(x(\tau))- \inf_{\mathcal H}\Phi) d\tau \leq {\mathcal E}_{\alpha,g} (t_0) + 2\int_{t_0}^{+\infty} \| x(\tau) - x^{*} + \frac{\tau}{\alpha-1} \dot{x}(\tau)\| \|\tau g(\tau) \| \rangle d\tau .$$ By (\[energy-10\]), we have $$\begin{aligned}
\sup_t \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \| <+\infty .\end{aligned}$$ As a consequence $$2 \frac{\alpha-3}{\alpha-1} \int_{t_0}^t \tau(\Phi(x(\tau))- \inf_{\mathcal H}\Phi) d\tau \leq {\mathcal E}_{\alpha,g} (t_0) + 2 \sup_t \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \|\int_{t_0}^{+\infty} \|\tau g(\tau) \| \rangle d\tau .$$ Since $\alpha >3$, we deduce that $$\label{basic-estim-2}
\int_{t_0}^{+\infty} \tau(\Phi(x(\tau))- \inf_{\mathcal H}\Phi) d\tau < + \infty .$$
**Step 2.** Let us show that $$\int_{t_0}^{\infty} t \|\dot{x}(t) \|^2 dt < +\infty.$$ To that end, we use the energy estimate which is obtained by taking the scalar product of (\[edo01\]) by $t^2 \dot{x}(t)$: $$\label{scale-energy1}
t^2\langle \ddot{x}(t), \dot{x}(t) \rangle +
\alpha t \|\dot{x}(t)\|^2 + t^2 \langle \nabla \Phi (x(t)), \dot{x}(t) \rangle = t^2 \langle g(t), \dot{x}(t) \rangle .$$ By the classical derivation chain rule, and Cauchy-Schwarz inequality, we obtain $$\label{scale-energy2}
\frac{1}{2}t^2 \frac{d}{dt}\|\dot{x}(t)\|^2 +
\alpha t \|\dot{x}(t)\|^2 + t^2 \frac{d}{dt}\Phi (x(t) \leq \| t g(t)\| \|t \dot{x}(t)\| .$$ After integration by parts $$\begin{aligned}
&\frac{t^2}{2}\|\dot{x}(t)\|^2 - \frac{{t_0}^2}{2}\|\dot{x}(t_0)\|^2- \int_{t_0}^t s \|\dot{x}(s)\|^2 ds + \alpha \int_{t_0}^t s \|\dot{x}(s)\|^2 ds \\
&+ t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi )
- {t_0}^2 (\Phi(x(t_0))- \inf_{\mathcal H}\Phi )
- 2\int_{t_0}^t s (\Phi(x(s))- \inf_{\mathcal H}\Phi )ds \leq \int_{t_0}^t \| s g(s)\| \|s \dot{x}(s)\| ds.\end{aligned}$$ As a consequence, for some constant $C\geq 0$, depending only on the Cauchy data, $$\label{scale-energy4}
\frac{t^2}{2}\|\dot{x}(t)\|^2 + (\alpha -1) \int_{t_0}^t s \|\dot{x}(s)\|^2 ds \leq C+
2\int_{t_0}^t s (\Phi(x(s))- \inf_{\mathcal H}\Phi )ds + \int_{t_0}^t \| s g(s)\| \|s \dot{x}(s)\| ds.$$ By (\[basic-estim-2\]) we have $\int_{t_0}^{\infty} s (\Phi(x(s))- \inf_{\mathcal H}\Phi )ds <+ \infty$. Moreover $\alpha >1$. As a consequence, from (\[scale-energy4\]) we deduce that, for some other constant $C$ $$\label{scale-energy5}
\frac{1}{2}\|t\dot{x}(t)\|^2 \leq C+
\int_{t_0}^t \| s g(s)\| \|s \dot{x}(s)\| ds.$$ Applying Gronwall-Bellman lemma \[GB-lemma\], we obtain $$\|t\dot{x}(t)\| \leq \sqrt{2C}+
\int_{t_0}^t \| s g(s)\| ds.$$ Since $\displaystyle{\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty}$, we infer $$\label{scale-energy7}
\sup_t \|t\dot{x}(t)\| < + \infty.$$ Returning to (\[scale-energy4\]), we deduce that $$\label{scale-energy8}
(\alpha -1) \int_{t_0}^t s \|\dot{x}(s)\|^2 ds \leq C+
2\int_{t_0}^{\infty} s (\Phi(x(s))- \inf_{\mathcal H}\Phi )ds + \sup_t \|t \dot{x}(t)\| \int_{t_0}^{\infty} \| s g(s)\| ds,$$ which gives $$\int_{t_0}^{\infty} t\| \dot{x}(t) \|^2 dt < + \infty .$$ Moreover, combining (\[energy-10\]), $$\begin{aligned}
\sup_t \| x(t) - x^{*} + \frac{t}{\alpha-1} \dot{x}(t) \| <+\infty ,\end{aligned}$$ with (\[scale-energy7\]), we deduce that $$\label{scale-energy9}
\sup_t \|x(t)\| < + \infty,$$ i.e., all the orbits are bounded.
**Step 3.** Our proof of the weak convergence property of the orbits of [$ \mbox{(AVD)}_{\alpha,g}$]{} relies on Opial’s lemma. Given $x^{*} \in {{\rm argmin}\kern 0.12em}\Phi$, let us define $h: [0, +\infty[ \rightarrow \mathbb R^+$ by $$\label{wconv-01}
h(t) = \frac{1}{2}\| x(t) - x^{*}\|^2 .$$ By the classical derivation chain rule $$\begin{aligned}
\label{wconv20}
& \dot{h}(t) = \langle x(t) - x^{*} , \dot{x}(t) \rangle,\\
& \ddot{h}(t) = \langle x(t) - x^{*} , \ddot{x}(t) \rangle + \| \dot{x}(t) \|^2 .\end{aligned}$$ Combining these two equations, and using (\[edo01\]) we obtain $$\begin{aligned}
\label{wconv30}
\ddot{h}(t) + \frac{\alpha}{t} \dot{h}(t) &= \| \dot{x}(t) \|^2 + \langle x(t) - x^{*} , \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) \rangle,\\
& = \| \dot{x}(t) \|^2 + \langle x(t) - x^{*} , -\nabla \Phi (x(t)) + g(t)\rangle .\end{aligned}$$ By monotonicity of $\nabla \Phi$ and $\nabla \Phi(x^{*}) = 0 $ $$\label{wconv40}
\langle x(t) - x^{*} , -\nabla \Phi (x(t)) \rangle \leq 0.$$ By (\[wconv30\]) and (\[wconv40\]) we infer $$\label{wconv50}
\ddot{h}(t) + \frac{\alpha}{t} \dot{h}(t) \leq \| \dot{x}(t) \|^2 + \| x(t) - x^{*} \| \| g(t) \|.$$ Equivalently $$\label{wconv60}
\ddot{h}(t) + \frac{\alpha}{t} \dot{h}(t) \leq k(t),$$ with $$k(t):= \| \dot{x}(t) \|^2 + \| x(t) - x^{*} \| \| g(t) \|.$$ By (\[scale-energy9\]) the orbit is bounded. Hence, for some constant $C\geq 0$ $$k(t)\leq \| \dot{x}(t) \|^2 + C \| g(t) \|.$$ By assumption $\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty$, and by (\[energy1\]) $\int_{t_0}^{\infty} t\| \dot{x}(t) \|^2 dt < + \infty$. Hence $t \mapsto tk(t) \in L^1 (t_0, +\infty)$. Applying Lemma \[basic-edo\], with $w(t)= \dot{h}(t)$, we deduce that $w^{+} \in L^1 (t_0, +\infty)$. Equivalently $\dot{h}^+(t) \in L^1 (t_0, +\infty)$, which implies that the limit of $h(t)$ exists, as $t \to + \infty$. This proves item $i)$ of the Opial’s lemma. We complete the proof by observing that item $ii)$ is satisfied too. Indeed, since $\Phi (x(t))$ converges to $\inf \Phi$, we have that every weak sequential cluster point of $x(\cdot)$ is a minimizer of $\Phi$.
Strong convergence results
--------------------------
Since the work of J.B. Baillon, we know that without additional assumptions, the trajectories of the gradient systems may not converge strongly. Let’s examine some practical interest situations where strong convergence of the trajectories of $ \mbox{{\rm(AVD)}}_{\alpha,g}$ is satisfied.
**Strong convergence under $int({{\rm argmin}\kern 0.12em}\Phi)\neq \emptyset $.**\
We will need the following result, see ([@APR1], Lemma 5.4).
\[strong-lem1\] Suppose that $\delta > 0$, and let $f : [\delta ;+\infty [ \to \mathcal H$ be a continuous function that satisfies $f \in L^{1}(\delta;+\infty ;\mathcal H)$. Suppose that $\alpha > 1$ and $x : [\delta ;+\infty [\to \mathcal H$ is a classical solution of $$t\ddot{x}(t)+\alpha\dot{x}(t)=f(t).$$ Then, $x(t)$ converges strongly in $\mathcal H$ as $t \to \infty$.
Suppose that $\alpha > 3$, $\displaystyle{\int_{t_0}^{+\infty}}tg(t)dt<+\infty$, and $\Phi$ satisfises $int({{\rm argmin}\kern 0.12em}\Phi)\neq \emptyset$. Let $x(\cdot)$ be a classical global solution of equation (\[edo01\]). Then, there exists some $x^{*} \in {{\rm argmin}\kern 0.12em}\,\Phi$ such that $x(t) \to x^{*}$ strongly as $t \to +\infty$.
We follow the same approach as that proposed in [@APR1 Theorem 3.1]. We first observe that the assumption $int({{\rm argmin}\kern 0.12em}\Phi)\neq \emptyset$ implies the existence of some $\bar{z}\in \mathcal H$ and $\rho > 0$ such that, for all $x\in \mathcal H$, $\langle\nabla\Phi (x),x-\bar{z} \rangle\geq\rho\Vert\nabla\Phi(x)\Vert$. In particular, for all $t \geq t_0$ $$\langle\nabla\Phi (x(t)),x(t)-\bar{z} \rangle\geq\rho\Vert\nabla\Phi(x(t))\Vert .$$ Combining this inequality with (\[rfastoche1\]) (that we recall below) $$\dot{\mathcal E}_{\alpha,g} (t)= \frac{4}{\alpha-1}t (\Phi(x(t))- \inf_{\mathcal H}\Phi ) - 2 t \langle x(t) - \bar{z} , \nabla \Phi (x(t)) \rangle$$ we obtain $$\label{Strong1}
\dot{\mathcal E}_{\alpha,g} (t)+2\rho t\Vert\nabla\Phi (x(t))\Vert\leq \frac{4}{\alpha-1}t (\Phi(x(t))- \inf_{\mathcal H}\Phi ).$$ Let us return to (\[basic-energy-Liap\]), which after integration, and using $\alpha >3$, gives $$\int_{t_{0}}^{\infty}t(\Phi (x(t))-\inf_{\mathcal H}\Phi ) dt<+\infty.$$ As a consequence, by integrating (\[Strong1\]), we deduce that $$\int_{t_{0}}^{\infty}t\Vert\nabla\Phi (x(t))\Vert dt<+\infty.$$ By setting $f(t)=tg(t)-t\nabla \Phi (x(t))$, we can rewrite equation (\[edo01\]) as $$t\ddot{x}(t)+\alpha\dot{x}(t)=f(t).$$ Since all assumptions of Lemma \[strong-lem1\] are satisfied, we can affirm that $x(t)$ converges strongly to some $x^{*}\in \mathcal H$. Recalling that $\Phi(x(t))\rightarrow\inf_{\mathcal H}\Phi$ and that $\Phi$ is continuous, we obtain $x^{*}\in {{\rm argmin}\kern 0.12em}\,\Phi$.
**Strong convergence in the case of an even function.**\
Recall that $\Phi:\mathcal H \rightarrow \mathbb R$ is an even function if $\Phi(-x)=\Phi(x)$ for all $x\in \mathcal H$. In this case, $0\in{{\rm argmin}\kern 0.12em}_{\mathcal{H}}\Phi$.
Suppose that $\alpha >3$, $\displaystyle{\int_{t_0}^{+\infty}}tg(t)dt<+\infty$, and $\Phi$ is an even function. Let $x(\cdot)$ be a classical global solution of equation (\[edo01\]). Then, there exists some $\bar{x}\in {{\rm argmin}\kern 0.12em}_{\mathcal{H}}\Phi$ such that $x(t)$ converges strongly to $\bar{x}$ as $t\to +\infty$.
Set, for $t_0 \leq \tau \leq r$, $$y(\tau )=\Vert x(\tau)\Vert^{2}-\Vert x(r)\Vert^{2}-\frac{1}{2}\Vert x(\tau ) -x(r)\Vert^{2}.$$ By derivating twice, we obtain $$\dot{y}(\tau)=\langle \dot{x}(\tau), x(\tau)+x(r) \rangle$$ and $$\ddot{y}(\tau)=\Vert \dot{x}(\tau)\Vert^{2}+\langle \ddot{x}(\tau), x(\tau)+x(r) \rangle.$$ From these two equations and (1), we deduce that $$\label{even}
\begin{array}{lll}
\ddot{y}(\tau)+\frac{\alpha}{\tau}\dot{y}(\tau)&=&\Vert \dot{x}(\tau)\Vert^{2}+\langle \ddot{x}(\tau)+\frac{\alpha}{\tau}\dot{x}(\tau), x(\tau)+x(r) \rangle \vspace{2mm}\\
&=& \Vert \dot{x}(\tau)\Vert^{2}+\langle g(\tau)-\nabla\Phi (x(\tau)), x(\tau)+x(r) \rangle .
\end{array}$$ Let us now consider the energy function, $W(\tau )=\frac{1}{2}\Vert \dot{x}(\tau)\Vert^{2}+\Phi(x(\tau))+\int_{\tau}^{\infty}\langle\dot{x}(t),g(t)dt\rangle$. We have $\frac{d}{d\tau}W(\tau)=-\frac{\alpha}{\tau}\Vert \dot{x}(\tau)\Vert^{2}$, and therefore $W$ is a nonincreasing function. As a consequence, $W(\tau)\geq W(r)$, which equivalently gives $$\frac{1}{2}\Vert \dot{x}(\tau)\Vert^{2}+\Phi(x(\tau))\geq \frac{1}{2}\Vert \dot{x}(r)\Vert^{2}+\Phi(x(r))-\int_{\tau}^{r}\langle\dot{x}(t),g(t)\rangle dt.$$ Using the convex differential inequality $\Phi (-x(r))\geq \Phi (x(\tau))-\langle\nabla\Phi (x(\tau)), x(\tau)+x(r)\rangle$, and the even property of $\Phi$, $\Phi(x(r))=\Phi(-x(r))$, we deduce that $$\frac{1}{2}\Vert \dot{x}(\tau)\Vert^{2}\geq -\langle\nabla\Phi (x(\tau)), x(\tau)+x(r)\rangle -\int_{\tau}^{r}\langle\dot{x}(t),g(t)\rangle dt.$$ Returning to (\[even\]), we finally obtain $$\ddot{y}(\tau)+\frac{\alpha}{\tau}\dot{y}(\tau)\leq\frac{3}{2}\Vert \dot{x}(\tau)\Vert^{2}+\langle g(\tau),x(\tau)+x(r)\rangle +\int_{\tau}^{r}\langle\dot{x}(t),g(t)\rangle dt .$$ Let us recall that, by Theorem \[Thm-weak-conv\], the trajectory $x(\cdot)$ is converging weakly, and hence bounded. Moreover, by (\[conv1\]), we have $\| \dot{x}(t) \| \leq \frac{C}{t} $. Hence, for some constant $C$ $$\label{even1}
\ddot{y}(\tau)+\frac{\alpha}{\tau}\dot{y}(\tau)\leq k(\tau):= \frac{3}{2}\Vert \dot{x}(\tau)\Vert^{2}+ C \| g(\tau)\| + C\int_{\tau}^{+ \infty}\frac{1}{t}\|g(t)\| dt.$$ Let us observe that the function $k$ does not depend on $r$. Let us verify that $\tau \mapsto \tau k(\tau) \in L^{1}(t_0,+\infty)$. By Theorem \[Thm-weak-conv\], we have $\int_{t_0}^{\infty} t\| \dot{x}(t) \|^2 dt < + \infty$. By assumption, $\int_{t_0}^{+\infty}tg(t)dt<+\infty$. Moreover, by Fubini theorem $$\int_{t_0}^{\infty}\tau \int_{\tau}^{\infty} \frac{1}{t}\|g(t)\| dt d\tau \leq \frac{1}{2} \int_{t_0}^{\infty} t\|g(t)\| dt < +\infty.$$ By integration of (\[even1\]), by a similar argument as in Lemma \[basic-edo\], we obtain $$\label{even2}
\dot{y}(\tau) \leq \frac{C}{\tau^{\alpha} } + \frac{1}{\tau^{\alpha} } \int_{t_0}^\tau u^{\alpha} k(u) du,$$ where $C={t_0}^{\alpha}\|\dot x(t_0)\|\,\|x\|_{\infty}$. Set $$K(t):= \frac{C}{\tau^{\alpha} } + \frac{1}{\tau^{\alpha} } \int_{t_0}^\tau u^{\alpha} k(u) du .$$ By using Fubini theorem once more, and the fact that $\tau \mapsto \tau k(\tau) \in L^{1}(t_0,+\infty)$, we deduce that $K\in L^{1}(t_0,+\infty)$. Integrating $ \dot{y}(\tau) \leq K(\tau) $ from $t$ to $r$, we obtain $$\frac{1}{2} \|x(t)- x(r) \|^2 \leq \| x(t) \|^2 -\| x(r) \|^2
+ \int_t^r K(\tau) d\tau.$$ Since $\Phi$ is even, we have $0 \in{{\rm argmin}\kern 0.12em}\Phi$. Hence $\lim_{t\to +\infty }\| x(t) \|^2$ exists (see the proof of Theorem \[Thm-weak-conv\]). As a consequence, $x(t)$ has the Cauchy property as $t \to + \infty$, and hence converges.
The case ${{\rm argmin}\kern 0.12em}\Phi =\emptyset .$
======================================================
\[Thm-armin-empty\] Suppose $\alpha >0 $, $\displaystyle{\int_{t_0}^{+\infty} \|g(t)\| dt < + \infty}$, and $\inf \Phi >-\infty$. Then, for any orbit $x: [t_0, +\infty[ \rightarrow \mathcal H$ of $ \mbox{{\rm(AVD)}}_{\alpha,g}$, the following minimizing property holds $$\lim_{t\rightarrow +\infty}\Phi (x(t))=\inf_{\mathcal H}\Phi.$$
We will use the following lemma, see [@APR1].
\[basic-int\] Take $\delta >0$, and let $f \in L^1 (\delta , +\infty)$ be nonnegative. Consider a nondecreasing continuous function $\psi:(\delta,+\infty)\to(0,+\infty)$ such that $\lim\limits_{t\to+\infty}\psi(t)=+\infty$. Then, $$\lim_{t \rightarrow + \infty} \frac{1}{\psi(t)} \int_{\delta}^t \psi(s)f(s)ds =0.$$
*Proof of Theorem* \[Thm-armin-empty\]. Let us first return to the proof of the energy estimates in Proposition \[energy-thm-1\]. Replacing $\inf \Phi$ by $\min \Phi$ in the expression of the energy function, we obtain by the same argument $$\begin{aligned}
\label{basic-est-empty}
& \sup_t \| \dot{x}(t) \| < + \infty , \\
& \int_{t_0}^{+\infty} \frac{1}{t} \| \dot{x}(t) \|^2 dt < + \infty . \end{aligned}$$ Consider the function $h(t)=\frac{1}{2}\Vert x(t)-z\Vert^{2}$, where this time, $z$ is an arbitrary element of $\mathcal H$. We can easily verify that $$\ddot{h}(t)+\frac{\alpha}{t}\dot{h}(t)= \Vert \dot{x}(t)\Vert^{2}-\langle \nabla \Phi(x(t)),x(t)-z\rangle+\langle g(t),x(t)-z\rangle.$$ By convexity of $\Phi$, we obtain $$\label{h1}
\ddot{h}(t)+\frac{\alpha}{t}\dot{h}(t)+\Phi (x(t))-\Phi (z)\leq \Vert \dot{x}(t)\Vert^{2}+\langle g(t),x(t)-z\rangle.$$ Consider the energy function $$W(t)=\frac{1}{2}\Vert \dot{x}(t)\Vert^{2}+\Phi (x(t))-\inf \Phi +\int_{t}^{\infty}\langle\dot{x}(s),g(s)\rangle ds.$$ By classical derivation rules, and (\[edo01\]) $$\begin{array}{lll}
\displaystyle{\frac{d}{dt}}W(t)&=& \langle \dot{x}(t),\ddot{x}(t)+\nabla\Phi(x(t))-g(t)\rangle\\
&=& -\dfrac{\alpha}{t}\Vert \dot{x}(t)\Vert^{2}\leq 0.
\end{array}$$ As a consequence, $W$ is a nonincreasing function. Moreover, $W$ is minorized by $-\Vert \dot{x}\Vert_{L^{\infty}}\int_{t_0}^{+\infty} \|g(s)\|ds.$ Hence, there exists some $W_{\infty} \in \mathbb R$ such that $W(t)\rightarrow W_{\infty}$ as $t\rightarrow\infty$. Let us take advantage of this property, and reformulate (\[h1\]) with the help of $W$: $$\label{h1b}
\ddot{h}(t)+\frac{\alpha}{t}\dot{h}(t)+W(t) + \inf \Phi -\Phi (z)\leq \frac{3}{2}\Vert \dot{x}(t)\Vert^{2}+\langle g(t),x(t)-z\rangle +\int_{t}^{\infty}\langle\dot{x}(s),g(s)\rangle ds .$$ For every $t>0$, $ W(t)\geq W_{\infty}$. Setting $B_{\infty}=W_{\infty} +\inf \Phi-\Phi(z) $, we obtain $$B_{\infty}\leq \frac{3}{2}\Vert \dot{x}(t)\Vert^{2}+\Vert g(t)\Vert\Vert x(t)-z\Vert+\Vert \dot{x}\Vert_{L^{\infty}}
\int_{t}^{\infty}\Vert g(s)\Vert ds-\frac{1}{t^{\alpha}}\frac{d}{dt}(t^{\alpha}\dot{h}(t)).$$ Multiplying this last equation by $\frac{1}{t}$, and integrating between two reals $0< t_{0}<\theta$, we get $$\label{h2}
B_{\infty} \ln (\frac{\theta}{t_{0}})\leq \frac{3}{2} \int_{t_{0}}^{\theta}\frac{1}{t}\Vert \dot{x}(t)\Vert^{2}dt+\int_{t_{0}}^{\theta}
\dfrac{\Vert g(t)\Vert\Vert x(t)-z\Vert}{t} dt+\Vert \dot{x}\Vert_{L^{\infty}} \int_{t_{0}}^{\theta} \left( \frac{1}{t}\int_{t}^{\infty}\Vert g(s)\Vert ds\right) dt
-\int_{t_{0}}^{\theta}\frac{1}{t^{\alpha+1}}\frac{d}{dt}(t^{\alpha}\dot{h}(t))dt.$$ Let us estimate the integrals in the second member of (\[h2\]):
1. By (\[basic-est-empty\]), $\int_{t_0}^{+\infty} \frac{1}{t} \| \dot{x}(t) \|^2 dt < + \infty $.
2. Exploiting the relation $\Vert x(t)-z\Vert \leq \Vert x(t_0)-z\Vert +\int_{t_0}^{t}\Vert\dot{x}(s)\Vert ds$, we obtain $$\int_{t_{0}}^{\theta}\dfrac{\Vert g(t)\Vert\Vert x(t)-z\Vert}{t} dt\leq \left( \frac{\Vert x_{0}-z\Vert}{t_{0}}+
\Vert \dot{x}\Vert_{L^{\infty}}\right) \int_{t_{0}}^{+\infty}\Vert g(t)\Vert dt< +\infty.$$
3. After integration by parts $$\int_{t_{0}}^{\theta} \left( \frac{1}{t}\int_{t}^{\infty}\Vert g(s)\Vert ds\right) dt = \ln \theta \int_{\theta}^{\infty}\Vert g(s)\Vert ds - \ln t_0 \int_{t_0}^{\infty}\Vert g(s)\Vert ds + \int_{t_{0}}^{\theta} \Vert g(t)\Vert\ln t \ dt.$$
4. Set $I=\int_{t_{0}}^{\theta}\frac{1}{t^{\alpha+1}}\frac{d}{dt}(t^{\alpha}\dot{h}(t))dt.$ By integrating by parts twice $$\begin{array}{lll}
I&=& \left[ \frac{1}{t}\dot{h}(t)\right]_{t_{0}}^{\theta} +(\alpha+1)\int_{t_{0}}^{\theta}\frac{1}{t^{2}}\dot{h}(t)dt\\
&=& C +\frac{1}{\theta}\dot{h}(\theta) +\frac{(1+\alpha )}{\theta^{2}}h(\theta )+2(1+\alpha)\int_{t_{0}}^{\theta}
\frac{1}{t^{3}}h(t)dt.
\end{array}$$ Since $h\geq 0$, we have $-I\leq -C -\frac{1}{\theta}\dot{h}(\theta).$ Then notice that $\vert\dot{h}(\theta )\vert=\vert\langle \dot{x}(\theta),x(\theta )-z\rangle\vert\leq \Vert\dot{x}\Vert_{L^{\infty}}(\Vert x(0)-z\Vert+\theta\Vert\dot{x}\Vert_{L^{\infty}})$.
Collecting the above results, we deduce from (\[h2\]) that $$\label{h2b}
B_{\infty} \ln (\frac{\theta}{t_{0}})\leq C +
\ln \theta \int_{\theta}^{\infty}\Vert g(s)\Vert ds +\Vert \dot{x}\Vert_{L^{\infty}} \int_{t_{0}}^{\theta} \Vert g(t)\Vert\ln t \ dt.$$ Dividing by $\ln (\frac{\theta}{t_{0}})$, and letting $\theta\rightarrow +\infty$, thanks to Lemma \[basic-int\] with $\psi (t) =\ln t$, we conclude that $B_{\infty} \leq 0.$ Equivalently, for every $z\in \mathcal H$, $W_{\infty}\leq \Phi (z)-\inf \Phi$, which leads to $W_{\infty} \leq 0.$
On the other hand, it is easy to see that $W(t)\geq \Phi(x(t))-\inf \Phi -\Vert\dot{x}\Vert_{L^{\infty}}\int_{t}^{+\infty}g(s)ds$. Passing to the limit, as $t\rightarrow+\infty$, we deduce that $$0\geq W_{\infty}\geq\limsup \Phi(x(t))- \inf \Phi .$$ Since we always have $\inf \Phi\leq \liminf \Phi(x(t))$, we conclude that $\lim_{t\rightarrow +\infty}\Phi(x(t))=\inf\Phi.$ $\square$
In [@APR], in the unperturbed case $g=0$, it has been observed that, when ${{\rm argmin}\kern 0.12em}\Phi =\emptyset$, the fast convergence property of the values, as given in Theorem \[fastconv-thm\], may fail to be satisfied. A fortiori, without making additional assumption on the perturbation term, we also loose the fast convergence property in the perturbed case (take $g=0$!).
From continuous to discrete dynamics and algorithms
===================================================
Time discretization of dissipative gradient-based dynamical systems leads naturally to algorithms, which, under appropriate assumptions, have similar convergence properties. This approach has been followed successfully in a variety of situations. For a general abstract discussion see [@Alv_Pey2], [@Alv_Pey3], and in the case or dynamics with inertial features see [@Al], [@AA1], [@aabr], [@APR], [@APR1]. To cover practical situations involving constraints and/or nonsmooth data, we need to broaden our scope. This leads us to consider the non-smooth structured convex minimization problem $$\label{algo1}
\min \left\lbrace \Phi (x) + \Psi (x): \ x \in \mathcal H \right\rbrace$$ where
$\bullet$ $\Phi: \mathcal H \to \mathbb R \cup \lbrace + \infty \rbrace $ is a convex lower semicontinuous proper function (which possibly takes the value $+ \infty$);
$\bullet$ $\Psi: \mathcal H \to \mathbb R $ is a convex continuously differentiable function, whose gradient is Lipschitz continuous.
The optimal solutions of (\[algo1\]) satisfy $$\partial \Phi (x) + \nabla \Psi (x) \ni 0,$$ where $\partial \Phi$ is the subdifferential of $\Phi$ in the sense of convex analysis. In order to adapt our dynamic to this non-smooth situation, we will consider the corresponding differential inclusion $$\label{algo2}
\ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \partial \Phi (x(t)) + \nabla \Psi (x(t)) \ni g(t).$$ This dynamic is within the following framework $$\label{algo2b}
\ddot{x}(t) + a(t) \dot{x}(t) + \partial \Theta (x(t)) \ni g(t),$$ where $\Theta: \mathcal H \to \mathbb R \cup \lbrace + \infty \rbrace $ is a convex lower semicontinuous proper function, and $a(\cdot)$ is a positive damping parameter.
The detailed study of this differential inclusion goes far beyond the scope of the present article, see [@ACR] for some results in the case of a fixed positive damping parameter, i.e., $a(t)= \gamma >0$ fixed, and $g=0$. A formal analysis of this sytem shows that the Lyapunov analysis, which has been developed in the previous sections, still holds, as long as one does not use the Lipschitz continuity property of the gradient (cocoercivity property). This is based on the fact that the convexity (subdifferential) inequalites are still valid, as well as the (generalized) derivation chain rule, see [@Bre1]. Thus, setting $\Theta (x)= \Phi (x) + \Psi (x)$, we can reasonably assume that, for $\alpha >3$, and $\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty$, for each trajectory of (\[algo2\]), there is rapid convergence of the values, $$\begin{aligned}
\Theta(x(t))- \min\Theta \leq \frac{C}{t^2} ,\end{aligned}$$ and weak convergence of the trajectory to an optimal solution.
Indeed, we are going to use these ideas as a guideline, and so introduce corresponding fast converging algorithms, making the link with Nesterov [@Nest1]-[@Nest4], Beck-Teboulle [@BT], and so extending the recent works of Chambolle-Dossal [@CD], Su-Boyd-Candès [@SBC], Attouch-Peypouquet-Redont [@APR] to the perturbed case. As a basic ingredient of the discretization procedure, in order to preserve the fast convergence properties of the dynamical system (\[algo2\]), we are going to discretize it *implicitely* with respect to the nonsmooth function $\Phi$, and *explicitely* with respect to the smooth function $\Psi$.
Taking a fixed time step size $h>0$, and setting $t_k= kh$, $x_k = x(t_k)$ the implicit/explicit finite difference scheme for (\[algo2\]) gives $$\label{algo3}
\frac{1}{h^2}(x_{k+1} -2 x_{k} + x_{k-1} ) +\frac{\alpha}{kh^2}( x_{k} - x_{k-1}) + \partial \Phi (x_{k+1} ) + \nabla \Psi (y_k) \ni g_k,$$ where $y_k$ is a linear combination of $x_k$ and $x_{k-1}$, that will be made precise further. After developing (\[algo3\]), we obtain $$\label{algo4}
x_{k+1} + h^2 \partial \Phi (x_{k+1}) \ni x_{k} + \left( 1- \frac{\alpha}{k}\right) ( x_{k} - x_{k-1}) - h^2 \nabla \Psi (y_k) + h^2 g_k .$$ A natural choice for $y_k$ leading to a simple formulation of the algorithm (other choices are possible, offering new directions of research for the future) is $$\label{algo5}
y_k= x_{k} + \left( 1- \frac{\alpha}{k}\right) ( x_{k} - x_{k-1}).$$ Using the classical proximal operator (equivalently, the resolvent operator of the maximal monotone operator $\partial \Phi$) $$\label{algo6}
\mbox{prox}_{ \gamma \Phi } (x)= {{{\rm argmin}\kern 0.12em}}_{\xi \in \mathcal H} \left\lbrace \Phi (\xi) + \frac{1}{2 \gamma} \| \xi -x \|^2
\right\rbrace = \left(I + \gamma \partial \Phi \right)^{-1} (x)$$ and setting $s=h^2$, the algorithm can be written as $$\label{algo7}
\left\{
\begin{array}{l}
y_k= x_{k} + \left( 1- \frac{\alpha}{k}\right) ( x_{k} - x_{k-1}); \\
\rule{0pt}{20pt}
x_{k+1} = \mbox{prox}_{s \Phi} \left( y_k- s (\nabla \Psi (y_k) -g_k) \right).
\end{array}\right.$$ For practical purpose, and in order to fit with the existing litterature on the subject, it is convenient to work with the following equivalent formulation $$\label{algo7b}
{\rm \mbox{(AVD)}_{\alpha,g}-algo} \ \left\{
\begin{array}{l}
y_k= x_{k} + \frac{k -1}{k + \alpha -1} ( x_{k} - x_{k-1}); \\
\rule{0pt}{20pt}
x_{k+1} = \mbox{prox}_{s \Phi} \left( y_k- s (\nabla \Psi (y_k) -g_k)\right).
\end{array}\right.$$ Indeed, we have $\frac{k -1}{k + \alpha -1} = 1- \frac{\alpha}{k + \alpha -1}$. When $\alpha$ is an integer, up to the reindexation $k\mapsto k + \alpha -1$, we obtain the same sequences $(x_k)$ and $(y_k)$. For general $\alpha >0$, we can easily verify that the algorithm $ {\rm \mbox{(AVD)}_{\alpha,g}-algo} $ is still associated with the dynamical system (\[algo2\]).
This algorithm is within the scope of the proximal-based inertial algorithms [@AA1], [@MO], [@LP], and forward-backward methods. In the unperturbed case, $g_k =0$, it has been recently considered by Chambolle-Dossal [@CD], Su-Boyd-Candès [@SBC], and Attouch-Peypouquet-Redont [@APR]. It enjoys fast convergence properties which are very similar to that of the continuous dynamic.
For $\alpha = 3$, $g_k =0$, we recover the classical algorithm based on Nesterov and Güler ideas, and developed by Beck-Teboulle (FISTA) $$\label{algo7c}
\ \left\{
\begin{array}{l}
y_k= x_{k} + \frac{k -1}{k + 2} ( x_{k} - x_{k-1}); \\
\rule{0pt}{20pt}
x_{k+1} = \mbox{prox}_{s \Phi} \left( y_k- s \nabla \Psi (y_k) \right).
\end{array}\right.$$ An important question regarding the (FISTA) method, as described in (\[algo7c\]), is the convergence of sequences $(x_k)$ and $(y_k)$. Indeed, it is still an open question. A major interest to consider the broader context of $ {\rm \mbox{(AVD)}_{\alpha,g}-algo}$ algorithms is that, for $\alpha >3$, these sequences converge, and they allow errors/perturbations, and using approximation methods. We will see that the proof of the convergence properties of $ {\rm \mbox{(AVD)}_{\alpha,g}-algo}$ algorithms can be obtained in a parallel way with the convergence analysis in the continuous case in Theorem \[Thm-weak-conv\].
\[Thm-algo\] Let $\Phi: \mathcal H \to \mathbb R \cup \lbrace + \infty \rbrace $ be a convex lower semicontinuous proper function, and $\Psi: \mathcal H \to \mathbb R $ a convex continuously differentiable function, whose gradient is $L$-Lipschitz continuous. Suppose that $S={{\rm argmin}\kern 0.12em}(\Phi + \Psi)$ is nonempty. Suppose that $\alpha \geq 3$, $ 0< s < \frac{1}{L} $, and $ \sum_{k \in \mathbb N} k \|g_k\| < + \infty $. Let $(x_k)$ be a sequence generated by the algorithm [${\rm \mbox{(AVD)}_{\alpha,g}-algo} $]{}. Then, $$(\Phi + \Psi)(x_k)- \min(\Phi + \Psi) =\mathcal O (\frac{1}{k^2}).$$ Precisely, $$\label{algo8}
(\Phi + \Psi)(x_k)- \min(\Phi + \Psi) \leq \frac{C (\alpha -1)}{2s\left( k + \alpha -2\right)^2 },$$ with $C$ given by $$C = \mathcal G (0) + 2s\left( \sum_{j=0}^{\infty} \left( j + \alpha -1\right) \| g_j\|\right) \left( \sqrt{\frac{\mathcal E (0)}{\alpha -1}} + \frac{2s}{\alpha -1} \sum_{j=0}^{\infty} \left( j + \alpha -1\right) \| g_j\| \right) ,$$ where $$\mathcal G (0) = \frac{2s}{\alpha -1} \left( \alpha -2\right)^2 (\Theta (x_0) - {\Theta}^*) + (\alpha -1)
\| y_0 - x^{*} \|^2 .$$
To simplify notations, we set $\Theta = \Phi + \Psi$, and take $x^{*} \in {{\rm argmin}\kern 0.12em}\Theta$, i.e., $\Theta (x^{*})= \inf \Theta $. In a parallel way to the continuous case, our proof is based on proving that $(\mathcal E (k))$ is a non-increasing sequence, where $\mathcal E (k)$ is the discrete version of the Lyapunov function ${\mathcal E}_{\alpha,g} (t)$ (we shall justify further that it is well defined), and which is given by $$\label{algo9b}
\mathcal E (k):= \frac{2s}{\alpha -1} \left( k + \alpha -2\right)^2 (\Theta (x_k) - \Theta(x^{*}) + (\alpha -1)
\| z_k -x^{*} \|^2 + \sum_{j=k}^{\infty} 2s\left( j + \alpha -1\right) \left\langle g_j, z_{j+1}- x^{*} \right\rangle ,$$ with $$\label{algo9c}
z_k := \frac{k + \alpha -1}{\alpha -1}y_k - \frac{k}{\alpha -1}x_k .$$ In the passage from the continuous to discrete, we recall that we must use the reindexing $k \mapsto k + \alpha -1$. Note that $\mathcal E (k)$ is equal to the Lyapunov function considered by Su-Boyd-Candès in [@SBC Theorem 4.3], plus a perturbation term.\
Let us introduce the function $\Psi_k : \mathcal H \rightarrow \mathbb R $ which is defined by $$\forall y \in \mathcal H, \ \Psi_k (y):= \Psi (y) - \left\langle g_k, y \right\rangle .$$ We also set $$\Theta_k = \Phi + \Psi_k .$$ We have $\nabla \Psi_k (y)= \nabla\Psi (y) -g_k$, and hence $\nabla \Psi_k$ is still $L$-Lipschitz continuous. We can reformulate our algorithm with the help of $\Psi_k $ as follows $$\label{algo12}
{\rm \mbox{(AVD)}_{\alpha,g}-algo} \ \left\{
\begin{array}{l}
y_k= x_{k} + \frac{k -1}{k + \alpha -1} ( x_{k} - x_{k-1}); \\
\rule{0pt}{20pt}
x_{k+1} = \mbox{prox}_{s \Phi} \left( y_k- s \nabla \Psi_k (y_k)\right).
\end{array}\right.$$ In order to analyze the convergence properties of the above algorithm, it is convenient to introduce the operator $G_{s,k}: \mathcal H \rightarrow \mathcal H$ which is defined by, for all $y \in \mathcal H$, $$G_{s,k} (y) = \frac{1}{s}\left( y - \mbox{prox}_{s \Phi} \left( y- s \nabla \Psi_k (y) \right) \right) .$$ Equivalently, $$\mbox{prox}_{s \Phi} \left( y- s \nabla \Psi_k (y) \right) = y - s G_{s,k} (y),$$ and the algorithm (\[algo12\]) can be formulated as $$\label{algo13}
{\rm \mbox{(AVD)}_{\alpha,g}-algo} \ \left\{
\begin{array}{l}
y_k= x_{k} + \frac{k -1}{k + \alpha -1} ( x_{k} - x_{k-1}); \\
\rule{0pt}{20pt}
x_{k+1} = y_k - s G_{s,k} (y_k).
\end{array}\right.$$ The variable $z_k$, which is defined in (\[algo9c\]) by $z_k = \frac{k + \alpha -1}{\alpha -1}y_k - \frac{k}{\alpha -1}x_k$, will play an important role. It comes naturally into play as a discrete version of the term $ \frac{t}{\alpha-1} \dot{x}(t) + x(t) - x^{*} $ which enters ${\mathcal E}_{\alpha,g}(t)$. Indeed, $$\begin{aligned}
\label{algo13-b}
\frac{k + \alpha -1}{\alpha -1} \left( x_{k+1}- x_{k} \right) + x_{k}& = \frac{k + \alpha -1}{\alpha -1} x_{k+1}
-\frac{k }{\alpha -1} x_k \\
& = z_{k+1} \nonumber\end{aligned}$$ where the last equality comes from (\[algo14b\]) below. Let us examine the recursive relation satisfied by $z_k$. We have $$\begin{aligned}
z_{k+1}&= \frac{k + \alpha}{\alpha -1}y_{k+1} - \frac{k+1}{\alpha -1}x_{k+1} \nonumber \\
&= \frac{k + \alpha}{\alpha -1}\left( x_{k+1} + \frac{k }{k + \alpha } ( x_{k+1} - x_{k}) \right) - \frac{k+1}{\alpha -1}x_{k+1} \nonumber\\
&= \frac{k + \alpha -1}{\alpha -1} x_{k+1}
-\frac{k}{\alpha -1}x_{k} \label{algo14b}\\
&= \frac{k + \alpha -1}{\alpha -1} \left(y_k - s G_{s,k} (y_k) \right) -\frac{k}{\alpha -1}x_{k} \nonumber\\
&= z_k -\frac{s}{\alpha -1} \left( k + \alpha -1\right) G_{s,k} (y_k) \label{algo14a}.\end{aligned}$$ We now use the classical formula in the proximal gradient (also called forward-backward) analysis (see [@BT], [@CD], [@PB], [@SBC]): for any $x, y\in \mathcal H$ $$\label{algo14}
\Theta_k (y - sG_{s,k} (y)) \leq \Theta_k (x) + \left\langle G_{s,k} (y), y-x \right\rangle -\frac{s}{2} \| G_{s,k} (y) \|^2 .$$ Note that this formula is valid since $s \leq \frac{1}{L}$, and $\nabla \Psi_k$ is $L$-lipschitz continuous. Let us write successively this formula at $y=y_k$ and $x= x_k$, then at $y=y_k$ and $x= x^{*}$. We obtain $$\begin{aligned}
&\Theta_k (y_k - sG_{s,k} (y_k)) \leq \Theta_k (x_k) + \left\langle G_{s,k} (y_k), y_k-x_k \right\rangle -\frac{s}{2} \| G_{s,k} (y_k) \|^2 \\
& \Theta_k (y_k - sG_{s,k} (y_k)) \leq \Theta_k (x^{*}) + \left\langle G_{s,k} (y_k), y_k-x^{*} \right\rangle -\frac{s}{2} \| G_{s,k} (y_k) \|^2 .\end{aligned}$$ Multiplying the first equation by $\frac{k}{k + \alpha -1}$, and the second by $\frac{\alpha -1}{k + \alpha -1}$, then adding the two resulting equations, and using $x_{k+1} = y_k - s G_{s,k} (y_k)$, we obtain $$\begin{aligned}
\label{algo15}
\Theta_k (x_{k+1}) \leq &\frac{k}{k + \alpha -1} \Theta_k (x_k) + \frac{\alpha -1}{k + \alpha -1}\Theta_k (x^{*})
-\frac{s}{2} \| G_{s,k} (y_k) \|^2 \\
& + \left\langle G_{s,k} (y_k), \frac{k}{k + \alpha -1}(y_k-x_k ) + \frac{\alpha -1}{k + \alpha -1} (y_k-x^{*})\right\rangle \label{algo15b}.\end{aligned}$$ Let us rewrite the scalar product in (\[algo15b\]) as follows: $$\begin{aligned}
\label{algo16}
\left\langle G_{s,k} (y_k), \frac{k}{k + \alpha -1}(y_k-x_k ) + \frac{\alpha -1}{k + \alpha -1} (y_k-x^{*})\right\rangle &= \frac{\alpha -1}{k + \alpha -1}
\left\langle G_{s,k} (y_k), \frac{k}{\alpha -1}(y_k-x_k ) + y_k-x^{*}\right\rangle \\
& =\frac{\alpha -1}{k + \alpha -1}
\left\langle G_{s,k} (y_k), \frac{k+\alpha -1}{\alpha -1}y_k- \frac{k}{\alpha -1} x_k -x^{*}\right\rangle \nonumber\\
& =\frac{\alpha -1}{k + \alpha -1}
\left\langle G_{s,k} (y_k), z_k -x^{*}\right\rangle \nonumber.\end{aligned}$$ Combining (\[algo15\])-(\[algo15b\]) with (\[algo16\]), we obtain $$\begin{aligned}
\label{algo17}
\Theta_k (x_{k+1}) \leq \frac{k}{k + \alpha -1} \Theta_k (x_k) + \frac{\alpha -1}{k + \alpha -1}\Theta_k (x^{*}) +
\frac{\alpha -1}{k + \alpha -1}
\left\langle G_{s,k} (y_k), z_k -x^{*}\right\rangle -\frac{s}{2} \| G_{s,k} (y_k) \|^2 .\end{aligned}$$ In order to write (\[algo17\]) in a recursive form, we use the relation (\[algo14a\]) satisfied by $z_k$, which gives $$z_{k+1} -x^{*} = z_k -x^{*} -\frac{s}{\alpha -1} \left( k + \alpha -1\right) G_{s,k} (y_k) .$$ After developing $$\| z_{k+1} -x^{*} \|^2 = \| z_{k} -x^{*} \|^2
-2\frac{s}{\alpha -1} \left( k + \alpha -1\right)
\left\langle z_{k} -x^{*}, G_{s,k} (y_k) \right\rangle + \frac{s^2}{(\alpha -1)^2} \left( k + \alpha -1\right)^2 \| G_{s,k} (y_k) \|^2 ,$$ and multiplying the above expression by $\frac{(\alpha -1)^2} {2s\left( k + \alpha -1\right)^2}$, we obtain $$\frac{(\alpha -1)^2} {2s\left( k + \alpha -1\right)^2}
\left( \| z_{k} -x^{*} \|^2 -\| z_{k+1} -x^{*} \|^2 \right) = \frac{\alpha -1}{k + \alpha -1}
\left\langle G_{s,k} (y_k), z_k -x^{*}\right\rangle -\frac{s}{2} \| G_{s,k} (y_k) \|^2 .$$ Replacing this expression in (\[algo17\]), we obtain $$\begin{aligned}
\label{algo18}
\Theta_k (x_{k+1}) \leq \frac{k}{k + \alpha -1} \Theta_k (x_k) + \frac{\alpha -1}{k + \alpha -1}\Theta_k (x^{*}) +
\frac{(\alpha -1)^2} {2s\left( k + \alpha -1\right)^2}
\left( \| z_{k} -x^{*} \|^2 -\| z_{k+1} -x^{*} \|^2 \right) .\end{aligned}$$ Equivalently $$\begin{aligned}
\label{algo19}
\Theta_k (x_{k+1})-\Theta_k (x^{*}) \leq \frac{k}{k + \alpha -1} \left( \Theta_k (x_k) -\Theta_k (x^{*}) \right) + \frac{(\alpha -1)^2} {2s\left( k + \alpha -1\right)^2}
\left( \| z_{k} -x^{*} \|^2 -\| z_{k+1} -x^{*} \|^2 \right) .\end{aligned}$$ Returning to $\Theta (y)= \Theta_k (y) + \left\langle g_k, y \right\rangle$, we obtain $$\begin{aligned}
\label{algo19b}
\Theta(x_{k+1})-\Theta (x^{*}) \leq &\frac{k}{k + \alpha -1} \left( \Theta (x_k) -\Theta(x^{*}) \right) + \frac{(\alpha -1)^2} {2s\left( k + \alpha -1\right)^2}
\left( \| z_{k} -x^{*} \|^2 -\| z_{k+1} -x^{*} \|^2 \right) \\
& + \left\langle g_k, x_{k+1}- x^{*} \right\rangle
- \frac{k}{k + \alpha -1} \left\langle g_k, x_{k}- x^{*} \right\rangle \nonumber.\end{aligned}$$ After reduction $$\begin{aligned}
\label{algo20}
\Theta(x_{k+1})-\Theta (x^{*}) \leq &\frac{k}{k + \alpha -1} \left( \Theta (x_k) -\Theta(x^{*}) \right) + \frac{(\alpha -1)^2} {2s\left( k + \alpha -1\right)^2}
\left( \| z_{k} -x^{*} \|^2 -\| z_{k+1} -x^{*} \|^2 \right) \\&+ \left\langle g_k, x_{k+1}- x_{k}
+ \frac{\alpha -1}{k + \alpha -1}(x_{k}- x^{*} )\right\rangle
\nonumber.\end{aligned}$$ Multiplying by $ \frac{2s}{\alpha -1}\left( k + \alpha -1\right)^2 $, we obtain $$\begin{aligned}
\label{algo21}
\frac{2s}{\alpha -1}\left( k + \alpha -1\right)^2 \left( \Theta(x_{k+1})-\Theta (x^{*})\right) \leq &\frac{2s}{\alpha -1} k\left( k + \alpha -1\right) \left( \Theta (x_k) -\Theta(x^{*}) \right) + (\alpha -1)
\left( \| z_{k} -x^{*} \|^2 -\| z_{k+1} -x^{*} \|^2 \right) \\&+ \frac{2s}{\alpha -1}\left( k + \alpha -1\right)^2 \left\langle g_k, x_{k+1}- x_{k}
+ \frac{\alpha -1}{k + \alpha -1}(x_{k}- x^{*} )\right\rangle
\nonumber.\end{aligned}$$ For $\alpha \geq 3$ one can easily verify that $$k\left( k + \alpha -1\right) \leq \left( k + \alpha -2\right)^2 .$$ More precisely $$k\left( k + \alpha -1\right) =\left( k + \alpha -2\right)^2 -k(\alpha -3) -(\alpha -2)^2 \leq \left( k + \alpha -2\right)^2 -k(\alpha -3).$$ As a consequence, from (\[algo21\]) we deduce that $$\begin{aligned}
\label{algo22}
\frac{2s}{\alpha -1}&\left( k + \alpha -1\right)^2 \left( \Theta(x_{k+1})-\Theta (x^{*})\right) + 2s \frac{\alpha -3}{\alpha -1}k \left( \Theta (x_k) -\Theta(x^{*}) \right) \leq \frac{2s}{\alpha -1} \left( k + \alpha -2\right)^2 \left( \Theta (x_k) -\Theta(x^{*}) \right) \\& +(\alpha -1)
\left( \| z_{k} -x^{*} \|^2 -\| z_{k+1} -x^{*} \|^2 \right) + \frac{2s}{\alpha -1}\left( k + \alpha -1\right)^2 \left\langle g_k, x_{k+1}- x_{k}
+ \frac{\alpha -1}{k + \alpha -1}(x_{k}- x^{*} )\right\rangle
\nonumber.\end{aligned}$$ Setting $$\label{algo23}
\mathcal G(k)= \frac{2s}{\alpha -1} \left( k + \alpha -2\right)^2 (\Theta (x_k) - {\Theta}^*) + (\alpha -1)
\| z_{k} -x^{*} \|^2 ,$$ we can reformulate (\[algo22\]) as $$\label{algo28}
\mathcal G (k+1) + 2s \frac{\alpha -3}{\alpha -1}k \left( \Theta (x_k) -\Theta(x^{*}) \right) \leq \mathcal G (k) + \frac{2s}{\alpha -1}\left( k + \alpha -1\right)^2 \left\langle g_k, x_{k+1}- x_{k}
+ \frac{\alpha -1}{k + \alpha -1}(x_{k}- x^{*} )\right\rangle.$$ Equivalently $$\label{algo29}
\mathcal G (k+1) + 2s \frac{\alpha -3}{\alpha -1}k \left( \Theta (x_k) -\Theta(x^{*}) \right) \leq \mathcal G (k) + 2s\left( k + \alpha -1\right) \left\langle g_k, \frac{k + \alpha -1}{\alpha -1} \left( x_{k+1}- x_{k} \right) + x_{k}- x^{*} \right\rangle.$$ Using (\[algo13-b\]) $$\begin{aligned}
z_{k+1} = \frac{k + \alpha -1}{\alpha -1} \left( x_{k+1}- x_{k} \right) + x_{k},\end{aligned}$$ we deduce that $$\label{algo30}
\mathcal G (k+1) + 2s \frac{\alpha -3}{\alpha -1}k \left( \Theta (x_k) -\Theta(x^{*}) \right) \leq \mathcal G (k) + 2s\left( k + \alpha -1\right) \left\langle g_k, z_{k+1}- x^{*} \right\rangle.$$ We now develop a similar analysis as in the continuous case. Given some integer $K$, set $$\mathcal E_K (k)= \mathcal G (k) + \sum_{j=k}^K 2s\left( j + \alpha -1\right) \left\langle g_j, z_{j+1}- x^{*} \right\rangle .$$ Then (\[algo30\]) is equivalent to $$\label{algo31}
\mathcal E_K (k+1) + 2s \frac{\alpha -3}{\alpha -1}k \left( \Theta (x_k) -\Theta(x^{*}) \right) \leq \mathcal E_K (k) .$$ Hence, the sequence $(\mathcal E_K (k))$ is nonincreasing. In particular $\mathcal E_K (k) \leq \mathcal E_K (0)$, which gives $$\mathcal G (k) + \sum_{j=k}^K 2s\left( j + \alpha -1\right) \left\langle g_j, z_{j+1}- x^{*} \right\rangle \leq \mathcal G (0) + \sum_{j=0}^K 2s\left( j + \alpha -1\right) \left\langle g_j, z_{j+1}- x^{*} \right\rangle .$$ As a consequence $$\label{algo31b}
\mathcal G (k) \leq \mathcal G (0) + \sum_{j=0}^{k-1} 2s\left( j + \alpha -1\right) \left\langle g_j, z_{j+1}- x^{*} \right\rangle .$$ By definition of $\mathcal G (k)$, neglecting some positive terms, and by Cauchy-Schwarz inequality, we infer $$(\alpha -1)
\| z_{k} -x^{*} \|^2 \leq \mathcal G (0) + 2s \sum_{j=0}^{k-1} \left( j + \alpha -1\right) \| g_j\| \|z_{j+1}- x^{*} \| .$$ Equivalently $$\label{algo32}
\| z_{k} -x^{*} \|^2 \leq \frac{1}{\alpha -1}\mathcal G (0) + \frac{2s}{\alpha -1} \sum_{j=1}^{k} \left( j + \alpha -2\right) \| g_{j-1}\| \|z_{j}- x^{*} \| .$$ We then use the following result, a discrete version of Gronwall’s lemma.
\[d-Gronwall\] Let $(a_k)$ be a sequence of positive real numbers such that $$a_k^2 \leq c + \sum_{j=1}^k \beta_j a_j$$ where $(\beta_j )$ is a sequence of positive real numbers such that $\sum_j \beta_j <+ \infty$, and $c$ is a positive real number. Then $$a_k \leq \sqrt{c} + \sum_{j=1}^{\infty} \beta_j .$$
Set $A_k := \sup_{1\leq j \leq k} a_j $. Then, for $1\leq l \leq k$ $$a_l^2 \leq c + \sum_{j=1}^l \beta_j a_j \leq c +
A_k \sum_{j=1}^{\infty} \beta_j$$ Passing to the supremum with respect to $l$, with $1\leq l \leq k$, we obtain $$A_k^2 \leq c +
A_k \sum_{j=1}^{\infty} \beta_j .$$ By elementary algebraic computation, it follows that $$A_k \leq \sqrt{c} + \sum_{j=1}^{\infty} \beta_j .$$
*Following the proof of Theorem \[Thm-algo\].* From (\[algo32\]), applying Lemma \[d-Gronwall\] with $a_k = \| z_{k} -x^{*} \|$, we deduce that $$\label{algo33}
\| z_{k} -x^{*} \| \leq M:= \sqrt{\frac{\mathcal G (0)}{\alpha -1}} + \frac{2s}{\alpha -1} \sum_{j=0}^{\infty} \left( j + \alpha -1\right) \| g_{j}\| .$$ Note that $M$ is finite, because of the assumption $ \sum_{k \in \mathbb N} k \|g_k \| < + \infty $. Returning to (\[algo31b\]) we obtain $$\label{algo34}
\mathcal G (k) \leq C:= \mathcal G (0) + 2s\left( \sum_{j=0}^{\infty} \left( j + \alpha -1\right) \| g_j\|\right) \left( \sqrt{\frac{\mathcal G (0)}{\alpha -1}} + \frac{2s}{\alpha -1} \sum_{j=0}^{\infty} \left( j + \alpha -1\right) \| g_j\| \right) .$$ By definition of $\mathcal G (k)$, and the positivity of its constitutive elements we finally obtain $$\frac{2s}{\alpha -1} \left( k + \alpha -2\right)^2 (\Theta (x_k) - {\Theta}^*) \leq C .$$ which gives (\[algo8\]).
In the particular case $\alpha = 3$, for a perturbed version of the classical FISTA algorithm, Schmidt, Le Roux, and Bach proved in [@SLB] a result similar to Theorem \[Thm-algo\] concerning the fast convergence of the values.
Let us now study the convergence of the sequence $(x_k)$.
\[Thm-algo2\] Let $\Phi: \mathcal H \to \mathbb R \cup \lbrace + \infty \rbrace $ be a convex lower semicontinuous proper function, and $\Psi: \mathcal H \to \mathbb R $ a convex continuously differentiable function, whose gradient is $L$-Lipschitz continuous. Suppose that $S={{\rm argmin}\kern 0.12em}(\Phi + \Psi)$ is nonempty. Suppose that $\alpha >3$, $ 0< s < \frac{1}{L} $, and $ \sum_{k \in \mathbb N} k \|g_k\| < + \infty $. Let $(x_k)$ be a sequence generated by the algorithm [${\rm \mbox{(AVD)}_{\alpha,g}-algo} $]{}. Then,
i\) $ \sum_k k \Big((\Phi + \Psi)(x_k) - \inf(\Phi + \Psi) \Big) < + \infty$;
ii\) $\sum k\|x_{k+1}-x_k\|^2<+\infty$ ;
iii) $ (x_k ) \ \mbox{converges weakly, as} \ k\to+\infty, \ \mbox{to some} \ x^*\in{{\rm argmin}\kern 0.12em}\Phi$.
The demonstration is parallel to that of Theorem \[Thm-weak-conv\].
**Step 1.** Let us return to (\[algo30\]), $$\mathcal G (k+1) + 2s \frac{\alpha -3}{\alpha -1}k \left( \Theta (x_k) -\Theta(x^{*}) \right) \leq \mathcal G (k) + 2s\left( k + \alpha -1\right) \left\langle g_k, z_{k+1}- x^{*} \right\rangle.$$ By (\[algo33\]), we know that the sequence $ (z_{k})$ is bounded. Summing the above inequalities, and using $\alpha >3$, we obtain $$\label{algo-conv1}
\sum_k k \left((\Phi + \Psi)(x_k) - \inf(\Phi + \Psi) \right) < + \infty ,$$ thats’ item $i)$.
**Step 2.** Now apply the fundamental inequality (\[algo14\]), which can be equivalently written as follows $$\label{algo-conv2}
\Theta_k (y -s G_{s,k} (y)) + \frac{1}{2s}\| y- sG_{s,k} (y)-x \|^2 \leq \Theta_k (x) + \frac{1}{2s} \| x-y \|^2 .$$ Take $y= y_k$, and $x=x_k$. Since $x_{k+1} = y_k - s G_{s,k} (y_k)$, and $y_k - x_{k} = \frac{k -1}{k + \alpha -1} ( x_{k} - x_{k-1}) $, we obtain $$\label{algo-conv3}
\Theta_k (x_{k+1}) + \frac{1}{2s}\| x_{k+1}-x_k \|^2 \leq \Theta_k (x_k) + \frac{1}{2s}\frac{(k -1)^2}{(k + \alpha -1)^2} \| x_{k} - x_{k-1} \|^2 .$$ Equivalently, by definition of $ \Theta_k $, $$\label{algo-conv4}
\Theta (x_{k+1}) + \frac{1}{2s}\| x_{k+1}-x_k \|^2 \leq \Theta (x_k) + \frac{1}{2s}\frac{(k -1)^2}{(k + \alpha -1)^2} \| x_{k} - x_{k-1} \|^2 + \left\langle
g_k , x_{k+1}-x_k \right\rangle .$$ To shorten notations, set $\theta_k = \Theta (x_k) -\Theta (x^{*}) $, $d_k = \frac{1}{2}\| x_{k} - x_{k-1} \|^2 $, $a=\alpha -1$. By Cauchy-Schwarz inequality, and with these notations, (\[algo-conv4\]) gives $$\label{algo-conv5}
\frac{1}{s}\left( d_{k+1} - \frac{(k -1)^2}{(k + a)^2} d_k \right) \leq \left( \theta_k - \theta_{k+1} \right) +
\| g_k\| \| x_{k+1}-x_k \| .$$ After multiplication by $(k + a)^2$, we obtain $$\label{algo-conv6}
\frac{1}{s}\left( (k + a)^2 d_{k+1} - (k -1)^2 d_k \right) \leq (k + a)^2\left( \theta_k - \theta_{k+1} \right) +
(k + a)^2 \| g_k\| \| x_{k+1}-x_k \| .$$ Summing from $k=1$ to $k= K$ gives $$\label{algo-conv7}
\sum_{k=1}^{K}\left( (k + a)^2 d_{k+1} - (k -1)^2 d_k \right) \leq s\sum_{k=1}^{K}(k + a)^2\left( \theta_k - \theta_{k+1} \right) +s\sum_{k=1}^{K}
(k + a)^2 \| g_k\| \| x_{k+1}-x_k \| .$$ By a similar computation as in Chambolle-Dossal [@CD Corollary 2], we equivalently obtain $$\begin{aligned}
\label{algo-conv8}
(K+a)^2 d_{K+1} + & \sum_{k=2}^{K}a\left( 2k +a -2\right)d_k \\
&\leq s \left( (a+1)^2 \theta_1 - (K+a)^2 \theta_{K+1} +\sum_{k=2}^{K}\left( 2k +2a -1 \right)\theta_k +\sum_{k=1}^{K}
(k + a)^2 \| g_k\| \| x_{k+1}-x_k \| \right) \nonumber.\end{aligned}$$ By (\[algo-conv1\]) we have $\sum_k \left( 2k +2a -1 \right)\theta_k < + \infty $. Hence there exists some constant $C$ such that, for all $ K \in \mathbb N$ $$\label{algo-conv9}
(K+a)^2 \| x_{K+1}-x_K \|^2
\leq C + 2s\sum_{k=1}^{K}
(k + a)^2 \| g_k\| \| x_{k+1}-x_k \| .$$ We now proceed to a parallel argument to that used in the proof of Theorem \[Thm-weak-conv\]. Let us write (\[algo-conv9\]) as follows, with $r_k:= (k + a)\| x_{k+1}-x_k \| $ $$\label{algo-conv10}
r_{k}^2
\leq C + 2s\sum_{j=1}^{k}
(j + a)\| g_j\| r_j .$$ We make appeal to the following discrete version of the Gronwall-Bellman lemma.
\[Gronw-dis\] Let $(r_k)$ be sequence of positive real numbers such that, for all $k\geq 1$ $$r_{k}^2
\leq C + \sum_{j=1}^{k}
\omega_j r_j$$ where $C$ is a positive constant, and $\sum_k \omega_j <+\infty$, with $\omega_j \geq 0$. Then the sequence $(r_k)$ is bounded with $$r_{k} \leq \sqrt{C} + \sum_{j \in \mathbb N} \omega_j .$$
For simplicity, let us assume $\omega_j >0$ (one can always reduce to this situation by adding some positive constant, arbitrarily small, see Brezis [@Bre1] for the proof of this lemma in the continuous case). Set $A_k := C + \sum_{j=1}^{k}
\omega_j r_j$, $A_0 = C$. We have $r_{k}^2 \leq A_k $, and $A_{k+1} -A_k = \omega_{k+1} r_{k+1}$. Equivalently $ r_{k+1} = \frac{A_{k+1} -A_k}{\omega_{k+1}}$, which gives $$\frac{A_{k+1} -A_k}{\omega_{k+1}} \leq \sqrt{A_{k+1}},$$ and hence $$\frac{A_{k+1}}{\sqrt{A_{k+1}}} - \frac{A_{k}}{\sqrt{A_{k+1}}}\leq \omega_{k+1}.$$ From this, and using that the sequence $(A_{k})$ is increasing, we deduce that $$\sqrt{A_{k+1}} - \sqrt{A_{k}} \leq \omega_{k+1}.$$ Summing this inequality, and using $r_{k} \leq \sqrt{A_{k}} $ gives the claim.
*Following the proof of Theorem \[Thm-algo2\].* Let us apply lemma \[Gronw-dis\] to inequality (\[algo-conv10\]) with $r_j = (j + a)\| x_{j+1}-x_j \|$, and $\omega_j = (j + a)\| g_j\| $. By using the assumption on the perturbation term $\sum_k k \|g_k\| <+\infty$, we deduce that $$\label{algo-conv12}
\sup_k k \| x_{k+1}-x_k \| < + \infty .$$ Injecting this information in (\[algo-conv8\]), we obtain $$\label{algo-conv13}
\sum_{k}a\left( 2k +a -2\right)d_k
\leq C + \sum_{k}\left( 2k +2a -1 \right)\theta_k +\sup_k ( (k+a) \| x_{k+1}-x_k \| ) \sum_{k}
(k + a) \| g_k \| .$$ From $a = \alpha -1 \geq 2$, (\[algo-conv1\]), and the definition of $d_k$, we deduce that $$\sum k\|x_{k+1}-x_k\|^2<+\infty,$$ which is our claim $ii)$.
**Step 3.** The last step consists in applying Opial’s lemma, whose discrete version is stated below.
\[Opial-discrete\] Let $S$ be a non empty subset of $\mathcal H$, and $(x_k)$ a sequence of elements of $\mathcal H$. Assume that $$\begin{aligned}
(i) & & \mbox{for every }z\in S,\>\lim_{k\to+\infty}\|x_k-z\|\mbox{ exists};\\
(ii) & &
\mbox{every weak sequential cluster point of the sequence} \ (x_k) \mbox{ belongs to }S.\end{aligned}$$ Then $$w-\lim_{k\to+\infty}x_k=x_{\infty}\ \ \mbox{ exists, for some element }x_{\infty}\in S.$$
We are going to apply Opial’s lemma with $S={{\rm argmin}\kern 0.12em}(\Phi + \Psi)$. By Theorem \[Thm-algo\], we have $(\Phi + \Psi)(x_k) \to
\min (\Phi + \Psi)$ (indeed, we have proved fast convergence). By the lower semicontinuity property of $\Phi + \Psi$ for the weak convergence of $\mathcal H$, we immediately obtain that item $(ii)$ of Opial’s lemma is satisfied. Thus the only point to verify is that $\lim \|x_k-x^{*}\|$ exists for any $x^{*} \in {{\rm argmin}\kern 0.12em}(\Phi + \Psi)$. Equivalently, we are going to show that $\lim h_k$ exists, with $h_k := \frac{1}{2} \|x_k - x^{*}\|^2$.
The beginning of the proof is similar to [@AA1], [@CD]. It consists in establishing a discrete version of the second-order differential inequality (\[wconv50\]) $$\ddot{h}(t) + \frac{\alpha}{t} \dot{h}(t) \leq \| \dot{x}(t) \|^2 + \| x(t) - x^{*} \| \| g(t) \|.$$ We use the parallelogram identity, which in an equivalent form can be written as follows: for any $a,b,c \in \mathcal H$ $$\label{algo-paral}
\frac{1}{2} \|a-b \|^2 + \frac{1}{2} \|a-c \|^2 =
\frac{1}{2} \|b -c \|^2 + \left\langle a-b, a-c \right\rangle .$$ Taking $b= x^{*}$, $a= x_{k+1}$, $c=x_k$, we obtain $$\frac{1}{2} \|x_{k+1}-x^{*} \|^2 + \frac{1}{2} \|x_{k+1}-x_k \|^2 =
\frac{1}{2} \| x_k - x^{*}\|^2 + \left\langle x_{k+1}-x^{*}, x_{k+1}-x_k\right\rangle .$$ Equivalently, $$\label{algo-conv14}
h_k - h_{k+1} = \frac{1}{2} \|x_{k+1}-x_k \|^2 + \left\langle x_{k+1}-x^{*}, x_k - x_{k+1}\right\rangle .$$ By definition of $y_k$ we have $$x_k - x_{k+1}= y_k - x_{k+1} - \frac{k -1}{k + \alpha -1}
(x_k - x_{k-1}).$$ Replacing in (\[algo-conv14\]), we obtain $$\label{algo-conv15}
h_k - h_{k+1} = \frac{1}{2} \|x_{k+1}-x_k \|^2 + \left\langle x_{k+1}-x^{*},y_k - x_{k+1}\right\rangle -
\frac{k -1}{k + \alpha -1}\left\langle x_{k+1}-x^{*},x_k - x_{k-1}\right\rangle .$$ Let us now use the monotonicity property of $\partial \Phi$. Since $- s\nabla \Psi (x^{*}) \in s\partial \Phi ( x^{*}) $, and $y_k - x_{k+1}- s \nabla \Psi (y_k) +sg_k\in s\partial \Phi (x_{k+1})$, we have $$\left\langle y_k - x_{k+1}- s \nabla \Psi (y_k) +sg_k +s\nabla \Psi (x^{*}) , x_{k+1} - x^{*} \right\rangle \geq 0.$$ Equivalently $$\left\langle y_k - x_{k+1}, x_{k+1} - x^{*} \right\rangle + s \left\langle \nabla \Psi (x^{*}) - \nabla \Psi (y_k) +g_k , x_{k+1} - x^{*} \right\rangle \geq 0.$$ Replacing in (\[algo-conv15\]) we obtain $$\label{algo-conv16}
h_{k+1}- h_k +\frac{1}{2} \|x_{k+1}-x_k \|^2 + s \left\langle \nabla \Psi (y_k) -\nabla \Psi (x^{*}) - g_k , x_{k+1} - x^{*} \right\rangle -
\frac{k -1}{k + \alpha -1}\left\langle x_{k+1}-x^{*},x_k - x_{k-1}\right\rangle \leq 0.$$ We now use the co-coercivity of $\nabla \Psi$ $$\begin{aligned}
\label{algo-conv17}
\left\langle \nabla \Psi (y_k) -\nabla \Psi (x^{*}) , x_{k+1} - x^{*} \right\rangle &= \left\langle \nabla \Psi (y_k) -\nabla \Psi (x^{*}) , x_{k+1} - y_k \right\rangle + \left\langle \nabla \Psi (y_k) -\nabla \Psi (x^{*}) , y_k -x^{*} \right\rangle
\nonumber\\
&\geq \frac{1}{L} \| \Psi (y_k) -\nabla \Psi (x^{*}) \| ^2 + \left\langle \nabla \Psi (y_k) -\nabla \Psi (x^{*}) , x_{k+1} - y_k \right\rangle \nonumber \\
&\geq \frac{1}{L} \| \Psi (y_k) -\nabla \Psi (x^{*}) \| ^2 - \| \nabla \Psi (y_k) -\nabla \Psi (x^{*}) \| \|x_{k+1} - y_k \|\\
&\geq -\frac{L}{2} \| x_{k+1} - y_k \| ^2
\nonumber.\end{aligned}$$ Combining (\[algo-conv16\]) and (\[algo-conv17\]) $$\label{algo-conv18}
h_{k+1}- h_k +\frac{1}{2} \|x_{k+1}-x_k \|^2 -\frac{sL}{2} \| x_{k+1} - y_k \| ^2 - s\|g_k \| \|x_{k+1} - x^{*}\| -
\frac{k -1}{k + \alpha -1}\left\langle x_{k+1}-x^{*},x_k - x_{k-1}\right\rangle \leq 0.$$ Let us use again (\[algo-paral\]) with $b= x^{*}$, $a= x_{k}$, $c=x_{k -1}$. We obtain $$\frac{1}{2} \|x_{k}-x^{*} \|^2 + \frac{1}{2} \|x_{k}-x_{k -1} \|^2 =
\frac{1}{2} \|x_{k -1} -x^{*} \|^2 + \left\langle x_{k}-x^{*}, x_{k}-x_{k -1} \right\rangle .$$ Equivalently $$\label{algo-paral-3}
h_{k-1}-h_{k} = \frac{1}{2} \|x_{k}-x_{k -1} \|^2
- \left\langle x_{k}-x^{*}, x_{k}-x_{k -1} \right\rangle .$$ Combining (\[algo-conv18\]) with (\[algo-paral-3\]) we obtain $$\begin{aligned}
\label{algo-conv19}
h_{k+1}- h_k - &\frac{k -1}{k + \alpha -1}\left( h_{k} -h_{k-1}\right) \leq
-\frac{1}{2} \|x_{k+1}-x_k \|^2 +\frac{sL}{2} \| x_{k+1} - y_k \| ^2 + s\|g_k \| \|x_{k+1} - x^{*}\|\\
& +\frac{k -1}{k + \alpha -1}\left(\frac{1}{2} \|x_k -x_{k -1} \|^2 + \left\langle x_k - x_{k-1}, x_{k+1}-x_k\right\rangle \right) .\nonumber\end{aligned}$$ By definition of $y_k = x_{k} + \frac{k -1}{k + \alpha -1} ( x_{k} - x_{k-1})$, we have $x_{k+1}- y_k = x_{k+1} - x_{k} - \frac{k -1}{k + \alpha -1} ( x_{k} - x_{k-1})$. Hence $$\|x_{k+1}- y_k \|^2= \| x_{k+1} - x_{k}\|^2 + \left( \frac{k -1}{k + \alpha -1} \right) ^2 \|x_{k} - x_{k-1}\|^2 -2\frac{k -1}{k + \alpha -1} \left\langle x_{k+1} - x_{k}, x_{k} - x_{k-1}\right\rangle$$ Substituting in (\[algo-conv19\]), we obtain $$\label{algo-conv20}
h_{k+1}- h_k - \gamma_k \left( h_{k} -h_{k-1}\right) \leq
-(1- \frac{sL}{2}) \| x_{k+1} - y_k \| ^2 + s\|g_k \| \|x_{k+1} - x^{*}\| + \left( \gamma_k + {\gamma_k}^2 \right)\|x_{k} - x_{k-1}\|^2 ,$$ where $\gamma_k= \frac{k -1}{k + \alpha -1} $. Since $0< s < \frac{1}{L} $, we have $(1- \frac{sL}{2}) >0$. On the other hand, since $\gamma_k <1$, we have $\gamma_k + {\gamma_k}^2 < 2 \gamma_k$. Hence $$\label{algo-conv21}
h_{k+1}- h_k - \gamma_k \left( h_{k} -h_{k-1}\right) \leq
s\|g_k \| \|x_{k+1} - x^{*}\| + 2 \gamma_k \|x_{k} - x_{k-1}\|^2 .$$ By (\[algo33\]), we know that the sequence $(z_k)$ is bounded. By (\[algo-conv12\]), we know that $
\sup_k k \| x_{k+1}-x_k \| < + \infty $ . Since $x_k = z_k - \frac{k + \alpha -1}{ \alpha -1} ( x_{k+1}-x_k )$, we deduce that the sequence $(x_k)$ is bounded. Returning to (\[algo-conv21\]), we have, for some constant $C$ $$\label{algo-conv22}
h_{k+1}- h_k - \gamma_k \left( h_{k} -h_{k-1}\right) \leq
C\|g_k \| + 2 \gamma_k \|x_{k} - x_{k-1}\|^2 .$$ We now use the estimation that we obtained in step 2, namely $ \sum_k k\|x_{k+1}-x_k\|^2<+\infty$. Combined with the assumption $ \sum_{k } k \|g_k\| < + \infty $, we deduce that $$\label{algo-conv23}
h_{k+1}- h_k - \gamma_k \left( h_{k} -h_{k-1}\right) \leq \omega_k ,$$ for some nonnegative sequence $(\omega_k) $ such that $ \sum_{k \in \mathbb N} k\omega_k < + \infty $. Taking the positive part, we obtain $$\label{algo-conv24}
\left( h_{k+1}- h_k\right)^+ - \gamma_k \left( h_{k} -h_{k-1}\right)^+ \leq \omega_k .$$ We are now using the following lemma, which is a discrete version of lemma \[basic-edo\].
\[diff-ineq-disc\] Let $(a_k)$ be sequence of nonnegative real numbers such that, for all $k\geq 1$ $$a_{k+1}
\leq \frac{k -1}{k + \alpha -1}a_k +
\omega_k$$ where $\alpha \geq 3$, and $\sum_k k\omega_k <+\infty$, with $\omega_k \geq 0$. Then the sequence $(a_k)$ is summable, i.e., $$\sum_{k \in \mathbb N} a_k < +\infty .$$
Since $\alpha \geq 3$ we have $\alpha -1\geq 2$, and hence $$a_{k+1}
\leq \frac{k -1}{k + 2}a_k + \omega_k .$$ Multiplying this expression by $(k+1)^2$, we obtain $$(k+1)^2 a_{k+1}
\leq \frac{(k -1)(k+1)^2}{k + 2}a_k + (k+1)^2\omega_k .$$ Then note that, for all integer $k$ $$\frac{(k -1)(k+1)^2}{k + 2} \leq k^2 .$$ Hence $$(k+1)^2 a_{k+1}
\leq k^2 a_k + (k+1)^2\omega_k .$$ Summing this inequality with respect to $j=1,2,...,k$, we obtain $$k^2 a_{k}
\leq a_1 + \sum_{j=1}^{k-1}(j+1)^2\omega_j .$$ Dividing by $k^2$, and summing with respect to $k$, we obtain $$\sum _k a_{k}
\leq a_1 \sum_k \frac{1}{k^2} + \sum_k \frac{1}{k^2}\sum_{j=1}^{k-1}(j+1)^2\omega_j .$$ Applying Fubini theorem to this last sum, we obtain $$\sum_k a_{k}
\leq a_1 \sum_k \frac{1}{k^2} + \sum_j \left( \sum_{k=j+1}^{\infty}\frac{1}{k^2} \right) (j+1)^2\omega_j .$$ We have $$\sum_{k=j+1}^{\infty}\frac{1}{k^2} \leq \int_{j}^{\infty}\frac{1}{t^2}dt = \frac{1}{j}.$$ Hence $$\sum_k a_{k}
\leq a_1 \sum \frac{1}{k^2} + \sum_j \frac{(j+1)^2}{j} \omega_j <+\infty,$$ which by $\frac{(j+1)^2}{j} \leq 4j$ for $j\geq 1$ gives the claim.
*End of the proof of Theorem \[Thm-algo2\].* Let us apply lemma \[diff-ineq-disc\] with $a_k = \left( h_{k} -h_{k-1}\right)^+$. We obtain $$\sum_{k }\left( h_{k} -h_{k-1}\right)^+ < +\infty ,$$ which, combined with $h_k$ nonnegative, gives the convergence of the sequence $(h_k)$, and ends the proof.
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[^1]: With the support of ECOS grant C13E03, Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA9550-14-1-0056.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Topological magnetic excitations called skyrmions exhibit striking spin configurations with deterministic chirality stabilized by the Dzyaloshinskii-Moriya interaction (DMI). These objects come in many forms depending on the symmetry and dimensionality of the system under consideration. Here, for the first time, we experimentally observe a new topological excitation called a magnetic domain wall (DW) skyrmion using Lorentz transmission electron microscopy (LTEM). LTEM contrast matches images simulated from micromagnetic calculations and their expected pinning behavior is observed through *in situ* application of a perpendicular magnetic field. Calculations of the energy barrier to DW skyrmion annihilation using the micromagnetic geodesic nudged elastic band (GNEB) model support this observed metastability of DW skyrmions at room temperature.'
author:
- Maxwell Li
- Arjun Sapkota
- Anish Rai
- Ashok Pokhrel
- Tim Mewes
- Claudia Mewes
- Di Xiao
- Marc De Graef
- Vincent Sokalski
bibliography:
- 'citations.bib'
title: Experimental observation of magnetic domain wall skyrmions
---
Magnetic skyrmions have attracted a great deal of attention in recent years from a combination of topological protection, which offers improvement to thermal stability, and their susceptibility to manipulation by spin-orbit torques (e.g. Dzyaloshinskii-Moriya interaction (DMI) and the spin Hall effect), making them of great interest for future spintronic applications including neuromorphic computing[@Jiang2015; @Huang2017; @Li2017; @Song2020]. Various forms of skyrmions have been predicted and observed in two-dimensional systems including Bloch[@Yu2010; @Huang2012] , Néel[@Woo2016; @Pollard2017; @Li2019] , and anti-skyrmions[@Heinze2011; @Nayak2017]. Reducing the size of these skyrmions remains a critical challenge and, although recent works have shown diameters as small as 10 nm, it is difficult to control distributions of 2D skyrmions at this lengthscale[@Caretta2018; @Meyer2019; @Legrand2020]. In this work we examine a markedly different topological spin texture called a magnetic domain wall (DW) skyrmion where the size is governed exclusively by exchange lengths both parallel and perpendicular to a magnetic domain wall. The DW skyrmions examined experimentally in this work, and described theoretically in prior work[@cheng2019], are 2-$\pi$ rotations of the internal magnetization of a Dzyaloshinskii domain wall (Fig. \[Fig:phase\]d)). This kind of feature is analogous to vertical Bloch lines (VBLs) (Fig. \[Fig:phase\]a,b)) which were once considered for universal computer memory[@Konishi1983], but waned in part because they were too large and difficult to control. The presence of a significant interfacial DMI greatly changes the energetics of DW excitations in the same way it did for magnetic bubble domains with respect to 2-D skyrmions; simply increasing DMI strength can decrease the size of a DW skyrmion to as small as 10 nm[@cheng2019]. Moreover, DW skyrmions are confined to move within a magnetic DW and are, therefore, not subject to edge pinning (like a conventional DW) and are not able to drift in unwanted directions as with 2D skyrmions (via the skyrmion Hall effect). Both DW skyrmions and 2-D skyrmions can be minimally defined as having an integer topological charge as calculated from $4\pi\,Q=\int\mathrm{d}x\mathrm{d}y\,\mathbf{m} \cdot \left( \partial_x \mathbf{m} \times \partial_y \mathbf{m} \right)$, where $\mathbf{m}$ is the unit magnetization vector. A DW skyrmion has a topological charge of $Q=\pm1$ whereas a VBL has a charge of $Q=\pm1/2$. In the case of a 2-$\pi$ VBL, the topological charge is equivalent to that of a DW skyrmion. Annihilation of a DW skyrmion yields a chiral Néel DW (Fig. \[Fig:phase\]c), $Q=0$), also referred to as a Dzyaloshinskii DW.
![\[Fig:phase\] Schematics depicting the internal magnetization of a a) 1-$\pi$ vertical Bloch line, b) 2-$\pi$ vertical Bloch line, c) chiral Néel domain wall, and d) domain wall skyrmion. e) Predicted phase diagram depicting conditions where the aforementioned magnetic textures can be expected to be observed with respect to DMI strength and film thickness. Stars and circles demark Pt/Co/Ni/Ir samples that have been examined. Fresnel mode Lorentz TEM micrographs of \[Pt/(Co/Ni)$_M$/Ir\]$_N$ samples demarked with circles in the phase diagram are depicted in f) ($M$=100, $N$=1), g) ($M$=10, $N$=1), and h) ($M$=2, $N$=20).](phasediagram.pdf)
![\[Fig:comparison\] a) Fresnel mode Lorentz TEM micrograph of domain wall skyrmions in a \[Pt/(Co/Ni)$_3$/Ir\]$_2$ sample. b) Magnetic profile and b) simulated Fresnel mode Lorentz TEM micrograph of domain wall skyrmion.](compare.pdf)
Despite their topological charge (and expected thermal stability), the experimental observation of DW skyrmions has remained elusive. This is likely due to a small window of DMI strength and film thickness where they can be stabilized. We leverage a highly tunable asymmetric multi-layer system based on (Pt/\[Co/Ni\]$_{M}$/Ir)$_{N}$ where a reduction in ‘$M$’ leads to a greater interfacial DMI from the Pt/Co and Ni/Ir interfaces and ‘$N$’ modulates the total film thickness to identify the optimal conditions where DW skyrmions exist[@Chen2013; @MoreauLuchaire2016; @Li2019]. Multi-layers of \[Pt(0.5 nm)/(Co(0.2 nm)/Ni(0.8 nm))$_M$/Ir(0.5 nm)\]$_N$ were prepared via rf (Ta layers) and dc (Pt, Co, Ni, Ir layers) magnetron sputtering on Si substrates and 10 nm thick amorphous Si$_3$N$_4$ TEM membranes (Norcada) in an Ar atmosphere fixed at $2.5 \times 10^{-3}$ Torr. All samples had a Ta(3 nm)/Pt(3nm) seed/adhesion layer and were capped with Ta(3 nm). Base pressure was maintained at less than $3\times 10^{-7}$ Torr. Magnetic properties were examined using alternating gradient field magnetometry (AGFM) and vibrating sample magnetometry (VSM)[@SupMat]. These films were imaged using Lorentz transmission electron microscopy (LTEM) using an aberration-corrected FEI Titan G2 80-300 at an accelerating voltage of 300 kV in Lorentz mode (objective lens off). LTEM employs the inherent in-plane magnetic induction of the sample to deflect the electron beam and form magnetic contrast. We have performed a systematic examination of 15 different iterations of this multi-layer system and qualitatively formulated a magnetic phase diagram (Fig. \[Fig:phase\]e)) that is divided into four distinct regimes: I. 1-$\pi$ VBLs, II. n-$\pi$ VBLs, III. chiral Néel DWs, and IV. DW skyrmions. In regions I and II, the DMI strength is negligible ($D<D_c$) and, as such, VBL physics dominates these regimes. In region II, samples examined were relatively thin and displayed a large number of VBL pile-ups, referred to as n-$\pi$ VBLs, as seen in a representative Lorentz micrograph of a $M=10$, $N=1$ sample (Fig. \[Fig:phase\]g)). However, the density of VBLs diminishes greatly when film thickness is increased as seen with samples in region I (Fig. \[Fig:phase\]f)). This is due to the formation of hybrid DWs, which are characterized by a twisting of the internal magnetization through the thickness of a film. This forms due to a competition between interlayer magnetostatic interactions and DMI leading to a flux-closure configuration of the DW[@Legrand2018; @Dovzhenko2018]. Such a configuration provides a zero energy path to annihilation for 2-$\pi$ VBLs, which are topologically equivalent to a DW skyrmion (although much larger). Towards the ultra-thin regime, the occurrence of hybrid DWs diminishes contributing to the increased density of n-$\pi$ VBLs observed. When DMI strength is increased above the critical value, chiral Néel DWs (called Dzyaloshinskii DWs) form preferentially over Bloch DWs (regions III and IV). In region III (with large thickness and DMI), we find only Dzyaloshinskii DWs with no internal DW excitations, which we attribute to the existence of hybrid DWs as with region I (Fig. \[Fig:phase\]h)). Finally, for the case of low thickness and high DMI (region IV), we find multiple instances of LTEM contrast marked by distinct dipole-like contrast along chiral Néel DWs (Fig. \[Fig:comparison\]a). This observation matches simulated Fresnel-mode LTEM images calculated from the micromagnetic profile of a DW skyrmion (Fig. \[Fig:comparison\]b,c). Moreover, these features were not observed in any other film examined excluding the possibility that the observed contrast originates from microstructural defects. We note that a significantly large DMI strength (even in the low thickness case) would also prevent the formation of such hybrid DWs[@Legrand2018], but would also suppress the formation of DW skyrmions as previously suggested and discussed later[@cheng2019].
In addition to the observed Fresnel-mode contrast, we provide further evidence for DW skyrmions based on their impact on DW motion. It has been well established that variations in the internal magnetization of any DW lead to a pinning effect due, in part, to a local increase in the DW energy[@Slonczewski1979; @Krizakova2019]. This can be considered in the context of creep motion of DW’s where the velocity depends exponentially on the elastic energy of the wall[@lemerle1998]. Fig. \[Fig:Pinning\] shows the evolution of the domain pattern of Fig. \[Fig:comparison\]a in response to an increasing perpendicular magnetic field applied *in situ* by exciting the objective lens of the microscope. It is clear that the associated magnetic contrast identified as DW skyrmions does not move with the rest of the DW, but rather remains pinned as the wall bows around them. In subsequent images, the DW skyrmion annihilates simultaneously with the rapid recovery of the wall from its pinned position. We note the absence of any contrast at the site where the DW skyrmion previously existed excluding the possibility that the contrast originated from a microstructural defect in the specimen (e.g., voids), further solidifying the magnetic origin of the contrast observed.
{width="90.00000%"}
In order to utilize these newfound DW skyrmions for possible spintronic applications, an understanding of their stability is necessary as thermal fluctuations can lead to their annihilation. Prior analytical examination of DW skyrmions was performed with $T=0$ K[@cheng2019]. To provide a realistic estimate of a DW skyrmion’s thermal stability an analysis performed at room temperature is necessary. Here, we have employed a geodesic nudged elastic band (GNEB) method[@Bessarab2015] in combination with a climbing image method[@Henkelman2000] implemented in the micromagnetic code M$^3$, a MATLAB code based on finite-differences[@M3]. The GNEB we use builds on the nudged elastic band model (NEB) but takes the constraint into account that the saturation magnetization of each cell in the simulation volume remains constant. For N magnetic moments this method results in an unconstrained optimization within a 2N dimensional Riemannian manifold, as is discussed in detail in reference [@Bessarab2015]. For the evaluation of the geodesic distance between two images we use Vincenty’s formula [@Vincenty1975]. In order to converge the images to the nearest minimum energy path we use a steepest descent method [@Exl2014] with a Barzilai-Borwein step length selection method [@Barzilai1988]. To determine their stability one has to find the activation barrier which separates the skyrmion state from lower energy states. In the case of conventional skyrmions this would be the skyrmion state and the simple ferromagnetic state. For the case of a DW skyrmion the corresponding lower energy state is a skyrmion-free domain wall. Since thermally activated magnetic transitions are rare events, dynamical simulations using a stochastic Landau-Lifshitz-Gilbert equation are not practical. Therefore the GNEB method is used to find the minimum energy path for the transition, which has been successfully applied to study the annihilation of conventional magnetic skyrmions[@Lobanov2016; @Sampaio2013]. To stabilize the DW skyrmion we consider an ultra-thin ferromagnetic film (2nm) with an interfacial DMI interaction and a uniaxial perpendicular anisotropy[@cheng2019]. The symmetric exchange and the dipole-dipole interaction are included in the micromagnetic simulations. The lateral resolution in the micromagnetic simulations is 0.5 nm. To calculate the minimum energy path between those two states, 30 images were created to represent the transition path between the two fixed endpoint images. Fig. \[Fig:NEB\]b) shows the domain wall with a DW skyrmion, and Fig. \[Fig:NEB\]c) shows the domain wall after the annihilation of the DW skyrmion. The transition between these states occurs through a sharp narrowing of the DW skyrmion before the center spin flips direction concurrent with a change in the topological charge. Most notably, even for the ultrathin film considered here, the energy barrier to annihilation is $> 60k_BT$ for $D < 1.0$ mJ/m$^2$ at room temperature. This energy barrier is directly rooted in the symmetric exchange (i.e., the exchange stiffness) as with 2D skyrmions. In the thin film approximation (i.e., uniform magnetization through the thickness), this value should scale linearly with thickness. However, as noted previously, it is expected that thicker samples would eventually form a hybrid DW structure where DW skyrmions cannot exist.
In summary, we demonstrate the first experimental observation of DW skyrmions in magnetic thin films that have sufficient DMI to stabilize Dzyaloshinskii DWs and are thin enough to suppress the formation of a hybrid DW. This conclusion is based on i) Fresnel-mode Lorentz TEM contrast that matches the simulated Fresnel-mode images calculated from micromagnetics, ii) the observation of a distinct pinning effect from the purported DW skyrmion, and iii) the GNEB calculations that predict metastable DW skyrmions over a range of material parameters including those matching the films studied experimentally. We expect that any film with comparable properties (particularly, thickness and interfacial DMI strength) should exhibit metastable DW skyrmions. We are hopeful this discovery will spark further investigation into the properties and potential application of DW skyrmions and other 1-D excitations for applications related to nanomagnonics and the broader use of domain walls as conduits for spin waves[@Jiang2015; @Huang2017; @Li2017; @Song2020].\
This research was supported by the Defense Advanced Research Projects Agency (DARPA) program on Topological Excitations in Electronics (TEE) under grant number D18AP00011. The authors also acknowledge use of the Materials Characterization Facility at Carnegie Mellon University supported by grant MCF-677785.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This letter studies an emerging wireless communication intervention problem at the physical layer, where a legitimate spoofer aims to spoof a malicious link from Alice to Bob, by replacing Alice’s transmitted source message with its target message at Bob side. From an information-theoretic perspective, we are interested in characterizing the maximum achievable spoofing rate of this new spoofing channel, which is equivalent to the maximum achievable rate of the target message at Bob, under the condition that Bob cannot decode the source message from Alice. We propose a novel combined spoofing approach, where the spoofer sends its own target message, combined with a processed version of the source message to cancel the source message at Bob. For both cases when Bob treats interference as noise (TIN) or applies successive interference cancelation (SIC), we obtain the maximum achievable spoofing rates by optimizing the power allocation between the target and source messages at the spoofer.'
author:
- |
Jie Xu, Lingjie Duan, and Rui Zhang\
[^1] [^2] [^3]
title: Fundamental Rate Limits of Physical Layer Spoofing
---
Wireless communication intervention, physical layer spoofing, achievable spoofing rate, power allocation.
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
Introduction
============
The emergence of infrastructure-free wireless communications (e.g., mobile ad hoc networks and unmanned aerial vehicle (UAV) communications) imposes new challenges on the public security, since they may be misused by malicious users to commit crimes or even terror attacks [@Mag]. To overcome this issue, authorized parties can launch legitimate information eavesdropping (see, e.g., [@XuDuanZhang1; @XuDuanZhang2; @ZengZhang; @ZengZhang2]) and jamming (see, e.g., [@ZouWangHanzo2015; @Medard; @Kashyap2004; @Liu2015]) on suspicious and malicious wireless communication links, so as to monitor and intervene in them for the purpose of detecting and preventing security attacks [@Mag].
We focus on the emerging wireless communication intervention at the physical layer. Different from the jamming intervention that can only disrupt or disable target links, we propose a new intervention via physical layer spoofing to change the communicated information over malicious links while keeping their operation. Such a physical layer spoofing has been first investigated in our previous work [@Xu2016a] by considering a three-node spoofing channel (see Fig. \[fig:1\]), where a legitimate spoofer aims to spoof an ongoing malicious link from Alice to Bob, by replacing Alice’s transmitted source message with its target message at Bob side. We have proposed a symbol-level spoofing approach in [@Xu2016a] for the spoofer to minimize the spoofing-symbol-error-rate of the target message at Bob under practical phase-shift keying modulations. Nevertheless, the fundamental information-theoretic limits of such a new spoofing channel remain unaddressed, thus motivating our study in this work.
In this letter, we are interested in characterizing the maximum achievable spoofing rate of the spoofing channel in Fig. \[fig:1\], which is equivalent to the maximum achievable rate of the target message at Bob, while ensuring that Bob cannot decode the source message. We propose a new combined spoofing approach, where the spoofer sends its own target message, combined with a processed version of the source message to cancel it at Bob. In particular, we assume Alice transmits with a constant rate, and consider two cases when Bob treats interference as noise (TIN) and applies successive interference cancelation (SIC), respectively. To successfully spoof in the former case, the spoofer should make the received target message at Bob be stronger than the source message; and in the latter case, the spoofer should make the maximum achievable rate of the source message (under different decoding orders) be strictly smaller than Alice’s communication rate. In both cases, we obtain the maximum achievable spoofing rates by optimizing the power allocation between the target and source messages at the spoofer. Numerical results show that our proposed combined spoofing approach with optimized power allocation outperforms other benchmark spoofing schemes.
System Model
============
=1 ![A three-node spoofing channel, where a legitimate spoofer aims to change the communicated information from Alice to Bob.[]{data-label="fig:1"}](SystemModel.eps "fig:"){width="6.5cm"}
As shown in Fig. \[fig:1\], we consider a three-node spoofing channel, where two malicious users Alice and Bob communicate to plan or commit crimes, and a legitimate spoofer aims to change the communicated data from Alice to Bob to defend against them. Practically, the malicious users can be identified [*a priori*]{} via, e.g., legitimate information eavesdropping [@XuDuanZhang1; @XuDuanZhang2; @ZengZhang; @ZengZhang2]. We define $h$ and $g$ as the complex channel coefficients of the malicious link from Alice to Bob and the spoofing link from the spoofer to Bob, respectively.
First, we consider the case without spoofing. Let $s$ denote the source message transmitted by Alice with unit power. The received signal by Bob is given by $$\begin{aligned}
\label{eqn:system:model}
y = h \sqrt{P} s + n,\end{aligned}$$ where $P$ denotes the constant transmit power of Alice, and $n$ denotes the receiver noise at Bob being a circularly symmetric complex Gaussian (CSCG) random variable with zero mean and unit variance. The capacity of the malicious link is given as $C \triangleq \log_2(1 + |h|^2P)$, which is achieved when Alice employs Gaussian signaling (i.e., setting $s$ as a CSCG random variable with zero mean and unit variance). Suppose that Alice communicates with Bob with a constant communication rate $R$ no greater than the channel capacity $C$, i.e., $R \le C$, where $R$ is chosen based on the quality of service (QoS) requirement.
Next, we consider the case with spoofing. The spoofer aims to change Bob’s decoded message from Alice’s source message $s$ to its desired target message. It is assumed that the spoofer has the perfect information of the source message $s$ and the channel coefficients $h$ and $g$. This assumption is made to help derive the spoofing rate upper bound, similar to that in the prior works in the information-theoretic literature (see, e.g., the correlated jamming in [@Kashyap2004] and the cognitive radio channel in [@Devroye2006]).[[^4]]{} In this case, the spoofer can use the same codebook of $s$ for sending the target message, such that Bob will decode the target message without awareness of being spoofed. Let $x$ denote the target message with unit power, which is in general independent of $s$. We consider a combined spoofing approach, where the spoofer designs its spoofing signal $z$ to be a combined version of both the source message $s$ and the target message $x$ with proper processing. Particularly, we have $z = \alpha s + \beta x$, where $\alpha$ and $\beta$ denote the complex transmit coefficients for the messages $s$ and $x$, respectively. In this case, the received signal $y$ at Bob can be expressed as $$\begin{aligned}
\label{eqn:system:model:2}
y = h\sqrt{P} s + g z + n = \left(h\sqrt{P} + g \alpha \right)s + g \beta x + n.\end{aligned}$$ By denoting $Q$ as the maximum spoofing power at the spoofer, then we have $$\begin{aligned}
\label{eqn:sum_power}
|\alpha|^2 + |\beta|^2 \le Q.\end{aligned}$$
In order to successfully spoof the malicious communication, the spoofer should design the spoofing signal (i.e., the transmit coefficients $\alpha$ and $\beta$) such that Bob is only able to successfully decode the target message $x$ but fails to decode the source message $s$. In this case, the successful spoofing critically depends on the decoding method employed by Bob. We consider two typical Bob receivers as follows, including the practical TIN receiver and the information-theoretically optimal SIC receiver. It is assumed that the spoofer is aware of which receiver being employed by Bob.
### TIN receiver at Bob
Bob does not know the coexistence of the two messages $s$ and $x$, and thus considers the stronger one between them to be its desired signal, and treats the other one (the co-channel interference) to be noise. In this case, the received message $x$ at Bob should have a stronger power than $s$ such that the spoofing is successful.
### SIC receiver at Bob
Bob is able to detect the coexistence of $s$ and $x$, and accordingly attempts to use SIC to decode both of them. From the successfully decoded ones (if any), Bob will decide which the desired message is. In particular, Bob first decodes one message ($x$ or $s$) by treating the other as noise, and then cancels it from the received message $y$ to decode the other one. Generally speaking, Bob can use two different decoding orders (first $x$ and then $s$, or first $s$ and then $x$). Under both receiver cases, we aim to characterize the maximum achievable spoofing rates of the target message $x$, provided that Bob cannot decode the source message $s$.[[^5]]{}
Spoofing TIN Receiver at Bob
============================
Problem Formulation for TIN Receiver
------------------------------------
When Bob employs the TIN receiver, the spoofer can successfully spoof the malicious communication link only when the received power of the target message $x$ is greater than that of the source message $s$. Mathematically, it must hold that $|g\beta|^2 > |h\sqrt{P} + g \alpha|^2$. Note that this strict inequality constraint may make the associated optimization problem ill-posed: an optimizer on the boundary of the feasible region may not be attainable. To address this issue, we revise it to be a non-strict inequality constraint as $$\begin{aligned}
|g\beta|^2 \ge |h\sqrt{P} + g \alpha|^2 + \delta_1 ,\label{eqn:successful:spoofing:nonstrict}\end{aligned}$$ where $\delta_1 > 0$ is a sufficiently small positive constant.
In this case, the received signal-to-interference-plus-noise-ratio (SINR) for the target message $x$ at Bob is $\gamma (\alpha,\beta)= \frac{|g\beta|^2}{|h\sqrt{P} + g \alpha|^2 + 1}$. Accordingly, the achievable spoofing rate (in bps/Hz) is expressed as follows by assuming $x$ is CSCG and $s$ is also CSCG as the “worst-case” noise. $$\begin{aligned}
r(\alpha,\beta) = \log_2\left( 1+ \frac{|g\beta|^2}{|h\sqrt{P} + g \alpha|^2 + 1} \right).\label{eqn:r:alpha}\end{aligned}$$ As a result, the achievable spoofing rate maximization problem is formulated as $$\begin{aligned}
\mathrm{(P1)}:~\max_{\alpha,\beta} ~& r (\alpha,\beta) \nonumber\\
\mathrm{s.t.}~& (\ref{eqn:sum_power})~{\rm and}~(\ref{eqn:successful:spoofing:nonstrict}).\nonumber\end{aligned}$$
Optimal Spoofing Solution to Problem (P1)
-----------------------------------------
First, we reformulate (P1) as an equivalent problem with a single real decision variable. It is evident that the optimality of (P1) is attained when the processed source message $s$ from the spoofer is destructively combined at Bob with that from Alice, and the sum-power constraint in (\[eqn:sum\_power\]) is tight. In other words, we have $$\begin{aligned}
\alpha =& -\frac{hg^*}{|h||g|}\tilde\alpha,\label{eqn:alpha:opt}\\
\beta =& \sqrt{Q - |\tilde\alpha|^2},\label{eqn:beta:opt}\end{aligned}$$ where the superscript $*$ denotes the conjugate operation, and $\tilde\alpha \ge 0$ denotes the magnitude of $\alpha$. Here, since both the objective function and constraints of (P1) are irrespective of the phase of $\beta$, in (\[eqn:beta:opt\]) we decide $\beta$ to be a real variable without loss of optimality. Therefore, (P1) is equivalently reformulated as follows to optimize an SINR function $\tilde\gamma(\tilde\alpha)$ with only a real decision variable $\tilde\alpha$. $$\begin{aligned}
&\mathrm{(P1.1)}:\max_{\tilde\alpha \ge 0} ~ \tilde\gamma(\tilde\alpha) \triangleq \frac{|g|^2(Q-\tilde\alpha^2)}{(|h|\sqrt{P} - |g|\tilde\alpha)^2 + 1} \nonumber\\
&\mathrm{s.t.} ~2|g|^2\tilde\alpha^2 - 2|h||g|\sqrt{P} \tilde\alpha+ |h|^2P - |g|^2Q + \delta_1 \le 0.\label{eqn:successful:spoofing:reform}\end{aligned}$$
Next, we check the feasibility of problem (P1.1) (and thus (P1)).
\[lemma1\] Problem (P1.1) (and thus (P1)) is feasible if and only if $Q \ge \frac{|h|^2P + 2\delta_1}{2|g|^2}$.
Note that the constraint in (\[eqn:successful:spoofing:reform\]) can be rewritten as $2|g|^2\left(\tilde\alpha - \frac{|h|\sqrt{P}}{2|g|}\right)^2 + \frac{|h|^2P}{2} - |g|^2Q + \delta_1 \le 0$, which specifies a nonempty feasible set if and only if $\frac{|h|^2P}{2} - |g|^2Q + \delta_1 \le 0$. Equivalently, problem (P1.1) is feasible if and only if $Q \ge \frac{|h|^2P + 2\delta_1}{2|g|^2}$. This proposition thus follows.
Finally, we obtain the optimal solutions to (P1.1) and (P1) when they are feasible. In this case, the constraint in (\[eqn:successful:spoofing:reform\]) is equivalently expressed as $$\begin{aligned}
\label{eqn:feasible}
{\underline{\omega}} \le \tilde\alpha \le {\overline{\omega}},\end{aligned}$$ where ${\underline{\omega}} = \frac{|h|\sqrt{P} - \sqrt{2|g|^2Q - |h|^2P - 2\delta_1}}{2|g|}$ and ${\overline{\omega}} = \frac{|h|\sqrt{P} + \sqrt{2|g|^2Q - |h|^2P - 2\delta_1}}{2|g|}$ denote the minimum and maximum values of $\tilde\alpha$ for the TIN spoofing to be successful, respectively. Furthermore, by checking its first-order derivative, we can show that there exist one local maximum point $\tilde\alpha_1$ and one local minimum point $\tilde\alpha_2$ for the SINR function $\tilde\gamma(\tilde\alpha)$, which are given by
$$\begin{aligned}
\label{eqn:alpha1}
\tilde\alpha_1 = \frac{|h|^2P + |g|^2Q + 1}{2|h||g|\sqrt{P}} - \frac{ \sqrt{\left(|h|^2P + |g|^2Q + 1\right)^2 - 4 |h|^2|g|^2 PQ}}{2|h||g|\sqrt{P}}\end{aligned}$$
and $\tilde\alpha_2 = \frac{|h|^2P + |g|^2Q + 1}{2|h||g|\sqrt{P}} + \frac{ \sqrt{\left(|h|^2P + |g|^2Q + 1\right)^2 - 4 |h|^2|g|^2 PQ}}{2|h||g|\sqrt{P}}$, respectively. In particular, $\tilde\gamma(\tilde\alpha)$ is first increasing over $\tilde\alpha \in [0,\tilde\alpha_1]$, then decreasing over $\tilde\alpha \in (\tilde\alpha_1,\tilde\alpha_2)$, and finally increasing over $\bar\alpha \in [\tilde\alpha_2,+\infty)$. Since $\lim_{\tilde\alpha \to \infty} \tilde\gamma(\tilde\alpha) = -1$ but $\tilde\gamma(\tilde\alpha) > 0, \forall \tilde\alpha \in [0,\tilde\alpha_1]$, it is evident that $\tilde\alpha_1$ is the globally optimal point to maximize $\tilde\gamma(\tilde\alpha)$ without any constraints. Then we have the following proposition.
\[proposition1\] The optimal solution to (P1.1) is given by $$\begin{aligned}
\label{eqn:solution:P1.1}
\tilde\alpha^{\star} = \max\left({\underline{\omega}},\tilde\alpha_1\right),\end{aligned}$$ and thus the optimal solution to (P1) is $\alpha^{\star} = -\frac{hg^*}{|h||g|}\tilde\alpha^\star$ and $\beta^\star = \sqrt{Q - |\alpha^{\star}|^2}$.
The optimal solution to (P1.1) can be easily verified based on the monotonic property of $\tilde\gamma(\tilde\alpha)$ together with the fact that $\tilde\alpha_1 \le {\overline{\omega}}$ and $\tilde\alpha_1 \le \sqrt{Q}$. Then, by substituting $\tilde\alpha^{\star}$ in (\[eqn:solution:P1.1\]) into (\[eqn:alpha:opt\]) and (\[eqn:beta:opt\]), the optimal solution to (P1) is derived. Therefore, this proposition is proved.
Spoofing SIC Receiver at Bob
============================
Problem Formulation for SIC Receiver
------------------------------------
When Bob employs the SIC receiver, the spoofer needs to design its spoofing signal such that Bob is able to decode the target message $x$ but fails to decode the source message $s$ for the purpose of successful spoofing. In general, the spoofer should consider the following two cases, depending on the decoding orders employed by Bob. Here, Bob can be viewed as a receiver of a two-user multiple-access channel (MAC) by considering Alice and the spoofer as the two transmitters.
In the first case, Bob first decodes $s$ by treating $x$ as noise, and then subtracts $s$ from the received signal $y$ to decode $x$. Accordingly, the maximum achievable rates of $s$ and $x$ at the receiver of Bob (under given $\alpha$ and $\beta$) are given as follows by assuming both $s$ and $x$ are CSCG. $$\begin{aligned}
r_s^{({\rm I})} & = \log_2\left(1+ \frac{|h \sqrt{P} + g \alpha|^2}{|g\beta|^2 + 1}\right),\\
r_x^{({\rm I})} & = \log_2\left(1+ |g\beta|^2\right).\label{eqn:r_2:1}
\end{aligned}$$ In order to prevent Bob from successfully decoding $s$, the spoofer should ensure that its maximum achievable rate is smaller than Alice’s communication rate, i.e., $$\begin{aligned}
\label{eqn:case1}
r_s^{({\rm I})} < R.\end{aligned}$$ Since Bob fails to decode $s$, the decoding of $x$ should suffer from the interference of $s$, and therefore the achievable spoofing rate is given as $r(\alpha,\beta)$ in (\[eqn:r:alpha\]).
In the second case, Bob first decodes $x$ by treating $s$ as noise, and then cancels $x$ from $y$ to decode $s$. Accordingly, the maximum achievable rates of $s$ and $x$ (under given $\alpha$ and $\beta$) at the receiver of Bob are respectively given by $$\begin{aligned}
r_s^{({\rm II})} & = \log_2\left(1+ |h \sqrt{P} + g \alpha|^2 \right),\\
r_x^{({\rm II})} & = \log_2\left(1+ \frac{|g\beta|^2}{|h \sqrt{P} + g \alpha|^2 + 1}\right).\label{eqn:r_2:2}
\end{aligned}$$ To prevent Bob from decoding $s$, the spoofer should ensure that $$\begin{aligned}
\label{eqn:case2}
r_s^{({\rm II})} < R.\end{aligned}$$ In this case, the rate $r_x^{({\rm II})}$ in (\[eqn:r\_2:2\]), which equals $r(\alpha,\beta)$ in (\[eqn:r:alpha\]), is the achievable spoofing rate.
By combining the two cases, the successful spoofing only requires (\[eqn:case2\]) to hold, since if it holds, (\[eqn:case1\]) will hold automatically. Note that in the above two cases, Bob cannot decode $s$ regardless of the decoding orders used with SIC; as a result, it can only treat the decoded target message $x$ as its desired message. Also note that (\[eqn:case2\]) is a strict inequality constraint. To address this issue, we revise (\[eqn:case2\]) as follows similarly as in (\[eqn:successful:spoofing:nonstrict\]). $$\begin{aligned}
\log_2(1+|h\sqrt{P} + g \alpha|^2) + \delta_2 \le R,\label{eqn:feasible:case2}\end{aligned}$$ where $\delta_2$ is a sufficiently small positive constant. The achievable spoofing rate maximization problem is formulated as $$\begin{aligned}
\mathrm{(P2)}:~\max_{\alpha,\beta} ~&r(\alpha,\beta)\nonumber\\
\mathrm{s.t.}~&(\ref{eqn:sum_power})~{\rm and}~(\ref{eqn:feasible:case2}).\nonumber\end{aligned}$$
Optimal Spoofing Solution to Problem (P2)
-----------------------------------------
Similar to (P1), it can be shown that the optimality of (P2) is attained when (\[eqn:alpha:opt\]) and (\[eqn:beta:opt\]) hold. In this case, (P2) is equivalently reformulated as $$\begin{aligned}
\mathrm{(P2.1)}:~\max_{0\le \tilde\alpha \le \sqrt{Q}} ~& \tilde\gamma (\tilde\alpha) \nonumber\\
\mathrm{s.t.}~& \underline{\chi} \le \tilde\alpha \le \overline{\chi},\label{eqn:feasibility:reform:2}\end{aligned}$$ where $\underline{\chi} = \frac{|h|\sqrt{P} - \sqrt{2^{R-\delta_2} - 1}}{|g|}$ and $\overline{\chi} = \frac{|h|\sqrt{P} + \sqrt{2^{R-\delta_2} - 1}}{|g|}$ denote the minimum and maximum values of $\tilde\alpha$ for the SIC spoofing to be successful, respectively.
Next, we check the feasibility of problem (P2.1) (and thus (P2)).
\[lemma2\] Problem (P2.1) (and thus (P2)) is feasible if and only if $Q \ge \underline{\chi}^2$.
The feasible condition of problem (P2.1) can be obtained by noting that $\tilde\alpha \le \sqrt{Q}$ and $\underline{\chi} \le \tilde\alpha$ should be satisfied at the same time.
Finally, when problems (P2.1) and (P2) are feasible, their optimal solutions are obtained in the following proposition.
\[proposition2\] The optimal solution to (P2.1) is given by $$\begin{aligned}
\label{eqn:propo2:eq1}
&\bar\alpha^{\star\star} = \max\left( \underline{\chi} ,\tilde\alpha_1\right),
$$ where $\tilde\alpha_1$ is the globally optimal point to maximize $\tilde\gamma(\tilde\alpha)$, as given in (\[eqn:alpha1\]). Then, the optimal solution to (P2) is given by $\alpha^{\star\star} = -\frac{hg^*}{|h||g|}\bar\alpha^{\star\star}$ and $\beta^{\star\star} = \sqrt{Q - |\alpha^{\star\star}|^2}$.
Similar to Proposition \[proposition1\] and based on the monotonic property of $\tilde\gamma(\tilde \alpha)$, the optimal solution to (P2.1) is obtained as $\bar\alpha^{\star\star}$ in (\[eqn:propo2:eq1\]). Substituting it into (\[eqn:alpha:opt\]) and (\[eqn:beta:opt\]), the optimal solution to (P2) is derived. Therefore, this proposition is proved.
It is interesting to compare the optimally designed spoofing signals for TIN and SIC receivers at Bob, respectively. First, it is observed from Lemmas \[lemma1\] and \[lemma2\] that the minimally required spoofing power for the TIN receiver is irrespective of the communication rate $R$ by Alice, while that for the SIC receiver is monotonically decreasing with respect to $R$. As a result, when $R$ is large (particularly when $2^R-1 > (1-\frac{1}{\sqrt{2}})^2|h|^2{P}$ by neglecting the sufficiently small $\delta_1$ and $\delta_2$), the minimally required spoofing power for the SIC receiver is smaller than that for the TIN receiver, and thus the SIC Bob receiver is easier to be spoofed than the TIN one in this case. Next, it is observed from Propositions \[proposition1\] and \[proposition2\] that if $\tilde\alpha_1 \ge \underline{\omega}$ and $\tilde\alpha_1 \ge \underline{\chi}$ both hold, then the designed spoofing signals become identical for both receivers. This happens when the spoofing power budget $Q$ becomes sufficiently large.
Numerical Results
=================
In this section, we provide numerical results to show the achievable spoofing rates of our proposed combined spoofing approach with optimal spoofing signals design. We compare our results with two benchmark schemes in the following.
- [*Heuristic combined spoofing with perfect source message cancelation*]{}: The spoofer tries to cancel all the source message by setting $\alpha = -\frac{hg^*\sqrt{P}}{|g|^2}$, and accordingly $\beta$ is given in (\[eqn:beta:opt\]). This scheme only works when $Q>\frac{|h|^2P}{|g|^2}$ for both TIN and SIC Bob receivers, where the minimally required spoofing power $\frac{|h|^2P}{|g|^2}$ is twice of that in Lemma \[lemma1\] for our proposed optimal combined spoofing.
- [*Naive spoofing*]{}: The spoofer uses all its transmit power to send the target message $s$, which corresponds to the case with $\alpha = 0$ and $\beta = \sqrt{Q}$. This scheme only applies to the case with the TIN receiver at Bob when $Q>\frac{|h|^2P}{|g|^2}$.
In the simulation, we normalize the channel coefficients to be $h=1$ and $g=1$ for the purpose of illustration, while our results can be easily extended to the other values of $h$ and $g$. We set $P=10$ dB, and $R = 2$ bps/Hz. Fig. \[fig:2\] shows the maximum achievable spoofing rate versus the spoofing power $Q$ at the spoofer. It is observed that the two benchmark schemes achieve positive spoofing rates (or successfully spoof) only when $Q > 10$ dB, while the optimal combined spoofing does so when $Q > 5$ dB for the TIN receiver at Bob and when $Q$ is larger than 3 dB for the SIC receiver. It is also observed that when $Q$ is larger than $7$ dB, the optimal combined spoofing achieves the same maximum achievable spoofing rate for both TIN and SIC receivers, and outperforms both benchmarks schemes. The heuristic combined spoofing is observed to achieve the same performance as the optimal one when $Q > 16$ dB. This shows that in this case, it is optimal for the spoofer to perfectly cancel the source message and then allocate the remaining power for the target message.
=1 ![The maximum achievable spoofing rate versus the spoofing power $Q$ at the spoofer.[]{data-label="fig:2"}](result.eps "fig:"){width="7cm"}
Conclusion
==========
This letter studied the achievable spoofing rates of the new wireless communication intervention via physical layer spoofing, where a legitimate spoofer sends a combined version of both the source and target messages to confuse a malicious link from Alice to Bob. We proposed optimal spoofing signal designs when Bob employs the TIN and SIC receivers, respectively. It is our hope that this work can provide new insights on the fundamental information-theoretic limits of the physical layer spoofing. How to extend the results to general multi-antenna and multiuser scenarios is an interesting research direction worth pursuing in the future work.
[1]{}
J. Xu, L. Duan, and R. Zhang, “Surveillance and intervention of infrastructure-free mobile communications: a new wireless security paradigm,” to appear in [*IEEE Wireless Commun.*]{}.
J. Xu, L. Duan, and R. Zhang, “Proactive eavesdropping via jamming for rate maximization over Rayleigh fading channels,” [*IEEE Wireless Commun. Letters*]{}, vol. 5, no. 1, pp. 80-83, Feb. 2016.
J. Xu, L. Duan, and R. Zhang, “Proactive eavesdropping via cognitive jamming in fading channels,” in [*Proc. IEEE ICC*]{}, 2016.
Y. Zeng and R. Zhang, “Active eavesdropping via spoofing relay attack,” in [*Proc. IEEE ICASSP*]{}, 2016.
Y. Zeng and R. Zhang, “Wireless information surveillance via proactive eavesdropping with spoofing relay,” [*IEEE J. Sel. Topics Signal Process.*]{}, vol. 10, no. 8, pp. 1449-1461, Dec. 2016.
Y. Zou, J. Zhu, X. Wang, and L. Hanzo, “A survey on wireless security: technical challenges, recent advances and future trends,” [*Proc. IEEE*]{}, vol. 104, no. 9, pp. 1727-1765, Sep. 2016.
M. Medard, “Capacity of correlated jamming channels,” in [*Proc. 35th Allerton Conf.*]{}, Monticello, IL, Oct. 1997, pp. 1043-1052.
A. Kashyap, T. Basar, and R. Srikant, “Correlated jamming on MIMO Gaussian fading channels,” [*IEEE Trans. Inf. Theory*]{}, vol. 50, no. 9, pp. 2119-2123, Sep. 2004.
Q. Liu, M. Li, X. Kong, and N. Zhao, “Disrupting MIMO communications with optimal jamming signal design,” [*IEEE Trans. Wireless Commun.*]{}, vol. 14, no. 10, pp. 5313-5325, Oct. 2015.
J. Xu, L. Duan, and R. Zhang, “Transmit optimization for symbol-level spoofing,” submitted to [*IEEE Trans. Wireless Commun.*]{}. \[Online\] Available: [<https://arxiv.org/abs/1608.00722>]{}
N. Devroye, P. Mitran, and V. Tarokh, “Achievable rates in cognitive radio channels,” [*IEEE Trans. Inf. Theory*]{}, vol. 52, no. 5, pp. 1813-1827, May 2006.
[^1]: J. Xu is with the School of Information Engineering, Guangdong University of Technology (e-mail: [email protected]). He is also with the Engineering Systems and Design Pillar, Singapore University of Technology and Design.
[^2]: L. Duan is with the Engineering Systems and Design Pillar, Singapore University of Technology and Design (e-mail: lingjie\[email protected]).
[^3]: R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail: [email protected]). He is also with the Institute for Infocomm Research, A\*STAR, Singapore.
[^4]: Though beyond the scope of this letter, please refer to [@Xu2016a] for a detailed example for the spoofer to practically obtain $s$, $h$ and $g$, and synchronize with Alice and Bob, where the spoofer can act as a fake relay in the malicious network in obtaining such information. Furthermore, the spoofer can work in a full-duplex mode (e.g., amplify and forward) to obtain $s$ via eavesdropping from Alice and at the same time spoof Bob.
[^5]: In practice, the spoofer can choose any rate (for $x$) no larger than the maximum achievable spoofing rate, provided with successful spoofing.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A new universality class distinct from the standard Wigner-Dyson ones is identified. This class is realized by putting a metallic quantum dot in contact with a superconductor, while applying a magnetic field so as to make the pairing field effectively vanish on average. A random-matrix description of the spectral and transport properties of such a quantum dot is proposed. The weak-localization correction to the tunnel conductance is nonzero and results from the depletion of the density of states due to the coupling with the superconductor. Semiclassically, the depletion is caused by a singular mode of phase-coherent long-range propagation of particles and holes.'
address: 'Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany'
author:
- 'Alexander Altland and Martin R. Zirnbauer'
date: 'February 20, 1996'
title: Random Matrix Theory of a Chaotic Andreev Quantum Dot
---
[2]{}
Wigner-Dyson level statistics is found in physical systems as diverse as highly excited molecules, atoms and nuclei, mesoscopic systems in the ballistic or diffusive regime, and chaotic Hamiltonian systems such as the stadium or Sinai’s billiard. The reason for the ubiquity and universality of Wigner-Dyson statistics is the relation of the Gaussian ensembles[@mehta] to attractive fixed points of the renormalization group flow for an effective field theory (nonlinear $\sigma$ model)[@efetov]. Depending on whether time reversal and/or spin rotation invariance is broken or not, the relevant ensemble has orthogonal, unitary, or symplectic symmetry.
Although the Wigner-Dyson ensembles are the generic ones, new universality classes may arise when additional symmetries or constraints are imposed. One example of this is provided by systems with A-B sublattice structure or chiral symmetry[@gade]. Another example, presented in this letter, are metallic systems in contact with a superconductor. Our considerations were inspired in part by work[@oppermann] on Anderson localization of normal excitations in a dirty superconductor or a superconducting glass.
The system we have in mind is depicted in Fig. 1: a metallic (normal-conducting) quantum dot (N) with the shape of, say, a stadium billiard, is surrounded by a superconductor (S) and contacted by a normal-metal lead (L) via a tunnel barrier (T). A Schottky (or potential) barrier forms at the NS-interface. The quantum dot is pierced by a weak magnetic flux of the order of one or several flux quanta. The temperature is so low that the phase-coherence length exceeds the system size by far. It will help our argument if the quantum dot is not perfectly shaped but contains some defects and/or impurities.
The unique feature that distinguishes our quantum dot from conventional mesoscopic systems is the process of Andreev reflection[@andreev], by which an electron incident on the NS-interface is retroreflected as a hole, and vice versa. The drastic consequences of this process for the thermodynamics and the transport of the quantum dot can be anticipated from a semiclassical argument. Consider the periodic orbit of Fig. 2: an electron starts out from point A, undergoes several normal reflections off the Schottky barrier and eventually returns to A, with the final velocity being roughly the negative of the initial one. At the point of return, the electron gets converted into a hole by Andreev reflection. Because the magnetic field is too weak to cause any significant bending of classical trajectories, the hole then simply tracks down the path laid out by the electron. The charge of the hole is opposite to that of the electron, so the magnetic phase accumulated along the total path is zero. Moreover, since a hole is not just the charge conjugate but also the time reverse of an electron, the dynamical phases cancel, too, provided that electron and hole are at the same energy (the Fermi energy). The two Andreev reflections add up to a total phase shift of $\pi$. Thus, the periodic orbit of Fig. 2 contributes to the periodic-orbit sum for the density of states [*with the negative sign*]{}. Orbits of this type reduce the mean density of states (“Weyl term”) and are expected to put our “Andreev quantum dot” in a universality class distinct from the standard Wigner-Dyson ones. We shall identify this universality class in the sequel.
In a microscopic mean-field treatment, we would start from the Bogoliubov-deGennes (BdG) Hamiltonian $${\cal H} = \pmatrix{
H_0 &\Delta \cr
\Delta^* &-H_0^* \cr}$$ where $H_0 = ({\bf p}-e{\bf A})^2/2m + V({\bf x}) -
\mu$ is a Hamiltonian for “particles”, $-H_0^*$ is the corresponding Hamiltonian for “holes”, and the pairing field $\Delta({\bf x})$, whose magnitude rises from zero inside the quantum dot to a nonzero value in the superconductor, converts particles into holes. The potential $V({\bf x})$ includes the Schottky barrier. Energy is measured relative to the chemical potential $\mu$.
Universality classes are characterized by their symmetries. Notice therefore that ${\cal H}$ obeys the relation $${\cal H} = - {\cal C} {\cal H}^{\rm T}
{\cal C}^{-1}, \quad {\cal C} = \pmatrix{
0 &{\bf 1} \cr
-{\bf 1}&0 \cr} .
\label{CT}$$ This symmetry originates from the electron’s being a spin-1/2 fermion and from spin-rotation invariance [@footnoteA]. The transformation ${\cal H} \mapsto
{\cal C} {\cal H}^{\rm T} {\cal C}^{-1}$ will be called ${\cal CT}$-conjugation as it combines time reversal (${\cal H} \mapsto {\cal H}^* = {\cal H}^{\rm T}$) with a kind of charge conjugation (${\cal H} \mapsto {\cal C
H C}^{-1}$).
By design, the classical motion of particles and holes in the billiard-shaped dot is chaotic and fills the available phase space ergodically. Now observe that every time a particle or hole is Andreev-reflected from the NS-interface, its wavefunction acquires an extra phase determined by the superconducting order parameter. In this context it is important that the applied magnetic field is screened by a supercurrent circulating along the NS-interface inside the superconductor. The supercurrent flow, in turn, is concomitant with a spatial variation of the phase $\phi$ of the order parameter[@footnoteB]. As a result of this and the chaotic dynamics, the extra phase picked up during Andreev reflection varies randomly along a typical semiclassical trajectory. So everything is quite random, and we expect some kind of random-matrix theory to apply. The question is now: what random-matrix theory?
Because the presence of the magnetic field makes the pairing field experienced by particles and holes vanish on average, $\Delta$ can be modelled by a stochastic variable with zero mean. Moreover, since the system has been designed to be chaotic, there exist no integrals of motion except for energy. The only symmetry (apart from hermiticity of the Hamiltonian) of relevance for the long-time[@footnote1] or ergodic limit we shall consider, is the ${\cal CT}$-oddness (\[CT\]). Experience with similar problems then tells us that we can model the ergodic limit by a Gaussian, or maximum-entropy, ensemble with probability density $\exp (-{\rm Tr} {\cal H}^2/2v^2) d{\cal H}$ subject to the constraint (\[CT\]). This implies that, in any orthonormal basis of states $\psi_a$ $(a=1,{\bar 1},
...,N,{\bar N})$ with ${\cal CT}$-conjugate basis $\psi_{\bar a} = {\cal C}\psi_a^*$, the variances of the random Hamiltonian matrix elements are given by the correlation law $$\langle {\cal H}_{ab}
{\cal H}_{cd} \rangle = v^2 (\delta_{ad}
\delta_{bc} - \delta_{a\bar c}\delta_{b\bar d}).
\label{correlator}$$ To complete the definition of our random-matrix model, we add to ${\cal H}$ a term $-i\Gamma$ which accounts for the coupling to the normal-metal lead and will be specified later.
As a first step, let us close off the contact with the lead $(\Gamma = 0)$. What can we say about the spectral statistics, the central characteristic of the ergodic [*isolated*]{} quantum dot? If $\psi_k$ is an eigenstate of ${\cal H}$ with eigenvalue $+E_k$ then, by (\[CT\]), so is ${\cal C}\psi_k^*$ with eigenvalue $-E_k$. Thus, there is an exact pairing between positive and negative eigenvalues. By diagonalizing ${\cal H}$ and computing the Jacobian of the transformation to diagonal form, we obtain the (unnormalized) joint probability density of the positive eigenvalues $E_k$ $(k=1,...,N)$: $$P(E_1,...,E_N) = \prod_{i<j}
(E_i^2-E_j^2)^2 \prod_{k=1}^N E_k^2 \
e^{-E_k^2/2v^2} ,
\label{ensemble}$$ which is manifestly invariant under $E_k \to
-E_k$[@nagao].
To calculate the spectral statistics, it is convenient to view (\[ensemble\]) as a Gaussian Unitary Ensemble (GUE) of $2N$ levels $E_1, E_{\bar 1}, ..., E_N,
E_{\bar N}$ with the mirror constraint $E_{\bar k} = -
E_k$. The correlation functions of the GUE are known to coincide with those of a one-dimensional gas of free fermions in the large-$N$ limit[@sutherland]. Now, when a fermion (i.e. an energy level) gets close to $E
= 0$, so does its mirror image. Because Fermi statistics makes the wavefunction vanish as two fermions approach, the constraint $E_{\bar k} = -E_k$ amounts to hard wall boundary conditions at $E = 0$. Hence, we can compute the eigenvalue density and its correlations for (\[ensemble\]) as the [*particle density and its correlations for a free Fermi gas with a hard wall at the origin*]{}. In this way we obtain $$\rho(E) = \langle
{\rm Tr} \delta(E-{\cal H})\rangle =
1/\delta - \sin (2\pi E/\delta) / 2\pi E ,
\label{density}$$ where $\delta$ is the level spacing for $E \gg \delta$. We see that the coupling to the superconductor depletes the mean density of states and makes it vanish quadratically at $E = 0$[@footnote2]. The states pushed away from $E = 0$ cause density oscillations, which ebb off as $1/E$. Note that the result (\[density\]) applies when both $E$ and $\delta$ are much smaller than the characteristic energy uncertainty set by the frequency of Andreev reflection.
To understand better the mechanism of depletion, we turn to diagrammatic perturbation theory. Let $G :=
\langle (E+i\varepsilon-{\cal H})^{-1}\rangle$ denote the ensemble average of the Gorkov Green’s function. We expand it in a geometric series with respect to ${\cal H}$ as usual. To do the ensemble average, we distinguish between two types of contraction, $\Pi_{ac,bd}^{\rm D} = v^2 \delta_{ad}\delta_{bc}$ and $\Pi_{ac,bd}^{\rm C} = -v^2 \delta_{a\bar c}
\delta_{b\bar d}$, corresponding to the first and second term in the basic law (\[correlator\]). Making this distinction is useful for organizing the perturbation series, since $\Pi^{\rm D}$ causes pure GUE behavior whereas $\Pi^{\rm C}$ generates the corrections to the GUE. By summing all nested $\Pi^{\rm D}$ self-energy graphs, we get Pastur’s equation, $G = (E+i\varepsilon - v^2 {\rm Tr}G)^{-1}$, which is exact for $E \gg \delta$ and $N\to\infty$. The solution, $G^0$, of this equation yields Wigner’s semicircle law for the density of states: $-{\rm Im
Tr}G^0/\pi = \sqrt{2N-(E/v)^2}/\pi v$. According to (\[density\]), corrections to this result, which is stationary and equal to $\sqrt{2N}/\pi v =: 1/\delta$ up to uninteresting terms of order $1/N$, should appear as we approach zero energy. It turns out that these arise from summing a geometric series of ladder graphs built solely from $\Pi^{\rm C}$-contractions. The ladder sum, $C_{ac,bd}$, satisfies Dyson’s equation $C=
C^0 + C^0 \Pi^{\rm C} C$ with $C_{ac,bd}^0 =
\delta_{ab} \delta_{cd} G_{aa}^0 G_{cc}^0$. Its solution $C = C^0({\bf 1} -\Pi^{\rm C}C^0)^{-1}$ is [*singular*]{} at $E = 0$: $$C = C^0 + C^0 \left(
\Pi^{\rm C} 2iN\delta/\pi E \right) C^0 +
{\cal O}(E/N)^0.$$ By evaluating the graph shown in Fig. 3 we get ${\rm
Tr}G/\pi = -i/\delta - 1/2\pi E + ...$, which are the leading terms in a $1/E$ expansion. (There is a renormalization by a factor of 1/2 coming from the possibility of connecting the external legs in Fig. 3 by a nonsingular $\Pi^{\rm D}$ ladder.) This perturbative result is to be compared with the exact formula ${\rm Tr}G/\pi = -i/\delta - [1-\exp(2\pi
iE/\delta)]/2\pi E$ reconstructed from (\[density\]) by causality. We see that diagrammatic perturbation theory properly reproduces the smooth part of the $1/E$ correction. (The oscillatory term is nonanalytic in the expansion parameter $1/E$ and cannot be recovered by the perturbative summation of graphs.)
What is the semiclassical meaning of the mode $C$? Our diagrammatic analysis suggests an interpretation as a mode of phase-coherent propagation of a particle-hole pair. To gain further insight, recall the periodic orbit shown in Fig. 2. In diagrammatic language, this orbit corresponds to connecting the Green’s function lines in Fig. 3 by just a single $\Pi^{\rm
C}$-contraction, and thus is the simplest semiclassical building block of the $C$-mode. More generally, an arbitrary number of Andreev reflections may be inserted into the loop. The only condition is that the [*particle-hole character of the states during the first and second traversal of the loop be ${\cal
CT}$-conjugate to each other*]{}. Because ${\cal
CT}$-conjugate states carry opposite charge and the loop is traversed twice [*in the same direction*]{}, the $C$-mode is ignorant of a weak magnetic field[@footnoteC]. Note, however, that the $C$-mode is sensitive to energy $E$ (or voltage), which breaks the phase relation between particles and holes.
With a solid understanding of the spectral properties in hand, we finally open up the tunnel barrier and turn to the prime experimental observable, the conductance $g$. Our treatment will be based on the linear response formula $g=4(e^2/h){\rm Tr}{S^\dagger}_{\rm ph}S_{\rm hp}
$[@takane] where $S_{\rm hp}$ is the part of the scattering matrix $S$ that maps incoming electrons onto outgoing holes. We suppress elastic phase shifts and parameterize the $S$-matrix by $
S = {\bf 1} - 2i{\tilde W}(i\varepsilon+i\Gamma
-{\cal H})^{-1} W
$ [@iwz] at $E = 0$. Here $W$ is a matrix coupling $2M$ channels (particles and holes) in the lead with $2N$ levels in the quantum dot, $\tilde W$ is its adjoint, and $\Gamma = W{\tilde W}$. $W$ is diagonal in particle-hole space. For a simple model, we take ${\tilde W}W$ to be a multiple of the identity in channel space: ${\tilde W}W = \gamma{\bf 1}$. Then $\Gamma = \gamma P$ where $P$ is a rank-$2M$ projector in level space. We assume $N \gg M$.
We begin by discussing the limit of an open quantum dot $(M\gg 1)$ where all structure in the density of states is washed out by the large level width. Let $T$ denote the probability for an electron in an incoming channel to be transmitted through the tunnel barrier, in this very limit. For our simple model, $T = 4\lambda\gamma/
(\lambda +\gamma)^2$ with $\lambda^2 = 2Nv^2$ [@iwz]. Having entered the quantum dot, the electron spends a long time there, undergoing many Andreev reflections, and finally returns to the lead either as a particle or as a hole, with equal probabilities. Hence $\langle {\rm Tr}{S^\dagger}_{\rm
ph}S_{\rm hp}\rangle = MT/2$, which we refer to as the “classical” value. To confirm this result diagrammatically, one has to sum a geometric series of $\Pi^{\rm D}$ ladder graphs, producing the diffuson[@footnote3]. On approaching the opposite (closed) limit, we expect a reduction relative to the classical value. The reason is simply this: the density of states in the isolated dot vanishes at $E =
0$, so that an electron trying to enter cannot find any state to go to and is immediately rejected into the lead. This effect should announce itself as a negative ${\cal O}(M^0)$ correction (“weak localization”). Such a correction in fact exists and is due to the graph shown in Fig. 4 featuring two diffusons and one $C$–mode. Evaluation of this graph leads to $$\langle g \rangle = (MT - 1 + T) \times 2e^2/h
+ {\cal O}(1/MT) .
\label{conductance}$$ Note that the weak localization correction in the corresponding N-system vanishes. A result similar to (\[conductance\]) was recently found in a related context[@brouwer].
Although the calculation of the graph of Fig. 4 is somewhat technical, the result can easily be checked and interpreted. Consider first the limit of many weakly coupled channels ($T\to 0$, $MT$ fixed). In this limit the absolute square of the Green’s function becomes independent of the level and particle-hole indices, on average[@mrznpa]. By conservation of probability, the conductance is then fully determined by a one-point function: $\langle g \rangle = -(4e^2/h)
\bar\gamma {\rm Im} \langle {\rm Tr} (i\bar\gamma-{\cal
H})^{-1}\rangle$ where $\bar\gamma = \gamma M/N$ is the mean level width. Analytic continuation of our result for the average Green’s function to $E = i\bar\gamma$ gives $\langle g\rangle = (MT-1+e^{-MT})\times 2e^2/h$, in agreement with (\[conductance\]).
To understand the other limit of strongly coupled channels $(T=1)$ it is easiest to argue directly at the $S$–matrix level. By its definition as a “sticking probability”, $T = 1 - |\langle S_{cc}\rangle |^2
+{\cal O}(1/MT)$. Putting $T = 1$ therefore amounts to assuming a fully random $S$-matrix that vanishes on average: $\langle S \rangle = 0$. The symmetries of $S$ are the same as those of $\exp i{\cal
H}$, so that from (\[CT\]) we deduce $S = {\cal C}
{S^{-1}}^{\rm T} {\cal C}^{-1}$, which is the defining equation of the symplectic group ${\rm Sp}(2M,{\bf
C})$. (${\cal C}$ is now meant to operate in channel space.) Unitarity fixes a subgroup ${\rm Sp}(2M)$. We are thus led to postulate an $S$-matrix ensemble with probability distribution $\langle\langle\bullet\rangle
\rangle$ given by the Haar (or invariant) measure of ${\rm Sp}(2M)$. This ensemble is Dyson’s Circular Unitary Ensemble[@mehta] adapted to fit our particle-hole symmetric situation. By elementary group theory, $\langle\langle S_{ab}\rangle\rangle = 0$ and $\langle
\langle S_{ab}^{\vphantom{*}} S_{cd}^* \rangle\rangle =
\delta_{ac}\delta_{bd}/2M$, so $\langle\langle g \rangle
\rangle = 2Me^2/h$, with no correction of order $M^0$.
In conclusion, we mention that the singular modes, the diffuson and the $C$-mode, translate into low-energy degrees of freedom of a corresponding nonlinear $\sigma$ model. They determine the attractive (metallic, or free) renormalization group fixed point of this field theory. Ultimately, the existence of such singular modes, which are forgetful of microscopic detail and saturate the long-time and long-distance physics, is the reason why we predict with confidence that the ergodic limit of the chaotic Andreev quantum dot is universal and our random-matrix theory applies.
[99]{} M.L. Mehta, [*Random Matrices*]{}, Academic Press 1991. K.B. Efetov, Adv. Phys. [**32**]{}, 53 (1983). R. Gade, Nucl. Phys. B[**398**]{}, 499 (1993); A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B[**432**]{}, 487 (1994). R. Oppermann, Physica A[**167**]{}, 301 (1990). A.F. Andreev, Zh. Eksp. Teor. Fiz. [**46**]{}, 1823 (1964). Spin-orbit scattering violates the symmetry (1) and drives the system into a separate universality class. Note that the choice of gauge $\phi =
{\rm const}$ is unnatural here as it forces $\int {\bf A}d{\bf x}$ across the lead to be very large. Here “long” means times greater than the mean time $t_A$ spent between successive Andreev reflections. In order for the ergodic limit to be nonempty, we require $t_A \gg \hbar
/\delta$ where $\delta$ is defined in the text below Eq. (\[density\]). Incidentally, the substitution $x_k = E_k^2$ turns this probability density into a member of the family of Laguerre ensembles studied recently by T. Nagao and K. Slevin, J. Math. Phys. [**34**]{}, 2075 (1993). B. Sutherland, J. Math. Phys. [**12**]{}, 246 (1971); see also B.D. Simons, P.A. Lee and B.L. Altshuler, Phys. Rev. Lett. [**72**]{}, 64 (1994). For lack of space the correlations are not recorded here. By this feature, the $C$-mode can be identified as a close relative of the usual (normal-metal) diffuson. It here appears as a separate mode because of our use of the BdG-formalism. Y. Takane and H. Ebisawa, J. Phys. Soc. Japan [**61**]{}, 1685 (1992). S. Iida, H.A. Weidenmüller and J.A. Zuk, Ann. Phys. [**200**]{}, 219 (1990). The diffuson is identical to the normal-metal diffuson except that in our case the opposite segments of the ladder are either two BdG-particles or two BdG-holes. P.W. Brouwer and C.W.J. Beenakker, Phys. Rev. B[**52**]{}, 3868 (1995). M.R. Zirnbauer, Nucl. Phys. A[**560**]{}, 95 (1993).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present the discovery of a rapidly evolving by the Korean Microlensing Telescope Network Supernova Program (KSP). KSP is a novel high-cadence supernova that offers deep ($\sim21.5$mag in $BVI$ bands) wide-field monitoring for the discovery of early and/or fast optical transients. , reported here, was discovered on 2015 September 27 during the KSP commissioning run in the direction of the nearby galaxy NGC 300, and stayed above detection limit for $\sim$ 22 days. We use our $BVI$ light-curves to constrain the ascent rate, $-3.7(7)$magday$^{-1}$ in $V$, decay time scale, $t^{V}_{2}=1.7(6)$days, and peak absolute magnitude, $-9.65\leq M_{V}\leq -9.25$mag. We evidence for a short-lived pre-maximum halt in all bands. . We discuss constraints on the nature of the progenitor and its environment using archival HST/ACS images and conclude with a broad discussion on the nature of the system.'
author:
- 'John Antoniadis, Dae-Sik Moon, Yuan Qi Ni, Dong-Jin Kim, Yongseok Lee, Hilding Neilson'
bibliography:
- 'author.bib'
title: 'Discovery of a rapid, luminous nova in NGC300 by the KMTNet Supernova Program'
---
Introduction {#sec:intro}
============
Over the past few years, wide-field variability surveys have significantly advanced our understanding of high-energy transients, from thermonuclear runaways and various types of supernovae (SNe) [e.g. @ptf1 and references therein], to $\gamma-$ray [@fermigrb] and fast radio bursts [@frbs].
In the optical regime, contemporary experiments are typically sensitive to two types of explosive phenomena: “local” optical transients (OTs) with small peak luminosities ($-10 \lesssim M_{\rm b} \lesssim
5$mag), such as classical and dwarf novae, and luminous OTs with $M_{\rm b} \lesssim -15$mag [e.g. @panstarrs1; @ptf1; @ptf2 and references therein]. The intermediate luminosity regime, which is as of yet poorly explored, is thought to be populated by rare cosmic explosions, such as rapid under-luminous SNe [@dsm+13], accretion-induced collapse of white dwarfs [WDs; @nk91; @mpqt09], fallback SNe [@dk13], electromagnetic counterparts to compact-object mergers [@bfc13], and orphan short-GRB afterglows [@tp02].
Exploring this parameter space is challenging for two reasons: first, our limited understanding of the underlying physical mechanisms makes it difficult to predict characteristic observational signatures and the extend to which these OTs blend with novae and SNe populations. Second, because these events are expected to be both rapid and rare, identification requires sampling of a sufficiently large volume with high temporal resolution, thereby driving the need for deep high-cadence surveys.
High-cadence experiments are additionally motivated by outstanding questions in long-standing astrophysical problems. For instance, our understanding of “infant” thermonuclear runaways and SNe is limited, with questions regarding trigger mechanisms, shock break-out emission, ejecta masses, progenitor structure, and asymmetries still remaining [@smartt09; @be08].
Motivated by those questions, we have secured $\sim$20% of the Korea Microlensing Telescope Network [KMTNet; @kmtnet2] observing time through 2020 for a dedicated survey focused on infant and/or rapidly-evolving OTs, which we call the KMTNet Supernova Program [KSP; see @ksp0]. [@kmtnet2]. KSP is capable of providing deep ($\lesssim$ 21.5mag[^1]), high-cadence continuous monitoring in $B$, $V$ and $I$ bands [@ksp0].
In this paper we present the discovery of , a rapidly-evolving OT found towards a spiral arm of the Sculptor galaxy NGC300. The transient stands out for its rapid decay rate and showcases the potential of KSP for providing well-sampled multi-color light-curves of rapidly-evolving eruptions. The paper is structured as follows. In §2, we provide a brief overview of the KSP data, and present the discovery of , alongside its multi-color evolution. we use HST archival images to place constraints on the progenitor of and its environment. Finally, we explore the ramifications of our result and conclude with a brief discussion on the prospects of the KSP in nova search in §5.
: Discovery and light-curve {#sec:dl}
===========================
Between 2015 July 1 and 2016 January 10, KSP monitored a 15deg$^2$ area towards the Sculptor group, including a 4deg$^2$ field around NGC 300 [$d=1.86(7)$Mpc[^2]; $M - m = 26.35$; @ngc300]. We collected $\sim1300$ frames per $BVI$ band, with a mean cadence of $\sim$3.5 h and .
We use 60-s exposures which typically yield a limiting magnitude of $\sim$21.5 mag at S/N=5 under 12 seeing for point sources. [@gsc2], The photon-limited astrometric precision is generally better than $\sim$05 under 12 seeing.
was discovered in the KSP data towards an NGC 300 spiral arm (Figure1) as a faint, rapidly-evolving OT. The source first appeared on an $I-$band image recorded on UT 270.76[^3] with an apparent magnitude of $m_{\rm I}$ = 20.7(3) mag, at $(\alpha,\delta)_{\rm J2000} =$(0:55:09.422,-37:42:16.5), 3373 ($\simeq 2$kpc) away from the centre of the galaxy.
{width="90.00000%"}
The original light-curve produced by our automated pipeline was contaminated by systematics, evident by the large-scale scatter ($\sim0.4$mag) around the dominant decay trend. For this reason we re-analysed the data using photometry of nearby stars. More specifically, we determined the local PSF by fitting a Moffat function plus a first degree polynomial for the background to 4 bright unsaturated stars within 35 of the transient. Figure\[fig:lc\] shows the $BVI$ light-curves of from first detection to its disappearance below the detection limit $\sim$ 22 days later.
{width="80.00000%"}
Rising Phase
------------
The initial phase of the transient (Figure\[fig:lc\]), sampled on 10–12 instances per each $BVI$ filter, is characterized by a mean ascent rate of –1.9(7) magday$^{-1}$ ($B$), –2.6(4) mag day$^{-1}$ ($V$), and –2.5(6) mag day$^{-1}$ ($I$), as determined by a linear fit to data taken prior to UT272. However, as can be seen in Figure\[fig:lc\], the ascent rate seems to decrease as the transient progresses and remains practically constant in all bands after UT270.67.
Excluding the data taken after the aforementioned time yields $-4.0(1.0)$, $-3.7(7)$ and $-3.8(9)$magday$^{-1}$ for the mean ascent rate in $B, V$ and $I$ respectively. Based on those estimates, the probability that the halt can be attributed to random noise is $<10^{-4}$. Given that KSPN2015-09a re-appears brighter two nights later, we thus interpret this as evidence for a pre-maximum halt (PMH), often seen in light-curves of novae [@hpk+14].
![The color evolution of after the onset of its decay phase. []{data-label="fig:lcm"}](Figure_3.pdf){width="50.00000%"}
Maximum and Decline Phase
-------------------------
was detected again on UT272.33 at $B$ =18.39(1), shortly after the onset of its decay phase (Figure \[fig:lc\]). An extrapolation of our best-fit models between the rising and early decline phases suggests that the peak luminosity was not missed by more than $\sim$ 0.6 days. We place the maximum light between 17.9–17.6mag, 17.1–16.7mag, and 17.5–17.2 mag in $B$, $V$, and $I$ band, respectively.
Assuming the transient is indeed associated with NGC300, the former correspond to peak absolute magnitudes between $-8.45$ and $-9.65$mag. We do not account for the negligible foreground extinction [$A_{\rm V}^{\rm f}=0.03$mag; @sfd98], nor for any reddening from NGC300, which should be of the same order since the galaxy is viewed nearly edge-on.
The post-maximum evolution is characterized by a rapid decay (Figure\[fig:lc\]). To quantify the decay rate, we adopt a decline law of the form $F_{\rm v}\propto t^{-\alpha}$, where $t$ is the time since maximum. For $B$ and $V$ we infer $\alpha=1.98(6)$ and $1.84(6)$ respectively. In the $I$ band, the decay rate evolves from $\alpha=1.41(6)$ at the onset of the decay phase to $\alpha=2.00(7)$ after $\sim$UT279. From the best-fit light-curves and the times of maximum-light derived above we infer $t^{V}_{2}=1.7(6)$ and $t^{V}_{3}=3.8(7)$days for the time required for the $V-$band light-curve to fade by 2 and 3 magnitudes, respectively.
The color evolution of is shown in Figure\[fig:lcm\]. The decay phase starts with $B-V\simeq0.5$mag and $V-I \simeq -0.3$mag. The color indexes then rapidly evolve to $B-V\simeq 0.0$mag and $V-I\simeq1.0$mag in less than 1 and 4 days, respectively. Subsequently, the excess in the $I$ band grows up to $V-I\simeq 1.5$mag within a few days while $B-V$ reverts to negative values.
The nature of
==============
[cf @psm+07; @wdb+15 and references therein].
HST Constraints on the progenitor and environment {#sec:hst}
=================================================
We analysed a set of archival HST/ACS frames towards NGC300 obtained using the f606w and f814w filters [see @ngc300hst for the original work]. The images were taken on 2014 July 2 with 850 and 611s exposure times, respectively.
We measured instrumental magnitudes and performed absolute calibration using DOLPHOT [@dolphot]. Pre-determined PSF models were used to extract instrumental magnitudes which were then corrected to infinite apertures using 12 bright isolated stars within 15 from . We determine the $5\sigma$ photometric limit to be $m_{\rm f606w} \leq 27.6$ $(M_{\rm f606w}\leq 0.84)$ and $m_{\rm f814w} \leq 26.5$ $(M_{\rm f814w}\leq 0.14)$mag. We used the default ACS astrometric calibration [see @acsastro] which provides a distortion-free system to a level of 5mas and then fitted for position offsets using the 4 common GSC sources nearest to .
Figure \[fig:image\] shows the f606w image around , with 05 and 10 error circles . The star nearest to the nominal KMTNet position of the source is located at ($\alpha$,$\delta$)$_{\rm J2000}$ = ($\rm 0^h55^m09.422^s, -37\degr42\arcsec16\farcs50$) and has $m_{\rm f606w} = 26.11(9)$ and $m_{\rm f814w} = 25.34(10)$mag. The brightest point-like source within the 95% error circle has $m_{\rm f606w} = 22.348(7)$ and $m_{\rm f814w} = 21.344(8)$mag, consistent with what one would expect for a super-giant at the distance of NGC300.
![HST image of NGC 300. The green circles, centred at the position of represent the $1$ and $2\sigma$ astrometric uncertainties of the KSP data. The location of the brightest star is also shown.[]{data-label="fig:image"}](figure_4.pdf){width="50.00000%"}
It is unlikely that either of the HST sources inside the KMTNet error box (Figure \[fig:image\]) is the progenitor of in quiescence, as one would expect $ 4 \leq M \leq 8$ for a typical main-sequence companion, which is well below the sensitivity of the HST data. The broader region, which is part of an NGC300 spiral arm, is characterized by a large number of bright stars (with $M_{\rm f606w}\simeq -5$). This indicates that is likely associated with a region with high star-forming activity.
Discussion {#sec:dis}
==========
Nova outbursts result from thermonuclear runaway eruptions on a white dwarf (WD). They occur in a binaries of the cataclysmic-variable type, in which matter is accreted from a non-degenerate companion [e.g. @drb+12]. Material accumulating on the WD surface eventually causes the envelope to become electron degenerate, leading to a runaway thermonuclear flash which ultimately gives rise to the nova phenomenon.
It is well established, both theoretically and observationally, that nova time scales, amplitudes and repentance rates depend sensitively on the WD mass, mass-accretion rate, envelope mass, companion type and wind power [@yps+05]. With few exceptions, bright and fast novae (hereafter FNe) occur in systems with massive WDs and high mass-accretion rates between $\dot{M}_{\rm acc} \simeq$ 10$^{-8}$ and 10$^{-4}$ M$_{\odot}$ yr$^{-1}$ [@pk95; @yps+05].
FNe rising phases last up to few days [e.g. @khc09; @ssh10]. During this time, the effective temperature increases dramatically (initially at constant radius) causing the surface brightness to rise by 10–20 magnitudes. Recent studies find no strong correlation between the ascent rate and peak brightness [e.g. @ckn+12], although no safe conclusions can be drawn from existing data. In addition, because of the rapid evolution time scales, very few infant FNe have been sampled with sufficient temporal cadence in multiple band to probe the underlying eruption mechanism in detail.
The work presented here provides an unprecedented multi-color view of an early FN eruption phase. We find that the brightness of increases at $-3.7$magday$^{-1}$ in $V$, indicating a total rising-phase duration between $\sim2.5$ and 5 days.
In addition, our data provide evidence for a short-lived pre-maximum halt (PMH) after UT270.67 (Figure\[fig:lc\]). While PMHs have been observed both in slow and fast novae [see @smei1; @smei2 and references therein], it is yet unclear if they reflect an intrinsic change of the WD.
For instance, some early studies suggest that they may be triggered by an external condition such as a sudden enhancement of mass loss from the donor. More recent theoretical work based on detailed 1D simulations [@hpk+14] finds that PMHs are explained naturally by a decrease in the convection-transport efficiency. The rise to peak brightness continues after the opacity decreases for radiation-transport to take over at the onset of the mass-loss phase. In late-rise and early-decline phases (see Figures \[fig:lc\] and \[fig:lcm\]). This is consistent with a transition in the emission mechanism expected in the latter scenario (see below).
In the early decline phase it is expected that the continuum spectrum is dominated by free-free scattering above the optically-thick photosphere. For the idealized case of pure optically-thin thermal Bremsstrahlung, [@khc09] find a universal decay law, $F_\nu$ $\propto t^{-1.75}$, which matches well the observed lignt-curve of (§ \[sec:dl\]). This in turn suggests that the transient evolution depends strongly on the wind rate and velocity, and less so on the WD mass [@khc09 and references therein]. At later phases, especially after UT 276.9, the transition to a steeper $I$-band decline rate suggests the presence of an additional thermal emission component [@hk16] which may the formation of a dust shell.
Considering that no spectroscopic information is available for , we resort to other historical FNe and theoretical studies to draw further conclusions on the WD mass and the nature of the donor star as below. Examples of well-studied bright ($-8.5 \leq M_{V} \leq - 10.5$) FNe in the Galaxy include [@novacat] V838Her ($t_{2}\simeq1$day), V1500Cyg ($t_{2}\simeq2$days), V2275Cyg ($t_{2}\simeq3$days) and more recently V2491Cyg [$t_{2}\simeq 2$days; @smei2] [$t_{2}\simeq2.5$days @ebh+17]. Spectra from the early eruption phases for these FNe indicate terminal wind velocities of $\sim 1000$ to 3000 kms$^{-1}$. In almost all cases, FNe are associated with massive ($\geqslant 1$M$_{\odot}$) WDs and a total wind mass-loss of few times 10$^{-6}$M$_{\odot}$. Given the observational similarities between and these FNe, we therefore conclude that the former most likely also hosts a massive WD of $\geqslant 1$M$_{\odot}$.
The discovery of in an early phase and dense multi-color monitoring of its light-curve through its entire eruption demonstrates the potential of KSP to provide an unprecedented view of novae and related phenomena. Based on the KSP early performance and sensitivity so far, of classical novae as well as other transients, including infant and nearby SNe [@ksp0].
We thank the anonymous referee for the useful suggestions. This research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) and the data were obtained at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. KSP is partly supported by an NSERC Discovery Grant to DSM. JA is a Dunlap Fellow at the Dunlap Institute for Astronomy and Astrophysics at the University of Toronto. The Dunlap Institute is funded by an endowment established by the David Dunlap family and the University of Toronto. We have made extensive use of NASA’s Astrophysics Data System. This research made use of Astropy, a community-developed core Python package for Astronomy.
[^1]:
[^2]: The numbers in the parentheses are equivalent to the 1-$\sigma$ uncertainty at the last quoted digits.
[^3]: All times are defined relevant to Jan12015.0 UT
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a further development of a method for accelerating the calculation of CMB power spectra, matter power spectra and likelihood functions for use in cosmological Bayesian inference. The algorithm, called [CosmoNet]{}, is based on training a multilayer perceptron neural network. We demonstrate the capabilities of [CosmoNet]{} by computing CMB power spectra (up to $\ell=2000$) and matter transfer functions over a hypercube in parameter space encompassing the $4\sigma$ confidence region of a selection of CMB (WMAP + high resolution experiments) and large scale structure surveys (2dF and SDSS). We work in the framework of a generic 7 parameter non-flat cosmology. Additionally we use [CosmoNet]{} to compute the WMAP 3-year, 2dF and SDSS likelihoods over the same region. We find that the average error in the power spectra is typically well below cosmic variance for spectra, and experimental likelihoods calculated to within a fraction of a log unit. We demonstrate that marginalised posteriors generated with [CosmoNet]{} spectra agree to within a few percent of those generated by [CAMB]{} parallelised over 4 CPUs, but are obtained 2-3 times faster on just a *single* processor. Furthermore posteriors generated directly via [CosmoNet]{} likelihoods can be obtained in less than 30 minutes on a single processor, corresponding to a speed up of a factor of $\sim 32$. We also demonstrate the capabilities of [CosmoNet]{} by extending the CMB power spectra and matter transfer function training to a more generic 10 parameter cosmological model, including tensor modes, a varying equation of state of dark energy and massive neutrinos. Finally we demonstrate that using [CosmoNet]{} likelihoods directly, the sampling strategy adopted by [CosmoMC]{} is highly sub-optimal. We find the generic [Bayesys]{} sampler [@Bayesys] sampler to be a further $\sim 10$ times faster, yielding 20,000 post burn-in samples in our 7 parameter model in just 3 minutes on a single CPU. [CosmoNet]{} and interfaces to both [CosmoMC]{} and [Bayesys]{} are publically available at [ www.mrao.cam.ac.uk/software/cosmonet]{}.'
author:
- |
T. Auld, M. Bridges and M.P. Hobson\
Astrophysics Group, Cavendish Laboratory, Magingley Road, Cambridge CB3 0HE, UK
bibliography:
- 'netpaperII.bib'
date: 'Accepted —. Received —; in original form '
title: '[CosmoNet]{}: fast cosmological parameter estimation in non-flat models using neural networks'
---
\[firstpage\]
cosmology: cosmic microwave background – methods: data analysis – methods: statistical.
Introduction
============
Bayesian inference in cosmology is normally carried out using sampling based methods as now required by the dimensionality of the models and increasingly high-precision of the data sets. Typically one requires the calculation of theoretical temperature and polarisation CMB power spectra $C_\ell^{\rm TT}$, $C_\ell^{\rm TE}$, $C_\ell^{\rm EE}$ and $C_\ell^{\rm BB}$ and/or the matter power spectrum $P(k)$ using codes such as [CMBfast]{} [@cmbfast] or [CAMB]{} [@camb]. These codes typically require of order 10 secs for spatially-flat models and 50 secs for non-flat models on a 2 GHz CPU. This approach is therefore computationally demanding, but does have the advantage that it is simple to generalise if one wishes to include new physics. [CosmoMC]{} [@cosmomc] currently represents the state of the art in cosmological Markov Chain Monte Carlo (MCMC) sampling and employs a number of strategies to improve performance, such as a division of the parameter space into ‘slow’ parameters (which determine the evolution of structure) and ‘fast’ parameters which determine the primordial power spectrum. Nonetheless, the technique is still computationally expensive.
A number of examples exist in the literature of methods reliant on generating (to some degree or other) grids of models, within which various interpolations are made to compute observable spectra at arbitrary parameter values. One example is [Dash]{} [@Kaplinghat] which requires a considerable investment of some 40 hours to generate a grid of transfer functions which can then be used to generate $C_\ell$ spectra for a given parameter combination about $30$ times faster than [CAMB]{}.
@Jimenez have built a less demanding method around the novel idea of transformation into the mostly uncorrelated physical parameterisation introduced by @Kosowsky. Since the $C_\ell$’s have a simple dependence on the input parameters they are then relatively easy to model. The algorithm, known as [CMBwarp]{} then uses polynomials to fit the spectra in which the polynomial coefficients are tied to the spectra at some, single point in the parameter space. This allows spectra to be generated $\sim 3000$ times faster than [CAMB]{}. Of course this method suffers from the drawback that the single model about which the polynomial fit is specified must be chosen carefully to lie close to the centre of the posterior distribution as accuracy decreases away from this point. Within a $3 \sigma$ region around the chosen model they estimate it gives better than 1% accuracy.
The advent of larger datasets have meant the time spent calculating model likelihoods is rapidly approaching the time necessary to generate the theoretical spectra. [CMBfit]{} [@Sandvik] proposes to remove the step of determining spectra altogether by providing a semi-analytic fit directly to the WMAP likelihood as a function of input cosmological parameters. Given the ubiquity of WMAP data in cosmological analyses the drawback of being tied to a single experiment is however not as limiting as one might think.
The methods just described, although useful, lack general applicability over a range of theoretical spectra and datasets. We have been motivated to generate a new method that can be applied, almost blindly to the problem of cosmological inference in order to remove the two largest bottlenecks of theoretical spectra generation and likelihood evaluation. Previously @Pico built a robust new method based on machine-learning called [Pico]{}. Their method requires the assembly of $\sim 10^4$ samples over the parameter space drawn uniformly from a desired region that could encompass any confidence region of a given experiment. This ‘training set’ is compressed via a principal component analysis (using Karhunen–Loève eigenmodes) which typically results in a reduction in the dimensionality of the training set by a factor of two. The training set is used to divide the parameter space into ($\sim 100$) regions using $k$-means clustering (see e.g. MacKay 1997) with the aim of each cluster encompassing a region of parameter space over which the power spectra vary equally. A polynomial fit is then used over each cluster providing a local interpolation of the power spectra within the cluster as a function of cosmological parameters. Crucially, the method fails to model the spectra accurately over the entire parameter space, hence the need for cluster division and thus making the algorithm difficult to extend.
Both [Pico]{} and [CMBWarp]{} provide similar improvements in efficiency, but [Pico]{} is an order of magnitude more accurate than both [DASh]{} and [CMBWarp]{}. It is generic enough to be extended to any observable spectra and is flexible enough to allow prediction of likelihood values, thus incorporating the benefits made by the [CMBfit]{} code. Given the current speed of the WMAP 3-year likelihood \[@Hinshaw, WMAP3\] code this particular facet of the method will become extremely important in future analyses.
We previously presented [@Auld] a new method that combined all of the advantages of [Pico]{} but in a simpler and more readily expandable form by training neural networks. The resulting algorithm is called [CosmoNet]{} and has some considerable additional benefits in terms of the scalability, accuracy and computational memory requirements. In addition, the training method we employ is sufficiently general and simple to apply that it allows the end user to generate their own trained nets over any chosen cosmological model. In this paper we extend the method to include more generic (non-flat) cosmological models, interpolations over matter transfer functions and two large scale structure likelihoods in addition to the suite of CMB power spectra and the WMAP3 likelihood. Additionally we extend the $\ell$ range of our CMB spectra interpolation to $\ell_{\rm max}=2000$. In Sec. \[sec:nn\] we briefly describe neural networks. In Sec. \[sec:results\] we describe the [CosmoNet]{} algorithm and training efficiency. In Sec. \[sec:cosmo\_param\] we present cosmological parameter estimates using the trained networks implemented as [CosmoNet]{}. In Sec. \[sec:10param\], we apply the [CosmoNet]{} training algorithm to a 10 parameter cosmological model, training the CMB power spectra and matter transfer functions and producing parameter estimates. Our discussions and conclusions are presented in Sec. \[sec:discuss\].
Neural network interpolation {#sec:nn}
============================
Neural networks are a methodology for computing loosely based around the structures found in animal brains. They consist of a number of interconnected processors called neurons. The neurons process information separately and pass information to one another via connections. Well-designed networks are able to ‘learn’ from training data and are able to make predictions when presented with new, possible incomplete, information. For an introduction to the science of neural networks the reader is directed to @Bailer.
Multilayer perceptron networks {#sec:mlp}
------------------------------
The perceptron [@Rosenblatt] is the simplest type of feed-forward neural network. It maps an input vector $\mathbf{x} \in \Re^n$ to a scalar output $f(\mathbf{x};\mathbf{w},\theta)$ via $$f(\mathbf{x};\mathbf{w},\theta) = \sum_i w_{i} x_i + \theta,
\label{eq:perceptron}$$ where $\{w_{i}\}$ and $\theta$ are the parameters of the perceptron, called the ‘weights’ and ‘bias’ respectively.
Multilayer perceptron neural networks (MLPs) are a type of feed-forward network composed of a number of ordered layers of perceptron neurons that pass scalar messages from one layer to the next. In this paper, we will work with 3-layer MLPs only. They consist of an input layer, a hidden layer and an output layer (Fig. \[fig:nn\]). In such a network, the outputs of the nodes in the hidden and output layers take the form $$\begin{aligned}
\mbox{hidden layer:} & h_j=g^{(1)}(f_j^{(1)}); &
f_j^{(1)} = \sum_l w^{(1)}_{jl}x_l +
\theta_j^{(1)}, \\
\mbox{output layer:} & y_i=g^{(2)}(f_i^{(2)}); & f_i^{(2)} =
\sum_j w^{(2)}_{ij}h_j + \theta_i^{(2)},\end{aligned}$$ where the index $l$ runs over input nodes, $j$ runs over hidden nodes and $i$ runs over output nodes. The functions $g^{(1)}$ and $g^{(2)}$ are called activation functions and are chosen to be bounded, smooth and monotonic. In this paper, we use $g^{(1)}(x)=\tanh x$ and $g^{(2)}(x)=x$, where the non-linear nature of the former is a key ingredient in constructing a viable network.
![\[fig:nn\] An example of a 3-layer neural network with seven input nodes, 3 nodes in the hidden layer and five output nodes. Each line represents one weight.](PLOTS/net.eps){width="0.5\linewidth"}
The weights $\mathbf{w}$ and biases $\mathbf{\theta}$ are the quantities we wish to determine, which we denote collectively by $\mathbf{a}$. As these parameters vary, a very wide range of non-linear mappings between the inputs and outputs are possible. In fact, according to a ‘universal approximation theorem’ (Leshno et al. 1993), a standard multilayer feed-forward network with a locally bounded piecewise continuous activation function can approximate any continuous function to [*any*]{} degree of accuracy if (and only if) the network’s activation function is not a polynomial. This result applies when activation functions are chosen apriori and held fixed as $\mathbf{a}$ varies. Accuracy increases with the number in the hidden layer and the above theorem tells us we can always choose sufficient hidden nodes to produce any accuracy. Since the mapping from cosmological parameter space to the space of CMB power spectra (and WMAP3 likelihood) is known to be continuous, a 3-layer MLP with an appropriate choice of activation function is an excellent candidate model for the replacement of the forward model provided by the CAMB package (and WMAP3 likelihood code).
The activation functions act as basic building blocks of non-linearity in a neural network model and should be as simple as possible. Additionally, the [MemSys]{} routines used in training (described below) require derivative information and so they should be differentiable. The universal approximation theorem thus motivates us to choose a monotonic (for simplicity), bounded and differentiable function that is not a polynomial and we choose the $\tanh$ function. Of course, this could be replaced by another such function, such as the sigmoid function, but the interpolation results would be almost identical.
Network training {#sec:training}
----------------
Let us consider building an empirical model of the [CAMB]{} mapping using a 3-layer MLP as described above (a model of the different likelihood codes can be constructed in an analogous manner). The number of nodes in the input layer will correspond to the number of cosmological parameters, and the number in the output layer will be the number of uninterpolated $C_\ell$ values output by [CAMB]{}. A set of training data ${\cal{D}} =
\{\mathbf{x}^{(k)},\mathbf{t}^{(k)}\}$ is provided by [CAMB]{} (the precise form of which is described later) and the problem now reduces to choosing the appropriate weights and biases of the neural network that best fit this training data.
As the [CAMB]{} mapping is exact, this is a deterministic problem, not a probabilistic one. We therefore wish to choose network parameters $\mathbf{a}$ that minimise the ‘error’ term $\chi^2(\mathbf{a})$ on the training set given by $$\chi^2(\mathbf{a}) = \frac{1}{2}\sum_k
\sum_i\left[t^{(k)}_i-y_i(\mathbf{x}^{(k)};\mathbf{a})\right]^2.$$ This is, however, a highly non-linear, multi-modal function in many dimensions whose optimisation poses a non-trivial problem. Despite the deterministic nature of the problem we use an extension of a Bayesian method provided by the [MemSys]{} package (Gull & Skilling 1999).
The [MemSys]{} algorithm considers the parameters $\mathbf{a}$ of the network to be probabilistic variables with prior probability distribution proportional to $\exp(-\alpha S(\mathbf{a}))$, where $S(\mathbf{a})$ is the positive-negative entropy functional (@Gull; @HobsonLasenby) and $\alpha$ is considered a hyper-parameter of the prior. The variable $\alpha$ sets the scale over which variations in $\mathbf{a}$ are expected, and is chosen to maximise its marginal posterior probability. Its value is inversely proportional to the standard deviation of the prior. For fixed $\alpha$, the log-posterior is thus proportional to $- \chi^2(\mathbf{a}) + \alpha
S(\mathbf{a})$. For each choice of $\alpha$ there is a solution $\hat{\mathbf{a}}$ that maximises the posterior. As $\alpha$ varies, the set of solutions $\hat{\mathbf{a}}$ is called the ‘maximum-entropy trajectory’. We wish to find the maximum of $-\chi^2$ which is the solution at the end of the trajectory where $\alpha=0$. It is difficult to recover results for $\alpha \neq \infty$ (for large $\alpha$ the solution is found at the maximum of the prior) when starting with a result that lies far from the trajectory. Thus for practical purposes, it is best to start from the point on the trajectory at $\alpha = \infty$ and iterate $\alpha$ downwards until either a Bayesian $\alpha$ is achieved, or in our deterministic case, $\alpha$ is sufficiently small that the posterior is dominated by $\chi^2$.
[MemSys]{} performs the algorithm using conjugate gradients at each step to converge to the maximum-entropy trajectory. The required matrix of second derivatives of $\chi^2$ is approximated using vector routines only. This avoids the need for the $O(N^3)$ operations required to perform exact calculations, that would be impractical for large problems. The application of [MemSys]{} to the problem of network training allows for the fast efficient training of relatively large network structures on large data sets that would otherwise be difficult to perform in a useful time-frame. The [MemSys]{} algorithms are described in greater detail in (Gull & Skilling 1999).
Results {#sec:results}
=======
We will demonstrate the approach of neural network training to cosmology by attempting to replace the [CAMB]{} generator for the computation of CMB power spectra up to $\ell=2000$, in both temperature and polarisation $C^{\rm
TT}_\ell$, $C^{\rm TE}_\ell$, $C^{\rm EE}_\ell$ and the matter power spectrum $P(k)$. In general [CAMB]{} does not compute the CMB spectra $C_\ell$ values for all $\ell$, instead it computes a set of 60 values (up to $\ell = 2000$) chosen at appropriately spaced intervals to ensure coverage over the main acoustic peaks. A cubic spline interpolation is then carried out internally in [CAMB]{} to produce a full compliment of $C_\ell$’s at each $\ell$ to compare with the data. In the case of flat geometries these chosen $\ell$ values are predetermined and fixed, but in non-flat cases they shift, as the features of the acoustic peak structure do with $\Omega_k$. [CAMB]{} choses the most appropriate $\ell$ set to ensure the main features are covered. This creates a difficulty for our training algorithm, as one would normally wish to learn how a set of observables changes with input parameters. In this case the observables are actually changing. In fact, as we shall demonstrate, if we fix the set of $\ell$’s to those used for flat geometries, although we see some degredation in the accuracy of the spectra we see minimal impact in the marginalised posteriors.
In addition to the CMB power spectra, [CAMB]{} also generates matter power spectra for comparison with large scale structure data. We chose not to train over the spectrum directly, but instead trained the matter transfer function $T(k)$ which can be used to generate $P(k)$ given the primordial spectrum. This has the advantage of allowing us to evaluate a number of derived parameters such as the age of the universe and $\sigma_8$ without the need for further trained networks [^1]. Since the acoustic peak structures that appear in the CMB also appear in the matter spectra and transfer functions, [CAMB]{} also likes to set appropriate scales on which to generate the spectrum in non-flat cosmologies. In the same manner in which we dealt with the CMB spectra we have trained the networks over a predetermined, but sufficiently dense set of fixed $k$ values (for example [CAMB]{} normally generates the function at $\sim 75$ such values; in this interpolation we have used $\sim 175$). Again this approximation has led to minimal impact on the posteriors obtained.
Current likelihood codes, such as the newly released WMAP3, now require similar computation times to the generation of spectra. This trend is not likely to improve in the future as larger datasets come on stream. Thus it is crucial if we are to improve the efficiency of cosmological inference to have a combined approach for the spectra generation as well as likelihoods. In this paper we have exploited the same network training algorithm used for spectra to predict WMAP likelihoods as well as large scale structure likelihoods from the 2dF and SDSS surveys. Replacement of these codes and [CAMB]{} thus alleviates both major bottlenecks in cosmological Bayesian inference.
Training Data
-------------
In order to replace the [CAMB]{} package in codes such as [CosmoMC]{} we need to decide upon an appropriate region within which to train the networks. Inside this region the regression codes reliant on the trained networks would predict the appropriate spectra and outside this region [CAMB]{} would need to be called in the normal fashion. Choosing too large a region will lead to longer training periods and a reduction in the interpolation accuracy. Too small and [CAMB]{} would be called so often by the MCMC sampler as to render any performance increase negligible. Training was thus carried out by uniformly sampling a $4
\sigma$ confidence region as determined using a typical mixture of CMB and large scale structure experiments: WMAP3 + higher resolution CMB observations (ACBAR; @ACBAR; BOOMERang; @BOOMI; @BOOMII; @BOOMIII; CBI; @CBIII; @CBIII and the VSA; @Dickinson) and galaxy surveys; 2dF; [@Percival] and SDSS; [@SDSSII].
To test the approach we performed training over a non-flat cosmology parameterised by: ($\Omega_{\rm b} h^2$, $\Omega_{\rm cdm} h^2$, $\Omega_k$, $\theta$, $\tau$, $n_s$, $A_s$). The physical parameters ($\Omega_{\rm b} h^2$, $\Omega_{\rm cdm} h^2$, $\theta$, $\tau$) were converted back to cosmological parameters ($\Omega_{\rm b}$, $\Omega_{\rm cdm}$, $H_0$, $z_{ \rm re}$) and used as input to CAMB to produce the training set of CMB power spectra and matter transfer functions. Ultimately we aim to train networks over a sufficiently general cosmological model (see Sec. \[sec:10param\]) so that the user could perform any analysis over a subset of the trained parameters, setting unwanted variables to whatever fixed value they choose. In this way the flat model computed previously in @Auld is superceded by the results of this paper.
Training Efficiency
-------------------
To investigate training efficiency with training data set size and number of hidden network nodes, we evaluate the testing error as the maximum entropy trajectory is traversed. The training was conducted on a single $2.2$ GHz processor. Asymptotic behaviour was observed. In particular the testing error appears to settle down, after a period of logarithmic decrease. For a network of this size with this amount of data it appears disproportionate to train past $\sim 100$ hours, indeed adequate results can be obtained in just a few hours. It is expected some tiny increase in accuracy could be achieved for [*much*]{} longer training periods. However, this would be disproportionate, unless there is a significant error propagated through to the parameter constraints generated by these networks.
For each of the neural networks, training was then performed with 5000 training data but using different numbers of hidden nodes. Fig. \[fig:trainingnodes\], shows the testing error evolution for networks with 10, 25, 50, 100 and 250 nodes in the hidden layer, for the $Cl^{TT}_{\ell}$ spectrum. It can be seen that increasing the numbers of nodes past 50 does not increase accuracy, but does increase the training time. Similar experiments were then performed to determine the optimal size of training set. Again it was observed that for each neural network, increasing the training set size past a certain value did not increase accuracy, but did slow training. The optimal numbers of hidden nodes and training set sizes obtained for all networks are displayed in Table \[tb:cosmonetworks\].
We note that in @Habib sub-percentage errors on the CMB spectra are achieved for a 6 parameter flat $\Lambda$CDM model over a much larger region of parameter space, using a Gaussian Process with just 128 training data. In this paper we have found that of order 1000 training data produce [*optimal*]{} results (for non-flat models) and we proceed on this basis. However, the reader should note that tests showed that [CosmoNet]{} also generated usable accuracies using 100’s rather than 1000’s of training data. We do not consider the use of more training data as a large overhead, however as the data need only be generated once, and [CosmoNet]{} training time scales linearly with data set size. We believe that the method presented in @Habib would become more accurate with more training data, but that training time may suffer, as the inversion of a matrix is needed that requires of order the cube of the data set size operations.
-- ------ ----
2000 50
2000 50
3000 50
-- ------ ----
: The optimal number of data, and hidden nodes in the neural network training.[]{data-label="tb:cosmonetworks"}
Training Results
----------------
Networks were trained on the optimal numbers of hidden nodes and training data for $\sim 100$ hours. The accuracy of each interpolated spectrum and likelihood was then evaluated on a test set of $10^4$ models drawn uniformly from the appropriate parameter hypercubes (see Fig. \[fig:spectraB\]). As discussed we would expect some error to be introduced in our interpolations for non-flat models owing to our use of a fixed (flat) set of $\ell$ values. We find a mean error of $\sim 5 \%$ of cosmic variance as compared to the $\sim 1 \%$ error found in our previous analysis of flat models [@Auld] which of course is still well below any possible experimental error. More importantly the 99 percentile errors are all comfortably below cosmic variance, showing that the networks will be usable even when analysing data from even a perfect experiment. A loss in accuracy is also observed for the matter transfer interpolation. Here we find a mean error of less than 0.2 %, representing a considerably larger drop in accuracy than with the CMB spectra. However 0.2 % still represents a small inaccuracy given the quality of current large scale structure datasets. The likelihood test set correlation coefficients were all $> 0.9999$ with errors of less than $0.2$ units close to the peak though with slightly larger deviations away from it.
Application to cosmological parameter estimation {#sec:cosmo_param}
================================================
To illustrate the usefulness of [CosmoNet]{} in cosmological inference we perform an analysis of the WMAP 3-year TT, TE, EE data and 2dF and SDSS surveys using [CosmoMC]{} in three separate ways: (i) using [CAMB]{} power spectra and the WMAP3, 2dF and SDSS likelihood codes; (ii) using [CosmoNet]{} power spectra and the WMAP3, 2dF and SDSS likelihood codes; (iii) using the [CosmoNet]{} likelihood nets alone and (iv) using [CosmoNet]{} likelihoods with the [Bayesys]{} sampler. The resulting marginalised parameter constraints using each method are shown in Fig. \[fig:B\_wlike\], and are clearly very similar with mean parameter values differing by less than 1 % of the value computed using the standard approach (i).
To determine the speed up introduced by using [CosmoNet]{} spectra and likelihood interpolations, 4 parallel MCMC chains were run on Intel Itanium 2 processors at the COSMOS cluster (SGI Altix 3700) at DAMTP, Cambridge using the basic [CosmoMC]{} sampling package. The time required to generate $\sim 20000$ post burn-in MCMC samples was recorded using methods (i)–(iv) described above [^2]. The results (see Table \[table:timings\]) illustrate that using [CosmoNet]{} spectra one can obtain reasonable posterior distributions in roughly 8 hours on a *single* CPU per chain whereas using [CAMB]{} not only took between 2-3 times longer but required 3 additional CPUs per chain. Using [CosmoNet]{}s likelihood interpolations alone produced dramatic time savings, with accurate results in roughly 30 minutes.
Cosmologists have invested considerable time in developing samplers that have as efficient a proposal distribution as possible. In [ CosmoMC]{}, the multi-variate Gaussian proposal distribution has a covariance matrix that is regularly updated using statistics from the samples gathered up to that point. This does lead to a higher acceptance rate and a corresponding lower number of likelihood evaluations, but is computationally intensive in its own right. However, when using a [CosmoNet]{} likelihoods directly there is no need to reduce the number of likelihood calls. The process of updating the proposal distribution slows the task considerably, as can be seen when comparing times with the efficient, yet likelihood intensive [Bayesys]{} sampler [@Bayesys] via method (iv), computing the relevent posteriors in just 3 minutes.
The reader should also note from both our previous work [@Auld], and that of the 10 parameter models below, that the timings for parameter estimation are roughly independent of the number of model parameters used. Our regression algorithm is indifferent to the complexity of the input cosmology.
-- ------------- --------------- ----------------- ----------------
4 4 4 4
4 1 1 1
$>$ 16 hrs. $\sim 8$ hrs. $\sim 30$ mins. $\sim 3$ mins.
-- ------------- --------------- ----------------- ----------------
: Time required to gather $\sim$ 20,000 post burn-in MCMC samples using different combinations of [ CAMB]{}, [CosmoNet]{}, the experimental likelihood codes and [Bayesys]{}. Note that [CAMB]{} is parallelised in method (i) over 4 CPUs per chain, if a single processor were used these timings would approach 4 $\times$ that quoted.[]{data-label="table:timings"}
Towards a 10 dimensional parameter space {#sec:10param}
========================================
In @Auld we presented trained networks capable of replacing [CAMB]{} and experimental likelihood codes for a 6 parameter flat cosmology. In this paper we have shown that this method is easily extendable to the more arduous computational demands of a non-flat cosmology. To test the scaleability to even higher dimensions we now examine a 10 dimensional cosmology including, in addition to the basic 7 given in Sec. \[sec:results\]: the equation of state of dark energy, $w$, the neutrino mass fraction, $\nu$ and the tensor to scalar ratio, $r$.
Training efficiency was examined as per the 7 parameter model (see Table \[table:training\_10param\]) and it was found that little increase in the quantity of training data or training time was needed for optimum results for the CMB power spectra and matter transfer function. An accurate tracer of the scaling of the training algorithm is given by the number of network hidden nodes as this determines the amount of computational resource required. In this case we find that at worst a $50 \%$ increase in the number of hidden nodes is needed for a $\sim 40 \%$ rise in the number of parameter dimensions (going from 7 to 10). This represents slightly more than a linear rise in resources and demonstrates our algorithm is easily scaleable to even higher dimensions if necessary. The accuracy of interpolated CMB spectra and matter transfer functions did not decrease *at all* when compared to the 7 parameter interpolations (see Fig. \[fig:spectraC\]), suggesting that the largest source of error in our method is introduced by fixing the set of $\ell$ and $k$ values at their flat positions. Providing an accurate interpolation for the three likelihood surfaces was however problematic. Using the order of 1000 training data provides very sparse coverage of the 10 dimensional hypercube. For CMB power spectra and the matter transfer function this is not a problem since they vary smoothly over a limited dynamical range. For likelihoods however, the dynamical range is much larger and to obtain parameter constraints we need very good accuracy within a region having a volume of the order of that of a $1 \sigma$ hypersphere. This hypersphere has a volume over 400,000 times less than the 10 parameter $4 \sigma$ hypercube over which we performed the training. This suggests that much more training data would be required. A potential solution to this problem would be to train likelihood networks only in some region over which the likelihood value was within (say) 50 log units of the peak value. The shape of this region could be determined by a classification net that returns an output that predicts whether a point lies inside or outside the desired region. This method will be explored in a future publication.
-- ------ ----
2000 75
2000 50
-- ------ ----
: The required number of data, and hidden nodes in the neural network training for optimum performance in a 10 dimensional parameterisation.[]{data-label="table:training_10param"}
\
\
Marginalised posteriors obtained from [CosmoNet]{} spectra were found to be accurate to within a few $\%$ of those computed via [CAMB]{} (see Fig. \[fig:C\]), and took roughly 8 hours on a single CPUs per chain to calculate (see Table \[table:timings\_10param\]). [CAMB]{} however required more than 20 hours of computational time with parallelisation over a further 4 CPUs per chain.
![\[fig:C\] The one-dimensional marginalised posteriors on the cosmological parameters within the 10-parameter non-flat $\Lambda$CDM model including tensor modes, varying equation of state of dark energy and massive neutrinos comparing: [CAMB]{} power-spectra and WMAP3, 2dF and SDSS likelihoods (red) with [CosmoNet]{} power spectra and WMAP3, 2dF and SDSS likelihoods (black).](PLOTS/param10_spectra.ps){width="3"}
-- -------------- ----------------
4 4
4 1
$>$ 20 hours $\sim 8$ hours
-- -------------- ----------------
: Time required to gather $\sim$ 20,000 post burn-in MCMC samples using different combinations of [CAMB]{}, [CosmoNet]{} and the experimental likelihood codes. Note that [CAMB]{} is parallelised in method (i) over 4 CPUs per chain, if a single processor were used these timings would approach 4 $\times$ that quoted.[]{data-label="table:timings_10param"}
Discussion and Conclusions {#sec:discuss}
==========================
We have extended our method of accelerating the estimation of CMB and matter power transfer functions, WMAP, 2dF and SDSS likelihood evaluations based on training a multilayer perceptron neural network to more generic non-flat cosmologies. We have demonstrated that the use of trained neural networks such as [CosmoNet]{} can replace the bulk of computational effort required by cosmological evolution codes such as [CAMB]{} and experimental likelihood codes, like that of WMAP3. [CosmoNet]{} shares all the improvements made by [Pico]{} in terms of accuracy on both spectral interpolation and parameter constraints, but has now been scaled to a more generic 7 parameter non-flat cosmology. Furthermore, although the training procedure requires the optimisation of a highly non-linear multi-dimensional function, the end user simply runs the [MemSys]{} package essentially as a ‘black box’. This means [CosmoNet]{} remains simple and efficient to train. We have found the biggest bottleneck in the procedure to be the generation of training and testing data using [CAMB]{}. Increasing the model complexity had limited impact on the necessary training time (all models taking about 100 hours to train) or interpolation accuracy. Moreover the increase in network hidden nodes was at worst linear with increasing parameter space. Thus we expect few resource difficulties in extending this method to even higher dimensions.
Although accurate likelihood interpolations in the 10 dimensional model interpolation are currently beyond the reach of our method, the corresponding CMB spectra and matter transfer functions *are* sufficiently accurate allowing a speed up over the standard performance of [CosmoMC]{}.
Finally, replacing the [CosmoMC]{} sampler entirely with [Bayesys]{} can produce further dramatic time savings of a factor of $\sim 10$, computing $\sim 20,000$ post burn-in samples in a few minutes on a single CPU.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
---------------
TA acknowledges a studentship from EPSRC. MB was supported by a Benefactors’ Scholarship at St. John’s College, Cambridge and an Isaac Newton Studentship. This work was conducted in cooperation with SGI/Intel utilising the Altix 3700 supercomputer at DAMTP Cambridge supported by HEFCE and PPARC. We thank S. Rankin and V. Treviso for their assistance.
\[lastpage\]
[^1]: A future goal is to train networks also over the transfer functions for CMB power spectra to achieve the same generality, but this involves substantial additional complications and will be explored in a subsequent publication.
[^2]: Note that [CAMB]{} was in fact parallelised over 3 additional processors per chain, therefore totalling 16 CPUs
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '[ The famous pinching problem says that on a compact simply connected $n$-manifold if its sectional curvature satisfies $K_{min} > \frac{1}{4}K_{max} > 0$, then the manifold is homeomorphic to the sphere. In \[8, problem 12\], S. T. Yau proposed the following problem: If we replace $K_{max}$ by the scalar curvature, can we deduce similar pinching theorems? In our present note we give an answer to this question in dimension $n = 4.$]{}'
author:
- Ezio de Araujo Costa
title: '**The scalar curvature and the biorthogonal curvature: A pinching problem**'
---
[**Mathematic subject classifications (2000): 53C25, 53C24. Key words: Scalar curvature, sectional curvature, biorthogonal curvature, isotropic curvature, 4-manifold**]{}\
\
Let $M = M^n$ be a connected differentiable $n$-manifold with scalar curvature $s$ and let $K$ be the sectional curvature. For each $x\in M$ lets consider $P_1, P_2$ two mutually orthogonal 2-planes in the tangent space $T_xM$. The [*biorthogonal (sectional) curvature* ]{} (see \[2\]) relative to $P_1$ and $P_2$ is the average $$K^\perp (
P_1, P_2) = \frac{K(P_1) + K(P_2)}{2}. \eqno [1.1]$$ If $n = 4$ and $P$ is a 2-plane in $T_xM$ we will write $$K^\perp (P) = \frac{K(P) + K(P^\perp) }{2}, \eqno [1.2]$$ where $P^\perp$ is the orthogonal complement of $P$ in $T_xM$. The sum of two sectional curvatures on two orthogonal planes appears in the works of W. Seeman \[6\] and M. H. Noronha \[5\]. In our presents article we consider manifolds $M$ of dimension four and the following functions:
$$K_1^\perp(x) = \textmd{min} \{K^\perp(P); P \textmd{ is a 2- plain in } T_xM \}, \eqno [1.3]$$
$$K_3^\perp(x) = \textmd{max}\{K^\perp (P); P \textmd{ is a 2- plain in } T_xM \}, \eqno [1.4]$$
$$K_2^\perp (x)= \frac{s(x)}{4} - K_1^\perp(x) - K_3^\perp(x). \eqno [1.5]$$
[**The biorthogonal curvature and the Weyl tensor** ]{}
Let $(M, g)$ be an oriented Riemannian 4-manifold. For each $x\in M$ the bundle of two-forms $\Lambda^2 $ of $M$ splits $\Lambda^2 = \Lambda^+ \bigoplus\Lambda^-$ into $\pm$-eigenspaces of the Hodge $\star$-operator: $\Lambda^\pm = \{\alpha \in \lambda^2 ; \star\alpha = \pm\alpha \}$. The Weyl tensor $W$ is an endomorphism of $\Lambda^2 $ such that $W = W^+\bigoplus W^-$, where $W^\pm : \Lambda^\pm
\longrightarrow \Lambda^\pm$ are self-adjoint with free traces and are called of the self-dual and anti-self-dual parts of $W$, respectively. Let $w_1^\pm \leq w_2^\pm \leq w_3^\pm$ be the eigenvalues of the tensors $W^\pm$, respectively. As was proved in \[2\], $$K_1^\perp - \frac{s}{12} = \frac{w_1^+ + w_1^-}{2}, \eqno [1.6]$$ $$K_2^\perp - \frac{s}{12} = \frac{w_2^+ + w_2^-}{2} \eqno [1.7]$$ and $$K_3^\perp - \frac{s}{12} = \frac{w_3^+ + w_3^-}{2} \eqno [1.8]$$
Based on a proposed question by Yau \[8, problem 12\], the authores of the article \[3, page 16\] proposed the following conjecture\
\
[*Let $(M, g)$ be a compact simply connected Riemannian n-manifold scalar curvature $s > 0$ and sectional curvature $K$. If $K > \frac{s}{n(n+2)}$ on $M$ then $M$ is diffeomorphic to the sphere $\mathbb{S}^4$.*]{}\
\
In dimension $n = 4$ we obtained the following\
\
[**Theorem 1**]{} - [*Let $(M,g)$ be a compact oriented 4-manifold with scalar curvature $s > 0$. Let $K_1^\perp$ and $K_3^\perp$ be the biorthogonal curvatures given by \[1.3\] and \[1.4\], respectively. If $K_1^\perp \geq \frac{s}{24}$ on $M$ or $K_3^\perp \leq \frac{s}{6}$ on $M$ then we have\
\
(1) $M$ is diffeomorphic to a connected sum $\mathbb{S}^4 \sharp(\mathbb{R} \times \mathbb{S}^3)/G_1\sharp ......\sharp(\mathbb{R} \times \mathbb{S}^3)/G_n$, where the $G_i$ are discrete subgroup of the isometry group of $\mathbb{R} \times \mathbb{S}^3)$ or\
\
(2) $(M, g)$ is conformal to a complex projective space $\mathbb{CP}^2$ with the Fubini-Study metric or\
\
(3) The universal covering of $M$ is isometric to product $\mathbb{R} \times N^3$, where $N^3$ is diffeomorphic to $\mathbb{S}^3$.* ]{}\
\
[**Remark 1.1**]{}- Compare the above Theorem 1 to Theorem 1.1 in \[3\] and the Conjecture A in page 17 of \[3\].
[**Proof of Theorem 1**]{}
Let $(M, g)$ be a compact oriented Riemannian 4-manifold. It is known that $M$ has nonnegative isotropic curvature if $w_3^\pm \leq s/6$ (see \[4\]), where $w_3^\pm$ are the largest eigenvalues of $W^\pm$, respectively. Equation \[1.6\] and the condition $K_1^\perp \geq \frac{s}{24}$ implies that $ w_1^+ \geq w_1^+ + w_1^- \geq -s/12$ and so $w_3^+ = -w_1^+ -w_2^+ \leq -2w_1^+ \leq s/6$. Similarly, $w_3^- \leq s/6$ and this proves that $M$ has nonnegative isotropic curvature. Notice that the condition $K_3^\perp \leq \frac{s}{6}$ also implies that $M$ has nonnegative isotropic curvature. On the other hand, it is easy to see that if $M = M_1^2 \times M_2^2$ then $M$ has $K_1^\perp = K_2^\perp = 0$ which contradicts the initial hypotheses. So, this proves that if $M$ is reducible then the universal covering of $M$ is isometric to product $\mathbb{R} \times N^3$, where $N^3$ is diffeomorphic to sphere $\mathbb{S}^3$. If $M$ is irreducible then the Theorem 1 follows from principal results in \[7\] and \[1\].
[1]{}
Chen, B.L., Tang, S. H. and Zhu, X.P. [*Complete classification of compact four-manifolds with positive isotropic curvature.*]{} J. Diff. Geom., [**91**]{}(2012), 41-80. Costa, E. A. [*A modified Yamabe invariant and a Hopf conjecture*]{}. arXiv: 1207.7107v1 \[math. DG\] 30 Jul 2012. Gu, Juan-Ru and Xu, Hong-Wei. [*The sphere theorems for manifolds with positive scalar curvature*]{}. arXiv: 11022424v1 \[math DG\] 11 Feb 2011 Micallef, M. and Moore, J. [*Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes.*]{} Ann. of Math.(2)[**127**]{} (1994), no.1, 3-12. Noronha, M. H. [*Positively curved 4-manifolds and the nonnegativity of isotropic curvatures*]{}. Michigan Math. J. [**44**]{} (1997),211-229 Seaman, W. [*Orthogonally pinched curvature tensors and applications*]{}. Math. Scand. [**69**]{} (1991), 5-14 Seshadri, H. [*Manifolds with nonnegative isotropic curvature.*]{} Comm. Anal. Geom. [**4**]{} (2009), 621-635. Yau, S. T. [*Open problems in geometry.*]{} Differential Geometry: Partial differential equations on manifolds. Proc. Symp. Pure Math. [bf 54]{}, Part 1, Amer. Math. Soc. (1993), 1-28.
Author’s address: Mathematics Department, Federal University of Bahia,\
zipcode: 40170110- Salvador -Bahia-Brazil\
Author’s email: [email protected] or [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a framework for learning disentangled and interpretable jointly continuous and discrete representations in an unsupervised manner. By augmenting the continuous latent distribution of variational autoencoders with a relaxed discrete distribution and controlling the amount of information encoded in each latent unit, we show how continuous and categorical factors of variation can be discovered automatically from data. Experiments show that the framework disentangles continuous and discrete generative factors on various datasets and outperforms current disentangling methods when a discrete generative factor is prominent.'
author:
- |
Emilien Dupont\
Schlumberger Software Technology Innovation Center\
Menlo Park, CA, USA\
`[email protected]`\
bibliography:
- 'biblio.bib'
title: Learning Disentangled Joint Continuous and Discrete Representations
---
Introduction
============
Disentangled representations are defined as ones where a change in a single unit of the representation corresponds to a change in single factor of variation of the data while being invariant to others (@bengio2013representation). For example, a disentangled representation of 3D objects could contain a set of units each corresponding to a distinct generative factor such as position, color or scale. Most recent work on learning disentangled representations has focused on modeling continuous factors of variation (@higgins2016beta [@kim2018disentangling; @chen2018isolating]). However, a large number of datasets contain inherently discrete generative factors which can be difficult to capture with these methods. In image data for example, distinct objects or entities would most naturally be represented by discrete variables, while their position or scale might be represented by continuous variables.
Several machine learning tasks, including transfer learning and zero-shot learning, can benefit from disentangled representations (@lake2017building). Disentangled representations have also been applied to reinforcement learning (@higgins2017darla) and for learning visual concepts (@higgins2017scan). Further, in contrast to most representation learning algorithms, disentangled representations are often interpretable since they align with factors of variation of the data. Different approaches have been explored for semi-supervised or supervised learning of factored representations (@kulkarni2015deep [@whitney2016understanding; @yang2015weakly; @reed2014learning]). These approaches achieve impressive results but either require knowledge of the underlying generative factors or other forms of weak supervision. Several methods also exist for unsupervised disentanglement with the two most prominent being InfoGAN and $\beta$-VAE (@chen2016infogan [@higgins2016beta]). These frameworks have shown promise in disentangling factors of variation in an unsupervised manner on a number of datasets.
InfoGAN (@chen2016infogan) is a framework based on Generative Adversarial Networks (@goodfellow2014generative) which disentangles generative factors by maximizing the mutual information between a subset of latent variables and the generated samples. While this approach is able to model both discrete and continuous factors, it suffers from some of the shortcomings of Generative Adversarial Networks (GAN), such as unstable training and reduced sample diversity. Recent improvements in the training of GANs (@arjovsky2017wasserstein [@gulrajani2017improved]) have mitigated some of these issues, but stable GAN training still remains a challenge (and this is particularly challenging for InfoGAN as shown in @kim2018disentangling). $\beta$-VAE (@higgins2016beta), in contrast, is based on Variational Autoencoders (@kingma2013auto [@rezende2014stochastic]) and is stable to train. $\beta$-VAE, however, can only model continuous latent variables.
In this paper we propose a framework, based on Variational Autoencoders (VAE), that learns disentangled continuous and discrete representations in an unsupervised manner. It comes with the advantages of VAEs, such as stable training, large sample diversity and a principled inference network, while having the flexibility to model a combination of continuous and discrete generative factors. We show how our framework, which we term JointVAE, discovers independent factors of variation on MNIST, FashionMNIST ([@xiao2017fashion]), CelebA (@liu2015faceattributes) and Chairs (@aubry2014seeing). For example, on MNIST, JointVAE disentangles digit type (discrete) from slant, width and stroke thickness (continuous). In addition, the model’s learned inference network can infer various properties of data, such as the azimuth of a chair, in an unsupervised manner. The model can also be used for simple image editing, such as rotating a face in an image.
Analysis of $\beta$-VAE {#beta-vae}
=======================
We derive our approach by modifying the $\beta$-VAE framework and augmenting it with a joint latent distribution. $\beta$-VAEs model a joint distribution of the data $\mathbf{x}$ and a set of latent variables $\mathbf{z}$ and learn continuous disentangled representations by maximizing the objective
$$\label{eq:1}
\mathcal{L}(\theta, \phi) = \mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}[\log p_{\theta}(\mathbf{x}|\mathbf{z})] - \beta D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x}) \ \| \ p(\mathbf{z}))$$
where the posterior or encoder $q_{\phi}(\mathbf{z}|\mathbf{x})$ is a neural network with parameters $\phi$ mapping $\mathbf{x}$ into $\mathbf{z}$, the likelihood or decoder $p_{\theta}(\mathbf{x}|\mathbf{z})$ is a neural network with parameters $\theta$ mapping $\mathbf{z}$ into $\mathbf{x}$ and $\beta$ is a positive constant. The loss is a weighted sum of a likelihood term $\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}[\log p_{\theta}(\mathbf{x}|\mathbf{z})]$ which encourages the model to encode the data $\mathbf{x}$ into a set of latent variables $\mathbf{z}$ which can efficiently reconstruct the data and a second term that encourages the distribution of the inferred latents $\mathbf{z}$ to be close to some prior $p(\mathbf{z})$. When $\beta=1$, this corresponds to the original VAE framework. However, when $\beta>1$, it is theorized that the increased pressure of the posterior $q_{\phi}(\mathbf{z}|\mathbf{x})$ to match the prior $p(\mathbf{z})$, combined with maximizing the likelihood term, gives rises to efficient and disentangled representations of the data (@higgins2016beta [@burgess2017beta]).
We can derive further insights by analyzing the role of the KL divergence term in the objective (\[eq:1\]). During training, the objective will be optimized in expectation over the data $\mathbf{x}$. The KL term then becomes (@makhzani2017pixelgan [@kim2018disentangling])
$$\begin{split}
\mathbb{E}_{p(\mathbf{x})}[D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x}) \ \| \ p(\mathbf{z}))] & = I(\mathbf{x}; \mathbf{z}) + D_{KL}(q(\mathbf{z}) \ \| \ p(\mathbf{z})) \\
& \geq I(\mathbf{x}; \mathbf{z}) \\
\end{split}$$
i.e., when taken in expectation over the data, the KL divergence term is an upper bound on the mutual information between the latents and the data (see appendix for proof and details). Thus, a mini batch estimate of the mean KL divergence is an estimate of the upper bound on the information $\mathbf{z}$ can transmit about $\mathbf{x}$. Penalizing the mutual information term improves disentanglement but comes at the cost of increased reconstruction error. Recently, several methods have been explored to improve the reconstruction quality without decreasing disentanglement (@burgess2017beta [@kim2018disentangling; @chen2018isolating; @gao2018auto]). @burgess2017beta in particular propose an objective where the upper bound on the mutual information is controlled and gradually increased during training. Denoting the controlled information capacity by $C$, the objective is defined as
$$\label{cap-increase}
\mathcal{L}(\theta, \phi) = \mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}[\log p_{\theta}(\mathbf{x}|\mathbf{z})] - \gamma | D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x}) \ \| \ p(\mathbf{z})) - C|$$
where $\gamma$ is a constant which forces the KL divergence term to match the capacity $C$. Gradually increasing $C$ during training allows for control of the amount of information the model can encode. This objective has been shown to improve reconstruction quality as compared to (\[eq:1\]) without reducing disentanglement (@burgess2017beta).
JointVAE Model
==============
We propose a modification to the $\beta$-VAE framework which allows us to model a joint distribution of continuous and discrete latent variables. Letting $\mathbf{z}$ denote a set of continuous latent variables and $\mathbf{c}$ denote a set of categorical or discrete latent variables, we define a joint posterior $q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x})$, prior $p(\mathbf{z},\mathbf{c})$ and likelihood $p_{\theta}(\mathbf{x}|\mathbf{z},\mathbf{c})$. The $\beta$-VAE objective then becomes
$$\mathcal{L}(\theta, \phi) = \mathbb{E}_{q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x})}[\log p_{\theta}(\mathbf{x}|\mathbf{z},\mathbf{c})] - \beta D_{KL}(q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x}) \ \| \ p(\mathbf{z},\mathbf{c}))$$
where the latent distribution is now jointly continuous and discrete. Assuming the continuous and discrete latent variables are conditionally independent[^1], i.e. $ q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x}) = q_{\phi}(\mathbf{z}|\mathbf{x})q_{\phi}(\mathbf{c}|\mathbf{x})$ and similarly for the prior $p(\mathbf{z},\mathbf{c}) = p(\mathbf{z})p(\mathbf{c})$ we can rewrite the KL divergence as
$$D_{KL}(q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x}) \ \| \ p(\mathbf{z},\mathbf{c})) = D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x}) \ \| \ p(\mathbf{z})) +
D_{KL}(q_{\phi}(\mathbf{c}|\mathbf{x}) \ \| \ p(\mathbf{c}))$$
i.e. we can separate the discrete and continuous KL divergence terms (see appendix for proof). Under this assumption, the loss becomes $$\mathcal{L}(\theta, \phi) = \mathbb{E}_{q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x})}[\log p_{\theta}(\mathbf{x}|\mathbf{z},\mathbf{c})] - \beta D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x}) \ \| \ p(\mathbf{z})) - \beta
D_{KL}(q_{\phi}(\mathbf{c}|\mathbf{x}) \ \| \ p(\mathbf{c}))$$
In our initial experiments, we found that directly optimizing this loss led to the model ignoring the discrete latent variables. Similarly, gradually increasing the channel capacity as in equation (\[cap-increase\]) leads to the model assigning all capacity to the continuous channels. To overcome this, we split the capacity increase: the capacities of the discrete and continuous latent channels are controlled separately forcing the model to encode information both in the discrete and continuous channels. The final loss is then given by $$\label{final-loss}
\mathcal{L}(\theta, \phi) = \mathbb{E}_{q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x})}[\log p_{\theta}(\mathbf{x}|\mathbf{z},\mathbf{c})] - \gamma | D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x}) \ \| \ p(\mathbf{z})) - C_z| - \gamma |
D_{KL}(q_{\phi}(\mathbf{c}|\mathbf{x}) \ \| \ p(\mathbf{c})) - C_c|$$
where $C_z$ and $C_c$ are gradually increased during training.
Parametrization of continuous latent variables
----------------------------------------------
As in the original VAE framework, we parametrize $q_{\phi}(\mathbf{z}|\mathbf{x})$ by a factorised Gaussian, i.e. $q_{\phi}(\mathbf{z}|\mathbf{x})=\prod_{i} q_{\phi}(z_i|\mathbf{x})$ where $q_{\phi}(z_i|\mathbf{x})=\mathcal{N}(\mu_i, \sigma_{i}^{2})$ and let the prior be a unit Gaussian $p(\mathbf{z})=\mathcal{N}(0,I)$. $\boldsymbol{\mu}$ and $\boldsymbol{\sigma^2}$ are both parametrized by neural networks.
Parametrization of discrete latent variables
--------------------------------------------
Parametrizing $q_{\phi}(\mathbf{c}|\mathbf{x})$ is more difficult. Since $q_{\phi}(\mathbf{c}|\mathbf{x})$ needs to be differentiable with respect to its parameters, we cannot parametrize $q_{\phi}(\mathbf{c}|\mathbf{x})$ by a set of categorical distributions. Recently, @maddison2016concrete and @jang2016categorical proposed a differentiable relaxation of discrete random variables based on the Gumbel Max trick ([@gumbel1954statistical]). If $c$ is a categorical variable with class probabilities $\alpha_1, \alpha_2, ..., \alpha_n$, then we can sample from a continuous approximation of the categorical distribution, by sampling a set of $g_k \sim \text{Gumbel}(0, 1)$ i.i.d. and applying the following transformation
$$y_k = \frac{\exp((\log \alpha_k + g_k)/\tau)}{\sum_i \exp((\log \alpha_i + g_i)/\tau)}$$
where $\tau$ is a temperature parameter which controls the relaxation. The sample $\mathbf{y}$ is a continuous approximation of the one hot representation of $\mathbf{c}$. The relaxed discrete distribution is called a Concrete or Gumbel Softmax distribution and is denoted by $g(\boldsymbol{\alpha})$ where $\boldsymbol{\alpha}$ is a vector of class probabilities.
We can parametrize $q_{\phi}(\mathbf{c}|\mathbf{x})$ by a product of independent Gumbel Softmax distributions, $q_{\phi}(\mathbf{c}|\mathbf{x})=\prod_{i} q_{\phi}(c_i|\mathbf{x})$ where $q_{\phi}(c_i|\mathbf{x})=g(\boldsymbol{\alpha}^{(i)})$ is a Gumbel Softmax distribution with class probabilities $\boldsymbol{\alpha}^{(i)}$. We let the prior $p(\mathbf{c})$ be equal to a product of uniform Gumbel Softmax distributions. This approach allows us to use the reparametrization trick (@kingma2013auto [@rezende2014stochastic]) and efficiently train the discrete model.
Architecture
------------
The final architecture of the JointVAE model is shown in Fig. \[architecture\]. We build the encoder to output the parameters of the continuous distribution $\boldsymbol{\mu}$ and $\boldsymbol{\sigma^2}$ and of each of the discrete distributions $\boldsymbol{\alpha^{(i)}}$. We then sample $z_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ and $c_i \sim g(\boldsymbol{\alpha}^{(i)})$ using the reparametrization trick and concatenate $\mathbf{z}$ and $\mathbf{c}$ into one latent vector which is passed as input to the decoder.
![JointVAE architecture. The input $\mathbf{x}$ is encoded by $q_{\phi}$ into the parameters of the latent distributions. Samples are drawn from each of the latent distributions using the reparametrization trick (indicated by dashed arrows on the diagram). The samples are then concatenated and decoded through $p_{\theta}$.[]{data-label="architecture"}](architecture.pdf){width="0.36\linewidth"}
Choice and sensitivity hyperparameters
--------------------------------------
The JointVAE loss in equation \[final-loss\] depends on the hyperparameters $\gamma$, $C_c$ and $C_z$. While the choice of these is ultimately empirical, there are various heuristics we can use to narrow the search. The value of $\gamma$, for example, is chosen so that it is large enough to maintain the capacity at the desired level (e.g. large improvements in reconstruction error should not come at the cost of breaking the capacity constraint). We found the model to be quite robust to changes in $\gamma$. As the capacity of a discrete channel is bounded, $C_c$ is chosen to be the maximum capacity of the channel, encouraging the model to use all categories of the discrete distribution. $C_z$ is more difficult to choose and is often chosen by experiment to be the largest value where the representation is still disentangled (in a similar way that $\beta$ is chosen as the lowest value where the representation is still disentangled in $\beta$-VAE).
Experiments {#experiments-section}
===========
We perform experiments on several datasets including MNIST, FashionMNIST, CelebA and Chairs. We parametrize the encoder by a convolutional neural network and the decoder by the same network, transposed (for the full architecture and training details see appendix). The code, along with all experiments and trained models presented in this paper, is available at <https://github.com/Schlumberger/joint-vae>.
### MNIST {#mnist .unnumbered}
Disentanglement results and latent traversals for MNIST are shown in Fig. \[mnist-disentanglement\]. The model was trained with 10 continuous latent variables and one discrete 10-dimensional latent variable. The model discovers several factors of variation in the data, such as digit type (discrete), stroke thickness, angle and width (continuous) in an unsupervised manner. As can be seen from the latent traversals in Fig. \[mnist-disentanglement\], the trained model is able to generate realistic samples for a large variety of latent settings. Fig. \[mnist-discrete-samples\] shows digits generated by fixing the discrete latent and sampling the continuous latents from the prior $p(\mathbf{z})=\mathcal{N}(0, 1)$, which can be interpreted as sampling from a distribution conditioned on digit type. As can be seen, the samples are diverse, realistic and honor the conditioning.
For a large range of hyperparameters we were not able to achieve disentanglement using the purely continuous $\beta$-VAE framework (see Fig. \[mnist-comparison\]). This is likely because MNIST has an inherently discrete generative factor (digit type), which $\beta$-VAE is unable to map onto a continuous latent variable. In contrast, the JointVAE approach allows us to disentangle the discrete factors while maintaining disentanglement of continuous factors. To the best of our knowledge, JointVAE is, apart from InfoGAN, the only framework which disentangles MNIST in a completely unsupervised manner and it does so in a more stable way than InfoGAN.
[0.32]{} ![Latent traversals of the model trained on MNIST with 10 continuous latent variables and 1 discrete latent variable. Each row corresponds to a fixed random setting of the latent variables and the columns correspond to varying a single latent unit. Each subfigure varies a different latent unit. As can be seen each of the varied latent units corresponds to an interpretable generative factor, such as stroke thickness or digit type.[]{data-label="mnist-disentanglement"}](mnist_slant.png "fig:"){width="1.0\linewidth"}
[0.32]{} ![Latent traversals of the model trained on MNIST with 10 continuous latent variables and 1 discrete latent variable. Each row corresponds to a fixed random setting of the latent variables and the columns correspond to varying a single latent unit. Each subfigure varies a different latent unit. As can be seen each of the varied latent units corresponds to an interpretable generative factor, such as stroke thickness or digit type.[]{data-label="mnist-disentanglement"}](mnist_stroke_thickness.png "fig:"){width="1.0\linewidth"}
[0.32]{} ![Latent traversals of the model trained on MNIST with 10 continuous latent variables and 1 discrete latent variable. Each row corresponds to a fixed random setting of the latent variables and the columns correspond to varying a single latent unit. Each subfigure varies a different latent unit. As can be seen each of the varied latent units corresponds to an interpretable generative factor, such as stroke thickness or digit type.[]{data-label="mnist-disentanglement"}](mnist_digit_type_thickness.png "fig:"){width="1.0\linewidth"}
[0.32]{} ![Latent traversals of the model trained on MNIST with 10 continuous latent variables and 1 discrete latent variable. Each row corresponds to a fixed random setting of the latent variables and the columns correspond to varying a single latent unit. Each subfigure varies a different latent unit. As can be seen each of the varied latent units corresponds to an interpretable generative factor, such as stroke thickness or digit type.[]{data-label="mnist-disentanglement"}](mnist_width.png "fig:"){width="1.0\linewidth"}
![Traversals of all latent dimensions on MNIST for JointVAE, $\beta$-VAE and $\beta$-VAE with controlled capacity increase (CC$\beta$-VAE). JointVAE is able to disentangle digit type from continuous factors of variation like stroke thickness and angle, while digit type is entangled with continuous factors for both $\beta$-VAE and CC$\beta$-VAE.[]{data-label="mnist-comparison"}](mnist-comparisons.pdf){width="0.8\linewidth"}
### FashionMNIST {#fashionmnist .unnumbered}
Latent traversals for FashionMNIST are shown in Fig. \[fashion-traversal\]. We also used 10 continuous and 1 discrete latent variable for this dataset. FashionMNIST is harder to disentangle as the generative factors for creating clothes are not as clear as the ones for drawing digits. However, JointVAE performs well and discovers interesting dimensions, such as sleeve length, heel size and shirt color. As some of the classes of FashionMNIST are very similar (e.g. shirt and t-shirt are two different classes), not all classes are discovered. However, a significant amount of them are disentangled including dress, t-shirt, trousers, sneakers, bag, ankle boot and so on (see Fig. \[fashion-discrete-samples\]).
[0.25]{} {width="1.0\linewidth"}
[0.25]{} {width="1.0\linewidth"}
[0.3]{} {width="1.0\linewidth"}
[0.32]{} ![Latent traversals of the model trained on CelebA. Each row corresponds to a fixed setting of the discrete latent variable and the columns correspond to varying a single continuous latent unit.[]{data-label="celeba-disentanglement"}](celeba_azimuth_paper.png "fig:"){width="1.0\linewidth"}
[0.32]{} ![Latent traversals of the model trained on CelebA. Each row corresponds to a fixed setting of the discrete latent variable and the columns correspond to varying a single continuous latent unit.[]{data-label="celeba-disentanglement"}](celeba_background_paper.png "fig:"){width="1.0\linewidth"}
[0.32]{} ![Latent traversals of the model trained on CelebA. Each row corresponds to a fixed setting of the discrete latent variable and the columns correspond to varying a single continuous latent unit.[]{data-label="celeba-disentanglement"}](celeba_age_paper.png "fig:"){width="1.0\linewidth"}
### CelebA {#celeba .unnumbered}
For CelebA we used a model with 32 continuous latent variables and one 10 dimensional discrete latent variable. As shown in Fig. \[celeba-disentanglement\], the JointVAE model discovers various factors of variation including azimuth, age and background color, while being able to generate realistic samples. Different settings of the discrete variable correspond to different facial identities. While the samples are not as sharp as those produced by entangled models, we can still see details in the images such as distinct facial features and skin tones (the trade-off between disentanglement and reconstruction quality is a well known problem which is discussed in @higgins2016beta [@burgess2017beta; @kim2018disentangling; @chen2018isolating]).
### Chairs {#chairs .unnumbered}
For the chairs dataset we used a model with 32 continuous latent variables and 3 binary discrete latent variables. JointVAE discovers several factors of variation such as chair rotation, width and leg style. Furthermore, different settings of the discrete variables correspond to different chair types and colors.
While there is a well defined discrete generative factor for datasets like MNIST and FashionMNIST, it is less clear what exactly would constitute a discrete factor of variation in datasets like CelebA and Chairs. For example, for CelebA, JointVAE maps various facial identities onto the discrete latent variable. However, facial identity is not necessarily discrete and it is possible that such a factor of variation could also be mapped to a continuous latent variable. JointVAE has a clear advantage in disentangling datasets where discrete factors are prominent (as shown in Fig. \[mnist-comparison\]) but when this is not the case using frameworks that only disentangle continuous factors may be sufficient.
Quantitative evaluation
-----------------------
We quantitatively evaluate our model on the dSprites dataset using the metric recently proposed by @kim2018disentangling. Since the dataset is generated from 1 discrete factor (with 3 categories) and 4 continuous factors, we used a model with 6 continuous latent variables and one 3 dimensional discrete latent variable. The results are shown in table \[quantitative-metric\]. Even though the discrete factor in this dataset is not prominent (in the sense that the different categories have very small differences in pixel space) our model is able to achieve scores close to the current best models. Further, as shown in Fig. \[quantitative-metric\], our model learns meaningful latent representations. In particular, for the discrete factor of variation, JointVAE is able to better separate the classes than other models.
Model Score
----------------- -------
**$\beta$-VAE** 0.73
**FactorVAE** 0.82
**JointVAE** 0.69
![*Left*: Disentanglement scores for various frameworks on the dSprites dataset. The scores are obtained by averaging scores over 10 different random seeds from the model with the best hyperparameters (removing outliers where the model collapsed to the mean). *Right*: Comparison of latent traversals on the dSprites dataset. There are 4 continuous factors and 1 discrete factor in the original dataset and only JointVAE is able to encode all information into 4 continuous and 1 discrete latent variables. Note that the final row of the JointVAE latent traversal corresponds to the discrete factor of dimension 3, which is why the patterns repeat with a period of 3.[]{data-label="quantitative-metric"}](dsprites-comparison.pdf){width="1.0\linewidth"}
Detecting disentanglement in latent distributions
-------------------------------------------------
As noted in Section \[beta-vae\], taken in expectation over data, the KL divergence between the inferred latents $q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x})$ and the priors, upper bounds the mutual information between the latent units and the data. Motivated by this, we can plot the KL divergence values for each latent unit averaged over a mini batch of data during training. As various factors of variation are discovered in the data, we would expect the KL divergence of the corresponding latent units to increase. This is shown in Fig. \[mnist-kl-training\]. As the capacities $C_z$ and $C_c$ are increased, the model is able to encode more and more factors of variation. For MNIST, the first factor to be discovered is digit type, followed by angle and width. This is likely because encoding digit type results in the largest reconstruction error reduction, followed by encoding angle and width and so on.
After training, we can also measure the KL divergence of each latent unit on test data and rank the latent units by their average KL values. This corresponds to ranking the latent units by how much information they are transmitting about $\mathbf{x}$. Fig. \[kl-order-mnist\] shows the ranked latent units for MNIST and Chairs along with a latent traversal of each unit. As can be seen, the latent units with large information content encode various aspects of the data while latent units with approximately zero KL divergence do not affect the output.
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[0.23]{} {width="1.0\linewidth"}
The inference network
---------------------
One of the advantages of JointVAE is that it comes with an inference network $q_{\phi}(\mathbf{z}, \mathbf{c}|\mathbf{x})$. For example, on MNIST we can infer the digit type on test data with 88.7% accuracy by simply looking at the value of the discrete latent variable $q_{\phi}(\mathbf{c}|\mathbf{x})$. Of course, this is completely unsupervised and the accuracy could likely be increased dramatically by using some label information.
Since we are learning several generative factors, the inference network can also be used to infer properties which we do not have labels for. For example, the latent unit corresponding to azimuth on the chairs dataset correlates well with the actual azimuth of unseen chairs. After training a model on the chairs dataset and identifying the latent unit corresponding to azimuth, we can test the inference network on images that were not used during training. This is shown in Fig. \[inferred-rotation\]. As can be seen, the latent unit corresponding to rotation infers the angle of the chair even though no labeled data was given (or available) for this task.
The framework can also be used to perform image editing or manipulation. If we wish to rotate the image of a face, we can encode the face with $q_{\phi}$, modify the latent corresponding to azimuth and decode the resulting vector with $p_{\theta}$. Examples of this are shown in Fig. \[face-manip\].
[0.48]{} {width="1.0\linewidth"}
[0.49]{} {width="1.0\linewidth"}
[0.37]{} ![Failure examples. (a) Background color is entangled with azimuth and hair length. (b) Various clothing items are entangled with each other.[]{data-label="failures"}](celeba_entangled.png "fig:"){width="1.0\linewidth"}
[0.37]{} ![Failure examples. (a) Background color is entangled with azimuth and hair length. (b) Various clothing items are entangled with each other.[]{data-label="failures"}](fashion_entangled.png "fig:"){width="1.0\linewidth"}
Robustness and sensitivity to hyperparameters
---------------------------------------------
While our framework is robust with respect to different architectures and optimizers, it is, like most frameworks for unsupervised disentanglement, fairly sensitive to the choice of hyperparameters (all hyperparameters needed to reproduce the results in this paper are given in the appendix). Even with a good choice of hyperparameters, the quality of disentanglement may vary based on the random seed. In general, it is easy to achieve some degree of disentanglement for a large set of hyperparameters, but achieving complete clean disentanglement (e.g. perfectly separate digit type and other generative factors) can be difficult. It would be interesting to explore more principled approaches for choosing the latent capacities and how to increase them, but we leave this for future work. Further, as mentioned in Section \[experiments-section\], when a discrete generative factor is not present or important, the framework may fail to learn meaningful discrete representations. We have included some failure examples in Fig. \[failures\].
Conclusion
==========
We have proposed JointVAE, a framework for learning disentangled continuous and discrete representations in an unsupervised manner. The framework comes with the advantages of VAEs such as stable training and large sample diversity while being able to model complex jointly continuous and discrete generative factors. We have shown that JointVAE disentangles factors of variation on several datasets while producing realistic samples. In addition, the inference network can be used to infer unlabeled quantities on test data and to edit and manipulate images.
In future work, it would be interesting to combine our approach with recent improvements of the $\beta$-VAE framework, such as FactorVAE (@kim2018disentangling) or $\beta$-TCVAE (@chen2018isolating). Gaining a deeper understanding of how disentanglement depends on the latent channel capacities and how they are increased will likely provide insights to build more stable models. Finally, it would also be interesting to explore the use of other latent distributions since the framework allows the use of any joint distribution of reparametrizable random variables.
### Acknowledgments {#acknowledgments .unnumbered}
The author would like to thank Erik Burton, José Celaya, Suhas Suresha, Vishakh Hegde and the anonymous reviewers for helpful suggestions and comments that helped improve the paper.
Supplementary material {#supplementary-material .unnumbered}
======================
Proofs
======
Expectation of KL divergence and Mutual Information {#kl-mi-proof}
---------------------------------------------------
We can define the joint distribution of the data and the encoding distribution as $q(\mathbf{z},\mathbf{x}) = p(\mathbf{x})q_{\phi}(\mathbf{z}|\mathbf{x})$. The distribution of the latent variables is then given by $q(\mathbf{z})=\mathbb{E}_{p(\mathbf{x})}[q_{\phi}(\mathbf{z}|\mathbf{x})]$. We can now rewrite the KL divergence between the posterior and the prior taken in expectation over the data as
$$\begin{split}
\mathbb{E}_{p(\mathbf{x})}[D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x}) \ \| \ p(\mathbf{z}))] & = \mathbb{E}_{p(\mathbf{x})}\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}\Big[\log\frac{q_{\phi}(\mathbf{z}|\mathbf{x})}{p(\mathbf{z})}\Big]\\
& = \mathbb{E}_{q(\mathbf{z}, \mathbf{x})}\Big[\log\frac{q_{\phi}(\mathbf{z}|\mathbf{x})}{p(\mathbf{z})}\frac{q(\mathbf{z})}{q(\mathbf{z})}\Big] \\
& = \mathbb{E}_{q(\mathbf{z}, \mathbf{x})}\Big[\log\frac{q_{\phi}(\mathbf{z}|\mathbf{x})}{q(\mathbf{z})}\Big] + \mathbb{E}_{q(\mathbf{z}, \mathbf{x})}\Big[\log\frac{q(\mathbf{z})}{p(\mathbf{z})} \Big]\\
& = \mathbb{E}_{q(\mathbf{z}, \mathbf{x})}\Big[\log\frac{q(\mathbf{z}, \mathbf{x})}{q(\mathbf{z})p(\mathbf{x})}\Big] + \mathbb{E}_{q(\mathbf{z})}\Big[\log\frac{q(\mathbf{z})}{p(\mathbf{z})} \Big]\\
& = I(\mathbf{x}; \mathbf{z}) + D_{KL}(q(\mathbf{z}) \ \| \ p(\mathbf{z}))\\
& \geq I(\mathbf{x}; \mathbf{z})
\end{split}$$
where the mutual information is defined under the joint distribution of the data and the encoding distribution.
Splitting the discrete and continuous KL divergence terms {#split-proof}
---------------------------------------------------------
Assuming the continuous and discrete latent variables are conditionally independent, i.e. $ q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x}) = q_{\phi}(\mathbf{z}|\mathbf{x})q_{\phi}(\mathbf{c}|\mathbf{x})$ and similarly for the prior $p(\mathbf{z},\mathbf{c}) = p(\mathbf{z})p(\mathbf{c})$ we can rewrite the joint KL divergence as $$\begin{split}
D_{KL}(q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x}) \ \| \ p(\mathbf{z},\mathbf{c})) & = \mathbb{E}_{q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x})}[\log \frac{q_{\phi}(\mathbf{z},\mathbf{c}|\mathbf{x})}{p(\mathbf{z},\mathbf{c})}]\\
& = \mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}\mathbb{E}_{q_{\phi}(\mathbf{c}|\mathbf{x})}[\log \frac{q_{\phi}(\mathbf{z}|\mathbf{x})q_{\phi}(\mathbf{c}|\mathbf{x})}{p(\mathbf{z})p(\mathbf{c})}] \\
& = \mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}\mathbb{E}_{q_{\phi}(\mathbf{c}|\mathbf{x})}[\log \frac{q_{\phi}(\mathbf{z}|\mathbf{x})}{p(\mathbf{z})}] + \mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}\mathbb{E}_{q_{\phi}(\mathbf{c}|\mathbf{x})}[\log \frac{q_{\phi}(\mathbf{c}|\mathbf{x})}{p(\mathbf{c})}]\\
& = \mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}[\log \frac{q_{\phi}(\mathbf{z}|\mathbf{x})}{p(\mathbf{z})}] + \mathbb{E}_{q_{\phi}(\mathbf{c}|\mathbf{x})}[\log \frac{q_{\phi}(\mathbf{c}|\mathbf{x})}{p(\mathbf{c})}] \\
& = D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x}) \ \| \ p(\mathbf{z})) +
D_{KL}(q_{\phi}(\mathbf{c}|\mathbf{x}) \ \| \ p(\mathbf{c}))
\end{split}$$
Model architecture
==================
The architecture of the model is shown in the table. The non linearities in both the encoder and decoder are ReLU except for the output layer of the decoder which is a sigmoid.
[c c]{} **Encoder $q_{\phi}$** & **Decoder $p_{\theta}$**\
\
32 Conv $4 \times 4$, stride 2 & Linear $\text{latent dimension} \times 256$\
32 Conv $4 \times 4$, stride 2 & Linear $256 \times 64 * 4 * 4$\
64 Conv $4 \times 4$, stride 2 & 64 Conv Transpose $4 \times 4$, stride 2\
64 Conv $4 \times 4$, stride 2 & 32 Conv Transpose $4 \times 4$, stride 2\
Linear $64 * 4 * 4 \times 256$ & 32 Conv Transpose $4 \times 4$, stride 2\
Linear layers for parameters of each distribution & 3 Conv Transpose $4 \times 4$, stride 2\
For 64 by 64 images (Chairs, CelebA and dSprites) the architecture shown in the table was used. For 32 by 32 images (MNIST and FashionMNIST which were resized from 28 by 28) we used the same architecture with the last conv layer in the encoder and first in the decoder removed.
Training details
================
Parameters and training details for each model.
MNIST
-----
- Latent distribution: 10 continuous, 1 10-dimensional discrete
- Optimizer: Adam with learning rate 5e-4
- Batch size: 64
- Epochs: 100
- $\gamma$: 30
- $C_z$: Increased linearly from 0 to 5 in 25000 iterations
- $C_c$: Increased linearly from 0 to 5 in 25000 iterations
FashionMNIST
------------
- Latent distribution: 10 continuous, 1 10-dimensional discrete
- Optimizer: Adam with learning rate 5e-4
- Batch size: 64
- Epochs: 100
- $\gamma$: 100
- $C_z$: Increased linearly from 0 to 5 in 50000 iterations
- $C_c$: Increased linearly from 0 to 10 in 50000 iterations
Chairs
------
- Latent distribution: 32 continuous, 3 binary discrete
- Optimizer: Adam with learning rate 1e-4
- Batch size: 64
- Epochs: 100
- $\gamma$: 300
- $C_z$: Increased linearly from 0 to 30 in 100000 iterations
- $C_c$: Increased linearly from 0 to 5 in 100000 iterations
CelebA
------
- Latent distribution: 32 continuous, 1 10-dimensional discrete
- Optimizer: Adam with learning rate 5e-4
- Batch size: 64
- Epochs: 100
- $\gamma$: 100
- $C_z$: Increased linearly from 0 to 50 in 100000 iterations
- $C_c$: Increased linearly from 0 to 10 in 100000 iterations
dSprites
--------
- Latent distribution: 6 continuous, 1 3-dimensional discrete
- Optimizer: Adam with learning rate 5e-4
- Batch size: 64
- Epochs: 30
- $\gamma$: 150
- $C_z$: Increased linearly from 0 to 40 in 300000 iterations
- $C_c$: Increased linearly from 0 to 1.1 in 300000 iterations
Note that since the KL divergence between a categorical variable and a uniform categorical variable is bounded, the discrete capacity is clipped during training if $C_c$ exceeds the maximum capacity. Let $P$ denote a categorical random variable and let $Q$ be a uniform categorical variable, then
$$\begin{split}
D_{KL}(P \| Q) & = \sum^{n}_{i=1} p_i \log \frac{p_i}{q_i} = \sum^{n}_{i=1} p_i \log \frac{p_i}{1/n} = -H(P) + \log n \leq \log n
\end{split}$$
During training $C_c$ is then clipped as $C_c = \min (C_c, \log n)$.
Things that didn’t work
=======================
We experimented with several things which we found did not improve disentanglement of joint continuous and discrete representations.
- Modifying the latent distribution in $\beta$-VAE to include a joint Gaussian and Gumbel-Softmax distribution without changing the loss. This generally resulted in the model ignoring the discrete codes.
- Changing the loss function to have a higher $\beta$ on the continuous KL term and a lower $\beta$ on the discrete KL term. For a large combination of $\beta$, we either found the model to ignore the discrete latent codes or to produce representations where continuous factors were encoded in the discrete latent variables.
- In $\beta$-VAE, there is a larger weight on the KL term than in a traditional VAE model. In most VAE models with a Gumbel-Softmax latent variable, the KL divergence between the Gumbel-Softmax variables is approximated by the KL divergence between the corresponding categorical variables. We hypothesized that the approximation error might be worse in $\beta$-VAE, since there is a larger weight on the KL term. Unfortunately, there is no closed form expression of the KL divergence between two Gumbel-Softmax variables. We used various approximations of this, but most estimates had very high variance and impeded learning in the model.
Choice of discrete dimensions
=============================
As discussed in the main section of the paper, it is not clear what exactly would constitute a discrete factor of variation for a dataset like CelebA for example. As such, the choice is somewhat arbitrary: when using a 10 dimensional discrete latent variable, the model encodes 10 facial identities and when using more than 10 dimensions it encodes more identities. This was generally found to be quite robust, except when the number of discrete dimensions was exceedingly large (>100), when the model would start to encode e.g. facial angles in the discrete dimensions. Similarly, the reason we choose a 10 dimensional discrete latent variable for MNIST is because we know a priori that there are 10 types of digits. When choosing less than 10 discrete dimensions on MNIST, the model tends to fuse digit types which look similar into one discrete dimension. For example, 4 and 9 or 5 and 8 may correspond to one discrete dimension instead of being separated. When using more than 10 dimensions, the model tends to separate different writing styles of digits into separate dimensions, e.g. 2’s with and without a curl at the bottom or 7’s with and without a middle stroke may be encoded into different categories.
Comparison with InfoGAN
=======================
We include comparisons with InfoGAN which can also disentangle joint continuous and discrete factors. InfoGAN successfully disentangles digit type, from angle and width. However, width and stroke thickness remain entangled. Further, InfoGAN models are typically less stable to train.
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[0.24]{}
Latent traversals
=================
In all figures latent traversals of continuous variables are from $\Phi^{-1}(0.05)$ to $\Phi^{-1}(0.95)$ where $\Phi^{-1}$ is the inverse cdf of a unit normal. Latent traversals of discrete variables are from 1 to the number of dimensions of the variable.
[^1]: $\beta$-VAE assumes the data is generated by a fixed number of independent factors of variation, so *all* latent variables are in fact conditionally independent. However, for the sake of deriving the JointVAE objective we only require conditional independence between the continuous and discrete latents.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Previous studies on the rotation of Sun-like stars revealed that the rotational rates of young stars converge towards a well-defined evolution that follows a power-law decay. It seems, however, that some binary stars do not obey this relation, often by displaying enhanced rotational rates and activity. In the Solar Twin Planet Search program, we observed several solar twin binaries, and found a multiplicity fraction of $42\% \pm 6\%$ in the whole sample; moreover, at least three of these binaries (HIP 19911, HIP 67620 and HIP 103983) clearly exhibit the aforementioned anomalies. We investigated the configuration of the binaries in the program, and discovered new companions for HIP 6407, HIP 54582, HIP 62039 and HIP 30037, of which the latter is orbited by a $0.06$ M$_\odot$ brown dwarf in a 1-month long orbit. We report the orbital parameters of the systems with well-sampled orbits and, in addition, the lower limits of parameters for the companions that only display a curvature in their radial velocities. For the linear trend binaries, we report an estimate of the masses of their companions when their observed separation is available, and a minimum mass otherwise. We conclude that solar twin binaries with low-mass stellar companions at moderate orbital periods do not display signs of a distinct rotational evolution when compared to single stars. We confirm that the three peculiar stars are double-lined binaries, and that their companions are polluting their spectra, which explains the observed anomalies.'
author:
- |
Leonardo A. dos Santos,$^{1}$[^1] Jorge Meléndez,$^{1}$ Megan Bedell,$^{2}$ Jacob L. Bean,$^{2}$ Lorenzo Spina,$^{1}$ Alan Alves-Brito,$^{3}$ Stefan Dreizler,$^{4}$ Iván Ramírez,$^{5}$ and Martin Asplund$^{6}$\
$^{1}$Universidade de São Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Rua do Matão 1226, São Paulo\
05508-090, Brazil\
$^{2}$University of Chicago, Department of Astronomy and Astrophysics, 5640 S. Ellis Ave, Chicago, IL 60637, USA\
$^{3}$Universidade Federal do Rio Grande do Sul, Instituto de Física, Av. Bento Gonçalves 9500, Porto Alegre, RS, Brazil\
$^{4}$University of Göttingen, Institut für Astrophysik, Germany\
$^{5}$Tacoma Community College, Washington, USA\
$^{6}$The Australian National University, Research School of Astronomy and Astrophysics, Cotter Road, Weston, ACT 2611, Australia\
bibliography:
- 'bib.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Spectroscopic binaries in the Solar Twin Planet Search program: from substellar-mass to M dwarf companions'
---
\[firstpage\]
stars: fundamental parameters – stars: solar-type – stars: rotation – binaries: spectroscopic – binaries: visual
Introduction
============
It is known that at least half of the stars in the Galaxy are multiple systems containing two or more stars orbiting each other [@2001ASPC..229...91K; @2017ApJ...836..139F], thus in many surveys and large samples of stars, binaries are ubiquitous. This is in contrast with the Sun, which is a single star, and attempts to find a faint stellar companion orbiting it rendered no results thus far [e.g., @2014ApJ...781....4L]. Many studies avoid contamination by binaries in their samples, the main reasons being because we do not understand well how binaries evolve and how the presence of a companion affects the primary star. However, with the development of instruments with higher spatial and spectral resolution and coronagraphs, it is now possible to better probe the secondary component of such systems.
We have been carrying out a radial velocity planet search focused on solar twins using HARPS . The definition of solar twin we use is a star with stellar parameters inside the ranges $5777 \pm 100$ K, $4.44 \pm 0.10$ dex(cgs) and $0.0 \pm 0.1$ dex, respectively, for $T_{\mathrm{eff}}$, $\log{g}$ and \[Fe/H\]. In total, 81 solar twins[^2] were observed on HARPS. As part of our survey we previously identified 16 clear spectroscopic binaries (SB) . We report here the identification of four additional SBs (HIP 14501, HIP 18844, HIP 65708 and HIP 83276) and the withdrawal of HIP 43297 and HIP 64673, which are unlikely to host stellar-mass companions, bringing the number of solar twin SBs to 18. Most of these SBs are single-lined – they do not contain a second component in their spectral lines –, meaning that their companions are either faint stars or located outside the $\sim$$1\arcsec$ aperture of the HARPS spectrograph. We confirm that there are three solar twins with spectra contaminated by a relatively bright companion (see discussion in Section \[peculiar\]). In our sample there are an additional 18 visual binaries[^3] or multiple systems, of which HIP 6407 and HIP 18844 have wide companions [see table 5 in @2014AJ....147...86T] as well as the spectroscopic companions reported here.
In we saw that the single or visual binary solar twins display a rotational evolution that can described with a relation between stellar age $t$ and rotational velocity $v_{\mathrm{rot}}$ in the form of a power law plus a constant: $v_{\mathrm{rot}} = v_{\mathrm{f}} + m\ t^{-b}$, where $v_{\mathrm{f}}$, $m$ and $b$ are free parameters to be fit with observations. This relation is explained by loss of angular momentum due to magnetized winds , and the index $b$ reflects the geometry of the stellar magnetic field [@1988ApJ...333..236K]. There are at least two solar twin binaries that display enhanced rotational velocities – above $2\sigma$ from the expected – and activity for their ages: HIP 19911 and HIP 67620; if we consider the revised age for HIP 103983 (Spina et al., in preparation), it can also be considered a fast rotator for its age.
Besides the enhanced rotational velocities and higher chromospheric activity (; @F16sub), some of these binaries also display peculiar chemical abundances (; ). As pointed out by , the ultra-depletion of beryllium, which is observed on HIP 64150, could be explained by the interaction of the main star with the progenitor of the white dwarf companion. In addition to HIP 64150, the confirmed binaries HIP 19911 and HIP 67620 also display clearly enhanced $\mathrm{[Y/Mg]}$ abundances .
One interesting aspect about stars with enhanced activity and rotation is that these characteristics were hypothesized to be the result of dynamo action from close-in giant planets [see @2016ApJ...830L...7K and references therein]. In fact, some of our early results pointed out that the star HIP 68468, for which we inferred two exoplanets candidates , had an enhanced rotational velocity when compared to other solar twins of the same age. However, a more careful analysis showed that the enhancement was instead a contribution of macroturbulence. Another explanation for these enhancements is due to magnetic interactions with either a close-in or an eccentric giant planet [@2000ApJ...533L.151C], but recent results obtained by, e.g., and @2017MNRAS.465.2734M show that they cannot explain such anomalies.
In light of these intriguing results, we sought to better understand the nature of these solar twin multiple systems by studying their orbital parameters, and use them to search for explanations of the observed anomalies, especially stellar rotation. The orbital parameters can be estimated from the radial velocity data of the stars [see, e.g., @2010exop.book...15M hereafter MC10], with the quality of the results depending strongly on the time coverage of the data.
Radial velocities
=================
Our solar twins HARPS data[^4] are completely described in . Their radial velocities (RV) are automatically measured from the HARPS Data Reduction Software **(see Table \[HARPS\_rvs\])**, and the noise limit of the instrument generally remains around 1 m s$^{-1}$. In order to broaden the coverage of our RV data, we also obtained more datasets that were available in the literature and public databases, including the HARPS archival data for other programs.
The mass and other stellar parameters of the solar twins were estimated with high precision using the combined HARPS spectra and differential analysis owing to their similarity with the Sun . The ages for the solar twins were obtained using Yonsei-Yale isochrones [@2001ApJS..136..417Y] and probability distribution functions as described in @2013ApJ...764...78R and in . The full description and discussion of the stellar parameters of the HARPS sample are going to be presented in a forthcoming publication (Spina et al., in preparation).
The additional radial velocities data obtained from online databases and the literature are summarized in Table \[add\_rvs\]. These are necessary to increase the time span of the observations to include as many orbital phases as possible at the cost of additional parameters to optimize for (see Section \[short\]).
[ccc]{} Julian Date & RV & $\sigma_{\mathrm{RV}}$\
(d) & (km s$^{-1}$) & (km s$^{-1}$)\
\
2455846.750257 & 6.816873 & 0.001020\
2455850.716200 & 6.811186 & 0.000997\
2455851.710847 & 6.806352 & 0.000886\
2455852.703837 & 6.801079 & 0.001140\
2456164.849296 & 6.204053 & 0.001059\
2456165.853256 & 6.203359 & 0.001049\
2456298.564449 & 5.954045 & 0.001029\
\
2452937.683821 & 7.024814 & 0.000475\
2452940.727854 & 7.025289 & 0.000516\
2453001.575609 & 7.026322 & 0.000658\
2453946.941856 & 7.023707 & 0.000708\
\[HARPS\_rvs\]
[lll]{} Instrument/Program & References & Data available for (HIP numbers)\
CfA Digital Speedometers & @2002AJ....124.1144L & 65708\
ELODIE^a^ & @1996AandAS..119..373B [@2004PASP..116..693M] & 43297, 54582, 62039, 64150, 72043, 87769\
SOPHIE^b^ & @2011SPIE.8151E..15P & 6407, 43297, 54582, 62039, 64150, 87769\
Lick Planet Search & @2014ApJS..210....5F & 54582, 65708\
AAT Planet Search & @2015MNRAS.453.1439J & 18844, 67620, 73241, 79578, 81746\
Various & @2016AJ....152...46W & 67620\
HIRES/Keck RV Survey^c^ & @2017AJ....153..208B & 14501, 19911, 62039, 64150, 72043, 103983\
\
\
\
\[add\_rvs\]
Methods
=======
The variation of radial velocities of a star in binary or multiple system stems from the gravitational interaction between the observed star and its companions. For systems with stellar or substellar masses, the variation of radial velocities can be completely explained by the Keplerian laws of planetary motion. For the sake of consistency, we will use here the definitions of orbital parameters as presented in .
To completely characterize the orbital motion of a binary system from the measured radial velocities of the main star, we need to obtain the following parameters: the semi-amplitude of the radial velocities $K$, the orbital period $T$, the time of periastron passage $t_0$, the argument of periapse $\omega$ and the eccentricity $e$ of the orbit. In order to estimate the minimum mass $m \sin{i}$ of the companion and the semi-major axis $a$ of the orbit, we need to know the mass $M$ of the main star.
Due to their non-negative nature, the parameters $K$ and $T$ are usually estimated in logarithmic scale in order to eliminate the use of search bounds. Additionally, for orbits that are approximately circular, the value of $\omega$ may become poorly defined. In these cases, a change of parametrization may be necessary to better constrain them. @2013PASP..125...83E, for instance, suggest using $\sqrt{e} \cos{\omega}$ and $\sqrt{e} \sin{\omega}$ (which we refer to as the EXOFAST parametrization) instead of $\omega$ and $e$ to circumvent this problem, which also can help improve convergence time.
One issue that affects the radial velocities method is the contamination by stellar activity [see, e.g., @2016MNRAS.457.3637H]. This activity distorts the spectral lines [@2005oasp.book.....G], which in turn produces artificial RV variations that can mimic the presence of a massive companion orbiting the star. More active stars are expected to have RV variations with larger amplitudes and a shorter activity cycle period [@2011arXiv1107.5325L]. For most binaries in our sample, the contamination by activity in the estimation of orbital parameters is negligible; the cases where this is not applicable are discussed in detail.
Binaries with well-sampled orbits {#short}
---------------------------------
For binaries with orbital periods $T \lesssim 15$ yr, usually there are enough RV data measured to observe a complete phase. In these cases, the natural logarithm of the likelihood of observing radial velocities $\mathbf{y}$ on a specific instrument, given the Julian dates $\mathbf{x}$ of the observations, their uncertainties $\mathbf{\sigma}$ and the orbital parameters $\mathbf{p}_{\mathrm{orb}}$ is defined as:
$$\ln{p \left(\mathbf{y} \mid \mathbf{x},\mathbf{\sigma}, \mathbf{p}_{\mathrm{orb}} \right)} = -\frac{1}{2} \sum_n \left[ \frac{ \left(y_\mathrm{n} - y_{\mathrm{model}} \right)^2}{\sigma_\mathrm{n}^2} + \ln{ \left(2 \pi \sigma_\mathrm{n}^2 \right)} \right] \mathrm{,}
\label{likelihood}$$
where $y_\mathrm{n}$ are the RV datapoints, $y_{\mathrm{model}}$ are the model RV points for a given set of orbital parameters, and $\sigma_\mathrm{n}$ are the RV point-by-point uncertainties. The RV models are computed from Eq. 65 in :
$$v_\mathrm{r} = \gamma + K (\cos{(\omega + f)} + e \cos{\omega}) \mathrm{,}$$
where $f$ is the true anomaly, and $\gamma$ is the systemic velocity (usually including the instrumental offset). The true anomaly depends on $e$ and the eccentric anomaly $E$:
$$\cos{f} = \frac{(1 - e^2)}{1 - e \cos{E}} - 1 \mathrm{;}$$
the eccentric anomaly, in turn, depends on $T$, $t_0$ and time $t$ in the form of the so called Kepler’s equation:
$$E - e \sin{E} = \frac{2 \pi}{T} \left(t - t_0 \right) \mathrm{.}$$
Eq. \[likelihood\] is minimized using the Nelder-Mead algorithm implementation from `lmfit` [@2016ascl.soft06014N version 0.9.5] to obtain the best-fit orbital parameters to the observed data. Because different instruments have different instrumental offsets, the use of additional RV data from other programs require the estimation of an extra value of $\gamma$ for each instrument.
The uncertainties of the orbital parameters are estimated using `emcee`, an implementation of the Affine Invariant Markov chain Monte Carlo Ensemble sampler [@2013PASP..125..306F version 2.2.1] using flat priors for all parameters in both and EXOFAST parametrizations. These routines were implemented in the Python package `radial`[^5], which is openly available online. The uncertainties in $m \sin{i}$ and $a$ quoted in our results already take into account the uncertainties in the stellar masses of the solar twins.
Binaries with partial orbits {#methods_long_period}
----------------------------
For the binary systems with long periods (typically 20 years or more), it is possible that the time span of the observations does not allow for a full coverage of at least one phase of the orbital motion. In these cases, the estimation of the orbital parameters renders a number of possible solutions, which precludes us from firmly constraining the configuration of the system. Nevertheless, RV data containing a curvature or one inflection allows us to place lower limits on $K$, $T$ and $m \sin{i}$, whilst leaving $e$ and $\omega$ completely unconstrained. When the RV data are limited but comprise two inflections, it may be possible to use the methods from Section \[short\] to constrain the orbital parameters, albeit with large uncertainties.
For stars with very large orbital periods ($T \gtrsim 100$ yr), the variation of radial velocities may be present in the form of a simple linear trend. In these cases, it is still possible to obtain an estimate of the mass of the companion – a valuable piece of information about it: @1999PASP..111..169T describes a statistical approach to extract the sub-stellar companion mass when the only information available from radial velocities is the inclination of the linear trend, provided information about the angular separation of the system is also available. In this approach, we need to adopt reasonable prior probability density functions (PDF) for the eccentricity $e$, the longitude of periastron $\varpi$, phase $\phi$ and the inclination $i$ of the orbital plane. As in @1999PASP..111..169T, we adopt the following PDFs: $p(i) = \sin{i}$, $p(e) = 2e$ and flat distributions for $\varpi$ and $\phi$.
We sample the PDFs using `emcee`, 20 walkers and 10000 steps; the first 500 burn-in steps are discarded. From these samples, we compute the corresponding companion masses and its posterior distribution. This posterior usually displays a very strong peak and long tails towards low and high masses which can be attributed to highly unlikely orbital parameters (see Fig. \[64150\_pdf\] for an example). In our results, we consider that the best estimates for the companion masses are the central bin of the highest peak of the distribution in a histogram with log-space bin widths of about 0.145 dex(M$_\odot$).
When no adaptive optics (AO) imaging data are available for the stars with a linear trend in their RVs, the most conservative approach is to provide the minimum mass for the putative companion. In the case of a linear trend, the lowest mass is produced when $e = 0.5$, $\omega = \pi/2$ and $\sin{i} = 1$ [@2015ApJ...800...22F], yielding
$$\label{feng_eq}
m_{\mathrm{min}} \approx \left( 0.0164\ \mathrm{M}_{\mathrm{Jup}}\right) \left( \frac{\tau}{\mathrm{yr}} \right)^{4/3} \left| \frac{dv/dt}{\mathrm{m\ s}^{-1}\ \mathrm{yr}^{-1}} \right| \left( \frac{M}{\mathrm{M}_\odot} \right)^{2/3} \mathrm{,}$$
where $\tau$ is 1.25 multiplied by the time span of the radial velocities and $dv/dt$ is the inclination of the linear trend.
Results
=======
We discovered new, short-period companions for the stars HIP 6407 and HIP 30037 (see Fig. \[orbits\_new\]) and new long-period companions for HIP 54582 and HIP 62039, and updated or reproduced the parameters of several other known binaries that were observed in our program (see Figs. \[orbits\_updated\] and \[long\_period\_rvs\]). We briefly discuss below each star, pointing out the most interesting results, inconsistencies and questions that are still open about each of them. The orbital parameters of the binaries with well-sampled orbits in their RV data are presented in Table \[short\_params\] and the systems with partial orbits are reported in Tables \[curvature\_results\] and \[linear\_trend\_results\].
Withdrawn binary candidates
---------------------------
In , we showed that HIP 43297 had a rotational velocity $v \sin{i}$ higher than expected for its age. Moreover, its radial velocities had variations that hinted for one or more companions orbiting it. We carefully analyzed the RVs and concluded that the periodic ($T = 3.8$ yr) signal observed is highly correlated (Pearson $R = 0.893$) with the activity *S*-index of the star [@F16sub]. In addition, we tentatively fitted a linear trend to the combined RVs from HARPS, ELODIE and SOPHIE, and obtained an inclination of $4.53 \pm 0.04$ m s$^{-1}$ yr$^{-1}$, but further monitoring of the system is required to infer the presence of a long-period spectroscopic companion. The revised stellar age for HIP 43297 yields $1.85 \pm 0.50$ Gyr (Spina et al., in preparation), which explains the high rotational velocity and activity.
The solar twin HIP 64673 displays significant fluctuations in its radial velocities, but they do not correlate with its activity index; the data covers approximately 5 years of RV monitoring and displays an amplitude $> 20$ m s$^{-1}$. If confirmed to be caused by massive companions, the RV variations of both HIP 43297 and HIP 64673 suggest substellar masses for the most likely orbital configurations. These stars are, thus, removed from the binaries sample of the Solar Twin Planet Search program.
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[llccccccc]{} & & $K$ & $T$ & $t_0$ & $\omega$ & $e$ & $m\sin{i}$ & $a$\
& & (km s$^{-1}$) & (days) & (JD-$2.45E6$ days) & ($^o$) & & (M$_\odot$) & (AU)\
& & $2.614$ & $1852.3$ & $5076.7$ & $-57.1$ & $0.682$ & $0.119$ & $3.070$\
& & $\pm 0.084$ & $^{+3.3}_{-3.1}$ & $^{+1.1}_{-1.3}$ & $\pm 0.9$ & $^{+0.009}_{-0.010}$ & $\pm 0.002$ & $\pm 0.005$\
& & $7.707$ & $2074.15$ & $4627.4$ & $40.92$ & $0.8188$ & $0.313$ & $6.16$\
& & $\pm 0.007$ & $\pm 0.09$ & $\pm 0.1$ & $\pm 0.06$ & $\pm 0.0003$ & $\pm 0.002$ & $\pm 0.02$\
& & $4.246$ & $31.61112$ & $5999.413$ & $-133.60$ & $0.30205$ & $0.0610$ & $0.1971$\
& & $\pm 0.003$ & $\pm 0.00006$ & $\pm 0.001$ & $\pm 0.02$ & $\pm 0.00008$ & $\pm 0.0002$ & $\pm 0.0003$\
& & $5.754$ & $207.273$ & $-3675.8$ & $-137.9$ & $0.311$ & $0.170$ & $0.851$\
& & $\pm 0.007$ & $\pm 0.004$ & $\pm 0.3$ & $\pm 0.2$ & $\pm 0.002$ & $\pm 0.001$ & $\pm 0.001$\
& & $6.311$ & $3803.3$ & $4945.7$ & $145.10$ & $0.3428$ & $0.578$ & $5.50$\
& & $\pm 0.002$ & $\pm 0.4$ & $\pm 0.7$ & $\pm 0.07$ & $\pm 0.0002$ & $\pm 0.002$ & $\pm 0.01$\
& & $1.125$ & $6681.8$ & $879.2$ & $138.9$ & $0.3322$ & $0.1014$ & $7.216$\
& & $\pm 0.007$ & $\pm 1.5$ & $\pm 1.9$ & $\pm 0.1$ & $\pm 0.0003$ & $\pm 0.0002$ & $\pm 0.007$\
& & $1.987$ & $3246.5$ & $5623.8$ & $-2.49$ & $0.6644$ & $0.1079$ & $4.387$\
& & $\pm 0.001$ & $\pm 0.7$ & $\pm 0.6$ & $\pm 0.06$ & $\pm 0.0005$ & $\pm 0.0002$ & $\pm 0.003$\
& & $2.100$ & $10278$ & $6659$ & $-51.6$ & $0.50$ & $0.210$ & $9.8$\
& & $\pm 0.018$ & $^{+274}_{-247}$ & $\pm 11$ & $\pm 1.5$ & $\pm 0.01$ & $\pm 0.005$ & $\pm 0.2$\
\
\[short\_params\]
Solar twins with new companions
-------------------------------
**HIP 6407:** This is a known binary system located 58 pc away from the solar system , possessing a very low-mass (0.073 M$_\odot$) L2-type companion separated by $44.8\arcsec$ (2222 AU), as reported by @2015ApJ...802...37B [and references therein]. In this study, we report the detection of a new close-in low-mass companion with $m\sin{i} = 0.12$ M$_\odot$ on a very eccentric orbit ($e = 0.67$) with $a = 3$ AU and an orbital period of approximately 5 years. As expected, the long-period companion does not appear in the RV data as a linear trend.
**HIP 30037:** The most compact binary system in our sample, hosting a brown dwarf companion orbiting the main star with a period of 31 days. The high precision of its parameters owes to the wide time span of observations, which covered several orbits. This is one of the first detections of a close-in brown dwarf orbiting a confirmed solar twin[^6]. HIP 30037 is a very quiet star, displaying no excessive jitter noise in its radial velocities. We ran stellar evolution models with `MESA`[^7] [@2011ApJS..192....3P; @2015ApJS..220...15P] to test the hypothesis of the influence of tidal acceleration caused by the companion on a tight orbit, and found that, for the mass and period of the companion, we should expect no influence in the rotational velocity.
**HIP 54582:** *RV Curvature only*. There are no reports of binarity in the literature. The slight curvature in the RVs of this star is only visible when we combine the HARPS data and the Lick Planet Search archival data. Owing to the absence of an inflection point, the orbital parameters of this system are highly unconstrained. We found that an orbit with $e \approx 0.2$ produces the least massive companion and shortest orbital period ($m \sin{i} = 0.03$ M$_\odot$ and $T = 102$ yr).
**HIP 62039:** *Linear trend*. There are no reports of visually detected close-in ($\rho < 2\arcsec$) companions around it. This can be attributed to: i) low luminosity companion, which is possible if it is a white dwarf or a giant planet, and ii) unfavorable longitude of periapse during the observation windows. By using Eq. \[feng\_eq\], we estimate that the minimum mass of the companion is 19 M$_{\mathrm{Jup}}$.
The peculiar binaries {#peculiar}
---------------------
### HIP 19911 {#19911_results}
This is one of the main outlier stars in the overall sample of solar twins in regards to its rotation and activity, which are visibly enhanced for both the previous and revised ages (; @F16sub; Spina et al., in preparation). For the estimation of orbital parameters reported below, we used only the LCES HIRES/Keck radial velocities, because there are too few HARPS data points to justify the introduction of an extra source of uncertainties (the HARPS points are, however, plotted in Fig. \[orbits\_updated\] for reference). When using the HARPS data, although the solution changes slightly, our conclusions about the system remain the same.
The orbital solution of HIP 19911 renders a 0.31 M$_\odot$ companion in a highly eccentric orbit ($e = 0.82$, the highest in our sample), with period $T = 5.7$ yr. Visual scrutiny reveals what seems to be another signal with large amplitude in the residuals of this fit ($> 250$ m s$^{-1}$, see Fig. \[orbits\_updated\]); the periodogram of the residuals shows a very clear peak near the orbital period of the stellar companion.
The cross-correlation function (CCF) plots for the HARPS spectra of HIP 19911 display a significant asymmetry – longer tail in the blue side – for the observations between October 2011 and February 2012, which suggests that the companion is contaminating the spectra. Upon visual inspection of the archival HIRES spectra[^8] taken on 17 January 2014, which is when we expect the largest RV difference between the main star and its companion, we saw a clear contamination of the spectrum by the companion (see Fig. \[SBII\]). This contamination could explain the large residuals of the orbital solution, as it introduces noise to the measured radial velocities. The double-lines also explain the inferred high rotational velocity of HIP 19911, since they introduce extra broadening to the spectral lines used to measure rotation. The presence of a bright companion may also affect estimates of chemical abundances, which elucidates the yttrium abundance anomaly . The double-lined nature of this system is not observed on the HARPS spectra due to an unfavorable observation window.
Even at the largest RV separation, we did not detect the Li I line at 6707.75 Å in the HIRES spectrum of the companion. This is expected because M dwarf stars have deeper convection zones, which means they deplete lithium much faster than Sun-like stars. This leads us to conclude that estimates of Li abundance on solar twin binaries using this line do not suffer from strong contamination by their companions; consequently, age estimates with lithium abundances may be more reliable for such binaries than isochronal or gyro ages.
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Another observation conundrum for this system is that @2015ApJ...799....4R, using AO imaging without a coronagraph, reports the detection of a visual companion with orbital period $\sim 12.4$ yr (roughly twice the one we estimated), a lower eccentricity ($e = 0.1677$) and similar semi-major axis of the orbit ($a = 6.17$ AU, if we consider a distance of 30.6 pc). Moreover, @2014AJ....147...86T [hereafter T14] reports that this visual companion has $m = 0.85$ M$_\odot$. The most likely explanation is that the observations of @2015ApJ...799....4R did in fact detect the spectroscopic companion, but the coarse timing of the observations produced a larger period; the lower eccentricity could be explained by a strong covariance between $e$ and the inclination $i$. If $i$ is lower, that means the mass of the companion is significantly higher than $m \sin{i} = 0.316$ M$_\odot$, and that would explain the value obtained by . A companion with a mass as large as 0.85 M$_\odot$ would likely pollute the spectra of HIP 19911, which agrees with our observation that this is a SB II system. If confirmed, this prominent $\sim$0.85 M$_\odot$ red dwarf companion could explain the observed activity levels for HIP 19911, since red dwarf stars are expected to be more active than Sun-like stars.
### HIP 67620 {#67620_results}
This is a well-known binary and the target with the largest amount of RV data available (see Fig. \[orbits\_updated\]). Its orbital parameters have been previously determined by @2006ApJS..162..207A and more recently by @2015MNRAS.453.1439J and @2016AJ....152...46W. The orbital parameters we obtained are in good agreement with @2016AJ....152...46W. It has one of the most peculiar rotation rates from our sample (2.77 km s$^{-1}$ for an age of 7.18 Gyr), an enhanced chromospheric activity [@F16sub] and an anomalous \[Y/Mg\] abundance . The orbital period of the system is far too long for gravitational interaction to enhance the rotation of the main star through tidal acceleration, thus we should expect that they evolve similarly to single stars from this point of view.
High-resolution imaging of HIP 67620 revealed a companion with $V_{\mathrm{mag}} \approx 10$ and separations which are consistent with the spectroscopic companion (@2012AJ....143...42H; ). As explained by @2015MNRAS.453.1439J, the presence of a companion with $m > 0.55$ M$_\odot$ can produce contaminations to the spectra that introduce noise to the measured RVs; our estimate of $m \sin{i}$ for this system is 0.58 M$_\odot$. These results suggest that, similarly to HIP 19911 but to a lesser degree, the companion of HIP 67620 may be offsetting our estimates for rotational velocity, stellar activity, chemical abundances and isochronal age.
We were unable to discern double-lines in the HARPS spectra, likely resulting from unfavorable Doppler separations (observations range from February 2012 to March 2013). However, an analysis of the CCF of this star shows slight asymmetries in the line profiles of the HARPS spectra, which indicates a possible contamination by the companion. @2017ApJ...836..139F reported HIP 67620 as a double-lined binary using spectra taken at high-resolution ($R \approx 60$$,$$000$) in February 2014 and July 2015. As expected due to the short time coverage of the HARPS spectra, we did not see any correlation between the bisector inverse slope and the radial velocities of HIP 67620.
@2015MNRAS.453.1439J found an additional signal on the periodogram of HIP 67620 at 532 d, which could be fit with a 1 M${_{\mathrm{Jup}}}$ planet, bringing down the $rms$ of the fit by a factor of 2. However, we did not find any significant peak in the periodogram of the residuals of the radial velocities for HIP 67620.
### HIP 103983 {#103983_results}
The revised age for HIP 103983 ($4.9 \pm 0.9$ Gyr; Spina et al., in preparation) renders this system as an abnormally fast rotator ($3.38$ km s$^{-1}$) for its age. However, upon a careful inspection of the HARPS data obtained at different dates, we identified that the spectrum from 2015 July 27 displays clearly visible double-lines, albeit not as well separated as those observed in the HIRES spectra of HIP 19911 (see Fig. \[SBII\]). No other anomalies besides enhanced rotation were inferred for this system. The CCF plots of the HARPS spectra show clear longer tails towards the blue side for most observations.
In we reported distortions in the combined spectra of HIP 103983; this is likely a result from the combination of the spectra at orbital phases in which the Doppler separation between the binaries is large. Since the observing windows of the HARPS spectra of HIP 19911 and HIP 67620 do not cover large RV separations (see Fig. \[orbits\_updated\]), the same effect is not seen in the combined spectra of these stars. This effect also explains why HIP 103983 is an outlier in fig. 4 of .
Although we have limited RV data, the `emcee` simulations converge towards a well-defined solution instead of allowing longer periods, as these produce larger residuals. It is important, however, to keep monitoring the radial velocities of this system in order to confirm that the most recent data points are in fact a second inflection in the radial velocities. The residuals for the fit for the HIRES spectra are on the order of 100 m s$^{-1}$, which is likely a result from the contamination by a bright companion. reported a $0.91$ M$_\odot$ visual companion at a separation of $0.093 \arcsec$, which is consistent with the spectrocopic semi-major axis we estimated: $0.149 \arcsec$ for a distance of 65.7 pc .
Other binaries with updated orbital parameters
----------------------------------------------
Among the known binaries in the solar twins sample, five of them display curvature in their RV data which allows the estimation of limits for their orbital parameters (see Table \[curvature\_results\] and Fig. \[long\_period\_rvs\]). Some of the linear trend binaries observed in our HARPS Solar Twin Planet Search program are targets with large potential for follow-up studies. For the companions with visual detection, we were able to estimate their most likely mass (see Table \[linear\_trend\_results\]).
**HIP 14501:** *Linear trend.* Its companion is reported by @2014ApJ...781...29C as the first directly imaged T dwarf that produces a measurable doppler acceleration in the primary star. Using a low-resolution direct spectrum of the companion, @2015ApJ...798L..43C estimated a model-dependent mass of 56.7 M$_{\mathrm{Jup}}$. Using the HARPS and HIRES/Keck RV data and the observed separation of $1.653\arcsec$ [@2014ApJ...781...29C], we found that the most likely value of the companion mass is 0.043 M$_\odot$ (45 M$_{\mathrm{Jup}}$), which agrees with the mass obtained by @2015ApJ...798L..43C. The most recent HARPS data hints of an inflection point in the orbit of HIP 14501 B (see Fig. \[rv\_14501\]), but further RV monitoring of the system is necessary to confirm it.
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![Radial velocities of HIP 14501. The RV shift in the $y$-axes is arbitrary. Time is given in $\mathrm{JD} - 2.45 \times 10^{6}$ d.[]{data-label="rv_14501"}](images/14501_rvs.pdf){width="47.00000%"}
**HIP 18844:** *Linear trend.* It is listed as a multiple system containing a closer-in low-mass stellar companion (estimated 0.06 M$\odot$, which agrees with our most likely mass) and orbital period $T = 6.5$ yr . For the companion farther away, @2015MNRAS.453.1439J reported a minimum orbital period of $\sim 195$ yr and $m\sin{i} = 0.33$ M$_\odot$, with a separation of $29\arcsec$ in 1941 ($\sim$$750$ AU for a distance of 26 pc).
**HIP 54102:** *RV curvature only.* It is listed as a proper motion binary by @2005AJ....129.2420M, but there are no other information about the companions in the literature. Its eccentricity is completely unconstrained due to lack of RV coverage. We estimate that its companion’s minimum mass is $12.6$ M$_\mathrm{Jup}$, with an orbital period larger than 14 years.
**HIP 64150:** *Linear trend.* The most likely companion mass obtained by the method explained in Section \[methods\_long\_period\] renders an estimate of 0.26 M$_\odot$, as seen in Fig. \[64150\_pdf\]. The higher mass (0.54 M$_\odot$) obtained by @2013ApJ...774....1C and can be attributed to less likely orbital configurations, but it is still inside the 1-$\sigma$ confidence interval of the RV+imaging mass estimate. The main star displays clear signals of atmosphere pollution caused by mass transfer from its companion during the red giant phase [@2011PASJ...63..697T], characterizing the only confirmed blue straggler of our sample. The measured projected separation of the binary system is 18.1 AU , which indicates that even for such a wide system the amount of mass transferred is still large enough to produce measurable differences in chemical abundances. It seems, however, that the amount of angular momentum transfer was not enough to produce significant enhancement in the rotation rate and activity of the solar twin. It is also important to note that the isochronal age measured for this system has a better agreement with the white dwarf (WD) cooling age estimated by than previous estimates, illustrating the importance of studying these Sirius-like systems to test the various methods of age estimation.
![Posterior probability distribution of the companion mass for HIP 64150. The mass obtained by @2014ApJ...783L..25M using SED fitting for the spectra of the WD companion is shown as a red vertical line.[]{data-label="64150_pdf"}](images/64150_torres.pdf){width="47.00000%"}
**HIP 65708:** This star has previously been reported as a single-lined spectroscopic binary with an orbital solution [@2002AJ....124.1144L]. Here we update this solution by leveraging the extremely precise radial velocities measured in the Lick Planet Search program and with the HARPS spectrograph. The minimum mass of the companion is 0.167 M$_\odot$, indicating it is a red dwarf, orbiting at less than 1 AU with a slight eccentricity of 0.31. Our results agree with the previous orbital solution, which was based solely on data with uncertainties two orders of magnitude higher than the most recent data from HARPS and the Lick Planet Search.
**HIP 72043:** *RV curvature only.* Similarly to HIP 54102, it is listed as a proper motion binary and we could not constrain its eccentricity. A fairly massive ($> 0.5$ M$_\odot$) companion is inferred at a very large period; this fit suggests that the longitude of periapse of the companion of HIP 72043 is currently at an unfavorable position for visual detection.
**HIP 73241:** *RV curvature only.* The companion’s orbit is eccentric enough to allow an estimation of the minimum eccentricity; its companion has been previously been confirmed by @2010ApJS..190....1R and visually detected by with a separation $0.318\arcsec$. In we listed this star as having an unusually high rotation, but here we revise this conclusion and list HIP 73241 as a candidate peculiar rotator because its $v \sin{i}$ is less than 2$\sigma$ above the expected value for its age. Similarly to HIP 67620, this peculiarity, if real, could also be explained by contamination by a bright companion, since we determined that the minimum companion mass $m \sin{i} > 0.49$ M$_\odot$.
**HIP 79578:** The companion is a well-defined 0.10 M$_\odot$ red dwarf orbiting the main star approximately every 18 years in a fairly eccentric orbit ($e = 0.33$). The orbital parameters we obtained differ significantly from the ones obtained by @2015MNRAS.453.1439J by more than $10\%$, except for the eccentricity; also in contrast, @2015MNRAS.453.1439J report it as a brown dwarf companion. The fit for this binary displays residuals of up to 30 m s$^{-1}$ for the AATPS radial velocities, and the periodogram of these residuals shows a peak near the period 725 days. When we fit an extra object with $m\sin{i} = 0.70$ M$_{\mathrm{Jup}}$ at this period ($a = 1.62$ AU and $e = 0.87$), it improves the general fit of the RVs by a factor of 7. It is important to mention, however, that there are only 17 data points for the AATPS dataset, and the HARPS dataset does not display large residuals for a single companion fit. We need thus more observations to securely infer the configuration of this binary system, and if it truly has an extra substellar companion at a shorter period.
**HIP 81746:** This is another high-eccentricity ($e = 0.7$) binary that does not display clear anomalies in its rotation and activity. Its companion is a 0.1 M$_\odot$ red dwarf orbiting the main star every 9 years. The orbital parameters we obtained are in good agreement with the ones reported by @2015MNRAS.453.1439J.
**HIP 83276:** *RV curvature only.* Although the HARPS radial velocities suggest the presence of a stellar mass companion, we do not have enough RV data points to infer any information about the orbital parameters of the system. Using radial velocities measured with the CORAVEL spectrograph, found the companion has $m\sin{i}=0.24$ M$_\odot$, $e=0.185$ and an orbital period of 386.72 days.
**HIP 87769:** *RV curvature only.* It is reported as a binary system by but, similarly to HIP 54102, lacks an inflection point in its RV data from HARPS, which spans 3.3 yr. There is a wide range of possible orbital solutions that suggest $m \sin{i}$ varying from brown dwarf masses to $\sim 1$ M$_\odot$. Higher eccentricities ($e > 0.8$) can be ruled out as unlikely because they suggest a companion with $m \sin{i} \approx 1$ M$_\odot$ at an orbital period of more than 500 yr and $a > 80$ AU.
------------------------ -------- --------------- ---------- ------------- ----------
$K$ $T$ $m \sin{i}$ $e$
(km s$^{-1}$) (yr) (M$_\odot$)
54102 96116 $> 0.182$ $> 14$ $> 0.012$ $\dots$
54582 97037 $> 0.193$ $> 102$ $> 0.03$ $\dots$
72043 129814 $> 2.11$ $> 104$ $> 0.40$ $\dots$
73241 131923 $> 5.93$ $> 21.0$ $> 0.49$ $> 0.72$
87769 163441 $> 1.90$ $> 81.5$ $> 0.30$ $\dots$
\[curvature\_results\]
------------------------ -------- --------------- ---------- ------------- ----------
: Lower limits of the orbital parameters of the spectroscopic binaries with curvature in their RV data.
[llcccc]{} & & $dv_r / dt$ & $\rho$ & Dist. & $m$\
& & (m s$^{-1}$ yr$^{-1}$) & (arcsec) & (pc)^d^ & (M$_{\mathrm{Jup}}$)\
14501 & 19467 & $-1.30 \pm 0.01$ & 1.653^a^ & 30.86 & 45\
18844^$\dag$^ & 25874 & $424 \pm 3$ & 0.140^b^ & 25.91 & 79\
62039 & 110537 & $7.25 \pm 0.03$ & $\dots$ & 42.68 & $> 19$\
64150 & 114174 & $61.72 \pm 0.02$ & 0.675^c^ & 26.14 & 270\
\
\
\
\
\
\[linear\_trend\_results\]
Considerations on multiplicity statistics
-----------------------------------------
Although planet search surveys are generally biased against the presence of binaries due to avoiding known compact multiple systems, the fraction of binary or higher-order systems in the whole sample of the Solar Twin Planet Search program is $42\% \pm 6\%$[^9]. This value agrees with previous multiplicity fractions reported by, e.g., @2010ApJS..190....1R and ; however, it is signficantly lower than the $58\%$ multiplicity factor for solar-type stars reported by @2017ApJ...836..139F, who argues that previous results are subject to selection effects and are thus biased against the presence of multiple systems.
The orbital period vs. mass ratio plot of companions in the Solar Twin Planet Search is shown in Fig. \[mratio\]. A comparison with the sample of solar-type stars from reveals two important biases in our sample: i) Mass ratios are mostly below 0.3 because of selection of targets that do not show large radial velocity variations in previous studies; ii) Orbital periods are mostly lower than 30 yr because longer values cannot be constrained from the recent RV surveys targeting solar-type stars with low-mass companions. In such cases, further monitoring of linear trend and RV curvature-only binaries may prove useful to understand the origins of the brown dwarf desert [@2006ApJ...640.1051G]. These targets are particularly appealing because the long periods mean that the separation from the main star is large enough to allow us to observe them directly using high-resolution imaging.
Previous studies on the period-eccentricity relation for binary stars found that systems with orbital periods below 10 days tend to have eccentricities near zero, while those between 10 and 1000 days follow a roughly flat distribution of eccentricities [@2016AJ....152..189K and references therein], an effect that is due to the timescales for circularization of orbits. In relation to our sample, with the exception of HIP 30037, HIP 65708 and HIP 83276, all of the binaries we observed have periods longer than 1000 days and eccentricities higher than 0.3, which agrees with the aforementioned findings. According to , the distribution of eccentricities on systems with $T > 1000$ d is a function of energy only, and does not depend on $T$ (see fig. 5 in ). Interestingly, HIP 30037, which hosts a brown dwarf companion with $T = 31.6$ d, falls inside the 25–35 days interval of orbital periods found by @2016AJ....152..189K that corresponds to a short stage of evolution of binaries undergoing a fast change in their orbits.
![Mass ratios in function of the orbital periods of binary stars or higher-order systems in the solar neighborhood. The purple circles are binaries in our sample with well defined period and $m \sin{i}$; the blue triangles correspond to the binaries in our sample for which we only have lower limits for the periods and $m \sin{i}$. The stars from are plotted as black dots (the darker ones are those with main star masses between 0.9 and 1.1 M$_\odot$).[]{data-label="mratio"}](images/mratio_period.pdf){width="47.00000%"}
Conclusions
===========
The Solar Twin Planet Search and several other programs observed 81 solar twins using the HARPS spectrograph. In total, 18 of these solar twins are spectroscopic binaries, 18 are visual binaries, and two intersect these categories. We found a multiplicity fraction of $42\% \pm 6\%$ in the whole sample, which is lower than the expected fraction ($\sim$$58\%$) because of selection effects that are generally seen in exoplanet search surveys.
We updated or reproduced the solutions of several known binaries, and determined all the orbital parameters of HIP 19911, HIP 65708, HIP 67620, HIP 79578, HIP 81746 and HIP 103983. The stars HIP 43297 and HIP 64673, which we previously reported as binaries, are likely to host long-period giant planets instead of stellar companions. For binaries with partial orbits, we were able to place lower limits for some of their orbital parameters owing to the presence of curvature or an inflection point in their RV data. We estimated the most likely mass of the companions of the binaries that display only linear trends in their RV data. Future work is needed on studying the long-period binaries using photometry data and high-resolution imaging in order to constrain the nature of their companions. These wide solar twin binaries are prime targets for detailed physical characterization of their companions owing to the favorable separation for AO imaging and the precision with which we can measure the stellar parameters of the main star – this is particularly important for fully convective red dwarf stars and very low-mass companions such as the T dwarf HIP 14501 B, whose evolution and structure is still poorly constrained.
Additionally, we reported the detailed discovery of new companions to the following solar twins: HIP 6407, HIP 30037, HIP 54582, and HIP 62039, for which we are able to determine an orbital solution for the first two using radial velocities. The latter two do not have enough RV data to obtain precise orbital parameters, but we can nonetheless estimate their minimum companions masses. We found that these new companions are likely very low-mass, ranging from 0.02 to 0.12 M$_\odot$ (although stressing that these are lower limits), which should be useful in understanding the origins of the brown dwarf desert in future research.
The anomalies and RV residuals observed on HIP 19911, HIP 67620 and HIP 103983 are likely due to contamination by the companion on the spectra of the main star. Although the peculiar stars in our sample are no longer considered blue straggler candidates, it is important to note that the detection of WD companions is particularly important for the study of field Sun-like stars because they allow the estimation of their cooling ages; these are more reliable than isochronal and chromospheric ages in some cases, providing thus robust tests for other age estimate methods. We do not expect that the presence of M dwarf companions contaminate lithium spectral lines in Sun-like stars, thus stellar ages derived from Li abundances may be more reliable for double-lined solar twins. We recommend a revision of the stellar parameters of the peculiar binary stars by analyzing high-resolution spectra at the highest Doppler separations possible, or using Gaussian processes to disentangle the contaminated spectra [see, e.g., @2017ApJ...840...49C].
We conclude that single-lined solar twin binaries with orbital periods larger than several months and moderate to low eccentricities do not display signals of distinct rotational evolution when compared to single solar twins. The most compact system in our sample, HIP 30037, which hosts a 0.06 M$_\odot$ brown dwarf companion at an orbital period of 31 days is, in fact, one of the quietest stars in the sample (in regards of its activity levels), and is thus a viable target for further efforts in detecting moderate- to long-period circumbinary planets.
Acknowledgements {#acknowledgements .unnumbered}
================
LdS acknowledges the financial support from FAPESP grants no. 2016/01684-9 and 2014/26908-1. JM thanks FAPESP (2012/24392-2) for support. LS acknowledges support by FAPESP (2014/15706-9). This research made use of SciPy [@scipy_ref], Astropy , Matplotlib [@Hunter:2007], and the SIMBAD and VizieR databases , operated at CDS, Strasbourg, France. We thank R. P. Butler, S. Vogt, G. Laughlin and J. Burt for allowing us to analyze the LCES HIRES/Keck data prior to publication. LdS also thanks B. Montet, J. Stürmer and A. Seifahrt for the fruitful discussions on the results and code implementation. We would also like to thank the anonymous referee for providing valuable suggestions to improve this manuscript.
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[^1]: E-mail: [email protected]
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{
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---
abstract: 'We study pairs $(V, V_{1})$, $V \subseteq V_1$, of models of $ZFC$ such that adding $\kappa-$many Cohen reals over $V_{1}$ adds $\lambda-$many Cohen reals over $V$ for some $\lambda> \kappa$.'
author:
- Moti Gitik and Mohammad Golshani
title: Adding a lot of Cohen reals by adding a few II
---
[^1]
Introduction
============
We continue our study from \[3\]. We study pairs $(V, V_{1})$, $V \subseteq V_1$, of models of $ZFC$ with the same ordinals, such that adding $\kappa-$many Cohen reals over $V_{1}$ adds $\lambda-$many Cohen reals over $V$ for some $\lambda> \kappa$[^2]. We are mainly interested when $V$ and $V_{1}$ have the same cardinals and reals. We prove that for such models, adding $\kappa-$many Cohen reals over $V_{1}$ cannot produce more Cohen reals over $V$ for $\kappa$ below the first fixed point of the $\aleph-$function, but the situation at the first fixed point of the $\aleph-$function is different. We also reduce the large cardinal assumptions from \[1, 3\] to the optimal ones.
Adding many Cohen reals by adding a few: a general result
==========================================================
In this section we prove the following general result.
Suppose $\kappa < \lambda$ are infinite (regular or singular) cardinals, and let $V_1$ be an extension of $V.$ Suppose that in $V_1:$
$(a)$ $\kappa < \lambda$ are still infinite cardinals[^3],
$(b)$ there exists an increasing sequence $\langle \kappa_n: n < \omega \rangle$ of regular cardinals, cofinal in $\kappa.$ In particular $cf(\kappa) = \omega,$
$(c)$ there is an increasing (mod finite) sequence $\langle
f_{\alpha}: \alpha < \lambda \rangle$ of functions in the product
$\hspace{.5cm}$$\prod_{n< \omega}(\kappa_{n+1}\setminus\kappa_n),$
$(d)$ there is a splitting $\langle S_{\sigma} : \sigma<\kappa \rangle$ of $\lambda$ into sets of size $\lambda$ such that for every countable
$\hspace{.5cm}$set $I \in V$ and every $\sigma<\kappa$ we have $|I\cap S_{\sigma}|< \aleph_0.$
Then adding $\kappa-$many Cohen reals over $V_1$ produces $\lambda-$many Cohen reals over $V.$
Condition $(c)$ holds automatically for $\lambda=\kappa^+$; given any collection $\mathcal{F}$ of $\kappa$-many elements of $\prod_{n<\omega}(\kappa_{n+1}\setminus\kappa_n),$ there exists $f$ such that for each $g \in \mathcal{F}, f(n)> g(n)$ for all large $n$ [^4]. Thus we can define by induction on $\alpha< \kappa^+$, an increasing (mod finite) sequence $\langle
f_{\alpha}: \alpha < \kappa^+ \rangle$ in $\prod_{n<\omega}(\kappa_{n+1}\setminus\kappa_n)$[^5].
Force to add $\k-$many Cohen reals over $V_1$. Split them into $\lan r_{i,\sigma} : i,\sigma<\k\operatorname{ran}$ and $\lan r'_{\sigma} : \sigma<\k\operatorname{ran}$. Also in $V,$ split $\kappa$ into $\kappa-$blocks $B_{\sigma}, \sigma<\kappa,$ each of size $\kappa,$ and let $\lan f_\a : \a<\l\operatorname{ran}\in V_{1}$ be an increasing (mod finite) sequence in $\prod_{n<\om}(\k_{n+1}
\setminus \k_{n})$. Let $\a<\l$. We define a real $s_\a$ as follows. Pick $\sigma<\kappa$ such that $\a\in S_{\sigma}.$ Let $k_{\a}=min\{k<\omega: r'_{\sigma}(k) \}=1$ and set
$\forall n<\om$, $s_\a(n)=r_{f_\a(n+k_{\a}),\sigma}(0)$.
The following lemma completes the proof.
$\lan s_\a:\a<\l\operatorname{ran}$ is a sequence of $\l-$many Cohen reals over $V$.
$(a)$ For a forcing notion $\mathbb{P}$ and $p,q\in \mathbb{P},$ we let $p\leq q$ mean $p$ is stronger than $q$.
$(b)$ For each set $I$, let ${\MCB}(I)$ be the Cohen forcing notion for adding $I-$many Cohen reals. Thus ${\MCB}(I)=\{p:p$ is a finite partial function from $I\times \om$ into 2 $\}$, ordered by $p\leq q$ iff $p \supseteq q$.
First note that $\lan\lan r_{i,\sigma} :
i,\sigma<\k\operatorname{ran}, \lan r'_{\sigma} : \sigma<\k\operatorname{ran}\operatorname{ran}$ is ${\MCB}(\k\times\k)\times \MCB(\k)-$generic over $V_1$. By the $c.c.c.$ of ${\MCB}(\l)$ it suffices to show that for any countable set $I\sse \l$, $I\in V$, the sequence $\lan s_\a:\a\in I\operatorname{ran}$ is ${\MCB}(I)-$generic over $V$. Thus it suffices to prove the following
$\hspace{1.5cm}$ For every $(p,q)\in {\MCB}(\k\times\k)\times\MCB(\k)$ and every open dense subset $D\in V$
(\*)$\hspace{1cm}$ of ${\MCB}(I)$, there is $(\ov{p}, \ov{q})\leq (p,q)$ such that $(\ov{p},\ov{q}) \vdash
\ulcorner \lan {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{s}}}}_\a : \a\in I\operatorname{ran}$ extends
$\hspace{1.5cm}$ some element of $D\urcorner. $
Let $(p, q)$ and $D$ be as above and for simplicity suppose that $p=q=\emptyset.$ Let $b\in D,$ and let $\a_1, ..., \a_m$ be an enumeration of the components of $b,$ i.e., those $\a$ such that $(\a,n)\in dom(b)$ for some $n.$ Also let $\sigma_1, ..., \sigma_m<\k$ be such that $\a_i \in S_{\sigma_i}, i=1, ..., m.$ By $(d)$ each $I\cap S_{\sigma_i}$ is finite, thus by $(c)$ we can find $n^*<\omega$ such that for all $n\geq n^*, 1\leq i \leq m$ and $\a^*_1<\a^*_2$ in $I\cap S_{\sigma_i}$ we have $f_{\a^*_1}(n)<f_{\a^*_2}(n).$ Let
$\ov{q}=\{\langle \sigma_i, n, 0 \rangle: 1\leq i \leq m, n< n^* \}.$
Then $\ov{q}\in \MCB(\k)$ and $(\emptyset, \ov{q})\vdash \ulcorner k_{\a_i}\geq n^* \urcorner$ for all $1\leq i \leq m.$ Let
$\ov{p}=\{ \langle f_{\a_i}(n+k_{\a_i}), \sigma_i, 0, b(\a_i, n) \rangle: 1\leq i\leq m, (\a_i, n)\in dom(b) \}.$
Then $\ov{p}\in \MCB(\k\times\k)$ is well-defined and for $(\a_i,n)\in dom(b), 1\leq i \leq m$ we have
$(\ov{p},\ov{q})\vdash \ulcorner {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{s}}}}_{\a_i}(n)= {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{r}}}}_{f_{\a_i}(n+k_{\a_i}),\sigma_i}(0)=\ov{p}(f_{\a_i}(n+k_{\a_i}), \sigma_i, 0)= b(\a_i, n) \urcorner$
and hence
$(\ov{p},\ov{q})\vdash \ulcorner \langle {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{s}}}}_{\a}: \a \in I \rangle$ extends $b \urcorner.$
(\*) follows and we are done.
The theorem follows.
Getting results from optimal hypotheses
=======================================
Suppose $GCH$ holds and $\kappa$ is a cardinal of countable cofinality and there are $\kappa-$many measurable cardinals below $\kappa.$ Then there is a cardinal preserving not adding a real extension $V_1$ of $V$ in which there is a splitting $\langle S_\sigma: \sigma<\k \rangle$ of $\k^+$ into sets of size $\k^+$ such that for every countable set $I\in V$ and every $\sigma<\k, |I\cap S_\sigma|<\aleph_0.$
Let $X$ be a set of measurable cardinals below $\kappa$ of size $\kappa$ which is discrete, i.e., contains none of its limit points, and for each $\xi \in X$ fix a normal measure $U_{\xi}$ on $\xi.$ For each $\xi \in X$ let $\MPB_{\xi}$ be the Prikry forcing associated with the measure $U_{\xi}$ and let $\MPB_{X}$ be the Magidor iteration of $\MPB_{\xi}$’s, $\xi\in X$ (cf. \[2, 5\]). Since $X$ is discrete, each condition in $\MPB_{X}$ can be seen as $p=\langle\langle
s_{\xi}, A_{\xi} \rangle: \xi \in X \rangle$ where for $\xi \in X, \langle s_{\xi}, A_{\xi} \rangle \in
\MPB_{\xi}$ and $supp(p)=\{\xi \in X: s_{\xi}\neq \emptyset
\}$ is finite. We may further suppose that for each $\xi \in X$ the Prikry sequence for $\xi$ is contained in $(sup(X \cap \xi),
\xi).$ Let $G$ be $\MPB_{X}-$generic over $V$. Note that $G$ is uniquely determined by a sequence $(x_{\xi}: \xi \in X)$, where each $x_{\xi}$ is an $\omega-$sequence cofinal in $\xi,$ $V$ and $V[G]$ have the same cardinals, and $GCH$ holds in $V[G].$
Work in $V[G]$. We now force $\langle S_\sigma: \sigma<\k\rangle$ as follows. The set of conditions $\mathbb{P}$ consists of pairs $p=\langle \tau, \lan s_\sigma: \sigma<\k \operatorname{ran}\rangle\in V[G]$ such that:
$(1)$ $\tau<\k^+,$
$(2)$ $\lan s_\sigma: \sigma<\k \operatorname{ran}$ is a splitting of $\tau$,
$(3)$ for every countable set $I\in V$ and every $\sigma<\k, |I\cap s_\sigma|<\aleph_0.$
$(a)$ Given a condition $p\in \mathbb{P}$ as above, $p$ decides an initial segment of $S_\sigma,$ namely $S_\sigma\cap \tau$, to be $s_\sigma$. Condition $(3)$ guarantees that each component in this initial segment has finite intersection with countable sets from the ground model.
$(b)$ Let $t_0=\bigcup_{\xi\in X}x_\xi.$ By genericity arguments, it is easily seen that $t_0$ is a subset of $\kappa$ of size $\kappa$ such that for all countable sets $I\in V, |I\cap t_0|<\aleph_0$. For each $i<\kappa$ set $t_i=t_0+i=\{\alpha+i: \alpha\in t_0 \}.$ Then again by genericity arguments, for every countable set $I\in V, |I\cap t_i|<\aleph_0$. Define $s_i, i<\kappa$ by recursion as $s_0=t_0$ and $s_i=t_i \setminus \bigcup_{j<i}t_j$ for $i>0.$ Then $p=\langle \kappa, \lan s_\sigma: \sigma<\k \operatorname{ran}\rangle\in \mathbb{P}$ (since again by genericity arguments, $\lan s_\sigma: \sigma<\k \operatorname{ran}$ is a splitting of $\k$), and hence $\mathbb{P}$ is non-trivial.
We call $\tau$ the height of $p$ and denote it by $ht(p).$ For $p=\langle \tau, \lan s_\sigma: \sigma<\k \operatorname{ran}\rangle$ and $q=\langle \nu, \lan t_\sigma: \sigma<\k \operatorname{ran}\rangle$ in $\MPB$ we define $p\leq q$ iff
$(1)$ $\tau\geq \nu,$
$(2)$ for every $\sigma<\k, s_{\sigma} \cap\nu=t_\sigma,$ i.e., each $s_\sigma$ end extends $t_\sigma.$
$(a)$ $\mathbb{P}$ satisfies the $\kappa^{++}-c.c,$
$(b)$ $\mathbb{P}$ is $<\kappa-$distributive.
$(a)$ is trivial, as $|\mathbb{P}| \leq 2^\k=\k^+$. For $(b)$, fix $\delta < \kappa, \delta$ regular, and let $p \in \mathbb{P}$ and ${\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{g}}}} \in V[G]^{\mathbb{P}}$ be such that
$p \vdash \ulcorner {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{g}}}}: \delta \rightarrow On \urcorner.$
We find $q \leq p$ which decides ${\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{g}}}}.$ Fix in $V$ a splitting of $\kappa$ into $\delta-$many sets of size $\kappa, \langle Z_i :i<\delta \operatorname{ran}$ [^6]. Let $\theta$ be a large enough regular cardinal. Pick an increasing continuous sequence $\langle M_{i}: i \leq \delta \rangle$ of elementary submodels of $\langle H(\theta), \in \rangle$ of size $\kappa$ such that [^7]:
1. $\langle M_{i}: i \leq \delta \rangle\in V[G],$
2. $p, \mathbb{P}, {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{g}}}}, \langle Z_i : i<\delta \operatorname{ran}\in M_{0},$
3. if $i < \delta$ is a limit ordinal, then $\langle M_j: j\leq i\rangle \in M_{i+1},$
4. $cf(M_{\delta} \cap \kappa^{+})=\delta,$
5. if $i$ is not a limit ordinal, then $cf^{V}(M_{i+1} \cap \kappa^{+})= \xi_{i}$ for a measurable $\xi_{i}$ of $V$ in $X$,
6. $i<j \Rightarrow \xi_{i} < \xi_{j},$
7. $\langle M_{i} \cap V: i \leq \delta \rangle \in V.$
For each non-limit $i< \delta, M_{i+1}\cap V$ is in $V$ by clause $(7),$ and so by clause $(5), cf^{V}(M_{i+1} \cap \kappa^{+})= \xi_{i}$, where $\xi_i\in X,$ so we can pick in $V$ a cofinal in $M_{i+1} \cap \kappa^{+}$ sequence $\langle \eta^{i}_{\a}: \alpha < \xi_{i} \rangle,$ where $\eta^i_\a > M_i\cap \k^+,$ for all $\a<\xi_i$ [^8].
Denote by $\xi_i^{'}$ the first element of the Prikry sequence of $\xi_{i}.$ We define a descending sequence $p_i=\langle \tau_i, \lan s_{i,\sigma}: \sigma<\k \operatorname{ran}\rangle$ of conditions by induction as follows:
[**i=0.**]{} Set $p_{0}=p.$
[**i=j+1.**]{} Assume $p_{j}$ is constructed such that $p_{j} \in
M_{j}$ if $j$ is not a limit ordinal, and $p_{j} \in
M_{j+1}$ if $j$ is a limit ordinal and $p_{j}$ decides ${\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{g}}}}\upharpoonright j.$ Fix a bijection $f_{j}:Z_j \rightarrow (ht(p_j),\eta_{\xi'_{j}}^{j})$ in $M_{j+1}$ and set [^9]
$p'_{j+1}=\langle \eta_{\xi'_{j}}^{j}, \langle s_{j,\sigma}\cup \{f_{j}(\sigma)\}:\sigma\in Z_j \rangle^{\frown}\lan s_{j,\sigma}: \sigma\in \k\setminus Z_j \operatorname{ran}\rangle$
Clearly $p'_{j+1} \in M_{j+1}.$ Let $p_{j+1} \in M_{j+1}$ be an extension of $p_{j+1}^{'}$ which decides ${\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{g}}}}(j).$
[**limit(i).**]{} Let $p_{i}= \langle sup_{j<i}ht(p_j), \lan \bigcup_{j<i}s_{j,\sigma}:\sigma<\k \operatorname{ran}\rangle.$
Let us show that the above sequence is well-defined. Thus we need to show that for each $i \leq \delta, p_{i} \in \mathbb{P}.$ We prove this by induction on $i$. The successor case is trivial. Thus fix a limit ordinal $i \leq \delta$. If $p_{i} \notin
\mathbb{P},$ we can find a countable set $I \in V, I \subseteq \k^+,$ and $\sigma<\k$ such that $I\cap s_{i,\sigma}$ is infinite. Define the sequence $\langle \alpha(j): j<i \rangle$ as follows:
- if $I \cap (M_{j+1}\setminus M_{j}) \neq \emptyset,$ then $\alpha(j)
\in [\sup(X\cap\xi_j), \xi_j]$ is the least such that $\eta_{\alpha(j)}^{j}> sup(I \cap (M_{j+1}\setminus M_{j})),$
- $\alpha(j)=\sup(X\cap \xi_j)$ otherwise. Note that in this case $\a(j) <\xi'_j$ (because the Prikry sequence for $\xi$ was chosen in the interval $(sup(X\cap \xi), \xi)$).
Clearly $\langle \alpha(j): j<i \rangle \in V.$
The set $K=\{j<i: \xi_{j}^{'} \leq \alpha(j) \}$ is finite.
Let $p \in \MPB_{X}, p=\langle\langle
s_{\xi}, A_{\xi} \rangle: \xi \in X \rangle.$ Extend $p$ to $q=\langle\langle
t_{\xi}, B_{\xi} \rangle: \xi \in X \rangle$ by setting
- $t_{\xi}=s_{\xi}$ and $B_{\xi}=A_{\xi}$ for $\xi
\in supp(p),$
- $t_{\xi}=\emptyset$ and $B_{\xi}=
A_{\xi} \backslash (\a(j)+1),$ if $\xi=\xi_j$ (some $j<i$) and $\xi\notin supp(p)$,
- $t_{\xi}=\emptyset$ and $B_{\xi}=
A_{\xi}$, otherwise.
Then $q \leq p$ and $q \vdash \ulcorner {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{K}}}} \subseteq \{j<i: \xi_j\in supp(p) \} \urcorner$, so $q \vdash \ulcorner {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{K}}}}$ is finite $ \urcorner.$
Take $i_0<i$ large enough so that no point $\geq i_0$ is in $K.$ Then for all $j\geq i_0$ we have $\xi_{j}^{'} > \alpha(j),$ hence $\eta_{\xi_{j}^{'}}^{j}> \sup(I \cap (M_{j+1}))$ [^10].
We have
$I\cap s_{i,\sigma}\subseteq I \cap (s_{i_0,\sigma}\cup \{f_{i_1}(\sigma) \})$
where $i_1$ is the unique ordinal less than $\delta$ so that $\sigma\in Z_{i_1}.$
Assume towards a contradiction that the inclusion fails, and let $t\in I\cap s_{i, \sigma}$ be such that $t\notin I \cap (s_{i_0,\sigma}\cup \{f_{i_1}(\sigma) \}).$ As $i$ is a limit ordinal, $I\cap s_{i,\sigma}=I \cap \bigcup_{j<i}s_{j,\sigma}$. Let $j<i$ be the least such that $t\in s_{j+1, \sigma}.$ Then as $t\in I\cap M_{j+1}$ and $j\geq i_0$ we have $t<\eta^j_{\xi'_j},$ so that by our definition of $p'_{j+1}, t$ must be of the form $f_j(\sigma),$ where $\sigma\in Z_j.$ But then $j=i_1$ and hence $t=f_{i_1}(\sigma).$ This is a contradiction, and the result follows.
Thus, as $I\cap s_{i,\sigma}$ is infinite, we must have $I\cap s_{i_0,\sigma}$ is also infinite, and this is in contradiction with our inductive assumption.
It then follows that $q=p_{\delta} \in \mathbb{P}$ and it decides ${\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{g}}}}.$
Let $H$ be $\mathbb{P}-$generic over $V[G]$ and set $V_{1}=V[G][H].$ It follows from Lemma 3.3 that all cardinals $\leq \k$ and $\geq \k^{++}$ are preserved. Also note that $\k^+$ is preserved, as otherwise it would have cofinality less that $\kappa$, which is impossible by the $<\k-$distributivity of $\mathbb{P}$. Hence $V_{1}$ is a cardinal preserving and not adding reals forcing extension of $V[G]$ and hence of $V$. For $\sigma<\k$ set $S_{\sigma}=\bigcup_{\langle \tau, \lan s_\sigma: \sigma<\k \operatorname{ran}\rangle\in H}s_{\sigma}.$
The sequence $\lan S_\sigma: \sigma<\k \operatorname{ran}$ is as required.
For each $\tau<\kappa^+,$ it is easily seen that the set of all conditions $p$ such that $ht(p)\geq \tau$ is dense, so $\lan S_\sigma: \sigma<\k \operatorname{ran}$ is a partition of $\k^+.$ Now suppose that $I\in V$ is a countable subset of $\k^+.$ Find $p=\langle \tau, \lan s_\sigma: \sigma<\k \operatorname{ran}\rangle\in H$ such that $\tau \supseteq I.$ Then for all $\sigma<\k, S_\sigma\cap I=s_\sigma\cap I,$ and hence $|S_\sigma\cap I|=|s_\sigma\cap I|<\aleph_0.$
Theorem 3.1 follows.
$(a)$The size of a set $I$ in $V$ can be changed from countable to any fixed $\eta < \kappa.$ Given such $\eta,$ we start with the Magidor iteration of Prikry forcings above $\eta$.[^11] The rest of the conclusions are the same.
$(b)$ It is possible to add a one element Prikry sequence to each $\xi \in X.$[^12] Then $V_1$ will be a cofinality preserving generic extension of $V$.
The next corollary follows from Theorem 3.1 and Remark 2.2.
Suppose that $GCH$ holds in $V,$ , $\kappa$ is a cardinal of countable cofinality and there are $\kappa-$many measurable cardinals below $\kappa$. Then there is a cardinal preserving not adding a real extension $V_1$ of $V$ such that adding $\k-$many Cohen reals over $V_1$ produces $\k^{+}-$many Cohen reals over $V$.
Assume that there is no sharp for a strong cardinal. Suppose $V_1\subseteq V_2$ have the same cardinals, same reals and there is an infinite set of ordinals $S$ in $V_2$ which does not contain an infinite subset which is in $V_1$. Then either
1. $S$ is countable, and then there is a measurable cardinal $\leq\sup(S)$ in $\mathcal{K}$,\
or
2. $S$ is uncountable, and then there is $\delta\leq\sup(S)$ which is a limit of $|S|$–many or $\delta-$many measurable cardinals of $\mathcal{K}$.
Given a model $V$, let $\mathcal{K}(V)$ denote the core model of $V$ below the strong cardinal. Note that $\mathcal{K}(V_1)=\mathcal{K}(V_2),$ since the models $V_1$ and $V_2$ agree about cardinals. We denote this common core model by $\mathcal{K}.$
Let us first assume that $S$ is countable. Suppose otherwise, i.e., there are no measurable cardinals $\leq\sup(S)$ in $\mathcal{K}$. Then by the Covering Theorem (see \[6\]) there is $Y \in \mathcal{K}$, $|Y|=\aleph_1$ which covers $S$. Fix some $f:\aleph_1 \leftrightarrow Y$ in $V_1$. Consider $Z = f^{-1''}S$. Then $Z$ also does not contain an infinite subset which is in $V_1$. But $Z$ is countable, hence there is $\eta<\omega_1$ with $Z \subseteq \eta$. Let $g:\omega \leftrightarrow \eta$ in $V_1$. Consider $X = g^{-1''}Z$. Then $X$ also does not contain an infinite subset which is in $V_1$. But this is impossible since $V_1,V_2$ have the same reals (and hence $X$ itself is in $V_1$). Contradiction.
Let us deal now with the uncountable case. Suppose otherwise, i.e., there is no $\delta\leq\sup(S)$ which is a limit of $|S|-$many or $\delta-$many measurable cardinals of $\mathcal{K}$. Pick a counterexample $S$ with $\sup(S)$ as small as possible. Denote $\sup(S)$ by $\delta$. By minimality, $\delta$ is a cardinal. Also, the measurable cardinals of $\mathcal{K}$ are unbounded in $\delta$. For otherwise, let $\xi$ be their supremum. Pick $S' \subseteq S$ of size $\xi$. By the Covering Theorem, $S'$ can be covered by a set in $\mathcal{K}$ of size $\xi<\delta$, and then we get a contradiction to the minimality of $\delta$, as witnessed by $\xi$ and $S'$ [^13].
Clearly, $\delta$ must be a singular cardinal and by the above, $\delta$ is a limit of measurable cardinals in $\mathcal{K}$. Fix a cofinal sequence $\langle \delta_i : i<\operatorname{cf}(\delta) \rangle$. Denote by $\eta $ the cardinality of the set $\{\alpha<\delta : \alpha \text{ is a measurable cardinal in } \mathcal{K} \}$. By the assumption, $|S|>\eta\geq \operatorname{cf}(\delta)$. But then there is $i^*<\operatorname{cf}(\delta) $ such that $S \cap \delta_{i^*}$ has size $>\eta$. This is impossible by the minimality of $\delta$. Contradiction.
The conclusions of the theorem are optimal. A Prikry sequence witnesses this in the countable case and the Magidor iteration of Prikry forcing witnesses this in the uncountable case.
Suppose that $V_1 \supseteq V$ are such that:
$(a)$ $V_1$ and $V$ have the same cardinals and reals,
$(b)$ $\kappa < \lambda$ are infinite cardinals of $V_1$,
$(c)$ there is no splitting $\lan S_{\sigma}:\sigma<\k \operatorname{ran}$ of $\l$ in $V_1$ as in Theorem 2.1$(d).$
Then adding $\kappa-$many Cohen reals over $V_1$ cannot produce $\lambda-$many Cohen reals over $V.$
Suppose not. Let $\langle r_{\alpha}: \alpha < \lambda \rangle$ be a sequence of $\lambda-$many Cohen reals over $V$ added after forcing with $\mathbb{C}(\kappa)$ over $V_1$. Let $G$ be $\mathbb{C}(\kappa)-$generic over $V_1$. For each $p\in \mathbb{C}(\kappa)$ set
$C_{p}=\{\alpha < \lambda: p$ decides $ {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{r}}}}_{\alpha}(0) \}.$
Then by genericity $\lambda = \bigcup_{p \in G} C_{p}.$ Fix an enumeration $\lan p_\xi:\xi<\k \operatorname{ran}$ of $G,$ and define a splitting $\lan S_{\sigma}:\sigma<\k \operatorname{ran}$ of $\l$ in $V_1[G]$ by setting $S_\sigma = C_{p_{\sigma}}\setminus \bigcup_{\xi<\sigma}C_{p_{\xi}}.$ By $(a)$ and $(c)$ we can find a countable $I\in V$ and $\sigma<\k$ such that $I\subseteq S_{\sigma}.$ [^14] Suppose for simplicity that $\forall \alpha \in S_{\sigma}, p_{\sigma} \vdash \ulcorner {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{r}}}}_{\alpha}(0)=0 \urcorner.$ Let $q \in \mathbb{C}(\k)$ be such that
$q \vdash^{V} \ulcorner I \in V$ is countable and $ \forall \alpha \in I, {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{r}}}}_{\alpha}(0)=0 \urcorner.$
Pick $\langle 0, \alpha \rangle \in \omega \times I$ such that $\langle 0, \alpha \rangle \notin supp(q).$ Let $\bar{q}=q \cup \{\langle \langle 0, \alpha \rangle, 1 \rangle \}.$ Then $\bar{q} \in \mathbb{C}(\k), \bar{q} \leq q$ and $\bar{q} \vdash \ulcorner {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{r}}}}_{\alpha}(0)=1 \urcorner$, which is a contradiction.
The following corollary answers a question from \[1\].
The following are equiconsistent:
$(a)$ There exists a pair $(V_1, V_2), V_1 \subseteq V_2$ of models of set theory with the same cardinals and reals and a cardinal $\kappa$ of cofinality $\omega$ (in $V_2$) such that adding $\kappa-$many Cohen reals over $V_2$ adds more than $\kappa-$many Cohen reals over $V_1.$
$(b)$ There exists a cardinal $\delta$ which is a limit of $ \delta-$many measurable cardinals.
Assume $(a)$ holds for some pair $(V_1, V_2)$ of models of set theory, $V_1 \subseteq V_2$ which have the same cardinals and reals. If there is a sharp for a strong cardinal, then clearly in $\mathcal{K},$ the core model for a strong cardinal, there is a cardinal $\delta$ which is a limit of $ \delta-$many measurable cardinals [^15]. So assume there is no sharp for a strong cardinal. Then by Theorem 3.10 there exists a splitting $\lan S_{\sigma}:\sigma<\k \operatorname{ran}$ of $\k^+$ in $V_2$ such that for every countable set $I\in V_1$ and $\sigma<\k, I\cap S_\sigma$ is finite. Take $S$ to be one of the sets $S_\sigma$ which has size $\k^+.$ So by Theorem 3.9, we get the consistency of $(b)$ [^16].
Conversely if $(b)$ is consistent, then by Corollary 3.8 the consistency of $(a)$ follows [^17].
Below the first fixed point of the $\aleph-$function
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Suppose that $V_1 \supseteq V$ are such that $V_1$ and $V$ have the same cardinals and reals. Suppose $\aleph_{\delta} < $ the first fixed point of the $\aleph-$function, $X \subseteq \aleph_{\delta}, X \in V_{1}$ and $|X| \geq \delta^{+}$ (in $V_1$). Then $X$ has a countable subset which is in $V$.
By induction on $\delta <$ the first fixed point of the $\aleph-$function.
[**Case 1.**]{} $\delta=0.$ Then $X \in V$ by the fact that $V_1$ and $V$ have the same reals.
[**Case 2.**]{} $\delta = \delta^{'}+1.$ We have $\delta^{'} < \aleph_{\delta^{'}},$ hence $\delta^{+} < \aleph_{\delta},$ thus we may suppose that $|X| \leq \aleph_{\delta^{'}}.$ Let $\eta = sup(X) < \aleph_{\delta}.$ Pick $f_{\eta}: \aleph_{\delta^{'}} \leftrightarrow \eta, f_{\eta} \in V.$ Set $Y=f_{\eta}^{-1''}X.$ Then $Y \subseteq \aleph_{\delta^{'}}, \delta^{'} < \aleph_{\delta^{'}}$ and $|Y| \geq \delta^{+} =\delta^{'+}.$ Hence by induction there is a countable set $B \in V$ such that $B \subseteq Y.$ Let $A = f_{\eta}^{''}B.$ Then $A \in V$ is a countable subset of $X$.
[**Case 3.**]{} $limit(\delta).$ Let $\langle \delta_{\xi}: \xi < cf\delta \rangle$ be increasing and cofinal in $\delta.$ Pick $\xi < cf\delta$ such that $|X \cap \aleph_{\delta_{\xi}}| \geq \delta^{+}.$ By induction there is a countable set $A \in V$ such that $A \subseteq X \cap \aleph_{\delta_{\xi}} \subseteq X.$
The following corollary gives a negative answer to another question from \[1\].
Suppose $V_{1}, V$ and $\delta$ are as in Theorem 4.1. Then adding $\aleph_{\delta}-$many Cohen reals over $V_1$ cannot produce $\aleph_{\delta+1}-$many Cohen reals over $V.$
Towards a contradiction suppose that adding $\aleph_{\delta}-$many Cohen reals over $V_1$ produces $\aleph_{\delta+1}-$many Cohen reals over $V.$ Then by Theorem 3.10, there exists $X \subseteq \aleph_{\delta+1}, X \in V_{1}$ such that $|X|=\aleph_{\delta+1} (\geq \delta^{+})$ and $X$ does not contain any countable subset from $V$ [^18], which is in contradiction with Theorem 4.1.
At the first fixed point of the $\aleph-$function
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The next theorem shows that Theorem 4.1 does not extend to the first fixed point of the $\aleph-$function.
Suppose $GCH$ holds and $\kappa$ is the least singular cardinal of cofinality $\omega$ which is a limit of $\kappa-$many measurable cardinals. Then there is a pair $(V[G], V[H])$ of generic extensions of $V$ with $V[G]\subseteq V[H]$ such that:
$(a)$ $V[G]$ and $V[H]$ have the same cardinals and reals,
$(b)$ $\kappa$ is the first fixed point of the $\aleph-$function in $V[G]$ ( and hence in V\[H\]),
$(c)$ in $V[H]$ there exists a splitting $\langle S_{\sigma}:\sigma<\k \rangle$ of $\k$ into sets of size $\k$ such that for every
$\hspace{.5cm}$countable $I\in V[G]$ and $\sigma<\k, |I\cap S_{\sigma}|<\aleph_0.$
We first give a simple observation.
Suppose there is $S \subseteq \k$ of size $\k$ in $V[H] \supseteq V[G]$ such that for every countable $A\in V[G], |A \cap S|<\aleph_0.$ Then there is a splitting $\langle S_{\sigma}:\sigma<\k \rangle$ of $\k$ as in $(c).$
Let $\lan \a_i: i<\k \operatorname{ran}$ be an increasing enumeration of $S.$ We may further suppose that $\a_0=0$, each $\a_i, i>0$ is measurable [^19] in $V$ and is not a limit point of $S$.[^20] Note that for all $i<\k, sup_{j<i}\a_j< \a_i\setminus sup_{j<i}\a_j.$ Now set:
$\hspace{1cm}$ $S_0=S,$
$\hspace{1cm}$ $S_{\sigma}=\{\a_{l}+\sigma: i\leq l <\k \},$ for $0<\sigma\in [sup_{j<i}\a_{j},\a_i ).$
Then $\langle S_{\sigma}:\sigma<\k \rangle$ is as required (note that for $\sigma>0, S_\sigma \subseteq S+\sigma = \{ \a+\sigma: \a\in S\},$ and clearly $S+\sigma,$ and hence $S_\sigma$, has finite intersection with countable sets from $V[G]$).
Thus it is enough to find a pair $(V[G], V[H])$ of generic extensions of $V$ satisfying $(a)$ and $(b)$ with $V[G] \subseteq V[H]$ such that in $V[H]$ there is $S \subseteq \k$ of size $\k$ composed of inaccessibles, such that for every countable $A\in V[G], |A\cap S|<\aleph_0.$
Let $X$ be a discrete set of measurable cardinals below $\kappa$ of size $\kappa,$ and for each $\xi \in X$ fix a normal measure $U_{\xi}$ on $\xi.$ For each $\xi \in X$ we define two forcing notions $\mathbb{P}_{\xi}$ and $\mathbb{Q}_{\xi}$ as follows.
In the following definitions we let $sup(X \cap \xi)=\omega$ for $\xi=minX.$
A condition in $\mathbb{P}_{\xi}$ is of the form $p=\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle$ where
1. $s_{\xi} \in [\xi \backslash sup(X \cap \xi)^{+}]^{<2}$,
2. if $s_{\xi} \neq \emptyset$ then $ s_{\xi}(0)$ is an inaccessible cardinal,
3. $A_{\xi} \in U_{\xi},$
4. $maxs_{\xi} < minA_{\xi},$
5. $s_{\xi}=\emptyset \Rightarrow f_{\xi} \in Col(sup(X \cap \xi)^{+}, < \xi),$ where $Col(sup(X \cap \xi)^{+}, < \xi)$ is the Lévy collapse for collapsing all cardinals less than $\xi$ to $sup(X \cap \xi)^{+},$ and making $\xi$ become the successor of $sup(X \cap \xi)^{+},$
6. $s_{\xi} \neq \emptyset \Rightarrow f_{\xi}=\langle f_{\xi}^{1}, f_{\xi}^{2} \rangle$ where $f_{\xi}^{1} \in Col(sup(X \cap \xi)^{+}, < s_{\xi}(0))$ and $f_{\xi}^{2} \in Col((s_{\xi}(0))^{+}, < \xi).$
For $p, q \in \mathbb{P}_{\xi}, p=\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle$ and $q=\langle t_{\xi}, B_{\xi}, g_{\xi} \rangle$ we define $p \leq q$ iff
1. $ s_{\xi}$ end extends $t_{\xi},$
2. $ A_{\xi} \cup (s_{\xi} \backslash t_{\xi}) \subseteq B_{\xi},$
3. $t_{\xi}=s_{\xi}=\emptyset \Rightarrow f_{\xi} \leq g_{\xi},$
4. $t_{\xi}=\emptyset$ and $s_{\xi} \neq \emptyset \Rightarrow sup(ran(g_{\xi})) < s_{\xi}(0)$ and $f_{\xi}^{1} \leq g_{\xi},$
5. $t_{\xi} \neq \emptyset \Rightarrow f_{\xi}^{1} \leq g_{\xi}^{1}$ and $f_{\xi}^{2} \leq g_{\xi}^{2}$ (note that in this case we have $s_{\xi}=t_{\xi}$).
We also define $p \leq^{*} q$ ($p$ is a Prikry or a direct extension of $q$) iff
1. $p \leq q,$
2. $s_{\xi}=t_{\xi}.$
The proof of the following lemma is essentially the same as in the proofs in \[2, 5\].
($GCH$) $(a)$ $\mathbb{P}_{\xi}$ satisfies the $\xi^{+}-c.c.$
$(b)$ Suppose $p=\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle \in \mathbb{P}_{\xi}$ and $l(s_{\xi})=1$ (where $l(s_{\xi})$ is the length of $s_\xi$). Then $\mathbb{P}_{\xi}/ p=\{q \in \mathbb{P}_{\xi}: q \leq p \}$ satisfies the $\xi-c.c.$
$(c)$ $(\mathbb{P}_{\xi}, \leq, \leq^{*})$ satisfies the Prikry property, i.e., given $p\in \mathbb{P}$ and a sentence $\sigma$ of the forcing language for $(\mathbb{P}, \leq),$ there exists $q\leq^* p$ which decides $\sigma.$
$(d)$ Let $G_{\xi}$ be $\mathbb{P}_{\xi}-$generic over $V$ and let $\langle s_{\xi}(0) \rangle$ be the one element sequence added by $G_{\xi}$. Then in $V[G_{\xi}], GCH$ holds, and the only cardinals which are collapsed are the cardinals in the intervals $(sup(X \cap \xi)^{++}, s_{\xi}(0))$ and $(s_{\xi}(0)^{++}, \xi),$ which are collapsed to $sup(X \cap \xi)^{+}$ and $s_{\xi}(0)^{+}$ respectively.
We now define the forcing notion $\mathbb{Q}_{\xi}.$ A condition in $\mathbb{Q}_{\xi}$ is of the form $p=\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle$ where
1. $s_{\xi} \in [\xi \backslash sup(X \cap \xi)^{+}]^{<3},$
2. if $s_{\xi} \neq \emptyset$ then for all $i < l(s_{\xi}), s_{\xi}(i)$ is an inaccessible cardinal,
3. $A_{\xi} \in U_{\xi},$
4. $maxs_{\xi} < minA_{\xi},$
5. $s_{\xi}=\emptyset \Rightarrow f_{\xi} \in Col(sup(X \cap \xi)^{+}, < \xi),$
6. $s_{\xi} \neq \emptyset \Rightarrow f_{\xi}=\langle f_{\xi}^{1}, f_{\xi}^{2} \rangle$ where, $f_{\xi}^{1} \in Col(sup(X \cap \xi)^{+}, < s_{\xi}(0))$ and $f_{\xi}^{2} \in Col((s_{\xi}(0))^{+}, < \xi).$
For $p, q \in \mathbb{Q}_{\xi}, p=\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle$ and $q=\langle t_{\xi}, B_{\xi}, g_{\xi} \rangle$ we define $p \leq q$ iff
1. $ s_{\xi}$ end extends $t_{\xi},$
2. $ A_{\xi} \cup (s_{\xi} \backslash t_{\xi}) \subseteq B_{\xi},$
3. $t_{\xi}=s_{\xi}=\emptyset \Rightarrow f_{\xi} \leq g_{\xi},$
4. $t_{\xi}=\emptyset$ and $s_{\xi} \neq \emptyset \Rightarrow sup(ran(g_{\xi})) < s_{\xi}(0)$ and $f_{\xi}^{1} \leq g_{\xi},$
5. $t_{\xi} \neq \emptyset$ and $s_{\xi}=t_{\xi} \Rightarrow f_{\xi}^{1} \leq g_{\xi}^{1}$ and $f_{\xi}^{2} \leq g_{\xi}^{2},$
6. $t_{\xi} \neq \emptyset$ and $s_{\xi} \neq t_{\xi} \Rightarrow sup(ran(g_{\xi}^{2})) < s_{\xi}(1), f_{\xi}^{1} \leq g_{\xi}^{1}$ and $f_{\xi}^{2} \leq g_{\xi}^{2}.$
We also define $p \leq^{*} q$ iff
1. $p \leq q,$
2. $s_{\xi}=t_{\xi}.$
As above we have the following.
($GCH$) $(a)$ $\mathbb{Q}_{\xi}$ satisfies the $\xi^{+}-c.c.$
$(b)$ Suppose $p=\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle \in \mathbb{Q}_{\xi}, l(s_{\xi})=2.$ Then $\mathbb{Q}_{\xi}/ p=\{q \in \mathbb{Q}_{\xi}: q \leq p \}$ satisfies the $\xi-c.c..$
$(c)$ $(\mathbb{Q}_{\xi}, \leq, \leq^{*})$ satisfies the Prikry property.
$(d)$ Let $H_{\xi}$ be $\mathbb{Q}_{\xi}-$generic over $V$ and let $\langle s_{\xi}(0), s_{\xi}(1) \rangle$ be the two element sequence added by $H_{\xi}$. Then in $V[H_{\xi}], GCH$ holds, and the only cardinals which are collapsed are the cardinals in the intervals $(sup(X \cap \xi)^{++}, s_{\xi}(0))$ and $(s_{\xi}(0)^{++}, \xi),$ which are collapsed to $sup(X \cap \xi)^{+}$ and $s_{\xi}(0)^{+}$ respectively.
Now let $\mathbb{P}$ be the Magidor iteration of the forcings $\mathbb{P}_{\xi}, \xi \in X,$ and $\mathbb{Q}$ be the Magidor iteration of the forcings $\mathbb{Q}_{\xi}, \xi \in X$. Since the set $X$ is discrete we can view each condition in $\mathbb{P}$ as a sequence $p=\langle \langle s_{\xi}, A_{\xi}, f_{\xi} \rangle: \xi \in X \rangle$ where for each $\xi \in X, \langle s_{\xi}, A_{\xi}, f_{\xi} \rangle \in \mathbb{P}_{\xi}$ and $supp(p)=\{\xi: s_{\xi} \neq \emptyset \}$ is finite. Similarly each condition in $\mathbb{Q}$ can be viewed as a sequence $p=\langle \langle s_{\xi}, A_{\xi}, f_{\xi} \rangle: \xi \in X \rangle$ where for each $\xi \in X, \langle s_{\xi}, A_{\xi}, f_{\xi} \rangle \in \mathbb{Q}_{\xi}$ and $supp(p)=\{\xi: s_{\xi} \neq \emptyset \}$ is finite (for more information see \[2, 4, 5\]).
If $p$ is as above, then we write $p(\xi)$ for $\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle.$
We also define
$\pi: \mathbb{Q} \rightarrow \mathbb{P}$
by
$\pi(\langle \langle s_{\xi}, A_{\xi}, f_{\xi} \rangle: \xi \in X \rangle)=\langle \langle s_{\xi}\upharpoonright 1, A_{\xi}, f_{\xi} \rangle: \xi \in X \rangle.$
It is clear that $\pi$ is well-defined.
$\pi$ is a projection, i.e.,
$(a)$ $\pi(1_{\mathbb{Q}})=1_{\mathbb{P}},$
$(b)$ $\pi$ is order preserving,
$(c)$ if $p \in \mathbb{Q}, q \in \mathbb{P}$ and $q \leq \pi(p)$ then there is $r \leq p$ in $\mathbb{Q}$ such that $\pi(r) \leq q.$
Now let $H$ be $\mathbb{Q}-$generic over $V$ and let $G=\pi^{''}H$ be the filter generated by $\pi^{''}H.$ Then $G$ is $\mathbb{P}-$generic over $V$.
$(a)$ if $\langle\tau_{\xi}: \xi \in X \rangle$ and $\langle \langle \eta_{\xi}^{0}, \eta_{\xi}^{1} \rangle: \xi \in X \rangle$ are the Prikry sequences added by $G$ and $H$ respectively, then $\tau_{\xi}=\eta_{\xi}^{0}$ for all $\xi \in X.$
$(b)$ The models $V[G]$ and $V[H]$ satisfy the $GCH,$ have the same cardinals and reals, and furthermore the only cardinals of $V$ below $\kappa$ which are preserved are $\{\omega, \omega_{1} \} \cup lim(X) \cup \{\tau_{\xi}, \tau_{\xi}^{+}, \xi, \xi^{+}: \xi \in X \}.$
$(c)$ $\kappa$ is the first fixed point of the $\aleph-$function in $V[G]$ (and hence in $V[H]$).
$(a)$ and $(b)$ follow easily from Lemmas 5.4 and 5.5 and the definition of the projection $\pi.$ Let’s prove $(c).$ It is clear that $\k$ is a fixed point of the $\aleph-$function in $V[G]$. On the other hand, by $(b)$, the only cardinals of $V$ below $\k$ which are preserved in $V[G]$ are $\{\omega, \omega_{1} \} \cup lim(X) \cup \{\tau_{\xi}, \tau_{\xi}^{+}, \xi, \xi^{+}: \xi \in X \},$ and so if $\l < \k$ is a limit cardinal in $V[G]$, then $\l\in lim(X).$ But by our assumption on $\k,$ if $\l\in lim(X),$ then $X\cap \l$ has order type less than $\l,$ and hence $(\{\omega, \omega_{1} \} \cup lim(X) \cup \{\tau_{\xi}, \tau_{\xi}^{+}, \xi, \xi^{+}: \xi \in X \}) \cap \l$ has order type less than $\aleph_\l.$ Thus $\l<\aleph_\l.$
Let $\mathbb{Q}/G =\{ p\ \in \mathbb{Q}: \pi(p) \in G \}.$ Then $V[H]$ can be viewed as a generic extension of $V[G]$ by $\mathbb{Q}/G.$
$\mathbb{Q}/G$ is cone homogenous: given $p$ and $q$ in $\mathbb{Q}/G$ there exist $p^{*} \leq p, q^{*} \leq q$ and an isomorphism $\rho:(\mathbb{Q}/G)/ p^{*} \rightarrow (\mathbb{Q}/G)/ q^{*}.$
Suppose $p,q \in \mathbb{Q}/G.$ Extend $p$ and $q$ to $p^{*}=\langle \langle s_{\xi}, A_{\xi}, f_{\xi} \rangle: \xi \in X \rangle$ and $q^{*}=\langle \langle t_{\xi}, B_{\xi}, g_{\xi} \rangle: \xi \in X \rangle$ respectively so that the following conditions are satisfied:
1. $supp(p^{*})=supp(q^{*}).$ Call this common support $K$.
2. For every $\xi \in K, l(s_{\xi})=l(t_{\xi})=2.$ Note that then for every $\xi \in K, s_{\xi}(0)=t_{\xi}(0)=\tau_{\xi},$ $f_{\xi}=\langle f_{\xi}^{1}, f_{\xi}^{2} \rangle$ and $g_{\xi}=\langle g_{\xi}^{1}, g_{\xi}^{2} \rangle$ where $f_{\xi}^{1}, g_{\xi}^{1} \in Col(sup(X \cap \xi)^{+}, < \tau_{\xi})$ and $f_{\xi}^{2}, g_{\xi}^{2} \in Col(\tau_{\xi}^{+}, < \xi).$
3. For every $\xi \in K, A_{\xi}=B_{\xi}.$
4. For every $\xi \in K, dom(f_{\xi}^{1})=dom(g_{\xi}^{1})$ and $dom(f_{\xi}^{2})=dom(g_{\xi}^{2}).$
5. For every $\xi \in K,$ there exists an automorphism $\rho_{\xi}^{1}$ of $Col(sup(X \cap \xi)^{+}, < \tau_{\xi})$ such that $\rho_{\xi}^{1}(f_{\xi}^{1})=g_{\xi}^{1}.$
6. For every $\xi \in K,$ there exists an automorphism $\rho_{\xi}^{2}$ of $Col(\tau_{\xi}^{+}, < \xi)$ such that $\rho_{\xi}^{2}(f_{\xi}^{2})=g_{\xi}^{2}.$
Note that clauses $(5)$ and $(6)$ are possible, as the corresponding forcing notions are homogeneous.
We now define $\rho:(\mathbb{Q}/G)/ p^{*} \rightarrow (\mathbb{Q}/G)/ q^{*}$ as follows. Suppose $r \in \mathbb{Q}/G, r \leq p^{*}.$ Let $r=\langle \langle r_{\xi}, C_{\xi}, h_{\xi} \rangle: \xi \in X \rangle.$ Then for every $\xi \in K, r_{\xi}=s_{\xi},$ and $h_{\xi}=\langle h_{\xi}^{1}, h_{\xi}^{2} \rangle$ where $h_{\xi}^{1} \in Col(sup(X \cap \xi)^{+}, < \tau_{\xi})$ and $h_{\xi}^{2} \in Col(\tau_{\xi}^{+}, < \xi).$ Let
$\rho(r)=\langle \langle t_{\xi}, C_{\xi}, \langle \rho_{\xi}^{1}(h_{\xi}^{1}), \rho_{\xi}^{2}(h_{\xi}^{2}) \rangle \rangle: \xi \in K \rangle ^{\frown} \langle \langle r_{\xi}, C_{\xi}, h_{\xi} \rangle: \xi \in X \setminus K \rangle.$
It is easily seen that $\rho$ is an isomorphism from $(\mathbb{Q}/G)/ p^{*}$ to $(\mathbb{Q}/G)/ q^{*}.$
The following lemma completes the proof.
Let $S=\{\eta_{\xi}^{1}:\xi \in X \}.$ Then $S$ is a subset of $\kappa$ of size $\kappa$ and $|A \cap S|< \aleph_{0}$ for every countable set $A \in V[G].$
$(a)$ Since $V[G]$ and $V[H]$ have the same reals, it suffices to prove the lemma for $A \subseteq S, A\in V[G].$ In fact suppose that the lemma is true for all countable $A \subseteq S, A\in V[G].$ If the lemma fails, then for some countable set $B\in V[G], |B\cap S|=\aleph_0.$ Let $g: \omega \rightarrow B$ be a bijection in $V[G]$. Then $g^{-1}[B\cap S]$ is a subset of $\omega$ which is in $V[H],$ and hence in $V[G].$ Thus $B\cap S\in V[G].$ Hence we find a countable subset $A \subseteq S$ in $V[G],$ namely $B\cap S,$ for which the lemma fails, which is in contradiction with our initial assumption.
$(b)$ In what follows we say $A$ codes $\xi$ (for $\xi \in X$), if $\eta_{\xi}^{1} \in A.$
Let ${\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{S}}}}$ be a $\mathbb{Q}/G-$name for $S.$ Also let $p_{0} \in H \cap \mathbb{Q}/G$ be such that $p_{o} \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner \check{A} \subseteq {\smash{\underset{\raisebox{1.2pt}[0cm][0cm]{$\sim$}}
{{S}}}} $ is countable$\urcorner.$
For every $p \in \mathbb{Q}/G$ and every $\xi \in X \setminus supp(p)$ there is $q \leq p$ in $\mathbb{Q}/G$ such that $\xi \in supp(q)$ and if $q(\xi)=\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle,$ then $l(s_{\xi})=2$ and $q \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner s_{\xi}(1) \notin \check{A} \urcorner.$
Let $p$ and $\xi$ be as in the claim. First pick $\langle\langle\langle t_{\xi}(0)\rangle, A_{\xi}, f_{\xi} \rangle\rangle \in G,$ and then let $q=p ^{\frown} \langle \langle s_{\xi}, A_{\xi}, f_{\xi} \rangle \rangle,$ where $s_{\xi}(0)=t_{\xi}(0)=\tau_{\xi}$ , $s_{\xi}(1)< \xi$ is large enough so that $s_{\xi}(1) \notin A,$ $sup(ran(f_{\xi}^{2}))<s_{\xi}(1)$ and $s_{\xi}(1)$ is inaccessible. Then $\pi( \langle \langle s_{\xi}, A_{\xi}, f_{\xi} \rangle \rangle)=\langle\langle\langle t_{\xi}(0)\rangle, A_{\xi}, f_{\xi} \rangle\rangle \in G.$ On the other hand $\pi(p) \in G.$ Let $r \in G, r \leq \pi(p), \langle\langle\langle t_{\xi}(0)\rangle, A_{\xi}, f_{\xi} \rangle\rangle.$ Then $r \leq \pi(q),$ hence $\pi(q) \in G.$ This implies that $q \in \mathbb{Q}/G.$ Clearly $q$ satisfies the requirements of the Claim.
It follows that the set
$D=\{p \in \mathbb{Q}/G: \forall \xi \in X\setminus supp(p)$ there exists $q \leq p$ as in the above Claim$ \}$
is dense open in $\mathbb{Q}/G.$ Let $p \in H \cap D.$ We can assume that $p \leq p_{0}.$ We show that $p \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner $if $\check{A}$ codes $\xi$ then $\xi \in supp(p) \urcorner.$ To see this suppose that $\xi \in X \setminus supp(p).$ Thus by Claim 5.12 we can find $q \leq p$ in $\mathbb{Q}/G$ such that $\xi \in supp(q)$ and if $q(\xi)=\langle s_{\xi}, A_{\xi}, f_{\xi} \rangle,$ then $l(s_{\xi})=2$ and $q \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner s_{\xi}(1) \notin \check{A} \urcorner.$ It then follows that $\sim p \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner s_{\xi}(1) \in \check{A} \urcorner.$ But then by the cone homogeneity of $\mathbb{Q}/G$ we have $p \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner s_{\xi}(1) \notin \check{A} \urcorner$ [^21]. Hence $p \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner \check{A}$ does not code $\xi \urcorner.$ This means that $p \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner \check{A} \subseteq \{s_{\xi}(1):\xi \in supp(p) \}=\{\eta_{\xi}^{1}:\xi \in supp(p) \} \urcorner.$ Lemma 5.10 follows by noting that $p \in H$ and since the Magidor iteration is used, the support of any condition is finite.
Theorem 5.1 follows.
The following theorem can be proved by combining the methods of the proofs of Theorems 3.1 and 5.1.
Suppose $GCH$ holds and $\kappa$ is the least singular cardinal of cofinality $\omega$ which is a limit of $\kappa-$many measurable cardinals. Also let $V[G]$ and $V[H]$ be the models constructed in the proof of Theorem 5.1. Then there is a cardinal preserving, not adding a real generic extension $V[H][K]$ of $V[H]$ such that in $V[H][K]$ there exists a splitting $\lan S_{\sigma}:\sigma<\k \operatorname{ran}$ of $\k^+$ into sets of size $\k^+$ such that for every countable set $I\in V[G]$ and $\sigma<\k, |I\cap S_{\sigma}|<\aleph_0.$
Work over $V[H]$ and force the splitting $\lan S_{\sigma}:\sigma<\k \operatorname{ran}$ as in the proof of Theorem 3.1, with $V, V[G]$ used there replaced by $V[G], V[H]$ here respectively. The role of the sequence $\bigcup_{\xi\in X}x_\xi$ in the proof of Theorem 3.1 is now played by the sequence $S=\{ \eta_\xi^1: \xi\in X\}$.
Suppose $GCH$ holds and there exists a cardinal $\kappa$ which is of cofinality $\omega$ and is a limit of $\kappa-$many measurable cardinals. Then there is pair $(V_1, V_2)$ of models of $ZFC, V_1 \subseteq V_2$ such that:
$(a)$ $V_1$ and $V_2$ have the same cardinals and reals.
$(b)$ $\kappa$ is the first fixed point of the $\aleph-$function in $V_1$ (and hence in $V_2$).
$(c)$ Adding $\kappa-$many Cohen reals over $V_2$ adds $\kappa^{+}-$many Cohen reals over $V_1.$
Let $V_1=V[G]$ and $V_2=V[H][K],$ where $V[G], V[H][K]$ are as in Theorem 5.13. The result follows using Remark 2.2 and Theorem 5.13.
[**Acknowledgement**]{}
The authors would like to thank the referee of the paper for his/her many helpful suggestions.
[xx]{} Gitik, Moti, Adding a lot of Cohen reals by adding a few. Unpublished paper.
Gitik, Moti, Prikry-type forcings. Handbook of set theory. Vols. 1, 2, 3, 1351–-1447, Springer, Dordrecht, 2010.
Gitik, Moti; Golshani, Mohammad, Adding a lot of Cohen reals by adding a few I, Trans. Amer. Math. Soc. 367 (2015), no. 1, 209-229.
Krueger, John, Radin forcing and its iterations. Arch. Math. Logic 46 (2007), no. 3–4, 223–-252.
Magidor, Menachem, How large is the first strongly compact cardinal? or A study on identity crises. Ann. Math. Logic 10 (1976), no. 1, 33-–57.
Mitchell, William J. The covering lemma. Handbook of set theory. Vols. 1, 2, 3, 1497-–1594, Springer, Dordrecht, 2010.
Moti Gitik, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel.
E-mail address: [email protected]
http://www.math.tau.ac.il/ gitik/
Mohammad Golshani, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran-Iran.
E-mail address: [email protected]
http://math.ipm.ac.ir/ golshani/
[^1]:
[^2]: By “$\lambda-$many Cohen reals” we mean “a generic object $\langle s_\a : \a < \l\rangle$ for the poset $\mathbb{C}(\l)$ of finite partial functions from $\l\times\omega$ to $2$”.
[^3]: $\lambda$ can be a regular or a singular cardinal, but by $(b)$, $\k$ is necessarily a singular cardinal of cofinality $\omega$.
[^4]: To see this let $\mathcal{F}=\bigcup_{n<\omega}\mathcal{F}_n,$ where $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq ...$ and $|\mathcal{F}_n|< \k_{n+1},$ and define $f$ so that $\sup\{g(n): g\in \mathcal{F}_n \} < f(n)\in \k_{n+1}\setminus \k_n$.
[^5]: Let $f_0$ be arbitrary. Given $\a<\k^+,$ we can apply the above to find $f_\a$ so that $f_\a(n) > f_\beta(n),$ for all large $n,$ and all $\beta<\a$.
[^6]: Note that this is possible, as $\delta<\kappa$ are cardinals in $V$. The splitting can also be chosen in $V[G].$
[^7]: Condition $(5)$ can be guaranteed using the fact that the set $K=\{\a<\k^+: cf^V(\a)\in X \}$ is a stationary subset of $\k^+$ in $V[G]$ (given $M_i$, build a suitable continuous increasing chain $\langle N_j: j<\k^+ \rangle$ consisting of models of size $\k.$ Then $\langle \sup(N_j\cap \k^+): j<\k^+ \rangle$ forms a club of $\k^+,$ and $M_{i+1}$ can be chosen to be one of those $N_j$ so that $\sup(N_j\cap \k^+)\in K$). $(7)$ can be guaranteed by the fact that $\mathbb{P}_X$ satisfies the $\k^+$-$c.c.$ and the models have size $\k$ (use the fact that given any model $N$ of size $\k,$ there exists a model in $V$ of the same size which contains $N\cap V$).
[^8]: Note that $\sup(M_{i+1} \cap \kappa^{+})=M_{i+1} \cap \kappa^{+}.$ This is because if $\xi<\k^+,$ and $\xi\in M_{i+1},$ then since $\k\cup\{\k\} \subseteq M_{i+1},$ and $M_{i+1}\models |\xi|=\k,$ we have $\xi \subseteq M_{i+1}$. Also, as the sequence of $M_i$’s in increasing continuous, $\sup(M_{i} \cap \kappa^{+})=M_{i} \cap \kappa^{+}$ holds for limit ordinals $i$.
[^9]: It is easily seen by induction on $j\leq i$ that $ht(p_j) < \eta^j_{\xi'_j}$: if $j=0$ or $j$ is a successor ordinal, then $p_{j}\in M_{j},$ so $ht(p_j) \in M_j\cap \k^+ < \eta^j_{\xi'_j}$. If $j$ is a limit ordinal, then $ht(p_j)=\sup_{k<j}ht(p_k) \leq \sup_{k<j}M_k\cap \k^+=M_j\cap \k^+ < \eta^j_{\xi'_j}.$
[^10]: This is trivial if $I \cap (M_{j+1}\setminus M_{j}) \neq \emptyset,$ as then $\eta_{\xi_{j}^{'}}^{j} > \eta_{\a(j)}^{j} > \sup(I \cap (M_{j+1}\setminus M_{j}))=\sup(I\cap M_{j+1}).$ If $I \cap (M_{j+1}\setminus M_{j})= \emptyset,$ then $\eta_{\xi_{j}^{'}}^{j} > M_{j}\cap \k^+=\sup(M_j\cap \k^+) \geq \sup(I\cap M_j)$ (as $I \subseteq \k^+$) and $\sup(I\cap M_{j+1})=\sup(I\cap M_j)$ (since $I$ has no points in $M_{j+1}\setminus M_{j}$), and hence again $\eta_{\xi_{j}^{'}}^{j}> \sup(I \cap (M_{j+1}))$.
[^11]: The reason for starting the iteration above $\eta$ is to add no subsets of $\eta$. This will guarantee that if $t_0$ is defined as in Remark 3.2$(b)$, then $t_0$ has finite intersection with sets from $V$ of size $\eta$. Using this fact we can show as before that there is a splitting of $\k$ into $\k$ sets, each of them having finite intersection with ground model sets of size $\eta$. This makes the second step of the above forcing construction well-behaved.
[^12]: Conditions in the forcing are of the form $\langle p_\xi: \xi\in X \rangle,$ where for each $\xi\in X, p_\xi$ is either of the form $A_\xi$ for some $A_\xi\in U_\xi$, or $\alpha_\xi$ for some $\alpha_\xi <\xi.$ We also require that there are only finitely many $p_\xi$’s of the form $\alpha_\xi$. When extending a condition, we allow either $A_\xi$ to become thinner, or replace it by some ordinal $\alpha_\xi \in A_\xi$.
[^13]: We then have $\sup(S')=\xi < \delta$ and $S'$ is a counterexample to our assumption of smaller supremum.
[^14]: In fact, by $(c)$ there exist a countable $I\in V$ and some $\sigma<\k$ such that $I\cap S_{\sigma}$ is infinite. By $(a)$, $V$ and $V_1$ have the same reals, and hence $I\cap S_{\sigma}\in V.$ So by replacing $I$ with $I\cap S_{\sigma},$ if necessary, we can assume that $I\subseteq S_{\sigma}$.
[^15]: In fact there are many such cardinals $\delta.$
[^16]: Note that necessarily case $(b)$ of Theorem 3.9 happens.
[^17]: If $cf(\delta)>\omega,$ then we can find $\delta^* < \delta$ of cofinality $\omega$ which is a limit of $\delta^*-$many measurable cardinals, so that Corollary 3.8 can be applied. To see such a $\delta^*$ exists, define an increasing sequence $\delta_n, n<\omega,$ of cardinals below $\delta,$ so that for any $n,$ there are at least $\delta_n-$many measurable cardinals below $\delta_{n+1},$ and let $\delta^*=\sup_n \delta_n.$
[^18]: In fact, there exists a splitting $\langle S_\sigma: \sigma< \aleph_\delta \rangle$ of $\aleph_{\delta+1}$ in $V_1$, consisting of sets of size $\aleph_{\delta+1}$ such that each $S_\sigma$ has finite intersection with any countable set from $V$. The set $X$ can be chosen to be any of $S_\sigma$’s.
[^19]: In fact it suffices for each $\a_i$ to be inaccessible in $V$.
[^20]: Let $f\in V$ be such that $f: \k \rightarrow X$ is a bijection, where $X$ is a discrete set of measurable cardinals of $V$ below $\kappa$ of size $\kappa$. Then if $S \subseteq \k$ satisfies the claim, so does $f[S],$ hence we can suppose all non-zero elements of $S$ are measurable in $V,$ and are not a limit point of $S$.
[^21]: If not, then for some $p'\leq p, p'\Vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner s_{\xi}(1) \in \check{A} \urcorner.$ By cone homogeneity of $\mathbb{Q}/G$ we can find $q^*\leq q, p^*\leq p'$ and an isomorphism $\rho:(\mathbb{Q}/G)/ p^{*} \rightarrow (\mathbb{Q}/G)/ q^{*}.$ But then by standard forcing arguments and the fact that $q^* \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner s_{\xi}(1) \notin \check{A} \urcorner,$ we can conclude that $p^* \vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner s_{\xi}(1) \notin \check{A} \urcorner,$ which is impossible, as $p^*\leq p'$ and $p'\Vdash_{\mathbb{Q}/G}^{V[G]} \ulcorner s_{\xi}(1) \in \check{A} \urcorner$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For any knot $T$ transverse to a given contact structure on a $3$-manifold, we exhibit a Legendrian two-component link ${\mathbb {L}}=L_1\sqcup L_2$ such that $T$ equals the transverse push-off of $L_1$ and contact $(+1)$-surgery on ${\mathbb {L}}$ has the same effect as a Lutz twist along $T$.'
address:
- 'Department of Mathematics, Peking University, Beijing 100871, P.R. China'
- 'Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany'
- 'A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Reáltanoda utca 13–15, H-1053, Hungary'
author:
- Fan Ding
- Hansjörg Geiges
- 'András I. Stipsicz'
title: Lutz twist and contact surgery
---
Introduction {#first}
============
The theorem of Lutz and Martinet [@mart71] asserts that any closed, oriented $3$-manifold $Y$ admits a contact structure in each homotopy class of tangent $2$-plane fields. Here $2$-plane fields are understood to be cooriented; a [*contact structure*]{} is understood to be cooriented and positive, that is, a $2$-plane field $\xi$ defined as the kernel of a global $1$-form $\alpha$ on $Y$ such that $\alpha\wedge d\alpha$ is a positive volume form.
The cited paper by Martinet only covers the existence of [*some*]{} contact structure on a given $Y$; for a proof of the existence of such a structure in every homotopy class of $2$-plane fields see [@geig], which gives a proof of that result along the lines of the original (and never fully published) argument by Lutz.
The key to that second step is what is nowadays known as a [*Lutz twist*]{}, a surgery on a knot $T$ in a given contact manifold $(Y,\xi )$ — with $T$ [*transverse*]{} to $\xi$ — that is topologically trivial (i.e., does not change $Y$), but transforms $\xi$ to a contact structure $\xi'$ in a different homotopy class of $2$-plane fields.
In a series of papers [@dige01; @dige04; @dgs04] we described a notion of contact $r$-surgery on [*Legendrian*]{} knots in a contact manifold (that is, knots tangent to the given contact structure), where $r\in{{\mathbb {Q}}}^*\cup\{\infty\}$ denotes the framing of the surgery relative to the natural contact framing of the Legendrian knot. This generalises the contact surgery introduced by Eliashberg [@elia90] and Weinstein [@wein91], which in our language is a contact $(-1)$-surgery. Amongst other things, we discussed explicit surgery diagrams for various contact manifolds and gave an alternative proof of the Lutz-Martinet theorem via such Legendrian surgeries. We did not, however, fully elucidate the relation between our surgery diagrams and the Lutz twist (although the principal connection was described in [@dige04], cf. [@etho01]). The intention of the present note is to give an explicit Legendrian surgery diagram for the Lutz twist. In particular, this yields surgery representations for all contact structures on $S^3$ analogous to [@dgs04] and provides concrete realisations for the considerations in Section 6 of [@dige04].
We shall henceforth assume that the reader is familiar with the basics of this notion of contact surgery; if not, the best place to start may well be [@dgs04], cf. also [@ozst]. One fact from [@dige01; @dige04] we should like to recall here is that contact $(+1)$-surgery is the inverse of contact $(-1)$-surgery.
The Lutz twist
==============
We briefly recall the definition of the Lutz twist, cf. [@geig]. Let $T$ be a knot transverse to a contact structure $\xi$ on a $3$-manifold $Y$. Then there is a tubular neighbourhood $\nu T$ of $T$ that is contactomorphic to the solid torus $S^1\times D^2_{\delta}$ (with $D^2_{\delta}$ denoting the $2$-disc of radius $\delta$) for some suitable $\delta >0$, with contact structure $\zeta =\ker (d\theta +r^2
\, d\varphi )$, where $\theta$ denotes the $S^1$-coordinate, and $r,\varphi$ are polar coordinates on $D^2_{\delta}$. For ease of notation we identify $(\nu T,\xi)$ with $(S^1\times D^2_{\delta},\zeta)$.
A [*simple Lutz twist*]{} along $T$ is the operation that replaces the contact structure $\xi$ on $Y$ by the one that coincides with $\xi$ outside $\nu T$, and on $S^1\times D^2_{\delta}$ is given by $$\zeta'=\ker \bigl( h_1(r)\, d\theta +h_2(r)\, d\varphi \bigr),$$ where $h_1,h_2\colon\, [0,\delta ]\rightarrow{{\mathbb {R}}}$ are smooth functions satisfying the following conditions:
- $h_1(r)=-1$ and $h_2(r)=-r^2$ for $r$ near $0$,
- $h_1(r)=1$ and $h_2(r)=r^2$ for $r$ near $\delta$,
- $(h_1(r),h_2(r))$ is never parallel to $(h_1'(r),h_2'(r))$ — in particular, neither of them is ever equal to (0,0) —,
- $h_1$ has exactly one zero on the interval $[0,\delta ]$.
The boundary conditions (i) and (ii) ensure that $\zeta'$ is defined around $r=0$ and coincides with $\zeta$ near $r=\delta$; (iii) is the condition for $\zeta'$ to be a contact structure; condition (iv) fixes the homotopy class (as $2$-plane field) of the new contact structure.
The contact structure $\zeta'$ is a so-called [*overtwisted*]{} contact structure in the sense of Eliashberg [@elia89], and as shown in that paper (specifically, Theorem 3.1.1), the classification of such overtwisted contact structures up to isotopy fixed near the boundary coincides with the classification of $2$-plane fields up to homotopy rel boundary. An immediate consequence of that classification is that the contact structure on $Y$ obtained from $\xi$ by a Lutz twist along $T$ is (up to isotopy) independent of any of the choices in the construction described above.
The surgery diagram for a Lutz twist
====================================
Let $(Y,\xi )$ be a given contact $3$-manifold and $T$ a knot in $Y$ transverse to $\xi$. In order to describe a Legendrian link ${\mathbb {L}}$ in $Y$ such that $(+1)$-contact surgery on ${\mathbb {L}}$ has the same effect as a Lutz twist along $T$, we may assume by [@dige04] that $(Y,\xi )$ has been obtained from $S^3$ with its standard contact structure $\xi_{st}$ by contact $(\pm 1)$-surgery on a Legendrian link in $(S^3,\xi_{st})$, and thus can be represented by the front projection (to the $yz$-plane) of this Legendrian link, considered as a link in ${{\mathbb {R}}}^3$ with its standard contact structure $\xi_{st}=\ker (dz+x\, dy)$, which is contactomorphic to $(S^3,\xi_{st})$ with a point removed.
For the representation of Legendrian and transverse knots via their front projection we refer to [@efm01; @etny; @gomp98]. Beware that these three papers use three different conventions for writing the standard contact structure on ${{\mathbb {R}}}^3$. We follow the one from [@gomp98] (which is also that of [@geig]). The positive transversality condition $\dot{z}+x\dot{y}>0$ for a curve $t\mapsto (x(t),y(t),z(t))$ implies that in the front projection of a positively transverse knot there can be no vertical tangencies going downwards ($\dot{y}=0$, $\dot{z}<0$), and all but the crossing shown in Figure \[figure:transverse-not\] are possible.
From these front projections it is easy to describe the positive transverse push-off of an oriented Legendrian knot: smooth the up-cusps and replace the down-cusps by kinks (there is only one possibility for the sign of the crossing in this kink). Similarly, one can easily describe an oriented Legendrian knot whose positive transverse push-off is a given transverse knot, cf. [@efm01]:
- In the front projection of the given transverse knot (oriented positively), replace vertical (upwards) tangencies by cusps.
- By Figure \[figure:transverse-not\], in those crossings of a positively transverse knot that cannot be interpreted as the front projection of a Legendrian knot, at least one of the strands is pointing up $(\dot{z}>0$). If one adds a zigzag to that strand (if both are going up, either can be chosen), it is possible to realise the given crossing by the front projection of a Legendrian curve.
Therefore, the following theorem gives a complete surgery description of Lutz twists.
\[thm\] Let $L_1$ be an oriented Legendrian knot in $(Y,\xi )$, represented by the front projection of a Legendrian knot in $({{\mathbb {R}}}^3,\xi_{st})$ disjoint from the link describing $(Y,\xi )$. Let $L_2$ be the Legendrian push-off of $L_1$ with two additional up-zigzags (see Figure \[figure:lutz\]). Let $\xi'$ be the contact structure on $Y$ obtained from $\xi$ by contact $(+1)$-surgery on both $L_1$ and $L_2$, and $\xi''$ the contact structure obtained from $\xi$ by a simple Lutz twist along the positive transverse push-off $T$ of $L_1$. Then $\xi'$ and $\xi''$ are isotopic via an isotopy fixed outside a tubular neighbourhood of $L_1$.
The proof of this theorem proceeds as follows: First of all, we verify that the described surgeries on $L_1$ and $L_2$ taken together do not change the manifold $Y$. Secondly, we check that the resulting contact structure is overtwisted by exhibiting an explicit overtwisted disc. Then, again by Eliashberg’s classification of overtwisted contact structures, and thanks to the fact that the two surgeries only change the contact structure in a tubular neighbourhood of $L_1$ (which contains the overtwisted disc just mentioned), it suffices to show that the described surgeries and the corresponding Lutz twist have the same effect on the homotopy class of the contact structure, regarded as a mere plane field.
\(1) Recall that contact $r$-surgery on a Legendrian knot $L$ means that topologically we perform surgery with coefficient $r\in{{\mathbb {Q}}}^*\cup\{\infty\}$ relative to the contact framing of $L$, which is determined by a vector field along $L$ transverse to the contact structure. In the front projection picture this corresponds to pushing $L$ in $z$-direction, and it is this what we mean by [*the*]{} Legendrian push-off of $L$. As shown in [@dige01], contact $(+1)$-surgery along $L$ and contact $(-1)$-surgery along its Legendrian push-off cancel each other, and in particular do not change the underlying manifold. Since adding two zigzags to a Legendrian knot adds two negative twists to its contact framing, we see that topologically the two contact $(+1)$-surgeries on $L_1$ and $L_2$ are the same as a contact $(+1)$-surgery along $L_1$ and a $(-1)$-surgery along its Legendrian push-off, and hence topologically trivial.
We can easily see this directly: Write $t$ for the Thurston-Bennequin invariant of $L_1$, so that the linking number between $L_1$ and its Legendrian push-off (or with $L_2$) is given by ${{\ell k}}(L_1,L_2)=t$. Then the topological framings (i.e., framings relative to the surface framing) of the surgeries are $n_1=t+1$ and $n_2=t-1$. After a handle slide (cf. [@gost99 Chapter 5]) we may replace the link $(L_1,L_2)$ by $(L_1,L_2-L_1)$, with linking number ${{\ell k}}(L_1,L_2-L_1)=
t-n_1=-1$ and framing of $L_2-L_1$ equal to $$(L_2-L_1)^2=L_2^2+L_1^2-2\,{{\ell k}}(L_2,L_1)=n_2+n_1-2t=0.$$ By construction, $L_2-L_1$ is an unknot, and the computation above shows that it is a $0$-framed meridian of $L_1$, which proves the claim that the composition of the two surgeries is topologically trivial.
\(2) We next exhibit the overtwisted disc in the manifold obtained by the contact surgeries along $L_1$ and $L_2$. Let $K$ be the knot indicated in Figure \[figure:lutz\], i.e., the Legendrian push-off of $L_1$ with one additional zigzag and with one extra negative linking with $L_2$. Alternatively, $L_2$ may be regarded as the Legendrian push-off of $K$ with one additional zigzag. The surface framing of $L_2$ determined by the Seifert surface of the oriented link $(-L_2)\sqcup K$ indicated in Figure \[figure:lutz\] is equal to $t-1$ (the contact framing of $K$), hence equal to the topological framing used for the surgery on $L_2$. So that Seifert surface glued to the meridional disc used for the surgery on $L_2$ defines a disc with boundary $K$ in the surgered manifold. The surface framing of $K$ determined by that disc equals the contact framing $t-1$, which is exactly the condition for an overtwisted disc.
The above verification that $K$ is the boundary of an overtwisted disc in the surgered manifold is completely straightforward. Nonetheless, it may be instructive to see that $K$ is not found by accident. Start with a meridian $K'$ to both $L_1$ and $L_2$, that is, an unknot with ${{\ell k}}(K',L_1)=
{{\ell k}}(K',L_2)=1$. If surgery along $L_1$ and $L_2$ has any chance of being a Lutz twist, we expect $K'$ to be isotopic to the boundary of an overtwisted disc.
There is an obvious pair of pants with boundary the oriented link $L_1\sqcup (-K')\sqcup (-L_2)$ that gives $K'$ the surface framing $n_{K'}=0$ and $L_1,L_2$ the framing $t+1, t-1$, respectively. Now perform a handle slide of $-K'$ over the $2$-handle attached to $L_1$ (corresponding to the surgery) to form, in the surgered manifold, the knot $L_1-K'$, which is the knot $K$ from above. We compute the linking numbers $$\begin{aligned}
{{\ell k}}(L_1-K',L_1) & = & n_1-1\; = \; t,\\
{{\ell k}}(L_1-K',L_2) & = & {{\ell k}}(L_1,L_2)-1\; = \; t-1,\end{aligned}$$ and the surface framing of $L_1-K'$, now with respect to the annulus with boundary $(L_1-K')\sqcup (-L_2)$: $$\begin{aligned}
(L_1-K')^2 & = & n_1+n_{K'}-2\,{{\ell k}}(L_1,K')\\
& = & (t+1)+0-2\cdot 1\\
& = & t-1,\end{aligned}$$ which is exactly what we had found for $K$ before.
\(3) It remains to be shown that the topological effect of the surgeries described in the theorem has the same effect on the homotopy class of the contact structure (regarded merely as a plane field) as a Lutz twist. By the neighbourhood theorem for Legendrian submanifolds (cf. [@geig]), the particular nature of $L_1$ is irrelevant for this consideration. It therefore suffices to consider specific examples for $L_1$, where the effect of the surgeries on the obstruction classes determining the homotopy type of the plane field can be computed explicitly.
For the following considerations cf. [@geig]. The tangent bundle of the solid torus $S^1\times D^2$ being trivial, (cooriented) tangent $2$-plane fields on $S^1\times D^2$ can be identified with maps $S^1\times D^2\rightarrow S^2$. Thus, the obstructions to homotopy of $2$-plane fields on $S^1\times D^2$ rel boundary $T^2$ are in $$H^2(S^1\times D^2,T^2;\pi_2(S^2))\cong {{\mathbb {Z}}}$$ and $$H^3(S^1\times D^2,T^2;\pi_3(S^2))\cong{{\mathbb {Z}}}.$$ The first obstruction corresponds to the extension of a given $2$-plane field along $T^2$ over a meridional disc of the solid torus and is detected by the (relative) first Chern class of the plane field (here the absence of $2$-torsion is crucial). The second obstruction relates to the extension of the plane field over the $3$-cell one needs to attach to $T^2\cup$ (meridional disc) to form the solid torus. This obstruction is captured by the Hopf invariant.
(3a) In order to deal with the first obstruction, we consider $Y=S^1\times S^2
\subset S^1\times{{\mathbb {R}}}^3$ with its standard tight contact structure $\xi =\ker (x\, d\theta +y\, dz-z\, dy)$, in obvious notation, and take $L_1$ to be an oriented Legendrian knot in the homology class of $S^1\times
\{ pt.\}$. The contact manifold $(S^1\times S^2,\xi )$ can be represented by contact $(+1)$-surgery on a Legendrian unknot $L_0$ with only two cusps, see [@dgs04]. For $L_1$ we take another such unknot linked once with $L_0$, and for $L_2$ its Legendrian push-off with additional zigzags as in the theorem. Write $\xi'$ for the contact structure on $Y$ obtained by performing contact $(+1)$-surgery on $L_1$ and $L_2$.
The contact structure $\xi$ has first Chern class $c_1(\xi )=0$. This follows from the observation that the vector field $$(z-y)\, \partial_x+x\,\partial_y-x\,\partial_z+(y+z)\,\partial_{\theta}$$ defines a trivialisation of $\xi$. Alternatively, this is a consequence of the homological computations in Section 3 of [@dgs04], given the fact that the rotation number ${\mbox{\rm rot}}(L_0)$ (with any orientation on $L_0$), which can be computed from the front projection as $(\# \mbox{\rm (down-cusps)}-\# \mbox{\rm (up-cusps)})/2$, is equal to $0$.
In the sequel we assume that the reader is familiar with those homological computations. Write $\mu_1,\mu_2$ for the meridional circles to $L_1,L_2$, respectively, as well as the homology classes they represent in the homology of the surgered manifold. Then, with $PD$ denoting the Poincaré duality isomorphism from cohomology to homology, $$\begin{aligned}
c_1(\xi') & = & {\mbox{\rm rot}}(L_1)PD^{-1}(\mu_1)+{\mbox{\rm rot}}(L_2)PD^{-1}(\mu_2)\\
& = & -2PD^{-1}(\mu_2).\end{aligned}$$ (This would be true even if ${\mbox{\rm rot}}(L_1)\neq 0$, since $\mu_1+\mu_2$ bounds a disc in $Y$ also after the surgery.)
Let $L_1'$ be a Legendrian push-off of $L_1$. Then the surgery along $L_1$ and $L_2$ may be assumed to occur in a tube containing $L_1$ and $L_2$, but not $L_1'$. This implies that $L_1'$ represents the same homology class in $H_1(Y)$ both before and after the surgery. Since ${{\ell k}}(L_1',L_2)=t$ and along $L_2$ we perform surgery with topological framing $t-1$, we have that $L_1'-\mu_2$ is homologically trivial in the surgered manifold. Hence $$\mu_2=[L_1']=[L_1]\in H_1(Y),$$ so that $$c_1(\xi')=-2PD^{-1}([L_1]),$$ which is the same as for a Lutz twist along the positive transverse push-off of $L_1$ (i.e., a transverse knot in the homology class of $L_1$), see [@geig Prop. 3.15].
Since $[L_1]$ generates $H_1(Y)$ in this example, this fully determines the effect of the surgery on the $2$-dimensional obstruction class.
(3b) Finally, in order to see that the effect that the surgery on the link ${\mathbb {L}}= L_1\sqcup L_2$ has on the $3$-dimensional obstruction is the same as that of a Lutz twist along a positive transverse push-off of $L_1$, it is sufficient to consider an arbitrary oriented Legendrian knot $L_1$ in $(S^3,\xi_{st})$. Set $r={\mbox{\rm rot}}(L_1)$, so that ${\mbox{\rm rot}}(L_2)=r-2$. As before we write $t$ for the Thurston-Bennequin invariant of $L_1$, so that the Thurston-Bennequin invariant of $L_2$ equals $t-2$. Let $X$ be the handlebody obtained from $D^4$ by attaching two $2$-handles corresponding to the two surgeries. Let $c\in H^2(X)$ be the cohomology class that evaluates to ${\mbox{\rm rot}}(L_i)$ on the surface in $X$ given by gluing a Seifert surface (with induced orientation) of $L_i$ in $D^4$ with the core disc of the corresponding handle, $i=1,2$. Since we perform $q=2$ contact $(+1)$-surgeries, Corollary 3.6 of [@dgs04] tells us that the $3$-dimensional invariant of the contact structure $\xi'$ obtained by these surgeries is given by $$\begin{aligned}
d_3(\xi') & = & \frac{1}{4}\bigl( c^2-3\sigma (X)-2\chi (X)\bigr) +q\\
& = & \frac{1}{4}c^2-\frac{3}{4}\sigma (X)+\frac{1}{2}.\end{aligned}$$ The signature $\sigma (X)$ is the signature of the matrix $\left(\begin{array}{cc}t+1&t\\t&t-1\end{array}\right)$, hence equal to $0$. Moreover, by that same formula we have $d_3(\xi_{st})=-1/2$. So the change in the $d_3$-invariant caused by the surgery is $$d_3(\xi')-d_3(\xi_{st})=\frac{1}{4}c^2+1.$$ As shown in Section 3 of [@dgs04], $c^2$ can be computed as $ar+b(r-2)$, where $(a,b)$ is the solution of $$\left(\begin{array}{cc}t+1&t\\t&t-1\end{array}\right)
\left(\begin{array}{c}a\\b\end{array}\right) =
\left(\begin{array}{c}r\\r-2\end{array}\right) .$$ This yields $a=r-2t$ and $b=2-r+2t$, hence $c^2=4r-4t-4$ and finally $d_3(\xi')-d_3(\xi_{st})=r-t$. This is exactly minus the so-called self-linking number $l(T)$ of the positive transverse push-off $T$ of $L_1$, cf. [@etny].
As shown in [@geig], the relative $d_3$-invariant $d_3(\xi'',\xi_{st})$, measuring the obstruction to homotopy over the $3$-skeleton between $\xi_{st}$ and the contact structure $\xi''$ obtained by a Lutz twist along $T$, equals $l(T)$. Thus, to conclude the proof one would need to verify that the absolute $d_3$-invariant of [@gomp98] and the relative $d_3$-invariant of [@geig] are, in the case at hand, related by $$d_3(\xi_1,\xi_2)=d_3(\xi_2)-d_3(\xi_1).
\label{eqn:d3}$$
This can be done by looking at explicit geometric models, though, as always, it is difficult to keep track of signs. So here is a more roundabout algebraic argument. Let $\xi_{\pm 1}$ be the contact structure obtained from $\xi_{st}$ by a Lutz twist along a transverse knot $T_{\mp 1}$ with self-linking number $l(T_{\mp 1})=\mp 1$ (this sign convention will be explained below); recall that the self-linking number is independent of the orientation of the transverse knot. For any natural number $n$, write $\xi_{\pm n}$ for the contact structure on $S^3$ given by taking the connected sum of $n$ copies of $(S^3,\xi_{\pm 1})$. The additivity of the relative $d_3$-invariant implies $d_3(\xi_{\pm n},\xi_{st})=\mp n$, which means that we get a contact structure on $S^3$ in each homotopy class of tangent $2$-plane fields.
The absolute $d_3$-invariant — for $2$-plane fields on $S^3$ — takes all the values in ${{\mathbb {Z}}}+1/2$, with $d_3(\xi_{st})=-1/2$. By [@dgs04 Lemma 4.2], it satisfies the additivity rule $$d_3(\eta_1\#\eta_2)=d_3(\eta_1)+d_3(\eta_2)+\frac{1}{2}.$$
These observations imply equation (\[eqn:d3\]) up to sign. We conclude $$d_3(\xi'',\xi_{st})=l(T)=d_3(\xi_{st})-d_3(\xi')=\pm d_3(\xi',\xi_{st}).$$ By the considerations in (3a), we know that the extension of the contact structure over a meridional disc is the same, up to homotopy, for surgery on ${\mathbb {L}}$ or Lutz twist along $T$. From the fact that there are standard models for the tubular neighbourhood of a Legendrian or transverse knot, respectively, we infer that $d_3(\xi',\xi_{st})$ and $d_3(\xi'',\xi_{st})$ can only differ by a constant term independent of the specific knot (corresponding to a different extension of the $2$-plane field over the $3$-cell attached to $T^2\cup$ (meridional disc)). Hence, the equation above can only hold if that constant is zero and the sign is the positive one. In turn, this yields equation (\[eqn:d3\]) in full generality.
(Our definition of $\xi_{\pm n}$ then entails $d_3(\xi_1)=1/2$ and $d_3(\xi_{-1})=-3/2$, which accords with our labelling of these structures in [@dgs04].)
This concludes the proof of the theorem.
If one defines $L_2$ by adding two down-zigzags instead of up-zigzags, in (3a) one obtains $c_1(\xi')=2PD^{-1}([L_1])$. This is the same as for a Lutz twist along the negative transverse push-off $T_-$ of $L_1$, since $T_-$ with the orientation that makes it positively transverse to $\xi$ represents the class $-[L_1]$. Similarly, with this $L_2$ we find in (3b) that $d_3(\xi')-d_3(\xi_{st})$ is equal to minus the self-linking number $t+r$ of the negative transverse push-off of $L_1$. Therefore, this choice of $L_2$ amounts to performing a Lutz twist along $T_-$.
F. D. is partially supported by grant no. 10201003 of the National Natural Science Foundation of China. H. G. is partially supported by grant no.GE 1254/1-1 of the Deutsche Forschungsgemeinschaft within the framework of the Schwerpunktprogramm 1154 “Globale Differentialgeometrie”. A. S. is partially supported by OTKA T034885.
[99]{}
, [*Symplectic fillability of tight contact structures on torus bundles*]{}, Algebr. Geom. Topol., 1 (2001), pp. 153–172.
, [*A Legendrian surgery presentation of contact $3$-manifolds*]{}, Math. Proc. Cambridge Philos. Soc., to appear.
, [*Surgery diagrams for contact $3$-manifolds*]{}, Turkish J. Math., to appear (Proceedings of the 10th Gökova Geometry-Topology conference).
, [*Classification of overtwisted contact structures on $3$-manifolds*]{}, Invent. Math., 98 (1989), pp. 623–637.
, [*Topological characterization of Stein manifolds of dimension $>2$*]{}, Internat. J. of Math., 1 (1990), pp. 29–46.
, [*Chekanov-Eliashberg invariants and transverse approximations of Legendrian knots*]{}, Pacific J. Math., 201 (2001), pp. 89–106.
, [*Legendrian and transversal knots*]{}, in Handbook of Knot Theory, Elsevier, to appear.
, [*On symplectic cobordisms*]{}, Math. Ann., 323 (2001), pp. 31–39.
, [*Contact geometry*]{}, in Handbook of Differential Geometry vol. 2, F.J.E. Dillen and L.C.A. Verstraelen, eds., Elsevier, to appear.
, [*Handlebody construction of Stein surfaces*]{}, Ann. of Math. (2), 148 (1998), pp. 619–693.
, [*$4$-Manifolds and Kirby Calculus*]{}, Grad. Stud. in Math. 20, American Mathematical Society, Providence, 1999.
, [*Formes de contact sur les variétés de dimension $3$*]{}, in Proceedings of the Liverpool Singularities Symposium II, Lecture Notes in Math. 209, Springer-Verlag, Berlin, 1971, pp. 142–163.
, [*Surgery on Contact $3$-Manifolds and Stein Surfaces*]{}, Bolyai Society Mathematical Studies, to appear.
, [*Contact surgery and symplectic handlebodies*]{}, Hokkaido Math. J., 20 (1991), pp. 241–251.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'It is shown that the second order moment of helicity distribution (the Levich-Tsinober invariant of inviscid hydrodynamics) can be preserved as an adiabatic invariant in magnetohydrodynamics. The distributed chaos approach and inertial range phenomenology have been developed for the magnetohydrodynamic processes dominated by this adiabatic invariant, and some consequences of the theoretical consideration have been compared with results of direct numerical simulations and measurements in solar wind.'
author:
- 'A. Bershadskii'
title: 'Helicity, low-dimensional distributed chaos and scaling in MHD turbulence'
---
Introduction
============
The multiplicity of regimes in fluids dynamics with active scalars and vectors (e.g. in the thermal convection and magnetohydrodynamics) is related to multiplicity of the invariants in such systems.
For instance, the helicity $$H = \int h({\bf x},t)~ d{\bf x}, \eqno{(1)}$$ where the helicity density $$h ({\bf x},t) = {\bf u} ({\bf x},t) \cdot {\boldsymbol \omega} ({\bf x}, t), \eqno{(2)}$$ ${\bf u} ({\bf x},t)$ is the velocity field and ${\boldsymbol \omega} ({\bf x}, t)= [\nabla \times {\bf u}]$ is the vorticity field, is a fundamental inviscid hydrodynamic invariant (see for a review Ref. [@mt]). In magnetohydrodynamics (MHD), however, the helicity Eq. (1) is not invariant and the magnetic helicity and the cross-helicity $$H_{cr} = \int {\bf u}({\bf x},t)\cdot {\bf B}({\bf x},t) d{\bf x}, \eqno{(3)}$$ where ${\bf B}({\bf x},t)$ is the magnetic filed, take its place (see the Ref. [@mt] and references therein).\
It was shown (see Refs. [@mt],[@lt] and references therein) that under certain rather general conditions the second order moment of helicity density is also an inviscid hydrodynamic invariant (the Levich-Tsinober invariant), and it will be shown in present paper that this invariant (unlike the helicity itself) can be preserved also in the magnetohydrodynamics as an adiabatic invariant.\
The MHD regimes dominated by this adiabatic invariant have been studied in present paper using the distributed chaos approach and inertial range phenomenology, and compared with the results of direct numerical simulations (DNS) and measurements in the solar wind.\
Second order moments of helicity distribution
=============================================
The magnetohydrodynamics of incompressible fluids is described by equations: $$\frac{\partial {\bf u}}{\partial t} = - {\bf u} \cdot \nabla {\bf u}
-\frac{1}{\rho} \nabla {\cal P} - {\bf F}_L + \nu \nabla^2 {\bf u} + {\bf f_u} \eqno{(4)}$$ $$\frac{\partial {\bf b}}{\partial t} = \nabla \times ( {\bf u} \times
{\bf b}) +\eta \nabla^2 {\bf b} + {\bf f_b} \eqno{(5)}$$ $$\nabla \cdot {\bf u}=0, ~~~~~~~~~~~\nabla \cdot {\bf b}=0, \eqno{(6)}$$ where the normalized magnetic field ${\bf b} = {\bf B}/\sqrt{\mu_0\rho}$ has the same dimension as velocity (the Alfvénic units), ${\bf F}_L = [{\bf b} \times (\nabla \times {\bf b})]$ is the Lorentz force, ${\bf f}_u$ and ${\bf f}_b$ are the external forcing functions (in the DNS these functions are usually concentrated in the large scales only).\
Dynamic equation for mean helicity can be readily obtained from the Eq. (4) and in the inviscid case ($\nu =0$) has the form $$\frac{d\langle h \rangle}{dt} = 2\langle {\boldsymbol \omega} \cdot (- {\bf F}_L + {\bf f_u}) \rangle \eqno{(7)}$$ where $\langle ... \rangle$ is an average over volume.
One can see from the Eq. (7) that the helicity is not an inviscid MHD invariant. However, if the correlation $\langle {\boldsymbol \omega} \cdot (- {\bf F}_L + {\bf f_u}) $ is considerable at large scales and it quickly becomes negligible for the smaller scales in a turbulent environment, then the second order moment of the helicity distribution (the Levich-Tsinober invariant of the inviscid hydrodynamics [@lt]) can be an inviscid quasi-invariant of the magnetohydrodynamics as well. To show this the volume of motion should be divided into the cells $V_j$ (with boundary conditions ${\boldsymbol \omega} \cdot {\bf n}=0$ on the surfaces of the cells - $S_j$), which move with the fluid [@mt]. For the cells with the small enough spatial scales (so that the correlation $\langle {\boldsymbol \omega} \cdot (- {\bf F}_L + {\bf f_u}) \rangle$ over the cell is negligible) the helicity density, averaged over the cell, can be considered as an adiabatic invariant in an inertial range of scales. We can use the following definition of the second order moment [@mt] $$I = \lim_{V \rightarrow \infty} \frac{1}{V} \sum_j H_{j}^2 \eqno{(8)}$$ with $$H_j = \int_{V_j} h({\bf x},t) ~ d{\bf x} \eqno{(9)}$$
For a strong MHD turbulence we can expect that the cells with the quasi-invariant $H_j^2$ determine the sum in the Eq. (8) (cf. Ref. [@bt] and references therein) and, as a consequence, $I$ is a quasi-invariant of the inviscid MHD dynamics and an adiabatic invariant in an inertial range of the viscid MHD turbulence.\
It should be also noted that in the non-dissipative case dynamics of the magnetic field ${\bf b}$ is described by equation $$\frac{\partial {\bf b}}{\partial t} = \nabla \times ( {\bf v} \times
{\bf b}) \eqno{(10)}$$ that is similar to the equation describing inviscid dynamics of vorticity ${\boldsymbol \omega}$ [@mt]. Therefore, the analogous consideration can be applied to the second order moment of the cross-helicity density $h_{cr} = {\bf v}({\bf x},t)\cdot {\bf b}({\bf x},t)$ (replacing ${\boldsymbol \omega}$ by ${\bf b}$) $$I_{cr} = \lim_{V \rightarrow \infty} \frac{1}{V} \sum_j H_{cr,j}^2, \eqno{(11)}$$ where $$H_{cr,j} = \int_{V_j} {\bf v}({\bf x},t)\cdot {\bf b}({\bf x},t) d{\bf x} \eqno{(12)}$$ Hence, the second order moment of the cross-helicity density $I_{cr}$ is an adiabatic invariant in an inertial range of the dissipative MHD turbulence.\
Taking into account the adiabatic invariance of the second order moment of helicity density $I$ we can obtain a relationship between characteristic velocity $u_c$ and characteristic wavenumber $k_c$ in an inertial range using dimensional considerations $$u_c \propto I^{1/4} k_c^{1/4} \eqno{(13)}$$
Analogously for the cross-helicity dominated case: $$u_c \propto I_{cr}^{1/4} k_c^{3/4}, \eqno{(14)}$$
Since in the Alfvénic units ${\bf b}$ has the same dimensionality as ${\bf u}$ the same dimensional considerations result in $$b_c \propto I^{1/4} k_c^{1/4} \eqno{(15)}$$ for the helicity dominated case and in $$b_c \propto I_{cr}^{1/4} k_c^{3/4}, \eqno{(16)}$$ for the cross-helicity dominated case.
Distributed chaos
=================
In fluid dynamics and in plasmas the exponential spectra $$E(k) \propto \exp-(k/k_c) \eqno{(17)}$$ are often appear at the onset of turbulent dynamics dominated by the deterministic chaos (see, for instance, Refs. [@mm],[@kds] and references therein). The further development of turbulent dynamics usually results in fluctuations of the parameter $k_c$ and one has to use an ensemble averaging $$E(k) \propto \int_0^{\infty} P(k_c) \exp -(k/k_c)dk_c \eqno{(18)}$$ in order to obtain the turbulent spectra. Therefore, an estimation of the probability distribution $P(k_c)$ is crucial for this purpose.\
A natural generalization of the exponential spectra Eq. (17) is the stretched exponential spectrum $$E(k) \propto \exp-(k/k_{\beta})^{\beta} \eqno{(19)}$$ Then, the problem is to find the parameter $\beta$ in the Eq. (19). For the stretched exponential spectra Eq. (19) an asymptotic of the $P(k_c)$ at large $k_c$ can be readily obtained form comparison of the Eqs. (18) and (19) [@jon] $$P(k_c) \propto k_c^{-1 + \beta/[2(1-\beta)]}~\exp(-\gamma k_c^{\beta/(1-\beta)}) \eqno{(20)}$$ If we will write the relationships Eqs. (13) and (14) in a general form $$u_c \propto k_c^{\alpha} \eqno{(21)}$$ and assume that $u_c$ has a Gaussian distribution with zero mean [@my] a relationship $$\beta = \frac{2\alpha}{1+2\alpha} \eqno{(22)}$$ is following from the Eqs. (20) and (21). Then for the helicity dominated turbulence the Eq. (13) provides $\alpha = 1/4$ and from the Eq. (22) we obtain $\beta = 1/3$, i.e the kinetic energy spectrum $$E(k) \propto \exp-(k/k_{\beta})^{1/3} \eqno{(23)}$$ whereas for the cross-helicity dominated turbulence the Eq. (14) provides $\alpha = 3/4$ and from the Eq. (22) we obtain $\beta = 3/5$, i.e. the kinetic energy spectrum $$E(k) \propto \exp-(k/k_{\beta})^{3/5} \eqno{(24)}$$ Analogous magnetic energy spectra are following from the Eqs. (15) and (16) respectively.\
It should be noted that the second order moments of the helicity (cross-helicity) distribution $I$ ($I_{cr}$) can be substantial even in the case of negligible [*mean*]{} helicity (cross-helicty). Therefore, it is interesting to compare this approach with the approach of the Ref. [@b2] where the mean cross-helicty plays a crucial role.
Direct numerical simulations of the MHD turbulence
==================================================
In Ref. [@farge] results of direct numerical simulations of incompressible MHD turbulence with zero mean magnetic field in a periodic domain were reported. The random external solenoidal forces were concentrated in low wavenumbers: $k < 2.5$ (see the Refs. [@farge],[@ya] for more details). The forcing did not generate mean magnetic and cross-helicity (the initial magnetic and cross-helicity were negligible). The initial kinetic and magnetic energy spectra $E_0^u(k)=E_0^b(k) = C k^4 \exp-(k^2/2)$. The Taylor-Reynolds number based on velocity $R_{\lambda} = 159$.\
A threshold method of orthogonal wavelets was used by the authors of the Ref. [@farge] in order to split the current density and vorticity fields into incoherent (with many degrees of freedom) and coherent (with a few percent of the degrees of freedom only) parts. Figures 1 and 2 show (in the log-log scales) the kinetic and magnetic energy spectra obtained in this simulation. The spectral data were taken from Figs. 5a and 5b of the Ref. [@farge]. One can see that the energy spectra corresponding to the coherent (low-dimensional) part are slightly different from the entire spectra for the small scales (large $k$) only.\
Figures 3 and 4 show the kinetic and magnetic energy spectra with the dashed lines indicating the stretched exponential law Eq. (23). Comparison with the Figs. 1 and 2 allows us to conclude that the helical low-dimensional distributed chaos dominates the both kinetic and magnetic spectra in this DNS (see Ref. [@l] about relation between the Levich-Tsinober invariant and coherent states in turbulence).\
In Ref. [@mpm] direct numerical simulations of decaying MHD turbulence with a superposition of deterministic (a Beltrami flow, cf. the Ref. [@l]) and random parts as initial conditions were performed. The deterministic part of the initial conditions consists of the ABC flows [@arn] in the large-scale wavenumber range $1 < k < 3$, whereas the random part is represented by small-scale Gaussian fluctuations. The initial mean cross-helicity is negligible.
Figure 5 shows the magnetic energy spectrum observed at $t=1.6$ (the max. time of the system’s evolution shown in the Fig. 2b of the Ref. [@mpm] where the spectral data were taken from). The dashed line indicates the stretched exponential law Eq. (23).\
It should be noted that the both Refs. [@farge] and [@mpm] emphasized crucial role of the current and vorticity sheets in the MHD turbulence.\
It is shown in recent Ref. [@bsb] that significant large-scale magnetic fields can be generated by helical turbulence in near incompressible conducting fluids despite the presence of strong small-scale dynamo effects. In the DNS reported in the Ref. [@bsb] a helical $\delta$-correlated in time forcing generates vortical motions in a large-scale wavenumber range around $k_f =4$. The Reynolds number $Re = 3300$ and the magnetic Prandtl number $Pr_m = 0.1$.\
Figures 6 and 7 show the kinetic and magnetic energy spectra with the dashed lines indicating the stretched exponential law Eq. (24). The spectral data were taken from Fig. 1 (final spectra) of the Ref. [@bsb]. The dotted vertical arrows indicate position of the scale $k_{\beta}$.
Helical scaling in solar wind
=============================
In hydrodynamics an analogy of the Komogorov phenomenology $$E(k) \propto \varepsilon^{2/3} k^{-5/3} \eqno{(25)}$$ (where $\varepsilon =|d\langle {\bf u}^2 \rangle/dt|$ is energy dissipation rate) was applied to helicity in an inertial range of scales [@fr] (see also Ref. [@bt]) $$E(k) \propto \varepsilon_h^{2/3} k^{-7/3} \eqno{(26)}$$ where $\varepsilon_h =|d\langle h \rangle/dt|$.\
Since in the magnetohydrodynamics the mean helicity is not an inviscid invariant one cannot expect that this phenomenology will work in the magnetohydrodynamics (though in a very strong external magnetic field it can be applied to the case of quasi two-dimensional MHD turbulence [@bkt]). However, if the second order moment of helicity $I$ is an inviscid MHD invariant (see above) one can try to generalise the Kolmogov phenomelogy on this case as well. Unlike energy or helicity the $I$-invariant is not a quadratic invariant. Therefore, the $\varepsilon_I =|dI^{1/2}/dt|$ should be used for the Kolmogorov-like phenomenology in this case:\
$$E(k) \propto \varepsilon_I^{2/3} k^{-4/3} \eqno{(27)}$$
The Kolmogorov-like spectra Eq. (25) for magnetic field were observed at 1 AU for different phases of solar cycle (see, for instance, Refs. [@mg],[@bs] and references therein). However, as it follows from the previous consideration, simultaneous spectra for bulk velocity and for magnetic field can provide a more valuable information. In the Ref. [@prg] the power spectra of the proton bulk velocity and corresponding magnetic field in the solar wind were computed using the data obtained by ’in situ’ measurements at 1 AU (in the ecliptic plane ) acquired by the Wind spacecraft. The measurements were made near solar minimum: from 23 May to 16 July 1995. Figures 8 and 9 show the kinetic and magnetic energy spectra (the spectral data were taken from Fig. 6 of the Ref [@prg]).\
The solar wind mean velocity $|\langle {\bf V} \rangle|$ (in the spacecraft frame) is usually considerably larger than the corresponding velocity fluctuations. Therefore, the Taylor “frozen-in” hypothesis can be applied in this case (see, for instance, Refs. [@hb]), i.e the measured by a probe temporal dynamics reflects the spatial structures convected past the probe by the mean velocity. Hence the corresponding frequency spectra reflect the wavenumber ones with the replacement $k \simeq 2\pi f/ |\langle {\bf V} \rangle|$.\
The straight lines in the Figs. 8 and 9 are drawn in order to indicate the spectral laws Eqs. (25) and (27) (in the log-log scales). One can see that for the kinetic energy spectrum the inertial range of scales is dominated by the helical scaling Eq. (27), while for the magnetic energy spectrum there are two subranges of the inertial range: the large-scales subrange is dominated by the helical scaling Eq. (27) and the small-scale subrange is dominated by the Kolmogorov-like scaling Eq. (25).
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{
"pile_set_name": "ArXiv"
}
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abstract: |
In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in $n$-dimensional space $$\begin{cases}
u_t - J\ast u +u+d(u(t,x))=
\int_{\mathbb{R}^n} f_\beta
(y) b(u(t-\tau,x-y)) dy, \\
u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n,
\end{cases}$$ where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_\beta(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $0<\alpha\le 2$. After establishing the existence for both the planar traveling waves $\phi(x\cdot{\bf e}+ct)$ for $c\ge c_*$ ($c_*$ is the critical wave speed) and the solution $u(t,x)$ for the Cauchy problem, as well as the comparison principles, we prove that, all noncritical planar wavefronts $\phi(x\cdot{\bf e}+ct)$ are globally stable with the exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau}$ for $\mu_\tau>0$, and the critical wavefronts $\phi(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/\alpha}$. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function. These rates are optimal and the stability results significantly develop the existing studies for nonlocal dispersion equations.
Nonlocal dispersion equations, traveling waves, global stability, the Fisher-KPP equation, time-delays, weighted energy, Fourier transform.
AMS: 35K57, 34K20, 92D25
author:
- |
Rui Huang$^a$, Ming Mei$^{b,c}$ and Yong Wang$^{d}$\
\
[*$^a$School of Mathematical Sciences, South China Normal University*]{}\
[*Guangzhou, Guangdong, 510631, China*]{}\
[[email protected]]{}\
[and ]{}\
[*$^b$Department of Mathematics, Champlain College Saint-Lambert*]{}\
[*Quebec, J4P 3P2, Canada* ]{}\
[*$^c$Department of Mathematics and Statistics, McGill University*]{}\
[*Montreal, Quebec, H3A 2K6, Canada*]{}\
[[email protected]]{}\
[and]{}\
[*$^d$Institute of Applied Mathematics, Academy of Mathematics and Systems Science*]{}\
[*Chinese Academy of Sciences, Beijing, 100190, China*]{}\
[[email protected]]{}\
date:
title: Planar Traveling Waves For Nonlocal Dispersion Equation With Monostable Nonlinearity
---
Introduction {#int}
============
For the gradient flow to an order parameter describing the state of a solid material, for example, a perfect crystal with two different orientations, it is usually described by a convolution model of phase transition in the form [@Bates-Fife-Ren-Wang; @Chasseigne-Chaves-Rossi; @Cortazar-Elgueta-Rossi-Wolanski; @Ignat-Rossi-1; @Ignat-Rossi-2] $$u_t = J\ast u-u +F(u), \ \ (x,t)\in \mathbb{R}^n\times \mathbb{R}_+,
\label{01}$$ where $x=(x_1,x_2,\cdots,x_n)\in \mathbb{R}^n$, $J(x)$ is a non-negative and radial kernel with unit integral, and $$(J\ast u) (t,x)=\int_{\mathbb{R}^n} J(x-y) u(t,y) dy.
\label{02}$$ As showed in [@Chasseigne-Chaves-Rossi; @Coville-Dupaigne-1], when the kernel $J(x)$ has a second momentum, for example, $J$ is compact-supported or Gaussian-like kernel $J\sim e^{-x^2}$, its Fourier transform looks like $${\hat J}(\xi)=1-\mathcal{K}|\xi|^2 + o(|\xi|^2), \ \ \mathcal{K}>0,$$ then the effect of the nonlocal dispersion $J\ast u-u$ is almost the same to the linear diffusion $\mathcal{K}\Delta u$: $$J\ast u-u \ \ \approx \ \ \mathcal{K}\Delta u,$$ which informs us to expect that the behaviors of the solutions to the nonlocal dispersion equation and the linear diffusion equation are almost identical [@Chasseigne-Chaves-Rossi; @Cortazar-Elgueta-Rossi-Wolanski; @Ignat-Rossi-1; @Ignat-Rossi-2] $$u_t= J\ast u-u \ \ \Leftrightarrow \ \ u_t=\mathcal{K} \Delta u.$$ Notice that, comparing with the heat equations, the solutions for the nonlocal dispersion equations usually loss the spatial regularity, but have much better regularity in time, see Remark \[remark\] below for details.
In general, $J(x)$ may not have a second momentum, let us say, $${\hat J}(\xi)= 1-\mathcal{K}|\xi|^\alpha +o(|\xi|^\alpha) \ \
\mbox{as}\ \xi\rightarrow0 \ \mbox{ for } \ \alpha \in (0, 2).$$ One example is the Cauchy law by taking $J(x)=\frac{1}{1+|x|^2}$ which implies its Fourier transform mentioned above with $\alpha=1$. In this case, the behavior of the solutions to the nonlocal dispersion equation is almost identical to the fractional diffusion equation [@Chasseigne-Chaves-Rossi; @Cortazar-Elgueta-Rossi-Wolanski; @Ignat-Rossi-1; @Ignat-Rossi-2] $$u_t= J\ast u-u \ \ \Leftrightarrow \ \ u_t=\mathcal{K} \Delta^{\alpha/2} u.$$
Equation represents also the dynamical population model of single species in ecology [@Fife], where $u(t,x)$ is the density of population at location $x$ and time $t$, and $J(x-y)$ is thought of as the probability distribution of jumping from location $y$ to location $x$, and $J*u=\int_{\mathbb{R}^n}J(x-y) u(t,y) dy$ is the rate at which individuals are arriving to position $x$ from all other places, while $-u(x,t)=-\int_{\mathbb{R}^n}J(x-y) u(t,x) dy$ stands the rate at which they are leaving the location $x$ to travel to all other places. In this case, under the consideration of the effects from birth rate and death rate, the equation is usually written as follows $$u_t = J\ast u -u +b(u(t-\tau,x))-d(u(t,x)), \ \ (x,t)\in \mathbb{R}^n\times \mathbb{R}_+,
\label{03}$$ where $b(u(t-\tau, x)$ is the birth rate function, $d(u(t,x))$ is the death rate function, and $\tau>0$ is the mature age of the single species, which is usually called the [*time-delay*]{}. Furthermore, if we consider the distribution of all matured population, the effect of birth rate is then involved in whole space $\mathbb{R}^n$ [@Gourley-Wu; @Mei-Ou-Zhao; @So-Wu-Zou], and the equation is expressed as $$\frac{\partial u}{\partial t} - J\ast u +u+d(u(t,x))=
\int_{\mathbb{R}^n} f_\beta
(y) b(u(t-\tau,x-y)) dy,
\label{04}$$ where $f_\beta(y)$, with $\beta>0$, is the heat kernel in the form of $$f_\beta (y)=\frac{1}{(4\pi \beta)^\frac n2}
e^{\frac{-|y|^2}{4\beta}} \ \ \mbox{ with } \ \ \int_{\mathbb{R}^n}
f_\beta (y) dy =1. \label{1.2}$$ Notice that, by using the property of heat kernel $$\lim_{\beta\to 0^+} \int_{\mathbb{R}^n} f_\beta
(y) b(u(t-\tau,x-y)) dy= b(u(t-\tau,x)),$$ we then derive the equation as a limit of the equation by taking $\beta\to 0^+$, and further derive the regular nonlocal dispersion equation from the equation by taking the time-delay $\tau=0$ and $F(u)=b(u)-d(u)$. In particular, if we set $d(u)=u^2$ and $b(u)=u$, then, from we get the classical Fisher-KPP equation with nonlocal dispersion $$u_t=J\ast u-u + u(1-u). \label{05}$$ So, the equations and and all are the special cases of the equation .
In this paper, we will concentrate ourselves to the Cauchy problem for the more generalized equation with non-locality of birth rate $$\begin{cases}
\displaystyle \frac{\partial u}{\partial t} - J\ast u +u+d(u(t,x))=
\int_{\mathbb{R}^n} f_\beta
(y) b(u(t-\tau,x-y)) dy, \\
u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n.
\end{cases}
\label{1.1}$$ When $\tau=0$ (no time-delay), then the above equation is reduced to $$\begin{cases}
\displaystyle \frac{\partial u}{\partial t} - J\ast u +u+d(u)=
\int_{\mathbb{R}^n} f_\beta
(y) b(u(t,x-y)) dy, \\
u(0,x)=u_0(x), \ \ x\in \mathbb{R}^n.
\end{cases}
\label{1.1-2}$$ We will also discuss how the time-delay $\tau$ effects the property of the solutions.
For the equation in 1D case, when $F(u)$ is bistable, namely, two constant equilibria $u_-$ and $u_+$ both are the stable nodes (the typical example is the Huxley equation with $F(u)=u(u-a)(1-u)$ for $0<a<1$), Bates [*et al*]{} [@Bates-Fife-Ren-Wang] and Chen [@XChen] proved that the traveling waves are globally stable as $t\to+\infty$. In this paper, we consider another important type of equations with monostable nonlinearity. The typical example in this case is Fisher-KPP equation with $F(u)=u(1-u)$. Hence, throughout this paper, we assume that the death rate $d(u)$ and birth rate $b(u)$ capture the following characters of monostable nonlinearity:
1. There exist $u_-=0$ and $u_+>0$ such that $d(0)=b(0)=0$, $d(u_+)= b(u_+)$, and $d(u),b(u)\in C^2[0,u_+]$;
2. $ b'(0)>d'(0)\ge 0$ and $0\le
b'(u_+)<d'(u_+)$;
3. For $0\le u\le u_+$, $d'(u)\ge 0$, $b'(u)\geq 0$, $d''(u)\ge 0$, $b''(u)\le 0$.
These characters are summarized from the classical Fisher-KPP equation, see also the monostable reaction-diffusion equations in ecology, for example, the Nicholson’s blowflies equation [@MLLS1; @MLLS2; @Mei-Ou-Zhao; @So-Wu-Zou] with $$d(u)=\delta u \ \mbox{ and } \ b(u)= p ue^{-a u}, \ p>0, \delta>0, a>0$$ and $u_-=0$ and $u_+=\frac{1}{a}\ln \frac{p}{\delta}>0$ under the consideration of $1<\frac{p}{\delta}\le e$; and the age-structured population model [@Gourley2; @Gourley-Wu; @Mei-Ou-Zhao; @Mei-Wong; @Pan-Li-Lin] with $$d(u)=\delta u^2 \ \mbox{ and } \ b(u)=pe^{-\gamma\tau}u, \ \ \delta>0, \ p>0, \ \gamma>0,$$ and $u_-=0$ and $u_+=\frac{p}{\delta}e^{-\gamma\tau}$.
Clearly, under the hypothesis (H$_1$)-(H$_3$), both $u_-=0$ and $u_+>0$ are constant equilibria of the equation , and $u_-=0$ is unstable and $u_+$ is stable for the spatially homogeneous equation associated with , this is why we call the equation , including and and , as monostable.
On the other hand, we also assume the kernel $J(x)$ satisfying:
1. $\displaystyle J(x)=\prod^n_{i=1}J_i(x_i)$, where $J_i(x_i)$ is smooth, and $J_i(x_i)=J_i(|x_i|)\ge 0$ and $\displaystyle \int_\mathbb{R} J_i(x_i) dx_i=1$ for $i=1,2\cdots,n$, and $\int_{\mathbb{R}}|y_1|J_1(y_1)e^{-\lambda_* y_1} dy_1<\infty$ for $\lambda_*>0$ defined in and ;
2. Fourier transform of $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha + o(|\xi|^\alpha)$ $\xi\rightarrow0$ with $\alpha \in (0,2]$ and $\mathcal{K}>0$.
A [*planar traveling wavefront*]{} to the equation for $\tau\ge 0$ is a special solution in the form of $u(t,x)=\phi(x\cdot {\bf e}+ct)$ with $\phi(\pm\infty)=u_\pm$, where $c$ is the wave speed, ${\bf e}$ is a unit vector of the basis of $\mathbb{R}^n$. Without loss of generality, we can always assume ${\bf e}={\bf e}_1=(1,0,\cdots,0)$ by rotating the coordinates. Thus, the planar traveling wavefront $\phi(x\cdot {\bf e}_1+ct)=\phi(x_1+ct)$ satisfies, for $\tau\ge 0$, $$\begin{cases}
c\phi'-J\ast\phi+\phi+d(\phi)=\displaystyle\int_{\mathbb{R}^n}f_\beta(y)
b(\phi(\xi_1-y_1-c\tau))dy, \\
\phi(\pm\infty)=u_\pm,
\end{cases}
\label{2.2}$$ where $'=\frac{d}{d\xi_1}$ and $\xi_1=x_1+ct$. Let $$f_{i\beta}(y_i):=\frac{1}{(4\pi\beta)^{1/2}}e^{-\frac{y_i^2}{4\beta}}.
\label{2.2-2}$$ Then $$f_\beta(y):=\prod^n_{i=1} f_{i\beta}(y_i), \ \mbox{ and }
\int_{\mathbb{R}}f_{i\beta}(y_i)dy_i=1, \ \ i=1,2,\cdots,n,
\label{2.2-3}$$ and is reduced to, for $\tau\ge 0$, $$\begin{cases}
c\phi'-J_1\ast\phi+\phi+d(\phi)=\displaystyle\int_{\mathbb{R}}f_{1\beta}(y_1)
b(\phi(\xi_1-y_1-c\tau))dy_1, \\
\phi(\pm\infty)=u_\pm.
\end{cases}
\label{2.2-4}$$ The main purpose of this paper is to study the global asymptotic stability of planar traveling wavefronts of the equations and with or without time-delay, respectively, in particular, in the case of the [*critical wave*]{} $\phi(x_1+c_*t)$. Here the number $c_*$ is called the [*critical speed*]{} (or the [*minimum speed*]{}) in the sense that a traveling wave $\phi(x_1+ct)$ exists if $c\ge c_*$, while no traveling wave $\phi(x_1+ct)$ exists if $c<c_*$.
The nonlocal dispersion equation has been extensively studied recently. Chasseigne [*et al*]{} [@Chasseigne-Chaves-Rossi] and Cortazar [*et al*]{} [@Cortazar-Elgueta-Rossi-Wolanski] showed that the linear nonlocal dispersion equation (with $F=0$) is almost equivalent to the linear diffusion equation, and the asymptotic behavior of the solutions to the linear equation of nonlocal dispersion is exactly the same to the corresponding linear diffusion equation. Ignat and Rossi [@Ignat-Rossi-1; @Ignat-Rossi-2] further obtained the asymptotic behavior of the solutions to the nonlinear equation . Garc$\acute{\mbox{i}}$a-Meli$\acute{\mbox{a}}$n and Quir$\acute{\mbox{o}}$s [@Garcia-Quiros] investigated the blow up phenomenon of the solution to the equation with $F(u)=u^p$, and gave the Fujita critical exponent. Regarding the structure of special solutions to like traveling wave solutions, early in 1997 Bates [*et al*]{} [@Bates-Fife-Ren-Wang] and Chen [@XChen] established the existence of the traveling waves for with bistable nonlinearity, and proved their global stability. For with monostable nonlinearity, recently Coville and his collaborators [@Coville; @Coville-Davila-Martinez; @Coville-Dupaigne-1; @Coville-Dupaigne-2] studied the existence and uniqueness (up to a shift) of traveling waves. See also the existence/nonexistence of traveling waves by Yagisita [@Y] and the existence of almost periodic traveling waves by Chen [@Chen]. However, the stability of traveling waves for the nonlocal equation (including and ) with monostable nonlinearity is almost not related, except a special case for the fast waves with large wave speed to the 1D age-structured population model by Pan [*et al*]{} [@Pan-Li-Lin]. As we know, such a problem is also very significant but challenging, because the equations of Fisher-KPP type possess an unstable node, different from the bistable case, this unstable node usually causes a serious difficulty in the stability proof, particularly, for the critical traveling waves. The main interest in this paper is to investigate the stability of traveling waves to with $\tau>0$ and with $\tau=0$. An easy to follow method will be introduced for the stability proof to the nonlocal dispersion equations.
In this paper, we will first investigate the linearized equation of , and derive the optimal decay rates of the solution to the linearized equation by means of Fourier transform. This is a crucial step for get the optimal convergence for the nonlocal stability of traveling waves. Then, we will technically establish the global existence and comparison principles of the solution to the $n$-D nonlinear equation with nonlocal dispersion . Inspired by [@Meot] for the classical Fisher-KPP equations and the further developments by [@Mei-Ou-Zhao; @Mei-Ou-Zhao2], by ingeniously selecting a weight function which is dependent on the critical wave speed $c_*$, and using the weighted energy method and the Green function method with the comparison principles together, we will further prove that, all noncritical planar traveling waves $\phi(x\cdot {\bf e} +ct)$ are exponentially stable in the form of $t^{-\frac{n}{\alpha}}e^{-\mu_\tau}$ for some constant $\mu_\tau=\mu(\tau)$ such that $0<\mu_\tau\le \mu_0$ for $\tau\ge 0$; and all critical planar traveling waves $\phi(x\cdot {\bf e} +c_*t)$ are algebraically stable in the form of $t^{-\frac{n}{\alpha}}$. These convergence rates are optimal and the stability results significantly develop the existing studies on the nonlocal dispersion equations. We will also show that the time-delay $\tau$ will slow down the convergence of the the solution $u(t,x)$ to the noncritical planar traveling waves $\phi(x\cdot{\bf e}+ct)$ with $c>c_*$, and cause the higher requirement for the initial perturbation around the wavefronts.
For the stability of traveling waves to other modeling equations, we refer to the classical and significant contributions in [@AW2; @Bramson; @Chen-Guo-Wu; @FM; @Ga; @HR; @Huang; @Ka; @K; @KPP; @Lau; @MR; @MNT; @MLLS1; @MLLS2; @Mei-Ou-Zhao; @Mei-Wong; @Meot; @S; @SmZ; @U; @VVV; @Xin; @X] for reaction-diffusion equations and [@FS; @Goodman; @Kawashima-Matsumura-1; @Kawashima-Matsumura-2; @Matsumura-Mei; @Matsumura-Nishihara-1; @Matsumura-Nishihara-2; @SX; @ZS] for fluid dynamical systems, and the references therein.
The paper is organized as follows. In section 2, we will state the existence of the traveling waves, and their stability. In section 3, we will give the solution formulas to the linearized dispersion equations of and , and derive the optimal decay rates by Fourier transform with energy method together. In section 4, we will prove the global existence of the solution to and establish the comparison principle. In section 5, based on the results obtained in sections 3 and 4, by using the weighted energy method, we will further prove the stability of planar traveling waves including the critical and noncritical waves. Finally, in section 6, we will give some particular applications of our stability theory to the classical Fisher-KPP equation with nonlocal dispersion and the Nicholson’s blowflies model, and make a concluding remark to a more general case.
Before ending this section, we make some notations. Throughout this paper, $C>0$ denotes a generic constant, while $C_i>0$ and $c_i>0$ ($i=0,1,2,\cdots$) represent specific constants. $j=(j_1,j_2,\cdots, j_n)$ denotes a multi-index with non-negative integers $j_i\ge 0$ ($i=1,\cdots,n$), and $|j|=j_1+j_2+\cdots +j_n$. The derivatives for multi-dimensional function are denoted as $$\partial^j_x f(x):= \partial^{j_1}_{x_1}\cdots \partial^{j_n}_{x_n}f(x).$$ For a $n$-D function $f(x)$, its Fourier transform is defined as $$\mathcal{F}[f](\eta)=\hat{f}(\eta):=\int_{\mathbb{R}^n}
e^{-\mbox{i}x\cdot \eta} f(x) dx, \ \ \ \ \ \mbox{i}:=\sqrt{-1},$$ and the inverse Fourier transform is given by $$\mathcal{F}^{-1}[\hat{f}](x):=\frac{1}{(2\pi)^n} \int_{\mathbb{R}^n}
e^{\mbox{i}x\cdot \eta} \hat{f}(\eta) d\eta.$$ Let $I$ be an interval, typically $I = \mathbb{R}^n$. $L^p(I)$ ($p\ge 1$) is the Lebesque space of the integrable functions defined on $I$, $W^{k,p}(I)$ ($k\ge 0, p\ge 1$) is the Sobolev space of the $L^p$-functions $f(x)$ defined on the interval $I$ whose derivatives $\partial^j_x f$ with $|j|=k$ also belong to $L^p(I)$, and in particular, we denote $W^{k,2}(I)$ as $H^k(I)$. Further, $L^p_w(I)$ denotes the weighted $L^p$-space for a weight function $w(x)>0$ with the norm defined as $$\|f\|_{L^p_w}=\Big(\int_{I} w(x) \left | f(x) \right | ^p
dx\Big)^{1/p},$$ $W^{k,p}_w(I)$ is the weighted Sobolev space with the norm given by $$\|f\|_{W^{k,p}_w}=\Big( \sum_{|j|=0}^k \int_{I} w(x) \left |
\partial^j_x f(x)\right |^p dx\Big)^{1/p},$$ and $H^k_w(I)$ is defined with the norm $$\|f\|_{H^k_w}=\Big( \sum_{|j|=0}^k \int_{I} w(x) \left |
\partial^j_x f(x)\right | ^2 dx\Big)^{1/2}.$$ Let $T > 0$ be a number and ${\cal B}$ be a Banach space. We denote by $C^0([0,T],{\cal B})$ the space of the $\cal B$-valued continuous functions on $[0,T]$, $L^2([0,T], {\cal
B})$ as the space of the ${\cal B}$-valued $L^2$-functions on $[0,T]$. The corresponding spaces of the ${\cal B}$-valued functions on $[0,\infty)$ are defined similarly.
Traveling Waves and Their Stabilities
=====================================
As we mentioned before, the existence and uniqueness (up to a shift) of traveling waves for the equation were proved in [@Coville; @Coville-Davila-Martinez; @Coville-Dupaigne-1; @Coville-Dupaigne-2], particular, in a recent work by Yagisita [@Y] for the existence and nonexistence of traveling waves, when the nonlinearity $F(u)$ is monostable. Without any difficulty, these results can be extended to the nonlocal equation with time-delay with the help of comparison principle established in Section 4, when $d(u)$ and $b(u)$ satisfy the monostable features (H$_1$)-(H$_3$). We state these results as follows without detailed proof.
Under the conditions (H$_1$)-(H$_3$) and (J$_1$)-(J$_2$), for the time-delay $\tau\ge 0$, there exist a minimum wave speed (also called the critical wave speed) $c_*>0$ such that
1. when $c\ge c_*$, there exits a monotone traveling wavefront $\phi(x_1+ct)$ of connecting $u_\pm$ exists;
2. when $c<c_*$, no traveling wave $\phi(x_1+ct)$ exists.
Here $(c_*,\lambda_*)$ with $c_*>0$ and $\lambda_*>0$ is given by $$H_{c_*}(\lambda_*)=G_{c_*}(\lambda_*), \ \ \ \
H'_{c_*}(\lambda_*)=G'_{c_*}(\lambda_*), \label{2.3}$$ where $$H_{c}(\lambda)= b'(0) e^{\beta\lambda^2-\lambda c \tau}, \
G_{c}(\lambda)=c\lambda-E_c(\lambda) +d'(0),\
E_c(\lambda)=\int_{\mathbb{R}}J_1(y_1)e^{-\lambda y_1}dy_1-1,
\label{2.4}$$ namely, $(c_*,\lambda_*)$ is the tangent point of $H_{c}(\lambda)$ and $G_{c}(\lambda)$ specified as $$\begin{aligned}
b'(0)
e^{\beta\lambda_*^2-\lambda_* c_* \tau}&=&c_*\lambda_* -\int_{\mathbb{R}}J_1(y_1)e^{-\lambda_* y_1} dy_1 +1 + d'(0),\label{2.4''} \\
b'(0)
(2\beta\lambda_*-c_*\tau)e^{\beta\lambda_*^2-\lambda_*c_*\tau}&=&c_* + \int_{\mathbb{R}}y_1 J_1(y_1)e^{-\lambda_* y_1} dy_1.
\label{2.4'}\end{aligned}$$ Furthermore, it can be verified:
1. In the case of $c>c_*$, there exist two numbers depending on $c$: $\lambda_{1}=\lambda_1(c)>0$ and $\lambda_{2}=\lambda_2(c)>0$ as the solutions to the equation $H_c(\lambda_{i})=G_c(\lambda_{i})$, i.e., $$b'(0)
e^{\beta\lambda_{i}^2-\lambda_{i} c \tau}= c\lambda_{i}
-\int_{\mathbb{R}}J_1(y_1)e^{-\lambda_i y_1} dy_1 + d'(0), \ \ \ i=1,2, \label{2.4'-new}$$ such that $$H_c(\lambda)<G_c(\lambda) \ \ \ \mbox{ for } \
\lambda_{1}<\lambda<\lambda_{2}, \label{2.5}$$ and particularly, $$H_c(\lambda_*)<G_c(\lambda_*) \ \mbox{ with } \
\lambda_{1}<\lambda_*<\lambda_{2}. \label{2.4'-newnew}$$
2. In the case of $c=c_*$, it holds $$H_{c_*}(\lambda_*)=G_{c_*}(\lambda_*) \ \mbox{ with } \
\lambda_{1}=\lambda_*=\lambda_{2}. \label{2.5-new}$$
3. When $\xi_1=x_1+ct \to \pm\infty$, for all $c\ge c_*$, the traveling wavefronts $\phi(x_1+ct)$ converge to $u_\pm$ exponentially as follows $$\label{newnew}
|\phi(\xi_1)-u_\pm|=O(1) e^{-\lambda^\pm |\xi_1|}.$$ Here $\lambda^{-}=\lambda_1(c)>0$ is given in , and $\lambda^+=\lambda^+(c)>0$ is the unique root determined by the following equation $$\label{newnewnew}
-c\lambda^+-\int_{\mathbb{R}}J_1(y_1)e^{-\lambda^+ y_1} dy_1+d'(u_+)=b'(u_+) e^{\beta
(\lambda^+)^2-\lambda^+ c\tau}.$$
\[TW\]
For easily understanding all cases mentioned in the above, we show them in Figure \[fig\].
(a)![(a): the case of $c>c_*$; (b): the case of $c=c_*$; and (c): the case of $c<c_*$.[]{data-label="fig"}](fig1 "fig:"){width="28.00000%"} (b)![(a): the case of $c>c_*$; (b): the case of $c=c_*$; and (c): the case of $c<c_*$.[]{data-label="fig"}](fig2 "fig:"){width="28.00000%"} (c)![(a): the case of $c>c_*$; (b): the case of $c=c_*$; and (c): the case of $c<c_*$.[]{data-label="fig"}](fig3 "fig:"){width="28.00000%"}
Before stating our main stability theorems, let us technically choose a weight function: $$w(x_1)=\begin{cases} e^{-\lambda_* (x_1-x_*)}, & \ \mbox{ for }
x_1\le
x_*, \\
1, & \ \mbox{ for } x_1>x_*,
\end{cases}
\label{2.8-new}$$ where $\lambda_*=\lambda_*(c_*)>0$ is given in and , and $x_*>0$ is a sufficiently large number such that, $$0<d'(\phi(x_*))-\int_{\mathbb{R}^n} f_{\beta}(y)
b'(\phi(x_*-y_1-c\tau)) dy< d'(u_+)-b'(u_+).
\label{nov4-new}$$ The selection of $x_*$ in is valid, because of $d'(u_+)-b'(u_+)>0$ (see(H$_2$)). In fact, we have $$\begin{aligned}
\lim_{\xi_1\to \infty} d'(\phi(\xi_1))&=&d'(u_+)\\
&>&b'(u_+)\\
&=& \int_{\mathbb{R}^n} f_\beta(y)\Big[\lim_{\xi_1\to\infty} b'(\phi(\xi_1-y_1-c\tau))\Big] dy\\
&=&\lim_{\xi_1\to\infty}\int_{\mathbb{R}^n} f_\beta(y)
b'(\phi(\xi_1-y_1-c\tau))dy,\end{aligned}$$ which implies that, by (H$_3$), there exists a unique $x_*\gg 1$ such that, for $\xi_1\in [x_*,\infty)$ $$\begin{aligned}
& &d'(u_+)-b'(u_+) \notag \\
& &> d'(\phi(\xi_1))-\int_{\mathbb{R}^n} f_\beta(y)
b'(\phi(\xi_1-y_1-c\tau)) dy \notag \\
& &\ge d'(\phi(x_*))-\int_{\mathbb{R}^n} f_{\beta}(y)
b'(\phi(x_*-y_1-c\tau)) dy \notag \\
&&>0. \label{01-18}\end{aligned}$$
Under assumptions $(H_1)$-$(H_3)$ and $(J_1)$-$(J_2)$, for a given traveling wave $\phi(x_1+ct)$ of the equation with $c\ge c_*$ and $\phi(\pm\infty)=u_\pm$, if the initial data $u_0(s,x)$ is bounded in $[u_-,u_+]$ and $u_0-\phi\in
C([-\tau,0]; H^{m}_w(\mathbb{R}^n)\cap L^1_w(\mathbb{R}^n))$ and $\partial_s (u_0-\phi) \in L^1([-\tau,0]; H^{m}_w(\mathbb{R}^n)\cap
L^1_w(\mathbb{R}^n))$ with $m>\frac{n}{2}$, then the solution of uniquely exists and satisfies:
1. When $c>c_*$, the solution $u(t,x)$ converges to the noncritical planar traveling wave $\phi(x_1+ct)$ exponentially $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+ct)|\le
C(1+t)^{-\frac{n}{\alpha}}e^{-\mu_\tau t}, \ \ t> 0, \label{2.12}$$ where $$\label{mu-tau}
0<\mu_\tau < \min\{d'(u_+)-b'(u_+), \ \varepsilon_1[G_c(\lambda_*)-H_c(\lambda_*)]\},$$ and $\varepsilon_1=\varepsilon_1(\tau)$ such that $0<\varepsilon_1<1$ for $\tau>0$, and $\varepsilon_1=\varepsilon_1(\tau)\to 0^+$ as $\tau\to+\infty$;
2. When $c=c_*$, the solution $u(t,x)$ converges to the critical planar traveling wave $\phi(x_1+c_*t)$ algebraically $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+c_*t)|\le
C(1+t)^{-\frac{n}{\alpha}}, \ \ t> 0. \label{2.12-2}$$
\[thm1\]
However, when the time-delay $\tau=0$, then we have the following stronger stability for the traveling waves but with a weaker condition on initial perturbation.
Under assumptions $(H_1)$-$(H_3)$ and $(J_1)$-$(J_2)$, for a given traveling wave $\phi(x_1+ct)$ of the equation with $c\ge c_*$ and $\phi(\pm\infty)=u_\pm$, if the initial data $u_0(x)$ is bounded in $[u_-,u_+]$ and $u_0-\phi\in H^{m}_w(\mathbb{R}^n)\cap L^1_w(\mathbb{R}^n)$ with $m>\frac{n}{2}$, then the solution of uniquely exists and satisfies:
1. When $c>c_*$, the solution $u(t,x)$ converges to the noncritical planar traveling wave $\phi(x_1+ct)$ exponentially $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+ct)|\le
C(1+t)^{-\frac{n}{\alpha}}e^{-\mu_0 t}, \ \ t> 0, \label{2.12-new-1}$$ where $$\label{mu-0}
0<\mu_0 < \min\{d'(u_+)-b'(u_+), \ G_c(\lambda_*)-H_c(\lambda_*)\};$$
2. When $c=c_*$, the solution $u(t,x)$ converges to the critical planar traveling wave $\phi(x_1+c_*t)$ algebraically $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+c_*t)|\le
C(1+t)^{-\frac{n}{\alpha}}, \ \ t> 0. \label{2.12-new-2}$$
\[thm2\]
1. Comparing Theorem \[thm1\] with time-delay and Theorem \[thm2\] without time-delay, we realize that, the sufficient condition on the initial perturbation around the wave in the case with time-delay is stronger than the case without time-delay, but the convergence rate to the noncritical waves $\phi(x_1+ct)$ for $c>c_*$ in the case with time-delay is weaker than the case without time-delay, see for $\mu_\tau \le \varepsilon_1 [G_c(\lambda_*)-H_c(\lambda_*)]< G_c(\lambda_*)-H_c(\lambda_*)$, and for $\mu_0 \le G_c(\lambda_*)-H_c(\lambda_*)$, and $\varepsilon_1=\varepsilon_1(\tau)\to 0^+$ as $\tau\to +\infty$. This means, the time-delay $\tau>0$ effects the stability of traveling waves a lot, not only the higher requirement for the initial perturbation, but also the slower convergence rate for the solution to the noncritical traveling waves.
2. The convergence rates showed both in Theorem \[thm1\] and Theorem \[thm2\] are explicit and optimal, particularly, the algebraic decay rates for the solution converging to the critical waves. Actually, all of them are derived from the linearized equations.
Linearized Nonlocal dispersion Equations {#reults}
========================================
In this section, we will derive the solution formulas for the linearized nonlocal dispersion equations with or without time-delay, as well as their optimal decay rates, which will play a key role in the stability proof in section 5.
Now let us introduce the solution formula for linear delayed ODEs [@KIK] and the asymptotic behaviors of the solutions [@Mei-Wang].
\[lemKIK\] Let $z(t)$ be the solution to the following linear time-delayed ODE with time-delay $\tau>0$ $$\label{p1}
\begin{cases}
\displaystyle\frac{d}{dt}z(t)+k_1 z(t) =k_2 z(t-\tau) \\
z(s)=z_0(s), \ \ \ s\in[-\tau,0].
\end{cases}$$ Then $$z(t)=e^{-k_1(t+\tau)} e^{{\bar k_2}t}_\tau z_0(-\tau)
+\int^0_{-\tau} e^{-k_1(t-s)}e^{{\bar k_2}(t-\tau-s)}_\tau
[z_0'(s)+k_1 z_0(s)]ds , \label{p2}$$ where $${\bar k_2}:=k_2 e^{k_1 \tau}, \label{k_2}$$ and $e^{{\bar k_2}t}_\tau$ is the so-called [delayed exponential function]{} in the form $$e^{{\bar k_2} t}_\tau =\begin{cases}
0, & -\infty<t<-\tau, \\
1, & -\tau\le t<0, \\
1+\frac{{\bar k_2} t}{1!}, & 0\le t<\tau, \\
1+\frac{{\bar k_2} t}{1!}+\frac{{\bar k_2}^2(t-\tau)^2}{2!}, & \tau\le t<2\tau, \\
\vdots & \vdots \\
1+\frac{{\bar k_2} t}{1!}+\frac{{\bar k_2}^2(t-\tau)^2}{2!}+\cdots + \frac{{\bar k_2}^m[t-(m-1)\tau]^m}{m!}, & (m-1)\tau\le t<m\tau, \\
\vdots & \vdots
\end{cases}
\label{p3}$$ and $e^{\bar{k}_2t}_\tau$ is the fundamental solution to $$\label{p4}
\begin{cases}
\displaystyle\frac{d}{dt}z(t) ={\bar k_2} z(t-\tau) \\
z(s)\equiv 1, \ \ \ s\in[-\tau,0].
\end{cases}$$
\[lemma3\] Let $k_1\ge 0$ and $k_2\ge 0$. Then the solution $z(t)$ to (or equivalently ) satisfies $$|z(t)|\le C_0 e^{-k_1 t} e^{{\bar k_2}t}_\tau, \label{12-26-1}$$ where $$C_0:= e^{-k_1\tau}|z_0(-\tau)| + \int^0_{-\tau} e^{k_1 s}|z'_0(s)+k_1 z_0(s)| ds, \label{12-26-2}$$ and the fundamental solution $e^{{\bar k_2} t}_\tau $ with ${\bar k_2}>0$ to satisfies $$e^{{\bar k_2} t}_\tau \le C(1+t)^{-\gamma} e^{{\bar k_2} t}, \ \ t>0,
\label{p5}$$ for arbitrary number $\gamma>0$.
Furthermore, when $k_1\ge k_2\ge 0$, there exists a constant $\varepsilon_1=\varepsilon_1(\tau)$ with $0< \varepsilon_1 <1$ for $\tau>0$, and $\varepsilon_1=1$ for $\tau=0$, and $\varepsilon_1=\varepsilon_1(\tau)\to 0^+$ as $\tau\to +\infty$, such that $$e^{-k_1 t}e^{{\bar k_2}t}_\tau \le C e^{-\varepsilon_1(k_1-k_2) t}, \ \ t>0,
\label{12-26-3}$$ and the solution $z(t)$ to satisfies $$|z(t)|\le C e^{-\varepsilon_1(k_1-k_2) t}, \ \ t>0.
\label{12-26-4}$$
Now, we consider the following linearized nonlocal time-delayed dispersion equation (which will be derived in section 5 for the proof of stability of traveling wavefronts) $$\begin{aligned}
\label{p12}
\begin{cases}
\displaystyle \frac{\partial {v}}{\partial t}
-\int_{\mathbb{R}^n}J(y)e^{-\lambda_\ast y_1}v(t,x-y)dy +c_1
v \vspace{2mm} \\
\qquad\qquad \displaystyle = c_2 \int_{\mathbb{R}^n} f_\beta(y)
e^{-\lambda_*(y_1+c\tau)}v(t-\tau,x-y) dy ,
\label{2010-13} \vspace{2mm} \\
v(s,x)=v_0(s,x), \ \ s\in [-\tau,0], \ x\in \mathbb{R}^n
\end{cases}\end{aligned}$$ for some given constant coefficients $c$, $c_1$ and $c_2$, where $c\ge c_*$ is the wave speed.
We are going to derive its solution formula as well as the asymptotic behavior of the solution. By taking Fourier transform to , and noting that, $$\begin{aligned}
& &\mathcal{F}\Big[\int_{\mathbb{R}^n} J(y)e^{-\lambda_* y_1} v(t,x-y) dy\Big](t,\eta) \notag \\
& &=\int_{\mathbb{R}^n} e^{-\mbox{i}x\cdot \eta}\Big(\int_{\mathbb{R}^n} J(y)e^{-\lambda_* y_1} v(t,x-y) dy\Big) dx \nonumber \\
& &=\int_{\mathbb{R}^n} J(y)e^{-\lambda_* y_1} \Big(\int_{\mathbb{R}^n} e^{-\mbox{i}x\cdot \eta} v(t,x-y)dx \Big) dy \notag \\
& &=\int_{\mathbb{R}^n} J(y)e^{-\lambda_* y_1} \Big(\int_{\mathbb{R}^n} e^{-\mbox{i}(x+y)\cdot \eta} v(t,x)dx \Big) dy \notag \\
& &=\Big(\int_{\mathbb{R}^n} e^{-\mbox{i}y\cdot
\eta}J(y)e^{-\lambda_* y_1}dy\Big) \hat{v}(t,\eta), \label{p13-0}\end{aligned}$$ and $$\begin{aligned}
\label{p15}
& & \mathcal{F}\Big[c_2 \int_{\mathbb{R}^n} f_\beta (y) e^{-\lambda_*(y_1+c \tau)} v(t-\tau,x-y) dy\Big](t-\tau,\eta) \notag \\
& &=c_2\int_{\mathbb{R}^n} e^{-\mbox{i}x\cdot \eta}
\Big(\int_{\mathbb{R}^n} f_\beta (y)
e^{-\lambda_*(y_1+c \tau)} v(t-\tau,x-y) dy\Big) dx \notag \\
& &=c_2 \int_{\mathbb{R}^n} f_\beta (y) e^{-\lambda_*(y_1+c \tau)}
\Big( \int_{\mathbb{R}^n} e^{-\mbox{i}x\cdot \eta} v(t-\tau,x-y)
dx\Big) dy
\notag \\
& &=c_2 \int_{\mathbb{R}^n} f_\beta (y) e^{-\lambda_*(y_1+c \tau)}
\Big( \int_{\mathbb{R}^n} e^{-\mbox{i}(x+y)\cdot \eta} v(t-\tau,x)
dx\Big) dy
\notag \\
& &=c_2 \int_{\mathbb{R}^n} f_\beta (y) e^{-\lambda_*(y_1+c \tau)} e^{-\mbox{i}y\cdot \eta}\Big( \int_{\mathbb{R}^n} e^{-\mbox{i}x\cdot \eta} v(t-\tau,x) dx\Big) dy \notag \\
& &=\Big(c_2 \int_{\mathbb{R}^n} f_\beta (y) e^{-\lambda_*(y_1+c \tau)} e^{-\mbox{i}y\cdot \eta}dy\Big) \hat{v}(t-\tau,\eta),
\end{aligned}$$ we have $$\label{p16}
\frac{d \hat{v}}{dt} + A(\eta)\hat{v}=B(\eta) \hat{v}(t-\tau,\eta),
\ \ \mbox{ with } \hat{v}(s,\eta)=\hat{v}_0(s,\eta), \
s\in[-\tau,0],$$ where $$\begin{aligned}
\label{p15-2}
A(\eta):=c_1-\int_{\mathbb{R}^n} J(y)e^{-\lambda_\ast
y_1}e^{-\mbox{i}y\cdot \eta}dy\end{aligned}$$ and $$B(\eta):=c_2 \int_{\mathbb{R}^n} f_\beta (y) e^{-\lambda_*(y_1+c
\tau)} e^{-\mbox{i}y\cdot \eta}dy. \label{p15-1}$$ By using the formula of the delayed ODE in Lemma \[lemKIK\], we then solve as follows $$\begin{aligned}
\hat{v}(t,\eta)&=& e^{-A(\eta)(t+\tau)}e^{\mathcal{B}(\eta)t}_\tau \hat{v}_0(-\tau,\eta) \notag \\
&
&+\int^0_{-\tau}e^{-A(\eta)(t-s)}e^{\mathcal{B}(\eta)(t-\tau-s)}_\tau
\Big[\partial_s \hat{v}_0(s,\eta) + A(\eta) \hat{v}_0(s,\eta)\Big]ds,
\label{p17}\end{aligned}$$ where $$\mathcal{B(\eta)}:=B(\eta) e^{A(\eta)\tau}. \label{12-26-7}$$ Then, by taking the inverse Fourier transform to , we get $$\begin{aligned}
v(t,x)&=& \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^n} e^{\mbox{i}x\cdot\eta} e^{-A(\eta)(t+\tau)}
e^{\mathcal{B(\eta)}t}_\tau \hat{v}_0(-\tau,\eta) d\eta
\notag \\
& &+\int^0_{-\tau} \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^n} e^{\mbox{i}x\cdot\eta} e^{-A(\eta)(t-s)}e^{\mathcal{B(\eta)}(t-\tau-s)}_\tau \notag \\
& & \ \ \times \Big[\partial_s\hat{v}_0(s,\eta) + A(\eta) \hat{v}_0(s,\eta)\Big] d\eta
ds, \label{p18-new}\end{aligned}$$ and its derivatives $$\begin{aligned}
\partial^k_{x_j}v(t,x)&=& \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^n} e^{\mbox{i}x\cdot\eta}(\mbox{i}\eta_j)^k
e^{-A(\eta)(t+\tau)}e^{\mathcal{B(\eta)}t}_\tau \hat{v}_0(-\tau,\eta) d\eta \notag \\
& &+\int^0_{-\tau} \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^n} e^{\mbox{i}x\cdot\eta}(\mbox{i}\eta_j)^k e^{-A(\eta)(t-s)}e^{\mathcal{B(\eta)}(t-\tau-s)}_\tau \notag \\
& & \ \ \ \ \ \ \ \ \ \ \ \times \Big[\partial_s\hat{v}_0(s,\eta)+
A(\eta)\hat{v}_0(s,\eta)\Big]d\eta ds \label{p18}\end{aligned}$$ for $k=0,1,\cdots$ and $ j=1,\cdots, n$.
Now we are going to derive the asymptotic behavior of $v(t,x)$.
\[lemma4\] Suppose that $v_0\in C([-\tau,0]; H^{m+1}(\mathbb{R}^n)\cap
L^1(\mathbb{R}^n))$ and $\partial_s v_0\in L^1([-\tau,0]; H^{m}(\mathbb{R}^n)\cap
L^1(\mathbb{R}^n))$ for $m\ge 0$, and let $$\label{p21}
\begin{cases}
\displaystyle \tilde{c}_1:=c_1-\int_{\mathbb{R}^n}J(y)e^{-\lambda_\ast y_1}dy, \vspace{2mm}\\
\displaystyle c_3:=c_2 \int_{\mathbb{R}^n} f_\beta (y)
e^{-\lambda_*(y_1+c\tau)}dy>0.
\end{cases}$$ If $\tilde{c}_1\geq c_3$, then there exists a constant $\varepsilon_1=\varepsilon_1(\tau)$ as showed in satisfying $0<\varepsilon_1<1$ for $\tau>0$, such that the solution of the linearized equation satisfies $$\| \partial^k_{x_j}v(t)\|_{L^2(\mathbb{R}^n)}\le C \mathcal{E}^k_{v_0}
t^{-\frac{n+2k}{2\alpha}}e^{-\varepsilon_1(\tilde{c}_1-c_3)t}, \
t>0, \label{4.14}$$ for $k=0,1, \cdots, [m]$ and $j=1,\cdots, n$, where $$\begin{aligned}
\mathcal{E}^k_{v_0}:&=& \|v_0(-\tau)\|_{L^1(\mathbb{R}^n)}+\|v_0(-\tau)\|_{H^{k}(\mathbb{R}^n)} \notag \\
&
&+\int_{-\tau}^{0}[\|(v'_{0s},v_0)(s)\|_{L^{1}(\mathbb{R}^n)}+\|(v'_{0s},v_0)(s)\|_{H^{k}(\mathbb{R}^n)}]ds.
\label{p20-1-1}\end{aligned}$$ Furthermore, if $m>\frac{n}{2}$, then $$\| v(t)\|_{L^\infty(\mathbb{R}^n)} \leq
C \mathcal{E}^m_{v_0}
t^{-\frac{n}{\alpha}}e^{-\varepsilon_1(\tilde{c}_1-c_3)t}, \ \ t>0.
\label{p20}$$ Particularly, when $\tilde{c}_1=c_3$, then $$\| v(t)\|_{L^\infty(\mathbb{R}^n)} \leq
C \mathcal{E}^m_{v_0}
t^{-\frac{n}{\alpha}}, \ \ t>0.
\label{p20-nnew}$$
[**Proof**]{}. Let $$\begin{aligned}
I_1(t,\eta):&=& (\mbox{i}\eta_j)^{k}e^{-A(\eta)(t+\tau)}e^{\mathcal{B(\eta)}t}_\tau \hat{v}_0(-\tau,\eta), \label{p22} \\
I_2(t-s,\eta):&=& (\mbox{i}\eta_j)^{k} e^{-A(\eta)(t-s)}e^{\mathcal{B(\eta)}(t-\tau-s)}_\tau
\Big[\partial_s\hat{v}_0(s,\eta)+ A(\eta)\hat{v}_0(s,\eta)\Big].
\label{p23}\end{aligned}$$ Then, is reduced to $$\partial^{k}_{x_j} v(t,x)=\mathcal{F}^{-1}[I_1](t,x) + \int^0_{-\tau} \mathcal{F}^{-1}[I_2](t-s,x) ds.
\label{p24}$$ So, by using Parseval’s equality, we have $$\begin{aligned}
\|\partial^{k}_{x_j} v(t)\|_{L^2(\mathbb{R}^n)}&\le
&\|\mathcal{F}^{-1}[I_1](t)\|_{L^2(\mathbb{R}^n)} + \int^0_{-\tau}
\|\mathcal{F}^{-1}[I_2](t-s)\|_{L^2(\mathbb{R}^n)} ds
\notag \\
&=&\|I_1(t)\|_{L^2(\mathbb{R}^n)}+\int^0_{-\tau}
\|I_2(t-s)\|_{L^2(\mathbb{R}^n)} ds. \label{p25}\end{aligned}$$ $$\begin{aligned}
\label{p26-1}
|e^{-A(\eta)t}|&=&e^{-c_1t} \Big|\exp\Big(t\int_{\mathbb{R}^n}
J(y)e^{-\lambda_\ast y_1}e^{-\mbox{i}y\cdot\eta}dy\Big)\Big|\nonumber\\
&=&e^{-{c}_1t}\exp\Big(t\int_{\mathbb{R}^n}
J(y)e^{-\lambda_\ast y_1} \cos (y\cdot \eta) dy\Big)\nonumber\\
& =& e^{-\tilde{c}_1t}\exp\Big(-t\int_{\mathbb{R}^n}
J(y)e^{-\lambda_\ast y_1} (1- \cos (y\cdot \eta)) dy\Big)\nonumber\\
&=: &e^{-k_1t}, \ \ \ \ \ \mbox{ with }
k_1:=\tilde{c}_1+\int_{\mathbb{R}^n} J(y)e^{-\lambda_\ast y_1} (1-
\cos (y\cdot \eta)) dy ,\end{aligned}$$ Note that, using , , and the facts $\frac{e^x+e^{-x}}{2}\ge 1$ for all $x\in \mathbb{R}$, and $\int_{\mathbb{R}^n}J(y)\sin(y\cdot \eta) dy =0$ because $J(y)$ is even and $\sin(y\cdot \eta)$ is odd, and $\int_{\mathbb{R}^n}J(y)dy=1$, we have $$\begin{aligned}
\label{p26}
&&\exp\Big(-t\int_{\mathbb{R}^n} J(y)e^{-\lambda_\ast y_1} (1- \cos
(y\cdot \eta)) dy\Big)\nonumber\\
&& = \exp\Big(-t\int_{\mathbb{R}^n} J(y)\frac{e^{-\lambda_\ast y_1}
+ e^{\lambda_\ast y_1} }{2} (1- \cos
(y\cdot \eta)) dy\Big)\nonumber\\
&& \leq \exp\Big(-t\int_{\mathbb{R}^n}
J(y)(1- \cos (y\cdot \eta)) dy\Big)\nonumber\\
&&= \exp\Big(-t\int_{\mathbb{R}^n}
J(y)[1- [\cos (y\cdot \eta)+\mbox{i}\sin (y\cdot\eta)]] dy\Big)\nonumber\\
&&= e^{(\hat{J}(\eta)-1)t}\end{aligned}$$ and $$|B(\eta)|\le c_2 \int_{\mathbb{R}^n} f_\beta (y)
e^{-\lambda_*(y_1+c\tau)}dy=c_3=:k_2, \label{p27}$$ and $$|\mathcal{B}(\eta)|=|B(\eta)e^{A(\eta) \tau}|\le c_3 e^{k_1\tau}=k_2
e^{k_1 \tau}=:{\bar k}_2, \label{p27-2}$$ and further $$|e^{\mathcal{B(\eta)}t}_\tau|\le e^{\bar k_2t}_\tau. \label{p28}$$ If $\tilde{c}_1\ge c_3$, from (J$_2$), namely, $1-\hat{J}(\eta)=\mathcal{K}|\eta|^\alpha-o(|\eta|^\alpha)>0\
\mbox{as}\ \eta\rightarrow0 $, then $k_1=\tilde{c}_1
+1-\hat{J}(\eta)\geq c_3=k_2$. Using , , and in Lemma \[lemma3\], we obtain $$\begin{aligned}
\|I_1(t)\|^2_{L^2(\mathbb{R}^n)}&=&\int_{\mathbb{R}^n}
|e^{-A(\eta)(t+\tau)}e^{\mathcal{B}(\eta)t}_\tau
\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k} d\eta\nonumber\\
&&\leq
C\int_{\mathbb{R}^n}(e^{-k_1(t+\tau)}e^{\bar{k}_2t}_\tau)^2|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta\nonumber\\
& &\le C\int_{\mathbb{R}^n}(e^{-\varepsilon_1(k_1-k_2)t})^2|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta\nonumber\\
&&= Ce^{-2\varepsilon_1(\tilde{c}_1-c_3)t}
\int_{\mathbb{R}^n}e^{-2\varepsilon_1(1-\hat{J}(\eta))t}|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta. \label{p29}\end{aligned}$$ Again from (J$_2$), there exist some numbers $0<\mathcal{K}_1<\mathcal{K}$, $0<\delta<1$ and $\tilde{a}>0$, such that $$\begin{aligned}
\label{4.11}
\begin{cases}
\mathcal{K}_1|\eta|^{\alpha}\le 1-\hat{J}(\eta) \leq \mathcal{K}|\eta|^{\alpha}, & \mbox{as}\
|\eta|\leq \tilde{a},\\
\delta:=\mathcal{K}_1{\tilde a}^{\alpha}\le 1-\hat{J}(\eta) \leq \mathcal{K}|\eta|^{\alpha}, & \mbox{as}\ |\eta|\geq \tilde{a}.
\end{cases}\end{aligned}$$ Therefore, we have $$\begin{aligned}
\label{4.12}
&&\int_{\mathbb{R}^n}e^{-2\varepsilon_1 (1-\hat{J}(\eta))t}|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta\nonumber\\
&&=\int_{|\eta|\leq
\tilde{a}}e^{-2\varepsilon_1(1-\hat{J}(\eta))t}|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta+\int_{|\eta|\geq
\tilde{a}}e^{-2\varepsilon_1(1-\hat{J}(\eta))t}|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta\nonumber\\
&&\leq\int_{|\eta|\leq
\tilde{a}}e^{-2\varepsilon_1\mathcal{K}_1|\eta|^{\alpha}t}|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta+\int_{|\eta|\geq \tilde{a}}e^{-2\varepsilon_1\delta
t}|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta\nonumber\\
&&\leq \|\hat{v}_0(-\tau)\|_{L^\infty(\mathbb{R}^n)}^2
t^{-\frac{n+2k}{\alpha}}\int_{|\eta|\leq \tilde{a}}e^{-2\varepsilon_1 \mathcal{K}_1|\eta
t^{\frac1\alpha}|^{\alpha}}|\eta_jt^{\frac1\alpha}|^{2k}
d(\eta t^{\frac{1}{\alpha}}) \notag \\
& & \ \ \ +e^{-2\varepsilon_1 \delta t}\int_{|\eta|\geq
\tilde{a}}|\hat{v}_0(-\tau,\eta)|^2|\eta_j|^{2k}
d\eta\nonumber\\
&&\leq C(\|v_0(-\tau)\|_{L^1(\mathbb{R}^n)}^2+\|v_0(-\tau)\|_{H^k(\mathbb{R}^n)}^2)
t^{-\frac{n+2k}{\alpha}}.\end{aligned}$$ Substitute into , we obtain $$\begin{aligned}
\label{4.13}
\|I_1(t)\|_{L^2(\mathbb{R}^n)}\leq
C(\|v_0(-\tau)\|_{L^1(\mathbb{R}^n)}+\|v_0(-\tau)\|_{H^k(\mathbb{R}^n)})t^{-\frac{n+2k}{2\alpha}}e^{-\varepsilon_1(\tilde{c}_1-c_3)t}\end{aligned}$$
Thus, in a similar way, we can also prove $$\begin{aligned}
\label{p32}
& &\|I_2(t-s)\|_{L^2(\mathbb{R}^n)}\nonumber\\
&&=\bigg(\int_{\mathbb{R}^n}
|e^{-A(\eta)(t-s)}e^{\mathcal{B(\eta)}(t-\tau-s)}_\tau|^2
\Big|\partial_s \hat{v}_0(s,\eta) + A(\eta)\hat{v}_0(s,\eta)\Big|^2\cdot|\eta_j|^{2k}d\eta\bigg)^{\frac12}\notag \\
& &\leq Ce^{-\varepsilon_1(\tilde{c}_1-c_3)t}
\bigg(\int_{\mathbb{R}^n}e^{-2\varepsilon_1(1-\hat{J}(\eta))t}
\Big(|\eta|^{2k}|\partial_s\hat{v}_0(s,\eta)|+|\eta|^{2k}|\hat{v}_0(s,\eta)|^2 \Big)d\eta\bigg)^{\frac12} \notag \\
& &\leq
Ct^{-\frac{n+2k}{2\alpha}}e^{-\varepsilon_1(\tilde{c}_1-c_3)t}
\Big(\|(\partial_s v_0,v_0)(s)\|_{L^{1}(\mathbb{R}^n)}+\|(\partial_s
v_0,v_0)(s)\|_{H^{k}(\mathbb{R}^n)}\Big).\end{aligned}$$ Substituting and to , we immediately obtain .
Similarly, we can prove . We omit the details. Thus, we complete the proof of Proposition \[lemma4\]. $\square$
For $\tau=0$, the equation is reduced to $$\begin{aligned}
\label{p12-new2}
\begin{cases}
\displaystyle \frac{\partial {v}}{\partial t}+c\frac{\partial v}{\partial x_1}
-\int_{\mathbb{R}^n}J(y)e^{-\lambda_\ast y_1}v(t,x-y)dy +c_1
v \vspace{2mm} \\
\qquad\qquad \displaystyle = c_2 \int_{\mathbb{R}^n} f_\beta(y)
e^{-\lambda_*(y_1+c\tau)}v(t,x-y-c\tau{\bf e}_1) dy ,
\label{2010-13-new} \vspace{2mm} \\
v(s,x)=v_0(x), \ \ \ x\in \mathbb{R}^n.
\end{cases}\end{aligned}$$ Taking Fourier transform to , as showed in , we have $$\label{p16-new2}
\frac{d \hat{v}}{dt} =[B(\eta)- A(\eta)]\hat{v},
\ \ \mbox{ with } \hat{v}(0,\eta)=\hat{v}_0(\eta),$$ where $A(\eta)$ and $B(\eta)$ are given in and with $\tau=0$, respectively. Integrating yields $$\hat{v}(t,\eta)=e^{-[A(\eta)-B(\eta)]t}\hat{v}_0(\eta).$$ Taking the inverse Fourier transform, we get the solution formula $$v(t,x)=\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} e^{\mbox{i}x\cdot \eta} e^{-[A(\eta)-B(\eta)]t}\hat{v}_0(\eta) d\eta.$$ Then, a similar analysis as showed before can derive the optimal decay of the solution in the case without time-delay as follows. The detail of proof is omitted.
\[lemma5\] Suppose that $v_0\in H^m(\mathbb{R}^n)\cap
L^1(\mathbb{R}^n))$ for $m\ge 0$, then the solution of the linearized equation satisfies $$\| \partial^k_{x_j}v(t)\|_{L^2(\mathbb{R}^n)}\le C (\|v_0\|_{L^1(\mathbb{R}^n)}+\|v_0\|_{H^k(\mathbb{R}^n)})
t^{-\frac{n+2k}{2\alpha}}e^{-(\tilde{c}_1-c_3)t}, \
t>0, \label{4.14-new2}$$ for $k=0,1, \cdots, [m]$ and $j=1,\cdots, n$, where the positive constants $\tilde{c}_1$ and $c_3$ are defined in for $\tau=0$.
Furthermore, if $m>\frac{n}{2}$, then $$\| v(t)\|_{L^\infty(\mathbb{R}^n)} \leq
C (\|v_0\|_{L^1(\mathbb{R}^n)}+\|v_0\|_{H^k(\mathbb{R}^n)})
t^{-\frac{n}{\alpha}}e^{-(\tilde{c}_1-c_3)t}, \ \ t>0.
\label{p20-new2}$$ Particularly, when $\tilde{c}_1=c_3$, then $$\| v(t)\|_{L^\infty(\mathbb{R}^n)} \leq
C (\|v_0\|_{L^1(\mathbb{R}^n)}+\|v_0\|_{H^k(\mathbb{R}^n)})
t^{-\frac{n}{\alpha}}, \ \ t>0.
\label{p20-nmmnew}$$
Global Existence and Comparison principle
=========================================
In this section, we prove the global existence and uniqueness of the solution for the Cauchy problem to the nonlinear equation with nonlocal dispersion , and then establish the comparison principle in $n$-D case by a different proof approach to the previous work [@Chen; @Coville-Dupaigne-2].
\[lemma6\] Let $u_0(s,x)\in C([-\tau,0]; C( \mathbb{R}^n))$ with $0=u_-\le u_0(s,x)\le u_+$ for $(s,x)\in [-\tau,0]\times \mathbb{R}^n$, then the solution to uniquely and globally exists, and satisfies that $u\in C^1([0,\infty); C( \mathbb{R}^n))$, and $ u_-\le u(t,x) \le u_+$ for $ (t,x)\in \mathbb{R}_+\times \mathbb{R}^n)$.
[**Proof**]{}. Multiplying by $e^{\eta_0 t}$ and integrating it over $[0,t]$ with respect to $t$, where $\eta_0>0$ will be technically selected in below, we then express in the integral form $$\begin{aligned}
\label{01-15}
u(t,x)&=&e^{-\eta_0 t}u(0,x)+\int_{0}^{t}e^{-\eta_0
(t-s)}\bigg[\int_{\mathbb{R}^n}J(x-y)u(s,y)dy+(\eta_0-1)
u(s,x)\nonumber\\
&& \ \ \ -d(u(s,x))+\int_{\mathbb{R}^n}f_\beta(y)b(u(s-\tau,x-y))dy\bigg]ds.\end{aligned}$$ Let us define the solution space as, for any $T\in[0,\infty]$, $$\begin{aligned}
\mathfrak{B}=\Big\{ u(t,x)|& u(t,x)\in C([0,T]\times \mathbb{R}^n) \mbox{ with }
u_-\le u \le u_+, \notag \\
& \qquad u(s,x)=u_0(s,x), (s,x)\in [-\tau,0] \times \mathbb{R}^n\Big\}, \label{01-13}\end{aligned}$$ with the norm $$\|u\|_{\mathfrak{B}}=\sup_{t\in[0,T]}e^{-\eta_0
t}\|u(t)\|_{L^\infty(\mathbb{R}^n)} ,$$ where $$\eta_0: =1+\eta_1 + \eta_2, \ \ \ \
\eta_1:=\max_{u\in[u_-,u_+]}|d'(u)|, \ \ \ \
\eta_2:=\max_{u\in[u_-,u_+]}|b'(u)|.
\label{01-14}$$ Clearly, $\mathfrak{B}$ is a Banach space.
Define an operator $\mathcal{P}$ on $\mathfrak{B}$ by $$\begin{aligned}
\mathcal{P}(u)(t,x):&=&
e^{-\eta_0 t}u_0(0,x)+\int_{0}^{t}e^{-\eta_0
(t-s)}\bigg[\int_{\mathbb{R}^n}J(x-y)u(s,y)dy+(\eta_0-1)
u(s,x)\nonumber\\
& & \ \
-d(u(s,x))+\int_{\mathbb{R}^n}f_\beta(y)b(u(s-\tau,x-y))dy\bigg]ds, \ \ \ \mbox{ for }\
\ 0\le t\le T, \label{01-32}\end{aligned}$$ and $$\mathcal{P}(u)(s,x):=
u_0(s,x), \ \ \ \mbox{for}\ \ s\in[-\tau,0].$$
Now we are going to prove that $\mathcal{P}$ is a contracting operator from $\mathfrak{B}$ to $\mathfrak{B}$.
Firstly, we prove that $\mathcal{P}: \mathfrak{B} \to \mathfrak{B}$. In fact, if $u\in \mathfrak{B}$, from (H$_1$)-(H$_3$), namely, $0=d(0)\le d(u)\le d(u_+)$, $0=b(0)\le b(u)\le b(u_+)$, and $d(u_+)=b(u_+)$, and using the facts $\int_{\mathbb{R}^n}J(x-y)dy=1$, $\int_{\mathbb{R}^n} f_\beta(y)dy=1$, and $$g(u):=(\eta_0-1)u-d(u) \mbox{ is increasing},
\label{g}$$ which implies $g(u_+)\ge g(u)\ge g(0)=0$ for $u\in [u_-,u_+]$, then we have $$\begin{aligned}
\label{4.1}
0=u_-\leq \mathcal{P}(u)&\leq& e^{-\eta_0 t}u_+ +\int_{0}^{t}e^{-\eta_0
(t-s)}\bigg[\int_{\mathbb{R}^n}J(x-y)u_+dy \notag \\
& & \ \ \ \ +(\eta_0-1)
u_+ -d(u_+) +\int_{\mathbb{R}^n}f_\beta(y)b(u_+)dy\bigg]ds \notag \\
&=&e^{-\eta_0 t}u_+ +\int_{0}^{t}e^{-\eta_0
(t-s)}[\eta_0 u_+ - d(u_+)+b(u_+)]ds \notag \\
&=& u_+.\end{aligned}$$ This plus the continuity of $\mathcal{P}(u)$ based on the continuity of $u$ proves $\mathcal{P}(u)\in\mathfrak{B}$, namely, $\mathcal{P}$ maps from $\mathfrak{B}$ to $\mathfrak{B}$.
Secondly, we prove that $\mathcal{P}$ is contracting. In fact, let $u_1, \ u_2 \in \mathfrak{B}$, and $v=u_1-u_2$, then we have $$\begin{aligned}
\label{4.2}
&&\mathcal{P}(u_1)-\mathcal{P}(u_2)\\
&&=\int_{0}^{t}e^{-\eta_0
(t-s)}\bigg[\int_{\mathbb{R}^n}J(x-y)v(s,y)dy+(\eta_0-1) v(s,x)
-[d(u_1(s,x))-d(u_2(s,x))]\nonumber\\
&&\qquad\quad+\int_{\mathbb{R}^n}f_\beta(y)[b(u_1(s-\tau,x-y))-b(u_2(s-\tau,x-y))]dy\bigg]ds.\end{aligned}$$ So, we have $$\begin{aligned}
\label{4.3}
|\mathcal{P}(u_1)-\mathcal{P}(u_2)|e^{-\eta_0 t}&\leq & \int_{0}^{t}e^{-2\eta_0
(t-s)}\Big(\eta_0+\max_{u\in[u_-,u_+]}|d'(u)|\Big)\|v\|_{\mathfrak{B}} ds \nonumber\\
&& +\max_{u\in[u_-,u_+]}|b'(u)|
\begin{cases}
\int_{0}^{t-\tau}e^{-2\eta_0 (t-s)}\|v\|_{\mathfrak{B}}ds, & \mbox{for}\ t\geq \tau \\
0, & \mbox{for}\ 0\leq t\leq \tau
\end{cases} \nonumber \\
&\leq& \frac{1}{2\eta_0}\Big((\eta_0+\eta_1)(1-e^{-2\eta_0
t})+\eta_2(e^{-2\eta_0
\tau}-e^{-2\eta_0 t})\Big)\|v\|_{\mathfrak{B}}\nonumber\\
&\le & \frac{\eta_0+\eta_1+\eta_2}{2\eta_0} \|v\|_{\mathfrak{B}} \notag \\
&=& \frac{2\eta_0-1}{2\eta_0}\|v\|_{\mathfrak{B}} \notag \\
&=:& \rho \|v\|_{\mathfrak{B}}\end{aligned}$$ for $0<\rho:=\frac{2\eta_0-1}{2\eta_0}<1$, namely, we prove that the mapping $\mathcal{P}$ is contracting: $$\label{4.4}
\|\mathcal{P}(u_1)-\mathcal{P}(u_2)\|_{\mathfrak{B}}\leq \rho \|u_1-u_2\|_{\mathfrak{B}}<\|u_1-u_2\|_{\mathfrak{B}}.$$
Hence, by the Banach fixed-point theorem, $\mathcal{P}$ has a unique fixed point $u$ in $\mathfrak{B}$, i.e, the integral equation has a unique classical solution on $[0,T]$ for any given $T>0$. Differentiating with respect to $t$, we get back to the original equation , i.e., $$\label{01-16}
u_t = J\ast u -u+d(u(t,x))+
\int_{\mathbb{R}^n} f_\beta
(y) b(u(t-\tau,x-y)) dy,$$ then we can easily confirm from the right-hand-side of that $u_t \in C([0,T]\times \mathbb{R}^n)$. This completes our proof. $\square$
\[remark\] From the proof of Proposition \[lemma6\], we realize that, when $u_0(s,x)\in C^k([-\tau,0]\times\mathbb{R}^n)$, then the solution of the time-delayed equation holds $u(t,x)\in C^{k+1}([0,\infty);C(\mathbb{R}^n))$; while for the non-delayed equation (i.e., $\tau=0$), if $u_0(x)\in C(\mathbb{R}^n)$, then the solution of the non-delayed equation holds $u(t,x)\in C^{\infty}([0,\infty);C(\mathbb{R}^n))$. This means that the solution to the nonlocal dispersion equation possesses a really good regularity in time. However, the solutions for lack the regularity in space.
Now we establish two comparison principle for . Although the comparison principle in 1D case were proved in [@Chen; @Coville-Dupaigne-2]. Here we give a comparison principle in $n$-D case with much weaker restriction on the initial data. The proof is also new and easy to follow. Different from the previous works [@Chen; @Coville-Dupaigne-2], instead of the differential equation , we will work on the integral equation , and sufficiently use the property of contracting operator $\mathcal{P}$.
Let $\bar{u}(t,x)$ be an upper solution to , namely $$\begin{cases}
\displaystyle \frac{\partial {\bar u}}{\partial t} - J\ast {\bar u} +{\bar u}+d({\bar u}(t,x))\ge
\int_{\mathbb{R}^n} f_\beta
(y) b({\bar u}(t-\tau,x-y)) dy, \\
{\bar u}(s,x) \ge u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n,
\end{cases}
\label{1.1-30}$$ where its integral form can be written as $$\begin{aligned}
\label{01-15-new-new}
&&{\bar u}(t,x)\ge e^{-\eta_0 t}{\bar u}(0,x)+\int_{0}^{t}e^{-\eta_0
(t-s)}\bigg[\int_{\mathbb{R}^n}J(x-y){\bar u}(s,y)dy+(\eta_0-1)
{\bar u}(x,s)\nonumber\\
&&\qquad\qquad\qquad\qquad-d({\bar u}(s,x))+\int_{\mathbb{R}^n}f_\beta(y)b({\bar u}(s-\tau,x-y))dy\bigg]ds, \ \ \mbox{ for } t>0\end{aligned}$$ and let $\underline{u}(t,x)$ be an lower solution to satisfying or conversely. Then we have the following comparison result.
\[lemma7\] Let $\underline{u}(t,x)$ and $\bar{u}(t,x)$ be the classical lower and upper solutions to , with $u_-\le \underline{u}(t,x), \ \bar{u}(t,x)\le u_+$, respectively, and satisfy $0\le \underline{u}(t,x) \le u_+$ and $0\le \bar{u}(t,x) \le u_+$ for $(t,x)\in \mathbb{R}_+\times \mathbb{R}^n$. Then $\underline{u}(t,x)\leq
\bar{u}(t,x)$ for $(t,x)\in [0,\infty)\times \mathbb{R}^n$.
[**Proof**]{}. We need to prove $\bar{u}(t,x)-\underline{u}(t,x)\ge 0$ for $(t,x)\in
[0,\infty)\times \mathbb{R}^n$, namely, $r(t):=\inf_{x\in
\mathbb{R}^n} v(t,x)\ge 0$, where $v(t,x):=\bar{u}(t,x)-\underline{u}(t,x)$.
If this is not true, then there exist some constants $\varepsilon>0$ and $T>0$ such that $r(t)>-\varepsilon e^{3\eta_0t}$ for $t\in[0,T)$ and $r(T)=-\varepsilon e^{3\eta_0T}$, where $\eta_0$ given in .
Since $\underline{u}(t,x)$ and ${\bar u}(t,x)$ are the lower and upper solutions to and $\bar{u}(s,x)-\underline{u}(s,x)\geq0$, for $s\in[-\tau,0]$, and using and , and noting ${\bar u}(t,x)-\underline{u}(t,x)\ge -\varepsilon e^{3\eta_0 T}$ for $(t,x)\in [0,T]\times \mathbb{R}^n$, then we have, for $0\le t\le T$, $$\begin{aligned}
&&\bar{u}(t,x)-\underline{u}(t,x)\nonumber\\
&&\geq e^{-\eta_0 t}[\bar{u}(0,x)-\underline{u}(0,x)] \notag \\
& & \ \ \ +\int^t_0 e^{-\eta_0(t-s)}\Big(\int_{\mathbb{R}^n} J(x-y) [\bar{u}(s,y)-\underline{u}(s,y)]dy \notag \\
& & \ \ \ + g(\bar{u}(s,x))-g(\underline{u}(s,x)) \notag \\
& & \ \ \ +\int_{\mathbb{R}^n} f_\beta(y)[b(\bar{u}(s-\tau,x-y))-b(\underline{u}(s-\tau,x-y))]dy\Big)ds \notag \\
&&\geq\int^t_0 e^{-\eta_0(t-s)}\Big(-\varepsilon e^{3\eta_0s}-\max_{\zeta\in[u_-,u_+]}g'(\zeta)\varepsilon e^{3\eta_0s}\Big)ds\nonumber\\
&&\ \ \ -\max_{u\in[u_-,u_+]}|b'(u)|
\begin{cases}
\int_{\tau}^{t}e^{-\eta_0 (t-s)}\varepsilon e^{3\eta_0(s-\tau)}ds, & \mbox{for}\ t\geq \tau \\
0, & \mbox{for}\ 0\leq t\leq \tau
\end{cases} \nonumber \\
&&\geq\begin{cases}
-(\eta_0+1)\varepsilon e^{-\eta_0t}\int^t_0 e^{4\eta_0s}ds-\eta_0\varepsilon e^{-3\eta_0\tau}e^{-\eta_0t}\int_{\tau}^{t} e^{4\eta_0s}ds, &\mbox{for}\ t\geq \tau \\
-(\eta_0+1)\varepsilon e^{-\eta_0t}\int^t_0 e^{4\eta_0s}ds, &
\mbox{for}\ 0\leq t\leq \tau
\end{cases}\nonumber\\
&&\geq-\frac{2\eta_0+1}{4\eta_0}\varepsilon e^{3\eta_0t}.\end{aligned}$$ Thus, from the assumption we know $$\begin{aligned}
-\varepsilon e^{3\eta_0T}=\inf_{x\in
\mathbb{R}^n}(\bar{u}(T,x)-\underline{u}(T,x))\geq-\frac{2\eta_0+1}{4\eta_0}\varepsilon
e^{3\eta_0T},\end{aligned}$$ which is a contradiction for $\eta_0>\frac{1}{2}$. Here, our $\eta_0$ defined in satisfies $\eta_0>1$. Thus the proof is complete. $\square$
Global Stability of Planar Traveling Waves
==========================================
The main purpose in this section is to prove Theorems \[thm1\] for all traveling waves including the critical traveling waves.
For given traveling wave $\phi(x_1+ct)$ with the speed $c\ge c_*$ and the given initial data $u_-\le u_0(s,x)\le u_+$ for $(s,x)\in [-\tau,0]\times \mathbb{R}^n$, let us define $U^+_0(s,x)$ and $U^-_0(s,x)$ as $$\begin{aligned}
U^-_0(s,x):&=& \min\{ \phi(x_1+cs), u_0(s,x)\} \notag \\
U^+_0(s,x):&=&\max\{ \phi(x_1+cs), u_0(s,x)\}
\label{2.15}\end{aligned}$$ for $(s,x)\in [-\tau,0]\times \mathbb{R}^n$. So, $$u_0-\phi=(U^+_0-\phi) + (U^-_0-\phi).$$
Since $u_0-\phi\in C([-\tau,0];H^{m+1}_w(\mathbb{R}^n)\cap L^1_w(\mathbb{R}^n))$ with $m>\frac{n}{2}$ and $w(x)\ge 1$ (see ), and noting Sobolev’s embedding theorem $H^m(\mathbb{R}^n)\hookrightarrow C(\mathbb{R}^n)$, we have $u_0-\phi \in C([-\tau,0];C(\mathbb{R}^n))$. On the other hand, the traveling wave $\phi(x_1+cs)$ is smooth, then we can guarantee $U_0^\pm(s,x) \in C([-\tau,0];C(\mathbb{R}^n))$. Thus, applying Proposition \[lemma6\], we know that the solutions of with the initial data $U^+_0(s,x)$ and $U^-_0(s,x)$ globally exist, and denote them by $U^+(t,x)$ and $U^-(t,x)$, respectively, that is, $$\begin{cases}
\displaystyle \frac{\partial U^\pm}{\partial t}-J\ast U^{\pm}+U^\pm+d( U^\pm)=
\int_{\mathbb{R}^n} f_\beta (y) b(U^\pm(t-\tau,x-y)) dy,\\
U^\pm(s,x)=U^\pm_0(s,x), \ \ \ x\in \mathbb{R}^n, \ s\in[-\tau,0].
\label{2.16-1}
\end{cases}$$ Then the comparison principle (Proposition \[lemma7\]) further implies $$\begin{aligned}
\label{2.18}
\begin{cases}
u_-\le U^-(t,x)\le u(t,x)\le U^+(t,x)\le u_+
\\
u_-\le U^-(t,x)\le \phi(x_1+ct)\le U^+(t,x)\le u_+
\end{cases}
\ \ \mbox{ for }(t,x)\in \mathbb{R}_+\times \mathbb{R}^n.\end{aligned}$$
In what follows, we are going to complete the proof for the stability in three steps.
[**Step 1. The convergence of $U^+(t,x)$ to $\phi(x_1+ct)$**]{}
Let $$V(t,x):=U^+(t,x)-\phi(x_1+ct), \ \ \ \
V_0(s,x):=U^+_0(s,x)-\phi(x_1+cs) . \label{2.19}$$ It follows from that $$V(t,x)\ge 0, \qquad \ V_0(s,x)\ge 0. \label{2.20}$$ We see from that $V(t,x)$ satisfies (by linearizing it around 0) $$\begin{aligned}
\label{2.21}
& & \displaystyle \frac{\partial V}{\partial t}
-\int_{\mathbb{R}^n}J(y) V(t,x-y)dy+V +d'(0) V \notag \\
& & \quad - b'(0) \int_{\mathbb{R}^n} f_\beta(y) V(t-\tau,x-y) dy \notag \\
& & = -Q_1(t,x)+ \int_{\mathbb{R}^n} f_\beta(y) Q_2(t-\tau, x-y)
dy +[d'(0)-d'(\phi(x_1+ct))]V
\notag \\
& &\quad + \int_{\mathbb{R}^n} f_\beta(y)
[b'(\phi(x_1-y_1+c(t-\tau))-b'(0)]V(t-\tau,x-y) dy \notag \\
& &=: I_1(t,x)+I_2(t-\tau,x)+I_3(t,x)+I_4(t-\tau,x),\end{aligned}$$ with the initial data $$V(s,x)=V_0(s,x), \ s\in [-\tau,0],
\label{2.21-1}$$ where $$Q_1(t,x)=d(\phi+V)-d(\phi)-d'(\phi)V \label{2.22-1}$$ with $\phi=\phi(x_1+ct)$ and $V=V(t,x)$, and $$Q_2(t-\tau,x-y)=b(\phi+V)-b(\phi)-b'(\phi)V \label{2.22}$$ with $\phi=\phi(x_1-y_1+c(t-\tau))$ and $V=V(t-\tau,x-y)$. Here $I_i, \ i=1,2,3,4$, denotes the $i$-th term in the right-side of line above (\[2.21\]).
From (H$_3$), i.e., $d''(u)\ge 0$ and $b''(u)\le 0$, applying Taylor formula to and , we immediately have $$Q_1(t,x)\ge 0 \ \mbox{ and } \ Q_2(t-\tau,x-y)\le 0,$$ which implies $$I_1(t,x)\le 0 \ \mbox{ and } \ I_2(t-\tau,x)\le 0. \label{2.22-new0}$$ From (H$_3$) again, since $d'(\phi)$ is increasing and $b'(\phi)$ is decreasing, then $d'(0)-d'(\phi(x_1+ct))\le 0$ and $b'(\phi(x_1-y_1+c(t-\tau)))-b'(0)\le 0$, which imply, with $V\ge 0$, $$I_3(t,x)\le 0 \ \mbox{ and } \ I_4(t-\tau,x)\le 0. \label{2.22-new1}$$ Thus, applying and to , we obtain $$\dfrac{\partial V}{\partial t}-J\ast V+V+d'(0) V-b'(0)
\int_{\mathbb{R}^n} f_\beta(y) V(t-\tau,x-y)dy\le 0. \label{2.23}$$
Let ${\bar V}(t,x)$ be the solution of the following equation with the same initial data $V_0(s,x)$: $$\label{2010-10}
\begin{cases}
\dfrac{\partial {\bar V}}{\partial t}-J\ast {\bar V}+\bar{V} +d'(0)
{\bar V} - b'(0) \displaystyle \int_{\mathbb{R}^n}
f_\beta(y) {\bar V}(t-\tau,x-y) dy = 0,\ \ \ \
(t,x)\in R_+\times \mathbb{R}^n,\\
{\bar V}(s,x)=V_0(s,x), \ \ \ s\in[-\tau,0], x\in \mathbb{R}^n.
\end{cases}$$ From Proposition \[lemma6\], we know that $\bar{V}(t,x)$ globally exists. Furthermore, is actually a linear equation, and its solution is as smooth as its initial data. By the comparison principle (Proposition \[lemma7\]), we have $$0\leq V(t,x) \le {\bar V}(t,x), \ \ \ \mbox{ for } (t,x)\in
\mathbb{R}_+\times \mathbb{R}^n. \label{2010-11}$$
Let $$v(t,x):=e^{-\lambda_*(x_1+ct-x_*)}{\bar V}(t,x). \label{2010-12}$$ From , $v(t,x)$ satisfies $$\begin{aligned}
&&\dfrac{\partial {v}}{\partial
t}-\int_{\mathbb{R}^n}J(y)e^{-\lambda_\ast y_1}v(t,x-y)dy +c_1
v\nonumber\\
&&\qquad\qquad = c_2 \int_{\mathbb{R}^n} f_\beta(y)
e^{-\lambda_*(y_1+c\tau)}v(t-\tau,x-y) dy , \label{2010-13-new-new}\end{aligned}$$ where $$c_1:=c\lambda_*+1+d'(0)>0, \
\mbox{ and } \ c_2:=b'(0). \label{2010-14}$$ When $\tau=0$, then is reduced to $$\dfrac{\partial {v}}{\partial t}
-\int_{\mathbb{R}^n}J(y)e^{-\lambda_\ast y_1}v(t,x-y)dy +c_1 v = c_2
\int_{\mathbb{R}^n} f_\beta(y) e^{-\lambda_* y_1}v(t,x-y) dy .
\label{2010-13-2}$$ Applying Proposition \[lemma4\] to for $\tau>0$ and Proposition \[lemma5\] to for $\tau=0$, we obtain the following decay rates: $$\begin{aligned}
& &\|v(t)\|_{L^\infty(\mathbb{R}^n)}\le C t^{-\frac{n}{\alpha}}
e^{-\varepsilon_1(\tilde{c}_1-c_3)t}, \ \ \ \mbox{ for } \ \tau>0, \label{new-1} \\
& &\|v(t)\|_{L^\infty(\mathbb{R}^n)}\le C t^{-\frac{n}{\alpha}}
e^{-(\tilde{c}_1-c_3)t}, \ \ \ \mbox{ for } \ \tau=0, \label{new-1-new}\end{aligned}$$ where $0<\varepsilon_1=\varepsilon_1(\tau)<1$, and $c_3$ is defined in , which can be directly calculated as, by using the property , $$\begin{aligned}
c_3&=&b'(0)\int_{\mathbb{R}^n} f_\beta(y) e^{-\lambda_*(y_1+c\tau)} dy \notag \\
&=&b'(0)\int_{\mathbb{R}} f_{1\beta}(y_1) e^{-\lambda_*(y_1+c\tau)} dy_1 \notag \\
&=&b'(0)e^{\beta\lambda_*^2-\lambda_* c \tau}>0. \label{new-2}\end{aligned}$$ and $$\begin{aligned}
\tilde{c}_1=c\lambda_*+1+d'(0)-\int_{\mathbb{R}}J(y_1)e^{-\lambda_\ast
y_1}dy_1=c\lambda_*+d'(0)-E_c(\lambda_\ast). \label{new-2-2}\end{aligned}$$
When $c>c_*$, namely, the wave $\phi(x_1+ct)$ is non-critical, from in Theorem \[TW\], we realize $$\tilde{c}_1:=c\lambda_*+d'(0)-E_c(\lambda_\ast)=G_c(\lambda_*)>H_c(\lambda_*)=b'(0)e^{\beta
\lambda_*^2-\lambda_* c\tau}=:c_3. \label{new-3}$$ Thus, and immediately imply the following exponential decay for $c>c_*$ $$\begin{aligned}
& &\|v(t)\|_{L^\infty(\mathbb{R}^n)}\le C t^{-\frac{n}{\alpha}}
e^{-\varepsilon_1\tilde{\mu} t}, \ \ \mbox{ for } \tau>0, \label{new-4} \\
& &\|v(t)\|_{L^\infty(\mathbb{R}^n)}\le C t^{-\frac{n}{\alpha}}
e^{-\tilde{\mu} t}, \ \ \mbox{ for } \tau=0, \label{new-4-new}\end{aligned}$$ where $$\tilde{\mu}:=\tilde{c}_1-c_3=G_c(\lambda_*)-H_c(\lambda_*)>0. \label{mu-1}$$ When $c=c_*$, namely, the wave $\phi(x_1+c_*t)$ is critical, from in Proposition \[TW\], we realize $$\tilde{c}_1:=c\lambda_*+d'(0)-E_c(\lambda_\ast)=G_c(\lambda_*)=H_c(\lambda_*)=b'(0)e^{\beta
\lambda_*^2-\lambda_* c\tau}:=c_3. \label{new-5}$$ Then, from and , we immediately obtain the following algebraic decay for $c=c_*$ $$\|v(t)\|_{L^\infty(\mathbb{R}^n)}\le C t^{-\frac{n}{\alpha}}, \ \ \ \mbox{ for all} \tau\ge 0.
\label{new-6}\\$$
Since $V(t,x)\leq {\bar V}(t,x)=e^{\lambda_*(x_1+ct-x_*)}v(t,\xi)$, and $0<e^{\lambda_*(x_1+ct-x_*)}\leq 1$ for $x_1\in
(-\infty,x_{*}-ct]$, we immediately obtain the following decay for $V$.
\[new-lemma3\] Let $V=V(t,x)$. Then
1. when $c>c_*$, then $$\begin{aligned}
& & \|V(t)\|_{L^\infty((-\infty, \ x_{*}-ct]\times\mathbb{R}^{n-1})} \leq C(1+t)^{-\frac{n}{\alpha}} e^{-\varepsilon_1\tilde{\mu} t},
\ \ \ \mbox{ for } \tau>0,\\
& & \|V(t)\|_{L^\infty((-\infty, \ x_{*}-ct]\times\mathbb{R}^{n-1})} \leq C(1+t)^{-\frac{n}{\alpha}} e^{-\tilde{\mu} t},
\ \ \ \mbox{ for } \tau=0;
\label{2010-27}\end{aligned}$$ Here $\tilde{\mu}:=\tilde{c}_1-c_3=G_c(\lambda_*)-H_c(\lambda_*)>0$ for $c>c_*$.
2. when $c=c_*$, then $$\|V(t)\|_{L^\infty((-\infty, \ x_{*}-ct]\times\mathbb{R}^{n-1})} \le C(1+t)^{-\frac{n}{\alpha}}, \ \ \ \mbox{ for all } \tau\ge 0.
\label{2010-28}$$
Next we prove $V(t,x)$ exponentially decay for $x \in
[x_*-ct,\infty) \times \mathbb{R}^{n-1}$.
\[new-lemma4\] For $\tau>0$, it holds that $$\begin{aligned}
&\|V(t)\|_{L^\infty([x_*-ct,\infty)\times \mathbb{R}^{n-1})} \leq C
t^{-\frac{n}{\alpha}}e^{-\mu_{\tau} t}, & \ \mbox{for}\ c>c_*
\label{new-7},\\
&\|V(t)\|_{L^\infty([x_*-ct,\infty)\times \mathbb{R}^{n-1})} \leq C
t^{-\frac{n}{\alpha}}, & \ \mbox{for}\ c=c_* \label{new-7-2},\end{aligned}$$ with some constant $0<\mu_{\tau}<\min\{d'(u_+)-b'(u_+),\varepsilon_1\tilde{\mu}\}$ for $c>c_*$.
[**Proof**]{}. From and , as set in $V(t,x):=U^+(t,x)-\phi(x_1+ct)$, we have $$\dfrac{\partial V}{\partial t} -J\ast V+V +d(\phi+V)-d(\phi) =
\int_{\mathbb{R}^n} f_\beta(y) [b(\phi+V)-b(\phi)] dy.
\label{new-7-1}$$ Applying Taylor expansion formula and noting (H$_3$) for $d''(u)\ge
0$ and $b''(u)\le 0$, we have $$\begin{aligned}
& &d(\phi+V)-d(\phi)=d'(\phi)V+d''({\bar\phi}_1)V^2\ge d'(\phi)V, \label{nov1}\\
& &b(\phi+V)-b(\phi)=b'(\phi)V+b''({\bar\phi}_2)V^2\le b'(\phi)V,
\label{nov2}\end{aligned}$$ where ${\bar\phi}_i$ ($i=1,2$) are some functions between $\phi$ and $\phi+V$. Substituting and into , and noticing Lemma \[new-lemma3\], we have $$\label{nov3}
\begin{cases}
\displaystyle \dfrac{\partial V}{\partial t}
-J\ast V+V +d'(\phi)V \le \int_{\mathbb{R}^n} f_\beta(y) b'(\phi(x_1-y_1+c(t-\tau))) V(t-\tau,x-y) dy, \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \
\mbox{ for } t>0, x\in \mathbb{R}^{n}\\
V|_{x_1\leq x_*-ct}\le C_2 (1+t)^{-\frac{n}{\alpha}}e^{-\varepsilon_1\tilde{\mu} t}, \qquad \ \ \mbox{ for } t>0, (x_2,\cdots,x_n)\in \mathbb{R}^{n-1}\\
V|_{t=s}=V_0(s,x), \qquad \qquad \qquad \qquad \quad \ \ \ \mbox{
for } s\in [-\tau,0], x\in \mathbb{R}^{n}
\end{cases}$$ for some positive constant $C_2$.
Let $$\tilde{V}(t)=C_3 (1+\tau+t)^{-\frac{n}{\alpha}}e^{-\mu_{\tau} t}
\label{nov5}$$ for $C_3\ge C_2\ge \max_{(s,x)\in[-\tau,0]\times\mathbb{R}^n}|V_0(s,x)|$. As in , for given $0<\varepsilon_0 <1$, we can select a sufficiently large number $x_*$ such that, for $\xi_1\ge x_*\gg 1$, $$d'(\phi(\xi_1))-\int_{\mathbb{R}^n} f_\beta(y)
b'(\phi(\xi_1-y_1-c\tau)) dy\geq \varepsilon_0 [d'(u_+)-b'(u_+)]>0.
\label{nov4}$$ Thus, we have $$\begin{aligned}
\label{01-20}
& & \displaystyle \frac{\partial \tilde{V}}{\partial t} -J\ast\tilde{V} + \tilde{V}
+d'(\phi)\tilde{V} - \int_{\mathbb{R}^n} f_\beta(y)
b'(\phi(\xi_1-y_1-c\tau))
\tilde{V}(t-\tau ) dy \notag \\
& &=-\frac{n}{\alpha}C_3(1+t+\tau)^{-\frac{n}{\alpha}-1} e^{-\mu_{\tau} t} -\mu_{\tau} C_3(1+t+\tau)^{-\frac{n}{\alpha}} e^{-\mu_{\tau} t} \notag \\
& &\ \ \ +C_3(1+t+\tau)^{-\frac{n}{\alpha}} e^{-\mu_{\tau} t}d'(\phi(\xi_1)) \notag \\
& &\ \ \ -C_3(1+t)^{-\frac{n}{\alpha}} e^{-\mu_{\tau} (t-\tau)} \int_{\mathbb{R}^n} f_\beta (y) b'(\phi(\xi_1-y_1-c\tau))dy \notag \\
& &=C_3(1+t+\tau)^{-\frac{n}{\alpha}} e^{-\mu_{\tau} t}\Big\{\Big[d'(\phi(\xi_1))-\int_{\mathbb{R}^n} f_\beta (y) b'(\phi(\xi_1-y_1-c\tau))dy \Big]-\mu_{\tau}
\notag \\
& & \ \ \ -\frac{n}{\alpha}(1+t+\tau)^{-1} -\left(e^{\mu_{\tau} \tau}\left(\frac{1+t}{1+t+\tau}\right)^{-\frac{n}{\alpha}}-1\right) \int_{\mathbb{R}^n} f_\beta (y) b'(\phi(\xi_1-y_1-c\tau))dy\Big\}
\notag \\
& &\ge C_3(1+t+\tau)^{-\frac{n}{\alpha}} e^{-\mu_{\tau} t}\Big\{\varepsilon_0 [d'(u_+)-b'(u_+)] -\mu_{\tau} -\frac{n}{\alpha}(1+t+\tau)^{-1}
\notag \\
& & \ \ \ -\left(e^{\mu_{\tau} \tau}\left(\frac{1+t}{1+t+\tau}\right)^{-\frac{n}{\alpha}}-1\right) \int_{\mathbb{R}^n} f_\beta (y) b'(\phi(\xi_1-y_1-c\tau))dy\Big\} \notag \\
& &\ge 0
\label{01-21}\end{aligned}$$ by selecting a sufficiently small number $$\begin{aligned}
&0<\mu_{\tau} <d'(u_+)-b'(u_+) & \mbox{ for } c>c_*, \label{01-23}\\
&\mu_{\tau}=0 & \mbox{ for } c=c_*, \label{01-24}\end{aligned}$$ and taking $t\ge l_0\tau$ for a sufficiently large integer $l_0\gg 1$. Hence, we proved that $$\begin{aligned}
\label{nov6}
\begin{cases}
\displaystyle \frac{\partial \tilde{V}}{\partial t} -J\ast\tilde{V} + \tilde{V}
+d'(\phi)\tilde{V} \geq \int_{\mathbb{R}^n} f_\beta(y)
b'(\phi(\xi_1-y_1-c\tau))
\tilde{V}(t-\tau) dy,\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \mbox{ for }
t>l_0\tau, \xi\in [x_*,+\infty)\times \mathbb{R}^{n-1}\\
\tilde{V}|_{\xi_1= x_*}=C_3
(1+\tau+t)^{-\frac{n}{\alpha}}e^{-\mu_{\tau} t} > C_2
(1+t)^{-\frac{n}{\alpha}}e^{-\varepsilon_1\tilde{\mu} t},
\ \mbox{ for } t>0, (\xi_2,\cdots,\xi_n)\in \mathbb{R}^{n-1}\\
\tilde{V}(t)=C_3 (1+\tau+t)^{-\frac{n}{\alpha}}e^{-\mu_{\tau}
t} > V_0(t,\xi), \qquad \ \ \ \mbox{ for } t\in
[-\tau,l_0\tau], \xi\in \mathbb{R}^{n}.
\end{cases}\end{aligned}$$
Denote $\Omega:=\{(x,t)|x_1\geq x_*-ct,\ t\geq l_0\tau,\
(x_2,\cdots,x_n)\in \mathbb{R}^{n-1}\}$. Noticing the construction of and , then similar to the proof of Proposition \[lemma7\] , we know that $$\label{4.15}
\tilde{V}(t)- V(t,x)\geq 0,\ \mbox{for}\
(x,t)\in\mathbb{R}^n\times[-\tau,\infty)\setminus \Omega.$$ Thus the proof is complete. $\square$
For $\tau=0$, it is easy to prove the corresponding results as follows.
\[new-lemma4-new\] For $\tau=0$, it holds that $$\begin{aligned}
&\|V(t)\|_{L^\infty([x_*-ct,\infty)\times \mathbb{R}^{n-1})} \leq C
t^{-\frac{n}{\alpha}}e^{-\mu_{\tau} t}, & \ \mbox{for}\ c>c_*
\label{new-7-new},\\
&\|V(t)\|_{L^\infty([x_*-ct,\infty)\times \mathbb{R}^{n-1})} \leq C
t^{-\frac{n}{\alpha}}, & \ \mbox{for}\ c=c_* \label{new-7-2-new},\end{aligned}$$ with some constant $0<\mu_{\tau}<\min\{d'(u_+)-b'(u_+),\varepsilon_1\tilde{\mu}\}$ for $c>c_*$.
Combing Lemma \[new-lemma3\]-Lemma \[new-lemma4-new\], we obtain the decay rates for $V(t,x)$ in $L^\infty(\mathbb{R}^n)$.
\[new-lemma5\] It holds that:
1. when $c>c_*$, then $$\begin{aligned}
& & \|V(t)\|_{L^\infty(\mathbb{R}^{n})} \le C(1+t)^{-\frac{n}{\alpha}} e^{-\mu_\tau t}, \quad \mbox{ for } \tau>0,
\label{2010-27-1} \\
& & \|V(t)\|_{L^\infty(\mathbb{R}^{n})} \le C(1+t)^{-\frac{n}{\alpha}} e^{-\mu_0 t}, \quad \mbox{ for } \tau=0,
\label{2010-27-1-new}\end{aligned}$$ where $0<\mu_\tau<\min\{d'(u_+)-b'(u_+),\varepsilon_1[G_c(\lambda_*)-H_c(\lambda_*)]\}$ with $0<\varepsilon_1<1$ for $\tau>0$, and $0<\mu_0< \min\{d'(u_+)-b'(u_+),G_c(\lambda_*)-H_c(\lambda_*)\}$ for $\tau=0$;
2. when $c=c_*$, $$\|V(t)\|_{L^\infty(\mathbb{R}^{n})} \le C(1+t)^{-\frac{n}{\alpha}}, \quad \mbox{ for all } \tau\ge 0.
\label{2010-28-1}$$
Since $V(t,x)=U^+(t,x)-\phi(x_1+ct)$, Lemma \[new-lemma5\] give directly the following convergence for the solution in the cases with time-delay.
\[new-lemma6\] It holds that:
1. when $c>c_*$, then $$\begin{aligned}
& & \sup_{x\in \mathbb{R}^{n}}|U^+(t,x)-\phi(x_1+ct)| \le C(1+t)^{-\frac{n}{\alpha}} e^{-\mu_\tau t}, \quad \mbox{ for } \tau>0,
\label{2010-27-2} \\
& & \sup_{x\in \mathbb{R}^{n}}|U^+(t,x)-\phi(x_1+ct)| \le C(1+t)^{-\frac{n}{\alpha}} e^{-\mu_0 t}, \quad \mbox{ for } \tau=0,
\label{2010-27-2-new}\end{aligned}$$ where $0<\mu_\tau<\min\{d'(u_+)-b'(u_+),\varepsilon_1[G_c(\lambda_*)-H_c(\lambda_*)]\}$ with $0<\varepsilon_1<1$ for $\tau>0$, and $0<\mu_0< \min\{d'(u_+)-b'(u_+),G_c(\lambda_*)-H_c(\lambda_*)\}$ for $\tau=0$;
2. when $c=c_*$, then $$\sup_{x\in \mathbb{R}^{n}}|U^+(t,x)-\phi(x_1+c_*t)| \le C(1+t)^{-\frac{n}{\alpha}}, \quad \mbox{ for all } \tau\ge 0.
\label{2010-28-2}$$
[**Step 2. The convergence of $U^-(t,x)$ to $\phi(x_1+ct)$**]{}
For the traveling wave $\phi(x_1+ct)$ with $c\ge c_*$, let $$V(t,x)=\phi(x_1+ct)-U^-(t,x), \ \ \
V_0(s,x)=\phi(x_1+cs)-U^-_0(s,x). \label{2.50}$$ As in Step 1, we can similarly prove that $U^-(t,x)$ converges to $\phi(x_1+ct)$ as follows.
\[new-lemma7\] It holds that:
1. when $c>c_*$, then $$\begin{aligned}
& & \sup_{x\in \mathbb{R}^{n}}|U^-(t,x)-\phi(x_1+ct)| \le C(1+t)^{-\frac{n}{\alpha}} e^{-\mu_\tau t}, \quad \mbox{ for } \tau>0,
\label{new2010-27-2} \\
& & \sup_{x\in \mathbb{R}^{n}}|U^-(t,x)-\phi(x_1+ct)| \le C(1+t)^{-\frac{n}{\alpha}} e^{-\mu_0 t}, \quad \mbox{ for } \tau=0,
\label{new2010-27-2-new}\end{aligned}$$ where $0<\mu_\tau<\min\{d'(u_+)-b'(u_+),\varepsilon_1[G_c(\lambda_*)-H_c(\lambda_*)]\}$ with $0<\varepsilon_1<1$ for $\tau>0$, and $0<\mu_0< \min\{d'(u_+)-b'(u_+),G_c(\lambda_*)-H_c(\lambda_*)\}$ for $\tau=0$;
2. when $c=c_*$, then $$\sup_{x\in \mathbb{R}^{n}}|U^-(t,x)-\phi(x_1+c_*t)| \le C(1+t)^{-\frac{n}{\alpha}}, \quad \mbox{ for all } \tau\ge 0.
\label{new2010-28-2}$$
[**Step 3. The convergence of $u(t,x)$ to $\phi(x_1+ct)$**]{}
Finally, we prove that $u(t,x)$ converges to $\phi(x_1+ct)$. Since the initial data satisfy $U^-_0(s,x)\le u_0(s,x) \le U^+_0(s,x)$ for $(s,x)\in [-\tau,0]\times \mathbb{R}^n$, then the comparison principle implies that $$U^-(t,x)\le u(t,x) \le U^+(t,x), \ \ \ \ (t,x)\in R_+\times
\mathbb{R}^n.$$ Thanks to Lemmas \[new-lemma6\] and \[new-lemma7\], by the squeeze argument, we have the following convergence results.
\[new-lemma8\] It holds that:
1. when $c>c_*$, then $$\begin{aligned}
& & \sup_{x\in \mathbb{R}^{n}}|u(t,x)-\phi(x_1+ct)| \le C(1+t)^{-\frac{n}{\alpha}} e^{-\mu_\tau t}, \quad \mbox{ for } \tau>0,
\label{nn2010-27-2} \\
& & \sup_{x\in \mathbb{R}^{n}}|u(t,x)-\phi(x_1+ct)| \le C(1+t)^{-\frac{n}{\alpha}} e^{-\mu_0 t}, \quad \mbox{ for } \tau=0,
\label{nn2010-27-2-new}\end{aligned}$$ where $0<\mu_\tau<\min\{d'(u_+)-b'(u_+),\varepsilon_1[G_c(\lambda_*)-H_c(\lambda_*)]\}$ with $0<\varepsilon_1<1$ for $\tau>0$, and $0<\mu_0< \min\{d'(u_+)-b'(u_+),G_c(\lambda_*)-H_c(\lambda_*)\}$ for $\tau=0$;
2. when $c=c_*$, then $$\sup_{x\in \mathbb{R}^{n}}|u(t,x)-\phi(x_1+c_*t)| \le C(1+t)^{-\frac{n}{\alpha}}, \quad \mbox{ for all } \tau\ge 0.
\label{nn2010-28-2}$$
Applications and Concluding Remark
==================================
In this section, we first give the direct applications of Theorem \[TW\]-\[thm1\] to the Nicholson’s blowflies type equation with nonlocal dispersion, and the classical Fisher-KPP equation with nonlocal dispersion. Then we point out that, the developed stability theory above can be also applied to the more general case.
Nicholson’s blowflies equation with nonlocal dispersion
-------------------------------------------------------
For the equation , by taking $d(u)=\delta u$ and $b(u)=pue^{-au}$ with $\delta>0$, $p>0$ and $a>0$, we get the so-called Nicholson’s blowflies equation with nonlocal dispersion
$$\begin{cases}
\displaystyle \frac{\partial u}{\partial t} - J\ast u +u+\delta u(t,x))=
p\int_{\mathbb{R}^n} f_\beta
(y) u(t-\tau,x-y)e^{-au(t-\tau,x-y)} dy, \\
u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n.
\end{cases}
\label{6.1}$$
Clearly, there exist two constant equilibria $u_-=0$ and $u_+=\frac{1}{a}\ln \frac{p}{\delta}$, and the selected $d(u)$ and $b(u)$ satisfy the hypothesis (H$_1$)-(H$_3$) automatically under the consideration of $1<\frac{p}{\delta}\le e$. Let $J(x)$ satisfy the hypothesis (J$_1$) and (J$_2$), from Theorem \[TW\] and Theorem \[thm1\], we have the following existence of monostable traveling waves and their stabilities.
Let $J(x)$ satisfy (J$_1$) and (J$_2$). For , there exists the minimal speed $c_*>0$, such that
1. when $c\ge c_*$, the planar traveling waves $\phi(x\cdot{\bf e}_1 +ct)$ exist uniquely (up to a shift);
2. when $c< c_*$, the planar traveling waves $\phi(x\cdot{\bf e}_1 +ct)$ do not exist;
Here $c_*>0$ and $\lambda_*>0$ are determined by $$H_{c_*}(\lambda_*)=G_{c_*}(\lambda_*) \ \mbox{ and } \ H'_{c_*}(\lambda_*)=G'_{c_*}(\lambda_*),$$ where $$H_c(\lambda)=pe^{\beta \lambda^2-\lambda c\tau} \ \mbox{ and } \ G_c(\lambda)=c\lambda -\int_\mathbb{R}J_1(y_1)e^{-\lambda y_1}dy_1 +1+\delta.$$ Particularly, when $c>c_*$, then $
H_c(\lambda_*)<G_c(\lambda_*).
$
Let $J(x)$ satisfy (J$_1$) and (J$_2$), and the initial data be $u_0-\phi\in C([-\tau,0]; H^m_w(\mathbb{R}^n)\cap L^1_w(\mathbb{R}^n))$ and $\partial_s(u_0-\phi)\in L^1([-\tau,0]; H^{m+1}_w(\mathbb{R}^n)\cap L^1_w(\mathbb{R}^n))$ with $m>\frac{n}{2}$, and $u_-\le u_0\le u_+$ for $(s,x)\in [-\tau,0]\times \mathbb{R}^n$. Then the solution of uniquely exists and satisfies:
1. when $c>c_*$, then $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+ct)|\le
C(1+t)^{-\frac{n}{\alpha}}e^{-\mu_\tau t}, \ \ t> 0, \label{2.12-new}$$ for $0<\mu_\tau < \min\{d'(u_+)-b'(u_+), \ \varepsilon_1[G_c(\lambda_*)-H_c(\lambda_*)]\}$, and $\varepsilon_1=\varepsilon_1(\tau)$ such that $0<\varepsilon_1<1$ for $\tau>0$ and $\varepsilon_1=1$ for $\tau=0$
2. when $c=c_*$, then $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+c_*t)|\le
C(1+t)^{-\frac{n}{\alpha}}, \ \ t> 0. \label{2.12-2-new}$$
Fisher-KPP equation with nonlocal dispersion
--------------------------------------------
For the equation , let $d(u)=u^2$, $b(u)=u$ and the delay $\tau=0$, and take the limit of as $\beta\to 0^+$, we get the classical Fisher-KPP equation with nonlocal dispersion without time-delay $$\begin{cases}
\displaystyle \frac{\partial u}{\partial t} - J\ast u +u=u(1-u) \\
u(0,x)=u_0(x), \ \ \ x\in \mathbb{R}^n.
\end{cases}
\label{6.2}$$ Then we have the existence of the monostable traveling waves and their stabilities from Theorem \[TW\] and Theorem \[thm1\].
Let $J(x)$ satisfy (J$_1$) and (J$_2$). For , there exists the minimal speed $c_*>0$, such that
1. when $c\ge c_*$, the planar traveling waves $\phi(x\cdot{\bf e}_1 +ct)$ exist uniquely (up to a shift);
2. when $c< c_*$, the planar traveling waves $\phi(x\cdot{\bf e}_1 +ct)$ do not exist;
Here $c_*:=\lambda_*^{-1} \int_{\mathbb{R}}J_1(y_1)e^{-\lambda_* y_1}dy_1$, and $\lambda_*>0$ is determined by $\int_{\mathbb{R}} (1+\lambda_* y_1) J_1(y_1) e^{-\lambda_* y_1} dy_1=0$. When $c>c_*$, then $
H_c(\lambda_*)<G_c(\lambda_*),
$ where $
H_c(\lambda_*)=1$ and $G_c(\lambda_*)=c\lambda_*-\int_{\mathbb{R}}J_1(y_1)e^{-\lambda_* y_1} dy_1 + 1$.
Let $J(x)$ satisfy (J$_1$) and (J$_2$), and the initial data be $u_0-\phi\in C([-\tau,0]; H^m_w(\mathbb{R}^n)\cap L^1_w(\mathbb{R}^n))$ with $m>\frac{n}{2}$, and $u_-\le u_0\le u_+$ for $x \in \mathbb{R}^n$. Then the solution of uniquely exists and satisfies:
1. when $c>c_*$, then $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+ct)|\le
C(1+t)^{-\frac{n}{\alpha}}e^{-\mu_0 t}, \ \ t> 0, \label{2.12-nnew}$$ for $0<\mu_0 < \min\{d'(u_+)-b'(u_+), \ G_c(\lambda_*)-H_c(\lambda_*)\}$;
2. when $c=c_*$, then $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+c_*t)|\le
C(1+t)^{-\frac{n}{\alpha}}, \ \ t> 0. \label{2.12-2-nnew}$$
Concluding Remark
-----------------
Here we give a remark on the wave stability to the generalized equations with nonlocal dispersion. Let us consider a more general monostable equation with nonlocal dispersion $$\begin{cases}
\displaystyle \frac{\partial u}{\partial t} - J\ast u +u+d(u(t,x))=
F\Big(\int_{\mathbb{R}^n} \kappa
(y) b(u(t-\tau,x-y)) dy \Big), \\
u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n,
\end{cases}
\label{6.3}$$ where $J(x)$ satisfies (J$_1$) and (J$_2$) as mentioned before, and $F(\cdot)$, $d(u)$, $b(u)$ and $g(x)$ satisfy
1. There exist $u_-=0$ and $u_+>0$ such that $d(0)=b(0)=F(0)=0$, $d(u_+)=F(b(u_+))$, $d\in C^2[0,u_+]$, $b\in C^2[0,u_+]$ and $F\in C^2[0,b(u_+)]$;
2. $F'(0)b'(0)>d'(0)\ge 0$ and $0<F'(b(u_+))b'(u_+)<d'(u_+)$;
3. $d'(u)\ge 0$, $b'(u)\ge 0$, $d''(u)\ge 0$ and $b''(u)\le 0$ for $u\in [0,u_+]$;
4. $F'(u)\ge 0$ and $F''(u)\le 0$ for $u\in [0,b(u_+)]$;
5. $\kappa(x)$ is a smooth, positive and radial kernel with $\int_{\mathbb{R}^n}\kappa(x) dx=1$ and $\int_{\mathbb{R}^n}\kappa(x)e^{-\lambda x_1} dx <+\infty$ for all $\lambda>0$.
Then, by a similar calculation, we can prove the existence of the traveling waves $\phi(x_1+ct)$ for $c\ge c_*$, where $c_*>0$ is a specified minimal wave speed, and that the noncritical traveling waves with $c>c_*$ are exponentially stable and the critical waves with $c=c_*$ are algebraically stable.
Assume that $(\mbox{J}_1)$-$(\mbox{J}_2)$ and $(\mathcal{H}_1)$-$(\mathcal{H}_5)$ hold. For , there exists the minimal speed $c_*>0$, such that
1. when $c\ge c_*$, the planar traveling waves $\phi(x\cdot{\bf e}_1 +ct)$ exist uniquely (up to a shift);
2. when $c< c_*$, the planar traveling waves $\phi(x\cdot{\bf e}_1 +ct)$ do not exist;
Here $c_*>0$ and $\lambda_*=\lambda_*(c_*)>0$ are determined by $$\mathcal{H}_{c_*}(\lambda_*)=\mathcal{G}_{c_*}(\lambda_*) \ \mbox{ and } \ \mathcal{H}'_{c_*}(\lambda_*)=\mathcal{G}'_{c_*}(\lambda_*),$$ where $$\mathcal{H}_c(\lambda)=F'(0)b'(0)\int_{\mathbb{R}^n} e^{-\lambda y_1} \kappa(y) dy, \ \ \
\mathcal{G}_c(\lambda)=c\lambda -\int_{\mathbb{R}} J_1(y_1) e^{-\lambda y_1} dy_1 +1+d'(0).$$ When $c>c_*$, then $$H_c(\lambda_*)<G_c(\lambda_*).$$
Assume that $(\mbox{J}_1)$-$(\mbox{J}_2)$ and $(\mathcal{H}_1)$-$(\mathcal{H}_5)$ hold. Let the initial data be $u_0-\phi\in C([-\tau,0]; H^{m+1}_w(\mathbb{R}^n)\cap L^1_w(\mathbb{R}^n))$ and $\partial_s(u_0-\phi)\in L^1([-\tau,0]; H^{m+1}_w(\mathbb{R}^n)\cap L^1_w(\mathbb{R}^n))$ with $m>\frac{n}{2}$, and $u_-\le u_0\le u_+$ for $x \in \mathbb{R}^n$. Then the solution of uniquely exists and satisfies:
1. when $c>c_*$, then $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+ct)|\le
C(1+t)^{-\frac{n}{\alpha}}e^{-\mu_0 t}, \ \ t> 0, \label{2.12-nnnew}$$ for $0<\mu_\tau < \min\{d'(u_+)-b'(u_+), \ \varepsilon_1[ G_c(\lambda_*)-H_c(\lambda_*)]\}$, and $0<\varepsilon_1<1$ for $\tau>0$ and $\varepsilon_1=1$ for $\tau=0$;
2. when $c=c_*$, then $$\sup_{x\in \mathbb{R}^n}|u(t,x)-\phi(x_1+c_*t)|\le
C(1+t)^{-\frac{n}{\alpha}}, \ \ t> 0. \label{2.12-2-nnnew}$$
[**Acknowledgments**]{} The research of MM was supported in part by Natural Sciences and Engineering Research Council of Canada under the NSERC grant RGPIN 354724-08. The research of RH was supported in part by NNSFC (No. 11001103) and SRFDP (No. 200801831002).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Tabular data is difficult to analyze and to search through, yielding for new tools and interfaces that would allow even non tech-savvy users to gain insights from open datasets without resorting to specialized data analysis tools or even without having to fully understand the dataset structure. The goal of our demonstration is to showcase answering natural language questions from tabular data, and to discuss related system configuration and model training aspects. Our prototype is publicly available and open-sourced (see <https://svakulenko.ai.wu.ac.at/tableqa>).'
author:
- Svitlana Vakulenko
- |
Vadim Savenkov\
[[email protected]]{}\
[[email protected]]{}
bibliography:
- 'emnlp2017.bib'
title: 'TableQA: Question Answering on Tabular Data'
---
Introduction
============
There is an abundance of tabular data on the web in the form of Open Data tables, which are regularly released by many national governments. Providing their data free of charge, publishing bodies seldom have dedicated resources to support the end users in finding and using it. In many open data portals the search facility remains limited: e.g., no search in the content of data tables is supported.
We attempt to remedy this situation through development of the information retrieval tools tailored specifically to the end users without technical background. Our Open Data Assistant chatbot [@us] offers an unconventional interface for cross-lingual data search via Facebook and Skype messaging applications enabling a quick overview of the available datasets collected from various open data portals. However, the current version of the chatbot supports only metadata-based search. In this paper, we work towards extending the chatbot to search within the content of open data tables and answering specific user questions using the values from these tables.
{width="80.00000%"}
Task Description
================
The task of question answering over tables is given an input table (or a set of tables) *T* and a natural language question *Q* (a user query), output the correct answer *A*.
Related Work
============
Recently, quite a few studies emerged that address the question-answering task on tables using deep neural networks. They involve search across tables [@DBLP:conf/www/SunMHYSY16] and learning to perform aggregation operations [@DBLP:conf/ijcai/YinLLK16; @DBLP:journals/corr/NeelakantanLAMA16]. However, all of the proposed systems are very complex, require significant computation resources and are engineered to work exclusively on tabular data.
We contribute to the growing body of work on question answering for tabular data by providing and evaluating a prototype based on the End-To-End Memory Networks architecture [@DBLP:conf/nips/SukhbaatarSWF15]. This architecture was originally designed for the question-answering tasks from short natural language texts (bAbI tasks) [@DBLP:journals/corr/WestonBCM15], which include testing elements of inductive and deductive reasoning, co-reference resolution and time manipulation. In this context the task of question answering over tables can be seen as an extension to the original bAbI tasks. It is very appealing to be able to apply the same type of architecture to querying semi-structured tables alongside the textual data for this could enable question answering on real-world documents that contain a mixture of both, e.g., user manuals and financial reports.
Architecture
============
The architecture of our system for table-based question answering is summarized in Figure \[fig:arch\]. Each of the individual components is described in further details below.
Table Representation
--------------------
Training examples consist of the input table decomposed into row-column-value triples and a question/answer pair, for instance:
------ ------------- ------------
Row1 City Klagenfurt
Row1 Immigration 110
Row1 Emmigration 140
Row2 City Salzburg
Row2 Immigration 170
Row2 Emmigration 100
------ ------------- ------------
\
**Question**: What is the immigration in Salzburg?\
**Answer**: 170
This representation preserves the row and column identifiers of the table values. In this way our system can also ingest and learn from multiple tables at once.
Learning Table Lookups
----------------------
Our method for question answering from tables is based on the End-To-End Memory Network architecture [@DBLP:conf/nips/SukhbaatarSWF15], which we employ to transform the natural-language questions into the table lookups. Memory Network is a recurrent neural network (RNN) trained to predict the correct answer by combining continuous representations of an input table and a question. It consists of a sequence of memory layers (3 layers in our experiments) that allow to go over the content of the input table several times and perform reasoning in multiple steps.
The data samples for training and testing are fed in batches (batch size is 32 in our experiments). Each of the data samples consists of the input table, a question and the correct answer that corresponds to one of the cells in the input table.
The input tables, questions and answers are embedded into a vector space using a bag-of-words models, which neglects the ordering of words. The output layer generates the predicted answer to the input question and is implemented as a softmax function in the size of the vocabulary, i.e. it outputs the probability distribution over all possible answers, which could be any of the table cells.
The network is trained using stochastic gradient descent with linear start to avoid the local minima as in [@DBLP:conf/nips/SukhbaatarSWF15]. The objective function is to minimize the cross-entropy loss between the predicted answer and the true answer from the training set.
{width="90.00000%"}
Query Disambiguation
--------------------
Since users may refer to the columns with words that differ from the labels used in the table headings, we employ a fastText model [@bojanowski2016enriching] pretrained on Wikipedia to compute similarity between the out-of-vocabulary (OOV) words from the user query and the words in our vocabulary, i.e. to align or ground the query in the local representation.
fastText provides continuous word representation, which reflects semantic similarity using both the word co-occurrence statistics and the sub-word-based similarity via the character n-grams. For each of the OOV words the query disambiguation module picks the most similar word from the vocabulary at query time and uses its embedding instead.
In our scenario this approach is particularly useful to match the paraphrases of the column headings, e.g., the word *emigration* is matched to the *emigration\_total* label. We empirically learned the similarity threshold of 0.8 that provides optimal precision/recall trade-off on our data.
Experiments
===========
Synthetic data
--------------
We produce synthetic training examples based on a real-world table by limiting the domain for each of the column-variables to n distinct values per column (10 in our experiments). From this vocabulary we generate sample tables and question/answer pairs using the predefined templates. Thus, the vocabulary size in our experiments was fixed to 65 words.
For each table we generate 4 unique rows and a question addressed towards a cell from one of these rows. The templates produce two types of questions that model functional dependencies in the table with
\(1) simple key (single column), e.g. What is the **immigration** in **Salzburg**?
\(2) composite key (combination of 2 columns), e.g. What was the **immigration** in **Salzburg** in **2011**?
We generate data samples using a randomized procedure. For the first task we select a unique value for the key-column in each row and pick all other values uniformly at random from the respective domains. In order to learn successfully for the second task we generate unique rows that partially overlap in their composite keys, for instance:
------ ------ --------------
Row2 City **Salzburg**
Row2 Year **2010**
Row3 City **Salzburg**
Row3 Year 2008
Row1 City Klagenfurt
Row1 Year **2010**
------ ------ --------------
\
These training examples explicitly require the model to attend to both columns that constitute the composite key. Otherwise, if a single column appears enough to uniquely identify the rows, the network ignores the second column of the composite key.
In order to avoid over-fitting when the network is memorizing the question template we provide 2 different question templates. At the data generation phase we select one of them uniformly at random for each training example. This aids the network in separating semantically important words (concepts from the table) from the connector words (*in*, *for*). This makes the model more flexible and robust in handling diverse question formulations which were not observed during the training phase.
Evaluation
----------
In order to test the robustness of the trained model we create a test set with a single batch, where we take the generated data samples and change the test question by perturbing the template and paraphrasing the original question. We provide several scenarios that explore the ability of the model to recover the correct answer. The template-based questions are modified by
**omitting words**: one or more words are removed from the original user query;
**changing the position of words** in the query;
**querying a different column** that did not appear in the questions from the training data set;
**inadequate questions**, for which data required to answer this question are not present in the input table.
In this way we obtained a test set with 32 samples (8 samples for each of the 4 corruption types) with the questions phrased the way they never appeared in the pattern-generated training examples but are semantically meaningful and could occur in the real-world settings.
---------------------------------------------------- -- -- --
**Task & **Test Error & **Training Set & **Epochs\
Simple key & 0.5 & 5,949 & 29\
Composite key & 0.59 & 18,953 & 88\
********
---------------------------------------------------- -- -- --
: \[results\] Evaluation results.
The evaluation results are summarized in Table \[results\]. The error analysis showed that both models failed to provide the correct answer for the columns that never appeared in the questions of the training set. Also, the models output false answers in response to the questions for which the correct answer is not contained in the input table often with a high confidence, when relying on a single column from the composite key.
Demonstration
=============
The aim of the demonstration is to showcase the power and limitations of the neural model trained to answer questions on semi-structured data. TableQA prototype is implemented as a Flask web application[^1] and is publicly available on our web-site (see <https://svakulenko.ai.wu.ac.at/tableqa>).
The user interface (Figure \[fig:ui\]) allows to enter a custom question for a sample table provided (alternatively, use one of the questions from the test set held-out during the training phase). The attention weights are visualized by highlighting the corresponding cells in the input table, which provides an insight on the data patterns learned by the neural network.
There is also an additional table below, which contains more details about the underlying prediction mechanism. It contains the triple-wise representation of the input table as consumed by the neural network and the attention weights for each of the three memory layers separately.
Discussion of Limitations and Outlook
=====================================
The query disambiguation module is disjoint from the training module, which makes different types of errors more transparent. However, it uses an over-simplifying assumption that each word in the user query corresponds to a single word from the model vocabulary. A sequence-to-sequence model [@DBLP:conf/emnlp/ChoMGBBSB14], which is a common approach for language translation, in place of this simple heuristic could make query disambiguation more robust. Also, the pre-trained word embeddings can be integrated within a single neural network architecture to make the computation more efficient.
Our experiments showed that the design of the training examples is very important especially when trying to teach attention over the composite key for the second task. Also, the second task (composite key look-up) turned out to be much more difficult requiring more examples and time to train.
Since the model is trained exclusively on positive examples, i.e. correct question/answer pairs, it is incapable of handling inadequate user queries, i.e. questions that can not be answered using the provided input. This observation makes an obvious application in the real-world settings by demonstrating the need to train neural networks to identify and correctly handle such questions.
Another question type, not covered in the current evaluation, are ambiguous questions, which may relate to several cells at the same time. The model has to be able to identify such a situation and prompt the user to disambiguate the query or fall-back to the predefined behavior, e.g., output all relevant data or only the most recent ones.
The task for the future work remains in evaluating the model on the joint task including other bAbI datasets. Also, extending the model to work on the new data that was not available during the training phase, i.e. to propagate the learned weights to the OOV words, will make the approach applicable for the real-world data. This may involve changes in the network architecture, e.g. towards learning a hierarchical representation of the table structure that will create the necessary layers of abstraction beyond the individual values [@DBLP:conf/ijcai/YinLLK16; @DBLP:journals/corr/NeelakantanLAMA16]. The challenge, however, is to keep the network architecture general enough to perform well on other bAbI tasks at the same time to be able to answer questions of various kind and on different types of data (tables and text).
Finally, the web application can be further extended to accommodate user feedback and collect new annotations of question/answer pairs towards enriching the training dataset beyond the template-generated examples and improving the model.
Conclusion
==========
In this paper we propose two new bAbI tasks for question answering from tables and provide an initial evaluation of the performance of the memory network architecture on them. These results can be used towards developing a natural-language interface that will support search in semi-structured data, such as Open Data tables.
The role of the demonstration is to provide an opportunity to interactively explore the performance and limitations of the trained model. It helps to understand which patterns the model has actually learned from the provided data samples. This tool will be useful for all who want to learn more about this family of models as well as for the researchers looking for directions to improve the neural network performance.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Austrian Research Promotion Agency (FFG) under the project CommuniData (grant no. 855407).
[^1]: Implementation based on <https://github.com/vinhkhuc/MemN2N-babi-python>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.'
address:
- 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India'
- 'Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain'
- 'Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain'
author:
- Indranil Biswas
- 'Tomás L. Gómez'
- Vicente Muñoz
title: Rationality and Brauer group of a moduli space of framed bundles
---
Introduction
============
Let $X$ be a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$. A framed bundle on $X$ is a pair of the form $(E\, ,\phi)$, where $E$ is a vector bundle on $X$, and $$\phi\, :\, E_{x_0}\,\longrightarrow\, {\mathbb C}^r$$ is a non–zero $\mathbb C$–linear homomorphism, where $r$ is the rank of $E$. The notion of a (semi)stable vector bundle extends to that for a framed bundle. But the (semi)stability condition depends on a parameter $\tau\,\in\, {\mathbb R}_{>0}$. Fix a positive integer $r$, and also fix a holomorphic line bundle $L$ over $X$. Also, fix a positive number $\tau\, \in\, \mathbb R$. Let ${\mathcal M}^\tau_L(r)$ be the moduli space of $\tau$–stable framed bundles of rank $r$ and determinant $L$.
In [@BGM], we investigated the geometric structure of the variety ${\mathcal M}^\tau_L(r)$. The following theorem was proved in [@BGM]:
*Assume that $\tau\, \in\, (0\, , \frac{1}{(r-1)! (r-1)})$. Then the isomorphism class of the Riemann surface $X$ is uniquely determined by the isomorphism class of the variety ${\mathcal M}^\tau_L(r)$.*
Our aim here is to investigate the rationality properties of the variety ${\mathcal M}^\tau_L(r)$. We prove the following (see Theorem \[thm1\] and Corollary \[cor1\]):
*The variety ${\mathcal M}^\tau_L(r)$ is rational.*
*If $\tau\, \in\, (0\, , \frac{1}{(r-1)! (r-1)})$, then $${\rm Br}({\mathcal M}^\tau_L(r))\,=\, 0\, ,$$ where ${\rm Br}({\mathcal M}^\tau_L(r))$ is the Brauer group of ${\mathcal M}^\tau_L(r)$.*
The rationality of ${\mathcal M}^\tau_L(r)$ is proved by showing that ${\mathcal M}^\tau_L(r)$ is birational to the total space of a vector bundle over the moduli space of stable vector bundles $E$ on $X$ together with a line in the fiber of $E$ over a fixed point. The rationality of ${\mathcal M}^\tau_L(r)$ also follows from [@Ho-extra], Example 6.9, taking $D$ to be the point $x_0$.
The Brauer group of ${\mathcal M}^\tau_L(r)$ is computed by considering the morphism to the usual moduli space that forgets the framing.
Rationality of moduli space
===========================
Let $X$ be a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$. Fix a holomorphic line bundle $L$ over $X$, and take an integer $r>0$. Fix a point $x_0\,\in\, X$. A framed coherent sheaf over $X$ is a pair of the form $(E\, ,\phi)$, where $E$ is a coherent sheaf on $X$ of rank $r$, and $$\phi\, :\, E_{x_0}\,\longrightarrow\, {\mathbb C}^r$$ is a non–zero $\mathbb C$–linear homomorphism. Let $\tau>0$ be a real number. A framed coherent sheaf is called $\tau$–*stable* (respectively, $\tau$–*semistable*) if for all proper subsheaves $E'\,\subset\, E$, we have $$\label{eq:st}
{\deg E' - \epsilon(E',\phi)\tau}
< {{\operatorname{rk}}E'} \, \, \frac{\deg E - \tau}{{\operatorname{rk}}E}$$ (respectively, ${\deg E' - \epsilon(E',\phi)\tau}
\leq {{\operatorname{rk}}E'}({\deg E - \tau})/{{\operatorname{rk}}E}$), where $$\epsilon(E',\phi)\,=\,\left\{
\begin{array}{lcl}
1 & \text{if} & \phi|_{E'_{x_0}}\,\neq\, 0, \\
0 & \text{if} & \phi|_{E'_{x_0}}\,=\, 0\, .
\end{array}
\right .$$ A framed bundle is a framed coherent sheaf $(E,\varphi)$ such that $E$ is locally free.
We remark that the framed coherent sheaves considered here are special cases of the objects considered in [@HL], and hence from [@HL] we conclude that the moduli space ${\mathcal M}^\tau_L(r)$ of $\tau$–stable framed bundles of rank $r$ and determinant $L$ is a smooth quasi–projective variety.
Let $(E\, ,\phi)$ be a $\tau$–semistable framed coherent sheaf. We note that if $\tau<1$, then $E$ is necessarily torsion–free, because a torsion subsheaf of $E$ will contradict $\tau$–semistability, hence in this case $E$ is locally free. But if $\tau$ is large, then $E$ can have torsion. In particular, the natural compactification of ${\mathcal M}^\tau_L(r)$ using $\tau$–semistable framed coherent sheaves could have points which are not framed bundles.
\[lem1\] There is a dense Zariski open subset $$\label{e1}
{\mathcal M}^\tau_L(r)^0\, \subset\, {\mathcal M}^\tau_L(r)$$ corresponding to pairs $(E\, ,\phi)$ such that $E$ is a stable vector bundle of rank $r$, and $\phi$ is an isomorphism.
The moduli space ${\mathcal M}^\tau_L(r)$ is irreducible.
From the openness of the stability condition it follows immediately that the locus of framed bundles $(E\, ,\phi)$ such that $E$ is not stable is a closed subset of the moduli space ${\mathcal M}^\tau_L(r)$ (see [@Ma p. 635, Theorem 2.8(B)] for the openness of the stability condition). It is easy to check that the locus of framed bundles $(E\, ,\phi)$ such that $\phi$ is not an isomorphism is a closed subset of ${\mathcal M}^\tau_L(r)$. Therefore, ${\mathcal M}^\tau_L(r)^0$ is a Zariski open subset of ${\mathcal M}^\tau_L(r)$.
We will now show that this open subset ${\mathcal M}^\tau_L(r)^0$ is dense. Let $(E,\varphi)$ be a $\tau$–stable framed bundle. The moduli stack of stable vector bundles is dense in the moduli stack of coherent sheaves, and both stacks are irreducible (see, for instance, [@Ho Appendix]). Therefore we can construct a family $\{E_t\}_{t\in T}$ of vector bundles parametrized by an irreducible smooth curve $T$ with a base point $0\, \in\, T$ such that the following two conditions hold:
1. $E_0\cong E$, and
2. the vector bundle $E_t$ is stable for all $t\,\not=\, 0$.
Shrinking $T$ if necessary (by taking a nonempty Zariski open subset of $T$), we get a family of frames $\{\phi_t\}_{t\in T}$ such that $\phi_0$ is the given frame $\phi$, and $\phi_t\,:\,E_{t,x_0}\,\longrightarrow\,
{\mathbb C}^r$ is an isomorphism for all $t\,\not=\, 0$. Since $E_t$ is stable, and $\phi_t$ is an isomorphism, it is easy to check that $(E_t,\phi_t)$ is $\tau$–stable. Therefore, ${\mathcal
M}^\tau_L(r)^0$ is dense in ${\mathcal M}^\tau_L(r)$.
To prove that ${\mathcal M}^\tau_L(r)$ is irreducible, first note that ${\mathcal M}^\tau_L(r)^0$ is irreducible because the moduli stack of stable vector bundles of fixed rank and determinant is irreducible. Since ${\mathcal M}^\tau_L(r)^0\,\subset\,
{\mathcal M}^\tau_L(r)$ is dense, it follows that ${\mathcal M}^\tau_L(r)$ is irreducible.
Let ${\mathcal N}_P$ be the moduli space of pairs of the form $(E\, ,\ell)$, where $E$ is a stable vector bundle on $X$ of rank $r$ with determinant $L$, and $\ell\, \subset\,
E_{x_0}$ is a line. Consider ${\mathcal M}^\tau_L(r)^0$ defined in . Let $$\label{e2}
\beta\, :\, {\mathcal M}^\tau_L(r)^0\, \longrightarrow\, {\mathcal N}_P$$ be the morphism defined by $(E\, ,\phi)\, \longmapsto\,
(E\, ,\phi^{-1}({\mathbb C}\cdot e_1))$, where the standard basis of ${\mathbb C}^r$ is denoted by $\{e_1\, ,\ldots\, ,e_r\}$.
\[prop1\] The variety ${\mathcal M}^\tau_L(r)^0$ is birational to the total space of a vector bundle over ${\mathcal N}_P$.
We will first construct a tautological vector bundle over ${\mathcal N}_P$. Let ${\mathcal N}_L(r)$ be the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $L$. Consider the projection $$\label{f}
f\, :\, {\mathcal N}_P\, \longrightarrow\, {\mathcal N}_L(r)$$ defined by $(E\, ,\ell)\,\longrightarrow\, E$. Let $P_{\text{PGL}}\, \longrightarrow\, {\mathcal N}_L(r)$ be the principal $\text{PGL}(r, {\mathbb C})$–bundle corresponding to $f$; the fiber of $P_{\text{PGL}}$ over any $E\, \in\, {\mathcal N}_L(r)$ is the space of all linear isomorphisms from $P({\mathbb C}^r)$ (the space of lines in ${\mathbb C}^r$) to $P(E_{x_0})$ (the space of lines in $E_{x_0}$); since the automorphism group of $E$ is the nonzero scalar multiplications (recall that $E$ is stable), the projective space $P(E_{x_0})$ is canonically defined by the point $E$ of ${\mathcal N}_L(r)$. Let $$Q\, \subset\, \text{PGL}(r, {\mathbb C})$$ be the maximal parabolic subgroup that fixes the point of $P({\mathbb C}^r)$ representing the line ${\mathbb C}\cdot e_1$. The principal $\text{PGL}(r, {\mathbb C})$–bundle $$f^*P_{\text{PGL}}\, \longrightarrow\, {\mathcal N}_P$$ has a tautological reduction of structure group $$\widetilde{E}_Q\, \subset\, f^*P_{\text{PGL}}$$ to the parabolic subgroup $Q$; the fiber of $\widetilde{E}_Q$ over any point $(E\, ,\ell)\, \in\, {\mathcal N}_P$ is the space of all linear isomorphisms $$\rho\, :\, P({\mathbb C}^r)\,\longrightarrow\, P(E_{x_0})$$ such that $\rho({\mathbb C}\cdot e_1) \,=\, \ell$. The standard action of $\text{GL}(r, {\mathbb C})$ on ${\mathbb C}^r$ defines an action of $Q$ on $({\mathbb
C}\cdot e_1)^*\bigotimes_{\mathbb C} {\mathbb C}^r$. Let $$\label{e3}
W\, :=\, f^*P_{\text{PGL}}(({\mathbb C}\cdot
e_1)^*\otimes {\mathbb C}^r)\, \longrightarrow\, {\mathcal N}_P$$ be the vector bundle over ${\mathcal N}_P$ associated to the principal $\text{PGL}(r, {\mathbb C})$–bundle $f^*P_{\text{PGL}}$ for the above $\text{PGL}(r, {\mathbb C})$–module $({\mathbb
C}\cdot e_1)^*\bigotimes_{\mathbb C} {\mathbb C}^r$. The action of $Q$ on $({\mathbb C}\cdot e_1)^*\bigotimes_{\mathbb C} {\mathbb C}^r$ fixes $$\text{Id}_{{\mathbb C}\cdot e_1}\, \in\,
({\mathbb C}\cdot e_1)^*\otimes_{\mathbb C} {\mathbb C}^r
\,=\, \text{Hom}({\mathbb C}\cdot e_1\, , {\mathbb C}^r)\, .$$ Therefore, the element $\text{Id}_{{\mathbb C}\cdot e_1}$ defines a nonzero section $$\label{e4}
\sigma\, \in\, H^0({\mathcal N}_P,\, W)\, ,$$ where $W$ is the vector bundle in . Note that the fiber of $W$ over $(E,\ell)$ is $\ell^* \otimes E_{x_0}$, and the evaluation of $\sigma$ at $(E,\ell)$ is $\text{Id}_\ell$.
The projective bundle $P(W)\, \longrightarrow\, {\mathcal N}_P$ parametrizing lines in $W$ is identified with the pullback $f^*{\mathcal N}_P$ of the projective bundle ${\mathcal N}_P$ to the total space of ${\mathcal N}_P$, where $f$ is constructed in . The tautological section ${\mathcal N}_P\, \longrightarrow\, f^*{\mathcal N}_P$ of the projection $f^*{\mathcal N}_P\, \longrightarrow\,{\mathcal N}_P$ coincides with the section given by $\sigma$ in .
Let $U\, \subset\, {\mathcal N}_P$ be some nonempty Zariski open subset such that there exists $$V\, \subset\, W\vert_{U}\, ,$$ a direct summand of the line subbundle of $W\vert_{U}$ generated by $\sigma$. Consider the vector bundle $${\mathcal W}\, :=\, V^*\otimes_{\mathbb C} {\mathbb C}^r
\, \longrightarrow\, U\, .$$ The total space of $\mathcal W$ will also be denoted by $\mathcal W$. Consider the map $\beta$ defined in . Let $$\gamma\,:\, {\mathcal M}^\tau_L(r)^0 \, \supset\,
\beta^{-1}(U)\, \longrightarrow\, \mathcal W$$ be the morphism that sends any $y\,:=\, (E\, ,\phi)\,\in\,
\beta^{-1}(U)$ to the homomorphism $$V_{\beta(y)}\, \longrightarrow\, {\mathbb C}^r$$ defined by $v\, \longmapsto\, \phi(v)/\lambda$, where $\lambda\, \in\, {\mathbb C}^*-\{0\}$ satisfies the identity $\phi(\sigma(\beta(y)))\,=\, \lambda\cdot e_1$. The morphism $\gamma$ is clearly birational.
\[thm1\] The moduli space ${\mathcal M}^\tau_L(r)$ is rational.
Since any vector bundle is Zariski locally trivial, the total space of a vector bundle of rank $n$ over ${\mathcal N}_P$ is birational to ${\mathcal N}_P\times {\mathbb A}^n$. Therefore, from Proposition \[prop1\] we conclude that ${\mathcal M}^\tau_L(r)^0$ is birational to ${\mathcal N}_P\times
{\mathbb A}^n$, where $n\,=\, \dim {\mathcal M}^\tau_L(r)^0
-\dim {\mathcal N}_P$.
The variety ${\mathcal N}_P$ is known to be rational [@BY p. 472, Theorem 6.2]. Hence ${\mathcal N}_P\times {\mathbb A}^n$ is rational, implying that ${\mathcal M}^\tau_L(r)^0$ is rational. Now from Lemma \[lem1\] we infer that ${\mathcal M}^\tau_L(r)$ is rational.
Brauer group of moduli of framed bundles
========================================
We quickly recall the definition of Brauer group of a variety $Z$. Using the natural isomorphism ${\mathbb C}^r\otimes
{\mathbb C}^{r'}\,\stackrel{\sim}{\longrightarrow}\,{\mathbb C}^{rr'}$, we have a homomorphism $\text{PGL}(r,{\mathbb C})\times \text{PGL}(r',{\mathbb C})
\,\longrightarrow\, \text{PGL}(rr',{\mathbb C})$. So a principal $\text{PGL}(r,{\mathbb C})$–bundle $\mathbb P$ and a principal $\text{PGL}(r',{\mathbb C})$–bundle ${\mathbb P}'$ on $Z$ together produce a principal $\text{PGL}(rr',{\mathbb C})$–bundle on $Z$, which we will denote by ${\mathbb P}\otimes {\mathbb P}'$. The two principal bundles $\mathbb P$ and ${\mathbb P}'$ are called *equivalent* if there are vector bundles $V$ and $V'$ on $Z$ such that the principal bundle ${\mathbb P}\otimes {\mathbb P}(V)$ is isomorphic to ${\mathbb P}'\otimes {\mathbb P}(V')$. The equivalence classes form a group which is called the *Brauer group* of $Z$. The addition operation is defined by the tensor product, and the inverse is defined to be the dual projective bundle. The Brauer group of $Z$ will be denoted by $\text{Br}(Z)$.
As before, fix $r$ and $L$. Define $$\tau(r) \, :=\, \frac{1}{(r-1)! (r-1)}\, .$$ Henceforth, we assume that $$\tau\, \in\, (0\, , \tau(r))\, ,$$ where $\tau$ is the parameter in the definition of a (semi)stable framed bundle. As before, let ${\mathcal M}^\tau_L(r)$ be the moduli space of $\tau$–stable framed bundles of rank $r$ and determinant $L$.
Let $\overline{\mathcal N}_L(r)$ be the moduli space of semistable vector bundles on $X$ of rank $r$ and determinant $L$. As in the previous section, the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $L$ will be denoted by ${\mathcal N}_L(r)$.
If $E$ is a stable vector bundle of rank $r$ and determinant $L$, then for any nonzero homomorphism $$\phi\, :\, E_{x_0}\, \longrightarrow\, {\mathbb C}^r\, ,$$ the framed bundle $(E\, ,\phi)$ is $\tau$–stable (see [@BGM Lemma 1.2(ii)]). Also, if $(E\, ,\phi)$ is any $\tau$–stable framed bundle, then $E$ is semistable [@BGM Lemma 1.2(i)]. Therefore, we have a morphism $$\label{delta}
\delta\, :\, {\mathcal M}^\tau_L(r)\, \longrightarrow\,
\overline{\mathcal N}_L(r)$$ defined by $(E\, ,\phi)\, \longrightarrow\,E$. Define $$\label{d2}
{\mathcal M}^\tau_L(r)'\, :=\, \delta^{-1}({\mathcal N}_L(r))\,
\subset\, {\mathcal M}^\tau_L(r)\, ,$$ where $\delta$ is the morphism in . From the openness of the stability condition (mentioned in the proof of Lemma \[lem1\]) it follows that ${\mathcal M}^\tau_L(r)'$ is a Zariski open subset of ${\mathcal M}^\tau_L(r)$.
\[lem2\] The Brauer group of the variety ${\mathcal M}^\tau_L(r)'$ vanishes.
We noted above that $(E\, ,\phi)$ is $\tau$–stable if $E$ is stable. Therefore, the morphism $$\delta_1\,:=\, \delta\vert_{{\mathcal M}^\tau_L(r)'}\, :\,
{\mathcal M}^\tau_L(r)'\, \longrightarrow\,{\mathcal N}_L(r)$$ defines a projective bundle over ${\mathcal N}_L(r)$, where $\delta$ is constructed in ; for notational convenience, this projective bundle ${\mathcal M}^\tau_L(r)'$ will be denoted by ${\mathcal P}$. The homomorphism $$\delta^*_1\, :\, \text{Br}({\mathcal N}_L(r))\,\longrightarrow
\, \text{Br}({\mathcal P})$$ is surjective, and the kernel of $\delta^*_1$ is generated by the Brauer class $$\text{cl}({\mathcal P})\, \in\, \text{Br}({\mathcal
N}_L(r))$$ of the projective bundle ${\mathcal P}$ (see [@Ga p. 193]). In other words, we have an exact sequence $$\label{es}
{\mathbb Z}\cdot \text{cl}({\mathcal P})\,\longrightarrow
\, \text{Br}({\mathcal N}_L(r))\,
\stackrel{\delta^*_1}{\longrightarrow}\,\text{Br}
({\mathcal M}^\tau_L(r)')\, \longrightarrow\, 0\, .$$
Let $${\mathbb P}\,:=\, {\mathcal N}_L(r)\times P({\mathbb C}^r)\,
\longrightarrow\, {\mathcal N}_L(r)$$ be the trivial projective bundle over ${\mathcal N}_L(r)$. Consider the projective bundle $$f\, :\, {\mathcal N}_P\, \longrightarrow\, {\mathcal N}_L(r)$$ in . Let $$({\mathcal N}_P)^*\, \longrightarrow\, {\mathcal N}_L(r)$$ be the dual projective bundle; so the fiber of $({\mathcal N}_P)^*$ over any point $z\, \in\,
{\mathcal N}_L(r)$ is the space of all hyperplanes in the fiber of ${\mathcal N}_P$ over $z$. It is easy to see that $$\label{f2}
{\mathcal P}\,=\, ({\mathcal N}_P)^*\otimes {\mathbb P}$$ (the tensor product of two projective bundles was defined at the beginning of this section).
Since ${\mathbb P}$ is a trivial projective bundle, from it follows that $$\text{cl}({\mathcal P})\, =\, \text{cl}(({\mathcal N}_P)^*)
\,=\, -\text{cl}({\mathcal N}_P)
\, \in \, \text{Br}({\mathcal N}_L(r))\, .$$ But the Brauer group $\text{Br}({\mathcal N}_L(r))$ is generated by $\text{cl}({\mathcal N}_P)$ [@BBGN Proposition 1.2(a)]. Hence $\text{cl}({\mathcal P})$ generates $\text{Br}({\mathcal N}_L(r))$. Now from we conclude that $\text{Br}({\mathcal M}^\tau_L(r)')\,=\, 0$.
\[cor1\] The Brauer group of the moduli space ${\mathcal M}^\tau_L(r)$ vanishes.
Since ${\mathcal M}^\tau_L(r)'$ is a nonempty Zariski open subset of ${\mathcal M}^\tau_L(r)$, the homomorphism $$\text{Br}({\mathcal M}^\tau_L(r))\, \longrightarrow\,
\text{Br}({\mathcal M}^\tau_L(r)')$$ induced by the inclusion ${\mathcal M}^\tau_L(r)'\,
\hookrightarrow\, {\mathcal M}^\tau_L(r)$ is injective. Therefore, from Lemma \[lem2\] it follows that $\text{Br}({\mathcal M}^\tau_L(r))\,=\, 0$.
[AAAA]{}
V. Balaji, I. Biswas, O. Gabber and D. S. Nagaraj, Brauer obstruction for a universal vector bundle. *Comp. Rend. Acad. Sci. Paris* **345** (2007), 265–268.
I. Biswas, T. Gómez and V. Muñoz, Torelli theorem for the moduli space of framed bundles, *Math. Proc. Camb. Phil. Soc.* **148** (2010), 409–423.
H. U. Boden and K. Yokogawa, Rationality of moduli spaces of parabolic bundles, *Jour. London Math. Soc.* **59** (1999), 461–478.
O. Gabber, Some theorems on Azumaya algebras, in: *The Brauer Group*, pp. 129–209, Lecture Notes in Math., Vol. 844, Springer, Berlin–New York, 1981.
N. Hoffmann, Rationality and Poincaré families for vector bundles with extra structure on a curve, *Int. Math. Res. Not.* 2007, no. 3, Art. ID rnm010, 30 pp.
N. Hoffmann, Moduli stacks of vector bundles on curves and the King-Schofield rationality proof, in: *Cohomological and geometric approaches to rationality problems*, pp. 133–148, Progr. Math., 282, Birkhäuser Boston, Inc., Boston, MA, 2010.
D. Huybrechts and M. Lehn, Framed modules and their moduli, *Int. Jour. Math.* **6** (1995), 297–324.
M. Maruyama, Openness of a family of torsion free sheaves, *Jour. Math. Kyoto Univ.* **16** (1976), 627–637.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the discovery of a radio-loud flat-spectrum QSO at z=5.47 with properties similar to those of the [*EGRET*]{} $\gamma$-ray blazars. This source is the brightest radio QSO at z$>$5, with a pc-scale radio jet and a black hole mass estimate $\ga 10^{10}M_\odot$. It appears to be the most distant blazar discovered to date. High energy observations of this source can provide powerful probes of the background radiation in the early universe.'
author:
- 'Roger W. Romani, David Sowards-Emmerd, Lincoln Greenhill & Peter Michelson'
title: 'Q0906+6930: The Highest-Redshift Blazar'
---
Introduction
============
AGN classification is still somewhat heuristic, but in the unified model, blazars are believed to be sources viewed close to the axis of a powerful relativistic jet [@up95 and references therein]. They thus have compact flat spectrum radio counterparts, with apparent superluminal motion at VLBI scales. Optically, the sources are often variable, exhibit significant polarization and show either broad emission lines (flat spectrum radio quasar) or continuum-dominated (BL Lac) spectra. Perhaps the most interesting consequence of the jet/line-of-sight alignment is the observability of a Compton scattered component at X-ray to GeV or (for nearby sources) TeV energies. The interaction of the high energy jet particles and radiation with surrounding photon fields allows unique probes of the extragalactic background light (EBL).
One of the principal discoveries of the [*EGRET*]{} experiment on the [*Compton Gamma Ray Observatory*]{} was the large population of blazars emitting at GeV energies. These represented the largest identified population in the third [*EGRET*]{} (3EG) catalog; still many high latitude sources remained unidentified. We have developed a method to extend identifications to fainter radio flux levels [@srm03; @srmu04], finding counterparts for 70% of the $|b|>10^\circ$ sources. Follow-on observations show that most of these are previously unidentified blazars. Upcoming GeV $\gamma$-ray missions, especially [*GLAST*]{}, will have much higher sensitivity and are expected to detect several thousand blazars. This substantially exceeds the total number of blazars cataloged to date [@lan01]. Thus in preparation for this mission, we have selected radio (& X-ray) sources with properties like the [*EGRET*]{} blazars and are obtaining optical identifications and spectroscopy [@srm04 SRM04].
To prioritize the optical observations, we have also developed a method for estimating the probability that a given position has an excess of $\gamma$-ray photons in the [*EGRET*]{} survey observations, including the effect of strong variability. The radio blazar candidates selected above show, as a set, a clear excess of $\gamma$-ray flux over random sky positions (SRM04); individually, they are well below the 4$\sigma$ criterion for inclusion in the 3EG catalog. For example, radio blazar candidates with $\ge$75% probability of [*EGRET*]{} $\gamma$-ray excess are found at $\sim 0.012/\circ^2$, and thus may represent 10-20% of the new sources detectable to [*GLAST*]{}. These are prime candidates for optical follow-up; we find that most are indeed flat spectrum radio quasars with redshifts 1-2.5. Q0609+6930, however, has a very high redshift of 5.47. Observations described below support a blazar identification for this source; this would be the highest redshift blazar discovered to date.
The discovery of Q0906+6930
===========================
In selecting $\gamma$-ray blazar candidates we start from compact 8.4GHz sources (from CLASS, Meyers et al. 2003 or from our own VLA snapshots) and identify flat/inverted spectrum sources using NVSS counterparts. Our selection algorithm also applies a weighting toward X-ray sources sources detected in the [*ROSAT*]{} All-Sky Survey (RASS); this is weak as many of the [*EGRET*]{} blazars are below the RASS survey threshold. This identification scheme differs from other blazar surveys [eg. DXRBS, @lan01] and gives a much higher correlation with the 3EG sources. We follow up with optical spectroscopy obtained with the Marcario Low Resolution Spectrograph [LRS; @hill98] at the Hobby-Eberly telescope , obtaining source classifications and redshifts.
Q0906+6930 has a strongly inverted spectrum $\alpha_{1.4/8.4}=-0.4$ $(S_\nu \propto \nu^{-\alpha})$ with a CLASS-epoch flux of $S_{8.4} = 190$mJy. The radio position is 09:06:30.75 +69:30:30.8 (J2000.0). This source was cataloged as compact at the $\sim 0.1^{\prime\prime}$ scale, but the short CLASS snapshots provide only limited $\sim 1$mJy sensitivity to flux on arcsecond scales. This source is just included in our ‘sub-threshold’ $\gamma$-ray target list, with a likelihood analysis of the $\gamma$-ray counts giving a 75% probability for emission in excess of background at this position.
HET/LRS optical spectrum
------------------------
We obtained 2$\times$300s exposure with the HET LRS on 1/18/04, under poor $\sim 2^{\prime\prime}$ conditions, which nonetheless showed a classic Ly$\alpha$ forest-absorbed QSO spectrum with high $z\approx 5.5$ redshift. Follow-on exposures of 2$\times$600s on 1/27/04 with a 300l/mm grating, $2^{\prime\prime}$ slit and 5150Å long-pass filter (to preclude 2nd order contamination) and improved $\sim 1^{\prime\prime}$ seeing produced the spectrum shown in figure 1. These data were subject to optimal extraction, calibration and telluric correction using standard IRAF routines. Uncorrectable fringing and imperfect sky subtraction introduce large noise beyond 9500Å. The resulting spectrum has a dispersion of 4Å/pixel and an effective resolution of 16Å, covering $\lambda\lambda$5200-10,000Å.
In the spectrum Ly$\alpha$ is strong, but absorbed; we estimate a pre-absorption rest equivalent width of 50-60Å. The redshift, z=5.47$\pm0.02$, is best constrained by the NV and OIV lines; the OIV rest EW is 22Å and the kinematic width is $W_{FWHM} = 5,000\pm 500$km/s. The UV continuum luminosity at 1350Å is $\lambda {\rm L_{1350}} \approx 5 \times 10^{47}$erg/s (we use $H_0=71$km/s/Mpc, $\Omega_m=0.27$, $\Omega_\Lambda=0.73$). CIV is not clearly detected, so if we use the OIV kinematic width in the @ves02 UV M-L relationship we obtain ${\rm Log}M = 6.2 + {\rm Log}[({\rm W_{FWHM}/10^3 km/s})^2(\lambda L_{1350}/10^{44}
{\rm erg/s})^{0.7}] \approx 10.2$. This is near the upper end of masses inferred for high z sources and formation of such a high $M$ black hole after $\sim 1$Gyr is difficult to understand. There are several candidate CIV absorption line systems redward of Ly$\alpha$. Two candidate systems with apparent damped Ly lines are marked. The presence of intervening absorption raises the possibility that Q0906+6930 is lensed, which could inflate our estimate of the BH mass. However, CLASS finds no radio lens, there is no extended emission visible on the POSS and the source appears point-like in a short pre-spectrum acquisition image. Improved optical/near IR spectroscopy is needed to tighten up the mass estimate, and to improve the identification of the absorption systems as strong CIV doublets.
VLBA Observations and Source SED
--------------------------------
To search for evidence of compact jet structure that would support the blazar designation of Q0906+6930, we obtained snapshots with the VLBA at 2cm and 7mm wavelengths (Figure 2). With 10 stations recording two 8-MHz bandpasses at 15.36GHz, each in two polarizations, we obtained an average of 60 minutes of on-source per baseline under nominal observing conditions on 2004 February 27 (MJD 53062). We used 0716+714 to calibrate phases and delays prior to self-calibration of the blazar, using AIPS. We constructed deconvolved images with natural weighting of the data (half-power beam width 1.55$\times$0.47mas at -2$^\circ$). The RMS background noise in the image was $\sim 10\%$ above the thermal noise limit of 0.18 mJy. The 7 mm observations recorded $\sim 41$ minutes on-source per baseline on 2004 March 3 (MJD 53067), tuned to 43.21 GHz. About 30% of the data could not be self-calibrated because of rapid ($\ll\,1$minute) tropospheric fluctuations. In addition, we obtained no usable data from the Los Alamos antenna, due to weather, and from one polarization-baseband combination recorded at the Saint Croix antenna. The half-power beamwidth was $0.55\times0.30$ mas at $-3.7^\circ$ for natural weighting.
The compact core was unresolved at both wavelengths. In the 2cm image a clear jet component is seen at PA 225$\pm 1^\circ$. For a two component Gaussian model we fit a core flux of 115$\pm 0.3$mJy and a jet flux of 6.3$\pm 0.4$mJy. The jet is marginally detected in our 7mm image with a maximum $\la 0.2$ beams from the position of the 2cm peak, although it is only comparable to the largest peaks in the image background. The two component fit gives a core flux 42$\pm 1.9$mJy and jet flux $4.1\pm 1.1$mJy, for a 3.7$\sigma$ detection.
Our combined [*EGRET*]{} likelihood analysis finds a nominal $\sim 1.5\sigma$ excess of $\gamma$-ray photons at the radio position. In the viewing period with the most significant detection, VP0220, the source produced 1.12$\pm 0.76 \times
10^{-7} {\gamma/ {\rm cm^2/s}}$(E$>$100MeV). However, if other sources beyond the standard 4$\sigma$ 3EG sources are present in this region the fit flux is even lower. At this point it is best to employ the mission-averaged upper limit $4 \times 10^{-8} {\gamma/ {\rm cm^2/s}}$(E$>$100MeV).
We can collect the existing data into a crude spectral energy distribution (SED, Figure 3). Comparing to the well known blazar 3C279 placed at z=5.47, Q0906+6930 has a brighter UV flux, fainter cm wavelength emission and (potentially) a GeV peak power nearly 10$\times$ greater. The source has by far the highest radio loudness $R = f_{5GHz}/f_{0.44\mu,rest} \approx 10^3$ of any $z > 5$ QSO. We also guide the eye with a simple synchrotron-Compton spectrum [@kra04] for a broken power-law approximating a cooled electron spectrum. The synchrotron component is broadly similar to that of 3C279. The compact core spectrum (inset) appears to turn over at cm wavelengths and does not connect to the falling optical spectrum. This is not however unexpected, as most blazars show extra radio components at cm wavelengths. If the 7mm jet detection is trusted, the $\alpha_{2cm/7mm} \approx 0.6$ of the jet component is somewhat flatter than that of the core $\alpha \approx 0.9$. This is unusual, although not unique for pc-scale jets (A. Marscher, private comm.); additional VLBI observations will be needed to see if this is temporary due to jet component brightening at 7mm. With only a marginal GeV detection and an X-ray upper limit, the amplitude of the Compton component is poorly constrained. However, it is interesting to note that the broad line region flux is much larger than that of 3C279 with with a $2-3\times$ brighter optical continuum and rest equivalent line widths $10\times$ larger. Since Comptonized BLR flux is generally believed to dominate the GeV emission, a bright $\gamma$-ray peak may be motivated, as shown by the dotted curves.
Prospects for High Energy Detection
===================================
We expect the Compton scattered emission of blazar jets to be highly beamed. Thus, few blazar X- and $\gamma$-ray sources are visible, but these may appear quite bright. The GLAST mission will have a large duty cycle and a survey sensitivity $\sim 2 \times 10^{-9} \gamma/{\rm cm^2/s}
\approx 10^{-12} {\rm erg/cm^2/s}$(E$>$100MeV), so even if the source is a flaring 3C279 analog, the prospects for detection are good. If the GeV spectrum of this source can be measured, then GLAST can observe the effect of pair attenuation of the $>$GeV flux by the optical-IR EBL [@mp96]. At its high redshift $\le$10GeV observations of Q0906+6930 will be probing $\le 3\mu$ photons from high $z$ star formation. This EBL is presently poorly understood, but an approximate attenuation curve based on the lower $z$ EBL is applied for the combined spectrum (full line). The cut-off is well within the GLAST range. It may be measured if, for example, a stable absorption cutoff is seen on a GeV spectrum whose slope and amplitude varies with time.
In the X-ray regime, core emission probes the low energy Compton component, with the best connection to the radio. However, most interesting is the possibility of detecting a resolved $\sim 1^{\prime\prime}$ ($\sim 10$kpc) scale X-ray jet produced by Compton up-scatter of CMB photons. Such extended ($\sim 2^{\prime\prime}$) jet structure has been seen in [*CXO*]{} observations of the QSO GB 1508+5714 [@sie03] at z=4.3. This feature is also seen as a faint radio jet [@che04] and the flux ratio $f_X/f_R\sim 10^2$ supports the CMB up-scatter hypothesis. Extended emission X-ray observed $\sim$20$^{\prime\prime}$ from the $z=5.99$ QSO J1306+0356 [@sch02b] may represent a second high-z jet, although radio emission is not seen. With a $(1+z)^4$ increase in the energy density of the CMB seed photons, such jets may be detectable to very high redshifts[@sch02a]. Q0906+6930 at z=5.47 experiences $1752 \times$ the local CMB energy density. As we already have evidence for an energetic jet, prospects for detection of a Compton up-scatter component are good and we are pursuing the required X-ray and radio observations to test this hypothesis. If seen, the link to the external seed field gives X-ray/radio measurements unique power to probe the particle and field populations in this high-z jet.
This work was supported in part by NASA grants NAGS-13344 (RWR) and NAS5-00147 (PFM) and SLAC/DOE contract DE-AC03-76SF00515.
The Hobby-Eberly Telescope (HET) is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen. The HET is named in honor of its principal benefactors, William P. Hobby and Robert E. Eberly. The Marcario Low Resolution Spectrograph is named for Mike Marcario of High Lonesome Optics who fabricated several optics for the instrument but died before its completion. The LRS is a joint project of the Hobby-Eberly Telescope partnership and the Instituto de Astronomia de la Universidad Nacional Autonoma de Mexico. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Statistical inference of genetic regulatory networks is essential for understanding temporal interactions of regulatory elements inside the cells. For inferences of large networks, identification of network structure is typically achieved under the assumption of sparsity of the networks. However, current approaches either have difficulty extending to networks with a large number of genes due to the computational constraints, or are difficult to interpret due to the use of module-based models. Also, in most previously proposed models, the choice of different parameters in the model is dealt with in a heuristic manner. For example, in LEARNe [@nam07], which used a system of differential equations to model the dynamics, an upper bound for the degree of connectivity of the network is assumed to be known.
When the number of time points in the expression experiment is not too small, we propose to infer the parameters of the ordinary differential equations using the techniques from functional data analysis (FDA) by regarding the observed time course expression data as continuous-time curves. The derivative of the expression curve with respect to time is easily calculated without using finite difference and thus the problem of missing observations or unequally spaced time points can be dealt with in a consistent way in this model. For networks with a large number of genes, we take advantage of the sparsity of the networks by penalizing the linear coefficients with a $L_1$ norm. The model is fitted using the efficient algorithm that finds the whole regularization path of the parameters which in turn produces the whole spectrum of possible performances in terms of sensitivity and positive predictive value (PPV). The smoothing parameters can be chosen via cross-validation which gives a reasonable trade-off between PPV and sensitivity. The algorithm is compared to the state-of-the-art program LEARNe in simulations for small to medium sized networks which shows the competitiveness of the new approach. The ability of the algorithm to infer network structure is demonstrated using the cell-cycle time course data for *Saccharomyces cerevisiae*.
author:
- |
Heng Lian\
Division of Mathematical Sciences,\
School of Physical and Mathematical Sciences,\
Nanyang Technological University, 637371,\
Singapore
bibliography:
- 'papers.bib'
- 'books.bib'
title: Inference of genetic networks from time course expression data using functional regression with lasso penalty
---
Introduction
============
The increasing amount of high-throughput time course data has provided biologists a window to the understanding of the biomolecular mechanism of different species. The expression of genes in these studies are indicative of the dynamic activities occurring inside the organism. Such regulatory activities involve complicated temporal interactions among different gene products, forming genetic networks indicating the causal relationships between different elements. It is the responsibilities of the statisticians to construct such networks using statistical models that uncovers such relationships. The utility of such models would be vital for the discovery of biological processes that are crucial for understanding of interactions of molecules involved in drug responses.
Different models have been proposed for the construction and analysis of such networks from time course data, including stochastic differential equations [@chen05] and graphical models [@friedman04]. Bayesian methods can be applied which explore hidden causal relationships between different nodes. In particular, dynamic Bayesian networks [@yu04] has achieved great success. For optimization of the networks, a large sample size is required for accurate estimation of the structure and the amount of computations required increases rapidly with the number of nodes, thus limiting the application to networks with a small number of genes.
Due to the obvious connection of the problem to the traditional time series analysis, multivariate autoregressive model has been used to fit the expression data. The utility of this model is constrained by the length of the time course data which is typically on the order of tens of times points, while the number of genes is much larger. Classical maximum likelihood estimator cannot be applied when the number of genes is greater than the number of time points.
To overcome the problem, several algorithms have been introduced, almost all of which applied some dimensionality reduction techniques. Subset selection [@gardner03] proceeds by searching for a subset of nodes with a fixed size that minimizes the least mean square error and use this subset for model fitting and inference. This typically leads to poor generalization performance due to overfitting because the number of time points is small. In addition, one needs to choose the size of the subset for searching and least square errors for subsets with different sizes are not directly comparable to each other. This choice would depend on one’s belief about the degree of connectivity of the networks, which is difficult to assess a priori in applications, and choosing a too small subset size obviously is disastrous to the performance. Alternatively, singular value decomposition (SVD), utilizing a parsimonious set of features underlying the data, can be used to reduce the dimension and infer the networks [@bansal06], but simulations showed that its performance is sensitive to the number of features selected, which is difficult to determine in practice. Based on similar principles, the state space approach uses low dimensional latent variables to reduce the number of parameters to be estimated, but the resulting latent variables are difficult to interpret [@hirose08]. The difficulty in interpreting the model also originates from the fact that the estimated networks do not have the sparsity property. Besides, it is difficult to apply the state space model to time course data with unequally spaced time points.
To address the previously mentioned overfitting effect of subset selection, [@nam07] proposed combining multiple models with least mean square error below a certain threshold. The motivation comes from the machine learning literature where model averaging or multiple voting is often observed to improve generalization performance. It was shown that this approach significantly outperforms simple subset selection and is at least comparable and sometimes better than SVD even when the number of features in SVD is optimally chosen in different situations, except when the number of time points is extremely small (six or smaller). The resulting algorithm named LEARNe, however, shares one common disadvantage with subset selection: an exhaustive search over all possible subsets with a fixed size is conducted to find the good performers, which is infeasible when the number of genes is large. Additionally, some arbitrary threshold should be used to find the final connectivity structure of the networks. Due to the heuristic nature of the algorithm, no statistical theory seems to exist for the choice of this threshold.
All the above mentioned approaches directly used expression data at discrete time points for fitting the model, which might be undesirable when the noise level is high. This is especially true when we use the ordinary differential equations to model the networks which requires estimation of the derivatives. In both LEARNe and SVD algorithms, the derivatives are replaced by the difference of expression level between consecutive time points, which might be a poor estimate of the derivative.
In this article, we propose a novel algorithm, using a penalized form of functional regression, by modelling the time course expressions levels over a certain period as continuous curves after smoothing the expression data. The sparsity of the networks is enforced by introducing $L_1$ penalty on the constant coefficients of the differential equations using another smoothing parameter. By varying the smoothing parameter, we can trace out the whole performance curve of our algorithm, which can be done using the modification of the least angle regression (LARS) program [@efron04]. The sparsity of the network can be inferred based on the data using cross-validation (CV) if desired. Unlike LEARNe, the algorithm is very efficient in computation and can deal with graphs with several hundred nodes when implemented in R on a personal computer. Our simulation shows that the algorithm has comparable and sometimes better performance than LEARNe, while sparsity is obtained without choosing an arbitrary threshold for the coefficient matrix.
Methods
=======
As a first step, we need to convert the time course expression data to continuous curves. This problem falls into the realm of functional data analysis (FDA) as studied extensively in the monograph [@ramsey05]. We use $g_{ij}, i=1,\ldots,n, j=1,\ldots,n_i$ to denote the expression level of gene $i$ at time points $t_j$ with $0\le
t_1<\ldots <t_{n_i}\le 1$. These expression levels have possibly been preprocessed and log transformed which usually results in better fit. Separately for each gene $i$, we search for the smooth function $g_i(t)$ that minimizes the following functional, $$\frac{1}{n_i}\sum_j(g_{ij}-g_i(t_j))^2+\lambda_1\int (g_i''(t))^2\,dt.$$ In the above, the first term enforces the closeness of the curve to the observed expression level at the discrete time points, the second term enforces the smoothness of the function $g_i$ with larger smoothing parameter $\lambda_1$ resulting in a smoother function. Note that with this approach, we don’t need to assume equally spaced time points or identical time points for different genes.
In practice, the above optimization is perform by assuming $g_i$ has an expansion in terms of a certain basis $$g_i(t)=\sum_{j=1}^K a_{ij}b_j(t).$$ After plugging in the above expansion, we only need to solve the $K$ dimensional parameters vector $\{a_{ij}\}$, which is an easy convex optimization problem.
The perhaps most popular basis used in this context is the B-spline basis with order $4$ (i.e. cubic splines). $K$ can be chosen large enough while smoothness of the function is control by the smoothing parameter $\lambda_1$. The automatic choice for $\lambda_1$ can be made using statistical methods like cross-validation. In our experience, we find that the result is quite robust to the choice of this parameter and we find it more convenient to use a fixed parameter in all experiments.
In general, the dynamics of regulatory networks can be written nonparametrically as $$g'_{i}(t)=f(g_1(t),\ldots,g_n(t)), \,t\in[0,1]$$ for a network with $n$ nodes. Inference of such general nonparametric model is difficult with limited amount of data. As a first order approximation, same as [@nam07], we model the regulatory networks using the system of linear ordinary differential equations $$g'_{i}(t)=\alpha_i+\sum_{j=1}^n\beta_{ij}g_j(t), \, t\in [0,1], \,i=1,\ldots,n$$ with $\beta_{ij}$ representing the regulatory effects of gene $j$ on gene $i$. The interpretation is that gene $j$ activates gene $i$ if $\beta_{ij}>0$ and gene $j$ depresses gene $i$ if $\beta_{ij}<0$.
From the estimate of $g_i(t)$, the derivative can be easily evaluated by $g'_i(t)=\sum_j a_{ij}b'_j(t)$. The network coefficients $\alpha_i, \beta_{ij}$ can be fitted by minimizing $$\int_0^1(g'_i(t)-\alpha_i-\sum_j\beta_{ij}g_j(t))^2dt$$ Unlike the discrete least squares, even when the original time points is smaller than the number of genes, there usually exists a unique minimizer of the above problem. However, overfitting still occurs when the number of genes is large. From biological considerations, the network is usually sparse with the evolution of the expression level of one gene only depending on the expression levels of a few other genes, which implies most of the interaction coefficients $\beta_{ij}$ are actually zero. The $L_1$-norm penalty, also commonly called lasso penalty, is well-known to produce sparse regression coefficients [@tibshiranni96]. Despite its popularity, we are unaware of its previous application in functional data analysis.
With the lasso penalty added, we will optimize $$\int_0^1(g'_i(t)-\alpha_i-\sum_j\beta_{ij}g_j(t))^2dt+\lambda_2|\beta_i|_1$$ where $\beta_i=\{\beta_{i1},\ldots,\beta_{in_i}\}$ is the network coefficients and $\lambda_2$ is a smoothing parameter with larger $\lambda_2$ producing sparser networks. By varying the smoothing parameter, we can produce a whole spectrum of networks with different degrees of sparsity. We approximate the integral $\int_0^1(g'_i(t)-\alpha_i-\sum_j\beta_{ij}g_j(t))^2dt$ by the discretized $$\frac{1}{T} \sum_{t=1}^T (g'_i(\frac{t}{T})-\alpha_i-\sum_j\beta_{ij}g_j(\frac{t}{T}))^2$$ $T$ is chosen to be large enough to approximate the integral well, and we find $T=20$ is sufficient for our simulations.
After discretization, the optimization problem becomes a standard linear regression with lasso penalty, which can be solved after converting to a quadratic programming problem [@tibshiranni96]. Computation with different values of $\lambda_2$ makes this algorithm less efficient. Fortunately, there exists an algorithm that computes the whole regularization path for the coefficients for all values of the smoothing parameter $\lambda_2$ which makes our approach very efficient computationally. This algorithm is a modification of least angle regression and takes advantage of the fact that the solution path is piecewise linear [@efron04; @rosset07].
If one desires to choose the parameter $\lambda_2$ based on the data, we can either use cross-validation within each gene which results in a different smoothing parameter for each gene, or we can use cross-validation on the whole dataset, which produces a single smoothing parameter for all the genes. Since the lasso penalty shrinks many networks coefficients to zero, we can infer the sparsity of the networks based on the data without using arbitrary thresholds for the coefficients as is commonly done in previous approaches when the structure of the network is unknown.
Since we will compare our approach to LEARNe, we will briefly describe that algorithm here. In LEARNe, one performs a least square regression for each subset $S$ of $\{1,\ldots,n\}$ of size $k$. That is, for each fixed gene $i$, one minimized $$\sum_{j=1}^{n_i-1} (\Delta g_{ij}-(\alpha_i+\sum_{s\in S}\beta_{is}g_{sj})\Delta_{j})^2$$ where $\Delta g_{ij}=g_{i(j+1)}-g_{ij}$ and $\Delta_{j}=t_{j+1}-t_j$. Thus LEARNe uses finite difference method to approximate the derivative of the expression level with respect to time. Each possible subset $S$ corresponds to a different linear regression model. Instead of using one single best model, which will overfit the data with a small number of observations, one collects the top $\mu\%$ models with the smallest sum of square above. This represents all the models that can fit the data reasonably well. Each model will vote independently on the signs of the network coefficients and the votes are collected into a $n\times(n+1)$ matrix $\Theta$, whose entries are integers with a large positive integer indicating a strong activating effect and large negative integer indicating a strong repressing effect. The final model is found with a thresholding procedure on $\Theta$. If the coefficients are desired, it can be calculated by least square regression with the final model. No suggestion was provided for choosing the threshold in [@nam07].
The authors of [@nam07] found by simulations that the result is robust to the choice of $\mu$ and the method consistently outperformed subset selection using a single model with the smallest least square error. It is also better than SVD unless the number of time points is unreasonably small. The most serious drawback of LEARNe in our opinion is the computational burden when $n$ is large. The author used $k=4$ in their simulation and this results in searching among ${n\choose 4}$ models. For example, when $n=100$, it contains close to 4,000,000 possible models! Empirically, even for $n=50$ with $k=2$, we find our algorithm is much faster than LEARNe, although this might be attributed to our poor implementation of LEARNe.
In [@nam07], no discussion is offered on the choice of $k$. This value obviously should depend on the sparsity of the networks. In practise, since the connectivity of the networks is unknown, it is difficult to choose appropriate $k$ especially considering the computational complexity that comes with large $k$.
Results
=======
We compare the performance of our functional analytical approach with LEARNe in simulations. The networks are generated as follows. For an even number of genes $n=2r$ and $m$ time points $\{1/m,2/m,\ldots,1\}$, The $n\times
n$ coefficient matrix $A$ is generated as follows. $$A_{2i-1,2i-1}=A_{2i,2i}=a_i, A_{2i-1,2i}=-A_{2i,2i-1}=b_i,$$ $$A_{i,j}=0 \mbox{ all other } i,j$$ $$a_i\stackrel{iid}{\sim} Uniform(-2,0), b_i\stackrel{iid}{\sim} Uniform(-5,5)$$ The structure of $A$ has the form $$A=\left[\begin{array}{ccccccc}
a_1&b_1&0&0&\cdots&&\\
-b_1&a_1&0&0&\cdots&&\\
0&0&a_2&b_2&\cdots&&\\
0&0&-b_2&a_2&\cdots&&\\
\vdots&\vdots&\vdots&\vdots&\ddots&&\\
&&&&&a_r&b_r\\
&&&&&-b_r&a_r
\end{array}\right]$$ and the system of differential equations is written in matrix form $$G'=AG$$ with $G(t)=(g_1(t),\ldots,g_n(t))^T$. Note the coefficients $\alpha_i$ do not appear in this simulation and also not used when fitting the model.
We generate the initial expression level $G(0)$ from Uniform distribution and solve the initial value differential equation problem using simple Euler method. The solution $G$ is evaluated at those $m$ time points and independent normal noise with variance $\sigma^2$ is added at each time point.
By the data generation mechanism, the evolution of one gene only depends on the expression level of itself as well as one other gene. The coefficient values $a_i$ are chosen to be negative so that the solution of the differential equations is asymptotically stable to avoid numerical problems.
We used several combinations of parameters $n,m,\sigma$ for our simulation. For each combination, we use 50 randomly generated time course expression matrix and calculate the average performance over these data. In our algorithm, by vary $\lambda_2$, we can reconstruct networks of varying degree of sparsity. By counting the number of connections in the reconstruction and the true network, the performance can be measured using the positive predictive value (PPV) versus sensitivity plots.
![PPV vs. sensitivity plots for simulated networks with 10 nodes and (a) 10 (b) 30 (c) 50 time points. The standard deviation of the noise is $\sigma$=0.1 []{data-label="fig:lownoise"}](pics/yetanother101001.ps){width="6cm"}
![PPV vs. sensitivity plots for simulated networks with 10 nodes and (a) 10 (b) 30 (c) 50 time points. The standard deviation of the noise is $\sigma$=0.1 []{data-label="fig:lownoise"}](pics/102001.ps){width="6cm"}
![PPV vs. sensitivity plots for simulated networks with 10 nodes and (a) 10 (b) 30 (c) 50 time points. The standard deviation of the noise is $\sigma$=0.1 []{data-label="fig:lownoise"}](pics/105001.ps){width="6cm"}
![PPV vs. sensitivity plots for simulated networks with 10 nodes and (a) 10 (b) 30 (c) 50 time points. The standard deviation of the noise is $\sigma$=0.3.[]{data-label="fig:highnoise"}](pics/101003.ps){width="6cm"}
![PPV vs. sensitivity plots for simulated networks with 10 nodes and (a) 10 (b) 30 (c) 50 time points. The standard deviation of the noise is $\sigma$=0.3.[]{data-label="fig:highnoise"}](pics/another102003.ps){width="6cm"}
![PPV vs. sensitivity plots for simulated networks with 10 nodes and (a) 10 (b) 30 (c) 50 time points. The standard deviation of the noise is $\sigma$=0.3.[]{data-label="fig:highnoise"}](pics/105003.ps){width="6cm"}
The simulation results are shown in Figure \[fig:lownoise\] and Figure \[fig:highnoise\]. These curves are produced by averaging over 50 randomly generated data sets with smoothing for visualization. That is, each curve is the result of using nonparametric smoothing over 50 PPV vs. sensitivity curves after applying a particular algorithm, either penalized functional approach or LEARNe. From those figures, it can be seen that for networks with 10 nodes, when the number of time points is small and with small noise level the performance of our algorithm is similar to LEARNe. But for simulation with a larger number of time points and larger noise level, the performance of our algorithm becomes better than LEARNe. This is possibly due to the fact that the finite different approximation to derivatives used in LEARNe comes into trouble in these situations. We also performed simulations with $n=30$ and $n=50$ and observed similar effects. More importantly, we are able to run our algorithm on networks with $n=500$ (taking about 20 minutes) while it is impossible to run LEARNe with $n$ bigger than $100$ in our implementation.
In the above simulations, we used $k=4$ when applying LEARNe to the simulated data, where $k$ is the subset size to search over in LEARNe. In these simulated data, it is known that the connection size is actually $2$ for each gene. Curiously, as shown in Figure \[fig:diffk\], using $k=2$ results in much worse performances. As expected, if we use $k=1$, the result is even worse. This shows that the performance of LEARNe depends critically on the size of subsets searched, which is in turn constrained by the computational resources available.
We demonstrate the performance of the our penalized functional model with the application to the cell cycle regulatory network of Saccharomyces cerevisiae. The dataset comes from [@spellman98] which provides a comprehensive list of cell cycle regulated genes identified by time course expression analysis. We use the 18 time points of the alpha factor synchronized expression data. This dataset has been used widely for evaluating a wide variety of statistical models. Same as [@nam07], we consider 20 genes including 4 transcription factors known to be involved in regulatory functions during different stages of the cell cycle.
We apply our approach to this dataset. The temporal evolution of these 20 genes are shown in Figure \[fig:fda\] after B-spline smoothing of the expression data. To get a final model with data-based inference of network structure, we use the smoothing parameter selected by cross-validation with the same smoothing parameter for all 20 genes. The final result with the interactions between each gene and four transcription factor is shown in Table \[tab:cycle\], and we compare the result with known interactions retrieved from the YEASTRACT database [@teixeira06]. For this submatrix, we get PPV=0.54 and sensitivity=0.80. Since all statistical models are merely mathematical approximations to the true world, it is plausible that automatically chosen model undersmoothes the coefficients matrix to provide a better fit to the data. One can also manually specify the smoothing parameter to achieved desired sparsity of the networks.
![PPV vs. sensitivity plots for simulated networks with 10 nodes and 10 time points. The standard deviation of the noise is $\sigma$=0.1. The algorithm LEARNe is applied with subset size $k=4$, $k=2$ and $k=1$.[]{data-label="fig:diffk"}](pics/anotherdifferentk.ps){width="7cm"}
![The expression levels of 20 genes from the cell cycle time course expression data, represented as continuous curves.[]{data-label="fig:fda"}](pics/expression.ps){width="7cm"}
Discussion
==========
We described a new algorithm for network construction using time course expression data. The algorithm is based on functional data analysis with connection coefficients regularized by the lasso penalty. This is a very powerful approach that makes inference for large networks possible due to the efficient optimization procedures previously proposed.
Our new algorithm provides several advantages over some previous approaches. First, it achieves noise reduction by regarding the expression data as continuous curves. High noise contained in the expression data usually disturbs inferences when one uses finite difference to approximate derivatives in the model. Second, the fitting procedure can be made fully automatic with little intervention from the user. The parameters in the model can be chosen using well studied statistical techniques such as cross-validation. Third, the existing optimization procedure can efficiently infer the network structure with the whole regularization path for the coefficients simultaneously and thus makes inference of large networks feasible.
It is well known that biological side information can reduce the number of false positives and false negatives. It would be interesting to take into account such information in future work. For example, prior knowledge on the interactions of genes can be incorporated into the smoothing parameter so that different interaction coefficients can be penalized differently, resulting in adaptive lasso penalty [@zou06]. We expect this strategy will achieve desired improvement on network prediction.
Funding {#funding .unnumbered}
=======
This research was funded by Singapore Ministry of Education Tier 1 SUG.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author thanks Dr. Dougu Nam for providing the MATLAB code for LEARNe.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We derive efficient and reliable goal-oriented error estimations, and devise adaptive mesh procedures for the finite element method that are based on the localization of a posteriori estimates. In our previous work \[SIAM J. Sci. Comput., 42(1), A371–A394, 2020\], we showed efficiency and reliability for error estimators based on enriched finite element spaces. However, the solution of problems on a enriched finite element space is expensive. In the literature, it is well known that one can use some higher-order interpolation to overcome this bottleneck. Using a saturation assumption, we extend the proofs of efficiency and reliability to such higher-order interpolations. The results can be used to create a new family of algorithms, where one of them is tested on three numerical examples (Poisson problem, p-Laplace equation, Navier-Stokes benschmark), and is compared to our previous algorithm.'
author:
- 'B. Endtmayer'
- 'U. Langer'
- 'T. Wick'
bibliography:
- './lit.bib'
title: 'Reliability and efficiency of DWR-type a posteriori error estimates with smart sensitivity weight recovering'
---
Introduction {#Section: Introduction}
============
Goal-oriented error estimation using adjoints was established in [@BeckerRannacher1995; @BeRa01], and is a current research topic as attested in many studies. Recent developments include Fractional-Step-$\theta$ Galerkin formulations [@MeiRi14], a partition-of-unity variational localization [@RiWi15_dwr], phase-field fracture problems [@Wi16_dwr_pff], surrogate constructions for stochastic inversion [@MATTIS201836], general adaptive multiscale modeling [@oden_2018], model adaptivity in multiscale problems [@MaiRa18], multigoal-oriented error estimation with balancing discretization and iteration errors [@EnLaWi18], nonstationary, nonlinear fluid-structure interaction [@FaiWi18], realizations on polygonal meshes using boundary-element based finite elements [@WeiWi18], realizations using the finite cell method [@StRaSchroe19], error estimation for sea ice simulations [@MeRi2020], discretization error estimation in computer assisted surgery [@DuBoBuBuChLlLoLoRoTo2020], and a unifying framework with inexact solvers covering discontinuous Galerkin and finite volume approaches [@MaVohYou20]. An open-source framework for linear, time-dependent, goal-oriented error estimation was published in [@KoeBruBau2019a]. Abstract convergence results of goal-oriented techniques were studied in [@HolPo2016] and [@FeiPraeZee16]. A worst-case multigoal-oriented error estimation was carried out in [@BruZhuZwie16]. Recently, using a saturation assumption, e.g., [@DoerflerNochetto2002; @CaGaGed16; @FeiPraeZee16; @BankParsaniaSauter2013], two-side error estimates, namely efficiency and robustness of the adjoint-based error estimator, could be shown [@EnLaWi20].
In most realizations, an adjoint problem is used in order to determine sensitivities that enter the error in a single target functional [@BeRa01; @RanVi2013] or multiple quantities of interest [@Ha08; @EnWi17; @EnLaWi18]. In [@RanVi2013; @EnLaWi18], balancing of discretization and nonlinear iteration errors for single and multiple goal functionals was considered, respectively. However, for nonlinear problems, sensitivity weights in the primal and adjoint variable are necessary. These weights must be of higher-order. Otherwise they yield zero sensitivities, and, therefore, the entire error estimator vanishes. The most straightforward way is to use higher-order finite element spaces [@BeRa01]. A computationally cheaper approach based on low-order finite elements and then a patch-wise higher-order interpolation was suggested as well in the early stages [@BeRa01] and combined in an elegant way to a weak localization in [@BraackErn02]. Rigorous proofs of effectivity measured in mesh-dependent norms, in which the true error and the estimator satisfy a common upper bound, were established in [@RiWi15_dwr]. Therein, saturation assumptions where not required, but the results of effectivity are weaker than in our recent work [@EnLaWi20]. More specifically, in [@EnLaWi20], the adjoint problem is solved in a higher-order space and then interpolated into the low-order space for calculating the interpolation error. For this procedure, two-side error estimates are proven, while using the previously mentioned saturation assumption.
The objective of this paper is now the other way around, namely efficiency proofs (under saturation assumptions) for interpolations from low-order finite element spaces into higher-order spaces, which are not present in the literature to date. We establish results when both the primal and adjoint problems are computed only in low-order finite element spaces. To this end, two additional error terms will be introduced. The resulting error estimator holds for nonlinear PDEs and nonlinear goal functionals, and accounts for the discretization error, nonlinear iteration error and the interpolation error. These derivations lead to a novel adaptive algorithm that works at low computational cost in goal-oriented frameworks. Our theoretical and algorithmic developments are then substantiated with the help of several numerical tests. It is known that the saturation assumption may be violated for transport or convection-dominated problems. We provide one numerical test in which this is the case. Furthermore, linear and nonlinear problem configurations are considered to cover most relevant classes of stationary, nonlinear settings.
The outline of this paper is as follows: In Section \[Section: The dual weighted residual method for nonlinear problems \], an abstract setting for the DWR method is introduced. Then, in Section \[Section: Efficiency and reliability results for the DWR estimator in enriched spaces\], the DWR approach with general approximations is stated. A discussion of all terms of the newly proposed error estimator is provided in Section \[Section: Localization and discussions on the error estimator parts\]. Based on these findings, an adaptive algorithm is designed in Section \[Section: Algorithms\]. Several numerical tests with linear PDEs and linear goal functionals, nonlinear settings, and a stationary Navier-Stokes configuration are carried out in Section \[Section: Numerical examples\]. We summarize our results in Section \[Section: Conclusions\].
An abstract setting and the dual weighted resiudal method {#Section: The dual weighted residual method for nonlinear problems }
=========================================================
In this section, we briefly introduce the notation and the settings that we consider in this work. These are similar to our previous studies [@EnLaWi18; @EnLaWi20].
An abstract setting
-------------------
Let $U$ and $V$ be reflexive Banach spaces, and let $\mathcal{A}: U \mapsto V^*$ be a nonlinear mapping, where $V^*$ denotes the dual space of the Banach space $V$. We consider the problem: Find $u \in U$ such that $$\label{Equation: Cont Primal Problem}
\mathcal{A}(u)=0 \qquad \text{ in } V^*.$$ This problem will be refereed as the *primal problem*. As mapping we have in mind some (possibly) nonlinear partial differential operator. Furthermore, we consider finite dimensional subspaces of $U_h \subset U$ and $V_h \subset V$. In this work, $U_h$ and $V_h$ are finite element spaces . This leads to the following finite dimensional problem: Find $u_h \in U_h$ such that $$\label{Equation: Discret Primal Problem}
\mathcal{A}(u_h)=0 \qquad \text{ in } V_h^*.$$ We assume that the problem (\[Equation: Cont Primal Problem\]) as well as the finite dimensional problem (\[Equation: Discret Primal Problem\]) are solvable. Further conditions will be imposed when needed. However, we do not aim for the solution $u$. The goal is to obtain one characteristic quantity (quantity of interest) $J(u)\in \mathbb{R}$, i.e., a functional evaluation, evaluated at the solution $u \in U$ , where $J: U \mapsto \mathbb{R}$.
The dual weighted residual method {#PU-DWR-NONLINEAR-ONEFUNTIONAL}
---------------------------------
In this section, we briefly review the Dual Weighted Residual (DWR) method for nonlinear problems. This work was extended to balance the discretization and iteration errors in [@RanVi2013; @RaWeWo10; @MeiRaVih109].
This paper forms together with our previous works [@EnLaWi18; @EnLaWi20] the basis of the current study. Since the DWR method is an adjoint based method, we consider the *adjoint problem*: Find $z \in V$ such that $$\label{Equation : cont adjoint Problem}
\left(\mathcal{A}'(u)\right)^* (z) = J'(u) \qquad \text{ in } U^*,$$ where $ \mathcal{A}'(u)$ and $J'(u)$ are the Fréchet-derivatives of the nonlinear operator and functional, respectively, evaluated at the solution of the primal problem $u$. In the following sections, we require a finite dimensional version of (\[Equation : cont adjoint Problem\]) that reads as follows: Find $z_h \in V_h$ such that $$\label{Equation : discrete adjoint Problem}
\left(\mathcal{A}'(u_h)\right)^* (z_h) = J'(u_h) \qquad \text{ in } U_h^*.$$ Similarly to the findings in [@RanVi2013; @BeRa01; @RaWeWo10; @EnLaWi20], we obtain an error representation in the following theorem:
\[Theorem: Error Representation\] Let us assume that $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in \mathcal{C}^3(U,\mathbb{R})$. If $u$ solves (\[Equation: Cont Primal Problem\]) and $z$ solves (\[Equation : cont adjoint Problem\]) for $u \in U$, then the error representation $$\begin{aligned}
\label{Error Representation}
\begin{split}
J(u)-J(\tilde{u})&= \frac{1}{2}\rho(\tilde{u})(z-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(u-\tilde{u})
-\rho (\tilde{u})(\tilde{z}) + \mathcal{R}^{(3)},\nonumber
\end{split}
\end{aligned}$$ holds true for arbitrary fixed $\tilde{u} \in U$ and $ \tilde{z} \in V$, where $\rho(\tilde{u})(\cdot) := -\mathcal{A}(\tilde{u})(\cdot)$, $\rho^*(\tilde{u},\tilde{z})(\cdot) := J'(u)-\mathcal{A}'(\tilde{u})(\cdot,\tilde{z})$, and the remainder term $$\begin{split} \label{Error Estimator: Remainderterm}
\mathcal{R}^{(3)}:=\frac{1}{2}\int_{0}^{1}\big[J'''(\tilde{u}+se)(e,e,e)
-\mathcal{A}'''(\tilde{u}+se)(e,e,e,\tilde{z}+se^*)
-3\mathcal{A}''(\tilde{u}+se)(e,e,e)\big]s(s-1)\,ds,
\end{split}$$ with $e=u-\tilde{u}$ and $e^* =z-\tilde{z}$.
The proof is given in [@EnLaWi18].
For the arbitrary elements $\tilde{u}$ and $\tilde{z}$, we think of approximations to the discrete solutions $u_h$ and $z_h$. The resulting error estimator reads as $$\label{FullErrorEstimator}
\eta=\frac{1}{2}\rho(\tilde{u})(z-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(u-\tilde{u})
+\rho (\tilde{u})(\tilde{z}) + \mathcal{R}^{(3)}.$$ This error estimator is exact, but it still depends on the unknown solutions $u$ and $z$. Therefore, it is not computable. To obtain a computable error estimator, one can replace the exact solutions $u$ and $z$ by an approximate solution on enriched finite dimensional spaces $U_h^{(2)}$ and $V_h^{(2)}$. This was already discussed in [@BeRa01; @BaRa03]. Some efficiency and reliability results for this replacement are discussed in [@EnLaWi20]. Other replacements will be covered by the theory in this work. This includes (patch-wise) reconstructions as suggested in [@BeRa01; @BaRa03; @BraackErn02] .
DWR error estimation using general approximations {#Section: Efficiency and reliability results for the DWR estimator in enriched spaces}
=================================================
In this key section, the DWR estimator is augmented to deal with general approximations such as interpolations. The latter allow for a very cost-efficient realization of the DWR estimator for both linear and nonlinear problems. These improvements are significant. However, it turns out that the governing proofs have the same structure as in Section 3 in our previous work [@EnLaWi20]. Therein, it was assumed that the solutions on enriched spaces are known.
In the current work, we consider that we just have some arbitrary approximations in the enriched spaces. Examples how this is done are by inaccurate or accurate solves on the enriched space or (patch-wise) higher-order interpolation operators [@BeRa01; @BaRa03]. We show efficiency and reliability for an alternate form of the error estimator using some saturation assumption for the goal functional on two different kind of approximations. In other words, this means that the approximations in the enriched spaces deliver a more accurate result in the quantity of interest than the approximation of $u_h$.
Results on discrete spaces
--------------------------
Let $\tilde{u}_h^{(2)} \in U_h^{(2)}$ be some arbitrary but fixed approximation of the primal problem: Find ${u}_h^{(2)} \in U_h^{(2)}$ such that $\mathcal{A}({u}_h^{(2)})=0$ in $(V_h^{(2)})^* $, and $\tilde{z}_h^{(2)} \in V_h^{(2)}$ be an approximation to the discretized adjoint problem : Find ${z}_h^{(2)} \in V_h^{(2)}$ $(\mathcal{A'}({u}_h^{(2)}))^*({z}_h^{(2)}) =J'({u}_h^{(2)})$ in $(U_h^{(2)})^*.$
\[Corrollary: discrete Error Representation\] Let us assume that $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in \mathcal{C}^3(U,\mathbb{R})$, and let $\tilde{u} \in U$ and $ \tilde{z} \in V$ be arbitrary but fixed. If $\tilde{u}_h^{(2)} \in U_h^{(2)}$ and $\tilde{z}_h^{(2)} \in V_h^{(2)}$ are some approximations of $u$ and $z$, then the error representation $$\begin{aligned}
\begin{split}
J(\tilde{u}_h^{(2)})-J(\tilde{u})&= \frac{1}{2}\rho(\tilde{u})(\tilde{z}_h^{(2)}-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(\tilde{u}_h^{(2)}-\tilde{u})
-\rho (\tilde{u})(\tilde{z}) \\ &-\rho(\tilde{u}_h^{(2)})(\frac{\tilde{z}_h^{(2)}+ \tilde{z}}{2})+\frac{1}{2}\rho^*(\tilde{u}_h^{(2)},\tilde{z}_h^{(2)})(\tilde{u}_h^{(2)}-\tilde{u})+ \tilde{\mathcal{R}}^{(3)(2)}
\end{split}
\end{aligned}$$ holds. In this error representation, the new terms in comparison to [@EnLaWi20] are $$\rho(\tilde{u}_h^{(2)})(\frac{\tilde{z}_h^{(2)} +\tilde{z}}{2})
\quad\text{and}\quad
\frac{1}{2}\rho^*(\tilde{u}_h^{(2)},\tilde{z}_h^{(2)})(\tilde{u}_h^{(2)}-\tilde{u}).$$ Furthermore, we have $\rho(\tilde{u})(\cdot) := -\mathcal{A}(\tilde{u})(\cdot)$ and $\rho^*(\tilde{u},\tilde{z})(\cdot) := J'(u)-\mathcal{A}'(\tilde{u})(\cdot,\tilde{z})$ as usual. Finally, the remainder term is given by $$\begin{split} \label{Error Estimator: Remainderterm1}
\tilde{\mathcal{R}}^{(3)(2)}:=\frac{1}{2}\int_{0}^{1}[J'''(\tilde{u}+se)(e,e,e)
-\mathcal{A}'''(\tilde{u}+se)(e,e,e,\tilde{z}+se^*)
-3\mathcal{A}''(\tilde{u}+se)(e,e,e)]s(s-1)\,ds,
\end{split}$$ with $e=\tilde{u}_h^{(2)}-\tilde{u}$ and $e^* =\tilde{z}_h^{(2)}-\tilde{z}$.
The proof is an extension of [@RanVi2013]. First we define a general approximation $x := (\tilde{u}_h^{(2)},\tilde{z}_h^{(2)}) \in X:=U \times V$ and $\tilde{x}:=(\tilde{u},\tilde{z}) \in X$. Assuming that $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in
\mathcal{C}^3(U,\mathbb{R})$, we know that the Lagrange functional, which is given by $$\mathcal{L}(\hat{x}):= J(\hat{u})-\mathcal{A}(\hat{u})(\hat{z}) \quad \forall (\hat{u},\hat{z})=:\hat{x} \in X,$$ belongs to $\mathcal{C}^3(X,\mathbb{R})$. This allows us to derive the following identity $$\mathcal{L}(x)-\mathcal{L}(\tilde{x})=\int_{0}^{1} \mathcal{L}'(\tilde{x}+s(x-\tilde{x}))(x-\tilde{x})\,ds.$$ Using the trapezoidal rule $$\int_{0}^{1}f(s)\,ds =\frac{1}{2}(f(0)+f(1))+ \frac{1}{2} \int_{0}^{1}f''(s)s(s-1)\,ds,$$ with $f(s):= \mathcal{L}'(\tilde{x}+s(x-\tilde{x}))(x-\tilde{x})$, cf. [@RanVi2013], we obtain $$\begin{aligned}
\mathcal{L}(x)-\mathcal{L}(\tilde{x}) =& \frac{1}{2}(\mathcal{L}'(x)(x-\tilde{x}) +\mathcal{L}'(\tilde{x})(x-\tilde{x})) + \mathcal{R}^{(3)}.
\end{aligned}$$ From the definition of $\mathcal{L}$, we observe that $$\begin{aligned}
J(\tilde{u}_h^{(2)})-J(\tilde{u})=&\mathcal{L}(x)-\mathcal{L}(\tilde{x}) +{A(\tilde{u}_h^{(2)})(\tilde{z}_h^{(2)}) }- A(\tilde{u})(\tilde{z}) \\=& \frac{1}{2}(\mathcal{L}'(x)(x-\tilde{x}) +\mathcal{L}'(\tilde{x})(x-\tilde{x})) +A(\tilde{u}_h^{(2)})(\tilde{z}_h^{(2)}) -A(\tilde{u})(\tilde{z})+ \tilde{\mathcal{R}}^{(3)(2)}.
\end{aligned}$$ It holds $$\begin{aligned}
\mathcal{L}'(x)(x-\tilde{x}) +\mathcal{L}'(\tilde{x})(x-\tilde{x}) = & \underbrace{J'(\tilde{u}_h^{(2)})(e)-\mathcal{A}'(\tilde{u}_h^{(2)})(e,\tilde{z}_h^{(2)})}_{=\rho^*(\tilde{u}_h^{(2)},\tilde{z}_h^{(2)})(\tilde{u}_h^{(2)}-\tilde{u})}-{A(\tilde{u}_h^{(2)})(e^*)}\\+ &\underbrace{J'(\tilde{u})(e)-\mathcal{A}'(\tilde{u})(e,\tilde{z})}_{=\rho^*(\tilde{u},\tilde{z})(\tilde{u}_h^{(2)}-\tilde{u})}-\underbrace{A(\tilde{u})(e^*)}_{=\rho(\tilde{u})(\tilde{z}_h^{(2)}-\tilde{z})}.
\end{aligned}$$ Further manipulations and rewriting, together with $$-\frac{1}{2}A(\tilde{u}_h^{(2)})(e^*) + {A(\tilde{u}_h^{(2)})(\tilde{z}_h^{(2)})}=
A(\tilde{u}_h^{(2)})(-\frac{1}{2}\tilde{z}_h^{(2)} +\frac{1}{2}\tilde{z}+\tilde{z}_h^{(2)}) )
= {A(\tilde{u}_h^{(2)})(\frac{\tilde{z}_h^{(2)} +\tilde{z}}{2}) },$$ yield $$\begin{aligned}
J(\tilde{u}_h^{(2)})-J(\tilde{u})=\frac{1}{2}\big(\rho^*(\tilde{u}_h^{(2)},\tilde{z}_h^{(2)})(\tilde{u}_h^{(2)}-\tilde{u})
-A(\tilde{u}_h^{(2)})(e^*)
+\rho(\tilde{u})(\tilde{z}_h^{(2)}-\tilde{z})+\rho^*(\tilde{u},\tilde{z})(\tilde{u}_h^{(2)}- \tilde{u})\big)\\ +{A(\tilde{u}_h^{(2)})(\tilde{z}_h^{(2)}) }) -A(\tilde{u})(\tilde{z})+ \tilde{\mathcal{R}}^{(3)(2)}\\
=\frac{1}{2}\big({{\rho^*(\tilde{u}_h^{(2)},\tilde{z}_h^{(2)})(\tilde{u}_h^{(2)}-\tilde{u})}+\rho(\tilde{u})(\tilde{z}_h^{(2)}-\tilde{z})+\rho^*(\tilde{u},\tilde{z})(\tilde{u}_h^{(2)}-\tilde{u})}\big) \\
+\underbrace{{A(\tilde{u}_h^{(2)})(\frac{\tilde{z}_h^{(2)} +\tilde{z}}{2}) })}_{=-\rho(\tilde{u}_h^{(2)})(\frac{\tilde{z}_h^{(2)}+ \tilde{z}}{2})} +A(\tilde{u})(\tilde{z})+ \tilde{\mathcal{R}}^{(3)(2)}.
\end{aligned}$$ These last statements prove the assertion.
If we use the solutions of the adjoint and the primal problem on the enriched spaces for computing $\tilde{z}_h^{(2)}$ and $\tilde{u}_h^{(2)}$, respectively, then $$\rho(\tilde{u}_h^{(2)})(\frac{\tilde{z}_h^{(2)}+ \tilde{z}}{2}) =0
$$ \^\*(\_h\^[(2)]{},\_h\^[(2)]{})(\_h\^[(2)]{}-)=0.
A practial choice for $\tilde{z}_h^{(2)}$ and $\tilde{u}_h^{(2)}$ is, that we can construct higher-order interpolations from $\tilde{z}_h$ and $\tilde{u}_h$. This allows to compute all subproblems with low-order finite elements and the error estimator can be constructed in a cheap way. Under a saturation assumption we prove some results for this well known interpolation techniques.
Theorem \[Corrollary: discrete Error Representation\] motivates the following choice of the error estimator. $$\label{Definition: Error Estimator}
\tilde{\eta}^{(2)}:=\frac{1}{2}\rho(\tilde{u})(\tilde{z}_h^{(2)}-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(\tilde{u}_h^{(2)}-\tilde{u})
-\rho (\tilde{u})(\tilde{z})-\rho(\tilde{u}_h^{(2)})(\frac{\tilde{z}_h^{(2)}+ \tilde{z}}{2})+\\ \frac{1}{2}\rho^*(\tilde{u}_h^{(2)},\tilde{z}_h^{(2)})(\tilde{u}_h^{(2)}-\tilde{u})+ \tilde{\mathcal{R}}^{(3)(2)}.$$
Two-sided estimates for DWR using a saturation assumption
---------------------------------------------------------
In the upcoming lemma, we derive two-sided bounds (efficiency and reliablity) for the error estimator $\tilde{\eta}^{(2)}$ defined by (\[Definition: Error Estimator\]).
\[Lemma: Error Estimatorboundsremainder\] If the assumptions of Theorem \[Theorem: Error Representation\] are fulfilled, then the computable error estimator $\tilde{\eta}^{(2)}$ can be bounded from below and above as follows: $$| J(u)-J(\tilde{u}) |-|J(u)-J(\tilde{u}_h^{(2)})| \leq |\tilde{\eta}^{(2)}| \leq | J(u)-J(\tilde{u}) |+ |J(u)-J(\tilde{u}_h^{(2)})|. \nonumber$$
We proof the bounds in the same way as in [@EnLaWi20]. We know that $ |\eta| = |\tilde{\eta}^{(2)} - (\tilde{\eta }^{(2)} - \eta)| $, and therefore, we can conclude that $$\begin{aligned}
|\eta| - |\eta - \tilde{\eta}^{(2)}| \leq |\tilde{\eta}^{(2)}| \leq |\eta|+|\eta - \tilde{\eta}^{(2)}|.
\end{aligned}$$ The statement of the lemma follows with the identities, $\eta-\tilde{\eta}^{(2)}=J(u)-J(\tilde{u})-J(\tilde{u}_h^{(2)})+J(\tilde{u}) = J(u)-J(\tilde{u}_h^{(2)})$ and $\eta= J(u)-J(\tilde{u})$.
In the sequel, we impose a saturation assumption, which is a common assumption in hierarchical based error estimation [@BankWeiser1985; @BoErKor1996; @BankSmith1993; @Verfuerth:1996a]. Even if the solutions of the primal and adjoint problem in the enriched spaces are used, the saturation assumption may be violated as shown in [@BoErKor1996; @EnLaWi20].
However, for particular problems, quantities of interest and refinements, it is possible to show the saturation assumption [@DoerflerNochetto2002; @Agouzal2002; @AchAchAgou2004; @FerrazOrtnerPraetorius2010; @BankParsaniaSauter2013; @CaGaGed16; @ErathGanterPraetorius2018; @Rossi2002]. It heavily depends on the quantity of interest, the finite dimensional spaces and the problem. We impose the following assumption, which is a slight generalization to [@EnLaWi20].
\[Assumption: Better approximation\] Let $\tilde{u}_h^{(2)}$ be an arbitrary, but fixed approximation in $U_h^{(2)}$, and let $\tilde{u}$ be some approximation in $U_h$. Then we assume that $$|J(u)-J(\tilde{u}_h^{(2)})| < b_h| J(u)-J(\tilde{u}) |$$ for some $b_h<b_0$ and some fixed $b_0 \in (0,1)$.
For gradient based functionals like the flux, one can use recovering techniques to reconstruct the gradient as in [@KoPeRe03]. Here, under certain conditions, the saturation assumption can be shown.
\[Remark: pointevaluation\] If $J(u)$ is just a point evaluation and the given point is a node in the mesh, then $|J(u)-J(\tilde{u}_h^{(2)})| =| J(u)-J(\tilde{u}) |$ in the case of higher-order interpolation as used in [@BeRa01; @RanVi2013; @BaRa03]. Therefore, the saturation assumption is never fulfilled. If the given point is not on the grid, then $|J(u)-J(\tilde{u}_h^{(2)})|$ converges to $| J(u)-J(\tilde{u}) |$ provided that the mesh is locally refine around the evaluation point.
\[Theorem: Efficiency and Reliability with remainder\] Let the Assumption \[Assumption: Better approximation\] be fulfilled. Then the computable error estimator $\tilde{\eta}^{(2)}$ is efficient and reliable, i.e. $$\label {Estimate: hEffektivity+Remainder}
\underline{c}_h|\tilde{\eta}^{(2)}| \leq | J(u)-J(\tilde{u}) | \leq \overline{c}_h|\tilde{\eta}^{(2)}|
\quad \mbox{and} \quad
\underline{c}|\tilde{\eta}^{(2)}| \leq | J(u)-J(\tilde{u}) | \leq \overline{c}|\tilde{\eta}^{(2)}|,$$ where $\underline{c}_h:= 1/(1+b_h)$, $\overline{c}_h:=1/( 1-b_h)$, $\underline{c}:= 1/(1+b_0)$, and $\overline{c}:=1/( 1-b_0)$.
The proof follows from the estimates given in the proof of Lemma \[Lemma: Error Estimatorboundsremainder\] and simple computations. We follow the same steps as in the proof of Theorem 3.3 in [@EnLaWi20].
Following [@RanVi2013; @EnLaWi20], we consider the practical discretization error estimator $$\begin{aligned}
\label{Error Estimator: practical discretization wo Remainder}
\tilde{\eta}_h^{(2)}:=\frac{1}{2}\rho(\tilde{u})(\tilde{z}_h^{(2)}-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(\tilde{u}_h^{(2)}-\tilde{u}) ,
\end{aligned}$$ that corresponds to the theoretical discretization error estimator $$\begin{aligned}
\label{Error Estimator: theoretical discretization wo Remainder}
\eta_h:=\frac{1}{2}\rho(\tilde{u})(z-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(u-\tilde{u}) .
\end{aligned}$$
\[Lemma: Difference Error Estimators\] Let $\eta_h$ and $\tilde{\eta}_h^{(2)}$ be as defined above, and let $\tilde{u} \in U$ and $ \tilde{z} \in V$ be arbitrary but fixed. Furthermore, we assume that $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in \mathcal{C}^3(U,\mathbb{R})$. If $\tilde{u}_h^{(2)} \in U_h^{(2)}$ and $\tilde{z}_h^{(2)} \in V_h^{(2)}$ are some approximations to $u\in U$ and $z\in V$, respectively, then, for the approximations $\tilde{u}_h^{(2)}$ and $\tilde{z}_h^{(2)}$ from the enriched spaces $U_h^{(2)}$ and $V_h^{(2)}$, the following estimates $$\label{Difference Error Estimators1}
\begin{split}
|J(u)-J(\tilde{u}_h^{(2)})|- |\mathcal{R}^{(3)} -\tilde{\mathcal{R}}^{(3)(2)} |- |\tilde{\eta}_{\tilde{z}_h^{(2)}}|- |\tilde{\eta}_{\tilde{u}_h^{(2)}}|\leq |\eta_h -\tilde{\eta}_h^{(2)}| ,\\
|\eta_h -\tilde{\eta}_h^{(2)}| \leq |J(u)-J(\tilde{u}_h^{(2)})|+ |\mathcal{R}^{(3)} -\tilde{\mathcal{R}}^{(3)(2)} |+ |\tilde{\eta}_{\tilde{z}_h^{(2)}}|+ |\tilde{\eta}_{\tilde{u}_h^{(2)}}|,
\end{split}$$ and $$\label{Difference Error Estimators2}
|J(u)-J(\tilde{u})|- |\rho (\tilde{u})(\tilde{z})|-|\mathcal{R}^{(3)}| \leq |\eta_h| \leq |J(u)-J(\tilde{u})|+|\rho (\tilde{u})(\tilde{z})|+|\mathcal{R}^{(3)}|$$ hold. Here, $\mathcal{R}^{(3)}$ is defined in (\[Error Estimator: Remainderterm\]), $\tilde{\mathcal{R}}^{(3)(2)}$ is from Theorem \[Corrollary: discrete Error Representation\], and we have $$\tilde{\eta}_{\tilde{u}_h^{(2)}}:= -\rho(\tilde{u}_h^{(2)})(\frac{\tilde{z}_h^{(2)}+ \tilde{z}}{2}), \qquad \tilde{\eta}_{\tilde{z}_h^{(2)}}:= \frac{1}{2}\rho^*(\tilde{u}_h^{(2)},\tilde{z}_h^{(2)})(\tilde{u}_h^{(2)}-\tilde{u}).$$
The inequality (\[Difference Error Estimators2\]) was already proven in [@EnLaWi20]. From Theorem \[Theorem: Error Representation\] and Theorem \[Corrollary: discrete Error Representation\], we get the identities $$J(u)-J(\tilde{u})= \underbrace{\frac{1}{2}\rho(\tilde{u})(z-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(u-\tilde{u}) }_{\eta_h}
+\rho (\tilde{u})(\tilde{z}) + \mathcal{R}^{(3)},$$ and $$J(\tilde{u}_h^{(2)})-J(\tilde{u})=\underbrace{ \frac{1}{2}\rho(\tilde{u})(\tilde{z}_h^{(2)}-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(\tilde{u}_h^{(2)}-\tilde{u}) }_{\tilde{\eta}_h^{(2)}}
+\tilde{\eta}_{\tilde{u}_h^{(2)}} + \tilde{\eta}_{\tilde{z}_h^{(2)}}+\rho (\tilde{u})(\tilde{z}) + \tilde{\mathcal{R}}^{(3)(2)}.$$ If we substract the equations from above, then we obtain $$J(u)- J(\tilde{u}_h^{(2)} )= \eta_h - {\tilde{\eta}_h^{(2)}} + \mathcal{R}^{(3)}- \tilde{\mathcal{R}}^{(3)(2)} -\tilde{\eta}_{\tilde{u}_h^{(2)}} - \tilde{\eta}_{\tilde{z}_h^{(2)}}.$$ From this equality, we conclude (\[Difference Error Estimators1\]) by using triangle inequality.
Indeed a refined analysis yields the inequalities $$|J(u)-J(\tilde{u}_h^{(2)})|- | \mathcal{R}^{(3)}- \tilde{\mathcal{R}}^{(3)(2)} -\tilde{\eta}_{\tilde{u}_h^{(2)}} - \tilde{\eta}_{\tilde{z}_h^{(2)}}|\leq |\eta_h -\tilde{\eta}_h^{(2)}| \leq |J(u)-J(\tilde{u}_h^{(2)})|+ | \mathcal{R}^{(3)}- \tilde{\mathcal{R}}^{(3)(2)} -\tilde{\eta}_{\tilde{u}_h^{(2)}} - \tilde{\eta}_{\tilde{z}_h^{(2)}}|.$$
Similar as in [@EnLaWi20], we can now show the following result:
\[Lemma: Bounds for Error Estimator\] If the conditions of Lemma \[Lemma: Difference Error Estimators\] are fulfilled, then the inequalities $$\begin{aligned}
\label{Bounds for Error Estimator}
|\tilde{\eta}_h^{(2)}| - \gamma(\mathcal{A},J,\tilde{u}_h^{(2)},\tilde{z}_h^{(2)},u,\tilde{u},\tilde{z}) \leq |J(u)-J(\tilde{u})| \leq |\tilde{\eta}_h^{(2)}| + \gamma(\mathcal{A},J,\tilde{u}_h^{(2)},\tilde{z}_h^{(2)},u,\tilde{u},\tilde{z})
\end{aligned}$$ are valid, where $$\label{twick_Nov_13_2018_eq_1}
\gamma(\mathcal{A},J,\tilde{u}_h^{(2)},\tilde{z}_h^{(2)},u,\tilde{u},\tilde{z}) :=|J(u)-J(\tilde{u}_h^{(2)})|+ |\mathcal{R}^{(3)} -\tilde{\mathcal{R}}^{(3)(2)} |+ |\tilde{\eta}_{\tilde{z}_h^{(2)}}|+ |\tilde{\eta}_{\tilde{u}_h^{(2)}}|+|\rho (\tilde{u})(\tilde{z})|+|\mathcal{R}^{(3)}|.$$
The proof follows the same idea as in the proof of Lemma 3.8 in [@EnLaWi20].
Computable error estimator under a strengthend saturation assumption
--------------------------------------------------------------------
Under a strengthened saturation assumption, the results from above can also be derived for the discretization error estimator $\tilde{\eta}_h^{(2)}$. We suppose the following:
\[Assumption: With Remainder\] Let $\tilde{u}_h^{(2)}$ be an arbitrary, but fixed approximation in $U_h^{(2)}$, and let $\tilde{u}$ be some approximation. Then the inequality $$\gamma(\mathcal{A},J,\tilde{u}_h^{(2)},\tilde{z}_h^{(2)},u,\tilde{u},\tilde{z}) < b_{h, \gamma}|
J(u)-J(\tilde{u}) |,$$ holds for some $b_{h, \gamma}<b_{0, \gamma}$ with some fixed $b_{0, \gamma} \in (0,1)$, where $\gamma(\cdot)$ is defined in .
Assumption \[Assumption: With Remainder\] implies Assumption \[Assumption: Better approximation\]. However, if vice versa, Assumption \[Assumption: Better approximation\] holds, then Assumption \[Assumption: With Remainder\] is fulfilled up to higher-order terms ($ |\mathcal{R}^{(3)} -\tilde{\mathcal{R}}^{(3)(2)} |$, $|\mathcal{R}^{(3)}|$), and by the parts $|\rho (\tilde{u})(\tilde{z})|$, $\tilde{\eta}_{\tilde{u}_h^{(2)}}$, $\tilde{\eta}_{\tilde{z}_h^{(2)}}$ which can be controlled by the accuracy of the approximations. [In particular, if the approximation $ \tilde{u}_h^{(2)}$ in the enriched space coincides with the finite element solution ${u}_h^{(2)}$, i.e. ${u}_h^{(2)} = \tilde{u}_h^{(2)}$, then $\tilde{\eta}_{\tilde{u}_h^{(2)}}=0$. If the same condition, i.e. ${z}_h^{(2)} = \tilde{z}_h^{(2)}$, is fulfilled for the adjoint problem, then $\tilde{\eta}_{\tilde{z}_h^{(2)}}=0.$]{}
\[Theorem: Efficiency and Reliability without remainder\] Let Assumption \[Assumption: With Remainder\] be satisfied. Then the practical error estimator $\tilde{\eta}_h^{(2)}$, defined in , is efficient and reliable, i.e. $$\label {Estimate: hEffektivity-Remainder}
\underline{c}_{h, \gamma}|\tilde{\eta}_h^{(2)}| \leq | J(u)-J(\tilde{u}) | \leq \overline{c}_{h, \gamma}|\tilde{\eta}_h^{(2)}|
\quad \mbox{and} \quad
\underline{c}_{\gamma}|\tilde{\eta}_h^{(2)}| \leq | J(u)-J(\tilde{u}) | \leq \overline{c}_{\gamma}|\tilde{\eta}_h^{(2)}| ,$$ with the positive constants $\underline{c}_{h, \gamma}:= 1/(1+b_{h, \gamma})$, $\overline{c}_{h, \gamma}:=1/( 1-b_{h, \gamma})$, $\underline{c}_{\gamma}:= 1/(1+b_{0, \gamma})$, $\overline{c}_{\gamma}:=1/( 1-b_{0, \gamma})$.
The proof follows the same steps as in [@EnLaWi20]. The result can be derived from Lemma \[Lemma: Bounds for Error Estimator\] and Assumption \[Assumption: With Remainder\]. For further information, we refer to the proof in [@EnLaWi20].
Bounds of the effectivity indices
---------------------------------
As in [@EnLaWi20] we derive bounds for the effectivity indices $I_{eff}$ and $I_{eff,h}$ which are defined by $$\label{Definition: Ieffs}
I_{eff}:= \frac{|\tilde{\eta}^{(2)}|}{|J(u)-J(\tilde{u})|}
\quad \mbox{and} \quad
I_{eff,h}:= \frac{|\tilde{\eta}_h^{(2)}|}{|J(u)-J(\tilde{u})|},$$ respectively. The quantities $ \tilde{\eta}^{(2)}$ and $\tilde{\eta}_h^{(2)}$ are defined as in (\[Definition: Error Estimator\]) and (\[Error Estimator: practical discretization wo Remainder\]). We then obtain:
\[Theorem: Ieffbounds\] Let us assume that $\mathcal{A} \in \mathcal{C}^3(U,V)$ and $J \in \mathcal{C}^3(U,\mathbb{R})$, and let $\tilde{u} \in U$ and $ \tilde{z} \in V$ be arbitrary, but fixed. Then the following two statements are true:
1. If Assumption \[Assumption: Better approximation\] is fulfilled, then $I_{eff} \in [1-b_0,1+b_0]$, and if additionally $b_h \rightarrow 0$, then $I_{eff} \rightarrow 1$.
2. If Assumption \[Assumption: With Remainder\] is fulfilled, then $I_{eff,h} \in [1-b_{0,\gamma},1+b_{0,\gamma}]$, and if additionally $b_{h,\gamma} \rightarrow 0$, then $I_{eff,h} \rightarrow 1$.
The bounds immediately follow from Lemma \[Lemma: Error Estimatorboundsremainder\] and Assumption \[Assumption: Better approximation\], and Lemma \[Lemma: Bounds for Error Estimator\] and Assumption \[Assumption: With Remainder\].
In practice, the efficiency indices $$I_{eff,\mathcal{R}}:=\frac{|\tilde{\eta}_h^{(2)} - \rho(\tilde{u})(\tilde{z})+\tilde{\mathcal{R}}^{(3)(2)}|}{|J(u)-J(\tilde{u})|}$$ and $I_{eff,h}$ almost coincide.
In this section, we omit errors coming from inexact data approximation and numerical quadrature.
Separation of the error estimator parts {#Section: Localization and discussions on the error estimator parts}
=======================================
In this section, we briefly describe the different parts of the error estimator. The error estimator, which is derived in the previous section, consists of five parts. The first three parts $ \tilde{\eta}_h^{(2)}$, $\eta_k$, $\eta^{(2)}_\mathcal{R}$ were already discussed in [@EnLaWi20]. We will now focus on the fourth and fifth part, which are novel. For completeness of presentations, we give a short recap about the other parts.
\[prop\_error\_est\] We split $\tilde{\eta}^{(2)}$ defined in (\[Definition: Error Estimator\]) into the following five parts $\tilde{\eta}_h^{(2)}$, $\eta_k$, $\eta^{(2)}_\mathcal{R}$, $\tilde{\eta}_{\tilde{u}_h^{(2)}}$, and $\tilde{\eta}_{\tilde{z}_h^{(2)}}$: $$\footnotesize
\tilde{\eta}^{(2)}:=\underbrace{\frac{1}{2}\rho(\tilde{u})(\tilde{z}_h^{(2)}-\tilde{z})+\frac{1}{2}\rho^*(\tilde{u},\tilde{z})(\tilde{u}_h^{(2)}-\tilde{u}) }_{:= \tilde{\eta}_h^{(2)}}
-\underbrace{\rho (\tilde{u})(\tilde{z})}_{:=\eta_k} + \underbrace{\tilde{\mathcal{R}}^{(3)(2)}}_{:=\eta^{(2)}_\mathcal{R}}\underbrace{-\rho({\tilde{u}_h^{(2)}})\left(\frac{\tilde{z}_h^{(2)}+\tilde{z}}{2}\right)}_ {:=\tilde{\eta}_{\tilde{u}_h^{(2)}}}+\underbrace{\rho^*(\tilde{u}_h^{(2)},\tilde{z}_h^{(2)})\left(\frac{\tilde{u}_h^{(2)}-\tilde{u}}{2}\right)}_{:=\tilde{\eta}_{\tilde{z}_h^{(2)}}}. \nonumber$$ If the approximations ${u}_h^{(2)} \in U_h^{(2)}$ and $\tilde{z}_h^{(2)} \in V_h^{(2)}$ are the solutions of the primal and the adjoint problem, respectively, then $\tilde{\eta}^{(2)}$ coincides with the error estimator from [@EnLaWi20], and the fourth and the fifth terms are zero.
#### The first part $\tilde{\eta}_h^{(2)}$:
Following [@RanVi2013; @EnLaWi18; @EnLaWi20], $\tilde{\eta}_h^{(2)}$ is related to the discretization error. We use the partition of unity approach developed in [@RiWi15_dwr] to localize $\tilde{\eta}_h^{(2)}$. Here the weak form of the estimator part can be used without applying integration by part.
#### The second part $\eta_k$:
The part $\eta_k=\rho(\tilde{u})(\tilde{z})$ measures the iteration error; cf. [@EnLaWi20; @RanVi2013; @EnLaWi18].
#### The third part $\eta^{(2)}_\mathcal{R}$:
The third part $R^{(3)(2)}$ is of higher order, and is often neglected in the literature. In [@EnLaWi20], we observed in several carefully designed studies that this part is indeed neglectable.
#### The fourth part $\tilde{\eta}_{\tilde{u}_h^{(2)}}$:
The forth part $\tilde{\eta}_{\tilde{u}_h^{(2)}}$ is a measure of the approximation quality of $\tilde{u}_h^{(2)}$. If $\tilde{u}_h^{(2)}$ solves the (discrete) primal problem on the enriched space $U_h^{(2)}$, then we obtain that $\tilde{\eta}_{\tilde{u}_h^{(2)}}=0$. The quantity indicates whether the problem needs to be solved with higher accuracy or the current approximation work sufficiently well. To this end, adaptive stopping criteria for both nonlinear and linear solvers can be designed as in [@EnLaWi20; @EnLaWi18; @EnLaNeiWoWi2020; @RanVi2013; @RaWeWo10; @MeiRaVih109].
#### The fifth part $\tilde{\eta}_{\tilde{z}_h^{(2)}}$:
The fifth and last part $\tilde{\eta}_{\tilde{z}_h^{(2)}}$ provides a quantity to estimate the approximation quality of $\tilde{z}_h^{(2)}$. In contrast to the fourth part, the fifth part $\tilde{\eta}_{\tilde{z}_h^{(2)}}=0$ if $\tilde{z}_h^{(2)}$ solves the (discrete) adjoint problem on the enriched space $V_h^{(2)}$. The quantity indicates whether we should solve the problem more accurate or it is fine to keep the current approximation. This can be used in an adaptive stopping rule criteria for the linear solver similar to [@RaWeWo10; @MeiRaVih109; @ErnVohral2013].
Algorithms {#Section: Algorithms}
==========
Based on the error estimator discussed in Proposition \[prop\_error\_est\], we now design an adaptive algorithm. We would like to mention that this is just one realization of several classes of algorithms, which can be constructed with this idea. Let us start with the initial mesh $\mathcal{T}_h^1$ and the corresponding finite element spaces $V_h^1$, $U_h^1$, $U_{h}^{1,(2)}$ and $V_{h}^{1,(2)}$, where $U_{h}^{1,(2)}$ and $V_{h}^{1,(2)}$ are the enriched finite element spaces. For the resulting adaptively refined meshes $\mathcal{T}_h^\ell$, with $\ell \geq 2$, we consider the following finite element spaces: $V_h^\ell$, $U_h^\ell$, $U_{h}^{\ell,(2)}$ and $V_{h}^{\ell,(2)}$, where $U_{h}^{\ell ,(2)}$ and $V_{h}^{\ell,(2)}$ are the enriched finite element spaces. To this end, we design Algorithm \[Outer DWR Algorithm\].
Start with some initial guess $u_h^{1}$, $\ell=1$, and some choice of $c_z$, $c_u \in \mathbb{R}$.\[Algorithm: Start\] Solve the primal problem: Find $u_h^\ell\in U_h^\ell$ such that $\mathcal{A}(u_h^\ell)(v_h^\ell)=0$ for all $v_h^\ell \in V_h^\ell$, using some nonlinear solver. \[Algorithm: small Primal\] Solve the adjoint problem: Find $z_h^\ell\in V_h^\ell$ such that $\mathcal{A}'(u_h^\ell)(z_h^\ell,v_h^\ell)= J'(u_h^\ell)(v_h^\ell)$ for all $v_h^\ell \in U_h^\ell$, using some linear solver.\[Algorithm: small Adjoint\] Compute the interpolations $u_h^{\ell,(2)}=I_{u}^{(2)}u_h^\ell\in U_h^{\ell,(2)}$ and $z_h^{\ell,(2)}=I_{z}^{(2)}z_h^\ell\in V_h^{\ell,(2)}$.\[Algorithm: Interpolations\] Compute the error estimators $\tilde{\eta}_h^{(2)},$ $\eta_k,$ $\eta^{(2)}_\mathcal{R}$, $\tilde{\eta}_{\tilde{z}_h^{(2)}}$, $\tilde{\eta}_{\tilde{u}_h^{(2)}}$ using $\tilde u_h^{(2)}=z_h^{\ell,(2)}$, $\tilde z_h^{(2)}=z_h^{\ell,(2)}$, $\tilde u=u_h^{\ell}$, and $\tilde z=z_h^{\ell}$. \[Algorithm: Compute\_Estimators\] \[Algorithm: Ifz\] Solve the adjoint problem on the enriched spaces: Find $z_h^{\ell,(2)}\in V_h^{\ell,(2)}$ such that $\mathcal{A}'(u_h^{\ell,(2)})(z_h^{\ell,(2)},v_h^{\ell,(2)})= J'(u_h^{\ell,(2)})(v_h^{\ell,(2)})$ for all $v_h^{\ell,(2)} \in U_h^{\ell,(2)}$, using some linear solver. \[Algorithm: Big Adjoint\] Go to Step \[Algorithm: Compute\_Estimators\]. \[Algorithm: Ifu\] Solve the primal problem on the enriched spaces: Find $u_h^{\ell,(2)}\in U_h^{\ell,(2)}$ such that $\mathcal{A}(u_h^{\ell,(2)})(v_h^{\ell,(2)})=~0$ for all $v_h^{\ell,(2)} \in V_h^{\ell,(2)}$, using some nonlinear solver. \[Algorithm: Big Primal\] Go to Step \[Algorithm: Compute\_Estimators\]. \[Algorithm: stopping Criteria\] Algorithm terminates with final **output** $J(u_h^\ell)$. Localize error estimator $\tilde{\eta_h^{(2)}}$, and $\eta^{(2)}_\mathcal{R}$ and mark elements. \[Algorithm: Localization and marking\] Refine marked elements:$ \mathcal{T}_h^\ell \mapsto \mathcal{T}_h^{\ell+1}$, $\ell=\ell+1$. \[Algorithm: Refinement\] Go to Step \[Algorithm: small Primal\]
We use the same interpolations as discussed in [@BeRa01; @BaRa03]. For further information, we refer the reader to [[@BaRa03]; see pp. 43-44.]{}
In Step \[Algorithm: small Primal\], we use a Newton method with adaptive stopping rule using the estimator part $\eta_k$. The initial guess for the Newton method was the solution on the previous grid. For further information about this Newton method we refer to [@EnLaWi20]. The arising linear systems were solved [by means of]{} the direct solver UMFPACK [@UMFPACK]. However, iterative solvers could also be used, where the ideas from [@RanVi2013; @RaWeWo10] can be exploited.
One can also use $|\tilde{\eta}_h|$ instead of $|\tilde{\eta}_h-\eta_k+\eta^{(2)}_\mathcal{R}|$. Indeed, in [@EnLaWi20], it was observed that $|\eta^{(2)}_\mathcal{R}|$ is of higher-order, and $\eta_k$ can be controlled by the choice of the accuracy of the solver. Furthermore, it is sufficient to use the localized ${\tilde\eta_h^{(2)}}$ instead of ${\tilde\eta_h^{(2)}}$ and $\eta^{(2)}_\mathcal{R}$.
It is not required that the problems in Step \[Algorithm: small Primal\] and Step \[Algorithm: small Adjoint\] are solved accurate. An estimate for this error is $\eta_k$, which is perturbed by higher order terms.
In Step \[Algorithm: small Adjoint\], we use a Newton method with an adaptive stopping role that is based on the estimator part $\eta_k$. We take the solution from the previous grid as initial guess for the Newton iteration. We refer the reader to [@EnLaWi20] for further information about this Newton method.
If the problems in Step \[Algorithm: Big Primal\] and Step \[Algorithm: Big Adjoint\] are solved exactly, then $\tilde{\eta}_{\tilde{z}_h^{(2)}}=\tilde{\eta}_{\tilde{u}_h^{(2)}}=0$. Therefore, the ‘if’ conditions in Step \[Algorithm: Ifz\] and \[Algorithm: Ifu\] are false.
The localization and marking techniques in Step \[Algorithm: Localization and marking\] coincide with those presented in [@EnLaWi20]. For more information on the localization, we refer to [@RiWi15_dwr].
In Step \[Algorithm: Ifz\] and \[Algorithm: Ifu\], we used the constants $c_u=c_z=0.5$. In general, one should choose these constants from the interval $(0,0.5]$.
For the choices $c_u<0$ and $c_z<0$, the resulting algorithm coincides with the algorithm presented in [@EnLaWi20]. Here the enriched problem needs to be solved at each level without any interpolations. On the other hand, if we choose $c_u=c_z=\infty$, then we never solve the enriched problem, and always use interpolations. This leads to a similar approach as in [@RanVi2013].
Numerical examples {#Section: Numerical examples}
==================
In this section, we discuss three different problems. We also vary the goal functionals. More precisely, the first example deals with the Poisson equation and the average of the solution over the computational domain $\Omega$ as simple linear model problem and quantity of interest, respectively. In the second test, we use a regularized $p$-Laplace equation, and in the third example, we consider a stationary Navier-Stokes benchmark problem. The programming code is based on the finite element library deal.II [@dealII90].
For the first two examples, we use continuous bi-linear ($Q_1^c$) finite elements for $V_h=U_h$, and continuous bi-quadratic ($Q_2^c$) finite elements for $V_h^{(2)}=U_h^{(2)}$ in sense of Ciarlet [@Ciarlet:2002:FEM:581834]. In the final example, we use the same configuration as in [@EnLaWi20], i.e., the finite element spaces $V_h=U_h$ and $V_h^{(2)}=U_h^{(2)}$ are based on $ \left[Q_2^c\right]^2 \times Q_1^c$ and $\left[Q_4^c\right]^2 \times Q_2^c$ finite elements, respectively.
We use the following abbreviations for the error estimators used in Algorithm \[Outer DWR Algorithm\]: **new*:* $c_u=c_z=0.5$, **full*:* $c_u=c_z=-1$, and **int*:* $c_u=c_z=\infty$ (=$10^{100}$) in the numerical experiments. The choice $c_u=c_z=-1$ means that we always solve the primal and adjoint problems. Therefore, for this case, the algorithm coincides with the algorithm presented in [@EnLaWi20] (up to the starting point of the Newton iteration). If we have $c_u=c_z=\infty$, then this results in the case where we always use higher-order interpolation to the approximate $u-u_h$ and $z-z_h$ as done in [@BeRa01].
Poisson equation
----------------
In the first example, we consider the Poisson equation on the unit square $\Omega = (0,1)^2$. The problem formally reads as: Find $u \in H^1(\Omega)$ such that $-\Delta u = 1$ in $\Omega$ and $u=0$ on $\partial \Omega$. The exact solution is given by $$\begin{aligned}
u(x,y)=\left(\frac{2}{\pi}\right)^4 \sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{sin\big((2k+1)\pi x\big)sin\big((2l+1)\pi y\big)}{(2k+1)(2l+1)\big((2k+1)^2+(2l+1)^2\big)}.
\end{aligned}$$ The quantity of interest is given by $J(u)=\int_{\Omega} u \text{d}x $. The evaluation at the solution yields $$\begin{aligned}
J(u)=&\left(\frac{2}{\pi}\right)^6 \sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{1}{(2k+1)^2(2l+1)^2\big((2k+1)^2+(2l+1)^2\big)}\\
=& \frac{1}{12} - \frac{31}{2\pi} \zeta(5) +\left(\frac{2}{\pi}\right)^5 \sum_{k=0}^{\infty}\frac{1}{(2k+1)^5\big(e^{(2k+1)\pi}+1\big)} \approx 0.03514425373878841,
\end{aligned}$$ where $\zeta$ is the Riemann zeta function.
set output “Figures/Example1aa\_archive.tex” set key bottom right set key opaque set datafile separator “|” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/Laplace/Mean/New/data.db “SELECT DISTINCT Refinementstep+1, Ieff from data WHERE Refinementstep <= 24 ”’ u 1:2 w lp lw 5 title ’ $I_{eff,h}$(*new*)’, ’< sqlite3 Data/Laplace/Mean/Full/data.db “SELECT DISTINCT Refinementstep+1, Ieff from data WHERE Refinementstep <= 24 ”’ u 1:2 w lp lw 3 title ’ $I_{eff,h}$ (*full*)’, ’< sqlite3 Data/Laplace/Mean/Interpolation/data.db “SELECT DISTINCT Refinementstep+1, Ieff from data WHERE Refinementstep <= 24 ”’ u 1:2 w lp lw 3 title ’ $I_{eff,h}$ (*int*)’, 1 lw 2 \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
\[Figure: LaplaceIeff\]
When we compare the errors in our quantity of interest for **new**, **full**, **int**, we observe that, for all three choices, we obtain almost the same error in comparison to the degrees of freedom (DOFs); cf. Figure \[Figure: LaplaceErrors\]. Furthermore, we see from Figure \[Figure: Laplacesolves\] that Algorithm \[Outer DWR Algorithm\] always decides to solve the primal problem on the enriched space on each level. However, the adjoint problem is never solved on the enriched space. If we compare the effectivity indices shown in Figure \[Figure: LaplaceIeff\], then we observe that, for **new** and **full**, the effectivity indices almost coincide. If we only use interpolation, then the result is slightly worse.
set output “Figures/Example1aaa\_archive.tex” set key bottom left set logscale set key opaque set datafile separator “|” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/Laplace/Mean/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE Refinementstep <= 24 ”’ u 1:2 w lp lw 5 title ’ Error (*new*)’, ’< sqlite3 Data/Laplace/Mean/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE Refinementstep <= 24 ”’ u 1:2 w lp lw 3 title ’ Error (*full*)’, ’< sqlite3 Data/Laplace/Mean/Interpolation/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE Refinementstep <= 24 ”’ u 1:2 w lp lw 3 title ’ Error (*int*)’, \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
\[Figure: LaplaceErrors\]
set output “Figures/Example1meansolves\_archive.tex” set key center set key opaque set datafile separator “|” set yrange \[-0.05:1.05\] set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/Laplace/Mean/New/data.db “SELECT DISTINCT Refinementstep, (1.0\*enrichedsolvesz)/(Refinementstep+1) from data ”’ u 1:2 w lp lw 3 title ’ $\mathfrak{z }_\ell$/$\ell$’, ’< sqlite3 Data/Laplace/Mean/New/data.db “SELECT DISTINCT Refinementstep, (1.0\*enrichedsolvesu)/(Refinementstep+1) from data ”’ u 1:2 w lp lw 3 title ’ $\mathfrak{u }_\ell$/$\ell$’, \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
\[Figure: Laplacesolves\]
Regularized p-Laplace equation
------------------------------
In the second numerical example, we consider the regularized $p$-Laplace equation with $\varepsilon =10^{-10}$ and $p=4$. The computational domain $\Omega$ is a slit domain given by $\Omega = (-1,1)^2 \setminus \{0\} \times (-1,0)$ as visualized in Figure \[Figure: Slit domain with BC\]. The given problem reads as: Find $u \in W^1_p(\Omega)$ such that $$-\text{div}((|\nabla u |^2+ \varepsilon^2)^{\frac{p-2}{2}} \nabla u) =1 \; \text{in $\Omega$},
\quad \mbox{and}\;
u=0 \; \text{on $\Gamma_D$},
\quad
(|\nabla u |^2+ \varepsilon^2)^{\frac{p-2}{2}})\nabla u \cdot \vec{n}=0 \; \text{on $\Gamma_N$}.$$ The boundary conditions are visualized in the left subfigure of Figure \[Figure: Slit domain with BC\]. We impose Neumann and homogeneous Dirichlet boundary conditions on the left side and on the right side of the slit, respectively.
In the right subfigure of Figure \[Figure: Slit domain with BC\], a plot of the solution is given. Even for the $p=4$, similarities to the distance function, which is the first eigenfunction of the $p$-Laplacian for $p= \infty$, described in [@KaHo2017a; @FaBo2016a], are visible.
### Integral evaluation
As first quantity of interest, we again consider $J(u) = \int_{\Omega} u(x) dx \approx 0.71755$. We observe in Figure \[Figure: Meal\_PLaplace\_Errors\] that we obtain a similar error for either solving the adjoint and primal problem each time (Error(*full*)), for using the interpolation on each level (Error(*int*)), and for Algorithm \[Outer DWR Algorithm\] (Error(*new*)).
As already noticed in [@EnLaWi20], we observe higher-order convergence of the remainder term. The rate is approximately in the order of $\mathcal{O}(\text{DOFs}^{-\frac{3}{2}})$. For the errors and additionally the error estimator $\eta^{(2)}_h$, which is plotted for Algorithm \[Outer DWR Algorithm\], the order of convergence is approximately $\mathcal{O}(\text{DOFs}^{-1})$.
In Figure \[Figure: Meal\_PLaplace\_Solves\], the number of solves in the enriched space using Algorithm \[Outer DWR Algorithm\] divided by the number of solves in the enriched space using the algorithm given in [@EnLaWi20] is shown. We conclude that, on the first seven levels, the same solves as from [@EnLaWi20] are required. Then Algorithm \[Outer DWR Algorithm\] decides that we just have to solve either the primal or the adjoint problem on the enriched space, and use the interpolation in the other. After level $\ell =15$ only interpolation is used. Going back to Figure \[Figure: Meal\_PLaplace\_Errors\], we observe that, on the levels $\ell = 8-12$, the error for just using interpolation is slightly worse than for the other approaches. However, on finer levels this effect does not appear anymore. Excellent effectivity indices are observed as visualized in Figure \[Figure: Meal\_PLaplace\_Ieff\]. For the full estimater, we observe almost no differences to the other versions proposed. In the case of Algorithm \[Outer DWR Algorithm\], the efficiency index $I_{eff}$ is approximately equal to $1.25$ when using interpolation only.
If we compare the different meshes, which are visualized in Figure \[Figure: Meal\_PLaplace\_Meshes\], then we detect that, even after $28$ adaptive refinements, we end up in almost coinciding meshes.
set output “Figures/Example3\_archive.tex” set key bottom left set key opaque set datafile separator “|” set logscale x set logscale y set xrange \[10:100000\] set yrange \[0.8e-8:1\] set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ Error (*new*)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 2 title ’ Error (*full*)’, ’< sqlite3 Data/P\_Laplace\_slit/Interpolate/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 2 title ’ Error (*int*)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$|\eta^{(2)}_h|$’, 1/x dt 3 lw 4 title ’$\mathcal{O}(\text{DOFs}^{-1})$’, 1/(x\*\*1.5) dt 3 lw 4 title ’$\mathcal{O}(\text{DOFs}^{-\frac{3}{2}})$’ \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
\[Figure: Meal\_PLaplace\_Errors\]
set output “Figures/Example1\_archive.tex” set key bottom left set key opaque set datafile separator “|” set yrange \[0:1\] set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT Refinementstep, (1.0\*enrichedsolvesz)/(Refinementstep+1) from data ”’ u 1:2 w lp lw 3 title ’ $\mathfrak{z }_\ell$ /$\ell$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT Refinementstep, (1.0\*enrichedsolvesu)/(Refinementstep+1) from data ”’ u 1:2 w lp lw 3 title ’ $\mathfrak{u }_\ell$ /$\ell$’, \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
\[Figure: Meal\_PLaplace\_Solves\]
set output “Figures/Example1b\_archive.tex” set key bottom set key opaque set datafile separator “|” set yrange \[0:2\] set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT Refinementstep, Ieff from data ”’ u 1:2 w lp lw 3 title ’ $I_{eff,h}$ (new)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT Refinementstep, Ieff from data ”’ u 1:2 w lp lw 3 title ’ $I_{eff,h}$ (*full*)’, ’< sqlite3 Data/P\_Laplace\_slit/Interpolate/data.db “SELECT DISTINCT Refinementstep, Ieff from data ”’ u 1:2 w lp lw 3 title ’ $I_{eff,h}$ (*int*)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT Refinementstep, Iefftilde from data ”’ u 1:2 w lp lw 3 title ’ $I_{eff}$ (*new*)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT Refinementstep, Iefftilde from data ”’ u 1:2 w lp lw 3 title ’ $I_{eff}$ (*full*)’, ’< sqlite3 Data/P\_Laplace\_slit/Interpolate/data.db “SELECT DISTINCT Refinementstep, Iefftilde from data ”’ u 1:2 w lp lw 3 title ’ $I_{eff}$ (*int*)’, \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
\[Figure: Meal\_PLaplace\_Ieff\]
![The resulting meshes on level $\ell=29$ for: solving always the enriched spaces (left), using Algorithm \[Outer DWR Algorithm\] (middle), using interpolations on all levels (right).[]{data-label="Figure: Meal_PLaplace_Meshes"}](Data/P_Laplace_slit/Full/mesh28 "fig:"){width="0.31\linewidth"}![The resulting meshes on level $\ell=29$ for: solving always the enriched spaces (left), using Algorithm \[Outer DWR Algorithm\] (middle), using interpolations on all levels (right).[]{data-label="Figure: Meal_PLaplace_Meshes"}](Data/P_Laplace_slit/New/mesh28 "fig:"){width="0.31\linewidth"}![The resulting meshes on level $\ell=29$ for: solving always the enriched spaces (left), using Algorithm \[Outer DWR Algorithm\] (middle), using interpolations on all levels (right).[]{data-label="Figure: Meal_PLaplace_Meshes"}](Data/P_Laplace_slit/Interpolate/mesh28 "fig:"){width="0.31\linewidth"}
### Point evaluation
In this part, we consider the point evaluation $J(u) = u(x_P) \approx 0.04501097$ as quantity of interest, where the point $x_P=-\frac{9}{10}(1,1)$ is also visualized in Figure \[Figure: Slit domain with BC\]. Inspecting Table \[Table: PLaplacePoint\], we observe that $\lim_{\ell \rightarrow \infty }I_{eff} =0$ due to the local refinement around the evaluation point as mentioned in Remark \[Remark: pointevaluation\].
Furthermore, Table \[Table: PLaplacePoint\] shows that the effectivity indices $I_{eff,h}$ and $I_{eff}$, defined in (\[Definition: Ieffs\]), are both better for Algorithm \[Outer DWR Algorithm\] than for using interpolation on every level. It is a bit surprising that the efficiency indices $I_{eff,h}$ perform equally well. Moreover, $\mathfrak{z }_\ell$ and $\mathfrak{u}_\ell$ show that Algorithm \[Outer DWR Algorithm\] decides to solve the enriched problems on several levels.
In comparison with the algorithm proposed in our previous work [@EnLaWi20], we save five times solving the primal problem on the enriched space, and 1 adjoint problem on the enriched space. In Figure \[Figure: P-LaplacePointMesh24\], we observe that heavy refinement occurs around our evaluation point.
The position of the point was motivated by the singularity in the distance function, which is the first eigenfunction of the $p$-Laplacian for $p= \infty$; see [@KaHo2017a; @FaBo2016a]. This singularity is also refined by our strategy, provided that it is sufficiently close to our point. The errors in the point evaluation are similar for interpolation and the new Algorithm \[Outer DWR Algorithm\].
![$p$-Laplace for $p=4$, $\varepsilon=10^{-10}$: Point evaluation: Mesh on level $\ell =25$.[]{data-label="Figure: P-LaplacePointMesh24"}](Data/P_Laplace_slit/Pointevaluation/Mesh24){width="1.0\linewidth"}
Navier-Stokes benchmark problem
-------------------------------
We now consider the stationary NS-benchmark problem NS2D-1[^1]; see [@TurSchabenchmark1996]. [The computational domain $\Omega$ is given by]{} $(0,H) \times (0,2.2)\setminus \mathcal{B}$, where $H =0.41$, and $\mathcal{B}:=B_\frac{1}{20}(0.2,0.2)$ is nothing but a circle with center at $(0.2,0.2)$ and radius $\frac{1}{20}$. [The problem reads as follows:]{} Find $\textbf{u}:=(u,p) \in [H^1(\Omega)]^2 \times L^2(\Omega)$ such that $$\begin{aligned}
- \nu \Delta u + (u \cdot \nabla) u - \nabla p=& 0 \qquad \qquad\text{in } \Omega,\nonumber\\
\nabla \cdot u =& 0\qquad \qquad\text{in } \Omega,\nonumber\\
u=&0 \qquad \qquad\text{on } \Gamma_{\text{no-slip}}, \\
u=&\hat{u}\qquad \qquad \text{on } \Gamma_{\text{inflow}}, \nonumber\\
\nu \frac{\partial u}{\partial \vec{n}} - p \cdot \vec{n }=& 0 \qquad \qquad\text{on } \Gamma_{\text{outflow}}, \nonumber
\end{aligned}$$ where $\nu = 10^{-3}$. The boundary parts are given by $\Gamma_{\text{outflow}} :=(\{x=2.2\} \cap \partial \Omega)\setminus \partial (\{x=2.2\} \cap \partial \Omega)$, $\Gamma_{\text{inflow}} :=\{x=0\} \cap \partial \Omega$, and $\Gamma_{\text{no-slip}} := \overline{\partial \Omega \setminus (\Gamma_{\text{inflow}} \cup \Gamma_{\text{outflow}})}$.
The inflow is described by $\hat{u}(x,y):=(3w(y)/10,0)$ with $w(y)=4y(H-y)/H^2$. The pressure is uniquely determined due to the do-nothing condition prescribed on $\Gamma_{\text{outflow}}$; see [@HeRaTu96]. Our quantity of interest is given by the lift which is defined as $$J(\textbf{u}):= 500\int_{\partial \mathcal{B}} \left[\nu \frac{\partial u}{\partial \vec{n}} - p \vec{n }\right]\cdot \vec{e}_2\,\text{ d}s_{(x,y)},$$ where $\vec{e}_2 = (0,1)$. The reference value $J(\textbf{u})=0.010618948146$ was taken from [@nabh1998high].
In the numerical simulations, we observed that the ‘if’ conditions (Step 6-7 and Step 9-10) in Algorithm \[Outer DWR Algorithm\] were entered, possibly multiple times, resulting in a significant improvement of the effectivity indices. In Table \[Table: NSAlgorithmusIeff\], these evaluations have the following correspondences to the previous algorithm:
1. Step 1 (Table \[Table: NSAlgorithmusIeff\]) $\quad\widehat{=}\quad$ Step 4 (Alg. \[Outer DWR Algorithm\]). For the computation of the estimators, we use $u_h^{\ell,(2)}=I_{u}^{(2)}u_h^\ell\in U_h^{\ell,(2)}$ and $z_h^{\ell,(2)}=I_{z}^{(2)}z_h^\ell\in V_h^{\ell,(2)}$.
2. Step 2 (Table \[Table: NSAlgorithmusIeff\]) $\quad\widehat{=}\quad$ Step 6-7 (Alg. \[Outer DWR Algorithm\]). For the computation of the estimators, we use $u_h^{\ell,(2)}=I_{u}^{(2)}u_h^\ell\in U_h^{\ell,(2)}$ and $z_h^{\ell,(2)}$ as the solution of the linear problem: Find $z_h^{\ell,(2)} \in V_h^{\ell,(2)}$ such that $\mathcal{A}'(u_h^{\ell,(2)})(z_h^{\ell,(2)},v_h^{\ell,(2)})= J'(u_h^{\ell,(2)})(v_h^{\ell,(2)})$.
3. Step 3 (Table \[Table: NSAlgorithmusIeff\]) $\quad\widehat{=}\quad$ Step 9-10 (Alg. \[Outer DWR Algorithm\]). For the computation of the estimators, we use $u_h^{\ell,(2)}$ as the solution of [the non-linear problem]{} Find $u_h^{\ell,(2)} \in U_h^{\ell,(2)}$ such that $\mathcal{A}(u_h^{\ell,(2)})(v_h^{\ell,(2)})=~0$ and $z_h^{\ell,(2)}$ as in the previous executed step.
4. Step 4 (Table \[Table: NSAlgorithmusIeff\]) $\quad\widehat{=}\quad$ Step 6-7 (Alg. \[Outer DWR Algorithm\]). For the computation of the estimators, we use $u_h^{\ell,(2)}$ as the solution of [the non-linear problem]{}: Find $u_h^{\ell,(2)} \in U_h^{\ell,(2)}$ such that $\mathcal{A}(u_h^{\ell,(2)})(v_h^{\ell,(2)})=~0$, and $z_h^{\ell,(2)}$ as the solution of [the linear problem]{}: Find $z_h^{\ell,(2)} \in V_h^{\ell,(2)}$ such that $\mathcal{A}'(u_h^{\ell,(2)})(z_h^{\ell,(2)},v_h^{\ell,(2)})= J'(u_h^{\ell,(2)})(v_h^{\ell,(2)})$.
Table \[Table: NSAlgorithmusIeff\] shows that we almost always improve the effectivity step by step. We notice that $I_{eff}$ does not depend on the choice of $z_h^{(2)}$ [since it coincides for Step 1 and Step 2 as well as for Step 3 and Step 4 provided that these steps are executed.]{} Furthermore, we observe that, in certain meshes, [the use of]{} interpolation leads to very bad effectivity indices. However, Algorithm \[Outer DWR Algorithm\] improves this during the [adaptive]{} process, [although]{} the saturation assumption is not fulfilled. The saturation assumption is also not fulfilled if $u_h^{(2)}$ is the exact discrete solution of the enriched space as shown in [@EnLaWi20].
In Figure \[Figure: NSIeffhIeff\], we monitor a similar quality of the effectivity indices for *new* and *full*. Here we observe that the interpolation error estimator delivers a worse result. The resulting meshes for $\ell =20$ are shown in Figure \[fig:Full\_New\_Int\]. Here also the meshes of *new* and *full* are more similar. This is not surprising since we perform more enriched solves. For the error, which is discussed in Figure \[Figure: NSError\], a particular conclusion could not be determined.
![Navier-Stokes benchmark problem: The corresponding meshes for $\ell=20$ and *new*(top), *full*(middle), *int*(bottom). []{data-label="fig:Full_New_Int"}](Data/NS/X01/Full_New_Int){width="0.95\linewidth"}
set output “Figures/Example1c\_archive.tex” set key opaque set logscale y set yrange \[0.01:100\] set datafile separator “|” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/NS/X01/New/data.db “SELECT DISTINCT Refinementstep+1, Ieff from data WHERE Refinementstep <= 20 ”’ u 1:2 w lp lw 3 title ’ $I_{eff,h}$(*new*)’, ’< sqlite3 Data/NS/X01/Full/data.db “SELECT DISTINCT Refinementstep+1, Ieff from data WHERE Refinementstep <= 20 ”’ u 1:2 w lp lw 3 title ’ $I_{eff,h}$(*full*)’, ’< sqlite3 Data/NS/X01/Interpolate/data.db “SELECT DISTINCT Refinementstep+1, Ieff from data WHERE Refinementstep <= 20 ”’ u 1:2 w lp lw 3 title ’ $I_{eff,h}$ (*int*)’, \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/NS/X01/New/data.db “SELECT DISTINCT Refinementstep, abs(Iefftilde) from data WHERE Refinementstep <= 20 ”’ u 1:2 w lp lw 3 title ’ Ieff (*new*)’, ’< sqlite3 Data/NS/X01/Full/data.db “SELECT DISTINCT Refinementstep,abs(Iefftilde) from data WHERE Refinementstep <= 20”’ u 1:2 w lp lw 3 title ’ Ieff (*new*)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
set output “Figures/Example1c2\_archive.tex” set key left set key opaque set logscale y set yrange \[0.01:100\] set datafile separator “|” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/NS/X01/New/data.db “SELECT DISTINCT Refinementstep+1, abs(Iefftilde) from data WHERE Refinementstep <= 20 ”’ u 1:2 w lp lw 3 title ’ $I_{eff}$(*new*)’, ’< sqlite3 Data/NS/X01/Full/data.db “SELECT DISTINCT Refinementstep+1,abs(Iefftilde) from data WHERE Refinementstep <= 20”’ u 1:2 w lp lw 3 title ’ $I_{eff}$(*full*)’, ’< sqlite3 Data/NS/X01/Interpolate/data.db “SELECT DISTINCT Refinementstep+1,abs(Iefftilde) from data WHERE Refinementstep <= 20”’ u 1:2 w lp lw 3 title ’ $I_{eff}$(*int*)’, \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/NS/X01/New/data.db “SELECT DISTINCT Refinementstep, abs(Iefftilde) from data WHERE Refinementstep <= 20 ”’ u 1:2 w lp lw 3 title ’ Ieff (*new*)’, ’< sqlite3 Data/NS/X01/Full/data.db “SELECT DISTINCT Refinementstep,abs(Iefftilde) from data WHERE Refinementstep <= 20”’ u 1:2 w lp lw 3 title ’ Ieff (*new*)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
\[Figure: NSIeffhIeff\]
set output “Figures/Example1r\_archive.tex” set key bottom left set key opaque set logscale set datafile separator “|” set grid ytics lc rgb “\#bbbbbb” lw 1 lt 0 set grid xtics lc rgb “\#bbbbbb” lw 1 lt 0 set xlabel ’’ plot ’< sqlite3 Data/NS/X01/New/data.db “SELECT DISTINCT DOFS\_primal , Exact\_Error from data WHERE Refinementstep <= 20 ”’ u 1:2 w lp lw 3 title ’ Error (*new*)’, ’< sqlite3 Data/NS/X01/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE Refinementstep <= 20”’ u 1:2 w lp lw 3 title ’ Error (*full*)’, ’< sqlite3 Data/NS/X01/Interpolate/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE Refinementstep <= 20 ”’ u 1:2 w lp lw 3 title ’ Error (*int*)’, \#plot ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data ”’ u 1:2 w lp lw 3 title ’ $|J(u)-J(u_h)|$ (a)’, ’< sqlite3 Data/P\_Laplace\_slit/Full/data.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data\_global ”’ u 1:2 w lp lw 2 title ’ $|J(u)-J(u_h)|$ (u)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_remainder from data ”’ u 1:2 w lp lw 2 title ’$\small|\eta^{(2)}_\mathcal{R}|$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, 0.5\*abs(Estimated\_Error\_adjoint+Estimated\_Error\_primal) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}_h$’, 1/x dt 3 lw 4 \#0.1/sqrt(x) lw 4, 0.1/(x\*sqrt(x)) lw 4, \# ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Exact\_Error from data WHERE DOFS\_primal <= 90000”’ u 1:2 w lp title ’Exact Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error from data”’ u 1:2 w lp title ’Estimated Error’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_primal from data”’ u 1:2 w lp title ’Estimated Error(primal)’, ’< sqlite3 dataSingle.db “SELECT DISTINCT DOFS\_primal, Estimated\_Error\_adjoint from data”’ u 1:2 w lp title ’Estimated(adjoint)’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)+abs(abs(Juh2-Juh) + Exact\_Error) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’, ’< sqlite3 Data/P\_Laplace\_slit/New/data.db “SELECT DISTINCT DOFS\_primal, abs(ErrorTotalEstimation)-abs(Juh2-Juh+ abs(Exact\_Error)) from data ”’ u 1:2 w lp lw 2 title ’$\small\eta^{(2)}$’,
\[Figure: NSError\]
Conclusions {#Section: Conclusions}
===========
[We derived adaptive algorithms]{} for computationally attractive low-order finite elements and interpolations to realize goal-oriented a posteriori error estimation using the DWR approach. Using saturation assumptions, we [rigorously proved]{} two-side error estimates showing the efficiency and robustness. These findings were [supported by means of]{} three numerical tests. Therein, the newly suggested error estimator was compared to the full estimator and a version in which only interpolations are used. For linear problems (Example 1), all three variants coincide with respect the error behaviour. For nonlinear problems (Example 2), differences can be observed. In the last numerical test (Example 3), a fluid-flow example was considered. Here, the PDE is semi-linear, but due to the convection term, the saturation assumption is not always fulfilled. This could be observed in terms of bad effectivity indices every now and then. Moreover, in the last example, the mechanism of our proposed adaptive algorithm is highlighted because the switch from interpolations to enriched spaces in some iterations significantly improves the effectivity indices. In future work, we plan to apply this algorithm to [ other applications, in particular, to multiphysics problems.]{}
Acknowledgments
===============
This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181 ‘Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems’. Furthermore, the first two authors would like to thank IfAM from the Leibniz Universtät Hannover (LUH) for the organization of their visit in Hannover in January 2020. The third author would like to thank RICAM for his supported visit in Linz in November 2019, and for funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122).
[^1]: <http://www.featflow.de/en/benchmarks/cfdbenchmarking/flow/dfg_benchmark1_re20.html>
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Using the generalized Lipatov-Altarelli-Parisi-Dokshitzer equations for the two-parton distribution functions we show numerically that the dynamical correlations contribute to these functions quite a lot in comparison with the factorized components. At the scale of CDF hard process ($\sim 5$ GeV) this contribution to the double gluon-gluon distribution is nearly 10$\%$ and increases right up to 30$\%$ at the LHC scale ($\sim 100$ GeV) for the longitudinal momentum fractions $x \leq 0.1$ accessible to these measurements. For the finite longitudinal momentum fractions $x \sim 0.2 \div 0.4$ the correlations are large right up to 90$\%$ in accordance with the predicted QCD asymptotic behaviour.'
---
=-15mm
\#1\#2[3.6pt]{}
[**Double parton correlations versus factorized distributions** ]{}\
V.L. Korotkikh, A.M. Snigirev\
M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics\
119992, Vorobievy Gory, Moscow, Russia\
E-mail: [email protected], [email protected]
$PACS$: 12.38.–t\
$Keywords$: many parton distributions, leading logarithm approximation
Recent CDF measurements [@cdf] of the inclusive cross section for double parton scattering have provided new and complementary information on the structure of the proton and possible parton-parton correlations. Both the absolute rate for the double parton process and any dynamics that correlations may introduce are therefore of interest. The possibility of observing two separate hard collisions has been proposed since long [@landshoff; @goebel], and from that has also developed in a number of works [@takagi; @paver; @humpert; @odorico; @sjostrand; @trelani; @trelani2; @del]. The Tevatron and specially LHC allow us to obtain huge data samples of these multiple interactions and to answer to many challenging questions of yet poorly-understood aspects of QCD. A brief review of the current situation and some progress in the modeling account of correlated flavour, colour, longitudinal and transverse momentum distributions can be found in ref. [@sjostrand2]. Multiple interactions require an ansatz for the structure of the incoming beams, i.e. correlations between the constituent partons. As a simple ansatz, usually, the two-parton distributions are supposed to be the product of two single-parton distributions times a momentum conserving phase space factor. In recent paper [@snig03] it has been shown that this hypothesis is in some contradiction with the leading logarithm approximation of perturbative QCD (in the framework of which a parton model, as a matter of fact, was established in the quantum field theories [@gribov; @lipatov; @dok]). Namely, the two-parton distribution functions being the product of two single distributions at some reference scale become to be dynamically correlated at any different scale of a hard process. The value of these correlations in comparison with the factorized components is the main purpose of this Letter.
In order to be clear and to introduce the denotations let us recall that, for instance, the differential cross section for the four-jet process (due to the simultaneous interaction of two parton pairs) is given by [@humpert; @odorico] $$\label{fourjet}
d \sigma = \sum \limits_{q/g} \frac{ d \sigma_{12} ~d \sigma_{34}}
{\sigma_{eff}}~ D_ p(x_1,x_3)~D_{\bar{p}}(x_2,x_4),$$ where $d \sigma_{ij}$ stands for the two-jet cross section. The dimensional factor $\sigma_{eff}$ in the denominator represents the total inelastic cross section which is an estimate of the size of the hadron, $\sigma_{eff} ~\simeq~2 \pi r_ p^2$ (the factor 2 is introduced due to the identity of the two parton processes). With the effective cross section measured by CDF, $(\sigma_{eff})_{CDF}=
(14.5 \pm 1.7^{+1.7}_{-2.3})$ mb [@cdf], one can estimate the transverse size $r_p~\simeq 0.5$ fm, which is too small in comparison with the proton radius $R_p$ extracted from $ep$ elastic scattering experiments. The relatively small value of $(\sigma_{eff})_{CDF}$ with respect to the naive expectation $2 \pi R_ p^2$ was, in fact, considered [@trelani; @trelani2] as evidence of nontrivial correlation effects in transverse space. But, apart from these correlations, the longitudinal momentum correlations can also exist and they were investigated in ref. [@snig03]. The factorization ansatz is just applied to the two-parton distributions incoming in eq. (\[fourjet\]): $$\label{factoriz}
D_ p(x_i,x_j)~ = ~ D_ p(x_i,Q^2)~ D_ p(x_j,Q^2)~(1-x_i-x_j),$$ where $D_ p(x_i,Q^2)$ are the single quark/gluon momentum distributions at the scale $Q^2$ (determined by a hard process).
However many parton distribution functions satisfy the generalized Lipatov-Altarelli-Parisi-Dokshitzer evolution equations derived for the first time in refs [@kirschner; @snig] as well as single parton distributions satisfy more known and cited Altarelli-Parisi equations [@lipatov; @dok; @altarelli]. Under certain initial conditions these generalized equations lead to solutions, which are identical with the jet calculus rules proposed originally for multiparton fragmentation functions by Konishi-Ukawa-Veneziano [@konishi] and are in some contradiction with the factorization hypothesis (\[factoriz\]). Here one should note that at the parton level this is the strict assertion within the leading logarithm approximation.
After introducing the natural dimensionless variable $$t = \frac{1}{2\pi b} \ln \Bigg[1 + \frac{g^2(\mu^2)}{4\pi}b
\ln\Bigg(\frac{Q^2}{\mu^2}\Bigg)\Bigg]~=~\frac{1}{2\pi b}\ln\Bigg
[\frac{\ln(\frac{Q^2}{\Lambda^2_{QCD} })}
{\ln(\frac{\mu^2}{\Lambda^2_{QCD}})}\Bigg]
,~~~~~b = \frac{33-2n_f}{12\pi}~~
{\rm {in~ QCD}},$$ where $g(\mu^2)$ is the running coupling constant at the reference scale $\mu^2$, $n_f$ is the number of active flavours, $\Lambda_{QCD}$ is the dimensional QCD parameter, the Altarelli-Parisi equations read [@lipatov; @dok; @altarelli] $$\label{e1singl}
\frac{dD_i^j(x,t)}{dt} =
\sum\limits_{j{'}} \int \limits_x^1
\frac{dx{'}}{x{'}}D_i^{j{'}}(x{'},t)P_{j{'}\to j}\Bigg(\frac{x}{x{'}}\Bigg).$$ They describe the scaling violation of the parton distributions $ D^j_i(x,t)$ inside a dressed quark or gluon ($i,
j~=~q/g$).
We will not write the kernels $P$ explicitly and derive the generalized equations for two-parton distributions $D_i^{j_1j_2}(x_1,x_2,t)$, representing the probability that in a dressed constituent $i$ one finds two bare partons of types $j_1$ and $j_2$ with the given longitudinal momentum fractions $x_1$ and $x_2$ (referring to [@snig03; @lipatov; @dok; @kirschner; @snig; @altarelli] for details), we give only their solutions via the convolution of single distributions [@kirschner; @snig] $$\begin{aligned}
\label{solution}
& D_i^{j_1j_2}(x_1,x_2,t) = \\
& \sum\limits_{j{'}j_1{'}j_2{'}} \int\limits_{0}^{t}dt{'}
\int\limits_{x_1}^{1-x_2}\frac{dz_1}{z_1}
\int\limits_{x_2}^{1-z_1}\frac{dz_2}{z_2}~
D_i^{j{'}}(z_1+z_2,t{'}) \frac{1}{z_1+z_2}P_{j{'} \to
j_1{'}j_2{'}}\Bigg(\frac{z_1}{z_1+z_2}\Bigg) D_{j_1{'}}^{j_1}(\frac{x_1}{z_1},t-t{'})
D_{j_2{'}}^{j_2}(\frac{x_2}{z_2},t-t{'}).\nonumber\end{aligned}$$ This convolution coincides with the jet calculus rules [@konishi] as mentioned above and is the generalization of the well-known Gribov-Lipatov relation installed for single functions [@gribov; @dok] (the distribution of bare partons inside a dressed constituent is identical to the distribution of dressed constituents in the fragmentation of a bare parton in the leading logarithm approximation). The solution (\[solution\]) shows that the distribution of partons is [*[correlated]{}*]{} in the leading logarithm approximation: $$\begin{aligned}
\label{nonfact}
D_i^{j_1j_2}(x_1,x_2,t) \neq D_{i}^{j_1}(x_1,t)
D_{i}^{j_2}(x_2,t).\end{aligned}$$
Of course, it is interesting to find out the phenomenological issue of this parton level consideration. This can be done within the well-known factorization of soft and hard stages (physics of short and long distances) [@collins]. As a result the equations (\[e1singl\]) describe the evolution of parton distributions in a hadron with $t ~(Q^2)$, if one replaces the index $i$ by index $h$ only. However, the initial conditions for new equations at $t=0 ~(Q^2=\mu^2)$ are unknown a priori and must be introduced phenomenologically or must be extracted from experiments or some models dealing with physics of long distances \[at the parton level: $D_{i}^{j}(x,t=0)~= ~\delta_{ij} \delta(x-1)$; $D_i^{j_1j_2}(x_1,x_2,t=0)~=~0$\]. Nevertheless the solution of the generalized Lipatov-Altarelli-Parisi-Dokshitzer evolution equations with the given initial condition may be written as before via the convolution of single distributions [@snig03; @snig] $$\begin{aligned}
\label{solution1}
& D_h^{j_1j_2}(x_1,x_2,t)~ = ~ D_{h(QCD)}^{j_1j_2}(x_1,x_2,t)~+\\
& \sum\limits_{j_1{'}j_2{'}}
\int\limits_{x_1}^{1-x_2}\frac{dz_1}{z_1}
\int\limits_{x_2}^{1-z_1}\frac{dz_2}{z_2}~
D_h^{j_1{'}j_2{'}}(z_1,z_2,0) D_{j_1{'}}^{j_1}(\frac{x_1}{z_1},t)
D_{j_2{'}}^{j_2}(\frac{x_2}{z_2},t) ~,\nonumber\end{aligned}$$ where $$\begin{aligned}
\label{solQCD}
& D_{h(QCD)}^{j_1j_2}(x_1,x_2,t) =\\
& \sum\limits_{j{'}j_1{'}j_2{'}} \int\limits_{0}^{t}dt{'}
\int\limits_{x_1}^{1-x_2}\frac{dz_1}{z_1}
\int\limits_{x_2}^{1-z_1}\frac{dz_2}{z_2}~
D_h^{j{'}}(z_1+z_2,t{'}) \frac{1}{z_1+z_2}P_{j{'} \to
j_1{'}j_2{'}}\Bigg(\frac{z_1}{z_1+z_2}\Bigg) D_{j_1{'}}^{j_1}(\frac{x_1}{z_1},t-t{'})
D_{j_2{'}}^{j_2}(\frac{x_2}{z_2},t-t{'})\nonumber\end{aligned}$$ are the dynamically correlated distributions given by perturbative QCD (compare (\[solution\]) with (\[solQCD\])).
The reckoning for the unsolved confinement problem (physics of long distances) is the unknown nonperturbative two-parton correlation function $ D_h^{j_1{'}j_2{'}}(z_1,z_2,0)$ at some scale $\mu^2$. One can suppose that this function is the product of two single-parton distributions times a momentum conserving factor at this scale $\mu^2$: $$\begin{aligned}
\label{fact}
D_h^{j_1j_2}(z_1,z_2,0) ~=~ D_{h}^{j_1}(z_1,0)
D_{h}^{j_2}(z_2,0)\theta(1-z_1-z_2).\end{aligned}$$ Then $$\begin{aligned}
\label{solution2}
& D_h^{j_1j_2}(x_1,x_2,t)~ =~ D_{h(QCD)}^{j_1j_2}(x_1,x_2,t)~+~
\theta (1-x_1-x_2)\bigg(D_{h}^{j_1}(x_1,t)D_{h}^{j_2}(x_2,t)~+~\\
& \sum\limits_{j_1{'}j_2{'}}
\int\limits_{x_1}^{1}\frac{dz_1}{z_1}
\int\limits_{x_2}^{1}\frac{dz_2}{z_2}~
D_h^{j_1{'}}(z_1,0)D_h^{j_2{'}}(z_2,0)
D_{j_1{'}}^{j_1}(\frac{x_1}{z_1},t)
D_{j_2{'}}^{j_2}(\frac{x_2}{z_2},t)[\theta(1-z_1-z_2)-1]
\bigg),
\nonumber \\\end{aligned}$$ where $$\label{1solution}
D_h^j(x,t) =
\sum\limits_{j{'}} \int \limits_x^1
\frac{dz}{z}~D_h^{j{'}}(z,0)~D_{j{'}}^j(\frac{x}{z},t)$$ is the solution of eq. (\[e1singl\]) with the given initial condition $D_h^j(x,0)$ for parton distributions inside a hadron expressed via distributions at the parton level.
This result (\[solution2\]) shows that if the two-parton distributions are factorized at some scale $\mu^2$, then the evolution violates this factorization [*[ inevitably]{}*]{} at any different scale ($Q^2 \neq \mu^2$), apart from the violation due to the kinematic correlations induced by the momentum conservation (given by $\theta$ functions)[[^1]]{}.
For a practical employment it is interesting to know the degree of this violation. Partly this problem was investigated theoretically in refs. [@snig; @snig2] and for the two-particle correlations of fragmentation functions in ref. [@puhala]. That technique is based on the Mellin transformation of distribution functions as $$\begin{aligned}
\label{mellin}
M_h^{j}(n,t) ~=~ \int\limits_{0}^{1}dx ~x^n~D_{h}^{j}(x,t). \end{aligned}$$ After that the integrodifferential equations (\[e1singl\]) become systems of ordinary linear-differential equations of first order with constant coefficients and can be solved explicitly [@snig; @snig2]. In order to obtain the distributions in $x$ representation an inverse Mellin transformation must be performed $$\begin{aligned}
\label{mellin in}
D_h^{j}(x,t) ~=~ \int\frac {dn}{2\pi i} ~x^{-n}~M_{h}^{j}(n,t), \end{aligned}$$ where the integration runs along the imaginary axis to the right from all $n$ singularities. This can be done numerically. However the asymptotic behaviour can be estimated. Namely, with the growth of $t~(Q^2)$ the first term in eq. (\[solution1\]) becomes [*[dominant]{}*]{} [[^2]]{} for finite $x_1$ and $x_2$ [@snig2]. Thus the two-parton distribution functions “forget” the initial conditions unknown a priori and the correlations perturbatively calculated appear.
The asymptotic prediction “teaches” us a tendency only and tells nothing about the values of $x_1,x_2, t(Q^2)$ beginning from which the correlations are significant (the more so since the asymptotic behaviour takes place over the double logarithm dimensionless variable $t$ as a function of $Q^2$). Naturally numerical estimations can give an answer to this specific question. We do it using the CTEQ fit [@cteq] for single distributions as an input in eq. (\[solQCD\]). The nonperturbative initial conditions $D_h^j(x,0)$ are specified in a parametrized form at a fixed low-energy scale $Q_0=\mu=1.3$ GeV. The particular function forms and the value of $Q_0$ are not crucial for the CTEQ global analysis at the flexible enough parametrization, which reads [@pumplin] $$\begin{aligned}
\label{paramet}
x D_p^{j}(x,0) ~=~ A_0^j x^{A_1^j} (1 - x)^{A_2^j} e^{A_3^j x} (1 + e^{A_4^j}
x)^{A_5^j}.\end{aligned}$$ The independent parameters $A_0^j,~ A_1^j,~ A_2^j,~ A_3^j, ~A_4^j, ~A_5^j$ for parton flavour combinations $u_v \equiv u-\bar{u}$, $d_v \equiv d-\bar{d}$, $g$ and $\bar{u}+\bar{d}$ are given in Appendix A of ref. [@pumplin]. To distinguish the $\bar{u}$ and $\bar{d}$ distributions the ratio $\bar{d} / \bar{u}$ is parametrized as a sum of two terms: $$\begin{aligned}
\label{paramet2}
D_p^{\bar{d}}(x,0) \big/ D_p^{\bar{u}}(x,0)~=
~ A_0 x^{A_1} (1 - x)^{A_2} ~+~ (1 + A_3 x) (1 - x)^{A_4}\end{aligned}$$ with the coefficients $A_0,~ A_1,~ A_2,~ A_3, ~A_4$ again from ref. [@pumplin]. The initial conditions for strange quarks are assumed: $$D_p^{\bar{s}}(x,0) ~=~ D_p^{s}(x,0)~=~
0.2\Big(D_p^{\bar{u}}(x,0)~+~D_p^{\bar{d}}(x,0)\Big).$$
The parton distribution functions $D_p^{j}(x,t)$ at all higher $Q(t)$ are determined from the input initial conditions $D_p^{j}(x,0)$ by the Altarelli-Parisi evolution equations. The CTEQ Evolution package [@Evolve] was used and adapted by us in order to obtain numerically single distributions $D_i^{j}(x,t)$ at all $t$ and at the parton level also. We fixed the fundamental parameter of perturbative QCD, $\Lambda_{QCD} = 0.281$ GeV, that is in accordance with the strong coupling constant, $\alpha_s(M_Z)~\simeq~0.2$, at the $Z$ resonance in one-loop approximation. Only the light quarks $u, d, s$ ($n_f = 3$) are taken into account in the evolution equations and are treated as massless, as usual. After that the triple integral (\[solQCD\]) was calculated numerically for three values of $Q = 5, 100, 250$ GeV as a function of $x = x_1 = x_2$. To be specific we considered the double gluon-gluon distribution function in the proton. In this case only the kernel $P_{g \to g g}$ can be taken into account as giving the main contribution to the perturbative double gluon-gluon distribution. The remnant terms of sum in eq. (\[solQCD\]) are relatively small and can only increase the effect under consideration because they are positive.
The results of numerical calculations are presented on fig. 1 for the ratio: $$\label{ratio}
R(x,t)~=~\Big(D_{p(QCD)}^{gg}(x_1,x_2,t) \Big/ D_p^{g}(x_1,t)D_p^{g}(x_2,t)(1 - x_1
- x_2)^2\Big)\Big|_{x_1=x_2=x}.$$ Here one should note that the momentum conserving phase space factor $(1 - x_1 - x_2)^2$ is introduced in eq. (\[ratio\]) instead of $(1 - x_1 - x_2)$ usually used. The reason is simple: this factor was introduced in eq. (\[factoriz\]), generally speaking, “by hand” in order to “save” the momentum conservation law, i.e. in order to make the product of two single distributions is equal to zero smoothly at $x_1 + x_2 = 1.$ However the generalized QCD evolution equations demand higher power of $(1 - x_1 - x_2)$ at $ x_1 + x_2 \rightarrow 1$: only the phase space integrals in eqs. (\[solution1\]) and (\[solQCD\]) give $$\int\limits_{x_1}^{1-x_2} dz_1
\int\limits_{x_2}^{1-z_1}dz_2~ =~(1 - x_1 - x_2)^2/2 .$$ In fact this power must depend on $t$ increasing with its growth as this takes place for single distributions at $ x \rightarrow 1$ [@dok; @dok2]. The asymptotic behaviour of two-particle fragmentation functions at $ x_1 + x_2 \rightarrow 1$ was investigated, for instance, in ref. [@vendramin] with the similar result. Our numerical calculations support this assertion also: the power of $(1 - x_1 - x_2)$ for the perturbative QCD gluon-gluon correlations is higher than 2 and increases with $t(Q)$ as one can see from fig. 1 However the introduced factor $(1 - x_1 - x_2)^2$ has not an influence practically on the ratio under consideration in the region of small $x_1, x_2$. And namely this region, in which multiple interactions can contribute to the cross section visibly, is interesting from experimental point of view. Fig. 1 shows that at the scale of CDF hard process ($\sim 5$ GeV) the ratio (\[ratio\]) is nearly 10$\%$ and increases right up to 30$\%$ at the LHC scale ($\sim 100$ GeV) for the longitudinal momentum fractions $x \leq 0.1$ accessible to these measurements. For the finite longitudinal momentum fractions $x \sim 0.2 \div 0.4$ the correlations are large right up to 90$\%$ . They become important in more and more $x$ region with the growth of $t$ in accordance with the predicted QCD asymptotic behaviour.
The correlation effect is strengthened insignificantly (up to 2$\%$) for the longitudinal momentum fractions $x \leq 0.1$ when starting from the slightly lower value $Q_0 = 1$ GeV (early used by CTEQ Collaboration). We conclude also that $R(x,t)\rightarrow const$ at $x \rightarrow 0$ most likely, calculating this ratio ($\simeq 0.1$) at $x_{min} = 10^{-4}$.
Seemingly the correction to the double gluon-gluon distributions at the CDF scale can be smoothly absorbed by uncertainties in the $\sigma_{eff}$ increasing the transverse effective size $r_p$ by a such way. But this augmentation is still not enough to solve a problem of the relatively small value of $r_p$ with respect to the proton radius without nontrivial correlation effects in transverse space [@trelani; @trelani2].
In summary, the numerical estimations show that the leading logarithm perturbative QCD correlations are quite comparable with the factorized distributions. With increasing a number of observable multiple collisions (statistic) the more precise calculations of their cross section (beyond the factorization hypothesis) will be needed also. In order to obtain the more delicate their characteristics (distributions over various kinematic variables) it is desirable to implement the QCD evolution of two-parton distribution functions in some Monte Carlo event generator as this was done for single distributions within, for instance, PYTHIA [@pythia].
[*Acknowledgements*]{}
It is pleasure to thank V.A. Ilyin drawing the attention to the problem of double parton scattering. Discussions with E.E. Boos, M.N. Dubinin, I.P. Lokhtin, S.V. Molodtsov, L.I. Sarycheva, V.I. Savrin, T. Sjostrand, D. Treleani and G.M. Zinovjev are gratefully acknowledged. Authors are specially thankful to A.S. Proskuryakov for the valuable help in adapting CTEQ Evolution package. This work is partly supported by grant N 04-02-16333 of Russian Foundation for Basic Research.
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[^1]: This is the analogue of the momentum conserving phase space factor in eq. (\[factoriz\])
[^2]: Such domination is the mathematical consequence of the relation between the maximum eigenvalues $\lambda(n)$ in the moments representation (after Mellin transformation): $\lambda(n_1+n_2)~>~\lambda(n_1)+\lambda(n_2)$ in QCD at the large $n_1,n_2$ (finite $x_1,x_2$), because $\lambda(n)\sim -\ln(n),
n \gg 1$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Strong gravitational lensing by galaxies provide us a unique opportunity to understand the nature of gravity on the galactic and extra-galactic scales. Unlike the traditional way to use the Einstein radius from electromagnetic domain, we propose a multi-messenger approach by combining data from both gravitational wave (GW) and the corresponding electromagnetic (EM) counterpart. The time-delays among multiple gravitational wave events and the multiple images of electromagnetic counterparts are the indicators of the same lensing mass. Hence, we can use the completely independent multi-messenger datasets to exam the consistency relationship arising in general relativity. To demonstrate the robustness of this approach, we calculate the different time-delay predictions between general relativity and some viable models of modified gravity. For the third generation of ground-based gravitational wave observatory, with one typical strongly lensed GW+EM event, the multi-messenger approach is able to distinguish $8\%$ modified gravity effect on tens of kiloparsec scale with $68\%$ confidence. Our results show that the gravitational wave multi-messenger approach can play an important role in revealing the nature of gravity on the galactic and extra-galactic scales.'
author:
- |
Tao Yang,$^{1}$ Bin Hu,$^{1}$[^1] Rong-Gen Cai,$^{2,3}$ Bin Wang,$^{4}$\
$^{1}$Department of Astronomy, Beijing Normal University, Beijing, 100875, China\
$^{2}$CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,\
Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China\
$^{3}$School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China\
$^{4}$Center for Gravitation and Cosmology, Yangzhou University, Yangzhou 225009, China\
bibliography:
- 'ref.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'A new probe of gravity: strongly lensed gravitational wave multi-messenger approach'
---
\[firstpage\]
gravitational waves, gravitational lensing: strong
Introduction
============
Einstein’s General Relativity (GR) has been precisely tested on Solar System scale [@Bertotti:2003rm; @Shapiro:2004zz], such as the Eddington’s measurement of light deflection during the solar eclipse of 1919 [@Dyson:1920cwa]; observation of the gravitational redshift [@Pound:1960zz]; the successful operation of the Global Positioning Satellites [@Ashby:2003vja]; measurements of the Shapiro time-delay [@Shapiro:1964uw]; and verification of energy loss via gravitational waves in the Hulse-Taylor pulsar [@Taylor:1979zz]. However, the long-range nature of gravity on the extra-galactic scale is still poorly understood. Testing gravity with higher accuracy has been continuously pursued during the past decades. The purpose of these activities are not only to examine a specific model, but also to reveal the nature of gravitational phenomena, such as dark matter and dark energy, on the cosmological scales. Parameterized Post-Newtonian (PPN) framework [@Thorne:1970wv] provides us a systematic way to quantify the deviation from GR. The scale independent post-Newtonian parameter $\gamma_{\texttt{PPN}}$ represents the ratio between dynamical mass and lensing mass. The former is the mass determining the motion of non-relativistic objects, such as the stellar velocity dispersion, while the latter is the mass determining the path deflection of relativistic particles, such as the Einstein radius of strong lensing image and Shapiro time-delay. GR predicts the equivalence of these two masses ($\gamma_{\texttt{PPN}}=1$), while the alternative models break it due to the extra relativistic scalar degree of freedom, which mediates the fifth force.
Strong gravitational lensings around galaxies provide us a unique opportunity to probe modifications to GR over a range of redshifts and on/above kiloparsec (kpc) scale [@Bolton:2006yz; @Smith:2009fn; @Schwab:2009nz; @Cao:2017nnq]. Recently, Collett [*et al.*]{} [@Collett:2018gpf] estimated $\gamma_{\texttt{PPN}}$ to be $0.97\pm0.09$ at $68\%$ confidence level by using a nearby lens, ESO 325-G004. In these analysis, the dynamical mass is estimated by the spectroscopic measurement of the stellar velocity dispersion of the lens galaxy; and the lensing mass is reconstructed by the measurement of Einstein radius. All these data are obtained by using the electromagnetic signals.
The first gravitational wave (GW) event GW150914 from the merger of binary black hole, detected by LIGO Scientific Collaboration$^a$ [^2] and Virgo Collaboration$^b$ [^3], opened a new observational window to explore our universe [@Abbott:2016blz]. Moreover, the following event GW170817 from a binary neutron star combined with the electromagnetic (EM) counterpart, announced the beginning of golden era of multi-messenger astronomy [@TheLIGOScientific:2017qsa]. The third generation of ground-based GW observatory, such as the Einstein Telescope$^c$ [@Punturo:2010zz] [^4], is expected to detect $10^4-10^5$ events per year and $50-100$ among them are strongly lensed [@Liao:2017ioi]. The time-delays of GW from different paths, are expected to be few months; while the duration of each signal is less than $0.1$ second. This leads to the fact that the uncertainties of time-delays from lensed GW events are almost negligible. Furthermore, the lensing mass reconstruction from the EM counterpart images has a systematic uncertainty around $0.6\%$ level [@Liao:2017ioi]. For the traditional lensed quasar system, the uncertainties of the time-delay and lensing mass reconstruction are both of order $\mathcal{O}(3\%)$ [@Liao:2014cka; @Suyu:2016qxx]. Recently, Liao [*et al.*]{} [@Liao:2017ioi] forecasted the robustness of Hubble parameter estimation by using the strongly lensed GW+EM signals. They found that $10$ such systems are able to provide a Hubble constant uncertainty of $0.68\%$ for a flat $\Lambda$ Cold Dark Matter ($\Lambda$CDM) universe by using the Einstein Telescope. This inspired us to consider the GW+EM system as a new probe to the nature of gravity.
In the limit of a weak gravitational field, consider the perturbed Friedmann-Lema\^ itre-Robertson-Walker (FLRW) metric in the conformal Newtonian gauge $ds^2=-\left(1+\frac{2\Psi}{c^2}\right)c^2 dt^2+a^2\left(1-\frac{2\Phi}{c^2}\right)d\vec{x}^2\,,$ where $a$ is the background scale factor, $\Psi$ and $\Phi$ are the Newtonian potential and spatial curvature perturbation. Non-relativistic particle motion only responds to spatial gradient of the Newtonian potential $\Psi$, due to the fact that its velocity is much less than the speed of light $c$. While relativistic particle feels the gradient of the Weyl potential, $\Phi_+=(\Phi+\Psi)/2$. GR predicts the equivalence of Newtonian and Weyl potentials, or the equivalence of Newtonian potential $\Psi$ and the spatial curvature perturbation $\Phi$, namely $\gamma_{\texttt{PPN}}\equiv\Phi/\Psi=1$. In contrast, the alternative models of gravity typically contain an additional relativistic scalar degree of freedom which mediates the fifth force, hence breaks this degeneracy.
On kpc scales, strong gravitational lensing, combined with stellar kinematics of the lens, allows a test of the weak field metric of gravity. Measurements of the stellar velocity dispersion determine the dynamical mass, whereas measurements of the image positions of the lens and time-delay between the multiple images determine the lensing mass. They reflect the nature of Newtonian and Weyl potentials of the underlining gravity, respectively. Based on the above analysis, we can conclude that GR uniquely predicts the equivalence of the dynamical and lensing masses. This equality ($\gamma_{\texttt{PPN}}=1$) provides us a way to test it. However, the analysis made in references [@Bolton:2006yz; @Smith:2009fn; @Schwab:2009nz; @Cao:2017nnq; @Collett:2018gpf] assumed a constant PPN parameter over the length scales relevant for their studies. We argue this is incomplete. The reasons are in two folds. On the one hand, the lens and source are well separated along line-of-sight. The corresponding length scale is well above the perpendicular lensing scale. On the other hand, the viable models of modified gravity (MG) have to shield the fifth force under a certain scale, namely screening scale [@Joyce:2014kja]. The modified gravity effect only arises above these scales. The photons and gravitons emitted from the source galaxies cumulate the modified gravity effect along the line-of-sight when they approach to us. Hence, a constant PPN parameter is not really physical. In particular, all those studies are limited to their measurements of the lensing mass in the EM domain, which is out of date in the richness of GW+EM multi-messenger observation today.
In this article, we deliver two new progresses in testing gravity, namely a new modelling of lensing potential and a new testing window. Let us introduce the first part. Since the lensing directly measures the Weyl potential, here we choose to parametrize the function $\Sigma(r)=\Phi_+/\frac{-GM}{r}$ in real space. It is related to the PPN parameter as $\Sigma=(1+\gamma_{\texttt{PPN}})/2$. For simplicity, we assume a spherical lens, but allow a scale dependent modification. The constraint from solar system tells us that, a successful alternative gravity model must screen the fifth force on these scales [@Joyce:2014kja]. Moreover, to avoid the inconsistency of the lens model with modified gravity, such as Singular Isothermal Sphere (SIS) model, we evade to modify the dynamics below kpc scale, [*i.e.*]{} we assume the GR is restored below kpc scale. Inspired by the viable screening mechanism [@Joyce:2014kja], we model $\Sigma$ via a step-like function with the transition scale from $10$ to $20$ kpc. Analogous to the parametric form given for $\mu$ and $\gamma$ which denote the modifications of $\Psi$ and $\Phi$ [@Bertschinger:2008zb; @Hojjati:2012rf], we set the parametric form of $\Sigma(r)$ as $$\Sigma(r)=
\begin{cases}
1\,, & 0<r<r_{01} \\
\frac{1+\alpha_1\left(\frac{r-r_{01}}{w_{01}}\right)^2}{1+\alpha_2\left(\frac{r-r_{01}}{w_{01}}\right)^2}\,, & r\geq r_{01}\,,
\end{cases}
\label{eq:phi}$$ where $\alpha_1$/$\alpha_2$ describes the magnitude, $r_{01}$ denotes the transition scale, and $w_{01}$ represents the width of transition step. We shall emphasize that if a sizeable deviation from unity $\Sigma$ is observed, we does not only rule out GR but also quite large amounts of currently viable scalar-tensor gravities [@Pogosian:2016pwr]. Hence, this kind of test is very crucial in the view of gravity examination.
As of the second innovation point, we propose a new multi-messenger approach to test gravity. This is inspired by two factors. The first is that both image and time-delay signals can be used to reconstruct the lensing mass. And the effect of modified gravity is involved into these two observables in different manners. Hence, GR gives a unique relationship between these two observables. We can verify this consistency relation. The second is that, compared with lensed quasar, GW+EM system provides a better uncertainties both in image reconstruction and time-delay measurement. This allows us to improve the estimation accuracy.
results
=======
Time delay from GW
------------------
For a given lensed GW+EM event by a galaxy with a specific mass profile, the time-delay between two images is given by $$\Delta t_{i,j}=\frac{1+z_l}{c}\frac{D_l D_s}{D_{ls}}\Delta \phi_{i,j}\,,
\label{eq:dt}$$ here $\Delta \phi_{i,j}=[(\theta_i-\beta)^2/2-\psi(\theta_i)]-[(\theta_j-\beta)^2/2-\psi(\theta_j)]$ is the Fermat potential difference between different images at angular positions $\theta_i$ and $\theta_j$. The source is located at angular $\beta$. $\psi$ is the effective two-dimensional lensing potential, which is the integral of the Weyl potential along the line-of-sight. Eq. (\[eq:dt\]) is a purely geometric relation, therefore is valid for both GR and modified gravity.
After getting the photometric measurement of the source and lens galaxy redshifts ($z_s$, $z_l$) as well as the spectroscopic measurement of the stellar velocity dispersion ($\sigma_v$) of the lens galaxy, we can calculate the time-delay of the two images for different source positions. The uncertainty of this lensed GW+EM system is dominated by the measurement uncertainty of velocity dispersions. With current lensing image reconstruction technique, for lensed quasars, the uncertainty is of order $\mathcal{O}(3\%)$ [@Liao:2014cka; @Suyu:2016qxx]. The velocity dispersion measurement uncertainty is of order $\mathcal{O}(10\%)$ [@Jee:2014uxa]. As demonstrated by Liao [*et al.*]{} [@Liao:2017ioi] and Wei [*et al.*]{} [@Wei:2017emo], the uncertainty of image reconstruction of EM counterparts of GW events ($\sigma_{\theta}$), such as short Gamma-ray burst and kilonovae, is of order $\mathcal{O}(0.6\%)$. Due to the better control of point spread function in the EM counterpart of GW event, the uncertainty of velocity dispersion measurement ($\sigma_{\sigma_v}$) could be improved to $\mathcal{O}(5\%)$ [@Liao:2017ioi]. In addition, other matter structures along the line-of-sight might contribute an extra systematic uncertainty ($\sigma_{\rm LOS}$) around $1\%$ level [@Suyu:2009by; @Suyu:2016qxx]. As we have demonstrated above, the uncertainty of time-delay measurement from the lensed GW events can be neglected. Thus the total uncertainty translated to the final time-delay is straight forward, $\sigma_{\Delta t}=\sqrt{\sigma^2_{\sigma_{v}}+\sigma_\theta^2+\sigma_{\rm LOS}^2}$, which is dominated by the measurement of stellar velocity dispersions.
![**Time-delay difference between GR and modified gravity for source position inside the Einstein radius.** The shaded area represents the total systematic uncertainty in time-delay. Only the modified gravity signals which are upon this uncertainty can be detected. $y=\beta/\theta_E$ is the source position in the unit of Einstein radius ($\theta_E$). Here we choose a typical strong lensing system with $z_l=0.3$, $z_s=1$, $\sigma_v=300$ km/s. The lens galaxy is a SIS model with $10$ kpc radius. The modified gravity effect is parametrized by the screening scale $r_{01}$, the width of screening shell $w_{01}$ and the amplitude $\delta \Sigma/\Sigma$. Several different parameter values of $r_{01}$ and $\delta \Sigma/\Sigma$ are shown in the legend and we fix the width $w_{01}=1.2$ kpc as a thin shell model.[]{data-label="fig:work1"}](delta_t_plot_1.pdf){width="70.00000%"}
As an example, we specify a typical strong gravitational lensing system according to the sample collected by Cao [*et al.*]{} [@Cao:2015qja] with $z_l=0.3$, $z_s=1$, and $\sigma_v=300$ km/s. The lens galaxy mass profile is set to be the SIS model with a $10$ kpc radius. The background model is fixed to $\Lambda$CDM model with $H_0=67.8$ km s$^{-1}$ Mpc$^{-1}$ and $\Omega_m=0.3$. Since the SIS lens model produces double images only when the source position is inside the Einstein radius, we plot the results with $\beta<\theta_E$. As shown in Fig. \[fig:work1\], when the modified gravity starts to take effect above $r_{01}=10$ ($20$) kpc scale, a $\alpha_1/\alpha_2=18\%$ ($50\%$) deviation is able to be discovered by one typical strong lens (red/blue curve). Comparing the purple curve ($r_{01}=10$ kpc, $\alpha_1/\alpha_2=50\%$) with the former two, we can conclude that a smaller screening scale corresponds to a larger deviation from GR, which can result in the easier detection of the deviation from GR.
Consistency relationship between time-delay and multiple images
---------------------------------------------------------------
Using the time-delay to test gravity is straight forward. However, it is limited by the measurement uncertainty of stellar velocity dispersions. For a lensed GW+EM system, we can get the precise time-delay ($0\%$ uncertainty) from GW domain and multiple images ($0.6\%$ uncertainty) from the EM domain. Both of them are the indicators of lensing mass, not the dynamical mass. Hence, we can use this consistency relation, Eq.(\[eq:dtGR\]), to test GR and successfully avoid dispersion measurement $$\Delta t_{i,j}=\frac{1+z_l}{2c}\frac{D_l D_s}{D_{ls}}(\theta_i^2-\theta_j^2)\,.
\label{eq:dtGR}$$
As we can read, by assuming GR and SIS lens model, the Fermat potential differences reduce into purely geometric relation $\Delta \phi_{i,j}=\frac{1}{2}(\theta_i^2-\theta_j^2)$. Eq.(\[eq:dtGR\]) is a unique prediction from GR, hence, can be tested. We call it $\Delta t-\theta$ consistency relation. If we assume the “true” Universe is governed by the law of modified gravity, applying a typical lensed GW+EM system as before, we can obtain the time-delay and positions of multiple images. Then we can compare the time-delay between the “ture” one with the one calculated from the $\Delta t-\theta$ relation under GR+SIS assumption. If the difference exceeds the systematic uncertainty, the modified gravity signal is detectable. Here the systematic uncertainties are only in the image position measurements and the mass along the line-of-sight contributions, $\sigma_{\Delta t}=\sqrt{\sigma_\theta^2+\sigma_{\rm LOS}^2}$.
![**Consistency relationship between time-delay and multiple images.** The parameter values are similar as Fig. \[fig:work1\]. Blue shaded area represents the total systematic uncertainty of $\Delta t-\theta$ relationship. The grey shaded area denotes the time-delay differences under GR+SIS assumption, if $H_0$ changes from $67$ km s$^{-1}$ Mpc$^{-1}$ to $73$ km s$^{-1}$ Mpc$^{-1}$. Here the stars in the end of each curve denote for the boundary outside which the $\Delta t-\theta$ relationship breaks, because the image lays outside of the lens galaxy mass distribution.[]{data-label="fig:work2"}](delta_t_plot_2.pdf){width="70.00000%"}
Fig. \[fig:work2\] shows a result using both time-delay and image positions. We can see that the time-delay difference is more sensitive to the magnitude than the screening scale. For a typical strongly lensed GW+EM event, a $8\%$ deviation of modified gravity from GR can be detected. Furthermore, since the time-delay can also arise due to the background expansion. We need to quantify the degeneracy between modified gravity effect and Hubble expansion. To do so, we choose two very different values of present Hubble constant, $H_0$, namely $67$ km s$^{-1}$ Mpc$^{-1}$ from Planck [@Ade:2015xua] and $73$ km s$^{-1}$ Mpc$^{-1}$ from SNe Ia [@Riess:2016jrr]. If the time-delay difference is larger than those from $H_0$ variation, we can conclude that this effect is unlikely due to the uncertainty of cosmic expansion measurement. As seen from Fig. \[fig:work2\], the latter can contribute a $8\%$ uncertainty at most. Thus, if we have a $60\%$ deviation from GR, we would have strong confidence to claim the deviation from $\Delta t-\theta$ consistency relation is caused by modified gravity rather than the inaccurate measurement of $H_0$.
Discussion
==========
To measure the GW time-delay, we need to record the time sequence when the maximum peak arrives. Compared with the standard sirens method [@Schutz:1986gp], our approach evades the complicated wave form calculation. Due to the huge hierarchy between the single GW event duration ($0.1$s) and the time-delay (a few months) among the multiple images, the uncertainty of the GW time-delay measurement can be safely neglected compared with those of quasar system ($3\%$). Furthermore, due to the fact that the appearance of EM counterpart of GW event, such as kilonovae, does not severely contaminate the source and lens galaxy images, the uncertainty of deflection angle measurement of GW+EM system is estimated around $0.6\%$. The corresponding uncertainty for lensed quasar is about $3\%$. Armed with these results, we can conclude that lensed GW+EM system can give a better estimation of lensing mass compared with the lensed quasar.
The time-delays from GW are the “multiple sounds” and the multiple images from EM counterpart are the “multiple images”. For a given system, the “multiple sounds” and “multiple images” can be derived simultaneously and respond to the same lensing mass. Thus, if we could measure the dynamical mass (via velocity dispersions) accurately, the constraint on modified gravity is straight forward. Both of the time-delay and image positions can be used, respectively, to estimate the difference between the lensing and dynamical mass.
In the first part of this article, we calculated the time-delay from different models of gravity for a given “double images” system. These theoretical predictions can be directly compared with the “double sounds”, hence can be used to constrain modified gravity. However, since the time-delay prediction asks for photometric and spectroscopic data, the uncertainties of these inputs will propagate into the final theoretical predictions. Hence, even though the GW time-delay data is very accurate, the robustness of this method is limited by the input data quality.
In the second part, we proposed a consistency relationship between “multiple sounds” and “multiple images”, namely $\Delta t-\theta$ relation. This is due to the fact that both “multiple sounds” and “multiple images” are the indicators of the same lensing mass. More importantly, under the assumption of GR+SIS, this relation is free of lensing potential. Hence, we do not need the stellar velocity dispersion measurement. Compared with the first method, this advantage can reduce the theoretical prediction uncertainty by a factor of $5$.
The PPN parameter is constrained up to the order of $10\%$ by Collett [*et al.*]{} using the ESO 325-G004. We should note that the uncertainty of the dynamical mass they derived is about $4\%$ which is smaller than the one we adopted, conservatively ($5\%$). Instead of assuming a constant PPN parameter, we use a scale-dependent $\Sigma$ function to parametrize the modified gravity effect. Since the gravitational lensing, in principle, is the redistribution of the relativistic radiation, the spatial derivative is essential. We argue that the constant PPN parameter is over simplified in the lensing data analysis.
In our calculation, we set the galaxy radius to be $10$ kpc. The SIS model $\rho(r)=\frac{\sigma_v^2}{2\pi Gr^2}$ with $\sigma_v=300$ km/s, gives the total mass $6.57\times 10^{11} M_{\odot}$. Most of the galaxy radius are about $0.5-50$ kpc, which correspond to the total mass $10^{10}-10^{12}M_{\odot}$. Thus the galaxy we set here is a typical one. We would emphasize here that our approach is more general and robust compared to the traditional way. The upcoming strongly lensed multi-messenger system, such as the GW+EM events, may not be well aligned with observer. Thus the traditional way to utilise the Einstein radius is impractical. In this paper, we take the SIS lens model as an example to demonstrate how much the modified gravity effect can be detected. For a more general lens model, such as singular isothermal ellipsoid (SIE), the calculation is straight forward. In summary, our approach is not limited by the specific source type or the lens model. We proposed a new methodology which can be adopted in the future test of gravity.
method
======
Model Setup
-----------
Unlike all the existing studies in the literature which estimate lensing mass by assuming a constant PPN parameter, we present a phenomenologically viable parametrization of the lensing potential, $\Phi_+=-\frac{GM}{r}\Sigma(r)$, for the screened modified gravity. In details, we force the modified gravity effect vanishing below galactic scale. This is well motivated by the current studies [@Joyce:2014kja]. A bonus is that we can adopt the standard lens modeling, such as SIS, without modifying the complicated galaxy dynamics. Hence, the modified gravity effect comes into this system via a very clean way, namely the line-of-sight integral. Take the deflection angle as an example, the modified gravity effect shows as the radial derivative of $\Sigma$ function in Eq. (\[eq:alpha\]). Notice that differential measurement of $dz$ here is not redshift, but the Cartesian coordinate infinitesimal increment along the “z” direction (line-of-sight) $$|\hat{\vec{\alpha}}| = \frac{4GM}{c^2}b\int^{\infty}_{0}\left(\frac{\Sigma}{r^3}-\frac{\Sigma_{,r}}{r^2}\right)dz.
\label{eq:alpha}$$ As demonstrated in Fig. \[fig:work3\], the integral can be split into two parts, which are labelled by white (GR) and blue (MG) regimes, respectively. The purple regime, denoted for galactic scale, is deeply located inside GR regime.
![**Sketch of the modified gravity (MG) modeling.** The MG effects are screened around the lens galaxy and arise in the far field. The photon and graviton emitted from the source galaxy go through the MG (blue) and GR (white) zones along their paths to us. The central purple regime denotes for the size of lens galaxy. Hence, the MG effects accumulated along the whole light path will lead to different predictions of image position and time-delay.[]{data-label="fig:work3"}](lensing.pdf){width="70.00000%"}
Calculations of the time-delay and image positions
--------------------------------------------------
Given lensing model and gravity theory, we can derive the multiple image positions and the time-delay. Under the assumption of GR, for a point mass lens, the two image positions are $\theta_\pm=\frac{1}{2}\left(\beta\pm\sqrt{\beta^2+4\theta_E^2}\right)$; for SIS model with $\beta<\theta_E$, two images are located at $\theta_{\pm}=\beta\pm \theta_E$. Using Eq. (\[eq:dtGR\]), the time-delay between multiple images is easily obtained.
The above calculation for the modified gravity case would be complicated. First of all, we need to derive the deflection angle for a point mass lens. Comparing to the GR value ($\frac{4GM}{c^2 b}$), the deflection angle $\frac{4GM}{c^2}I(b)$ in the modified gravity relies on the derivative of the lensing potential $$I(b)=b\int_0^{\infty}\left(\frac{\Sigma}{r^3}-\frac{\Sigma_{,r}}{r^2}\right)dz\;,\;\; r=\sqrt{b^2+z^2}\;.$$ If $\Sigma=1$, then $I(b)=1/b^2$, the deflection angle for a point mass returns to the GR value. The modified gravity theory with $\Sigma$ deviating from unity shall lead a modification to the deflection angle. For a general lens model, the deflection angle is, basically, the integration of the point mass over the corresponding lensing mass distribution. Hence, for any specific lens model with a given mass profile, we can calculate the deflection angle for a given impact parameter $b$. Unlike the GR case, the time-delay in the modified gravity case does not only depend on the geometric term, but also the effective lensing potential. Hence, we need restore to Eq. (\[eq:dt\]). The deviation of $\Sigma$ from unity represents the size of modified gravity effect. It produces both modifications to deflection angle and effective lensing potential, hence the final anomaly of time-delay. After propagating all of the systematic uncertainties to the errors of time-delay, we can compare the anomaly of time-delay caused by modified gravity to the systematic uncertainties to see how precisely we can distinguish modified gravity theory from GR.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Ran Li for helpful discussions and comments. TY and BH are supported by the Beijing Normal University Grant under the reference No. 312232102 and by the National Natural Science Foundation of China Grants No. 210100088. TY is also supported by China Postdoctoral Science Foundation under Grants No. 2017M620662. BH is also partially supported by the Chinese National Youth Thousand Talents Program under the reference No. 110532102 and the Fundamental Research Funds for the Central Universities under the reference No.310421107. BW acknowledges the support by NNSFC No.11835009. RGC was supported by the National Natural Science Foundation of China Grants No.11690022, No.11375247, No.11435006, and No.11647601, and by the Strategic Priority Research Program of CAS Grant No.XDB23030100 and by the Key Research Program of Frontier Sciences of CAS.
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[^1]: E-mail: [email protected]
[^2]: $^a$<https://ligo.org>
[^3]: $^b$<http://www.virgo-gw.eu>
[^4]: $^c$<http://www.et-gw.eu>
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Optimal measurement scheme with an efficient data processing is important in quantum-enhanced interferometry. Here we prove that for a general binary-outcome measurement, the simplest data processing based on inverting the average signal can saturate the Cramér-Rao bound. This idea is illustrated by binary-outcome homodyne detection, even-odd photon counting (i.e., parity detection), and zero-nonzero photon counting that have achieved super-resolved interferometric fringe and shot-noise limited sensitivity in coherent-light Mach-Zehnder interferometer. The roles of phase diffusion are investigated in these binary-outcome measurements. We find that the diffusion degrades the fringe resolution and the achievable phase sensitivity. Our analytical results confirm that the zero-nonzero counting can produce a slightly better sensitivity than that of the parity detection, as demonstrated in a recent experiment.'
author:
- 'X. M. Feng'
- 'G. R. Jin'
- 'W. Yang'
title: 'Quantum interferometry with binary-outcome measurements in the presence of phase diffusion'
---
Introduction
============
Optical phase measurement in a Mach-Zehnder interferometer (MZI) consists of three steps \[see e.g., Refs. [@Kay; @Helstrom], and also Fig. \[fig1\]\]. First, a probe state $\hat{\rho}$ is prepared and is injected into the MZI. Second, it undergoes a dynamical process described by a unitary operator $\hat{U}(\phi )$ and evolves into a phase-dependent state $\hat{\rho}(\phi )=\hat{U}(\phi )\hat{\rho}\hat{U}^{\dagger }(\phi )$, where $\phi $ is a dimensionless phase shift. Finally, a detection $\hat{\mu}$ and a specific data processing are made at the output ports to obtain an interferometric signal $\langle \hat{\mu}(\phi )\rangle =\mathrm{Tr}[\hat{\rho}(\phi )\hat{\mu}]$, which shows an oscillatory pattern with the resolution determined by the full width at half maximum $(\mathrm{FWHM})$ of the signal and the wavelength $\lambda$, i.e., $\Delta x\propto \mathrm{FWHM}\times \lambda /(2\pi )$ [@Born; @Boto]. In addition, the phase sensitivity is determined by the error-propagation formula $\delta \phi =\Delta\hat{\mu}/|\partial \langle \hat{\mu}\rangle /\partial \phi |$, with the square root of the variance $\Delta \hat{\mu}\equiv \sqrt{\langle \hat{\mu}^{2}\rangle-\langle \hat{\mu}\rangle ^{2}}$ [@Bevington].
An optimal measurement scheme with a proper choice of data processing is important to improve the resolution and the sensitivity [@Bevington; @Dowling]. For instance, in a coherent-light MZI, the intensity measurement at one of the two output ports produces the signal $\langle \hat{\mu}\rangle \propto \sin ^{2}(\phi /2)$ or $\cos ^{2}(\phi /2)$, which exhibits the $\mathrm{FWHM}=\pi $ and hence the fringe resolution $\Delta x\sim \lambda /2$, known as the Rayleigh resolution limit [@Born; @Boto]. Resch *et al.* [@Resch] demonstrated that coherent light can provide a better resolution beyond the Rayleigh limit (i.e., super-resolution); however, the achievable sensitivity is much worse than the shot-noise limit $1/\sqrt{N}$, where $N$ is average number of photons. Pezzé *et al.* [@Pezze] have proposed that coincidence photon counting with a Bayesian estimation strategy results in the shot-noise limited sensitivity over a broad phase interval. However, the visibility of the coincidence rates decays quickly with the increased number of photons being detected [@Afek2010].
The parity measurement gives binary outcomes, dependent upon even or odd number of photons at one of two output ports. It originates from atomic spectroscopy with an ensemble of trapped ions [@Bollinger] and was discussed in the context of optical interferometry by Gerry [@Gerry2] and subsequently by others [@Anisimov; @Seshadreesan]. Recently, Gao *et al.* [@Gao] proposed that the parity measurement can lead to the super-resolution in the coherent-light MZI. Using a binary-outcome homodyne detection, Distante *et al.* [@Andersen] have demonstrated a super-resolution with the $\mathrm{FWHM}\sim\pi/\sqrt{N}$ and a phase sensitivity close to the shot-noise limit. Most recently, Cohen *et al.* [@Eisenberg] have realized the parity measurement in a polarization version of the MZI. In contrast to the previous theory [@Gao], they found that the peak height of signal decreases as the average photon number $N$ increases, which in turn leads to divergent phase sensitivity at certain phase shifts. More surprisingly, they found that the zero-nonzero photon counting (hereinafter, called the $Z$ detection) can saturate the shot-noise limit and gives a slightly better sensitivity than that of the parity measurement. Since both the parity and the $Z$ detections are simply two kinds of photon counting, the reason the $Z$ detection prevails is still lacking.
In this paper, we present a unified description to the above coherent-light MZI experiments [@Andersen; @Eisenberg], using general expressions of conditional probabilities for detecting an outcome in homodyne detection and in photon counting measurement. We first show that for a general binary-outcome measurement, the simplest data processing based on inverting the average signal can saturate the Cramér-Rao (CR) bound. For such measurements, more complicated data processing techniques such as maximal likelihood estimation or Bayesian estimation are not necessary. This conclusion is independent of the input states and the presence of noises. Next, we investigate the role of phase diffusion [@Qasimi; @Teklu; @Liu; @Brivio; @Genoni11; @Genoni12; @Escher12; @Zhong; @Bardhan] on the binary-outcome homodyne detection, the parity measurement, and the $Z$ measurement. Our analytical results show that the diffusion plays a role in a form of $N\gamma$, rather than the phase-diffusion rate $\gamma$ and the mean photon number $N$ alone. When $N\gamma \ll 1$, the effect of phase noise can be negligible; both the resolution and the best sensitivity almost follow the shot-noise scaling $\sim 1/\sqrt{N}$. As $N\gamma$ increases, both the resolution and the sensitivity deviate from the scaling. Analytically, we confirm that the $Z$ detection gives a better sensitivity than that of the parity detection, as demonstrated recently by Cohen *et al.* [@Eisenberg].
![(Color online) Three steps of quantum interferometry: (i) A probe state $\hat{\rho}$ is prepared and is injected into the interferometer; (ii) an unknown phase shift is accumulated during a unitary process $\hat{U}(\phi)$; (iii) the phase information is extracted via a detection $\hat{\mu}$ and a proper choice of data processing. A Mach-Zehnder interferometer fed with a coherent-state light is considered to investigate the role of phase diffusion (indicated by $\Delta\phi$) in the binary-outcome measurements.[]{data-label="fig1"}](fig1.eps){width="0.95\columnwidth"}
Quantum phase measurements with binary outcomes
===============================================
We first briefly review quantum phase measurement in a standard MZI fed with coherent-state light (see Fig. \[fig1\]). Similar to the experimental setup [@Andersen], a coherent state $|\alpha \rangle $ with amplitude $\alpha =\sqrt{N}$ is injected into one port of the MZI and the other port is left in vacuum $|0\rangle $. After a 50:50 beamsplitter [@Bs; @Gerrybook], the photon state becomes a product of coherent states $|\alpha /\sqrt{2}\rangle _{a}\otimes |\alpha /\sqrt{2}\rangle _{b}$, where the subscript $a$ ($b$) denotes the path or the polarization mode. Second, the phase shift $\phi $ is accumulated in one arm of the interferometer [@Andersen] through an unitary evolution $\hat{U}(\phi )=\exp (-i\phi\hat{N}_{a})$, with the number operator $\hat{N}_{a}=\hat{a}^{\dag }\hat{a}$. The phase accumulation results in the photon state $\hat{\rho}(\phi )=\hat{\rho}_{a}(\phi )\otimes \hat{\rho}_{b}(0)$, where the density operators for the two modes are given by $\hat{\rho}_{a}(\phi)\equiv \hat{U}(\phi )\hat{\rho}_{a}(0)\hat{U}^{\dag }(\phi )=|\alpha e^{-i\phi }/\sqrt{2}\rangle_{aa}\langle \alpha e^{-i\phi}/\sqrt{2}|$ and $\hat{\rho}_{b}(0)=|\alpha /\sqrt{2}\rangle _{bb}\langle \alpha /\sqrt{2}|$, respectively. After the second 50:50 beamsplitter, the phase-encoded state becomes $$\hat{\rho}_{\mathrm{out}}(\phi )=\hat{B}_{1/2}\hat{\rho}(\phi )\hat{B}_{1/2}^{\dag }=|\psi _{\mathrm{out}}(\phi )\rangle \langle \psi _{\mathrm{out}}(\phi )|, \label{output0}$$ where $\hat{B}_{1/2}$ denotes the beam-splitter operator [@Bs; @Gerrybook], and $|\psi _{\mathrm{out}}\rangle =|\alpha (e^{-i\phi }-1)/2\rangle_{c}\otimes |\alpha (e^{-i\phi }+1)/2\rangle _{d}$ is the output state for the optical modes $c$ and $d$. Finally, a measurement and data processing are performed to obtain phase-sensitive output signal, which determine the fringe resolution and the phase sensitivity.
For instance, a homodyne detection of the phase quadrature $\hat{p}=(\hat{c}-\hat{c}^{\dag })/2i$ at the output port $c$ is described by the projection operators $\{|p\rangle \langle p|\}$ with $\hat{p}|p\rangle =p|p\rangle $. The probability for detecting an outcome $p$ is simply given by $P(p|\phi)\equiv \mathrm{Tr}[\hat{\rho}_{\mathrm{out}}(\phi )|p\rangle \langle p|]$, with its explicit form, $$P(p|\phi )=\sqrt{\frac{2}{\pi }}\,{\exp }\left[ {-2}\left( {p+\frac{\sqrt{N}{\sin \phi }}{2}}\right) ^{2}\right] , \label{homodyne}$$ where we have used the wave function of a coherent state $\langle p|\alpha\rangle \equiv (2/\pi )^{1/4}\exp [-(p-y_{0})^{2}-2ix_{0}p+ix_{0}y_{0}]$, with $x_{0}=\mathrm{Re}(\alpha )$ and $y_{0}=\mathrm{Im}(\alpha )$. The output signal is then given by $\langle \hat{p}(\phi )\rangle=\int_{\mathbb{R}}\!dppP(p|\phi )\propto \sqrt{N}\sin \phi $, which exhibits the $\mathrm{FWHM}\approx \pi $ and hence the Rayleigh limit in fringe resolution. The Fisher information of the homodyne measurement is given by $$F(\phi )=\int_{\mathbb{R}}\!\!dp\frac{1}{P(p|\phi )}\left[ \frac{\partial P(p|\phi )}{\partial \phi }\right] ^{2}=N\cos ^{2}\phi , \label{FH}$$ which yields the CR bound $\delta \phi _{\mathrm{CRB}}=1/(\sqrt{N}|\cos\phi |)$, dependent upon the true value of phase shift. Only at $\phi _{\min}=k\pi $ for integers $k$, the lower bound of phase sensitivity can reach the shot-noise limit.
Next, let us consider a general photon counting characterized by a set of projection operators $\{|n,m\rangle \langle n,m|\}$, with the two-mode Fock states $|n,m\rangle \equiv |n\rangle _{c}\otimes |m\rangle _{d}$. The probability for detecting $n$ photons at the output port $c$ and $m$ photons at the port $d$, i.e., the coincidence rate $P(n,m|\phi )\equiv \langle n,m| \hat{\rho}_{\mathrm{out}}(\phi )|n,m\rangle $ [@Afek2010], is given by $$P(n,m|\phi )=\frac{e^{-N}}{n!m!}\left( N\sin ^{2}\frac{\phi }{2}\right)^{n}\left( N\cos ^{2}\frac{\phi }{2}\right) ^{m}, \label{coincidence}$$ with the mean photon number $N=\alpha ^{2}$. For a light intensity measurement at the output port $d$, we obtain the signal that is proportional to $\langle \hat{N}_{d}(\phi )\rangle =\sum_{m}mP(m|\phi)=N\cos ^{2}(\phi /2)$, where $\hat{N}_{d}=\hat{d}^{\dag }\hat{d}$ and the probability $P(m|\phi )=\sum_{n}P(n,m|\phi )$. It is easy to find that the $\mathrm{FWHM}=\pi $ and the resolution $\Delta x\propto \lambda /2$, as discussed before. In addition, we obtain the Fisher information of this light-intensity detection, $$F(\phi )=\sum_{m}\frac{1}{P(m|\phi )}\left[ \frac{\partial P(m|\phi )}{\partial \phi }\right] ^{2}=N\sin ^{2}\frac{\phi }{2}, \label{Fisher}$$ and hence the lower bound $\delta \phi _{\mathrm{CRB}}=1/[\sqrt{N}|\sin(\phi /2)|]$, which reaches the shot-noise limit at $\phi _{\min }=(2k+1)\pi$ for integers $k$.
From Eq. (\[coincidence\]), one can note that the coincidence rate with $ nm\neq 0$ shows multifold oscillations as a function of $\phi $, leading to an enhanced phase resolution beyond the Rayleigh limit. As demonstrated by Afek *et al*. [@Afek2010], however, the visibility of the multifold oscillations decays quickly with the increased number of photons being detected. Using path-entangled NOON states, the super-resolution of interferometric fringe $\Delta x\propto \lambda /(2N)$ is possible [@Mitchell; @Walther; @Chen], but with $N\lesssim 5$ [@Afek].
The homodyne detection and the photon counting with a proper choice of data processing can improve the phase resolution. Recently, it has been shown that the binary-outcome homodyne detection [@Andersen] and photon counting [@Eisenberg] in the coherent light MZI lead to the super-resolution $\Delta x\propto \lambda /(2\sqrt{N})$ and the sensitivity close to the shot-noise limit. We show below that the above observations can be understood by proper data processing to Eqs. (\[homodyne\]) and (\[coincidence\]). Moreover, we verify that the CR bound of *any* binary-outcome measurements can be saturated by the simplest data processing based on inverting the average signal.
In quantum interferometry, any measurement of the Hermitian operator $\hat{\mu}$ with respect to arbitrary phase-encoded state $\hat{\rho}(\phi )$ can be modeled by projection onto the orthonormalized eigenstates $\{|\mu\rangle \langle \mu |\}$ of operator $\hat{\mu}$, with $\hat{\mu}|\mu\rangle =\mu |\mu \rangle $. Here, we focus on projection measurements with binary outcomes $\mu _{\pm }$. The associated probabilities are defined as $P(\pm |\phi )\equiv \mathrm{Tr}[\hat{\rho}(\phi )|\mu _{\pm }\rangle \langle \mu _{\pm }|]=\langle \mu _{\pm }|\hat{\rho}(\phi )|\mu _{\pm }\rangle $. The simplest data processing is based on the average signal and its second moment, i.e., $\langle \hat{\mu}^{k}(\phi )\rangle \equiv \mu_{+}^{k}P(+|\phi )+\mu _{-}^{k}P(-|\phi )$ for $k=1$, $2$. Using the normalization condition $P(+|\phi )+P(-|\phi )=1$, we obtain the variance $ (\Delta \hat{\mu})^{2}=(\mu _{+}-\mu _{-})^{2}P(+|\phi )P(-|\phi )$. For phase-independent outcomes $\mu _{\pm }$, we further obtain the slope of signal $|\partial \langle \hat{\mu}\rangle /\partial \phi |=|(\mu _{+}-\mu_{-})\partial P(+|\phi )/\partial \phi |$. Therefore, the error-propagation formula gives the phase sensitivity $$\delta \phi =\frac{\Delta \hat{\mu}}{|\partial \langle \hat{\mu}\rangle/\partial \phi |}=\frac{\sqrt{P(+|\phi )P(-|\phi )}}{|\partial P(+|\phi )/\partial \phi |}=\frac{1}{\sqrt{F\left( \phi \right) }}, \label{sensitivity}$$ where the Fisher information for the binary-outcome detection is given by Eq. (\[Fisher\]) with $m=\pm $. Obviously, the sensitivity obtained from the error-propagation formula can saturate the CR bound over the entire phase interval. Actually, this conclusion holds not only for the projection measurements, but also for the most general kind of quantum measurement (i.e., positive operator-valued measure). In addition, Eq. (\[sensitivity\]) remains valid for arbitrary input state and is independent from the presence of noises. Previously, Seshadreesan *et al.* [@Seshadreesan] found that for the parity measurement, the inversion estimator can reach the CR bound. This measurement is a special case of photon counting at one of two output ports with the outcomes $\mu _{\pm }=\pm 1$ (see below).
Binary-outcome homodyne detection
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Recently, Distante *et al.* [@Andersen] demonstrated a binary-outcome homodyne detection by dividing the total data of phase quadrature into binary outcomes: $|p|\leq p_{0}$ and $|p|>p_{0}$, denoted by $+$“ and $-$”, respectively, with the associated probabilities $$P(+|\phi )=\int_{-p_{0}}^{p_{0}}\!\!dpP(p|\phi )\text{, \ \ \ }P(-|\phi)=1-P(+|\phi ),$$ where $P(p|\phi )$ is given by Eq. (\[homodyne\]). Note that the conditional probability for detecting the outcome $+$ is the same to the interferometric signal ${\langle \hat{p}_{+}(\phi )\rangle =}\mathrm{Tr}[\hat{\rho}_{\mathrm{out}}{(\phi )}\hat{p}_{+}]$, with the observable $\hat{p}
_{+}=\int_{-p_{0}}^{p_{0}}\!dp|p\rangle \langle p|$ and $\hat{p}_{+}^{2}=\hat{p}_{+}$. Moreover, this kind of data processing results in a super-resolved fringe pattern [@Andersen], which can be understood by considering the limit $p_{0}\rightarrow 0$, corresponding to a detection of the phase quadrature with $p=0$. In this case, the observable becomes ${\hat{p}_{+}=}|p=0\rangle \langle p=0|$ and the interferometric signal is therefore given by ${\langle \hat{p}_{+}\rangle =}P(p=0|\phi )$, with its explicit form \[cf. Eq. (\[homodyne\])\] $${\langle \hat{p}_{+}(\phi )\rangle }=\,\sqrt{\frac{2}{\pi }}\exp \left( -\frac{N}{2}\sin ^{2}\left( \phi \right) \right) , \label{p-homody}$$ which, as illuminated by the red solid line of Fig. \[fig2\](a), becomes narrowing in a comparison with that of the intensity detection \[$\propto \cos ^{2}(\phi /2)$, the gray dotted line\]. To understand this behavior, we now analyze Eq. (\[p-homody\]) near the phase origin $\phi \sim 0$, and obtain ${\langle \hat{p}_{+}\rangle }\propto \exp (-N\phi ^{2}/2)$, which gives the $\mathrm{FWHM}=2\sqrt{(2\ln 2)/N}<\pi /\sqrt{N}$ and hence the resolution $\Delta x<\lambda /(2\sqrt{N})$. Clearly, the resolution is improved by a factor $\sqrt{N}$ beyond the Rayleigh limit $\lambda /2$ [@Andersen].
![(Color online) Normalized output signal (a) and phase sensitivity (b) for the binary-outcome homodyne detection with a fixed number of photons $N=200$ and various phase-diffusion rates $\gamma=0$ (red solid), $10^{-4}$ (blue dashed), and $10^{-3}$ (green dot-dashed). The signals in (a), normalized by $\sqrt{2/\pi}$, and the sensitivities in (b) are the exact results, obtained by numerically integrating Eq. (\[rhoout-2HD\]). Gray dotted line in (a) is the normalized signal of the intensity measurement, i.e., $\cos^{2}(\phi/2)$; while in (b), it indicates the best sensitivity $1.03/\sqrt{N}$, given by Eq. (\[bestsensitivity-2HD\]) for $\gamma=0$. Insets: zoomed output signal and the phase sensitivity near the phase origin $\phi=0 $. []{data-label="fig2"}](fig2.eps){width="0.9\columnwidth"}
According to the error-propagation formula, i.e., Eq. (\[sensitivity\]), we further obtain the phase sensitivity $$\delta \phi _{H}=\frac{1}{\sqrt{F(\phi )}}=\frac{2}{N}\frac{\left( \sqrt{\frac{\pi }{2}}e^{(N/2)\sin ^{2}\phi }{-1}\right) ^{1/2}}{|\sin (2\phi)|}, \label{delphi}$$ which saturates the CR bound. As shown by the red solid line of Fig. \[fig2\](b), one can find that the sensitivity diverges at the phase origin $\phi =0$, due to the nonzero variance of ${\langle \hat{p}_{+}\rangle }$ and the vanishing slope of signal as $\phi \rightarrow 0$. One can also see this from Eq. (\[delphi\]). It is interesting to note that local minimum of $\delta \phi _{H}$ (i.e., the best sensitivity $\delta \phi _{H,\min }$) can be obtained by maximizing the slope $|\partial {\langle \hat{p}_{+}\rangle }/\partial \phi |=N|\sin (2\phi )|{\langle \hat{p}_{+}\rangle }/2$, where $\langle \hat{p}_{+}\rangle $ is given by Eq. (\[p-homody\]). As a result, one can obtain the optimal phase shift $\phi _{\mathrm{\min }}$, obeying $N\sin^{2}(2\phi _{\mathrm{\min }})=4\cos (2\phi _{\mathrm{\min }})$. In the large-$N$ limit, it gives $\exp [N\sin ^{2}(\phi _{\mathrm{\min }})/2]\approx\sqrt{e}$, due to $\phi _{\mathrm{\min }}\approx 0$ and $N\phi_{\mathrm{\min }}^{2}\approx 1$. Finally, from Eq. (\[delphi\]), we obtain the best sensitivity, $$\delta \phi _{H,\min }=\frac{\left( \sqrt{\frac{\pi }{2}}e^{(N/2)\sin ^{2}\phi _{\mathrm{\min }}}-{1}\right) ^{1/2}}{|N\cos (2\phi _{\mathrm{\min }})|^{1/2}}\approx \frac{(\sqrt{e\pi /2}-{1)}^{1/2}}{\sqrt{N}},
\label{delmin_2HD}$$ in agreement with Distante *et al.* [@Andersen]. For nonzero $p_{0}$ ($=1/2$), it has been demonstrated that the best sensitivity can reach $\delta \phi _{H,\min }\approx 1.37/\sqrt{N}$, approaching the shot-noise limit [@Andersen].
Parity detection
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According to Gao *et al.* [@Gao], the parity detection in the standard MZI also results in the super-resolution. The parity measurement [@Bollinger; @Gerry2; @Anisimov; @Seshadreesan] at the output port $c$, described by the parity operator $\hat{\Pi}=(-1)^{\hat{c}^{\dag}\hat{c}}$, groups the photon counting $\{n,m\}$ into binary outcomes $\pm 1$, according to even or odd number of photons $n$ at that port $c$. Such a kind of data processing provides an optimal phase estimator for the input path-symmetric states [@Seshadreesan]. The conditional probabilities $P(\pm 1|\phi )$ are obtained by a sum of $P(n,m|\phi )$ over the even or the odd $n$’s, namely $$P(+1|\phi )=\sum_{m,n}^{\mathrm{even}\text{ }n}P(n,m|\phi )=\frac{1}{2}\left( 1+e^{-2N\sin ^{2}(\phi/2)}\right) ,$$ and $P(-1|\phi )=1-P(+1|\phi )$. In deriving the above result, we have used the identity $\sum_{n}^{\mathrm{even}\text{ }n}x^{n}/n!=\cosh x$. The signal corresponds to the expectation value of the parity operator $\hat{\Pi}$ with respect to $\hat{\rho}_{\mathrm{out}}$, given by $$\langle \hat{\Pi}(\phi )\rangle =P(+1|\phi )-P(-1|\phi )=e^{-2N\sin^{2}(\phi /2)}, \label{P+}$$ which coincides with Gao *et al.* [@Gao]. Near $\phi=0$, the signal $\langle \hat{\Pi}\rangle \approx \exp(-N\phi^{2}/2)$, similar to the binary-outcome homodyne detection of Eq. (\[p-homody\]), results in the super-resolution with $\Delta x\sim\lambda/(2\sqrt{N})$. Due to $\hat{\Pi}^{2}=1$, the phase sensitivity is given by $$\delta \phi _{\Pi }=\frac{\sqrt{e^{4N\sin ^{2}(\phi/2)}-1}}{N\left\vert \sin \phi \right\vert }\approx \frac{1}{\sqrt{N}}\left( 1+\frac{1+2N}{8}\phi ^{2}\right) , \label{delphi_P}$$ which saturates the CR bound over the entire phase interval. In the last step, the sensitivity is expanded up to the second order of $\phi$ [@Gao]. Obviously, the sensitivity can reach the shot-noise limit at $\phi =0$ \[see the red dashed line of Fig. \[fig3\](a)\].
![(Color online) Phase sensitivities $\delta\phi$ against dimensionless phase shift $\phi$ (in units of $1/\pi$ ) for the parity detection (a) and the $Z$ detection (b). Solid (red dashed) lines: the exact solutions of $\delta\phi$ with (without) the phase diffusion for the mean photon number $N=200$ and the phase-diffusion rate $\gamma=10^{-4}$. Horizontal dotted lines: the best sensitivities, predicted by Eqs. (\[sen\_P\]) and (\[sen\_Z\]). The arrows indicate the positions of the optimal phase shift $\phi_{\min}$. Shaded areas: the shot-noise limit $1/\sqrt{N}$ and below.[]{data-label="fig3"}](fig3.eps){width="1\columnwidth"}
$Z$ detection
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As the final example, we consider the zero-nonzero photon counting (named as the $Z$ detection) at the output port $c$, following Ref. [@Eisenberg]. In this scheme, the counting data $\{n,m\}$ are classified into binary outcomes: $0$ for $n=0$ and $\emptyset $ for $n\neq 0$, with the associated probabilities $$P(0|\phi )=\sum_{m}P(n=0,m|\phi )=e^{-N\sin ^{2}(\phi/2)}, \label{Zout}$$and $P(\emptyset |\phi )=1-P(0|\phi )$. The output signal $\langle\hat{Z}(\phi)\rangle=P(0|\phi)$ corresponds to the expectation value of the observable $\hat{Z}=|0\rangle _{cc}\langle 0|$ with respect to $\hat{\rho}_{\mathrm{out}}$. Note that at $\phi =0$, the output state is indeed $|\psi_{\mathrm{out}}\rangle=|0\rangle_{c}\otimes |\alpha\rangle_{d}$, so no photon is detected at output port $c$ and $P(0|\phi =0)=\langle\hat{Z}(0)\rangle=1$. Near the phase origin, one can find that the signal $\langle \hat{Z}\rangle \approx \exp(-N\phi^{2}/4)$ and hence the $\mathrm{FWHM}=4\sqrt{\ln 2/N}\sim\pi/\sqrt{N}$, leading to a super-resolved fringe pattern. However, the resolution becomes worse by a factor $\sqrt{2}$ than that of the previous detections [@Eisenberg]. Using $\hat{Z}^{2}=\hat{Z}$ and the error-propagation formula, we obtain the phase sensitivity $$\delta \phi _{Z}=\frac{2\sqrt{e^{N\sin ^{2}(\phi/2)}-1}}{N\left\vert \sin \phi \right\vert }\approx \frac{1}{\sqrt{N}}\left( 1+\frac{1+N/2}{8} \phi ^{2}\right) , \label{deltaphi_Z}$$ which can also saturate the CR bound and reach the shot-noise limit at $\phi =0$, as shown by the red dashed line of Fig. \[fig3\](b).
With the above binary-outcome detections, one can note that both the phase resolution and the best sensitivity exhibit the shot-noise scaling $\sim 1/
\sqrt{N}$. Specially, the parity and the $Z$ detections show the best sensitivity at $\phi =0$ due to the peak heights $\langle \hat{\Pi}(0)\rangle =\langle \hat{Z}(0)\rangle =1$. The finite value of $\delta \phi $ at $\phi =0$ (i.e., $\delta \phi _{\min }=1/\sqrt{N}$) can be understood by the fact that as $\phi \rightarrow 0$, both the variance and the slope of signal for each detection approach zero. In real experiment, however, Cohen *et al.* [@Eisenberg] observed that the peak height decreases exponentially as a function of $N$ and the sensitivity diverges at $\phi =0$ [@Bs]. They attributed these observations to the dark counts and the imperfect visibility due to the background counts [@Eisenberg]. In the next section, we investigate the roles of photon loss and phase diffusion noise on the binary-outcome detections.
Binary-outcome detections under the phase diffusion
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We first consider the role of photon loss by introducing a fictitious beam splitter in one of two arms after the phase accumulation [@Enk; @Rubin; @Dorner; @Ma; @Escher2011; @Zhang]. Only the absorption of photons that carries the phase information is important, so the phase-dependent state can be rewritten as $|\psi (\phi )\rangle =\hat{B}_{T}|\alpha e^{-i\phi }/\sqrt{2}\rangle _{a}\otimes |\alpha /\sqrt{2}\rangle _{b}\otimes |0\rangle _{E}$, where the beam splitter $\hat{B}_{T}$ couples the interferometric mode $a$ and the environment mode $E$ with photon transmission rate $T$ [@Bs; @Gerrybook]. Tracing over the environment mode, one obtains the phase-dependent state $$\hat{\rho}(\phi )=\left\vert \frac{\alpha \sqrt{T}e^{-i\phi }}{\sqrt{2}},\frac{\alpha \sqrt{T}}{\sqrt{2}}\right\rangle \left\langle \frac{\alpha \sqrt{T}e^{-i\phi }}{\sqrt{2}},\frac{\alpha \sqrt{T}}{\sqrt{2}}\right\vert ,$$where the transmission rate $T=1$ means no loss and $T=0$ corresponds to a complete photon loss. Comparing with the noiseless case, one can find that the photon loss leads to a replacement $\alpha \rightarrow \alpha \sqrt{T}$ (i.e., $N\rightarrow NT$) in the output signals. Such a trivial influence cannot explain the imperfect visibility observed by Cohen *et al.* [@Eisenberg].
Next, we consider the phase-diffusion process after the phase accumulation, which produces a phase fluctuation in one of the two paths (as depicted by Fig. \[fig1\]). Formally, the presence of phase noise can be modeled by the following master equation [@Qasimi; @Teklu; @Liu; @Brivio; @Genoni11; @Genoni12; @Escher12; @Zhong; @Bardhan]: $\partial \hat{\rho}/\partial t=\Gamma _{p}(2\hat{N}_{a}\hat{\rho}\hat{N}_{a}-\hat{N}_{a}^{2}\hat{\rho}-\hat{\rho}\hat{N}_{a}^{2})$, with $\hat{N}_{a}=\hat{a}^{\dag }\hat{a}$ and the phase-diffusion rate $\Gamma_{p}$. Following Refs. [@Genoni12; @Brivio], the solution of $\hat{\rho}$ is given by an integration $\hat{\rho}_{\gamma }(\phi )\propto \int_{\mathbb{R}}d\xi e^{-\xi^{2}/(4\gamma)}\hat{U}(\xi )\hat{\rho}(\phi )\hat{U}^{\dag}(\xi )$, where $\gamma=\Gamma_{p}t$ is a dimensionless diffusion rate. Note that for the noiseless case, the phase-encoded state $\hat{\rho}(\phi)$ obeys $\hat{U}(\xi)\hat{\rho}(\phi)\hat{U}^{\dag}(\xi)=\hat{\rho}(\xi+\phi)$. Replacing $\xi\rightarrow\xi-\phi$ and performing the second beam-splitter operation to $\hat{\rho}_{\gamma}(\phi)$, we obtain the final state, $$\hat{\rho}_{\gamma ,\mathrm{out}}(\phi )\!\!=\!\!\int_{\mathbb{R}}\!d\xi \frac{1}{\sqrt{4\pi \gamma }}\exp \left[ -\frac{(\xi -\phi )^{2}}{4\gamma }\right] \hat{\rho}_{\mathrm{out}}(\xi ),$$ where $\hat{\rho}_{\mathrm{out}}(\phi)$ has been given by Eq. (\[output0\]). For a very weak diffusion rate (i.e., $\gamma\rightarrow 0$), using $\mathrm{lim}_{\gamma\rightarrow 0}e^{-(\xi-\phi)^{2}/(4\gamma)}/\sqrt{4\pi\gamma}=\delta (\xi-\phi)$, one can easily obtain the final state $\hat{\rho}_{\gamma, \mathrm{out}}(\phi)=\hat{\rho}_{\mathrm{out}}(\phi)$, recovering the noiseless case.
Under the phase diffusion, all the relevant quantities, such as the output signal and the conditional probabilities, can be obtained by integrating the Gaussian with the quantities without diffusion. For instance, the binary-outcome homodyne detection gives the output signal $${\langle \hat{p}_{+}(\phi )\rangle }_{\gamma }\!\!=\!\!\int_{\mathbb{R}}\!d\xi \frac{1}{\sqrt{4\pi \gamma }}\exp \left[ -\frac{(\xi -\phi )^{2}}{4\gamma }\right] \langle {\hat{p}_{+}}(\xi )\rangle, \label{rhoout-2HD}$$ where $\langle{\hat{p}_{+}}(\xi)\rangle$ has been given by Eq. (\[p-homody\]). Integrating it with the Gaussian, one can obtain the exact numerical result of the output signal \[see Fig. \[fig2\](a)\], as well as the conditional probability $P(p|\phi)$ for detecting the phase quadrature $p=0$. Due to $\hat{p}_{+}^{2}=\hat{p}_{+}$, one can also obtain the variance $(\Delta {\hat{p}_{+}})^{2}={\langle\hat{p}_{+}\rangle}_{\gamma}(1-{\langle \hat{p}_{+}\rangle}_{\gamma})$ and hence the phase sensitivity $\delta \phi_{H}$ \[see Fig. \[fig2\](b)\]. According to Eq. (\[sensitivity\]), the phase sensitivity can also saturate the CR bound over the whole phase interval. The numerical results in Fig. \[fig2\] show that the phase diffusion degrades the fringe resolution and the best sensitivity (see the green dot-dashed lines). In real experiment [@Andersen], however, the exact results of the signal and the sensitivity without any noise agree very well with their experimental data, implying that the diffusion rate is very small (e.g., $\gamma\lesssim 10^{-4}$). One can also see this from the blue dashed lines of Fig. \[fig2\]; the numerical results for $\gamma=10^{-4}$ almost merge with that of the noiseless case. Even for such a small diffusion rate, the phase diffusion has a dramatic influence in the binary-outcome photon counting measurements (see below).
![(Color online) Log-log plot of the best sensitivity for the binary-outcome homodyne detection with $\gamma =0$ (crosses) and $\gamma =10^{-4}$ (open circles). The open circles and the crosses are obtained by numerically integrating Eq. (\[rhoout-2HD\]). Solid (dashed) line: analytical result of $\delta\phi_{H,\min }$, given by Eq. (\[bestsensitivity-2HD\]). Inset: the exact and the analytical results of $\delta\phi_H$ as a function of $ \phi $ (in units of $1/\pi$) for $N=200$ and $\gamma =10^{-4} $. Gray dotted lines at $\phi=\pm\Delta/\sqrt{N}$ indicate the location of the best sensitivity. []{data-label="fig4"}](fig4.eps){width="0.9\columnwidth"}
To present a unified description of the above measurements, we first analyze the role of phase noise in the binary-outcome homodyne detection. Without any noise, the signal can be approximated as $\langle{\hat{p}_{+}}({\phi})\rangle\approx\sqrt{2/\pi}\exp(-N{\phi}^{2}/2)$. Inserting it into Eq. (\[rhoout-2HD\]), we obtain the output signal $${\langle \hat{p}_{+}(\phi)\rangle}_{\gamma}\approx\sqrt{\frac{2}{\pi\Delta^{2}}}\exp\left(-\frac{N\phi ^{2}}{2\Delta^{2}}\right) , \label{appro-2HD}$$ where we have introduced $\Delta\equiv\sqrt{1+2N\gamma}>1$. Clearly, the phase diffusion leads to the degradation of the fringe resolution, due to the $\mathrm{FWHM}\approx 2\Delta \sqrt{(2\ln 2)/N}$. In addition, we obtain the sensitivity $$\delta \phi _{H}\approx \frac{\Delta ^{2}}{N|\phi |}\sqrt{\Delta \sqrt{\frac{\pi }{2}}e^{N\phi ^{2}/(2\Delta ^{2})}-\,1}, \label{sensitivity-2HD}$$ which shows a good agreement with the exact numerical result at $\phi\sim0$. By maximizing the slope of signal $|\partial{\langle\hat{p}_{+}\rangle}_{\gamma}/\partial\phi|$, we further obtain the optimal phase shift $N\phi_{\mathrm{\min}}^{2}\approx\Delta^{2}$ or $\phi_{\mathrm{\min}}\approx\pm\Delta/\sqrt{N}$ (see the inset of Fig. \[fig4\]), which is valid for a small diffusion rate $\gamma$ and large enough $N$. Therefore, from Eq. (\[sensitivity-2HD\]) we have $$\begin{aligned}
\delta \phi _{H,\min }\!\! &\approx &\frac{\Delta (\Delta \sqrt{e\pi /2}-{1})^{1/2}}{\sqrt{N}} \notag \\ &\!\!=\!\!&\frac{\eta }{\sqrt{N}}\left\{ 1+\frac{3\eta ^{2}+1}{2\eta ^{2}} N\gamma +O[(N\gamma )^{2}]\right\} , \label{bestsensitivity-2HD}\end{aligned}$$ where, for brevity, we introduce $\eta\equiv(\sqrt{e\pi/2}-{1)}^{1/2}\approx 1.03$. Note that for the noiseless case, i.e., $\gamma=0$, the best sensitivity is simply given by $\eta/\sqrt{N}$, recovering Eq. (\[delmin\_2HD\]). As shown by Fig. \[fig4\], one can find that for large enough $N$ ($\gtrsim 10$), the series expansion of $\delta\phi_{H,\min}$ up to order of $(N\gamma)^1$ works well to predict the best sensitivity (the crosses and the open circles).
Next, we perform similar analysis to the binary-outcome photon counting detections. For the parity detection, the signal under the phase diffusion is given by $$\langle \hat{\Pi}\rangle _{\gamma }=\int_{\mathbb{R}}\!d\xi \frac{e^{-(\xi -\phi )^{2}/4\gamma }}{\sqrt{4\pi \gamma }}\langle \hat{\Pi}(\xi )\rangle
\approx \frac{1}{\Delta }e^{-N\phi ^{2}/(2\Delta ^{2})}, \label{Pgamma}$$ where we have approximated $\langle \hat{\Pi}(\xi)\rangle\approx \exp(-N\xi^{2}/2)$ and introduced $\Delta=\sqrt{1+2N\gamma}$ as done before. For the $Z$ detection, the signal reads $$\langle \hat{Z}\rangle _{\gamma }=\int_{\mathbb{R}}\!d\xi \frac{e^{-(\xi -\phi )^{2}/4\gamma }}{\sqrt{4\pi \gamma }}\langle \hat{Z}(\xi )\rangle \approx \frac{1}{\Delta _{0}}e^{-N\phi ^{2}/(2\Delta _{0})^{2}}, \label{Zgamma}$$ where $\Delta_{0}\equiv \sqrt{1+N\gamma}$. Similar to the binary-outcome homodyne detection, the fringe resolution of each detection degrades by a factor $\Delta $ or $\Delta _{0}$, as the $ \mathrm{FWHM}\approx 2\Delta \sqrt{(2\ln 2)/N}$ for the parity detection and $4\Delta_{0}\sqrt{(\ln 2)/N}$ for the $Z$ detection. Actually, the role of phase diffusion in the above three measurements is the same; it is uniquely determined by the product $N\gamma$, instead of $\gamma$ or $N$ alone. When $N\gamma\ll 1$, the diffusion is negligible and the resolution $\Delta x\sim \lambda /(2\sqrt{N})$. With the increase of $N\gamma$, the resolution degrades and its scaling undergoes a transition from $ N^{-1/2}$ to $N^{0}$, due to the $\mathrm{FWHM}\rightarrow 4\sqrt{\gamma\ln 2}$ as $N\gamma\gg 1$.
Unlike the homodyne detection, the phase diffusion has a dramatic influence on the sensitivity for the binary-outcome photon counting. Even for a small dephasing rate $\gamma \sim 10^{-4}$, one can find that the peak heights $ 1/\Delta $ and $1/\Delta _{0}$ decrease as $N$ increases, similar to Ref. [@Eisenberg]. The degradation of the peak heights results in nonzero variance at $\phi=0$, while the slope of signal tends to zero as $\phi \rightarrow 0$. Therefore, the sensitivities of the two detections diverge at the phase origin. As shown in Fig. \[fig3\], one can find that the best sensitivity occurs at $\phi \neq 0$. A similar result has been observed by Cohen *et al.* [@Eisenberg]. To understand this behavior, we now calculate the best sensitivity by minimizing analytical result of $\delta\phi$. For the parity measurement, we find that the best sensitivity occurs at $\phi _{\min }\approx \pm \Delta \sqrt{\lbrack 1+w(-e^{-1}\Delta ^{-2})]/N}$, where $w(z)$ denotes the Lambert W function (also called the product logarithm), defined by the principal solution of $w$ in the equation $z=we^{w}$. With the optimal phase $\phi_{\min}$, we obtain the best sensitivity, $$\begin{aligned}
\delta \phi _{\Pi ,\min } &\approx &\frac{1}{\sqrt{N}}\Delta ^{2}\exp \left[ \frac{1+w\left( -e^{-1}\Delta ^{-2}\right) }{2}\right] \notag \\ &\approx &\frac{1}{\sqrt{N}}\left( 1+\sqrt{N\gamma }+\frac{11N\gamma }{6} \right) , \label{sen_P}\end{aligned}$$ where, in the last step, we have expanded $\delta \phi _{\Pi ,\min }$ up to order of $(N\gamma )^{1}$. Note that the first line of Eq. (\[sen\_P\]) is valid in the parameter regime $z=e^{-1}\Delta ^{-2}\in (0,e^{-1})$, for which the W function obeys the inequality $0<-w(-z)<1$ and hence $1+w(-z)>0$. The second line of Eq. (\[sen\_P\]), as shown by the dot-dashed curve of Fig. \[fig5\], agrees quite well with the exact numerical result (the crosses).
Similarly, for the $Z$ detection, we obtain the best sensitivity $\delta\phi_{Z,\mathrm{\min}}\approx \Delta_{0}/\sqrt{-Nw(-e^{-1}\Delta_{0}^{-1})}$, which can be further approximated as $$\delta \phi _{Z,\mathrm{\min }}\approx \frac{1}{\sqrt{N}}\left( 1+\frac{ \sqrt{N\gamma }}{2}+\frac{17N\gamma }{24}\right) . \label{sen_Z}$$ From Fig. \[fig5\], one can find that for $N\lesssim 10^{2}$ and $\gamma\sim 10^{-4}$ (i.e., $N\gamma\lesssim 10^{-2}$), the best sensitivities almost follow the shot-noise scaling. When $N\gamma >10^{-2}$, both $\delta\phi_{\Pi, \mathrm{\min}}$ and $\delta\phi_{Z,\mathrm{\min}}$ become worse. However, the $Z$ detection gives a slightly better sensitivity than that of the parity detection, which can be easily understood by comparing Eqs. (\[sen\_P\]) and (\[sen\_Z\]).
![(Color online) Log-log plot of the best sensitivities for the parity measurement (crosses) and the $Z$ detection (open circles) with a given phase-diffusion rate $\gamma =10^{-4}$. Dot-dashed (red solid) line: the analytical result of $\delta \phi _{\Pi ,\min }$ ($ \delta \phi _{Z,\min }$), given by Eqs. (\[sen\_P\]) and (\[sen\_Z\]), respectively. Dotted line: the shot-noise limit $\delta \phi _{\min }=1/\sqrt{N}$. The exact results, indicated by the crosses and the open circles, are obtained by numerically integrating the output signals for the two measurement schemes.[]{data-label="fig5"}](fig5.eps){width="0.9\columnwidth"}
Finally, we should point out that our result, Eq. (\[sensitivity\]), remains valid even in the presence of the noise. This can be easily found from explicit forms of the variances $(\Delta \hat{\Pi})^{2}=4P(+1|\phi )P(-1|\phi )$ and $(\Delta \hat{Z})^{2}=P(0|\phi )P(\emptyset |\phi )$, where $P(\pm 1|\phi )=(1\pm \langle \hat{\Pi}\rangle _{\gamma })/2$ and $ P(0|\phi )=\langle \hat{Z}\rangle _{\gamma }$ are the conditional probabilities for the parity and the $Z$ detections. Indeed, the phase sensitivity of any binary-outcome detection always saturates the CR bound.
Conclusion
==========
In summary, we have shown that phase sensitivity with a general binary-outcome measurement always saturates the Cramér-Rao bound. Its validity is demonstrated by the recent experiments based on coherent-light Mach-Zehnder interferometer [@Andersen; @Eisenberg]. The observed super-resolution in fringe pattern and the shot-noise limited sensitivity can be well understood as suitable data processing over the conditional probabilities $P(p|\phi)$ and $P(n,m|\phi)$ for detecting a phase quadrature $p$ in a homodyne detection and a pair of photons $\{n,m\}$ in a photon counting measurement, respectively.
We consider the performance of the binary-outcome homodyne measurement, the parity, and the $Z$ detections in the presence of phase diffusion. Interestingly, we find that the role of phase diffusion is uniquely determined by a product of the mean photon number $N$ and the diffusion rate $\gamma $. When $N\gamma \ll 1$, both the resolution and the sensitivity almost exhibit the shot-noise scaling $\sim 1/\sqrt{N}$. Except for the experimental imperfections, we show that a very weak phase diffusion can dramatically change the behavior of the sensitivity in the binary-outcome photon counting. Our analytical results also confirm that the $Z$ detection gives a better sensitivity than that of the parity detection, in agreement with the experiment observation [@Eisenberg].
We thank Professor J. P. Dowling for helpful discussions. This work was supported by the NSFC (Contracts No. 11174028, No. 11274036, and No. 11322542), the MOST (Contract No. 2014CB848701), and the FRFCU (Contract No. 2011JBZ013).
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Throughout this work, we consider the action of the beam splitter as $\hat{B}_{T}=\exp [-i\theta \hat{J}_{y}^{(k,l)}]$, where $\hat{J}_{y}^{(k,l)}=(\hat{a}_{k}^{\dag}\hat{a}_{l}-$H.c.$)/2i$. The BS couples two optical modes $k$ and $l$ with the photon transmission rate $T$ and the mixing angle $\theta =2\arccos (\sqrt{T})$. For a 50:50 beam splitter, the photon transmission rate $T=1/2$ and $\theta =\pi /2$, corresponding to a $\pi /2$ pulse in atomic spectroscopy. If one adopts another kind of the beam splitter, $\hat{B}_{1/2}=\exp [-i\pi \hat{J}_{x}^{(k,l)}/2]$, where $\hat{J}_{x}^{(k,l)}=(\hat{a}_{k}^{\dag }\hat{a}_{l}+$H.c.$)/2$, the phase should be shifted as $\phi \rightarrow \phi +\pi $, as shown in Refs. [@Eisenberg; @Gerrybook].
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the invariant measure is unique.
[**Keywords:**]{} quasi-periodic measures, invariant measures, random dynamical systems, random quasi-periodic paths, Markovian random dynamical system, Markovian semigroup.
author:
- Chunrong Feng
- Baoyou Qu
- Huaizhong Zhao
title: 'Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations'
---
Introduction
============
Quasi-periodic oscillation of a dynamical system is a motion given by a quasi-periodic function $F$ such that $$\label{quasi-periodic function}
F(t)=f(t,t,\cdots,t),$$ for some continuous function $f(t_1,t_2,\cdots,t_m), \ (t_1,t_2,\cdots,t_m)\in {\mathbb R}^m $ $(m\geq 2)$ which is periodic in $t_1,t_2,\cdots,t_m$ with periods $\tau_1,\tau_2,\cdots,\tau_m$ respectively, where $\tau_1,\tau_2,\cdots,\tau_m$ are strictly positive and their reciprocals are rationally linearly independent i.e. for any nonzero integer-valued vector $k=(k_1,k_2,\cdots,k_m)$, $$k_1\frac{1}{\tau_1}+k_2\frac{1}{\tau_2}+\cdots+k_m\frac{1}{\tau_m}\neq 0.$$ This topic has been subject to many important studies including Kolmogorov-Arnold-Moser (KAM) theory on Hamiltonian systems ([@Kolmogorov1954],[@Moser1962],[@Arnold1963]).
Quasi-periodic motion is a common phenomenon in nature, e.g. arising in describing the movement of planets around the sun. The existence of a quasi-periodic motion for the nearly integrable regimes of the three-body problem with some transversality condition is given by the KAM theory. However many problems in nature are mixture of randomness and quasi-periodic motions. For example the temperature process which is random has one year periodicity due to the revolution of the earth around the sun and one day-night periodicity due to the rotation of the earth. Similarly, the energy demands should have similar nature. Thus to provide a rigorous mathematical theory is key in modelling random quasi-periodic phenomena in real world. As far as we know, such a concept still does not exist and the current paper is the first attempt in this direction.
The concepts of random periodic paths and periodic measures were introduced recently ([@Zhao-Zheng2009],[@Feng-Zhao-Zhou2011],[@Feng-Zhao2012],[@Feng-Wu-Zhao2016],[@Feng-Zhao2018]). They are two different indispensable ways to describe random periodicity. The theory has led to progress in the study of bifurcations ([@Wang2014]), random attractors ([@Bates-Lu-Wang2014]), stochastic resonance ([@Cherubini-Lamb-Rasmussen-Sato2017]), strange attractors ([@Huang-Lian2016]) and modelling the El Nîno phenomenon ([@Chekroun-Simonnet-Ghil2011]).
In this paper, we study random quasi-periodicity of random dynamical systems or semi-flows over a metric dynamical system $(\Omega,\mathcal{F},P, (\theta_t)_{t\in\mathbb{R}})$. First we define random quasi-periodic path $\varphi$ of the stochastic-flows $u(t,s): \Omega\times\mathbb{R}^d\rightarrow \mathbb{R}^d, t\geq s$ as a random path satisfying $$u(t,s,\varphi(s,\omega),\omega)=\varphi(t,\omega), t\geq s,s\in \mathbb{R} \text{ a.s.},$$ and the pull-back random path $$t\longmapsto \varphi(t,\theta_{-t}\omega)$$ is a quasi-periodic function for almost every sample path $\omega\in\Omega$.
For a Markovian semi-flow, let $p(t,s,x,\cdot), t\geq s,$ be its transition probability. Then a measure-valued function $\rho: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{R}^d)$ is called a quasi-periodic measure if $\rho$ is an entrance measure i.e. $$\int_{\mathbb{R}^d}P(t,s,x,\Gamma)\rho_s(dx)=\rho_t(\Gamma)\ \ {\rm \ for \ all}\ \Gamma \in {\cal B}({\mathbb R}^d),$$ and the measure-valued map $$s\longmapsto \rho_s$$ is a quasi-periodic function.
We will give a sufficient condition for the existence and uniqueness of random quasi-periodic path for a stochastic differential equation on $\mathbb{R}^d$ $$\label{SDE}
\begin{cases}
dX(t)=b(t, X(t))dt+\sigma(t, X(t))dW_t, \quad t\geq s,\\
X(s)=\xi,
\end{cases}$$ where $b,\sigma$ are quasi-periodic in the time variable t. As this is the first paper in this area, the main purpose here is to establish basic mathematical concepts and useful tools. We do not strike to technical details to try to provide best possible sufficient conditions in the current paper.
We will prove the law of random quasi-periodic path is a quasi-periodic measure. We further give a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation.
For simplicity, we only consider quasi-periodicity with two periods: $\tau_1$ and $\tau_2$ in the current paper. Our results also apply to general cases with any periods $\tau_1,\tau_2,\cdots,\tau_m$ without any extra difficulties.
Solving the reparameterised SDE is a key step in the analysis of finding random quasi-periodic paths. Let $\tilde{b}, \tilde{\sigma}$ be two functions such that $$\tilde{b}(t,t,x)=b(t,x), \tilde{\sigma}(t,t,x)=\sigma(t,x)$$ where $\tilde{b}(t_1,t_2,x), \tilde{\sigma}(t_1,t_2,x)$ are periodic in $t_1,t_2$ with periods $\tau_1$ and $\tau_2$ respectively. Define $$\tilde{b}^{r_1,r_2}(t,x)=\tilde{b}(t+r_1,t+r_2,x)$$ $$\tilde{\sigma}^{r_1,r_2}(t,x)=\tilde{\sigma}(t+r_1,t+r_2,x),$$ then the solution $K^{r_1,r_2}$ of SDE (\[SDE\]) when $b,\sigma$ are replaced by $\tilde{b}^{r_1,r_2}, \tilde{\sigma}^{r_1,r_2}$, where $r_1,r_2$ are regarded as parameters, satisfies $$K^{r,r}(t,s,x,\omega)=u(t+r,s+r,x,\theta_{-r}\omega)$$ where $u(t,s,\cdot,\omega)$ is the semi-flow generated by (\[SDE\]). Moreover we can prove under a dissipative condition about the drifts $b$ and $\tilde{b}^{r_1,r_2}$, $$\lim_{s\rightarrow -\infty}K^{r_1,r_2}(t,s,x,\omega)={\varphi}^{r_1,r_2}(t,\omega) \text{ exists a.s.}$$ and $$\varphi(r,\omega)={\varphi}^{r,r}(0,\theta_{-r}\omega)$$ is a random quasi-periodic path of (\[SDE\]).
Note the re-paramerterised SDE enjoys the following property: for all $r_1,r_2,r\in \mathbb{R}$, $t\geq s$, $$\begin{aligned}
\label{2019aug3}
K^{r_1,r_2}(t+r,s+r,x,\theta_{-r}\omega)=K^{r_1+r,r_2+r}(t,s,x,\omega), \ P-a.s. \text{ on }\omega.
\end{aligned}$$ This is a very useful observation in our analysis, but the original time dependent SDE (\[SDE\]) does not have such a convenient relation.
Lifting the semi-flow to $\tilde{\mathbb X}=[0, \tau_1) \times [0, \tau_2)\times\mathbb{R}^d$ is key to obtain an invariant measure from the quasi-periodic measure. Define $$\tilde{\Phi}(t,\omega)(s_1,s_2,x)=(t+s_1 \mod\tau_1,\ t+s_2 \mod\tau_2,\ K^{s_1,s_2}(t,0,x,\omega))$$ and $$\tilde{Y}(s,\omega)=(s\mod \tau_1,\ s\mod\tau_2,\ \varphi(s,\omega)).$$ Then $\tilde{Y}$ is a random quasi-periodic path of the cocycle $\tilde{\Phi}$. Moreover we will prove that $\tilde{P}(t,(s_1,s_2,x), \tilde{\Gamma})=P\{\omega: \tilde{\Phi}(t,\omega)(s_1,s_2,x)\in \tilde{\Gamma}\}, \tilde{\Gamma}\in\mathcal{B}(\tilde{\mathbb X})$ is Feller and $$\tilde{\mu}_s(\tilde{\Gamma})=P\{\omega:\tilde{Y}(s,\omega)\in\tilde{\Gamma}\}=[\delta_{s\mod \tau_1}\times \delta_{s\mod \tau_2}\times \rho_s](\tilde{\Gamma})$$ is a quasi-periodic measure with respect to $\tilde{P}$. We will show that $$\{\bar{\tilde{\mu}}_T=\frac{1}{T}\int_{0}^{T}\tilde{\mu}_sds: T\in \mathbb{R}^+\}$$ is tight and a weak limit $\bar{\tilde{\mu}}$ is an invariant measure with respect to $\tilde{P}^*$. Moreover, we will further show the invariant measure is unique and is given by the average $$\frac{1}{\tau_1\tau_2}\int_0^{\tau_1}\int_0^{\tau_2}\delta_{s_1}\times\delta_{s_2}\times\tilde{\rho}_{s_1,s_2}ds_1ds_2.$$
Random path and entrance measure
================================
Existence and uniqueness of random path
---------------------------------------
In the stochastic differential equation (\[SDE\]), $b: \mathbb{R}\times \mathbb{R}^d\rightarrow \mathbb{R}^d, \ \sigma: \mathbb{R}\times \mathbb{R}^d\rightarrow \mathbb{R}^{d\times d}$ are continuous functions, $W_t$ is a two-sided $\mathbb{R}^d$-valued Brownian motion on probability space $(\Omega,\mathcal{F},P)$ with $W_0=0$ and $W_t-W_s$ being a Gussian distribution $\mathcal{N}(0, \Sigma(t-s))$, $\Sigma \in \mathcal{S}_d$ is a symmetric nondegenerate matrix, $\xi$ is a $\mathbb{R}^d$-valued $\mathcal{F}_{-\infty}^s$-measurable random variable, where $\mathcal{F}_a^b$ is the natural filtration generated by $(W_u-W_v)_{a\leq u,v\leq b}$. Now we consider the following assumptions.
\[Dissipative\] There exists a constant $\alpha>0$ such that for all $t\in \mathbb{R}, x,y\in \mathbb{R}^{d}$ $$((x-y), (b(t,x)-b(t,y)))\leq -\alpha (x-y)^2$$
\[Lipschitz and bounded\] The diffusion matrix $\sigma(t, x)$ is continuous with respect to t and Lipschitz continuous with Lipschitz constant $\beta$, i.e. for all $t\in \mathbb{R}, x,y\in \mathbb{R}^{d}$ $$\|\sigma(t, x)-\sigma(t, y)\|\leq \beta|x-y|,$$ and there exists $M>0$ such that $$\sup_t|b(t, 0)|+\sup_t\|\sigma(t, 0)\|\leq M.$$ Here $\|\sigma\|=(Tr(\sigma\cdot\sigma^T))^{\frac{1}{2}}$.
Under Condition \[Dissipative\] and Condition \[Lipschitz and bounded\], the solution of (\[SDE\]) exists, denoted by $X(t,s,\xi)$ satisfying for $P-a.e.$ $\omega\in\Omega$ $$X(t,s,\xi(\omega),\omega)=X(t,r,\omega)\circ X(r,s,\xi(\omega), \omega), \text{ for all } s\leq r\leq t.$$ We call $u:\Delta \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}^d$ with $u(t,s,\omega)x=X(t,s,x,\omega)$ a stochastic semi-flow, where $\Delta=\{(t,s): t\geq s, t,s\in\mathbb{R}\}$.
A random path of a semi-flow $u:\Delta \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}^d$ is a measurable map $\varphi: \mathbb{R}\times \Omega \rightarrow \mathbb{R}^d$ such that for P-a.e. $\omega\in \Omega$ $$\label{Random Path}
u(t+s, s, \varphi(s, \omega), \omega)=\varphi(t+s, \omega), \text{ for all } t\geq 0, s\in\mathbb{R}.$$ In addition, if $u$ is generated by an SDE, we say $\varphi$ is a random path of this SDE.
In the following, we will always use $\|\cdot\|_2$ to denote the norm in the $L^2(\Omega, dP)$ space.
\[Existence and uniqueness of random path\] Assume Conditions \[Dissipative\], \[Lipschitz and bounded\] and $\alpha >\frac{\beta^2}{2}$. Then there exists a unique uniformly $L^2$-bounded random path $\varphi$ of SDE (\[SDE\]), i.e. $\sup_{t\in \mathbb{R}}\|\varphi(t)\|_2<\infty$.
First we give some lemmas before we prove Theorem \[Existence and uniqueness of random path\].
\[Bounded solution\] Assume Conditions \[Dissipative\], \[Lipschitz and bounded\] and $\alpha >\frac{\beta^2}{2}$. Let $X_t^{s,\xi}$ be the solution of SDE (\[SDE\]) with initial condition $(s, \xi)$, where $\xi \in L^2(\Omega)$. Then there exists a constant $C=C(\alpha, \beta, M)$ such that for all $t\geq s$, $\|X_t^{s, \xi}\|_2^2\leq C(1+\|\xi\|_2^2)$.
For any fixed $\lambda$, applying It$\hat{\rm o}$’s formula to $e^{2\lambda t}|X_t^{s, \xi}|^2$, we have $$\begin{aligned}
e^{2\lambda t}|X_t^{s, \xi}|^2&=&e^{2\lambda s}|\xi|^2+\int_{s}^{t}\left(2\lambda e^{2\lambda r}|X_r^{s, \xi}|^2+2e^{2\lambda r}X_r^{s, \xi}\cdot b(r, X_r^{s, \xi})+e^{2\lambda r}\|\sigma(r, X_r^{s, \xi})\|^2\right)dr\\
&&+\int_{s}^{t}2e^{2\lambda r}X_r^{s, \xi}\sigma(r, X_r^{s, \xi})dW_r.
\end{aligned}$$ In Condition \[Dissipative\], let $y=0$. Then for arbitrary $\epsilon >0$, by Young inequality and Condition \[Lipschitz and bounded\] $$\begin{aligned}
x\cdot b(t, x) &\leq -\alpha|x|^2+x\cdot b(t, 0)\\
&\leq -(\alpha-\epsilon)|x|^2+\frac{M^2}{4\epsilon},
\end{aligned}$$ and $$\begin{split}
\|\sigma(t, x)\|^2 &\leq (\|\sigma(t,x)-\sigma(t,0)\|+\|\sigma(t,0)\|)^2\\
&\leq (\beta|x|+\|\sigma(t,0)\|)^2\\
&\leq (\beta^2+\epsilon)|x|^2+(\frac{\beta^2}{\epsilon}+1)M^2.
\end{split}$$ We choose $\epsilon$ small enough such that $\alpha>\frac{\beta^2}{2}+2\epsilon$. Let $\lambda=\alpha-\epsilon$. Thus there exists a constant $C(\alpha,\beta,M)$ depending on $\alpha,\beta,M$ such that $$\begin{split}
e^{2(\alpha-\epsilon) t}|X_t^{s, \xi}|^2\leq
&e^{2(\alpha-\epsilon) s}|\xi|^2+ \int_{s}^{t}\left(e^{2(\alpha-\epsilon)r}(\frac{1}{2\epsilon}+\frac{\beta^2}{\epsilon}+1)M^2+e^{2(\alpha-\epsilon) r} (\beta^2+\epsilon)|X_r^{s, \xi}|^2\right)dr\\
&+\int_{s}^{t}2e^{2(\alpha-\epsilon) r}X_r^{s, \xi}\sigma(r, X_r^{s, \xi})dW_r\\
\leq &e^{2(\alpha-\epsilon) s}|\xi|^2+C(\alpha, \beta, M)e^{2(\alpha-\epsilon)t} +(\beta^2+\epsilon)\int_{s}^{t}e^{2(\alpha-\epsilon)r}|X_r^{s, \xi}|^2dr\\
&+\int_{s}^{t}2e^{2(\alpha-\epsilon) r}X_r^{s, \xi}\sigma(r, X_r^{s, \xi})dW_r.
\end{split}$$ Taking expectation of both sides, we have $$e^{2(\alpha-\epsilon)t}\mathbb{E}|X_t^{s,\xi}|^2\leq e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2 +C(\alpha, \beta, M)e^{2(\alpha-\epsilon)t}+ (\beta^2+\epsilon)\int_{s}^{t}e^{2(\alpha-\epsilon)r}\mathbb{E}|X_r^{s,\xi}|^2dr.$$ By Gronwall inequality $$\begin{split}
e^{2(\alpha-\epsilon) t}\mathbb{E}|X_t^{s,\xi}|^2\leq &e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2+C(\alpha, \beta, M)e^{2(\alpha-\epsilon)t}\\
&\int_{s}^{t}\left(e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2+C(\alpha, \beta, M)e^{2(\alpha-\epsilon)r}\right)(\beta^2+\epsilon)\exp\left(\int_{r}^{t}(\beta^2+\epsilon)d\gamma\right)dr\\
=&e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2+C(\alpha, \beta, M)e^{2(\alpha-\epsilon)t} +\int_{s}^{t}e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2(\beta^2+\epsilon)e^{(\beta^2+\epsilon)(t-r)}dr\\
&+\int_{s}^{t}C(\alpha, \beta, M)e^{2(\alpha-\epsilon)r}(\beta^2+\epsilon)e^{(\beta^2+\epsilon)(t-r)}dr\\
=&e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2+C(\alpha, \beta, M)e^{2(\alpha-\epsilon)t} +e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2\left(e^{(\beta^2+\epsilon)(t-s)}-1\right)\\
&+C(\alpha, \beta, M)e^{(\beta^2+\epsilon)t}\frac{\beta^2+\epsilon}{2\alpha-\beta^2-3\epsilon} \left(e^{(2\alpha-\beta^2-3\epsilon)t}-e^{(2\alpha-\beta^2-3\epsilon)s}\right)\\
\leq &e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2+C(\alpha, \beta, M)e^{2(\alpha-\epsilon)t} +e^{2(\alpha-\epsilon)s}\mathbb{E}|\xi|^2\left(e^{(\beta^2+\epsilon)(t-s)}\right).
\end{split}$$ Here and in the following, $C(\alpha,\beta,M)$ is constant, which may be different from line to line. Then $$\begin{split}
\mathbb{E}|X_t^{s,\xi}|^2
&\leq e^{-2(\alpha-\epsilon)(t-s)}\mathbb{E}|\xi|^2+C(\alpha, \beta, M) +e^{-2(2\alpha-\beta^2-3\epsilon)(t-s)}\mathbb{E}|\xi|^2\\
&\leq C(\alpha, \beta, M)(1+\mathbb{E}|\xi|^2).
\end{split}$$
\[Exponential decay\] Assume Conditions \[Dissipative\], \[Lipschitz and bounded\] and $\alpha>\frac{\beta^2}{2}$. Let $X_t^{s,\xi}$ and $X_t^{s,\eta}$ be two solutions of SDE (\[SDE\]) with initial values $\xi$ and $\eta$ respectively, where $\xi, \eta \in L^2(\Omega)$. Then $$\|X_t^{s,\xi}-X_t^{s,\eta}\|_2\leq e^{-(\alpha-\beta^2/2)(t-s)}\|\xi-\eta\|_2.$$
Note $$X_t^{s,\xi}-X_t^{s,\eta}=\xi-\eta+\int_{s}^{t}\left(b(r,X_r^{s,\xi})-b(r,X_r^{s,\eta})\right)dr +\int_{s}^{t}\left(\sigma(r,X_r^{s,\xi})-\sigma(r,X_r^{s,\eta})\right)dW_r.$$ Applying It$\hat{\rm o}$’s formula to $e^{2\alpha t}|X_t^{s,\xi}-X_t^{s,\eta}|^2$, by the dissipative Condition \[Dissipative\], we have $$\begin{split}
&e^{2\alpha t}|X_t^{s,\xi}-X_t^{s,\eta}|^2\\
=&e^{2\alpha s}|\xi-\eta|^2
+\int_{s}^{t}\left[2\alpha e^{2\alpha r}|X_r^{s,\xi}-X_r^{s,\eta}|^2+2e^{2\alpha r}(X_r^{s,\xi}-X_r^{s,\eta})\cdot\left(b(r,X_r^{s,\xi})-b(r,X_r^{s,\eta})\right) \right]dr\\
&+\int_{s}^{t}e^{2\alpha r}\left(\sigma(r,X_r^{s,\xi})-\sigma(r,X_r^{s,\eta})\right)^2dr+\int_{s}^{t}e^{2\alpha r}(X_r^{s,\xi}-X_r^{s,\eta})\left(\sigma(r,X_r^{s,\xi})-\sigma(r,X_r^{s,\eta})\right)dW_r\\
\leq &e^{2\alpha s}|\xi-\eta|^2+\int_{s}^{t}\beta^2 e^{2\alpha r}|X_r^{s,\xi}-X_r^{s,\eta}|^2dr\\ &+\int_{s}^{t}e^{2\alpha r}(X_r^{s,\xi}-X_r^{s,\eta})\left(\sigma(r,X_r^{s,\xi})-\sigma(r,X_r^{s,\eta})\right)dW_r.
\end{split}$$ Taking expectation on both sides, we have $$e^{2\alpha t}\|X_t^{s,\xi}-X_t^{s,\eta}\|_2^2\leq e^{2\alpha s}\|\xi-\eta\|_2^2+\int_{s}^{t}\beta^2 e^{2\alpha r}\|X_r^{s,\xi}-X_r^{s,\eta}\|_2^2dr.$$ By Gronwall inequality, we have $$e^{2\alpha t}\|X_t^{s,\xi}-X_t^{s,\eta}\|_2^2\leq e^{2\alpha s}\|\xi-\eta\|_2^2 e^{\beta^2(t-s)},$$ thus the lemma follows.
Now we give the proof of Theorem \[Existence and uniqueness of random path\]
Existence: For any fixed $\xi \in L^2(\Omega)$. Let $s_1<s_2<t$, then $$X_t^{s_1,\xi}=X_t^{s_2, X_{s_2}^{s_1, \xi}}.$$ Now consider $\|X_t^{s_1, \xi}-X_t^{s_2, \xi}\|_2$. Applying Lemma \[Bounded solution\] and Lemma \[Exponential decay\], we have $$\begin{split}
\|X_t^{s_1, \xi}-X_t^{s_2, \xi}\|_2=&\|X_t^{s_2, X_{s_2}^{s_1, \xi}}-X_t^{s_2, \xi}\|_2\\
\leq &e^{-(\alpha-\beta^2/2)(t-s_2)}\|X_{s_2}^{s_1, \xi}-\xi\|_2\\
\leq &e^{-(\alpha-\beta^2/2)(t-s_2)}\left(\|X_{s_2}^{s_1, \xi}\|_2+\|\xi\|_2\right)\\
\leq &C(\alpha, \beta, \xi)e^{-(\alpha-\beta^2/2)(t-s_2)}.
\end{split}$$ Thus there exists a $L^2$-limit of $\left(X_t^{s,\xi}\right)_{s\leq t}$ as $s\rightarrow -\infty$. By Lemma \[Exponential decay\], we know that this limit is independent of $\xi$. Define $$\varphi(t):=L^2-\lim_{s\rightarrow -\infty}X_t^{s, \xi},$$ then $$\|\varphi(t)\|_2\leq \limsup_{s\rightarrow -\infty}\|X_t^{s, \xi}\|\leq C(\alpha, \beta, M, \xi)\leq C(\alpha,\beta,M).$$ Next we will prove that $\varphi$ is a random path of SDE (\[SDE\]). For any $t\geq s\geq r$, we have $$u(t, s, X_s^{r,\xi})=X_t^{r,\xi}.$$ By Lemma \[Exponential decay\], we know that $\|u(t, s, X_s^{r,\xi})-u(t, s, \varphi(s))\|\leq e^{-(\alpha-\beta^2/2)(t-s)}\|X_s^{r,\xi}-\varphi(s)\|$. It follows that $$L^2-\lim_{r\rightarrow -\infty}u(t, s, X_s^{r,\xi})=u(t, s, \varphi(s))=\varphi(t)=L^2-\lim_{r\rightarrow -\infty}X_t^{r,\xi},$$ i.e. $$\label{Equation random path of semi-flow u}
u(t, s, \varphi(s, \omega),\omega)=\varphi(t, \omega), \quad P-a.s.$$ Thus $\varphi$ is a $L^2$-bounded random path of SDE (\[SDE\]).
Uniqueness: If there are two uniformly $L^2$-bounded random path $\varphi_1, \varphi_2$ of SDE (\[SDE\]), by Lemma \[Exponential decay\], we have $$\begin{split}
\|\varphi_1(t)-\varphi_2(t)\|_2\leq& e^{-(\alpha-\beta^2/2)(t-s)}\|\varphi_1(s)-\varphi_2(s)\|_2\\
\leq &e^{-(\alpha-\beta^2/2)(t-s)}(\sup_{r\in \mathbb{R}}\|\varphi_1(r)\|_2+\sup_{r\in \mathbb{R}}\|\varphi_2(r)\|_2) \rightarrow 0 \text{ as } s\rightarrow -\infty.
\end{split}$$ Then $\varphi_1(t)=\varphi_2(t),$ for all $t\in \mathbb{R}$, $P-a.e.$.
It is worth noticing that in the part of (\[Equation random path of semi-flow u\]), the pathwise continuity of $u(t,s,\cdot): \mathbb{R}^d\rightarrow \mathbb{R}^d$ was not used.
Existence and uniqueness of entrance measure
--------------------------------------------
For a semi-flow $u: \triangle\times \mathbb{R}^d\times \Omega\rightarrow \mathbb{R}^d$ with $u(t,s,x,\omega)=X_t^{s,x}(\omega)$, we define the transition $P: \triangle\times \mathbb{R}^d\times \mathcal{B}(\mathbb{R}^d)\rightarrow \mathbb{R}^+$ by $P(t,s,x,\Gamma)=P(X_t^{s,x}\in \Gamma)$ for all $t\geq s$, $x\in\mathbb{R}^d$ and $\Gamma\in\mathcal{B}(\mathbb{R}^d)$. We further define $P^*(t,s): \mathcal{P}(\mathbb{R}^d)\rightarrow\mathcal{P}(\mathbb{R}^d)$ by $$\label{Define measure transition P^*}
P^*(t,s)\mu(\Gamma)=\int_{\mathbb{R}^d}P(t,s,x,\Gamma)\mu(dx), \text{ for all } \mu\in\mathcal{P}(\mathbb{R}^d),\Gamma\in\mathcal{B}(\mathbb{R}^d).$$ Here $$\mathcal{P}(\mathbb{R}^d):=\{\text{all probability measures on } (\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))\}.$$
We say a measure-valued map $\mu: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{R}^d)$ is an entrance measure of SDE(\[SDE\]) if $P^*(t,s)\mu_s=\mu_t$ for all $t\geq s, s\in \mathbb{R}$.
Set $$\mathcal{M}^2:=\{\mu:\mathbb{R}\rightarrow \mathcal{P}(\mathbb{R}^d)| \sup_{t\in \mathbb{R}}\int_{\mathbb{R}^d}x^2\mu_t(dx)<\infty\}.$$
\[Existence and uniqueness of entrance measure\] Assume Conditions \[Dissipative\], \[Lipschitz and bounded\] and $\alpha>\frac{\beta^2}{2}$. Then there exists a unique entrance measure of SDE (\[SDE\]) in $\mathcal{M}^2$.
Before we prove Theorem \[Existence and uniqueness of entrance measure\], we need the following lemma.
\[Two measure are the same\] Assume $\mu_1$ and $\mu_2$ are two probability measures on $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$, and for any open set $\mathcal{O}$ we have $\mu_1(\mathcal{O})\leq \mu_2(\mathcal{O})$. Then $\mu_1=\mu_2$.
Let $\mathcal{C}:=\{\text{all open sets on }\mathbb{R}^d\}$. We know that $\mu_1\leq \mu_2$ on $\mathcal{C}$.
For any given $\mathcal{O}\in \mathcal{C}$, $\mathcal{O}^c=\mathbb{R}^d\setminus \mathcal{O}$ is a closed set. Define $$\mathcal{O}^c_{\delta}:=\{x: dist(x, \mathcal{O}^c)<\delta\},$$ where $dist(x, \mathcal{O}^c)=\inf_{y\in \mathcal{O}^c}|x-y|$. Then we know that $\mathcal{O}^c_{\delta}$ is open set and $\mathcal{O}^c_{\delta}\downarrow \mathcal{O}^c$ as $\delta\downarrow 0$. Further more $$\mu_1(\mathcal{O}^c)=\lim_{\delta\downarrow 0}\mu_1(\mathcal{O}^c_{\delta})\leq \lim_{\delta\downarrow 0}\mu_2(\mathcal{O}^c_{\delta})=\mu_2(\mathcal{O}^c).$$ Since $\mu_1$ and $\mu_2$ are probability measures, we have $$1-\mu_1(\mathcal{O})\leq 1-\mu_2(\mathcal{O}),$$ which implies $\mu_1(\mathcal{O})\geq \mu_2(\mathcal{O})$. Hence $\mu_1\geq \mu_2$ on $\mathcal{C}$. This leads to $\mu_1=\mu_2$ on $\mathcal{C}$.
Since $\mathcal{C}$ is a $\pi$-system and $\sigma(\mathcal{C})=\mathcal{B}(\mathbb{R}^d)$, thus $\mu_1=\mu_2$ on $\mathcal{B}(\mathbb{R}^d)$.
Now we give the proof of Theorem \[Existence and uniqueness of entrance measure\].
Existence: Applying Theorem \[Existence and uniqueness of random path\], we know that there exists a uniformly $L^2$-bounded random path $\varphi$ of SDE (\[SDE\]). Let $\rho_t=\mathcal{L}(\varphi(t))$ be the law of $\varphi(t)$. Since $\varphi$ is the random path of SDE (\[SDE\]), then for any $\Gamma\in\mathcal{B}(\mathbb{R}^d)$, we have $$\label{Law of random path adapted}
\begin{split}
P^*(t,s)\rho_s(\Gamma)&=\int_{\mathbb{R}^d}P(t,s,x,\Gamma)\rho_s(dx)\\
&=\int_{\mathbb{R}^d}P(X_t^{s,x}\in\Gamma)P(\varphi(s)\in dx)\\
&=P(X_t^{s,\varphi(s)}\in\Gamma)\\
&=P(\varphi(t)\in\Gamma)\\
&=\rho_t(\Gamma).
\end{split}$$ Thus $\rho$ is an entrance measure of SDE (\[SDE\]). And since $\varphi$ is uniformly $L^2$-bounded, then $$\sup_{t\in \mathbb{R}}\int_{\mathbb{R}^d}x^2\rho_t(dx)=\sup_{t\in \mathbb{R}}\mathbb{E}|\varphi(t)|^2<\infty,$$ which means $\rho\in \mathcal{M}^2$.
Uniqueness: We aim to prove that for any entrance measure $\mu$ of SDE (\[SDE\]) in $\mathcal{M}^2$, $\mu_t=\rho_t$ for all $t\in \mathbb{R}$. By Lemma \[Two measure are the same\], we just need to prove $\rho_t(\mathcal{O})\leq \mu_t(\mathcal{O})$ for any open set $\mathcal{O}\subset \mathbb{R}^d$. Since for any $s<t$, we have $$\begin{split}
\rho_t(\mathcal{O})-\mu_t(\mathcal{O})
&=\rho_t(\mathcal{O})-\int_{\mathbb{R}^d}P(t,s,x,\mathcal{O})\mu_s(dx)\\
&=\int_{\mathbb{R}^d}\left(\rho_t(\mathcal{O})-P(X_t^{s,x}\in\mathcal{O})\right)\mu_s(dx)\\
&=\int_{\mathbb{R}^d}\left(P(\varphi(t)\in \mathcal{O})-P(X_t^{s,x}\in\mathcal{O})\right)\mu_s(dx).
\end{split}$$ Define $$\mathcal{O}_{\delta}:=\{x: dist(x, \mathcal{O}^c)>\delta\}.$$ Then $\mathcal{O}_{\delta}\uparrow \mathcal{O}$ as $\delta\downarrow 0$ and $$\begin{split}
P(X_t^{s,x}\in \mathcal{O})
&=P(X_t^{s,x}-\varphi(t)+\varphi(t)\in \mathcal{O})\\
&\geq P(\varphi(t)\in \mathcal{O}_{\delta}, |X_t^{s,x}-\varphi(t)|<\delta)\\
&\geq P(\varphi(t)\in \mathcal{O}_{\delta})-P(|X_t^{s,x}-\varphi(t)|\geq \delta).
\end{split}$$ Thus it turns out from the above and the Chebyshev inequality that $$\begin{split}
&P(\varphi(t)\in \mathcal{O})-P(X_t^{s,x}\in \mathcal{O})\\
&\leq P(\varphi(t)\in \mathcal{O}\setminus\mathcal{O}_{\delta})+P(|X_t^{s,x}-\varphi(t)|\geq\delta)\\
&\leq P(\varphi(t)\in \mathcal{O}\setminus\mathcal{O}_{\delta})+\frac{1}{\delta^2}\mathbb{E}|X_t^{s,x}-\varphi(t)|^2.
\end{split}$$ Applying Lemma \[Bounded solution\] and Lemma \[Exponential decay\], we have $$\begin{split}
\mathbb{E}|X_t^{s,x}-\varphi(t)|^2
&=\lim_{r\rightarrow -\infty}\mathbb{E}|X_t^{s,x}-X_t^{r,x}|^2\\
&\leq \limsup_{r\rightarrow -\infty, r<s}\mathbb{E}|X_t^{s,x}-X_t^{s,X_s^{r,x}}|^2\\
&\leq \limsup_{r\rightarrow -\infty, r<s}e^{-2(\alpha-\beta^2/2)(t-s)}\mathbb{E}|x-X_s^{r,x}|^2\\
&\leq \limsup_{r\rightarrow -\infty, r<s}C(1+x^2)e^{-2(\alpha-\beta^2/2)(t-s)}\\
&= C(1+x^2)e^{-2(\alpha-\beta^2/2)(t-s)}.
\end{split}$$ Here $C=C(\alpha,\beta, M)$. Then for any $\delta>0$ and $s<t$, we have $$\begin{split}
\rho_t(\mathcal{O})-\mu_t(\mathcal{O})
&=\int_{\mathbb{R}^d}\left(P(\varphi(t)\in \mathcal{O})-P(X_t^{s,x}\in\mathcal{O})\right)\mu_s(dx)\\
&\leq \int_{\mathbb{R}^d}\left(P(\varphi(t)\in \mathcal{O}\setminus\mathcal{O}_{\delta})+\frac{1}{\delta^2}\mathbb{E}|X_t^{s,x}-\varphi(t)|^2\right)\mu_s(dx)\\
&\leq P(\varphi(t)\in \mathcal{O}\setminus\mathcal{O}_{\delta})+\frac{C}{\delta^2}e^{-2(\alpha-\beta^2/2)(t-s)}\int_{\mathbb{R}^d}(1+x^2)\mu_s(dx).
\end{split}$$ Hence for any $\delta>0$, we have $$\begin{split}
\rho_t(\mathcal{O})-\mu_t(\mathcal{O})
&\leq P(\varphi(t)\in \mathcal{O}\setminus\mathcal{O}_{\delta})+\limsup_{s\rightarrow -\infty}\frac{C}{\delta^2}\left(1+\sup_{r\in \mathbb{R}}\int_{\mathbb{R}^d}x^2\mu_r(dx)\right)e^{-2(\alpha-\beta^2/2)(t-s)}\\
&\leq P(\varphi(t)\in \mathcal{O}\setminus\mathcal{O}_{\delta})=\rho_t(\mathcal{O}\setminus\mathcal{O}_{\delta}).
\end{split}$$ Since $\mathcal{O}_{\delta}\uparrow \mathcal{O}$ as $\delta\downarrow 0$, we have $$\rho_t(\mathcal{O})-\mu_t(\mathcal{O})\leq \lim_{\delta\downarrow 0}\rho_t(\mathcal{O}\setminus\mathcal{O}_{\delta})=0,$$ which implies $\rho_t(\mathcal{O})\leq \mu_t(\mathcal{O})$.
By the proof of Theorem \[Existence and uniqueness of random path\], we know that $\varphi(t)=L^2-\lim_{s\rightarrow -\infty}X_t^{s,x}$. Then we have the following proposition. We use $C_b(\mathbb{R}^d)$ to be the linear space of all continuous and bounded functions on $\mathbb{R}^d$.
\[Weakly convergence of P(t,s)\] The entrance measure $\rho_t$ is the limit of $P(t,s,x,\cdot)$ in $\mathcal{P}(\mathbb{R}^d)$ with weak topology, i.e. for all $f\in C_b(\mathbb{R}^d)$, we have $$\lim_{s\rightarrow -\infty}\int_{\mathbb{R}^d}f(y)P(t,s,x,dy)=\int_{\mathbb{R}^d}f(y)\rho_t(dy).$$
Since $\int_{\mathbb{R}^d}f(y)P(t,s,x,dy)=\mathbb{E}f(X_t^{s,x})$ and $\int_{\mathbb{R}^d}f(y)\rho_t(dy)=\mathbb{E}f(\varphi(t))$, we need to prove that for all $f\in C_b(\mathbb{R}^d)$, $$\lim_{s\rightarrow -\infty}\mathbb{E}f(X_t^{s,x})=\mathbb{E}f(\varphi(t)).$$ First we prove $\limsup_{s\rightarrow -\infty}\mathbb{E}f(X_t^{s,x})\leq\mathbb{E}f(\varphi(t))$. Otherwise there exists a sequence $s_n\downarrow -\infty$ as $n \rightarrow \infty$ and a constant $\lambda=\limsup_{s\rightarrow -\infty}\mathbb{E}f(X_t^{s,x})>\mathbb{E}f(\varphi(t))$ such that $\lim_{n\rightarrow \infty}\mathbb{E}f(X_t^{s_n,x})=\lambda$. Since $\lim_{n\rightarrow \infty}\mathbb{E}[|X_t^{s_n,x}-\varphi(t)|^2]=0$, we know that there exists a subsequence $\{s_{n_k}\}\subseteq \{s_n\}$ such that $X_t^{s_{n_k},x}\xrightarrow{a.s.} \varphi(t)$ as $k \rightarrow \infty$. Thus $f(X_t^{s_{n_k},x})\xrightarrow{a.s.} f(\varphi(t))$. Then by Lebesgue’s dominated convergence theorem, we have $$\lim_{k\rightarrow \infty}\mathbb{E}f(X_t^{s_{n_k},x})=\mathbb{E}f(\varphi(t)),$$ which contradicts that $$\lim_{k\rightarrow \infty}\mathbb{E}f(X_t^{s_{n_k},x})=\lambda>\mathbb{E}f(\varphi(t)).$$ Hence $$\limsup_{s\rightarrow -\infty}\mathbb{E}f(X_t^{s,x})\leq\mathbb{E}f(\varphi(t)).$$ Similarly we can also prove that $$\liminf_{s\rightarrow -\infty}\mathbb{E}f(X_t^{s,x})\geq\mathbb{E}f(\varphi(t)),$$ which completes our proof.
Random quasi-periodic path, quasi-periodic measure and invarant measure
=======================================================================
Existence and uniqueness of random quasi-periodic path {#Section of Quasi-periodic path}
------------------------------------------------------
In SDE (\[SDE\]), if we assume the coefficients $b, \sigma$ are quasi-periodic functions in time $t$, can we obtain a kind of random quasi-periodic path? What should the “quasi-periodicity" of a random path be defined? We give the following definition.
\[Quasi-periodic random path\] A measurable path $\varphi: \mathbb{R}\times\Omega\rightarrow \mathbb{R}^d$ is called random quasi-periodic path of periods $\tau_1, \tau_2$ of a semi-flow $u$, where the reciprocals of $\tau_1$ and $\tau_2$ are rationally linearly independent, if it is a random path, i.e. $u(t, s, \varphi(s, \omega), \omega)=\varphi(t, \omega)$ for $t\geq s$, and there exists $\tilde{\varphi}: \mathbb{R}\times\mathbb{R}\times\Omega\rightarrow\mathbb{R}^d$ such that $\tilde{\varphi}(t,t,\omega)=\varphi(t,\theta_{-t}\omega)$ and $$\tilde{\varphi}(t+\tau_1,s,\omega)=\tilde{\varphi}(t,s,\omega), \
\tilde{\varphi}(t,s+\tau_2,\omega)=\tilde{\varphi}(t,s,\omega).$$ We also say $\varphi$ is a random quasi-periodic path of an SDE if u is generated by this SDE.
We give the quasi-periodic condition.
\[Quasi-periodic condition\] Assume that $b, \sigma$ in SDE (\[SDE\]) are quasi-periodic functions with periods $\tau_1, \tau_2$, where the reciprocals of $\tau_1$ and $\tau_2$ are rationally linearly independent, which means there exists $\tilde{b}:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^d\rightarrow\mathbb{R}^d$ and $\tilde{\sigma}:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^d\rightarrow\mathbb{R}^{d\times n}$ such that $\tilde{b}(t,t,x)=b(t,x)$, $\tilde{\sigma}(t,t,x)=\sigma(t,x)$ for all $t\in\mathbb{R}, x\in\mathbb{R}^d$ satisfying $$\tilde{b}(t+\tau_1,s,x)=\tilde{b}(t,s,x), \
\tilde{b}(t,s+\tau_2,x)=\tilde{b}(t,s,x),$$ and $$\tilde{\sigma}(t+\tau_1,s,x)=\tilde{\sigma}(t,s,x), \
\tilde{\sigma}(t,s+\tau_2,x)=\tilde{\sigma}(t,s,x).$$
\[Quasi-dissipative\] Assume $\tilde{b}, \tilde{\sigma}$ in Condition \[Quasi-periodic condition\] satisfy the following conditions:
(1)
: $(x-y)\left(\tilde{b}(t,s,x)-\tilde{b}(t,s,y)\right)\leq -\alpha(x-y)^2$ for all $x,y\in \mathbb{R}^d$ and $t,s\in \mathbb{R}$;
(2)
: $\|\tilde{\sigma}(t,s,x)-\tilde{\sigma}(t,s,y)\|\leq \beta |x-y|$ for all $x,y\in \mathbb{R}^d$ and $t,s\in \mathbb{R}$;
(3)
: $\sup_{t,s\in \mathbb{R}}|\tilde{b}(t,s,0)|+\sup_{t,s\in \mathbb{R}}\|\tilde{\sigma}(t,s,0)\|<\infty$.
Conditions \[Quasi-periodic condition\], \[Quasi-dissipative\] imply Conditions \[Dissipative\], \[Lipschitz and bounded\]. Now we give the following main theorem.
\[Existence and uniqueness of quasi-periodic random path\] Assume Conditions \[Quasi-periodic condition\], \[Quasi-dissipative\] and $\alpha>\frac{\beta^2}{2}$. Then there exists a unique uniformly $L^2$-bounded random quasi-periodic path of SDE (\[SDE\]).
Uniqueness: Applying Theorem \[Existence and uniqueness of random path\], we know that if there exists a uniformly $L^2$-bounded random quasi-periodic path, it must be the random path $\varphi$ defined in Theorem \[Existence and uniqueness of random path\]. So uniqueness holds.
Existence: The solution of SDE (\[SDE\]) $u(t,s,x,\omega)$ can be written as $$u(t,s,x,\omega)=x+\int_s^t b(r,u(r,s,x,\omega))dr+(\omega)\int_s^t \sigma(r,u(r,s,x,\cdot))dW_r.$$ Then $$\label{Solution u shift time}
\begin{split}
u(t+r,s+r,x,\omega)
=&x+\int_{s+r}^{t+r}b(v,u(v,s+r,x,\omega))dv+(\omega)\int_{s+r}^{t+r}\sigma(v,u(v,s+r,x,\cdot))dW_v\\
=&x+\int_s^tb(v+r,u(v+r,s+r,x,\omega))dv\\
&+(\omega)\int_s^t\sigma(v+r,u(v+r,s+r,x,\cdot))d\tilde{W}^r_v.
\end{split}$$ Here $\tilde{W}^r_v=\theta_rW_v$. Replacing $\omega$ by $\theta_{-r}\omega$ in equation (\[Solution u shift time\]), we have $$\begin{split}
u(t+r,s+r,x,\theta_{-r}\omega)
=&x+\int_s^tb(v+r,u(v+r,s+r,x,\theta_{-r}\omega))dv\\
&+(\theta_{-r}\omega)\int_s^t\sigma(v+r,u(v+r,s+r,x,\cdot))d\tilde{W}^r_v\\
=&x+\int_s^tb(v+r,u(v+r,s+r,x,\theta_{-r}\omega))dv\\
&+(\omega)\int_s^t\sigma(v+r,u(v+r,s+r,x,\theta_{-r}\cdot))dW_v,
\end{split}$$ then $$\label{Solution u shift time 2}
\begin{split}
u(t+r,s+r,x,\theta_{-r}\omega)
=&x+\int_s^t\tilde{b}(v+r,v+r,u(v+r,s+r,x,\theta_{-r}\omega))dv\\
&+(\omega)\int_s^t\tilde{\sigma}(v+r,v+r,u(v+r,s+r,x,\theta_{-r}\cdot))dW_v.
\end{split}$$ Denote $u^r(t,s,x,\omega):=u(t+r,s+r,x,\theta_{-r}\omega)$, $\tilde{b}^{r_1,r_2}(t,x):=\tilde{b}(t+r_1,t+r_2,x)$ and $\tilde{\sigma}^{r_1,r_2}(t,x):=\tilde{\sigma}(t+r_1,t+r_2,x)$, then equation (\[Solution u shift time 2\]) can be written as $$\label{Solution u_r}
\begin{split}
u^r(t,s,x,\omega)
=x+\int_s^t\tilde{b}^{r,r}(v,u^r(v,s,x,\omega))dv+(\omega)\int_s^t\tilde{\sigma}^{r,r}(v,u^r(v,s,x,\cdot))dW_v.
\end{split}$$ Since Condition \[Quasi-dissipative\] holds, then for all $r_1,r_2\in \mathbb{R}$, $\tilde{b}^{r_1,r_2}$ and $\tilde{\sigma}^{r_1,r_2}$ satisfy $$(x-y)\left(\tilde{b}^{r_1,r_2}(t,x)-\tilde{b}^{r_1,r_2}(t,y)\right)\leq -\alpha(x-y)^2,$$ and $$\|\tilde{\sigma}^{r_1,r_2}(t,x)-\tilde{\sigma}^{r_1,r_2}(t,y)\|\leq \beta |x-y|,$$ for all $t\in \mathbb{R}$, $x,y\in\mathbb{R}^d$. Thus the following equation $$\label{Solution K_r_1,r_2}
\begin{split}
K^{r_1,r_2}(t,s,x,\omega)
=x+\int_s^t\tilde{b}^{r_1,r_2}(v,K^{r_1,r_2}(v,s,x,\omega))dv+(\omega)\int_s^t\tilde{\sigma}^{r_1,r_2}(v,K^{r_1,r_2}(v,s,x,\cdot))dW_v,
\end{split}$$ has a unique solution, denoted by $K^{r_1,r_2}(t,s,x,\omega)$. Since $\alpha>\frac{\beta^2}{2}$, similar to the proof of Theorem \[Existence and uniqueness of random path\], we know there exist $\varphi^r(t), \varphi^{r_1,r_2}(t)$ such that $$\label{L2-lim K u}
\begin{cases}
\varphi^r(t)=L^2-\lim_{s\rightarrow -\infty}u^r(t,s,x)\\
\varphi^{r_1,r_2}(t)=L^2-\lim_{s\rightarrow -\infty}K^{r_1,r_2}(t,s,x),
\end{cases}$$ Compairing (\[Solution u\_r\]) and (\[Solution K\_r\_1,r\_2\]), obviously we know that for all $r,t\in \mathbb{R}$ $$\begin{aligned}
\label{2019aug1}
K^{r,r}=u^r\ \ a.s.
\end{aligned}$$ and thus $\varphi^{r,r}(t)=\varphi^r(t)$ $a.s.$ for all $r,t\in \mathbb{R}$. By quasi-periodicity of $\tilde{b}$ and $\tilde{\sigma}$, we know that $\tilde{b}^{r_1+\tau_1, r_2}=\tilde{b}^{r_1,r_2}=\tilde{b}^{r_1,r_2+\tau_2}$, $\tilde{\sigma}^{r_1+\tau_1, r_2}=\tilde{\sigma}^{r_1,r_2}=\tilde{\sigma}^{r_1,r_2+\tau_2}$. Thus it turns out that almost surely $$K^{r_1+\tau_1,r_2}(t,s,x)=K^{r_1,r_2}(t,s,x)=K^{r_1,r_2+\tau_2}(t,s,x),$$ for all $t\geq s, s\in\mathbb{R}$, then almost surely $$\varphi^{r_1+\tau_1,r_2}(t)=\varphi^{r_1,r_2}(t)=\varphi^{r_1,r_2+\tau_2}(t),$$ for all $t\in\mathbb{R}$. Let $\varphi$ be the unique random path of SDE (\[SDE\]), by Theorem \[Existence and uniqueness of random path\] we have $$\varphi^r(t,\omega)=\varphi(t+r,\theta_{-r}\omega).$$ Let $\tilde{\varphi}(r_1,r_2,\omega):=\varphi^{r_1,r_2}(0,\omega)$, then $$\tilde{\varphi}(r_1+\tau_1,r_2,\omega)=\tilde{\varphi}(r_1,r_2,\omega)=\tilde{\varphi}(r_1,r_2+\tau_2,\omega),$$ and $$\tilde{\varphi}(r,r,\omega)=\varphi^{r,r}(0,\omega)=\varphi^r(0,\omega)=\varphi(r,\theta_{-r}\omega).$$ Therefore, the unique random path $\varphi$ can be written as $$\varphi(t,\omega)=\tilde{\varphi}(t,t,\theta_t\omega), \text{ for a.e. } \omega\in\Omega,$$ where $\tilde{\varphi}(t+\tau_1,s,\omega)=\tilde{\varphi}(t,s,\omega)$, $\tilde{\varphi}(t,s+\tau_2,\omega)=\tilde{\varphi}(t,s,\omega)$, for $a.e. $ $\omega\in\Omega$. This means $\varphi$ is a random quasi-periodic path.
\[Solution K shift time in r\] We can conduct similar operations as in (\[Solution u shift time\]) and (\[Solution u shift time 2\]) to re-parameterised equation (\[Solution K\_r\_1,r\_2\]). Noticing $$\begin{aligned}
\label{2019aug2}
{\tilde b}^{r_1,r_2}(v+r,\cdot)={\tilde b}^{r_1+r,r_2+r}(v,\cdot), \ \ {\tilde \sigma}^{r_1,r_2}(v+r,\cdot)={\tilde \sigma}^{r_1+r,r_2+r}(v,\cdot)
\end{aligned}$$ and using the same argument as in the proof of (\[2019aug1\]), we can conclude important property (\[2019aug3\]). This property is similar to the shift property of the autonomous stochastic differential equations which leads to their cocycle property with a perfection argument. Though there is nothing similar to be said about the original SDEs due to the time dependency of the coefficients, this property holds due to “time-invariance” of the re-parameterised coefficients in the sense of (\[2019aug2\]).
Existence and uniqueness of quasi-periodic measure
--------------------------------------------------
First we give the definition of the quasi-periodic probability measure as follows.
\[Definition of quasi-periodic measure\] We say a map $\rho: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{R}^d)$ is a quasi-periodic probability measure of periods $\tau_1, \tau_2$ of SDE (\[SDE\]), where the reciprocals of $\tau_1$ and $\tau_2$ are rationally linearly independent, if $P^*(t,s)\rho_s=\rho_t$ for all $t\geq s$, and there exists $\tilde{\rho}:\mathbb{R}\times\mathbb{R}\rightarrow \mathcal{P}(\mathbb{R}^d)$ with $\tilde{\rho}_{t,t}=\rho_t$ such that $$\tilde{\rho}_{t+\tau_1,s}=\tilde{\rho}_{t,s}, \
\tilde{\rho}_{t,s+\tau_2}=\tilde{\rho}_{t,s},$$ for all $t,s\in\mathbb{R}$.
\[Existence of quasi-periodic measure\] Assume Conditions \[Quasi-periodic condition\], \[Quasi-dissipative\] and $\alpha>\frac{\beta^2}{2}$. Then there exists a unique quasi-periodic probability measure of periods $\tau_1, \tau_2$ of SDE (\[SDE\]) in $\mathcal{M}^2$.
Uniqueness: Applying the proof of Theorem \[Existence and uniqueness of entrance measure\], we know that if there exists a quasi-periodic probability measure with periods $\tau_1, \tau_2$ of SDE (\[SDE\]) in $\mathcal{M}^2$, it must be the unique entrance measure of SDE (\[SDE\]) defined by the law of the random path.
Existence: by Theorem \[Existence and uniqueness of quasi-periodic random path\], we know that SDE (\[SDE\]) has a uniformly $L^2$-bounded random quasi-periodic path $\varphi:\mathbb{R}\times\Omega \rightarrow \mathbb{R}^d$ with periods $\tau_1$ and $\tau_2$, i.e. there exists a $\tilde{\varphi}: \mathbb{R}\times\mathbb{R}\times\Omega \rightarrow \mathbb{R}^d$ such that $\varphi(t, \theta_{-t}\cdot)=\tilde{\varphi}(t,t,\cdot)$ and $\tilde{\varphi}(t+\tau_1,s)=\tilde{\varphi}(t,s)=\tilde{\varphi}(t,s+\tau_2)$ for all $t,s\in\mathbb{R}$. Let $$\label{Define quasi-periodic measure by quasi-periodic path}
\rho_t=\mathcal{L}(\varphi(t)), \ \tilde{\rho}_{t,s}=\mathcal{L}(\tilde{\varphi}(t,s))$$ be the laws of $\varphi(t)$ and $\tilde{\varphi}(t,s)$ respectively. Since $\varphi$ is the random path of SDE (\[SDE\]), then by equation (\[Law of random path adapted\]) we have $P^*(t,s)\rho_s=\rho_t$ for all $t\geq s$. Since $\theta_{-t}$ preserves probability measure $P$, then $\rho_t=\mathcal{L}(\varphi(t))=\mathcal{L}(\tilde{\varphi}(t,t))=\tilde{\rho}_{t,t}$. By the construction of $\tilde{\rho}$, we have $$\tilde{\rho}_{t+\tau_1,s}=\mathcal{L}(\tilde{\varphi}(t+\tau_1,s))=\mathcal{L}(\tilde{\varphi}(t,s))=\tilde{\rho}_{t,s}$$ and $$\tilde{\rho}_{t,s+\tau_2}=\mathcal{L}(\tilde{\varphi}(t,s+\tau_2))=\mathcal{L}(\tilde{\varphi}(t,s))=\tilde{\rho}_{t,s}.$$ Also since $\varphi$ is uniformly $L^2$-bounded, then $$\sup_{t\in \mathbb{R}}\int_{\mathbb{R}^d}x^2\rho_t(dx)=\sup_{t\in \mathbb{R}}\mathbb{E}|\varphi(t)|^2<\infty,$$ which means $\rho\in \mathcal{M}^2$.
We include the following example with a number of reasons. First, O-U process is one of the simplest stochastic process that one would analyse for new concepts. Second, it is instructive and does illustrate clearly the idea of random quasi-periodicity and two kinds of formulations as well as their relation. Third, the formulae for its random quasi-periodic path and quasi-periodic measure can be written down explicitly. Last, but not least, this equation is relevant in various different applications e.g. modelling energy consumptions or temperature variants with two obvious daily and seasonal periodicities.
The Ornstein-Uhlenbeck process with mean reversion of single-period was used in modelling electricity prices ([@BKM07],[@LS02]), daily temperature ([@BB07]), biological neurons ([@IDL14]) etc. The quasi-periodic O-U process we introduce here allows a feature of multiple periods which is natural in many real world situations e.g energy consumptions, temperature, business cycles, economics cycles. While it is not the purpose of this paper to study these interesting applied problems in their specific contexts, our work in this paper provides a mathematical theory of random quasi-periodicity for this purpose.
Here we consider the following mean reversion multidimensional Ornstein-Uhlenbeck equation on $\mathbb{R}^d$ $$\label{O-U equation}
dX_t=(S(t)-AX_t)dt+\sigma(t)dW_t$$ where $S(t), \sigma(t)$ are deterministic quasi-periodic functions with periods $\tau_1,\tau_2$ and $A\in S_d$ with $A>0$, which means that $A$ is a symmetrical matrix with positive eigenvalues $\{\lambda_n\}_{n=1}^{d}$. The analysis is given as follows.
Applying It$\hat{o}$’s formula to $e^{tA}X_t$, we have $$X_t=e^{-(t-s)A}X_s+\int_{s}^{t}e^{-(t-r)A}S(r)dr+\int_{s}^{t}e^{-(t-r)A}\sigma(r)dW_r \quad t\geq s.$$ Let $$\begin{aligned}
\label{2019aug4}
\varphi(t):=\int_{-\infty}^{t}e^{-(t-r)A}S(r)dr+\int_{-\infty}^{t}e^{-(t-r)A}\sigma(r)dW_r.
\end{aligned}$$ Then we have $$\varphi(t)=e^{-(t-s)A}\varphi(s)+\int_{s}^{t}e^{-(t-r)A}S(r)dr+\int_{s}^{t}e^{-(t-r)A}\sigma(r)dW_r,$$ which means that $\varphi$ is a random path of SDE (\[O-U equation\]). Next we will show that $\varphi$ is also a random quasi-periodic path. We first rewrite $\varphi(t)$ by $$\begin{split}
\varphi(t, \omega)
&=\int_{-\infty}^{t}e^{-(t-r)A}S(r)dr+\left[\int_{-\infty}^{t}e^{-(t-r)A}\sigma(r)dW_r\right](\omega)\\
&=\int_{-\infty}^{0}e^{rA}S(r+t)dr+\left[\int_{-\infty}^{0}e^{rA}\sigma(r+t)dW_r\right](\theta_t\omega).
\end{split}$$ Since $S,\sigma$ are quasi-periodic functions with periods $\tau_1,\tau_2$, then there exist $\tilde{S},\tilde{\sigma}$ such that $S(t)=\tilde{S}(t,t)$ and $\sigma(t)=\tilde{\sigma}(t,t)$ and $$\begin{cases}
\tilde{S}(t+\tau_1,s)=\tilde{S}(t,s)=\tilde{S}(t,s+\tau_2)\\
\tilde{\sigma}(t+\tau_1,s)=\tilde{\sigma}(t,s)=\tilde{\sigma}(t,s+\tau_2).
\end{cases}$$ Then we have $$\begin{split}
\varphi(t, \omega)
&=\int_{-\infty}^{0}e^{rA}\tilde{S}(r+t,r+t)dr+\left[\int_{-\infty}^{0}e^{rA}\tilde{\sigma}(r+t,r+t)dW_r\right](\theta_t\omega).
\end{split}$$ Let $$\begin{aligned}
\label{2019aug5}
\tilde{\varphi}(t,s,\omega)=\int_{-\infty}^{0}e^{rA}\tilde{S}(r+t,r+s)dr+\left[\int_{-\infty}^{0}e^{rA}\tilde{\sigma}(r+t,r+s)dW_r\right](\omega).
\end{aligned}$$ Then we have $\varphi(t,\theta_{-t}\omega)=\tilde{\varphi}(t,t,\omega)$ and $$\tilde{\varphi}(t+\tau_1,s,\omega)=\tilde{\varphi}(t,s,\omega), \
\tilde{\varphi}(t,s+\tau_2,\omega)=\tilde{\varphi}(t,s,\omega),$$ which shows that $\varphi$ is a random quasi-periodic path of SDE (\[O-U equation\]) with periods $\tau_1,\tau_2$.
Let $\rho_t=\mathcal{L}(\varphi(t))$. By Theorem \[Existence of quasi-periodic measure\], we know that $\rho_t$ is the unique quasi-periodic probability measure with periods $\tau_1,\tau_2$ of SDE (\[O-U equation\]). Moreover, from (\[2019aug4\]), we know that $$\rho_t(\cdot)=\mathcal{N}\left(\int_{-\infty}^{t}e^{-(t-r)A}S(r)dr, \int_{-\infty}^{t}e^{-(t-r)A}\sigma(r)\sigma(r)^Te^{-(t-r)A}dr\right)(\cdot),$$ where $\mathcal{N}$ is the multivariate normal distribution. Let $\tilde \rho_{t,s}=\mathcal{L}(\tilde \varphi(t,s))$. Then from (\[2019aug5\]), we know that $$\tilde \rho_{t,s}(\cdot)=\mathcal{N}\left(\int_{-\infty}^{0}e^{rA}\tilde S(r+t,r+s)dr, \int_{-\infty}^{0}e^{rA}(\tilde \sigma\tilde
\sigma^T)(r+t,r+s)e^{rA}dr\right)(\cdot).$$ It is obvious that $\rho_t=\tilde \rho_{t,t}$.
In Subsection \[2019aug6\], we will develop a way to lift a quasi-periodic stochastic flow to the cylinder $[0,\tau_1)\times [0,\tau_2)\times {\mathbb R}^d$ and prove $\tilde \mu_{t,s}=\delta_t\times\delta_s\times \tilde \rho_{t,s}$ is a quasi-periodic measure. This setup will enable us to prove that the average ${1\over {\tau_1\tau_2}}\int _0^{\tau_1}\int_0^{\tau_2}\tilde \mu_{t,s}dtds$ is an invariant measure on the cylinder. Our result also implies that for this particular case, it is the unique invariant measure for the lifted quasi-periodic Ornstein-Uhlenbeck process.
The lift and invariant measure {#2019aug6}
------------------------------
In Section \[Section of Quasi-periodic path\], we have the existence and uniqueness of random quasi-periodic path, and in this case, we will lift the semi-flow $u$ and obtain an invariant measure. Consider the cylinder $\tilde{\mathbb X}=[0, \tau_1) \times [0, \tau_2)\times\mathbb{R}^d$ with the following metric $$d(\tilde{x}, \tilde{y})=d_1(t_1,s_1)+d_2(t_2,s_2)+|x-y|, \text{ for all } \tilde{x}=(t_1,t_2,x), \tilde{y}=(s_1,s_2,y) \in \tilde{\mathbb X},$$ where $d_1,d_2$ are the metrics on $[0,\tau_1), [0,\tau_2)$ defined by $$d_i(t_i,s_i)=\min(|t_i-s_i|, \tau_i-|t_i-s_i|), \text{ for all } t_i,s_i\in [0,\tau_i), i=1,2.$$ Denote by $\mathcal{B}(\tilde{\mathbb X})$ the Borel measurable set on $\tilde{\mathbb X}$ deduced by metric $d$. Then we have the following lemma.
Assume Conditions \[Quasi-periodic condition\], \[Quasi-dissipative\] and $\alpha>\frac{\beta^2}{2}$. We lift the semi-flow $u: \triangle\times \mathbb{R}^d\times \Omega\rightarrow \mathbb{R}^d$ to a random dynamical system on a cylinder $\tilde{\mathbb X}=[0, \tau_1) \times [0, \tau_2)\times\mathbb{R}^d$ by the following: $$\tilde{\Phi}(t,\omega)(s_1,s_2,x)=(t+s_1\mod \tau_1,\ t+s_2 \mod\tau_2,\ K^{s_1,s_2}(t,0,x,\omega)),$$ where $K^{r_1,r_2}$ is the solution of (\[Solution K\_r\_1,r\_2\]). Then $\tilde{\Phi}: \mathbb{R}^+\times \tilde{\mathbb X}\times\Omega\rightarrow \tilde{\mathbb X}$ is a cocycle on $\tilde{\mathbb X}$ over the metric dynamical system $(\Omega, \mathcal{F}, P, (\theta_t)_{t\in \mathbb{R}})$.
Moreover, assume $\varphi: \mathbb{R}\times \Omega\rightarrow \mathbb{R}^d$ is a random path of the semi-flow u. Then $\tilde{Y}: \mathbb{R} \times \Omega\rightarrow \tilde{\mathbb X}$ defined by $$\tilde{Y}(s,\omega)=(s\mod \tau_1,\ s\mod\tau_2,\ \varphi(s,\omega))$$ is a random path of the cocycle $\tilde{\Phi}$ on $\tilde{\mathbb X}$.
We first prove that $\tilde{\Phi}$ is a cocycle on $\tilde{\mathbb X}$. Note $K^{r_1,r_2}$ is periodic in $r_1,r_2$ with periods $\tau_1,\tau_2$. It follows that for any $(s_1,s_2,x)\in \tilde{\mathbb X}, t,s\in \mathbb{R}^+$, we have $$\begin{aligned}
&&\tilde{\Phi}(t, \theta_s\omega)\circ\tilde{\Phi}(s,\omega)(s_1,s_2,x)\\
&=&\tilde{\Phi}(t,\theta_s\omega)(s+s_1\mod\tau_1,\ s+s_2\mod\tau_2,\ K^{s_1,s_2}(s,0,x,\omega))\\
&=&(t+s+s_1\mod\tau_1,\ t+s+s_2\mod\tau_2,\ K^{s+s_1, s+s_2}(t,0,K^{s_1,s_2}(s,0,x,\omega), \theta_s\omega)).
\end{aligned}$$ Now we compute the $K^{s+s_1, s+s_2}(t,0,K^{s_1,s_2}(s,0,x,\omega), \theta_s\omega)$ term. By equation (\[Solution K\_r\_1,r\_2\]), we know that $$\begin{aligned}
\label{Solution K_s_1,s_2}
&&K^{s_1,s_2}(s,0,x,\omega)\\\nonumber
&=&x+\int_0^s\tilde{b}^{s_1,s_2}(v,K^{s_1,s_2}(v,0,x,\omega))dv+(\omega)\int_0^s\tilde{\sigma}^{s_1,s_2}(v,K^{s_1,s_2}(v,0,x,\cdot))dW_v\\\nonumber
&=&x+\int_0^sb(v+s_1,v+s_2,K^{s_1,s_2}(v,0,x,\omega))dv\\\nonumber
&&+(\omega)\int_0^s\sigma(v+s_1,v+s_2,K^{s_1,s_2}(v,0,x,\cdot))dW_v,
\end{aligned}$$ and $$\begin{aligned}
\label{semi-group of K_r_1,r_2}
&&K^{s+s_1, s+s_2}(t,0,K^{s_1,s_2}(s,0,x,\omega), \theta_s\omega)\\
&=&K^{s_1,s_2}(s,0,x,\omega)+\int_0^tb(v+s+s_1,v+s+s_2,K^{s+s_1,s+s_2}(v,0,K^{s_1,s_2}(s,0,x,\omega),\theta_s\omega))dv\nonumber\\
&&+(\theta_s\omega)\int_0^t\sigma(v+s+s_1,v+s+s_2,K^{s+s_1,s+s_2}(v,0,K^{s_1,s_2}(s,0,x,\theta_{-s}\cdot),\cdot))dW_v\nonumber\\
&=&x+\int_0^sb(v+s_1,v+s_2,K^{s_1,s_2}(v,0,x,\omega))dv\nonumber\\
&&+(\omega)\int_0^s\sigma(v+s_1,v+s_2,K^{s_1,s_2}(v,0,x,\cdot))dW_v\nonumber\\
&&+\int_s^{t+s}b(v+s_1,v+s_2,K^{s+s_1,s+s_2}(v-s,0,K^{s_1,s_2}(s,0,x,\omega),\theta_s\omega))dv\nonumber\\
&&+(\omega)\int_s^{t+s}\sigma(v+s_1,v+s_2,K^{s+s_1,s+s_2}(v-s,0,K^{s_1,s_2}(s,0,x,\cdot),\theta_s\cdot))dW_v \quad P-a.e. \text{ on } \omega.\nonumber
\end{aligned}$$ Let $$Q(r,0,x,\omega)=
\begin{cases}
K^{s_1,s_2}(r,0,x,\omega), \quad 0\leq r \leq s,\\
K^{s+s_1,s+s_2}(r-s,0,K^{s_1,s_2}(s,0,x,\omega), \theta_s\omega), s<r \leq t+s.
\end{cases}$$ From equation (\[Solution K\_s\_1,s\_2\]) and (\[semi-group of K\_r\_1,r\_2\]), we know that for all $0\leq r\leq t+s$, $Q(r, 0, x, \omega)$ solves the following equation $$Q(r, 0, x, \omega)
=x+\int_0^r\tilde{b}^{s_1,s_2}(v,Q(v,0,x,\omega))dv +(\omega)\int_0^r\tilde{\sigma}^{s_1,s_2}(v,Q(v, 0, x, \cdot))dW_v.$$ which implies $Q(r, 0, x, \omega)=K^{s_1,s_2}(r,0,x,\omega) \quad P-a.s. \text{ on } \omega$ for all $0\leq r\leq t+s$. In particular, $$K^{s+s_1, s+s_2}(t,0,K^{s_1,s_2}(s,0,x,\omega), \theta_s\omega)=Q(t+s, 0, x, \omega) =K^{s_1,s_2}(t+s,0,x,\omega).$$ Hence $$\begin{split}
&\tilde{\Phi}(t, \theta_s\omega)\circ\tilde{\Phi}(s,\omega)(s_1,s_2,x)\\
=&(t+s+s_1\mod\tau_1,t+s+s_2\mod\tau_2, K^{s+s_1, s+s_2}(t,0,K^{s_1,s_2}(s,0,x,\omega), \theta_s\omega))\\
=&(t+s+s_1\mod\tau_1,t+s+s_2\mod\tau_2, K^{s_1,s_2}(t+s, 0, x, \omega))\\
=&\tilde{\Phi}(t+s,\omega)(s_1,s_2,x),
\end{split}$$ which implies the cocycle property of $\tilde{\Phi}$.
Next, since $\varphi$ is a random path of the semi-flow $u$ and $\tilde{Y}(s,\omega)=(s\mod \tau_1, s\mod\tau_2, \varphi(s,\omega))$, then $$\begin{split}
\tilde{\Phi}(t,\theta_s\omega)\tilde{Y}(s,\omega)
=&(t+s\mod \tau_1, t+s\mod\tau_2, K^{s, s}(t, 0, \varphi(s, \omega), \theta_s\omega))\\
=&(t+s\mod \tau_1, t+s\mod\tau_2, u^s(t, 0, \varphi(s, \omega), \theta_s\omega))\\
=&(t+s\mod \tau_1, t+s\mod\tau_2, u(t+s, s, \varphi(s, \omega), \theta_{-s}\theta_s\omega))\\
=&(t+s\mod \tau_1, t+s\mod\tau_2, \varphi(t+s,\omega))\\
=&\tilde{Y}(t+s,\omega),
\end{split}$$ which means $\tilde{Y}$ is a random path of the cocycle $\tilde{\Phi}$ on $\tilde{\mathbb X}$.
Consider the Markovian transition $\tilde{P}:\mathbb{R}^+\times \tilde{\mathbb X}\times \mathcal{B}(\tilde{\mathbb X})\rightarrow [0,1]$ generated by the cocycle $\tilde{\Phi}$, i.e., $$\tilde{P}(t, (s_1,s_2,x), \tilde{\Gamma})=P(\omega: \tilde{\Phi}(t,\omega)(s_1,s_2,x)\in \tilde{\Gamma}),$$ for all $t\in \mathbb{R}^+, (s_1,s_2,x)\in \tilde{\mathbb X}, \tilde{\Gamma}\in \mathcal{B}(\tilde{\mathbb X})$. Similarly, for any $\tilde{\mu}\in \mathcal{P}(\tilde{\mathbb X})$, we define $$\tilde{P}^*_t\tilde{\mu}(\tilde{\Gamma})=\int_{\tilde{\mathbb X}}\tilde{P}(t,(s_1,s_2,x),\tilde{\Gamma})\tilde{\mu}(ds_1\times ds_2\times dx).$$ Then we have the following theorem.
If $\rho: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{R}^d)$ is the entrance measure of semi-group $P^*$, i.e. $P^*(t,s)\rho_s=\rho_t$, then $\tilde{\mu}: \mathbb{R}\rightarrow \mathcal{P}(\tilde{\mathbb X})$ defined by $$\tilde{\mu}_t=\delta_{t\mod \tau_1}\times \delta_{t\mod \tau_2}\times \rho_t$$ is an entrance measure of semi-group $\tilde{P}^*$, i.e., $$\tilde{P}^*_t\tilde{\mu}_s=\tilde{\mu}_{t+s}.$$ Moreover, $\tilde{\mu}$ is also a quasi-periodic measure.
For any $\tilde{\Gamma}\in \mathcal{B}(\tilde{\mathbb X})$, let $\tilde{\Gamma}_s:=\{x\in \mathbb{R}^d| (s\mod \tau_1, s\mod \tau_2,x)\in \tilde{\Gamma}\}$. Then we have $$\begin{split}
\tilde{P}^*_t\tilde{\mu}_s(\tilde{\Gamma})
&=\int_{\tilde{\mathbb X}}\tilde{P}(t, (s_1,s_2,x), \tilde{\Gamma})\tilde{\mu}_s(ds_1\times ds_2\times dx)\\
&=\int_{\mathbb{R}^d}\tilde{P}(t, (s\mod \tau_1,\ s\mod \tau_2,x), \tilde{\Gamma})\rho_s(dx)\\
&=\int_{\mathbb{R}^d}P(\omega: \tilde{\Phi}(t,\omega)(s\mod \tau_1,\ s\mod \tau_2,x)\in \tilde{\Gamma})\rho_s(dx)\\
&=\int_{\mathbb{R}^d}P(\omega: (t+s\mod \tau_1,\ t+s\mod \tau_2, u^s(t,0,x,\omega))\in \tilde{\Gamma})\rho_s(dx)\\
&=\int_{\mathbb{R}^d}P(\omega: u(t+s,s,x,\theta_{-s}\omega)\in \tilde{\Gamma}_{t+s})\rho_s(dx)\\
&=\int_{\mathbb{R}^d}P(\omega: u(t+s,s,x,\omega)\in \tilde{\Gamma}_{t+s})\rho_s(dx)\\
&=\int_{\mathbb{R}^d}P(t+s,s,x, \tilde{\Gamma}_{t+s})\rho_s(dx)\\
&=P^*(t+s,s)\rho_s(\tilde{\Gamma}_{t+s})\\
&=\rho_{t+s}(\tilde{\Gamma}_{t+s})=\tilde{\mu}_{t+s}(\tilde{\Gamma}).
\end{split}$$ Moreover, let $$\label{Quasi-periodic measure after lift}
\hat{\mu}_{s_1,s_2}=\delta_{s_1\mod\tau_1}\times\delta_{s_2\mod\tau_2}\times\tilde{\rho}_{s_1,s_2},$$ we know that $\tilde{\mu}_s=\hat{\mu}_{s_1,s_2}$ and $$\hat{\mu}_{s_1+\tau_1,s_2}=\hat{\mu}_{s_1,s_2}, \
\hat{\mu}_{s_1,s_2+\tau_2}=\hat{\mu}_{s_1,s_2},$$ which completes our proof.
For the above entrance measure $\tilde{\mu}$, set $$\bar{\tilde{\mu}}_T:=\frac{1}{T}\int_{0}^{T}\tilde{\mu}_sds$$ and $$\label{Tight measure set}
\mathcal{M}:=\{\bar{\tilde{\mu}}_T: T\in \mathbb{R}^+\}.$$
We have the following lemma.
\[Lemma of tight measure set\] Assume Conditions \[Quasi-periodic condition\], \[Quasi-dissipative\] and $\alpha>\frac{\beta^2}{2}$. Then $\mathcal{M}$ is tight, and hence is weakly compact in $\mathcal{P}(\tilde{\mathbb X})$.
We just need to prove that for any $\epsilon>0$, there exists a compact set $\tilde{\Gamma}_{\epsilon}\in \mathcal{B}(\tilde{\mathbb X})$ such that for all $T\in \mathbb{R}^+$, we have $$\bar{\tilde{\mu}}_T({\tilde{\Gamma}_{\epsilon}})>1-\epsilon.$$ Since the entrance measure $\rho_t$ is the law of the $L^2$-bounded random path $\varphi(t)$, then $\{\rho_t: t\in \mathbb{R}\}$ is tight because $$\begin{split}
\rho_t(B_{N}(0))&=P(|\varphi(t)| < N)\\
&=1-P(|\varphi(t)| \geq N)\\
&\geq 1-\frac{\|\varphi(t)\|_2^2}{N^2}\\
&\geq 1-\frac{\sup_{t\in \mathbb{R}}\|\varphi(t)\|_2^2}{N^2}.
\end{split}$$ Then for the given $\epsilon>0$, there exists a compact set $\Gamma_{\epsilon}\subset \mathbb{R}^d$ such that for all $t\in \mathbb{R}$, $$\rho_t(\Gamma_{\epsilon})>1-\epsilon.$$ It is well-known that $[0,\tau_1), [0,\tau_2)$ are both homeomorphic to the circle $S^1$ under metrics $d_1,d_2$ respectively. Hence they are compact and $\tilde{\Gamma}_{\epsilon}=[0,\tau_1)\times [0,\tau_2) \times \Gamma_{\epsilon}$ is compact on $\tilde{\mathbb X}$. Moreover $$\begin{split}
\bar{\tilde{\mu}}_T(\tilde{\Gamma}_{\epsilon})=\frac{1}{T}\int_0^T\tilde{\mu}_s(\tilde{\Gamma}_{\epsilon})ds=\frac{1}{T}\int_0^T\rho_s(\Gamma_{\epsilon})ds>1-\epsilon,
\end{split}$$ which completes our proof.
For any $f\in C^0(\tilde{\mathbb X})$, which is defined as the collection of $\mathcal{B}(\tilde{\mathbb X})$ measurable functions, we define $$\label{P^*_t act on functions}
\tilde{P}_tf(\tilde{x})=\int_{\tilde{\mathbb X}}\tilde{P}(t,\tilde{x},d\tilde{y})f(\tilde{y}), \text{ for any } \tilde{x}\in \tilde{\mathbb X}.$$ We have the following Feller property of the semi-group $\tilde{P}_t, t\geq 0$.
\[Feller property of P\^\*\] Assume Conditions \[Quasi-periodic condition\], \[Quasi-dissipative\] and $\alpha>\frac{\beta^2}{2}$. In addition, we assume that $\tilde{b}(t,s,x), \tilde{\sigma}(t,s,x)$ are continuous with respect to $(t,s)$ uniformly in $x$. Then the semi-group $\tilde{P}_t, t\geq 0$, defined by (\[P\^\*\_t act on functions\]), is Feller, i.e. for all $f\in C_b(\tilde{\mathbb X})$, $\tilde{P}_tf\in C_b(\tilde{\mathbb X})$.
Obviously $\|\tilde{P}_tf\|_{\infty}\leq \|f\|_{\infty}$, then we just need to prove that $\tilde{P}_tf$ is continuous. It is sufficient to prove that for any sequence $\tilde{x}_n=(r_1^n,r_2^n,x_n), \tilde{x}=(r_1,r_2,x)\in \tilde{\mathbb X}$ with $\tilde{x}_n\xrightarrow{n\rightarrow \infty} \tilde{x}$, we have $\tilde{P}_tf(\tilde{x}_n)\xrightarrow{n\rightarrow \infty} \tilde{P}_tf(\tilde{x})$. Since $$\begin{split}
\tilde{P}_tf(\tilde{x})
&=\int_{[0, \tau_1) \times [0, \tau_2)\times\mathbb{R}^d}\tilde{P}(t,(r_1,r_2,x),ds_1\times ds_2\times dy)f(s_1,s_2,y)\\
&=\int_{[0, \tau_1) \times [0, \tau_2)\times\mathbb{R}^d}P(\tilde{\Phi}(t,\cdot)(r_1,r_2,x) \in ds_1\times ds_2\times dy)f(s_1,s_2,y)\\
&=\int_{\mathbb{R}^d}P(K^{r_1,r_2}(t,0,x) \in dy)f(t+r_1\mod \tau_1,t+r_2\mod \tau_2,y)\\
&=\mathbb{E}f(t+r_1\mod \tau_1,t+r_2\mod \tau_2,K^{r_1,r_2}(t,0,x)).
\end{split}$$ Let $f_t(r_1,r_2,x):=f(t+r_1\mod \tau_1,t+r_2\mod \tau_2, x)$. Then we have $$\begin{split}
|\tilde{P}_tf(\tilde{x}_n)-\tilde{P}_tf(\tilde{x})|
=&|\mathbb{E}f_t(r_1^n,r_2^n,K^{r_1^n,r_2^n}(t,0,x_n))-\mathbb{E}f_t(r_1,r_2,K^{r_1,r_2}(t,0,x))|\\
\leq& |\mathbb{E}f_t(r_1^n,r_2^n,K^{r_1^n,r_2^n}(t,0,x_n))-\mathbb{E}f_t(r_1^n,r_2^n,K^{r_1,r_2}(t,0,x))|\\
&+|\mathbb{E}f_t(r_1^n,r_2^n,K^{r_1,r_2}(t,0,x))-\mathbb{E}f_t(r_1,r_2,K^{r_1,r_2}(t,0,x))|\\
=:&A_1^n+A_2^n.
\end{split}$$ Since $f\in C_b(\tilde{\mathbb X})$, then $f_t\in C_b(\tilde{\mathbb X})$ and $f_t(r_1^n,r_2^n,K^{r_1,r_2}(t,0,x))\xrightarrow{a.s.} f_t(r_1,r_2,K^{r_1,r_2}(t,0,x))$ as $n\rightarrow \infty$. By Lebesgue’s dominated convergence theorem, we have $$\begin{split}
A_2^n=|\mathbb{E}f_t(r_1^n,r_2^n,K^{r_1,r_2}(t,0,x))-\mathbb{E}f_t(r_1,r_2,K^{r_1,r_2}(t,0,x))| \xrightarrow{n\rightarrow \infty} 0.
\end{split}$$ Furthermore, by the uniformly continuous of $\tilde{b}, \tilde{\sigma}$, we know that $\tilde{b}^{r_1^n,r_2^n}\xrightarrow[n\rightarrow \infty]{uniformly} \tilde{b}^{r_1,r_2}$ and $\tilde{\sigma}^{r_1^n,r_2^n}\xrightarrow[n\rightarrow \infty]{uniformly} \tilde{\sigma}^{r_1,r_2}$. Let $$b_n=\sup_{t\in \mathbb{R}, x\in \mathbb{R}^d}|\tilde{b}^{r_1^n,r_2^n}(t,x)-\tilde{b}^{r_1,r_2}(t,x)|$$ and $$\sigma_n=\sup_{t\in \mathbb{R}, x\in \mathbb{R}^d}|\tilde{\sigma}^{r_1^n,r_2^n}(t,x)-\tilde{\sigma
}_{r_1,r_2}(t,x)|.$$ Then $\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\sigma_n=0$. Set $K_n(t)=K^{r_1^n,r_2^n}(t,0,x_n)$ and $K(t)=K^{r_1,r_2}(t,0,x)$. Applying It$\hat{\rm o}$ formula to $|K_n(t)-K(t)|^2$, we have $$\begin{aligned}
\label{Eq of K_n and K 1}
&&|K_n(t)-K(t)|^2\nonumber\\
&=&|x_n-x|^2+
\int_{0}^{t}2(K_n(s)-K(s))(\tilde{b}^{r_1^n,r_2^n}(s,K_n(s))-\tilde{b}^{r_1,r_2}(s,K(s)))ds\nonumber\\
&&+\int_{0}^{t}\|\tilde{\sigma}^{r_1^n,r_2^n}(s,K_n(s))-\tilde{\sigma}^{r_1,r_2}(s,K(s))\|^2ds\nonumber\\
&&+\int_{0}^{t}2(K_n(s)-K(s))(\tilde{\sigma}^{r_1^n,r_2^n}(s,K_n(s))-\tilde{\sigma}^{r_1,r_2}(s,K(s)))dW_s.
\end{aligned}$$ Note $$\begin{aligned}
\label{Eq of K_n and K 2}
&&2(K_n(s)-K(s))(\tilde{b}^{r_1^n,r_2^n}(s,K_n(s))-\tilde{b}^{r_1,r_2}(s,K(s)))\nonumber\\
&=&2(K_n(s)-K(s))(\tilde{b}^{r_1^n,r_2^n}(s,K_n(s))-\tilde{b}^{r_1,r_2}(s,K_n(s)))\nonumber\\
&&+2(K_n(s)-K(s))(\tilde{b}^{r_1,r_2}(s,K_n(s))-\tilde{b}^{r_1,r_2}(s,K(s)))\nonumber\\
&\leq& 2b_n|K_n(s)-K(s)|-2\alpha|K_n(s)-K(s)|^2\nonumber\nonumber\\
&\leq& \frac{b_n^2}{\lambda}+\lambda|K_n(s)-K(s)|^2-2\alpha|K_n(s)-K(s)|^2,
\end{aligned}$$ and $$\begin{aligned}
\label{Eq of K_n and K 3}
&&\|\tilde{\sigma}^{r_1^n,r_2^n}(s,K_n(s))-\tilde{\sigma}^{r_1,r_2}(s,K(s))\|^2\nonumber\\
&=&\|\tilde{\sigma}^{r_1^n,r_2^n}(s,K_n(s))-\tilde{\sigma}^{r_1,r_2}(s,K_n(s))+\tilde{\sigma}^{r_1,r_2}(s,K_n(s))-\tilde{\sigma}^{r_1,r_2}(s,K(s))\|^2\nonumber\\
&\leq& (\|\tilde{\sigma}^{r_1^n,r_2^n}(s,K_n(s))-\tilde{\sigma}^{r_1,r_2}(s,K_n(s))\|+\|\tilde{\sigma}^{r_1,r_2}(s,K_n(s))-\tilde{\sigma}^{r_1,r_2}(s,K(s))\|)^2\nonumber\\
&\leq& (\sigma_n+\beta|K_n(s)-K(s)|)^2\nonumber\\
&\leq& \sigma_n^2(1+\frac{\beta^2}{\lambda})+(\lambda+\beta^2)|K_n(s)-K(s)|^2,
\end{aligned}$$ where $\lambda>0$ be the number such that $\alpha-\frac{\beta^2}{2}>\lambda$. Comparing with (\[Eq of K\_n and K 2\]) and (\[Eq of K\_n and K 3\]), we take expectation both side on (\[Eq of K\_n and K 1\]) to have $$\|K_n(t)-K(t)\|_2^2\leq |x_n-x|^2+\frac{b_n^2}{\lambda}+\sigma_n^2(1+\frac{\beta^2}{\lambda}) \xrightarrow{n\rightarrow \infty} 0.$$ Then we have $K^{r_1^n,r_2^n}(t,0,x_n)\xrightarrow[n\rightarrow \infty]{L^2} K^{r_1,r_2}(t,0,x)$. Let $$R_N=\{\omega: |K^{r_1,r_2}(t,0,x,\omega)|\leq N\}$$ and $$R_N^n=\{\omega: |K^{r_1^n,r_2^n}(t,0,x_n,\omega)|\leq N\}.$$ Then by Chebyshev inequality we have $\lim_{N\rightarrow \infty}(\inf_{n\in \mathbb{N}}P(R_N^n\cap R_N))=1$. Since $f$ is continuous, then it is uniformly continuous on all compact subset of $\tilde{\mathbb X}$. Then for arbitrary $\epsilon>0$, there exists $\delta_N^{\epsilon}>0$ such that when $(t_1,t_2,x), (s_1,s_2,y)\in [0,\tau_1)\times [0,\tau_2) \times B_N(0)$, where $B_{N}(0)$ is a ball centered at 0 with radius $N$ in $\mathbb{R}^d$, and $d_1(t_1,s_1)+d_2(t_2,s_2)+|x-y|<\delta_N^{\epsilon}$, we have $|f((t_1,t_2,x))-f((s_1,s_2,y))|<\epsilon$. Set $$C^n_{\delta_N^{\epsilon}}=\{\omega: |K^{r_1^n,r_2^n}(t,0,x_n)-K^{r_1,r_2}(t,0,x)|<\delta_N^{\epsilon}\}.$$ Then also by Chebyshev inequality $\lim_{n\rightarrow \infty}P(C^n_{\delta_N^{\epsilon}})=1$. Hence for all $\omega\in C^n_{\delta_N^{\epsilon}}\cap R_N^n\cap R_N$, $$|f_t(r_1^n,r_2^n,K^{r_1^n,r_2^n}(t,0,x_n))-f_t(r_1^n,r_2^n,K^{r_1,r_2}(t,0,x))|<\epsilon.$$ Therefore $$\begin{split}
\limsup_{n\rightarrow \infty}A_1^n
=&\limsup_{n\rightarrow \infty} |\mathbb{E}f_t(r_1^n,r_2^n,K^{r_1^n,r_2^n}(t,0,x_n))-\mathbb{E}f_t(r_1^n,r_2^n,K^{r_1,r_2}(t,0,x))|\\
\leq&\epsilon+2\|f\|_{\infty}\limsup_{n\rightarrow \infty} [(1-P(C^n_{\delta_N^{\epsilon}}))+(1-P(R_N^n\cap R_N))]\\
=&\epsilon.
\end{split}$$ Since $\epsilon>0$ is arbitrary, we have $A_1^n\xrightarrow{n\rightarrow \infty} 0$. We complete the proof of $\tilde{P}_tf(\tilde{x}_n)\xrightarrow{n\rightarrow \infty} \tilde{P}_tf(\tilde{x})$.
From Lemma \[Lemma of tight measure set\] and Proposition \[Feller property of P\^\*\], we have the existence of invariant measure under $\tilde{P}^*$.
\[Invariant measure\] Assume Conditions \[Quasi-periodic condition\], \[Quasi-dissipative\] and $\alpha>\frac{\beta^2}{2}$. In addition, we assume that $\tilde{b}(t,s,x), \tilde{\sigma}(t,s,x)$ are continuous with respect to $(t, s)$ uniformly in $x$. Then there exists a uniqueness invariant probability measure with respect to the semi-group $\tilde{P}^*$ which is given by $$\frac{1}{\tau_1\tau_2}\int_0^{\tau_1}\int_0^{\tau_2}\delta_{s_1}\times\delta_{s_2}\times\tilde{\rho}_{s_1,s_2}ds_1ds_2.$$
Existence: From Lemma \[Lemma of tight measure set\], we know that $\mathcal{M}$ defined by (\[Tight measure set\]) is tight and hence weakly compact. This means that there exists a sequence $\{T_n\}_{n\geq 1}$ with $T_n\uparrow \infty$ as $n\rightarrow \infty$ and a probability measure $\bar{\tilde{\mu}}\in \mathcal{P}(\tilde{\mathbb X})$ such that $\bar{\tilde{\mu}}_{T_n}\xrightarrow{w} \bar{\tilde{\mu}}$. Moreover, for any fixed $t>0$, since $$\begin{split}
\tilde{P}^*_t\bar{\tilde{\mu}}_{T_n}-\bar{\tilde{\mu}}_{T_n}
&=\frac{1}{T_n}\int_{0}^{T_n}\tilde{P}^*_t\tilde{\mu}_sds-\frac{1}{T_n}\int_{0}^{T_n}\tilde{\mu}_sds\\
&=\frac{1}{T_n}\int_{0}^{T_n}\tilde{\mu}_{t+s}ds-\frac{1}{T_n}\int_{0}^{T_n}\tilde{\mu}_sds\\
&=\frac{1}{T_n}\int_{t}^{t+T_n}\tilde{\mu}_sds-\frac{1}{T_n}\int_{0}^{T_n}\tilde{\mu}_sds\\
&=\frac{1}{T_n}\int_{T_n}^{t+T_n}\tilde{\mu}_sds-\frac{1}{T_n}\int_{0}^{t}\tilde{\mu}_sds,\\
\end{split}$$ so $$\begin{split}
\limsup_{n\rightarrow \infty}\|\tilde{P}^*_t\bar{\tilde{\mu}}_{T_n}-\bar{\tilde{\mu}}_{T_n}\|_{BV}
&\leq \limsup_{n\rightarrow \infty}\frac{1}{T_n}(\int_0^t\|\tilde{\mu}_s\|_{BV}ds+\int_{T_n}^{T_n+t}\|\tilde{\mu}_s\|_{BV}ds)\\
&\leq \limsup_{n\rightarrow \infty}\frac{2t}{T_n}=0.
\end{split}$$ Hence $\tilde{P}^*_t\bar{\tilde{\mu}}_{T_n}\xrightarrow{w} \bar{\tilde{\mu}}$. On the other hand, for any $f\in C_b(\tilde{\mathbb X})$, by Proposition \[Feller property of P\^\*\], we have $\tilde{P}_tf\in C_b(\tilde{\mathbb X})$, and therefore $$\begin{split}
\lim_{n\rightarrow \infty}\int_{\tilde{\mathbb X}}f(\tilde{y})\tilde{P}^*_t\bar{\tilde{\mu}}_{T_n}(d\tilde{y})
&=\lim_{n\rightarrow \infty}\int_{\tilde{\mathbb X}} \int_{\tilde{\mathbb X}}f(\tilde{y})\tilde{P}(t, \tilde{x}, d\tilde{y}) \bar{\tilde{\mu}}_{T_n}(d\tilde{x})\\
&=\lim_{n\rightarrow \infty}\int_{\tilde{\mathbb X}}\tilde{P}_tf(\tilde{x})\bar{\tilde{\mu}}_{T_n}(d\tilde{x})\\
&=\int_{\tilde{\mathbb X}}\tilde{P}_tf(\tilde{x})\bar{\tilde{\mu}}(d\tilde{x})\\
&=\int_{\tilde{\mathbb X}}f(\tilde{y})\tilde{P}^*_t\bar{\tilde{\mu}}(d\tilde{y}).
\end{split}$$ This means $\tilde{P}^*_t\bar{\tilde{\mu}}_{T_n}\xrightarrow{w} \tilde{P}^*_t\bar{\tilde{\mu}}$. Summarizing above we have that $\tilde{P}^*_t\bar{\tilde{\mu}}=\bar{\tilde{\mu}}$.
Moreover, using the same method as in the proof of Proposition \[Feller property of P\^\*\], we know that the quasi-periodic path $\tilde{\varphi}$ of SDE (\[SDE\]) is continuous under $L^2$ norm, i.e. $$\lim_{(t,s)\rightarrow (t_0,s_0)}\|\tilde{\varphi}(t,s)-\tilde{\varphi}(t_0,s_0)\|_2^2=0.$$ Then similar to the proof of Proposition \[Weakly convergence of P(t,s)\], we know that $\tilde{\rho}$ is continuous under the weak topology in $\mathcal{P}(\mathbb{R}^d)$, i.e. for all $f\in C_b(\mathbb{R}^d)$, $$\lim_{(t,s)\rightarrow (t_0,s_0)}\int_{\mathbb{R}^d}f(x)\tilde{\rho}_{t,s}(dx)=\int_{\mathbb{R}^d}f(x)\tilde{\rho}_{t_0,s_0}(dx).$$ Let $\hat{\mu}$ defined by (\[Quasi-periodic measure after lift\]). It is easy to check that $\hat{\mu}$ is also continuous under the weak topology in $\mathcal{P}(\tilde{\mathbb X})$. Since $\frac{1}{\tau_1}$ and $\frac{1}{\tau_2}$ are rationally linearly independent, by definition 5.1 in [@P.Walters1982], $T_t: [0,\tau_1)\times [0,\tau_2) \rightarrow [0,\tau_1)\times [0,\tau_2)$ defined by $$T_t(s_1,s_2)=(t+s_1\mod \tau_1,\ t+s_2\mod \tau_2), \text{ for all } s_1,s_2\in [0,\tau_1)\times [0,\tau_2)$$ is a minimal ratation. Then applying Theorem 6.20 in [@P.Walters1982], we know that $\frac{1}{\tau_1\tau_2}L$ is a unique ergodic probability measure on $[0,\tau_1)\times [0,\tau_2)$, where $L$ present the Lebesgue measures. Hence by Birkhoff’s ergodic theory, $$\begin{split}
\bar{\tilde{\mu}}_T&=\frac{1}{T}\int_{0}^{T}\tilde{\mu}_tdt\\
&=\frac{1}{T}\int_{0}^{T}\hat{\mu}_{T_t(0,0)}dt\\
&\xrightarrow{T\rightarrow \infty}\int_{[0, \tau_1) \times [0, \tau_2)}\hat{\mu}_{s_1,s_2}\frac{1}{\tau_1\tau_2}ds_1ds_2.
\end{split}$$ So $$\bar{\tilde{\mu}}=\int_{[0, \tau_1) \times [0, \tau_2)}\hat{\mu}_{s_1,s_2}\frac{1}{\tau_1\tau_2}ds_1ds_2=\frac{1}{\tau_1\tau_2}\int_0^{\tau_1}\int_0^{\tau_2}\delta_{s_1}\times\delta_{s_2}\times\tilde{\rho}_{s_1,s_2}ds_1ds_2$$ is an invariant measure with respect to $\tilde{P}^*.$
Uniqueness: We need to prove that for any invariant probability measure $\upsilon$, we have $\upsilon=\bar{\tilde{\mu}}$. By Lemma \[Two measure are the same\], we only need to prove that for any open set $\tilde{\mathcal{O}}\in \mathcal{B}(\tilde{\mathbb{X}})$, we have $\upsilon(\tilde{\mathcal{O}})\geq \bar{\tilde{\mu}}(\tilde{\mathcal{O}})$. Define $$\tilde{\mathcal{O}}^{r_1,r_2}=\{x\in \mathbb{R}^d: (r_1\mod \tau_1, \ r_2\mod \tau_2, x)\in \tilde{\mathcal{O}}\},$$ $$\tilde{\mathcal{O}}^{r_1,r_2}_{\delta}=\{x: dist(x, (\tilde{\mathcal{O}}^{r_1,r_2})^c)>\delta\},$$ and $$\tilde{\mathcal{O}}^{\delta}=\bigcup_{(s_1,s_2)\in [0,\tau_1)\times [0,\tau_2)}(s_1,s_2)\times\tilde{\mathcal{O}}^{s_1,s_2}_{\delta}.$$ We know that $\tilde{\mathcal{O}}^{r_1,r_2}, \tilde{\mathcal{O}}^{r_1,r_2}_{\delta}$ and $\tilde{\mathcal{O}}^{\delta}$ are open sets, $\tilde{\mathcal{O}}^{r_1,r_2}_{\delta}\uparrow \tilde{\mathcal{O}}^{r_1,r_2}$ and $\tilde{\mathcal{O}}^{\delta}\uparrow \tilde{\mathcal{O}}$ as $\delta \downarrow 0$. Then $$\label{Equation of invariant measure v}
\begin{split}
\upsilon\left(\tilde{\mathcal{O}}\right)
=&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T\tilde{P}^*_t\upsilon\left(\tilde{\mathcal{O}}\right)dt\\
=&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T\int_{\tilde{\mathbb X}}\tilde{P}\left(t,(s_1,s_2,x),\tilde{\mathcal{O}}\right)\upsilon(d\tilde{x})dt\\
=&\lim_{T\rightarrow \infty}\int_{\tilde{\mathbb X}} \frac{1}{T}\int_0^T P\left(K^{s_1,s_2}(t,0,x,\cdot)\in \tilde{\mathcal{O}}^{t+s_1,t+s_2}\right)dt\upsilon\left(d\tilde{x}\right).
\end{split}$$ Applying Remark \[Solution K shift time in r\] and measure preserving transformation $\theta_t$, it follows that $$\upsilon\left(\tilde{\mathcal{O}}\right)=\lim_{T\rightarrow \infty}\int_{\tilde{\mathbb X}} \frac{1}{T}\int_0^TP\left(K^{t+s_1,t+s_2}(0,-t,x,\cdot)\in \tilde{\mathcal{O}}^{t+s_1,t+s_2}\right)dt\upsilon(d\tilde{x}).$$ Similar to the proof of Theorem \[Existence and uniqueness of random path\], Lemma \[Bounded solution\] and Lemma \[Exponential decay\], it can be shown that the solution $K^{r_1,r_2}$ of (\[Solution K\_r\_1,r\_2\]) has the following estimate $$\|K^{r_1,r_2}(t,s,x)-\tilde{\varphi}^{r_1,r_2}(t)\|_2\leq Ce^{-(\alpha-\beta^2/2)(t-s)},$$ for all $r_1,r_2\in\mathbb{R}, t\geq s$, where $C=C(\alpha,\beta,\tilde{M})$ only depends on $\alpha,\beta,\tilde{M}$ with $\tilde{M}=\sup_{t,s\in \mathbb{R}}(|\tilde{b}(t,s,0)|+\|\tilde{\sigma}(t,s,0)\|).$ Then for all $\delta>0$, by Chebyshev’s inequality, we have $$\label{Equation invariant measure v part 2}
\begin{split}
&P\left(K^{t+s_1,t+s_2}(0,-t,x,\cdot)\in \tilde{\mathcal{O}}^{t+s_1,t+s_2}\right)\\
\geq&P\left(\tilde{\varphi}^{t+s_1,t+s_2}(0)\in \tilde{\mathcal{O}}^{t+s_1,t+s_2}_{\delta}, \ |K^{t+s_1,t+s_2}(0,-t,x)-\tilde{\varphi}^{t+s_1,t+s_2}(0)|<\delta\right)\\
\geq&P\left(\tilde{\varphi}^{t+s_1,t+s_2}(0)\in \tilde{\mathcal{O}}^{t+s_1,t+s_2}_{\delta}\right)-P\left(|K^{t+s_1,t+s_2}(0,-t,x)-\tilde{\varphi}^{t+s_1,t+s_2}(0)|\geq\delta\right)\\
\geq&\tilde{\rho}_{t+s_1,t+s_2}\left(\tilde{\mathcal{O}}^{t+s_1,t+s_2}_{\delta}\right)-\frac{C^2}{\delta^2}e^{-2(\alpha-\beta^2/2)t}\\
=&\hat{\mu}_{t+s_1,t+s_2}\left(\tilde{\mathcal{O}}^{\delta}\right)-\frac{C^2}{\delta^2}e^{-2(\alpha-\beta^2/2)t}.\\
\end{split}$$ Thus it turns out from (\[Equation of invariant measure v\]), (\[Equation invariant measure v part 2\]) and Fatou’s Lemma that $$\begin{split}
\upsilon\left(\tilde{\mathcal{O}}\right)
\geq&\liminf_{T\rightarrow \infty}\int_{\tilde{\mathbb X}} \frac{1}{T}\int_0^T\left(\hat{\mu}_{t+s_1,t+s_2}\left(\tilde{\mathcal{O}}^{\delta}\right)-\frac{C^2}{\delta^2}e^{-2(\alpha-\beta^2/2)t}\right)dt\upsilon(d\tilde{x})\\
% \geq&\liminf_{T\rightarrow \infty}\int_{\tilde{\mathbb X}} \frac{1}{T}\int_0^T\left(\hat{\mu}_{t+s_1,t+s_2}\left(\tilde{\mathcal{O}}^{\delta}\right)-\frac{C^2}{\delta^2}e^{-2(\alpha-
%\beta^2/2)t}\right)dt\upsilon(d\tilde{x})\\
\geq&\int_{\tilde{\mathbb X}}\left(\liminf_{T\rightarrow \infty} \frac{1}{T}\int_0^T\hat{\mu}_{t+s_1,t+s_2}\left(\tilde{\mathcal{O}}^{\delta}\right)dt-\lim_{T\rightarrow \infty}\frac{C^2}{2\delta^2(\alpha-\beta^2/2)T}\right)\upsilon(d\tilde{x})\\
\geq&\int_{\tilde{\mathbb X}}\left(\liminf_{T\rightarrow \infty} \frac{1}{T}\int_0^T\hat{\mu}_{t+s_1,t+s_2}\left(\tilde{\mathcal{O}}^{\delta}\right)dt\right)\upsilon(d\tilde{x}).
\end{split}$$ Again by Birkhoff’s ergodic theory, we know that for all $(s_1,s_2)\in \mathbb{R}^2$ $$\frac{1}{T}\int_0^T\hat{\mu}_{t+s_1,t+s_2}dt\xrightarrow{T\rightarrow\infty} \bar{\tilde{\mu}}.$$ Then since $\mathcal{O}^{\delta}$ is open, and by Proposition 2.4 in [@N.; @Ikeda-S.; @Watanabe], we have $$\upsilon\left(\tilde{\mathcal{O}}\right) \geq \bar{\tilde{\mu}}\left(\mathcal{O}^{\delta}\right).$$ Since $\mathcal{O}^{\delta}\uparrow \mathcal{O}$ as $\delta\downarrow 0$, the desired result follows from the continuity of measures with respect to an increasing sequence of sets.
It is not obvious how to check directly that $\frac{1}{\tau_1\tau_2}\int_0^{\tau_1}\int_0^{\tau_2}\delta_{s_1}\times\delta_{s_2}\times\tilde{\rho}_{s_1,s_2}ds_1ds_2$ is an invariant measure with respect to $\tilde{P}^*$ without appealing to the tightness argurement.
By a similar proof of Lemma \[Lemma of tight measure set\], Proposition \[Feller property of P\^\*\] and Theorem \[Invariant measure\], it is not difficult to derive a general theorem. Here we denote by $\mathbb{X}$ a metric space, $\mathcal{B}(\mathbb{X})$ the Borel $\sigma$-algebra on $\mathbb{X}$, $B_b(\mathbb{X})$ the linear space of all $\mathcal{B}(\mathbb{X})$-bounded measurable functions and $\mathcal{P}(\mathbb{X})$ the collection of all probability measures on $(\mathbb{X}, \mathcal{B}(\mathbb{X}))$. Assume that $P(t,x,\Gamma), t\geq 0, x\in\mathbb{X}, \Gamma \in \mathcal{B}(\mathbb{X})$, is a Markovian transition function on $\mathbb{X}$. Denote by $P_t, t\geq 0: B_b(\mathbb{X}) \rightarrow B_b(\mathbb{X})$ and $P^*_t, t\geq 0: \mathcal{P}(\mathbb{X}) \rightarrow \mathcal{P}(\mathbb{X})$, the Markovian semi-groups associated with $P(t,x,\cdot)$. We say $\rho: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{X})$ is an entrance measure with respect to $P^*$ if $P^*_t\rho_s=\rho_{t+s}$ for all $t\in \mathbb{R}^+, s\in \mathbb{R}$. We say $\rho$ is quasi-periodic if exists a measure-valued function $\tilde{\rho}_{s_1,s_2}$ satisfying the same relation with $\rho_s$ as in Definition \[Definition of quasi-periodic measure\]. However we do not have the uniqueness of invariant measure in the general case.
Assume the entrance measure $\rho$ with respect to $P_t^*, t\geq 0$, is a quasi-periodic measure with periods $\tau_1$ and $\tau_2$, where the reciprocals of $\tau_1$ and $\tau_2$ are rationally linearly independent. If $\{\bar{\rho}_T=\frac{1}{T}\int_0^T\rho_sds: T\in \mathbb{R}^+ \}$ is tight and the Markovian semi-group $P_t, t\geq 0$, is Feller, then there exists one invariant measure given by $$\frac{1}{\tau_1\tau_2}\int_0^{\tau_1}\int_0^{\tau_2}\tilde{\rho}_{s_1,s_2}ds_1ds_2.$$
Density of entrance measure and quasi-periodic measure
======================================================
In this section, we will give a sufficient condition to guarantee the existence of the density of the entrance measure. We need an extra condition.
\[Sigma invertible and bounded\] Assume that $b, \sigma$ in SDE (\[SDE\]) satisfy the following conditions:
(1)
: $\sigma$ is invertible and $\sup_{t\in \mathbb{R}}\|\sigma^{-1}(t,x)\|<\infty$;
(2)
: $b(t,x)$ is continuous with respect to $t,x$.
We now give the definition of the well-known BMO space and some lemmas which will used in this section.
Denote by BMO(s,t) the space of all $(\mathcal{F}_s^r)_{s\leq r\leq t}$-adapted $\mathbb{R}^d$-valued process $M$ with $$\|M\|_{BMO(s,t)}:=\sup_{T\in \mathcal{T}_s^t}\left\|\left(\mathbb{E}\left[\int_{T}^{t}|M_r|^2dr|\mathcal{F}_s^T\right]\right)^{\frac{1}{2}}\right\|_{L^{\infty}}<\infty,$$ where $s<t$ and $\mathcal{T}_s^t$ is the set of stopping times taking their values in $[s,t]$.
Then we have the following lemma.
\[BMO implies Lp\] Let $M\in BMO(s,t)$. Then there exists $p>1$ such that $$\mathbb{E}\left[\left(\mathcal{E}\left(\int_{s}^{t}M_rdW_r\right)\right)^p\right]<\infty,$$ where $\mathcal{E}\left(\int_{s}^{t}M_rdW_r\right):=\exp\{\int_{s}^{t}M_rdW_r-\frac{1}{2}\int_{s}^{t}|M_r|^2dr\}$.
By Theorem 3.1 in [@Kazamaki94], we know that if $\|M\|_{BMO(s,t)}\leq \Phi(p)$ for some $p>1$, where $\Phi$ is a continuous monotone function from $(1,\infty)$ to $\mathbb{R}_+$ with $\Phi(1+)=\infty$ and $\Phi(\infty)=0$, then $\mathcal{E}\left(\int_{s}^{t}M_rdW_r\right)$ is in $L^p$.
We also need the following lemma which is almost the same as Lemma 4.1 in [@FZZ19].
\[Equivalent law\] Assume Conditions \[Dissipative\], \[Lipschitz and bounded\] and \[Sigma invertible and bounded\] hold. Let $X_t^{s,x}$ be the solution of SDE (\[SDE\]) and $Z_t^{s,x}$ be the solution of the following SDE $$\label{Diffusion SDE}
\begin{cases}
dZ_t=\sigma(t, Z_t)dW_t, \quad t\geq s,\\
Z_s=x\in \mathbb{R}^{d}.
\end{cases}$$ Then the laws of $X_t^{s,x}$ and $Z_t^{s,x}$ are equivalent, i.e. $$P^{X_t^{s,x}}(B)=\tilde{P}^{Z_t^{s,x}}(B), \text{ for all } B\in \mathcal{B}(\mathbb{R}^d),$$ where $\frac{d\tilde{P}}{dP}=\mathcal{E}\left(\int_{s}^{t}\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right)$
This lemma can be proved by almost the same proof as them of Lemma 4.1 in [@FZZ19].
Now we have the following theorem.
\[Existence of density\] Assume Conditions \[Dissipative\], \[Lipschitz and bounded\] and \[Sigma invertible and bounded\] hold. If $\alpha>\frac{\beta^2}{2}$, then $P(t,s,x,\cdot)$ and the entrance measure $\rho_t$ are absolutely continuous with respect to the Lebesgue measure $L$ on $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$, and hence have the density $p(t,s,x,y)$ and $q(t,y)$ respectively.
First we prove that $P(t,s,x,\cdot)$ is absolutely continuous with respect to $L$, i.e. for any $\Gamma\in \mathcal{B}(\mathbb{R}^d)$, $L(\Gamma)=0$ implies $P(t,s,x,\Gamma)=P(X_t^{s,x}\in \Gamma)=0$. By Lemma \[Equivalent law\], we know that $$\label{equation of Girsonov}
\begin{split}
P(X_t^{s,x}\in \Gamma)&=\tilde{P}(Z_t^{s,x}\in \Gamma)
=\mathbb{E}_{\tilde{P}}[1_{\Gamma}(Z_t^{s,x})]\\
&=\mathbb{E}\left[\mathcal{E}\left(\int_{s}^{t}\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right)1_{\Gamma}(Z_t^{s,x})\right],
\end{split}$$ where $Z_t^{s,x}$ is the solution of SDE (\[Diffusion SDE\]). Set $T_n:=\inf_{t\geq s}\{|Z_t^{s,x}|\geq n\}$. Since $\mathbb{E}_{\tilde{P}}[\sup_{r\in [s,t]}|Z_r^{s,x}|^2]<\infty$, then we have $$\tilde{P}(T_n>t)=\tilde{P}(\sup_{r\in [s,t]}|Z_r^{s,x}|\leq n)\rightarrow 1 \text{ as } n\rightarrow \infty.$$ Thus $$\begin{split}
P(X_t^{s,x}\in \Gamma)&=\mathbb{E}_{\tilde{P}}[1_{\Gamma}(Z_t^{s,x})]\\
&=\mathbb{E}_{\tilde{P}}[1_{\Gamma}(Z_t^{s,x})1_{[s,T_n]}(t)]+\mathbb{E}_{\tilde{P}}[1_{\Gamma}(Z_t^{s,x})1_{(T_n, \infty)}(t)]\\
&\leq \lim_{n\rightarrow \infty}[\mathbb{E}_{\tilde{P}}[1_{\Gamma}(Z_t^{s,x})1_{[s,T_n]}(t)]+\tilde{P}(T_n<t)]\\
&=\lim_{n\rightarrow \infty}\mathbb{E}_{\tilde{P}}[1_{\Gamma}(Z_t^{s,x})1_{[s,T_n]}(t)]\\
&=\lim_{n\rightarrow \infty}\mathbb{E}\left[1_{[s,T_n]}(t)\mathcal{E}\left(\int_{s}^{t}\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right)1_{\Gamma}(Z_t^{s,x})\right].\\
\end{split}$$ Since $$1_{[s,T_n]}(t)\mathcal{E}\left(\int_{s}^{t}\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right) \leq \mathcal{E}\left(\int_{s}^{t}1_{[s,T_n]}(r)\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right),$$ we have $$\begin{split}
P(X_t^{s,x}\in \Gamma)
&\leq \liminf_{n\rightarrow \infty}\mathbb{E}\left[\mathcal{E}\left(\int_{s}^{t}1_{[s,T_n]}(r)\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right)1_{\Gamma}(Z_t^{s,x})\right].
\end{split}$$ We only need to prove that if $L(\Gamma)=0$, then for all $n$ $$\mathbb{E}\left[\mathcal{E}\left(\int_{s}^{t}1_{[s,T_n]}(r)\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right)1_{\Gamma}(Z_t^{s,x})\right]=0.$$ Let $a_n(r)=1_{[s,T_n]}(r)\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})$. By Condition \[Sigma invertible and bounded\], we know that there exists $C>0$ such that $\sup_{r\in \mathbb{R}}|a_n(r)|\leq C$. Then $$\sup_{T\in \mathcal{T}_s^t}\left\|\left(\mathbb{E}\left[\int_{T}^{t}|a_n(r)|^2dr|\mathcal{F}_s^T\right]\right)^{\frac{1}{2}}\right\|_{L^{\infty}}\leq C\sqrt{t-s},$$ which means $a_n \in BMO(s,t)$. By Lemma \[BMO implies Lp\], there exists $p>1$ such that $$\gamma_n:=\left(\mathbb{E}\left[\left(\mathcal{E}\left(\int_{s}^{t}a_n(r)dW_r\right)\right)^p\right]\right)^{\frac{1}{p}}<\infty.$$ Since $Z_t^{s,x}=x+\int_{s}^{t}\sigma(r,Z_r^{s,x})dW_r$, note that $\int_{s}^{t}\sigma(r,Z_r^{s,x})dW_r$ is in law a Brownian motion with time $\hat{\sigma}_t=\int_{s}^{t}\|\sigma(r,Z_r^{s,x})\|^2dr$, i.e. there exists a standard Brownian motion $\tilde{W}$ such that $\int_{s}^{t}\sigma(r,Z_r^{s,x})dW_r\stackrel{d}{=}\tilde{W}_{\hat{\sigma}_t}$. Also notice $$\sqrt{d}=\|\sigma(t,x)\sigma^{-1}(t,x)\|\leq \|\sigma(t,x)\| \|\sigma^{-1}(t,x)\|,$$ thus $$\|\sigma(t,x)\|\geq \frac{\sqrt{d}}{\|\sigma^{-1}(t,x)\|}\geq \frac{\sqrt{d}}{\sup_{t\in \mathbb{R}, x\in \mathbb{R}^d}\|\sigma^{-1}(t,x)\|}=:\underline{\sigma},$$ which suggests that $\hat{\sigma}_t\geq \underline{\sigma}(t-s)$. Using Proposition 6.17 in Chapter 2 in [@Karatzas91], we have $$\begin{split}
\mathbb{E}\left[1_{\Gamma}(Z_t^{s,x})\right]
&=\mathbb{E}\left[1_{\Gamma}(x+\tilde{W}_{\hat{\sigma_t}})\right]\\
&=\mathbb{E}\left[\mathbb{E}\left[1_{\Gamma}(x+\tilde{W}_{\hat{\sigma_t}})|\mathcal{F}_{\hat{\sigma}_t-\underline{\sigma}(t-s)}\right]\right]\\
&=\mathbb{E}\left[\mathbb{E}\left[1_{\Gamma}(x+y+\tilde{W}_{\underline{\sigma}(t-s)})\right]\left|_{y=\tilde{W}_{\hat{\sigma}_t-\underline{\sigma}(t-s)}}\right.\right].
\end{split}$$ Note $$\begin{split}
\mathbb{E}\left[1_{\Gamma}(x+y+\tilde{W}_{\underline{\sigma}(t-s)})\right]
&=\frac{1}{(2\pi\underline{\sigma}(t-s))^{d/2}|\det\Sigma|^{1/2}}\int_{\mathbb{R}^d}1_{\Gamma}(x+y+z)e^{-(1/2\underline{\sigma}(t-s))|\Sigma^{-1/2} z|^2}dz\\
&\leq \frac{1}{(2\pi\underline{\sigma}(t-s))^{d/2}|\det\Sigma|^{1/2}}L(\Gamma),
\end{split}$$ where $W_1\sim \mathcal{N}(0, \Sigma)$. Then $$\mathbb{E}\left[1_{\Gamma}(Z_t^{s,x})\right]\leq \frac{1}{(2\pi\underline{\sigma}(t-s))^{d/2}|\det\Sigma|^{1/2}}L(\Gamma).$$ Let $q$ be the dual number of $p$. Then by Cauchy-Schwarz inequality, $$\begin{split}
\mathbb{E}\left[\mathcal{E}\left(\int_{s}^{t}1_{[s,T_n]}(r)\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right)1_{\Gamma}(Z_t^{s,x})\right]
&\leq \gamma_n \{\mathbb{E}[1_{\Gamma}(Z_t^{s,x})]\}^{\frac{1}{q}}\\
&\leq C_n\cdot L(\Gamma)^{\frac{1}{q}},
\end{split}$$ where $C_n=\gamma_n\cdot \left(\frac{1}{(2\pi\underline{\sigma}(t-s))^{d/2}|\det\Sigma|^{1/2}}\right)^{\frac{1}{q}}$.
So if $L(\Gamma)=0$, then $\mathbb{E}\left[\mathcal{E}\left(\int_{s}^{t}1_{[s,T_n]}(r)\sigma^{-1}(r,Z_r^{s,x})b(r,Z_r^{s,x})dW_r\right)1_{\Gamma}(Z_t^{s,x})\right]=0$, and hence $P(t,s,x,\Gamma)=P(X_t^{s,x}\in \Gamma)=0$. Thus $P(t,s,x,\cdot)$ is absolutely continuous with respect to the Lebesgue measure and by Radon-Nikodym theorem, the density of $P(t,s,x,\cdot)$ with respect to the Lebesgue measure exists.
For the entrance measure $\rho_t$, since $$\label{Entrance measure of markov kernel}
\rho_t(\Gamma)=P^*(t,s)\rho_s(\Gamma)=\int_{\mathbb{R}^d}P(t,s,x,\Gamma)\rho_s(dx),$$ then if $L(\Gamma)=0$, we have $\rho_t(\Gamma)=0$. This also suggests that $\rho_t$ is absolutely continuous with respect to $L$ and thus its density exists.
We already know the conditions to guarantee the existence of the density $p(t, s, x, y)$ and $q(t,y)$ of the two- parameter Markov transition kernel $P(t,s,x,\cdot)$ and entrance measure $\rho_t$ respectively. By Fubini theorem, we know that $$\rho_t(\Gamma)=\int_{\mathbb{R}^d}P(t,s,x,\Gamma)\rho_s(dx)=\int_{\Gamma}\int_{\mathbb{R}^d}p(t,s,x,y)\rho_s(dx)dy=\int_{\Gamma}\int_{\mathbb{R}^d}p(t,s,x,y)q(s,x)(dx)dy.$$ Then it is obvious that $$\label{Equation of entrance measure density}
q(t,y)=\int_{\mathbb{R}^d}p(t,s,x,y)q(s,x)(dx).$$ Moreover it is well-known that $p(\cdot,s,x,\cdot)$ satisfies the following Fokker-Planck equation $$\label{Fokker-Planck equation}
\partial_tp(t,s,x,y)=\mathcal{L}^*(t)p(t,s,x,y),$$ where $\mathcal{L}^*(t)p$ is the Fokker Planck operator given by $$\label{Fokker-Planck operator}
\mathcal{L}^*(t)p=-\sum_{i=1}^d\partial_{x_i}(b_i(t,y)p)+\frac{1}{2}\sum_{i,j=1}^d\partial_{x_ix_j}^2 \left(\sigma\sigma^T_{ij}(t,y)p\right).$$
Now we have the following theorem.
\[Theorem Fokker-Planck\] Assume the same assumptions as in Theorem \[Existence of density\]. Let $q \in C_+^{1,2}(\mathbb{R}\times \mathbb{R}^d)\bigcap L^1(\mathbb{R}^d)$ with $\|q(t,\cdot)\|_{L^1(\mathbb{R}^d)}=1$ for all $t$, and define $\rho: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{R}^d)$ by $$\rho_t(\Gamma)=\int_{\Gamma}q(t,y)dy, \text{ for all } t\in \mathbb{R}.$$ Then $\rho$ is an entrance measure if and only if $$\label{Equation of Fokker-Planck}
\partial_tq=\mathcal{L}^*(t)q.$$ Hence the solution of (\[Equation of Fokker-Planck\]) and the entrance measure have one to one correspondence.
Assume first that $\rho$ is an entrance measure. We already know that $p,q$ satisfy (\[Equation of entrance measure density\]) and $p(t,s,x,y)$ satisfies Fokker-Planck equation (\[Fokker-Planck equation\]). We take the derivative with respect to t on both side of (\[Equation of entrance measure density\]) to have $$\begin{split}
\partial_tq(t,x)=&\int_{\mathbb{R}^d}\partial_tp(t,s,y,x)q(s,y)dy\\
=&\int_{\mathbb{R}^d}\mathcal{L}^*(t)p(t,s,y,x)q(s,y)dy\\
=&\int_{\mathbb{R}^d}-\sum_{i=1}^d\partial_{x_i}(b_i(t,x)p(t,s,y,x))q(s,y)dy\\
&+\int_{\mathbb{R}^d}\frac{1}{2}\sum_{i,j=1}^d\partial_{x_ix_j}^2 \left(\sigma\sigma^T_{ij}(t,x)p(t,s,y,x)\right)q(s,y)dy\\
=:& I+\uppercase\expandafter{\romannumeral2}.
\end{split}$$ For the first part, we have $$\begin{split}
I&=-\sum_{i=1}^d\int_{\mathbb{R}^d}[\partial_{x_i}(b_i(t,x))p(t,s,y,x) +b_i(t,x)\partial_{x_i}(p(t,s,y,x))]q(s,y)dy\\
&=-\sum_{i=1}^d\partial_{x_i}(b_i(t,x))\int_{\mathbb{R}^d}p(t,s,y,x)q(s,y)dy -\sum_{i=1}^d b_i(t,x)\partial_{x_i}\int_{\mathbb{R}^d}p(t,s,y,x)q(s,y)dy\\
&=-\sum_{i=1}^d\partial_{x_i}(b_i(t,x))q(t,x) -\sum_{i=1}^d b_i(t,x)\partial_{x_i}q(t,x)\\
&=-\sum_{i=1}^d\partial_{x_i}(b_i(t,x)q(t,x)).
\end{split}$$ Similarly, for the second part, we have $$II=\frac{1}{2}\sum_{i,j=1}^d\partial_{x_ix_j}^2 \left(\sigma\sigma^T_{ij}(t,x)q(t,x)\right).$$ Hence the density function $q(t,x)$ of entrance measure $\rho_t$ satisfies $$\partial_tq=\mathcal{L}^*(t)q.$$
Conversely, if $q$ is the solution of (\[Fokker-Planck equation\]), then by Fubini’s theorem, we have for all $\Gamma\in \mathcal{B}(\mathbb{R}^d)$ $$\begin{split}
P^*(t,s)\rho_s(\Gamma)&=\int_{\mathbb{R}^d}P(t,s,y,\Gamma)\rho_s(dy)\\
&=\int_{\mathbb{R}^d}\int_{\Gamma}p(t,s,y,x)dxq(s,y)dy\\
&=\int_{\Gamma}\int_{\mathbb{R}^d}p(t,s,y,x)q(s,y)dydx\\
&=\int_{\Gamma}q(t,x)dx\\
&=\rho_t(\Gamma)
\end{split}$$ which means $\rho$ is a entrance measure.
Since the entrance measure is unique under the assumption in Theorem \[Theorem Fokker-Planck\], we know that the solution of the following Fokker-Planck equation $$\begin{cases}
\partial_tq=\mathcal{L}^*(t)q\\
q(0,\cdot)\in C_+^2(\mathbb{R}^d)\bigcap L^1(\mathbb{R}^d), \|q(0,\cdot)\|_{L^1(\mathbb{R}^d)}=1.
\end{cases}$$ is unique.
Now assume that $u^r(t,s,x)$ and $K^{r_1,r_2}(t,s,x)$ are the solutions of equation (\[Solution u\_r\]) and (\[Solution K\_r\_1,r\_2\]) respectively, and the corresponding semi-groups $P^r, P^{r_1,r_2}$ defined as $$\begin{cases}
P^r(t,s,x,\Gamma):=P(u^r(t,s,x)\in\Gamma)\\
P^{r_1.r_2}(t,s,x,\Gamma):=P(K^{r_1,r_2}(t,s,x)\in\Gamma).
\end{cases}$$ We can also define $P^{t,*}(t,s) \ (resp. \ P^{r_1,r_2,*}(t,s))$ as in (\[Define measure transition P\^\*\]) when we replace $\{P^*(t,s), P(t,s,x,\Gamma)\}$ by $\{P^{r,*}(t,s), P^r(t,s,x,\Gamma)\} \ (resp. \ \{P^{r_1,r_2,*}(t,s), P^{r_1.r_2}(t,s,x,\Gamma)\})$. Let $\varphi^r(t), \varphi^{r_1,r_2}(t)$ be defined as in (\[L2-lim K u\]), and $\rho^r_t, \rho^{r_1,r_2}_t$ be the laws of $\varphi^r(t), \varphi^{r_1,r_2}(t)$ respectively. Then we have $$P^{r,*}(t,s)\rho^r_s=\rho^r_t, \ P^{r_1,r_2,*}(t,s)\rho^{r_1,r_2}_s=\rho^{r_1,r_2}_t.$$ Similar to Condition \[Sigma invertible and bounded\], we give the following condition.
\[Quasi sigma invertible and bounded\] Assume that $\tilde{b}, \tilde{\sigma}$ in Condition \[Quasi-periodic condition\] satisfy the following conditions:
(1)
: $\tilde{\sigma}$ is invertible and $\sup_{t,s\in \mathbb{R}}\|\tilde{\sigma}^{-1}(t,s,x)\|<\infty$;
(2)
: $\tilde{b}(t,s,x)$ is continuous with respect to $t,s,x$.
Then by Theorem \[Existence and uniqueness of entrance measure\] and Theorem \[Existence of density\], we can directly deduce the following theorem
Assume Conditions \[Quasi-periodic condition\], \[Quasi-dissipative\] and \[Quasi sigma invertible and bounded\] hold. If $\alpha>\frac{\beta^2}{2}$, then $\rho^r, \rho^{r_1,r_2}$ are the entrance measures of equation (\[Solution u\_r\]) and (\[Solution K\_r\_1,r\_2\]) respectively. Moreover $P^r(t,s,x,\cdot), P^{r_1,r_2}(t,s,x,\cdot)$ and the entrance measures $\rho^r_t, \rho^{r_1,r_2}_t$ are absolutely continuous with respect to the Lebesgue measure $L$ on $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$, and hence have the density $p^r(t,s,x,y)$, $p^{r_1,r_2}(t,s,x,y)$, $q^r(t,y)$, $q^{r_1,r_2}(t,y)$ respectively.
Similarly, we know that $$q^r(t,x)=\int_{\mathbb{R}^d}p^r(t,s,y,x)q^r(s,y)(dy)$$ and $$q^{r_1,r_2}(t,x)=\int_{\mathbb{R}^d}p^{r_1,r_2}(t,s,y,x)q^{r_1,r_2}(s,y)(dy).$$ Moreover, $q^r$, $q^{r_1,r_2}$ satisfy the following Fokker-Planck equations $$\partial_tq^r=\mathcal{L}^{r,*}(t)q^r, \quad \partial_tq^{r_1,r_2}=\mathcal{L}^{r_1,r_2,*}(t)q^{r_1,r_2},$$ where $\mathcal{L}^{r,*}$ and $\mathcal{L}^{r_1,r_2,*}$ are given by (\[Fokker-Planck operator\]) where $b,\sigma$ are replaced by $\tilde{b}^r, \tilde{\sigma}^r$ and $\tilde{b}^{r_1,r_2}, \tilde{\sigma}^{r_1,r_2}$ respectively.
By the proof of Theorem \[Existence and uniqueness of quasi-periodic random path\], we know that $u^r(t,s,x,\cdot)=u(t+r,s+r,x,\theta_{-r}\cdot)$ and $\varphi^{r}(t,\cdot)=\varphi(t+r,\theta_{-r}\cdot)$. Since $\theta_{-r}$ preserves the probability measure $P$, then $P^r(t,s,x,\cdot)=P(t+r,s+r,x,\cdot)$ and $\rho^r_t=\rho_{t+r}$. Hence their densities have the following relations $$p^r(t,s,x,y)=p(t+r,s+r,x,y), \quad q^r(t,x)=q(t+r,x).$$
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Kening Lu and Hans Crauel for raising our interests to consider random quasi-periodicity in various occasions. We acknowledge the financial support of a Royal Society Newton Fund (Ref NA150344) and an EPSRC Established Career Fellowship to HZ (Ref EP/S005293/1).
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J. J. Lucia and E. Schwartz, Electricity prices and power derivatives: Evidence from the Nordic Power Exchange, [*E.S. Review of Derivatives Research*]{}, Vol 5 (2002), 5-50.
J. K. Moser, On invariant curves of area-preserving mappings of an annulus, [*Nach. Akad. Wiss. Göttingen, Math. Phys.*]{} (1962) Kl. II 1 : 1-20.
P. Walters, [*An Introduction to Ergodic Theory, Graduate Tests in Mathematics*]{}, 79, [Springer-Verlag New York]{} (1982).
B.X. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, [*Nonlinear Analysis*]{}, Vol. 103 (2014), 9-25.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The BCS-BEC crossover is studied in a systematic way in the broken-symmetry phase between zero temperature and the critical temperature. This study bridges two regimes where quantum and thermal fluctuations are, respectively, important. The theory is implemented on physical grounds, by adopting a fermionic self-energy in the broken-symmetry phase that represents fermions coupled to superconducting fluctuations in weak coupling and to bosons described by the Bogoliubov theory in strong coupling. This extension of the theory beyond mean field proves important at finite temperature, to connect with the results in the normal phase. The order parameter, the chemical potential, and the single-particle spectral function are calculated numerically for a wide range of coupling and temperature. This enables us to assess the quantitative importance of superconducting fluctuations in the broken-symmetry phase over the whole BCS-BEC crossover. Our results are relevant to the possible realizations of this crossover with high-temperature cuprate superconductors and with ultracold fermionic atoms in a trap.'
author:
- 'P. Pieri, L. Pisani, and G.C. Strinati'
title: 'BCS-BEC crossover at finite temperature in the broken-symmetry phase'
---
Introduction
============
In the BCS to Bose-Einstein condensation (BEC) crossover [@Eagles-69; @Leggett-80; @NSR-85; @Randeria-90; @Haussmann-93; @PS-94; @PS-96; @Levin-97; @Zwerger-97; @Pi-S-98], largely overlapping Cooper pairs smoothly evolve into non-overlapping composite bosons as the fermionic attraction is progressively increased. These two physical situations (Cooper pairs vs composite bosons) correspond to the weak- and strong-coupling limits of the theory, while in the interesting intermediate-coupling regime neither the fermionic nor the bosonic properties are fully realized. Under these circumstances, the theory is fully controlled on the weak- and strong-coupling sides, while at intermediate coupling an interpolation scheme results (as for all crossover approaches). These physical ideas are implemented, in practice, by allowing for a strong decrease of the chemical potential [*at a given temperature*]{} when passing from the weak- to the strong-coupling limit.
The BCS-BEC crossover can be considered both below (broken-symmetry phase) and above (normal phase) the superconducting critical temperature. In particular, in the normal phase preformed pairs exist in the strong-coupling limit up to a temperature $T^*$ corresponding to the breaking of the pairs, while coherence among the pairs is established when the temperature is lowered below the superconducting critical temperature $T_c$. This framework could be relevant to the evolution of the properties of high-temperature cuprate superconductors from the overdoped (weak-coupling) to the underdoped (strong-coupling) regions of their phase diagram [@Damascelli]. The BCS-BEC crossover can be also explicitly realized with ultracold fermionic atoms in a trap, by varying their mutual effective attractive interaction via a Fano-Feshbach resonance [@expcross].
The BCS-BEC crossover has been studied extensively in the past, either at $T=0$ or for $T\ge T_c$. At $T=0$, the solution of the two coupled BCS (mean-field) equations for the order parameter $\Delta$ and the chemical potential $\mu$ has been shown to cross over smoothly from a BCS weak-coupling superconductor with largely overlapping Cooper pairs to a strong-coupling superconductor where tightly-bound pairs are condensed in a Bose-Einstein (coherent) ground state [@Eagles-69; @Leggett-80; @MPS]. For this reason, the BCS mean field has often been considered to be a reliable approximation for studying the whole BCS-BEC crossover at $T=0$. At finite temperature, the increasing importance in strong coupling of the thermal excitation of collective modes (corresponding to noncondensed bosons) was first pointed out by Nozières and Schmitt-Rink [@NSR-85]. By their approach, the expected result that the superconducting critical temperature should approach the Bose-Einstein temperature $T_{{\rm BE}}$ in strong coupling was obtained (coming from [*above $T_c$*]{}) via a (first-order) inclusion of the $t$-matrix self-energy in the fermionic single-particle Green’s function. The same type of $t$-matrix approximation (also with the inclusion, by some authors, of self-consistency) has then been widely adopted to study the BCS-BEC crossover above $T_c$, both for continuum [@Haussmann-93] and lattice models [@Fresard; @Micnas; @Randeria-97-1; @KKK].
Despite its conceptual importance, a systematic study of the BCS-BEC crossover in the temperature range $0<T<T_c$ is still lacking. A diagrammatic theory for the BCS-BEC crossover that extends below $T_c$ the self-consistent $t$-matrix approximation was proposed some time ago by Haussmann [@Haussmann-93]. The ensuing coupled equations for the order parameter and chemical potential were, however, solved explicitly only at $T_c$, [@Haussmann-2] leaving therefore unsolved the problem of the study of the whole temperature region below $T_c$. The work by Levin and coworkers [@levin98], on the other hand, even though based on a “preformed-pair scenario”, has focused mainly on the weak-to-intermediate coupling region, where the fermionic chemical potential remains inside the single-particle band. An extension of the self-consistent $t$-matrix approximation to the superconducting phase for a two-dimensional lattice model was considered in Ref. . In that paper, however, the shift of the chemical potential associated with the increasing coupling strength was ignored, by keeping it fixed at the noninteracting value.[@footnote-1] The results of Ref. are thus not appropriate to address the BCS-BEC crossover, for which the renormalization of the chemical potential (that evolves from the Fermi energy in weak coupling to half the binding energy of a pair in strong coupling) plays a crucial role [@Eagles-69; @Leggett-80; @NSR-85]. Additional studies have made use of a fermion-boson model [@ranninger], especially in the context of trapped Fermi gases [@ohashi].
Purpose of the present paper is to study the BCS-BEC crossover in the superconducting phase over the whole temperature range from $T=0$ to $T=T_c$, thus filling a noticeable gap in the literature. We will consider a three-dimensional continuum model, for which the fermionic attraction can be modeled by a point-contact interaction. As noted in Refs. and , with this model the structure of the diagrammatic theory for the single-particle fermionic self-energy simplifies considerably, since only limited sets of diagrammatic structures survive the regularization of the contact potential in terms of the fermionic two-body scattering length $a_F$. [@Randeria-93; @Pi-S-98] The dimensionless interaction parameter $(k_{F} a_{F})^{-1}$ (where the Fermi wave vector $k_{F}$ is related to the density via $n=k_{F}^{3}/(3\pi^{2})$) then ranges from $-\infty$ in weak coupling to $+\infty$ in strong coupling. The crossover region of interest is, however, restricted in practice by $(k_{F} |a_{F}|)^{-1} \lesssim 1$.
For this model, a systematic theoretical study of the evolution of the single-particle spectral function in the normal phase from the BCS to BEC limits has been presented recently [@PPSC-02]. Like in Ref. , also in Ref. the coupling of a fermionic single-particle excitation to a (bosonic) superconducting fluctuation mode was taken into account by the $t$-matrix self-energy. This approximation embodies the physics of a dilute Fermi gas in the weak-coupling limit and reduces to a description of independent composite bosons in the strong-coupling limit. In this way, single-particle spectra were obtained in Ref. as functions of coupling strength and temperature.
In the present paper, the $t$-matrix approximation for the self-energy is suitably extended below $T_c$. In particular, the *same* superconducting fluctuations, that in Refs. and were coupled to fermionic independent-particle excitations above $T_{c}$, are now coupled to fermionic BCS-like single-particle excitations below $T_{c}$. In the strong-coupling limit, it turns out that these superconducting fluctuations merge in a nontrivial way[@APS-02] into a state of condensed composite bosons described by the Bogoliubov theory, and evolve consistently into a state of independent composite bosons above $T_{c}$ (as the Bogoliubov theory for point-like bosons does [@Bassani-GCS-01]). In this way, a direct connection is established between the structures of the single-particle fermionic self-energy above *and* below $T_{c}$, as they embody the same kind of bosonic mode which itself evolves with temperature.
A comment on the validity of the Bogoliubov theory at finite temperature (and, in particular, close to the Bose-Einstein transition temperature $T_{BE}$) might be relevant at this point. A consistent theory for a *dilute* condensed Bose gas was developed long ago in terms of a (small) gas parameter [@Beliaev-58; @Popov-87], of which the Bogoliubov theory [@FW] is only an approximate form valid at low enough temperatures (compared with $T_{BE}$). That theory correctly describes also the dilute Bose gas in the normal phase [@Popov-87], whereas the Bogoliubov theory (when extrapolated above the critical temperature) recovers the independent-boson form (albeit in a non-monotonic way, with a discontinuous jump affecting the bosonic condensate [@Bassani-GCS-01]). It would therefore be desirable to identify (at least in principle) a fermionic theory that, in the strong-coupling limit of the fermionic attraction, maps onto a more sophisticated bosonic theory, overcoming the apparent limitations of the Bogoliubov theory. In practice, however, it should be considered already a nontrivial achievement of the present approach the fact that the bosonic Bogoliubov approximation can be reproduced from an originally fermionic theory. For these reasons, and also because it is actually the intermediate-coupling (crossover) region to be of most physical interest, in the following we shall consider the Bogoliubov approximation as a reasonable limiting form of our fermionic theory.
As it is always the case for the BCS-BEC crossover approach, implementation of the theory developed in the present paper rests on solving two coupled equations for the order parameter $\Delta$ and the chemical potential $\mu$. The equations here considered for $\Delta$ and $\mu$ generalize the usual equations already considered at the mean-field level [@Eagles-69; @Leggett-80; @NSR-85], by including fluctuation corrections. Our equations reproduce the expected physics in the strong-coupling limit, at least at the level of approximation here considered. Their solution provides us with the values of $\Delta$ and $\mu$ as functions of coupling strength $(k_{F} a_{F})^{-1}$ and temperature $T$, thus extending results obtained previously at the mean-field level. In particular, the order parameter is now found to vanish at a temperature (close to) $T_{c}$ even in the strong-coupling limit, while it would had vanished close to $T^{*}$ at the mean-field level [@PE-00].
The analytic continuation of the fermionic self-energy to the real frequency axis is further performed to obtain the single-particle spectral function $A({\mathbf k},\omega)$, that we study in a systematic way as a function of wave vector ${\mathbf k}$, frequency $\omega$, coupling strength $(k_{F} a_{F})^{-1}$, and temperature $T$. In this context, two novel sum rules (specific to the broken-symmetry phase) are obtained, which provide compelling checks on the numerical calculations. In addition, the numerical calculations are tested against analytic (or semi-analytic) approximations obtained in the strong-coupling limit. The study of a dynamical quantity like $A({\bf k},\omega)$ enables us to attempt a comparison with the experimental ARPES and tunneling spectra for cuprate superconductors below $T_c$, for which a large amount of data exists showing peculiar features for different doping levels and temperatures. As in Ref. above $T_c$, this comparison concerns especially the experimental data about the M points in the Brillouin zone of cuprates, where pairing effects are supposed to be stronger than along the nodal lines.
Our main results are the following. About thermodynamic quantities, we will show that fluctuation corrections over and above mean field are especially important at finite temperature $T\lesssim T_c$ when approaching the strong-coupling limit. At zero temperature, fluctuation corrections to thermodynamic quantities turn out to be of some relevance only in the intermediate-coupling region. This supports the expectation [@Leggett-80] that the BCS mean field at zero temperature should describe rather well the BCS-BEC crossover essentially for all couplings. Regarding instead dynamical quantities like $A({\bf k},\omega)$, our calculation based on a “preformed-pair scenario” reveals two distinct spectral features for $\omega < 0$. These features, which have different temperature and doping dependences, together give rise to a peak-dip-hump structure which is actively debated for the ARPES spectra of cuprate superconductors. Our results differ from those previously obtained by other calculations [@levin98] also based on a “preformed-pair scenario”, where a single feature was instead obtained in the spectral function for $\omega < 0$. An explanation of this discrepancy between the two calculations will be provided. It will also turn out from our calculation that the coherent part of $A({\bf k},\omega)$ for $\omega < 0$ follows essentially a BCS-like behavior as far as its wave-vector dependence is concerned, albeit with a gap value which contains an important contribution from fluctuations at finite temperature. The same BCS-like behavior is not found, however, by our calculation for the dependence of the spectral weight of the coherent peak on temperature and coupling. This evidences a dichotomy in the behavior of $A({\bf k},\omega)$, according to which of its dependences one is after. Such a dichotomy is clearly observed in experiments on cuprate superconductors, in good qualitative agreement with the results obtained by our calculations. [@pps04]. A detailed quantitative comparison of our results with the experimental data on cuprates would, however, require a more refined theoretical model, as to include the quasi-two-dimensional lattice structure, the $d$-wave character of the superconducting gap, and also a fermionic attraction that depends effectively on doping (and possibly on temperature). Future work on this subject should address these additional issues.
The present theory could be improved in several ways. In the present approach, the effective interaction between the composite bosons is treated within the Born approximation. For a dilute system of composite bosons one knows how to improve on this result, as shown in Ref. (see also Ref. ). In addition, the Bogoliubov description for the composite bosons could be also improved, for instance, by extending to the composite bosons the Popov treatment for point-like bosons [@Popov-87]. Finally, on the weak-coupling side of the crossover the BCS theory could be modified by including the contributions shown by Gor’kov and Melik-Barkhudarov [@gmb] to yield a finite renormalization of the critical temperature and of the gap function [*even*]{} in the extreme weak-coupling limit. Work along these lines is in progress.
The plan of the paper is as follows. In Sec. II we discuss our choice for the fermionic self-energy in the superconducting phase, from which the order parameter $\Delta$ and the chemical potential $\mu$ are obtained as functions of temperature and coupling strength, and the spectral function $A({\mathbf k},\omega)$ also results. Analytic results are presented in the strong-coupling limit, where the order parameter is shown to be connected with the bosonic condensate density of the Bogoliubov theory. In addition, the analytic continuation of our expressions for the fermionic self-energy and spectral function is carried out in detail. In Sec. III we present our numerical calculations, and discuss the results for the single-particle spectral function in the context of the available experimental data for high-temperature cuprate superconductors. Section IV gives our conclusions. In Appendix A two sum rules are derived for the superconducting phase, which are used as checks of the numerical results.
Diagrammatic theory for the BCS-BEC crossover in the superconducting phase
==========================================================================
In this section, we discuss the choice of the fermionic single-particle self-energy in the superconducting phase for a (three-dimensional) continuum system of fermions mutually interacting via an attractive point-contact potential, with an $s$-wave order parameter. We shall place special emphasis to the strong-coupling limit of the theory, where composite bosons forms as bound fermion pairs. We extend in this way *below* $T_{c}$ an analogous treatment for the self-energy, made previously in the normal phase to calculate the single-particle spectral function.[@PPSC-02]
Knowledge of the detailed form of the attractive interaction is not generally required when studying the BCS-BEC crossover. Accordingly, one may consider the simple form $v_{0} \delta ({\mathbf r})$ of a “contact” potential, where $v_{0}$ is a negative constant. This choice entails a suitable regularization in terms, e.g., of a cutoff $k_{0}$ in wave-vector space. In three dimensions, this is achieved via the scattering length $a_{F}$ of the associated fermionic two-body problem, by choosing $v_{0}$ as follows [@Pi-S-98]: $$v_{0} \, = \, - \, \frac{2 \pi^{2}}{m k_{0}} \, - \,
\frac{\pi^{3}}{m a_{F} k_{0}^{2}} \,\,\,
\label{v0}$$ $m$ being the fermion mass. With this choice, the classification of the (fermionic) many-body diagrams is considerably simplified not only in the normal phase [@Pi-S-98] but also in the broken-symmetry phase [@APS-02], since only specific diagrammatic substructures survive when the limit $k_{0} \rightarrow \infty$ (and thus $v_{0} \rightarrow 0$) is eventually taken.
In particular, the *particle-particle ladder* depicted in Fig. 1(a) survives the regularization of the potential.[@footnote-Nambu-arrows] It is obtained by the matrix inversion: $$\begin{aligned}
\left(
\begin{array}{cc}
\Gamma_{11}(q)&\Gamma_{12}(q)\\
\Gamma_{21}(q)&\Gamma_{22}(q)\end{array}\right)
&=&
\left(
\begin{array}{cc}
\chi_{11}(-q)&\chi_{12}(q)\\
\chi_{12}(q)&\chi_{11}(q)\end{array}\right)\nonumber\\
&\times& [\chi_{11}(q) \chi_{11}(-q) - \chi_{12}(q)^{2}]^{-1}
\label{Gamma-solution}\end{aligned}$$ with the notation $$\begin{aligned}
- \chi_{11}(q) &=& \frac{m}{4\pi a_F} + \int \! \frac{d {\mathbf p}}{(2\pi)^{3}} \left[
\frac{1}{\beta} \sum_{n}
{\mathcal G}_{11}(p+q) {\mathcal G}_{11}(-p)
\right.\nonumber \\
& &\phantom{\frac{m}{4\pi a_F}}\phantom{\frac{m}{4\pi a_F}}
\phantom{\frac{m}{4\pi a_F}} - \left.\frac{m}{|{\bf p}|^2}\right]
\label{A-definition}\\
\chi_{12}(q) & = & \int \! \frac{d {\mathbf p}}{(2\pi)^{3}} \,
\frac{1}{\beta} \, \sum_{n} \,
{\mathcal G}_{12}(p+q) \,{\mathcal G}_{21}(-p) \,\,\, .
\label{B-definition}\end{aligned}$$ In these expressions, $q=({\mathbf q},\Omega_{\nu})$ and $p=({\mathbf p},\omega_{n})$, where ${\mathbf q}$ and ${\mathbf p}$ are wave vectors, and $\Omega_{\nu}=2\pi\nu/\beta$ ($\nu$ integer) and $\omega_{n}=(2n+1)\pi/\beta$ ($n$ integer) are bosonic and fermionic Matsubara frequencies, respectively (with $\beta=(k_{B}T)^{-1}$, $k_{B}$ being the Boltzmann’s constant); $$\begin{aligned}
{\mathcal G}_{1 1}({\mathbf p},\omega_n) \, & = & \, - \frac{\xi({\mathbf p})
+ i \omega_n}
{E({\mathbf p})^2 + \omega_n^2} \, = \, - \, {\mathcal G}_{2
2}({\mathbf -p},-\omega_n) \nonumber \\
{\mathcal G}_{2 1}({\mathbf p},\omega_n) \, & = & \, \frac{\Delta}
{E({\mathbf p})^2 + \omega_n^2} \, = \, {\mathcal G}_{1
2}({\mathbf p},\omega_n)
\label{BCS-Green-function}\end{aligned}$$ are the BCS single-particle Green’s functions in Nambu notation, with $\xi({\mathbf p})={\mathbf p}^{2}/(2m) - \mu$ and $E({\mathbf p})=\sqrt{\xi({\mathbf p})^{2}+\Delta^{2}}$ for an isotropic ($s$-wave) order parameter $\Delta$. \[Hereafter, we shall take the order parameter to be real with no loss of generality.\]
The expressions (\[A-definition\]) and (\[B-definition\]) for $\chi_{11}(q)$ and $\chi_{12}(q)$ considerably simplify *in the strong-coupling limit* (that is, when $\beta \mu \rightarrow - \infty$ and $\Delta \ll |\mu|$). In this limit, one then obtains for the matrix elements (\[Gamma-solution\]) [@Haussmann-93; @APS-02]: $$\Gamma_{11}(q) \, = \, \Gamma_{22}(-q) \, \simeq \, \frac{8 \pi}{m^{2}
a_{F}} \,
\frac{\mu_{B} \, + \, i \Omega_{\nu} \, + \, {\mathbf q}^{2}/(4m)}
{E_{B}({\mathbf q})^{2} \, - \, (i \Omega_{\nu})^{2}}
\label{Gamma-11-approx}$$ and $$\Gamma_{12}(q) \, = \, \Gamma_{21}(q) \, \simeq \, \frac{8 \pi}{m^{2} a_{F}} \,
\frac{\mu_{B}}{E_{B}({\mathbf q})^{2} \, - \, (i \Omega_{\nu})^{2}} \,\,\,
, \label{Gamma-12-approx}$$ where $$E_{B}({\mathbf q}) \, = \, \sqrt{ \left( \frac{{\mathbf q}^{2}}{2 m_{B}} \, +
\, \mu_{B}\right)^{2}
\, - \, \mu_{B}^{2}}
\label{Bogoliubov-disp}$$ has the form of the Bogoliubov dispersion relation [@FW] ($m_{B}=2m$ being the bosonic mass, $\mu_{B} = \Delta^{2}/(4 |\mu|) = 2 \mu + \epsilon_{0}$ the bosonic chemical potential, and $\epsilon_{0} = (m a_{F}^{2})^{-1}$ the bound-state energy of the associated fermionic two-body problem). The above relation between the fermionic and bosonic chemical potentials holds provided $\mu_{B} \ll \epsilon_{0}$ (cf. also Sec. IID). Note that $\mu_{B}$ can be cast in the Bogoliubov form $$\mu_{B} \, = \, v_{2}(0) \,\, n_{0}(T) \label{pot-chim-Bog}$$ where $v_{2}(0)=4 \pi a_{F}/m$ is the residual bosonic interaction [@Haussmann-93; @Pi-S-98] and $n_{0}(T)=\Delta^2(T) m^2 a_F/(8\pi)$ is the [*condensate density*]{}. The relation (\[pot-chim-Bog\]) is formally obtained already at the (BCS) mean-field level [@APS-02], albeit with an unspecified dependence of $n_{0}(T)$ on temperature. Within our fluctuation theory, the temperature dependence of $n_{0}(T)$ will coincide in strong coupling with the expression given by the Bogoliubov theory (see Sec. IID). In particular, at zero temperature and at the lowest order in the residual bosonic interaction[@APS-02], $n_{0}$ reduces to the bosonic density $n_{B}=n/2$ and $\mu_{B}$ is given by $2
k_{F}^{3} a_{F}/(3 \pi m)$.
![ (a) Particle-particle ladder in the broken-symmetry phase. Conventions for four-momenta and Nambu indices are specified. Dots delimiting the potential (broken line) represent $\tau_3$ Pauli matrices. Only combinations with $\ell_{L}=\ell'_{L}$ and $\ell_{R}=\ell'_{R}$ occur owing to the regularization we have adopted for the potential. (b) Fermionic self-energy diagram associated with the expression (\[Sigma-normal-phase\]) in the normal phase. (c) Fermionic self-energy diagram associated with the expressions (\[Sigma-broken-11\]) and (\[Sigma-broken-12\]) in the broken-symmetry phase. (d) BCS contribution (15) to the self-energy.](fig1.eps)
Note further that the above result for $v_2(0)$ can be cast in the bosonic form $v_2(0) = 4\pi a_B/m_B$ with $a_B = 2 a_F$. The present theory thus describes the effective interaction between the composite bosons within the Born approximation, while improved theories[@Pi-S-98; @petrov] for $a_B$ would give smaller values for the ratio $a_B/a_F$. These improvements will not be considered in the present paper.
Apart from the overall factor $- 8 \pi/(m^{2} a_{F})$ (and a sign difference in the off-diagonal component [@APS-02]), the expressions (\[Gamma-11-approx\]) and (\[Gamma-12-approx\]) coincide with the normal and anomalous non-condensate bosonic Green’s functions within the Bogoliubov approximation [@FW], respectively. These expressions will be specifically exploited in Sec. IID, where the strong-coupling limit of the fermionic self-energy will be analyzed in detail.
In the normal phase, on the other hand, the BCS single-particle Green’s functions are replaced by the bare single-particle propagator ${\mathcal G}_0(p) = [i\omega_{n} - \xi({\mathbf p})]^{-1}$, while for arbitrary coupling the particle-particle ladder acquires the form: $$\begin{aligned}
&&\Gamma_0(q) = - \left\{\frac{m}{4 \pi a_{F}} + \int \!
\frac{d{\mathbf p}}{(2\pi)^{3}}\right. \nonumber\\
&&\times \left[\frac{\tanh(\beta \xi({\mathbf p})/2)
+\tanh(\beta
\xi({\mathbf p-q})/2)}{2(\xi({\mathbf p})+\xi({\mathbf p-q})-i\Omega_{\nu})}
\left. - \frac{m}{{\mathbf p}^{2}} \right] \right\}^{-1}
\, . \label{most-general-pp-sc}\end{aligned}$$ In particular, in the strong-coupling limit the expression (\[most-general-pp-sc\]) reduces to $$\Gamma_0(q) \, \simeq \, - \, \frac{8 \pi}{m^{2} a_{F}} \, \frac{1}{i
\Omega_{\nu} - {\mathbf q}^{2}/(4m)}\, ,
\label{Gamma-o-approx}$$ which coincides (apart again from the overall factor $- 8 \pi/(m^{2}
a_{F})$) with the free-boson Green’s function.
The above quantities constitute the essential ingredients of our theory for the fermionic self-energy and related quantities in the broken-symmetry phase. As shown in Ref. , they also serve to establish a *mapping* between the fermionic and bosonic diagrammatic structures in the broken-symmetry phase, in a similar fashion to what was done in the normal phase [@Pi-S-98].
Choice of the self-energy
-------------------------
In a recent study [@PPSC-02] of the single-particle spectral function in the normal phase based on the BCS-BEC crossover approach, the fermionic self-energy was taken of the form: $$\Sigma_0(k) \, = \, - \, \frac{1}{\beta {\mathcal V}} \, \sum_{q} \,
\Gamma_0(q) \,\, {\mathcal G}_0(q-k)
\label{Sigma-normal-phase}$$ where ${\mathcal V}$ is the quantization volume and $k=({\bf k},\omega_s)$ is again a four-vector notation with wave vector ${\bf k}$ and fermionic Matsubara frequency $\omega_s$ ($s$ integer). In this expression, $\Gamma_0(q)$ is given by Eq. (\[most-general-pp-sc\]) for arbitrary coupling and ${\mathcal G}_0(k)$ is the bare single-particle propagator. The self-energy diagram corresponding to the expression (\[Sigma-normal-phase\]) is depicted in Fig. 1(b). The fermionic single-particle excitations are effectively coupled to a (bosonic) superconducting fluctuation mode, which reduces to a free composite boson in the strong-coupling limit. Physically, the choice (\[Sigma-normal-phase\]) for the self-energy entails the presence of a pairing interaction above $T_{c}$, which can have significant influence on the single-particle (as well as other) properties.
In the present paper, we choose the self-energy in the broken-symmetry phase below $T_c$, with the aim of recovering the expression (\[Sigma-normal-phase\]) when approaching $T_c$ from below and the Bogoliubov approximation for the composite bosons in the strong-coupling limit. To this end, we adopt the *simplest* approximations to describe fermionic *as well as* bosonic excitations in the broken-symmetry phase, which reduce to bare fermionic and free bosonic excitations in the normal phase, respectively. These are the BCS single-particle Green’s functions (\[BCS-Green-function\]) (in the place of the bare single-particle propagator ${\mathcal G}_0$) and the particle-particle ladder (\[Gamma-solution\]) (in the place of its normal-phase counterpart $\Gamma_0$). By this token, the fermionic self-energy (\[Sigma-normal-phase\]) is replaced by the following $2 \times 2$ matrix: $$\begin{aligned}
\!\!\!\!\!\!\Sigma^{L}_{11}(k) = -\Sigma^{L}_{22}(-k) =
- \frac{1}{\beta {\mathcal V}} \sum_{q}
\Gamma_{11}(q) {\mathcal G}_{11}(q-k)
\label{Sigma-broken-11}\\
\Sigma^{L}_{12}(k) = \Sigma^{L}_{21}(k) =
- \frac{1}{\beta {\mathcal V}} \sum_{q}
\Gamma_{12}(q) {\mathcal G}_{12}(q-k)\phantom{111}
\label{Sigma-broken-12}\end{aligned}$$ where the label $L$ refers to the particle-particle ladder. The corresponding self-energy diagram is depicted in Fig. 1(c).[@footnote-Nambu-arrows]
The choice (\[Sigma-broken-11\]) and (\[Sigma-broken-12\]) for the self energy is made on physical grounds. A formal “ab initio” derivation of these expressions can also be done in terms of “conserving approximations” in the Baym-Kadanoff sense, that hold even in the broken-symmetry phase [@Baym-62]. In such a formal derivation, however, the single-particle Green’s functions entering Eqs.(\[Sigma-broken-11\]) and (\[Sigma-broken-12\]) (also through the particle-particle ladder (\[Gamma-solution\])) would be required to be self-consistently determined with the *same* self-energy insertions. In our approach, we take instead the single-particle Green’s functions to be of the BCS form (\[BCS-Green-function\]). The order parameter $\Delta$ and chemical potential $\mu$ are obtained, however, via two coupled equations (to be discussed in Sec. IIC) that include the self-energy insertions (\[Sigma-broken-11\]) and (\[Sigma-broken-12\]). In this way, we will recover the Bogoliubov form (\[Gamma-11-approx\]) and (\[Gamma-12-approx\]) for the particle-particle ladder not only at zero temperature but also at finite temperatures (and, in particular, close to the Bose-Einstein transition temperature).
The choice (\[Sigma-broken-11\]) and (\[Sigma-broken-12\]) for the self energy is not exhaustive. In the broken-symmetry phase there, in fact, exists an additional self-energy contribution that survives the regularization (\[v0\]) of the interaction potential in the limit $k_{0} \rightarrow \infty$, even though it does not contain particle-particle rungs[@Haussmann-footnote]. This additional self-energy diagram is the ordinary BCS contribution depicted in Fig. 1(d), with the associated expression $$\Sigma^{BCS}_{12}(k) \, = \, \Sigma^{BCS}_{21}(k) \, = \, - \, \Delta \,\,
, \label{Sigma-12-BCS}$$ while the corresponding (Hartree-Fock) diagonal elements vanish with the regularization we have adopted. Relating the expression (\[Sigma-12-BCS\]) to the diagram of Fig. 1(d) rests on the validity of the BCS gap equation \[Eq. (\[BCS-gap\_equation\]) below\], for *arbitrary* values of the chemical potential. For this, as well as for an additional reason (cf. Sec. IID), we shall consistently consider that equation to hold for the order parameter $\Delta$.
The choice (\[Sigma-12-BCS\]) alone would be appropriate to describe the system in the weak-coupling (BCS) limit, where the superconducting fluctuation contributions (\[Sigma-broken-11\]) and (\[Sigma-broken-12\]) represent only small corrections. In the intermediate- and strong-coupling regions, on the other hand, both contributions (\[Sigma-broken-11\])-(\[Sigma-broken-12\]) *and* (\[Sigma-12-BCS\]) might become equally significant (depending on the temperature range below $T_{c}$). We thus consider both contributions *simultaneously* and write the fermionic self-energy in the matrix form: $$\begin{aligned}
\left(\begin{array}{cc} \Sigma_{11}(k) & \Sigma_{12}(k) \\ \Sigma_{21}(k)
& \Sigma_{22}(k) \end{array} \right)\phantom{11111111111111111111111111}
\nonumber\\
= \left( \begin{array}{cc} \Sigma^{L}_{11}(k) & \Sigma^{L}_{12}(k) +
\Sigma^{BCS}_{12}(k) \\
\Sigma^{L}_{21}(k) + \Sigma^{BCS}_{21}(k) & \Sigma^{L}_{22}(k) \end{array}
\right)\, . \label{total-self-energy}\end{aligned}$$ In the following, however, we shall neglect $\Sigma^{L}_{12}$ in comparison to $\Sigma^{BCS}_{12}$. It will, in fact, be proved in Sec. IID that, in strong coupling, $\Sigma^{L}_{12}$ is subleading with respect to both $\Sigma^{BCS}_{12}$ and $\Sigma^{L}_{11}$. Inclusion of $\Sigma^{L}_{12}$ is thus not required to properly recover the Bogoliubov description for the composite bosons in the strong-coupling limit.
To summarize, the fermionic single-particle Green’s functions are obtained in terms of the bare single-particle propagator ${\mathcal G}_0(k)$ and of the self-energy (\[Sigma-broken-11\]) and (\[Sigma-12-BCS\]) via the Dyson’s equation in matrix form: $$\begin{aligned}
& &\left( \begin{array}{cc} G_{11}^{-1}(k) & G_{12}^{-1}(k) \\ G_{21}^{-1}(k)
& G_{22}^{-1}(k) \end{array} \right)
= \left( \begin{array}{cc} {\mathcal G}_0(k)^{-1} & 0 \\ 0 & -
{\mathcal G}_0(-k)^{-1} \end{array} \right) \nonumber \\
&& -
\left( \begin{array}{cc} \Sigma^{L}_{11}(k) &
\Sigma^{BCS}_{12}(k) \\
\Sigma^{BCS}_{21}(k) & \Sigma^{L}_{22}(k) \end{array}
\right) \label{Dyson-equation} \, . \end{aligned}$$ If only the BCS contribution (\[Sigma-12-BCS\]) to the self-energy were retained, the fermionic single-particle Green’s functions $G_{ij}(k)$ ($i,j=1,2$) would reduce to the BCS form (\[BCS-Green-function\]). Upon including, in addition, the fluctuation contribution (\[Sigma-broken-11\]) to the self-energy, modified single-particle Green’s functions result, which we are going to study as functions of coupling strength and temperature.
Comparison with the Popov approximation for dilute superfluid fermions
----------------------------------------------------------------------
The choice of the self-energy (\[Sigma-broken-11\]) and (\[Sigma-12-BCS\]) resembles the approximation for the self-energy introduced by Popov[@Popov-87] for superfluid fermions in the dilute limit $k_F |a_F| \ll 1$ (with $a_F < 0$). There is, however, an important difference between the Popov fermionic approximation and our theory. We include in Eq. (\[Sigma-broken-11\]) the full $\Gamma_{11}$ obtained by the matrix inversion of Eq. (\[Gamma-solution\]); Popov instead neglects $\chi_{12}$ therein and approximate $\Gamma_{11}$ by $1/\chi_{11}$, thus removing the feedback of the Bogoliubov-Anderson mode on the diagonal fermionic self-energy $\Sigma_{11}$. Retaining this mode is essential when dealing with the BCS-BEC crossover, to describe the composite bosons in the strong-coupling limit by the Bogoliubov approximation, as discussed in Sec. IIA. Approaching the weak-coupling limit, on the other hand, the presence of the Bogoliubov-Anderson mode becomes progressively irrelevant and the self-energies coincide in the two theories. As a check on this point, we have verified that, in the weak-coupling limit and at zero temperature, $\Sigma_{11}$ obtained by our theory (using the numerical procedures discussed in Sec. III) reduces to $4\pi a_F n /(2 m)$, which is the expression obtained also with the Popov approximation[@Popov-87] in the absence of the Bogoliubov-Anderson mode.
There is another difference between the Popov fermionic approximation and our theory as formulated in Sec. IIA, which concerns the off-diagonal fermionic self-energy $\Sigma_{12}$. Our expression (\[Sigma-12-BCS\]) for $\Sigma_{12}$ was obtained from the diagram of Fig. 1(d), where the single particle line represents the off-diagonal BCS Green’s function of Eq. (\[BCS-Green-function\]) with no insertion of the diagonal self-energy $\Sigma_{11}$. Within the Popov approximation, on the other hand, $\Sigma_{12}$ is defined formally by the same diagram of Fig. 1(d), but with the single-particle line being fully self-consistent (and thus including $\Sigma_{11}$). Since $\Sigma_{11}$ turns out to approach a constant value $\Sigma_0$ in the weak-coupling limit (as discussed above), inclusion of $\Sigma_{11}\simeq\Sigma_0$ can be simply made by a shift of the chemical potential (such that $\mu \to \mu - \Sigma_0$). This shift affects, however, the value of the gap function $\Delta$ in a non-negligible way even in the extreme weak-coupling limit. Neglecting this shift, in fact, results in a reduction by a factor $e^{1/3}$ of the BCS asymptotic expression $(8\epsilon_F/e^2) \exp[\pi/(2 k_F a_F)]$ for $\Delta$ (where $\epsilon_F=k_F^2/(2m)$). Inclusion of the shift $\Sigma_0$ is thus important to recover the BCS value for $\Delta$ in the (extreme) weak-coupling limit.
The need to include the constant shift $\Sigma_0$ on the weak-coupling side of the crossover was also discussed in Ref. while studying the spectral function $A({\bf k},\omega)$ in the normal phase with the inclusion of pairing fluctuations. In that context, inclusion of the shift $\Sigma_0$ proved necessary to have the pseudogap depression of $A({\bf k},\omega)$ centered about $\omega=0$. Inclusion of the shift $\Sigma_0$ in the broken-symmetry phase (at least when approaching the critical temperature from below) is thus also necessary to connect the spectral function $A({\bf k},\omega)$ with continuity in the weak-coupling side of the crossover.
Combining the above needs for $\Delta$ and $A({\bf k},\omega)$, we have introduced the constant shift $\Sigma_0$ for all temperatures below $T_c$, by replacing $\mu$ with $\mu-\Sigma_0$ in the BCS Green’s functions (\[BCS-Green-function\]) entering the convolutions (\[A-definition\]) and (\[B-definition\]). The same replacement is made in the gap equation \[Eq. (\[BCS-gap\_equation\]) below\]. In the Dyson’s equation (\[Dyson-equation\]), however, $\mu$ is left unchanged since the constant shift $\Sigma_0$ is already contained in $\Sigma_{11}(k)$ as soon as its $k$-dependence is irrelevant. Accordingly, we have included this constant shift in the calculation of both thermodynamic and dynamical quantities in the weak-coupling side for $(k_F a_F)^{-1}\le -0.5 $, and neglected it for larger couplings when $\Sigma_{11}(k)$ can no longer be approximated by a constant.
![Self-energy shift $\Sigma_0$ (in units of $\epsilon_F$) vs temperature $T$ (units of $T_c$) and coupling $(k_F a_F)^{-1}$.[]{data-label="shift"}](fig2.eps)
It turns out that the temperature dependence of $\Sigma_0$ is rather weak in the above coupling range. A plot of $\Sigma_0$ vs $T/T_c$ and $(k_F a_F)^{-1}$ is shown in Fig. \[shift\]. Here, the critical temperature $T_c$ is obtained by applying the Thouless criterion from the normal phase as was done in Ref. (this procedure to obtain $T_c$ will be used in the rest of the paper). In this plot, the constant shift $\Sigma_0$ is obtained as $\Sigma_0={\rm Re} \Sigma_{11}^R(|{\bf k}|=
\sqrt{ 2 m (\mu - \Sigma_0)},\omega=0)$, in analogy to what was also done in Ref. . Here, $\Sigma_{11}^R({\bf k},\omega)$ is the analytic continuation to the real frequency axis of the Matsubara self-energy $\Sigma_{11}({\bf k},\omega_s)$ discussed in Sec. IIE.
Coupled equations for the order parameter and the chemical potential
--------------------------------------------------------------------
Thermodynamic quantities, such as the order parameter $\Delta$ and the chemical potential $\mu$, are obtained directly in terms of the Matsubara single-particle Green’s functions, without the need of resorting to the analytic continuation to the real frequency axis.
Quite generally, the order parameter $\Delta$ is defined in terms of the “anomalous” Green’s function $G_{12}({\mathbf k},\omega_{s})$ via $\Delta=v_{0}\langle\psi_{\uparrow}({\mathbf r})\psi_{\downarrow}({\mathbf r})
\rangle$ \[cf. Eq. (\[equation-of-motion\])\], where the strength $v_{0}$ of the contact potential is kept to comply with a standard definition of BCS theory [@FW]. One obtains: $$\Delta \, = \, - \, v_{0} \, \int \! \frac{d {\mathbf k}}{(2\pi)^{3}} \,
\frac{1}{\beta} \sum_{s}
\, G_{12}({\mathbf k},\omega_{s}) \,\, .
\label{Delta-G-12}$$ By the same token, the chemical potential $\mu$ can be obtained in terms of the “normal” Green’s function $G_{11}({\mathbf k},\omega_{s})$ via the particle density $n$: $$n \, = \, 2 \, \int \! \frac{d {\mathbf k}}{(2\pi)^{3}} \, \frac{1}{\beta}
\sum_{s} \, e^{i\omega_{s}\eta}
\, G_{11}({\mathbf k},\omega_{s})
\label{n-G-11}$$ where $\eta=0^+$. The two equations (\[Delta-G-12\]) and (\[n-G-11\]) are coupled, since the Green’s functions depend on both $\Delta$ and $\mu$. The results of their numerical solution will be presented in the next section for various temperatures and couplings.
In the following treatment, we shall deal with the two equations (\[Delta-G-12\]) and (\[n-G-11\]) *on a different footing*. Specifically, we will enter in the density equation (\[n-G-11\]) the expression for the normal Green’s function obtained from Eq. (\[Dyson-equation\]), that includes both BCS *and* fluctuation contributions (see Eq. (\[G-11-Matsubara\]) below). We will use instead in the gap equation (\[Delta-G-12\]) the BCS anomalous function (\[BCS-Green-function\]), that includes only the BCS self-energy (\[Sigma-12-BCS\]). In this way, the gap equation (\[Delta-G-12\]) reduces to the form $$\frac{m}{4 \pi a_{F}} \, + \, \int \! \frac{d{\mathbf k}}{(2\pi)^{3}} \, \left[
\frac{\tanh(\beta E({\mathbf k})/2)}{2 E({\mathbf k})} \, - \,
\frac{m}{{\mathbf k}^{2}} \right] \, = \, 0
\label{BCS-gap_equation}$$ where the regularization of the contact potential in terms of the scattering length $a_{F}$ has been introduced. This equation has the same *formal* structure of the BCS gap equation, although the numerical values of the chemical potential entering Eq. (\[BCS-gap\_equation\]) differ from those obtained by the BCS density equation. This procedure ensures that the bosonic propagators (\[Gamma-solution\]) in the broken-symmetry phase are *gapless*, as shown explicitly by the Bogoliubov-type expressions (\[Gamma-11-approx\]) and (\[Gamma-12-approx\]) in the strong-coupling limit. In general, in fact, there is no *a priori* guarantee that a given (conserving) approximation for fermions would result into a “gapless” approximation [@HM-65] for the composite bosons in the strong-coupling limit of the fermionic attraction.
Including fluctuation corrections to the BCS density equation as in Eq. (\[n-G-11\]), on the other hand, results in the emergence of important effects in the strong-coupling limit of the theory, as discussed next.
Analytic results in the strong-coupling limit
---------------------------------------------
We proceed to show that the original fermionic theory, as defined by the Dyson’s equation (\[Dyson-equation\]), maps onto the Bogoliubov theory for the composite bosons which form as bound-fermion pairs in the strong-coupling limit. To this end, we shall exploit the conditions $\beta|\mu|\gg 1$ and $\Delta\ll|\mu|$ ($\mu<0$) (which *define* the strong-coupling limit) in the (Matsubara) expressions (\[Sigma-broken-11\]) and (\[Sigma-broken-12\]) for $\Sigma^{L}(p)$, thus also verifying that $\Sigma_{12}^L$ can be neglected.
These expressions are calculated by performing the wave-vector and frequency convolutions with the approximate expressions (\[Gamma-11-approx\]) and (\[Gamma-12-approx\]) for the particle-particle ladder and the expressions (\[BCS-Green-function\]) for the BCS single-particle Green’s functions.
Upon neglecting contributions that are subleading under the above conditions, we obtain in this way for the diagonal part of the self-energy: $$\begin{aligned}
\Sigma_{11}^{L}({\mathbf k},\omega_{s}) & \simeq & \frac{8 \pi}{m^{2} a_{F}}
\int \! \frac{d {\mathbf q}}{(2 \pi)^{3}}
\left[ \frac{u_{B}^{2}({\mathbf q})
b(E_{B}({\mathbf q}))}
{i\omega_{s} + E({\mathbf q}-{\bf k}) - E_{B}({\mathbf q})} \right.\nonumber \\
& - & \left. \frac{v_{B}^{2}({\mathbf q})
b(-E_{B}({\mathbf q}))}
{i\omega_{s} + E({\mathbf q}-{\bf k}) + E_{B}({\mathbf q})} \right] \, .
\label{Sigma-11-strong-coupling}\end{aligned}$$ In this expression, $E({\mathbf k})$ is the BCS dispersion of Eqs. (\[BCS-Green-function\]), $E_{B}({\mathbf q})$ is the Bogoliubov dispersion relation (\[Bogoliubov-disp\]), $b(x) = [\exp (\beta x) - 1]^{-1}$ is the Bose distribution, and $$v_{B}^{2}({\mathbf q}) \, = \, u_{B}^{2}({\mathbf q}) - 1 \, = \,
\frac{\frac{{\mathbf q}^{2}}{2m_{B}} \, + \mu_{B} \, - \,
E_{B}({\mathbf q})}{2 E_{B}({\mathbf q})} \label{u-v-Bogoliubov}$$ are the standard bosonic factors of the Bogoliubov transformation [@FW].
In the numerators of the expressions within brackets in Eq. (\[Sigma-11-strong-coupling\]), the Bose functions are peaked at about ${\mathbf q}=0$ and vary over a scale ${\mathbf q}^{2}/(2m_{B}) \approx T \ll |\mu|$. Similarly, the factors $u_{B}^{2}({\mathbf q})$ and $v_{B}^{2}({\mathbf q})$ are also peaked at about ${\mathbf q}=0$ and vary over a scale ${\mathbf q}^{2}/(2m_{B}) \approx \mu_{B} \ll |\mu|$. The denominators in the expression (\[Sigma-11-strong-coupling\]), on the other hand, vary over the much larger scale $|\mu|$. For these reasons, we can further approximate the expression (\[Sigma-11-strong-coupling\]) as follows: $$\Sigma_{11}^{L}({\mathbf k},\omega_{s}) \, \simeq \, \frac{8 \pi}{m^{2}
a_{F}} \,\,
\frac{1}{i\omega_{s} + \xi({\mathbf k})} \,\, n'_{B}(T)
\label{Sigma-11-n-prime}$$ where $$n'_{B}(T) =
\int \! \frac{d {\mathbf q}}{(2 \pi)^{3}} \left[
u_{B}^{2}({\mathbf q}) b(E_{B}({\mathbf q}))
- v_{B}^{2}({\mathbf q}) b(-E_{B}({\mathbf q})) \right]
\label{noncondensate-density}$$ identifies the bosonic *noncondensate density* according to Bogoliubov theory [@FW]. Note that in the normal phase (when the condensate density $n_0(T)$ of Eq. (\[pot-chim-Bog\]) vanishes), the noncondensate density (\[noncondensate-density\]) becomes the full bosonic density $n_{B}=n/2$, and Eq. (\[Sigma-11-n-prime\]) reduces to the expression obtained in Ref. directly from the form (\[Sigma-normal-phase\]) of the fermionic self-energy.
The off-diagonal self-energy $\Sigma_{12}^{L}(k)$ can be analyzed in a similar way. Since its magnitude is supposed to be the largest at zero temperature, we estimate it correspondingly for ${\mathbf k}=0$ and $\omega_{s}=0$ as follows: $$\begin{aligned}
\Sigma_{12}^{L}({\mathbf k}=0,\omega_{s}=0) &\simeq& \frac{8 \pi}{m^{2}
a_{F}} \frac{\mu_{B} \Delta }{2}
\int \! \frac{d {\mathbf k'}}{(2 \pi)^{3}} \frac{1}{E({\mathbf k'})
E_{B}({\mathbf k'})}\nonumber\\
&\times& \frac{1}{(E({\mathbf k'}) \, +\, E_{B}({\mathbf k'}))} \,\, .
\label{Sigma-12-strong-coupling}\end{aligned}$$ At the leading order, we can neglect both $\Delta$ and $\mu_{B}$ in the integrand, where the energy scale $|\mu|$ dominates. We thus obtain $\Sigma_{12}^{L}(k=0) \approx \Delta (\Delta^{2}/(2
|\mu|^{2}))$ in the strong-coupling limit (where the relation $\mu_{B}=\Delta^{2}/(4|\mu|)$ - see below - has been used). This represents a subleading contribution in the small dimensionless parameter $\Delta/|\mu|$ with respect to both the BCS contribution $\Sigma_{12}^{{\rm BCS}}(k)=-\Delta$ and the diagonal fluctuation contribution $\Sigma_{11}^{L}(k)$. It can accordingly be neglected.
Within the above approximations, the inverse (\[Dyson-equation\]) of the fermionic single-particle Green’s function reduces to: $$\begin{aligned}
\left( \begin{array}{cc} G_{11}^{-1}(k) & G_{12}^{-1}(k) \\ G_{21}^{-1}(k)
& G_{22}^{-1}(k) \end{array} \right)
\phantom{1111111111111111111111111}\nonumber\\
\simeq \left( \begin{array}{cc} i\omega_{s}-\xi({\mathbf k}) -
\frac{\Delta_0^{2}}{i\omega_{s}+\xi({\mathbf k})} & \Delta \\
\Delta & i\omega_{s}+\xi({\mathbf k}) -
\frac{\Delta_0^{2}}{i\omega_{s}-\xi({\mathbf k})} \end{array} \right)
\label{G-inverse-strong-coupling}\end{aligned}$$ with the notation $$\Delta_0^{2} \, \equiv \, \frac{8 \pi}{m^{2} a_{F}} \,\, n'_{B}(T) \;
. \label{Delta-o}$$ From Eq. (\[G-inverse-strong-coupling\]) we get the desired expression for $G_{11}({\mathbf k},\omega_{s})$ in the strong-coupling limit: $$G_{11}({\mathbf k},\omega_{s}) \, \simeq \, \frac{1}{i\omega_{s} \, - \,
\xi({\mathbf k}) \, - \,
\frac{\Delta^{2} \, + \, \Delta_0^{2}}{i\omega_{s} \, + \,
\xi({\mathbf k})}} \label{G-11-strong-coupling}$$ where we have discarded a term of order $\Delta_0^{2}/|\mu|$ with respect to $|\mu|$. Note that Eq. (\[G-11-strong-coupling\]) has the same formal structure of the corresponding BCS expression (\[BCS-Green-function\]), with the replacement $E({\mathbf k})
\rightarrow
\tilde{E}({\mathbf k})=\sqrt{\xi({\mathbf k})^{2}+(\Delta^{2}+\Delta_0^{2})}
$. We rewrite it accordingly as: $$G_{11}({\mathbf k},\omega_{s}) \, \simeq \,
\frac{\tilde{u}^{2}({\mathbf k})}{i\omega_{s} \, - \, \tilde{E}({\mathbf k})}
\, + \, \frac{\tilde{v}^{2}({\mathbf k})}{i\omega_{s} \, + \,
\tilde{E}({\mathbf k})}
\label{G-11-strong-coupling-figo}$$ with the modified BCS coherence factors $\tilde{v}^{2}({\mathbf k}) = 1 -
\tilde{u}^{2}({\mathbf k}) =
(1 - \xi({\mathbf k})/\tilde{E}({\mathbf k}))/2$.
Before making use of the asymptotic expression (\[G-11-strong-coupling-figo\]) in the density equation (\[n-G-11\]), it is convenient to manipulate suitably the gap equation (\[BCS-gap\_equation\]) in the strong-coupling limit. Expanding $1/E({\mathbf k})$ therein as $[1-\Delta^{2}/(2\xi({\mathbf k})]/\xi({\mathbf k})$ and evaluating the resulting elementary integrals, one obtains: $$\frac{\Delta^{2}}{4|\mu|} \, \simeq \, 2 \, \left( \sqrt{2 \, |\mu| \,
\epsilon_0} \, - \, 2 \, |\mu| \right)\, .
\label{Delta-mu-strong-coupling}$$ Setting further $2 \mu = - \epsilon_0 + \mu_{B}$, one gets the relation $\Delta^{2}/(4|\mu|)=\mu_{B}$ quoted already after Eqs. (\[Bogoliubov-disp\]) and (\[Sigma-12-strong-coupling\]).
Let’s now consider the density equation (\[n-G-11\]). With the BCS-like form (\[G-11-strong-coupling-figo\]) one obtains immediately: $$n \, \simeq \, 2 \, \int \! \frac{d {\mathbf k}}{(2\pi)^{3}} \,
\tilde{v}^{2}({\mathbf k}) \label{n-G-11-strong-coupling}$$ that holds for $T \ll \epsilon_{0}$, at temperatures well below the dissociation threshold of the composite bosons. Similarly to what was done to get the gap equation (\[Delta-mu-strong-coupling\]), in Eq. (\[n-G-11-strong-coupling\]) one expands $1/\tilde{E}({\mathbf k})$ as $[1-(\Delta^{2}+\Delta_0^{2})/(2\xi({\mathbf k})]/\xi({\mathbf k})$ and evaluates the resulting elementary integrals, to obtain: $$n \, \simeq \, \frac{m^{2} \, a_{F}}{4 \pi} \,\, \left( \Delta^{2} \, + \,
\Delta_0^{2} \right) \,\, . \label{final-n}$$ Recalling the definition (\[Delta-o\]) for $\Delta_{0}^{2}$, as well as the expressions (\[Delta-mu-strong-coupling\]) and (\[pot-chim-Bog\]) for the order parameter, which we rewrite in the form $$\Delta^{2} \, = \, \frac{8 \pi}{m^{2} a_{F}} \,\, n_0(T)
\label{Delta-n-o}$$ in analogy to Eq. (\[Delta-o\]), the result (\[final-n\]) becomes eventually: $$n \, = \, 2 \left( n'_{B}(T) \, + \, n_0(T) \right)
\label{n-n-B-n-o}$$ that holds asymptotically for $T \ll \epsilon_0$.
These results imply that, in the strong-coupling limit, the original fermionic theory recovers the Bogoliubov theory for the composite bosons, not only at zero temperature but also *at any temperature* in the broken-symmetry phase. Accordingly, the noncondensate density $n'_{B}(T)$ is given by the expression (\[noncondensate-density\]), the bosonic factors $v_{B}^{2}({\mathbf q})$ and $u_{B}^{2}({\mathbf q})$ are given by Eq. (\[u-v-Bogoliubov\]), and the dispersion relation $E_{B}({\mathbf q})$ is given by Eq. (\[Bogoliubov-disp\]). In the strong-coupling limit, the present fermionic theory thus inherits all virtues and shortcomings of the Bogoliubov theory for a weakly-interacting Bose gas [@Bassani-GCS-01]. The present fermionic theory at arbitrary coupling then provides an interpolation procedure between the Bogoliubov theory for the composite bosons and the weak-coupling BCS theory plus pairing fluctuations. Both these analytic limits will constitute important checks on the numerical calculations reported in Sec. III. Note that inclusion of the off-diagonal fluctuation contribution $\Sigma_{12}^L(k)$ to the self-energy is not required to recover the Bogoliubov theory in strong coupling. For this reason, we will not consider $\Sigma_{12}^L(k)$ altogether in the numerical calculations presented in Sec. III, as anticipated in Eq. (\[Dyson-equation\]).
The above analytic results enable us to infer the main features of the temperature dependence of the order parameter in the strong-coupling limit. In particular, the low-temperature behavior $n_0(T) = n_0(0) - m_{B}(k_{B} T)^{2}/(12 c)$( where $c = \sqrt{n_0v_{2}(0)/m_{B}}$ is the sound velocity) within the Bogoliubov approximation, implies that $\Delta(T)$ decreases from $\Delta(0)$ with a $T^{2}$ behavior, in the place of the exponential behavior obtained within the BCS theory (with an $s$-wave order parameter) [@FW]. In addition, in the present theory the order parameter vanishes over the scale of the Bose-Einstein transition temperature $T_{BE}$, while in the BCS theory it would vanish over the scale of the bound-state energy $\epsilon_0$ of the composite bosons.
Note finally that the fermionic quasi-particle dispersion $\tilde{E}({\mathbf k})$, entering the expression (\[G-11-strong-coupling-figo\]) of the diagonal Green’s function in the strong-coupling limit, contains the sum $\Delta^{2}+\Delta_0^{2}$ instead of the single term $\Delta^{2}$ of the BCS dispersion $E({\mathbf k})$.
Spectral function and sum rules
-------------------------------
We pass now to identify the form of the spectral function $A({\mathbf k},\omega)$ associated with the approximate choice of the Matsubara self-energy of Eq. (\[Dyson-equation\]). To this end, we need to perform the *analytic continuation* in the complex frequency plane, thus determining the [*retarded*]{} fermionic single-particle Green’s functions from their Matsubara counterparts. The approach developed in this subsection holds specifically for the approximate choice for the self-energy of Eq. (\[Dyson-equation\]). It thus differs from the general analysis presented in the Appendix which holds for the exact Green’s functions, irrespective of any specific approximation.
In general, the process of analytic continuation to the real frequency axis from the numerical Matsubara Green’s functions proves altogether nontrivial, as it requires in practice recourse to approximate numerical methods such as, e.g., the method of Padé approximants [@Serene-1977]. We then prefer to rely on a procedure whereby the analytic continuation to the real frequency axis is achieved by avoiding numerical extrapolations from the Matsubara Green’s functions.
The fermionic normal and anomalous Matsubara single-particle Green’s functions are obtained at any given coupling from matrix inversion of Eq. (\[Dyson-equation\]): $$\begin{aligned}
&&G_{11}({\mathbf k},\omega_{s}) = \left[\phantom{\frac{1}{1}}\!\!\!
i\omega_{s}-\xi({\mathbf k})-\Sigma_{11}({\mathbf k},\omega_{s})\right.
\nonumber\\
&&\phantom{111111111}-\left.
\frac{\Delta^2}{i\omega_{s} + \xi({\mathbf k})
- \Sigma_{22}({\mathbf k},\omega_{s})}\right]^{-1}
\label{G-11-Matsubara} \\
&&G_{12}({\mathbf k},\omega_{s})=
\Delta [\,(i\omega_{s}-\xi({\mathbf k})-
\Sigma_{11}({\mathbf k},\omega_{s}))\nonumber\\
&&\phantom{1111} \times (i\omega_{s}+\xi({\mathbf k})-
\Sigma_{22}({\mathbf k},\omega_{s})) - \Delta^{2}]^{-1}\; .
\label{G-12-Matsubara} \end{aligned}$$
Consider first the normal Green’s function (\[G-11-Matsubara\]), which we rewrite in the compact form $$G_{11}({\mathbf k},\omega_{s}) \, = \,
\frac{1}{i\omega_{s}-\xi({\mathbf k})-\sigma_{11}({\mathbf k},\omega_{s})}
\label{G-11-compact}$$ with the short-hand notation $$\sigma_{11}({\mathbf k},\omega_{s}) \, \equiv \,
\Sigma_{11}({\mathbf k},\omega_{s}) \, + \,
\frac{\Delta^2}{i\omega_{s} \,\, + \,\, \xi({\mathbf k})
- \,\, \Sigma_{22}({\mathbf k},\omega_{s})} \,\, .
\label{sigma-11-compact}$$ To perform the analytic continuation of this expression, we look for a function $\sigma_{11}({\mathbf k},z)$ of the complex frequency $z$ which satisfies the following [*requirements*]{} at any given ${\mathbf k}$:\
(i) It is analytic off the real axis;\
(ii) It reduces to $\sigma_{11}({\mathbf k},\omega_{s})$ given by Eq. (\[sigma-11-compact\]) when $z$ takes the discrete values $i \omega_{s}$ on the imaginary axis;\
(iii) Its imaginary part is negative (positive) for $\mathrm{Im} \, z>0$ ($\mathrm{Im} \, z<0$) ;\
(iv) It vanishes when $|z|\to\infty$ along any straight line parallel to the real axis with $\mathrm{Im} \, z \neq 0$.
Once the function $\sigma_{11}({\mathbf k},z)$ is obtained, the expression $$G^{R}({\mathbf k},\omega) \, = \, \frac{1}{\omega \, + \, i\eta \, - \,
\xi({\mathbf k}) \, - \,
\sigma_{11}({\mathbf k},\omega + i\eta)}
\label{G-R-compact}$$ ($\eta$ being a positive infinitesimal) represents the *retarded* ($R$) single-particle Green’s function (for real $\omega$) associated with the Matsubara Green’s function (\[G-11-compact\]), since it satisfies the requirements of the Baym-Mermin theorem [@Baym-Mermin-61] for the analytic continuation from the Matsubara Green’s function.
The first step of the above program is to find the analytic continuation of $\Sigma_{11}({\mathbf k},\omega_s)$ (and $\Sigma_{22}({\mathbf k},\omega_s)$) off the real axis in the complex $z$-plane. To this end, it is convenient to express $\Sigma_{11}({\mathbf k},\omega_s)$ via the spectral form: $$\Sigma_{11}({\mathbf k},\omega_s)=
\int_{-\infty}^{+\infty}\frac{d\omega'}{\pi}
\frac{h({\mathbf k},\omega')}{i\omega_s-\omega'}
\label{spect}\;\, .$$ With the replacement $i \omega_s\to z$, the spectral representation (\[spect\]) defines an analytic function $\Sigma_{11}({\mathbf k},z)$ off the real axis. In the case of interest with $\Sigma_{11}({\mathbf k},\omega_s)$ given by Eq. (\[Sigma-broken-11\]), the function $h({\mathbf k},\omega)$ of Eq. (\[spect\]) reads: $$\begin{aligned}
h({\bf k},\omega)&=&-\int \frac{d {\bf q}}{(2\pi)^3}
\left\{u^2_{{\bf q}-{\bf k}}
{\rm Im}\,\Gamma_{11}^R({\bf q},\omega+E({\bf q}-{\bf k}))\right.\nonumber\\
&\times& \left[f(E({\bf q}-{\bf k}))+b(\omega+E({\bf q}-{\bf k}))\right]
\nonumber
\\
&+ & v^2_{{\bf q}-{\bf k}} {\rm Im}\,\Gamma_{11}^R({\bf q},
\omega-E({\bf q}-
{\bf k}))\nonumber\\
&\times& \left.\left[f(-E({\bf q}-{\bf k}))+b(\omega-E({\bf q}-{\bf k}))
\right] \right\}
\label{imsig11}\end{aligned}$$ where $f(x)=[\exp(\beta x)+1]^{-1}$ is the Fermi distribution while $u^2_{{\bf k}}$ and $v^2_{{\bf k}}$ are the BCS coherence factors. To obtain the expression (\[imsig11\]), a spectral representation has been also introduced for $\Gamma_{11}$ entering Eq. (\[Sigma-broken-11\]), by writing: $$\Gamma_{11}({\mathbf q},\Omega_{\nu}) \, = - \frac{1}{\pi}
\int_{-\infty}^{+\infty} \! d\omega' \,
\frac{{\rm Im}\, \Gamma_{11}^R({\mathbf q},\omega')}{i\Omega_{\nu} -
\omega'}\; .
\label{spectral-representation}$$ Here, the spectral function $\Gamma_{11}^R({\mathbf q},\omega)$ is *defined* by $\Gamma_{11}({\mathbf q},i\Omega_{\nu} \rightarrow \omega + i \eta)$, which is obtained from the definitions (\[Gamma-solution\])-(\[B-definition\]) with the replacement $i\Omega_{\nu} \rightarrow \omega + i \eta$ [*after*]{} the sum over the internal frequency $\omega_n$ has been performed therein. Even in the absence of an explicit Lehmann representation for $\Gamma_{11}$, in fact, it can be shown that the spectral representation (\[spectral-representation\]) holds provided the function $\Gamma_{11}({\mathbf q},i\Omega_{\nu}\to z)$ of the complex variable $z$ is analytic off the real axis. The crucial point is to verify that the denominator in Eq. (\[Gamma-solution\]) with the replacement $i \Omega_{\nu}\to z$ never vanishes off the real axis. This property can be explicitly verified in the strong-coupling limit, as discussed below. For arbitrary coupling, we have checked it with the help of numerical calculations. For the validity of the expression $(\ref{spectral-representation})$, it is also required that $\Gamma_{11}({\mathbf q},z)$ vanishes for $|z|\to\infty$. This property can be proved directly from Eqs. (\[Gamma-solution\])-(\[B-definition\]), according to which $\Gamma_{11}({\mathbf q},z)$ has the asymptotic expression $$\Gamma_{11}({\bf q},z) \simeq \frac{-1}{\frac{m}{4 \pi a_{F}}-
\frac{m^{3/2}}{4 \pi} \sqrt{-z +{q^2 \over 4 m} -
2 \mu}}
\label{gammasy}$$ and thus vanishes for $|z|\to \infty$. Once $\Sigma_{11}({\mathbf k},z)$ has been explicitly constructed according to the above prescriptions, $\Sigma_{22}({\mathbf k},z)$ is obtained as $-\Sigma_{11}({\mathbf k},-z)$ in accordance with Eq. (\[Sigma-broken-11\]).
From the spectral representation (\[spect\]) for $\Sigma_{11}({\mathbf k},z)$, it can be further shown that $\Sigma_{11}({\mathbf k},z)$ vanishes when $|z|\to\infty$ along any straight line parallel to the real axis with ${\rm Im}\,\, z \neq 0$. It can also be shown that ${\rm Im}\,\Sigma_{11}({\mathbf k},z) < 0$ (${\rm Im}\,\Sigma_{11}({\mathbf k},z) > 0$) when ${\rm Im}\, z > 0$ (${\rm Im}\, z < 0$). This property follows from the spectral representation of $\Sigma_{11}({\mathbf k},z)$, provided $h({\bf k},\omega)\ge 0$ in Eq. (\[spect\]). For arbitrary coupling, we have verified that $h({\bf k},\omega)\ge 0$ with the help of numerical calculations. In the strong-coupling limit, this condition can be explicitly proved, as discussed below.
From these properties of $\Sigma_{11}({\mathbf k},z)$ (and $\Sigma_{22}({\mathbf k},z)$) it can then be verified that the function $$\begin{aligned}
\sigma_{11}({\mathbf k},z) & = &
\Sigma_{11}({\mathbf k},z) \nonumber\\
&+&
\frac{\Delta^{2}}{z \, + \, \xi({\mathbf k})
+ \, \Sigma_{11}({\mathbf k},-z)} ,
\label{sigma-tilde-compact}\end{aligned}$$ satisfies the requirements (i)-(iv) stated after Eq. (\[sigma-11-compact\]). With the replacement $z \to \omega + i\eta$, Eq. (\[G-R-compact\]) follows eventually on the real frequency axis for the retarded Green’s function $G^R({\bf k},\omega)$.
For later convenience, we introduce the following notation on the real frequency axis : $$\Sigma_{11}^R({\mathbf k},\omega) \equiv
\Sigma_{11}({\mathbf k},\omega + i\eta)$$ such that $\Sigma_{11}({\mathbf k},-\omega - i\eta) =
\Sigma_{11}^R({\mathbf k},-\omega)^*$ and $$\begin{aligned}
\sigma_{11}^R({\mathbf k},\omega)\equiv
\sigma_{11}({\mathbf k},\omega + i\eta)\phantom{1111111111111111111}\nonumber\\
=\Sigma_{11}^R({\mathbf k},\omega) \, + \,
\frac{\Delta^{2}}{\omega + i \eta \, + \, \xi({\mathbf k})
+ \, \Sigma_{11}^R(-{\mathbf k},-\omega)^*}\; .
\label{sigma-tilde-real}\end{aligned}$$ From Eq. (\[spect\]) it is also clear that ${\rm Im}\,\Sigma_{11}^R({\mathbf k},\omega)= -h({\mathbf k},\omega)$, and that ${\rm Re}\Sigma_{11}^R({\mathbf k},\omega)$ and ${\rm Im}\,\Sigma_{11}^R({\mathbf k},\omega)$ are related by a Kramers-Kronig transform.
As anticipated, the properties of the function $\Sigma_{11}({\bf k},z)$, required above to obtain the retarded Green’s function (\[G-R-compact\]) on the real axis, can be explicitly verified in the strong-coupling limit without recourse to numerical calculations. In this case, the approximate expression (\[Gamma-11-approx\]) can be used for $\Gamma_{11}$. This can be cast in the form (\[spectral-representation\]), with $$\begin{aligned}
{\rm Im}\, \, \Gamma_{11}^R({\bf q},\omega)&=& -\frac{8 \pi^2}{m^{2}
a_{F}}
[v_{B}^2({\bf q})\delta(\omega + E_{B}({\mathbf q}))\nonumber\\
&&\phantom{11111}- u_{B}^2({\bf q})\delta(\omega - E_{B}({\mathbf q}))]\; .
\label{imgamma}\end{aligned}$$ Entering the expression (\[imgamma\]) into Eq. (\[imsig11\]) and the resulting expression into Eq. (\[spect\]), one obtains for $\Sigma_{11}({\bf k},\omega_s)$ the sum of four terms: $$\begin{aligned}
&&\Sigma_{11}({\bf k},\omega_s)=-\frac{8 \pi}{m^{2} a_{F}}\nonumber\\
&&\times
\int \! \frac{d {\mathbf q}}{(2 \pi)^{3}}
\left\{ u_{B}^{2}({\mathbf q}) u^2_{{\bf q}- {\bf k}}
\frac{b(E_{B}({\mathbf q})) + f(E({\bf q}-{\bf k}))}{E_B({\bf q}) -
E({\bf q} - {\bf k}) - i\omega_{s}} \right. \nonumber \\
& + & \left. u_{B}^{2}({\mathbf q}) v^2_{{\bf q}- {\bf k}}
\frac{b(E_{B}({\mathbf q})) + f(- E({\bf q}-{\bf k}))}{E_B({\bf q}) +
E({\bf q} - {\bf k}) - i\omega_{s}} \right. \nonumber\\
& + & \left.
v_{B}^{2}({\mathbf q})u^2_{{\bf q}- {\bf k}}
\frac{b(-E_{B}({\mathbf q})) + f(E({\bf q}-{\bf k}))}{E_B({\bf q}) +
E({\bf q} - {\bf k}) + i\omega_{s}} \right. \nonumber\\
& + & \left.
v_{B}^{2}({\mathbf q})v^2_{{\bf q}- {\bf k}}
\frac{b(-E_{B}({\mathbf q})) + f(-E({\bf q}-{\bf k}))}{E_B({\bf q}) -
E({\bf q} - {\bf k}) + i\omega_{s})}\right\}\; .
\label{Sig11}\end{aligned}$$ Since in strong coupling $f(E({\bf k}))\to 0$, $u^2_{{\bf k}}\to 1$, and $v^2_{{\bf k}} \to 0$, the second and fourth term within braces on the right-hand side of the Matsubara expression (\[Sig11\]) may be dropped. The simplified expression (\[Sigma-11-strong-coupling\]) then results from Eq. (\[Sig11\]). In the strong-coupling limit, one would then be tempted to perform the analytic continuation $i\omega_s\to z$ directly from the expression (\[Sigma-11-strong-coupling\]). Care must, however, be exerted on this point since [*the processes of taking the strong-coupling limit and performing the analytic continuation may not commute*]{}. By performing the analytic continuation $i\omega_s \to z$ directly in Eq. (\[Sig11\]) one, in fact, obtains two additional terms with respect to the analytic continuation of Eq. (\[Sigma-11-strong-coupling\]). These two additional terms cannot be dropped [*a priori*]{} by the presence of the small factor $v^2_{{\bf q} - {\bf k}}$ in the strong-coupling limit, because for real $z$ the corresponding energy denominators may vanish. Retaining properly these two additional terms indeed affects in a qualitative way the spectral function $A({\bf k},\omega)$ in the strong-coupling limit, as discussed in Sec. III.
With the expression obtained by the analytic continuation $i\omega_s \to z$ of Eq. (\[Sig11\]), one can prove explicitly that $\Sigma_{11}({\bf k},z)$ is analytic off the real axis and vanishes like $z^{-1}$ along any straight line parallel to the real axis with ${\rm Im}\, \, z \neq 0$, and that ${\rm sgn} [{\rm Im}\,\Sigma_{11}({\bf k},z)]=
-{\rm sgn} [{\rm Im}\,\, z]$. In this way, the properties of the function $\Sigma_{11}({\bf k},z)$, required to obtain the retarded Green’s function (\[G-R-compact\]) on the real axis, are explicitly verified in the strong-coupling limit.
Once the retarded Green’s function has been obtained in the form (\[G-R-compact\]) according to the above prescriptions, its imaginary part defines the spectral function $$A({\mathbf k},\omega) \equiv - (1/\pi)
\mathrm{Im} \, G^{R}({\mathbf k},\omega)
\label{akw}$$ which will be calculated numerically in Sec. III for a wide range of temperatures and couplings. In the Appendix, it is shown at a formal level that $A({\mathbf k},\omega)$ satisfies the sum rule (\[sum-rule-G-R\]). This sum rule will be considered an important test for the numerical calculations of Sec. III. To this end, it is necessary to prove that the sum rule (\[sum-rule-G-R\]) holds even for our approximate theory based on the Dyson’s equation (\[Dyson-equation\]).
To prove the sum rule (\[sum-rule-G-R\]) for the approximate theory, it is sufficient that the approximate $G_{11}({\bf k},z)$ (from which the retarded Green’s function (\[G-R-compact\]) results when $z=\omega + i \eta$) behaves like $z^{-1}$ for large $|z|$. This property is verified by our theory, as shown above. As a consequence: $$\begin{aligned}
\int_{-\infty}^{+\infty} \! d\omega \, A({\mathbf k},\omega) =
- \frac{1}{\pi}\,
\mathrm{Im} \left[ \int_{-\infty}^{\infty} \! d\omega \,
G^{R}({\mathbf k},\omega)\right]\nonumber\\
= - \frac{1}{\pi}\,
\mathrm{Im} \left[ - \oint_{C} \! d\omega \, G_{11}({\mathbf k},z)
\right] = 1 \; \label{sum-rule-A}\end{aligned}$$ where the contour $C$ is a half-circle in the upper-half complex plane with center in the origin, large radius (such that the approximation $G_{11}({\mathbf k},z) \sim z^{-1}$ is valid), and counterclockwise direction.
Finally, the analytic continuation of the anomalous Matsubara single-particle Green’s function (\[G-12-Matsubara\]) can be obtained by following the same procedure adopted for the normal Green’s function (\[G-11-Matsubara\]). One writes for the retarded anomalous Green’s function $$\begin{aligned}
&& F^R({\bf k},\omega)=
\Delta [(\omega + i\eta -\xi({\mathbf k})-
\Sigma_{11}^R({\mathbf k},\omega)) \nonumber\\
&&\times (\omega + i\eta+\xi({\mathbf k})+
\Sigma_{11}^R(-{\mathbf k},-\omega)^*) - \Delta^{2}]^{-1}\; .
\label{FRet} \end{aligned}$$ in the place of Eqs. (\[G-R-compact\]) and (\[sigma-tilde-real\]). In this case, the analytic properties of $\Sigma_{ii}({\mathbf k},z)$ ($i=1,2$) discussed above imply that $G_{12}({\mathbf k},z) \sim -
\Delta/ z^{2}$ asymptotically for large $|z|$. As a consequence, the imaginary part of $F^R({\bf k},\omega)$ $$B({\mathbf k},\omega) \equiv - (1/\pi)
{\mathrm Im}\,F^{R}({\mathbf k},\omega)$$ satisfies the two following sum rules: $$\int_{-\infty}^{+\infty} \! d\omega \, B({\mathbf k},\omega) \, = \, 0
\label{sum-rule-B-1}$$ and $$\int_{-\infty}^{+\infty} \! d\omega \, B({\mathbf k},\omega) \, \omega \, =
\, - \, \Delta \,\, . \label{sum-rule-B-2}$$ These sum rules can be verified by introducing the contour $C$ as in Eq. (\[sum-rule-A\]). Note again that these sum rules (which are proved on general grounds in the Appendix for the exact anomalous retarded single-particle Green’s function) follow here from our approximate form of $F^{R}({\mathbf k},\omega)$ only on the basis of the properties of analyticity. Verifying numerically the sum rules (\[sum-rule-A\]), (\[sum-rule-B-1\]), and (\[sum-rule-B-2\]) at any coupling and temperature will, in practice, constitute an important check on the validity of the above procedure for the analytic continuation.
An additional numerical check on the validity of the whole procedure at intermediate-to-weak coupling will be provided by the merging of the results, obtained by calculating the spectral function $A({\mathbf k},\omega)$ when approaching $T_{c}$ from below, with the results previously obtained in the normal phase[@PPSC-02] when approaching $T_{c}$ from above.
Numerical results and discussion
================================
In this section we present the numerical results based on the formal theory developed in Sec. II. Specifically, in Sec. IIIA we present the results obtained by solving the coupled equations (\[n-G-11\]) and (\[BCS-gap\_equation\]) for the order parameter and chemical potential. Section IIIB deals instead with the numerical calculation of the spectral function (\[akw\]) in the broken-symmetry phase, over the whole coupling range from weak to strong.
Order parameter and chemical potential
--------------------------------------
Before presenting the numerical results for $\Delta$ and $\mu$, it is worth outlining briefly the numerical procedure we have adopted.
At given temperature and coupling, the coupled equations (\[n-G-11\]) and (\[BCS-gap\_equation\]) for the unknowns $\Delta$ and $\mu$ are solved via the Newton’s method. This requires knowledge of the self-energy $\Sigma_{11}({\bf k},\omega_{s})$ of Eq. (\[Sigma-broken-11\]), with $\Gamma_{11} (q)$ obtained from Eqs. (\[Gamma-solution\])-(\[B-definition\]). \[As anticipated, in the numerical calculations we neglect $\Sigma_{12}^L$ in comparison to $\Sigma_{12}^{{\rm BCS}}$, since inclusion of $\Sigma_{12}^L$ is not required to recover the Bogoliubov results in the strong-coupling limit, as shown in Sec. IID.\]
To this end, the frequency sums in Eqs. (\[A-definition\]) and (\[B-definition\]) are evaluated analytically, while the remaining wave-vector integral is calculated numerically by the Gauss-Legendre method. In particular, the radial wave-vector integral extending up to infinity is partitioned into an inner and an outer region, with the transformation $|{\bf p}|\to 1/|{\bf p}|$ exploited in the outer region.
The bosonic frequency sum in Eq. (\[Sigma-broken-11\]) requires special care, owing to its slow convergence and the lack of an intrinsic energy cutoff within our continuum model. We have accordingly partitioned this frequency sum into three regions, separated by the frequency scales $\Omega_{c_1}$ and $\Omega_{c_2}$ (with $0<\Omega_{c_1}<\Omega_{c_2})$. For $|\Omega_{\nu}|<\Omega_{c_1}$, the frequency sum is calculated explicitly. For $\Omega_{c_1}<|\Omega_{\nu}|<\Omega_{c_2}$, the frequency sum is approximated with great accuracy by the corresponding numerical integral, owing to the slow dependence of $\Gamma_{11}$ on $\Omega_{\nu}$. Finally, the tail of the frequency sum for $\Omega_{c_2}<|\Omega_{\nu}|$ (where the asymptotic expression (\[gammasy\]) yields $\Gamma_{11}
\propto (i\Omega_{\nu})^{-1/2}$) is evaluated analytically. Typically, $\Omega_{c_1}$ is taken of the order of the largest among the energy scales $|\omega_{s}|,\Delta,|\mu|,{\bf q}^2/(2m)$, and ${\bf k}^2/(2m)$; $\Omega_{c_2}$ is then taken at least ten times $\Omega_{c_1}$. It turns out that it is most convenient to apply this procedure to the frequency sum in Eq. (\[Sigma-broken-11\]) after the integration over the two angular variables of the wave vector ${\bf q}$ has been performed analytically; the remaining radial wave-vector integration is then performed numerically, with a cutoff much larger than the wave-vector scales $|{\bf k}|, \sqrt{2m \Delta}$, and $\sqrt{|\mu|}$.
![Order parameter $\Delta$ (in units of $\epsilon_F$) vs temperature (in units of $T_c$) for different values of the coupling $(k_F a_F)^{-1}$. Results obtained by the inclusion of fluctuations (full lines) are compared with mean-field results (dashed lines). []{data-label="gapvsT"}](fig3.eps)
Finally, the frequency sum in the particle number equation (\[n-G-11\]) is evaluated by adding and subtracting the BCS Green’s function ${\cal G}_{11}$ on the right-hand side of that equation, in order to speed up the numerical convergence. Matsubara frequencies are here summed numerically up to a cutoff frequency, beyond which the sum is approximated by the corresponding numerical integral. The radial part of the wave-vector integral in Eq. (\[n-G-11\]) is also calculated numerically up to a cutoff scale beyond which a power-law decay sets in, so that the contribution from the tail can be calculated analytically.
With the above numerical prescriptions, we have obtained the behavior of $\Delta$ and $\mu$ vs temperature and coupling reported in Figs. \[gapvsT\]-\[mut0\].
Specifically, Fig. \[gapvsT\] shows the order parameter $\Delta$ vs temperature for different couplings \[$(k_F a_F)^{-1}=-0.5,0.5,1.2$, from top to bottom\], in the window $-1 \lesssim (k_F a_F)^{-1}\lesssim +1$ where the crossover from weak to strong coupling is exhausted. Comparison is made with the corresponding curves obtained within mean field (dashed lines), when the BCS Green’s function ${\mathcal G}_{11}$ enters Eq. (\[n-G-11\]) in the place of the dressed $G_{11}$. In these plots, the temperature is normalized with respect to the critical temperature $T_c$ for the given coupling. This comparison shows that fluctuation corrections on top of mean field get progressively important at given coupling as the temperature is raised toward $T_c$. Close to $T_c$, fluctuation corrections become even more important upon approaching the strong-coupling limit. Near zero temperature, on the other hand, fluctuation corrections become negligible when approaching strong coupling. This confirms the expectation that, near zero temperature, the BCS mean field should be rather accurate both in the weak- and strong-coupling limits [@Leggett-80].
Note from Fig. \[gapvsT\] that $\Delta$ jumps discontinuously close to the critical temperature when fluctuations are included on top of mean field. This jump becomes more evident as the coupling is increased. It reflects an analogous behavior of the condensate density near the critical temperature as obtained by the Bogoliubov theory for point-like bosons [@Luban] . In the present theory this jump is carried over to the composite bosons, even at fermionic couplings \[as in the middle panel of Fig. \[gapvsT\] \] when the composite bosons are not yet fully developed. When the fermionic coupling increases beyond the values reported in Fig. \[gapvsT\], however, the residual interaction between the composite bosons decreases further and the jump becomes progressively smaller. More refined theories for point-like bosons (see, e.g., Ref. ) remove the jump of the bosonic condensate density, which thus should be considered as an artifact of the Bogoliubov approximation. Apart from this jump, note that when the temperature is decreased below $T_c$ the order parameter $\Delta$ grows more rapidly with the inclusion of fluctuations than within mean field.
![Chemical potential $\mu$ (in units of $\epsilon_F$) vs temperature (in units of $T_c$), for the same values of the coupling $(k_Fa_F)^{-1}$ as in Fig. \[gapvsT\].[]{data-label="muvsT"}](fig4.eps)
Figure \[muvsT\] shows the chemical potential $\mu$ vs temperature for the same coupling values of Fig. \[gapvsT\]. Note that in weak coupling the chemical potential decreases slightly upon moving deep in the superconducting phase from $T_c$ to $T=0$, in agreement with the BCS behavior. In strong coupling the opposite occurs, reflecting the behavior of the bosonic chemical potential $\mu_B=2\mu+\epsilon_0$ within the Bogoliubov theory. It should be, however, mentioned that with improved bosonic approximations [@griffin], the bosonic chemical potential would rather decrease upon entering the condensed phase.
![Order parameter $\Delta$ at $T=0$ (in units of $\epsilon_F$) vs the coupling $(k_Fa_F)^{-1}$ . Results obtained by the inclusion of fluctuations (full line) are compared with mean-field results (dashed line).[]{data-label="gapt0"}](fig5.eps)
Figure \[gapt0\] shows the order parameter $\Delta$ at zero temperature (full line) and the corresponding mean-field value (dashed line) vs the coupling $(k_F a_F)^{-1}$. While $\Delta$ increases monotonically in absolute value from weak to strong coupling (as expected on physical grounds), the relative importance of the fluctuation corrections to the order parameter at zero temperature (over and above mean field) reaches a maximum in the intermediate-coupling region, never exceeding about 30%. This results confirms again that the BCS mean field is a reasonable approximation to the ground state for all couplings.
Figure \[mut0\] shows the chemical potential $\mu$ at zero temperature vs the coupling parameter $(k_F a_F)^{-1}$. The results obtained by the inclusion of fluctuations (full lines) are compared with mean field (dashed lines). Even for this thermodynamic quantity the fluctuation corrections to the mean-field results appear to be not too important at zero temperature.
Note, finally, that the values for $\Delta$ and $\mu$ obtained from our theory at $T=0$ with the coupling value $(k_F a_F)^{-1}=0$ are in remarkable agreement with a recent Quantum Monte Carlo calculation [@carlson] performed for the same coupling. Our calculation yields, in fact, $\Delta/\epsilon_F=0.53$ and $\mu/\epsilon_F = 0.445$, to be compared with the values $\Delta/\epsilon_F=0.54$ and $\mu/\epsilon_F = 0.44 \pm 0.01$ of Ref. . \[In contrast, BCS mean field yields $\Delta/\epsilon_F=0.69$ and $\mu/\epsilon_F = 0.59$.\]
![Chemical potential $\mu$ at $T=0$ (in units of $\epsilon_F$ for $\mu >0$ and of $\epsilon_0/2$ for $\mu<0$) vs the coupling $(k_Fa_F)^{-1}$. Results obtained by the inclusion of fluctuations (full line) are compared with mean-field results (dashed line).[]{data-label="mut0"}](fig6.eps)
In summary, the above results have shown that, for thermodynamic quantities like $\Delta$ and $\mu$, fluctuation corrections to mean-field values in the broken-symmetry phase are important only as far as the temperature dependence is concerned, while at zero temperature the mean-field results are reliable.
Spectral function
-----------------
For a generic value of the coupling, calculation of the imaginary part of the retarded self-energy ${\rm Im} \Sigma_{11}^R({\bf k},\omega)=-h({\bf k},\omega)$ (with $h({\bf k},\omega)$ given by Eq. $(\ref{imsig11})$) requires us to obtain the imaginary part of the particle-particle ladder $\Gamma_{11}^R$ on the real-frequency axis, as determined by the formal replacement $i\Omega_\nu \rightarrow \omega+i\eta$ in the Matsubara expressions (\[Gamma-solution\])-(\[B-definition\]). After performing the frequency sum therein, the wave-vector integrals of Eqs. (\[A-definition\]) and (\[B-definition\]) for the functions $\chi_{ij}({\bf q},i\Omega_\nu \rightarrow \omega+i\eta)$ ($(i,j)=1,2$) are evaluated numerically, by exploiting the properties of the delta function for the imaginary part and keeping a finite albeit small value of $\eta \, (=10^{-8}\sqrt{\mu^2+\Delta^2})$ for the real part.
![Dispersion $\omega({\bf q})$ of the pole of $\Gamma^R_{11}({\bf q},\omega)$ at $T=0$ (full lines) and boundary of the particle-particle continuum (dashed lines) for three characteristic couplings.[]{data-label="gamma"}](fig7.eps)
Direct numerical calculation of the imaginary part of the particle-particle ladder fails, however, when this part has the structure of a delta function for real $\omega$ at given ${\bf q}$. This occurs when the determinant in the denominator of Eq. (\[Gamma-solution\]) vanishes for real $\omega$. To deal with this delta function, let’s first consider the case $T=0$ for which three cases can be distinguished, according to: (i) $a_F<0$ and $\mu>0$ (weak-to-intermediate coupling); (ii) $a_F>0$ and $\mu>0$ (intermediate coupling); (iii) $a_F>0$ and $\mu<0$ (intermediate-to-strong coupling). The curves $\omega({\bf q})$ where the (analytic continuation of the) determinant in the denominator of Eq. (\[Gamma-solution\]) vanishes are shown (full lines) for these three cases in Figs. \[gamma\] (a), (b), and (c), respectively. In these figures we also show the boundaries (dashed lines) delimiting the particle-particle continuum, where the imaginary part of the particle-particle ladder is nonvanishing and regular (in the sense that it does not have the structure of a delta function). At finite temperature, the sharp boundary of the particle-particle continuum smears out, owing to the presence of Fermi functions after performing the sum over the Matsubara frequencies in Eqs. (\[A-definition\]) and (\[B-definition\]). The Fermi functions produce, in fact, a finite (albeit exponentially small with temperature) imaginary part of the particle-particle ladder also below the (dashed) boundaries of Fig. \[gamma\], resulting in a Landau-type damping of the Bogoliubov-Anderson mode $\omega({\bf q})$. In addition, the finite imaginary part broadens the delta-function structure centered at the curves $\omega({\bf q})$ of Fig. \[gamma\], turning it into a Lorentzian function. In practice, our numerical calculation takes advantage of this broadening occurring at finite temperature, and deals with smooth Lorentzian functions instead of the delta-function peaks. [@footnotegamma]
As a further consistency check on our numerical calculations, we have sistematically verified that the three sum rules (\[sum-rule-A\]), (\[sum-rule-B-1\]), (\[sum-rule-B-2\]) are satisfied within numerical accuracy, for all temperatures and couplings we have considered.
![Imaginary part of the self-energy $\sigma^R_{11}$ for $|{\bf k}|=k_{\mu'}$ vs frequency (in units of $\epsilon_F$) at different temperatures for the coupling values $(k_F a_F)^{-1}=$ -0.5 (a), 0.1 (b), and 0.5 (c).[]{data-label="sigmaim"}](fig8.eps)
The imaginary and real parts of the retarded self-energy $\sigma_{11}^R({\bf k},\omega)$ obtained from Eq. (\[sigma-tilde-real\]) are shown, respectively, in Figs. \[sigmaim\] and \[sigmareal\] as functions of frequency at different temperatures and for different couplings (about the crossover region of interest). The magnitude of the wave vector ${\bf k}$ is taken in Figs. \[sigmaim\] and \[sigmareal\] at a special value (denoted by $k_{\mu'}$), which is identified from the behavior of the ensuing spectral function $A({\bf k},\omega)$ when performing a scanning over the wave vector (see Fig. 12 below). Accordingly, $k_{\mu'}$ is chosen to minimize the gap in the spectral function, in agreement with a standard procedure in the ARPES literature. On the weak-coupling side (when the the self-energy shift $\Sigma_0$ discussed in Sec. IIB is included in our calculation), $k_{\mu'}$ coincides with $\sqrt{ 2 m (\mu - \Sigma_0)}$. On the strong-coupling side (when $\mu$ becomes negative) one takes instead $k_{\mu'} = 0$.
![Real part of the self-energy $\sigma^R_{11}$ for $|{\bf k}|=k_{\mu'}$ vs frequency (in units of $\epsilon_F$) at different temperatures for the coupling values $(k_F a_F)^{-1}=$ -0.5 (a), 0.1 (b), and 0.5 (c).[]{data-label="sigmareal"}](fig9.eps)
![Spectral function for $|{\bf k}|=k_{\mu'}$ vs frequency (in units of $\epsilon_F$) at different temperatures for the coupling values $(k_F a_F)^{-1}=$ -0.5 (a), 0.1 (b), and 0.5 (c).[]{data-label="spectT"}](fig10.eps)
For all couplings here considered, the progressive evolution found in $A({\bf k},\omega)$ (from the presence of a pseudogap about $\omega=0$ at $T_c$ to the occurrence of a superconducting gap near zero temperature) stems from the interplay of the two contributions in Eq. (\[sigma-tilde-real\]) to the imaginary part of $\sigma^R_{11}({\bf k},\omega)$ about $\omega=0$. Specifically, for intermediate-to-weak coupling (with $\mu>0$) the first term on the right-hand side of Eq. (\[sigma-tilde-real\]) (which is responsible for the pseudogap suppression in $A({\bf k},\omega)$ at $T_c$) would produce a narrow peak structure in $A(k_{\mu'},\omega)$ about $\omega=0$ upon lowering $T$, since ${\rm Re} \Sigma_{11}^{R}(k_{\mu'},\omega)-\Sigma_0$ vanishes while $|{\rm Im}\,\Sigma_{11}^R(k_{\mu'},\omega)|$ becomes progressively smaller. The presence of the second term on the right-hand-side of Eq. (\[sigma-tilde-real\]), however, gives rise to a narrow peak in ${\rm Im}\, \sigma^R_{11}(k_{\mu'},\omega)$ about $\omega=0$, as seen from Fig. \[sigmaim\] (a), resulting in a depression of $A(k_{\mu'},\omega)$ about $\omega=0$. \[This occurs barring a small temperature range close to $T_c$, where the second term on the right-hand-side of Eq. (\[sigma-tilde-real\]) is not yet well developed.\] At larger couplings (when $\mu < 0 $), the first term on the right-hand-side of Eq. (\[sigma-tilde-real\]) would not produce a peak in $A({\bf k}=0,\omega)$ about $\omega=0$ upon lowering the temperature, because $|\mu|+{\rm Re} \Sigma_{11}^R(k=0,\omega)$ does not correspondingly vanish in this case even though ${\rm Im}\,\Sigma_{11}^R({\bf k}=0,\omega)$ does. In addition, in this case the second term on the right-hand-side of Eq. (\[sigma-tilde-real\]) does not produce a peak in $A({\bf k}=0,\omega)$ about $\omega=0$.
![Temperature dependence of the total weight of the spectral function at negative frequencies for different coupling values about the crossover region.[]{data-label="weightneg"}](fig11.eps)
Figure \[spectT\] shows the resulting spectral function $A({\bf k},\omega)$ vs $\omega$ for $|{\bf k}|=k_{\mu'}$ at different temperatures and couplings. In all cases, at $T_c$ there occurs only [*a broad pseudogap feature*]{} both for $\omega >0$ and $\omega <0$. \[For photoemission experiments only the case $\omega <0$ is relevant, so that we shall mostly comment on this case in the following.\] A [*coherent peak*]{} is seen to grow on top of this broad pseudogap feature as the temperature is lowered below $T_c$. When zero temperature is eventually reached, the pseudogap feature is partially suppressed in favor of the coherent peak, which thus absorbs a substantial portion of the spectral intensity. This interplay between the broad pseudogap feature and the sharp coherent peak results in a characteristic [*peak-dip-hump structure*]{}, which is best recognized from the features for weak-to-intermediate coupling. Generally speaking, this coherent peak (and its corresponding counterpart at positive frequencies) for intermediate-to-weak coupling is associated with the two dips in ${\rm Im}\, \sigma^R_{11}$ symmetrically located about zero frequency \[cf. Figs. \[sigmaim\] (a) and (b)\]. In strong coupling, instead, the coherent peak results from a delicate balance between the real and imaginary parts of $\sigma^R_{11}$ near the boundary of the region where ${\rm Im}\,\Sigma^R_{11}=0$.
An interesting fact is that the weights of the negative and positive frequency parts of the spectrum turn out to be separately (albeit approximatively) constant as functions of temperature for given coupling, as shown in Fig. \[weightneg\] for three characteristic couplings. This implies that, for a given coupling, the coherent peak for $\omega < 0$ grows at the expenses of the accompanying broad pseudogap feature upon decreasing the temperature.
The result that the total area for [*negative*]{} $\omega$ should be (approximately) constant as a function of temperature can be realized also from the analytic results in the extreme strong-coupling limit discussed in Sec. IID. Taking the analytic continuation of the Matsubara Green’s function (\[G-11-strong-coupling-figo\]) (which is appropriate in the strong-coupling limit as far as this total area is concerned, as it will be shown below) results, in fact, in the total weight $\tilde{v}^2({\bf k})$ of the $\omega < 0$ region being independent of temperature, since the combination $\Delta^2+\Delta_0^2$ entering the expression of $\tilde{E}({\bf k})$ is proportional to the total density in this limit \[cf. Eq. (\[final-n\])\].
Returning to Fig. \[spectT\], it is also interesting to comment on the positions of the pseudogap feature and the coherent peak as functions of temperature for given coupling. The position of the coherent peak depends markedly on temperature, shifting progressively toward more negative frequencies as the temperature is lowered. In particular, for weak-to-intermediate coupling the position of the coherent peak about coincides with (minus) the value of the order parameter $\Delta$. In the strong-coupling region (where $\mu < 0$), on the other hand, its position is about at $-\sqrt{\Delta^2+\mu^2}$. This remark entails the possibility of extracting two important quantities from the temperature evolution of the coherent peak in the spectral function: (i) The frequency position of this peak when approaching $T_c$ determines whether $\mu$ is positive (when the peak position approaches $\omega=0$) or negative (when the peak position approaches $-|\mu|$), corresponding to weak-to-intermediate coupling and strong coupling, respectively; (ii) In both cases, the temperature dependence of the order parameter can be extracted from the frequency position of the coherent peak.
The above results for the coherent peak contrast somewhat with the position of the pseudogap feature by decreasing temperature below $T_c$, also determined from Fig. \[spectT\]. The broad pseudogap feature does not depend sensitively on temperature for all couplings shown in this figure. This indicates that the broad pseudogap feature does not relate to the order parameter below $T_c$.
As far as the spectral function is concerned, one of the key results of our theory is thus the presence of [*two structures*]{} (coherent peak and pseudogap), which behave rather independently from each other as functions of temperature and coupling. This result, which is also evidenced by the behavior of the experimental spectra in tunneling experiments on cuprates [@kugler], originates in our theory from the presence of two distinct contributions to the self-energy, namely, the BCS and fluctuation contributions of Eq. (\[Dyson-equation\]). While the broad pseudogap feature at $T<T_c$ develops with continuity from the only feature present at $T>T_c$, the coherent peak [*per se*]{} would be present in a BCS approach even in the absence of the fluctuation contribution. This remark, of course, does not imply that the two contributions to the self-energy of Eq. (\[Dyson-equation\]) are totally independent from each other. They both depend, in fact, on the value of the order parameter $\Delta$ which is, in turn, determined by both self-energy contributions via the chemical potential.
![Spectral function at different wave vectors about $k_{\mu'}$ for $T=0.6 T_c$ vs frequency (in units of $\epsilon_F$) for the coupling value $(k_F a_F)^{-1}=-0.5$.[]{data-label="momentumevol"}](fig12.eps)
A further important feature that can be extracted from our calculation of the spectral function is the evolution of the coherent peak for varying wave vector at fixed temperature and coupling. Figure \[momentumevol\] reports $A({\bf k},\omega)$ vs $\omega$ for different values of the ratio $k/k_{\mu'}$ about unity when $(k_F a_F)^{-1}=-0.5$ and $T/T_c=0.6$. Here, $k/k_{\mu'}=1$ identifies the underlying Fermi surface that represents the “locus of minimum gap”. When $k/k_{\mu'} < 1$, there is a strong asymmetry between the two coherent peaks at $\omega < 0$ and $\omega > 0$, with the peak at $\omega < 0$ absorbing most of the total weight. The situation is reversed when $k/k_{\mu'} > 1$. When $k/k_{\mu'} = 1$ the spectrum is (about) symmetric between $\omega$ and $-\omega$. In addition, when following the position of the coherent peak at $\omega < 0$ starting from $k/k_{\mu'} < 1$, one sees that this position moves toward increasing $\omega$, reaches a minimum distance from $\omega=0$, and bounces eventually back to more negative values of $\omega$. The value of the minimum distance from $\omega=0$ identifies an energy scale $\Delta_m$. At the same time, the weight of the coherent peak at $\omega < 0$ progressively decreases for increasing $k/k_{\mu'}$ starting from $k/k_{\mu'} < 1$. When $k/k_{\mu'}$ becomes larger than unity, the weight of the coherent peak is transferred from negative to positive frequencies. This situation is characteristic of the BCS theory, where only the coherent peaks are present without the accompanying broad pseudogap features. Our calculation shows that this situation persists also for couplings values inside the crossover region, where the presence of the pseudogap feature is well manifest due to strong superconducting fluctuations. \[Sufficiently far from the underlying Fermi surface, the coherent peak and the pseudogap feature merge into a single structure, as it is evident from Fig. \[momentumevol\]. In this case, the above as well as the following considerations apply to the structure as a whole and not to its individual components.\]
![(a) Positions of the coherent peaks (in units of $\epsilon_F$) vs the wave vector as extracted from Fig. \[momentumevol\] . Positive (squares) and negative (circles) branches are compared with BCS-like dispersions (full lines), as explained in the text. (b) Corresponding weights vs the wave vector, with particle-like (full line) and hole-like (dashed line) contributions.[]{data-label="bcslike"}](fig13.eps)
Figure \[bcslike\](a) summarizes this finding for the dispersion of the coherent peaks, by showing the positions of the two coherent peaks as extracted from Fig. \[momentumevol\] vs $k/k_{\mu'}$. These positions are compared with the two branches $\pm \sqrt{\xi({\bf k})^2+\Delta_m^2}$ of a BCS-like dispersion, where $\Delta_m$ is also identified from Fig. \[momentumevol\]. \[The value of $\Delta_m$ turns out to about coincide with the value of the order parameter $\Delta$ at the same temperature, see below.\] The corresponding evolution of the weights of these peaks is shown in Fig. \[bcslike\](b), where the characteristic feature of an avoided crossing is evidenced. The dispersion of the positions and weights of the coherent peaks shown in Fig. \[bcslike\] compare favorably with those recently obtained experimentally [@matsui] for slightly overdoped Bi2223 samples below the critical temperature (for $T/T_c\simeq 0.6$).
An additional outcome of our calculation is reported in Fig. \[deltam\], where the distance $\Delta_m$ of the coherent peak in $A({\bf k,\omega})$ from $\omega=0$ at $|{\bf k}|=k_{\mu'}$ is compared at low temperature with the order parameter $\Delta$ when $\mu > 0$ and with $\sqrt{\mu^2+\Delta^2}$ when $\mu < 0$. This plot thus compares dynamical and thermodynamic quantities. The good agreement between the two curves confirms our identification of the coherent-peak position in $A({\bf k},\omega)$ with the minimum value of the excitations in the single-particle spectra according to a BCS-like expression (where the value of the order parameter $\Delta$ is, however, obtained by including also fluctuation contributions).
![Position $\Delta_m$ (in units of $\epsilon_F$) of the quasi-particle peak at $T=0.1 T_c$ vs the coupling $(k_Fa_F)^{-1}$ (full line). The dashed line corresponds to the value of the order parameter $\Delta$ when $\mu > 0$ and of $\sqrt{\Delta^2 +{\mu}^2}$ when $\mu < 0$.[]{data-label="deltam"}](fig14.eps)
Finally, it is interesting to comment on the strong-coupling result (\[G-11-strong-coupling\]) for the diagonal Green’s function, with a characteristic double-fraction structure. The corresponding spectral function $A({\bf k},\omega)$, obtained from that expression after performing the analytic continuation $i\omega_n\to\omega + i\eta$, shows only a [*single*]{} feature for $\omega < 0$, with a temperature-independent position. This contrasts the numerical results we have presented \[cf. in particular Fig. \[spectT\]\]. This difference is due to the fact that, in our numerical calculation, the analytic continuation has been properly performed [*before*]{} taking the strong-coupling limit, as emphasized in Sec. IIE. With this procedure, in fact, the pseudogap structure and the coherent peak remain distinct from each other even in the strong-coupling limit, without getting lumped into a single feature. Such a noncommutativity of the processes of taking the analytic continuation and the strong-coupling limit was noted already in a previous paper [@PPSC-02] when studying the spectral function above $T_c$. More generally, the occurrence of this noncommutativity is expected whenever one considers approximate expressions in the Matsubara representation and takes the analytic continuation of these expressions to real frequency.
To make evident the noncommutativity of the two processes, we show in Fig. \[fignoncomm\] the spectral function $A({\bf k}=0,\omega)$ for $(k_F a_F)^{-1}=2.0$ and $T/T_c=0.1$, obtained by two alternative methods, namely: (i) Using the analytic continuation of the expression (\[Sig11\]) for $\Sigma_{11}$ where $i \omega_s \to \omega + i\eta$ (full line); (ii) Taking the strong-coupling expression (\[Sigma-11-n-prime\]) for $\Sigma_{11}$, in which the analytic continuation $i\omega_s\to \omega + i \eta$ is performed (broken line). Method (i) results in the presence of [*two*]{} distinct structures in $A({\bf k},\omega)$ for $\omega < 0$, corresponding to the coherent (delta-like) peak and the broad pseudogap feature. Method (ii) gives instead a [*single*]{} delta-like peak. It is interesting to note that the total spectral weight of the two peaks for $\omega < 0$ obtained by method (i) (=0.049 for the coupling of Fig. \[fignoncomm\]) about coincides with the weight of the delta-like peak (=0.044) obtained by method (ii). \[We have verified that this correspondence between the spectral weights persists also at stronger couplings.\]
![Spectral function vs frequency for $(k_F a_F)^{-1}=2.0$ and $T/T_c=0.1$, obtained by taking alternatively the analytic continuation of $\Sigma_{11}$ from the expression (\[Sig11\]) (full line) or from the expression (\[Sigma-11-n-prime\]) (broken line).[]{data-label="fignoncomm"}](fig15.eps)
These remarks explain the occurrence of a single feature in the spectral function as obtained by a different theory based on a preformed-pair scenario [@levin98]. In that theory, a single-particle Green’s function with a double-fraction structure is considered in the Matsubara representation for any coupling, and correspondingly a single feature in the spectral function is obtained for real frequencies [@footnotelevin]. Our theory shows instead the appearance of two distinct energy scales (pseudogap and order parameter) in the spectral function below $T_c$.
We are thus led to conclude that the occurrence of two distinct energy scales below $T_c$ in photoemission and tunneling spectra should not be necessarily associated with the presence of an “extrinsic” pseudogap due to additional non-pairing mechanisms, as sometimes reported in the literature [@levin02].
Concluding remarks
==================
In this paper, we have extended the study of the BCS-BEC crossover to finite temperatures below $T_c$. This has required us to include (pairing) fluctuation effects in the broken-symmetry phase on top of mean field. Our approximations have been conceived to describe both a system of superconducting fermions in weak coupling and a system of condensed composite bosons in strong coupling, via the simplest theoretical approaches valid in the two limits. These are the BCS mean field (plus superconducting fluctuations) in weak coupling and the Bogoliubov approximation in strong coupling. To this end, analytic results have been specifically obtained in strong coupling from our general expression of the fermionic self-energy.
Results of numerical calculations have been presented both for thermodynamic and dynamical quantities. The latter have been defined by a careful analytic continuation in the frequency domain. In this context, a noncommutativity of the analytic continuation and the strong-coupling limit has been pointed out.
Results for thermodynamic quantities (like the order parameter and chemical potential) have shown that the effects of pairing fluctuations over and above the BCS mean field become essentially irrelevant in the zero-temperature limit, even in strong coupling. Results for a dynamical quantity like $A({\bf k},\omega)$ have shown, in addition, that two structures (a broad pseudogap feature that survives above $T_c$ and a strong coherent peak which emerges only below $T_c$) are present simultaneously, and that their temperature and coupling behaviors are rather (even though not completely) independent from each other.
These features produced in the spectral function by our theory originate from a totally [*intrinsic*]{} effect, namely, the occurrence of a strong attractive interaction (irrespective of its origin). Additional features produced by other [*extrinsic*]{} effects could obviously be added on top of the intrinsic effects here considered.
Similar results have recently been obtained in Ref. , using a boson-fermion model for precursor pairing below $T_c$. In that reference, a two-peak structure for $A({\bf k},\omega)$ has also been obtained, although with a self-energy correction introduced by a totally different method.
The attractive interaction adopted in this paper is the simplest one that can be considered, depending on a single parameter only. Detailed comparison of the results of this theory with experiments on cuprates would then require one to specify the dependence of this effective parameter on temperature and doping.
The simplified model that we have adopted in this paper should instead be considered realistic enough for studying theoretically the BCS-BEC crossover for Fermi atoms in a trap. The occurrence of this crossover in these systems is being rather actively studied experimentally at present. [@expcross] In this case, the calculation should also take into account the external trapping potential by considering, e.g., a local version of our theory with local values of the density and chemical potential in the trap. [@trap].
Financial support from the Italian MIUR under contract COFIN 2001 Prot.2001023848 is gratefully acknowledged.
Analytic continuation for the fermionic retarded single-particle Green’s functions and sum rules below the critical temperature
===============================================================================================================================
In this appendix, we extend [*below*]{} the critical temperature a standard procedure for obtaining [*at a formal level*]{} the fermionic retarded single-particle Green’s functions via analytic continuation from their Matsubara counterparts. This is done in terms of the Lehmann representation [@FW] and of the Baym-Mermin theorem[@Baym-Mermin-61]. In this context, besides the usual sum rule that holds also above the critical temperature [@FW], we will obtain two additional sum rules that hold specifically below the critical temperature.
The results proved in this appendix hold *exactly*, irrespective of the approximations adopted for the Matsubara self-energy. To satisfy the above three sum rules with an approximate choice of the self-energy, however, it is *not* required for the ensuing approximation to the fermionic single-particle Green’s functions to be “conserving” in the Baym’s sense [@Baym-62]. Rather, it is sufficient that the analytic continuation from the Matsubara frequencies to the real frequency axis is taken properly, as demonstrated in Sec. IIE with the specific choice (\[total-self-energy\]) of the self-energy.
We begin by considering the fermionic “normal” and “anomalous” *retarded* single-particle Green’s functions in the broken-symmetry phase, defined respectively by $$\begin{aligned}
G^{R}({\mathbf r},t;{\mathbf r'},t') = - i \theta(t-t')
\langle \left\{
\psi_{\uparrow}({\mathbf r},t),\psi_{\uparrow}^{\dagger}({\mathbf r'},t')
\right\} \rangle \label{G-retarded} \\
F^{R}({\mathbf r},t;{\mathbf r'},t') = - i \theta(t-t')
\langle \left\{
\psi_{\uparrow}({\mathbf r},t),\psi_{\downarrow}({\mathbf r'},t') \right\}
\rangle\, . \label{F-retarded}\end{aligned}$$ Here, $\theta(t)$ is the unit step function, $\psi_{\sigma}({\mathbf r},t)$ is the fermionic field operator with spin $\sigma=(\uparrow,\downarrow)$ at position ${\mathbf r}$ and (real) time $t$ (such that $\psi_{\sigma}({\mathbf r},t) = \exp (iKt) \psi_{\sigma}({\mathbf r}) \exp
(-iKt)$ with $ K = H - \mu N$ in terms of the system Hamiltonian $H$ and the particle number $N$), the braces represent an anticommutator, and $\langle\cdots\rangle$ stands for the grand-canonical thermal average.
The Matsubara counterparts of (\[G-retarded\]) and (\[F-retarded\]) are similarly defined by $$\begin{aligned}
G({\mathbf r},\tau;{\mathbf r'},\tau') & = & -
\langle T_{\tau} \left[ \psi_{\uparrow}({\mathbf r},\tau)
\psi_{\uparrow}^{\dagger}({\mathbf r'},\tau') \right] \rangle
\label{G-Matsubara} \\
F({\mathbf r},\tau;{\mathbf r'},\tau') & = & -
\langle T_{\tau} \left[ \psi_{\uparrow}({\mathbf r},\tau)
\psi_{\downarrow}({\mathbf r'},\tau') \right] \rangle \, ,
\label{F-Matsubara}\end{aligned}$$ where now $\psi_{\sigma}({\mathbf r},\tau) = \exp (K\tau)
\psi_{\sigma}({\mathbf r}) \exp (-K\tau)$, $\psi_{\sigma}^{\dagger}({\mathbf r},\tau) = \exp (K\tau)
\psi_{\sigma}^{\dagger}({\mathbf r}) \exp (-K\tau)$, and $T_{\tau}$ is the time-ordering operator for imaginary time $\tau$.
The Lehmann analysis for the normal function $G^{R}$ in the broken-symmetry phase proceeds along similar lines as for the normal phase [@FW]. The result is that (for a homogeneous system) the wave-vector and (real) frequency Fourier transform can be obtained by the spectral representation $$G^{R}({\mathbf k},\omega) \, = \, \int_{-\infty}^{+\infty} \, d \omega' \,
\frac{A({\mathbf k},\omega')}{\omega \, - \, \omega' \, + \, i
\, \eta} \label{spectral-repres-G-R}$$ $\eta$ being a positive infinitesimal. Here, the real and positive definite *spectral function* $A({\mathbf k},\omega) = - (1/\pi) \mathrm{Im}\, G^{R}({\mathbf k},\omega)$ satisfies the sum rule $$\int_{-\infty}^{+\infty} \, d \omega \, A({\mathbf k},\omega) \, = \, 1
\label{sum-rule-G-R}$$ for any given ${\mathbf k}$, as a consequence of the canonical anticommutation relation of the field operators.
A similar analysis for the Matsubara normal Green’s function leads to the spectral representation $$G({\mathbf k},\omega_{s}) \, = \, G_{11}({\mathbf k},\omega_{s}) \, = \,
\int_{-\infty}^{+\infty} \, d \omega' \,
\frac{A({\mathbf k},\omega')}{i\omega_{s} \, - \, \omega'} \,\, ,
\label{spectral-repres-G-Matsubara}$$ in terms of the *same* spectral function $A({\mathbf k},\omega)$ of Eq. (\[spectral-repres-G-R\]), where $\omega_{s}=(2s+1)\pi/\beta$ ($s$ integer) is a fermionic Matsubara frequency and the diagonal Nambu Green’s function has been introduced. The spectral representations (\[spectral-repres-G-R\]) and (\[spectral-repres-G-Matsubara\]), together with knowledge of the asymptotic behavior $G^{R}({\mathbf k},\omega) \sim
\omega^{-1}$ for large $|\omega|$, are sufficient to guarantee that the retarded normal function is the correct analytic continuation of its Matsubara counterpart in the upper-half of the complex frequency plane [@FW], in accordance with the Baym-Mermin theorem [@Baym-Mermin-61].
The above Lehmann analysis can be extended to the anomalous function (\[F-retarded\]) as well. One obtains $$F^{R}({\mathbf k},\omega) \, = \, \int_{-\infty}^{+\infty} \, d \omega' \,
\frac{B({\mathbf k},\omega')}{\omega \, - \, \omega' \, + \, i
\, \eta} \label{spectral-repres-F-R}$$ in the place of Eq. (\[spectral-repres-G-R\]). The new spectral function $B({\mathbf k},\omega)$ vanishes for large $|\omega|$ but, in general, is no longer real and positive definite. \[One obtains for $B({\mathbf k},\omega)$ the same formal expression [@FW] for $A({\mathbf k},\omega)$ in terms of the eigenstates $|n>$ of the operators $H$ and $N$, apart from the replacement of $|\langle n'|\psi_{\uparrow}({\mathbf r}=0)|n \rangle|^{2}$ with $\langle n|\psi_{\downarrow}({\mathbf r}=0)|n' \rangle$ $\langle n'
|\psi_{\uparrow}({\mathbf r}=0)|n\rangle$.\][@footnote-2] It can then be readily verified that $B({\mathbf k},\omega)$ satisfies the sum rule $$\int_{-\infty}^{+\infty} \, d \omega \, B({\mathbf k},\omega) \, = \, 0
\,\, , \label{sum-rule-F-R}$$ which is again a consequence of the canonical anticommutation relation of the field operators. The above properties guarantee that $F^{R}({\mathbf k},\omega)$ vanishes faster than $\omega^{-1}$ for large $|\omega|$.
By a similar token, considering the Matsubara anomalous Green’s function leads to the spectral representation $$F({\mathbf k},\omega_{s}) \, = \, G_{12}({\mathbf k},\omega_{s}) \, = \,
\int_{-\infty}^{+\infty} \, d \omega' \,
\frac{B({\mathbf k},\omega')}{i\omega_{s} \, - \, \omega'} \,\, ,
\label{spectral-repres-F-Matsubara}$$ where the off-diagonal Nambu Green’s function has been introduced. These considerations suffice again to guarantee that the retarded anomalous function is the correct analytic continuation of its Matsubara counterpart in the upper-half complex frequency plane, in accordance with the Baym-Mermin theorem [@Baym-Mermin-61].
Finally, an additional sum rule for $B({\mathbf p},\omega)$ can be obtained by using the relation $$\begin{aligned}
&&\int_{-\infty}^{+\infty} d \omega \, B({\mathbf k},\omega) \, \omega =
i \, \int_{-\infty}^{+\infty}
\frac{d \omega}{2 \pi} \, F^{R}({\mathbf k},\omega) \, \omega \, e^{- i
\omega \eta} \nonumber \\
& &= i \, \int \! d{\mathbf r} \, e^{- i {\mathbf k}\cdot{\mathbf r}} \,
\langle \left\{\frac{\partial \psi_{\uparrow}({\mathbf r},t=0^+)}{\partial
t},\psi_{\downarrow}(0)\right\}\rangle
\label{3rd-sum-rule-initial}\end{aligned}$$ and exploiting the equation of motion for the field operator. For the contact potential we are considering throughout this paper, we write $$\begin{aligned}
\langle \left\{\frac{\partial \psi_{\uparrow}({\mathbf r},t=0^+)}
{\partial t},\psi_{\downarrow}(0) \right\}\rangle
&=& - v_{0} \delta({\mathbf r}) \langle
\psi_{\uparrow}({\mathbf r})\psi_{\downarrow}({\mathbf r})\rangle\nonumber\\
&=& - \delta({\mathbf r}) \Delta
\label{equation-of-motion}\end{aligned}$$ in terms of the order parameter $\Delta$. The expression (\[3rd-sum-rule-initial\]) thus becomes:
$$\int_{-\infty}^{+\infty} \, d \omega \, B({\mathbf k},\omega) \, \omega \, =
\, - \, \Delta \,\, . \label{3rd-sum-rule-final}$$
This constitutes a third sum rule for the spectral functions in the broken-symmetry phase.
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J. Carlson, S.-Y. Chang, V.R. Pandharipande, and K.E. Schmidt, Phys. Rev. Lett. [**91**]{}, 050401 (2003).
The boundary (dashed) lines in Fig. \[gamma\] are determined by the conditions: (i) $\Delta^2 + ({\bf q}^2/8m -\mu)^2 -
\omega^2/4=0$ for all ${\bf q}$ when $\mu < 0$ and for $|{\bf q}| > \sqrt{8 m \mu}$ when $\mu > 0$; (ii) $\Delta^2 - \omega^2/4=0$ for $|{\bf q}| < \sqrt{8 m \mu}$ when $\mu > 0$. Note also that the Bogoliubov-Anderson mode (full line) terminates at $|\omega|=2 \Delta$ in the upper panel of Fig. \[gamma\], while it remains below the particle-particle continuum (dashed lines) in the middle and lower panels of Fig. \[gamma\].
M. Kugler, O. Fischer, Ch. Renner, S. Ono, and Y. Ando, Phys. Rev. Lett. [**86**]{}, 4911 (2001).
H. Matsui, T. Sato, T. Takahashi, S.-C. Wang, H.-B. Yang, H. Ding, T. Fujii, T. Watanabe, and A. Matsuda, Phys. Rev. Lett. [**90**]{}, 217002 (2003).
Besides the difference mentioned in the text about the relative order of performing the analytic continuation to real frequency and of considering approximate expressions for the self-energy in the Matsubara representation, it is also worth mentioning some additional differences between our theory and the theory of Ref. based on a “preformed-pair scenario” below $T_c$. In our theory, the fluctuation propagator (\[Gamma-solution\]) is built on the BCS Green’s functions (\[BCS-Green-function\]) and is obtained via the inversion of a 2 $\times$ 2 matrix. In the theory of Ref. , the corresponding propagator is instead built on a non-interacting Green’s function ${\cal G}_0$ and on a dressed Green’s function with the functional form of Eq. (\[G-11-strong-coupling\]). We have verified that, in the extreme strong-coupling limit (where the residual interaction between the composite bosons becomes irrelevant) and for temperatures $T\ll T_c$, the two theories give essentially the same results in the Matsubara representation. Differences show up, however, at weaker couplings, when the interaction between composite bosons matters. Specifically, our theory for the composite bosons reproduces the Bogoliubov results for point-like bosons with $\mu_B \neq 0$ and $n_0(T=0)\neq n$, owing to the boson-boson interaction and the depletion of the condensate. In the theory of Ref. , on the other hand, $\mu_B$ for the composite bosons is taken to vanish for $T\le T_c$ and also $n_0(T=0)=n$ for all couplings. More generally for any coupling, in our theory (quantum) fluctuation corrections to mean-field quantities survive even at $T=0$ while in the theory of Ref. fluctuation corrections vanish identically at $T=0$.
See the discussion on this topic reported in Y.-J. Kao, A.P. Iyengar, J. Stajic, and K. Levin, Phys. Rev. B [**66**]{}, 214519 (2002).
T. Domański and J. Ranninger, Phys. Rev. Lett. [**91**]{}, 255301 (2003).
A. Perali, P. Pieri, L. Pisani, and G.C. Strinati, Phys. Rev. Lett. (in press).
Quite generally, the function $B({\mathbf k},\omega)$ introduced through the Lehmann representation (\[spectral-repres-F-R\]) can be cast in the form $B({\mathbf k},\omega) = [F^{A}({\mathbf k},\omega)
- F^{R}({\mathbf k},\omega)]/(2 \pi i)$, in terms of the *advanced* ($A$) and *retarded* ($R$) “anomalous” functions. In particular, when $F^{A}({\mathbf k},\omega) =
F^{R}({\mathbf k},\omega)^{*}$ for *real* $\omega$, $B({\mathbf k},\omega)$ can be identified with $-(1/\pi) \mathrm{Im} F^{R}({\mathbf k},\omega)$. This is the case for the “anomalous” retarded and advanced Green’s functions resulting from the approximate expression (\[G-12-Matsubara\]) of the text with a *real* order parameter $\Delta$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We performed a multiwavelength study toward infrared dark cloud (IRDC) G34.43+0.24. New maps of $^{13}$CO $J$=1-0 and C$^{18}$O $J$=1-0 were obtained from the Purple Mountain Observatory (PMO) 13.7 m radio telescope. At 8 $\mu$m (Spitzer - IRAC), IRDC G34.43+0.24 appears to be a dark filament extended by 18$^{\prime}$ along the north-south direction. Based on the association with the 870 $\mu$m and C$^{18}$O $J$=1-0 emission, we suggest that IRDC G34.43+0.24 should not be 18$^{\prime}$ in length, but extend by 34$^{\prime}$. IRDC G34.43+0.24 contains some massive protostars, UC H II regions, and infrared bubbles. The spatial extend of IRDC G34.43+0.24 is about 37 pc assuming a distance of 3.7 kpc. IRDC G34.43+0.24 has a linear mass density of $\sim$ 1.6$\times$10$^{3}$ $M_{\odot}$ $ \rm pc^{-1}$, which is roughly consistent with its critical mass to length ratio. The turbulent motion may help stabilizing the filament against the radial collapse. Both infrared bubbles N61 and N62 show a ringlike structure at 8 $\mu$m. Particularly, N61 has a double-shell structure, which has expanded into IRDC G34.43+0.24. The outer shell is traced by 8 $\mu$m and $^{13}$CO $J$=1-0 emission, while the inner shell is traced by 24 $\mu$m and 20 cm emission. We suggest that the outer shell (9.9$\times10^{5}$ yr) is created by the expansion of H II region G34.172+0.175, while the inner shell (4.1$\sim$6.3$\times$10$^{5}$ yr) may be produced by the energetic stellar wind of its central massive star. From GLIMPSE I catalog, we selected some Class I sources with an age of $\sim$$10^{5}$ yr. These Class I sources are clustered along the filamentary molecular cloud.'
author:
- 'Jin-Long Xu, Di Li, Chuan-Peng Zhang, Xiao-Lan Liu, Jun-Jie Wang, Chang-Chun Ning, and Bing-Gang Ju'
title: 'Gas Kinematics and Star Formation in the Filamentary IRDC G34.43+0.24'
---
INTRODUCTION
============
Because infrared dark clouds (IRDCs) are considered the precursors of massive stars and star clusters (Egan et al. 1998; Carey et al. 2000; Rathborne et al. 2006). They have attracted much attention particularly in the last decade. IRDCs are seen as dark absorption features against the Galactic background at mid-infrared wavelengths (Egan et al. 1998; Hennebelle et al. 2001; Simon et al. 2006; Peretto & Fuller 2009). Egan et al. (1998) initially suggested that IRDCs are isolated objects, possibly left over after their parental molecular clouds have been dispersed. Subsequent studies of IRDCs suggest that they are cold ($<$25 K), dense ($\thicksim$10$^{3}$-10$^{5}$ cm$^{-3}$), and are thought to host very early stages of massive star formation (Carey et al. 2000; Simon et al. 2006; Pillai et al. 2006; Ragan et al. 2011; Liu et al. 2014). One striking feature of IRDCs is their filamentary shape. Jackson et al. (2010) mentioned that such filaments probably resulted from the passage of a spiral shock. Wang et al. (2015) found that some filaments are closely associated with the Milky Way spiral substructures. In addition, compact cores are found in IRDCs (Rathborne et al. 2006). Some are found to be candidates for massive starless objects (Tackenberg et al. 2012; Beuther et al. 2013), other show the signposts of massive star formation (Rathborne et al. 2011), such as hot molecular cores, ultra-compact H [II]{} (UC H [II]{}) regions, and infall and outflow (Rathborne et al. 2007, 2008). Another important feature of massive star formation is when a massive star formed inside an IRDC, UV radiation and stellar winds would ionize the surrounding gas and create an infrared bubble (Churchwell et al. 2006), and even disrupt the natal IRDC (Jackson et al. 2010).
Infrared bubbles are the bright 8.0 $\mu$m emission surrounding O and early-B stars, which show the full or partial ring structures (Churchwell et al. 2006). Churchwell et al. (2006 & 2007) have compiled a list of $\sim$600 infrared bubbles, while Simpson et al. (2012) have created a new catalogue of 5106 infrared bubbles through visual inspection via the online citizen science website ‘The Milky Way Project (MWP)’. Deharveng et al. (2010) studied 102 bubbles. They show that 86$\%$ of these bubbles enclose H [II]{} regions ionized by O and B stars. Since the bubbles are usually found in or near massive star-forming regions, which can provide important information on the dynamical processes and physical environments of their surrounding cloud.
G34.43+0.24 is a filamentary IRDC at a kinematic distance of 3.7 kpc (Rathborne et al. 2006; Faúndez et al. 2004; Simon et al. 2006). VLBI parallax measurements of H$_{2}$O maser sources within the IRDC determined a distance of 1.56 kpc (Kurayama et al. 2011). Although a parallax measurement would seem to be the more reliable distance determination, Foster et al. (2012) suggested that the parallax determinations to the same sources are incorrect. Foster et al. (2014) considered that the Kurayama et al. (2011) measurement relied on only a single reference background source, and the measurements are inherently difficult due to the low declination of this target. They thus adopted the kinematic/extinction distance of 3.9 kpc for this cloud. IRDC G34.43+0.24 spans by 9$^{\prime}$ from north to south in equatorial projection to a total mass of 1000 M$_{\odot}$ from the ammonia observation (Miralles et al. 1994). Famous UC H [II]{} complex G34.26+0.15 is roughly 11$^{\prime}$ south of IRDC G34.43+0.24 (Shepherd et al. 2007). Rathborne et al. (2006) identified nine cores in this IRDC (see Fig. 1). The MM1 core appears to be a massive B2 protostar in an early stage of evolution, based on the weak 6 cm continuum emission and the lack of detection at NIR wavelengths (Shepherd et al. 2004). Rathborne et al. (2011) confirmed that the MM1 core is a hot molecular core with a rotating structure surrounding the central protostar (Rathborne et al. 2008). From CO observations with Owens Valley Radio Observatory (OVRO) array of six 10.4 m antenna, two massive outflows were discovered in the MM1 core (Shepherd et al. 2007). The MM2 core is associated with UC H [II]{} G34.4+0.23 (Miralles et al. 1994; Molinari et al. 1998), IRAS 18507+0121 source (Bronfman et al. 1996), an H$_{2}$O maser (Miralles et al. 1994), and an CH$_{3}$OH maser (Szymczak & Kus 2000). Shepherd et al. (2007) also detected three massive outflows centered on or near UC H [II]{} G34.4+0.23. The MM2 core is also undergoing infalling motion with a mass infall rate of about 1.8$\times$10$^{-3}$ M$_{\odot}$ yr$^{-1}$. The MM3 core is located close to the northern edge of the filament IRDC G34.43+0.24, about 3.5$^{\prime}$ north of the MM2. Sanhueza et al. (2006) reported an outflow in CO $J$=3-2 toward the MM3 core. From ALMA observations, the 1.3 mm continuum and several molecular lines reveal a highly collimated outflow with a dynamic timescale of less than 740 yr and a hot molecular core toward the MM3 core (Sakai et al. 2013). Although star formation has already started in the MM3 core (Rathborne et al. 2005), the MM3 core is thought to be in an earlier evolutionary stage than the MM1 and MM2 cores. The rest of the cores (MM4, MM5, MM6, MM7, MM8, and MM9) do not have obvious sign of massive star formation.
To investigate the gas kinematics and star formation in IRDC G34.43+0.24, we performed a multi-wavelength study toward the IRDC. Combining the NRAO VLA Sky survey, ATLASGAL survey, and GLIMPSE survey, we aim to construct a comprehensive larger-scale picture of IRDC G34.43+0.24. The observations and data reduction are described in Sect.2, and the results are presented in Sect.3. In Sect.4, we will discuss the star formation scenrio. The conclusions are summarized in Sect.5.
OBSERVATIONS AND DATA REDUCTION
===============================
Purple Mountain Data
--------------------
We made the mapping observations of IRDC G34.43+0.24 and its adjacent regions in the transitions of $^{13}$CO $J$=1-0 and C$^{18}$O $J$=1-0 lines using the Purple Mountain Observation (PMO) 13.7 m radio telescope at De Ling Ha in the west of China at an altitude of 3200 meters, in 2012 May and 2013 Jan. The new 3$\times$3 beam array receiver system in single-sideband (SSB) mode was used as front end. The back end is a Fast Fourier Transform Spectrometer (FFTS) of 16384 channels with a bandwidth of 1 GHz, corresponding to a velocity resolution of 0.17 km s$^{-1}$ for $^{13}$CO $J$=1-0 and C$^{18}$O $J$=1-0. The half-power beam width (HPBW) was 53$^{\prime\prime}$ at 115 GHz and the main beam efficiency was 0.5. The pointing accuracy of the telescope was better than 5$^{\prime\prime}$, which was derived from continuum observations of planets. The source W51D (19.2 K) was observed once per hour as flux calibrator. The system noise temperature (Tsys) in SSB mode varied between 150 K and 400. The mean rms noise level of the calibrated brightness temperature was 0.3 K for $^{13}$CO $J$=1-0 and C$^{18}$O $J$=1-0. Mapping observations were centered at RA(J2000)=$18^{\rm h}53^{\rm m}16.52^{\rm s}$, DEC(J2000)=$01^{\circ}12'43.3^{\prime\prime}$ using the on-the-fly mode with a constant integration time of 6 second at each point. The standard chopper wheel calibration technique is used to measure antenna temperature $T_{\rm A}$$^{\ast}$ corrected for atmospheric absorption. The final data was recorded in brightness temperature scale of $T_{\rm mb}$ (K). The data were reduced using the GILDAS/CLASS [^1] package.
Archival Data
--------------
The 1.4 GHz radio continuum emission data were obtained from the NRAO VLA Sky Survey (NVSS; Condon et al. 1998) which is a 1.4 GHz continuum survey covering the entire sky north of -40$^{\circ}$ declination with a noise of about 0.45 mJy/beam and a resolution of 45$^{\prime\prime}$.
We extract 870 $\mu$m data from the ATLASGAL survey, which is one of the first systematic surveys of the inner Galactic plane in the submillimetre band. The survey was carried out with the Large APEX Bolometer Camera (LABOCA; Siringo et al. 2009), an array of 295 bolometers observing at 870 $\mu$m (345 GHz). The APEX telescope has a full width at half-maximum (FWHM) beam size of 19$^{\prime\prime}$ at this frequency and has a positional accuracy of 2$^{\prime\prime}$. The initial survey covered a Galactic longitude region of 300$^{\circ}$$<$$\ell$$<$60$^{\circ}$ and $|b|$$<$1.5$^{\circ}$ but this was later extended to include 280$^{\circ}$$<$$\ell$$<$300$^{\circ}$ and -2$^{\circ}$$<$$b$$<$1$^{\circ}$ to take account of the warp present in this part of the Galactic plane (Schuller et al. 2009).
We also utilize NIR data in four bands from the Spitzer GLIMPSE survey and MIR data in two bands. GLIMPSE survey (Benjamin et al. 2003) observed the Galactic plane (65$^{\circ}$ $<$ $|l|$ $<$ 10$^{\circ}$ for $|b|$ $<$ 1$^{\circ}$) with the four IR bands (3.6, 4.5, 5.8, and 8.0 $\mu$m) of the Infrared Array Camera (IRAC) (Fazio et al. 2004) on the Spitzer Space Telescope. The resolution ranges from 1.5$^{\prime\prime}$ (3.6 $\mu$m) to 1.9$^{\prime\prime}$ (8.0 $\mu$m). MIPSGAL is a survey of the same region as GLIMPSE, using MIPS instrument (24 and 70 $\mu$m) on Spitzer (Rieke et al. 2004). The MIPSGAL resolution at 24 $\mu$m is 6$^{\prime\prime}$.
RESULTS
=======
Infrared and Radio Continuum Images of IRDC G34.43+0.24
-------------------------------------------------------
Figure 1a shows the Spitzer-IRAC 8 $\mu$m emission of IRDC G34.43+0.24. At 8 $\mu$m, IRDC G34.43+0.24 displays a dark extinction feature from north to south. Adjacent to the south of IRDC G34.43+0.24, we find some bright 8 $\mu$m emission and smaller dark filaments, which is marked in the white dashed lines and seems to be smaller IRDCs. The bright Spitzer-IRAC 8 $\mu$m emission is attributed to polycyclic aromatic hydrocarbons (PAHs) (Leger & Puget 1984). The PAHs molecules can be destroyed inside the ionized gas, but are excited in the photodissociation region (PDR) by the UV radiation within H [II]{} region (Pomarès et al. 2009). Hence, the Spitzer-IRAC 8 $\mu$m emission can be used to delineate an infrared bubble. Infrared bubbles N61 and N62 are firstly identified by Churchwell et al. (2006). N61 is located to the north of the H [II]{} complex G34.26+0.15, while N62 is to the south. Both N61 and N62 show a ringlike structure opened at the south. From the Simpson et al. (2012) catalogue, we identified another six infrared bubbles in the observed region, named at B1-B6. We use the green circles to represent bubbles B1-B6 in Fig. 1a. The parameters of these bubbles are listed in Table 1. From the position and radius of each bubble, we note that B1-B4 overlap with each other and are situated on the bright 8 $\mu$m clump, while B5 and B6 are located on the rim of N61 in Fig. 1a.
Figure 1b presents the Spitzer-MIPSGAL 24 $\mu$m image of IRDC G34.43+0.24. IRDC G34.43+0.24 mostly appears as dark extinction at 24 $\mu$m. Rathborne et al. (2006) identified nine dust cores (MM1, MM2, MM3, MM4, MM5, MM6, MM7, MM8, and MM9) in this IRDC (see Fig. 1). MM1-MM3 are associated with the bright 24 $\mu$m point sources, which are generally considered protostars (Chambers et al. 2009). B1-B6 exhibit the strong 24 $\mu$m emission. Because B1-B4 overlap with each other in the line of sight, we cannot distinguish the morphology of the hot dust of each bubble from each other. In Fig. 1b, both N61 and N62 contain 24 $\mu$m emission in their inner regions. Moreover, the 24 $\mu$m emission also delineate the ringlike shells of N61 and N62, which is similar to those in the 8 $\mu$m emission. The 24 $\mu$m emission inside N61 shows a semi-ringlike shape, as also shown in the right panel of Fig. 10.
Figure 1c shows the ATLASGAL 870 $\mu$m emission of IRDC G34.43+0.24. The 870 $\mu$m emission traces the distribution of cold dust (Beuther et al. 2012). The long dark filament of IRDC G34.43+0.24 and other smaller IRDC fragments coincide well with the 870 $\mu$m emission. MM1-MM4 cores, UC H [II]{} region G34.24+0.13, and H [II]{} complex G34.26+0.15 are associated with bright 870 $\mu$m clumps. UC H [II]{} region G34.24+0.13 hosts a massive protostellar object that coincides with a methanol maser (Hunter et al. 1998). Taking into consideration the morphology in both 8 $\mu$m and 870 $\mu$m, we put forth the hypothesis that IRDC G34.43+0.24 and the few smaller IRDCs south of IRDC G34.43+0.24 are one continuous structure lying behind the H [II]{} region G34.172+0.175, with a spatial extent of $\sim$28$^{\prime}$, much longer than the 9$^{\prime}$ listed in Miralles et al. (1994).
The 21 cm emission is mainly from free-free emission, which can be used to trace the ionized gas of H [II]{} regions. Figure 1d presents the 21 cm continuum emission map. The infrared bubbles B1-B6 are associated with ionized gas, while there is no 21 cm emission from IRDC G34.43+0.24. In a GBT X-band survey (Anderson et al. 2011), 448 previously unknown Galactic H [II]{} regions were detected in the Galactic zone 343$^{\circ}$$\leq$$\ell$$\leq$67$^{\circ}$ and $|b|$$\leq$1$^{\circ}$. H [II]{} regions G34.172+0.175 and G34.325+0.211 are located in our investigative region from the catalog of Anderson et al. (2011). The hydrogen radio recombination line (RRL) velocities of the two H [II]{} regions are 57.3 $\pm$ 0.1 km s$^{-1}$ and 62.9 $\pm$ 0.1 km s$^{-1}$, which are associated with N61 and N62, respectively (Anderson et al. 2011). In Fig. 1d, the ionized gas of G34.325+0.211 shows a compact structure, while it displays a ringlike shape for G34.172+0.175, as shown in a blue dashed circle.
CO Molecular Emission of IRDC G34.43+0.24
-----------------------------------------
IRDCs are dense with the high-volume densities of $\thicksim$10$^{3}$-10$^{5}$ cm$^{-3}$ (Carey et al. 2000; Simon et al. 2006; Ragan et al. 2011). Comparing the median optically thick $^{13}$CO line, the optically thin and relatively abundant C$^{18}$O line (Yonekura et al (2005), whose emission is more suited to trace IRDCs. Figure 2 shows the channel maps of C$^{18}$O $J$=1-0 in step of 1 km s$^{-1}$. The channel maps are superimposed on the Spitzer-IRAC 8 $\mu$m emission. C$^{18}$O is detected in a velocity range between 53 and 65 km s$^{-1}$ and are well correlated with IRDC G34.43+0.24. Figure 3 displays the integrated intensity map of C$^{18}$O $J$=1-0 overlaid on the Spitzer-IRAC 8 $\mu$m emission. From the perspective of large scale, the C$^{18}$O $J$=1-0 emission shows a filamentary structure extended by $\sim$28$^{\prime}$ from north to south. Figure 4 presents the velocity-field (Moment 1) map of C$^{18}$O $J$=1-0 of IRDC G34.43+0.24 overlaid with its integrated intensity contours. From the velocity-field map, we can discern that the filament is in the velocity interval of 53–63 km s$^{-1}$, indicating that the filament is a single coherent object. To investigate spatial correlation between gas and dust of the filament, we made the $^{13}$CO $J$=1-0 and C$^{18}$O $J$=1-0 integrated intensity maps overlaid on the ATLASGAL 870 $\mu$m emission (Figure 5). The $^{13}$CO $J$=1-0 emission is more extended than that of the ATLASGAL 870 $\mu$m and C$^{18}$O $J$=1-0, whereas the spatial distribution of the C$^{18}$O $J$=1-0 emission is similar to that of the 870 $\mu$m dust emission.
In addition, the Spitzer-IRAC 8 $\mu$m emission can be divided into three regions, named as regions I, II, and III (see Fig. 3). We do not have C$^{18}$O data for the 6$^{\prime}$ north-south dark filament in region I. Region II contains IRDC G34.43+0.24, which coincides well with the C$^{18}$O $J$=1-0 emission ($\sim$12$^{\prime}$ length). In region III, a C$^{18}$O $J$=1-0 molecular clump has a dimension of about 16$^{\prime}$ and is correlated with the bright 8 $\mu$m emission and some small IRDCs. The molecular clump contains UC H [II]{} region G34.24+0.13 and H [II]{} complex G34.26+0.15. Moreover, the molecular clump also shows two elongated structures along its northwest. Liu et al. (2013) also identified these elongated structure in the Spitzer-IRAC \[4.5\]/\[3.6\] flux ratio map. As depicted by the long white dashed lines, we find that the two elongated structures are coincident with two small IRDCs in the northwest. To the southwestern edge of the molecular clump, the C$^{18}$O $J$=1-0 emission shows a half molecular shell with an opening toward the south. The half shell is associated well with the infrared bubble N61.
To estimate the mass of the coherent filament, we use the optical thin C$^{18}$O emission. Based on the mapped region of the filament in C$^{18}$O $J$=1-0, we only calculate the gas mass from region II to region III. Assuming the local thermodynamical equilibrium (LTE), the column density are determined by (Scoville et al. 1986) $$\mathit{N_{\rm C^{18}O}}=4.75\times10^{13}\frac{T_{\rm ex}+0.88}{exp(-5.27/T_{\rm ex})}\int T_{\rm mb}dv ~\rm cm^{-2},$$ where $dv$ is the velocity range in km s$^{-1}$, $T_{\rm mb}$ is the corrected main-beam temperature of C$^{18}$O $J$=1-0, and $T_{\rm ex}$ is the excitation temperature of molecular gas. Region II is a IRDC, we adopt an excitation temperature of 10 K in this region (Simon et al. 2006). For the region III containing H [II]{} regions, an excitation temperature of 20 K is adopted (Dirienzo et al. 2012). The C$^{18}$O abundance of $N(\rm H_{2})/N(C^{18}O)$ is about 7$\times10^{6}$ (Castets et al. 1982). The mean number density of $\rm H_{2}$ is estimated to be $$\mathit{n(\rm H_{2})}=8.1\times10^{-20}N(\rm H_{2})/r,$$ where $\rm r$ is the averaged radiuses of regions II and III in parsecs (pc). Their mass is given by $$\mathit{M_{\rm H_{2}}}=\frac{4}{3}\pi
r^{3}\mu_{g}m(\rm H_{2})n(\rm H_{2}),$$ where $\mu_{g}$=1.36 is the mean atomic weight of the gas, and $m(\rm
H_{2})$ is the mass of a hydrogen molecule. The obtained parameters of region II and region III are listed in Table 1. The total mass of gas is $\sim$4.8$\times10^{4}$$\rm M_{ \odot}$ from region II to region III.
Molecular Clump with H [II]{} Complex G34.26+0.15
-------------------------------------------------
The molecular clump in region III contains H [II]{} regions and infrared bubbles. We analyze the gas dynamics in the H [II]{} regions and bubbles using $^{13}$CO $J$=1-0. Figure 6 shows the channel maps of $^{13}$CO $J$=1-0 in the velocity range of 48–67 km s$^{-1}$ with interval 1 km s$^{-1}$. By comparison with the Spitzer-IRAC 8 $\mu$m emission, we identify two half shells of $^{13}$CO $J$=1-0 gas at 48–53 km s$^{-1}$ and 56–60 km s$^{-1}$. There are also two dense cores that can be identified by the channel maps. The peak positions of the identified cores are marked by the white crosses in Fig. 6. The velocity range of one core is from 52 km s$^{-1}$ to 56 km s$^{-1}$, the other is in the 53–66 km s$^{-1}$ interval. Adopting above four velocity ranges, we made the integrated $^{13}$CO $J$=1-0 maps overlaid on the Spitzer-IRAC 8 $\mu$m image, shown in Figures 7(a), 7(b), 7(c) and 7(d), respectively. In Fig. (7a), comparing the scales of infrared bubbles B1, B2, B3, and B4 with that of the half shell, B1 is consistent with the half shell of the molecular gas at 48–53 km s$^{-1}$. Because the $^{13}$CO $J$=1-0 component of the central velocity at 51 km s$^{-1}$ also show clearly a half shell (green contours) in Fig. 6, we adopt 51 km s$^{-1}$ as the velocity of B1. In Fig. 7(b), the northwest core is associated with H [II]{} complex G34.26+0.15. We did not find the associated objects with the southeast core, but find that B2 is located between two cores. Hence, B2 may be coincident with the molecular gas at 52–56 km s$^{-1}$. In Fig. 7(c), the ATLASGAL 870 $\mu$m emission of G34.26+0.15 shows a compact core, which is well coincident with the northwest core identified in Fig. 7(b). Previous centimeter observations showed that the complex G34.26+0.15 consists of two hypercompact H [II]{} regions called at A and B, a cometary ultracompact H [II]{} region named C, and an extended ringlike H [II]{} region called component D (Reid & Ho 1985). In Fig. 7(d), the $^{13}$CO $J$=1-0 half shell is associated well with N61. The molecular gas shows two integrated intensity gradients toward G34.172+0.175 and infrared bubble B3, as marked by the white arrows. Anderson et al. (2011) found that the RRL velocity of G34.172+0.175 is 57.3 $\pm$ 0.1 km s$^{-1}$, which is just located between 56 and 60 km s$^{-1}$. Hence, G34.172+0.175 is interacting with the half molecular shell. Figure 6 shows that N62 is consistent with the molecular gas of 56 to 60 km s$^{-1}$ in the morphology.
Figure 8 displays the spectra of $^{13}$CO $J$=1-0 and C$^{18}$O $J$=1-0 toward H [II]{} complex G34.26+0.15. The $^{13}$CO $J$=1-0 line shows asymmetric profile with double peaks, which may be caused by large optical depths (Allen et al. 2004). The C$^{18}$O $J$=1-0 line with a single emission peak can be considered as optically thin, hence it can be used to determine the systemic velocity. We measure a systemic velocity of $\sim$ 58.2 $\pm$ 0.1 km $\rm s^{-1}$ for G34.26+0.15, which is associated with the hydrogen RRL velocity (57.3 $\pm$ 0.1 km s$^{-1}$) of G34.172+0.175 (Anderson et al. 2011). The $^{13}$CO $J$=1-0 spectrum shows a blue-profile signature (Wu et al. 2007), and the blue peak of $^{13}$CO $J$=1-0 is stronger than its red peak with an absorption dip near the systemic velocity, which may be produced by infall motion or accretion. In order to identify which one creates the spectral signature, we plot the map grids of H [II]{} complex G34.26+0.15 using $^{13}$CO $J$=1-0, presented in Fig. 9. The map grids indeed exhibit the large-scaled blue asymmetric feature. Moreover, Liu et al. (2013) detected an infall motion with the HCN $J$=3-2, HCO$^{+}$ $J$=1-0, and CN $J$=2-1 lines for G34.26+0.15 from the JCMT and SMA observations. They obtained the mass infall rate of 1.2$\times$10$^{-2}$ M$_{\odot}$ yr$^{-1}$. Hence, the infall motion in G34.26+0.15 may be responsible for signature of the detected $^{13}$CO $J$=1-0 spectrum.
Infrared Bubbles N61 and N62
----------------------------
Infrared bubbles N61 and N62 lie on the borders of a filamentary molecular cloud in the line of sight. To clearly analyze the morphology of N61 and N62, we present the zoomed three color images from the Spitzer 3.6 $\mu$m, 8 $\mu$m, and 24 $\mu$m bands for the two bubbles, as shown in Fig. 10. The white contours correspond to the 21 cm continuum emission extracted from NVSS. In Fig. 10, both the left and right panels show the PDR visible in the 8 $\mu$m emission, which originates mainly in the PAHs. The PAHs emission of both N61 and N62 displays an almost ringlike shape with an opening towards the south. The openings of the two bubbles resemble a typical ‘champagne flow’ (Stahler & Palla 2005) with lower density material at the south. Both the 24 $\mu$m and 21 cm continuum emission are visible within the two bubbles. The 21 cm continuum emission traces the ionized gas from H [II]{} regions G34.172+0.175 and G34.325+0.211 (Anderson et al. 2011). Interestingly, the 24 $\mu$m and 21 cm continuum emission in N61 presents a central cavity. Deharveng et al. (2010) studied 102 bubbles. Ninety-eight percent of these bubbles exhibit 24 $\mu$m emission in their central regions. Only bubble N49 show a central hole in both the radio continuum emission and the 24 $\mu$m emission (Watson et al. 2008). They suggested that a central hole could be attributed to a stellar wind emitted by the central massive star. The CO molecular shell associated with N61 exhibits an integrated intensity gradient toward the filamentary molecular cloud. We suggest that H [II]{} region G34.172+0.175 has expanded into the molecular gas, and the produced shocks have collected the molecular gas into a molecular shell.
To estimate the age of the molecular shell in the filamentary molecular cloud, we calculate the dynamical age of H [II]{} region G34.172+0.175. Because H [II]{} region G34.325+0.211 that created N62 is close to the filamentary molecular cloud, we also computed the dynamical age of this H [II]{} region. Assuming an H [II]{} region expanding in a homogeneous medium, the dynamical age is estimated by (Dyson & Williams 1980) $$\mathit{t_{\rm H {\small II}}}=7.2\times10^{4}(\frac{R_{\rm H {\small II}}}{\rm pc})^{4/3}(\frac{Q_{\rm Ly}}{10^{49} \rm ph~s^{-1}})^{-1/4}(\frac{n_{\rm i}}{10^{3}\rm cm^{-3}})^{-1/2} \rm ~yr,$$ where $R_{\rm H {\small II}}$ is the radius of H [II]{} region, $n_{\rm i}$ is the initial number density of gas, and $Q_{\rm Ly}$ is the ionizing luminosity.
Assuming the radio continuum emission is optically thin, the ionizing luminosity $Q_{\rm Ly}$ is computed by Condon (1992) $$\mathit{Q_{\rm Ly}}=7.54\times10^{46}(\frac{\nu}{\rm GHz})^{0.1}(\frac{T_{e}}{\rm K})^{-0.45}(\frac{S_{\nu}}{\rm Jy})(\frac{D}{\rm kpc})^{2}\rm ~s^{-1},$$ Where $\nu$ is the frequency of the observation, $S_{\nu}$ is the observed specific flux density, and $D$ is the distance to the H [II]{} region. H [II]{} regions G34.172+0.175 and G34.325+0.211 have the flux density of 1.7$\pm$0.7 Jy and (3.1$\pm$0.4)$\times$10$^{-1}$ Jy at 9 GHz (Anderson et al. 2011), respectively. We adopt an effective electron temperature of 10$^{4}$ K, and a distance of 3.7 kpc (see Sect.4.1). Finally, we obtain $Q_{\rm Ly}$$\backsimeq$(3.5$\pm$1.5)$\times10^{46}$ ph s$^{-1}$ and (6.3$\pm$0.8)$\times10^{45}$ ph s$^{-1}$ for G34.172+0.175 and G34.325+0.211. Inoue et al. (2001) suggested that only half of Lyman continuum photons from the central source in a Galactic H [II]{} region ionizes neutral hydrogen, remainder being absorbed by dust grains within the ionized region. Using Smith et al. (2002), we determined the spectral type of the ionizing star of G34.172+0.175 is between B0.5V and B1V, while between B1V and B1.5V for G34.325+0.211.
G34.172+0.175 and G34.325+0.211 are located on the borders of a filamentary cloud. We assume that the ionized stars of the two H [II]{} regions may form in a filamentary IRDC. Simon et al. (2006) has mapped 379 IRDCs in the $^{13}$CO $J$=1-0 molecular line with the Boston University-Five College Radio Observatory Galactic Ring Survey. They obtained a volume-averaged H$_{2}$ densities of $\sim$2$\times$10$^{3}$ cm$^{-3}$ for all the IRDCs. We take the volume-averaged H$_{2}$ as the initial number density of the gas. Adopting the radius of $\sim$3.2$\pm$0.5 pc and $\sim$1.5$\pm$0.1 pc for G34.172+0.175 and G34.325+0.211 obtained from Anderson et al. (2011), we derived that the dynamical ages of H [II]{} regions G34.172+0.175 and G34.254+0.14 are (9.9$\pm$0.2)$\times10^{5}$ yr and 5.6$\pm$0.1)$\times10^{5}$ yr, respectively. Because H [II]{} regions spend a significant amount of time in the ultra-compact and compact phases, our obtained ages for the H [II]{} regions may be lower limits.
In H [II]{} region G34.172+0.175, there is a cavity of ionized gas created by the stellar wind from its central massive stars. Using the stellar type of the central massive star, we can estimate the timescale of the ionized cavity. Applying the equation (McCray et al. 1983) $$\mathit{R_{W}}=4.3\times10^{-10}(L_{W}/n_{0})^{1/5}t^{3/5}_{W},$$ where $n_{0}~[\rm cm^{-3}]$ is the gas medium density, $L_{W}~[\rm ergs/s]$ is the wind luminosity, and $R_{W}~[\rm pc]$ is the cavity radius. The wind luminosity can be expressed as (Castor et al. 1975) $$\mathit{L_{W}}=3.16\times10^{-35} \dot{M}_{W}v_{W}^{2},$$ where $\dot{M}_{W}$ and $v_{W}$ are the mass-loss rate and velocity of the stellar wind, respectively. The spectral type of the central star is B1V$\sim$B0.5V for G34.172+0.175. Adopting a mass-loss rate and velocity of the stellar wind of B0.5V$\sim$B1V star summarized by Chen et al. (2013), we obtain the wind luminosity ($L_{W}$) of 0.9$\sim$3.3$\times$10$^{33}$ erg/s. We measured that the cavity radius is about 1.2 pc (1.1$^{\prime}$ at 3.7 kpc) from Fig. 10 (left panel). Using the medium density of $\sim$2$\times$10$^{3}$ cm$^{-3}$, we infer that the timescale of the ionized cavity is 4.1$\sim$6.3$\times$10$^{5}$ yr, which is much less than the main-sequence (MS) lifetime of such stars (1.3$\sim$1.6$\times$10$^{7}$ yr for B1V$\sim$B0.5V stars, Chen et al. 2013). Comparing the timescale of the ionized cavity with that of the infrared bubble N61 (9.9$\pm$0.2$\times10^{5}$ yr), we suggest that the ionized cavity has been blown via the energetic stellar wind after H [II]{} region G34.172+0.175 begin to expand in a filamentary molecular cloud. Similarly, after some time an ionized cavity will emerge in infrared bubble N62.
Distribution of Young Stellar Objects
-------------------------------------
A population of low-mass stars with an age of about a few Myr were detected in IRDC G34.43+0.24 (Shepherd et al. 2007). In this section, we examine the young stellar object (YSO) populations by GLIMPSE I catalog toward IRDC G34.43+0.24. From the catalog, we selected 11872 near-infrared sources with the 3.6, 4.5, 5.8, and 8.0 $\mu$m, within a circle of 23$^{\prime}$ in radius centered on R.A.=18$^{\rm h}53^{\rm m}15.054^{\rm s}$ (J2000), Dec=$+01^{\circ}19'19.81^{\prime\prime}$ (J2000). The size of this region completely covers the extension of IRDC G34.43+0.24 and its south region. Figure 11 shows the $[5.8]-[8.0]$ versus $[3.6]-[4.5]$ color-color (CC) diagram. The regions in the figure indicate the stellar evolutionary stages based on the criteria of Allen et al. (2004), Paron et al (2009), and Petriella et al (2010). YSOs are generally classified according to their evolutionary stage. These near-infrared sources are classified into three regions: Class I sources are protostars with circumstellar envelopes, Class II sources are disk-dominated objects, and other sources. Class I sources occur in a period on the order of $\sim$$10^{5}$ yr, while the age of Class II sources is $\sim$$10^{6}$ yr (André & Montmerle 1994). Using this criteria, we find 256 Class I sources and 419 Class II sources. Here Class I and Class II sources are chosen to be YSOs.
Figure 12 (left panel) shows the spatial distribution of both Class I and Class II sources. From Fig. 12 (left panel), we note that Class I sources (red dots) are asymmetrically distributed across the whole selected region, and are mostly concentrated in the large-scale filamentary molecular cloud, while Class II sources (yellow dots) are dispersively distribution. Foster et al. (2014) only detected a population low-mass protostars in IRDC G34.43+0.24, and find that the population appears to be distributed along IRDC G34.43+0.24 rather than exclusively associated with the dense clumps. We found that some clustered Class I sources not only coincide with the dense clumps of IRDC G34.43+0.24, but also with its southern molecular cloud. Regarding the geometric distribution of the Class I and Class II sources, we can plot the map of star surface density. Because Class II sources are uniform distribution, we only plot the map for Class I sources. This map was obtained by counting all Class I sources with a detection in the 3.6 $\mu$m, 4.5 $\mu$m, 5.8 $\mu$m, and 8.0 $\mu$m bands in squares of $4'\times 4'$, as shown in Fig. 12 (right panel). From Fig. 12 (right panel), we can see that there are clear signs of clustering along the filamentary IRDCs, where IRDC G34.43+0.24 MM1-MM9 (Rathborne et al. 2006), as well as H [II]{} regions G34.26+0.15 and G34.24+0.13 (Hunter et al. 1998) are located on the peak position of each clustering stars.
DISCUSSIONS
===========
Dynamic Structure of IRDC G34.43+0.24
-------------------------------------
From the Spitzer-IRAC 8 $\mu$m emission, IRDC G34.43+0.24 shows a filamentary dark extinction feature. Based on the association with the C$^{18}$O $J$=1-0 emission, the Spitzer-IRAC 8 $\mu$m emission could be divided into regions I, II, and III. Previous ammonia observation indicates that IRDC G34.43+0.24 extends by 9$^{\prime}$ from north to south (Miralles et al. 1994). We found that IRDC G34.43+0.24 should extend by 18$^{\prime}$ from region I to region II. In region III, we find not only some bright PAHs emission, but also some small IRDCs. The bright PAHs emission is coincident with infrared bubbles B1-B6 (Simpson et al. 2012), and N61 and N62 (Churchwell et al. 2006 & Deharveng et al. 2010). From the perspective of large scale, the C$^{18}$O $J$=1-0 emission of 54 to 65 km s$^{-1}$ shows a filamentary structure extended by $\sim$28$^{\prime}$ from region II to region III. The cold dust emission traced by the ATLASGAL 870 $\mu$m further confirms the existence of the filamentary molecular cloud. We suggest that the previous identified IRDC G34.43+0.24 (9$^{\prime}$) is only part of the larger filamentary cloud. From the velocity-field map of C$^{18}$O $J$=1-0 for IRDC G34.43+0.24, we conclude that the filament is a single coherent object, which should extend by 34$^{\prime}$ from region I to region III.
Moreover, the molecular clump in region III displays two elongated structures in the $^{13}$CO line along its northwest. Based on the Spitzer-IRAC \[4.5\]/\[3.6\] flux ratio, Liu et al. (2013) identified several elongated structures in the same position. They suggested that if the \[4.5\]/\[3.6\] ratio traces the distribution of shocked gas, these elongated structures may indicate the multiple jets generated from the G34.26+0.15 complex. Since they also identified the broad line wings in the HCN $J$=3-2 line toward the complex, indicating that there seem to exist the energetic outflow motions. We found that the two elongated structures are well correlated with the 8 $\mu$m dark extinction emission and the 870 $\mu$m cold dust emission (see Fig. 3 and Fig. 1 (c)). Figure 13 shows the position-velocity (PV) diagrams constructed from the $^{13}$CO $J$=1-0 and C$^{18}$O $J$=1-0 emission along the long filamentary IRDC G34.43+0.24. The elongated structures have obvious velocity gradients with respect to the systemic velocity. The direction of velocity gradient is shown by the red dashed lines. If the elongated structures with the velocity gradient indicate the outflow gas, which is heated, then we cannot detect the dark gas and cold dust emission. Hence, we conclude that the two elongated structures may be two small IRDCs with the high velocity. The observations of more molecular tracer with higher spatial resolutions are needed. For the filamentary IRDC G34.43+0.24, we did not detect the 8 $\mu$m dark extinction emission that connects with its southern part, but identify some small IRDCs and infrared bubbles. If the infrared bubbles emerge, it means that the adjacent IRDC would be segmented and disrupted, and then evolved to the terminal stage (Jackson et al. 2010). From the positions of the small IRDCs, we suggest that the south part of IRDC G34.43+0.24 in region III may be disrupted into some small IRDCs by adjacent star formation activity, such as the outflow motion.
Table 2 lists the name (1) of the infrared bubbles and H [II]{} regions in the region III of IRDC G34.43+0.24, (2)-(3) equatorial coordinates, (4) velocity ($V_{\rm LSR}$), and (5) the associated velocity ranges. Because N61 and N62 are associated with H [II]{} regions 34.325+0.211 and 34.172+0.175, we assume that the bubble and H [II]{} region are consistent with the same $^{13}$CO emission, respectively. The $^{13}$CO $J$=1-0 emission in velocity interval 56 to 60 km s$^{-1}$ is associated with H [II]{} region G34.172+0.175. G34.172+0.175 has the RRL velocity of 62.9 $\pm$ 0.1 km s$^{-1}$ (Anderson et al. 2011), which is not located between this velocity range. Hence, we did not detect the $^{13}$CO $J$=1-0 emission associated with G34.172+0.175 and N62. Hunter et al. (1998) detected a cool dust core associated with H [II]{} region G34.24+0.13, but we did not find the associated $^{13}$CO $J$=1-0 emission. Because the sizes of B4-B6 are smaller, we did not identify the velocity component associated with the $^{13}$CO $J$=1-0 emission. According to the Galactic rotation model of Fich et al. (1989) together with $R_{\odot}$ = 8.5 kpc and $V_{\odot}$ = 220 km s$^{-1}$, where $V_{\odot}$ is the circular rotation speed of the Galaxy, we derive a kinematic distance of 3.7 kpc based on the average for the whole cloud. Here we will adopt the near kinematic distance. At the distance of 3.7 kpc, the filamentary IRDC G34.43+0.24 has a length of $\sim$37 pc. Nessie IRDC is rare object so far. Jackson et al. (2010) obtained that the length of Nessie IRDC is 80 pc, but Goodman et al. (2014) suggested that Nessie is probably longer than 80 pc. A similar filament of more than 50 pc length has been presented by Kainulainen et al. (2011). We obtained the total mass of the filamentary IRDC G34.43+0.24 are $\sim$4.8$\times10^{4}$$\rm M_{ \odot}$ from region II to region III. The stability of the filament can be estimated by the virial parameter $\alpha$=$M_{\rm vir}/M$=$2\sigma_{v}^{2}l/(GM)$ (Fiege & Pudritz 2000), where $\sigma_{v}$=$\triangle V_{\rm FWHM}/(2\sqrt{2\rm ln2})$ is the average velocity dispersion of C$^{18}$O $J$=1-0, $l$ is the length of the filament, and G is the gravitational constant. $\triangle V_{\rm FWHM}$ is the mean full width at half-maximum (FWHM) of C$^{18}$O $J$=1-0 emission. In the filament, we found the mean FWHM to be 4.9 km $\rm s^{-1}$, while its length is 30 pc from region II to region III. The mean thermal broadening can be given by $V_{\rm therm}$ =$\sqrt{k T_{\rm ex}/(\mu_{g}m(\rm H_{2})) }$. Region II has an excitation temperature of 10 K, while an excitation temperature of 20 K is adopted for region III. Adopting $T_{\rm ex}$=20 K, we obtain the maximum $V_{\rm therm}$ $\approx$ 0.3 km s$\rm ^{-1}$, which is less than the mean FWHM of 4.9 km $\rm s^{-1}$. This confirms the supersonic nature of this filament. The virial parameter $\alpha$ is thus estimated to be 1.5, indicating that the filament is stable as a whole, despite infall motion found in H [II]{} complex G34.26+0.15.
If the turbulence is dominant, a critical mass to length ratio can be estimated as ($M/l$)$_{\rm crit}$=84$(\Delta V)^{2}M_{\odot}$ $ \rm pc^{-1}$ (Jackson et al. 2010). Then, we obtain ($M/l$) $_{\rm crit}$ $\approx$ 2.0$\times$10$^{3}$ $M_{\odot}$ $ \rm pc^{-1}$. Besides, using the total mass ($\sim$4.8$\times10^{4}$$\rm M_{ \odot}$) and length ($\sim$37 pc) of the filament from region II to region III, we derived a linear mass density $M/l$ of $\sim$ 1.6$\times$10$^{3}$ $M_{\odot}$ $ \rm pc^{-1}$, which is roughly consistent with ($M/l$) $_{\rm crit}$. Considering the uncertainties of the FWHM, the turbulent motion may be helping stabilize the filament against the radial collapse.
Star Formation Scenario
-----------------------
IRDCs have been proposed to be the birthplace of massive stars and their host clusters (Carey et al. 2000; Egan et al. 1998; Rathborne et al. 2006). Rathborne et al. (2006) identified nine clumps in IRDC G34.43+0.24. Three of these cores exist the massive star formation activity, the remaining cores are the starless cores. In region III, UC H II region G34.24+0.13 and H II complex G34.26+0.15 are embedded in a filamentary molecular clump. H II complex 34.26+0.15 contains four small H [II]{} region (Reid & Ho 1985). The cometary G34.26+0.15C is not only associated with a hot molecular core (Hunter et al. 1998; Campbell et al. 2004; Mookerjea et al. 2007), but also has an infall motion (Liu et al. 2013), indicating that UC H [II]{} complex G34.26+0.15 is an ongoing massive star formation region. Additionally, several infrared bubbles are located in region III. The bubbles can be produced by stellar wind and overpressure by ionization and heating by stellar UV radiation from O and early-B stars. The infrared bubbles emerge, suggesting that several massive stars have formed in this region. Especially, N61 has a double-shell structure, the outer traced by 8 $\mu$m and $^{13}$CO $J$=1-0 emission and the inner traced by 24 $\mu$m and 21 cm emission. The outer shell (9.9$\times10^{5}$ yr) is created by the expansion of H II region G34.172+0.175 in the filamentary molecular clump, while the inner shell (4.1$\sim$6.3$\times$10$^{5}$ yr) may be produced by the energetic stellar wind from its central massive star. We determined that the spectral type of the ionizing star of G34.172+0.175 is between B0.5V and B1V. Since the timescales of the double shells are much less than the main-sequence lifetime of such stars (1.3$\sim$1.6$\times$10$^{7}$ yr for B1V$\sim$B0.5V stars, Chen et al. 2013), we infer that the central massive star of N61 is still in the main-sequence stage. To summarize, region I shows a filamentary dark extinction, but no star formation activity are detected; Region II with IRDC G34.43+00.24 presents a filamentary dark extinction associated with C$^{18}$O emission. A UC H II region and three massive protostars are identified in this region; Region III shows a filamentary C$^{18}$O molecular cloud, but no filamentary dark extinction is detected except for some small IRDCs. Moreover, some massive protostars, UC H II regions, and even infrared bubbles are detected in region III. We suggest that IRDC G34.43+0.24 is in different evolutionary stages from region I to region III. The star-forming filamentary clouds represent a later evolutionary stage in the life of an IRDC (Jackson et al. 2010). IRDC G34.43+0.24 in region III appear in a later evolutionary stage such as Orion and NGC 6334 (Kraemer et al. 1999).
Furthermore, Shepherd et al. (2007) identified some massive protostars with an age of about 10$^{5}$ yr after they detected a low-mass population of stars ($\sim$ 10$^{6}$ yr) around IRDC G34.43+0.24 MM2 (Shepherd et al. 2004). They suggested that the stars in this region may have formed in two stages: first lower mass stars formed and then more massive stars began to form. Because Foster et al. (2014) detected a population low-mass protostars that are distributed along filamentary IRDC G34.43+0.24 rather than exclusively associated with the dense clumps, they concluded that massive stars predominantly from in the most bound parts of the filaments, while the low-mass protostars form in less bound portions of the filament. From the GLIMPSE I catalog, we selected some Class I sources with an age of $\sim$$10^{5}$ yr on the basis of infrared color indices. It is interesting note that these Class I sources are clustered distribution along the filamentary molecular cloud. The north star cluster is associated with IRDC G34.43+0.24, while the south star cluster is well correlated with the filamentary molecular clump in region III. In addition, IRDC G34.43+0.24 MM1-MM9 (Rathborne et al. 2006), as well as H [II]{} regions G34.26+0.15 and G34.24+0.13 (Hunter et al. 1998) are located on the peak position of each clustering stars. These results are consistent with previous suggestions that IRDCs are the precursors of massive stars and star clusters. If these selected Class I sources are most low-mass protostars as the observations of Foster et al. (2014), the low-mass protostars may form contemporaneously with high-mass protostars in such a filament. Based on the presence of distributed low-mass star formation, Foster et al. (2014) suggest that the low-mass and massive protostars certainly may form by the “sausage” instability in IRDC G34.43+0.24. The “sausage” instability model in IRDCs is proposed by Jackson et al. (2010). Comparing the age of H [II]{} region G34.172+0.175 (9.9$\times10^{5}$ yr) with that of Class I sources ($\sim$10$^{5}$ yr), we infer that the ionizing stars of G34.172+0.175 may be the first generation of massive stars that formed in such a filament.
CONCLUSIONS
===========
We present the molecular $^{13}$CO $J$=1-0 and C$^{18}$O $J$=1-0, infrared, and radio continuum observations toward IRDC G34.43+0.24 ob a large scale. The main results are summarized as follows:
1\. At 8 $\mu$m (Spitzer - IRAC), IRDC G34.43+0.24 appears to be a dark filament along the north-south direction. Both the ATLASGAL 870 $\mu$m and C$^{18}$O $J$=1-0 images exhibit bright filamentary structures containing massive protostars, UC H II regions, and infrared bubbles. The spatial extend of IRDC G34.43+0.24 is about 37 pc (34$^{\prime}$) at a distance of 3.7 kpc. IRDC G34.43+0.24 has a linear mass density of $\sim$ 1.6$\times$10$^{3}$ $M_{\odot}$ $ \rm pc^{-1}$, which is roughly consistent with its critical mass to length ratio, suggesting that the turbulent motion may be helping stabilize the filament against the radial collapse.
2\. IRDC G34.43+0.24 could be divided into three portions. Each portion may be in different evolutionary stage. The southern portion of IRDC G34.43+0.24 containing the strong star-forming activity and several smaller IRDCs, may evolute a later stage in the life of an IRDC.
3\. Infrared bubble N61 has expanded into IRDC G34.43+0.24, which has a double-shell structure, the outer traced by 8 $\mu$m and $^{13}$CO $J$=1-0 emission and the inner traced by 24 $\mu$m and 20 cm emission. N61 is associated with H [II]{} region G34.172+0.175. We conclude that the outer shell with a timescale of 9.9$\times10^{5}$ yr is created by the expansion of H [II]{} region G34.172+0.175, while the inner shell with an age of 4.1$\sim$6.3$\times$10$^{5}$ yr may be produced by the energetic stellar wind from its central massive star. N62 shows a ringlike structure at 8 $\mu$m, which is associated with H [II]{} region G34.325+0.211 with an age of (5.6$\pm$0.1)$\times10^{5}$ yr.
4\. The selected Class I sources with an age of $\sim$$10^{5}$ yr are clustered along the filamentary molecular cloud. The north star cluster is associated with IRDC G34.43+00.24, while the south star cluster is well correlated with the filamentary molecular clump. IRDC G34.43+0.24 MM1-MM9, as well as H [II]{} regions G34.26+0.15 and G34.24+0.13 are located on the peak positions of each clustering stars. Comparing the age of H [II]{} region G34.172+0.175 (9.9$\times10^{5}$ yr) with that of Class I sources ($\sim$10$^{5}$ yr), we infer that the ionizing stars of G34.172+0.175 may be the first generation of massive stars that formed in IRDC G34.43+0.24.
We are very grateful to the anonymous referee for his/her helpful comments and suggestions. This work is made use of data from the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. The ATLASGAL project is a collaboration between the Max-Planck-Gesellschaft, the European Southern Observatory (ESO) and the Universidad de Chile. It includes projects E-181.C-0885, E-078.F-9040(A), M-079.C-9501(A), M-081.C-9501(A) plus Chilean data. We are also grateful to the staff at the Qinghai Station of PMO for their assistance during the observations. Thanks for the Key Laboratory for Radio Astronomy, CAS to partly support the telescope operating. This work was supported by the National Natural Science Foundation of China (Grant No. 11363004 and 11403042). This paper is also partly supported by National Key Basic Research Program of China(973 Program) 2015CB857100. DL acknowledges the support from the Guizhou Scientific Collaboration Program (20130421).
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![(a) the Spitzer-IRAC 8 $\mu$m emission. (b) the Spitzer-MIPSGAL 24 $\mu$m emission. (c) the ATLASGAL 870 $\mu$m emission. (d) the NVSS 21 cm emission. The unit of color bar is MJy sr$^{-1}$ for 8 and 24$\mu$m, while Jy beam$^{-1}$ for 870 $\mu$m and 21 cm. In each panel, the white dashed lines mark the IRDCs. The nine green pluses show the positions of IRDC G34.43+0.24 MM1-MM9 (Rathborne et al. 2006). The two blue pluses present the positions of H [II]{} regions G34.26+0.15 and G34.24+0.13 (Hunter et al. 1998). The six green circles represent the bubbles identified by Simpson et al. (2012). ](f01.pdf)
![Contours of C$^{18}$O $J$=1-0 emission superimposed on the 8 $\mu$m emission map (grey). The integrated velocity is from 53 to 65 km s$^{-1}$. The contour levels are from 2.8 to 23.1 by a step of 1.7 K km s$^{-1}$ . The six [**green**]{} circles show the bubbles identified by Simpson et al. (2012). I, II, and III may represent the evolutive stage of IRDC. The [**red** ]{}dot symbols mark the mapping points. The right color bar is the unit in MJy sr$^{-1}$. ](f03.pdf)
5.8mm
[ccccccccc]{} Name & Trace& Area &$N_{\rm H_{2}}$ & $n(\rm H_{2})$& $M$\
& & (arcmin$^{2}$) &(cm$^{-2}$) &(cm$^{-3}$) & ($\rm M_{ \odot}$)\
Region II &C$^{18}$O& 33.6 &2.4$\times10^{22}$ & 5.5$\times10^{2}$ & 6.0$\times10^{3}$\
Region III &C$^{18}$O& 109.3 &4.5$\times10^{22}$ & 5.7$\times10^{2}$ & 4.2$\times10^{4}$\
[rcccccccccr]{} Source name & R.A. & Decl. & $V_{\rm LSR}$ & Velocity range\
& (J2000.) & (J2000.) & km s$^{-1}$ & km s$^{-1}$\
\
IRDC G34.43+0.24 &18 53 18.9 & 01 26 38.6 &57.6$\pm$0.1$^{a}$ &(53.0–65.0)\
H[II]{} G34.325+0.211 &18 53 13.8 & 01 20 08.0 &62.9$\pm$0.1$^{b}$ &(56.0–60.0)\
H[II]{} G34.26+0.15 & 18 53 20.1 & 01 14 37.1 &54.6$^{c}$ &(53.0–66.0)\
H[II]{} G34.24+0.13 & 18 53 21.5 & 01 13 45.3 &57.0$^{d}$ &–\
H[II]{} G34.172+0.175 & 18 53 04.7 & 01 11 02.0 &57.3$\pm$0.1$^{b}$ &– &\
\
N61 & 18 53 10.5 & 01 09 14.9 & – & (56.0–60.0)\
N62 & 18 53 13.6 & 01 20 46.7 & – & –\
[MWP G034246+001023(B1)]{} & 18 53 28.3 & 01 12 58 & 51.0$\pm$0.5 & (48.0–53.0)\
[MWP G034261+001357(B2)]{} & 18 53 22.7 & 01 14 42 &– & (52.0–56.0)\
[MWP G034262+001267(B3)]{} & 18 53 24.7 & 01 14 30 &– & (56.0–60.0)\
[MWP G034240+001200s(B4)]{} & 18 53 23.8 & 01 13 08 &– & –\
[MWP G034210+001200s(B5)]{} & 18 53 20.5 & 01 11 32 &– &–\
[MWP G034200+001000s(B6)]{} & 18 53 23.7 & 01 10 27 &– & –\
Notes.(a) Miralles et al. 1994; The NH$_{3}$ line. (b) Anderson et al. 2011; The radio recombination line. (c) Kolpak et al. 2003; The radio recombination line. (d) Hunter et al. 1998; The H$_{2}$CO line.
![1.4 GHz radio continuum emission contours (white) overlayed on the three color image of the bubble N61 and N62 composed from the Spitzer 3.6 $\mu$m, 8 $\mu$m, and 24 $\mu$m bands in blue, green, and red, respectively. The blue circle indicates that the ionized gas of H [II]{} G34.325+0.211 has a central hole.](f010.pdf)
![GLIMPSE color–color diagram \[5.8\]–\[8.0\] versus \[3.6\]–\[4.5\] for sources. The regions indicate the stellar evolutionary stage as defined by Allen et al. (2004). Class I sources are protostars with circumstellar envelopes and Class II are disk-dominated objects.](f011.pdf)
![Left panel: Positions of Class I and II sources relative to the $^{13}$CO $J=1-0$ emission (blue contours) overlaid on the 8 $\mu$m emission. The Class I sources are labeled as the red dots, and the Class II sources as the yellow dots. Two dashed ellipses mark the YSOs associated with IRDC G34.43+0.24. Right panel: Stellar-surface density map (red contours) of Class I candidates are superimposed on the 8 $\mu$m emission. Contours range from 4 to 15 stars (4arcmin)$^{-2}$ in steps of 2 stars (4arcmin$)^{-2}$. 1$\sigma$ is 1.3 (4arcmin)$^{-2}$ (background stars). The nine green pluses show the positions of IRDC G34.43+0.24 MM1-MM9 (Rathborne et al. 2006), while the two blue pluses present the positions of H [II]{} regions G34.26+0.15 and G34.24+0.13 (Hunter et al. 1998), also shown in Fig. 1. ](f012.pdf)
[^1]: http://www.iram.fr/IRAMFR/GILDAS/
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the 2-d setting, given an $H^1$ solution $v(t)$ to the linear Schrödinger equation $i\partial_t v +\Delta v =0$, we prove the existence (but not uniqueness) of an $H^1$ solution $u(t)$ to the defocusing nonlinear Schrödinger (NLS) equation $i\partial_t u + \Delta u -|u|^{p-1}u=0$ for nonlinear powers $2<p<3$ and the existence of an $H^1$ solution $u(t)$ to the defocusing Hartree equation $i\partial_t u + \Delta u -(|x|^{-\gamma}\star|u|^{2})u=0$ for interaction powers $1<\gamma<2$, such that $\|u(t)-v(t)\|_{H^1} \to 0$ as $t\to +\infty$. This is a partial result toward the existence of well-defined continuous wave operators $H^1 \to H^1$ for these equations. For NLS in 2-d, such wave operators are known to exist for $p\geq 3$, while for $p\leq 2$ it is known that they cannot exist. The Hartree equation in 2-d only makes sense for $0<\gamma<2$, and it was previously known that wave operators cannot exist for $0<\gamma\leq 1$, while no result was previously known in the range $1<\gamma<2$. Our proof in the case of NLS applies a new estimate of Colliander-Grillakis-Tzirakis [@CGT] to a strategy devised by Nakanishi [@N]. For the Hartree equation, we prove a new correlation estimate following the method of [@CGT].'
address:
- 'University of California, Berkeley'
- 'University of Illinois, Urbana-Champaign'
author:
- Justin Holmer
- Nikolaos Tzirakis
title: 'Asymptotically linear solutions in $H^1$ of the 2-d defocusing nonlinear Schrödinger and Hartree equations'
---
Introduction
============
In this paper we consider the asymptotic behavior of $H^1$ solutions on $\mathbb{R}^n$ to the following two equations: the defocusing nonlinear Schrödinger equation (NLS) $$\label{nls}
i\partial_tu + \Delta u -|u|^{p-1}u=0,$$ with nonlinear exponents $1<p<(n+2)/(n-2)$, and the defocusing Hartree equation $$\label{hnls}
i\partial_t u+ \Delta u -(|x|^{-\gamma} \star |u|^{2})u=0,$$ with interaction exponents $0<\gamma <\min(n,4)$. For both equations, there is a local well-posedness theory (see [@tc][@tt] for exposition) for the initial-value problem yielding solutions $u\in C(t\in(-T,T); H_x^1(\mathbb{R}^n))$, $T=T(\|u_0\|_{H^1})$ that satisfy mass conservation: $$\|u(t)\|_{L^{2}}=\|u_{0}\|_{L^{2}}$$ and energy conservation: $$E(u)(t)=\frac{1}{2}\int |\nabla u(t)|^{2}dx+\frac{1}{p+1}\int |u(t)|^{p+1}dx=E(u_{0})$$ for NLS (1.1) and $$E(u)(t)=\frac{1}{2}\int |\nabla u(t)|^{2}dx+\frac{1}{4}\int (|x|^{-\gamma} \star |u(t)|^{2})|u(t)|^2dx=E(u_{0}).$$ for Hartree .
The conservation laws provide an $H^1$ *a priori* bound on the solutions guaranteeing that the local-in-time solutions are in fact global in $H^1$. In this situation, we seek to describe the asymptotic behavior of solutions in time. The comparison between the dynamics of the nonlinear equations and their linear counterparts gives rise to the following two questions. Fix a data class $X$ (defined by some spatial norm). We note that since $u(x,t)$ solves NLS/Hartree $\implies$ $\bar u(x,-t)$ solves NLS/Hartree, studying asymptotics at $+\infty$ is equivalent to studying them at $-\infty$. (But of course, a given nonlinear solution $u(t)$ will in general have a different limiting behavior at $+\infty$ than at $-\infty$.) Let $U(t)=e^{it\Delta}$ denote the linear Schrödinger evolution group.
*Existence of wave operators*. Given a linear solution $v_+(t)=U(t)u_+$ with $u_+\in X$, does there exists a global nonlinear solution $u(t)$ to NLS or Hartree such that $U(-t)u(t)\in X$ for all $t$ and $$\label{E:convergence}
\|U(-t)u(t)-u_+ \|_X \to 0$$ as $t\to +\infty$?[^1] Is the solution $u(t)$ the unique solution satisfying in a certain space-time function class $Y$? If so, then we can define an operator $\Omega_+:X\to X$ sending $u_+\mapsto u(0)$, called the *wave operator*. Is $\Omega_+$ continuous?
*Asymptotic completeness*. Given a global solution $u(t)$ to NLS or Hartree such that $U(-t)u(t)\in X$ for all $t\in \mathbb{R}$, does there exist $u_+\in X$ such that holds? Such a $u_+$ is necessarily unique.
In this paper, we consider the above problems for $X=H^{1}$. There is a large amount of literature dealing with the weighted space $\Sigma$, where $\|\phi\|_\Sigma = \|xu\|_{L^2}+ \|u\|_{H^1}$. However, to remain focused in our discussion, we will not mention these results here. Also, some results are available for the focusing analogues ($+$ sign in front of the nonlinearities), but we will also not discuss this case either.
The analysis of these problems breaks into different ranges of $p$ and $\gamma$. The NLS equation has scaling $u(x,t) \mapsto \lambda^\frac{2}{p-1}u(\lambda x,\lambda^2 t)$ and the Hartree equation has scaling $u(x,t) \mapsto \lambda^\frac{n-\gamma+2}{2}u(\lambda x,\lambda^2t)$. The $H^1$ subcritical restrictions are thus $1<p<(n+2)/(n-2)$ and $0<\gamma<\min(n,4)$. The $L^2$ critical exponent for NLS is $p=1+\frac4n$ and for Hartree is $\gamma=2$. The basic heuristic is that for $L^2$ subcritical exponents one has good short-time control, for $L^2$ supercritical exponents one has good long-time control, and in the $L^2$ critical case the scaling shows that short times and long times are equivalent. The expectation therefore is that the questions of asymptotic completeness and existence of wave operators should be easier in the $L^2$ supercritical case (although not necessarily false in the $L^2$ subcritical case). It turns out that an important boundary is $p_0=1+\frac2n$ for NLS and $\gamma_0=1$ for Hartree. For nonzero solutions $u(t)$ to NLS with $p\leq p_0$ or Hartree with $\gamma\leq \gamma_0$, $U(-t)u(t)$ does not converge in $L^2$ (proved by Strauss [@ws], Barab [@Barab], and Tsutsumi [@Tsut85] for NLS and by Hayashi-Tsutsumi [@HT] and Glassey [@G77] for Hartree). Thus wave operators cannot exist in any reasonable sense, and in this regime the positive results involve the construction of *modified* wave operators, a topic we do not discuss here. The region $p<p_0$ and $\gamma<\gamma_0$ is called long-range and the region $p>p_0$ and $\gamma>\gamma_0$ is called short-range.
In the $L^2$ supercritical case $p>1+\frac4{n}$, $\gamma>2$, a fixed point argument using the Strichartz estimates (similar to the one used to solve the local Cauchy problem) establishes both the existence of continuous wave operators and asymptotic completeness for small initial data. To extend the asymptotic completeness results to large initial data, one usually proceeds through the derivation of *a priori* estimates for general solutions to furnish the needed decay estimates as $t\to \infty$. These estimates take advantage of the momentum conservation law $$\vec{p}(t)={\operatorname{Im}}\int_{\Bbb R^n}\bar{u}\nabla u dx.$$ We can establish, for example, the generalized virial identity (Lin-Strauss [@ls]), valid for $n \geq 3$, $$\label{linstr}
\int_{0}^{T}\int_{\Bbb R^n}(-\Delta \Delta |x|)|u(x,t)|^2dxdt+\frac{2(p-1)}{p+1}\int_{0}^{T}\int_{\Bbb R^n}\frac{|u(x,t)|^{p+1}}{|x|}dxdt \lesssim
\sup_{[0,T]}|M_{a}(t)|$$ where $u$ is a solution to (1.1) and $M_{a}(t)$ is the Morawetz action defined by $$\label{mora}
M_{a}(t)=2\int_{\Bbb R^n}\nabla a(x) \cdot {\operatorname{Im}}(\bar{u}(x)\nabla u(x))dx.$$ Using this point of view, one can establish *a priori* estimates that completely answer the questions above in the $L^2$ supercritical case $p>1+\frac{4}{n}$. In the case of NLS, Ginibre-Velo [@gv1] proved asymptotic completeness for $n \geq 3$ using and subsequently, Nakanishi [@kn] settled the case of low dimensions $n=1,2$ using special modifications of . These results are very technical but nowadays we can simplify the proofs using correlation estimates – see Colliander-Grillakis-Tzirakis [@CGT] and the references therein. Corresponding results for Hartree were obtained in Ginibre-Velo [@GV] and Nakanishi [@Nak].
As we have mentioned, in the $L^2$ supercritical case, the existence of continuous wave operators can be established by a fixed point argument using the Strichartz estimates. However, for a given $u_+$, the corresponding $u(t)$ is only uniquely given (by this argument) in $Y=L_t^qL_x^r$, a Strichartz space – this is a statement of *conditional uniqueness*. However, the above global decay estimates plus the use of the Strichartz estimates can be used to establish that any solution to NLS or Hartree in $C(\mathbb{R}; H_x^1)$ must also belong to $Y$ thereby establishing *unconditional uniqueness*. We point out, however, that conditional uniqueness is adequate to give well-defined wave operators and conditional as opposed to unconditional uniqueness statements are anyway quite standard in low regularity well-posedness theory.
In the $L^2$ critical case, one can again establish the existence of continuous wave operators with conditional uniqueness by a direct fixed point argument using the Strichartz estimates. However, the Morawetz global decay estimates are alone insufficient to establish unconditional uniqueness for $X=H^1$. Asymptotic completeness for $X=H^1$ is an interesting and difficult problem. In the case of NLS, Killip-Tao-Visan-Zhang [@tv][@ktv] recently showed asymptotic completeness for $n \geq 2$ assuming radial initial data (their result is true even for $L^2$ radial data). Following their work, Miao-Xu-Zhao [@MXZ] established the analogous result for radial solutions to Hartree with $\gamma=2$, $n\geq 3$.
The $L^2$ subcritical case $p_0=1+\frac{2}{n}<p<1+\frac{4}{n}$ and $\gamma_0=1<\gamma<2$ in $X=H^1$ is a rather open area. There are satisfactory results available for $X=\Sigma$ that we shall not discuss here; see Cazenave [@tc] Chapter 7 for an overview and references.[^2] Regarding the energy space $X=H^1$, a result that essentially goes back to Segal [@is] states that for NLS, one has *weak* asymptotic completeness. This means that for any solution $u(t)$ to NLS, $U(-t)u(t)$ converges weakly in $H^1$ as $t\rightarrow \pm \infty$. Presumably this result carries over to Hartree. The obvious drawback of this result is that the weak topology cannot even distinguish standing waves from the zero solution. Nakanishi [@N] showed, by a compactness argument for dimensions $n \geq 3$ and $X=L^{2}$ or $X=H^{1}$, a result in the direction of the existence of wave operators: Given any solution $v_+\in C(\Bbb R ; X)$ of the linear Schrödinger equation, there exists a (strong) solution $u \in C(\Bbb R ; X)$ of NLS such that $\lim_{t \rightarrow \infty}\|u(t)-v_+(t)\|_{X}=0$. The uniqueness (even in the conditional sense) of such a solution would give well-defined wave operators in $X$, but it is unknown if uniqueness holds or not. Nakanishi’s proof adapts to establish an analogous result for Hartree for $n\geq 3$.[^3]
In this paper we try to complete the picture in low dimensions and follow closely the argument in Nakanishi [@N]. We have the following theorem:
\[Theorem1\] Let $n=2$ and $2<p<3$. Then, for any solution $v\in C(\Bbb R ; H^1)$ of the linear Schrödinger equation, there exists a (strong) solution $u \in C(\Bbb R ; H^1)$ of NLS satisfying $$\lim_{t \rightarrow \infty}\|u(t)-v(t)\|_{H^1}=0.$$ The same result holds for Hartree in the case $n=2$ and $1<\gamma<2$.
Nakanishi’s proof is restricted to dimensions 3 and higher because 3 is the minimum dimension where the time decay of the linear dispersive estimate is integrable. In two dimensions, the decay just fails to be integrable, but we can still obtain the result by making use of a gain introduced by the global *a priori* Morawetz estimates. In the case of NLS, the needed estimate appears in Colliander-Grillakis-Tzirakis [@CGT] [^4], and for Hartree we supply here in §\[S:Morawetz\] a proof of an analogous estimate. We note that, like Nakanishi, we do not know how to prove a uniqueness statement that would give the existence of a well-defined wave operator. Presumably, a uniqueness proof would adapt to establish continuity of the wave operator. In addition we cannot extend the result to dimension one due to the very weak decay of the linear solution, even though Morawetz estimates are now available in that setting.
[**Acknowledgement.**]{} We thank Kenji Nakanishi for useful discussions and for pointing his work in [@N] to us. We also thank the anonymous referees for their comments and suggestions.
Global decay estimates {#S:Morawetz}
======================
In this section, we discuss the following theorem.
\[theorem3\] Let $u$ be an $H^{\frac{1}{2}}$ solution to NLS or Hartree on the space-time slab $I\times {\Bbb R}^n$, $n \geq 2$. Then $$\label{22d}
\|D_x^{-\frac{n-3}{2}}(|u|^2)\|_{L_I^2L_{x}^{2}} \lesssim \|u\|_{L_I^{\infty}\dot{H}_{x}^{\frac{1}{2}}}\|u\|_{L_I^{\infty}L_x^2}$$
The proof of Theorem \[theorem3\] in the case of NLS can be found in [@CGT]. Here, we prove the Hartree case, following the method developed in [@CGT]. We will prove in details the 2d estimate. The computations for the general case are almost identical. The derivation of the estimate with the Hartree nonlinearity is more complicated because the time derivative of the pressure is not any more a divergence of a vector field. On the other hand, we still can view the evolution equation as describing the evolution of a compressible dispersive fluid whose pressure is a function of the density. Morally, this fact and the defocusing character of the nonlinearity is enough to guarantee the existence of positive terms in the expansion of the time derivative of the Morawetz action. Note in addition that if $u_1(x_{1},t)$ and $u_2(x_{2},t)$ are solutions of $iu_t+\Delta u=f(u)$, then the tensor product $u(x,t)=(u_1\otimes u_2)(x,t)=u_1(x_1,t)u_2(x_2,t)$ satisfies the same equation with nonlinearity $$f(x,t)=f(u_1(x_1,t))\otimes u_2(x_2,t)+f(u_2(x_2,t))\otimes u_1(x_1,t) \, .$$ Here, $x=(x_1,x_2)$ and $\Delta =\Delta_{x_1}+\Delta_{x_2}$. Sometimes we write $x \in \Bbb R^n \otimes \Bbb R^n$ and we mean $x=(x_1,x_2)$ where $x_1 \in \Bbb R^n$ and $x_2 \in \Bbb R^n$. In particular if the equation is defocusing, the tensor product also satisfies a defocusing equation. We can thus derive correlation estimates by applying the Morawetz and Lin-Strauss method to tensor products of solutions.
The reader will notice that we use Einstein’s summation convention throughout the paper. According to this convention, when an index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position, it implies that we are summing over all of its possible values.
The main technique is to use commutator vector operators acting on the local conservation laws of the equation. We define $$M_{a}^{\otimes_{2}}(t)= \int_{\Bbb R^{n}\otimes \Bbb R^{n}}\nabla a(x)
\cdot {\operatorname{Im}}\left(\overline{u_1\otimes u_2}(x)\nabla (u_1\otimes u_2(x))\right )dx$$ which is the Morawetz action for the tensor product of two solutions $u:=(u_{1}\otimes u_{2})(x,t)$ where $x=(x_1,x_2) \in \Bbb R^n \otimes \Bbb R^n$. If we specialize to the case $u_1=u_2$, $a(x)=a(x_1,x_2)=|x_1-x_2|$, $n=2$, and observe that $$\partial_{x_1}a(x_1,x_2)=\frac{x_1-x_2}{|x_1-x_2|}=-\frac{x_2-x_1}{|x_1-x_2|}=-\partial_{x_2}a(x_1,x_2) \, ,$$ we can view $M(t) := M_{a}^{\otimes_{2}}(t)$ as $$\label{commute}
M(t)=2\int_{\Bbb R^2 \otimes \Bbb R^2}\frac{x_1-x_2}{|x_1-x_2|}\cdot \left( \vec{p}(x_1,t)\rho(x_2,t)-\vec{p}(x_2,t)\rho(x_1,t)\right) \, dx_1dx_2 \,,$$ where $\rho=\frac{1}{2}|u|^2$ is the mass density and $p_{j}={\operatorname{Im}}(\bar{u}\partial_{j}u)$ is the momentum density. Now define the integral operator $$D^{-1}f(x)=\int_{\Bbb R^2}\frac{1}{|x-y|}f(y)dy \, ,$$ where $D$ stands for the derivative. This is indeed justified because for $n=2$ the distributional Fourier transform of $\frac{1}{|x|}$ is $\frac{1}{|\xi|}$. The main observation is that we can write the action term $M(t)$ using a commutator in the following manner: $$\label{E:M}
M(t)=2\langle [x;D^{-1}]\rho(t) \ | \ \vec{p}(t)\rangle.$$ This equation follows from an elementary rearrangement of the terms of . This suggests that the estimate is derived using the vector operator $\vec{X}$ defined by $$\label{E:X}
\vec{X}=[x;D^{-1}] \, .$$ We change notation and write $x_1:=x$ and $x_2:=y$. The crucial property is that the derivatives of this operator $\partial_{j}X^{k}$ form a positive definite operator. Note that in physical space $$\label{E:Xint}
\vec{X}f(x)=\int_{\Bbb R^2}\frac{x-y}{|x-y|}f(y)dy \, ,$$ and a calculation shows that
$$\label{E:dX}
(\partial_{j}X^{k})f(x)=\int_{\Bbb R^2}\eta_{j}^{k}(x,y)f(y)dy \, ,$$
where $$\eta_{j}^{k}(x,y)=\frac{\delta_{j}^{k}}{|x-y|}-\frac{(x_j-y_j)(x^k-y^k)}{|x-y|^3} \, .$$ Thus, $\partial_j X^k$ is a positive definite operator, meaning that for any $\alpha(x)=(\alpha_1(x),\alpha_2(x))$, we have $$\int \partial_j X^kf(x) \; \alpha^j(x) \alpha_k(x) \, dx = \int \beta(y) \,f(y) \, dy \quad \text{with} \quad \beta\geq 0 \,.$$ Note also from that the divergence of the vector field $\vec{X}$ is given by $$\nabla \cdot \vec{X}=D^{-1}.$$ Now, if we differentiate (using ), we obtain that $$\label{dermor}
\frac{1}{2}\partial_{t}M(t)=\langle \vec{X}\partial_{t}\rho(t) \ | \ \vec{p}(t)\rangle-\langle \vec{X}\cdot \partial_{t}\vec{p}(t) \ | \ \rho(t)\rangle \, ,$$ where we have used the fact that $\vec{X}$ is an antisymmetric operator. Now recall the local conservation of mass $$\label{localmass}
\partial_{t}\rho+\partial_{j}p^{j}=0 \,.$$ Set $V(x)=|x|^{-\gamma}$. An explicit calculation shows that if we differentiate in time the pressure $p_{k}$, we obtain $$\label{localmom}
\partial_{t}p_k+\partial_{j}\big( \sigma_{k}^{j}+\delta_{k}^{j}\left( -\Delta \rho \right)\big)+4\rho \; \partial_{k}( V \star \rho)=0 \, ,$$ where $$\label{E:sigma}
\sigma_{k}^{j}=\frac{1}{\rho}(p^jp_k+\partial^j \rho \partial_k \rho) \, .$$ Applying the operator $X$ to the equation and contracting with $p_{k}$, and similarly applying the operator $X$ to equation and contracting with $\rho$, we obtain from that $$\begin{aligned}
\frac{1}{2}\partial_{t}M &=
\begin{aligned}[t]
& \langle \sigma_{k}^{j} \ | \ (\partial_{j}X^{k})\rho \rangle -\langle p^{j} \ | \ (\partial_{j}X^{k})p_k\rangle \\
&- \langle \Delta \rho \ | \ (\partial_{j}X^{j})\rho\rangle+4\langle X^{k}[ \rho \, \partial_{k}(V \star \rho)] \ | \ \rho \rangle
\end{aligned}
\\
&=
\begin{aligned}[t]
&\langle \sigma_{k}^{j} \ | \ (\partial_{j}X^{k})\rho\rangle - \langle p^{j} \ | \ (\partial_{j}X^{k})p_k\rangle \\
&-\langle \Delta \rho \ | \ (\partial_{j}X^{j})\rho\rangle -4\langle \rho \, \partial_{k}(V \star \rho) \ | \ X^{k}\rho \rangle \,,
\end{aligned}\end{aligned}$$ where we used again the fact that $\vec{X}$ is antisymmetric. Substituting , we have that $$\frac{1}{2}\partial_{t}M=P_{1}+P_{2}+P_{3}+P_{4}$$ where $$\begin{aligned}
& P_1:=\langle \rho^{-1}\partial_{k}\rho \partial^{j}\rho \ | \ (\partial_{j}X^{k})\rho\rangle \\
& P_2:=\langle \rho^{-1}p_kp^j \ | \ (\partial_{j}X^{k})\rho\rangle-\langle p^j \ | \ (\partial_{j}X^{k})p_k\rangle \\
& P_3:=\langle (-\Delta \rho) \ | \ (\partial_{j}X^{j})\rho\rangle=\langle (-\Delta \rho) \ | \ (\nabla \cdot \vec{X})\rho\rangle \\
& P_4:=-4\langle \rho \, \partial_{k}(V \star \rho) \ | \ X^{k}\rho \rangle \end{aligned}$$ The term $P_1$ is clearly positive since $\partial_{j}X^{k}$ is a positive definite operator. Let’s analyze $P_3$. Noting that $-\Delta=D^2$, we have $$P_3=\langle (-\Delta \rho) \ | \ (\nabla \cdot \vec{X})\rho\rangle=\langle D^2\rho \ | \ D^{-1}\rho\rangle=
\langle D^{\frac{1}{2}}\rho \ | \ D^{\frac{1}{2}}\rho\rangle=\frac{1}{2}\|D^{\frac{1}{2}}(|u|^2)\|_{L^2}^2 \, .$$ There are two terms for which their positivity is not immediate–the terms $P_{2}$ and $P_{4}$. To check $P_2$, recall , where the kernel $\eta_{kj}(x,y)$ is symmetric. Then $$P_2=\int_{\Bbb R^2 \times \Bbb R^2}\left( \frac{\rho(y)}{\rho(x)}p_k(x)p^j(x)-p_k(y)p^j(x) \right) \eta_{j}^{k}(x,y)\, dxdy \, .$$ By symmetry, we get $$P_2=\int_{\Bbb R^2 \times \Bbb R^2}\left( \frac{\rho(x)}{\rho(y)}p_k(y)p^j(y)-p_k(x)p^j(y) \right)\eta_{j}^{k}(x,y) \, dxdy \, .$$ and thus, $$\begin{aligned}
P_2 &= \frac{1}{2}\int_{\Bbb R^2 \times \Bbb R^2}\left( \frac{\rho(y)}{\rho(x)}p_k(x)p^j(x)+\frac{\rho(x)}{\rho(y)}p_k(y)p^j(y)-p^j(x)p_k(y)-p^j(y)p_k(x) \right) \eta_{j}^{k}(x,y) \, dxdy \\
& =\frac{1}{2}\int_{\Bbb R^n \times \Bbb R^n}\left( \sqrt{\frac{\rho(y)}{\rho(x)}}p_k(x)-\sqrt{\frac{\rho(x)}{\rho(y)}}p_k(y)\right)
\left( \sqrt{\frac{\rho(y)}{\rho(x)}}p^j(x)-\sqrt{\frac{\rho(x)}{\rho(y)}}p^j(y)\right) \eta_{j}^{k}(x,y) \, dxdy.\end{aligned}$$ Thus, if we define the two point momentum vector $$\vec{J}(x,y)=\sqrt{\frac{\rho(y)}{\rho(x)}}\vec{p}(x)-\sqrt{\frac{\rho(x)}{\rho(y)}}\vec{p}(y) \, ,$$ we can write $$P_2=\frac{1}{2}\langle J^{j}J_{k} \ | \ (\partial_{j}X^{k})\rangle \geq 0 \, ,$$ since $\partial_{j}X^{k}$ is positive definite.
Now we discuss $P_4$. Recalling the definition of $V$ and $\vec{X}$, we expand the form and obtain $$P_4=4\gamma \int_{\Bbb R^2}\int_{\Bbb R^2}\int_{\Bbb R^2}\rho(x)\rho(y)\rho(z)\frac{(x-y)\cdot (x-z)}{|x-y|\, |x-z|^{\gamma +2}} \, dxdydz$$ By symmetry, this term becomes $$P_4=2\gamma \int_{\Bbb R^2}\int_{\Bbb R^2}\int_{\Bbb R^2}\frac{\rho(x)\rho(y)\rho(z)}{|x-z|^{\gamma +2}}\left( \frac{(x-y)\cdot (x-z)}{|x-y|}+\frac{(z-y)\cdot (z-x)}{|z-y|}\right) \, dxdydz \, .$$ But $$\begin{aligned}
{\hspace{0.3in}&\hspace{-0.3in}}\frac{(x-y)\cdot (x-z)}{|x-y|}+\frac{(z-y)\cdot (z-x)}{|z-y|} \\
&=|x-y|+\frac{(x-y)\cdot (y-z)}{|x-y|}+|z-y|+\frac{(y-x)\cdot (z-y)}{|z-y|} \end{aligned}$$ Since the 2nd term is dominated by the 3rd and the 4th is dominated by the 1st, the sum is greater than $0$. Thus, $P_4\geq 0$.
Since all the terms are positive, we keep only $P_3$, and after integrating in time we have $$\|D^{\frac{1}{2}}(|u|^2)\|_{L_{t}^{2}L_{x}^2}^2 \lesssim \sup_{t} M(t).$$ It remains to show that $M(t)$ is bounded by the appropriate norms. But $$\begin{aligned}
\frac{1}{2}M(t) &= \langle [x;D^{-1}]^{j}\rho(t) \ | \ p_{j}(t)\rangle \lesssim \|p_{j}\|_{L^1}\|[x;D^{-1}]_j\rho(t)\|_{L^{\infty}} \\
&\lesssim \|p_{j}\|_{L^1}\|\rho\|_{L^1}\|[x;D^{-1}]_j\|_{L^1\rightarrow L^{\infty}} \, .\end{aligned}$$ We clearly have that $\|p_{j}\|_{L^1}\lesssim \|u\|_{\dot{H}^{\frac{1}{2}}}^2$ and $\|\rho\|_{L^1}=\frac{1}{2}\|u\|_{L^{2}}^2$. Finally, the operator norm $\|[x;D^{-1}]_j\|_{L^1\rightarrow L^{\infty}}$ is bounded by $1$, which is immediate from the definition of $X$.
Thus, all in all, we have .
In this paper, for both the nonlinear Schrödinger and Hartree equations, we will use the following estimate, which is an easy consequence of Theorem \[theorem3\] and interpolation.
\[C:Morawetz\_interpolated\] Let $u$ be a solution to NLS with $n=2$, $p>1$ or Hartree with $n=2$, $0<\gamma<2$. Then for $$\frac3q+\frac2r=1, \quad 4\leq q \leq \infty, \quad 2\leq r \leq 8\,,$$ we have the *a priori* bound $$\|u\|_{L_t^qL_x^r} \lesssim E(u_{0})^{\frac{r-2}{6r}}\|u_{0}\|_{L^2}^{\frac{2(r+1)}{3r}}$$
By Sobolev imbedding and Theorem \[theorem3\], we have $$\| u\|_{L_t^4L_x^8}^2 = \| |u|^2 \|_{L_t^2L_x^4} \lesssim \|D^{1/2} |u|^2 \|_{L_t^2L_x^2} \lesssim \|u\|_{L_t^\infty \dot H_x^{1/2}} \|u\|_{L_t^\infty \dot L_x^2} \lesssim E(u_{0})^{\frac{1}{4}}\|u_{0}\|_{L^2}^{\frac{3}{2}}$$ We conclude by using the interpolation (Hölder) inequality $$\|u\|_{L_t^qL_x^r} \leq \|u\|_{L_t^\infty L_x^2}^{1-\theta} \|u\|_{L_t^4L_x^8}^\theta \,, \quad \theta = \frac43-\frac8{3r}.$$
Weak time-decay bounds
======================
In this section we establish the key estimates required to implement Nakanishi’s argument. These estimates are consequences of the Morawetz bounds derived in §\[S:Morawetz\] (in particular, Corollary \[C:Morawetz\_interpolated\]) and the $L_x^p$ time-decay estimates for the linear group $U(t)$.
\[L:NLSwkdecay\] Suppose $n=2$, $2<p<\infty$, and $u(t)$ is a (global) $H^1$ solution to the defocusing NLS. Then there exists $\alpha,\beta>0$ such that for any $\psi \in C_c^\infty$ and any $s<t$, we have $$\left| \int_s^t {\langle}|u(\sigma)|^{p-1}u(\sigma), U(\sigma)\psi {\rangle}\, d\sigma \right| \lesssim \begin{cases} |t-s|^\alpha & \\ s^{-\beta} & \text{if }s\geq 1 \end{cases}$$ This notation means that both estimates hold. The brackets ${\langle}\cdot, \cdot {\rangle}$ represent the $L^2_x$ spatial pairing. The implicit constant in this estimate depends on $\|\psi\|_{L^1}$, $\|\psi\|_{L^2}$, and $\|u\|_{L_{t\in (-\infty,+\infty)}^\infty H_x^1}$. We recall that $\|u\|_{L_{t\in (-\infty,+\infty)}^\infty H_x^1}$ is controlled by the mass and energy.
We begin by proving the first estimate. By the unitarity $\|U(\sigma)\psi\|_{L^2} = \|\psi\|_{L^2}$, we have $$\left| \int_s^t {\langle}|u(\sigma)|^{p-1}u(\sigma), U(\sigma)\psi {\rangle}\, d\sigma \right| \lesssim \int_s^t \|u(\sigma)\|_{L^{2p}}^p \, d\sigma$$ Employing the Gagliardo-Nirenberg estimate $\|u(\sigma)\|_{L_x^{2p}}^p \leq \|u\|_{L^2} \|\nabla u\|_{L^2}^{p-1}$, we obtain that the above display is controlled by $|s-t|$.
Now we derive the second estimate. Suppose $s\geq 1$. By the 2-d space-time decay estimate $\|U(\sigma)\psi\|_{L_x^\infty} \leq \sigma^{-1}\|\psi\|_{L^1}$, we obtain $$\left| \int_s^t {\langle}|u(\sigma)|^{p-1}u(\sigma), U(\sigma) \psi {\rangle}\, d\sigma \right| \lesssim \int_s^t \sigma^{-1} \|u(\sigma)\|_{L_x^p}^p\, d\sigma$$ Let $q$ be such that $\frac3q=1-\frac2p$. First, suppose $p<5$. Then $q>p$, and we can apply Hölder in $\sigma$ with conjugate pair $(\frac{3}{5-p},\frac{q}{p})$ to obtain that the above display is controlled by $s^{-\frac{p-2}3}\|u\|_{L_{[s,t]}^qL_x^p}^p$. Corollary \[C:Morawetz\_interpolated\] furnishes the required bound. Suppose now that $p\geq 5$. Then, by the Gagliardo-Nirenberg inequality, $\|u(\sigma)\|_{L_x^p}^p \leq \|u(\sigma)\|_{L_x^5}^5\|\nabla u(\sigma)\|_{L_x^2}^{p-5}$, and therefore $$\int_s^t \sigma^{-1} \|u(\sigma)\|_{L_x^p}^p\, d\sigma \leq s^{-1}\|\nabla u\|_{L_t^\infty L_x^2}^{p-5}\|u\|_{L_t^5L_x^5}^5 \,.$$ We now appeal to Corollary \[C:Morawetz\_interpolated\] with $r=q=5$.
\[L:Hwkdecay\] Suppose $n=2$, $1<\gamma <2$, and $u(t)$ is a (global) $H^1$ solution to the Hartree equation. Then there exists $\alpha,\beta>0$ such that for any $\psi \in C_c^\infty$ and any $s<t$, we have $$\left| \int_s^t {\langle}(|x|^{-\gamma}\star|u(\sigma)|^2)u(\sigma), U(\sigma)\psi {\rangle}\, d\sigma \right| \lesssim \begin{cases} |t-s|^\alpha & \\ s^{-\beta} & \text{if }s\geq 1 \end{cases}$$ This notation means that both estimates hold. The brackets ${\langle}\cdot, \cdot {\rangle}$ represent the $L^2_x$ spatial pairing. The implicit constant in this estimate depends on $\|\psi\|_{L^1}$, $\|\psi\|_{L^\frac43}$, and $\|u\|_{L_{t\in (-\infty,+\infty)}^\infty H_x^1}$. We recall that $\|u\|_{L_{t\in (-\infty,+\infty)}^\infty H_x^1}$ is controlled by the mass and energy.
By the Hardy-Littlewood-Sobolev (HLS) inequality, [@es], we have that $$\label{E:HLS}
\| |x|^{-\gamma}\star |u|^2 \|_{L^2} \lesssim \| |u|^2 \|_{L^\frac{2}{3-\gamma}} = \|u\|_{L^\frac{4}{3-\gamma}}^2$$ HLS in 2-d requires $\gamma<2$, and the restriction to $\gamma>1$ implies that $\frac{2}{3-\gamma}>1$, which is also required for the validity of HLS.
To derive the first estimate, we apply Hölder with partition $(2,4,4)$, and use the space-time estimate $\|U(\sigma)\psi\|_{L^4} \lesssim |\sigma|^{-1/2} \|\psi\|_{L^{4/3}}$ to obtain $$\left| \int_s^t {\langle}(|x|^{-\gamma}\star|u(\sigma)|^2)u(\sigma), U(\sigma)\psi {\rangle}\, d\sigma \right| \lesssim \int_s^t \|(|x|^{-\gamma}\star|u(\sigma)|^2)\|_{L_x^2}\|u(\sigma)\|_{L_x^4} |\sigma|^{-1/2} \,d \sigma$$ We now follow through with and Gagliardo-Nirenberg to obtain the bound $\int_s^t |\sigma|^{-1/2} \, d\sigma$. If $0\leq s<t$, this evaluates to $t^{1/2}-s^{1/2} \leq (t-s)^{1/2}$. If $s\leq t\leq 0$, then we similarly obtain a bound of $|t-s|^\frac12$. If $s<0<t$, then the integral evaluates to $|s|^\frac12+t^\frac12 \leq 2|t-s|^\frac12$.
To derive the second estimate, we apply Hölder with partition $(2,2,\infty)$, and use the space-time estimate $\|U(\sigma)\psi\|_{L^\infty} \lesssim |\sigma|^{-1} \|\psi\|_{L^1}$ to obtain $$\left| \int_s^t {\langle}(|x|^{-\gamma}\star|u(\sigma)|^2)u(\sigma), U(\sigma)\psi {\rangle}\, d\sigma \right| \lesssim \int_s^t \|(|x|^{-\gamma}\star|u(\sigma)|^2)\|_{L_x^2}\|u(\sigma)\|_{L_x^2} |\sigma|^{-1} \,d \sigma$$ Following through with , we obtain $$\lesssim \int_s^t |\sigma|^{-1}\|u(\sigma)\|_{L_x^\frac{4}{3-\gamma}}^2 \, d\sigma$$ Set $r=\frac{4}{3-\gamma}$, note that ($1<\gamma<2 \implies 2<r<4$) and define $q$ so that $\frac{3}{q}+\frac2{r}=1$. Since $q>2$ (in fact $q>6$) we can apply Hölder in $\sigma$ with partition $(\frac{3}{4-\gamma}, \frac{q}{2})$ to obtain that the above is bounded by $s^{-\frac{\gamma-1}{3}}\|u\|_{L_t^qL_x^r}^2$. Corollary \[C:Morawetz\_interpolated\] furnishes the required bound.
Basic uniqueness and regularity
===============================
A key step in the proof of the main theorem is the statement that if $u$ solves NLS or Hartree as a distribution with some additional regularity, then $u$ is in fact the unique (strong) solution satisfying the mass and energy conservation laws. In this section we record precise statements. While all the material in this section appears in the literature (see e.g. Prop. 4.2.9, Lemma 4.2.8, and Theorem 3.3.9 in [@tc] and Theorems I, III in Kato [@tk]), we believe that the $H^1$ *unconditional* uniqueness statements are less well-known than the $H^1$ local well-posedness statements that come equipped with *conditional* uniqueness. Therefore, we will effectively assume the standard $H^1$ local existence and carefully discuss the unconditional uniqueness claims.
We will need the Strichartz estimates in the following form. Suppose that $(r,q)$ satisfy the admissibility condition $$\frac{2}{q}+\frac{n}{r}=\frac{n}{2}, \qquad 2\leq q\leq \infty$$ excluding the case $n=2$ and $q=2$. Then $$\|e^{it\Delta}\phi \|_{L_t^qL_x^r} \lesssim \|\phi\|_{L^2}$$ $$\left\|\int_0^t e^{i(t-t')\Delta}f(\cdot,t') \,dt' \right\|_{L_t^qL_x^r} \lesssim \|f\|_{L_t^{q'}L_x^{r'}}$$ where $q'$, $r'$ are the Hölder duals to $q$, $r$, respectively.
Let $I\subset \mathbb{R}$ be a bounded[^5] open interval of time. Since we are discussing distributional solutions to NLS and Hartree with $u\in L_I^\infty H_x^1$, we first off need to make sure that the nonlinearities belong to $L_{\text{loc}}^1$ (and are thus well-defined distributions) and that the interaction potential term in the energy is well-defined.
For NLS, we assume that $p$ is energy-subcritical, i.e. $$\label{E:energysub}
1<p<\frac{n+2}{n-2}$$ Then by Sobolev imbedding, $u\in L_I^\infty L_x^r$ for all $2\leq r < \frac{2n}{n-2}$. Thus, the nonlinearity $|u|^{p-1}u\in L_{\text{loc}}^1$, and furthermore $u\in L_I^\infty L_x^{p+1}$, which implies that the energy is a well-defined quantity for a.e. $t$.
For Hartree, suppose $0<\gamma<n$. Note that by the Hardy-Littlewood-Sobolev inequality, if $u(t)\in H_x^1$, then $|x|^{-\gamma}\star |u(t)|^2 \in L_x^p$ for all $p$ such that $$\max\left( 1-\frac2n-\frac{n-\gamma}{n}, 0\right) < \frac1p < 1-\frac{n-\gamma}{n}$$ Thus, we certainly have that the nonlinearity $(|x|^{-\gamma}\star|u|^2)u \in L_{\text{loc}}^1$ is a well-defined distribution.
\[L:interp\] Suppose that $I$ is an open interval of time, $u\in L_I^\infty H_x^1$ and $\partial_t u \in L_I^\infty H_x^{-1}$. (We are not asserting here that $u$ solves any equation.) Then $u$ can be redefined on a set of zero measure in time such that $u \in C_I^{0,\frac12-\frac{s}{2}}(I; H_x^s)$ for all $-1\leq s<1$, with $$\|u(t_2)-u(t_1)\|_{H_x^s} \lesssim |t_2-t_1|^{\frac12-\frac{s}{2}}\|\partial_t u\|_{L_I^\infty H_x^{-1}}^{\frac12-\frac{s}{2}}\|u\|_{L_I^\infty H_x^1}^{\frac12+\frac{s}{2}} \,.$$ Note that we do not claim that it follows that $u\in C(I;H_x^1)$ under these hypotheses alone (the case $s=1$ is excluded), nor do we claim that $u\in C^1(I;H_x^{-1})$. It is true, however, that $u\in C^{0,1}(I;H_x^{-1})$, i.e. $u$ is Lipschitz continuous as a map from time into $H_x^{-1}$ (the case $s=-1$ is included in the above estimate).
For $s=-1$ the result is immediate by the fundamental theorem of calculus and for $s=1$, the statement is trivial. The result follows by interpolation.
\[L:NLS\_uniqueness\] Suppose $u$ solves NLS (in the sense of distributions) with $1<p<\frac{n+2}{n-2}$ on a bounded open time interval $I$ and $u\in L_I^\infty H_x^1$. Then $u(t)$ can be redefined on a set of times $t$ of zero measure such that $u\in C(I;H_x^1)\cap C^1(I;H_x^{-1})\cap L_I^qW_x^{1,p+1}$, and $u$ satisfies the mass and energy conservation laws. Here, $2<q\leq \infty$ is defined by $$\frac{2}{q}+\frac{n}{p+1}=\frac{n}{2} \,.$$ Also, if $v$ is another distributional solution to NLS on $I$, $v\in L_I^\infty H_x^1$, and $v(t)$ and $u(t)$ agree at some $t_0\in I$ (after being redefined as above so that $v,u\in C(I;H_x^1)$), then $u\equiv v$.
Clearly, $\Delta u \in L_I^\infty H_x^{-1}$. By Sobolev imbedding, $$\| |u|^{p-1}u \|_{H_x^{-1}} \lesssim \| |u|^{p-1}u \|_{L_x^\alpha} = \|u\|_{L_x^{\alpha p}}^p \,$$ for any $\alpha$ such that $\max( \frac{2n}{n+2}, \,1+\, ) \leq \alpha \leq 2$. Now if $2\leq \alpha p \leq \frac{2n}{n-2}$, then by Gagliardo-Nirenberg-Sobolev, we’ll have $\|u\|_{L_x^{\alpha p}} \lesssim \|u \|_{H_x^1}$. But it is clear that for $1<p<\frac{n+2}{n-2}$, the two intervals $\max( \frac{2n}{n+2}, \, 1+\, ) \leq \alpha \leq 2$ and $\frac{2}{p} \leq \alpha \leq \frac{2n}{(n-2)p}$ have a nontrivial intersection.
Consequently, $\partial_t u = -\Delta u +|u|^{p-1}u \in L_I^\infty H_x^{-1}$, and we can apply Lemma \[L:interp\] to $u$ to conclude that $u\in C^{0,\frac12-\frac{s}{2}}(I; H_x^s)$ for all $-1\leq s <1$. In particular, $u(t)$ is now well-defined as an element of $L_x^2$ for *all* $t$. Moreover, since $u\in L_I^\infty H_x^1$, we have that $u(t)\in H_x^1$ for all $t$ except possibly on a set of measure zero. Select some $t_0\in I$ such that $u(t)\in H_x^1$. By the standard local theory (see Kato [@tk], Theorems I, III), there exists an interval $I'$ containing $t_0$ with length depending only on $\|u(t_0)\|_{H_x^1}$, and a solution $v\in C^1(I';H_x^{-1})\cap C(I';H_x^1) \cap L_{I'}^qW_x^{1,p+1}$ of NLS such that $v(t_0)=u(t_0)$ and $\|v\|_{C(I';H_x^1)} \lesssim \|u(t_0)\|_{H_x^1}$. Kato [@tk] also shows the solution $v$ satisfies the mass and energy conservation laws. By appealing to the claim below (with $I$ replaced by $I'$), we conclude that $u=v$ on $I'$. This completes the proof, since we can carry out this argument on subintervals $I'$ that fill out $I$.
*Claim*. If $u$ and $v$ have the properties given in the theorem statement and there exists $t_0\in I$ such that $v(t_0)=u(t_0)$, then necessarily $u=v$ on $I$. (Here, the statement $u(t_0)=v(t_0)$ is well-defined since we know by the above argument that both $u,v\in C(I; L_x^2)$).
To prove the claim, let $w=u-v$. We will show that on some interval $I'$ containing $t_0$ with $|I'|$ depending only on $\max(\|u\|_{L_I^\infty H_x^1}, \|v\|_{L_I^\infty H_x^1})$, we have $w\equiv 0$. The argument can then be iterated to obtain that $u=v$ on all of $I$. Since $U(t)$ is a strongly continuous unitary group on $H_x^{-1}$, $w(t)$ satisfies the integral equation $$w(t) = i \int_{t_0}^t U(t'-t_0) (|u(t')|^{p-1}u(t')-|v(t')|^{p-1}v(t')) \, dt' \,.$$ Applying the Strichartz estimates with $r=p+1$, $$\begin{aligned}
\|w\|_{L_{I'}^qL_x^{p+1}}
&\lesssim \| |u|^{p-1} u - |v|^{p-1}v \|_{L_{I'}^{q'}L_x^{\frac{p+1}{p}}} \\
& \lesssim |I'|^{\frac{1}{q'}-\frac1{q}}(\|u\|_{L_{I'}^\infty L_x^{p+1}}^{p-1}+\|v\|_{L_{I'}^\infty L_x^{p+1}}^{p-1}) \|w\|_{L_{I'}^qL_x^{p+1}}\end{aligned}$$ Taking $|I'|$ small depending only on $\max(\|u\|_{L_I^\infty H_x^1}, \|v\|_{L_I^\infty H_X^1})$, we obtain that $w\equiv 0$ on $I'$, as claimed.
\[L:Hartree\_uniqueness\] Suppose $u$ solves Hartree (in the sense of distributions) with $0<\gamma<\min(n,4)$ on a bounded open time interval $I$ and $u\in L_I^\infty H_x^1$. Then $u(t)$ can be redefined on a set of times $t$ of zero measure such that $u\in C(I;H_x^1)\cap C^1(I;H_x^{-1})\cap L_I^qW_x^{1,r}$, and $u$ satisfies the mass and energy conservation laws. Here, $r$ and $q$ are defined by $$\frac1r = \frac12-\frac{\gamma}{4n}, \qquad \frac{2}{q}+\frac{n}{r}=\frac{n}{2} \,.$$ Also, if $v$ is another distributional solution to Hartree on $I$, $v\in L_I^\infty H_x^1$, and $v(t)$ and $u(t)$ agree at some $t_0\in I$ (after being redefined as above so that $u,v\in C(I;H_x^1)$), then $u\equiv v$.
By Sobolev imbedding, $\| (|x|^{-\gamma}\ast |u|^2)u\|_{H_x^{-1}} \leq \| (|x|^{-\gamma}\ast |u|^2)u\|_{L_x^\alpha}$ for any $\alpha$ such that $$\frac12 \leq \frac1{\alpha} \leq \min\Big( \frac12+\frac1n \, , \; 1-\Big)$$ Select an $\alpha$ meeting this condition and, in addition, the following: $$\max\Big(\frac12+\frac{\gamma}{n}-\frac{3}{n} \, , \; \frac12-\frac{\gamma}{2n}+ \, \Big) \leq \frac1{\alpha}\leq \frac{1}{2}+\frac{\gamma}{n} .$$ This is possible since $0<\gamma< \min(\gamma, 4)$. Let $$\frac1{\beta} = \frac13\Big(1-\frac{\gamma}{n}+\frac1{\alpha}\Big), \qquad \frac{1}{\tilde \beta} = \frac{2}{\beta}-1+\frac{\gamma}{n} \,.$$ We note that $2\leq \beta \leq \frac{2n}{n-2}$, $\tilde \beta <\infty$, and $\frac{1}{\tilde \beta} + \frac{1}{\beta} = \frac{1}{\alpha}$. Applying Hölder and the Hardy-Littlewood-Sobolev theorem on fractional integration, we have $$\| (|x|^{-\gamma}\ast |u|^2)u\|_{L_x^\alpha} \leq \|(|x|^{-\gamma}\ast |u|^2)\|_{L_x^{\tilde \beta}} \|u\|_{L_x^\beta} \lesssim \|u\|_{L_x^\beta}^3 \,.$$ By Sobolev imbedding, we have $\|u\|_{L_x^\beta} \lesssim \|u\|_{H_x^1}$. Thus, $(|x|^{-\gamma}\ast |u|^2) u \in L_I^\infty H_x^{-1}$. It is also clear that $\Delta u \in L_I^\infty H_x^{-1}$, and therefore from the Hartree equation it follows that $\partial_t u \in L_I^\infty H_x^{-1}$. We can thus appeal to Lemma \[L:interp\], and conclude that $u$ can be redefined on a set of times $t$ of zero measure such that $u\in C^{0,\frac12-\frac{s}{2}}(I;H_x^1)$ for $-1\leq s<1$. In particular, $u(t)$ is well-defined for *all* $t$ as an element of $L_x^2$. Moreover, since $u\in L_I^\infty H_x^{-1}$, we have, except on a set of measure zero, that $u(t)\in H_x^1$. Take any $t_0\in I$ for which $u(t)\in H_x^1$. By the local theory for the Hartree equation (see Ginibre-Velo [@gv-hartree-lwp]), there exists an interval $I'$, with center $t_0$ and $|I'|$ depending only on $\|u(t_0)\|_{H_x^1}$, and $v\in C^1(I'; H_x^{-1})\cap C(I'; H_x^1) \cap L_{I'}^{q}W_x^{1,r}$ solving the Hartree equation, with $v(t_0)=u(t_0)$. Moreover, $v$ satisfies the energy and mass conservation laws. The proof is completed upon appealing to the following uniqueness claim.
*Claim*: Suppose that $u$, $v$ are given functions satisfying the conditions in the theorem statement and there exists $t_0\in I$ such that $u(t_0)=v(t_0)$. (The statement $u(t_0)=v(t_0)$ is well-defined since the above argument established that $u,v\in C(I;L_x^2)$). Then $u\equiv v$ on $I$.
Let $w=u-v$. By iteration, it suffices to show that $w\equiv 0$ on an interval $I'$ centered at $t_0$ with length depending only on $\max(\|u\|_{L_I^\infty H_x^1}, \|v\|_{L_I^\infty H_x^1})$.
We note that with $r$ as defined in the statement of the theorem, $$\max\Big(\frac14,\frac12-\frac1n\Big) < \frac1r<\frac12 \,,$$ and so $(r,q)$ is a Strichartz admissible pair. Let $\tilde r$ be defined by $$\frac1{\tilde r} = \frac2r - \frac{n-\gamma}{n}=\frac{\gamma}{2n}\, .$$ By the Hardy-Littlewood-Sobolev inequality, $$\label{E:uniqueHartree1}
\| |x|^{-\gamma} \ast (u_1 u_2) \|_{L_x^{\tilde r}} \lesssim \| u_1u_2 \|_{L_x^{\frac{r}{2}}} \leq \| u_1\|_{L_x^r}\|u_2\|_{L_x^r}$$ Moreover, $\frac1{r}+\frac1{\tilde r} = \frac1{r'}$. (Recall $r'$ is the Hölder dual to $r$.) Consequently, by Hölder and we have $$\label{E:uniqueHartree2}
\| (|x|^{-\gamma}*(u_1\bar u_2))u_3 \|_{L_x^{r'}} \lesssim \|u_1\|_{L_x^r} \|u_2\|_{L_x^r} \|u_3\|_{L_x^r}$$ Since $U(t)$ is a strongly continuous unitary group on $H_x^{-1}$, we have that $w$ satisfies the associated integral equation. Applying the Strichartz estimates, we obtain $$\|w\|_{L_{I'}^q L_x^r} \lesssim \| (|x|^{-\gamma}*|u|^2)u - (|x|^{-\gamma}*|v|^2)v \|_{L_{I'}^{q'}L_x^{r'}}$$ Applying , $$\|w\|_{L_{I'}^q L_x^r} \lesssim |I|^{\frac{1}{q'}-\frac{1}{q}} (\| u\|_{L_{I'}^\infty L_x^r}^2 + \|v\|_{L_{I'}^\infty L_x^r}^2)\|w\|_{L_{I'}^qL_x^r} \,.$$ Select $|I'|$ sufficiently small, depending only on $\max( \|u\|_{L_I^\infty H_x^1}, \|v\|_{L_I^\infty H_x^1})$, to obtain $w\equiv 0$ on $I'$.
Nakanishi’s argument
====================
We will prove Theorem \[Theorem1\]. The only difference between the NLS case and the Hartree case is that, for the former, we use Lemma \[L:NLSwkdecay\], \[L:NLS\_uniqueness\] and for the latter, we use Lemma \[L:Hwkdecay\], \[L:Hartree\_uniqueness\]. Therefore, we shall only present the details for the NLS case.
We are given an $H_x^1$ solution $v(t)$ to the linear Schrödinger equation and want to find an $H_x^1$ solution $u(t)$ to NLS such that $\|u(t)-v(t)\|_{H_x^1} \to 0$ as $t\to +\infty$. For any time $T>0$, we have a unique solution $w_T=w \in C(\Bbb R ; H^{1})$ to the initial value problem $$\left\{
\begin{matrix}
iw_{t}+ \Delta w -|w|^{p-1}w=0, & x \in {\mathbb R^2}, & t\in {\mathbb R},\\
w(x,T)=v(T)\in H^{1}({\mathbb R^2}).
\end{matrix}
\right.$$ We start by constructing $u(t)$ as a weak limit of $w_T(t)$ as $T\to +\infty$ and establish that $U(-t)u(t)$ converges weakly to $v(0)$ as $t\to \infty$ in $H_x^1$. This is accomplished by an argument using the Arzela-Ascoli theorem and the (uniform in $n$) weak decay estimate in Lemma \[L:NLSwkdecay\]. Then, by using the conservation laws, we upgrade the weak convergences to strong convergences, and obtain in particular that $U(-t)u(t) \to v(0)$ strongly in $H_x^1$ as $t\to \infty$.
We now turn to the details. Let $\{ \psi_j \}_{j=1}^\infty \subset C_c^\infty(\mathbb{R}^n)$ be a basis of $H^{-1}(\mathbb{R}^n)$ and let $f_{j,T}(t) = {\langle}U(-t)w_T(t),\psi_j {\rangle}$.
**Assertion 1**. $\forall \, j\in \mathbb{N}$, $\lim_{t\to \pm \infty} f_{j,T}(t)$ exists (and is a number in $\mathbb{C}$). Thus, $f_{j,T}$ is defined on the compact interval $[-\infty,+\infty]$.
*Proof of Assertion 1*. By writing the equivalent integral representation of the solution $w_{T}(t)$ we have that $$w_{T}(t)=U(t-T)v(T)-i\int_{T}^{t}U(t-\sigma)\{|w_{T}|^{p-1}w_{T}(\sigma)\}d\sigma$$ where $U(t)=e^{it\Delta}$ is the linear semigroup. Then $$\begin{aligned}
U(-t)w_{T}(t) & =U(-T)v(T)-i\int_{T}^{t}U(-\sigma)\{|w_{T}|^{p-1}w_{T}(\sigma)\}d\sigma \\
&= v(0)-i\int_{T}^{t}U(-\sigma)\{|w_{T}|^{p-1}w_{T}(\sigma)\}d\sigma\end{aligned}$$ and thus $$\langle U(-t)w_{T}(t), \psi_{j}\rangle=\langle v(0), \psi_{j}\rangle-i\int_{T}^{t}\langle U(-\sigma)\{|w_{T}|^{p-1}w_{T}(\sigma)\}, \psi_{j} \rangle d\sigma.$$ Now for any $T \geq 1$, by the proof of Lemma \[L:NLSwkdecay\], we have that the limit of $f_{j,T}(t) = {\langle}U(-t)w_T(t),\psi_j {\rangle}$ exists as $t \rightarrow \pm \infty$ and it is a finite number. *End proof of Assertion 1*.
**Assertion 2**. $\forall \, j\in \mathbb{N}$, the family $\{f_{j,T}\}_T$ is equicontinuous in $C([-\infty,+\infty])$ (here $C([-\infty,+\infty])$ is given the sup norm over $t\in[-\infty,+\infty]$). Moreover, $\forall \, j\in \mathbb{N}$, the family $\{f_{j,T}\}_T$ is uniformly bounded.
*Proof of Assertion 2*. The equicontinuity of $\{f_{j,T}\}_T$ follows immediately from Lemma 3.1. Moreover we have that $$\begin{aligned}
|f_{j,T}(t)| = |{\langle}& U(-t)w_T(t),\psi_j {\rangle}| \lesssim \|w_{T}(t)\|_{H^1} \lesssim E^{\frac{1}{2}}(w_{T})(t) \\
& =E^{\frac{1}{2}}(w_T)(T) = E^{\frac12}(v)(T) \\
& = \left ( \frac{1}{2}\int |\nabla v(T)|^{2}dx+\frac{1}{p+1}\int |v(T)|^{p+1}dx \right )^{\frac{1}{2}} \\
&\lesssim \|v(0)\|_{H^1} + \|v(0)\|_{H^1}^\frac{p+1}{2} \,.\end{aligned}$$ The last inequality follows because $\|v(T)\|_{L^2}=\|v(0)\|_{L^2}$, $\|\nabla v(T)\|_{L^2} = \|\nabla v(0)\|_{L^2}$, and by Gagliardo-Nirenberg, we have that $$\|v(T)\|_{L^{p+1}}\lesssim \|v(T)\|_{L^2}^{\frac{2}{p+1}}\|\nabla v(t)\|_{L^{2}}^{\frac{p-1}{p+1}} \lesssim \|v(0)\|_{H^{1}}.$$ Thus $\{f_{j,T}\}_T$ is uniformly in $T$, bounded. *End proof of Assertion 2*.
Now, by Assertions 1–2 and the Arzela-Ascoli theorem and a diagonal argument, there exists a sequence $T_n\to +\infty$ such that $\forall \, j\in \mathbb{N}$, the sequence $f_{j,T_n}$ converges strongly in $C([-\infty,+\infty])$ as $n\to \infty$ (uniformly in $t$). This is the key ingredient in the proof of Assertions 3 and 6 below.
**Assertion 3**. For each $t\in (-\infty,+\infty)$, the sequence $w_{T_n}(t)$ converges weakly in $H^1$ as $n \rightarrow \infty$. Let $u(t)$ denote this weak limit.
It is important to note one strength of Assertion 3: there is a *single* sequence $T_n$ such that *for all* $t$, the weak convergence along this sequence $T_n$ holds. In contrast, it is a simple statement that for each $t$, there is a sequence $T_n$ for which the weak convergence holds, with the sequence $T_n$ depending on $t$.
*Proof of Assertion 3*. . We know, by the above Arzela-Ascoli argument, that $\forall \; j$, ${\langle}w_{T_{n}}(t), U(t)\psi_j {\rangle}$ converges as $n \rightarrow \infty$ to a limit; let’s call it $L(U(t)\psi_{j})$. $L$ clearly extends to a linear functional on the finite span $E\subset H^{-1}$ of $\{ U(t)\psi_j \}_j$, and $L: E\to \mathbb{C}$ is bounded since $w_{T_{n}}(t)$ is bounded in $H^{1}$ uniformly in $T_{n}$. Since $E$ is dense in $H^{-1}$, $L$ extends to a bounded linear functional on all of $H^{-1}$. Then by the Riesz Representation Theorem we know that there exists some $u(t) \in H^1$ such that $L(U(t)\psi_{j})={\langle}u(t), U(t)\psi_{j}{\rangle}$. Thus $$\lim_{n \rightarrow \infty}{\langle}U(-t)w_{T_{n}}(t),\psi_j {\rangle}={\langle}U(-t)u(t), \psi_{j}{\rangle}\,,$$ i.e. $U(-t)w_{T_n}(t)$ converges weakly in $H^1$ to $U(-t)u(t)$ as $n\to +\infty$. Since $U(t)$ is unitary on $H^{-1}$, $w_{T_n}(t)$ converges weakly in $H^1$ to $u(t)$ as $n\to +\infty$. *End proof of Assertion 3*.
The quantities $\|w_{T_n}(t) \|_{H^1}$ are uniformly in $t$ controlled by the mass and energy of $w_{T_n}$, and hence uniformly in $n$ bounded (see the more detailed argument in the proof of Assertion 2), say by $M$. Thus, $\|u(t)\|_{L_t^\infty H_x^1} \leq M$. The next goal is to confirm that $u(t)$ solves NLS in the sense of distributions (Assertion 5 below), and thus we can appeal to the Lemma \[L:NLS\_uniqueness\] giving that $u(t)$ is a strong solution to NLS and satisfies the mass and energy conservation laws. In order to do this, we need Assertion 4 below.
**Assertion 4**. For the same sequence $T_n$ in Assertion 3, we have that for all $t$, $w_{T_n}(t)\to u(t)$ strongly in $L_{\text{loc}}^{p+1}$.
*Proof of Assertion 4*. Fix $t\in \mathbb{R}$, and pick any subsequence $T_{n_k}$ of $T_n$. By the Rellich theorem, there exists a subsequence $T_{\tilde n_k}$ of $T_{n_k}$ such that $w_{T_{\tilde n_k}}(t)$ converges strongly in $L^{p+1}_{\text{loc}}$. Since $w_{T_{\tilde n_k}}(t)$ converges weakly to $u(t)$ in $H^1$, we must have that $w_{T_{\tilde n_k}}(t) \to u(t)$ strongly in $L^{p+1}_{\text{loc}}$. It follows[^6] that $w_{T_n}(t) \to u(t)$ in $L^{p+1}_{\text{loc}}$.
Now that we know that we have one distinguished sequence $T_n$ such that $\forall \; t$, $w_{T_n}(t) \to u(t)$ weakly in $H^1$ and strongly in $L^{p+1}_{\text{loc}}$, we can prove:
**Assertion 5**. $u$ solves NLS in the distributional sense.
*Proof of Assertion 5*. Let $\phi\in C_c^\infty(\mathbb{R}^{n+1})$ be a test function. We must show that (with ${\langle}\cdot, \cdot {\rangle}$ denoting the $H_x^1$–$H_x^{-1}$ pairing) $$\label{E:u_solves_NLS}
\int_t {\langle}u(t), -i\partial_t\phi(t) + \Delta\phi(t) {\rangle}\, dt+ \int_t {\langle}|u(t)|^{p-1}u(t),\phi(t) {\rangle}\, dt =0 \, .$$ We know that for each $n$, $$\label{E:wn_solves_NLS}
\int_t {\langle}w_{T_n}(t), -i\partial_t\phi(t) + \Delta\phi(t) {\rangle}\, dt+ \int_t {\langle}|w_{T_n}(t)|^{p-1}w_{T_n}(t),\phi(t) {\rangle}\, dt =0 \, .$$ For each fixed $t$, we have $${\langle}w_{T_n}(t), -i\partial_t\phi(t) + \Delta\phi(t) {\rangle}\to {\langle}u(t), -i\partial_t\phi(t) + \Delta\phi(t) {\rangle}\text{ as }n\to \infty$$ since $w_{T_n}(t) \to u(t)$ weakly; also we have $${\langle}|w_{T_n}(t)|^{p-1}w_{T_n}(t),\phi(t) {\rangle}\to {\langle}|u(t)|^{p-1}u(t),\phi(t) {\rangle}\text{ as }n\to \infty$$ since $w_{T_n}(t) \to u(t)$ strongly in $L^{p+1}_{\text{loc}}$. By dominated convergence (using that $\|w_{T_n}\|_{L_t^\infty H_x^1} \leq c$ independent of $n$), we can send $n\to \infty$ in to obtain . *End proof of Assertion 5*.
At this point we know that $u\in L_t^\infty H_x^1$ solves NLS as a distribution and thus by Lemma \[L:NLS\_uniqueness\], $u\in C(\mathbb{R}; H_x^1)$ and $u$ satisfies the mass and energy conservation laws.
It follows easily from Assertion 3 that for each $t\in (-\infty,+\infty)$, $U(-t)w_{T_n}(t)$ converges weakly to $U(-t)u(t)$ in $H^1$ as $n\to +\infty$. We also have:
**Assertion 6**. $U(-t)u(t) \to v(0)$ weakly in $H^1$ as $t\to +\infty$.
*Proof of Assertion 6*. As in Assertion 1 we write $$\label{E:A}
\langle U(-t)w_{T_{n}}(t), \psi_{j}\rangle=\langle v(0), \psi_{j}\rangle-i\int_{T_{n}}^{t}\langle U(-\sigma)\{|w_{T_{n}}|^{p-1}w_{T_{n}}(\sigma)\}, \psi_{j} \rangle d\sigma.$$ By the above Arzela-Ascoli argument, we know that ${\langle}U(-t)w_{T_n}(t),\psi_j {\rangle}\to {\langle}U(-t)u(t),\psi_j {\rangle}$ *uniformly in $t$* as $n\to +\infty$. Let $\epsilon>0$. It follows that $\exists \; N$ such that $n\geq N$ implies that $$\label{E:B}
\forall \; t, \qquad |{\langle}U(-t)w_{T_n}(t),\psi_j {\rangle}- {\langle}U(-t)u(t),\psi_j {\rangle}| < \frac{\epsilon}{2}$$ But by Lemma \[L:NLSwkdecay\], for any $n$, $$\label{E:C}
\forall \; t\geq T_n, \quad \left| \int_{T_n}^t {\langle}U(-\sigma)|w_{T_n}(\sigma)|^{p-1}w_{T_n}(\sigma), \psi_j {\rangle}\, d\sigma \right| \lesssim T_n^{-\beta}$$ Fix $n\geq N$ such that $T_n^{-\beta} \leq \frac{\epsilon}{2}$. By combining , , and , we obtain that for all $t\geq T_n$, $$| {\langle}U(-t)u(t),\psi_j{\rangle}-{\langle}v(0),\psi_j {\rangle}| \leq \epsilon$$ *End proof of Assertion 6*.
Now we are ready to finish the proof. The key facts that we have established thus far are: (1) $u(t)\in C(\mathbb{R}; H_x^1)$ solves NLS and satisfies mass and energy conservation; (2) for fixed $t$, $w_{T_n}(t) \to u(t)$ weakly in $H_x^1$ as $n\to \infty$; (3) $U(-t)u(t) \to v(0)$ weakly in $H_x^1$ as $n\to \infty$. The main remaining goal is to upgrade the weak convergence in (3) to strong convergence. In the process, we will end up upgrading the weak convergence in (2) to strong convergence.
Since $w_{T_n}(t)$ converges weakly to $u(t)$ in $H^1$ as $n\to \infty$, and using the $L^2$ conservation of NLS and the free Schrödinger equation, we have $$\|u(t)\|_{L^2} \leq \liminf_{n\to +\infty} \|w_{T_n}(t) \|_{L^2} \leq \limsup_{n\to +\infty} \|w_{T_n}(t) \|_{L^2} = \|v(0)\|_{L^2}$$ However, since $U(-t)u(t)$ converges weakly to $v(0)$ in $H^1$ as $t\to +\infty$, we have $$\|v(0)\|_{L^2} \leq \liminf_{t\to \infty} \|U(-t)u(t)\|_{L^2} \leq \limsup_{t\to \infty} \|U(-t)u(t)\|_{L^2} = \|u(t)\|_{L^2}$$ Combining the two inequalities, we learn that all of the inequalities are equalities and thus that $U(-t)u(t) \to v(0)$ strongly in $L^2$ as $t\to +\infty$ and that for each $t\in (-\infty,+\infty)$, $w_{T_n}(t)\to u(t)$ strongly in $L^2$. To prove the above statements we use the fact that if $f_{j} \rightarrow f$ weakly in $L^2$ and $\|f_{j}\|_{L^2} \rightarrow \|f\|_{L^2}$ then $f_{j} \rightarrow f$ strongly in $L^2$. By interpolation inequalities, the fact that these limits are strong in $L^2$ and bounded in $H^1$, we learn that these limits are strong in $L^{p+1}$.
Since $U(-t)u(t)$ converges to $v(0)$ weakly in $H^1$ as $t\to +\infty$, we have $$\label{E:wkbd101}
\|\nabla v(0)\|_{L^2}^2 \leq \liminf_{t\to +\infty} \|\nabla \; U(-t)u(t)\|_{L^2}^2 = \liminf_{t\to +\infty} \|\nabla u(t)\|_{L^2}^2$$ Since for each $t\in (-\infty,+\infty)$, $w_{T_n}(t)$ converges weakly to $u(t)$ in $H^1$, we have $$\begin{aligned}
E[u(t)] &= \frac12 \|\nabla u(t)\|_{L^2}^2 + \frac1{p+1}\|u(t)\|_{L^{p+1}}^{p+1}\\
&\leq \liminf_{n\to +\infty} \; \frac12\|\nabla w_{T_n}(t)\|_{L^2}^2 \; + \frac1{p+1}\|u(t)\|_{L^{p+1}}^{p+1}\\
&= \liminf_{n\to +\infty} \left( E[w_{T_n}(t)] - \frac1{p+1}\|w_{T_n}(t)\|_{L^{p+1}}^{p+1}\right) + \frac1{p+1}\|u(t)\|_{L^{p+1}}^{p+1}\\
&= \liminf_{n\to +\infty} E[w_{T_n}(T_n)] \,,
\intertext{where we have used that $w_{T_n}(t)$ converges strongly to $u(t)$ in $L^{p+1}$ and the conservation of energy for $w_{T_n}$. Since $w_{T_n}(T_n)=v(T_n)$, we have,}
&= \liminf_{n\to +\infty} \left( \frac12\|\nabla v(T_n)\|_{L^2}^2 - \frac1{p+1}\|v(T_n)\|_{L^{p+1}}^{p+1} \right)\\
&= \frac12\|\nabla v(0)\|_{L^2}^2 \,,\end{aligned}$$ where we have used that the free Schrödinger evolution is unitary on $\dot H^1$ and the space-time decay estimate. Thus, we have $$\label{E:wkbd100}
E[u(t)]\leq \frac12\|\nabla v(0)\|_{L^2}^2$$ Since $U(-t)u(t)$ converges strongly to $v(0)$ in $L^2$ as $t\to +\infty$, we have that $\|u(t)-v(t)\|_{L^2} \to 0$ as $t\to +\infty$. By interpolation and the boundedness of $u(t)$ and $v(t)$ in $H^1$, we have that $\|u(t)-v(t)\|_{L^{p+1}} \to 0$ as $t\to +\infty$. But the space-time decay estimate for free Schrödinger propagation implies that $v(t)\to 0$ in $L^{p+1}$ and thus $u(t)\to 0$ in $L^{p+1}$ as $t\to +\infty$. By , we have $$\limsup_{t\to +\infty} \|\nabla u(t)\|_{L^2}^2 \leq \|\nabla v(0)\|_{L^2}^2 \,.$$ Combining this with , we see that the inequalities must be equalities and thus $U(-t)u(t) \to v(0)$ strongly in $H^1$.
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[^1]: For some spaces $X$, such as $X=\Sigma$, the linear flow is not unitary and it is not known whether the statement $\|U(-t)u(t)-u_+\|_X\to 0$ is equivalent to $\|u(t)-U(t)u_+\|_X \to 0$. On this question see Begout [@pb]. In this paper, we consider $X=H^1$ on which the linear flow is unitary.
[^2]: In weighted spaces, the existence of continuous wave operators is known for the whole eligible $L^2$ subcritical range and asymptotic completeness is known to hold for $p>p_S$ and $\gamma>\gamma_S$, where $p_S>p_0$ and $\gamma_S>\gamma_0$ are certain exponents that emerge from the use of the pseudoconformal conservation law. The main remaining open problem (as far as we know) in this setting is to determine whether or not asymptotic completeness holds for $p_0<p<p_S$ and $\gamma_0<\gamma<\gamma_S$.
[^3]: K. Nakanishi, personal communication.
[^4]: For the case of the nonlinear Schrödinger equation these estimates have been obtained independently and simoultaneously by Planchon and Vega [@pv].
[^5]: For convenience in the discussion that follows, we require $I$ to be bounded. Of course, one can concatenate intervals to draw global in time conclusions.
[^6]: Recall the basic fact: Given $h$ and a sequence $h_n$ in a metric space, suppose that for every subsequence $h_{n_k}$ of $h_n$, there exists a subsequence $h_{\tilde n_k}$ of $h_{n_k}$ such that $h_{\tilde n_k} \to h$. Then $h_n \to h$, i.e. the original sequence converges to $h$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Dynamic nuclear polarization (DNP) enhances nuclear magnetic resonance (NMR) signals by transferring electron spin polarization to nuclei. As DNP requires microwave magnetic fields $\mathrm{B_1}$ strong enough to saturate electron spins, microwave resonators are generally used to achieve a sufficient $\mathrm{B_1}$, at the expense of restricting the sample size. Higher fields improve NMR sensitivity and resolution. However, resonators at 9 T for example can only hold nano-liters (nL). Larger volumes are possible by avoiding resonators, but the higher power needed to reach $\mathrm{B_1}$ is likely to evaporate the sample. Here, we demonstrate a breakthrough in liquid state DNP at 9 T, boosting the sample size to the $\mu$L range. We could use high-power (70 W) microwaves thanks to a planar probe designed to alleviate dielectric heating. We enhanced the ${}^1$H NMR signal intensity of 2 $\mu$L of liquid water by a factor of 14, while maintaining the water temperature below 40 ${}^{\circ}$C.'
author:
- Dongyoung Yoon
- 'Alexandros I. Dimitriadis'
- Murari Soundararajan
- Christian Caspers
- 'J$\acute{\text{e}}$r$\acute{\text{e}}$my Genoud'
- Stefano Alberti
- Emile de Rijk
- 'Jean-Philippe Ansermet'
title: |
Enormous sample scale-up from nL to $\mu$L\
in high field liquid state dynamic nuclear polarization
---
Nuclear magnetic resonance (NMR) is a powerful analytical tool in chemistry and biochemistry. However, the low sensitivity of NMR, which is directly linked to the strength of the nuclear magnetic moments, generally requires samples containing a large number of nuclear spins, more than $10^{17}$ spins, or long measurement times. Dynamic nuclear polarization (DNP) can increase NMR signal intensity by transferring the much higher polarization of unpaired electron spins to bulk nuclei[@Abragam]. DNP requires microwaves (MW) of sufficiently strong magnetic fields ($\mathrm{B_1}$) to saturate the electrons spin resonance. At X band ($\simeq$ 9 GHz at $\simeq$ 3 kG), this can be easily achieved by using microwave resonators with a high quality factor (Q)[@Armstrong2009; @Hofer2008; @Tuerke2012]. The sample is located at a position of low electric field ($\mathrm{E_1}$), thus preventing dielectric heating of the sample under test. However, the size of the microwave resonators decreases as the magnetic field and the frequency increase. As a consequence, the sample volume contained at the $\mathrm{B_1}$ maximum in microwave resonators is inherently limited by the microwave wavelength. This can restrict the sample volume for DNP at 9 T to nanoliters only. Furthermore, it becomes difficult to combine it with the resonator used for NMR at radio frequencies (RF).
Therefore, microwave resonators are generally not used for high-field DNP ($>$ 5 T) and instead, high-power microwave generators such as gyrotrons or extend interaction klystron (EIK) are used[@Becerra1993; @Rosay2010; @Kemp2016]. A power level of a few watts is sufficient in solid state DNP with magic angle spinning (MAS) at 9 T [@Bruker] to saturate at least partially the electron spin resonance in glassy frozen solutions because, in these samples, the electron spin-lattice relaxation time $T_{1e}$ is quite long, about 0.1-1 ms[@Thurber2010]. In addition, dielectric heating is weak in solid state DNP because of the relatively low loss tangent (tan $\mathrm{\delta}$ $\simeq$ 0.01) of frozen samples[@Nanni2011]. Furthermore, the samples are thermalized thanks to the high cooling power of the cold $\mathrm{N_2}$ used to spin the rotors. Signal enhancement in solid state DNP with MAS can be optimized by choosing suitable radicals[@Hu2004; @Zagdoun2013].
Saturating electron spins of radicals in liquids at high field is more challenging because $T_{1e}$ is much shorter (*e.g.* $\sim$ 120 ns for TEMPOL radicals)[@Prandolini2009]. Therefore, a much larger $\mathrm{B_1}$ is required in liquids than in solids. Since aqueous or other polar solutions generally have larger dielectric loss than frozen solutions (tan $\mathrm{\delta}$ $\geq$ 0.8), more cooling power is needed. Prisner *et al.* conducted liquid state DNP at 9 T using an $e-{}^1$H double resonance probe with a Q $\simeq$ 400 for *e*, and the sample was restricted to only a few nL [@Prandolini2008; @Prandolini2009; @Gafurov2012; @Neugebauer2013]. The microwave resonator allowed them to achieve 90 % of the full electron spin saturation with 100 mW of microwave power. However, dielectric heating large enough to heat the sample over the boiling point could occur, so that active cooling obtained by blowing $\mathrm{N_2}$ gas on the sample was necessary to keep the sample temperature under control.
In this study, we report a new approach that can boost the possible sample volume to microliters for liquid state DNP at 9 T. We used a high-power gyrotron ($\simeq$ 150 W), and obtained DNP without relying on a microwave resonator to saturate electron spins. Such a high MW power would boil off solutions held in conventional NMR probes. An innovative DNP-NMR probe design (Pat. Pend. [@annino2013]) prevents heating of the sample because the liquid is located where $\mathrm{E_1}$ is minimum and because the liquid is also well thermalized with the metal backing of the probe. The thin sample (d = 100 $\mu$m $\ll$ $\lambda$/4) can be of as much as 10 $\mu$L. We succeeded in an enhancement of ${}^1$H NMR by a factor of -14 for an aqueous solution and an enhancement of ${}^{31}$P NMR by a factor of 200 for a solution of triphenylphosphine in fluorobenzene. All DNP experiments here were performed without active cooling devices.
Results
=======
**DNP spectrometer and experiment procedure.** Figure \[Experiment-set-up\] (a) shows the liquid state DNP spectrometer used in this study, which is composed of a high-power gyrotron, two corrugated polarizing $\lambda$/4 & $\lambda$/8 mirrors that allow us to control the polarization of the millimeter wave, a 5 m-long corrugated waveguide, a miter bend, a superconducting NMR magnet (9 T), and the planar probe. The gyrotron was designed using a triode-magnetron-injection gun that enables an independent control of the anode voltage[@Alberti2012; @Alberti2013]. This allows for unique features such as fast frequency tunability and fast switchability[@Yoon2016]. The gyrotron is tunable within about 1 GHz around 260 GHz by changing the magnetic field. The gyrotron power shows a maximum of about 150 W near the lowest frequency of its tuning range, and decreases with increasing frequency. Modulation of the anode voltage by an external trigger enables the MW to be switched on and off rapidly, as depicted in Fig. \[Experiment-set-up\] (a). NMR signals with MW were obtained as follows: first, MW is turned on for 1-5 s, then a free induction decay (FID) is acquired after MW is turned off. MW is kept off for a while after the NMR signal acquisition to allow the sample to return to the initial temperature.
![Overhauser ${}^{31}$P DNP enhancements of the liquid sample as a function of microwave power in the planar probe and in a solenoid coil.[]{data-label="enhancement_power"}](enhancement_power.pdf){width="55.00000%"}
**Planar probe.** As illustrated in Fig. \[Experiment-set-up\] (b), the planar probe is composed of 1) a grid that consists of a thin layer of 150 parallel copper wires with a width of 50 $\mu$m and a period of 100 $\mu$m etched on a 300 $\mu$m-thick fused silica substrate, and 2) a 3 mm-thick copper support that has a square indentation (1 cm $\times$ 1 cm $\times$ 100 $\mu$m) in which the sample is placed. The width and period of the wires of the grid were designed to have a high transmission of over 90 % at about 260 GHz when the MW is polarized with the $\mathrm{E_1}$ field perpendicular to the wires. The rectangular and triangular strip contacts at both ends of the wires of the grid are designed so as to feed homogenous RF-currents to the wires. The rectangular and triangular strip contacts are connected to the support and to the central core of a coaxial cable, respectively. The grid and capacitors (not shown) form the RF resonator used for NMR. The grid sits on the support and is gently pressed down by three Cu-Be springs(Fig. \[Experiment-set-up\] (b)). After clamping the grid and its support together, liquid samples can be injected from the holes at the bottom of the support into the indentation. The support is gold-coated to protect from chemical reactions with the sample.
The support acts as a ground plane for MW and a heat sink. With nearly ideal boundary condition (i.e. $\mathrm{E_1}$ = 0) imposed by the conductive ground plane, the sample is located at a maximum of $\mathrm{B_1}$, which is a node of $\mathrm{E_1}$. This geometry results in reducing $\mathrm{E_1}$ and, hence, the dielectric heating in the sample. As shown in the simulation of $\mathrm{E_1}$ amplitude above the Cu support (COMSOL Multiphysics) in Fig. \[simulation\] (a) and (b), $\mathrm{E_1}$ is minimized at the ground plane, and becomes stronger farther away from the Cu support. As the skin depth of pure water at 260 GHz is about 200 $\mu$m[@water_dielectric] larger than 100 $\mu$m of the sample thickness, all nuclear spins in the sample can be hyperpolarized by DNP. The sample is also in the thermal contact with the support that has a high thermal conductivity and a large thermal capacity. This enables rapid heat transport from the sample to the support. For the above two reasons, we expect to be able to apply high power MW without producing severe sample heating.
**Quasi optical units to produce circularly polarized microwaves.** The quasi optical $\lambda$/4 and $\lambda$/8 mirrors for 260 GHz are placed after the gyrotron output window, and are tailored to transform the linearly polarized gyrotron output to circular polarization. The circular polarization produced by the mirrors is twice as effective at saturating electron spins as linearly polarized MW. We previously showed that circularly polarized microwaves are able to induce a larger enhancement in frozen solutions than linearly polarized microwaves at the same power[@Yoon2016]. However, the grid on the probe itself acts as a linear polarizer, rejecting half the power. An alternative probe configuration uses a coil loosely wound around the Cu support and the fused silica, in place of the wire grid. While this produces somewhat inhomogeneous $\mathrm{B_1}$ in the sample place, it is less polarization selective and the average $\mathrm{B_1}$ intensity is higher when circularly polarized MW beams are used. This method was used for ${}^1$H DNP, while the grid was used for ${}^{31}$P DNP.\
**Liquid state ${}^1$H DNP at 9 T in water.** We obtained liquid state ${}^1$H DNP at 9.2 T (${}^1$H = 395 MHz) and at room temperature with water containing 80 mM TEMPOL (4-hydroxy-2,2,6,6-tetramethylpiperidin-1-oxyl). The DNP mechanism in liquid state is the Overhauser effect[@PhysRev.102.975]. Recently, the Overhauser effect was also observed in the solid phase of glass-forming media, such as ortho-tetraphenyl or polystrene, doped with radicals at high fields (over 9 T) and even at room temperature[@Can2014; @Lelli2015]. Polarization transfer from electron to nuclear spins is mediated by cross relaxation due to time-dependent scalar or dipolar interactions. The coupling factor that represents the transfer efficiency becomes larger as the spectral density of the time-dependent interactions contains larger component near the EPR frequency. As the EPR frequency increases and becomes comparable to the inverse correlation time of translational or tumbling motions of the molecules, the component near the EPR frequency in the spectral density rapidly decreases. This implies that the coupling factor becomes smaller as the magnetic field increases, which results in less efficient liquid state DNP at high field[@Hofer2008; @Prandolini2008]. In particular, the coupling factor between water protons and radicals via dipolar relaxation was predicted to be negligible at high field. However, Prisner *et al.* observed surprisingly larger dipolar relaxation in water with ${}^{14}$N-TEMPOL at 9 T. They obtained a ${}^1$H enhancement of about -14 at about 40 ${}^{\circ}$C. The enhancement increased up to about -40 with temperature increasing up to 95 ${}^{\circ}$C as a result of faster diffusion and shorter correlation time of water molecules at higher temperatures[@Neugebauer2013]. Here, we irradiated circularly polarized MW of about 70 W, whose magnetic field strength is estimated to be about $\simeq$ 1.3 G in free space and $\simeq$ 2.5 G on the ground plane of the Cu support. In a previous report, the Prisner group obtained a saturation factor of over 0.9 in ${}^{14}$N-TEMPOL dissolved in water with a $\mathrm{B_1}$ of about 1.4 G (100 mW with a conversion factor of 0.45 mT$\cdot$W${}^{-1/2}$)[@Neugebauer2013]. Fig. \[simulation\] (c) and (d) present our simulation results on the $\mathrm{B_1}$ distribution in the sample region of the planar probe with tan $\mathrm{\delta}$ = 0.8 (water at $\simeq$ 5 ${}^{\circ}$C) and tan $\mathrm{\delta}$ = 1.2 (water $\simeq$ 30 ${}^{\circ}$C), respectively[@water_dielectric]. The loss tangent increases rapidly from $\simeq$ 0.7 to $\simeq$ 1.4 with temperature in the range from 0 ${}^{\circ}$C to 50 ${}^{\circ}$C[@water_dielectric; @Ellison2007]. Therefore, the sample heating by MW induces an increase in loss tangent, which results in shorter skin depth, and smaller $\mathrm{B_1}$. As shown in Fig. \[simulation\] (c) for tan $\mathrm{\delta}$ = 0.8 with linearly polarized MW of 70 W, the $\mathrm{B_1}$ at the center shows a maximum of about 2.8 G right at the ground plane, and decreases down to 1.8 G at the bottom of the fused silica substrate. This implies that the $\mathrm{B_1}$ at the center in the middle of the probe is sufficiently large at any depth of the sample. $\mathrm{B_1}$ decreases laterally because of the Gaussian power distribution in the MW beam. Thus, $\mathrm{B_1}$ is insufficient to saturate the electron spins far away from the middle of the probe. Therefore, we used only 2 $\mu$L in order to place all the sample around the center, thus ensuring hyperpolarization of all the nuclei. This sample volume is about 700-fold greater than previously reported ($\sim$ 3 nL) using a microwave resonator for 260 GHz[@Neugebauer2013]. If the temperature (tan $\mathrm{\delta}$) increases to 30 ${}^{\circ}$C (1.2), the skin depth is expected to decrease down to 85 $\mu$m that is smaller than the sample thickness, most of the power is absorbed above the ground plane. This short skin depth causes the $\mathrm{B_1}$ maximum to occur underneath the grid, and also to reduce its strength to 1.9 G, as shown in Fig. \[simulation\] (d).
On the contrary, $\mathrm{E_1}$ is constantly minimized at the ground plane in both tan $\mathrm{\delta}$, as shown in Fig. \[simulation\] (a) and (b). The $\mathrm{B_1}$ for tan $\mathrm{\delta}$ = 1.2 in most of the sample region is lower than that for tan $\mathrm{\delta}$ = 0.8, which implies that a larger power is required at higher temperature in order to saturate electron spins. Therefore, a loose NMR coil and circularly polarized MW were used for ${}^1$H DNP in this experiment in order to obtain saturation. The DNP frequency was set to the center of the electron paramagnetic resonance (EPR) spectrum of ${}^{14}$N-TEMPOL, which was obtained using our own high field EPR spectrometer[@Caspers2016].
We turned on MW for 1 sec and turned off MW for 5 sec to cool the sample. Figure \[DNPspectrum\] (a) shows the ${}^1$H Overhauser DNP spectrum and the NMR spectrum without MW. The negative enhancement indicates dipolar relaxation between electrons and ${}^1$H, consistent with previous results. The enhancement $\epsilon$ was calculated as $\epsilon = I/I_0 -1$, where $I_0$ and *I* are the integrated areas of the DNP and NMR signals. The enhancement is estimated to be about -14, which is similar to the previous results that Prisner *et al.* obtained at 40 ${}^{\circ}$C[@Neugebauer2013]. The large broadening about 10 ppm in the spectra comes from the inhomogeneity of the NMR magnet of $\sim$ 2 ppm/$\mathrm{mm}^2$. We were unable to use ${}^1$H NMR thermometry in order to estimate the sample temperature after MW irradiation because the ${}^1$H NMR frequency shift as a function of temperature (-0.012 ppm/${}^{\circ}$C)[@Gafurov2012] is much smaller than our NMR magnet inhomogeneity.
However, the enhancement found, similar to that at 40 ${}^{\circ}$C in ref.[@Neugebauer2013], implies a similar sample temperature after MW irradiation because we can assume that the electron spins were almost fully saturated in our case also. Our estimate of the temperature increase is also supported by the NMR thermometry experiments described below. The DNP signal intensity remains constant between experimental cycles, which also indicates that no sample loss occurs due to boiling.\
**Liquid state ${}^{31}$P DNP at 9 T in fluorobenzene solution.** We also performed liquid state ${}^{31}$P DNP with a polar solution of triphenylphosphine ($\mathrm{Ph_3P}$) dissolved in fluorobenzene ($\mathrm{C_6H_5F}$) with 80 mM BDPA ($\alpha$,$\gamma$- bisdiphenylene-$\beta$-phenylallyl). The $\epsilon^{''}$ for $\mathrm{C_6H_5F}$ at 3 GHz was estimated to be 1.5 in a previous report[@Poley1955], while that for water is about 29 at the same frequency. BDPA was reported to have a longer $T_{1e}$ than TEMPOL, so a smaller microwave power is needed. Griffin *et al.* performed liquid state ${}^{31}$P DNP at 5 T with $\mathrm{Ph_3P}$ dissolved in a non-polar solvent of benzene with BDPA, and obtained a high enhancement of about 180 using 0.5 W of MW power[@Loening2002]. The solvent used here ($\mathrm{C_6H_5F}$) is polar, and much larger dielectric heating is expected than in the benzene solutions that Griffin *et al.* used[@Loening2002]. Figure \[DNPspectrum\] (b) shows the DNP and NMR spectra obtained by the planar probe with 17 W and a sample volume of 10 $\mu$L. MW was irradiated at the center of the EPR spectrum of BDPA. A positive enhancement was observed, which indicates scalar relaxation consistent with ref[@Loening2002]. Although smaller enhancement was expected since we are in a higher field (9 T vs. 5 T), we obtained a similar enhancement of a factor of about 200, which shows effective scalar relaxation even at 9.3 T. Liquid state ${}^{31}$P DNP was also demonstrated in a standard solenoid coil with 2 mm inner diameter. While the planar probe allows high power MW irradiation without severe sample heating, the DNP signal with the solenoid coil had to be obtained with MW powers of less than 8 W in order to prevent sample loss due to boiling, as depicted in Fig. \[enhancement\_power\]. The enhancement in the solenoid coil increases up to about 50 at 4.5 W and begins to decrease with further power due to large sample loss. On the contrary, the enhancement obtained by the planar probe initially increases rapidly as the power increases, and stays constant around 200 beyond the saturation power of 8 W. The sample in the solenoid coil has smaller enhancement than that in the planar probe because only the fraction of the sample located within the skin depth can be directly enhanced. A solenoid coil with smaller diameter showed smaller enhancement than the solenoid coil used in this study because it had smaller heat capacity, causing even more severe sample loss.\
**Measurement of sample temperature by NMR thermometer.** To quantify the decrease in dielectric heating in the planar probe, we performed NMR thermometry using potassium hexacyanocobaltate ($\mathrm{K_3[Co(CN_6)]}$) dissolved in water, as the ${}^{59}$Co chemical shift has been reported as having a large dependence on temperature($\approx$1.504 ppm/${}^{\circ}$C)[@Levy1980]. We obtained ${}^{59}$Co NMR spectra in the solenoid coil and the planar probe with increasing MW irradiation time from 1 sec to 5 sec. Figure \[Co\_spectrum\] (a) shows the ${}^{59}$Co NMR spectra obtained in the solenoid coil as a function of irradiation time with 17 W of MW power. The spectrum without MW in the solenoid coil has a broadening of about 10 ppm due to the magnetic field inhomogeneity of the NMR magnet used in this study, while that in the planar probe has a larger inhomogeneous broadening of about 25 ppm due to the larger sample area. The sample volume in the solenoid coil was about 40 $\mu$L. The spectrum is shifted by about 15 ppm at an irradiation time of 1 sec. As the irradiation time increases, the spectrum not only shows a larger shift, but also becomes broadened and distorted due to temperature gradients in the sample. The spectrum at an irradiation time of 5 sec is shifted to about 110 ppm, and becomes remarkably broadened, which indicates a severe sample heating and an inhomogeneous temperature in the sample. We found that more than half of the sample had evaporated after the experiment. On the contrary, the planar probe shows much smaller shifts and negligible broadening increase with irradiation time, as depicted in Fig. \[Co\_spectrum\] (b). A maximum shift of about 23 ppm is observed after an irradiation time of 5 sec without a significant broadening or distortion in the spectrum. This implies that dielectric heating is kept under control in the planar probe and that the temperature gradient remains negligible.
‘
![The maximum temperature increases in the planar probe and in the solenoid coil.[]{data-label="maximum_temperature"}](maximum_temperature.pdf){width="53.00000%"}
We also tried a much higher MW power of 100 W. This MW power is strong enough to bring a frozen solution in a normal solenoid coil at 20 K to the liquid state at room temperature within 1.5 sec[@Yoon2016142]. Starting from a sample at room temperature, a large sample loss would occur in the solenoid coil due to evaporation even with a small irradiation time, so this experiment was performed only with the planar probe, as shown in Fig. \[Co\_spectrum\] (c). The spectra show larger shifts compared to those in Fig. \[Co\_spectrum\] (b), and become more distorted because the temperature in the sample becomes inhomogeneous. However, the spectrum is shifted only to about 45 ppm ($\simeq$ 30${}^{\circ}$C) even with an irradiation time of 5 sec, showing an efficient suppression of dielectric heating. No sample loss was observed even with an irradiation time of 5 sec.
Since a large temperature gradient occurs in the solenoid coil, we compared only maximum temperature increases in the planar probe and in the solenoid coil, as displayed in Fig. \[maximum\_temperature\]. The maximum temperature increases $\mathrm{\triangle T_{max}}$ are estimated by the positions at half maximum of the peak height. For the solenoid coil, some part of the sample were estimated to reach temperatures as high as 100 ${}^{\circ}$C, which is consistent with the large sample loss observed after the experiment. On the other hand, the temperature of the sample in the planar probe is found to increase by about 20 ${}^{\circ}$C with 17 W of MW power. Even at 100 W, the planar probe shows an increase of only about 25 ${}^{\circ}$C for an irradiation time of 1 sec. We expect a roughly similar increase in the sample temperature after MW irradiation of 70 W in the liquid state ${}^1$H DNP.
Discussion
==========
In this study, we showed a new methodology for increasing the sample volume for liquid state DNP at 9 T from nL to $\mu$L by using a high power gyrotron and a planar probe. The increase in sample volume has several benefits; first, easy sample handling (injection into the planar probe, or dropping on the Cu support), second, negligible interference from background signals, and third, ${}^1$H NMR signal intensity large enough for a single scan. Even though sufficient MW power was produced by the gyrotron, the ${}^1$H enhancement was limited by the small coupling factor. The enhancement could be improved by using radicals with narrower EPR lines such as ${}^{15}$N-TEMPOL and Fremy’s Salt[@Prandolini2009; @Gafurov2012] or using supercritical fluids, whose correlation time for molecular motion would be much shorter than that of water[@VanBentum2016; @Wang2015]. Dielectric heating in the planar probe can be further reduced by using thinner samples while still holding $\mu$L of liquid. This new method can be extended to higher or lower magnetic fields by modifying the thickness of the sample region and the grid.
Methods
=======
**Sample preparation** The sample for liquid state ${}^1$H DNP was prepared by dissolving 80 mM ${}^{14}$N-TEMPOL (Sigma Aldrich) in de-ionized water without degassing. The prepared sample of about 2 $\mu$L was dropped onto the center of the Cu support and closed by a silica cover. The sample for liquid state ${}^{31}$P DNP was prepared in the following manner: first, $\mathrm{N_2}$ was bubbled in fluorobenzene in a glove box, second, a 2 M $\mathrm{Ph_3P}$ solution was dissolved in the $\mathrm{N_2}$ bubbled-fluorobenzene, and third, 80 mM BDPA (Sigma Aldrich) was dissolved in the solution prepared at the preceded step. 1M $\mathrm{K_3[Co(CN_6)]}$ dissolved in de-ionized water was used for the ${}^{59}$Co NMR thermometry experiments.\
[50]{}
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We gratefully acknowledge financial support by the Swiss National Science Foundation, Requip (No. 206021-121303/1), FN($\mathrm{200021{-}153230}$), and CTI-Project no. 15617.1 PFNM-NM.
Author contributions
====================
D.Y. carried out the DNP experiments, A.D. and E.D.R. contributed with simulations, C.C. and M.S. with high field EPR measurements, J.G. and S. A. implemented the gyrotron settings, D.Y. and M.S. prepared the samples, D.Y. and J-Ph. A. designed the experiment.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In a previous paper Crater and Van Alstine applied the Two Body Dirac equations of constraint dynamics to quark-antiquark bound states using a relativistic extention of the Adler-Piran potential and compared their spectral results to those from other approaches which also considered meson spectroscopy as a whole and not in parts. In this paper we explore in more detail the differences and similarities in an important subset of those approaches, the quasipotential approach. In the earlier paper, the transformation properties of the quark-antiquark potentials were limited to a scalar and an electromagnetic-like four vector, with the former accounting for the confining aspects of the overall potential, and the latter the short range portion. The static Adler-Piran potential was first given an invariant form and then apportioned between those two different types of potentials. Here we make a change in this apportionment that leads to a substantial improvement in the resultant spectroscopy by including a time-like confining vector potential over and above the scalar confining one and the electromagnetic-like vector potential. Our fit includes 19 more mesons than the earlier results and we modify the scalar portion of the potential in such a way that allows this formalism to account for the isoscalar mesons $\eta $ and$~\eta ^{\prime }$ not included in the previous work. Continuing the comparisons of formalisms and spectral results made in the previous paper with other approaches to meson spectroscopy we examine in this paper the quasipotential approach of Ebert, Faustov, and Galkin.'
author:
- |
Horace W. Crater[^1] and James Schiermeyer\
The University of Tennessee Space Institute
title: 'Applications of Two-Body Dirac Equations to the Meson Spectrum with Three Versus Two Covariant Interactions, SU(3) Mixing, and Comparison to a Quasipotential Approach'
---
Introduction
============
There are a number of strategies in computational treatments of quantum chromodynamics that emerge in the study of meson spectroscopy. One is to set up a discrete lattice analog of the full quantum field theory. A second is to first make analytic approximations which replace the quantum field theoretic problem by a classical variational problem involving an effective Lagrange function and action. The latter approach has been exploited by Adler and Piran [@adl] and in a previous paper Crater and Van Alstine gave a detailed account of applications of the Two Body Dirac equations (TBDE) of constraint dynamics to the meson quark-antiquark bound states [@crater2] using a relativistic extension of the Adler-Piran potential. That paper also included a comparison of this approach to others [@wisc]-[@isgr] who, like ours, considered the whole spectrum instead of just selected parts.
Here we update the results presented in [@crater2] in four ways. First, we include 19 more mesons not included in the previous work. Second, we obtain a substantial improvement in our fit to most all of the mesons by allowing the confining interaction, pure scalar in [@crater2], to take on a timelike vector portion. We still include the electromagnetic-like vector potential used previously. Third, we extend the relativistic Schrödinger-like form of the TBDE to include isoscalar mixing, thus incorporating the isoscalar mesons $\eta $ and $~\eta ^{\prime }$. And finally we critically examine, by comparison with the TBDE, aspects of quasipotential approaches including a recent one presented in [@rusger] as well as in [@isgr].
In Sec. 2 we give a short review of the relativistic two-body constraint formalism, distinguishing between our new approach used for confining given in this paper and the one presented in [@crater2] and including a discussion of the closely related quasipotential approach. In Sec. 3 we review the static Adler-Piran potential and how we apportion it between the three invariant potential functions $A(r),\ S(r),$ and $V(r)$ used in our TBDE. In Sec. 4 we present our main new results on meson spectroscopy. In Sec. 5 we include our treatment of SU(3) mixing and in Sec. 6 we discuss the meson spectral results of [@rusger] including the advantages and shortcomings of their quasipotentials bound state formalism.
Review of Relativistic Two-Body Formalisms
==========================================
Two-Body Constraint Approach
----------------------------
When the interaction and the masses are known, a common starting point in describing the relativistic two-body bound state problem is the Bethe-Salpeter equation [@bse]. The Bethe-Salpeter equation is, however, usually not considered in its full four-dimensional form due to the difficulty of treating the relative time coordinate [@nak]. Numerous truncations of the Bethe-Salpeter equation have been proposed for the relativistic two-body problem [@quasi; @yaes]. Some of these types of approximate methods have previously been applied with considerable success to the $q\bar{q}$ meson spectrum [@prl84]-[@lu], [@crater2]. The ladder approximation and the instantaneous approximation of the Salpeter equation have been widely used. It should be noted, however, that the simple ladder approximation and the Salpeter equation do not lead to the correct one-body limit [@oneb], and do not respect gauge invariance [@gauge]. Crossed-ladder diagrams must be included to insure gauge-invariant scattering amplitudes.
The Two Body Dirac equations of constraint dynamics provide a covariant three-dimensional truncation of the Bethe-Salpeter equation. Sazdjian [saz]{} has shown that the Bethe-Salpeter equation can be algebraically transformed into two independent equations. The first yields a covariant three-dimensional eigenvalue equation which for spinless particles takes the form $$\biggl(\mathcal{H}_{10}+\mathcal{H}_{20}+2\Phi _{w}\biggr )\Psi
(x_{1},x_{2})=0, \label{eq:sum}$$where $\mathcal{H}_{i0}=p_{i}^{2}+m_{i}^{2}$ . The quasipotential $\Phi _{w}$ is a modified geometric series in the Bethe-Salpeter kernel $K$ such that in lowest order in $K$ $$\Phi _{w}=\pi iw\delta (P\cdot p)K, \label{bsen}$$where $P=p_{1}+p_{2}$ is the total momentum, $p=\eta _{2}p_{1}-\eta _{1}p_{2}
$ is the relative momentum, $w$ is the invariant total center of momentum (c.m.) energy with $P^{2}=-w^{2}$. $\ $The $\eta _{i}$ must be chosen so that the relative coordinate $x=x_{1}-x_{2}$ and $p$ are canonically conjugate, i.e. $\eta _{1}+\eta _{2}=1$. The second equation overcomes the difficulty of treating the relative time in the center of momentum system by setting an invariant condition on the relative momentum $p$, $$(\mathcal{H}_{10}-\mathcal{H}_{20})\Psi (x_{1},x_{2})=0=2P\cdot p\Psi
(x_{1},x_{2}). \label{eq:dif}$$Note that this implies $p^{\mu }\Psi =p_{\perp }^{\mu }\Psi \equiv (\eta
^{\mu \nu }+\hat{P}^{\mu }\hat{P}^{\nu })p_{\nu }\Psi $ in which $\hat{P}^{\mu }=P^{\mu }/w$ is a time like unit vector $(\hat{P}^{2}=-1)$ in the direction of the total momentum.
One can further combine the sum and the difference of Eqs. (\[eq:sum\]) and (\[eq:dif\]) to obtain a set of two relativistic equations one for each particle with each equation specifying two generalized mass-shell constraints $$\mathcal{H}_{i}\Psi (x_{1},x_{2})=(p_{i}^{2}+m_{i}^{2}+\Phi _{w})\Psi
(x_{1},x_{2})=0,~i=1,2, \label{dir}$$including the interaction with the other particle. These constraint equations are just those of Dirac’s Hamiltonian constraint dynamics[^2] [dirac,cnstr]{}. In order for the two simultaneous wave equations of (\[dir\]) to have solutions other than zero, Dirac’s constraint dynamics stipulate that these two constraints must be compatible among themselves, $[\mathcal{H}_{1},\mathcal{H}_{2}]\Psi =0$, that is, they must be first class. With no external potentials, the coordinate dependence of the quasipotential $\Phi
_{w}$ $\ $would be through $x$ and the compatibility condition becomes $[p_{1}^{2}-p_{2}^{2},\Phi _{w}]\Psi =P^{\mu }\partial \Phi _{w}/\partial
x^{\mu }=0$. In order for this to be true in general, $\Phi _{w}$ must depend on the relative coordinate $x$ only through its component, $x_{\perp
},$ perpendicular to $P,$$$x_{\perp }^{\mu }=(\eta ^{\mu \nu }+\hat{P}^{\mu }\hat{P}^{\nu
})(x_{1}-x_{2})_{\nu }. \label{ti}$$Since the total momentum is conserved, the single component wave function $\Psi ~$in coordinate space is a product of a plane wave eigenstate of $P$ and an internal part $\psi $ [@cra87], depending on this $x_{\perp }.$[^3]
We find a plausible structure for the quasipotential $\Phi _{w}$ by observing that the one-body Klein-Gordon equation $(p^{2}+m^{2})\psi =(\mathbf{p}^{2}-\varepsilon ^{2}+m^{2})\psi =0$ takes the form $(\mathbf{p}^{2}-\varepsilon ^{2}+m^{2}+2mS+S^{2}+2\varepsilon A-A^{2})\psi =0~$when one introduces a scalar interaction and timelike vector interaction via $m\rightarrow m+S~$and $\varepsilon \rightarrow \varepsilon -A$. In the two-body case, separate classical [@fw] and quantum field theory [saz97]{} arguments show that when one includes world scalar and vector interactions then $\Phi _{w}$ depends on two underlying invariant functions $S(r)$ and $A(r)$ through the two-body Klein-Gordon-like potential form with the same general structure, that is$$\Phi _{w}=2m_{w}S+S^{2}+2\varepsilon _{w}A-A^{2}. \label{em}$$Those field theories further yield the c.m. energy dependent forms $$m_{w}=m_{1}m_{2}/w, \label{mw}$$and$$\varepsilon _{w}=(w^{2}-m_{1}^{2}-m_{2}^{2})/2w, \label{ew}$$ones that Tododov [@cnstr] introduced as the relativistic reduced mass and effective particle energy for the two-body meson system. Similar to what happens in the nonrelativistic two-body problem, in the relativistic case we have the motion of this effective particle taking place as if it were in an external field (here generated by $S$ and $A$). The two kinematical variables (\[mw\]) and (\[ew\]) are related to one another by the Einstein condition $$\varepsilon _{w}^{2}-m_{w}^{2}=b^{2}(w),$$where the invariant $$b^{2}(w)\equiv
(w^{4}-2w^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2})/4w^{2},
\label{bb}$$is the c.m. value of the square of the relative momentum expressed as a function of $w$. One also has$$b^{2}(w)=\varepsilon _{1}^{2}-m_{1}^{2}=\varepsilon _{2}^{2}-m_{2}^{2},$$in which $\varepsilon _{1}$ and $\varepsilon _{2}$ are the invariant c.m. energies of the individual particles satisfying$$\ \varepsilon _{1}+\varepsilon _{2}=w,\ \varepsilon _{1}-\varepsilon
_{2}=(m_{1}^{2}-m_{2}^{2})/w. \label{es}$$In terms of these invariants, the relative momentum appearing in Eq. ([bsen]{}) and (\[eq:dif\]) is given by$$p^{\mu }=(\varepsilon _{2}p_{1}^{\mu }-\varepsilon _{1}p_{2}^{\mu })/w\mathrm{,} \label{relm}$$so that $\eta _{1}+\eta _{2}=(\varepsilon _{1}+\varepsilon _{2})/w=1$. In [@tod] the forms for these two-body and effective particle variables are given sound justifications based solely on relativistic kinematics, supplementing the dynamical arguments of [@fw] and [@saz97].
Originally, the Two Body Dirac equations of constraint dynamics arose from a supersymmetric treatment of two pseudoclassical constraints (with Grassmann variables in place of gamma matrices) which were then quantized [@cra82]-[@cww]. Sazdjian later derived [@saz] different forms of these same equations, just as with their spinless counterparts above, as covariant and three-dimensional truncation of the Bethe-Salpeter equation. The forms of the equations are varied (see Appendix A), but the one that is the most familiar is the “external potential” form similar in structure to the ordinary Dirac equation. For two particles interacting through world scalar and vector interactions they are $$\begin{aligned}
\mathcal{S}_{1}\psi & \equiv \gamma _{51}(\gamma _{1}\cdot (p_{1}-\tilde{A}_{1})+m_{1}+\tilde{S}_{1})\Psi =0, \notag \\
\mathcal{S}_{2}\psi & \equiv \gamma _{52}(\gamma _{2}\cdot (p_{2}-\tilde{A}_{2})+m_{2}+\tilde{S}_{2})\Psi =0. \label{tbde}\end{aligned}$$Here $\Psi $ is a 16 component wave function consisting of an external plane wave part that is an eigenstate of $P$ and an internal part $\psi =\psi
(x_{\perp })$. The vector potential$\ \tilde{A}_{i}^{\mu }$ was taken to be an electromagnetic-like four-vector potential with the time and spacelike portions both arising from a single invariant function $A.~$[^4] The tilde on these four-vector potentials as well as on the scalar ones $\tilde{S}_{i}$ indicate that they are not only position dependent but also spin-dependent by way of the gamma matrices. In this paper we allow for the presence of a timelike portion arising from an independent invariant function $V.~$[^5] In either case, the operators $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ must commute or at the very least $[\mathcal{S}_{1},\mathcal{S}_{2}]\psi =0$ since they operate on the same wave function. [^6] This compatibility condition gives restrictions on the spin dependence which the vector and scalar potentials$$\tilde{A}_{i}^{\mu }=\tilde{A}_{i}^{\mu }(A(r),V(r),p_{\perp },\hat{P},w,\gamma _{1},\gamma _{2}),~\ \tilde{S}_{i}=\tilde{S}_{i}(S(r),A(r),p_{\perp },\hat{P},w,\gamma _{1},\gamma _{2}). \label{paul1}$$are allowed to have [^7] in addition to requiring that they depend on the invariant separation $r\equiv \sqrt{x_{\perp }^{2}}$ through the invariants $A(r),V(r),$ and $S(r)$ . The covariant constraint (\[eq:dif\]) can also be shown to follow from Eq. (\[tbde\]). We give the explicit connections between $\tilde{A}_{i}^{\mu },\tilde{S}_{i}$ and the invariants $A(r),V(r),$ and $S(r)$ in Appendix A. The Pauli reduction of these coupled Dirac equations lead to a covariant Schrödinger-like equation for the relative motion with an explicit spin-dependent potential $\Phi _{w},$[^8] $${\bigg(}p_{\perp }^{2}+\Phi _{w}(A(r),V(r),S(r),p_{\perp },\hat{P},w,\sigma
_{1},\sigma _{2}){\bigg)}\psi _{+}=b^{2}(w)\psi _{+}\ , \label{schlike}$$with $b^{2}(w)$ playing the role of the eigenvalue.[^9] This eigenvalue equation can then be solved for the four-component effective particle spinor wave function $\psi _{+}$ related to the 16 component spinor $\psi (x_{\perp })$ in Appendix A.
The set of equations (\[tbde\]) and the equivalent Schrödinger-like equation (\[schlike\]) possess a number of important and desirable features. First, they reduce to the correct one-body Dirac form when one of the two constituents becomes very massive. (The Salpeter equation does not have this important property.) Second, the generalized three-dimensional Schrödinger equation (\[schlike\]) is quite similar to the nonrelativistic Schrödinger equation and it indeed goes over to the correct nonrelativistic Schrödinger equation in the limit of weak binding. One can thus employ familiar techniques to obtain its solutions. Third, Eq. ([schlike]{}) can be solved nonperturbatively for both QED bound states (e.g. positronium and muonium) and QCD bound states (i.e. bound states obtained from two-body relativistic potential models for mesons) since every term in $\Phi _{w}$ is nonsingular in the sense that they are less attractive than $-1/4r^{2}$ (no delta functions or attractive $1/r^{3}$ potentials for example). Thus, unlike with the $1/m$ non-relativistic and semirelativistic expansions, the covariant Dirac formalism itself introduces natural cutoff factors that smooth out singular spin-dependent interactions, there being no need to introduce them by hand (- see [@cra82],[@cra84],[@becker], [@yoon] and Sec. 6.4) as in other approaches. Fourth, the relativistic potentials appearing in these equations are directly related through Eq. (\[bsen\]) to the interactions of perturbative quantum field theory, while for QCD bound states they may be introduced semiphenomenolgically through $A(r)$ and $S(r)$ (and in this paper $V(r)$). Fifth, these equations have been tested analytically [@exct] and numerically [@becker] against the known perturbative fine and hyperfine structures of QED bound states and related field theoretic bound states. The (nonperturbative) successes with these QED bound states provide strong motivation for applying the constrained Dirac formalism to meson bound states as in [@crater2]. Sixth, these equations provide a covariant three-dimensional framework in which the local potential approximation consistently fulfills the requirements of gauge invariance in QED [@saz96]. Finally, the same general structures of the Darwin, spin-orbit, spin-spin and tensor terms in $\Phi _{w}(A,V,S)$ of Eq. (\[schlike\]) and (\[57\]), responsible for the accurate hyperfine structures of QED bound states arising naturally from the TBDE formalism when $A=-\alpha /r$ and $S=V=0$, are used with the only alteration in its application to the QCD bound states being that $A,V,S$ are apportioned appropriately from the Adler-Piran potential.
We emphasize the importance of the nonperturbative QED bound state test. (See Appendix C for a review of the application of the TBDE to QED.) Many of the wave equations used in the standard approaches to QED bound states have been modified to include QCD inspired potentials and then applied to nonperturbative numerical calculations of QCD bound states without first testing those approaches nonperturbatively in QED. By this we mean that the accepted perturbative results of those equations (QED spectral results correct through order $\alpha ^{4})$ have not been replicated using numerical methods. Sommerer $et~al.$ [@iowa] have shown that the Blankenbecler-Sugar equation and the Gross equations fail this test. This indicates a danger in applying such three-dimensional truncations of the Bethe-Salpeter equation: if failure occurs in their applications to QED bound states this brings into question the spectral results of similar nonperturbative (i.e. numerical) approaches based on the same truncations when applied to QCD bound states. This would be true especially when the only difference between the vector portions of the QED and QCD potentials would be the replacing of the QED $-\alpha /r$ by a similar $A(r)$ from QCD.
In [@crater2] we presented details of the application of this formalism to meson spectroscopy using a covariant version of the Adler-Piran static quark potential. Note especially that the equations used there displayed a single $\Phi (A(r),S(r),p_{\perp },\hat{P},w,\sigma _{1},\sigma _{2},)$ in Eq. (\[schlike\]). It depends on the quark masses through factors such as those that appear in Eq. (\[em\]). However its dependence is the same for all quark mass ratios - hence a single structure for all the $Q\bar{Q},\ q\bar{Q},$ and $q\bar{q}$ mesons in a single overall fit. We found that the fit provided by the TBDE for the entire meson spectrum (from the pion to the excited bottomonium states) competes with the best fits to partial spectra provided by other approaches and does so with the smallest number of interaction functions (just $A(r)$ and $S(r)$) without additional cutoff parameters necessary to make those approaches numerically tractable. We also found that the pion bound state displays some characteristics of a Goldstone boson. That is, as the quark mass tends to zero, the pion mass (unlike the $\rho $ and the excited $\pi $) vanishes, in contrast to almost every other relativistic potential model. (For more discussion on this see footnote 23 below).
Two Body Quasipotential Approaches
----------------------------------
Also presented in [@crater2] was a detailed comparison between the meson spectroscopy results of our model and those of several other approaches: one based on the Breit equation [@bry], two on truncated versions of the Bethe-Salpeter equation [@wisc], [@iowa], and one on a quasipotential approach [@isgr]. We explore in this section in more detail the differences and similarities between our approach and the quasipotential approach. The quasipotential equation was first introduced by Logunov and Tavkhelidze [@quasi]. In its homogeneous form, that equation describes a two-particle relativistic composite system with its c.m. momentum space form (in the notation used here $\mathbf{p}$ is the relative momentum given in Eq. (\[relm\])) for spinless particles given by$$(w-\sqrt{\mathbf{p}^{2}+m_{1}^{2}}-\sqrt{\mathbf{p}^{2}+m_{2}^{2}})\Psi _{w}(\mathbf{p)=}\int V(\mathbf{p,q,}w)\Psi _{w}(\mathbf{q)}\frac{d^{3}q}{(2\pi
)^{3}}, \label{log}$$where $\Psi _{w}(\mathbf{p)}$ is the quasipotential wave function projected onto positive-frequency states and $V(\mathbf{p,q,}w)$ is the quasipotential calculated by means of the off-energy shell scattering amplitude (so that the respective c.m. energies of the two particles are not given by the above square roots but by Eq. (\[es\])). The corresponding inhomogeneous quasipotential equation is of the general form$$T(\mathbf{p,q,}w)+V(\mathbf{p,q,}w)+\int V(\mathbf{p,k,}w)G_{w}(\mathbf{k)}V(\mathbf{k,q,}w)=0, \label{quasi}$$a linear integral equation of the Lippmann-Schwinger type relating the quasipotential to the off-energy shell extrapolation of the Feynman scattering amplitude. The choice of this equation and the accompanying homogeneous equation is not unique [@yaes]. For example, the Green function $G_{w}(\mathbf{k)}$ has only its imaginary part determined by requiring the condition of elastic unitarity on Eq. (\[quasi\]). [^10] Todorov [@quasi] took advantage of this nonuniqueness to write down a local version of the corresponding homogeneous equation of the form$$\left( \mathbf{p}^{2}-b^{2}\right) \phi (\mathbf{p)+}\frac{2\varepsilon
_{1}\varepsilon _{2}}{w}\int V_{w}(\mathbf{p,k)}\phi (\mathbf{k)}\frac{d^{3}k}{(2\pi )^{3}}=0, \label{tod0}$$with$~\phi (\mathbf{p})$ the wave function in momentum space. In [@cra84] Crater and Van Alstine showed that the spinless version of Eq. (\[schlike\]) in the case of QED ($V=S=0$) has, in the c.m. frame, the form (see Eq. (66b) and discussion below Eq. (73b) of that paper) $$(\mathbf{p}^{2}-(\varepsilon _{w}-A)^{2}+m_{w}^{2}+\frac{1}{2}\mathbf{\nabla
}^{2}\mathcal{G}+\frac{1}{4}\left( \mathbf{\nabla }\mathcal{G}\right)
^{2})\psi =0, \label{cra84}$$where$$\begin{aligned}
\mathcal{G} &\mathcal{=}&\ln G, \notag \\
G &=&\frac{1}{(1-2A/w)^{1/2}}.\end{aligned}$$As was pointed out in that paper, for $A=-\alpha /r,$ this reduces for weak potentials to the minimal or gauge structure form postulated by Todorov, $$\lbrack (\mathbf{p-A)}^{2}-(\varepsilon _{w}-A^{0})^{2}+m_{w}^{2}]\psi =0.
\label{gage}$$Although not noticed at the time, Eq. (\[cra84\]) does in fact additionally have this minimal structure not only for arbitrary strength couplings but also for potentials not restricted to Coulomb potentials, provided just that$$\begin{aligned}
A^{0} &=&A, \notag \\
\mathbf{A} &\mathbf{=}&\mathbf{-}\frac{i}{2}\mathbf{\nabla }\mathcal{G~}I_{s},\end{aligned}$$where $I_{s}$ is the space reflection operator satisfying$$I_{s}f(\mathbf{r)=}f(-\mathbf{r).}$$
Aneva, Karchev, and Rizov [@akr] developed the weak potential version of this two-body Klein-Gordon equation for two spin combinations: for one spin-zero and one spin-one half particle and for a spin-one-half particle-antiparticle pair. For the latter it has the form, $$\left( \mathbf{p}^{2}-b^{2}\right) \phi _{\lambda _{1}\lambda _{2}}(\mathbf{p)+}\frac{2\varepsilon _{1}\varepsilon _{2}}{w}\int \bar{u}_{\lambda
_{1}^{\prime }}(\mathbf{p)}\bar{v}_{\lambda _{2}}(-\mathbf{k)}\mathbb{V}_{w}(\mathbf{p,k)}v_{\lambda _{2}^{\prime }}(-\mathbf{p)}u_{\lambda _{1}}(\mathbf{k)}\phi _{\lambda _{1}^{\prime }\lambda _{2}^{\prime }}(\mathbf{k)}\frac{d^{3}k}{(2\pi )^{3}}=0. \label{riz}$$Expressing the on shell free four-component Dirac spinors in terms of two-component Pauli spinors and assuming the local quasipotentials $$\mathbb{V}_{w}(\mathbf{p,k)=}\mathbb{V}_{w}(\mathbf{p-k)=}\mathcal{A}(\mathbf{p-k})\gamma _{1}^{\mu }\gamma _{2\mu }+\mathcal{V}(\mathbf{p-k})\beta _{1}\beta _{2}+\mathcal{S}(\mathbf{p-k})1_{1}1_{2}, \label{qua}$$Eq. (\[riz\]) can be brought to a four-component wave equation form superficially similar to Eq. (\[schlike\]) in the c.m. frame. We write it as$${\bigg(}\mathbf{p}^{2}+\mathcal{V}_{w}(A(r),V(r),S(r),\mathbf{p},w,\mathbf{\sigma }_{1}\mathbf{,\sigma }_{2}){\bigg)}\phi (\mathbf{r)}=b^{2}(w)\phi (\mathbf{r).} \label{vw}$$However, there are distinct differences. First of all, the spin structure of $\Phi _{w}$ is not identical to that of $\mathcal{V}_{w}$ even if the functions $A(r),V(r),S(r)$ are the same. The reason is that the spin dependence of the vector and scalar potentials $\tilde{A}_{i}^{\mu }~$ and $\ \tilde{S}_{i}$ and in particular the minimal type of context in which they appear in Eqs. (\[tbde\]) arise from (nonlinear) hyperbolic functions (see Appendix A and [@jmath], [@long]) of matrices such as appear in Eq. (\[qua\]). Without that hyperbolic structure the external potential forms in which the minimal structures appear would be absent. To reproduce the effects of those nonlinear functions in Eq. (\[vw\]) one would have to supplement Eq. (\[qua\]) with types of invariants other than just scalar and vector [@saz86], [@saz94], [@saz97] (a pseudovector invariant for example). Secondly, the desirable minimal scalar structures as appear in the first line on the right hand side of Eq. (\[57\]) below would not appear in $\mathcal{V}_{w}$ without including higher order diagrams that would again require invariants other than just scalar and vector. These minimal scalar structures are not only desirable, they arise naturally and strictly from $O(1/c^{2})$ expansions of classical and quantum field theoretic potentials [@fw], [@saz97] and from gauge invariance considerations (see Todorov in [@quasi], and also [@akr], [crstr]{}, and [@cra87]). In a later section we discuss the recent work of [@rusger], which uses a quasipotential equation similar to Eq. (\[vw\]) in meson spectroscopy calculations.
The Adler-Piran Potential for the Two Body Dirac Equations.
===========================================================
In this section we use the relativistic Schrödinger-like Eq.([schlike]{}) to construct a relativistic naive quark model by choosing the three invariant functions $A,V$ and $S$ to incorporate the Adler-Piran static quark potential [@adl]. This potential was originally obtained from the QCD field theory through a nonlinear effective action model for heavy quark statics. Adler and Piran used the renormalization group approximation to obtain both total flux confinement and a linear static potential at large distances. Their model uses nonlinear electrostatics with displacement and electric fields related through a nonlinear constitutive equation with the effective dielectric constant given by a leading log-log model which fixes all parameters in their model apart from a mass scale $\Lambda .$ Their static potential also contains an unknown “integration constant” $U_{0}$ in the final form of their potential (hereafter called $V_{AP}(r)$). We insert into Eq.(\[schlike\]) invariants $A,V$ and $S$ with forms determined so that the sum $A+V+S$ appearing as the potential in the nonrelativistic limit of our equations becomes the Adler-Piran nonrelativistic $Q\bar{Q}$ potential (which depends on two parameters $\Lambda $ and $U_{0})$ plus the Coulomb interaction between the quark and antiquark. That is, $$V_{AP}(r)+V_{coul}=\Lambda (U(\Lambda r)+U_{0})+\frac{e_{1}e_{2}}{r}=A+V+S\ .
\label{asap}$$As determined by Adler and Piran, the short and long distance behaviors of $U(\Lambda r)$ generate known lattice and continuum results through the explicit appearance of an effective running coupling constant in coordinate space. That is, the Adler-Piran potential incorporates asymptotic freedom through $$\Lambda U(\Lambda r<<1)\sim 1/(r\ln \Lambda r),$$and linear confinement through $$\Lambda U(\Lambda r>>1)\sim \Lambda ^{2}r.$$In addition to obtaining these leading behavior analytic forms for short and long distances, they converted the numerically obtained values of the potential at all distances (short, intermediate, and long distances) to compact analytic expressions. The explicit closed form expressions [@adl] for $U(\Lambda r)$ are different for each of the four regions, $\ $and are linked continuously. Letting $x=\Lambda r,$$$\begin{aligned}
U(x) &=&-(16\pi /27)(1+a_{1}x^{a_{2}})f(w_{p})/w_{p},~~~~~0<x\leq 0.0125,
\notag \\
w_{p} &=&1/(a_{3}x)^{2}, \notag \\
U(x) &=&K+\alpha (x/0.125)^{E},~~~~~0.0125\leq x\leq 0.125, \notag \\
E &=&\beta +\gamma \ln (1/x)+\delta (\ln (1/x))^{2}+\varepsilon (\ln
(1.x))^{3} \notag \\
U(x) &=&K^{\prime }+\alpha ^{\prime }\ln x+\beta ^{\prime }(\ln
x)^{2}+\gamma ^{\prime }(lnx)^{3}+\delta ^{\prime }(\ln x)^{4}+\varepsilon
^{\prime }(lnx)^{4},~~~~0.125\leq \Lambda r\leq 2, \notag \\
U(x) &=&\Lambda (c_{1}x+c_{2}\ln x+\frac{c_{3}}{\sqrt{x}}+\frac{c_{4}}{x}+c_{5}),~~~~2\leq \Lambda r<\infty .\end{aligned}$$The function $f$ is defined by $w_{p}=f(\ln f+\zeta \ln \ln f);~\zeta
=2(51-19n_{f}/3)/(11-2n_{f}/3)^{2}=64/81$ for $n_{f}=3~.$ The various constants $~a_{1}$ to $a_{3},~K,\alpha ,\beta ,\gamma ,\delta ,\varepsilon
,K^{\prime },\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime },\delta
^{\prime },\varepsilon ^{\prime },$and $c_{1}$ to$~c_{5}$ are given by the Adler-Piran leading log-log model [@adl] and are not adjustable parameters. We modify these closed forms so that the connections between different regions are continuous in second derivatives. The nonrelativistic analysis used by Adler and Piran, however, does not determine the relativistic transformation properties of the potential. How this potential is apportioned between vector and scalar is therefore somewhat, although not completely, arbitrary.
In earlier work [@cra88] we divided the potential in the following way among three relativistic invariants $A,V$ and $S$ for all $x=\Lambda r$. (In our former construction, the additional invariant $V$ was responsible for a possible independent timelike vector interaction.)
$$\begin{aligned}
S& =\eta (\Lambda (c_{1}x+c_{2}\ln (x)+\frac{c_{3}}{\sqrt{x}}+c_{5}+U_{0}),
\notag \\
V& =(1-\eta )\Lambda (c_{1}x+c_{2}\ln (x)+\frac{c_{3}}{\sqrt{x}}+c_{5}+U_{0}), \notag \\
A& =U(x)-\Lambda (c_{1}x+c_{2}\ln (x)+\frac{c_{3}}{\sqrt{x}}+c_{5}),
\label{old}\end{aligned}$$
in which $\eta ={\frac{1}{2}}$. That is, we assumed that (with the exception of the Coulomb-like term ($c_{4}/x$)) the long distance part was equally divided between scalar and a proposed timelike vector.
In the present investigation, we compute the best fit meson spectrum for the following apportionment of the Adler-Piran potential: $$\begin{aligned}
A &=&\exp (-\beta \Lambda r)[V_{AP}-\frac{c_{4}}{r}]+\frac{c_{4}}{r}+\frac{e_{1}e_{2}}{r}, \notag \\
\ \ V+S &=&V_{AP}+\frac{e_{1}e_{2}}{r}-A=(V_{AP}-\frac{c_{4}}{r})(1-\exp
(-\beta \Lambda r))\equiv \mathcal{U}, \label{aps}\end{aligned}$$In order to covariantly incorporate the Adler-Piran potential into our equations, we treat the short distance portion as purely electromagnetic-like (in the sense of the transformation properties of the potential). The attractive ($c_{4}=-0.58$) QCD-Coulomb-like portion (not to be confused with the electrostatic $V_{coul}=e_{1}e_{2}/r$) is assigned completely to the electromagnetic-like part $A$. That is, the constant portion of the running coupling constant corresponding to the exchange diagram is expected to be electromagnetic-like ($\sim \gamma _{1\mu }\gamma
_{2}^{\mu }$). Through the additional parameter $\beta $, the exponential factor gradually turns off the electromagnetic-like contribution (i.e. $A)$ to the potential at long distance except for the $1/r$ portion mentioned above, while the scalar and timelike portions (i.e. $S$ and $V$) gradually turn on, becoming fully responsible for the linear confining and subdominant terms at long distance. We choose not to consider an apportionment function with a large number of parameters as the simple exponential gives a single length scale for the turning of the potential from electromagnetic-like to scalar and timelike. Altogether our three invariant potential functions depend on three parameters: $\Lambda ,U_{0},$ and $\beta $. We furthermore let a free parameter $\xi $ divide the relative portions of $\mathcal{U}$ as follows $$\begin{aligned}
S &=&\xi \mathcal{U=\xi }(V_{AP}-\frac{c_{4}}{r})(1-\exp (-\beta \Lambda r)),
\notag \\
V &=&\mathcal{U-S=(}1-\xi \mathcal{)}(V_{AP}-\frac{c_{4}}{r})(1-\exp (-\beta
\Lambda r)). \label{usv}\end{aligned}$$This differs from the division in the earlier work [@cra88]. Also, the earlier work did not include the effects of the tensor interaction or spin-orbit difference terms or the $u-d$ quark mass differences[^11] (see Eq. (\[57\]) below). In [@crater2] , Crater and Van Alstine chose $\xi =1$ and thus assumed that the scalar interaction is solely responsible for the long distance confining terms.
When inserted into the constraint equations, $V,\ S$ and $A$ become relativistic invariant functions of the invariant separation $r=\sqrt{x_{\perp }^{2}}$ . The covariant structures of the constraint formalism then automatically determine the exact forms by which the central static potential is supplemented with accompanying relativistic spin-dependent and recoil terms.
Meson Spectroscopy from the Schrödinger-like form of the Two-Body Dirac Equations
=================================================================================
Center of Momentum form of Covariant Pauli-Schrödinger Reduction of the Two-Body Dirac Equations
------------------------------------------------------------------------------------------------
In Appendix A we outline the steps needed to obtain the explicit c.m. form of Eq. (\[schlike\]). That form is [@liu], [@saz94], [crater2]{}, $$\begin{aligned}
& \{\mathbf{p}^{2}+\Phi (\mathbf{r,}m_{1},m_{2},w,\mathbf{\sigma }_{1},\mathbf{\sigma }_{2})\}\psi _{+}= \notag \\
=& \{\mathbf{p}^{2}+2m_{w}S+S^{2}+2\varepsilon _{w}A-A^{2}+2\varepsilon
_{w}V-V^{2}+\Phi _{D} \notag \\
& +\mathbf{L\cdot (\sigma }_{1}\mathbf{+\sigma }_{2}\mathbf{)}\Phi _{SO}+\mathbf{\sigma }_{1}\mathbf{\cdot \hat{r}\sigma }_{2}\mathbf{\cdot \hat{r}L\cdot (\sigma }_{1}\mathbf{+\sigma }_{2}\mathbf{)}\Phi _{SOT} \notag \\
& +\mathbf{\sigma }_{1}\mathbf{\cdot \sigma }_{2}\Phi _{SS}+(3\mathbf{\sigma
}_{1}\mathbf{\cdot \hat{r}\sigma }_{2}\mathbf{\ \cdot \hat{r}-\sigma }_{1}\mathbf{\cdot \sigma }_{2})\Phi _{T} \notag \\
& +\mathbf{L\cdot (\sigma }_{1}\mathbf{-\sigma }_{2}\mathbf{)}\Phi _{SOD}+i\mathbf{L\cdot \sigma }_{1}\mathbf{\times \sigma }_{2}\Phi _{SOX}\}\psi _{+}
\notag \\
& =b^{2}\psi _{+}. \label{57}\end{aligned}$$The detailed forms of the separate quasipotentials $\Phi _{i}$ are given in Appendix A together with their forms for weak potential and in the static limit. In Appendix B we give the radial forms of Eq. (\[57\]). The subscripts of most of the quasipotentials are self explanatory. [^12] After the eigenvalue $b^{2}$ of (\[57\]) is obtained, the invariant mass of the composite two-body system $w$ can then be obtained by inverting Eq. (\[bb\]). It is given explicitly by $$w=\sqrt{b^{2}+m_{1}^{2}}+\sqrt{b^{2}+m_{2}^{2}}.$$The structure of the linear and quadratic terms in Eq. (\[57\]) as well as the Darwin and spin-orbit terms, are plausible in light of the discussion given above Eq. (\[em\]), and in light of the static limit Dirac structures that come about from the Pauli reduction of the Dirac equation (see Eqs. (\[571\]) below for the two-component Pauli reduction of the Dirac equation). Their appearances as well as that of the remaining spin structures are direct outcomes of the Pauli reductions of the simultaneous TBDE Eq. (\[tbde\]).
Spectral Results
----------------
Theory1 (abbreviated by Th1) has two invariant interaction functions: $A(r)$ for the short distance behavior and $S(r)$ for scalar confinement. Theory 2 (abbreviated Th2) has three invariant interaction functions including the previous two plus $V(r)~$for$~$ timelike vector confinement. In Appendix A.1 we outline how these invariant interaction functions are related to what we call vertex invariants ($\mathcal{J}(r),~\mathcal{G(}r),\mathcal{L}(r)$), which define the covariant gamma matrix interaction structures which enter into the hyperbolic form of the TBDE as seen in Eqs. (\[hyp1\], \[hyp2\], \[hyp3\]-\[th2\]), and the related energy and mass potentials $E_{1,2}$ and $M_{1,2}$, which characterize the external potential forms of the TBDE as seen in Eqs. (\[extd\]-\[d3\]). The distinction between Th1 and Th2 includes more than the partial cancellation $S^{2}-V^{2}=\mathcal{U}^{2}(2\xi -1)$ between the quadratic scalar and timelike vector interactions. It also includes their partial cancellations for spin-orbit and Darwin terms (see Appendix A.4). In this section we present spectral results with (Th2) and without (Th1) the added timelike invariant function $V(r).$
We display our results in Tables \[1\]-\[15.5\]. The first table lists the best fit values for the quark masses and the potential parameters $\Lambda ,\ \Lambda U_{0},\ \beta $ for Th1 (scalar only confinement) and Th2. The ratio that optimized the fit for Th2 is $\xi =0.704,$ (see Eq. ([usv]{})). In the first two columns of Tables \[2\]-\[9\], we list quantum numbers and experimental rest mass values (in GeV) and experimental errors listed parenthetically (in MeV) for 105 known mesons. We include all well known and plausible candidates listed in the standard reference ([@prtl]). We omit only those mesons with substantial flavor mixing, like the $\eta $ and $\eta ^{\prime }$ mesons . In the tables, the quantum numbers listed are those of the $\psi _{+}$ part of the 16-component wave function. In the third and fourth columns are the theoretical values, with Th1 referring to the results without the timelike vector interaction and Th2 with the timelike vector interaction. In the fifth and sixth columns we give the differences between our theoretical results and the experimental and in the last two columns the contributions of each theoretically computed values to the total $\chi ^{2}$ of 237 for Th1 and 173 for Th2 [^13].
To generate the fits, in addition to varying the five quark masses we vary the parameters $\Lambda ,\ U_{0},$ $\beta ,$ and $\xi $ in the apportioned static Adler-Piran potential in $A,V,$ and $S$. Those invariants are put into our relativistic wave equations just as we have inserted the invariant Coulomb potential $A=-\alpha /r$ (but with $V=S=0$) to obtain the results of QED bound states [@exct; @becker]. Note especially that we use a single $\Phi (A,S)$ for Th1 and a single $\Phi (A,V,S)$ for Th2 for all quark mass ratios. Hence in each theory we use a single structure for all the $Q\bar{Q},\ q\bar{Q},$ *and* $q\bar{q}$ mesons in a single overall fit. The entire confining part of the potential transforms as a world scalar for Th1 and combined timelike and scalar for Th2. Since $\xi >0.5$ in our equations, this structure leads in both models to linear confinement at long distances and quadratic confinement at extremely long distances (where the quadratic contribution $S^{2}$ outweighs the linear term $2m_{w}S$ in Th1 and $S^{2}-V^{2}$ outweighing the linear terms $2m_{w}S+2\varepsilon _{w}V$ in Th2). At distances at which $\exp (-\beta \Lambda r)<<1,~$the corresponding fine and hyperfine structures producing spin-orbit, Thomas, Darwin, spin-spin, and tensor terms (the last two are relatively small in that domain) are dominated by the confining interaction, while at short distances ($\exp (-\beta \Lambda r)\sim 1)$ the electromagnetic-like portion of the interaction gives the dominant contribution to the fine and hyperfine structures. Furthermore because the signs of each of the spin-orbit and Darwin terms in the Pauli-form of our TBDE are opposite for the scalar and vector interactions (see Appendix A.4), the spin-orbit contributions of those parts of the interaction produce opposite effects with degrees of cancellation depending on the size of the quarkonia atom. Another point to make is that because of the various sizes of the quarkonia atoms and the c.m. energy dependence the behavior of $\Phi _{w}$ is sharply different for the light mesons compared with the heavy ones. This may possibly account for the ability of our formalism to obtain good fits for the light meson hyperfine splittings while at the same time giving good overall fits to the heavy mesons.
We obtain the meson masses given in columns three and four as the result of a least squares fit using the known experimental errors from the Particle Date Group (PDG) tables [@prtl] and an assumed calculational error of 1.0 MeV. We employ the calculational error not to represent the uncertainty of our algorithm but more to prevent the mesons that are stable with respect to the strong interaction from being weighted too heavily. Our $\chi ^{2}$ is per datum (105) minus parameters (8 or 9). In Table \[1\], the value of $\beta $ for Th1 implies that (in the best fit) as the quark separation increases, our apportioned Adler-Piran potential switches from primarily vector to scalar at about ($\beta \Lambda )^{-1}\sim $0.60 fermi. This shift is a relativistic effect since the effective nonrelativistic limit of the potential ($\mathcal{A}+S$) exhibits no such shift (i.e., by construction $\beta $ drops out). For Th2, this distance is substantially less, $(\beta
\Lambda )^{-1}\sim $0.22 fermi.
$\begin{tabular}{|llll|}
\hline
Parameter & Th1 & & Th2 \\ \hline
$m\_[b]{}$ & $4.917 $ & & $4.953 $ \\
$m\_[c]{}$ & $1.546 $ & & $1.585 $ \\
$m\_[s]{}$ & $0.2874 $ & & $0.3079 $ \\
$m\_[u]{}$ & $0.0713 $ & & $0.0985 $ \\
$m\_[d]{}$ & $0.0771 $ & & $0.1045 $ \\
$$ & $0.2213 $ & & $0.2255 $ \\
$U\_[0]{}$ & $1.815$ GeV & & $1.770$ GeV \\
$$ & $1.502$ & & $4.408$ \\
$$ & $1$ & & $0.704$ \\ \hline
\end{tabular}$
$\bigskip $
$u\bar{d}$ mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
--------------------------------------- ------------- ------- ------- ----------- ----------- ------------------ ------------------ --
$\pi :u\overline{d}\ 1\,{}^{1}S_{0}$ 0.140(0.0) 0.141 0.134 -0.002 0.006 0.0 0.3
$\rho :u\overline{d}\ 1\,{}^{3}S_{1}$ 0.775(0.4) 0.790 0.781 -0.015 -0.005 1.9 0.2
$b_{1}:u\overline{d}\ 1\,{}^{1}P_{1}$ 1.230(3.2) 1.283 1.243 -0.053 -0.014 2.5 0.2
$a_{1}:u\overline{d}\ 1\,{}^{3}P_{1}$ 1.230(40.) 1.425 1.320 -0.195 -0.090 0.2 0.1
$\pi :u\overline{d}\ 2\,{}^{1}S_{0}$ 1.300(100) 1.493 1.435 -0.193 -0.135 0.0 0.0
$a_{2}:u\overline{d}\ 1\,{}^{3}P_{2}$ 1.318(0.6) 1.276 1.310 0.042 0.008 12.9 0.5
$\rho :u\overline{d}\ 2\,{}^{3}S_{1}$ 1.465(25.) 1.745 1.684 -0.280 -0.219 1.3 0.8
$a_{0}:u\overline{d}\ 1\,{}^{3}P_{0}$ 1.474(19.) 1.165 1.024 0.309 0.450 2.6 5.6
$b_{2}:u\overline{d}\ 1\,{}^{1}D_{2}$ 1.672(3.2) 1.815 1.763 -0.143 -0.090 18.2 7.2
$a_{3}:u\overline{d}\ 1\,{}^{3}D_{3}$ 1.689(2.1) 1.663 1.718 0.026 -0.029 1.2 1.5
$a_{1}:u\overline{d}\ 1\,{}^{3}D_{1}$ 1.720(20.) 1.944 1.847 -0.224 -0.127 1.2 0.4
$a_{2}:u\overline{d}\ 2\,{}^{3}P_{2}$ 1.732(16.) 2.025 2.009 -0.293 -0.277 3.3 3.0
$\pi :u\overline{d}\ 3\,{}^{1}S_{0}$ 1.816(14.) 2.090 2.037 -0.274 -0.221 3.8 2.5
$b_{2}:u\overline{d}\ 2\,{}^{1}D_{2}$ 1.895(16.) 2.300 2.267 -0.405 -0.372 6.4 5.4
$a_{4}:u\overline{d}\ 1\,{}^{3}F_{4}$ 2.011(12.) 1.984 2.057 0.027 -0.046 0.1 0.1
$b_{2}:u\overline{d}\ 3\,{}^{1}D_{2}$ 2.090(29.) 2.704 2.700 -0.614 -0.610 4.5 4.4
$\rho :u\overline{d}\ 3\,{}^{3}S_{1}$ 2.149(17.) 2.281 2.326 -0.132 -0.177 0.6 1.1
$a_{3}:u\overline{d}\ 2\,{}^{3}D_{3}$ 2.250(45.) 2.275 2.290 -0.025 -0.040 0.0 0.0
$a_{5}:u\overline{d}\ 1\,{}^{3}G_{5}$ 2.330(35.0) 2.258 2.349 0.072 -0.019 0.0 0.0
$a_{6}:u\overline{d}\ 1\,{}^{3}H_{6}$ 2.450(130) 2.500 2.609 -0.050 -0.159 0.0 0.0
: $u\bar d$ Mesons, Theory 1 and 2 - In this table and the ones below, the meson masses are in units of GeV, with experimental errors given parenthetically in units of MeV.[]{data-label="2"}
$s\bar{u},~s\bar{d}~$Mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
--------------------------------------------------- ------------ ------- ------- ----------- ----------- ------------------ ------------------ --
$K\,{}^{-}:s\overline{u}\ 1\,{}^{1}S_{0}$ 0.494(0.0) 0.485 0.519 0.008 -0.025 0.7 6.4
$K\,{}^{0}:s\overline{d}\ 1\,{}^{1}S_{0}$ 0.498(0.0) 0.488 0.520 0.010 -0.022 1.0 5.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}S_{1}$ 0.892(0.3) 0.917 0.896 -0.025 -0.004 6.0 0.2
$K^{\ast }\,{}^{-}:s\overline{d}\ 1\,{}^{3}S_{1}$ 0.896(0.3) 0.919 0.897 -0.023 -0.001 4.8 0.0
$K\,{}^{-}:s\overline{u}\ 1\,{}^{1}P_{1}$ 1.272(7.0) 1.356 1.339 -0.084 -0.067 1.4 0.9
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}P_{1}$ 1.403(7.0) 1.419 1.359 -0.016 0.044 0.1 0.4
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}S_{1}$ 1.414(15.) 1.759 1.706 -0.345 -0.292 5.3 3.8
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}P_{0}$ 1.425(50.) 1.132 1.079 0.293 0.346 0.3 0.5
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}P_{2}$ 1.426(50.) 1.379 1.404 0.047 0.022 6.8 1.4
$K^{\ast }\,{}^{-}:s\overline{d}\ 1\,{}^{3}P_{2}$ 1.432(1.3) 1.380 1.405 0.052 0.027 10.2 2.8
$K\,{}^{-}:s\overline{u}\ 2\,{}^{1}S_{0}$ 1.460(40.) 1.523 1.476 -0.063 -0.016 0.0 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}D_{1}$ 1.717(27.) 1.922 1.837 -0.205 -0.120 0.6 0.2
$K\,{}^{-}:s\overline{u}\ 1\,{}^{1}D_{2}$ 1.773(8.0) 1.835 1.803 -0.062 -0.030 0.6 0.1
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}D_{3}$ 1.776(7.0) 1.740 1.792 0.036 -0.016 0.3 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}D_{2}$ 1.816(13.) 1.824 1.795 -0.008 0.021 0.0 0.0
$K\,{}^{-}:s\overline{u}\ 3\,{}^{1}S_{0}$ 1.830(13.) 2.115 2.081 -0.285 -0.251 0.5 0.4
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}P_{2}$ 1.973(33.) 2.078 2.060 -0.105 -0.087 0.1 0.1
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}F_{4}$ 2.045(9.0) 2.045 2.117 0.000 -0.072 0.0 0.6
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}D_{2}$ 2.247(17.) 2.326 2.313 -0.079 -0.066 0.2 0.1
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}F_{3}$ 2.324(24.) 2.642 2.600 -0.318 -0.276 1.7 1.3
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}G_{5}$ 2.382(14.) 2.309 2.401 0.073 -0.019 0.3 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}F_{4}$ 2.490(20.) 2.555 2.600 -0.065 -0.110 0.1 0.3
: $s\bar u$ and $s\bar d$ Mesons, Theory 1 and 2[]{data-label="3"}
$s\bar{s}$ Mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
--------------------------------------- ------------ ------- ------- ----------- ----------- ------------------ ------------------ --
$\phi :s\overline{s}\ 1\,{}^{3}S_{1}$ 1.019(0.0) 1.050 1.013 -0.031 0.006 9.3 0.4
$\phi :s\overline{s}\ 1\,{}^{3}P_{0}$ 1.370(100) 1.211 1.175 0.159 0.195 0.0 0.0
$\phi :s\overline{s}\ 1\,{}^{3}P_{1}$ 1.518(5.0) 1.480 1.437 0.038 0.081 0.5 2.5
$\phi :s\overline{s}\ 1\,{}^{3}P_{2}$ 1.525(5.0) 1.496 1.506 0.029 0.019 0.3 0.1
$\phi :s\overline{s}\ 2\,{}^{3}S_{1}$ 1.680(20.) 1.811 1.875 -0.131 -0.195 0.4 0.9
$\phi :s\overline{s}\ 1\,{}^{3}D_{3}$ 1.854(7.0) 1.839 1.879 0.015 -0.025 0.0 0.1
$\phi :s\overline{s}\ 2\,{}^{3}P_{2}$ 2.011(70) 2.149 2.128 -0.138 -0.117 0.0 0.0
$\phi :s\overline{s}\ 3\,{}^{3}P_{2}$ 2.297(28.) 2.612 2.603 -0.315 -0.306 1.3 1.2
: $s\bar s$ Mesons, Theory 1 and 2[]{data-label="4"}
$c\bar{u},~c\bar{d}~$Mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
-------------------------------------------- ------------ ------- ------- ----------- ----------- ------------------ ------------------ --
$D^{0}:c\overline{u}\ 1\,{}^{1}S_{0}$ 1.865(0.2) 1.865 1.876 0.000 -0.011 0.0 1.1
$D^{+}:c\overline{d}\ 1\,{}^{1}S_{0}$ 1.870(0.2) 1.872 1.883 -0.003 -0.013 0.1 1.7
$D^{\ast 0}:c\overline{u}\ 1\,{}^{3}S_{1}$ 2.007(0.2) 2.013 2.007 -0.006 0.000 0.4 0.0
$D^{\ast +}:c\overline{d}\ 1\,{}^{3}S_{1}$ 2.010(0.2) 2.019 2.013 -0.008 -0.002 0.7 0.1
$D^{\ast 0}:c\overline{u}\ 1\,{}^{3}P_{0}$ 2.352(50.) 2.224 2.221 0.128 0.131 0.1 0.1
$D^{\ast +}:c\overline{d}\ 1\,{}^{3}P_{0}$ 2.403(14.) 2.232 2.230 0.171 0.173 1.5 1.5
$D^{+}:c\overline{d}\ 1\,{}^{3}P_{2}$ 2.460(3.0) 2.398 2.414 0.062 0.046 3.9 2.1
$D^{\ast 0}:c\overline{u}\ 1\,{}^{3}P_{2}$ 2.461(1.6) 2.393 2.409 0.069 0.052 13.2 7.7
: $c\bar u$ and $c\bar d$ Mesons, Theory 1 and 2[]{data-label="5"}
$c\bar{s}$ Mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
----------------------------------------------- ------------ ------- ------- ----------- ----------- ------------------ ------------------ --
$D_{s}:c\overline{s}\ 1\,{}^{1}S_{0}$ 1.968(0.3) 1.972 1.974 -0.004 -0.006 0.1 0.3
$D_{s}^{\ast }:c\overline{s}\ 1\,{}^{3}S_{1}$ 2.112(0.5) 2.138 2.119 -0.026 -0.007 5.4 0.4
$D_{s}^{\ast }:c\overline{s}\ 1\,{}^{3}P_{0}$ 2.318(0.6) 2.348 2.340 -0.031 -0.022 6.9 3.5
$D_{s}:c\overline{s}\ 1\,{}^{1}P_{1}$ 2.535(0.3) 2.505 2.499 0.030 0.036 8.3 11.6
$D_{s}^{\ast }:c\overline{s}\ 1\,{}^{3}P_{2}$ 2.573(0.9) 2.534 2.532 0.039 0.040 8.4 8.9
$D_{s}^{\ast }:c\overline{s}\ 2\,{}^{3}S_{1}$ 2.690(7.0) 2.714 2.702 -0.024 -0.012 0.1 0.0
: $c\bar s$ Mesons, Theory 1 and 2[]{data-label="6"}
$c\bar{c}$ Mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
--------------------------------------------- ------------- ------- ------- ----------- ----------- ------------------ ------------------ --
$\eta _{c}:c\overline{c}\ 1\,{}^{1}S_{0}$ 2.980(1.2) 2.965 2.973 0.015 0.007 1.0 0.2
$J/\psi (1S):c\overline{c}\ 1\,{}^{3}S_{1}$ 3.097(0.0) 3.131 3.128 -0.034 -0.031 11.4 9.7
$\chi _{0}:c\overline{c}\ 1\,{}^{3}P_{0}$ 3.415(0.3) 3.395 3.397 0.020 0.018 3.7 3.0
$\chi _{1}:c\overline{c}\ 1\,{}^{3}P_{1}$ 3.511(0.1) 3.506 3.505 0.005 0.006 0.2 0.4
$h_{1}:c\overline{c}\ 1\,{}^{1}P_{1}$ 3.526(0.3) 3.522 3.523 0.004 0.003 0.2 0.1
$\chi _{2}:c\overline{c}\ 1\,{}^{3}P_{2}$ 3.556(0.1)) 3.562 3.557 -0.006 -0.001 0.4 0.0
$\eta _{c}:c\overline{c}\ 2\,{}^{1}S_{0}$ 3.637(4.0) 3.606 3.602 0.031 0.035 0.6 0.7
$\psi (2S):c\overline{c}\ 2\,{}^{3}S_{1}$ 3.686(0.0) 3.688 3.689 -0.002 -0.002 0.0 0.1
$\psi (1D):c\overline{c}\ 1\,{}^{3}D_{1}$ 3.773(0.4) 3.807 3.807 -0.034 -0.034 0.9 0.9
$\chi _{2}:c\overline{c}\ 2\,{}^{3}P_{2}$ 3.929(5.0) 3.980 3.983 -0.051 -0.054 1.0 1.1
$\psi (3S):c\overline{c}\ 3\,{}^{3}S_{1}$ 4.039(10.) 4.086 4.092 -0.047 -0.053 0.2 0.3
$\psi (2D):c\overline{c}\ 2\,{}^{3}D_{1}$ 4.153(3.0) 4.164 4.169 -0.011 -0.016 0.1 0.3
$\psi (4S):c\overline{c}\ 4\,{}^{3}S_{1}$ 4.421(4.0) 4.410 4.426 0.011 -0.005 0.1 0.0
$\psi (3D):c\overline{c}\ 3\,{}^{3}D_{1}$ 4.421(4.0) 4.467 4.483 -0.046 -0.062 1.2 2.3
$\psi (5S):c\overline{c}\ 5\,{}^{3}S_{1}$ 4.800(100) 4.690 4.719 0.110 0.081 0.0 0.0
$\psi (4D):c\overline{c}\ 4\,{}^{3}D_{1}$ 4.880(100) 4.735 4.764 0.145 0.116 0.0 0.0
$\psi (6S):c\overline{c}\ 6\,{}^{3}S_{1}$ 5.180(100) 4.940 4.983 0.203 0.197 0.0 0.0
$\psi (5D):c\overline{c}\ 5\,{}^{3}D_{1}$ 5.290(100) 4.977 5.020 0.350 0.270 0.1 0.1
: $c\bar c$ Mesons, Theory 1 and 2[]{data-label="7"}
$b\bar{u},~b\bar{d}$ $b\bar{s}~$Mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
---------------------------------------------------- ------------ ------- ------- ----------- ----------- ------------------ ------------------ --
$B^{-}:b\overline{u}\ 1\,{}^{1}S_{0}$ 5.279(0.3) 5.281 5.283 -0.002 -0.004 0.0 0.2
$B^{0}:b\overline{d}\ 1\,{}^{1}S_{0}$ 5.280(0.3) 5.282 5.284 -0.003 -0.005 0.1 0.2
$B^{\ast -}:b\overline{u}\ 1\,{}^{3}S_{1}$ 5.325(0.5) 5.335 5.333 -0.010 -0.008 0.8 0.5
$B^{\ast -}:b\overline{u}\ 1\,{}^{3}P_{2}$ 5.747(2.9) 5.671 5.687 0.076 0.059 6.2 3.8
$B_{s}^{0}:b\overline{s}\ 1\,{}^{1}S_{0}$ 5.366(0.6) 5.373 5.367 -0.007 -0.001 0.3 0.0
$B_{s}^{\ast 0}:b\overline{s}\ 1\,{}^{3}S_{1}$ 5.413(1.3) 5.441 5.430 -0.029 -0.017 3.0 1.0
$B_{s}^{\ast 0}:b\overline{s}\ 1\,{}^{3}P_{1}$ 5.829(0.7) 5.789 5.792 0.040 0.037 10.9 9.4
$B_{s}^{\ast 0}:b\overline{s}\ 1\,{}^{3}P_{2}$ 5.840(0.6) 5.805 5.805 0.035 0.035 8.9 9.0
$B_{c}^{-}:b\overline{c}\ 1\,{}^{1}S_{0}$ 6.276(21.) 6.249 6.251 0.027 0.025 0.4 0.4
: $b\bar u$ , $b\bar d$ and $b\bar s$ Mesons, Theory 1 and 2
[8]{}
$b\bar{b}$ Mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
----------------------------------------------- ------------- -------- -------- ----------- ----------- ------------------ ------------------ --
$\eta _{b}:b\overline{b}\ 1\,{}^{1}S_{0}$ 9.389(4.0) 9.337 9.330 0.052 0.059 1.6 2.0
$\Upsilon (1S):b\overline{b}\ 1\,{}^{3}S_{1}$ 9.460(0.3) 9.444 9.444 0.016 0.017 2.4 2.6
$\chi _{b0}:b\overline{b}\ 1\,{}^{3}P_{0}$ 9.859(0.4) 9.836 9.834 0.023 0.026 4.6 5.6
$\chi _{b1}:b\overline{b}\ 1\,{}^{3}P_{1}$ 9.893(0.3) 9.886 9.886 0.007 0.007 0.4 0.4
$\chi _{b2}:b\overline{b}\ 1\,{}^{3}P_{2}$ 9.912(0.3) 9.922 9.920 -0.010 -0.008 0.9 0.6
$\Upsilon (2S):b\overline{b}\ 2\,{}^{3}S_{1}$ 10.023(0.3) 10.022 10.022 0.001 0.002 0.0 0.0
$\Upsilon (D):b\overline{b}\ 2\,{}^{3}D_{2}$ 10.161(0.6) 10.178 10.179 -0.017 -0.018 2.1 2.3
$\chi _{b0}:b\overline{b}\ 2\,{}^{3}P_{0}$ 10.232(0.4) 10.230 10.229 0.002 0.003 0.1 0.1
$\chi _{b1}:b\overline{b}\ 2\,{}^{3}P_{1}$ 10.255(0.5) 10.261 10.262 -0.006 -0.007 0.3 0.4
$\chi _{b2}:b\overline{b}\ 2\,{}^{3}P_{2}$ 10.269(0.4) 10.284 10.286 -0.015 -0.017 1.9 2.5
$\Upsilon (3S):b\overline{b}\ 3\,{}^{3}S_{1}$ 10.355(0.6) 10.366 10.368 -0.011 -0.013 0.8 1.2
$\Upsilon (4S):b\overline{b}\ 4\,{}^{3}S_{1}$ 10.579(1.2) 10.626 10.633 -0.046 -0.053 8.8 11.7
$\Upsilon (5S):b\overline{b}\ 5\,{}^{3}S_{1}$ 10.865(8.0) 10.844 10.857 0.021 0.008 0.1 0.0
$\Upsilon (6S):b\overline{b}\ 6\,{}^{3}S_{1}$ 11.019(8.0) 11.036 11.055 -0.017 -0.036 0.0 0.2
: $b\bar b$ Mesons, Theory 1 and 2[]{data-label="9"}
------------------------------------------------- --------- --------- --------------- --------- ---------
**Meson Family** **Th1** **Th2** **\# Mesons** **Th1** **Th2**
$u\overline{d}$ 62.1 30.5 20 3.1 1.5
$s\overline{u}~,~s\overline{d}$ 37.7 26.5 22 1.8 1.3
$s\overline{s}$ 11.0 4.5 8 1.4 0.6
$c\overline{u}~,~c\overline{d}$ 20.9 14.6 8 2.6 1.8
$c\overline{s}$ 29.4 26.6 6 4.9 4.4
$c\overline{c}$ 28.2 23.4 18 1.6 1.3
$b\overline{u}~,~b\overline{s}~,~b\overline{c}$ 31.2 25.9 9 3.5 2.9
$b\overline{b}$ 24.8 25.1 14 1.8 1.8
Total 245.3 177.1 105 2.4 1.7
------------------------------------------------- --------- --------- --------------- --------- ---------
: $\chi^2$ by Family for Th1 and Th2[]{data-label="10"}
Table \[10\] lists the 8 meson families, their respective $\chi ^{2}$ contributions and their averages. The most striking feature is that as the quark masses increase from the lightest to the heaviest, the differences of the respective $\chi ^{2}$ shifts from about a factor of 2 to almost even. The heaviest mesons are also the smallest in mean radius. This means that they are less likely to experience the effects of the $S^{2}$ and $-V^{2}$ portions of the confining interactions. As the mesons become large, they experience more of the effects of these parts of the potentials. The most dramatic improvement from the inclusion of the timelike vector confining potential $V(r)$ (Th2) is with the light quark $u\bar{d}$ family. Referring now to Tables \[2\]-\[9\] most of the improvement comes from that of the fits to the $a_{2},b_{2}$ and $b_{1}$ mesons. For the $s\bar{u},s\bar{d}$ family the largest improvement comes from the lowest lying $^{3}P_{2}$ mesons. The ground state singlet-triplet splitting changes from an overestimation to an underestimation. For the $s\bar{s}$ family the most significant improvement is in the ground state, although somewhat off-balanced by a worse fit for the lowest lying $^{3}P_{1}$ state. In the case of the $c\bar{u},c\bar{d}$ family the main improvement is from the $^{3}P_{2}$ mesons. For the $c\bar{s}$ mesons there is a slight overall improvement for Th2 with offsetting changes for the $^{3}P_{0}$ and $^{1}P_{1}$ mesons. There is only a very slight improvement for the $c\bar{c}$ mesons. For both Th1 and Th2 the worst fit is to the $J/\psi $ meson with a mass too large by about 30 MeV. It cannot be adjusted downward by lowering the charm mass due to the fact that other mesons in this family would be pushed further from the data. With the heavy-light family, a single $b$ quark, there again is not much overall change and even less in the $b\bar{b}$ family although another significant improvement is in the $^{3}P_{2}$ $b\bar{u}$ state. An oddity with the $b\bar{b}$ is the sudden increase in $\chi ^{2}$ at the $4^{3}S_{1}$ meson, the worst fit of all the mesons in terms of the incremental $\chi ^{2}$. Since that meson is closest to threshold, its mass will be most affected by it, whereas our theoretical model does not take threshold effects into account.
A possible explanation of why most improvements come for the $^{3}P_{2}$ states is that the effect on the spin-orbit coupling due to the Thomas terms is opposite in sign for the timelike vector and scalar mesons. Without the balancing effect of the timelike vector confining interaction, the scalar interaction enforces an inversion of the spin-orbit splittings of the light mesons that are far too distorted for the $u\bar{d}$ and $s\bar{u}$ multiplets[^14]. Also, the long range scalar parts contribute oppositely in sign from the short range vector part attributable to the $A(r)$ potential.
Family Exp. Th1. Th2.
----------------- ----------- ------ ----------- ------ ---------- ------
$u\overline{d}$ 635 649 647
$s\overline{u}$ 398 432 377
$s\overline{d}$ 398 431 377
$c\overline{u}$ 142 148 131
$c\overline{d}$ 140 147 130
$c\overline{s}$ 144 166 145
$c\overline{c}$ 117 166 155
$b\overline{u}$ 46 54 50
$s\overline{s}$ 47 68 63
$b\overline{b}$ 71 107 114
: Ground State Singlet-Triplet Splittings (MeV)
\[11\]
We now examine another important feature of our method: the goodness with which our equations account for spin-dependent effects (both fine- and hyperfine- splittings). Table \[11\] shows the best fit versus experimental ground state singlet-triplet splittings and six versus four of the ten hyperfine splittings are improved using Th2 over Th1. Both give good fits for all hyperfine ground state splittings except for the $\eta
_{c}-\psi $ system and $\eta _{b}-\Upsilon $ system which for the latter over estimate the splittings by about 50%![^15] One problem with the fit for the $c\bar{c}$ system of mesons may be due to the fact that the $D^{\ast }\ ^{3}P_{2}$, $^{1}P_{1}$ and $D_{s}^{\ast }\
^{3}P_{2}$ fits are significantly low while the $J/\psi $ fit is significantly high. Lowering the $c$ quark mass corrects the $J/\psi $ mass while raising the $D^{\ast },D_{s}^{\ast }\ P$ state masses would require raising the $c$ quark mass. Reducing one discrepancy would worsen the other, at least in our three invariant function approach
Family Exp. Th1. Th2.
--------------------------------------- ----------- ------- ----------- ------- ---------- -------
$u\overline{d}$ -0.36 -0.57 -0.03
$s\overline{u}$ -1.05 -0.14 0.16
$s\overline{s}$ 0.05 0.06 0.26
$c\overline{c}$ 0.47 0.50 0.48
$b\overline{b}\ \ \ (1^{3}P_{2,1,0})$ 0.56 0.72 0.65
$b\overline{b}\ \ \ (2^{3}P_{2,1,0})$ 0.61 0.74 0.73
: Spin-Orbit Splitting R Ratios
\[12\]
For the spin-orbit splittings Table \[12\] gives the $R$ ratios $(^{3}P_{2}-^{3}P_{1})/(^{3}P_{1}-^{3}P_{0}))$. Both sets of fits are very poor for the two lightest multiplets. The fact that Th1 has the same sign for the $u\bar{d}$ as the experiment values is not an indication that it gives reasonable results since the negative sign originates from the numerator instead of the denominator. Of the four remaining multiplets, Th2 gives a better fit on 3. It must be said, however, that none of the better fits are very good except for the $c\bar{c}.$ From the experimental point of view the poor $R$ value for the $u\bar{d}$ and $u\bar{s}$ may be the uncertain status of the $^{3}P_{0}$ light quark meson bound states, or theoretically our low $^{3}P_{0}$ theoretical meson masses. Also, the lack of any mechanism in our model to account for the effects of decay rates on level shifts undoubtedly has an effect. Another likely cause for the poorer performance of Th1 as one goes from heavier to light mesons is that the radial size of the meson grows so that the long distance interactions, in which the scalar interaction becomes dominant, play a more important role. This effect is blunted by Th2 as seen in the $u\bar{d}$ numbers. In Table \[2\] The spin-orbit terms due to scalar interactions are opposite in sign and tend (at long distance) to dominate the spin-orbit terms due to vector interactions for Th1, but less so in Th2. This results in partial to full multiplet inversions as we proceed from the $s\bar{s}$ to the $u\bar{d}$ mesons. This inversion mechanism is less for Th2 than for Th1 because of the value of $\xi $.
Family Exp. Th1. Th2.
----------------- ----------- ------ ----------- ------ ---------- ------
$u\overline{d}$ 76 30 36
$s\overline{u}$ 146 9 14
$c\overline{c}$ -1 3 -1
: Splitting Between $^1P_1$ and weighted triplet states (MeV)
\[13\]
The hyperfine structure of our equations also influences the splitting between the $^{1}P_{1}$ and the weighted sum $[5(^{3}P_{2})+3(^{3}P_{1})+1(^{3}P_{0})]/9$ of bound states. Table \[13\] indicates the agreement of the theoretical and experimental mass differences is excellent for the $c\bar{c}$ system, too small but of the right sign for the $u\bar{s}$ system and $u\bar{d}$ systems. The agreement, however, for the light systems is nevertheless considerably better than that in the case of the fine structure splitting $R$ ratios. Note that in the case of unequal mass $P$ states, our calculations of the two values incorporate the effects of $\mathbf{L}\cdot (\mathbf{\sigma }_{1}-\mathbf{\sigma }_{2})$ and $\mathbf{L}\cdot \mathbf{\sigma }_{1}\times \mathbf{\sigma }_{2}~$which mix spin.
Family Exp. Th1. Th2.
---------------------------------------------- ----------- ------ ----------- ------ ---------- ------
$u\overline{d}$ 255 199 163
$s\overline{u}$ 303 163 131
$c\overline{c}\ \ \ (1^{3}D_{1}-2^{3}S_{1})$ 87 119 118
$c\overline{c}\ \ \ (2^{3}D_{1}-3^{3}S_{1})$ 114 78 77
: Mixing Due to the Tensor Term Between Orbital D and Radial S Excitations of the Spin-Triplet Ground States (in MeV)
\[14\]
Next consider the mixing due to the tensor term between orbital $D$ and radial $S~$excitations of the spin-triplet ground states. This mixing occurs most notably in the $c\bar{c},u\bar{s}$ and $u\bar{d}$ systems. Table [14]{} show that Th1 is better than Th2 although both are pretty far off the mark. For the charmonium system, the lower doublet results are high whereas the higher doublet results are low.
Family Exp. Difference Th1. Difference Th2. Difference
-------- -- ------------------------- -- ------------------------- -- ------------------------
1160, 516 1352, 597 1301, 602
690, 684 955, 536 903, 642
966, 370 1038, 592 957, 605
522 842 810
661 761 862
657 641 629
589, 353 557, 398 561, 403
563, 332, 224, 286, 154 578, 344, 260, 218, 192 578, 346, 265, 224,198
: Radial Excitations (MeV)
\[15\]
Next we consider the effects of the change from Th1 to Th 2 on the radial excitations. Table \[15\] shows that for the most part Th1 gives slightly better results for the radially excited states, although where both theories are furthest off (the $u\bar{d}$ states) Th2 gives better results. The radially excited $u\bar{d}$ mesons have a larger mean radius than for the heavier meson and thus the temporizing effects of the $-V^{2}$ term tends to counteract more the increased confining potential for large $r$ from linear to quadratic due to the $S^{2}$ terms.
Finally we comment on the isospin splittings shown in table \[15.5\]. There are two effects we must consider here: the positive $d-u$ mass differences of about $6$ MeV for both theories and the Coulomb interaction between the quarks on the order of $\alpha \times 197$ MeV or less depending on the meson sizes. The Coulomb interaction is counter to the $d-u$ mass difference for the $s\bar{d}-s\bar{u}$ and $b\bar{d}-b\bar{u}$ splittings while enhancing the $d-u$ mass difference for the $c\bar{d}-c\bar{u}$ splittings. These alternatively competing and enhancing effects are seen in the sizes of the splittings for both theories as you read down the table from the $s\bar{d}-s\bar{u}$ through the $c\bar{d}-c\bar{u}$ to the $b\bar{d}-b\bar{u}$ splitting. For the $K-K^{\ast }$ family the values for the isospin splittings are $3~$and $0$ MeV for Th1 and 2 vs the experimental value 4 MeV for the singlet ground states while for the triplet the isospin splittings are $2~$and $1$ MeV vs the experimental value 4 MeV. The experimental splitting grows for the orbital excitation ($K_{2}^{\ast }$) to 6 MeV. The probable reason for the increase is that at the larger distances, the influence of the Coulomb differences becomes small so that only the $d-u$ mass difference influences the result. Our theories do not show a similar increase for the orbital excitations. In the case of the $D^{+}-D^{0}$ splitting our mass differences for Th1 and Th2 are 7 and 7 MeV respectively versus the experimental mass difference of just 5 MeV. Here we see the opposite overall effect between the combined effects of the $d-u$ mass difference and the slightly increased electromagnetic binding present in the case of the $D^{0}$ and the slightly decreased binding in the case of the $D^{+}$. Whereas in the kaon system the results are too small, for the $D$ the results are too large . This can be partially understood since the Coulomb and $d-u$ mass differences work in concert with the Coulomb potential for these doublets. These effects work in the same way for the spin-triplet splitting resulting in the theoretical values of 6 and 6 MeV for the two theories compared with the experimental value 3 MeV. For the $^{3}P_{2}$ isodoublet we obtain -5 and -5 MeV versus about 1 for the experimental value again showing the expected opposite trend from that of the kaon system. The experimental splitting between the $^{3}P_{0}$ isodoublet of 51 MeV appears incomprehensibly large. Our two values are 8 and 9 MeV. The isospin splittings that we obtain for the spin singlet $B$ meson system are 2 and 1 MeV for Th1 and Th2 versus 1 MeV. Here the competing effects cancel as in the kaon system only more so since the mesons are smaller and thus the Coulomb parts play a stronger role than for the kaon.
Family Exp Theory 1 Theory 2
---------------------------------------------- ----------- ----- ----------- ---------- ---------- ----------
$s\bar{d}-s\overline{u}:\ 1^{1}S_{0}$ 4 3 0
$s\overline{d}-s\overline{u}:\ 1^{3}S_{1}$ 4 2 1
$s\overline{d}-s\overline{u}:\ 1^{3}P_{2}$ 6 1 1
$c\overline{d}-c\overline{u}:\ 1^{1}S_{0}\ $ 5 7 7
$c\overline{d}-c\overline{u}:\ 1^{3}S_{1}\ $ 3 6 6
$c\overline{d}-c\overline{u}:\ 1^{3}P_{0}\ $ 51 8 9
$c\overline{d}-c\overline{u}:\ 1^{3}P_{2}\ $ 1 -5 -5
$b\bar{d}-b\overline{u}:\ 1^{1}S_{0}$ 1 2 1
$d-u\ $ mass 5.8 6.0
: Isospin Splitting (MeV)[]{data-label="15.5"}
The Effective Relativistic Schrödinger Equation with Flavor Mixing for Spin-Zero Isoscalar Mesons.
==================================================================================================
Consider the general eigenvalue equation (\[57\]) for an isoscalar meson, one with quark structure $q\bar{q}$. As seen in Appendix A the mass dependence appearing in $\Phi _{w}$ directly or indirectly through $m_{w},\varepsilon _{w},\varepsilon _{1},\varepsilon _{2},$ is of four types: $m_{1}m_{2},$ $m_{1}^{2}\ $and $m_{2}^{2}$, $m_{1}^{2}+m_{2}^{2}$ and $m_{1}^{2}-m_{2}^{2}.$ The actual isoscalar mesons consist of mixtures of three equal mass quark-antiquark pairs. We write the three separate equal mass versions of (\[57\]), using Eq. (\[bb\]), together in shorthand as $$\lbrack \mathbf{p}^{2}+\Phi _{w}(\mathbf{r,}m_{1}=m_{2},w,\mathbf{\sigma }_{1},\mathbf{\sigma }_{2})]\begin{bmatrix}
\psi _{u\bar{u}} \\
\psi _{d\bar{d}} \\
\psi _{s\bar{s}}\end{bmatrix}\equiv \lbrack \mathbf{p}^{2}+\Phi _{w}(\mathbf{r,}\mathbb{M})]\begin{bmatrix}
\psi _{u\bar{u}} \\
\psi _{d\bar{d}} \\
\psi _{s\bar{s}}\end{bmatrix}=\frac{1}{4}(w^{2}-4\mathbb{M}^{2})\begin{bmatrix}
\psi _{u\bar{u}} \\
\psi _{d\bar{d}} \\
\psi _{s\bar{s}}\end{bmatrix}. \label{mx}$$in which$$\mathbb{M=}\begin{bmatrix}
m_{u} & 0 & 0 \\
0 & m_{d} & 0 \\
0 & 0 & m_{s}\end{bmatrix}.$$Eq. (\[mx\]) does not include mixing between the pairs. Motivated by ideas presented by Brayshaw [^16] [@bry], we model the effects of $q_{i}\bar{q}_{i}\rightarrow q_{j}\bar{q}_{j}$ via two gluon annihilation and creation as an effective scalar potential by postulating a symmetric matrix $\mathbb{M}$ that is not diagonal, $$\begin{bmatrix}
m_{u} & 0 & 0 \\
0 & m_{d} & 0 \\
0 & 0 & m_{s}\end{bmatrix}\rightarrow \mathbb{M=}\begin{bmatrix}
m_{u} & 0 & 0 \\
0 & m_{d} & 0 \\
0 & 0 & m_{s}\end{bmatrix}+\begin{bmatrix}
\delta m_{u} & \sqrt{\delta m_{u}\delta m_{d}} & \sqrt{\delta m_{u}\delta
m_{s}} \\
\sqrt{\delta m_{u}\delta m_{d}} & \delta m_{d} & \sqrt{\delta m_{d}\delta
m_{s}} \\
\sqrt{\delta m_{u}\delta m_{s}} & \sqrt{\delta m_{d}\delta m_{s}} & \delta
m_{s}\end{bmatrix}. \label{53}$$Suppose that an orthogonal matrix $\mathbb{R}$ diagonalizes $\mathbb{M~}$[^17] $$\mathbb{RMR}^{-1}=\mathbb{M}_{D}.$$Then Eq. (\[mx\]) becomes $$\lbrack \mathbf{p}^{2}+\Phi _{w}(\mathbf{r,}\mathbb{M}_{D})]\psi =\frac{1}{4}(w^{2}-4\mathbb{M}_{D}^{2})\psi .$$This gives us, in essence, three new effective families of equal quark-anti-quark mesons, like ones that contain $b,c,s,u,d$ except that mixtures are involved. In this paper we see if this idea is successful for the ground state pseudoscalar isoscalar family of mesons alone. With the three parameters, one obtains three different effective quark masses, one for each isoscalar family. The three $\delta m_{i}$ are adjusted to give the best fit to the correct $\pi ^{0},\ \eta ,\ \eta ^{\prime }$ masses. [^18] Table \[16\] gives the values of $\delta m_{u},\ \delta m_{d},\ \delta m_{s}$ together with the three effective quark masses, the eigenvalues of $\mathbb{M}$ which we call $m_{q(\pi ^{0})},\ m_{q(\eta )},\ m_{q(\eta ^{\prime })}$.
$\begin{tabular}{|lccc|}
\hline
Parameter & Th1 & \ \ \ \ \ & Th2 \\ \hline
$m\_[u]{}$ & $0.1004$ & & $0.1070$ \\
$m\_[d]{}$ & $0.1378$ & & $0.1055$ \\
$m\_[s]{}$ & $0.0468$ & & $0.0578$ \\
$m\_[q(\^[0]{})]{}$ & $0.0737$ & & $0.1015$ \\
$m\_[q()]{}$ & $0.2175$ & & $0.2261$ \\
$m\_[q(\^)]{}$ & $0.4297$ & & $0.4536$ \\ \hline
\end{tabular}$
\[16\]
Paralleling the earlier Tables \[2\]-\[9\], Table \[17\] gives the best fit values for the $\pi ^{0},~\eta ,$ and $\eta ^{\prime }$ mesons. The predicted quark content becomes a further test of our model.
Mesons Exp. Th1. Th2. Exp.-Th1. Exp.-Th2. $\chi ^{2}$-Th1. $\chi ^{2}$-Th2.
------------------------------------ ------------ ------- ------- ----------- ----------- ------------------ ------------------ --
$\pi ^{0}:1\,{}^{1}S_{0}$ 0.135(0.0) 0.139 0.134 -0.004 0.001 0.2 0.0
$\eta :\ 1\,{}^{1}S_{0}$ 0.548(0.0) 0.548 0.548 0.000 0.000 0.0 0.0
$\eta ^{\prime }:\ 1\,{}^{1}S_{0}$ 0.958(0.2) 0.958 0.958 0.000 0.000 0.0 0.0
: $c\bar s$ Mesons,Theory 1 and 2 (GeV)[]{data-label="17"}
The corresponding eigenvectors are quite close to the mixtures$$\begin{aligned}
|\pi ^{0}\rangle &=&\frac{1}{\sqrt{2}}\left[
\begin{array}{c}
1 \\
-1 \\
0\end{array}\right] \equiv |\pi _{3}\rangle , \notag \\
~|\eta \rangle &=&\frac{\cos \theta }{\sqrt{6}}\left[
\begin{array}{c}
1 \\
1 \\
-2\end{array}\right] -\frac{\sin \theta }{\sqrt{3}}\left[
\begin{array}{c}
1 \\
1 \\
1\end{array}\right] \equiv \cos \theta ~|\eta _{8}\rangle -\sin \theta ~|\eta
_{1}\rangle , \notag \\
~|\eta ^{\prime }\rangle &=&\frac{\sin \theta }{\sqrt{6}}\left[
\begin{array}{c}
1 \\
1 \\
-2\end{array}\right] +\frac{\cos \theta }{\sqrt{3}}\left[
\begin{array}{c}
1 \\
1 \\
1\end{array}\right] \equiv \sin \theta |\eta _{8}\rangle +\cos \theta ~|\eta _{1}\rangle
,\end{aligned}$$With the three eigenvectors in matrix form we find$$\begin{bmatrix}
|\pi ^{0} & |\eta \rangle & |\eta ^{\prime }\rangle\end{bmatrix}=\begin{bmatrix}
0.770 & 0.470 & 0.431 \\
-0.638 & 0.574 & 0.514 \\
-0.006 & -0.671 & 0.742\end{bmatrix},$$corresponding to $\theta =-12.6$ degrees for Th1 and$$\begin{bmatrix}
|\pi ^{0} & |\eta \rangle & |\eta ^{\prime }\rangle\end{bmatrix}=\begin{bmatrix}
0.717 & 0.542 & 0.438 \\
-0.697 & 0.565 & 0.442 \\
-0.008 & -0.622 & 0.783\end{bmatrix},$$corresponding to $\theta =-16.3$ degrees for Th2. The Th2 value is consistent with chiral perturbation theory results corresponding to using the formula [@prtl]. $$\tan ^{2}\theta _{\lbrack quad]}=\frac{4m_{K}^{2}-m_{\pi }^{2}-3m_{\eta }^{2}}{m_{\pi }^{2}+3m_{\eta ^{\prime }}^{2}-4m_{K}^{2}}. \label{mix}$$With the meson masses listed in Tables \[2\] and \[3\] for Th2 we obtain $\theta _{\lbrack quad]}=-17.2$ degrees, reasonably close to our Th2 result of $-16.3$ degrees. On the other hand, using the values listed there for Th1 gives $\theta _{\lbrack quad]}=-7.3~$degrees which is significantly different from $-12.6$ degrees. Using the experimental masses gives $\theta _{\lbrack quad]}=-11.5$ degrees. Radiative vector meson decays give an angle between $-10$ and $-20$ degree while fits to tensor decay widths give $-17~$degrees. Note that even though the chiral symmetry and its breaking are not built into our model as in for example it is in [est]{}, it is of interest that the use of Eq. (\[mix\]) does produce a consistent result by use of our theoretically computed masses[^19].
The Quasipotential Equation of Ebert, Faustov and Galkin
========================================================
The Model and Comparisons with TBDE
------------------------------------
The model which we critically examine here gives excellent fits to the meson spectrum as well as numerous meson decay rates. The quasipotential approach of Ebert, Faustov and Galkin (EFG) [@rusger] is a local one very similar to that of Todorov [@quasi] and Aneva, Karchev, and Rizov [akr]{}, discussed in Sec. 2 (see Eq. (\[riz\])). Insofar as our discussion given in that section is concerned, the main difference between the TBDE and EFG approach is the replacement of the timelike vector confining interaction in Eq. (\[qua\]) with a different confining vector interaction: $$\begin{aligned}
\mathcal{V}(\mathbf{p-k})\beta _{1}\beta _{2} &\rightarrow &\mathcal{V}(\mathbf{p-k})\Gamma _{1\mu }\Gamma _{2}^{\mu }, \notag \\
\Gamma _{i\mu } &=&\gamma _{i\mu }-\frac{i\kappa (p-k)_{\nu }\sigma _{i\mu
\nu }}{2m_{i}},~i=1,2.\end{aligned}$$They include as do we a scalar confining potential. In coordinate space their choice is$$\begin{aligned}
V(r) &=&(1-\varepsilon )(Ar+B), \notag \\
S(r) &=&\varepsilon (Ar+B).\end{aligned}$$For their electromagnetic-like vector interaction they use the Coulomb gauge (instead of the Feynman gauge used in the TBDE).$$\frac{4}{3}\alpha _{s}D_{\mu \nu }(\mathbf{p-k)}\gamma _{1}^{\mu }\gamma
_{2}^{\nu }.$$Its momentum space form is$$D^{00}(\mathbf{p-k)}\mathbf{=-}\frac{4\pi }{\left( \mathbf{p-k}\right) ^{2}},~D^{ij}=-\frac{4\pi }{\left( p-k\right) ^{2}}(\delta ^{ij}-\frac{(p-k)^{i}(p-k)^{j}}{\left( \mathbf{p-k}\right) ^{2}}).$$ The addition of the Pauli term with their value $\kappa =-1$ has the effect of cancelling the lowest order spin-dependent contributions in each factor of $\Gamma _{i\mu }$ when sandwiched between on energy shell spinors. In the nonrelativistic limit their scalar and vector confining interactions combine to $(Ar+B)$ with $A=0.18$ GeV$^{2}$, B=$-0.3$GeV$.$ In essence their approach embodies a modified version of the Cornell potential into the local quasipotential approach. Their choice of $\kappa
$ was fixed by an analysis of the fine structure splitting of heavy quarkonia $^{3}P_{J}$ states [@rusger1] and their choice of $\varepsilon
=-1$ is determined from considerations of charmonium radiative decay.
Old Mesons Exp. TBDE EFG Exp.- TBDE Exp. - EFG. $\chi ^{2}$(TBDE) $\chi ^{2}$(EFG).
--------------------------------------------------- ------- ------- ------- ------------ ------------- ------------------- ------------------- --
$\pi :u\overline{d}\ 1\,{}^{1}S_{0}$ 0.140 0.134 0.154 0.005 -0.014 0.4 2.5
$\rho :u\overline{d}\ 1\,{}^{3}S_{1}$ 0.775 0.779 0.776 -0.003 -0.001 0.1 0.0
$b_{1}:u\overline{d}\ 1\,{}^{1}P_{1}$ 1.230 1.237 1.258 -0.007 -0.028 0.1 0.9
$a_{1}:u\overline{d}\ 1\,{}^{3}P_{1}$ 1.230 1.311 1.254 -0.081 -0.024 0.0 0.0
$\pi :u\overline{d}\ 2\,{}^{1}S_{0}$ 1.300 1.426 1.292 -0.126 0.008 0.0 0.0
$a_{2}:u\overline{d}\ 1\,{}^{3}P_{2}$ 1.318 1.303 1.317 0.015 0.001 2.0 0.0
$\rho :u\overline{d}\ 2\,{}^{3}S_{1}$ 1.465 1.674 1.486 -0.209 -0.021 0.8 0.0
$a_{0}:u\overline{d}\ 1\,{}^{3}P_{0}$ 1.474 1.015 1.176 0.459 0.298 7.0 3.0
$b_{2}:u\overline{d}\ 1\,{}^{1}D_{2}$ 1.672 1.752 1.643 -0.080 0.029 6.8 0.9
$a_{3}:u\overline{d}\ 1\,{}^{3}D_{3}$ 1.689 1.706 1.714 -0.017 -0.025 0.7 1.4
$a_{1}:u\overline{d}\ 1\,{}^{3}D_{1}$ 1.720 1.836 1.742 -0.116 -0.022 0.4 0.0
$a_{2}:u\overline{d}\ 2\,{}^{3}P_{2}$ 1.732 1.997 1.779 -0.265 -0.047 3.3 0.1
$\pi :u\overline{d}\ 3\,{}^{1}S_{0}$ 1.816 2.022 1.788 -0.206 0.028 2.6 0.0
$b_{2}:u\overline{d}\ 2\,{}^{1}D_{2}$ 1.895 2.252 1.960 -0.357 -0.065 6.0 0.2
$a_{4}:u\overline{d}\ 1\,{}^{3}F_{4}$ 2.011 2.042 2.018 -0.031 -0.007 0.1 0.0
$b_{2}:u\overline{d}\ 3\,{}^{1}D_{2}$ 2.090 2.682 2.216 -0.592 -0.126 5.0 0.2
$\rho :u\overline{d}\ 3\,{}^{3}S_{1}$ 2.149 2.309 1.921 -0.160 0.228 1.1 2.2
$a_{6}:u\overline{d}\ 1\,{}^{3}H_{6}$ 2.450 2.590 2.475 -0.140 -0.025 0.0 0.0
$K\,{}^{-}:s\overline{u}\ 1\,{}^{1}S_{0}$ 0.494 0.528 0.482 -0.034 0.012 14.2 1.6
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}S_{1}$ 0.892 0.898 0.897 -0.007 -0.005 0.5 0.3
$K\,{}^{-}:s\overline{u}\ 1\,{}^{1}P_{1}$ 1.272 1.336 1.294 -0.064 -0.022 1.0 0.1
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}P_{1}$ 1.403 1.354 1.412 0.049 -0.009 0.6 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}S_{1}$ 1.414 1.698 1.675 -0.284 -0.261 4.3 3.6
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}P_{0}$ 1.425 1.075 1.362 0.350 0.063 0.6 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}P_{2}$ 1.426 1.401 1.424 0.025 0.002 0.0 0.0
$K\,{}^{-}:s\overline{u}\ 2\,{}^{1}S_{0}$ 1.460 1.414 1.538 0.046 -0.078 0.0 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}D_{1}$ 1.717 1.828 1.699 -0.111 0.018 0.2 0.0
$K\,{}^{-}:s\overline{u}\ 1\,{}^{1}D_{2}$ 1.773 1.795 1.709 -0.022 0.064 0.1 0.8
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}D_{3}$ 1.776 1.784 1.789 -0.008 -0.013 0.0 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}D_{2}$ 1.816 1.787 1.824 0.029 -0.008 0.1 0.0
$K\,{}^{-}:s\overline{u}\ 3\,{}^{1}S_{0}$ 1.830 2.069 2.065 -0.239 -0.235 4.1 3.9
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}P_{2}$ 1.973 2.050 1.896 -0.077 0.077 0.1 0.1
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}F_{4}$ 2.045 2.106 2.096 -0.061 -0.051 0.5 0.4
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}D_{2}$ 2.247 2.301 2.163 -0.054 0.084 0.1 0.3
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}F_{3}$ 2.324 2.585 2.348 -0.261 -0.024 1.4 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 1\,{}^{3}G_{5}$ 2.382 2.387 2.356 -0.005 0.026 0.0 0.0
$K^{\ast }\,{}^{-}:s\overline{u}\ 2\,{}^{3}F_{4}$ 2.490 2.585 2.436 -0.095 0.054 0.3 0.1
$\phi :s\overline{s}\ 1\,{}^{3}S_{1}$ 1.019 1.017 1.038 0.002 -0.019 0.1 4.1
$\phi :s\overline{s}\ 1\,{}^{3}P_{0}$ 1.370 1.175 1.420 0.195 -0.050 0.0 0.0
$\phi :s\overline{s}\ 1\,{}^{3}P_{1}$ 1.518 1.436 1.464 0.082 0.054 3.1 1.4
$\phi :s\overline{s}\ 1\,{}^{3}P_{2}$ 1.525 1.505 1.529 0.020 -0.004 0.2 0.0
$\phi :s\overline{s}\ 2\,{}^{3}S_{1}$ 1.680 1.868 1.698 -0.188 -0.018 1.1 0.0
$\phi :s\overline{s}\ 1\,{}^{3}D_{3}$ 1.854 1.874 1.950 -0.020 -0.096 0.1 2.2
$\phi :s\overline{s}\ 2\,{}^{3}P_{2}$ 2.011 2.120 2.030 -0.109 -0.019 0.0 0.0
$\phi :s\overline{s}\ 3\,{}^{3}P_{2}$ 2.297 2.590 2.412 -0.293 -0.115 1.3 0.2
: Comparison of Th2($A,S,V$) Fits with Ebert et al [@rusger], [@rusger1]-[@rusger5][]{data-label="18"}
------------------------------------------------ -------- -------- -------- ------------- ---------- ------------ ----------------- --
New Mesons Exp. TBDE. EFG Exp. - TBDE Exp.-EFG $\chi $\chi ^{2}$-EFG
^{2}$-TBDE
$D^{+}:c\overline{d}\ 1\,{}^{1}S_{0}$ 1.870 1.881 1.871 -0.012 -0.001 1.5 0.0
$D^{\ast +}:c\overline{d}\ 1\,{}^{3}S_{1}$ 2.010 2.010 2.010 0.000 0.000 0.0 0.0
$D^{\ast +}:c\overline{d}\ 1\,{}^{3}P_{0}$ 2.403 2.224 2.406 0.179 -0.003 2.0 0.0
$D^{\ast 0}:c\overline{u}\ 1\,{}^{3}P_{2}$ 2.460 2.408 2.460 0.052 0.000 3.3 0.0
$D_{s}:c\overline{s}\ 1\,{}^{1}S_{0}$ 1.968 1.976 1.969 -0.007 -0.001 0.5 0.0
$D_{s}^{\ast }:c\overline{s}\ 1\,{}^{3}S_{1}$ 2.112 2.120 2.111 -0.008 0.001 0.7 0.0
$D_{s}^{\ast }:c\overline{s}\ 1\,{}^{3}P_{0}$ 2.318 2.338 2.509 -0.020 -0.191 3.7 320
$D_{s}:c\overline{s}\ 1\,{}^{1}P_{1}$ 2.535 2.498 2.536 0.038 -0.001 15.4 0.0
$D_{s}^{\ast }:c\overline{s}\ 1\,{}^{3}P_{2}$ 2.573 2.531 2.571 0.042 0.002 11.8 0.0
$D_{s}^{\ast }:c\overline{s}\ 2\,{}^{3}S_{1}$ 2.690 2.698 2.731 -0.008 -0.041 0.0 0.4
$\eta _{c}:c\overline{c}\ 1\,{}^{1}S_{0}$ 2.980 2.972 2.978 0.008 0.002 0.3 0.0
$J/\psi (1S):c\overline{c}\ 1\,{}^{3}S_{1}$ 3.097 3.126 3.097 -0.029 0.000 10.3 0.0
$\chi _{0}:c\overline{c}\ 1\,{}^{3}P_{0}$ 3.415 3.393 3.423 0.022 -0.008 5.2 0.7
$\chi _{1}:c\overline{c}\ 1\,{}^{3}P_{1}$ 3.511 3.500 3.509 0.010 0.002 1.3 0.0
$h_{1}:c\overline{c}\ 1\,{}^{1}P_{1}$ 3.526 3.519 3.525 0.007 0.001 0.6 0.0
$\chi _{2}:c\overline{c}\ 1\,{}^{3}P_{2}$ 3.556 3.553 3.556 0.004 0.000 0.1 0.0
$\eta _{c}:c\overline{c}\ 2\,{}^{1}S_{0}$ 3.637 3.597 3.663 0.040 -0.026 1.1 0.5
$\psi (2S):c\overline{c}\ 2\,{}^{3}S_{1}$ 3.686 3.683 3.684 0.004 0.002 0.1 0.1
$\psi (1D):c\overline{c}\ 1\,{}^{3}D_{1}$ 3.773 3.801 3.795 -0.028 -0.022 8.4 5.2
$\chi _{2}:c\overline{c}\ 2\,{}^{3}P_{2}$ 3.929 3.975 3.972 -0.046 -0.043 1.0 0.9
$\psi (3S):c\overline{c}\ 3\,{}^{3}S_{1}$ 4.039 4.083 4.088 -0.044 -0.049 0.2 0.3
$\psi (2D):c\overline{c}\ 2\,{}^{3}D_{1}$ 4.153 4.160 4.194 -0.007 -0.041 0.1 2.0
$B^{-}:b\overline{u}\ 1\,{}^{1}S_{0}$ 5.279 5.285 5.280 -0.006 -0.001 0.4 0.0
$B^{0}:b\overline{d}\ 1\,{}^{1}S_{0}$ 0.000 0.000 0.000 0.000 0.000 0.0 0.0
$B^{\ast -}:b\overline{u}\ 1\,{}^{3}S_{1}$ 5.325 5.334 5.326 -0.009 -0.001 0.8 0.0
$B^{\ast -}:b\overline{u}\ 1\,{}^{3}P_{2}$ 5.747 5.687 5.741 0.060 0.006 4.7 0.0
$B_{s}^{0}:b\overline{s}\ 1\,{}^{1}S_{0}$ 5.366 5.370 5.372 -0.004 -0.006 0.1 0.3
$B_{s}^{\ast 0}:b\overline{s}\ 1\,{}^{3}S_{1}$ 5.413 5.432 5.414 -0.019 -0.001 1.7 0.0
$B_{s}^{\ast 0}:b\overline{s}\ 1\,{}^{3}P_{1}$ 5.829 5.793 5.831 0.037 -0.002 11.0 0.0
$B_{s}^{\ast 0}:b\overline{s}\ 1\,{}^{3}P_{2}$ 5.840 5.805 5.842 0.034 -0.002 10.5 0.0
$\eta _{b}:b\overline{b}\ 1\,{}^{1}S_{0}$ 9.389 9.334 9.400 0.055 -0.011 2.1 0.1
$\Upsilon (1S):b\overline{b}\ 1\,{}^{3}S_{1}$ 9.460 9.447 9.460 0.014 0.000 2.1 0.0
$\chi _{b0}:b\overline{b}\ 1\,{}^{3}P_{0}$ 9.859 9.835 9.864 0.024 -0.005 6.1 0.2
$\chi _{b1}:b\overline{b}\ 1\,{}^{3}P_{1}$ 9.893 9.887 9.892 0.006 0.001 0.4 0.0
$\chi _{b2}:b\overline{b}\ 1\,{}^{3}P_{2}$ 9.912 9.921 9.912 -0.009 0.000 0.8 0.0
$\Upsilon (2S):b\overline{b}\ 2\,{}^{3}S_{1}$ 10.023 10.021 10.020 0.002 0.003 0.0 0.1
$\Upsilon (1D):b\overline{b}\ 1\,{}^{3}D_{2}$ 10.161 10.178 10.157 -0.017 0.004 2.4 0.1
$\chi _{b0}:b\overline{b}\ 2\,{}^{3}P_{0}$ 10.232 10.228 10.232 0.005 0.001 0.2 0.0
$\chi _{b1}:b\overline{b}\ 2\,{}^{3}P_{1}$ 10.255 10.261 10.253 -0.005 0.002 0.3 0.1
$\chi _{b2}:b\overline{b}\ 2\,{}^{3}P_{2}$ 10.269 10.284 10.267 -0.015 0.002 2.4 0.0
$\Upsilon (3S):b\overline{b}\ 3\,{}^{3}S_{1}$ 10.355 10.366 10.355 -0.011 0.000 1.0 0.0
$\Upsilon (4S):b\overline{b}\ 4\,{}^{3}S_{1}$ 10.579 10.628 10.604 -0.049 -0.025 11.7 3.0
------------------------------------------------ -------- -------- -------- ------------- ---------- ------------ ----------------- --
: Comparison of Th2($A,S,V$) Fits with Ebert et al [@rusger], [@rusger1]-[@rusger5][]{data-label="19"}
We compare in Tables \[18\] and \[19\] their results to our Th 2 by listing the deviations from the experimental results and respective $\chi
^{2}$. Of the 86 common mesons fit to the respective models, the collected results of quasipotential approach of [@rusger], [@rusger1]-[rusger5]{} (EFG) are more accurate in 69 of the fits (61 when comparing $\chi
^{2}$)[^20]. This includes the difficult singlet-triplet splittings for the ground and excited states of charmonium and the ground state of bottomonium, as well as good fits to most of the radial and orbital excitations of the ground states of the light mesons. Particularly interesting examples for the $\pi -\rho $ family are the$~^{3}P_{1}-^{1}P_{1}$ splitting and the radial excitation of the singlet and triplet $S-$states, and for the $K$-$K^{\ast }$ families the $~^{3}P_{1}-^{1}P_{1}$ splitting and the $^{3}P_{0}$ $\ $mass. These three areas of their spectrum are noteworthy improvements over the TBDE approach. It is hard, however, to give an even theoretical comparison between the two approaches for a number of different reasons. It is worthwhile, however, to point out the differences in the two approaches, summarized in Table \[20\]. First of all in our approach we give an overall fit to the entire spectrum. The approach of EFG to the spectrum is spread over several papers and it is not clear that a uniform parametrization would yield the same results as given in the tables (summarized here) from their separate papers. They do use the same values for the constants $A,\ B,\ \kappa ,$ and $\varepsilon $ as well as the quark masses in the various papers. However, the $\Lambda $ parameter they use in the parametrization of the short distance QCD-Coulomb part of the potential is different [^21]. Another difference is that the static QCD potential in the Adler-Piran model displays explicitly the asymptotic freedom behavior by its radial dependence [^22] whereas in the Coulomb potential used by EFG, the asymptotic freedom behavior is displayed indirectly in their $\alpha _{s}$ by the quark mass dependence as seen in footnote 20. On the other hand, for their heavy quark bound states a radial modification displaying short distance QCD asymptotic freedom corrections was used but was not for the light quark bound states. Both the quasipotential approach of EFG and the TBDE used here, extending earlier work of Crater and Van Alstine, have three invariant interaction functions. Both use electromagnetic-like four-vector interactions; in the Feynman gauge for the TBDE and the Coulomb gauge for the quasipotential approach. In the EFG quasipotential approach the third interaction is a Pauli-modified vector interaction whereas in our approach it is timelike vector interaction. The potentials used in each of the three parts are of course different in the two approaches. The spin dependence, although similar for the most part have distinctly different origins. In the 16 component TBDE the kinematics and dynamics and spinors are tied together in one wave equation (see. e.g. Eq. (\[schlike\])). Its off shell dependence is fixed by the wave equation and the spinors are all interacting. In the quasipotential approach the potential is constructed in part from the actions of free particle spinors. This leaves substantial leeway in how the off-shell behavior is fixed.
Properties TBDE Quasipotential Approach of EFG
------------------------------------------- ----------------------------------------- ---------------------------------------------
Invariant Interactions 3-$A($EM-like vector)$,\ S$(scalar), 3-$A($EM-like vector)$,\ S$(scalar),
$V$(timelike vector) $V$(Pauli-modified vector)
QCD Coupling Coordinate Space Dependent Quark Mass Dependent
Meson Fits Overall Spectrum, Same Spectrum in Parts with Different
Parametrizations Parametrizations
Spin- Dependence Fixed by the TBDE given $A,\ S,\ V$ Fixed by the Quasipotential and $A,\ S,\ V$
Kinematics Exact Exact
Singular potentials Avoided by Dirac Equation Formalism Avoided by Adhoc Substitutions
Numerical evaluations Yes Not for Heavy Mesons
Chiral Symmetry Zero Pion Mass for $m_{q}\rightarrow 0$ Not Tested
QED Spectral Tests Both Perturbative and Nonperturbative Perturbative Only
Static Limit ($m_{2}\rightarrow \infty )$ Reduces to One-Body Dirac Eq. Does Not Reduce to One-Body Dirac Eq.
: Comparison of TBDE with EFG[]{data-label="20"}
Both approaches have exact relativistic kinematics (from the use of $b^{2}(w) $) and do not use either $v/c$ or $1/m_{q}$ expansions. Both approaches lead to non-linear eigenvalue equations and give good values of the pion and kaon masses. Both approaches avoid singular effective potentials that would otherwise prevent nonperturbative spectral calculations. In the quasipotential approach of EFG, those singularities are avoided in an adhoc though plausible fashion (see Eq. (12) in [rusger]{}). In the case of the TBDE, natural smoothing mechanisms appear in the Dirac formalism allowing one to avoid these adhoc assumptions [cra82]{}, [@becker] [^23]. In addition, in the approach of the TBDE, strictly nonperturbative (i.e. numerical techniques) are used for the spectral evaluations. This does not appear to be the case for the work of EFG, particularly for the heavy mesons, where use of perturbation theory is required because of singular potentials. It may very well be that their adhoc substitutions used in the later paper [@rusger] will render the use of perturbation theory unnecessary for those mesons. But in that case clear tests must ensure that not only do the adhoc substitutions give the same results as the perturbative treatments, but that this hold in the sensitive testing grounds of QED [@iowa], [@becker] ground states and those of related field theories . Both approaches display chiral symmetry breaking through the appearance of quark masses. It has been demonstrated in the TBDE, however, that the pion mass vanishes in the limit in which the quark mass vanishes [@crater2]. That is not demonstrated in the EFG formalism nor for any other potential model formalism that we know of for the mesons (In an exception, Sazdjian has demonstrated this using pseudoscalar interactions for the TBDE [@saz86]).[^24]
The final point we want to make about the differences is that the wave equation arising from the TBDE has been tested in perturbative QED and related field theories. Todorov, and others [@quasi], [@akr] and [@tod75] showed using perturbative methods how their local version of the quasipotential equation displays the accepted QED spectral results through order $\alpha ^{4}$ for two oppositely charged particles with arbitrary mass ratios. The work by Crater and Van Alstine [@exct] and others [@becker] go beyond this and show that not only do the TBDE display the correct fine and hyperfine spectral results when treated perturbatively, but those same results can be recovered when the equations are treated nonperturbatively. In [@mfa], a local quasipotential equation closely related to that used by EFG in the meson spectrum was shown to also reproduce perturbatively the spectral results through order $\alpha
^{4}$ of QED for two oppositely charged particles with arbitrary mass ratios. However, the important nonperturbative tests as done in [becker]{} of the bound state formalism for QED has not been carried out with the local quasipotential equation of EFG.
It is our contention that any relativistic potential model that includes four-vector interactions should, when the vector interaction is replaced by its QED counterpart, and confining potentials are set $=0$, reproduce the the standard QED spectral results. There are two models for which we carry out this test. We limit our test to the singlet positronium ground state. The fundamental question we ask is do these two approaches, [@isgr] and [@rusger], which are used quite successfully for meson spectroscopy give the correct spectral results when restricted to QED. The first one we examine is that of Ebert, Faustov and Galkin [@rusger].
Positronium Ground State Spectral Test of the Quasipotential Equation of Ebert, Faustov and Galkin.
---------------------------------------------------------------------------------------------------
The effective Schrödinger equation of this approach restricted to an equal mass bound system for vector interactions is given in Eqs. (1-4), (13-22) of [@rusger]. For that restriction we have, in the notation of the present paper,$$\begin{aligned}
\left( \frac{\mathbf{p}^{2}}{2\mu _{R}}+V(r)\right) \Psi _{w} &=&\frac{b^{2}(w)}{2\mu _{R}}\Psi _{w}, \notag \\
\mu _{R} &=&\frac{\varepsilon _{1}\varepsilon _{2}}{w}=\frac{w}{4}, \notag
\\
b^{2}(w) &=&\frac{1}{4}(w^{2}-4m^{2}), \notag \\
V(r) &=&A(r)[1+(\frac{w-2m}{w})^{2}]+(\frac{2}{w(w+2m)}+\frac{8}{3w^{2}}\mathbf{S}_{1}\cdot \mathbf{S}_{2})\nabla ^{2}A, \notag \\
A &=&-\frac{\alpha }{r}. \label{eff}\end{aligned}$$The unperturbed effective Hamiltonian is$$H_{0}=\frac{\mathbf{p}^{2}}{2\mu _{R}}-\frac{\alpha }{r},$$and the perturbation is for singlet states$$H_{1}=-(\frac{w-2m}{w})^{2}\frac{\alpha }{r}-(\frac{8\pi \alpha }{w(w+2m)}-\frac{8\pi \alpha }{w^{2}})\delta ^{3}(\mathbf{r).} \label{prt}$$Comparing with the nonrelativistic hydrogenic Schrödinger equation $$\left( \frac{\mathbf{p}^{2}}{2\mu }-\frac{\alpha }{r}\right) \psi =\mathcal{E}\psi _{w}, \label{s1}$$with ground state energy$$\mathcal{E=-}\frac{\mu \alpha ^{2}}{2}, \label{s2}$$we see that for the eigenvalue equation (\[eff\]), the total c.m. invariant energy from $H_{0}$ is determined by analogy with Eqs. (\[s1\], \[s2\]) from$$\frac{b^{2}(w)}{2\mu _{R}}=-\frac{\mu _{R}\alpha ^{2}}{2}. \label{ev}$$Thus with $$w^{2}-4m^{2}=-4\mu _{R}^{2}\alpha ^{2}=-\frac{w^{2}}{4}\alpha ^{2}$$we find that $$w=2m-\frac{m\alpha ^{2}}{4}+\frac{3m\alpha ^{4}}{64}.$$Substituting this into Eq. (\[prt\]) at the appropriate order gives $$\begin{aligned}
H_{1} &=&-(\frac{m\alpha ^{2}}{8m})^{2}\frac{\alpha }{r}+\frac{\pi \alpha }{m^{2}}\delta ^{3}(\mathbf{r)} \notag \\
&\rightarrow &\frac{\pi \alpha }{m^{2}}\delta ^{3}(\mathbf{r)},\end{aligned}$$with their spin-spin terms partially canceling their spin independent contact (Darwin) term while the first term is of higher order. The ground state unperturbed wave function is$$\begin{aligned}
\Psi _{w} &=&\frac{\exp (-r/a_{eff})}{\left( \pi a_{eff}^{3}\right) ^{1/2}},
\notag \\
a_{eff} &=&\frac{1}{\mu _{R}\alpha }\rightarrow \frac{2}{m\alpha }.\end{aligned}$$The expectation value is$$\langle H_{1}\rangle =\frac{\alpha }{m^{2}\left( \frac{2}{m\alpha }\right)
^{3}}=\frac{m\alpha ^{4}}{8},$$and so Eq. (\[ev\]), including $\langle H_{1}\rangle $ then becomes, to the appropriate order$$\begin{aligned}
\frac{b^{2}(w)}{2\mu _{R}} &=&-\frac{\mu _{R}\alpha ^{2}}{2}+\frac{m\alpha
^{4}}{8}, \notag \\
w^{2}-4m^{2} &=&-\frac{w^{2}}{4}\alpha ^{2}+\frac{m^{2}\alpha ^{4}}{2},\end{aligned}$$so that $$w=2m-\frac{m\alpha ^{2}}{4}+\frac{3m\alpha ^{4}}{64}+\frac{m\alpha ^{4}}{8}=2m-\frac{m\alpha ^{2}}{4}+\frac{11m\alpha ^{4}}{64}. \label{evq}$$
This result is in disagreement with the accepted fine structure result of $$w=2m-\frac{m\alpha ^{2}}{4}-\frac{21m\alpha ^{4}}{64}. \label{exct}$$In the Two-Body Dirac equation this spectrum results from an exact solution of the Schrödinger-like form [@exct], [@becker], [@crstr] [^25] $$(\mathbf{p}^{2}+2\varepsilon _{w}A-A^{2})\psi =b^{2}\psi ,$$which yields $$w=m\sqrt{2+2/\sqrt{1+\frac{\alpha ^{2}}{\left( 1+\sqrt{\frac{1}{4}-\alpha
^{2}}-\frac{1}{2}\right) ^{2}}}}=2m-\frac{m\alpha ^{2}}{4}-\frac{21m\alpha
^{4}}{64}+.., \label{ect}$$obtained by steps similar to those outlined in Eq.(\[evq\]).
The local quasipotential approach of [@rusger] does not include the term $-A^{2}=$ $-\alpha ^{2}/r^{2}$ which gauge invariance considerations would demand. It is of interest that if their approach includes this terms then the added contribution from this potential is$$\frac{1}{2\mu }\langle -\frac{\alpha ^{2}}{r^{2}}\rangle =\frac{1}{m}\langle
-\frac{\alpha ^{2}}{r^{2}}\rangle =-\frac{m\alpha ^{4}}{2}.$$This, together with the fact that combining this with the earlier results gives the correct added $O(\alpha ^{4})$ correction,$$\frac{11m\alpha ^{4}}{64}-\frac{m\alpha ^{4}}{2}=-\frac{21m\alpha ^{4}}{64},
\label{df}$$points strongly to the lack of this term as being the cause of the incorrect QED spectral prediction in this approach.
We should emphasize that the closely related formalism of [@mfa] *does* produce the correct $-\frac{21m\alpha ^{4}}{64}$ relativistic correction. The difference between the formalism of EFG and that of [mfa]{} is that the former does not include two loop and iterated Born diagrams[^26] contained in the latter. Those combined diagrams to lowest order in $\alpha $ do produce the $-A^{2}=$ $-\alpha ^{2}/r^{2}$ $\ $term which would account for the spectral difference seen in Eq. (\[df\]) (see also [@tod75]). Since that $-A^{2}$ term is not included in the EFG meson spectral formalism it is likely that possibly important relativistic corrections for their meson spectrum will be missing. In a private communication, Faustov stated that the reason they did not include the contributions of two and more gluon exchange diagrams within QCD in calculations of the meson spectra, is that the effects of these diagrams would be contained in the confining , long range potential, the origin of which is not known and which is thus added phenomenologically. However the $-A^{2}$ contribution from those two-gluon exchange diagrams due to the Coulomb-like potential $A$ is short range and therefore would not by itself contribute to the confining potential. In other words we claim that since its effects are short range, it should be considered apart from the phenomenologically added confining interaction.
Positronium Test of the Approach of Godfrey and Isgur.
------------------------------------------------------
Although Crater and Van Alstine carried out an earlier comparison [crater2]{} with this approach [@isgr], in light of the problem with the above quasipotential approach it is instructive to include a parallel perturbative treatment on their different quasipotential equation. Their relativistic Schrödinger equation (see their Eqs. (1-4)) relevant for the case considered here has the Hamiltonian which include the spin-spin term in addition to the modified Coulomb term (see their Eq. (A15) [^27]). Their equation was of the quasipotential type given in Eq. (\[log\]) extended to include spin. Here we consider its semirelativistic expansion.$$\begin{aligned}
H &=&2\sqrt{\mathbf{p}^{2}+m^{2}}-\frac{\alpha }{r}+\frac{2}{3m^{2}}\mathbf{S}_{1}\cdot \mathbf{S}_{2}\nabla ^{2}A-\frac{\alpha }{2m^{2}}\{\mathbf{p}^{2},\frac{1}{r}\} \notag \\
&\rightarrow &2m+H_{0}+H_{1}, \notag \\
H_{0} &=&\frac{\mathbf{p}^{2}}{m}-\frac{\alpha }{r}, \notag \\
H_{1} &=&-\frac{\left( \mathbf{p}^{2}\right) ^{2}}{4m^{3}}+\frac{2\pi \alpha
}{m^{2}}\delta ^{3}(\mathbf{r)}-\frac{\alpha }{2m^{2}}\{\mathbf{p}^{2},\frac{1}{r}\}.\end{aligned}$$The ground state unperturbed wave function is$$\begin{aligned}
\Psi &=&\frac{\exp (-r/a)}{\left( \pi a^{3}\right) ^{1/2}}, \notag \\
a &=&\frac{2}{m\alpha }.\end{aligned}$$We find that$$\begin{aligned}
\langle H_{1}\rangle &=&-\frac{1}{4m}\langle \Psi \frac{\mathbf{p}^{2}}{m}\frac{\mathbf{p}^{2}}{m}\Psi \rangle +\langle \Psi \frac{2\pi \alpha }{m^{2}}\delta ^{3}(\mathbf{r)}\Psi \rangle -\frac{\alpha }{2m^{2}}\langle \Psi \{\mathbf{p}^{2},\frac{1}{r}\}\Psi \rangle \\
&=&-\frac{1}{4m}\langle \Psi (\mathcal{E+}\frac{\alpha }{r})^{2}\Psi \rangle
+\frac{m\alpha ^{4}}{4}-\frac{\alpha }{2m^{2}}\langle \Psi \{\mathbf{p}^{2},\frac{1}{r}\}\Psi \rangle ,\end{aligned}$$and with $\mathcal{E}\mathcal{=-}\frac{m\alpha ^{2}}{4}$ we have $$\begin{aligned}
-\frac{1}{4m}\mathcal{E}^{2} &=&-\frac{m\alpha ^{4}}{64}, \notag \\
-\frac{\mathcal{E}}{2m}\langle \Psi \frac{\alpha }{r}\Psi \rangle &=&\frac{\alpha ^{2}}{8}(-2\mathcal{E)=}\frac{m\alpha ^{4}}{16}, \notag \\
-\frac{1}{4m}\langle \Psi \frac{\alpha ^{2}}{r^{2}}\Psi \rangle &=&-\frac{1}{4m}\frac{\alpha ^{2}}{\pi a^{3}}4\pi \int_{0}^{\infty }dr\exp (-2r/a)=-\frac{m\alpha ^{4}}{8},\end{aligned}$$and using$$\begin{aligned}
-\frac{\alpha }{2m^{2}}\langle \Psi \{\mathbf{p}^{2},\frac{1}{r}\}\Psi
\rangle &=&-\frac{2(2\mu )\alpha }{2m^{2}}\langle \Psi \frac{1}{r}(\mathcal{E+}\frac{\alpha }{r})\Psi \rangle \notag \\
&=&-\frac{\mathcal{E}}{m}\langle \Psi \frac{\alpha }{r}\Psi \rangle -\frac{1}{m}\langle \Psi \frac{\alpha ^{2}}{r^{2}}\Psi \rangle =\frac{m\alpha ^{4}}{8}-\frac{m\alpha ^{4}}{2}\end{aligned}$$and find that$$\langle H_{1}\rangle =\frac{m\alpha ^{4}}{4}(-\frac{1}{16}+\frac{1}{4}-\frac{1}{2}+1+\frac{1}{2}-2),$$and so $$w=2m+\langle H_{0}\rangle +\langle H_{1}\rangle =2m-\frac{m\alpha ^{2}}{4}-\frac{13m\alpha ^{4}}{64}.$$This also does not agree with the accepted result of Eq. (\[exct\]). Since the addition of the $-\alpha ^{2}/r^{2}$ term would drive this to the other side of the accepted value it is not clear where the error is in this approach.
Comparison of Two Approaches to Dirac Equation in the Static Limit
------------------------------------------------------------------
The dynamics of the heavy-light $q\bar{Q}$ bound states, particularly the $u\bar{b}$ and $d\bar{b},$ should be well approximated by the ordinary one-body Dirac equation. In the limit when say $m_{2}\rightarrow \infty $ details outlined in Appendix A.5 show that our TBDE reduce to the single particle Dirac equation for a spin-one-half particle in an external scalar and vector potential,$$(\mathbf{\gamma }\cdot \mathbf{p-}\beta (\varepsilon -A)+m+S)\psi =0,
\label{obde}$$in which $\varepsilon $ is the total energy of the single particle of mass $m $. In this same limit Eq. (\[57\]) (see Appendix A5 ) becomes$$\begin{aligned}
&&(\mathbf{p}^{2}+2mS+S^{2}+2\varepsilon A-A^{2}+\frac{1}{2}\frac{\nabla
^{2}A-\nabla ^{2}S}{m+S+\varepsilon -A}+\frac{3}{4}\left( \frac{S^{\prime
}-A^{\prime }}{m+S+\varepsilon -A}\right) ^{2}+\frac{A^{\prime }-S^{\prime }}{m+S+\varepsilon -A}\frac{\mathbf{L\cdot \sigma }_{1}}{r})\psi _{+} \notag
\\
&=&(\varepsilon ^{2}-m^{2})\psi _{+}, \label{571}\end{aligned}$$which agrees with the Pauli reduction of the Dirac equation (\[obde\]) for a single particle in an external scalar and vector potential when the first order momentum terms are scaled away (see for example [@yoon]). The wave function $\psi _{+}$ is the upper two-component spinor. From the point of view of the single particle Dirac equation the quadratic $S^{2}$ and $-A^{2}$ terms above are not put in by hand but arise naturally from the Pauli reduction.
We compare Eq. (\[571\]) with the corresponding equations from the two quasipotential approaches, including scalar and vector interactions[^28]. Referring to Eqs. (1-4), (13-22) of [@rusger] we have the $m_{2}\rightarrow \infty $ limit of$$(\mathbf{p}^{2}+2\varepsilon (A+S)+\frac{1}{2}\frac{\nabla ^{2}A}{m+\varepsilon }+\frac{A^{\prime }-S^{\prime }}{m+\varepsilon }\frac{\mathbf{L\cdot \sigma }_{1}}{r})\psi =(\varepsilon ^{2}-m^{2})\psi , \label{rgst}$$in which we have used $\mu _{R}\rightarrow \varepsilon .~$It is evident that for the vector interaction alone ($S=0$), this equation will not yield a spectrum perturbatively equivalent to the Dirac spectrum for $A=-\alpha /r$ and for the same reason as with positronium, that is, the lacking of the $-A^{2}$ term. This would result in incorrect fine structure for hydrogen-like atoms (see discussions and footnotes below Eq. (\[df\]) for the reason for this omission). Also, there are three parts for the scalar potential interaction that differ from the Dirac equation: the appearance of $2\varepsilon S$ instead of the expected $2mS$, the lacking of the $S^{2}$ term, and the absence of any scalar Darwin term. The appearance of $\varepsilon $ instead of $m$ can be traced to the use of a common reduced mass for multiplying both vector and scalar interactions. Beyond that is the absence of the potential energy terms in the denominators of the spin-orbit and Darwin terms. The authors correct this by hand, but their correction does not match the forms in the Dirac equation. On the other hand those potential energy terms in the denominators of the Darwin and spin-orbit terms of Eq. (\[571\]) provide a natural smoothing mechanism that eliminates such singular potentials as delta functions and $1/r^{3}$ potentials. For example take the case of $S=0,~A=-\alpha /r$. The Laplacian term would produce $4\pi \delta ^{3}(\mathbf{r)}$. However, the $A$ term in the denominator would then be evaluated at the origin and completely cancel the effects of the delta function. Its perturbative effects are reproduced by the adjacent $3/4$ term. Similarly the $1/r^{3}$ behavior of the spin-orbit weak potential form in which the $A$ in the denominator is ignored is modified to very near the origin to a less singular $1/r^{2}$ potential by the effect of the $A$ term in the denominator as well as the $3/4$ term. Similar smoothing mechanics naturally built in to the Pauli structure of the Dirac equation occur in the Pauli reduction of the TBDE ( see [@cra82],[@cra84],[@becker], [@yoon]).
Referring to A-15, 16 of [@isgr] we have the $m_{2}\rightarrow \infty $ limit of$$(\sqrt{\mathbf{p}^{2}+m^{2}}+A+S+\frac{A^{\prime }-S^{\prime }}{4m^{2}}\frac{\mathbf{L\cdot \sigma }_{1}}{r})\psi =w\psi .$$This Hamiltonian form is missing Darwin terms for not only scalar interactions, but also for vector interactions as well. Those are as important for spectral studies as the spin-orbit terms and their lack is a serious defect in both these equations. The lack of the vector Darwin term would result in incorrect fine structure for $L=0$ hydrogen levels.
Conclusion and Future Directions
================================
The application of Dirac’s constraint dynamics applied to the relativistic two-body problem leads quite naturally to the Two Body Dirac equations of constraint dynamics when both particles have spin-one-half. This paper follows many earlier ones analyzing the structures and applications of those equations. It has several sets of aims and results. First we showed that when the interaction structure used in these equations is extended from two invariant functions (generated by what we have called $A(r)$ and $S(r)$) to three (not only the two that generate an electromagnetic-like interactionand a confining world scalar interaction but also the $V(r)$ that generates a confining timelike vector interaction), that the fit to the meson spectrum is improved substantially. However, there is still a considerable amount of improvement that is needed, primarily in the radial and orbital excitations of the singlet and triplet ground states. Work in progress seeks to extend invariant functions to ones that generate covariant pseudoscalar, pseudovector, and tensor interactions. Second, this paper also included 19 mesons not included in earlier work [@crater2] where only two invariant functions were used. Among those 19 were the isoscalar $\eta $ and $\eta
^{\prime }$ mesons. Here we developed an approach motivated by some work of Brayshaw [@bry] which introduces a constant symmetric but nondiagonal mass matrix that couples isoscalar $q\bar{q}$ channels. The three parameters introduced are adjusted (when possible) to fit the $\pi ^{0},~\eta $ and $\eta ^{\prime }$ meson masses and then used to predict accurately, at least in Th2, the $SU(3)$ pseudoscalar mixing angle . Missing is an attempt to connect this mass matrix to the compatibility condition between the constraints $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$.
A glance at Ref. [@yaes] shows that there are no shortages of attempts to stake claims of which relativistic two-body truncation of the Bethe-Salpeter equation is most successful. One of the purposes of [crater2]{} was to clarify some guidelines and important benchmarks that such equations should have when applied to any relativistic two-body problem be it for QED bound states, QCD bound states, or nucleon-nucleon scattering. This clarification continues in this paper with a fairly detailed analysis of the local quasipotential approach of Ebert, Faustov, and Galikin. This approach was chosen among numerous others for two reasons: a) the close connection between the minimal dynamical structure of the constraint approach and the early work done on the local quasipotential approach by Todorov and his coauthors and b) the extensive phenomenological studies by the local quasipotential approach of EFG in meson spectral and decay studies.
The two quasipotential methods that we discussed have three weaknesses. Neither method uses a wave equation to uncover their respective spin-dependent corrections. Rather, they use an on shell version of the scattering amplitude and the quasipotential equation for the potential. However, as discussed in both papers, they each must make assumptions that allow them to include expected off shell effects. In contrast, the TBDE of constraint dynamics include automatically by their mathematically consistent construction, off mass shell effects. The second weakness, is that both approaches, when applied to QED, do not produce the correct hyperfine structure. This, in our opinion, is a serious but easily correctable drawback in both approaches. Let us be precise here about our concerns. In both of these two quasipotential approaches to the relativistic QCD potential model, if one turns off the confining interaction and replaces the nonconfining vector potential by the Coulomb potential their resultant QED spectrum will be incorrect whether computed perturbatively or numerically. This calls into question their QCD spectral results since the resultant relativistic corrections which the omitted term $-A^{2}$ would have contributed is of the same order as the spin-dependent corrections (dependent on $A^{\prime }$ and $\nabla ^{2}A$) which, of course, are not omitted. Even if that correction is made, the wave equation should be shown to have (just as can be shown with the one-body Dirac equation) the same spectral results whether treated perturbatively or nonperturbatively. In addition, their modeling of the scalar interactions does not conform with the approach of the classical and quantum field theories used by Crater, Van Alstine, Yang, Sazdjian and Jolluli which supports our choice of vector interaction structures[^29]. The third weakness is the lack of agreement in the $m_{2}\rightarrow \infty $ limit with the Pauli-form of the one-body Dirac equation. In particular, the lack of scalar Darwin terms in both approaches and vector Darwin terms in the approach of [@isgr] is a serious weakness.
The constraint approach has been tested against both classical and quantum field theories for both scalar and vector interactions. In our construction of the vector potential [@cra84], [@cra87], [becker]{} three primary guideposts were used beyond that of a minimal structure. The first is the use of just the barest (lowest order) input from field theory, the nonrelativistic Coulomb potential in QED.
The second is the gauge-like minimal coupling structure Eq. (\[gage\]) of the potential which Todorov postulated and was later confirmed in three independent ways: a) Rizov, Todorov, and Aneva [@tod75] demonstrated how the gauge structure (particularly the form$~(\varepsilon
_{w}-A)^{2}-m_{w}^{2}$ ) arises in perturbation theory at higher order than the Born approximation [^30], b) by a comparison of the Fokker-Tetrode classical field theory $O(1/c^{2})$ expansion for the Hamiltonian with the general quasipotential structure of $\mathbf{p}^{2}+\Phi (\mathbf{r,}w)=(\varepsilon _{w}^{2}-m_{w}^{2})~$[@fw] [^31], and c) Jollouli and Sazdjian [@saz97] who found a similar structure from nonperturbative quantum field theoretic arguments both for scalar and vector interactions. These three arguments demonstrate that the Born approximation structures (particularly the first term of (\[qua\]) used to model the QCD potentials in [@isgr]) cannot possibly yield that gauge structure and thus cannot yield the correct positronium spectrum when applied to QED. The higher order structures in [@saz97] and [tod75]{} as well as the nonlinear hyperbolic structures in Eqs. (\[hyp1\], \[hyp2\], \[hyp3\], \[one\]-\[three\]) argue that the Born structure of the first term in (\[qua\]) are insufficient and must be supplemented by other invariant couplings. The papers by Sazdjian and collaborators [@saz86], [@saz94], [@saz97] demonstrate this explicitly, showing that pseudovector coupling is essential if done perturbatively in order to get the Dirac equation into an external field form in which the minimal structure can be demonstrated. The work in [@jmath], [long]{} as well as by Sazdjian and collaborators show that the vector coupling (see Eq. (\[vecc\]) ) alone will, when placed in the nonlinear context of the hyperbolic parametrization given in Eqs. (\[hyp1\],[hyp2]{}), yield that external field form in which the minimal structure can be demonstrated.
The third guidepost is the use of the relativistic reduced mass $m_{w}$ and energy $\varepsilon _{w}$ (see Eqs. (\[mw\]) and (\[ew\])). In addition to the discussions given in [@tod], their appearance in the forms $2m_{w}S+2\varepsilon _{w}A$ as part of the minimal structure is also dictated by the same field theory mechanisms discussed in item b) and c) above. Note that even though in Eq. (\[tod0\]) there is the appearance of another relativistic reduced mass ($\varepsilon
_{1}\varepsilon _{2}/w$), in working out the quasipotential using the spinors as in Eq. (\[riz\]) that reduced mass is replaced by $m_{w}$ in the scalar and $\varepsilon _{w}$ in the vector case. This replacement does not appear in the work of [@rusger].
So, while the works of Ebert, Faustov, and Galkin, and Godfrey and Isgur are quite impressive in terms of spectral and decay agreements, it would be of value for adherents of those approaches to consider the criticisms presented in this paper, related to the ability of those equations to reproduce the static limit Dirac equations structures (which would include the $-A^{2}$ term) and general short range structure related to the $-A^{2}$ gauge term. These criticisms are not about their choice of QCD inspired potentials, but rather about how their relativistic wave equations translate the physics of those potentials into spectral results. These thus concern the impact on their QCD meson spectral results those two approaches would have based on their field theory connections to QED bound states .
Finally a few words about future direction lines of research related to this paper. The approach given in our paper views the meson as a two-body bound state in a first quantized formalism. In place of the nonrelativistic Schrödinger equation for two interacting particles, we use the TBDE of constraint dynamics. There are systematic corrections that should follow the completion of this first step which would end with the inclusion of covariant pseudoscalar, pseudovector, and tensor interactions, in addition to the scalar and vector interactions we have included in this paper. First is a second quantized version the TBDE similar that what has been accomplished for nonrelativistic second quantized formalisms by the Cornell group [@cornell], Tornqvist [@trv], and more recently by Barnes and Swanson [@bsw]. The latter includes a microscopic theory [@bsa] of the $^{3}P_{0}$ model which includes pair production. The aim, as in their recent paper, would be to gain a measure of the effects of two-body meson decays on the observed rest mass of the decaying meson. Beyond that would be many body formalisms which view the meson more generally as a linear combination of $q\bar{q}+q\bar{q}g+...~$
The primary weakness in our results of this paper are: 1) the radial and orbital excitations of the old meson spectroscopy; 2) the less than good hyperfine splittings of the $c\bar{c}$ and $b\bar{b}$ families compared with the good splitting results we obtained with all the other families, including the lightest and most highly relativistic $u\bar{d}~$; 3) the failure to account for the light ($^{3}P_{0})$ scalar mesons; 4) Failure to reproduce the square root Gell-Mann-Oakes-Renner relation. It is too early to say if the completion of the first quantization program will rectify any of these problems. We point out, however, that should the weakness of our model, relating to the radial excitations of the light $q\bar{q}$ bound states be substantially improved by including pseudoscalar, pseudovector, and tensor interactions, this would offer the opportunity of allowing the hypothesis discussed in footnote 13 to be actively considered. Whether including these other interactions lead to substantial improvements or not, it will be essential to follow in parallel with our relativistic formalism, the second quantized nonrelativistic formalisms developed earlier.
Relativistic Schrödinger Equation Details
=========================================
Connections Between the TBDE and Eq. ( \[57\]) and Forms for $\tilde{A}_{i}^{\protect\mu },\tilde{S}_{i}$ in Terms of the Invariants $A(r),V(r),$ and $S(r)$
-------------------------------------------------------------------------------------------------------------------------------------------------------------
Here we present an outline of some details of Eq. (\[tbde\]) and its Pauli-Schrödinger reduction given in full elsewhere (see [cra87,jmath,long,liu]{}). Each of the two Dirac equations in (\[tbde\]) has a form similar to a single particle Dirac equation in an externalfour-vector and scalar potential but here acting on a 16 component wave function $\Psi $ which is the product of an external part (being a plane wave eigenstate of $P)~$multiplying the internal wave function $\psi $ $$\psi =\begin{bmatrix}
\psi _{1} \\
\psi _{2} \\
\psi _{3} \\
\psi _{4}\end{bmatrix}.$$The four $\psi _{i}$ are each four-component spinor wave functions. To obtain the actual general spin-dependent forms of those $\tilde{A}_{i}^{\mu
},\tilde{S}_{i}$ potentials which were required by the compatibility condition $[\mathcal{S}_{1},\mathcal{S}_{2}]\psi =0$ was a most perplexing problem, involving the discovery of underlying supersymmetries in the case of scalar and timelike vector interactions[@cra82], [@cra87]. Extending those external potential forms to more general covariant interactions necessitated an entirely different approach leading to what is called the hyperbolic form of the TBDE. The most general hyperbolic form for compatible TBDE is $$\begin{aligned}
\mathcal{S}_{1}\psi &=&(\cosh (\Delta )\mathbf{S}_{1}+\sinh (\Delta )\mathbf{S}_{2})\psi =0\mathrm{,} \notag \\
\mathcal{S}_{2}\psi &=&(\cosh (\Delta )\mathbf{S}_{2}+\sinh (\Delta )\mathbf{S}_{1})\psi =0, \label{hyp1}\end{aligned}$$where $\Delta $ represents any invariant interaction singly or in combination. It has a matrix structure in addition to coordinate dependence. Depending on that matrix structure we have either vector, scalar or more general tensor interactions [@jmath]. The operators $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$ are auxiliary constraints satisfying $$\begin{aligned}
\mathbf{S}_{1}\psi &\equiv &(\mathcal{S}_{10}\cosh (\Delta )+\mathcal{S}_{20}\sinh (\Delta )~)\psi =0, \notag \\
\mathbf{S}_{2}\psi &\equiv &(\mathcal{S}_{20}\cosh (\Delta )+\mathcal{S}_{10}\sinh (\Delta )~)\psi =0, \label{hyp2}\end{aligned}$$in which the $\mathcal{S}_{i0}$ are the free Dirac operators $$\mathcal{S}_{i0}=\frac{i}{\sqrt{2}}\gamma _{5i}(\gamma _{i}\cdot
p_{i}+m_{i}). \label{es0}$$This, in turn leads to the two compatibility conditions [cww,jmath,saz86]{} $$\lbrack \mathcal{S}_{1},\mathcal{S}_{2}]\psi =0,$$and $$\lbrack \mathbf{S}_{1},\mathbf{S}_{2}]\psi =0,$$provided that $\ \Delta (x)=\Delta (x_{\perp }).$ These compatibility conditions do not restrict the gamma matrix structure of $\Delta $. That matrix structure is determined by the type of vertex-vertex structure we wish to incorporate in the interaction. The three types of invariant interactions $\Delta $ that we use in this paper are$$\begin{aligned}
\Delta _{\mathcal{L}}(x_{\perp }) &=&-1_{1}1_{2}\frac{\mathcal{L}(x_{\perp })}{2}\mathcal{O}_{1},\ \mathcal{O}_{1}=-\gamma _{51}\gamma _{52},~~~\text{scalar}\mathrm{,} \notag \\
\Delta _{\mathcal{J}}(x_{\perp }) &=&\beta _{1}\beta _{2}\frac{\mathcal{J}(x_{\perp })}{2}\mathcal{O}_{1},~~~\text{timelike\ vector}\mathrm{,} \notag
\\
\Delta _{\mathcal{G}}(x_{\perp }) &=&\gamma _{1\perp }\cdot \gamma _{2\perp }\frac{\mathcal{G}(x_{\perp })}{2}\mathcal{O}_{1},~~\text{spacelike\ vector}{,} \label{hyp3}\end{aligned}$$where$$\begin{aligned}
\gamma _{i\perp }^{\mu } &=&(\eta ^{\mu \nu }+\hat{P}^{\mu }\hat{P}^{\nu
})\gamma _{\nu i}, \notag \\
\gamma _{5i} &=&\gamma _{i}^{0}\gamma _{i}^{1}\gamma _{i}^{2}\gamma _{i}^{3},
\notag \\
\beta _{i} &=&-\gamma _{i}\cdot \hat{P},~\ i=1,2. \label{beta}\end{aligned}$$$~$For For general independent scalar, timelike vector, and spacelike vector interactions we have$$\Delta (x_{\perp })=\Delta _{\mathcal{L}}+\Delta _{\mathcal{J}}+\Delta _{\mathcal{G}}. \label{Delta}$$The special case of an electromagnetic-like interaction (in the Feynman gauge) corresponds to $\mathcal{J}=-\mathcal{G}$ or $$\begin{aligned}
\Delta _{\mathcal{J}}+\Delta _{\mathcal{G}} &\equiv &\Delta _{\mathcal{EM}}=(-\gamma _{1}\cdot \hat{P}\gamma _{2}\cdot \hat{P}+\gamma _{1\perp }\cdot
\gamma _{2\perp })\frac{\mathcal{G}(x_{\perp })}{2}\mathcal{O}_{1} \notag \\
&=&\gamma _{1}\cdot \gamma _{2}\frac{\mathcal{G}(x_{\perp })}{2}\mathcal{O}_{1}. \label{vecc}\end{aligned}$$Our Th1 corresponds to a scalar and electromagnetic interaction, $$\Delta (x_{\perp })=\Delta _{\mathcal{L}}+\Delta _{\mathcal{EM}}.
\label{A10}$$Our Th2 corresponds to a modification of the timelike portion of $\Delta _{\mathcal{EM}}$ to $$\begin{aligned}
\Delta (x_{\perp }) &=&\Delta _{\mathcal{L}}+\Delta _{\mathcal{J}}+\Delta _{\mathcal{G}}=(-1_{1}1_{2}\mathcal{L}(x_{\perp })+\beta _{1}\beta _{2}\mathcal{J}(x_{\perp })+\gamma _{1\perp }\cdot \gamma _{2\perp }\mathcal{G}(x_{\perp }))\frac{\mathcal{O}_{1}}{2}, \notag \\
\mathcal{J}\mathcal{\neq -G} &&. \label{th2}\end{aligned}$$ This leads to[^32] [jmath,long]{} $\left( \partial _{\mu }=\partial /\partial x^{\mu }\right) $$$\begin{aligned}
\mathcal{S}_{1}\psi & =\big(-G\beta _{1}\Sigma _{1}\cdot \mathcal{P}_{2}+E_{1}\beta _{1}\gamma _{51}+M_{1}\gamma _{51}-G\frac{i}{2}\Sigma
_{2}\cdot \partial (\mathcal{L}\beta _{2}\mathcal{-J}\beta _{1})\gamma
_{51}\gamma _{52}\big)\psi =0, \notag \\
\mathcal{S}_{2}\psi & =\big(G\beta _{2}\Sigma _{2}\cdot \mathcal{P}_{1}+E_{2}\beta _{2}\gamma _{52}+M_{2}\gamma _{52}+G\frac{i}{2}\Sigma
_{1}\cdot \partial (\mathcal{L}\beta _{1}\mathcal{-J}\beta _{2})\gamma
_{51}\gamma _{52}\big)\psi =0, \label{extd}\end{aligned}$$ with $$\begin{aligned}
G &=&\exp \mathcal{G}, \notag \\
\mathcal{P}_{i} &\equiv &p_{\perp }-\frac{i}{2}\Sigma _{i}\cdot \partial
\mathcal{G}\Sigma _{i}. \label{d2}\end{aligned}$$The connections between what we call the vertex invariants $\mathcal{L},\mathcal{J},\mathcal{G}$ and the mass and energy potentials $M_{i},E_{i}$ are found to be $$\begin{aligned}
M_{1} &=&m_{1}\ \cosh \mathcal{L}\ +m_{2}\sinh \mathcal{L}, \notag \\
M_{2} &=&m_{2}\ \cosh \mathcal{L}\ +m_{1}\ \sinh \mathcal{L}, \notag \\
E_{1} &=&\varepsilon _{1}\ \cosh \mathcal{J}\ +\varepsilon _{2}\sinh
\mathcal{J}, \notag \\
E_{2} &=&\varepsilon _{2}\ \cosh \mathcal{J}+\varepsilon _{1}\sinh \mathcal{J}. \label{d2b}\end{aligned}$$Eq. (\[extd\]) depends on the standard Pauli-Dirac representation of gamma matrices in block forms (see Eq. (2.28) in [@crater2] for their explicit forms) and where[^33] $$\Sigma _{i}=\gamma _{5i}\beta _{i}\gamma _{\perp i}. \label{d3}$$Comparing Eq. (\[extd\]) with Eq. (\[tbde\]) we find that the spin-dependent vector interactions of Eq. (\[tbde\]) are [cra87,becker]{}$$\begin{aligned}
\tilde{A}_{1}^{\mu }& =\big((\varepsilon _{1}-E_{1})-i\frac{G}{2}\left(
\gamma _{2}\cdot \partial \mathcal{J}\right) \gamma _{2}\cdot \hat{P}\big )\hat{P}^{\mu }+(1-G)p_{\perp }^{\mu }-\frac{i}{2}\partial G\cdot \gamma
_{2}\gamma _{2\perp }^{\mu }, \notag \\
A_{2}^{\mu }& =\big((\varepsilon _{2}-E_{2})+i\frac{G}{2}\left( \gamma
_{1}\cdot \partial \mathcal{J}\right) )\gamma _{1}\cdot \hat{P}\big )\hat{P}^{\mu }-(1-G)p_{\perp }^{\mu }+\frac{i}{2}\partial G\cdot \gamma _{1}\gamma
_{1\perp }^{\mu }. \label{vecp}\end{aligned}$$Note that the first portion of the vector potentials is timelike (parallel to $\hat{P}^{\mu })$ while the next portion is spacelike (perpendicular to $\hat{P}^{\mu })$. The spin-dependent scalar potentials $\tilde{S}_{i}$ are $$\begin{aligned}
\tilde{S}_{1}& =M_{1}-m_{1}-\frac{i}{2}G\gamma _{2}\cdot \partial \mathcal{L}, \notag \\
\tilde{S}_{2}& =M_{2}-m_{2}+\frac{i}{2}G\gamma _{1}\cdot {\partial }\mathcal{L}{.} \label{scalp}\end{aligned}$$Eq. (\[vecp\]) simplifies to$$\begin{aligned}
\tilde{A}_{1}^{\mu }& =\big((\varepsilon _{1}-E_{1})\big )\hat{P}^{\mu
}+(1-G)p_{\perp }^{\mu }-\frac{i}{2}\partial G\cdot \gamma _{2}\gamma
_{2}^{\mu }, \notag \\
A_{2}^{\mu }& =\big((\varepsilon _{2}-E_{2})\big )\hat{P}^{\mu
}-(1-G)p_{\perp }^{\mu }+\frac{i}{2}\partial G\cdot \gamma _{1}\gamma
_{1}^{\mu },\end{aligned}$$for electromagnetic-like interactions.
We have chosen a parametrization for $\mathcal{L},~\mathcal{J},$ and $\mathcal{G}$ that takes advantage of the Todorov effective external potential forms and at the same time will display the correct static limit form for the Pauli reduction (see Eq. (\[571\])). The choice for these parametrizations is fixeded due to the fact that for classical [@fw] or quantum field theories [@saz97] for separate scalar and vector interactions the spin independent part of the quasipotential $\Phi _{w}~$ involves the difference of squares of the invariant mass and energy potentials ($M_{i}$ and $E_{i}$ respectively)$$M_{i}^{2}=m_{i}^{2}+2m_{w}S+S^{2};\ E_{i}^{2}=\varepsilon
_{i}^{2}-2\varepsilon _{w}V+V^{2}, \label{kg1}$$so that $$M_{i}^{2}-E_{i}^{2}=2m_{w}S+S^{2}+2\varepsilon _{w}V-V^{2}-b^{2}(w).
\label{kg}$$
Strictly speaking, the forms in Eq. (\[kg1\]) and Eq. (\[kg\]) are for scalar and timelike vector interactions. Eqs. (\[tbde\]) and (\[extd\]) involve combined scalar, electromagnetic-like, and separate timelike vector interactions. Without the separate timelike interactions this amounts to working in the Feynman gauge with the simplest relation between space- and timelike parts, (see Eqs. (\[vecc\]), (\[A10\]), and [@cra88; @crater2]). In the general case the mass and energy potentials in place of Eq. (\[kg1\]) are respectively $$\begin{aligned}
M_{i}^{2} &=&m_{i}^{2}+\exp (2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2}),
\label{one} \\
E_{i}^{2} &=&\exp (2\mathcal{G(A))(}\left( \varepsilon
_{i}-A)^{2}-2\varepsilon _{w}V+V^{2}\right) , \label{two}\end{aligned}$$so that from Eq. (\[d2b\]), $$\begin{aligned}
\exp (\mathcal{L}) &=&\exp (\mathcal{L}(S,A))=\frac{\sqrt{m_{1}^{2}+\exp (2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2})}+\sqrt{m_{2}^{2}+\exp (2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2})}}{m_{1}+m_{2}},\ \label{1.5} \\
\exp (\mathcal{J}) &=&\exp (\mathcal{J}(V,A))=\exp (\mathcal{G)}\frac{\sqrt{(\varepsilon _{1}-A)^{2}-2\varepsilon _{w}V+V^{2}}+\sqrt{(\varepsilon
_{2}-A)^{2}-2\varepsilon _{w}V+V^{2}}}{\varepsilon _{1}+\varepsilon _{2}},
\notag\end{aligned}$$with $$\exp (2\mathcal{G(}A\mathcal{))=}\frac{1}{(1-2A/w)}\equiv G^{2}.
\label{three}$$
Below we present, the consequent connections to the invariant interaction functions $\ A,V,$and $S$.
a\) In the case of electromagnetic interactions $(V=0$) with scalar confinement (Th1), we have $$\begin{aligned}
\mathcal{J} &=&-\mathcal{G=}\frac{1}{2}\log (1-2A/w)=\log \frac{E_{1}+E_{2}}{w}, \notag \\
E_{i}^{2} &=&\exp (2\mathcal{G)(\varepsilon }_{i}-A)^{2}, \notag \\
M_{i}^{2} &=&m_{i}^{2}+\exp (2\mathcal{G)}\left( 2m_{w}S+S^{2}\right) ,\end{aligned}$$and the spin-independent ($SI)$ minimal coupling comes from$$\begin{aligned}
\exp (2\mathcal{G)}p^{2}+M_{i}^{2}-E_{i}^{2} &\rightarrow &p^{2}+\exp (-2\mathcal{G)(}M_{i}^{2}-E_{i}^{2}) \notag \\
&=&p^{2}+\Phi _{SI}-b^{2}\end{aligned}$$and appears as$$\Phi _{SI}-b^{2}=2m_{w}S+S^{2}+m_{i}^{2}(1-2A/w)-(\varepsilon
_{i}-A)^{2}=2m_{w}S+S^{2}+2\varepsilon _{w}A-A^{2}-b^{2}.$$
b\) In the case of pure timelike vector interactions and scalar interactions (with no electromagnetic-like interactions this model is not appropriate for meson spectroscopy) we have$$\begin{aligned}
\mathcal{J} &=&\mathcal{J}_{0}\equiv \log \frac{E_{10}+E_{20}}{w}, \notag \\
E_{i0}^{2} &\equiv &\varepsilon _{i}^{2}-2\varepsilon _{w}V+V^{2}, \notag \\
\mathcal{G} &\mathcal{=}&0,\end{aligned}$$and the spin-independent minimal coupling appears as$$\Phi _{SI}=2m_{w}S+S^{2}+2\varepsilon _{w}V-V^{2}.$$
c\) When we include independent timelike and electromagnetic-like simultaneously together with scalar interactions (Th2) then we have$$\begin{aligned}
-\mathcal{G} &\mathcal{=}&\frac{1}{2}\log (1-2A/w), \notag \\
\mathcal{J} &=&\log \frac{E_{1}+E_{2}}{w}, \notag \\
E_{i}^{2} &=&\exp (2\mathcal{G)(}\left( \varepsilon _{i}-A)^{2}-2\varepsilon
_{w}V+V^{2}\right) =\exp (2\mathcal{G)(}E_{i0}^{2}-2\varepsilon _{i}A+A^{2})\end{aligned}$$and the spin-independent minimal coupling appears like$$\Phi _{SI}=2m_{w}S+S^{2}+2\varepsilon _{w}A-A^{2}+2\varepsilon _{w}V-V^{2}.$$
Details on Eq. (\[57\])
-----------------------
The Klein-Gordon like potential energy terms appearing at the beginning of the Pauli form (\[57\]) arise from$$M_{i}^{2}-E_{i}^{2}=\exp (2\mathcal{G)[}2m_{w}S+S^{2}+2\varepsilon
_{w}A-A^{2}+2\varepsilon _{w}V-V^{2}-b^{2}(w)].$$To obtain the symbolic Pauli form of Eq. (\[schlike\]) and the subsequent detailed form in Eq. (\[57\]) involves steps similar to those used in the Pauli reduction of the single particle Dirac equation [@yoon] but with the combinations $\phi _{\pm }=\psi _{1}\pm \psi _{4}$ and $\chi _{\pm
}=\psi _{2}\pm \psi _{3}$ instead of the individual single particle wave function. This reduces the Pauli forms to 4 uncoupled 4 component relativistic Schrödinger equations [@saz94; @cra94; @long; @crater2; @liu]. We work in the c.m. frame in which $\hat{P}=(1,\mathbf{0)}$ and $\hat{r}=(0,\mathbf{\hat{r}).}$ The final four-component wave functions $\psi _{\pm
},\eta _{\pm }$ that appear in Eq. (\[57\]) are defined by [@liu] $$\begin{aligned}
\phi _{\pm }& =\exp (\mathcal{F}+\mathcal{K}\boldsymbol{\sigma }_{1}\mathbf{\cdot \hat{r}}\boldsymbol{\sigma }_{2}\mathbf{\cdot \hat{r}})\psi _{\pm
}=(\exp \mathcal{F})(\cosh \mathcal{K}+\sinh \mathcal{K}\boldsymbol{\sigma }_{1}\mathbf{\cdot \hat{r}}\boldsymbol{\sigma }_{2}\mathbf{\cdot \hat{r}})\psi _{\pm }, \notag \\
\chi _{\pm }& =\exp (\mathcal{F}+\mathcal{K}\boldsymbol{\sigma }_{1}\mathbf{\cdot \hat{r}}\boldsymbol{\sigma }_{2}\mathbf{\cdot \hat{r}})\eta _{\pm
}=(\exp \mathcal{F})(\cosh \mathcal{K}+\sinh \mathcal{K}\boldsymbol{\sigma }_{1}\mathbf{\cdot \hat{r}}\boldsymbol{\sigma }_{2}\mathbf{\cdot \hat{r}})\eta _{\pm }, \label{fk}\end{aligned}$$in which $$\begin{aligned}
\mathcal{F}& =\frac{1}{2}\log \frac{\mathcal{D}}{\varepsilon
_{2}m_{1}+\varepsilon _{1}m_{2}}-\mathcal{G}, \notag \\
\mathcal{D}& \mathcal{=}E_{2}M_{1}+E_{1}M_{2}, \notag \\
\mathcal{K}& =\frac{(\mathcal{L}-\mathcal{J})}{2}. \label{kf}\end{aligned}$$In analogy to what occurs in the decoupled form of the Schrödinger equation for the individual single particle wave function, this substitution has the convenient property that in the resultant bound state equation, the coefficients of the first order relative momentum terms vanish.
Using the results in [@liu] and [@yoon] we obtain for the general case of unequal masses the relativistic Schrödinger equation (\[57\]) that is a detailed c.m. form of Eq. (\[schlike\]). In that equation we have introduced the abbreviations
$$\begin{aligned}
\Phi _{D}& =-\frac{2(\mathcal{F}^{\prime }+1/r)(\cosh 2\mathcal{K}-1)}{r}+\mathcal{F}^{\prime 2}+\mathcal{K}^{\prime 2}+\frac{2\mathcal{K}^{\prime
}\sinh 2\mathcal{K}}{r}-\mathbf{\nabla }^{2}\mathcal{F}+m(r), \notag \\
\Phi _{SO}& =-\frac{\mathcal{F}^{\prime }}{r}-\frac{(\mathcal{F}^{\prime
}+1/r)(\cosh 2\mathcal{K}-1)}{r}+\frac{\mathcal{K}^{\prime }\sinh 2\mathcal{K}}{r}, \notag \\
\Phi _{SOD}& =(l^{\prime }\cosh 2\mathcal{K}-q^{\prime }\sinh 2\mathcal{K}),
\notag \\
\Phi _{SOX}& =(q^{\prime }\cosh 2\mathcal{K}-l^{\prime }\sinh 2\mathcal{K}),
\notag \\
\Phi _{SS}& =k(r)+\frac{2\mathcal{K}^{\prime }\sinh 2\mathcal{K}}{3r}-\frac{2(\mathcal{F}^{\prime }+1/r)(\cosh 2\mathcal{K}-1)}{3r}+\frac{2\mathcal{F}^{\prime }\mathcal{K}^{\prime }}{3}-\frac{\mathbf{\nabla }^{2}\mathcal{K}}{3}, \notag \\
\Phi _{T}& =\frac{1}{3}[n(r)+\frac{(3\mathcal{F}^{\prime }-\mathcal{K}^{\prime }+3/r)\sinh 2\mathcal{K}}{r}+\frac{(\mathcal{F}^{\prime }-3\mathcal{K}^{\prime }+1/r)(\cosh 2\mathcal{K}-1)}{r}+2\mathcal{F}^{\prime }\mathcal{K}^{\prime }-\mathbf{\nabla }^{2}\mathcal{K}], \notag \\
\Phi _{SOT}& =-\mathcal{K}^{\prime }\frac{\cosh 2\mathcal{K}-1}{r}-\frac{\mathcal{K}^{\prime }}{r}+\frac{(\mathcal{F}^{\prime }+1/r)\sinh 2\mathcal{K}}{r}, \label{54}\end{aligned}$$
where$$\begin{aligned}
k(r)& =\frac{1}{3}\nabla ^{2}(\mathcal{K}+\mathcal{G)}-\frac{2\mathcal{F}^{\prime }(\mathcal{G}^{\prime }+\mathcal{K}^{\prime })}{3}\mathcal{-}\frac{1}{2}\mathcal{G}^{\prime 2}, \notag \\
n(r)& =\frac{1}{3}[\nabla ^{2}\mathcal{K}-\frac{1}{2}\nabla ^{2}\mathcal{G}+\frac{3(\mathcal{G}^{\prime }-2\mathcal{K}^{\prime })}{2r}+\mathcal{F}^{\prime }(\mathcal{G}^{\prime }-2\mathcal{K}^{\prime })], \notag \\
m(r)& =-\frac{1}{2}\nabla ^{2}\mathcal{G+}\frac{3}{4}\mathcal{G}^{\prime 2}+\mathcal{G}^{\prime }\mathcal{F}^{\prime }-\mathcal{K}^{\prime 2},
\label{knm}\end{aligned}$$and
$$\begin{aligned}
l^{\prime }(r)& =-\frac{1}{2r}\frac{E_{2}M_{2}-E_{1}M_{1}}{E_{2}M_{1}+E_{1}M_{2}}(\mathcal{L}^{\prime }+\mathcal{J}^{\prime }), \notag
\\
q^{\prime }(r)& =\frac{1}{2r}\frac{E_{2}M_{1}-E_{1}M_{2}}{E_{2}M_{1}+E_{1}M_{2}}(\mathcal{L}^{\prime }+\mathcal{J}^{\prime }).
\label{A32}\end{aligned}$$
(The prime symbol stands for $d/dr,$ and the explicit forms of the derivatives are given in Eq. (\[der\])$\,$below). For $L=J$ states, the hyperbolic terms cancel and the spin-orbit difference terms in general produce spin mixing except for equal masses or $J=0$. For ease of use we have listed in Appendix A3 the explicit forms that appear in the above $\Phi
$s in Eqs. (\[54\]) - (\[knm\]) in terms of the general invariant potentials $A(r),V(r),$ and $S(r).~$ The radial components of Eq. (\[57\]) are given in Appendix B.
Explicit Expressions for terms in the Relativistic Schrödinger Equation (\[57\]) from $A(r),V(r)$ and $S(r)$
------------------------------------------------------------------------------------------------------------
Given the functions $A(r)$ , $V(r),$ and $S(r)$ for the interaction, users of the relativistic Schrödinger equation (\[57\]) will find it convenient to have an explicit expression in an order that would be useful for programing the terms in the associated equation (\[54\])-(\[A32\]). We use the definitions above given in Eqs. (\[one\] )-(\[three\]), and (\[kf\]). In order that the terms in Eq. (\[54\]) be reduced to expressions involving just $A(r),V(r),~S(r)$ and their derivatives, we list the following formulae $$\begin{aligned}
\mathcal{F}^{\prime } &=&\frac{(\mathcal{L}^{\prime }+\mathcal{J}^{\prime
})(E_{2}M_{2}+E_{1}M_{1})}{2(E_{2}M_{1}+E_{1}M_{2})}-\mathcal{G}^{\prime },
\notag \\
\mathcal{G}^{\prime } &=&\frac{A^{\prime }}{w-2A}, \notag \\
\mathcal{L}^{\prime } &=&\frac{M_{1}^{\prime }}{M_{2}}=\frac{M_{2}^{\prime }}{M_{1}}=\frac{w}{M_{1}M_{2}}\left( \frac{S^{\prime }(m_{w}+S)}{w-2A}+\frac{(2m_{w}S+S^{2})A^{\prime }}{(w-2A)^{2}}\right) , \notag \\
\mathcal{J}^{\prime } &=&\frac{E_{1}^{\prime }}{E_{2}}=\frac{E_{2}^{\prime }}{E_{1}}=-\frac{(\mathcal{G}^{\prime }[(\varepsilon _{1}-A)(\varepsilon
_{2}-A)+2\varepsilon _{w}V-V^{2}]+(\varepsilon _{w}-V)V^{\prime })}{E_{1}E_{2}(w-2A)/w}, \notag \\
.\mathcal{K}^{\prime } &=&\frac{(\mathcal{L}^{\prime }-\mathcal{J}^{\prime })}{2}. \label{der}\end{aligned}$$Also needed are $$\begin{aligned}
\cosh 2\mathcal{K} &=&\frac{1}{2}\left( \frac{(\varepsilon _{1}+\varepsilon
_{2})(M_{1}+M_{2})}{(m_{1}+m_{2})(E_{1}+E_{2})}+\frac{(m_{1}+m_{2})(E_{1}+E_{2})}{(\varepsilon _{1}+\varepsilon _{2})(M_{1}+M_{2})}\right) , \notag \\
\sinh 2\mathcal{K} &=&\frac{1}{2}\left( \frac{(\varepsilon _{1}+\varepsilon
_{2})(M_{1}+M_{2})}{(m_{1}+m_{2})(E_{1}+E_{2})}-\frac{(m_{1}+m_{2})(E_{1}+E_{2})}{(\varepsilon _{1}+\varepsilon _{2})(M_{1}+M_{2})}\right) ,\end{aligned}$$and $$\begin{aligned}
\mathbf{\nabla }^{2}\mathcal{F} &=&\frac{(\mathbf{\nabla }^{2}\mathcal{L}+\mathbf{\nabla }^{2}\mathcal{J})(E_{2}M_{2}+E_{1}M_{1})}{2(E_{2}M_{1}+E_{1}M_{2})}-(\mathcal{L}^{\prime }+\mathcal{J}^{\prime })^{2}\frac{(m_{1}^{2}-m_{2}^{2})^{2}}{2\left( E_{2}M_{1}+E_{1}M_{2}\right) ^{2}}-\mathbf{\nabla }^{2}\mathcal{G}, \notag \\
\mathbf{\nabla }^{2}\mathcal{L} &=&\frac{-\mathcal{L}^{\prime
2}(M_{1}^{2}+M_{2}^{2})}{M_{1}M_{2}} \notag \\
&&+\frac{w}{M_{1}M_{2}}\left( \frac{\mathbf{\nabla }^{2}S(m_{w}+S)+S^{\prime
2}}{w-2A}+\frac{4S^{\prime }(m_{w}+S)A^{\prime }+(2m_{w}S+S^{2})\mathbf{\nabla }^{2}A}{(w-2A)^{2}}+\frac{4(2m_{w}S+S^{2})A^{\prime 2}}{(w-2A)^{3}}\right) , \notag \\
\mathbf{\nabla }^{2}\mathcal{J} &\mathcal{=}&\mathcal{-[(}\frac{E_{1}^{2}+E_{2}^{2}}{E_{1}E_{2}}\mathcal{)J}^{\prime }-2\mathcal{G}^{\prime
}]\mathcal{J}^{\prime }-\frac{\exp (2\mathcal{G)}}{E_{1}E_{2}}\{\nabla ^{2}\mathcal{G}[(\varepsilon _{1}-A)(\varepsilon _{2}-A)+2\varepsilon
_{w}V-V^{2}]+(\varepsilon _{w}-V)\mathbf{\nabla }^{2}V \notag \\
&&-\mathcal{G}^{\prime 2}(w-2A)^{2}-V^{\prime 2}+2V^{\prime }\mathcal{G}^{\prime }(\varepsilon _{w}-V)\} \notag \\
\mathbf{\nabla }^{2}\mathcal{G} &=&\frac{\mathbf{\nabla }^{2}A}{w-2A}+2\mathcal{G}^{\prime 2}.\end{aligned}$$The expressions for $k(r),m(r),$ and $n(r)$ that appear in Eqs. (\[54\])) are given in Eqs. (\[knm\]). They can be evaluated using the above expressions plus $$\mathbf{\nabla }^{2}\mathcal{K}=\frac{\mathbf{\nabla }^{2}\mathcal{L}-\mathbf{\nabla }^{2}\mathcal{J}}{2}. \label{blw}$$The only remaining parts of Eq. (\[54\]) that need expressing are those for $l^{\prime }$ and $q^{\prime }~$given in Eq. (\[A32\]) Using Eq. (\[kf\]) they can be obtained in terms of the above formulae.
Weak Potential Limits of Quasipotentials
----------------------------------------
The weak potential forms of the quasipotentials are needed for working out perturbative spectral corrections. For weak potentials, we take $\varepsilon _{i}=m_{i}$ wherever they appear in the potentials and assume $|A|,|V|,|S|<<m_{i}$. Thus$$\begin{aligned}
\mathcal{L} &\mathcal{\rightarrow }&\frac{m_{w}S}{m_{1}m_{2}}\rightarrow
\frac{S}{m_{1}+m_{2}}, \notag \\
\mathcal{J} &\mathcal{\rightarrow }&\mathcal{-}\frac{A}{w}-\frac{\varepsilon
_{w}V}{\varepsilon _{1}\varepsilon _{2}}\rightarrow -\frac{A+V}{m_{1}+m_{2}},
\notag \\
\mathcal{G} &\mathcal{\rightarrow }&\frac{A}{w}\rightarrow \frac{A}{m_{1}+m_{2}}\end{aligned}$$Among other results, this will allow us to see how the scalar interaction contributes oppositely to the spin-orbit and Darwin terms from both vector interactions to lowest order. In that same limit, the dominant portions of $\Phi ^{\prime }s$ in Eq. (\[57\]) are$$\begin{aligned}
\Phi _{D}& \rightarrow -\nabla ^{2}\mathcal{F\rightarrow }-\frac{1}{4r\left(
m_{1}+m_{2}\right) }(\frac{\nabla ^{2}(\mathcal{S}-A-V)(m_{2}^{2}+m_{1}^{2})}{m_{2}m_{1}}-\nabla ^{2}A), \notag \\
\Phi _{SO}& \rightarrow -\frac{\mathcal{F}^{\prime }}{r}\rightarrow -\frac{1}{4r\left( m_{1}+m_{2}\right) }(\frac{(\mathcal{S}^{\prime }-A^{\prime
}-V^{\prime })(m_{2}^{2}+m_{1}^{2})}{m_{2}m_{1}}-A^{\prime }) \notag \\
\Phi _{SOD}& \rightarrow l^{\prime }\rightarrow -\frac{1}{4r\left(
m_{1}+m_{2}\right) }\frac{(\mathcal{S}^{\prime }-A^{\prime }-V^{\prime
})(m_{2}^{2}-m_{1}^{2})}{m_{1}m_{2}} \notag \\
\Phi _{SOX}& \rightarrow q^{\prime }\rightarrow 0 \notag \\
\Phi _{SS}& \rightarrow \frac{1}{3}\nabla ^{2}\mathcal{G\rightarrow }\frac{1}{6\left( m_{1}+m_{2}\right) }\nabla ^{2}A, \notag \\
\Phi _{T}& \rightarrow -\mathbf{\nabla }^{2}\frac{(S+A+V)}{9(m_{1}+m_{2})},
\notag \\
\Phi _{SOT}& \rightarrow -\frac{\mathcal{K}^{\prime }}{r}\rightarrow -\frac{(S^{\prime }+A^{\prime }+V^{\prime })}{2r(m_{1}+m_{2})}. \label{wp1}\end{aligned}$$Note how in the Darwin and spin-orbit terms the $\mathcal{S}^{\prime
}-V^{\prime }$ dependence shows how the scalar and timelike confining effects tend to cancel for $\xi <1$. As anticipated, only the Darwin and spin-orbit terms survive in the static limit when one of the two particles becomes very massive. In that limit (say $m_{2}\rightarrow \infty $), the two spin-orbit terms $\Phi _{SO}$ and $\Phi _{SOD}~$combine to$$-\mathbf{L\cdot (\sigma }_{1}+\mathbf{\sigma }_{2})\frac{\mathcal{S}^{\prime
}-A^{\prime }-V^{\prime }}{4r}-\mathbf{L\cdot (\sigma }_{1}-\mathbf{\sigma }_{2})\frac{\mathcal{S}^{\prime }-A^{\prime }-V^{\prime }}{4r}=-\mathbf{L\cdot \sigma }_{1}\frac{\mathcal{S}^{\prime }-A^{\prime }-V^{\prime }}{2r}.
\label{wp2}$$
Single Particle Limit ($m_{2}\rightarrow \infty $) of the TBDE
--------------------------------------------------------------
In addition to using $p_{2}\rightarrow (m_{2},\mathbf{0})$, $p_{1}=(\varepsilon _{1},\mathbf{p}$)$,$ the single particle limit of the TBDE is obtained by substituting $m_{2}\rightarrow \infty $ in the expressions for the various potentials listed in Appendices A1, A2, and A3$.$ $\ $To determine that limit, we use that the total c.m. energy $w=\varepsilon _{2}+\varepsilon _{1}\rightarrow m_{2}+\varepsilon _{1}$. In that case using the expressions for $m_{w}$ and $\varepsilon _{w}$ in Eqs. (\[mw\]) and (\[ew\]) we find$$\begin{aligned}
m_{w} &\rightarrow &m_{1}\equiv m, \notag \\
\varepsilon _{w} &\rightarrow &\varepsilon _{1}\equiv \varepsilon , \notag
\\
G &=&(1-2A/(m_{2}+\varepsilon ))^{-1/2}\rightarrow 1, \label{a45}\end{aligned}$$and thus for the quantities related to the mass potentials, we have$$\begin{aligned}
M_{1} &\rightarrow &\sqrt{\left( m+S\right) ^{2}}=m+S, \notag \\
M_{2}-m_{2} &\rightarrow &\sqrt{m_{2}^{2}+2mS+S^{2}}-m_{2}\rightarrow 0,
\notag \\
\exp (\mathcal{L)} &\mathcal{=}&\exp \mathcal{(}\frac{M_{1}+M_{2}}{m_{1}+m_{2}})\rightarrow 1, \notag \\
\tilde{S}_{1} &\rightarrow &S, \notag \\
\tilde{S}_{2} &\rightarrow &0.\end{aligned}$$For the energy potentials we have, since $\hat{P}\rightarrow (1,\mathbf{0})$ that$$\begin{aligned}
E_{1} &\rightarrow &\sqrt{\varepsilon ^{2}-2\varepsilon (A+V)+A^{2}+V^{2}}\equiv \varepsilon -U \notag \\
\varepsilon _{2}-E_{2} &\rightarrow &\varepsilon _{2}-\sqrt{\varepsilon
_{2}^{2}-2\varepsilon _{2}A-2\varepsilon V+A^{2}+V^{2}} \notag \\
&\rightarrow &\varepsilon _{2}-\varepsilon _{2}\sqrt{1-2A/\varepsilon _{2}}\rightarrow A, \notag \\
\exp (\mathcal{J)} &\mathcal{=}&\exp \mathcal{(}\frac{E_{1}+E_{2}}{\varepsilon _{1}+\varepsilon _{2}})\rightarrow 1, \notag \\
\tilde{A}_{1}^{\mu } &\rightarrow &U(1,\mathbf{0}), \notag \\
\tilde{A}_{2}^{\mu } &\rightarrow &A(1,\mathbf{0}). \label{a47}\end{aligned}$$ The TBDE (\[tbde\]) thus reduce to$$\begin{aligned}
(\mathbf{\gamma }_{1}\cdot \mathbf{p-}\beta _{1}(\varepsilon -U)+m+S)\psi
&=&0, \notag \\
(-\beta _{2}(m_{2}-A)+m_{2})\psi &\rightarrow &-m_{2}(\beta _{2}-1)\psi =0,\end{aligned}$$effectively to a single ordinary one-body Dirac equation (\[obde\]) for a particle in combined external scalar and time only component vector potential. Note that $U\neq A+V.$ [^34] Doing the usual Pauli-reduction yields for the upper component wave function the Schrödinger-like form which is the same as the $m_{2}\rightarrow \infty $ limit of Eq. (\[57\]). That limit can be readily seen from Eq. (\[kf\]) and Eqs. (\[a45\]-\[a47\]) which shows $$\begin{aligned}
\mathcal{F} &\rightarrow &\frac{1}{2}\log \frac{m+S+\varepsilon -U}{m+\varepsilon }, \notag \\
\mathcal{F}^{\prime } &=&\frac{1}{2}\frac{S^{\prime }-U^{\prime }}{m+S+\varepsilon -A}, \notag \\
\nabla ^{2}\mathcal{F} &\mathcal{=}&\frac{1}{2}\frac{\nabla ^{2}S-\nabla
^{2}U}{m+S+\varepsilon -U}-\frac{1}{2}\left( \frac{S^{\prime }-U^{\prime }}{m+S+\varepsilon -U}\right) ^{2}, \notag \\
\mathcal{L}^{\prime },\mathcal{J}^{\prime },\mathcal{G}^{\prime },\mathcal{K}^{\prime } &\rightarrow &0, \notag \\
\Phi _{D} &\rightarrow &\mathcal{F}^{\prime 2}-\mathbf{\nabla }^{2}\mathcal{F}, \notag \\
\Phi _{SO} &\rightarrow &-\frac{\mathcal{F}^{\prime }}{r}, \notag \\
\Phi _{SOD} &\rightarrow &l^{\prime }=-\frac{1}{2r}\frac{E_{2}M_{2}-E_{1}M_{1}}{E_{2}M_{1}+E_{1}M_{2}}(\mathcal{L}^{\prime }+\mathcal{J}^{\prime }) \notag \\
&\rightarrow &-\frac{1}{2r}\frac{E_{2}M_{2}}{E_{2}M_{1}+E_{1}M_{2}}(\mathcal{L}^{\prime }+\mathcal{J}^{\prime })\rightarrow -\frac{\mathcal{F}^{\prime }}{r}.\end{aligned}$$From the last line we obtain$$\Phi _{SO}\mathbf{L\cdot (\sigma }_{1}+\mathbf{\sigma }_{2})+\Phi _{SOD}\mathbf{L\cdot (\sigma }_{1}-\mathbf{\sigma }_{2})=-\frac{2\mathcal{F}^{\prime }}{r}\mathbf{L\cdot \sigma }_{1}=-\frac{S^{\prime }-U^{\prime }}{m+S+\varepsilon -U}\mathbf{L\cdot \sigma }_{1}.$$and hence Eq. (\[57\]) reduces to Eq. (\[571\]). In Section 6.4 we display our static limit equations for Th1.
Radial Equations
================
The following are radial eigenvalue equations, [@liu], [@yoon], corresponding to Eq. (\[57\]) . For a general singlet $^{1}J_{J}$ wave function $v_{LSJ}=v_{J0J}\equiv v_{0}$ coupled to a general triplet $^{3}J_{J}$ wave function $v_{J1J}\equiv v_{1}$, the wave equation
$$\begin{aligned}
& \{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon
_{w}A-A^{2}+2\varepsilon _{w}V-V^{2}+\Phi _{D}\mathbf{-}3\Phi _{SS}\}v_{0}
\notag \\
& +2\sqrt{J(J+1)}(\Phi _{SOD}-\Phi _{SOX})v_{1} \notag \\
& =b^{2}v_{0}, \label{ss}\end{aligned}$$
is coupled to
$$\begin{aligned}
& \{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon
_{w}A-A^{2}+2\varepsilon _{w}V-V^{2}+\Phi _{D} \notag \\
& -2\Phi _{SO}+\Phi _{SS}+2\Phi _{T}-2\Phi _{SOT}\}v_{1}+2\sqrt{J(J+1)}(\Phi
_{SOD}+\Phi _{SOX})v_{0} \notag \\
& =b^{2}v_{1}. \label{pp}\end{aligned}$$
For a general $S=1,$ $J=L+1$ wave function $v_{J-11J}\equiv v_{+}~$coupled to a general $S=1,$ $J=L-1~$wave function $v_{J+11J}\equiv v_{-}$ the equation
$$\begin{aligned}
& \{-\frac{d^{2}}{dr^{2}}+\frac{J(J-1)}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon
_{w}A-A^{2}+2\varepsilon _{w}V-V^{2}+\Phi _{D} \notag \\
& +2(J-1)\Phi _{SO}+\Phi _{SS}+\frac{2(J-1)}{2J+1}(\Phi _{SOT}-\Phi
_{T})\}v_{+} \notag \\
& +\frac{2\sqrt{J(J+1)}}{2J+1}\{3\Phi _{T}-2(J+2)\Phi _{SOT}\}v_{-} \notag
\\
& =b^{2}v_{+}, \label{pl}\end{aligned}$$
$\allowbreak \allowbreak \allowbreak $is coupled to $$\begin{aligned}
& \{-\frac{d^{2}}{dr^{2}}+\frac{(J+1)(J+2)}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon _{w}A-A^{2}+2\varepsilon _{w}V-V^{2}+\Phi _{D} \notag \\
& -2(J+2)\Phi _{SO}+\Phi _{SS}+\frac{2(J+2)}{2J+1}(\Phi _{SOT}-\Phi
_{T})\}v_{-} \notag \\
& +\frac{2\sqrt{J(J+1)}}{2J+1}\{3\Phi _{T}+2(J-1)\Phi _{SOT}\}v_{+} \notag
\\
& =b^{2}v_{-}. \label{mi}\end{aligned}$$For the uncoupled $^{3}P_{0}$ states the single equation is$$\begin{aligned}
& \{-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon
_{w}A-A^{2}+2\varepsilon _{w}V-V^{2}+\Phi _{D} \notag \\
& -4\Phi _{SO}+\Phi _{SS}+4(\Phi _{SOT}-\Phi _{T})\}v_{-}=b^{2}v_{-}.\end{aligned}$$
Two Body Dirac Equations for QED
================================
The Schrödinger-like form Eq. (\[57\]) of the TBDE given in Eq. ([tbde]{}) can be used for QCD bound state (meson spectroscopy) and for QED bound states (positronium, muonium, and hydrogen-like systems). In our meson spectroscopy work presented in this paper we use the three invariant functions $S(r),~A(r),$ and $V(r)$ specified in Sec. 3. Once these are specified then so are the three vertex invariants $\mathcal{L(}r),~\mathcal{G(}r),$ and $\mathcal{J(}r).$ They, in turn, fix the quasipotentials given in Eq. (\[54\]) and below to (\[blw\]) that appear in Eq. (\[57\]) and its radial forms of Appendix B. To make the transition from QCD meson bound states to QED bound states we simply take $S(r)=V(r)=0$ and $A(r)=-\alpha /r$ (we remind the reader that only in the c.m. system is the invariant $r=\sqrt{x_{\perp }^{2}}$ equal to $\left\vert \mathbf{r}\right\vert ).$ Our QED spectral results then follow from solving nonperturbatively (i.e. numerically or analytically) the radial eigenvalue equations of Appendix B. For example, since $\Phi _{D}=3\Phi _{SS}$ for $S=V=0,$ the equal mass spin singlet equation (\[ss\]) collapses to
$$\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}-\frac{2\varepsilon _{w}\alpha }{r}-\frac{\alpha ^{2}}{r^{2}}\}v_{0}=b^{2}v_{0},$$
which has the analytic spectral solution given in Eq. (\[ect\]) for $J=0$. As long as $J(J+1)-\alpha ^{2}$ $>-1/4,$ since the above equation take the limiting form of$$\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}-\frac{\alpha ^{2}}{r^{2}}\}v_{0}=0,$$it is clear that the effective potential is nonsingular, thus allowing a well defined eigenvalue solution[^35]. The paper [@yoon] demonstrated that the corresponding short distance behavior of the potentials in the other radial equations of Appendix B for the other QED bound states also allow well defined eigenvalue solutions[^36]. Beyond that, numerical solutions of these eigenvalue equations yield spectra (not limited just to the singlet ground states) in agreement with standard perturbative $O(\alpha ^{4})$ results [@becker]. In electron volts the numerical binding energy for the singlet ground state of positronium from Eq. (\[ss\]) is -6.8033256279 versus the perturbative result of Eq. (\[exct\]) or $m(-\alpha
^{2}/4-21\alpha ^{4}/64)=$ -6.8033256719. The difference in units of $m\alpha ^{4}/2$ is -6.08E-05 which is on the order of $\alpha ^{2}$, so that, as expected from (\[exct\]) the difference is on the order of $m\alpha ^{6}.~$The corresponding numerical binding energy in electron volts for the triplet ground state from the coupled equations Eqs. (\[pl\]) and (\[mi\]) is -6.8028426132 versus the perturbative result [@becker] of $m(-\alpha ^{2}/4+\alpha ^{4}/192)=~$-6.8028426636. These results do not include the effects of the annihilation diagram. The difference in units of $m\alpha ^{4}/2$ is 6.97E-05 which is on the order of $\alpha ^{2}$, so that the difference is also on the order of $m\alpha ^{6}$. These results (from a very extensive list of numerically computed spectral results in [becker]{}) taken together, represent crucial tests of this formalism, ones that have not been demonstrated in any other relativistic bound state formalism. In fact, the authors of [@iowa] have found a particular quasipotential formalism that does give such agreement, but only for the ground state. They also demonstrate that several well know two-body relativistic bound state formalisms (including the Blankenbecler-Sugar formalism and the formalism of Gross) fail this test. Let us be explicit about the implications of either the failure of this test or the lack of performing this test. When one proposes a new bound state formalism such as Dirac did in 1928, it was essential that at the very least it reproduce the nonrelativistic hydrogenic spectrum. Beyond that it provides two other remarkable results. First of all, by way of an order $1/c^{2}$ expansion, it gave an order $\alpha ^{4}$ perturbative correction to the nonrelativistic spectral results. Remarkably, an exact solution of the same equation later by Darwin gave spectral results which when truncated to order $\alpha ^{4}$ agreed precisely with the perturbative treatment, and, at the time, with experimental fine structure measurements. Breit, in the development of his two-body equation gave us an equation with interactions beyond the Coulomb potential that ultimately reproduced, when treated perturbatively, spectral results for two-body systems that agreed through terms of order $\alpha ^{4}$ with experiment. Unlike the one-body Dirac equation which has an exact spectral solution which agrees, at order $\alpha
^{4}$, with its perturbative solution, the Breit equation has no exact or for that matter numerical solution which agrees, at order $\alpha ^{4}$, with its perturbative solution. The same could be said for most all other two-body relativistic treatments proposed since then, with the two exceptions ([@becker] and [@iowa]) noted above. One’s two-body formalism having an agreement of its nonperturbative treatment with its perturbative treatment of the spectra in a well established field theory such as QED, in our opinion, should be regarded as a necessary hurdle to pass before going on to apply these formalisms to potential models for meson spectroscopy.
The authors wish to thank Dr. C. Y. Wong, Prof. J. H. Yoon, and Joshua Whitney for helpful suggestions and discussions.
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[^1]: [email protected]
[^2]: These equations were originally proposed in the form of classical generalized mass shell first class constraints $\mathcal{H}_{i}=(p_{i}^{2}+m_{i}^{2}+\Phi _{i})\approx 0$, and their quantization $\mathcal{H}_{i}\Psi =0$ without reference to a quantum field theory. For the classical $\mathcal{H}_{i}$ to be compatible, their Poisson bracket with one another must either vanish strongly or depend on the constraints themselves, $\{\mathcal{H}_{1},\mathcal{H}_{2}\}\approx 0$. The simplest solution of this equation is $\Phi _{1}=\Phi _{2}$, a kind of relativistic third law condition, together with their common transverse coordinate dependence $\Phi _{w}(x_{\perp }),$ just as with its quantum version.
[^3]: We use the same symbol $P$ for the eigenvalue so that the $w$ dependence in Eq. (\[em\]) is regarded as an eigenvalue dependence. The wave function $\Psi $ can be viewed either as a relativistic 2-body wave function (similar in interpretation to the Dirac wave function) or, if a close connection to field theory is required, related directly to the Bethe Salpeter wave function $\chi \mathbf{~}$by [@saz] $\Psi =-\pi i\delta (P\cdot p)\mathcal{H}_{10}\chi =-\pi i\delta (P\cdot p)\mathcal{H}_{20}\chi $.
[^4]: In a perturbative context, i.e. for weak potentials, that would mean that this aspects of $\tilde{A}_{i}^{\mu }$ is regarded as arising from a Feynman gauge vertex coupling of a form proportional to $\gamma _{1}^{\mu }\gamma
_{2\mu }A$ (see Appendix A).
[^5]: In a perturbative or weak potential context, that would mean that this aspect of $\tilde{A}_{i}^{\mu }$ is regarded as arising from an additional vertex coupling proportional to $-\gamma _{1}\cdot \hat{P}~\gamma _{2}\cdot
\hat{P}V.$ Similarly in a perturbative or weak potential context $\tilde{S}_{i}$ is regarded as arising from a vertex coupling proportional to $1_{1}1_{2}S.$ (See Appendix A).
[^6]: The $\gamma _{5}$ matrices for each of the two particles are designated by $\gamma _{5i}$ $i=1,2$. The reason for putting these matrices out front of the whole expression is that including them facilitates the proof of the compatibility condition, see [@cra82], [@cra87].
[^7]: The dependence of the scalar potentials $\tilde{S}_{i}$ on the invariant $A(r)$ responsible for the electromagnetic-like potential is seen in [cra87]{} and [@saz97] to result from the way the scalar and vector fields combine. That combination without the presence of the independent time-like portion leads to a two-body Klein-Gordon-like potential portion of $\Phi _{w}
$ to be of the form given in Eq. (\[em\]).
[^8]: In the presence of an additional and independent time-like vector interaction $V$, we assume the scalar and vector fields combine in such a way that leads to a two-body Klein-Gordon-like potential portion of $\Phi
_{w}$ of the form $2m_{w}S+S^{2}+2\varepsilon _{w}A-A^{2}+2\varepsilon
_{w}V-V^{2}$ instead of that given in Eq. (\[em\]).
[^9]: Due to the dependence of $\Phi _{w}$ on $w,$ this is a nonlinear eigenvalue equation.
[^10]: For hermitian potentials, that condition has the symbolic form of $T-T^{\dag
}=T^{\dag }(G-G^{\dag })T.$
[^11]: In the present treatment, we treat the entire interaction present in our equations, thereby keeping each of these effects. In our former treatment [@cra88] we also performed a decoupling between the upper-upper and lower-lower components of the wave functions for spin-triplet states which turned out to be defective but which we subsequently corrected in our numerical test of our formalism for QED [@becker].
[^12]: The subscript on quasipotential $\Phi _{D}$ refers to Darwin. It consist of what are called Darwin terms, those that are the two-body analogue of terms that accompany the spin-orbit term in the one-body Pauli reduction of the ordinary one-body Dirac equation, and ones related by canonical transformations to Darwin interactions [@fw; @sch73], momentum dependent terms arising from retardation effects. The subscripts on the other quasipotentials refer respectively to $SO$ (spin-orbit), $SOD$ (spin-orbit difference), $SOX$ (spin-orbit cross terms), $SS$ (spin-spin), $T$ (tensor), $SOT$ (spin-orbit-tensor)
[^13]: The reader of [@crater2] may notice that the total $\chi ^{2}$ of the model there of 101 (corresponding to Th1 here) was substantially lower than the 237 that we found here. There are several reasons for this difference. The main reason is that in this paper we do not include a 5% addition to the calculational uncertainty based on the total meson widths. Also, there are 19 more mesons in the present model, many of which were difficult to fit. Thirdly, the experimental errors changed. Fourth, many of the newer mesons (for example the $\eta _{b}$) not only added more to the $\chi ^{2}$ from their own fit, but also indirectly to the older ones (e.g. the $1^{3}S_{1}$ $\Upsilon $ meson). There is no difference in the parameter sets and potentials in Th1 and those used in [@crater2] although the values are different.
[^14]: Another possible source of the strange multiplet inversion, is that the observed $^{3}P_{0}$ states of 1450 for the $u\bar{d}$ and 1430 for the $u\bar{s}$ systems are, in fact not zero node states, but rather one node excited states. This may, in our formalism, give room to an interpretation of the $u\bar{d}(980)$ and the $\kappa (700-900)$ mesons as possible candidates for the zero node states. Our tables support that more than the identification of a zero node $1474$ for the $u\bar{d}$ and $1425$ for the $u\bar{s}$ systems. However, using the parameters on our model, we would obtain the $2^{3}P_{0}$ values of $1800$ and $1850$ for those one node states, well above the $1474$ and $1425$ experimental values. So this does not appear to be a plausible alternative for either Th1 or Th2.
[^15]: We point out, however, that our model does much better in simultaneously working with heavy and light $q\bar{q}$ hyperfine splittings than that obtained by typical constituent (non-chiral) quark models (a major exception discussed here in detail is the quasipotential model of [@rusger])
[^16]: Brayshaw considered the modification of the meson mass by $w\mathbf{1}\rightarrow w\mathbf{1}+|G\rangle Z\langle G|$ with $Z$ a fixed parameter and $|G\rangle $ in the $ns$ subspace. This is equivalent to the mixing matrix $$\mathbb{T=}Z\begin{bmatrix}
\langle n\bar{n}|G\rangle \langle G|n\bar{n}\rangle & \langle n\bar{n}|G\rangle \langle G|s\bar{s}\rangle \\
\langle s\bar{s}|G\rangle \langle G|n\bar{n}\rangle & \langle s\bar{s}|G\rangle \langle G|s\bar{s}\rangle\end{bmatrix}\equiv
\begin{bmatrix}
t_{11} & t_{12} \\
t_{21} & t_{22}\end{bmatrix},$$with the property that (for real matrix elements)$$t_{11}t_{22}=t_{12}t_{21}=\left\vert t_{12}\right\vert ^{2}=t_{12}^{2},$$just as with the choice of Eq.(\[53\]).
[^17]: We ignore here the coupling that would result from this diagonalization that would be brought on by the Coulomb interactions between the equal mass $q\bar{q}$ pairs.
[^18]: These fits are simultaneous with the fits of the earlier 105 mesons, with the same quark masses and potential parameters used in the generation of Tables (\[2\]-\[9\]). Oddly, a precise fit to the $\pi ^{0}$ was not possible in either theory, in spite of the three extra parameters $\delta
m_{u},\delta m_{d},$ and $\delta m_{s}$ available.
[^19]: There are at least three other additional hints of a possible emergent chiral symmetry and its breaking in the TBDE model . a) the relatively small quark $u,d,$masses $\sim 60-100$ MeV compared with $\sim 300$ MeV with most other quark models b) Our $\pi $ mass decreases toward zero as$~m_{q}\rightarrow 0$, c) the matrix element of the divergence of the axial vector current being proportional to the quark mass[@saz86],[@cra88]
[^20]: Due to their large descrepancy on the (controversial) $^{3}P_{0}$ $D_{s}(2370)$ meson, their $\chi ^{2}$ of 389 is actually larger than our 255 (our fit here is altered to include less mesons and $m_{u}=m_{d}$ so there are slight differences from the previous tables). If one eliminates that meson from the fit our $\chi ^{2}$ reduces to 251 while their $\chi ^{2}
$ reduces to 69. It should be pointed out that their fit did not appear to be a least $\chi ^{2}$ one like our fit. It also was not an overall fits like ours.
[^21]: In their ealier papers where the fits to the heavier mesons are given they use $\alpha _{s}=4\pi /(\beta _{0}\ln (\mu ^{2}/\Lambda ^{2}))$ where $\mu $ is the reduced mass and in the recent papers where the fits to lighter mesons are given they use $\alpha _{s}=4\pi /(\beta _{0}\ln ((\mu
^{2}+M_{B}^{2})/\Lambda ^{2})).$ In the former papers they use $\Lambda =169$ or $178$ MeV and in the recent ones $\Lambda =413$ MeV. It is not clear how using just one form for all the mesons would affect the overall fit.
[^22]: In [@yoon] the Adler-Piran potential was replaced by a form that displays asymptotic freedom in the QCD coupling via $\left( 8\pi /27\right)
/\ln (K+B/(\Lambda r)^{2})$. Although the model used there does not give as good a fit to the meson spectrum as the Adler-Piran model it does display asymptototic freedom in a simpler form.
[^23]: In [@yoon], Crater, Yoon, and Wong described some unusual singularity structures of the effective potentials and wave functions that show up in Eq. (\[schlike\]) for singlet and triplet states, in both QED and QCD. In these cases the TBDE lead to effective potentials and wave functions that are nevertheless not singular. The most noteworthy case was for the coupled ${{}^{3}}S_{1}$-${{}^{3}}D_{1}~$ triplet system, when the tensor coupling is properly taken into account. There it was shown that including the tensor coupling is essential in order that the effective potentials and wave functions are well behaved at short distances, with the $S$-state and $D$-state wave functions becoming simply proportional to each other at short distance (see Appendix C).
[^24]: H. Sazdjian has considered chiral symmetry and its breaking in the context of a closely related version of the TBDE. This was later discussed by Crater and Van Alstine, [@cra88]. In particular, it is found that the matrix element of the divergence of the axial vector current is proportional to the quark mass. This demsonstrates that the quark masses in the TBDE play the same role as the quark masses in QFT. In addition, Sazdjian shows that the pion decay constant in the context of the TBDE does not vanish in the limit of $m_{q}\rightarrow 0$. Sazdjian shows analytically in the context of a pseudoscalar confining potential, the existence of a massless pseudoscalar meson when $m_{q}\rightarrow 0$. These are two of the main effects of the spontaneous breakdown of chiral symmetry. In our earlier work with Van Alstine [@cra88], we showed numerically for the case of scalar confinining intereractions that the calculated pion mass tends to zero as $m_{q}\rightarrow 0$. In a later work ([@crater2]) we showed, however, that the behavior is not of the square-root relation ($m_{\pi }\sim \sqrt{m_{q}}/F_{\pi }$). The same behavior appears to hold with the present calculations.
[^25]: This equation and its gauge structure can also be seen to result from the equal mass singlet equation version of (\[57\]) for $S=V=0,$ $A=-\alpha /r$ or its radial version given by Eq. (\[ss\]). (It is noted that under these conditions, $\Phi _{D}=3\Phi _{SS},$and $\Phi _{SOD}=\Phi _{SOX}=0$)
[^26]: The authors are grateful to Professor R. N. Faustov for pointing out to us the results of [@mfa] and for their reason that the bound state equation used in [@rusger] did not include the $-A^{2}$ term.
[^27]: In their appendix A Godfrey and Isgur modify the Coulomb and contact spin-spin term used here with smearing functions and extra non-local parts in order to account for the off mass shell effects not present in the on shell scattering amplitudes from which they extract their potentials. We do not include the effects of the gaussian smoothing factors in the determination of the modification of the Coulomb term from their Eq. (A15).
[^28]: In [@faustdrc] the static limit Dirac equation was recovered from a two-body quasipotential equation by techniques with some similarity to the Gross equation[@gross1]. In that equation the relative energy is constrained by restricting one of the spin-one half particles to its positive energy mass shell. This differs from the TBDE which treats both particles off shell but yet constraining the relative energy covariantly through $P\cdot p\psi =0.$ The EFG equation also has this constraint, but unlike the equation derived in [@faustdrc], the Gross equation and the TBDE, it does not have the Dirac equation as a static limit as seen by a comparison of Eq. (\[rgst\]) with Eq. (\[571\]).
[^29]: In particular, their approach does not include the quadratic structure $2m_{w}S+S^{2}$ implied by the those authors’ approaches to scalar field theories.
[^30]: In particular, the quasipotential equation Eq. (\[quasi\]) for the potential to be used in Eq. (\[tod0\]) must be iterated to second order ($V^{(2)}=T_{1}GT_{1}-T_{2}$). It is remarkable that this this gauge like structure postulate Eq. (\[gage\]) anticipates the systematic inclusion of higher order terms by the quasipotential formalism.
[^31]: In this, not only does the gauge like minimal structure $(\varepsilon
_{w}-A)^{2}$ appear to be a natural outgrowth of classical $O(1/c^{2})$ expansion to at least order $1/c^{4},$ but also a minimal scalar interaction structure appears so that combined they yield $(\varepsilon
_{w}-A)^{2}-(m_{w}+S)^{2}$ , again, at least through order $1/c^{4}$.
[^32]: In short, one inserts Eq. (\[hyp2\]) into (\[hyp1\]) and brings the free Dirac operator (\[es0\]) to the right of the matrix hyperbolic functions. Using commutators and $\cosh ^{2}\Delta -\sinh ^{2}\Delta =1$ one arrives at Eq. (\[extd\]). The structure of these equations are very much the same as that of a Dirac equation for each of the two particles, with $M_{i}$ and $E_{i}$ playing the roles that $m+S$ and $\varepsilon -A$ do in the single particle Dirac equation (\[obde\]). Over and above the usual kinetic part, the spin-dependent modifications involving $G\mathcal{P}_{i}$ and the last set of derivative terms are two-body recoil effects essential for the compatibility (consistency) of the two equations
[^33]: Just as $x^{\mu }$ is a four vector, so are $\gamma ^{\mu }$ (in the sense of Dirac) and $P^{\mu }.$ Thus, the matrix structures of the time-like and space-like interactions in Eq. (\[hyp3\]) are $\gamma _{1}^{0}\gamma
_{2}^{0}$ and $\mathbf{\gamma }_{1}\cdot \mathbf{\gamma }_{2}$ only in the c.m. system due to the fact that from Eq. (\[beta\]), $\beta _{i}=\gamma
_{i}^{0}$ only in the c.m. frame. Likewise, $\Sigma _{i}^{\mu }=(0,\mathbf{\Sigma )}$ only in the c.m. frame just as is $x_{\perp }^{\mu }=(0,\mathbf{r)}$ in that frame only.
[^34]: Strictly speaking, in the static limit, the square roots forms obtained from Eq. (\[one\]) imply $M_{1}\rightarrow \left\vert m+S\right\vert >0.~$For for large distances, $A\rightarrow 0,$ and so Eq. (\[two\]) implies $E_{1}\rightarrow |\varepsilon -V|$ and for short distances $V\rightarrow 0$, $E_{1}\rightarrow |\varepsilon -A|.$ These could possibly be in opposition to the forms with indefinite sign of $m+S$ and $\varepsilon -V$ or $\varepsilon $ $-A$ that would appear in the one body Dirac equation. In our applications to meson spectroscopy $S$ is always positive so we need not worry about the use of the square root form of $M_{i}$ in the computation of $M_{i}$. Since $V$ is positive and increasing with distance, it is possible that at large enough distances $\varepsilon -V<0.$ In that case use of the positive root for the square root would not agree with the sign of $\varepsilon -V$. We found that for the $^{1}S_{0}$ $b\bar{u}$ and $b\bar{d}
$ mesons that does occur near the very end of the integration cutoff distance of about $1.92$ Fermis. That is so close to the end, however, that use of the positive square root is unlikely to have any effect on the spectral results (for the $^{3}P_{2}$ $b\bar{u}$ and $b\bar{d}$ mesons $\varepsilon -V$ remained positive throughout the integration region.) Future work may address the theoretical problem of how to make the approach to the static limit of the square root forms smooth and exact.
[^35]: In particular, the short distance radial behavior is $v_{J0J}\rightarrow r^{\sqrt{J(J+1)-\alpha ^{2}}\text{.}}$
[^36]: The short distance radial dependencies of the wave functions that arise from Eqs. (\[pl\]) and (\[mi\]) are the well behaved forms $v_{J-11J}(r)=r^{(1/2+\sqrt{J(J+1)-\alpha ^{2}})}$ and $v_{J+11J}(r)=\frac{J}{\sqrt{J(J+1)}}v_{J-11J}(r)~$[@yoon]. Such behaviors as represented here and in the previous footnote arise because the effective potentials that appear in our bound state equations are well defined (no delta function potentials for example). We refer the interested reader to Eq. (42-43) of section V of [@yoon] for the intriging details of the cancellation of otherwise singular potentials due to the presence of the tensor coupling terms.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A distributed optimal control problem with final observation for a three-dimensional Lagrange averaged Navier-Stokes-$\alpha$ model is studied. The solvability of the optimal control problem is proved and the first-order optimality conditions are established. Moreover, by using the Lagrange multipliers method, an optimality system in a weak and a strong form is derived.'
address:
- '(Corresponding Author) Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia.'
- ' Departamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta-Chile.'
author:
- 'E.J. Villamizar-Roa'
- 'E. Ortega-Torres'
title: 'A distributed control problem for a three-dimensional Lagrangian averaged Navier-Stokes-$\alpha$ model '
---
[^1]
Introduction
============
We are interested in the study of an optimal control problem for a Lagrange averaged Navier-Stokes-$\alpha$ model (also known as LANS-$\alpha$ or the viscous Camassa-Holm system). The LANS-$\alpha$ model, introduced by S. Chen, C. Foias, D.D. Holm, E. Oslon, E.S. Titi, and S. Wynne in [@chen1], is the first one to use Lagrangian averaging to address the turbulence closure problem, that is, the problem of capturing the physical phenomenon of turbulence at computably low resolution. This model provides closure by modifying the nonlinearity in the Navier-Stokes equations to stop the cascading of turbulence at scales smaller than a certain length, but without introducing any extra dissipation (c.f. [@lans2; @chen1; @chen2; @chen3; @holm; @lans4]). The mathematical model is obtained by regularizing the 3D Navier-Stokes equations through a filtration of the fluid motion that occurs below a certain length scale $\alpha^{1/2};$ the length scale is the filter width derived from inverting the Helmholtz operator $I-\alpha\Delta.$ Explicitly, LANS-$\alpha$ model can be written in the following form: $$\left\{
\begin{array}
[c]{rcl}%
&&(I-\alpha \Delta) u_t+\nu (I-\alpha \Delta)
Au+(u\cdot\nabla)\,(I-\alpha \Delta)u
-\alpha (\nabla u)^*\cdot \Delta u + \nabla p= f \mbox{ in } Q, \\
&& \nabla \cdot u =0 \mbox{ in } Q,\\
&& u=0, \quad Au=0 \mbox{ on } \Gamma \times (0,T),\\
&& u(x,0)= u_0(x) \mbox{ in } \Omega,
\end{array}
\right. \label{eq1}$$ where $u$ and $p$ are unknown, representing respectively, the large-scale (or averaged) velocity and the pressure, in each point of $Q=\Omega \times (0,T), 0 < T <\infty.$ Here, $\Omega$ is a bounded domain in $\mathbb{R}^3$ (with boundary $\Gamma$of class $C^2$) where the fluid is occurring, and $(0,T)$ is a time interval. The operator $A$ denotes the known Stokes operator. Moreover, the right hand side $f$ is a fixed external force and $u_0$ a given initial velocity field. The positive constant $\nu$ represents the kinematic viscosity of the fluid.
The interest of studying the LANS-$\alpha$ models arises principally in the approximation of many problems relating to turbulent flows because it preserves the properties of transport for circulation and vorticity dynamics of the Navier-Stokes equations. One of the main reasons justifying its use is the high-computational cost that the Navier-Stokes model requires [@lans2]. Notice that when $\alpha=0$ the LANS-$\alpha$ model reduces to the classical Navier-Stokes system. We refer [@lans2; @chen1; @chen2; @chen3; @lans7; @lans8; @holm; @lans4] and references therein, for a complete description of the development of the LANS-$\alpha$ model, as well as, a discussion about the physical significance, namely, in turbulence theory.
From a mathematical point of view, several advances related the well-posedness, long time behavior, decay rates of the velocity and the vorticity, the connection between the solutions of the LANS-$\alpha$ model and the 3D Navier-Stokes system and Leray-$\alpha$ model, the existence and uniqueness of solutions for stochastic versions, have been developed in last years, see for instance [@lans9; @lans2; @marquez-duran; @lans5; @Cheskidov; @coutand; @Fo-Ho-Ti; @lans7; @lans8; @lans4; @lans3] and references therein. In particular, opposed to three dimensional Navier-Stokes equations, for LANS-$\alpha$ model, the existence and uniqueness of weak solutions is known (see for instance [@lans7]). This point is relevant in control problems because it permits to guarantee that the reaction of the flow produced by the action of a control is unique.
In this paper we are interested in an optimal control problem for the LANS-$\alpha$ model (\[eq1\]) where the body force is regarded as the control and a final observation is considered; in this sense we say that it is an optimal control problem for a distributed parameter system with final observation. More precisely, we wish to minimize the functional $$J(u,v)=\frac{\gamma_1}{2}\int_0^T\Vert u(t)-u_d(t)\|_{D(A)}^2
dt+\frac{\gamma_2}{2}\int_\Omega\vert u(x,T)-u_T(x)\vert^2dx
+\frac{\gamma_3}{2}\int_0^T \|v(t)\|_{2}^2 dt,$$ where the velocity field is subject to state system (\[eq1\]) where $f$ is now replaced by the distributed control field $v.$ The functions $u_d,
u_T$ are given and denote the desired state, and the parameters $\gamma_1,\gamma_2,\gamma_3>0$ stand the cost coefficients for the control. The exact mathematical formulation will be given in Section 3. We will prove the solvability of the optimal control problem and state the first-order optimality conditions. By using the Lagrange multipliers method we derive an optimality system in a weak and strong formulation. To the best of our knowledge, this paper is the first work dealing with optimal control problems where the state variable satisfies the 3D LANS-$\alpha$ model (\[eq1\]). However, in the recent papers [@tian2; @tian1] the authors studied the problem of optimal control of the viscous Camassa-Holm equation in one dimension. The models treated in [@tian2; @tian1] can be viewed as one dimensional versions of the three dimensional LANS-$\alpha$ model.
Related to the nonstationary Navier-Stokes system, there are many results available in the literature concerned with the study of optimal control problems (see [@fursikov] and references therein). In particular, necessary conditions for optimal control of 2D-Navier-Stokes model can be found in [@abergel; @gunzburger1; @gunzburger2; @hinze]. Necessary conditions for optimal control of 3D Navier-Stokes were obtained in [@casas].
The paper is organized as follows. In section 2 we establish the notation to be used and recall some known results for the LANS-$\alpha$ model. In section 3 we setting the precise optimal control problem and prove the existence of optimal solutions. In section 4 we derive the first-order optimality conditions, and by using the Lagrange multipliers method we derive an optimality system.
Preliminaries
=============
Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ with boundary $\Gamma$ of class $C^2$. We denote by $\mathcal{D}(\Omega)$ and $\mathcal{D}^{\prime}(\Omega)$ the space of functions of class $C^{\infty}(\Omega)$ with compact support, and the space of distributions on $\Omega,$ respectively. Throughout this paper we use standard notations for Lebesgue and Sobolev spaces. In particular, the $L^2(\Omega)$-norm and the $L^2(\Omega)$-inner product, will be represented by $\Vert \cdot\Vert$ and $(\cdot,\cdot),$ respectively. We consider the solenoidal Banach spaces $H$ and $V$ defined, respectively, as the closure in $(L^2(\Omega))^3$ and $(H^1(\Omega))^3$ of $$\begin{aligned}
\mathcal{V}&=&\{u \in (\mathcal{D}(\Omega))^3 : \nabla \cdot u=0
\mbox{ in } \Omega\}.\end{aligned}$$ Here, $\nabla \cdot u$ denotes the divergence of the field $u.$ The norm and the inner product in $V$ will be denoted by $\Vert
u\Vert_V$ and $(\nabla u, \nabla v),$ respectively. Throughout this paper, if $X$ is a Banach space with dual space $X'$, the duality paring between $X'$ and $X$ will be denoted by $\langle\cdot,
\cdot\rangle_{X',X}.$ To simplify the notation, we will use the same notation for vectorial valued and scalar valued spaces.
For $X$ Banach space, $\|\cdot\|_X$ denotes its norm and $L^p(0,T;X)$ denotes the standard space of functions from $[0,T]$ to $X,$ endowed with the norm $$\|u\|_{L^p(0,T;X)}= \bigg( \int_0^T\|u\|_X^p dt\bigg)^{1/p}, \ 1\leq p <\infty,\qquad
\|u\|_{L^\infty(0,T;X)}= \sup_{t\in (0,T)} \|u(t)\|_X.$$ In the sequel we will identify the spaces $L^p(0,T;X):= L^p(X)$ and $L^p(0,T;L^p(\Omega)) :=L^p(Q)$.We recall the following compactness result:
([@Simon])\[lema1\] Let $B_0, B$ and $B_1$ be Banach spaces with $B_0\hookrightarrow B\hookrightarrow
B_1$ continuously and $B_0\hookrightarrow B$ compact. For $1\leq p\leq
\infty$ and $T <\infty$ consider the Banach space $$\label{eq5}
W=\{ u \in L^{p} (0,T;B_0), \ u_t\in L^{1} (0,T;B_1)\}.$$ Then $W \hookrightarrow L^p(0,T;B)$ compactly.
Let $P:L^2(\Omega) \rightarrow H$ be the Leray projector, and denote by $A= - P\Delta$ the Stokes operator with domain $D(A)=H^2(\Omega)\cap V$. It is known that $A$ is a self-adjoint positive operator with compact inverse. Since $\Gamma$ is of class $C^2$, the norms $\|A u\|$ and $\|u\|_{H^2}$ are equivalent. For $u\in D(A)$ and $v\in L^2(\Omega)$ we define the element of $H^{-1}(\Omega)\equiv (H^1_0(\Omega))'$ by $$\langle(u\cdot\nabla)v,w\rangle_{H^{-1},H^1_0}=\sum_{i,j=1}^3\langle\partial_i
v_j,u_iw_j\rangle_{H^{-1},H^1_0},\ \forall w\in H^1_0(\Omega).$$ In particular, if $v\in H^1(\Omega),$ the definition of $\langle(u\cdot\nabla)v,w\rangle_{H^{-1},H^1_0}$ coincides with the definition of $$((u\cdot\nabla)v,w)=\sum_{i,j=1}^3\int_\Omega(u_i\partial_i
v_j)w_jdx.$$ Let us denote by $(\nabla u)^*$ the transpose of $\nabla u.$ Thus, if $u \in D(A)$ then $(\nabla u)^* \in H^1(\Omega) \subset
L^6(\Omega)$. Consequently, for $v\in L^2(\Omega)$ we have that $(\nabla u)^*\cdot v \in L^{3/2}(\Omega)\subset H^{-1}(\Omega)$ with $$\langle(\nabla u)^*\cdot
v,w\rangle_{H^{-1},H^1_0}=\sum_{i,j=1}^3\int_\Omega(\partial_ju_i)
v_iw_jdx,\ \forall w\in H^1_0(\Omega).$$ One can check that for $u,w \in D(A), v\in L^2(\Omega),$ the following equality holds $$\label{eq7}
\langle(u\cdot \nabla) v, w\rangle_{H^{-1},H^1_0}= -\langle(\nabla
w)^*\cdot v,u\rangle_{H^{-1},H^1_0}.$$ We consider the nonlinear operator $ B: D(A)\times D(A)\rightarrow
D(A)'$ defined by $$\begin{aligned}
\label{eq8}
\langle B(u,v),w\rangle_{D(A)',D(A)}=\langle(u\cdot \nabla)(v-\alpha
\Delta v), w\rangle_{V',V}+\langle(\nabla u)^*\cdot (v-\alpha\Delta
v),w\rangle_{V',V}.\end{aligned}$$ Thus, from (\[eq7\]) we have $$\label{eq9}
\langle B(u,v),u \rangle_{D(A)',D(A)}= 0, \ \forall \, u,v \in D(A).$$ Also, we have that $$\begin{aligned}
|\langle B(u,v),w\rangle_{D(A)',D(A)}|&\leq &C\|u\|\|\nabla v\|\|A
w\|+ C\,\alpha (\|u\|_{L^6}\|\nabla w\|_{L^3}+
\|\nabla u\|\|w\|_{L^\infty})\|\Delta v\|\\
&\leq & C\|\nabla u\|\|A v\|\|A w\|+C\alpha \|\nabla u\|\|A w\|\|A
v\| \leq C\|\nabla u\|\|A v\|\|A w\|.\end{aligned}$$ Therefore, $$\label{eq10}
\|B(u,v)\|_{D(A)'} \leq C\,\|\nabla u\|\|A v\|\leq C\Vert
u\Vert_V\Vert v\Vert_{D(A)}, \quad \forall \, u, v \in D(A),$$ and thus, for all $u, v \in L^\infty(V)\cap L^2(D(A))$ it holds $B(u,v) \in L^2(D(A)')$. Denoting by $ \Delta_\alpha= I-\alpha \Delta$, one gets $$\Delta_\alpha u \in L^\infty(V') \cap L^2(H)\quad \mbox{ and }
\quad \Delta_\alpha Au \in L^2(D(A)') \quad \forall u\in
L^2(D(A))\cap L^\infty(V).$$ With the above notations, the system (\[eq1\]) can be rewritten as $$\left\{
\begin{array}
[c]{rcl}%
&&\Delta_\alpha u_t +\nu \Delta_\alpha Au+ B(u,u) + \nabla p
= f \mbox{ in } Q,\\
&&\nabla\cdot u=0 \mbox{ in } Q, \\
&& u=0, \quad Au=0 \mbox{ on } \Gamma \times (0,T),\\
&& u(x,0)= u_0(x) \mbox{ in } \Omega.
\end{array}
\right. \label{eq11}$$ Now we are in position to establish the definition of weak solution of Problem (\[eq1\]) (equivalently (\[eq11\])).
(*Weak solution*) For $f\in L^2(Q)$ and $u_0\in V,$ a *weak solution* of the problem (\[eq11\]) is a field $u\in L^2(D(A)) \cap
L^\infty(V)$ with $u_t \in L^2(H)$ satisfying
$$\left\{
\begin{array}
[c]{rcl}%
&&\frac{d}{dt}((u, w) +\alpha(\nabla u, \nabla w))
+\nu (Au,w+\alpha Aw)\\
&&\ \ \ \ \ \ +\langle B(u,u), w\rangle_{D(A)',D(A)}
=(f, w), \quad \forall \, w\in D(A),\\
&&\ u(x,0)= u_0(x) \mbox{ in } \Omega,
\end{array}
\right. \label{eq16}$$
or equivalently, $$\left\{
\begin{array}
[c]{rcl}%
&&\Delta_\alpha u_t+\nu \Delta_\alpha Au+ B(u,u)
= f \mbox{ in } D(A)', \\
&& A u=0 \mbox{ in } \Gamma \times (0,T),\\
&& u(x,0)= u_0(x) \mbox{ in } \Omega.
\end{array}
\right. \label{eq17}$$
\[teor2\] (*Existence and uniqueness of weak solution*) Assuming that $f\in L^2(Q)$ and $u_0 \in V$, there exists a unique weak solution of (\[eq11\]).
*Proof.* The existence of weak solutions follows from the classical Galerkin approximations and energy estimates; the uniqueness follows from a standard Gronwall argument (see e.g. [@BRV; @real1; @lans2; @lans5; @lans7; @lans3]).
A distributed control problem: Existence of optimal solution
============================================================
We start by establishing the control in the system. We denote the control by $v\in L^2(Q)$ which will be use as a source term in (\[eq11\])$_1$; thus the right-hand side of equality (\[eq17\])$_1$ will be defined as: $$(f,w) =(v,w), \quad \forall w\in D(A).$$In order to specify exactly the problem, we make some considerations. We define the Banach space $$\mathbb{W}=\{ u\in L^2(D(A))\cap L^\infty(V)\,:\, u_t\in L^2(H)\},$$ with norm given by $$\Vert w\Vert_{\mathbb{W}}:=\max \{\Vert
u\Vert_{L^2(D(A))},\Vert u\Vert_{L^\infty(V)}, \Vert u_t\Vert_{L^2(H)}\}.$$
Since $D(A)\hookrightarrow V \hookrightarrow H$, and $D(A)\hookrightarrow V $ compactly, from Lemma \[lema1\] we have $\mathbb{W}\hookrightarrow L^2(V)$ compactly; furthermore, and as $D(A),V,H$ are Hilbert spaces, $\mathbb{W} \hookrightarrow C([0,T];V)$ (cf. [@lions]). We also consider the subspace $\mathbb{W}_0$ of $\mathbb{W}$ defined by $$\mathbb{W}_0:=\{ u\in \mathbb{W} : Au=0 \mbox{ on } \Gamma\times (0,T)\}.$$
In order to establish the control problem, we assume the following general hypotheses:
- The regularization parameters $\gamma_1, \gamma_2$ and $\gamma_3,$ which measures the cost of the control, are fixed positive numbers.
- The initial data $ u_0 \in V$, the desired velocity $u_d\in L^2(D(A))$ and the function $u_T \in H$.
- The set of admissible controls $\mathcal{U}_{ad}$ is defined by: $$\label{eq22}
\mathcal{U}_{ad}=\{v \in L^2(Q): v_{a,i}(x,t) \leq v_i(x,t)\leq v_{b,i}(x,t)
\ a.e. \mbox{ on } Q,\ i=1,2,3\},$$ where the control constraints $v_a, v_b$ are required to be in $L^2(Q)$ with $ v_{a,i}(x,t) \leq v_{b,i}(x,t)$ a.e. on $Q.$ Notice that $\mathcal{U}_{ad}$ is a non-empty, convex and closed set in $L^2(Q)$.
Under the hypotheses (H1)-(H3), for $(u,v)\in \mathbb{W}_0\times
\mathcal{U}_{ad}$ we define the following objective functional $$\label{eq23}
J(u,v)=\frac{\gamma_1}{2}\int_0^T\|u(t)-u_d(t)\|_{D(A)}^2
dt+\frac{\gamma_2}{2}\int_\Omega |u(x,T)-u_T(x)|^2 dx
+\frac{\gamma_3}{2}\int_0^T \|v(t)\|^2dt.$$ Thus, we consider the following distributed optimal control problem: $$\left \{
\begin{array}
[c]{rcl}%
\mbox{Minimize}&& J(u,v)\\
&&\ \ ({u},{v}) \in \mathbb{W}_0\times\mathcal{U}_{ad},
\end{array}
\right.\label{eq24}$$ subject to the state equation $$\left\{
\begin{array}
[c]{rcl}%
&& \Delta_\alpha u_t +\nu \Delta_\alpha Au+ B(u,u)
= v \mbox{ in } L^2(D(A)'),\\
&& u(x,0)= u_0(x) \mbox{ in } V.
\end{array}
\right. \label{eq25}$$ Then, the set of admissible solutions to (\[eq24\])-(\[eq25\]) is defined as: $$\label{eq28}
\mathcal{S}_{ad}=\{(u,v) \in \mathbb{W}_0\times \mathcal{U}_{ad}:
J(u,v) < \infty \mbox{ and } (u,v)
\mbox{ satisfies } (\ref{eq25})\}.$$
Existence of solution
---------------------
We will show that the optimal control problem (\[eq24\])-(\[eq25\]) has a solution.
Under the assumptions $(H1)$-$(H4)$, there exists a solution $(\hat{u}, \hat{v}) \in \mathcal{S}_{ad}$ to the optimal control problem (\[eq24\])-(\[eq25\]).
By Theorem \[teor2\], the pair $(u,v_a)\in \mathcal{S}_{ad}$; thus the set $\mathcal{S}_{ad}\neq \phi$. Now, since $J$ is bounded from below ($J(u,v) \geq 0$), there exists a minimizing sequence $(u^m, v^m)$ in $\mathcal{S}_{ad}$ such that $$\lim_{m\rightarrow\infty} J(u^m,v^m)=\inf\{J(u,v): (u,v)\in\mathcal{S}_{ad}\},$$ and for all $w\in D(A)$ it holds: $$\label{eq29}
\langle\Delta_\alpha
u_t^m,w\rangle_{D(A)',D(A)}+\nu\langle\Delta_\alpha Au^m,
w\rangle_{D(A)',D(A)}+\langle B(u^m,u^m),w\rangle_{D(A)',D(A)} =(
v^m,w).$$ From (\[eq29\]), using integration by parts on $\Omega$, we have $$\label{eq30}
( u_t^m,w)+\alpha(\nabla u^m_t,\nabla w) +\nu(\nabla u^m, \nabla w)
+\nu \alpha( Au^m, A w)+\langle B(u^m, u^m),w\rangle=( v^m,w).$$ Then, setting $w=u^m(t)$ in (\[eq30\]) and taking into account (\[eq9\]), we get $$\frac12\frac{d}{dt}(\|u^m\|^2+\alpha\|\nabla u^m\|^2)+\nu \|\nabla
u^m\|^2 +\nu \alpha \|A u^m\|^2=(v^m,u^m).$$ Using the Hölder and Young inequalities it holds $$|( v^m,u^m) | \leq \|v^m\|\|u^m\|\leq C_\nu \,\|v^m\|^2 + \frac{\nu}{2}\|\nabla u^m\|^2.$$ Therefore, $$\label{eq31}
\frac{d}{dt}(\|u^m\|^2+\alpha\|\nabla u^m\|^2)+\nu \|\nabla u^m\|^2
+2\nu \alpha \|A u^m\|^2\leq C_\nu \,\|v^m\|^2.$$ Integrating (\[eq31\]) from $0$ to $t\in [0,T]$, we obtain $$\begin{aligned}
\label{eq32}
&&\|u^m(t)\|^2+\alpha\|\nabla u^m(t)\|^2+\nu \int_0^t (\|\nabla
u^m(s)\|^2
+\alpha\|A u^m(s)\|^2) ds \nonumber\\
&&\hspace*{1cm} \leq C\,\int_0^t \|v^m(s)\|^2 ds +
\|u_0\|^2+\alpha\|\nabla u_0\|^2.\end{aligned}$$ Since $v^m \in \mathcal{U}_{ad}$ and $u_0\in V,$ from (\[eq32\]) we conclude that $$\begin{aligned}
\label{eq33}
\{u^m\}_{m\geq 1} \ \mbox{ is uniformly bounded in} \ L^\infty(V)\cap
L^2(D(A))\\
\{v^m\}_{m\geq 1} \ \mbox{ is uniformly bounded in} \ L^2(Q).\label{eq34}\end{aligned}$$ On the other hand, from (\[eq29\]) and by applying integration by parts on $\Omega$ we get $$\langle\Delta_\alpha u^m_t,w\rangle_{D(A)',D(A)}=-\nu(\nabla u^m, \nabla
w) -\nu \alpha(Au^m, A w)-\langle B(u^m,
u^m),w\rangle_{D(A)',D(A)}+( v^m,w),$$ and then, by using the Hölder inequality together inequalities (\[eq10\]) and (\[eq33\]), we obtain $$\begin{aligned}
|\langle\Delta_\alpha u^m_t,w\rangle_{D(A)',D(A)}|
&\leq & C_{\nu,\alpha}(\|\nabla u^m\|+\|Au^m\|+\|B(u^m, u^m)\|_{D(A)'}+\|v^m\|)\|w\|_{D(A)}\\
&\leq &C_{\nu,\alpha}(\|\nabla u^m\|+\|Au^m\|+\|\nabla u^m\|\|A u^m\|+\|v^m\|)\|w\|_{D(A)}\\
&\leq &C_{\nu,\alpha}(\|\nabla u^m\|+\|Au^m\|+\|v^m\|)\|w\|_{D(A)}.\end{aligned}$$ Since $\langle \Delta_\alpha u_t,w\rangle_{D(A)',D(A)}= \langle
u_t+\alpha A u_t,w\rangle_{D(A)',D(A)} $ for all $w\in D(A)$, the last inequality implies $$\|u_t^m+\alpha A u^m_t\|_{D(A)'}\leq C\,(\|\nabla u^m\|+\|Au^m\|+\|v^m\|),$$ and by using the Young inequality $$\label{eq35}
\|u_t^m+\alpha A u^m_t\|^2_{D(A)'} \leq C\,(\|\nabla
u^m\|^2+\|Au^m\|^2+\|v^m\|^2).$$ By integrating (\[eq35\]) from $0$ to $t\in [0,T]$ and taking into account (\[eq33\])-(\[eq34\]) we have $$\label{eq36}
\int_0^t \|u_t^m(s)+\alpha A u^m_t(s)\|^2_{D(A)'}ds \leq C.$$ Since the operator $A$ is self adjoint and positive, the following inequality holds (see [@lans3]) $$\label{eq37}
\|v\|^2_{D(A)'} \leq \|v+\alpha A v\|_{D(A)'}^2 \ \mbox{for each }\
v\in D(A)'.$$ Then, by using triangular inequality and (\[eq37\]), we get $$\label{eq38}
\|\alpha Au^m_t\|_{D(A)'}^2\leq \|u^m_t+\alpha Au^m_t\|_{D(A)'}^2+
\|u^m_t\|^2_{D(A)'} \leq C\,\|u^m_t+\alpha Au^m_t\|_{D(A)'}^2.$$ Thus, from (\[eq36\]) and (\[eq38\]), we conclude that
$$\label{eq39}
\{Au^m_t\}_{m\geq 1} \ \mbox{ is uniformly bounded in} \ L^2(D(A)'),$$
$$\label{eq40}
\{u^m_t\}_{m\geq 1} \ \mbox{ is uniformly bounded in} \ L^2(Q).$$
Moreover, from (\[eq33\]) and (\[eq40\]) we have $$\label{eq41}
\{u^m\}_{m\geq 1} \ \mbox{ is uniformly bounded in } \ \mathbb{W},$$ with $\mathbb{W}$ compactly imbedded in $L^2(V)$ (cf. Lemma \[lema1\]). Then, from (\[eq33\]), (\[eq34\]), (\[eq39\]), (\[eq40\]) and (\[eq41\]), there exists a subsequence, which again we denote by $(u^m, v^m),$ converging to some limit $(\hat{u},\hat{v}) \in
\mathbb{W} \times\mathcal{U}_{ad}$ such that as $m \rightarrow
\infty$, $$\begin{aligned}
&& u^m \rightarrow \hat{u} \mbox{ weakly in } L^2(D(A)) \mbox{ and strongly in } L^2(V),\\
&& u^m_t \rightarrow \hat{u}_t \mbox{ weakly in } L^2(Q),\\
&& A u^m_t \rightarrow A\hat{u}_t\mbox{ weakly in } L^2((D(A)'),\\
&& v^m \rightarrow \hat{v} \mbox{ weakly in } L^2(Q).\end{aligned}$$ Passing to the limit as $m\rightarrow \infty$ in (\[eq29\]), we can obtain that $(\hat{u}, \hat{v})$ satisfies (\[eq25\]), with $A \hat{u}=0$ on $\Gamma\times (0,T),$ and $J(\hat{u}, \hat{v})<\infty.$ Since $u_0=u^m(0)$ in $V$ for all $m$, and $u^m\in \mathbb{W},$ it holds $u_0=\hat{u}(0)$ in $V$. Consequently $\hat{u}\in \mathbb{W}_0$ and thus we get that $(\hat{u},
\hat{v}) \in \mathcal{S}_{ad}.$ Then we obtain $$\label{eq42}
\lim_{m\rightarrow \infty} J(u^m, v^m) = \inf\{J(u,v): (u,v)
\in \mathcal{S}_{ad}\}\leq J(\hat{u}, \hat{v}).$$ As the functional $J: \mathcal{S}_{ad} \rightarrow \mathbb{R}$ is weakly lower semicontinuous, we have that (cf. [@brez]) $$\label{eq43}
J(\hat{u}, \hat{v}) \leq \lim_{m\rightarrow \infty}\inf J(u^m, v^m).$$ Finally, from (\[eq42\]) and (\[eq43\]) we conclude that $\displaystyle J(\hat{u},\hat{v}) =\inf\{J(u,v): (u,v) \in
\mathcal{S}_{ad}\}$.
First order optimality conditions
=================================
In order to derive an optimal system by using the Lagrange multiplier method, we formulate an abstract Lagrange multiplier principle. Let $X$ and $Y$ be two Banach spaces, $\mathcal{J}:X\rightarrow \mathbb{R}$ and $\mathcal{G}:X\rightarrow
Y.$ Consider the problem $$\label{eq43a}
\min_{z\in X}\mathcal{J}(z) \qquad \mbox{subject to } \
\mathcal{G}(z)=0.$$ The Lagrange function corresponding to the problem (\[eq43a\]) is defined by $$\mathcal{L}(z,\lambda_0,\lambda)= \lambda_0 \mathcal{J}(z)-\langle \lambda, \mathcal{G}(z)\rangle_{Y',Y},$$ where $\lambda_0\in \mathbb{R}$ and $\lambda \in Y'$ are called Lagrange multipliers. Then the following result is known (see e.g. [@gunz; @iofi]).
[@gunz; @iofi]\[ioffe\](The Lagrange multiplier rule). Let $\hat{z}$ be a solution of (\[eq43a\]). Assume that the functional $\mathcal{J}$ and the mapping $\mathcal{G}$ are continuously differentiable at the point $\hat{z}$ and that the rang of the mapping $\mathcal{G}_z(\hat{z}):X
\rightarrow Y$ is closed. Then there exists a nonzero Lagrange multiplier $(\lambda_0,\lambda)\in \mathbb{R}^+\times Y'$, such that $$\begin{aligned}
\mathcal{L}_z(\hat{z},\lambda_0, \lambda)h &=&\lambda_0 \mathcal{J}_z(\hat{z})h-\langle \lambda, \mathcal{G}_z(\hat{z})h\rangle_{Y',Y}=0\quad \forall \, h \in X,\\
\mathcal{L}(\hat{z},\lambda_0, \lambda)&\leq & \mathcal{L}({z},\lambda_0, \lambda),\ \forall z\in X,\end{aligned}$$ where $\mathcal{L}_z(\cdot,\cdot,\cdot)$ denotes the Fréchet derivative of $\mathcal{L}.$ Furthermore, if $\mathcal{G}_z(\hat{z}):X \rightarrow Y$ is an epimorphism, then $\lambda_0\neq 0$ and $\lambda_0$ can be taken as 1.
In order to derive the first-order optimality conditions for the problem (\[eq24\])-(\[eq25\]), we will apply Theorem \[ioffe\].
Observing (\[eq25\]), we define the operator $$\begin{aligned}
F: \mathbb{W}_0\times L^2(D(A)') & \rightarrow & L^2(D(A)')\\
(u,v) \ \ & \rightarrow & F(u,v):=\Delta_\alpha u_t+\nu
\Delta_\alpha Au+ B(u,u)- v.\end{aligned}$$
\[w21\] The operator $F$ is Fréchet differentiable with respect to $u$.
Using (\[eq8\]), $B(u+w,u+w)=B(u,u)+B(u,w)+B(w,u) +B(w,w)$. Then $$\label{eq44}
F(u+w,v) - F(u,v) =\Delta_\alpha w_t+\nu \Delta_\alpha Aw +B(u,w)
+B(w,u)+B(w,w).$$ Denoting by $Lw=\Delta_\alpha w_t+\nu \Delta_\alpha Aw +B(u,w)
+B(w,u)$, from (\[eq44\]) we get $$\label{eq45}
\|F(u+w,v) - F(u,v)-Lw\|_{L^2(D(A)')}=\|B(w,w)\|_{L^2(D(A)')}.$$ Since $w\in \mathbb{W}_0$, from (\[eq10\]) we obtain $$\|B(w,w)\|_{L^2(D(A)')}\leq C\|w\|_{L^\infty(V)}
\|w\|_{L^2(D(A))}\leq C\|w\|^2_{\mathbb{W}_0},$$ and then from (\[eq45\]) we have $$\|F(u+w,v) - F(u,v)-Lw\|_{L^2(D(A)')}\leq C\|w\|^2_{\mathbb{W}_0}.$$ Thus, $$\lim_{\|w\|_{\mathbb{W}_0}\rightarrow 0}\frac{\|F(u+w,v)
- F(u,v)-(\Delta_\alpha w_t +\nu \Delta_\alpha
Aw+B(u,w)+B(w,u))\|_{L^2(D(A)')}} {\|w\|_{\mathbb{W}_0}}=0.$$ Therefore, the Fréchet derivative of $F$ with respect to $u$ in an arbitrary $(u,v)$ is given by the operator $F_u(u,v):\mathbb{W}_0\rightarrow
L^2(D(A)')$ such that for each $w \in \mathbb{W}_0$, $$\label{eq46}
F_u(u,v)w=\Delta_\alpha w_t+\nu \Delta_\alpha Aw +B_u(u,u)w,$$ where $B_u(u,u)w=B(u,w)+B(w,u)$ is the Fréchet derivative of $B$ with respect to $u$ in an arbitrary point $(u,u)$.
Now let us to consider the closed linear subspace $\mathbb{Y}_0$ of $\mathbb{W}_0$ defined by $$\mathbb{Y}_0=\{ w \in \mathbb{W}_0 : w(x,0)=0 \ \forall x\in \Omega\}.$$ The following preliminary result holds:
\[l1\] Let $(u,v)\in \mathbb{W}_0\times L^2(Q)$ and $g\in L^2(D(A)')$ be given. Then there exists a unique solution $w\in \mathbb{Y}_0$ of the linear problem $$\begin{aligned}
\label{e1}
F_u(u, v)w=g.\end{aligned}$$
The proof follows by using the classical Galerkin approximations and energy estimates (see [@BRV; @real1]).
The functional $J$ is Fréchet differentiable with respect to $u.$
From definition of the functional $J$ we get: $$\begin{aligned}
J(u+w,v)-J(u,v)&=&\gamma_1\int_0^T (Aw,
Au-Au_d)dt + \frac{\gamma_1}{2}\int_0^T\|Aw\|^2 dt\\
&&+\gamma_2(w(T),u(T)-u_T)+\frac{\gamma_2}{2}\Vert w(T)\Vert^2.\end{aligned}$$ Then $$\begin{aligned}
&&\vert J(u+w,v)-J(u,v)- \gamma_1\int_0^T (Aw, Au-
Au_d)dt-\gamma_2(w(T),u(T)-u_T)\vert\\
&&\ \ \ \leq C (\|w\|^2_{L^2(D(A))}+\Vert w(T) \Vert^2)\leq
C\|w\|^2_{\mathbb{W}_0},\end{aligned}$$ which implies that $$\begin{aligned}
\lim_{\|w\|_{\mathbb{W}_0 \rightarrow 0}}\frac{|J(u+w,v)-J(u,v)
-\gamma_1\int_0^T (Aw,
Au-Au_d)dt-\gamma_2(w(T),u(T)-u_T)|}{\|w\|_{\mathbb{W}_0} }=0.\end{aligned}$$ Thus, the Fréchet derivative of $J$ with respect to $u$ in an arbitrary $(u,v)$ is the operator $J_u(u,v):\mathbb{W}_0\rightarrow
\mathbb{R}$ defined by: $$\label{eq49}
J_u(u,v)w=\gamma_1\int_0^T (Aw, Au-Au_d)dt+\gamma_2(w(T),u(T)-u_T),\
w \in \mathbb{W}_0.$$ With the above notations, let us define the Lagrange functional $$\begin{aligned}
\label{l2}
\mathcal{L}(u,v,\lambda)=J(u,v)-\langle F(u,v),\lambda\rangle_{L^2(D(A)'),L^2(D(A))},\end{aligned}$$ where $\lambda\in L^2(D(A)).$ Thus, the Fréchet derivative of $\mathcal{L}$ with respect to $u$ is $$\begin{aligned}
\label{l3}
\mathcal{L}_u(u,v,\lambda)w=J_u(u,v)w-\langle
F_u(u,v)w,\lambda\rangle_{L^2(D(A)'),L^2(D(A))},\ \forall w\in \mathbb{W}_0.\end{aligned}$$
Now we will state and prove the necessary first-order optimality conditions
\[cond\](Necessary conditions) Let $(\hat{u},\hat{v})\in S_{ad}$ be a solution of the optimal control problem (\[eq24\])-(\[eq25\]). Then, there exists a $\lambda\in
L^2(D(A))$ such that $$\begin{aligned}
\label{e2}
\mathcal{L}_u(\hat{u}, \hat{v},\lambda)h=J_u(\hat{u},
\hat{v})h-\langle F_u(\hat{u}, \hat{v})h,\lambda)\rangle_{L^2(D(A)'),L^2(D(A))}=0, \
\forall h\in \mathbb{Y}_0.\end{aligned}$$ Moreover, the minimum principle holds $$\label{e5}
\mathcal{L}(\hat{u}, \hat{v},\lambda)\leq \mathcal{L}(\hat{u},
v,\lambda) \quad \forall v\in \mathcal{U}_{ad}.$$
We will apply Theorem \[ioffe\]; for that, in particular, we need to prove the surjectivity of the operator $F_u(\hat{u}, \hat{v})$. Thus, in order to simplify the calculations, we rewrite the problem (\[eq24\])-(\[eq25\]) in an equivalent optimal control problem. For this purpose, by considering $z\in \mathbb{Y}_0,$ we use the change of variable $u=\hat{u}+z.$ Thus, by replacing $u$ in (\[eq25\]) we get $$\begin{aligned}
\label{e3}
\Delta_\alpha z_t +\nu \Delta_\alpha Az+
B(\hat{u},z)+B(z,\hat{u})+B(z,z)=0.\end{aligned}$$ Therefore we obtain the following equivalent optimal control problem: $$\label{eq24b}
\min_{{z}\in \mathbb{Y}_0} \widetilde{J}(z):=\min_{{z}\in \mathbb{Y}_0} J(\hat{u}+z,\hat{v}),$$ subject to the state equation $$\begin{aligned}
G(z)=\Delta_\alpha z_t +\nu \Delta_\alpha Az+ B(\hat{u},z)+B(z,\hat{u})+B(z,z)=0.\label{eq25b}\end{aligned}$$ Observe that ${z}=0$ is the optimal solution of the control problem (\[eq24b\])-(\[eq25b\]) provided $(\hat{u}, \hat{v})$ minimizes $J$.
Thus, we will apply Theorem \[ioffe\] for the problem (\[eq24b\])-(\[eq25b\]). For that, we will verify all its conditions. $-$ ***Step one.** The operator $G$ is continuously differentiable with respect to $z$.*Following the proof of Lemma \[w21\], it is not difficult to obtain that the derivative of the operator $G:\mathbb{Y}_0\rightarrow L^2(D(A)')$ at a point $\hat{z}$ is given by the linear and continuous operator $G_z(\hat{z}):\mathbb{Y}_0\rightarrow L^2(D(A)')$ defined by $$\begin{aligned}
\label{e4}
G_z(\hat{z})h=\Delta_\alpha h_t +\nu \Delta_\alpha Ah+
B(\hat{u},h)+B(h,\hat{u})+B(h,\hat{z})+B(\hat{z},h).\end{aligned}$$
\[r1\] Notice that at the optimal solution $\hat{z}=0$ of $\widetilde{J}$ it holds $G_z(0)h=F_u(\hat{u},\hat{v})h$ for all $h\in \mathbb{Y}_0,$ being $(\hat{u},\hat{v}) \in S_{ad}$ the optimal solution of the control problem (\[eq24\])-(\[eq25\]).
$-$ [***Step two.** The functional $\widetilde{J}$ is continuously differentiable with respect to $z$.*]{}Notice that from definition of $J$ we have $$\begin{aligned}
{J}(\hat{u}+z+\epsilon h,\hat{v})-J(\hat{u}+z,\hat{v})&=&\gamma_1\epsilon\int_0^T ( Ah,
A(\hat{u}+z)-Au_d)dt + \frac{\epsilon^2\gamma_1}{2}\int_0^T\|Ah\|^2 dt\nonumber\\
&+&\epsilon\gamma_2(h(T),(\hat{u}+z)(T)-u_T)+\frac{\epsilon\gamma_2}{2}\Vert
h(T)\Vert^2.\label{w1}\end{aligned}$$ Since $$\begin{aligned}
\widetilde{J}_z(z)h=\lim_{\epsilon \rightarrow
0}\frac{\widetilde{J}(z+\epsilon h)-\widetilde{J}(z)}{\epsilon}=\lim_{\epsilon \rightarrow
0}\frac{J(\hat{u}+z+\epsilon h,\hat{v})-J(\hat{u}+z,\hat{v})}{\epsilon},\end{aligned}$$ then, from (\[w1\]), the derivative of $\widetilde{J}$ is given by the linear and continuous operator $\widetilde{J}_z(z):\mathbb{Y}_0\rightarrow
\mathbb{R}$ defined by $$\label{eq49z}
\widetilde{J}_z(z)h=\gamma_1\int_0^T (Ah,
A\hat{u}+Az-Au_d)dt+\gamma_2(h(T),\hat{u}(T)+z(T)-u_T),\ \forall h
\in \mathbb{Y}_0.$$
\[r2\] Notice that at the optimal solution $\hat{z}=0$ of $\widetilde{J}$ it holds $\widetilde{J}_z(0)h=J_u(\hat{u},\hat{v})h,$ for all $h\in
\mathbb{Y}_0,$ being $(\hat{u},\hat{v}) \in S_{ad}$ the optimal solution of the control problem (\[eq24\])-(\[eq25\]).
$-$***Step three.** $G_z(0):\mathbb{Y}_0\rightarrow L^2(D(A)')$ is surjective.*
Observing (\[e4\]), the derivative of $G$ at the optimal solution $\hat{z}=0$ of $\widetilde{J}$ is $$\begin{aligned}
G_z(0)h=\Delta_\alpha h_t +\nu \Delta_\alpha Ah+
B(\hat{u},h)+B(h,\hat{u}).\end{aligned}$$ Then, by using Lemma \[l1\] together Remark \[r1\], for each $g\in L^2(D(A)')$ there exists a unique $h\in \mathbb{Y}_0$ such that $G_z(0)h=g.$ Notice that the range of the mapping $G_z(0):\mathbb{Y}_0\rightarrow L^2(D(A)')$ is a closed set. Thus, the conditions of Theorem \[ioffe\] are verified. Consecuently, by defining the Lagrange functional $$\label{eq51b}
\widetilde{\mathcal{L}}(z,\lambda)= \widetilde{J}(z)-\langle
G(z),\lambda\rangle_{L^2(D(A)'),L^2(D(A))},$$ Theorem \[ioffe\] guarantees the existence of a $\lambda\in L^2(D(A))$ such that $$\widetilde{\mathcal{L}}_z(0,\lambda)h= \widetilde{J}_z(0)h -\langle
G_z(0)h,\lambda\rangle_{L^2(D(A)'),L^2(D(A))}=0,\ \forall h\in
\mathbb{Y}_0.\label{eq51c}$$ From Remark \[r1\], Remark \[r2\], (\[l3\]) and (\[eq51c\]), we obtain $$\label{51d}
\widetilde{\mathcal{L}}_z(0,\lambda)h=\mathcal{L}_u(\hat{u},\hat{v})h
\ \ \forall h\in \mathbb{Y}_0,$$ with $(\hat{u},\hat{v})\in S_{ad}$ the optimal solution of $J.$ Therefore, from (\[eq51c\]) and (\[51d\]), the equality (\[e2\]) is verified. The inequality (\[e5\]) follows directly from Theorem \[ioffe\].
Following Theorem 1.5 in [@fursikov] (see also [@iofi]), since $\mathcal{U}_{ad}$ is a convex set, the minimum principle (\[e5\]) implies $$\label{eq56}
\mathcal{L}_v(\hat{u},\hat{v},\lambda)(v-\hat v) \geq 0 \quad
\forall \,v\in \mathcal{U}_{ad}.$$
The weak formulation of an optimality system.
---------------------------------------------
The optimality system will be obtained from the necessary optimality conditions given in Theorem \[cond\].
From (\[eq46\]), (\[eq49\]) and (\[e2\]), we obtain the adjoint equation in a weak formulation
$$\begin{aligned}
\int_0^T\langle\Delta_\alpha h_t+\nu \Delta_\alpha A h
+B_u(\hat{u},\hat{u})h,\lambda \rangle_{D(A)',D(A)} dt=\gamma_1\int_0^T (Ah, A\hat{u}-Au_d)dt&&\nonumber\\
+\gamma_2(h(T),\hat{u}(T)-u_T), \forall h\in
\mathbb{Y}_0.&&\label{eq57}\end{aligned}$$
From the minimum principle (\[e5\]) we have $$\begin{aligned}
0\leq J(\hat{u},{v})-J(\hat{u},\hat{v})+\langle
F(\hat{u},\hat{v})-F(\hat{u},{v}),\lambda \rangle_{L^2(D(A)'),
L^2(D(A))},\end{aligned}$$ which implies that $$\begin{aligned}
\label{e6}
0\leq \frac{\gamma_3}{2}\int_0^T(\Vert v\Vert^2-\Vert
\hat{v}\Vert^2)dt+( v-\hat{v},\lambda).\end{aligned}$$ Using the equality $\Vert v\Vert-\Vert \hat{v}
\Vert^2=2(v-\hat{v},\hat{v})+\Vert v- \hat{v}\Vert^2,$ from (\[e6\]) we get $$\begin{aligned}
\label{e7bb}
0\leq \int_0^T\gamma_3 ( v-\hat{v},\hat{v}) dt+\int_0^T (
v-\hat{v},\lambda) dt+\frac{\gamma_3}{2}\int_0^T\Vert
v-\hat{v}\Vert^2 dt.\end{aligned}$$ From (\[e7bb\]) we can extract the following optimality condition $$\begin{aligned}
\int_0^T( \gamma_3\hat{v}+\lambda, v-\hat{v}) dt\geq 0 \quad
\forall\, v\in \mathcal{U}_{ad}.\end{aligned}$$ Thus we have the variational inequality $$\label{eq79}
(\hat{v} + \frac{1}{\gamma_3}\lambda, v-\hat{v}) \geq 0 \quad
\mbox{a.e. in }\ Q, \, \forall\, v\in \mathcal{U}_{ad}.$$ Moreover, since $\mathcal{U}_{ad}$ is a convex and closed set in $L^2(Q),$ by the theorem of the projection onto a closed convex set (see [@brez]), the control $\hat{v}$ in the inequality (\[eq79\]) can be characterized as a projection; thus we have the optimality condition $$\label{eq80}
\hat{v}= Proj_{\mathcal{U}_{ad}}\big( -\frac{1}{\gamma_3}
\lambda\big) \ \mbox{ a.e. in } \ Q.$$ Consequently, the equations (\[eq25\]), (\[eq57\]) and the condition (\[eq80\]) form an optimality system in a weak formulation for the optimal control problem considered.
Taking into account the definition of $\mathcal{U}_{ad},$ the projection representation (\[eq80\]) for $\hat{v}=(\hat{v}_1,\hat{v}_2,\hat{v}_3)$ is in each component $$\hat{v}_i=Proj_{[v_{ai}, v_{bi}]}\big( -\frac{1}{\gamma} \lambda_i\big) \ \mbox{a.e. in } \ Q, \ i=1,2,3.$$
The strong form of the optimality system
----------------------------------------
We wish to represent the optimality system as a system of partial differential equations with boundary, initial and terminal conditions. Since we do not know at this moment whether the $\lambda_t$ exists, we need to analyze the regularity of $\lambda.$
Using integration by parts, for $\lambda \in L^2(D(A))$ and $h\in \mathbb{Y}_0$ we get $$\begin{aligned}
\langle\Delta_\alpha
h_t,\lambda\rangle_{D(A)',D(A)}&=&(h_t,\lambda)+\alpha(\nabla h_t,
\nabla\lambda) =(h_t,\lambda)-\alpha(h_t, \Delta\lambda)
=(\Delta_\alpha\lambda,h_t),\end{aligned}$$ and then $$\begin{aligned}
\label{eq72}
\langle\Delta_\alpha h_t, \lambda\rangle_{L^2(D(A)'),L^2(D(A))}&=&
(\Delta_\alpha\lambda,h_t)_{L^2(Q),L^2(Q)}.\end{aligned}$$
Also, by using integration by parts in (\[eq57\]), for $ h\in
\mathbb{Y}_0$ and $ \lambda \in L^2(D(A))$, we have $$\begin{aligned}
\langle\nu \Delta_\alpha A h,\lambda\rangle_{D(A)',D(A)}&=&\nu(A h,
\lambda)-\alpha \nu\langle\Delta A h, \lambda\rangle_{D(A)',D(A)}
=\nu (A h, \lambda) -\alpha\nu( A h, \Delta \lambda)\\
&=&\nu (A h,\Delta_\alpha \lambda)=\langle\nu A\Delta_\alpha
\lambda, h\rangle_{D(A)',D(A)}.\end{aligned}$$ Then for all $h\in \mathbb{Y}_0$ we can write $$\label{eq58}
\langle\nu \Delta_\alpha A
h,\lambda\rangle_{L^2(D(A)'),L^2(D(A))}=\langle \nu A\Delta_\alpha
\lambda, h\rangle_{L^2(D(A)'),L^2(D(A))}.$$ Since $B_u(\hat{u},\hat{u}):\mathbb{W}_0\rightarrow L^2(D(A)')$ and $\mathbb{Y}\subset \mathbb{W},$ then the adjoint operator of $B_u(\hat{u},\hat{u}),$ denoted by $B_u^*(\hat{u},\hat{u}),$ is defined by: $$\label{eq64}
\langle B^*_u(\hat{u}, \hat{u})\lambda, h
\rangle_{\mathbb{Y}'_0,\mathbb{Y}_0}=\langle B_u(\hat{u},\hat{u}) h,
\lambda \rangle_{L^2(D(A)'),\,L^2(D(A))}.$$ On the other hand, since $A=-P\Delta,$ we can obtain $$\begin{aligned}
\gamma_1\int_0^T (Ah, A\hat{u}-Au_d)dt &=&-\gamma_1\int_0^T (\Delta
h, A(\hat{u}-u_d))dt\nonumber\\
&=&-\gamma_1\int_0^T \langle\Delta
A(\hat{u}-u_d),h\rangle_{D(A)',D(A)}dt.\label{e8bb}\end{aligned}$$ Thus, from (\[eq57\]), (\[eq72\]), (\[eq58\]), (\[eq64\]) and (\[e8bb\]), we get
$$\begin{aligned}
(\Delta_\alpha\lambda,h_t)_{L^2(Q),L^2(Q)}=-\langle\nu
A\Delta_\alpha \lambda, h\rangle_{L^2(D(A)'),L^2(D(A))}
-\langle B^*_u(\hat{u}, \hat{u})\lambda, h\rangle_{\mathbb{Y}'_0,\mathbb{Y}_0}&&\nonumber\\
-\gamma_1\langle\Delta
A(\hat{u}-u_d),h\rangle_{L^2(D(A)'),L^2(D(A))}+\gamma_2(h(T),\hat{u}(T)-u_T).&&\label{ops2}\end{aligned}$$
In order to obtain a representation of the weak time derivative of $\Delta_\alpha\lambda$ we analyze the regularity of $B^*_u(\hat{u},
\hat{u})\lambda$ in (\[ops2\]).
Notice that from (\[eq8\]) and (\[eq7\]) we get $$\begin{aligned}
\label{eq61}
\langle B_u(\hat{u},\hat{u})h,\lambda
\rangle_{D(A)',D(A)}&=&\langle\hat{u}\cdot\nabla \Delta_\alpha h,
\lambda\rangle_{V',V}
-\alpha((\nabla \hat{u})^*\cdot \Delta h, \lambda)\nonumber \\
&&+\langle h\cdot\nabla\Delta_\alpha\hat{u}, \lambda\rangle_{V',V}
-\alpha((\nabla h)^*\cdot \Delta\hat{u}, \lambda)\nonumber\\
&=&-(\hat{u}\cdot\nabla \lambda,\Delta_\alpha h)
-\alpha(\lambda \cdot\nabla \hat{u}, \Delta h)\nonumber\\
&&-(h\cdot\nabla \lambda,\Delta_\alpha \hat{u}) -\alpha(\lambda
\cdot\nabla h, \Delta\hat{u}).\end{aligned}$$ We bound the terms in (\[eq61\]). From Hölder and Sobolev inequalities we obtain $$\begin{aligned}
|(\hat{u}\cdot\nabla \lambda,\Delta_\alpha h)|&\leq &
C\|\hat{u}\|_{L^6}\|\nabla \lambda\|_{L^3}\|\Delta_\alpha h\|
\leq C\|\hat{u}\|_V \|\lambda\|_{D(A)} \|h\|_{D(A)},\label{eq65}\\
|(\lambda \cdot\nabla \hat{u}, \Delta h)| &\leq &
C\|\lambda\|_{L^\infty}\|\nabla \hat{u}\|\| \Delta h\| \leq
C\|\lambda\|_{D(A)}\|\hat{u}\|_V \|h\|_{D(A)}.\label{eq66}\end{aligned}$$ By observing that $ w\cdot \nabla v =0$ on $\Gamma$ if $w,v \in
D(A)$, and using integration by parts on $\Omega$, for $w, v, z \in
D(A)$ we have $$\label{eq67}
(w\cdot \nabla v, \Delta z)= (\nabla (w\cdot \nabla v), \nabla z)
=(\nabla v \nabla w, \nabla z) + (w \nabla(\nabla v), \nabla z),$$ where $w \nabla(\nabla v)=\sum_{i=1}^3w_i\frac{\partial}{\partial
x_i}\nabla v.$ Then, by using (\[eq67\]), the fact that $\|\nabla
v\|_{L^4}\leq C\|v\|_{D(A)}$ and $D(A) \subset L^\infty(\Omega)$, we obtain $$\begin{aligned}
|(h\cdot\nabla \lambda,\Delta_\alpha \hat{u})|&=&
|(h\cdot\nabla \lambda,\hat{u})-\alpha (h\cdot\nabla \lambda,\Delta \hat{u})|\nonumber\\
&\leq & |(h\cdot\nabla \lambda,\hat{u})| +\alpha |(\nabla
\lambda\nabla h,\nabla \hat{u})|
+\alpha |(h \nabla(\nabla \lambda),\nabla \hat{u})|\nonumber \\
&\leq & C\|h\|_{L^3}\|\nabla \lambda\|\|\hat{u}\|_{L^6} +C(\|\nabla
\lambda\|_{L^4}\|\nabla h\|_{L^4}
+\|h\|_{L^\infty}\| \nabla(\nabla \lambda)\|)\|\nabla \hat{u}\|\nonumber \\
&\leq &C \|h\|_{D(A)}\|\lambda\|_{D(A)}\|\hat{u}\|_V,\label{eq68}\\
|(\lambda\cdot\nabla h,\Delta\hat{u})| &\leq & |(\nabla h\nabla
\lambda,\nabla \hat{u})|
+|( \lambda\nabla(\nabla h),\nabla \hat{u})|\nonumber \\
&\leq & C(\|\nabla h\|_{L^4}\|\nabla \lambda\|_{L^4}
+\|\lambda\|_{L^\infty}\| \nabla(\nabla h)\|)\|\nabla \hat{u}\|\nonumber\\
&\leq &C \|h\|_{D(A)}\|\lambda\|_{D(A)}\|\hat{u}\|_V.\label{eq69}\end{aligned}$$ From (\[eq64\]), (\[eq61\])-(\[eq66\]), (\[eq68\]) and (\[eq69\]), and by using the Hölder inequality, for $\lambda \in
L^2(D(A))$, $\hat{u}\in \mathbb{W}_0$ and $h \in \mathbb{Y}_0$, we have $$\begin{aligned}
|\langle B^*_u(\hat{u}, \hat{u})\lambda,
h\rangle_{\mathbb{Y}'_0,\mathbb{Y}_0}|\leq C
\|\hat{u}\|_{L^\infty(V)} \|\lambda\|_{L^2(D(A))}\|h\|_{L^2(D(A))},\end{aligned}$$ which implies $$\label{eq70}
B^*_u(\hat{u}, \hat{u})\lambda \in L^2(D(A)').$$ Then, for all $h\in \mathbb{Y}_0$ we can rewrite (\[ops2\]) as the following equality $$\begin{aligned}
(\Delta_\alpha\lambda,h_t)_{L^2(Q),L^2(Q)}&=&\langle -\nu
A\Delta_\alpha \lambda - B^*_u(\hat{u}, \hat{u})\lambda -
\gamma_1\Delta
A(\hat{u}-u_d),h\rangle_{L^2(D(A)'),L^2(D(A))}\\
&&+\gamma_2(h(T),\hat{u}(T)-u_T).\end{aligned}$$ Since $h(T)$ can be arbitrary, when $\Delta_\alpha\lambda(T)=\gamma_2(\hat{u}(T)-u_T),$ we have the existence of a representation of $\Delta_\alpha \lambda_t$ in distributional sense as being $$\Delta_\alpha \lambda_t=\nu A\Delta_\alpha
\lambda+B^*_u(\hat{u}, \hat{u})\lambda+\gamma_1 \Delta
A(\hat{u}-u_d).$$ Thus we obtain that $\lambda \in L^2(D(A))$ is the solution of $$\left\{
\begin{array}[c]{rcl}
&&\Delta_\alpha\lambda_t-\nu A\Delta_\alpha \lambda-B^*_u(\hat{u},
\hat{u})\lambda=\gamma_1 \Delta A(\hat{u}-u_d)\ \mbox{in}\ L^2(D(A)'),\label{e17}\\
&&\Delta_\alpha\lambda(T)=\gamma_2(\hat{u}(T)-u_T).
\end{array}
\right.$$ From (\[eq64\]), (\[eq61\]) and (\[eq70\]) we have $$\begin{aligned}
\label{eq75}
\langle B^*_u(\hat{u}, \hat{u})\lambda,
h\rangle_{D(A)',D(A)}&=&-(\hat{u}\cdot\nabla \lambda,h)
+\alpha(\hat{u}\cdot\nabla \lambda,\Delta h)
-\alpha(\lambda \cdot\nabla \hat{u}, \Delta h)\nonumber \\
&&-(h\cdot\nabla \lambda, \Delta_\alpha \hat{u}) +\alpha(\lambda
\cdot\nabla \Delta\hat{u},h).\end{aligned}$$ Observing that $ v\cdot \nabla w =0$ on $\Gamma$ if $v, w\in D(A)$, and using integration by parts on $\Omega$, for $\lambda, h \in
D(A)$ we get $$\begin{aligned}
\label{eq76}
\alpha(\hat{u}\cdot\nabla \lambda,\Delta h)- \alpha (\lambda
\cdot\nabla \hat{u}, \Delta h) &=&-\alpha(\nabla(\hat{u}\cdot\nabla
\lambda),\nabla h) +\alpha(\nabla(\lambda\cdot\nabla
\hat{u}),\nabla h)\nonumber\\
&=&\alpha(\Delta(\hat{u}\cdot\nabla\lambda),
h)-\alpha(\Delta(\lambda\cdot\nabla\hat{u}), h) .\end{aligned}$$ Taking into account (\[eq7\]), we get $$\label{eq77}
-(h\cdot\nabla \lambda, \Delta_\alpha \hat{u}) =\langle h\cdot\nabla
\Delta_\alpha \hat{u},\lambda\rangle_{V',V} =-((\nabla
\lambda)^*\cdot \Delta_\alpha \hat{u},h).$$ Thus, from (\[eq75\])-(\[eq77\]) we obtain $$\begin{aligned}
\langle B^*_u(\hat{u}, \hat{u})\lambda, h\rangle_{D(A)', D(A)}
&=&\langle-\hat{u}\cdot\nabla \lambda+\alpha\Delta(\hat{u}\cdot\nabla
\lambda)
-\alpha \Delta(\lambda\cdot\nabla \hat{u}), h\rangle_{D(A)', D(A)}\\
&&-\langle (\nabla \lambda)^*\cdot \Delta_\alpha \hat{u}
+\alpha\lambda \cdot\nabla \Delta\hat{u},h\rangle_{D(A)', D(A)},\end{aligned}$$ which implies the following equality in $L^2(D(A))':$ $$\label{eq78}
B^*_u(\hat{u}, \hat{u})\lambda=-\hat{u}\cdot\nabla \lambda
+\alpha\Delta(\hat{u}\cdot\nabla \lambda) -\alpha(\lambda\cdot\nabla
\hat{u}) -(\nabla \lambda)^*\cdot \Delta_\alpha \hat{u}
+\alpha\lambda \cdot\nabla \Delta\hat{u}.$$
Therefore, from (\[e17\]) and (\[eq78\]), we have $$\label{eq76}
\left\{
\begin{array}{rcl}
\Delta_\alpha \lambda_t-\nu \Delta_\alpha
A\lambda+\hat{u}\cdot\nabla \lambda
&+&\alpha\Delta(\hat{u}\cdot\nabla \lambda+\lambda\cdot\nabla
\hat{u})
+(\nabla \lambda)^*\cdot \Delta_\alpha \hat{u}\\
-\alpha\lambda \cdot\nabla \Delta\hat{u}
&=&\gamma_1\Delta A(\hat{u}-u_d )\quad \mbox{ in } \ L^2(D(A)'),\\
\nabla\cdot \lambda&=& 0 \ \mbox{in } Q,\\
\lambda&=&0 \ \mbox{on } \Gamma \times (0,T),\\
\Delta_\alpha \lambda(T)&=&\gamma_2 (\hat{u}(T)-u_T)\quad \mbox{ in
} \Omega.
\end{array}
\right.$$
Summarizing the state equation (\[eq25\]), the adjoint equation (\[eq76\]) and the optimality condition (\[eq80\]) we get the optimality system, in the strong form, as desired.
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[^1]: E. Ortega-Torres was supported by Fondecyt-Chile, Grant 1080399
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We construct a Schrödinger-like equation for the longitudinal wave function of a meson in the valence $q\bar{q}$ sector, based on the ’t Hooft model for large-$N$ two-dimensional QCD, and combine this with the usual transverse equation from light-front holographic QCD, to obtain a model for mesons with massive quarks. The computed wave functions are compared with the wave function [*ansatz*]{} of Brodsky and De Téramond and used to compute decay constants and parton distribution functions. The basis functions used to solve the longitudinal equation may be useful for more general calculations of meson states in QCD.'
author:
- 'Sophia S. Chabysheva'
- 'John R. Hiller'
title: |
Dynamical model for longitudinal wave functions\
in light-front holographic QCD
---
Introduction {#sec:Introduction}
============
Light-front holographic QCD [@LFhQCD] exploits an approximate AdS$_5$/QCD duality to obtain a Schrödinger-like equation for the transverse wave functions of hadrons, for massless quarks. Although a true QCD dual theory is unknown, the approximate duality with AdS$_5$ can be obtained by altering the geometry of AdS$_5$ at infrared scales corresponding to the QCD confinement scale, $1/\Lambda_{\rm QCD}$. The modifications incorporate confinement, and, in the modified AdS$_5$ theory, one obtains the point-like behavior of QCD partons and dimensional counting rules [@PolchinskiStrassler]. The approximate duality leads to a boost-independent light-front equation for the valence state of a hadron [@LFhQCD; @formfactormatch; @LFhQCD2]. This equation provides a first-order approximation to the light-front QCD eigenvalue problem for hadrons in the valence Fock sector, but only for massless quarks.
From a modeling perspective, light-front holographic QCD generates an effective potential for valence quarks from the choice of the warping of the AdS$_5$ dual, rather than modeling the effective potential directly. This leads to a light-front equation for the transverse wave function for massless quarks. The longitudinal wave function is determined by correspondence with a form factor in the AdS$_5$ dual [@formfactormatch], instead of a dynamical equation. The soft-wall model [@softwall] in particular admits analytic solutions for the spectrum and the transverse wave functions, and, what is more, yields a spectrum [@Gershtein; @Vega; @Ebert; @Branz; @Kelley] consistent with linear Regge trajectories. However, because the duality relies on the approximate conformal limit of zero-mass quarks, the transverse and longitudinal wave functions do not include dependence on quark masses.
For realistic calculations, we need to make an extension to massive quarks. In light-front coordinates, the mass is associated with the longitudinal part of the kinetic energy, as given in (\[eq:invariantmass\]), not the transverse. Thus the introduction of massive quarks requires specification of either the longitudinal wave function or a dynamical equation for it. Brodsky and De Téramond have provided an [*ansatz*]{} for the former [@ansatz]; the purpose of this paper is to consider the latter possibility and compare with the Brodsky–De Téramond [*ansatz*]{}.
The Brodsky–De Téramond [*ansatz*]{} extends the transverse momentum dependence to include the full invariant mass. For meson states, where the transverse wave function can be a gaussian, this extends an exponentiation of $k_\perp^2/x(1-x)$, where $\vec k_\perp$ is the transverse momentum and $x$ is the longitudinal momentum fraction, to $k_\perp^2/x(1-x)+\mu_1^2/x+\mu_2^2/(1-x)$. To be consistent with the zero-mass limit, the quark masses $\mu_1$ and $\mu_2$ are current quark masses, the parameters in the QCD Lagrangian. Constituent quark masses are nonzero even when the current quark masses are zero.
An alternative, which we consider here, is to assume separation of variables for the meson wave function and then provide a light-front equation for the longitudinal part that includes quark masses and a model potential. The longitudinal kinetic term will be just $m_1^2/x+m_2^2/(1-x)$; here, to be consistent with nonrelativistic quark models, the quark masses $m_1$ and $m_2$ are constituent masses. The potential term should be confining and should yield longitudinal wave functions consistent with the dual AdS$_5$/QCD form-factor analysis, in the zero-current-mass limit. A potential model that achieves this, and is directly related to QCD, is the ’t Hooft model obtained in the large-$N$ limit of two-dimensional QCD [@tHooft]. It is just such a instantaneous gluon-exchange potential that appears in four-dimensional QCD in light-cone gauge. The structure of our quark model, with its longitudinal/transverse separation, is then a direct analog of transverse-lattice QCD [@TransLatticeBP; @TransLatticeBPR], where the ’t Hooft model provides the longitudinal connection between transverse lattice planes.
In the remaining sections, we explore this potential-model approach.[^1] In Sec. \[sec:motivation\] we provide some background details and the motivations for our choice of longitudinal potential. The details of the model and its solution are discussed in Sec. \[sec:model\]; sample applications are illustrated in Sec. \[sec:sample\]. A summary and some additional remarks are included in Sec. \[sec:summary\]. Appendices contain details of the conventions for light-front coordinates, of the dual form-factor analysis, and of the numerical solution for the model.
Motivation {#sec:motivation}
==========
To define our model, we begin from an effective light-front Schrödinger equation for the quark-antiquark wave function $\psi(x,\vec k_\perp)$ of a meson \[eq:fullLFSE\] =M\^2, where the first three terms are the kinematic invariant mass, as discussed in \[sec:LFcoords\], and $\widetilde U$ is an effective potential. A transverse Fourier transform to a relative coordinate $\vec b_\perp$ yields =M\^2, with $\nabla_\perp^2=\frac{\partial^2}{\partial b_\perp^2}
+\frac{1}{b_\perp}\frac{\partial}{\partial b_\perp}
+\frac{1}{b_\perp^2}\frac{\partial^2}{\partial \varphi^2}$ the transverse Laplacian and $\varphi$ the polar angle. The choice of a new coordinate $\zeta\equiv\sqrt{x(1-x)}b_\perp$ is then convenient. Combined with the factorization =e\^[iL]{}X(x)()/ and the natural assumption that $\widetilde U$ conserves the angular momentum component $L_z$, the light-front equation (\[eq:fullLFSE\]) reduces to [@LFhQCD] \[eq:reducedLFSE\] X(x)()=M\^2X(x)(). For zero-mass quarks, $\widetilde U$ becomes just $U(\zeta)$, a function of $\zeta$ only, and the longitudinal wave function $X$ is no longer determined by Eq. (\[eq:reducedLFSE\]), which leaves a one-dimensional equation for the transverse wave function $\phi(\zeta)$ \[eq:transverse\] ()=M\^2(). The assumed duality with AdS$_5$ can suggest models for $U$, through a correspondence between the transverse Schrödinger equation (\[eq:transverse\]) and the equation of motion for a spin-$J$ field in AdS$_5$ [@LFhQCD]. Confinement is introduced by a dilaton profile $\widetilde\phi(z)$, where $z$ is the holographic coordinate of AdS$_5$. With the identification of $z$ with $\zeta$ [@LFhQCD], the corresponding effective potential is [@effU] U()=12”()+14’()\^2 +’(). For the soft-wall model [@softwall], the dilaton profile is $\widetilde\phi(z)=e^{\pm \kappa^2 z^2}$, with $\kappa$ a parameter, and the effective potential reduces to an oscillator potential U()=\^4\^2+2\^2(J-1). For this potential, the spectrum of masses is $M^2=4\kappa^2\left(n+(J+L)/2\right)$, with $n$ the radial quantum number, and the transverse wave functions are the two-dimensional oscillator eigenfunctions. The spectrum of the model provides for a linear Regge trajectory and a good fit to light meson masses [@spectrum].
The longitudinal wave function $X$ remains unspecified. It can, however, be constrained by the duality in an analysis of the meson form factor [@LFhQCD]. As summarized in \[sec:dualFF\], this leads to the conclusion that $X(x)=\sqrt{x(1-x)}$ for massless quarks.
For massive quarks, something needs to be assumed, beyond the AdS$_5$ correspondence. One approach, as already mentioned, is the [*ansatz*]{} by Brodsky and De Téramond [@ansatz]. Since the transverse wave functions are harmonic oscillator eigenfunctions, the ground state is a simple Gaussian $e^{-\kappa^2\zeta^2/2}=e^{-\kappa^2 x(1-x)b_\perp^2/2}$. The transform to transverse momentum is, of course, also a Gaussian, $\frac{4\pi^2}{\kappa^2}\frac{1}{x(1-x)}e^{-k_\perp^2/(2\kappa^2 x(1-x))}$. The [*ansatz*]{} replaces $k_\perp^2/(x(1-x))$ with $k_\perp^2/x(1-x)+\mu_1^2/x+\mu_2^2/(1-x)$. The form of $X(x)$ is then \[eq:ansatz\] X\_[BdT]{}(x)=N\_[BdT]{}e\^[-(\_1\^2/x+\_2\^2/(1-x))/2\^2]{}, with $N_{\rm BdT}$ a normalization factor. The $\sqrt{x(1-x)}$ form is recovered in the zero-mass limit.
In our approach, we start from the full light-front Schrödinger equation (\[eq:fullLFSE\]) and replace the effective potential $\widetilde U$ by $U(\zeta)+U_\parallel$ and the current masses $\mu_i$ by constituent masses $m_i$. The potential $U_\parallel$ is an integral operator which acts on functions of momentum fraction $x$. The combination of this longitudinal potential and the change in mass is meant to represent the longitudinal effects of the original (unknown) effective potential $\widetilde U$; the specific choice of constituent masses is driven by consistency with nonrelativistic quark models. The equation then factorizes into the transverse equation ()=(M\^2-M\_\^2)(), which differs from (\[eq:transverse\]) only by the separation constant $M_\parallel^2$, and the longitudinal equation \[eq:longitudinal\] X(x)=M\_\^2 X(x). The effective longitudinal potential $U_\parallel$ can be adjusted to make $M_\parallel^2$ equal to zero for the ground state; this allows the fit of $M^2$ to the meson mass spectrum to remain unaffected. Also, because the transverse spectrum is a good fit, we consider only the ground state for the longitudinal equation.
Our choice for the longitudinal potential $U_\parallel$ is the ’t Hooft model [@tHooft] obtained in the large-$N$ limit of two-dimensional QCD. The selection is motivated by two factors. First, it is the natural choice for a confining potential in one spatial dimension, particularly for modeling the longitudinal part of three-dimensional QCD. When QCD is quantized in light-cone gauge, such a potential appears automatically as an instantaneous Coulomb-like interaction between quark currents. For this reason, it is also part of the longitudinal interaction included in transverse lattice gauge theory [@TransLatticeBP; @TransLatticeBPR], where fields on transverse nodes and links are coupled longitudinally by a continuum model, a transverse/longitudinal separation not unlike the situation for light-front holographic QCD.
Second, there exists a nearly exact analytic solution for the ground state, which can be improved easily with numerical calculations [@Bergknoff; @MaHiller; @MoPerry] and which can be arranged to be consistent with the expected $X(x)$ in the zero-current-mass limit. As is known from the work of ’t Hooft [@tHooft] and Bergknoff [@Bergknoff], the approximate analytic solution is of the form $x^{\beta_1}(1-x)^{\beta_2}$, with $\beta_i$ determined by the quark masses and the longitudinal coupling. For equal constituent masses, the coupling can be adjusted to obtain the desired $\beta_i=1/2$ for zero current masses. We also use this condition to fix the value of the longitudinal coupling.
The model {#sec:model}
=========
With the longitudinal potential taken from the ’t Hooft model [@tHooft], the longitudinal equation (\[eq:longitudinal\]) becomes \[eq:longitudinaleqn\] X(x) +dy -C X(x)=M\_\^2 X(x), with ${\cal P}$ indicating the principal value and $C$ a constant. Because the transverse equation already introduces enough quantum numbers, we consider only the ground state of the longitudinal equation; any additional quantum number associated with longitudinal excitations would represent double counting. Also, since the light-meson spectrum is already represented by the transverse equation, the constant $C$ is used to set $M_\parallel$ to zero. The net effect is that the additional longitudinal equation is only for determination of the longitudinal wave function and has nothing to say about the spectrum.
As discussed in the previous section, the ground-state wave function $X(x)$ is well approximated by the form $x^{\beta_1}(1-x)^{\beta_2}$. From the dual form-factor analysis [@formfactormatch], the light-meson wave function should have this form with $\beta_1=\beta_2=1/2$. Analysis of the endpoint behavior for the solution of (\[eq:longitudinaleqn\]) shows that $\beta_i$ should satisfy the transcendental equation [@tHooft; @Bergknoff] -1+\_i \_i=0. If we take the up and down quark masses, $m_u$ and $m_d$, to be equal, the square-root behavior is obtained if $g^2/\pi=m_u^2$, consistent with $\cot\pi/2=0$. This fixes the value of the coupling constant. Although the model could be more flexible if $g$ were flavor dependent, we do not consider this.
The exact solution for the wave function is not analytic. However, a numerical solution, as presented in \[sec:numerical\], is straightforward.
For the ground state, the complete wave function is given by a normalized product of the longitudinal wave function $X$ and a transverse Gaussian (x,)=N X(x) e\^[-\^2\^2/2]{}, or, in terms of the transverse coordinate $b_\perp$, (x,b\_)=N X(x) e\^[-\^2 x(1-x)b\_\^2/2]{}. The factor $N$ is fixed by the normalization P\_[q|[q]{}]{}=\_0\^1 dx \_0\^db\_\^2 |(x,b\_)|\^2, where $P_{q\bar{q}}$ is the probability of the quark-antiquark valence state. If $X(x)$ is separately normalized such that \_0\^1 dx =1, then $N=\frac{\kappa}{\pi}\sqrt{P_{q\bar{q}}}$.
The wave functions can be used to compute decay constants and parton distributions. A decay constant is given by [@decayconstant] f\_M=2\_0\^1 dx \_0\^ (x,k\_). As discussed in [@Vega] and shown in [@pdf], the parton distribution $f(x)$ is given by f(x)=x(1-x) \^2(x), if the wave function takes the form (x,k\_)=(x) e\^[-k\_\^2/(2 \^2 x(1-x))]{}. Applying this to our model, we obtain f(x)=P\_[q|[q]{}]{}. For the [*ansatz*]{}, we have f\_[BdT]{}(x)=N\_[BdT]{}\^2 P\_[q|[q]{}]{} e\^[-(\_1\^2/x+\_2\^2/(1-x))/\^2]{}, with the normalization of the [*ansatz*]{} given by N\_[BdT]{}=\^[-1/2]{}.
Sample calculations {#sec:sample}
===================
To see the implications of our model, we compare the form of the longitudinal wave function with the [*ansatz*]{} by Brodsky and De Téramond [@ansatz], both directly and through the computation of decay constants and parton distribution functions, for the pion, kaon, and J/$\Psi$. Where parameter values are needed, we use the current-quark parameterization of Vega [*et al*]{}. [@Vega] with no additional fits or adjustments. The parameter values are listed in Table \[tab:parameters\].
---------- ------- ------- --------- --------- ---------------- ---------- ------- -------------- --------
meson $m_1$ $m_2$ $\mu_1$ $\mu_2$ $P_{q\bar{q}}$ $\kappa$ model [*ansatz*]{} exper.
pion 330 330 4 4 0.204 951 131 132 130
kaon 330 500 4 101 1 524 160 162 156
J/$\Psi$ 1500 1500 1270 1270 1 894 267 238 278
---------- ------- ------- --------- --------- ---------------- ---------- ------- -------------- --------
: \[tab:parameters\] Meson parameters and decay constants. All dimensionful parameters are in units of MeV. Results are compared between our model, which uses constituent-quark masses, and the [*ansatz*]{} of Brodsky and De Téramond [@ansatz], with current-quark masses. Parameter and experimental values are from Vega [*et al*]{}. [@Vega] and the Particle Data Group [@PDG].
Figure \[fig:longwf\] compares the [*ansatz*]{} with the $X(x)$ computed in our model. For the pion, the two wave functions are essentially the same, since both involve only tiny variations from the wave function $\sqrt{x(1-x)}$ for quarks with zero current mass.
[cc]{}\
\
\
\
![\[fig:longwf\]Longitudinal wave functions $X(x)$ for the (a) pion, (b) kaon, and (c) J/$\Psi$. The solid lines are wave functions from our model; the dashed lines show the [*ansatz*]{} by Brodsky and De Téramond [@ansatz]. ](kaonlongwf.eps "fig:"){width="6cm"} & ![\[fig:longwf\]Longitudinal wave functions $X(x)$ for the (a) pion, (b) kaon, and (c) J/$\Psi$. The solid lines are wave functions from our model; the dashed lines show the [*ansatz*]{} by Brodsky and De Téramond [@ansatz]. ](JPsilongwf.eps "fig:"){width="6cm"}\
(b) & (c)
The results for decay constants are included in Table \[tab:parameters\]. The values for the pion and kaon are consistent and in agreement with experiment. Our model value for the J/$\Psi$ is significantly closer to experiment than the value obtained from the longitudinal [*ansatz*]{}.
The parton distributions are plotted for comparison in Figure \[fig:pdf\]. The rough similarity of the wave functions translates into similar parton distributions.
[cc]{}\
\
\
\
![\[fig:pdf\] Same as Fig. \[fig:longwf\] but for parton distributions $f(x)$ multiplied by $x$. ](kaonpdf.eps "fig:"){width="6cm"} & ![\[fig:pdf\] Same as Fig. \[fig:longwf\] but for parton distributions $f(x)$ multiplied by $x$. ](JPsipdf.eps "fig:"){width="6cm"}\
(b) & (c)
Concluding remarks {#sec:summary}
==================
We have constructed and solved a relativistic light-front equation for the longitudinal wave functions of mesons with massive quarks, to be used in tandem with the transverse equation of light-front holographic QCD. Comparisons with the [*ansatz*]{} [@ansatz] (\[eq:ansatz\]) show that for lighter mesons, the longitudinal wave functions are quite similar. However, for the J/$\Psi$, there is a notable difference, which translates into a better estimate of the decay constant, as listed in Table \[tab:parameters\]. Wave functions for the J/$\Psi$, and also the pion and kaon, are shown in Fig. \[fig:longwf\], and parton distributions are shown in Fig. \[fig:pdf\]. The similarity of the longitudinal wave functions provides the [*ansatz*]{} with a connection to the fundamental interactions of QCD.
Perhaps the most broadly useful outcome of this exercise is to illustrate that there is a convenient set of basis functions for longitudinal wave functions of meson valence states. These are the $f_n$ defined in (\[eq:basis\]), with the parameters $\beta_i$ to be optimized as needed for a given application. One could, of course, use the eigenfunctions of the ’t Hooft model, but these do not have an analytic form and would add an extra layer of complication to any calculation.
The importance of the choice of basis functions is in the rate of convergence as the basis is expanded. For the alternative of discrete light-cone quantization, it is known that convergence can be much slower [@vandesande]. Thus, these basis functions may prove useful for calculations, such as those described in [@Vary], which use the transverse light-front holographic eigenfunctions.
This work was supported in part by the Department of Energy through Contract No. DE-FG02-98ER41087. We thank G.F. de Téramond and S.J. Brodsky for helpful comments.
Light-front coordinates {#sec:LFcoords}
=======================
Our conventions for light-front coordinates [@Dirac; @DLCQreviews] are as follows. We define light-front time $x^+=t+z$ and the longitudinal light-front spatial coordinate $x^-=t-z$. The transverse coordinates are collected as $\vec x_\perp=(x,y)$. The corresponding light-front energy and momentum are $p^-=E-p_z$, $p^+=E+p_z$, and $\vec p_\perp=(p_x,p_y)$. From the invariant mass relation $m^2=p^2=p^+p^--\vec p_\perp^2$, we obtain $p^-=(m^2+p_\perp^2)/p^+$ for an on-shell particle.
For a system of particles, with total momentum $(P^+,\vec P_\perp)$, the longitudinal momentum fraction $x_i=p_i^+/P^+$ and relative transverse momentum $\vec k_{\perp i}=\vec p_{\perp i}-x_i\vec P_\perp$, for the ith particle, are boost invariant and therefore a convenient choice for independent variables in the description of the system. They sum to one and zero, respectively: \_i x\_i=1, \_ik\_[i]{}=0. The invariant mass for the system is \[eq:invariantmass\] P\^+\_i p\_i\^–P\_\^2 =\_i -P\_\^2 =\_i . For a two-particle system, the internal variables reduce to $x=x_1$, $x_2=1-x$, $\vec k_\perp=\vec k_{\perp 1}$, and $\vec k_{\perp 2}=-\vec k_\perp$, and the invariant mass becomes + =++.
Dual form-factor analysis {#sec:dualFF}
=========================
In the Drell–Yan–West frame [@DrellYan], the form factor for momentum transfer $q^2$ can be written in terms of Fock-state wave functions $\psi_n$ as [@BrodskyDrell] \[eq:formfactor\] F(q\^2)=\_n\_j e\_j \_n\^\*(x\_i,k\_[i]{}\^)\_n(x\_i,k\_[i]{}), where $n$ denotes the Fock sector, $e_j$ the charge of the $j$th quark, \_[i=1]{}\^n dx\_i (1-\_j x\_j), (\_[i=1]{}\^n ) 16 \^3 (\_j k\_[j]{}), and, with $j$ the index of the quark that absorbed the photon, k\_[i]{}\^={
[ll]{} k\_[i]{}+(1-x)q\_, & i=j\
k\_[i]{}-xq\_, & ij.
. Substitution of the wave function as a transverse Fourier transform in impact space, \_n(x\_i,k\_[i]{})=(4)\^[(n-1)/2]{} \_[i=1]{}\^[n-1]{} d\^2b\_[i]{} e\^[i\_[j=1]{}\^[n-1]{}b\_[j]{}k\_[j]{}]{} \_n(x\_i,b\_[i]{}), converts the expression (\[eq:formfactor\]) for the form factor to F(q\^2)=\_n \_[j=1]{}\^[n-1]{}dx\_j d\^2b\_[j]{} e\^[iq\_\_[j=1]{}\^[n-1]{}x\_jb\_[j]{}]{} |\_n(x\_i,b\_[i]{})|\^2. For the quark-antiquark valence sector alone, with the wave function given by $\widetilde\psi_2=e^{iL\varphi}X(x)\phi(\zeta)/\sqrt{2\pi\zeta}$, this reduces to F(q\^2)= d J\_0(q\_)|()|\^2, with $J_0$ the Bessel function of order zero. This is to be compared with the form computed in AdS$_5$ [@PolchinskiStrassler] F(q\^2)=dx d J\_0(q\_)|()|\^2. Thus, the conclusion [@LFhQCD] that $X(x)=\sqrt{x(1-x)}$, when the quarks have zero current mass.
Numerical solution {#sec:numerical}
==================
We solve (\[eq:longitudinaleqn\]) numerically by expanding the wave function $X$ in terms of orthonormal basis functions $f_n$, chosen to include the analytic approximation explicitly. As discussed by Mo and Perry [@MoPerry], these basis functions are \[eq:basis\] f\_n(x)=N\_n x\^[\_1]{}(1-x)\^[\_2]{}P\_n\^[(2\_2,2\_1)]{}(2x-1), with $P_n^{(2\beta_2,2\beta_1)}$ the Jacobi polynomial of order $n$. The normalization factor $N_n$ is given by [@AbramowitzStegun] N\_n=. The solution is then represented as X(x)=\_n c\_n f\_n(x). For equal-mass cases, the longitudinal equation obeys an $x\leftrightarrow(1-x)$ symmetry, and only the even-$n$ terms will contribute. In general, we find that only a few terms are needed; the $n=0$ term, for which the Jacobi polynomial is constant and $f_0\propto x^{\beta_1}(1-x)^{\beta_2}$, represents 90% or more of the probability.
The expansion coefficients $c_n$ are obtained by diagonalizing the longitudinal equation in the $f_n$ basis. The matrix representation is ( A\_1+A\_2+B)c=c, with $\xi\equiv C/m_u^2$ and matrices $A_1$, $A_2$, and $B$ defined by (A\_1)\_[nm]{}&=&\_0\^1 f\_n(x) f\_m(x), (A\_2)\_[nm]{}=\_0\^1 f\_n(x) f\_m(x),\
B\_[nm]{}&=&\_0\^1 dx [P]{}\_0\^1 dy f\_n(x). Following ’t Hooft [@tHooft], the matrix representation is made explicitly symmetric by rewriting the potential term as B\_[nm]{}=12\_0\^1 dx \_0\^1 dy . In addition to the explicit symmetry, which simplifies the matrix diagonalization, this rearrangement also resolves the principal value prescription. The matrices are small because the number of terms needed in the expansion are few; the diagonalization is then straightforward.
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[^1]: This is quite different from a holographic description of the ’t Hooft model itself [@KatzOkui], where one considers an AdS$_3$ dual to two-dimensional QCD.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'K. Sellgren$^1$'
title: 'Aromatic Hydrocarbons, Diamonds, and Fullerenes in Interstellar Space: Puzzles to be Solved by Laboratory and Theoretical Astrochemistry'
---
**Abstract**
New research is presented, and previous research is reviewed, on the emission and absorption of interstellar aromatic hydrocarbons. Emission from aromatic hydrocarbons dominate the mid-infrared emission of many galaxies, including our own Milky Way galaxy. Only recently have aromatic hydrocarbons been observed in absorption in the interstellar medium, along lines of sight with high column densities of interstellar gas and dust. Much work on interstellar aromatics has been done, with astronomical observations and laboratory and theoretical astrochemistry. In many cases the predictions of laboratory and theoretical work are confirmed by astronomical observations, but in other cases clear discrepancies exist which provide problems to be solved by a combination of astronomical observations, laboratory studies, and theoretical studies. The emphasis of this paper will be on current outstanding puzzles concerning aromatic hydrocarbons which require further laboratory and theoretical astrochemistry to resolve. This paper will also touch on related topics where laboratory and theoretical astrochemistry studies are needed to explain astrophysical observations, such as a possible absorption feature due to interstellar “diamonds” and the search for fullerenes in space.
Introduction
============
Interstellar atomic gas, molecular gas, and dust grains, where grains are solid particles with sizes of 1 – 1000 nm, pervade the space in between the stars known as the interstellar medium (ISM). The grains and molecules in the lower density regions of our Galaxy, called the diffuse ISM, are believed to form primarily in the outflowing gas from dying stars with masses similar to that of our own Sun. The final stage of this type of stellar death is a shell of ejected gas, known as a planetary nebula, which surrounds the stellar corpse, known as a white dwarf. Grains and molecules find it difficult to form in the general ISM, which has typical gas densities of 1 – 10 atoms cm$^{-3}$, and thus extremely low chemical reaction rates. The outflows from dying stars have much higher densities ($>$ 10$^6$ atoms cm$^{-3}$), which make chemical reactions leading to formation of molecules and grains much more probable.
In our universe, the pattern of cosmic elemental abundances is determined first by the hydrogen and helium formed in the Big Bang, when our universe began, and secondly by nuclear reactions within stars, which account for virtually all the other elements. As a result of the pattern of common nuclear reactions within stars, only a very restricted set of elements are important in interstellar chemistry. Figure 1 shows the cosmic abundance, relative to hydrogen, of all elements which have an abundance larger than 10$^{-6}$ relative to hydrogen \[1\]. It can be seen that the most important elements for interstellar chemistry are H, O, C, N, Mg, Si, Fe, S, Al, Ca, Na, and Ni. The inert gases He, Ne, and Ar are also relatively abundant but do not interact chemically. Other elements exist in the universe, but their scarcity leads to their playing a minor role in interstellar chemistry. By far the most abundant interactive elements are H, O, C, and N, and thus it is no surprise that the overwhelming majority of interstellar molecules detected to date are primarily composed of these elements.
The interstellar infrared emission features (IEFs), at 3.3, 6.2, 7.7, 8.6, 11.3, and 12.7 $\mu$m (790, 890, 1160, 1300, 1610, and 3040 cm$^{-1}$), dominate the mid-infrared emission of our own and other galaxies. Figure 2 shows the 600 – 2000 cm$^{-1}$ spectrum of one region within our own Galaxy, NGC 7023 \[2\]. This figure shows most of the IEFs. Note that this figure is plotted versus cm$^{-1}$, to be helpful to chemists, but that the astronomical convention is to plot spectra versus $\mu$m and that the astronomical convention will be followed for the remainder of this paper.
The IEFs, which have widths broader than natural broadening mechanisms for lines in atomic gas or simple molecular gas, were mainly first discovered by astronomical observations in the 1970’s \[3 – 6\], with the 12.7 $\mu$m IEF discovered only after higher spectral resolution observations \[7\] were able to distinguish it from Ne$^+$ emission at 12.8 $\mu$m (780 cm$^{-1}$). The IEFs have been referred to by a variety of names in the literature, with “feature” and “band” used interchangeably: the “unidentified emission features,” the “unidentified infrared bands,” the “infrared emission features,” the “aromatic infrared bands,” and the “polycyclic aromatic hydrocarbon (PAH) bands.” The origin of the last two names will become apparent in later paragraphs.
The IEFs are believed to arise in either interstellar grains or interstellar molecules, and because their emission can account for up to 30 – 40% of the overall emission of galaxies, solving problems related to their composition, size, and origin are essential for understanding the overall energy budget of radiation from galaxies. Within our galaxy, the IEFs are observed in a wide variety of environments. One important clue to the composition of the carrier of the IEFs, however, is the astronomical observation that the strength of the 7.7 $\mu$m IEF shows a strong correlation with the ratio of carbon to oxygen in planetary nebulae \[8 – 10\]. Ordinarily there is more oxygen than carbon in interstellar gas and in stars, which results in an interstellar chemistry in which common elements most often combine with hydrogen and oxygen to form molecules and grains. The heavier elements in such an oxygen-rich environment generally condense into solid dust grains composed of oxides of these elements, such as silicate grains which are commonly observed in the ISM. In some dying stars, however, convection within the star dredges up so much carbon, produced in nuclear reactions at the star’s center, that the composition of the outflowing gas from a dying star changes to a peculiar situation in which the gas has more carbon than oxygen. This carbon-rich gas alters the chemistry of grain and molecule formation in the outflowing gas, preferentially forming molecules and grains which are rich in hydrogen and carbon. The observed correlation between the 7.7 $\mu$m IEF strength and the C/O ratio in planetary nebulae strongly suggests that the IEFs form from dust and/or molecules condensed in the outflows from dying stars with carbon-rich outflows. Once the IEFs, like other grains and molecules, are formed, the gas ejected from the dying star slowly mixes with the surrounding gas and dust within the galaxy and becomes part of the general ISM. The ISM in galaxies, therefore, contains a mixture of molecules and grains formed in oxygen-rich environments and carbon-rich environments.
Astronomical observations show strong emission from the IEFs in many regions of star formation, where dust and gas are interacting with the ultraviolet (UV) and visible radiation emanating from young stars or from stars in the process of forming (protostars). Two examples of star formation regions with strong IEF emission are reflection nebulae, where grains within a cloud of gas and dust scatters the light of a nearby young star, and photodissociation regions surrounding H II regions (see reviews by \[11, 12\]). An H II region is the astronomical name for a cloud of ionized hydrogen gas surrounding a young star hot enough to ionize hydrogen; a photodissociation region is the astronomical name for the surface of a cloud of molecular hydrogen gas, where the UV radiation from a hot star is in the process of photodissociating the molecular hydrogen. Regions of the ISM are commonly referred to by the state of their hydrogen gas – neutral (H I), ionized (H II), or molecular (H$_2$) – because 71% of the mass of interstellar gas is composed of hydrogen. The hydrogen gas in reflection nebulae can be either neutral or molecular, but not ionized.
Astronomical mapping and images of the IEFs in the H II regions and adjacent photodissociation regions in the star-forming Orion Nebula and the planetary nebula NGC 7027 suggest that the IEFs are concentrated in a narrow zone between the ionization front and the photodissociation front \[13, 14\]. The ionization front is where ionized hydrogen abruptly changes to neutral hydrogen at a distance from the ionizing star where all of the ionizing photons from the star has been absorbed by intervening hydrogen gas and re-radiated as several lower energy, non-ionizing photons. The ionization front can be delineated by astronomical observations of recombination lines of neutral hydrogen; such lines are strong in ionized hydrogen but absent from neutral hydrogen. The photodissociation front can be traced by astronomical observations of H$_2$ emission; the same UV photons which can dissociate H$_2$ have an alternate path in which they cascade through the electronic levels of H$_2$ to populate vibrational-rotational levels of H$_2$, which are then observed as H$_2$ emission lines in the near-infrared (1 – 2.5 $\mu$m or 4000 – 10,000 cm$^{-1}$).
The IEFs are also observed in regions of low UV intensity, such as the surfaces of molecular clouds and the general ISM where the illumination source is the diffuse interstellar radiation field \[15 – 17\]. The ubiquity of IEF emission within our Galaxy, within star formation regions, within carbon-rich planetary nebulae, and within the general ISM, makes understanding their carriers an essential part of understanding how our Galaxy emits, absorbs, and reprocesses radiation. The prevalence of IEF emission in star formation regions, and the recent detection of IEF absorption toward protostars \[18 – 23\], also illuminates the important role that the IEF carriers play in the material from which stars and planets form.
The IEFs were first proposed to be due to aromatic hydrocarbons by Duley & Williams \[24\], based on wavelength coincidences between the IEFs and laboratory spectra of aromatic hydrocarbons. Another clue to the composition of the IEF emitters was contributed by Sellgren et al. \[25\] and Sellgren \[26\], when we discovered 1 – 5 $\mu$m (2000 – 10,000 cm$^{-1}$) continuum emission in reflection nebula that was not due to scattered starlight. Starlight which is scattered from grains is strongly polarized, and this infrared continuum emission in reflection nebulae was unpolarized \[27\]. This continuum emission had a color temperature around 1000 K, and was spatially associated with IEF emission in these sources. The color temperature of the continuum emission showed no decrease with distance from the exciting star, as would be expected for thermal emission by dust in equilibrium with the star’s radiation field. Furthermore the stellar radiation field was far too weak to heat grains to equilibrium temperatures of 1000 K at the observed nebular distances. Temperatures of $\sim$ 30 – 60 K were expected instead, and in fact dust in this temperature range had already been observed \[28, 29\] at far infrared wavelengths (40 – 400 $\mu$m or 25 – 250 cm$^{-1}$). We proposed a new paradigm for interstellar dust, in which the dust size distribution contained a component of 1 nm sized grains which were so small that a single UV photon absorbed from a nearby star could briefly heat the grain to temperatures as high as 1000 K. Tiny particles of this size has been proposed earlier \[30, 31\]. These tiny grains spend the vast majority of their time at low temperature, but when they undergo stochastic heating they can for a microsecond or millisecond radiate emission at high temperatures \[25, 26\]. This model explains the high color temperature of the continuum emission, the constancy of the color temperature with distance from the exciting star, and the lack of polarization. Once we had proposed this model, several groups quickly realized the importance of this emission mechanism to the newly emerging data from the Infrared Astronomical Satellite (IRAS), launched in 1984, because the tiny grains, as they cooled radiatively, would be characterized by progressively lower color temperatures with time and thus contribute significantly to emission in the IRAS 12 and 25 $\mu$m (400 and 830 cm$^{-1}$) broad-band photometric filters \[32 – 34\].
The real breakthrough in understanding the IEFs, however, was made by Léger & Puget \[35\] and Allamandola et al. \[36\]. They were the first to combine the aromatic hydrocarbon idea \[24\] with a 1 nm particle size \[26\], and to propose that the IEFs were due to polycyclic aromatic hydrocarbon (PAH) molecules with a size of roughly 1 nm or smaller. Other aromatic materials have been proposed for the IEFs, including tiny grains composed of hydrogenated amorphous carbon, quenched carbonaceous composite, and coal-like materials \[37 – 42\]. Virtually all astronomers working on the IEFs agree on the aromatic nature of the IEFs. There is still debate over the exact nature of the interstellar aromatics, but many favor the PAH hypothesis because it so neatly ties together well-studied aromatic materials with the 1 nm sizes suggested by the continuum emission associated with the IEFs.
The goal of the astronomical observational research of myself and my collaborators has been to test specific predictions of the PAH hypothesis, and to urge chemical theorists and laboratory astrochemists to make modifications in the PAH hypothesis when the predictions and observations do not match. This interplay between the observations, theory, and laboratory work has proven very fruitful in refining our understanding of the size, composition, ionization state, and hydrogenation of the material responsible for the IEFs in the ISM.
A Challenge: Approximating Interstellar Conditions in the Laboratory
--------------------------------------------------------------------
One of the most difficult challenges in matching astronomical observations and laboratory measurements of candidate materials for the IEFs is the vast gulf between interstellar conditions and laboratory conditions. The densities in interstellar space, even in the densest interstellar clouds, are so low that they exceed the capabilities of the best vacuum systems on Earth. The vast majority of published laboratory work has been measurements of solid-phase, room temperature aromatic materials in absorption, which are then compared to astrophysical observations of $\sim$1000 K aromatic materials in emission in extremely low density environments. Some progress has been made in approximating the low densities of interstellar conditions by use of inert-gas matrix isolation techniques, but even in these cases matrix interactions introduce frequency shifts in the wavelengths of absorption lines of unknown magnitude and direction. Other progress has been made in measuring aromatic molecules in emission in the laboratory, which has helped to quantify the temperature dependence of wavelength shifts in PAHs. Recent evidence, discussed below, suggests that the PAHs that have been best-studied in the laboratory are too small to approximate interstellar PAHs. Furthermore, astrophysical theoretical calculations predict that PAHs should be ionized in the best-observed regions of the ISM. Both chemical quantum calculations and laboratory measurements demonstrate marked changes in the relative intensities and central wavelengths of PAH bands with ionization state. All of these mismatches between interstellar conditions and laboratory conditions make it difficult to achieve a firm identification of the material(s) responsible for the IEFs.
The IEFs and the Hardness of the UV Radiation Field
===================================================
One of the early predictions of the PAH model is that PAH molecules should only be excited by UV radiation, since laboratory absorption curves of small, neutral PAHs show a sharp cutoff in their absorption at UV wavelengths with little or no absorption at visible wavelengths. Sellgren et al. \[43\] tested this prediction by using observations from the IRAS satellite to search for infrared emission in reflection nebulae excited by stars of different effective temperature, $T_{\rm eff}$. We first verified that each reflection nebula was indeed excited by the central star of the nebula, by requiring that the dust temperature derived from the the ratio of 60 $\mu$m (170 cm$^{-1}$) intensity to 100 $\mu$m (100 cm$^{-1}$) intensity reach a peak value at the star. The emission at 60 and 100 $\mu$m is thought to primarily come from grains, with a size around 100 nm, which are in equilibrium with the radiation field, and whose temperature therefore will increase with proximity to the heating star. We then examined the 12 $\mu$m emission of each reflection nebula, at a nebular position offset from the star so that stellar emission would not contaminate the 12 $\mu$m data, and compared it to the total infrared emission at the same spatial location. For each nebula we then measured $R(12/{\rm total})$, the ratio of the total flux in the IRAS 12 $\mu$m band to the total infrared flux. The IRAS 12 $\mu$m band we assumed to be dominated by IEF emission and its associated continuum emission, as had been shown by mid-infrared spectroscopy for some of the sources in our sample \[34\]. We were expecting $R(12/{\rm total})$ to show a precipitous drop for cool stars ($T_{\rm eff}$ $<$ 10,000 K), where the fraction of total stellar radiation radiated in the UV was small. Much to our surprise, the value of $R(12/{\rm total})$ was independent of $T_{\rm eff}$, over the range $T_{\rm eff}$ = 5,000 K – 22,000 K, as illustrated in Figure 3. Our observational results required that the material responsible for emitting the IRAS 12 $\mu$m emission in reflection nebulae absorb not only at UV but also at visible wavelengths. This discovery altered the astronomical community’s perception of PAHs as being small in size and neutrally charged, and drove theorists and laboratory astrochemists to consider both larger PAHs and ionized PAHs in their models and laboratory work. This is because increasing the PAH size and ionizing PAHs both have the effect of extending the absorption cross-section of PAHs out to visible wavelengths.
The 12 $\mu$m IRAS filter bandpass is wide and encompasses several IEFs as well as their associated continuum. The material responsible for the continuum has never been identified, and the possibility exists that it is not due to the same material that produces the IEFs. In this case, it might be possible that the IRAS 12 $\mu$m emission we observed in reflection nebulae illuminated by cool stars is due to continuum emission alone without any contribution from the IEFs. Thus, when the Infrared Space Observatory (ISO) was launched in 1995, we embarked on an observational program to obtain spectra of some of the reflection nebulae studied by Sellgren et al. \[43\], particularly reflection nebulae illuminated by both cool and hot stars which had similar values of $R(12/{\rm total})$ . These observations have been made with ISO’s mid-infrared camera (ISOCAM) combined with its circular variable filter (CVF), which allow us to obtain low-resolution ($\lambda / \Delta \lambda $ = 40) spectra, at 5 – 15 $\mu$m (600 – 2000 cm$^{-1}$), simultaneously at roughly a thousand spatial positions across each nebula. The ISO spectra of these reflection nebulae, therefore, should unambiguously determine whether the IEFs are present in these sources.
The first ISO results from Uchida et al. \[44\] was our discovery that IEFs are clearly detected in vdB 133, a reflection nebula illuminated by a binary system with very little UV radiation. The binary system, a luminous star with $T_{\rm eff}$ = 6,800 K plus a fainter star with $T_{\rm eff}$ = 12,000 K, provides a ratio of UV ($\lambda$ $<$ 400 nm) to total stellar flux which is a factor of four lower than more typical reflection nebulae which are illuminated by stars with $T_{\rm eff}$ $\sim$ 20,000 K. Yet, despite the softer radiation environment, the IEF spectrum in this UV-poor environment is very similar to IEF spectra observed in sources with much harsher UV environments, as shown in Figure 4.
New Results: Laboratory Analogs and the IRAS Data on R(12/total)
----------------------------------------------------------------
I am currently collaborating with J. Pizagno, K. Uchida, and M. Werner on a comparison of various laboratory and theoretical candidates for the IEF carriers with the IRAS observations \[43\] of $R(12/{\rm total})$, to quantify which materials can reproduce the lack of dependence of $R(12/{\rm total})$ on $T_{\rm eff}$ (Pizagno et al. 2000, in preparation). We are convolving the UV and visible laboratory and theoretical absorption curves of different materials with the energy distributions emitted by stars of different $T_{\rm eff}$, and comparing the results to our IRAS observations. Figure 3 shows one sample result of our calculations, where we compare the astronomical observations to the predicted results for two small PAH molecules, neutral naphthalene \[45\] and singly ionized naphthalene (from F. Salama 1999, private communication). These new results show that small neutral PAHs are completely unable to provide enough visible absorption to explain the IEF emission observed around UV-poor sources. They also show that while ionized PAHs have more visible absorption than their neutral counterparts, small PAHs even when ionized also cannot explain the IRAS observations.
Need for Data and Theory for Larger PAHs and Varying Ionization
---------------------------------------------------------------
It seems likely that a combination of ionization and larger PAH size will be required to explain the astronomical observations. We strongly encourage laboratory astrochemists to measure parameters for larger PAHs, both neutral and ionized. We also strongly encourage theoretical astrochemists to calculate parameters for larger PAHs, both neutral and ionized, particularly for PAH sizes and/or ionization states that are not currently accessible by laboratory techniques.
PAH Ionization
==============
Uchida et al. \[46\] have just completed a more comprehensive ISO study of IEFs in reflection nebulae, in which we have made unexpected observations which have important ramifications for models of PAH ionization. PAH ionization models predict that PAHs should be primarily positively charged in regions of high UV radiation, such as reflection nebulae, and primarily neutral or negatively charged in regions of low UV radiation, such as the diffuse ISM and cirrus clouds \[47 – 50\]. The UV radiation intensity is generally characterized by the variable $G_0$, where $G_0$ = 1 corresponds to the UV radiation field in the solar neighborhood. Laboratory and theoretical studies show that the ratio of PAH emission at 6 – 10 $\mu$m (1000 – 1700 cm$^{-1}$) to the PAH emission at 10 – 14 $\mu$m (700 – 1000 cm$^{-1}$) is a sensitive function of PAH ionization state, with this ratio varying by at least a factor of 10 between ionized and neutral PAHs \[51 – 59\].
We have measured the ratio of IEF emission at 6 – 10 $\mu$m to the IEF emission at 10 – 14 $\mu$m, observed in reflection nebulae with $G_0$ varying from 20 to 6$\times$10$^4$. Figure 5 shows that we see little or no evidence for any change in this ratio with $G_0$ (and thus PAH ionization state). This provides a major puzzle for the PAH hypothesis.
PAH Ionization: Astrochemical Theory and Laboratory Data Needed
---------------------------------------------------------------
There are several ways to resolve the discrepancy between the observations and theory for the predicted dependence of the PAH ionization state on the UV field. The calculated ionization states of PAHs are somewhat sensitive to the assumed PAH size. The amount by which the ratio of fluxes at 6 – 10 $\mu$m to 10 – 14 $\mu$m changes also depends on PAH size, with smaller changes in the ratio for larger PAHs. The calculated ionization states of PAHs are also very sensitive to the uncertain PAH recombination rate, which requires better laboratory measurements and/or better theoretical predictions for the recombination rates, particularly as a function of PAH size. These are all areas in which astrochemistry in the laboratory and theoretical calculations can really make a difference in interpreting the astronomical observations.
The Full-Width at Half Maximum of the 7.7 $\mu$m IEF
====================================================
Another unexpected discovery Uchida et al. \[46\] have made with our recent ISO observations concerns spectral changes in the IEFs at low levels of illumination. We have quantitatively compared the IEF spectra (Fig. 4) of three sources we observed ourselves, vdB 133, vdB 17 (NGC 1333), and vdB 59 (NGC 2068), to the IEF spectra of published spectra of NGC 7023 (\[2\]; Fig. 1) and $\rho$ Oph \[15\]. We have found no evidence for any systematic spectral differences with $T_{\rm eff}$. Our new ISO observations, however, find that the full width at half maximum (FWHM) of the 7.7 $\mu$m IEF is dependent on the distance between star and nebula in vdB 17 \[46\]. Figures 6 and 7 show that in the most distant regions of vdB 17, corresponding to the lowest UV illumination levels, $G_0$ = 20 – 60, the 7.7 $\mu$m IEF becomes significantly broader and begins to blend with the 8.6 $\mu$m IEF. This effect has now been observed in a second reflection nebula, Ced 201 \[60\], although they do not give the $G_0$ values at which the broadening of the 7.7 $\mu$m IEF becomes significant. None of the other IEFs appear to change width with stellar distance in either source. We also find no broadening of the 7.7 $\mu$m IEF or other IEFs in vdB 59, down to $G_0$ = 200. More significantly, the spectrum of $\rho$ Oph \[15\] at $G_0$ = 40 shows no sign of a broader 7.7 $\mu$m IEF or any blending of the 7.7 $\mu$m IEF with the 8.6 $\mu$m.
The unexpected broadening of the 7.7 $\mu$m IEF at low UV illumination levels, in some sources but not others, is very much a mystery. One possibility is that the intermittent broadening of the 7.7 $\mu$m feature depends on two parameters, one being $G_0$, for instance, and the other being some factor such as $T_{\rm eff}$. Stellar $T_{\rm eff}$ seems like a good starting point for investigation, because vdB 17 and Ced 201, which are illuminated by stars with $T_{\rm eff}$ = 10,000 – 11,000 K, show the broadening of the 7.7 $\mu$m IEF at low $G_0$ values, while $\rho$ Oph, which shows no such broadening at $G_0$ = 40, is illuminated by a pair of stars with $T_{\rm eff}$ = 22,000 K.
Two other ideas seem like promising avenues to explore. One is the recent ISO observations of Moutou et al. \[61\], illustrated in Figure 8. We find that the 7.7 $\mu$m IEF, when observed at higher spectral resolution ($\lambda / \Delta \lambda$ = 1800), appears to break up into at least three different features \[61\], at 7.45, 7.6, and 7.8 $\mu$m (1280, 1320, and 1340 cm$^{-1}$). Previous astronomical spectra at $\lambda / \Delta \lambda$ = 100 – 200 \[62, 63\] resolved the 7.7 $\mu$m IEF into two features at 7.6 and 7.8 $\mu$m. Other astronomical observations \[9, 10\] at a lower resolution of $\lambda / \Delta \lambda$ = 50, where only the central wavelength of the 7.7 $\mu$m IEF could be measured, demonstrated that the 7.7 $\mu$m IEF central wavelength depends on the physical conditions in the emitting region (whether or not the hydrogen gas is ionized, for instance). Since the results of \[9, 10\] almost surely result from subtle changes in the relative strengths of the subcomponents of the 7.7 $\mu$m IEF, then the broadening of the 7.7 $\mu$m IEF observed at low resolution \[46\] could reflect more obvious changes in the relative strengths of the different subcomponents observed at higher resolution \[61 – 63\].
The other idea deserving further exploration is related to the fundamental reason why the IEFs are so much broader than gas molecules or grains moving at the Doppler speeds (1 km s$^{-1}$ or less) typical of these interstellar regions. There is no clear consensus on whether the IEF widths are due to physical processes (such as timescales for internal molecular processes, which depend on the size and temperature of the molecule) or to mixtures of aromatics with slightly different compositions, sizes, and central wavelengths. Observations of the IEFs in absorption suggest that the mixture of IEF carriers may be dominant in determining the central wavelength and width of the IEF emission, as discussed in the next section, and if this is the case, then the astronomical observations of changes in the central wavelength or FWHM of the 7.7 $\mu$m IEF could reflect changes in the size distribution or composition of interstellar aromatics under different physical conditions.
The 7.7 Micron IEF Width: Need for Laboratory and Theoretical Work
------------------------------------------------------------------
Astrochemical work is desperately needed to formulate some theoretical framework for why the 7.7 $\mu$m width changes, so that astronomical observations can be made to test this framework. As described above, the intermittent broadening of the 7.7 $\mu$m IEF could depend on two parameters, one being $G_0$, for instance, and the other being some factor such as stellar temperature. Can laboratory or theoretical work provide any insights into why these two parameters, or some other set of parameters, might affect the 7.7 $\mu$m width? Does the broadening of the 7.7 $\mu$m IEF observed at low resolution reflect changes in the relative strengths of the different subcomponents at 7.45, 7.6, and 7.8 $\mu$m observed at higher resolution \[61\]? Or do the relative strengths of the different subcomponents of the 7.7 $\mu$m IEF stay the same, but the width of each subcomponent change? What physical change would be needed in the IEF carriers for the width to change? Could it be that the size distribution or composition of interstellar aromatics change with different physical conditions? Or is it some other phenomenon entirely? The answer to the questions posed by observations of the 7.7 $\mu$m IEF are rooted in the ongoing debate as to the origin of the IEF widths themselves.
Aromatics in Absorption
=======================
Until recently, all astronomical observations of aromatic hydrocarbons in the ISM were of the IEFs in emission. The absorption signatures of the IEFs had not been seen. We have begun a program to try to trace the abundance and chemical changes of aromatic hydrocarbons during their residence time in molecular clouds. A broad 3.25 $\mu$m (3080 cm$^{-1}$) absorption feature along the line-of-sight to the protostar Mon R2/IRS-3 was tentatively detected, and its discovery confirmed with better spectra, by Sellgren et al. \[18, 19\]. We proposed that this feature is due to the C–H stretch absorption of cold aromatic hydrocarbons within the molecular cloud. Laboratory work on PAHs \[64\] has shown that the central wavelength of PAH features shifts to increasing wavelength with increasing temperature. This predicts that aromatics in absorption should be blue-shifted by about the observed amount, relative to aromatics in emission, because aromatics in emission are emitting at temperatures of $\sim$1000 K, while aromatics in absorption are absorbing at much lower temperatures, $\sim$ 10 – 80 K. In Brooke et al. \[21, 22\] we have now detected the 3.25 $\mu$m absorption feature toward a total of five protostars embedded in molecular clouds.
New Results: Comparing Aromatics in Emission and Absorption
-----------------------------------------------------------
The current observations of aromatics in absorption, compared to aromatics in emission, at first glance seem to have a simple interpretation. But there is an interesting complication. Ground-based \[65\] and ISO \[66\] spectra of sources near the Galactic Center reveal a narrow absorption feature near 3.28 $\mu$m (3050 cm$^{-1}$), which these authors identify with aromatic hydrocarbons. If this feature is due to the 3.29 $\mu$m (3040 cm$^{-1}$) IEF emitter, observed in absorption through the diffuse ISM in our Galaxy, it should spend the vast majority of its time at cold temperatures and thus have a wavelength blue-shifted relative to the 3.29 $\mu$m IEF in emission, similar to the blue-shifted wavelength (3.25 $\mu$m; 3080 cm$^{-1}$) observed in absorption toward molecular clouds. The longer wavelength of the 3.28 $\mu$m absorption suggests either that the aromatics are at a higher temperature, perhaps due to localized heating by sources in the Galactic Center, or that the composition, sizes, or ionization state of the aromatics observed in the diffuse ISM are different from the aromatics observed in molecular clouds.
The differences in observed width are also confusing. Laboratory studies \[64\] have shown that as individual PAHs are heated, their central wavelengths both shift to longer wavelengths and the width of the PAH band broadens. Figure 9, which has not been published elsewhere, illustrates the contrast between the short wavelength and broad width of the 3.25 $\mu$m aromatic absorption in molecular clouds \[19\], and the longer wavelength and narrow width of the 3.28 $\mu$m aromatic absorption towards the Galactic Center \[66\]. Both spectra have $\lambda / \Delta \lambda$ = 1000. Figure 9 also shows a spectrum of the 3.29 $\mu$m IEF in emission in the reflection nebula NGC 7023, from the work in progress of myself, C. Moutou, A. Léger, L. Verstraete, M. Werner, M. Giard, and D. Rouan. This spectrum, with $\lambda / \Delta \lambda$ = 1000, demonstrates that the observed width of the 3.29 $\mu$m IEF in emission (Sellgren et al. 2000, in preparation) is similar to or narrower than the 3.25 $\mu$m feature in absorption \[19\], again in contradiction to the temperature model. This suggests that the observed wavelength shifts and widths are not due to temperature effects but rather to composition, size distribution, ionization, or other effects.
This mystery is underscored by recent observations of an absorption feature at 6.18 $\mu$m (1620 cm$^{-1}$), believed to be the aromatic C–C stretch, toward five hot, mass-losing stars and two Galactic Center sources in ISO spectra \[67\]. A possible absorption feature at 6.24 $\mu$m (1600 cm$^{-1}$) toward several protostars has also been tentatively identified as the C–C stretching mode of aromatic hydrocarbons \[20, 23\]. If the 6.18 $\mu$m absorption toward the sources observed by Schutte et al. \[67\] arises from aromatics in the diffuse ISM, the aromatics should again be cold with blue-shifted, narrow absorption features. The 6.24 $\mu$m absorption feature observed by Keane et al. \[23\] toward protostars in molecular clouds, if due to aromatics, should also arise from cold aromatics with blue-shifted, narrow absorption features. The astronomical observations, however, do not fit this picture. The 6.18 $\mu$m absorption feature observed through the diffuse ISM \[67\] is blue-shifted relative to the 6.22 $\mu$m (1610 cm$^{-1}$) IEF in emission \[68\], as expected, but the 6.24 $\mu$m absorption in molecular clouds is observed at a similar or slightly longer wavelength as the 6.22 $\mu$m IEF in emission. Furthermore, the 6.18 $\mu$m absorption feature observed through the diffuse ISM is significantly narrower than the 6.24 $\mu$m absorption feature observed towards molecular clouds, as shown in Figure 10 of Keane et al. \[23\]. Again this is strong evidence that both the central wavelengths and widths of aromatics observed in absorption and emission in the ISM are determined not by temperature or molecular physics of a single molecule, but rather by the composition mixture, size distribution, or mix of ionization states of aromatics observed along a particular line-of-sight.
Need for Laboratory Work on PAHs in Water Ice Matrices
------------------------------------------------------
One possible way to reconcile the temperature picture of the wavelength and width of aromatic absorption and emission with the astronomical observations is to consider whether aromatics in molecular clouds might be embedded in ice mantles on grains. Solid H$_2$O ice is observed toward all sources in molecular clouds with detected 3.3 or 6.2 $\mu$m aromatic absorption features. New astrochemical laboratory measurements at NASA’s Ames Research Center are currently underway to measure the effect of a surrounding ice matrix on PAH absorption (M. Bernstein 2000, private communication). These results, once published, may clarify our understanding of aromatic absorption in molecular clouds. Measurements are needed for both neutral and ionized PAHs.
Interstellar “Diamond-like” Carbon
==================================
The broad 3.47 $\mu$m (2880 cm$^{-1}$) absorption feature (FWHM $\approx$ 0.1 $\mu$m or 80 cm$^{-1}$) was first noted in ground-based spectra of four protostars by Allamandola et al. \[69\]. They suggested that the feature might be due to the C–H stretch absorption of solo hydrogens attached to $sp^3$ bonded carbon clusters, the “diamond”-like form of carbon. The feature was present in every molecular cloud source looked at by Brooke et al. \[21\]. Interstellar microdiamonds, formed outside our solar system, have been identified in meteorites \[70, 71\] and so must therefore exist and survive in the diffuse ISM. Our observations \[21, 22\] of the 3.47 $\mu$m absorption feature, however, reached the startling conclusion that the optical depth of the 3.47 $\mu$m absorption feature is not correlated with the depth of silicate absorption, as might be expected for two refractory minerals such as diamonds and silicates (Figure 10). Instead, we found that the 3.47 $\mu$m optical depth is strongly correlated with the depth of the H$_2$O ice band, a very volatile ice that cannot exist outside molecular clouds (Figure 11).
Need for Laboratory or Theoretical Frequencies for C–H Bonds on the Surfaces of Microdiamonds
---------------------------------------------------------------------------------------------
Under the assumption that the 3.47 $\mu$m feature is due to C–H bonds, Brooke et al. \[21\] interpreted the correlation with H$_2$O ice as indicating that both C–H bonds and H$_2$O ice form in step on molecular cloud dust by hydrogen addition reactions. However, we noted that other identifications of the 3.47 $\mu$m feature are also possible. One fertile region for astrochemical research is to better establish the formation mechanism and absorption spectrum of interstellar “diamond”-like carbon of interstellar size (1 nm; \[70\]) with surface hydrogen atoms. Predictions or measurements of the complete vibrational spectrum, including additional modes which could be searched for in astronomical spectra, would be particularly useful. It would also be useful to search for alternative identifications for the 3.47 $\mu$m feature with simple molecules, composed of cosmically abundant elements (Fig. 1), which would have sublimation temperatures close to that of H$_2$O ice. Again, predictions or measurements of other vibrational modes that could be used to test the identification would be helpful.
C$_{60}^+$ in the interstellar medium
=====================================
The fullerene C$_{60}$ was first discovered in the laboratory by Kroto et al. \[72\]. They proposed that this novel form of aromatic carbon could potentially play an important role in the ISM. Theoretical studies of dust formation in carbon-rich stellar mass-loss \[73 – 75\] suggested that fullerenes could be formed in such carbon-rich environments and expelled into the ISM. Laboratory studies have also shown that one product of the photoerosion of hydrogenated amorphous carbon grains is fullerenes \[76\]. The existence of fullerenes such as C$_{60}$ and C$_{70}$ in our solar system, for instance in meteoritic samples, is hotly debated \[77, 78\].
Searches for C$_{60}$ in the ISM, through its UV absorption band at 386 nm, have placed stringent limits of $<$ 0.01% of the cosmic carbon abundance in C$_{60}$ \[79, 80\]. The dominant ionization state of C$_{60}$ in the ISM, however, is predicted to be C$_{60}^+$ \[47, 49, 50\]. Foing & Ehrenfreund \[81, 82\] detected diffuse interstellar bands (see \[83\] for a review of the diffuse interstellar bands) at 958 and 963 nm, which they argued were due to C$_{60}^+$, based on a comparison to laboratory data \[84\].
This inspired Moutou et al. \[61\] to use ISO data to search for the 7.1 and 7.5 $\mu$m (1331 and 1406 cm$^{-1}$) vibrational emission lines of C$_{60}^+$ in NGC 7023, a reflection nebula with strong IEF emission. Such a search requires high signal-to-noise and high spectral resolution ($\lambda / \Delta \lambda$ = 1800), because these features, if present, would appear as weak bumps on the blue wing of the strong aromatic 7.7 $\mu$m IEF. From our ISO spectra (Figure 8), we place an upper limit on C$_{60}^+$ in NGC 7023 of $<$ 0.3% of interstellar carbon \[61\].
The central star of NGC 7023, HD 200775, is known to have abnormally weak diffuse interstellar bands \[85\]. If the diffuse interstellar bands observed at 958 and 963 nm are due to C$_{60}^+$, then the general weakness of the diffuse interstellar bands in NGC 7023 might naturally lead to a failure to detect C$_{60}^+$ in the mid-infrared toward this source. Astronomers can place more robust limits on the abundance of fullerenes in the ISM by searching for the 7.1 and 7.5 $\mu$m C$_{60}^+$ lines towards a larger number of lines of sight, particularly those in which the diffuse interstellar bands are strong.
Laboratory and Theoretical Work Needed for Fullerenes
-----------------------------------------------------
Laboratory astrochemists can also contribute greatly to the search for interstellar fullerenes by further experiments. The assignment of the interstellar bands at 958 and 963 nm to C$_{60}^+$ was based on a comparison of gas-phase C$_{60}^+$ in the ISM with C$_{60}^+$ measured in inert-gas matrices, with the effects of matrix shifts on the wavelengths being unknown. The 7.1 and 7.5 $\mu$m C$_{60}^+$ bands were also measured in inert-gas matrices, again introducing an unknown amount of matrix shift to the wavelengths. Laboratory measurements of all these bands in the gas phase would help immensely to determine whether detections of astronomical absorption features near the laboratory wavelengths of C$_{60}^+$ prove or disprove the existence of C$_{60}^+$ in the ISM. Finally, by symmetry, there should be two longer wavelength vibrational modes of C$_{60}^+$ which were not measured by Fulara et al. \[84\]. Laboratory wavelengths for these two additional modes would open up new avenues for astronomical observations of C$_{60}^+$.
Conclusions
===========
I have emphasized in this paper a number of puzzling astronomical observations of aromatics in the ISM, and related molecules, where understanding of these problems would greatly benefit from new experiments in laboratory astrochemistry and progress in chemical theory. The study of the ISM is an area where the interaction of astronomy, physics, and chemistry has proven particular fruitful in reaching new conclusions about the universe that surrounds us, and I strongly encourage astrochemistry groups to tackle these experimental and theoretical problems as part of this on-going interaction.
I would like to thank Tim Brooke for contributions to the text on the 3.47 $\mu$m feature. I am grateful to Max Bernstein, Jean Chiar and Farid Salama for communicating data in advance of publication, and to Jacquie Keane for sharing a preprint before the refereeing process was complete. I also thank James Pizagno for speedy assistance with Figure 3.
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**Figure Captions**
[**Figure 1**]{}— The cosmic abundances of elements from Cox \[1\], presented as log$_{10}$ of the elemental abundance relative to hydrogen versus the atomic number of the element. Only elements with abundances greater than 10$^{-6}$ relative to hydrogen are illustrated.
— The ISOCAM + CVF spectrum of the reflection nebula NGC 7023 from Cesarsky et al. \[2\], plotted as relative flux density versus frequency in cm$^{-1}$. The spectral resolution is $\nu / \Delta \nu$ = 40. Five of the six interstellar emission features (IEFs) can be seen, at 790, 890, 1160, 1300, and 1610 cm$^{-1}$ (6.2, 7.7, 8.6, 11.3, and 12.7 $\mu$m).
— IRAS broad-band photometric observations ([*filled squares*]{}) of $R(12/{\rm total})$, the ratio of the flux in the IRAS 12 $\mu$m filter to the total infrared flux, plotted against $T_{\rm eff}$(star), the effective stellar temperature of the star which illuminates each reflection nebula, from Sellgren et al. \[43\]. Error bars are $\pm$1-$\sigma$. Upper limits are 3-$\sigma$. The curves are the predicted values of $R(12/{\rm total})$ from Pizagno et al. (2000, in preparation), derived by convolving the UV and visible absorption curves of neutral naphthalene ([*dashed curve*]{}), from Salama & Allamandola \[45\], and ionized naphthalene ([*solid curve*]{}), from F. Salama (1999, private communication), with theoretical energy distributions of stars as a function of $T_{\rm eff}$(star).
— Scaled ISOCAM + CVF spectra of three reflection nebulae, vdB 17 (NGC 1333; [*dotted line*]{}), vdB 59 (NGC 2068; [*thick solid line*]{}), and vdB 133 ([*thin solid line*]{}), plotted versus wavelength in $\mu$m, from Uchida et al. \[44, 46\]. The spectral resolution is $\lambda / \Delta \lambda$ = 40. The stars which illuminate each reflection nebula respectively have $T_{\rm eff}$ = 11,000 K for vdB 17, 19,000 K for vdB 59, and 6,800 + 12,000 K for vdB 133 (which is illuminated by a binary star).
— Ratio of the 5.5 – 9.75 $\mu$m flux to the 10.25 – 14.0 $\mu$m flux versus the UV radiation field, $G_0$, from Uchida et al. \[46\]. All data are derived from ISOCAM + CVF spectra. Different nebular positions are illustrated for vdB 17 ([*filled diamonds*]{}; \[46\]), and vdB 59 ([*filled circles*]{}; \[46\]). Results at a single nebular position are shown for vdB 133 ([*open circle*]{}; \[46\]), NGC 7023 ([*cross*]{}; from Cesarsky et al. \[2\]), and $\rho$ Oph ([*open square*]{}; from Boulanger et al. \[15\]). Error bars are $\pm$1-$\sigma$.
— ISOCAM + CVF spectra of two reflection nebulae, plotted versus wavelength in $\mu$m, from Uchida et al. \[46\]. The right panel is vdB 59 (NGC 2068), while the left panel is vdB 17 (NGC 1333). Spectra at different nebular positions are labeled by their values of the UV intensity, $G_0$. The spectral resolution is $\lambda / \Delta \lambda$ = 40. Note the broadening of the 7.7 $\mu$m IEF at the lowest $G_0$ values in vdB 17, which is unexplained by current models.
— FWHM in $\mu$m of the 6.2, 8.6, and 7.7 $\mu$m IEFs as a function of the UV radiation field, $G_0$, within vdB 17, from Uchida et al. \[46\]. Data are derived from ISOCAM + CVF spectra. Error bars are $\pm$1-$\sigma$. No correction has been made for the instrinsic spectral resolution of the instrument ($\lambda / \Delta \lambda$ = 40).
— Spectra at 7.0 – 8.7 $\mu$m of NGC 7023, plotted versus wavelength in $\mu$m, observed with ISO’s Short Wavelength Spectrometer (SWS) at $\lambda / \Delta \lambda$ = 1800 by Moutou et al. \[61\]. Solid vertical lines mark the laboratory positions of C$_{60}^+$ bands \[84\]. The data labeled “(up – down)/2” provide an estimate of the noise of the spectrum. The emission line in NGC 7023 at 8.02 $\mu$m is the 0–0 S(4) line of molecular hydrogen. The SWS responsivity calibration is shown at the bottom of the plot, offset for clarity.
— ([*top*]{}) Optical depth, plotted versus wavelength in $\mu$m, of a possible aromatic absorption feature toward the protostar Mon R2/IRS–3, embedded in a molecular cloud, from ground-based spectra ($\lambda / \Delta \lambda$ = 1000) obtained by Sellgren et al. \[19\] at the United Kingdom Infrared Telescope Facility. ([*middle*]{}) Optical depth, plotted versus wavelength in $\mu$m, of a possible aromatic absorption feature toward the enigmatic Galactic Center source GCS3, which is obscured primarily by the diffuse interstellar medium, from Short Wavelength Spectrometer spectra ($\lambda / \Delta \lambda$ = 1000) from ISO observed by Chiar et al. \[66\]. ([*bottom*]{}) Normalized intensity of the 3.3 $\mu$m IEF toward the reflection nebula NGC 7023 (Sellgren et al. 2000, in preparation), from the Short Wavelength Spectrometer on ISO ($\lambda / \Delta \lambda$ = 1000). The emission line at 3.234 $\mu$m is the 1–0 O(5) line of molecular hydrogen.
— The optical depth of the 3.47 $\mu$m absorption band in molecular clouds, possibly due to “diamond”-like carbon, plotted versus the optical depth of 9.7 $\mu$m silicate absorption, a refractory grain component. Data are from Brooke et al. \[22\] ([*filled circles*]{}), Sellgren et al. and Brooke et al. \[18, 21\] ([*open circles*]{}), and Chiar et al. \[82\] ([*crosses*]{}).
— The optical depth of the 3.47 $\mu$m absorption band in molecular clouds, possibly due to “diamond”-like carbon, plotted versus the optical depth of 3.1 $\mu$m H$_2$O ice absorption, a volatile grain component. Data are from Brooke et al. \[22\] ([*filled circles*]{}), Sellgren et al. and Brooke et al. \[18, 21\] ([*open circles*]{}), and Chiar et al. \[82\] ([*crosses*]{}).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For the $K=2$ user MISO BC, i.e., the wireless broadcast channel where a transmitter equipped with $K=2$ antennas sends independent messages to $K=2$ receivers each of which is equipped with a single antenna, the sum generalized degrees of freedom (GDoF) are characterized for arbitrary channel strength and channel uncertainty levels for each of the channel coefficients. The result is extended to $K>2$ users under additional restrictions which include the assumption of symmetry.'
author:
- |
Arash Gholami Davoodi, Bofeng Yuan and Syed A. Jafar\
[Center for Pervasive Communications and Computing (CPCC)]{}\
[University of California Irvine, Irvine, CA 92697]{}\
[*Email: {gholamid, bofengy, syed}@uci.edu*]{}
bibliography:
- 'Thesis.bib'
title: 'GDoF of the MISO BC: Bridging the Gap between Finite Precision and Perfect CSIT'
---
Introduction
============
As the first steps in the path towards progressively refined capacity approximations, degrees of freedom (DoF) and generalized degrees of freedom (GDoF) studies of wireless networks have turned out to be surprisingly useful. By exposing large gaps where they exist in our understanding of the capacity limits, these studies have been the catalysts for numerous discoveries over the past decade [@Jafar_FnT]. Some of the most interesting unresolved questions brought to light by recent DoF and GDoF studies have to do with channel uncertainty and the diversity of channel strengths. Consider the wireless network with $K$ transmitters and $K$ receivers, which could represent the $K$ user interference channel, the $K\times K$ $X$ channel, or the $K$ user MISO BC, i.e., the broadcast channel formed by allowing full cooperation among the transmitters in a $K$ user interference channel. Consider, first the issue of channel uncertainty. If the channel state information at the transmitter(s) (CSIT) is perfect then the $K$ user interference channel has $K/2$ DoF, the $K\times K$ $X$ channel has $K^2/(2K-1)$ DoF, and the MISO BC has $K$ DoF almost surely.[^1] The optimal DoF are achieved by interference alignment for the interference and $X$ channel settings, and by transmit zero-forcing in the MISO BC. However, if the CSIT is available only to finite precision, then the MISO BC has only $1$ DoF, i.e., the DoF collapse as conjectured by Lapidoth et al. nearly a decade ago [@Lapidoth_Shamai_Wigger_BC]. The conjecture was proved recently in [@Arash_Jafar_GC14]. Since the MISO BC contains within it the $K$ user interference and $X$ channels, the collapse of DoF under finite precision CSIT implies that neither zero-forcing nor interference alignment is robust enough to provide a DoF advantage under finite precision CSIT, i.e., the DoF collapse for the interference and $X$ channels as well. Now consider the diversity of channel strengths which is explored through the studies of generalized degrees of freedom (GDoF). If all cross channels are much weaker relative to the direct channels then the GDoF do not collapse even with finite precision CSIT, e.g., the collapse of GDoF is avoided in the interference channel simply by treating the weak interference as noise [@Geng_TIN_opt]. Since the $X$ and BC settings include the interference channel, the collapse of DoF is avoided there as well. If some cross-channels are strong while others are so weak that they can be ignored entirely, as in the topological interference management problem [@Jafar_TIM], then even under finite precision CSIT, interference alignment plays a key role, albeit in a more robust form that does not depend on actual channel realizations.
Through these isolated and somewhat extreme data points, the DoF and GDoF studies have established that the capacity of wireless networks in the high SNR regime is quite sensitive, separately, to the level of channel uncertainty and relative channel strengths. To build upon this progress, here we initiate a study that 1) spans the space between the extremes studied so far, and 2) unifies the isolated elements of the picture. To venture between the extremes we allow a range of channel knowledge spanning from perfect to absent, and a range of channel strengths spanning from weak to strong. To present a unified view, we study the combined impact of both channel uncertainty and channel strengths by simultaneously incorporating both into our system model.
Arguably the main hurdle in expanding GDoF studies thus far has been the difficulty of obtaining good outer bounds. This is exemplified by the conjecture of Lapidoth et al. which remained unresolved for nearly a decade. DoF outer bounds under channel uncertainty have until recently been limited mostly to compound channel arguments [@Weingarten_Shamai_Kramer]. Compound channel arguments produce tight outer bounds in several settings of interest that have been successfully explored in prior work. For example, it is known that in order to maintain the full DoF (i.e., the same as with perfect CSIT), the channel estimation error should scale as $O(SNR^{-1})$ [@Caire_Jindal_Shamai; @Jindal; @Kobayashi_Caire_Jindal]. Compound channel arguments also produce tight outer bounds for various settings involving retrospective [@Gou_Jafar] and blind interference alignment [@Jafar_corr]. However, outer bounds based on compound channel arguments are evidently not strong enough to bridge the gap between perfect CSIT and finite precision CSIT. For instance, although the collapse of DoF of the MISO BC was originally conjectured under the compound setting by Weingarten et al. in [@Weingarten_Shamai_Kramer], this conjecture was settled in the negative by [@Gou_Jafar_Wang] and [@Maddah_Compound].
The reason that a broader study now seems feasible, is because of a new approach based on combinatorial accounting of the size of Aligned Image Sets (in short, the AIS approach), that was introduced in [@Arash_Jafar_GC14] to settle the conjectured collapse of DoF under finite precision CSIT. Reference [@Arash_Jafar_GC14] also showed that the AIS approach could be used to address *partial* CSIT. Consider the $K=2$ user MISO BC. For this setting the DoF are characterized in [@Arash_Jafar_GC14] with channel knowledge ranging from perfect to absent. In particular, if the channel estimation error terms scale as SNR$^{-\beta}$, so that $\beta$ values between $0$ and $1$ capture the full range of channel uncertainties, from essentially no channel knowledge ($\beta=0$) to perfect channel knowledge ($\beta=1$), then it is shown that this channel has $1+\beta$ DoF. However, the study in [@Arash_Jafar_GC14] ignores the *diversity* of channel knowledge since all channels are assumed to have the same $\beta$ parameter. Moreover, since this study is limited to DoF, it also does not capture the diversity of channel strengths[^2].
On the other hand, our recent work in [@Arash_Jafar_GC15] expands the AIS approach to study the diversity of channel strengths. Consider again the $K=2$ user MISO BC. The sum GDoF for this setting are characterized under arbitrary channel strength levels for each of the channel coefficients in [@Arash_Jafar_GC15]. However, the study in [@Arash_Jafar_GC15] is limited to the extreme setting of $\beta=0$ for all channel coefficients, i.e., it does not capture the range and diversity of channel uncertainty parameters.
Given these recent indicators that the AIS approach can be applied to study partial channel knowledge or the diversity of channel strengths individually, this work takes the next natural step, by jointly studying partial channel knowledge and the diversity of channel strengths in the same channel model. As a result for the $K=2$ user setting, the sum generalized degrees of freedom (GDoF) are characterized for *arbitrary* channel strength and channel uncertainty levels for each of the channel coefficients. Extensions to $K>2$ users are obtained under additional assumptions of symmetry. The results are presented and discussed in Section \[sec:mainresult\].
System Model
============
[\[sec-sys\]]{}
The Channel
-----------
Under the GDoF framework, the channel model for the $K$ user MISO BC is defined by the following input-output equations. $$\begin{aligned}
Y_k(t)=\sum_{l=1}^{K}\sqrt{P^{\alpha_{kl}}}G_{kl}(t)X_l(t)+Z_k(t),~~ \forall k\in[K].\end{aligned}$$ The channel uses are indexed by $t\in\mathbb{N}$, $X_l(t)$ is the symbol sent from transmit antenna $l$ subject to a unit power constraint, $Y_k(t)$ is the symbol observed by Receiver $k$, $Z_k(t)$ is the zero mean unit variance additive white Gaussian noise (AWGN) at Receiver $k$, and $G_{kl}(t)$ are the channel fading coefficients between transmit antenna $l$ and Receiver $k$. $P$ is the nominal $SNR$ parameter that is allowed to approach infinity. The channel strengths are represented in $\alpha_{kl}$ parameters.
Bounded Density Assumption
--------------------------
An important definition for this work is the notion of a “bounded density" assumption.
A set of random variables, $\mathcal{A}$, is said to satisfy the bounded density assumption if there exists a finite positive constant $f_{\max}$, \[def1\] $$\begin{aligned}
0<f_{\max}<\infty\end{aligned}$$ such that for all finite cardinality disjoint subsets $\mathcal{A}_1, \mathcal{ A}_2$ of $\mathcal{A}$, $$\begin{aligned}
\mathcal{ A}_1\subset \mathcal{A}, \mathcal{A}_2\subset \mathcal{A}, \mathcal{A}_1\cap\mathcal{A}_2=\phi, \mathcal|{A}_1|<\infty, \mathcal|\mathcal{ A}_2|<\infty\end{aligned}$$ the conditional probability density functions exist and are bounded as follows, $$\begin{aligned}
\forall A_1, A_2, ~~f_{\mathcal{A}_1|\mathcal{A}_2}(A_1|A_2)&\leq&f_{\max}^{|\mathcal{A}_1|}.\end{aligned}$$
Partial CSIT
------------
Under partial CSIT, the channel coefficients may be represented as $$\begin{aligned}
G_{kl}(t)&=&\hat{G}_{kl}(t)+\sqrt{P^{-\beta_{kl}}}\tilde{G}_{kl}(t)
$$ where $\hat{G}_{kl}(t)$ are the channel estimate terms and $\tilde{G}_{kl}(t)$ are the estimation error terms. To avoid degenerate conditions, the ranges of values are bounded away from zero and infinity as follows, i.e., there exist constants $\Delta_1, \Delta_2$ such that $0<\Delta_1\leq
|{G}_{kl}(t)|$, and $|\tilde{G}_{kl}(t)|,|\tilde{G}_{kl}(t)|<\Delta_2<\infty$. The channel variables $\hat{G}_{kl}(t), \tilde{G}_{kl}(t)$, $\forall k,l\in\{1,2\}, t\in\mathbb{N}$, are subject to the bounded density assumption with the difference that the actual realizations of $\hat{G}_{kl}(t)$ are revealed to the transmitter, but the realizations of $\tilde{G}_{kl}(t)$ are not available to the transmitter. Note that under the partial CSIT model, the variance of the channel coefficients $G_{kl}(t)$ behaves as $\sim P^{-\beta_{kl}}$ and the peak of the probability density function behaves as $\sim\sqrt{P^{\beta_{kl}}}$. In order to span the full range of partial channel knowledge at the transmitters, the corresponding range of $\beta_{kl}$ parameters, assumed throughout this work, is $$\begin{aligned}
0\leq \beta_{kl}\leq\alpha_{kl}\end{aligned}$$ Note that $\beta_{kl}=0$ and $\beta_{kl}=\alpha_{kl}$ correspond to the two extremes where the channel knowledge is essentially absent and perfect, respectively.
GDoF
----
The definitions of achievable rates $R_i(P)$ and capacity region $\mathcal{C}(P)$ are standard. The GDoF region is defined as $$\begin{aligned}
\mathcal{D}&=&\{(d_1,\cdots,d_K): \exists (R_1(P),\cdots, R_K(P))\in\mathcal{C}(P), \mbox{ s.t. } d_k=\lim_{P\rightarrow\infty}\frac{R_k(P)}{C_o(P)}, \forall k\in[K]\} \label {region}\end{aligned}$$ where $C_o(P)$ is a reference capacity of an additive white Gaussian noise channel $Y=X+N$ with transmit power $P$ and unit variance additive white Gaussian noise. For real settings, $C_o(P)=1/2\log(P)+o(\log(P))$ and for complex settings $C_o(P)=\log(P)+o(\log(P))$.
Main Results {#sec:mainresult}
============
$K=2$ Users
-----------
The first result is for the $K=2$ user MISO BC, where we allow arbitrary channel strength parameters $\alpha_{kl}$ and channel uncertainty parameters $\beta_{kl}$ for each channel coefficient. The sum GDoF for this setting is characterized in the following theorem.
\[Theorem2\] The sum GDoF value of the $2$-user MISO BC is $$\begin{aligned}
\mathcal{D}_\Sigma&=&\min(D_1,D_2)\end{aligned}$$ where $$\begin{aligned}
D_1&=&\max(\alpha_{11},\alpha_{12})+\max(\alpha_{21}-\alpha_{11}+\min(\beta_{11},\beta_{12}),\alpha_{22}-\alpha_{12}+\min(\beta_{11},\beta_{12}),0)\\
D_2&=&\max(\alpha_{21},\alpha_{22})+\max(\alpha_{11}-\alpha_{21}+\min(\beta_{21},\beta_{22}),\alpha_{12}-\alpha_{22}+\min(\beta_{21},\beta_{22}),0)
\label{aad}\end{aligned}$$
Several observations can be made from Theorem \[Theorem2\].
1. [**Recovering Prior Results:**]{} Since the current setting is a generalization of the $K=2$ user settings considered in [@Arash_Jafar_GC14] and [@Arash_Jafar_GC15], naturally the corresponding results from [@Arash_Jafar_GC14] and [@Arash_Jafar_GC15] can be recovered as special cases of Theorem \[Theorem2\]. For example, setting $\alpha_{ij}=1, \beta_{ij}=\beta$ for all $i,j\in\{1,2\}$, recovers the sum DoF result of [@Arash_Jafar_GC14], i.e., $$\begin{aligned}
\mathcal{D}_\Sigma&=&1+\beta\end{aligned}$$ Setting $\beta_{ij}=0$ for all $i,j\in\{1,2\}$, and allowing arbitrary $\alpha_{ij}$ values recovers the sum GDoF result of [@Arash_Jafar_GC15], i.e., $$\begin{aligned}
\mathcal{D}_\Sigma&=&\min(D_1,D_2)\\
D_1&=&\max(\alpha_{11},\alpha_{12})+\max((\alpha_{21}-\alpha_{11})^+,(\alpha_{22}-\alpha_{12})^+)\\
D_2&=&\max(\alpha_{21},\alpha_{22})+\max((\alpha_{11}-\alpha_{21})^+,(\alpha_{12}-\alpha_{22})^+)\end{aligned}$$
2. [**Redundancy of Strongest CSIT:**]{} The sum GDoF value depends only on $\min(\beta_{11},\beta_{12})$ and $\min(\beta_{21},\beta_{22})$, i.e., it does not depend on the strongest CSIT parameter associated with each receiver. While for $K=2$ we can equivalently state that the GDoF *depend only on the weakest* CSIT parameter for each receiver, it is easy to see[^3] that such an interpretation does not extend beyond $K=2$ users. We expect that the insight that potentially generalizes to $K>2$ users is that the GDoF value does *not depend on the strongest* CSIT parameter for each receiver. Intuitively, this is because the receivers, with their full channel knowledge, have the ability to normalize one of the channel coefficients so that it is essentially known to the transmitter. Evidently, such a normalization can only be done for the channel coefficient with the strongest CSIT parameter without affecting the CSIT levels of the remaining coefficients. For compact notation let us define $$\begin{aligned}
\beta_1&\triangleq&\min(\beta_{11},\beta_{12})\\
\beta_2&\triangleq&\min(\beta_{21},\beta{22})\end{aligned}$$ Note that $D_1, D_2$ may be equivalently expressed as $$\begin{aligned}
D_1&=&\max(\alpha_{11},\alpha_{12},\alpha_{21}+(\alpha_{12}-\alpha_{11})^++\beta_1,\alpha_{22}+(\alpha_{11}-\alpha_{12})^++\beta_1)\\
D_2&=&\max(\alpha_{22},\alpha_{21},\alpha_{12}+(\alpha_{21}-\alpha_{22})^++\beta_2,\alpha_{11}+(\alpha_{22}-\alpha_{21})^++\beta_2)\end{aligned}$$
3. [**Optimality of Single User Transmission:**]{} From the sum GDoF we note that if and only if both of the following conditions are satisfied $$\begin{aligned}
\alpha_{11}&\geq&\alpha_{21}+\beta_1\\
\alpha_{12}&\geq&\alpha_{22}+\beta_1\end{aligned}$$ then it is optimal to serve only user $1$, and the sum GDoF value is $\max(\alpha_{11},\alpha_{12})$. Note that the value of $\beta_2$ is irrelevant here. In words, it is optimal to serve only User 1, if and only if each transmit antenna ‘prefers’ User 1 to User 2 (i.e., has a stronger connection to User 1 than User 2) by at least $\beta_1$. The corresponding conditions for optimality of serving only user $2$ are obtained by switching the indices. Note that this is the only setting where all the available CSIT is useless.
4. [**GDoF vs CSIT Budget:**]{} Since Theorem \[Theorem2\] simultaneously allows arbitrary levels of CSIT and arbitrary channels strengths, it offers insights into the optimal allocation of CSIT resources as a function of given channel strengths, which may be arbitrary, to maximize the sum GDoF. The CSIT budget formulation depends on the relative costs of acquiring CSIT for each link, which may depend on the feedback mechanism employed. As a simple example, suppose the total CSIT budget is $$\begin{aligned}
\beta=\beta_{11}+\beta_{12}+\beta_{21}+\beta_{22}\end{aligned}$$ Then, given the value of $\beta$, it should be optimally allocated among $\beta_{11}, \beta_{12}, \beta_{21}, \beta_{22}$, as a function of all the channel strength parameters $\alpha_{11}, \alpha_{12}, \alpha_{21}, \alpha_{22}$, in order to maximize the sum GDoF value $\mathcal{D}_\Sigma(\beta)$. This can be done easily based on Theorem \[Theorem2\]. For example, consider again the setting where *each transmit antenna prefers the same user*, i.e., $$\begin{aligned}
\alpha_{11}&\geq&\alpha_{21}\\
\alpha_{12}&\geq&\alpha_{22}\end{aligned}$$ Further, without loss of generality, let us assume that $$\begin{aligned}
\alpha_{11}+\alpha_{22}&\geq&\alpha_{21}+\alpha_{12}\end{aligned}$$ Note that there is no loss of generality in this assumption because the transmit antennas can always be labeled in a way that this assumption is true. Then, based on Theorem \[Theorem2\], the sum GDoF with the optimal allocation of CSIT are shown in Figure \[fig:GDoF(beta)\].
![Sum GDoF $\mathcal{D}_\Sigma(\beta)$ with optimal allocation of CSIT Budget $\beta$ when $\alpha_{11}\geq\alpha_{21}, \alpha_{12}\geq\alpha_{22}, \alpha_{11}+\alpha_{22}\geq\alpha_{21}+\alpha_{12}$.[]{data-label="fig:GDoF(beta)"}](gdof){width="5.4in"}
5. [**When each transmit antenna prefers a different user:**]{} Consider the setting where each transmit antenna prefers a different user. Without loss of generality, suppose the first transmit antenna prefers User 1 and the second transmit antenna prefers User 2, i.e., $$\begin{aligned}
\alpha_{11}&>&\alpha_{21}\\
\alpha_{22}&>&\alpha_{12}\end{aligned}$$ In this case, the sum GDoF value simplifies to $$\begin{aligned}
\mathcal{D}_\Sigma&=&\min(\alpha_{22}+(\alpha_{11}-\alpha_{12})^++\beta_1, \alpha_{11}+(\alpha_{22}-\alpha_{21})^++\beta_2)\end{aligned}$$ Note that in this case, increasing the CSIT budget $\beta$ always increases the sum GDoF under optimal CSIT allocation. In particular, if both ‘direct’ channels are stronger than both ’cross’ channels, i.e., $$\begin{aligned}
\min(\alpha_{11},\alpha_{22})&\geq&\max(\alpha_{12},\alpha_{21})\end{aligned}$$ then $$\begin{aligned}
\mathcal{D}_\Sigma&=&\alpha_{11}+\alpha_{22}-\max(\alpha_{12}-\beta_1, \alpha_{21}-\beta_2)\end{aligned}$$ To see the sum GDoF with optimal allocation of CSIT resources, assume without loss of generality that $\alpha_{12}\geq\alpha_{21}$. The optimized sum GDoF $\mathcal{D}_\Sigma(\beta)$ in this case are shown in Figure \[fig:GDoF(beta)2\].
![Sum GDoF $\mathcal{D}_\Sigma(\beta)$ with optimal allocation of CSIT Budget $\beta$ when $\min(\alpha_{11},\alpha_{22})\geq\max(\alpha_{12},\alpha_{21}), \alpha_{12}\geq\alpha_{21}$.[]{data-label="fig:GDoF(beta)2"}](gdof2){width="5.4in"}
Extension to $K$ Users
----------------------
The second result is an extension to the MISO BC with arbitrary number of users ($K>2$), albeit under the following restrictions which include assumptions of symmetry to limit the number of parameters. For all $k,l\in[K]$, we set $$\begin{aligned}
\alpha_{kl}&=&\left\{
\begin{array}{ll}
\alpha, & k\neq l\\
1, &k=l.
\end{array}
\right., ~~
\alpha\in[0,1]\label{eq:cond1}\\
\beta_{kl}&=&\beta, ~~\beta\in[0,\alpha]\label{eq:cond2}\end{aligned}$$ The GDoF characterization in this setting is presented in the following theorem.
\[Theorem3\] The sum GDoF value of the $K$-user MISO BC that satisfies conditions (\[eq:cond1\]) and (\[eq:cond2\]) is $$\begin{aligned}
\mathcal{D}_\Sigma&=&(\alpha-\beta)+K(1-(\alpha-\beta))\end{aligned}$$
Recall that the DoF are obtained as a special case of GDoF, by setting $\alpha=1$. With this specialization, we note that Theorem \[Theorem3\] shows that the $K$ user MISO BC has $1-\beta+K\beta$ DoF, matching the outer bound shown in [@Arash_Jafar_GC14]. This covers the extremes of perfect CSIT ($\beta=1$) where the DoF become equal to $K$ and finite precision CSIT ($\beta=0$) where the DoF collapse to $1$. It also shows that $\beta\geq 1$ is necessary to achieve the full $K$ DoF, thus matching the results of [@Jindal]. However, more significantly, it *bridges* these divergent extremes by characterizing the DoF for all intermediate values of $\beta$ as well.
The DoF value of $1-\beta+K\beta$ has a simple intuitive interpretation. Using terminology analogous to [@ADT_FnT], the signal power levels split into the bottom $\beta$ levels where CSIT is perfect and the remaining top $1-\beta$ levels where CSIT is only available to finite precision. This is because transmission in a direction orthogonal to estimated channel vector of undesired user (zero-forcing) with power up to $\sim P^\beta$ leaks no power above the noise floor at the undesired receiver. Due to essentially perfect zero-forcing, the bottom $\beta$ levels contribute $K\beta$ DoF. The top $1-\beta$ levels, which cannot be zero-forced, contribute the remaining $1-\beta$ DoF.[^4]
Beyond DoF, which implicitly assume all channels are equally strong, by allowing $\alpha<1$, the GDoF setting allows us in this work to characterize the impact of different channel strengths (albeit restricted within assumptions of symmetry). Remarkably here we find that cross-channel strength parameters $\alpha$ and channel uncertainty parameters $\beta$ counter each other on equal terms, so that only their difference $(\alpha-\beta)$ matters. The sum GDoF value $(\alpha-\beta)+K(1-(\alpha-\beta))$ reflects essentially perfect CSIT over $1-(\alpha-\beta)$ dimensions which yield $K(1-(\alpha-\beta))$ GDoF through zero-forcing, while the remaining $(\alpha-\beta)$ dimensions cannot conceal interference and contribute only $(\alpha-\beta)$ GDoF.
Finally, regarding the regime $\alpha>1$ which is not addressed in Theorem \[Theorem3\], we believe that this regime includes new challenges, both in terms of achievability and outer bounds, which go beyond the insights available so far.
Proof of Theorem \[Theorem2\]
=============================
The most interesting aspect of the proof is the outer bound, for which we will generalize the Aligned Image Sets (AIS) argument of [@Arash_Jafar_GC14]. Since many of the details are repetitive we will focus primarily on the distinct aspects. Furthermore, we will present the proof only for the real setting here. Since the extension to complex settings follows along the lines of similar extensions in [@Arash_Jafar_GC14; @Arash_Jafar_GC15] it does not bear repeating.
Outer Bound
-----------
For notational convenience, let us define $$\begin{aligned}
\bar{P}&=&\sqrt{P}\end{aligned}$$ The first step in the AIS approach is the transformation into a deterministic setting such that a GDoF outer bound on the deterministic setting is also a GDoF outer bound on the original setting. Since the derivation of the deterministic setting is identical to [@Arash_Jafar_GC14], we directly present the deterministic model as follows.
### Deterministic Channel Model
The deterministic channel model has inputs $\bar{X}_i(t) \in\mathbb{Z}$ and outputs $\bar{Y}_i(t) \in\mathbb{Z},~ \forall t\in\mathbb{N}, i \in \{1,2\}$, such that $$\begin{aligned}
\bar{Y}_1(t)&=&\lceil \bar{P}^{\alpha_{11}-\max(\alpha_{11},\alpha_{12})} G_{11}(t)\bar{X}_1(t)\rceil+\lceil \bar{P}^{\alpha_{12}-\max(\alpha_{11},\alpha_{12})}G_{12}(t)\bar{X}_2(t)\rceil\\
\bar{Y}_2(t)&=&\lceil\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}G_{21}(t)\bar{X}_1(t)\rceil+\lceil \bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}G_{22}(t)\bar{X}_2(t)\rceil~~~~~~~\end{aligned}$$ and $\bar{X}_1(t)\in\{0,1,\cdots,\lceil \bar{P}^{\max(\alpha_{11},\alpha_{12})} \rceil\}$, $\bar{X}_2(t)\in\{0,1,\cdots,\lceil \bar{P}^{\max(\alpha_{21},\alpha_{22})} \rceil\}$.
### Functional Dependence and Aligned Image Sets
Following directly along the AIS approach [@Arash_Jafar_GC14], and omitting $o(\log(P))$ and $o(n)$ terms that are inconsequential for GDoF, we have $$\begin{aligned}
n(R_1+R_2)&\leq& H(\bar{Y}_1^{[n]}|W_2,G^{[n]})+H(\bar{Y}_2^{[n]}|G^{[n]})-H(\bar{Y}_2^{[n]}|G^{[n]},W_2)\\
&\leq& n\max(\alpha_{21},\alpha_{22})\log(\bar{P})+H(\bar{Y}_1^{[n]}|W_2,G^{[n]})-H(\bar{Y}_2^{[n]}|G^{[n]},W_2)\label{eq:fd1}$$ As in [@Arash_Jafar_GC14], for the outer bound there is no loss of generality in fixing $W_2$ as a constant, and assuming the following functional dependence $$\begin{aligned}
(\bar{X}_1^{[n]},\bar{X}_2^{[n]})&=&f_1(\bar{Y}_1^{[n]},G_{11}^{[n]},G_{12}^{[n]})\\
\Rightarrow\bar{Y}_2^{[n]}&=&f_2(\bar{Y}_1^{[n]},G^{[n]})\end{aligned}$$ The aligned image sets are defined as $$\begin{aligned}
S_{\nu^{[n]}}(G^{[n]})&=\{\bar{Y}_1^{[n]}\mbox{ s. t. } f_2(\bar{Y}_1^{[n]},G^{[n]})=f_2(\nu^{[n]},G^{[n]})\}\end{aligned}$$ i.e., $S_{\nu^{[n]}}(G^{[n]})$ is the set of distinct images — one of which is $\nu^{[n]}$ — cast at Receiver 1, which correspond to the same image at Receiver 2.
Following the AIS approach, the sum-rate bound in (\[eq:fd1\]) leads to the following bound expressed in terms of the expected cardinality of the aligned image sets. $$\begin{aligned}
n(R_1+R_2)&\leq& n\max(\alpha_{21},\alpha_{22})\log(\bar{P})+\log\mbox{E}\left|S_{\nu^{[n]}}(G^{[n]})\right|\label{eq:Fano}\end{aligned}$$
### Bounding the Probability that Images Align
Given $G_{11}^{[n]}, G_{12}^{[n]}$, consider two distinct realizations of User 1’s output sequence $\bar{Y}_1^{[n]}$, denoted as $\lambda^{[n]}$ and $\nu^{[n]}$, which are produced by the corresponding realizations of the codeword $(X_1^{[n]}, X_2^{[n]})$ denoted by $(\lambda_1^{[n]},\lambda_2^{[n]})$ and $(\nu_1^{[n]}, \nu_2^{[n]})$, respectively. $$\begin{aligned}
\lambda(t)&=&\lfloor \bar{P}^{\alpha_{11}-\max(\alpha_{11},\alpha_{12})} G_{11}(t)\lambda_1(t)\rfloor+\lfloor \bar{P}^{\alpha_{12}-\max(\alpha_{11},\alpha_{12})}G_{12}(t)\lambda_2(t)\rfloor\label{eq:lambdanu1}\\
\nu(t)&=&\lfloor \bar{P}^{\alpha_{11}-\max(\alpha_{11},\alpha_{12})}G_{11}(t)\nu_1(t)\rfloor
+\lfloor \bar{P}^{\alpha_{12}-\max(\alpha_{11},\alpha_{12})}G_{12}(t)\nu_2(t)\rfloor\label{eq:lambdanu2}~~~~~~~\end{aligned}$$ We wish to bound the probability that the images of these two codewords align at User 2, i.e., $\nu^{[n]}\in S_{\lambda^{[n]}}$. For simplicity, consider first the single channel use setting, $n=1$. For $\nu\in S_\lambda$ we must have, $$\begin{aligned}
&&\lfloor\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}G_{21}\nu_1\rfloor+\lfloor \bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}G_{22}\nu_2\rfloor\nonumber\\
&=&\lfloor \bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}G_{21}\lambda_1\rfloor+\lfloor \bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})} G_{22}\lambda_2\rfloor\label{xe}\end{aligned}$$ So for fixed value of $G_{22}$ the random variable $\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}G_{21}(\nu_1-\lambda_1)$ must take values within an interval of length no more than 4. If $\nu_1\neq\lambda_1$, then $G_{21}$ must take values in an interval of length no more than $\frac{4}{\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}|\nu_1-\lambda_1|}$, the probability of which is no more than $\frac{4f_{\max}{\bar{P}}^{\beta_{21}}}{\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}|\nu_1-\lambda_1|}$. Similarly, for fixed value of $G_{21}$ the random variable $\bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}G_{22}(\nu_2-\lambda_2)$ must take values within an interval of length no more than 4. If $\nu_1=\lambda_1$ then, because $\nu \neq \lambda$, we must have $\nu_2\neq\lambda_2$, and the probability of alignment is similarly bounded by $\frac{4f_{\max}{\bar{P}}^{\beta_{22}}}{\bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}|\nu_2-\lambda_2|}$. Thus, based on $(\ref{xe})$, either the probabilty of alignment is zero or we have, $$\begin{aligned}
\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}\Delta_1|\nu_1-\lambda_1|&\leq& \bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})} \Delta_2|\nu_2-\lambda_2|+2\\
\bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}\Delta_1|\nu_2-\lambda_2|&\leq& \bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}\Delta_2|\nu_1-\lambda_1|+2\end{aligned}$$ Next we will bound the max of $ \bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}|\nu_1-\lambda_1|$ and $ \bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}|\nu_2-\lambda_2|$. From (\[eq:lambdanu1\]) and (\[eq:lambdanu2\]) we have $$\begin{aligned}
|\lambda-\nu|&\leq&2+ \bar{P}^{\alpha_{11}-\max(\alpha_{11},\alpha_{12})}|G_{11}||\lambda_1-\nu_1|+ \bar{P}^{\alpha_{12}-\max(\alpha_{11},\alpha_{12})}|G_{12}||\lambda_2-\nu_2|~~~\\
&\leq&2+2\Delta_2 \max(\bar{P}^{\alpha_{11}-\max(\alpha_{11},\alpha_{12})}|\nu_1-\lambda_1|,\bar{P}^{\alpha_{12}-\max(\alpha_{11},\alpha_{12})} |\nu_2-\lambda_2|)~~~\\
&\leq&2+2\Delta_2 \max(\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{12})}|\nu_1-\lambda_1|,\bar{P}^{\alpha_{22}-\max(\alpha_{11},\alpha_{12})} |\nu_2-\lambda_2|)~~~\nonumber\\
&&\times \bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})}\end{aligned}$$ so, if $|\lambda-\nu|>\frac{4\Delta_2\bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})}}{\Delta_1}+2$, the probability of $\nu\in S_\lambda$ is no more than $$\begin{aligned}
&&\frac{4\Delta_2f_{\max}{\bar{P}}^{\beta_{22}}}{\bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}\Delta_2|\nu_2-\lambda_2|}\\
&\leq&\frac{4\Delta_2f_{\max}{\bar{P}}^{\beta_{22}}}{\max(\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}\Delta_1|\nu_1-\lambda_1|-2,\bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}\Delta_2|\nu_2-\lambda_2|)}~~~~~\\
&\leq&\frac{4\Delta_2f_{\max}{\bar{P}}^{\beta_{22}}}{\Delta_{1}\max(\bar{P}^{\alpha_{21}-\max(\alpha_{21},\alpha_{22})}|\nu_1-\lambda_1|,\bar{P}^{\alpha_{22}-\max(\alpha_{21},\alpha_{22})}|\nu_2-\lambda_2|)-2}~~~~~\\
&\leq&\frac{4\Delta_2f_{\max}{\bar{P}}^{\beta_{22}}}{\Delta_1\frac{|\lambda-\nu|-2}{2\Delta_2\bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})}}-2}\\
&\leq&\frac{8\frac{\Delta_2^2}{\Delta_1}f_{\max}{\bar{P}}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})+\beta_{22}}}{|\lambda-\nu|-\frac{4\Delta_2\bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})}}{\Delta_1}-2}\end{aligned}$$ Define $\Delta=\frac{4\Delta_2\bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})}}{\Delta_1}+2$, ($\Delta$ scales as $\bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})}$). Now let us return to the case of general $n$, where we similarly have, $$\begin{aligned}
\mathbb{P}(\lambda^{[n]}\in S_{\nu^{[n]}})&\leq&\prod_{t:|\lambda(t)-\nu(t)|\leq\Delta} 1\times \prod_{t:|\lambda(t)-\nu(t)|>\Delta}\frac{8\frac{\Delta_2^2}{\Delta_1}f_{\max}\bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})+\beta_{22}}}{|\lambda(t)-\nu(t)|-\Delta}\nonumber\end{aligned}$$
### Bounding the Expected Size of Aligned Image Sets.
$$\begin{aligned}
\mbox{E}(|S_{\nu^{[n]}}|)&=&\sum_{\lambda^n\in\{\bar{Y_1}^{[n]}\}}\mathbb{P}\left(\lambda^n\in S_{\nu^{[n]}}\right)\nonumber\\
&\leq&\prod_{t=1}^n\left(\sum_{\lambda(t):|\lambda(t)-\nu(t)|\leq\Delta}1+\sum_{\lambda(t):|\lambda(t)-\nu(t)|>\Delta}\frac{8\frac{\Delta_2^2}{\Delta_1}f_{\max}\bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})+\beta_{22}}}{|\lambda(t)-\nu(t)|-\Delta}\right)\nonumber\\
&\leq&\prod_{t=1}^n\left(2\Delta+1+{8\frac{\Delta_2^2}{\Delta_1}f_{\max}\bar{P}^{\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})+\beta_{22}}} \times 2(1+ \max(\alpha_{11},\alpha_{12})\log (1+2\Delta_2\bar{P}))\right)\nonumber\\
&\leq&(8\frac{\Delta_2^2}{\Delta_1}f_{\max})^n\bar{P}^{n(\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})+\beta_{22})^+}\times\left(\max(\alpha_{11},\alpha_{12})\log(\bar{P})+o(\log(\bar{P}))\right)^n\label{eq:aveSK}\nonumber\end{aligned}$$
### The GDoF Bound
Substituting back into (\[eq:Fano\]) we have $$\begin{aligned}
n(R_1+R_2)&\leq &n\max(\alpha_{21},\alpha_{22})\log(\bar{P})+\log\mbox{E}|S_{\nu^{[n]}}|\nonumber\\
&\leq &n\left(\max(\alpha_{21},\alpha_{22})+(\max(\alpha_{11}-\alpha_{21},\alpha_{12}-\alpha_{22})+\beta_{22})^+\right)\log(\bar{P})\nonumber\end{aligned}$$ So that we obtain the GDoF bound $$\begin{aligned}
d_1+d_2&\leq&\max(\alpha_{21},\alpha_{22})+\max(\alpha_{11}-\alpha_{21}+\beta_{22},\alpha_{12}-\alpha_{22}+\beta_{22},0)\end{aligned}$$ By symmetry we also have the GDoF bounds, $$\begin{aligned}
d_1+d_2&\leq&\max(\alpha_{21},\alpha_{22})+\max(\alpha_{11}-\alpha_{21}+\beta_{21},\alpha_{12}-\alpha_{22}+\beta_{21},0)\\
d_1+d_2&\leq&\max(\alpha_{11},\alpha_{12})+\max(\alpha_{21}-\alpha_{11}+\beta_{11},\alpha_{22}-\alpha_{12}+\beta_{11},0)\\
d_1+d_2&\leq&\max(\alpha_{11},\alpha_{12})+\max(\alpha_{21}-\alpha_{11}+\beta_{12},\alpha_{22}-\alpha_{12}+\beta_{12},0)\end{aligned}$$ Note that $\min(M+\max(A,0),M+\max(B,0))=M+\max(\min(A,B),0)$, so, together these bounds give us $d_1+d_2\leq \min(D_1,D_2)$, completing the proof of the outer bound for Theorem \[Theorem2\].
Achievability
-------------
Since the GDoF depend only on the worst channel uncertainty of each receiver, i.e., $\min(\beta_{11},\beta_{12})$ for receiver $1$ or $\min(\beta_{21},\beta_{22})$ for receiver $2$, for the achievability proof, we can assume without loss of generality that $\beta_{11},\beta_{12}$ are equal to $\beta_1$, and, $\beta_{21},\beta_{22}$ are equal to $\beta_2$. With this assumption we will prove that $\min(D_1,D_2)$ is achievable. Without loss of generality, we ignore measure zero events such as channel rank-deficiencies. This is because the channels are generated according to bounded densities, so that the probability mass that can be placed in a space whose measure approaches zero, must also approach zero.
Without loss of generality assume $\alpha_{11}$ as the maximum of $\alpha_{ij}$, $\forall i,j \in \{1,2\}$. The achievability proof is presented separately for the three cases of $(\alpha_{21}>\alpha_{22}, \alpha_{11}-\alpha_{12}>\alpha_{21}-\alpha_{22})$; $(\alpha_{21}>\alpha_{22}, \alpha_{11}-\alpha_{12}\leq \alpha_{21}-\alpha_{22})$; and $(\alpha_{21}\leq\alpha_{22})$.
1. [$\alpha_{21}>\alpha_{22}, \alpha_{11}-\alpha_{12}>\alpha_{21}-\alpha_{22}$.]{}\
We wish to achieve the sum-DoF value of $d_1 + d_2 =\min(\alpha_{11}+(\alpha_{22}-\alpha_{12}+\beta_1)^+,\alpha_{11}+\beta_2)$. As $ \alpha_{11}-\alpha_{12}>\alpha_{21}-\alpha_{22}$, the first antenna can transmit $\alpha_{21}-\alpha_{22}$ DoF using highest power level as it will be decoded at the receivers without any interference from the second antenna. So, decreasing both $\alpha_{11},\alpha_{21}$ by $\alpha_{21}-\alpha_{22}$, we have a new channel with channel coefficients $\alpha'_{11}=\alpha_{11}-\alpha_{21}+\alpha_{22},\alpha'_{12}=\alpha_{12},\alpha'_{21}=\alpha_{22},\alpha'_{22}=\alpha_{22}$ where, we need to achieve the sum-DoF value of $d_1 + d_2 =\min(\alpha'_{11}+(\alpha'_{21}-\alpha'_{12}+\beta_1)^+,\alpha'_{11}+\beta_2)$ through the tuple $d_1 = \alpha'_{11}, d_2 =m$, where $m=\min((\alpha'_{21}-\alpha'_{12}+\beta_1)^+,\beta_2)$. The case $m=0$ is obviously achieved, So, lets consider the case where $m>0$ i.e. $\alpha'_{21}-\alpha'_{12}+\beta_1>0$. To achieve $\alpha'_{11}+m$ DoF, let us split User 1’s message as $W_1 = (W_c,W_{1z},W_{1p})$ and User 2’s message as $W_{2} = W_{2z}$, where $W_{1z},W_{1p}$ act as private sub-messages to be decoded only by user 1, $W_{2z}$ acts as a private sub-message to be decoded only by User 2, while $W_c$ acts as a common submessage that can be decoded by both users. $W_c$, $W_{1z}$, $W_{2z}$ and $W_{1p}$ carry $\alpha'_{21}-m,m,m,\alpha'_{11}-\alpha'_{21}$ DoF respectively. Messages $W_c,W_{1z},W_{2z},W_{1p}$ are encoded into independent Gaussian codebooks $X_c,X_{1z},X_{2z},X_{1p}$, with unit powers, producing the transmitted symbols as follows. $$\begin{aligned}
\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]=&~~c_o{\bf V}_cX_c+c_o\sqrt{P^{-\alpha'_{21}}}{\bf V}_{1p}X_{1p}+c_o\sqrt{P^{m-\alpha'_{21}}}{\bf V}_{1z} X_{1z}+c_o{\bf V}_{2z} X_{2z}\end{aligned}$$ Here ${\bf V}_{c}$, ${\bf V}_{1p}$, ${\bf V}_{1z}$, ${\bf V}_{2z}$ are vectors as follows $$\begin{aligned}
{\bf V}_{c}&=&\left[\begin{array}{c}
1\\
0
\end{array}
\right]\\
{\bf V}_{1p}&=&\left[\begin{array}{c}
1\\
0
\end{array}
\right]\\
{\bf V}_{1z}&=&\left[\begin{array}{c}
\hat{G}_{22}\\
-\hat{G}_{21}
\end{array}
\right]\\
{\bf V}_{2z}&=&\left[\begin{array}{c}
\hat{G}_{12}\sqrt{P^{m+\alpha'_{12}-\alpha'_{21}-\alpha'_{11}}}\\
-\hat{G}_{11}\sqrt{P^{m-\alpha'_{21}}}
\end{array}
\right]\end{aligned}$$ In words, ${\bf V}_{1z}$ is a unit vector orthogonal to the estimated channel vector of User $2$, and ${\bf V}_{2z}$ is a unit vector orthogonal to the estimated channel vector of User 1. Thus, $X_{1z},X_{2z}$ are zero-forced to the estimated channels of the undesired users. $c_o$ is a scaling factor, $O(1)$ in $P$, chosen to ensure that the transmit power constraint is satisfied. The signal seen at Receiver 1 is, $$\begin{aligned}
{Y}_1&=&\left[\begin{array}{cc}\bar{P}^{\alpha'_{11}}\hat{G}_{11}&\bar{P}^{\alpha'_{12}}\hat{G}_{12}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+
\left[\begin{array}{cc}\bar{P}^{\alpha'_{11}-\beta_1}\tilde{G}_{11}&\bar{P}^{\alpha'_{12}-\beta_1}\tilde{G}_{12}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+Z_1\nonumber\\
&=&\bar{P}^{\alpha'_{11}}c_1 X_c+\bar{P}^{\alpha'_{11}-\alpha'_{21}}c_2X_{1p}+\bar{P}^{\alpha'_{11}+m-\alpha'_{21}}c_3X_{1z}+\bar{P}^{m+\alpha'_{12}-\beta_1-\alpha'_{21}}c_4X_{2z}+Z_1\nonumber\end{aligned}$$ where the $c_i$ are non-zero and bounded, i.e., $O(1)$ functions of $P$. Note that, $m+\alpha'_{12}-\beta_1-\alpha'_{21}\leq 0$. User 1 first decodes $X_c$ while treating all other signals as white noise. This is possible because $X_c$ is received with power $\sim {P}^{\alpha'_{11}}$, the effective noise has power $\sim P^{\alpha'_{11}+m-\alpha'_{21}}$, and $X_c$ carries $\alpha'_{21}-m$ DoF. After decoding $X_c$, the receiver subtracts its contribution from its received signal and then proceeds to decode $X_{1z}$ while treating remaining signals as noise. This is possible since $X_{1z}$ is received with power $\sim {P}^{\alpha'_{11}+m-\alpha'_{21}}$, the effective noise has power $\sim {P}^{\alpha'_{11}-\alpha'_{21}}$, and $X_{1z}$ carries $m$ DoF. After decoding $X_{1z}$, the receiver subtracts its contribution from its received signal and then proceeds to decode $X_{1p}$ while treating remaining signals as noise. Since $X_{1p}$ is received with power $\sim {P}^{\alpha'_{11}-\alpha'_{21}}$, the remaining signals and noise are received with only $O(1)$ power, and $X_{1p}$ carries $\alpha'_{11}-\alpha'_{21}$ DoF, this decoding is successful as well. The signal seen at Receiver 2 is, $$\begin{aligned}
{Y}_2&=&\left[\begin{array}{cc}\bar{P}^{\alpha'_{21}}\hat{G}_{21}&\bar{P}^{\alpha'_{22}}\hat{G}_{22}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+
\left[\begin{array}{cc}\bar{P}^{\alpha'_{21}-\beta_2}\tilde{G}_{21}&\bar{P}^{\alpha'_{22}-\beta_2}\tilde{G}_{22}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+Z_2\nonumber\\
&=&\bar{P}^{\alpha'_{21}}e_1 X_c+e_2X_{1p}+\bar{P}^{m-\beta_2}e_3X_{1z}+\bar{P}^{m}e_4X_{2z}+Z_1\nonumber\end{aligned}$$ where the $e_i$ are non-zero and bounded, i.e., $O(1)$ functions of $P$. Note that, $m-\beta_2\leq 0$. User 1 first decodes $X_c$ while treating all other signals as white noise. This is possible because $X_c$ is received with power $\sim {P}^{\alpha'_{21}}$, the effective noise has power $\sim P^{m}$, and $X_c$ carries $\alpha'_{21}-m$ DoF. After decoding $X_c$, the receiver subtracts its contribution from its received signal, and then decodes $X_{2z}$. Since $X_{2z}$ is received with power $\sim {P}^{m}$, the remaining signals and noise are received with only $O(1)$ power, and as $X_{2z}$ carries $m$ DoF, this decoding is successful as well.
2. [$\alpha_{21}>\alpha_{22}, \alpha_{11}-\alpha_{12}\leq \alpha_{21}-\alpha_{22}$.]{}
We wish to achieve the sum-DoF value of $d_1 + d_2 =\min(\alpha_{11}+(\alpha_{21}-\alpha_{11}+\beta_1)^+,\alpha_{21}+\alpha_{12}-\alpha_{22}+\beta_2)$. Since $ \alpha_{11}-\alpha_{12}\leq\alpha_{21}-\alpha_{22}$, the first antenna can transmit $\alpha_{11}-\alpha_{12}$ DoF using its highest power levels and it will be decoded at the receivers without any interference from the second antenna. So, decreasing both $\alpha_{11},\alpha_{21}$ by $\alpha_{11}-\alpha_{12}$, we have a new channel with channel coefficients $\alpha'_{11}=\alpha_{12},\alpha'_{12}=\alpha_{12},\alpha'_{21}=\alpha_{21}-\alpha_{11}+\alpha_{12},\alpha'_{22}=\alpha_{22}$, where we need to achieve the sum-DoF value of $d_1 + d_2 =\min(\alpha'_{11}+(\alpha'_{21}-\alpha'_{11}+\beta_1)^+,\alpha'_{21}+\alpha'_{11}-\alpha'_{22}+\beta_2)$ through the tuple $d_1 = \alpha'_{11}, d_2 =m$, where $m=\min((\alpha'_{21}-\alpha'_{11}+\beta_1)^+,\alpha'_{21}-\alpha'_{22}+\beta_2)$. Note that the case $m=0$ is obviously achievable, so lets consider the case where $m>0$. To achieve $\alpha'_{11}+m$ DoF, similar to first case, let us split User 1’s message as $W_1 = (W_c,W_{1z},W_{1p})$ and User 2’s message as $W_{2} = W_{2z}$, where $W_{1z},W_{1p}$ act as private sub-messages to be decoded only by user 1, $W_{2z}$ acts as a private sub-message to be decoded only by User 2, while $W_c$ acts as a common submessage that can be decoded by both users. $W_c$, $W_{1z}$, $W_{2z}$ and $W_{1p}$ carry $\alpha'_{21}-m,m,m,\alpha'_{11}-\alpha'_{21}$ DoF respectively. Messages $W_c,W_{1z},W_{2z},W_{1p}$ are encoded into independent Gaussian codebooks $X_c,X_{1z},X_{2z},X_{1p}$, with unit powers, producing the transmitted symbols as follows. $$\begin{aligned}
\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]=&~~c_o{\bf V}_cX_c+c_o\sqrt{P^{-\alpha'_{21}}}{\bf V}_{1p}X_{1p}+c_o{\bf V}_{1z} X_{1z}+c_o\sqrt{P^{m-\alpha'_{21}}}{\bf V}_{2z} X_{2z}\end{aligned}$$ Here ${\bf V}_{c}$, ${\bf V}_{1p}$, ${\bf V}_{2p}$ are vectors as follows $$\begin{aligned}
{\bf V}_{c}&=&\left[\begin{array}{c}
1\\
0
\end{array}
\right]\\
{\bf V}_{1p}&=&\left[\begin{array}{c}
1\\
0
\end{array}
\right]\\
{\bf V}_{1z}&=&\left[\begin{array}{c}
\hat{G}_{22}\sqrt{P^{m+\alpha'_{22}-2\alpha'_{21}}}\\
-\hat{G}_{21}\sqrt{P^{m-\alpha'_{21}}}
\end{array}
\right]\\
{\bf V}_{2z}&=&\left[\begin{array}{c}
\hat{G}_{12}\\
-\hat{G}_{11}
\end{array}
\right]\end{aligned}$$ Thus, ${\bf V}_{1z}$ is a unit vector orthogonal to the estimated channel vector of User $2$, and ${\bf V}_{2z}$ is a unit vector orthogonal to the estimated channel vector of User 1. The private messages carried by symbols $X_{1z},X_{2z}$ are zero-forced to the estimated channels of the undesired users, whereas the common message is heard by both users. $c_o$ is a scaling factor, $O(1)$ in $P$, chosen to ensure that the transmit power constraint is satisfied. The signal seen at Receiver 1 is, $$\begin{aligned}
{Y}_1&=&\left[\begin{array}{cc}\bar{P}^{\alpha'_{11}}\hat{G}_{11}&\bar{P}^{\alpha'_{12}}\hat{G}_{12}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+
\left[\begin{array}{cc}\bar{P}^{\alpha'_{11}-\beta_1}\tilde{G}_{11}&\bar{P}^{\alpha'_{12}-\beta_1}\tilde{G}_{12}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+Z_1\nonumber\\
&=&\bar{P}^{\alpha'_{11}}c_1 X_c+\bar{P}^{\alpha'_{11}-\alpha'_{21}}c_2X_{1p}+\bar{P}^{\alpha'_{11}+m-\alpha'_{21}}c_3X_{1z}+\bar{P}^{m-\alpha'_{21}-\beta_1+\alpha'_{11}}c_4X_{2z}+Z_1\nonumber\end{aligned}$$ where the $c_i$ are non-zero and bounded, i.e., $O(1)$ functions of $P$. Note that, $m-\alpha'_{21}-\beta_1+\alpha'_{11}\leq 0$. Similar to the first case, with the similar approach and similar SINR values, Receiver 1 can decode $X_c,X_{1z},X_{1p}$ successfully. The signal seen at Receiver 2 is, $$\begin{aligned}
{Y}_2&=&\left[\begin{array}{cc}\bar{P}^{\alpha'_{21}}\hat{G}_{21}&\bar{P}^{\alpha'_{22}}\hat{G}_{22}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+
\left[\begin{array}{cc}\bar{P}^{\alpha'_{21}-\beta_2}\tilde{G}_{21}&\bar{P}^{\alpha'_{22}-\beta_2}\tilde{G}_{22}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+Z_2\nonumber\\
&=&\bar{P}^{\alpha'_{21}}e_1 X_c+e_2X_{1p}+\bar{P}^{m-\alpha_{21}+\alpha_{22}-\beta_2}e_3X_{1z}+\bar{P}^{m}e_4X_{2z}+Z_1\nonumber\end{aligned}$$ where the $e_i$ are non-zero and bounded, i.e., $O(1)$ functions of $P$. Note that, $m-\alpha_{21}+\alpha_{22}-\beta_2\leq 0$. Similar to the first case, with similar SINR values, Receiver 2 can decode $X_c,X_{2z}$ successfully.
3. [$\alpha_{21}\leq\alpha_{22}$.]{}\
We wish to achieve the sum-DoF value of $d_1 + d_2 =\min(\alpha_{11}+(\alpha_{22}-\alpha_{12}+\beta_1)^+,\alpha_{22}+\alpha_{11}-\alpha_{21}+\beta_2)$ through the tuple $d_1 = \alpha_{11}, d_2 =m$, where $m=\min((\alpha_{22}-\alpha_{12}+\beta_1)^+,\alpha_{22}-\alpha_{21}+\beta_2)$. The case $m=0$ is obviously achievable, so lets consider the case where $m>0$. Note that $m<\alpha_{22}$. To achieve $\alpha_{11}+m$ DoF, similar to first case, let us split User 1’s message as $W_1 = (W_c,W_{1z},W_{1p})$ and User 2’s message as $W_{2} = W_{2z}$, where $W_{1z},W_{1p}$ act as private sub-messages to be decoded only by user 1, $W_{2z}$ acts as a private sub-message to be decoded only by User 2, while $W_c$ acts as a common submessage that can be decoded by both users. $W_c$, $W_{1z}$, $W_{2z}$ and $W_{1p}$ carry $\alpha_{22}-m,m,m$ and $\alpha_{11}-\alpha_{22}$ DoF, respectively. Messages $W_c,W_{1z},W_{2z},W_{1p}$ are encoded into independent Gaussian codebooks $X_c,X_{1z},X_{2z},X_{1p}$, with unit powers, producing the transmitted symbols as follows. $$\begin{aligned}
\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]=&~~c_o{\bf V}_cX_c+c_o\sqrt{P^{-\alpha_{22}}}{\bf V}_{1p}X_{1p}+c_o{\bf V}_{1z} X_{1z}+c_o{\bf V}_{2z} X_{2z}\end{aligned}$$ Here ${\bf V}_{c}$, ${\bf V}_{1p}$, ${\bf V}_{2p}$ are vectors as follows $$\begin{aligned}
{\bf V}_{c}&=&\left[\begin{array}{c}
1\\
1
\end{array}
\right]\\
{\bf V}_{1p}&=&\left[\begin{array}{c}
1\\
1
\end{array}
\right]\\
{\bf V}_{1z}&=&\left[\begin{array}{c}
\hat{G}_{22}\sqrt{P^{m-\alpha_{22}}}\\
-\hat{G}_{21}\sqrt{P^{m+\alpha_{21}-2\alpha_{22}}}
\end{array}
\right]\\
{\bf V}_{2z}&=&\left[\begin{array}{c}
\hat{G}_{12}\sqrt{P^{m-\alpha_{11}+\alpha_{12}-\alpha_{22}}}\\
-\hat{G}_{11}\sqrt{P^{m-\alpha_{22}}}
\end{array}
\right]\end{aligned}$$ So ${\bf V}_{1z}$ is a unit vector orthogonal to the estimated channel vector of User $2$, and ${\bf V}_{2z}$ is a unit vector orthogonal to the estimated channel vector of User 1. The private messages $X_{1z},X_{2z}$ are zero-forced to the estimated channels of the undesired users, whereas the common message is heard by both users. $c_o$ is a scaling factor, $O(1)$ in $P$, chosen to ensure that the transmit power constraint is satisfied. The signal seen at Receiver 1 is, $$\begin{aligned}
{Y}_1&=&\left[\begin{array}{cc}\bar{P}^{\alpha_{11}}\hat{G}_{11}&\bar{P}^{\alpha_{12}}\hat{G}_{12}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+
\left[\begin{array}{cc}\bar{P}^{\alpha_{11}-\beta_1}\tilde{G}_{11}&\bar{P}^{\alpha_{12}-\beta_1}\tilde{G}_{12}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+Z_1\nonumber\\
&=&\bar{P}^{\alpha_{11}}c_1 X_c+\bar{P}^{\alpha_{11}-\alpha_{22}}c_2X_{1p}+\bar{P}^{\alpha_{11}+m-\alpha_{22}}c_3X_{1z}+\bar{P}^{m-\alpha_{22}-\beta_1+\alpha_{12}}c_4X_{2z}+Z_1\nonumber\end{aligned}$$ where the $c_i$ are non-zero and bounded, i.e., $O(1)$ functions of $P$. Note that, $m-\alpha_{22}-\beta_1+\alpha_{12}\leq 0$. Similar to the first case, with SINR values of $P^{\alpha_{22}-m}$, $P^{m}$, and $P^{\alpha_{11}-\alpha_{22}}$, Receiver 1 can decode $X_c,X_{1z},X_{1p}$ respectively. The signal seen at Receiver 2 is, $$\begin{aligned}
{Y}_2&=&\left[\begin{array}{cc}\bar{P}^{\alpha_{21}}\hat{G}_{21}&\bar{P}^{\alpha_{22}}\hat{G}_{22}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+
\left[\begin{array}{cc}\bar{P}^{\alpha_{21}-\beta_2}\tilde{G}_{21}&\bar{P}^{\alpha_{22}-\beta_2}\tilde{G}_{22}\end{array}\right]\left[\begin{array}{c}
X_1\\
X_2
\end{array}
\right]+Z_2\nonumber\\
&=&\bar{P}^{\alpha_{22}}e_1 X_c+e_2X_{1p}+\bar{P}^{m+\alpha_{21}-\alpha_{22}-\beta_2}e_3X_{1z}+\bar{P}^{m}e_4X_{2z}+Z_1\nonumber\end{aligned}$$ where the $e_i$ are non-zero and bounded, i.e., $O(1)$ functions of $P$. Note that, $m+\alpha_{21}-\alpha_{22}-\beta_2\leq 0$. Similar to the first case, with the similar approach and similar SINR value, Receiver 2 can decode $X_c,X_{2z}$ successfully.
Proof of Theorem \[Theorem3\]
=============================
Outer Bound
-----------
The generalization of the proof to the $K$ user setting requires only a few extra steps for initial set up before the problem decomposes into the equivalent of what has been shown for the $K=2$ case. Here we describe the additional setup steps. Starting with the deterministic model, for the $k^{th}$ user we bound the rate as $$\begin{aligned}
nR_k&\leq&I(W_k; \bar{Y}_k^{[n]}|G^n,W_{k+1}, W_{k+2}, \cdots, W_K)+o(n)\\
&\leq& H(\bar{Y}_k^{[n]}|G^n,W_{k+1}, \cdots, W_K)-H(\bar{Y}_k^{[n]}|G^n,W_k, W_{k+1}, \cdots, W_K)+o(n)\end{aligned}$$ where $G^n$ includes all channel realizations. Adding the rate bounds we obtain $$\begin{aligned}
n\sum_{k=1}^KR_k&\leq&n\log(\bar{P})+\sum_{k=2}^K\left(H(\bar{Y}_{k-1}^{[n]}|G^n,W_{k},\cdots, W_K)-H(\bar{Y}_k^{[n]}|G^n,W_k, \cdots, W_K)\right)\nonumber\end{aligned}$$ From this point on, the process of bounding the difference of entropy terms follows the proof of Theorem \[Theorem2\], so that we arrive at the bound $$\begin{aligned}
n\sum_{k=1}^KR_k&\leq&n\log(\bar{P})+(1-\alpha+\beta)(K-1)n\log(\bar{P})\end{aligned}$$ which bounds the total DoF by $1+(K-1)(1-\alpha+\beta)=(\alpha-\beta)+K(1-(\alpha-\beta))$.
Achievability
-------------
Let us prove that the sum-GDoF value of $\sum_{k=1}^Kd_k = 1+(K-1)(1-\alpha+\beta)$ is achievable through the $K$-tuple $d_1 = 1, d_k =1-\alpha+\beta$ for $k=2,3,\cdots, K$. To do this, let us similarly split User 1’s message as $W_{1} = (W_c,W_{1p})$, where $W_{1p}$ acts as a private sub-message to be decoded only by User $1$, while $W_c$ acts as a common message that can be decoded by all users. The remaining messages $W_2, W_3, \cdots, W_K$ are all private, intended to be decoded only by their desired users. The common message $W_c$ carries $\alpha-\beta$ GDoF, whereas all private messages $W_{1p}, W_2, W_3, \cdots, W_K$ carry $1-\alpha+\beta$ GDoF each. Messages $W_c,W_{1p}, W_2, W_3, \cdots, W_K$ are encoded into unit power independent Gaussian codebooks $X_c,X_{1p}, X_{2p}, \cdots, X_{Kp}$, respectively.
The transmitted symbols are constructed as follows. $$\begin{aligned}
\left[\begin{array}{c}
X_1\\
\vdots\\
X_K
\end{array}
\right]&=a\sqrt{1-P^{\beta-\alpha}}{\bf V}_cX_c
+a\sqrt{P^{\beta-\alpha}}\sum_{k=1}^K{\bf V}'_{kp}X_{kp}
$$ Let $\hat{\bf G}$ be the $K\times K$ matrix whose $(k,l)^{th}$ term is defined as $$\begin{aligned}
\hat{\bf G}(k,l)&=&\left\{
\begin{array}{ll}
\hat{G}_{k,k},& k=l\\
\sqrt{P^{\alpha-1}}\hat{G}_{k,l},&k\neq l
\end{array}
\right.\end{aligned}$$ The ${\bf V}'_{kp}$ are unit vectors chosen so that $$\begin{aligned}
\hat{\bf G}
\left[\begin{array}{lll}{\bf V}'_{1p}&\cdots&{\bf V}'_{Kp}\end{array}\right]\end{aligned}$$ is a diagonal matrix. In other words, the $k^{th}$ private message is sent in a direction orthogonal to the estimated channel vector of every user except the $k^{th}$ user. As before, ${\bf V}_c$ is a generic vector and $a$ is a scaling factor that is $O(1)$ in $P$, chosen to ensure that the transmit power constraint is satisfied.
Let us take a closer look at the vectors ${\bf V}'_{kp}$. Consider, e.g., the product of the second row of $\hat{\bf G}$ and ${\bf V}'_{1p}$, scaled by $\sqrt{P^{1-\alpha}}$, $$\begin{aligned}
\hat{G}_{2,1}{\bf V}'_{1p}(1) + \sqrt{P^{1-\alpha}}\hat{G}_{2,2}{\bf V}'_{1p}(2)+\hat{G}_{2,3}{\bf V}'_{1p}(3)+\cdots+\hat{G}_{2,K}{\bf V}'_{1p}(K)&=&0.\end{aligned}$$ which implies that ${\bf V}'_{1p}(2)$ cannot be more than $O(\sqrt{P^{\alpha-1}})$ in $P$. Similarly, considering the product of the $m^{th}$ row of $\hat{\bf G}$ and ${\bf V}'_{1p}$, scaled by $\sqrt{P^{1-\alpha}}$, we note that ${\bf V}'_{1p}(m)$ cannot be more than $O(\sqrt{P^{\alpha-1}})$ in $P$, for $m\neq 1$. Proceeding similarly for the vector ${\bf V}'_{kp}$, considering the product of the $m^{th}$ row of $\hat{\bf G}$ and ${\bf V}'_{kp}$, scaled by $\sqrt{P^{1-\alpha}}$, we note that ${\bf V}'_{kp}(m)$ cannot be more than $O(\sqrt{P^{\alpha-1}})$ in $P$, for $m\neq k$. With this observation, we can define vectors ${\bf V}_{kp}$ whose elements are $O(1)$, such that $$\begin{aligned}
{\bf V}'_{kp}&={\bf M}_k{\bf V}_{kp}, ~~~~\forall k\in[K]\end{aligned}$$ and ${\bf M}_k$ is a $K\times K$ diagonal matrix with a $1$ as the $(k,k)^{th}$ element, all of whose remaining diagonal terms are equal to $\sqrt{P^{\alpha-1}}$.
The signal seen at Receiver 1 is, $$\begin{aligned}
{Y}_1&=&\sqrt{P}\left[\begin{array}{lll}\hat{G}_{11}&\cdots&\hat{G}_{1K}\end{array}\right]{\bf M}_1\left[\begin{array}{c}
X_1\\
\vdots\\
X_K
\end{array}
\right]+\sqrt{P^{1-\beta}}
\left[\begin{array}{lll}\tilde{G}_{11}&\cdots&\tilde{G}_{12}\end{array}\right]{\bf M}_1\left[\begin{array}{c}
X_1\\
\vdots\\
X_K
\end{array}
\right]+Z_1~~~~~\\
&=&\sqrt{P}a_o X_c+\sqrt{P^{1+\beta-\alpha}}a_1X_{1p}+\sum_{k=2}^Ka_kX_{kp}+Z_1\end{aligned}$$ where the $a_k$ are $O(1)$ in $P$.
User 1 first decodes $X_c$ while treating all other signals as white noise. This is possible because $X_c$ is received with power $\sim P$, the effective noise has power $\sim P^{1+\beta-\alpha}$, and $X_c$ carries $1-(1+\beta-\alpha)=\alpha-\beta$ GDoF. After decoding $X_c$, the receiver subtracts its contribution from its received signal and then proceeds to decode $X_{1p}$ while treating remaining signals as noise. Since $X_{1p}$ is received with power $\sim P^{1+\beta-\alpha}$, the remaining signals and noise are received with only $O(1)$ power, and $X_{1p}$ carries $1+\beta-\alpha$ DoF, this decoding is successful as well. Thus, User 1 achieves $\alpha-\beta+1-\beta+\alpha=1$ GDoF. All other users proceed similarly to achieve $1-\beta+\alpha$ DoF, so that the total GDoF achieved equal $1+(K-1)(1-\beta+\alpha)$.
Conclusion
==========
Because of the coarse and asymptotic character of DoF and GDoF metrics, even small gaps in our understanding of these coarse approximations can hide the most consequential ideas. Numerous discoveries around interference alignment emerged from efforts to find new achievable schemes to bridge the gap between the best inner and outer bounds. Following in the same spirit, this work bridges the extremes of known DoF results between perfect and finite precision CSIT. In the process, it expands our understanding of a relatively new idea – the aligned image sets (AIS) approach. Interference alignment and AIS can be seen as two sides of the same coin. In the pursuit of DoF and GDoF characterizations, just as interference alignment enables powerful achievable schemes to close the gap from below, the AIS approach enables powerful outer bounds to close the gap from above. Whether these ideas are enough to close the GDoF gaps for all channels and regimes of interest, if so then what new insights emerge from the new GDoF characterizations, and if not, then what new ideas hide in the remaining gaps, are exciting questions for the future.
[^1]: Channel state information at the receivers (CSIR) is assumed perfect throughout this work.
[^2]: In the DoF model, any non-zero channel is capable of carrying only 1 DoF regardless of its strength.
[^3]: For instance, consider the $K=3$ user MISO BC where we have $3$ antennas at the transmitter and $\alpha_{ij}=1$ for all $i,j\in\{1,2,3\}$. Suppose there is no CSIT for all the channel coefficients associated with the first transmit antenna ($\beta_{11}=\beta_{21}=\beta_{31}=0$) and perfect CSIT ($\beta_{ij}=\alpha_{ij}$ for all $i\in\{1,2,3\}, j\in\{2,3\}$) for the rest. If the sum GDoF were limited by the worst case, then this setting would be equivalent to the case where all $\beta_{ij}=0$, i.e., $\mathcal{D}_\Sigma=1$ according to [@Arash_Jafar_GC14]. But we know that $2$ DoF are achievable simply by ignoring the first antenna and the first user, reducing it to a $2$ user MISO BC with perfect CSIT.
[^4]: The achievability argument extends naturally to other settings. For example, it similarly follows that in the corresponding $K$ user interference channel the DoF value of $1-\beta+\frac{K}{2}\beta$ is achievable.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Several general properties, concerning reduction algebras – rings of definition and algorithmic efficiency of the set of ordering relations – are discussed. For the reduction algebras, related to the diagonal embedding of the Lie algebra $\gl_n$ into $\gl_n\oplus\gl_n$, we establish a stabilization phenomenon and list the complete sets of defining relations.'
---
[**Diagonal reduction algebras of $\gl$ type**]{}
\
$^\circ$[*Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia*]{}\
$^\star$[*Centre de Physique Théorique[^1], Luminy, 13288 Marseille, France*]{}
Introduction
============
Reduction algebras were introduced [@AST2; @M] for a study of representations of a Lie algebra with the help of the restriction to a subalgebra.
Let $\g$ be a Lie algebra, $\f\subset \g$ its reductive Lie subalgebra; that is, the adjoint action of $\k$ on $\g$ is completely reducible (in particular, $\k$ is reductive). Suppose $\k$ is given with a triangular decomposition \[intro1\] =++ .Denote by $\I_+$ the left ideal of $\Ar :=\U(\g)$ generated by elements of $\nplus$, $\I_+ :=\Ar\nplus$. Then the reduction algebra $\S(\g,\k)$, related to the pair $(\g,\k)$, is defined as the quotient $\mathrm{Norm}(\I_+)/\I_+$ of the normalizer of the ideal $\I_+$ over $\I_+$ (one should keep in mind that the notation $\S(\g,\k)$ is abbreviated: the data needed for the definiton of the reduction algebra includes, in addition to the pair $(\g,\k)$, the triangular decomposition ). The space $\S(\g,\k)$ is equipped with a natural structure of the associative algebra. By definition, for any $\g$-module $V$ the space $V^{\nplus}$ of vectors, annihilated by $\nplus$, is a module over $\S(\g,\k)$. If $V$ decomposes, as an $\f$-module, into a direct sum of irreducible $\k$-modules $V_i$ with finite- dimensional multiplicities, then the $\g$-module structure on $V$ can be uniquely restored from the $\S(\g,\k)$-module structure on $V^{\nplus}$.
The reduction algebra simplifies after the localization over the multiplicative set generated by elements $h_\gamma+k$, where $\gamma$ ranges through the set of roots of $\k$, $k\in{{\mathbb Z}}$; here $h_\gamma$ is the coroot corresponding to $\gamma$. Let $\Uh$ be the localization of the universal enveloping algebra $\U(\h)$ of the Cartan sub-algebra $\h$ of $\k$ over the above multiplicative set. The localized reduction algebra $\Z(\g,\k)$ is an algebra over the commutative ring $\Uh$; the principal part of the defining relations is quadratic but the relations may contain linear terms or degree 0 terms, see [@Zh; @KO]. Besides, the reduction algebra admits another description as a (localized) double coset space $\Ar/(\Ar \nplus +\nminus\Ar)$ endowed with the multiplication map defined with the help of the extremal projector [@KO] of Asherova–Smirnov–Tolstoy [@AST].
The general theory of reduction algebras [@Zh] provides the set of quadratic-linear-constant ordering relations over $\UUh$, the field of fractions of $\U (\h )$, for natural generators of any reduction algebra $\Z(\g,\f)$. However there are two subtle questions concerning these relations. The first question is: are these ordering relations defined over the smaller ring $\Uh$? Secondly, is it possible to use these ordering relations for an algorithmic ordering of polynomial expressions in the reduction algebra? In the first part of the paper we give affirmative answers to these questions for any reduction algebra $\Zgk$.
The main theme of the second part of the paper is the special restriction problem, when $\g$ is the direct sum of two copies of the Lie algebra ${\gl_n}$ and $\f$ is the diagonally embedded ${\gl_n}$. The resulting reduction algebra we call [*diagonal reduction*]{} algebra of $\gl_n$ and denote by $\Z_n$. A finite-dimensional irreducible module over $\g=\gl_n\oplus\gl_n$ is the tensor product of two irreducible $\gl_n$-modules; restricting the $\g$-module to $\k$ we obtain the decomposition of the tensor product into the direct sum of irreducible $\gl_n$-modules. One of the main results of the paper is the explicit description of the diagonal reduction algebra $\Z_n$. Some examples and applications of the diagonal reduction algebras are given in [@KO3].
We present a list of defining relations for natural generators of $\Z_n$. The derivation of these relations uses heavily the Zhelobenko automorphisms [@KO] of reduction algebras and is given in the work [@KO2]. In the present paper we formulate and prove the stabilization property of the algebras $\Z_n$. The stabilization phenomenon provides a natural way of extending relations for $\Z_{n}$ to relations for $\Z_{n+1}$ ($\Z_{n}$ is not a subalgebra of $\Z_{n+1}$). The stabilization principle is the second essential ingredient for the derivation of the set of defining relations.
We also prove that our list of defining relations is equivalent over $\Uh$ to the list of the canonical ordering relations. The proof is not difficult once we treat the algebras over $\UUh$: the arguments for the equivalence are based on certain asymptotic considerations. The proof of the equivalence over $\Uh$ is more delicate, it uses the stabilization phenomenon and calculations of certain determinants of Cauchy type.
Reduction algebras related to a reductive pair {#section2new}
==============================================
Let $\g$ be a finite-dimensional Lie algebra and $\k\subset\g$ its reductive subalgebra. Assume that the embedding $\k\subset\g$ is also reductive, that is the adjoint action of $\k$ in $\g$ is semi-simple. Let $\p$ be an $\ad_\k$-invariant complement of $\k$ in $\g$. Choose a triangular decomposition of Lie algebra $\k$; here $\h$ is a Cartan subalgebra of $\k$ while $\nplus$ and $\nminus$ are nilradicals of two opposite Borel subalgebras $\bb_\pm\subset\k$. Let $\De\in\h^*$ be the root system of $\f\ts$. The subsets of $\De$ consisting of the positive and negative roots will be denoted by $\De_+$ and $\De_-$ respectively. Let $\Q$ be the root lattice, $\Q:=\{\gamma\in\h^*\,|\,\gamma=\sum_{\alpha\in\Delta_+, n_\a\in{{\mathbb Z}}}n_\a \a\}$. It contains the positive cone $\Q_+$, \_+:={\^\*|=\_[\_+, n\_[[Z]{}]{}, n\_0]{}n\_} .\[cqp\]For $\lambda,\mu\in\h^*$, the notation >\[paor\]means that the difference $\lambda-\mu$ belongs to $\Q_+$, $\lambda-\mu\in\Q_+$. This is a partial order in $\h^*$.
Let $\Sym$ be the Weyl group of the root system $\De\ts$. Let $\si_1\lcd\si_r\in\Sym$ be the reflections in $\h^*$ corresponding to the simple roots $\al_1\lcd\al_r\ts$. We also use the induced action of the Weyl group $\Sym$ on the vector space $\h\ts$. It is defined by setting $\la\ts(\si(H))=\si^{-1}(\la)(H)$ for all $\si\in\Sym$, $H\in\h$ and $\la\in\h^*$. We assume that this action is extended to the action of a cover of the group $\Sym$ by automorphisms of the Lie algebra $\g$. In other words, there are automorphisms $\sy_i:\g\to\g$ which satisfy the same braid group relations as $\si_i$, preserve the subspaces $\h$ and $\k$, and coincide with $\si_i$ being restricted to $\h$. We denote by the same symbols the canonical extensions of $\sy_i$ to automorphisms of $\U(\g)$.
Let $\rho$ be the half-sum of the positive roots of $\k$. Then the *shifted action* $\circ$ of the group $\Sym$ on the vector space $\h^*$ is defined by setting \[shifted\] =(+)-. With the help of we induce the action $\circ$ of $\Sym$ on the commutative algebra $\U(\h)\ts$ by regarding the elements of this algebra as polynomial functions on $\h^\ast$. In particular, then $(\si\circ H)(\la)=H(\si^{-1}\circ\lambda)$ for $H\in\h\ts$.
For each $i=1\lcd r$ let $h_{\a_i}=\al_i^\vee\in\h$ be the coroot vector corresponding to the simple root $\al_i\ts$, so that the value $\al_j\ts(H_i)$ equals the $(i\com j)$ entry $\kar_{ij}$ of the Cartan matrix $\kar\ts$ of $\k$. Here $h_{\a_i}$ belongs to the semi-simple part of $\f\ts$. Let $e_{\a_i}\in\nplus$ and $e_{-\a_i}\in\nminus$ be the Chevalley generators of that subalgebra corresponding to the roots $\al_i$ and $-\al_i\ts$ so that $$[e_{\a_i}\com e_{-\a_j}]=\de_{ij}\,h_{\al_i}\,, \quad [h_{\al_i},e_{\al_j}]=
\kar_{ij}\,e_{\al_j}\,,
\quad [h_{\al_i},e_{-\al_j}]=-\kar_{ij}\,e_{-\al_j}\, .$$ For each $\al\in\De$ let $h_\al=\al^\vee\in\h$ be the corresponding coroot vector. Denote by $\Uh$ the ring of fractions of the commutative algebra $\U(\h)$ relative to the set of denominators $$\label{M2} \{\,h_\al+l\ |\ \al\in\Delta\ts,\ l\in{{\mathbb Z}}\,\ts\}\,.$$ The elements of this ring can also be regarded as rational functions on the vector space $\h^\ast\ts$. The elements of $\U(\h)\subset\,\overline{\!\U(\h)\!\!\!}\,\,\,$ are then regarded as polynomial functions on $\h^\ast\ts$. Let $\overline{\!\U(\k)\!\!\!}\,\,\,\subset \Ab= \overline{\!\U(\g)\!\!\!}\,\,\,$ be the rings of fractions of the algebras $\U(\k)$ and $\Ar=\U(\g)$ relative to the set of denominators . These rings are well defined, because both $\U(\f)$ and $\U(\g)$ satisfy the Ore condition relative to . Since $\si_i$ preserve the set of denominators , the automorphisms $\sy_i$ admit a natural extension to $\Ab$.
Define $\Z(\g,\k)$ to be the double coset space of $\Ab$ by its left ideal $\Ib:=\Ab\nplus$, generated by elements of $\nplus$, and the right ideal $\Jb:=\nminus\Ab$, generated by elements of $\nminus$, $\Z(\g,\k):=\Ab/(\Ib+\Jb)$. The space $\Z(\g,\k)$ is an associative algebra with respect to the multiplication map $$\label{not5a}a\mult b:=aP b\ .$$ Here $P$ is the extremal projector [@AST] of the Lie algebra $\k$ corresponding to the triangular decomposition . We call $\Z(\g,\k)$ the [*reduction*]{} algebra associated to the pair $(\g,\k)$. The assignment $\,x\mapsto x\mod \Ib+\Jb\,$ establishes an injective homomorphism of the algebra $\S(\g,\k)$ (see Introduction for the definition) to $\Zgk$, see [@KO]. Moreover, the localization of the image of $\S(\g,\k)$ with respect to $\Uh$ coincides with $\Zgk$.
The algebra $\Z(\g,\k)$ can be equipped with the action of Zhelobenko automorphisms [@KO]. Denote by $\q_{i}$ the Zhelobenko automorphism $\q_i:\Zgk\to\Zgk$ corresponding to the simple root $\alpha_i$, $i=1\lcd r$. It is defined as follows [@KO]. First we define a map $\q_i:\Ar\to \Ab/\Ib$ by $$\label{not7}\q_i(x):=\sum_{k\geq 0}\frac{(-1)^k}{k!}\hat{e}_{\a_i}^k(\sy_i(x))
e_{-\a_i}^k\
\ds\prod_{j=1}^k(h_{\a_i}-j+1)^{-1}
\quad \mod\Ib\ .$$ Here $\hat{x}$ stands for the adjoint action of the element $x$, so that $\hat{x}(y)=xy-yx$ for $x\in\k$ and $y\in\Ab$. The operator $\q_i$ has the property \[not2\]\_i(hx)=(\_ih)\_i(x)for any $x\in\Ar$ and $h\in\h$; $\si\circ h$ is defined in . With the help of (\[not2\]), the map $\q_i$ can be extended to the map (denoted by the same symbol) $\q_i:\Ab\to \Ab/\Ib$ by the setting $\q_i(\phi x)=(\si_i\circ \phi )\q_i(x)$ for any $x\in\Ar$ and $\phi\in\Uh$. One can further prove that $\q_i(\Ib)=0$ and $\q_i(\Jb)\subset (\Jb+\Ib)/\Ib$, so that $\q_i$ can be viewed as a linear operator $\q_i:\Zgk\to\Zgk$. Due to [@KO], this is an algebra automorphism, satisfying . The operators $\q_i$ satisfy the same braid group relations as $\si_i$ and the inversion relation [@KO]: $$\label{invr}\q_i^2(x)=(h_{\a_i}+1)^{-1}\ \sy_i^2(x)\
(h_{\a_i}+1)\ ,\qquad x\in\Zgk\ .$$
Let $\p$ be an $\ad_\k$-invariant complement of $\k$ in $\g$, as above. Choose a linear basis $\{p_K\}$ of $\p$ and equip it with a total order $\prec$. For an arbitrary element $a\in\Ab$ let $\widetilde{a}$ be its image in the reduction algebra; in particular, $\pp_K$ is the image in $\Zgk$ of the basic vector $p_K\in\p$.
.2cm The general theory of reduction algebras, see [@Zh] for the statements (a)-(c), says:
- Since $\h$ normalizes both $\nplus$ and $\nminus$, the algebra $\Zgk$ is a $\Uh$-bimodule with respect to the multiplication by elements of $\Uh$. It is free as a left $\Uh$-module and as a right $\Uh$-module. As a generating (over $\Uh$) subspace one can take a projection of the space $\mathrm{S}(\p)$ of symmetric tensors on $\p$ to $\Zgk$, that is a subspace of $\Zgk$, formed by linear combinations of images of the powers $p^\nu$, where $p\in\p$ and $\nu\geq 0$.
- Assignments $\deg (\widetilde{\hspace{.05cm}X\hspace{.05cm}})=l$ for the image of any product of $l$ elements from $\p$, $X=p_{K_1}p_{K_2}\cdots p_{K_l}$, and $\deg (Y)=0$ for any $Y\in\Uh$ define the structure of a filtered algebra on $\Zgk$. The subspace $\Zgk^{(k)}$ of elements of degree not greater than $k$ is a free left $\Uh$-module and a free right $\Uh$-module, with a generating subspace formed by linear combinations of images of the powers $p^\nu$, where $p\in\p$ and $k\geq\nu\geq 0$.
- In the sequel we will choose for $\{p_K\}$ a weight ordered basis; that is, each $p_K$ has a certain weight $\mu_K$, =\_K(h)p\_K\[weir\]for all $h\in\h$. The total order $\prec$ will be compatible with the partial order $<$ on $\h^*$, see , in the sense that $\mu_K<\mu_L\ \Rightarrow p_K\prec p_L\ .$ Then the images $\pp_{\bar{L}}$ of the monomials ($\bar{L}$ is understood as the multiindex) p\_[|[L]{}]{}:=p\_[L\_1]{}\^[n\_1]{}p\_[L\_2]{}\^[n\_2]{}p\_[L\_m]{}\^[n\_m]{},p\_[L\_1]{}p\_[L\_2]{}… p\_[L\_m]{},k=n\_1+…+n\_m ,\[inimp\]in $\Zgk^{(k)}$ are linearly independent over $\Uh$ and their projections to the quotient $\Zgk^{(k)}/ \Zgk^{(k-1)}$ form a basis of the left $\Uh$-module $\Zgk^{(k)}/ \Zgk^{(k-1)}$. The structure constants of the algebra $\Zgk$ in the basis $\{\pp_{\bar{L}}\}$ belong to the ring $\Uh$.
Choosing the PBW basis of $\Ar$ induced by any ordered basis of $\k +\p$, which starts from a basis in $\nminus$ and ends by a basis in $\nplus$, we see that the statement about the monomials in (c) is valid without any condition on the order $\prec$. However, the compatibility of the order $\prec$ with the partial order $<$ on $\h^*$ will be crucial for most of the statements below.
- The algebra $\Zgk$ is the unital associative algebra, generated by $\Uh$ and all $\{\pp_L\}$, with the weight relations and the ordering relations \[not3\] \_I\_J=\_[IJKL]{} \_K\_L+\_M\_[IJL]{}\_L+ \_[IJ]{} ,p\_Ip\_J , where ${\mathrm{B}}_{IJKL}$, ${\mathrm{C}}_{IJL}$ and ${\mathrm{D}}_{IJ}$ are certain elements of $\Uh$.
Let $\UUh$ be the field of fractions of the ring $\U (\h )$. In [@Zh], sections 4.2.3 - 4.2.4 and 6.1.5, it is proved that the reduction algebra $\Zgk$ is generated by the elements $\pp_L$ with the defining ordering relations as an algebra over $\UUh$. We shall now show that the statement (d) holds over the smaller ring $\Uh$; in other words, the relations are defined over $\Uh$ and the elements $\pp_L$ generate over $\Uh$ the algebra $\Zgk$.
.2cm We first prove that the structure constants ${\mathrm{B}}_{IJKL}$, ${\mathrm{C}}_{IJL}$ and ${\mathrm{D}}_{IJ}$ belong actually to $\Uh$. This fact can be understood with the help of the factorized formula [@AST] for the extremal projector $P$. Indeed, decomposing the product, we represent the projector $P$, after some reorderings, as a sum of terms $\xi e_{-\gamma_1}\cdots e_{-\gamma_m}e_{\gamma'_{1}}\cdots e_{\gamma'_{m'}}$, where $\xi\in\Uh$, $\gamma_1,\dots ,\gamma_m$ and $\gamma'_1,\dots ,\gamma'_{m'}$ are positive roots of $\f$; the denominator of $\xi$ is a product of linear factors of the form $h_\gamma+\rho(h_\gamma)+\ell$, where $\gamma$ is a positive root of $\f$ and $\ell$ a positive integer, $\ell >0$. We calculate the product $a\mult b$ in the following way. In the summand $a\xi e_{-\gamma_1}\cdots e_{-\gamma_m}e_{\gamma'_{1}}\cdots e_{\gamma'_{m'}}b$ of $a\mult b$, we move $\xi$ and all $e_{-\gamma}$’s to the left through $a$ by taking multiple commutators with $a$ and, similarly, all $e_{\gamma'}$’s to the right through $b$. Proceeding this way, we write \_I\_J=\_[IJKL]{}\[matrM\](we recall that $\widetilde{a}$ denotes the image of an element $a\in\Ar$ in the reduction algebra) where the (uniquely defined by the method of calculation) matrix ${\mathrm{M}}$ with entries in $\Uh$ has a triangular structure (even more is true: ${\mathrm{M}}_{IJKL}\neq 0$ $\Rightarrow$ $p_I\succ p_K$) with 1’s on the diagonal; denominators of entries of the matrix ${\mathrm{M}}$ are of the form $h_\gamma
+\rho(h_\gamma)+ \pi(h_\gamma)+\ell$, where $\pi$ is the weight, with respect to $\h$, of the corresponding $p_I$ (the summand $\pi (h_\gamma )$ appeared when, in calculating $\pp_I\mult \pp_J$ as above, we first moved $\xi\in\Uh$ to the left through $\pp_I$; taking further multiple commutators, we do not change the denominators any more). Take the formal (in the sense that for the moment we do not pay attention to possible dependencies between $\widetilde{\phantom{\,}p_Ip_J}$ or between $\pp_K\mult \pp_L$ in the algebra) inverse: $\widetilde{\phantom{\,}p_Ip_J}=
{\mathrm{M}}_{IJKL}^{-1}\pp_K\mult \pp_L$; the inverse matrix ${\mathrm{M}}^{-1}$ is triangular as well, its entries are in $\Uh$ and it has 1’s on the diagonal; the determinant of ${\mathrm{M}}$ is thus 1 and it follows that the above described structure of denominators of the entries of the matrix ${\mathrm{M}}$ remains the same for the matrix ${\mathrm{M}}^{-1}$. The commutation relation $p_Ip_J=p_Jp_I+\varUpsilon$, $p_I\succ p_J$, $\varUpsilon\in\g$, in $\U (\g )$ becomes $\widetilde{\phantom{\,}p_Ip_J}=
\widetilde{\phantom{\,}p_Jp_I}+\widetilde{\phantom{\,}\varUpsilon\phantom{\,}}$, $\widetilde{\phantom{\,}\varUpsilon\phantom{\,}}\in\p+\h$, in the reduction algebra. Translate this into the ordering rule for the product $\mult$, expressing the projections $\widetilde{\phantom{\,}pp\phantom{\,}}$’s in terms of the products $\pp\mult\pp\ \, $’s with the help of the matrix ${\mathrm{M}}^{-1}$ in both, left and right, hand sides: the right hand side, being rewritten in terms of the multiplication $\mult$, consists of ordered terms only, the left hand side is $\pp_I\mult \pp_J+\dots$, where dots stand for terms with $\pp_{I'}\mult \pp_{J'}$, $p_I\succ p_{I'}$; such term is either ordered or, by induction in $I$, can be rewritten in the ordered form as in . The coefficient in front of $\pp_{I'}\mult \pp_{J'}$ is from $\Uh$, so the reordering of the products $\pp_{I'}\mult \pp_{J'}$ may force the coefficient of degree 1 or degree 0 term in to belong to $\Uh$.
.2cm In the same manner we prove by induction on the filtration (described in the statement (b)) degree, that the algebra $\Zgk$ is generated over $\Uh$ by the elements $\{\pp_L\}$. To see this, consider the weight basis, described in the statement (c), that is, the basis $\pp_{\overline{L}}$ ($\overline{L}$ is the multi-index) of the free $\Uh$-module $\Zgk^{(k)}/\Zgk^{(k-1)}$, composed by images in $\Zgk$ of products $p_{L_1}^{n_1}p_{L_2}^{n_2}\cdots p_{L_m}^{n_m}$, where $ p_{L_1}\prec p_{L_2}\prec\ldots\prec p_{L_m}$ and $k=n_1+n_2+\dots +n_m$. Equip the set of these basic elements with a total order $\prec$ compatible with the partial order $<$ on $\h^*$; the compatibility has the same meaning as for the elements $\{\pp_L\}$: \[$\h$-weight of $\pp_{\overline{K}}$\] $<$ \[$\h$-weight of $\pp_{\overline{L}}$\] $\Rightarrow$ $\pp_{\overline{K}}\prec
\pp_{\overline{L}}$. By the same, as above, arguments, referring to the structure of the projector $P$, we have the following generalization of : \_I\_=\_[IK]{} ,\[matrM2\]where the matrix ${\mathrm{M}}$, with entries in $\Uh$, has again a triangular structure with 1’s on the diagonal. Therefore, the matrix ${\mathrm{M}}$ is invertible and its inverse matrix ${\mathrm{M}}^{-1}$ has entries in $\Uh$. The formula $\widetilde{\phantom{\,}p_Ip_{\overline{J}}}
={\mathrm{M}}_{I\overline{J}K\overline{L}}^{-1}\pp_K\mult \pp_{\overline{L}}$ implies the induction step: the subspace $\Zgk^{(k+1)}$ is generated by products in $\Zgk$ of elements from $\Zgk^{(1)}$. ${\relax\ifmmode\else\unskip\quad\fi{\hbox{\rlap{$\sqcap$}$\sqcup$}}}$
.2cm Note that, before the localization, the algebra $\S(\g,\k)={\rm Norm}(\Ar\nplus)/\Ar\nplus$, as well as its image in $\Zgk$, is not generated by the elements of degree 1. The subalgebra of $\S(\g,\k)$, generated by the elements of degree 1 (”step algebra”), was the original subject of Mickelsson’s investigation [@M].
- [The following monomials]{} form a basis of the left $\Uh$-module $\Zgk$: \_[I\_1]{}\_[I\_2]{}\_[I\_a]{},p\_[I\_1]{}p\_[I\_2]{} …p\_[I\_a]{} .\[inimpb\]
Before the proof of (e) we prove a more subtle statement.
Any expression in $\Zgk$ can be written in the ordered form by a repeated application of as instructions “replace the left hand side by the right hand side”.
[*[Proof]{}*]{} of Proposition. To save the space in the proof of this proposition we take a liberty to sometimes write $I\prec J$ instead of $p_I\prec p_J$ (the same reservation concerns the use of $\preceq$, $\succ$ and $\succeq$).
.2cm Consider the homogeneous quadratic part of the relations : \[not3h\] \_[I\_1]{}\_[I\_2]{}=… \_[I\_1’]{}\_[I\_2’]{} ,I\_1I\_2 ,where dots stand for coefficients from $\Uh$. Denote by ${\cal{I}}(\pp_{I_1}\mult\pp_{I_2})$ the right hand side of . We understand as the set of instructions $\pp_{I_1}\mult\pp_{I_2}\leadsto {\cal{I}}(\pp_{I_1}\mult\pp_{I_2})$ ($\leadsto$ stands for “replace”) in the free algebra with the weight generators $\pp_I$.
Let us prove the statement for a cubic monomial $\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3}$. For such a monomial one can apply the instructions to $\pp_{I_1}\mult\pp_{I_2}$ if $I_1\succ I_2$ and to $\pp_{I_2}\mult\pp_{I_3}$ if $I_2\succ I_3$. Denote the results by ${\cal{I}}_{12}(\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3})$ and ${\cal{I}}_{23}(\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3})$ respectively.
For an element $\psi\in\h^*$, $\psi =\sum l_i\alpha_i$, where $\alpha_i$ are the simple roots, let $d(\psi ):=\sum l_i$. The function $d$ is compatible with the partial order $<$ on $\h^*$ in the sense that $d(\alpha )<d(\beta )$ if $\alpha <\beta$. Denote by the same letter $d$ the function on the set of indices, labeling the weight base of $\p$; it is defined by $d(I):=d(\mu_I)$, where $\mu_I$ is the weight of $\pp_I$.
We have $d(I_1')+d(I_2')=d(I_1)+d(I_2)$ for any monomial $\pp_{I_1'}\mult\pp_{I_2'}$ appearing in the right hand side of (and the difference $d(I_1)-d(I_1')$ is an integer). Since $I_1\succ I_2$ and $I_1'\preceq I_2'$, it follows that $d(I_1)\geq d(I_2)$ and $d(I_1')\leq d(I_2')$; therefore, $d(I_1')\leq d(I_1)$ and $d(I_2')\geq d(I_2)$.
Associate to a monomial $\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3}$, that is, to an ordered triple $(I_1,I_2,I_3)$ of indices, the number ${\mathfrak{d}}(I_1,I_2,I_3):=2d(I_1)+d(I_2)$. When we apply the ordering instructions ${\cal{I}}_{12}$ or ${\cal{I}}_{23}$ to $\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3}$, the function ${\mathfrak{d}}$ does not increase; that is, the value of ${\mathfrak{d}}$ on any of the appearing monomials is not greater than ${\mathfrak{d}}(I_1,I_2,I_3)$. Indeed, if we replace $\pp_{I_1}\mult\pp_{I_2}$ by $\pp_{I_1'}\mult\pp_{I_2'}$ then $2d(I_1')+d(I_2')=d(I_1')+
\bigl( d(I_1')+d(I_2')\bigr)=d(I_1')+\bigl( d(I_1)+d(I_2)\bigr)\leq d(I_1)+
\bigl( d(I_1)+d(I_2)\bigr)=2d(I_1)+d(I_2)$; and if we replace $\pp_{I_2}\mult\pp_{I_3}$ by $\pp_{I_2'}\mult\pp_{I_3'}$ then simply $d(I_2')\leq d(I_2)$ and $d(I_1')=d(I_1)$.
For a linear combination $X=\sum c_{I_1I_2I_3}\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3}$ of cubic monomials, with coefficients $c_{I_1I_2I_3}\in\Uh$, denote the maximal value of ${\mathfrak{d}}$ on the monomials, appearing in $X$, by the same symbol ${\mathfrak{d}}$; that is, ${\mathfrak{d}}(X):={\displaystyle{\max_{(I_1,I_2,I_3):
c_{I_1I_2I_3}\neq 0}}}{\mathfrak{d}}(I_1,I_2,I_3)$.
.2cm Assume that the assertion is false and there exists a cubic monomial which cannot be ordered by the instructions . Since $\k +\p$ is finite-dimensional, the set of values of the function ${\mathfrak{d}}$ on cubic monomials is bounded from below. So the minimal value ${\mathfrak{d}}_{\min}$ of the function ${\mathfrak{d}}$ on the set of cubic monomials which cannot be ordered is finite, ${\mathfrak{d}}_{\min} >-\infty$. Let $\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3}$ be a monomial, which cannot be ordered, with ${\mathfrak{d}}(I_1,I_2,I_3)={\mathfrak{d}}_{\min}$. The application of the ordering instructions cannot strictly decrease the value of ${\mathfrak{d}}$, this would contradict to the minimality of ${\mathfrak{d}}(I_1,I_2,I_3)$. Therefore, among the appearing monomials, there is at least one monomial $\pp_{I_1'}\mult\pp_{I_2'}\mult\pp_{I_3'}$ with the same value of ${\mathfrak{d}}$. If $\pp_{I_1'}\mult\pp_{I_2'}\mult\pp_{I_3'}$ appears in ${\cal{I}}_{12}(\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3})$ then $2d(I_1')+d(I_2')=2d(I_1)+d(I_2)$, ${ {d(I_1)\geq d(I_2),\ d(I_1')\leq d(I_2')}}$ and $I_3'=I_3$; since the total weight is conserved, $d(I_1')=d(I_1)$, [ and]{} ${{d(I_2')=d(I_2)}}$, [ so]{} $d(I_1)=d(I_2)=d(I_1')=d(I_2')$. If $\pp_{I_1'}\mult\pp_{I_2'}\mult\pp_{I_3'}$ appears in ${\cal{I}}_{23}(\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3})$ then ${ {d(I_2)\geq d(I_3),\ d(I_2')\leq d(I_3')}}$, $d(I_2')=d(I_2)$ and $I_1'=I_1$; [ by the same arguments we have again]{}, $d(I_2)=d(I_3)=d(I_2')=d(I_3')$. Due to the structure of the matrix ${\mathrm{M}}$, defined in , and the arguments used in the proof of the statement (d), ${\cal{I}}(\pp_{I}\mult\pp_{J})$ with $d(I)=d(J)$ contains exactly one monomial $\pp_{I'}\mult\pp_{J'}$ with $d(I')=d(I)$ and this monomial is $\pp_{J}\mult\pp_{I}$. Therefore, up to monomials with the value of ${\mathfrak{d}}$ smaller than ${\mathfrak{d}}_{\min}$ (they can be ordered by assumption) and up to a coefficient from $\Uh$, the operation ${\cal{I}}_{12}$, $I_1\succ I_2$, is simply $\pp_{I_1}\mult\pp_{I_2}\leadsto\pp_{I_2}\mult\pp_{I_1}$; the operation ${\cal{I}}_{23}$, $I_2\succ I_3$, is $\pp_{I_2}\mult\pp_{I_3}\leadsto\pp_{I_3}\mult\pp_{I_2}$. The transpositions (12) and (23) of neighbors generate all permutations of three letters. The orbit of $\pp_{I_1}\mult\pp_{I_2}\mult\pp_{I_3}$ under the group of permutations of three letters $I_1,I_2$ and $I_3$ contains the ordered monomial, the contradiction.
The degree 0 or 1 terms, contained in the full instructions , may only cause an appearance of linear or quadratic terms in the process of ordering of a cubic polynomial. So, any cubic polynomial can be ordered by as well.
More generally, to a monomial $X=\pp_{I_1}\mult\pp_{I_2}\mult\cdots\mult\pp_{I_k}$ of an arbitrary degree $k$ we associate the number ${\mathfrak{d}}(I_1,\ldots,I_k):=(k-1)d(I_1)+(k-2)d(I_2)+\ldots+d(I_{k-1})$, and, in the minimal situation, conclude that up to terms smaller than $X$ in an appropriate sense, the instructions essentially reduce to transpositions $(i,i+1)$ of neighbors, which generate the whole symmetric group on $k$ letters, and thus an ordered expression is in the orbit. ${\relax\ifmmode\else\unskip\quad\fi{\hbox{\rlap{$\sqcap$}$\sqcup$}}}$
.2cm [*[Proof]{}*]{} of statement (e). By the statement (d) above, the algebra $\Zgk$ is generated by $\pp_I$ and, due to the form of relations, has a filtration by the $\mult$-degree. Let $\Zgk^{(\mult k)}$ be the subspace of elements of degree not greater than $k$ with respect to the product $\mult$. Since $\pp_{I_1}\mult\pp_{I_2}\mult\dots\mult\pp_{I_k}=\pp_{I_1}P\pp_{I_2}P\dots P\pp_{I_k}$, it follows that $\Zgk^{(\mult k)}\subset \Zgk^{(k)}$. The opposite inclusion holds as well because the algebra $\Zgk$ is generated by $\pp_I$. We conclude that the two filtrations coincide.
.2cm Therefore, every element $\widetilde{\phantom{\,}p_{I_1}\cdots p_{I_k}}$, $I_1\preceq \ldots\preceq I_k$, is in $\Zgk^{(\mult k)}$ and, by proposition above, can be ordered. The cardinalities of the sets $\{\widetilde{p_{I_1}\cdots p_{I_k}}\mid I_1\preceq \
\ldots \preceq I_k\}$ and $\{\pp_{I_1}\mult\ldots\mult\pp_{I_k}\mid I_1\preceq \ldots\preceq I_k\}$ are equal, so due to the set $\{\pp_{I_1}\mult\ldots\mult\pp_{I_k}\mid
I_1\preceq \ldots\preceq I_k\}$ is a basis of $\Zgk^{(\mult k)}/\Zgk^{(\mult (k-1))}$.${\relax\ifmmode\else\unskip\quad\fi{\hbox{\rlap{$\sqcap$}$\sqcup$}}}$
.2cm Note that for an order which is not compatible with the partial order $<$ on $\h^*$, the ordering relations of the form may exist but the statement (e) does not necessarily hold. For instance, the ordering relations can be written for a lexicographical order for the generators $\z_{ij}$ and $t_i$ (with $\z_{ii}=t_i$) of the algebra $\Z_n$, defined in the next Section, but the ordering procedure loops for cubic monomials, already for $n=2$ (we don’t give details; it is an explicit calculation).
Diagonal reduction algebra of $\gl_n$ {#section-notation}
=====================================
-.2cm Let $\gl_n$ be the Lie algebra of the general linear group of $n$-dimensional complex linear space. Consider the reductive pair $(\g,\k)$ with $\g=\gl_n\oplus\gl_n$ and $\k=\gl_n$ diagonally embedded into $\gl_n\oplus\gl_n$. The corresponding reduction algebra we call ’diagonal reduction algebra’ and denote it by $\Z_n$.
We fix the following notations for generators of these algebras $\g$ and $\k$. Let $E_{ij}^{(1)}$ and $E_{ij}^{(2)}$, $i,j=1\lcd n$, be the standard generators of the two copies of the Lie algebra $\gl_n$ in $\gl_n\oplus\gl_n$, $$[E_{ij}^{(a)},E_{kl}^{(b)}]=\delta_{ab}\left(\delta_{jk}E_{il}^{(a)}-
\delta_{il}E_{kj}^{(a)}\right)\ ,$$ where $\delta_{ab}$ and $\delta_{ij}$ are the Kronecker symbols. Set e\_[ij]{}:=(E\_[ij]{}\^[(1)]{}+E\_[ij]{}\^[(2)]{}) ,E\_[ij]{}:=(E\_[ij]{}\^[(1)]{}- E\_[ij]{}\^[(2)]{}) .The elements $e_{ij}$ span the diagonally embedded Lie algebra $\k\simeq\gl_n$, while $E_{ij}$ form an adjoint $\k$-module $\mathfrak{p}$. The Lie algebra $\k$ and the space $\mathfrak{p}$ constitute a symmetric pair, that is, $[\k,\k]\subset \k$, $[\k,\p]\subset \p$, and $[\p,\p]\subset \k$:
[ll]{}\[e\_[ij]{},e\_[kl]{}\]&=\_[jk]{}e\_[il]{}-\_[il]{}e\_[kj]{} , =\_[jk]{}E\_[il]{}-\_[il]{}E\_[kj]{} ,\
\[E\_[ij]{},E\_[kl]{}\]&=\_[jk]{}e\_[il]{}-\_[il]{}e\_[kj]{} .
In the sequel, $h_a$ means the element $e_{aa}$ of the Cartan subalgebra $\h$ of the subalgebra $\k\in\gl_n\oplus\gl_n$ and $h_{ab}$ the element $e_{aa}-e_{bb}$.
.2cm Let $\{\ve_a\}$ be the basis of $\h^*$ dual to the basis $\{ h_a\}$ of $\h$, $\ve_a(h_b)=\delta_{ab}$. We shall use as well the root notation $h_\alpha$, $e_{\alpha}$, $e_{-\alpha}$ for elements of $\k$, and $H_\alpha$, $E_{\alpha}$, $E_{-\alpha}$ for elements of $\p$. The Lie sub-algebra $\nplus$ in the triangular decomposition is spanned by the root vectors $e_{ij}$ with $i<j$ and the Lie sub-algebra $\nminus$ by the root vectors $e_{ij}$ with $i>j$. Let $\bb_+$ and $\bb_-$ be the corresponding Borel sub-algebras, $\bb_+=\h\oplus\nplus$, $\bb_-=\h\oplus\nminus$. The system $\Delta_+$ of positive roots of $\k$ consists of roots $\ve_i-\ve_j$ with $i<j$ and the system $\Delta_-$ consists of roots $\ve_i-\ve_j$ with $i>j$.
We fix the following action of the cover of the symmetric group $\S_n$ (the Weyl group of the diagonal $\k$) on the Lie algebra $\gl_n\oplus\gl_n$ by automorphisms \_i(x):=\_[(e\_[i,i+1]{})]{}\_[(-e\_[i+1,i]{})]{}\_[(e\_[i,i+1]{})]{}(x) ,so that $\sy_i(e_{kl})=(-1)^{\delta_{ik}+\delta_{il}}e_{\sigma_i(k)\sigma_i(l)}$ and $
\sy_i(E_{kl})=(-1)^{\delta_{ik}+\delta_{il}}E_{\sigma_i(k)\sigma_i(l)}$. Here $\sigma_i=(i,i+1)$ is an elementary transposition in the symmetric group. We extend naturally the above action of the cover of $\S_n$ to the action by automorphisms on the associative algebra $\Ar\equiv\Ar_n:=\U(\gl_n)\otimes \U(\gl_n)$. The restriction of this action to $\h$ coincides with the natural action $\si(h_{k})=h_{\si(k)}$, $\si\in\S_n$, of the Weyl group on the Cartan sub-algebra. The shifted action of the Weyl group on $\h$ looks as: \[not1\]h\_k:= h\_[(k)]{}+k-(k) ,k=1,...,n ;§\_n .It becomes the usual action for the variables \_k:=h\_k-k ,\_[ij]{}:=\_i-\_j ;\[hrond\]so that for any $\si\in\S_n$ we have $\si\circ\th_k=\th_{\si(k)}$ and $\si\circ\th_{ij}=\th_{\si(i)\si(j)}$ . The set of denominators, defining the localizations $\Uh$ and $\Ab$ consists of elements h\_[ij]{}+l ,l[[Z]{}]{} , 1i<jn .\[musel\] We choose the set of vectors $E_{ij}$, $i,j=1,...,n$, as a basis of the space $\p$. The weight of $E_{ij}$ is $\ve_i-\ve_j$. The compatibility of a total order $\prec$ with the partial order $<$ on $\h^*$ means the condition $$\label{not4a}E_{ij}\prec E_{kl}\qquad \text{if}\qquad i-j>k-l\ .$$ The order in each subset $\{E_{ij}|i-j=a\}$ with a fixed $a$ can be chosen arbitrarily. For instance, we can set $$\label{not4}E_{ij}\prec E_{kl}\quad \text{if}\quad i-j>k-l\quad
\text{or}\quad i-j=k-l\quad\text{and}\quad i>k\ .$$ .2cm Denote the images of the elements $E_{ij}$ in $\Z_n$ by $\s_{ij}$. We use also the notation $t_i$ for the elements $\s_{ii}$ and $t_{ij}:=t_i-t_j$ for the elements $\s_{ii}-\s_{jj}$. The order induces as well the order on the generators $\s_{ij}$ of the algebra $\Z_n$: \[not4b\]\_[ij]{}\_[kl]{} E\_[ij]{}E\_[kl]{} .The statement implies an existence of structure constants $\mathrm{B}_{(ab),(cd),(ij),(kl)} \in \Uh$ and $\mathrm{D}_{(ab),(cd)}\in\Uh$ such that for any $a,b,c,d=1,\ldots,n$ we have \[not6\]\_[ab]{}\_[cd]{}=\_[i,j,k,l:\_[ij]{} \_[kl]{}]{}\_[(ab),(cd),(ij),(kl)]{}\_[ij]{}\_[kl]{}+\_[(ab),(cd)]{} .Linear terms in the right hand side of are absent since here $(\g,\k)$ form a symmetric pair. The relations together with the weight conditions =(\_a-\_b)(h) \_[ab]{}\[wede\]are the defining relations for the algebra $\Z_n$.
.2cm The structure of denominators of entries of the matrices ${\mathrm{M}}$ and ${\mathrm{M}}^{-1}$, mentioned in the proof of above, shows that for the algebra $\Z_n$ the denominators of the structure constants $\mathrm{B}_{(ab),(cd),(ij),(kl)}$ and $\mathrm{D}_{(ab),(cd)}$ are products of linear factors of the form $\th_{ij}+\ell$, $i<j$, where $\ell\geq -1$ is an integer. This is because in our situation the $\sl_2$–sub-algebra (of the diagonal $\gl_n$), corresponding to an arbitrary positive root $\ve_i-\ve_j$, $i<j$, has only 1, 2- and 3-dimensional representations in $\p$, so the numbers $\ell$’s in the denominators of the summands of the projector can drop at most by 2 due to the presence of the term $(\pi ,\gamma )$.
The Chevalley anti-involution $\epsilon$ in $\U(\gl_n\oplus\gl_n)$, $\epsilon(e_{ij}):=e_{ji}$, $\epsilon(E_{ij}):=E_{ji}$, induces the anti-involution $\epsilon$ in the algebra $\Z_n$: (\_[ij]{})=\_[ji]{} ,(h\_k)=h\_k .\[anep\]Besides, the outer automorphism of the Dynkin diagram of $\gl_n$ induces the involutive automorphism $\omega$ of $\Z_n$, \[not2a\](\_[ij]{})=(-1)\^[i+j+1]{}\_[j’i’]{} ,(h\_k)=-h\_[k’]{} ,where $i'=n+1-i$. The operations $\epsilon$ and $\omega$ commute, $\epsilon\omega =\omega\epsilon$.
.2cm Central elements of the sub-algebra $\U(\gl_n)\otimes 1\subset \Ar$, generated by $n$ Casimir operators of degrees $1\lcd n$, as well as central elements of the sub-algebra $1\otimes \U(gl_n)\subset \Ar$ project to central elements of the algebra $\Z_n$. In particular, central elements of degree $1$ project to central elements h\_1+…+h\_n\[clit\]t\_1+…+t\_nof the algebra $\Z_n$. The difference of central elements of degree two projects to the central element \[drclit\]\_[i=1]{}\^n(h\_i-2i)t\_iof the algebra $\Z_n$. The images of other Casimir operators are more complicated.
Change of variables
-------------------
We shall use the following elements of $\Uh$: $$\!\! \tphi_{ij}:=\frac{\th_{ij}}{\th_{ij}-1}\ ,\ \tpsi_{ij}:=\frac{\th_{ij}-1}{\th_{ij}}\ ,
\ \ttphi_{ij}:=\frac{\th_{ij}-1}{\th_{ij}-2}\ ,\ \ttpsi_{ij}:=
\frac{\th_{ij}-2}{\th_{ij}-1}\ ,\ \Cprime_{ij}:=\frac{\th_{ij}-3}{\th_{ij}-2}\ ,$$ the variables $\th_{ij}$ are defined in . Note that $\ \tphi_{ij}\tpsi_{ij}=\ttphi_{ij}\ttpsi_{ij}=1$.
.2cm Define elements $\tt_1,\ldots, \tt_n\in\Z_n$ by \[3.1\]\_1:=t\_1 ,\_2:=\_[1]{}(t\_1) ,\_3:=\_[2]{}\_[1]{}(t\_1) , …,\_n:=\_[n-1]{}\_[2]{}\_[1]{}(t\_1) .Using we find the relations
[ll]{}&\
&\
& ,
\[acwot\]which can be used to convert the definition into a linear over the ring $\Uh$ change of variables:
[rl]{}\[3.1a\]&=\
&=
In terms of the new variables $\tt$’s, the linear in $t$ central element (\[clit\]) reads $$\sum t_i=\sum\tt_i\prod_{a:a\neq i}\frac{\th_{ia}+1}{\th_{ia}}\ .$$
In the following, we use the notion of [*coefficient-bounded*]{} formulas and relations. It means the following. Given a family of formulas for each $n$ (expressing some action, relations [*etc.*]{}) with coefficients in $\Uh$, we say that it is coefficient-bounded if the degrees of the numerators and denominators (in the reduced form, with no common factors) of the coefficients do not grow with $n$.
For example, the set of relations for $\Z_n$, which we shall exhibit, will have coefficient-bounded terms with respect to a certain set of generators. In this sense the action is coefficient-bounded while the change of variables is however not coefficient-bounded.
Braid group action {#brgac}
------------------
Since $\q_{i}^2(x)=x$ for any element $x$ of zero weight, the braid group acts as its symmetric group quotient on the space of weight 0 elements. Although the change of variables is not coefficient-bounded in the sense of Section \[section2new\], the action of the transformations $\q_{i}$ on the new variables $\tt$’s is coefficient-bounded: it follows from and $\q_{i}(t_1)=t_1$ for all $i>1$ that \[3.2\]\_(\_i)=\_[(i)]{}§\_n.
The action of the Zhelobenko automorphisms on the generators $\s_{kl}$ looks as follows: $$\begin{aligned}
&\q_i(\s_{ik})=-\s_{i+1,k}\tphi_{i,i+1}\ ,&& \notag
\q_i(\s_{ki})=-\s_{k,i+1}\ ,&&k\not=i,i+1\ ,\\ \label{3.5}
&\q_i(\s_{i+1,k})=\s_{i,k}\ ,&&\q_i(\s_{k,i+1})=\s_{k,i}\tphi_{i,i+1}\ ,&&k\not=i,i+1\ ,\\
&\q_i(\s_{i,i+1})=-\s_{i+1,i}\tphi_{i,i+1}\ttphi_{i,i+1}\ ,&& \notag
\q_i(\s_{i+1,i})=-\s_{i,i+1}\ ,\\ \notag
&\q_{i}(\s_{j,k})=\s_{j,k}\ ,\ \ j,k\neq i,i+1\ .&&\end{aligned}$$
.1cm Denote $i'=n+1-i$, as before. The braid group action is compatible with the anti-involution $\epsilon$ and the involution $\omega$ (note that $\omega (\th_{ij})=\th_{j'i'}$), see and , in the following sense: $$\begin{aligned}
\epsilon\, \q_i&=\q_i^{-1}\epsilon\ \ ,\label{qeps} &\omega \q_i&=\q_{i'-1}\omega\ .\end{aligned}$$
.1cm Let $w_0$ be the longest element of the Weyl group of $\gl_n$, the symmetric group $\S_n$. Similarly to the squares of the transformations corresponding to the simple roots, see , the action of $\q_{w_0}^2$ is the conjugation by a certain element of $\Uh$. Moreover, one can observe by a direct calculation, that
\[3.6\]\_[w\_0]{}(\_[ij]{})=(-1)\^[i+j]{} \_[i’j’]{}\_[a:a<i’]{}\_[ai’]{}\_[b:b>j’]{}\_[j’b]{} ,\_[w\_0]{}(\_i)=\_[i’]{} .
The formula implies the existence of the ordering relations for the generators $\s_{ij}$ in the inverse to - order.
. There exist $\mathrm{B}'_{(ab),(cd),(ij),(kl)}$ and $\mathrm{D}'_{(ab),(cd)}\in\Uh$ such that for any $\s_{ab}$ and $\s_{cd}$ we have $$\label{not6a}\s_{ab}\mult\s_{cd}=\sum_{i,j,k,l:\s_{kl}
\preceq\s_{ij}}\mathrm{B}'_{(ab),(cd),(ij),(kl)}
\s_{ij}\mult\s_{kl}+\mathrm{D}'_{(ab),(cd)}\ .$$ \[opor\]
Indeed, we apply the transformation $\q_{w_0}$ to the equalities and substitute . This gives the relations since the assignment $(i,j)\mapsto(i',j')$ reverses the order $\prec$.
Defining relations {#section3.3}
------------------
To save space we omit in this section the symbol $\mult$ for the multiplication in the algebra $\Z_n$. It should not lead to any confusion since no other multiplication is used in this section.
.2cm Each relation which we will derive will be of a certain weight, equal to a sum of two roots. From general considerations the upper estimate for the number of terms in a quadratic relation of weight $\lambda=\alpha+\beta$ is the number $|\lambda|$ of quadratic combinations $\s_{\alpha'}\s_{\beta'}$ with $\alpha'+\beta'=\lambda$. There are several types of relation weights, excluding the trivial one, $\lambda=2(\ve_i-\ve_j)$, $|\lambda|=1$:
1. $\lambda=\pm(2\ve_i-\ve_j-\ve_k)$, where $i,j$ and $k$ are pairwise distinct. Then $|\lambda|=2$.
2. $\lambda=\ve_i-\ve_j+\ve_k-\ve_l$ with pairwise distinct $i,j,k$ and $l$. Then $|\lambda|=4$.
3. $\lambda=\ve_i-\ve_j$, $i\neq j$. For $\s_{\alpha'}\s_{\beta'}$, there are $2(n-2)$ possibilities (subtype 3a) with $\alpha'=\ve_i-\ve_k$, $\beta'=\ve_k-\ve_j$ or $\alpha'=\ve_k-\ve_j$, $\beta'=\ve_i-\ve_k$ with $k\neq i,j$ and $2n$ possibilities (subtype 3b) with $\alpha'=0$, $\beta'=\ve_i-\ve_j$ or $\alpha'=\ve_i-\ve_j$, $\beta'=0$. Thus $|\lambda|=4(n-1)$.
4. $\lambda=0$. There are $n^2$ possibilities (subtype 4a) with $\alpha'=0$, $\beta'=0$ and $n(n-1)$ possibilities (subtype 4b) with $\alpha'=\ve_i-\ve_j$, $\beta'=\ve_j-\ve_i$, $i\neq j$. Here $|\lambda|=n(2n-1)$.
Below we write down relations for each type (and subtype) separately. The relations of types 1 and 2 have a simple form in terms of the original generators $\s_{ij}$. To write the relations of types 3 and 4, it is convenient to renormalize the generators $\s_{ij}$ with $i\not=j$. Namely, we set $$\label{not8}\hs_{ij}=\s_{ij}\prod_{k=1}^{i-1}\tphi_{ki}\ .$$
In terms of the generators $\hs_{ij}$, the formulas for the action of the automorphisms $\q_i$ translate as follows: $$\begin{array}{lll}\q_{i}(\hs_{ik})=-\hs_{i+1,k}\ ,&
\q_{i}(\hs_{i+1,k})=\hs_{i,k}\tphi_{i+1,i}\ ,&k\not=i,i+1\ ,\\[.3 em]
\q_{i}(\hs_{ki})=-\hs_{k,i+1}\ ,&
\q_{i}(\hs_{k,i+1})=\hs_{k,i}\tphi_{i,i+1}=\tpsi_{i+1,i}\hs_{k,i}\ ,&k\not=i,i+1\ ,\\[.3 em]
\q_{i}(\hs_{i,i+1})=-\tpsi_{i+1,i}\hs_{i+1,i}\ ,&\q_{i}(\hs_{i+1,i})=-\hs_{i,i+1}\tphi_{i+1,i}\
,&\\[.3 em]
\q_{i}(\hs_{j,k})=\hs_{j,k}\ , \ \ j,k\neq i,i+1\ .&&\end{array}$$ Although the renormalization is not coefficient-bounded, the action of the braid group stays coefficient-bounded.
#### 1.
The relations of the type 1 are: $$\label{relation1}\s_{ij}\s_{ik}=\s_{ik}\s_{ij}\tphi_{kj}\ ,\qquad
\s_{ji}\s_{ki}=\s_{ki}\s_{ji}\tpsi_{kj}\ ,\qquad \text{for}\quad j<k\ ,\ i\not=j,k\ .$$
#### 2.
Denote $D_{ijkl}:= \th_{ik}^{-1}-\th_{jl}^{-1}$. Then, for any four pairwise different indices $i,j,k$ and $l$, we have the following relations of the type 2: $$\label{relation2}\begin{split} [ \s_{ij},\s_{kl} ]&=\s_{kj}\s_{il}D_{ijkl}\ ,
\qquad i<k\ ,\ j<l \ ,\\[.4em] \s_{ij}\s_{kl}-\s_{kl}\s_{ij}\tphi_{jl}'\tphi_{lj}'&=\s_{kj}\s_{il}
D_{ijkl}\ ,\qquad i<k\ ,\ j>l\ .\end{split}$$
#### 3a.
Let $i\neq k\neq l\neq i$. Denote $$\mathring{E}_{ikl}:=-\left((\tt_i-\tt_k)\frac{\th_{il}+1}{\th_{ik}\th_{il}}+(\tt_k-\tt_l)
\frac{\th_{il}-1}{\th_{kl}\th_{il}}\right)\hs_{il}+\sum_{a:a\neq i,k,l}
\hs_{al}\hs_{ia}\frac{\ttphi_{ai}}{\th_{ka}+1}\ .$$ With this notation the first group of the relations of the type 3 is: $$\begin{aligned}
\notag\hs_{ik}\hs_{kl}\tpsi_{ik} -\hs_{kl}\hs_{ik}\ttphi_{ki}
&=\mathring{E}_{ikl}\ , && i<k<l\ ,\\[.4em] \notag
\hs_{ik}\hs_{kl}\tpsi_{ik}\tpsi_{lk}\ttphi_{lk} -\hs_{kl}\hs_{ik}\ttphi_{ki}
&=\mathring{E}_{ikl}\ , && i<l<k\ ,\\[.4em] \label{relation3}
\hs_{ik}\hs_{kl}\tphi_{ki} -\hs_{kl}\hs_{ik}\ttphi_{ki}&=\mathring{E}_{ikl}\ , && k<i<l\ ,\\[.4em]
\notag
\hs_{ik}\hs_{kl}\tphi_{ki}\tphi_{li}\ttpsi_{li} -\hs_{kl}\hs_{ik}\ttphi_{ki}
&=\mathring{E}_{ikl}\ ,&& k<l<i\ ,\\[.4em] \notag
\hs_{ik}\hs_{kl}\tpsi_{ik}\tpsi_{lk}\ttphi_{lk}\tphi_{li}\ttpsi_{li}-\hs_{kl}\hs_{ik}\ttphi_{ki}
&=\mathring{E}_{ikl}\ , && l<i<k\ ,\\[.4em] \notag
\hs_{ik}\hs_{kl}\tphi_{ki}\tpsi_{lk}\ttphi_{lk}\tphi_{li}\ttpsi_{li}
-\hs_{kl}\hs_{ik}\ttphi_{ki}&=\mathring{E}_{ikl}\ , && l<k<i\ .\end{aligned}$$
The relations can be written in a more compact way with the help of both systems, $\s_{ij}$ and $\hs_{ij}$, of generators. Let now $$E_{ikl}:=-\left((\tt_i-\tt_k)\frac{\th_{il}+1}{\th_{ik}\th_{il}}+(\tt_k-\tt_l)
\frac{\th_{il}-1}{\th_{kl}\th_{il}}\right)\s_{il}+\sum_{a:a\neq i,k,l}
\hs_{al}\s_{ia}\frac{\ttphi_{ai}}{\th_{ka}+1}\ .$$ Then
[rcll]{} \_[ik]{}\_[kl]{}\_[ik]{} -\_[kl]{}\_[ik]{}\_[ki]{}&=&E\_[ikl]{} , & k<l ,\
\_[ik]{}\_[kl]{}\_[ik]{}\_[lk]{}\_[lk]{} -\_[kl]{}\_[ik]{}\_[ki]{}&=&E\_[ikl]{} , & l<k .
\[shfo3a\]Moreover, after an extra redefinition: $\!\mathring{\hspace{.1cm}z}_{kl}\!\!\!\!\!\!\mathring{\phantom{z}}\ \ =\hs_{kl}
\ttphi_{lk}$ for $k>l$, the left hand side of the second line in becomes, up to a common factor, the same as the left hand side of the first line, namely, it reads $(\s_{ik}
\mathring{\hspace{.1cm}z}_{kl}\!\!\!\!\!\!\mathring{\phantom{z}}\ \ \tpsi_{ik} -
\mathring{\hspace{.1cm}z}_{kl}\!\!\!\!\!\!\mathring{\phantom{z}}\ \ \s_{ik}\ttphi_{ki})
\tpsi_{lk}$.
#### 3b.
Let $l\neq j$. The second group of relations of the type 3 reads: $$\begin{aligned}
\notag\hs_{ij}\tt_{i}=&\ \tt_i\hs_{ij}\Cprime_{ji}-\tt_j\hs_{ij}
\frac{1}{\th_{ij}+2}-
\sum_{a:a\not=i,j}\hs_{aj}\hs_{ia}\frac{1}{\th_{ia}+2}\ ,\\[.4em]
\label{3.12}\hs_{ij}\tt_{j}=&-\tt_i\hs_{ij}\frac{\Cprime_{ji}}{\th_{ij}-1}+
\tt_j\hs_{ij}\tphi_{ij}\tpsi_{ji}\ttphi_{ji}+
\sum_{a:a\not=i,j}\hs_{aj}\hs_{ia}\tphi_{ij}\tpsi_{ji}\frac{\ttphi_{ai}}{\th_{ja}+1}\ ,\\[.4em]
\notag
\hs_{ij}\tt_{k}=&\,\tt_i\hs_{ij}\frac{(\th_{ij}+3)\ttphi_{ji}}{(\th_{ik}^2-1)(\th_{jk}-1)}+
\tt_j\hs_{ij}\frac{(\th_{ij}+1)\ttphi_{ji}}{(\th_{ik}-1)(\th_{jk}-1)^2}+
\tt_k\hs_{ij}\tphi_{ik}\tphi_{ki}\tphi_{jk}\ttpsi_{jk}\\[.4em] \notag
-&\hs_{kj}\hs_{ik}\frac{(\th_{ij}+1)\ttphi_{ki}}{(\th_{ik}-1)(\th_{jk}-1)}
-\sum_{a:a\not=i,j,k}\hs_{aj}\hs_{ia}\frac{\th_{ij}+1}{(\th_{ik}-1)(\th_{jk}-1)}
\cdot\frac{\ttphi_{ai}}{\th_{ka}+1}\ .\end{aligned}$$
#### 4a.
The relations of the weight zero (the type 4) are also divided into 2 groups. This is the first group of the relations: \[relation4a\]\[\_i,\_j\]=0Note that the relations hold for the diagonal reduction algebra for an arbitrary reductive Lie algebra: the images of the generators, corresponding to the Cartan sub-algebra, commute.
#### 4b.
The second group of the relations of the type 4 is (here $i\not=j$) \[relation4\]\[\_[ij]{},\_[ji]{}\]=\_[ij]{}-(\_i-\_j)\^2+ \_[a:a=i,j]{}(\_[ai]{}\_[ia]{}-\_[aj]{}\_[ja]{} ) .The list of relations is completed.
.3cm Denote by $\mathfrak{R}$ the system – , and – of the relations.
. The relations $\mathfrak{R}$ are the defining relations for the weight generators $\s_{ij}$ and $t_i$ of the algebra $\Z_n$. In particular, the set of ordering relations follows over $\Uh$ from (and is equivalent to) $\mathfrak{R}$. \[mthe\]
The derivation of the relations is given in [@KO2]; the proof of Theorem is in Section \[sectionproofs\].
.2cm The relations , (a straightforward verification), as well as , , and , have coefficient-bounded terms with respect to the generators $\hs_{ij}$ and $\tt_i$; there is no coefficient-boundedness with respect to the original generators $\s_{ij}$ and $t_i$. We think that the set of ordering relations is not coefficient-bounded.
Stabilization {#subsection3.4}
-------------
Consider an embedding of $\gl_n$ to $\gl_{n+1}$, given by an assignment $e_{ij}\mapsto e_{ij}$, $i,j=1,\ldots,n$, where $e_{ij}$ in the source are the generators of $\gl_n$ and target $e_{ij}$ are in $\gl_{n+1}$. The same rule $E_{ij}\mapsto E_{ij}$ defines an embedding of the Lie algebra $\gl_n\oplus\gl_n$ to the Lie algebra $\gl_{n+1}\oplus\gl_{n+1}$ and of the enveloping algebra $\Ar_n=\U(\gl_n\oplus\gl_n)$ to $\Ar_{n+1}=\U(\gl_{n+1}\oplus\gl_{n+1})$. This embedding clearly maps nilpotent sub-algebras of $\gl_n$ to the corresponding nilpotent sub-algebras of $\gl_{n+1}$ and thus defines an embedding $\iota_n:\Z_n\to \Z_{n+1}$ of the corresponding double coset spaces. However, the map $\iota_n$ is not a homomorphism of algebras. This is because the multiplication maps are defined with the help of projectors, which are different for $\gl_n$ and $\gl_{n+1}$.
.2cm Nevertheless, there is an important connection between the two multiplication maps. Namely, let $\V_{n+1}$ be the left ideal of the algebra $\Z_{n+1}$, generated by elements $\s_{i,n+1}$, $i=1,\ldots,n$, and $\V_{n+1}'$ be the right ideal of the algebra $\Z_{n+1}$, generated by elements $\s_{n+1,i}$, $i=1,\ldots, n$. For a moment denote by $\mmult_{(n)}:\Z_n\otimes\Z_n\to\Z_n$ and $\mmult_{(n+1)}:\Z_{n+1}\otimes\Z_{n+1}$ the multiplication maps in $\Z_n$ and $\Z_{n+1}$ (instead of the default notation $\mult$, see ).
.2cm Let $\pi_{n+1}:\Z_{n+1}\to \Z_{n+1}$ be any linear operator in $\Z_{n+1}$, which projects $\Z_{n+1}$ onto $\iota_n(\Z_n)$; assume that either the ideal $\V_{n+1}$ or the ideal $\V'_{n+1}$ is in the kernel of $\pi_{n+1}$, $$\pi_{n+1}(x)=x\ , \quad x\in\iota_n(\Z_n)\ ,\qquad\text{and}\quad
\pi_{n+1}(\V_{n+1})=0\quad\text{or}\quad
\pi_{n+1}(\V'_{n+1})=0\ .$$ Define a map $\tilde{\mult}_{(n)}: \iota_n(\Z_n)\otimes \iota_n(\Z_n)\to\iota_n(\Z_n)$ as a composition $$\tilde{\mmult}_{(n)}=\pi_{n+1}\mmult_{(n+1)}\ .$$
\[proposition1\]. We have a commutative diagram of maps \_n \_[(n)]{}=\_[(n)]{}(\_n\_n) .
More precisely, for $i,j,k,l\leq n$ the difference $\iota_n(z_{ij}\mmult_{(n)}z_{kl})-z_{ij}\mmult_{(n+1)}z_{kl}$ in $\Z_{n+1}$ can be written in the form $\sum_{a=1}^n \s_{n+1,a}\mmult_{(n+1)}z_{i+k-j-l+a,n+1}\xi^{(a)}$, where $\xi^{(a)}\in \Uh$.
For the proof of Proposition, we need the following
\[lemma1\]. The left ideal of $\Z_n$, generated by all $\s_{in}$, $i=1\lcd n-1$, consists of images in $\Z_n$ of sums $\sum_i X_i E_{in}$ with $X_i\in\Ab$, $i=1\lcd n-1$.
.2cm The right ideal of $\Z_n$, generated by all $\s_{ni}$, $i=1\lcd n-1$, consists of images in $\Z_n$ of sums $\sum_i E_{ni}Y_i$ with $Y_i\in\Ab$, $i=1\lcd n-1$.
[*[Proof of Lemma]{}*]{}. We follow the arguments used in the proof of the relations . Present the projector $P$ as a sum of terms $\xi e_{-\gamma_1}\cdots e_{-\gamma_m}e_{\gamma'_{1}}\cdots e_{\gamma'_{m'}}$, where $\xi\in\Uh$, $\gamma_1,\dots ,\gamma_m$ and $\gamma'_1,\dots ,\gamma'_{m'}$ are positive roots of $\f$. For any $\lambda\in Q_+$ denote by $P_\lambda$ the sum of above elements with $\gamma_1+\dots +\gamma_{m}=
\gamma'_1+\dots +\gamma'_{m'}=\lambda$. Then $P=\sum_{\lambda\in Q_+}P_\lambda$. For any $X,Y\in\Ab$ define an element $X\mult_\lambda Y$ as the image of $XP_\lambda Y$ in the reduction algebra. We have $X\mult Y=\sum_{\lambda\in Q_+} X\mult_\lambda Y$.
For any $X\in\Ab$ and $i<n$ consider the product $X\mult_\lambda \s_{in}$. Let $\lambda=\sum\nolimits_{k=1}^n\lambda_k\ve_k$. The product $X\mult_\lambda \s_{in}$ is zero if $\lambda_n\not=0$. Indeed, in this case in each summand of $P_\lambda$ one of $e_{\gamma'_{k'}}$ is equal to some $e_{jn}$. We can order all the monomials in $\U(\n_+)$ in such a way that all $e_{jn}$ stand on the right. Since $[e_{jn},E_{in}]=0$, the product $e_{jn}E_{in}$ belongs to the left ideal $\Ib$ and thus $X\mult_\lambda \s_{in}=0$ in $\Z_n$. If $\lambda_n=0$, then by PBW arguments, $P_\lambda$ can be written as a sum of monomials composed of generators $e_{ij}$, $1\leq i<j<n$, and thus their adjoint action leaves the space, spanned by all $E_{in}$, $i<n$, invariant, so $X\mult_\lambda \s_{in}$ is presented as an image of the sum $\sum_jX_j E_{jn}$ with $X_j\in\Ab$, $j<n$. Thus, the left ideal, generated by $\s_{in}$ is contained in the vector space of images in $\Z_n$ of sums $\sum_i X_i E_{in}$.
.2cm Moreover, $X\mult \s_{in}$ is the image of $XE_{in}+\sum_{m<i}X^{(m)}E_{mn}$ for some $X^{(m)}$ and the induction on $i$ proves the inverse inclusion.
.2cm The second part of lemma is proved similarly.${\relax\ifmmode\else\unskip\quad\fi{\hbox{\rlap{$\sqcap$}$\sqcup$}}}$
.2cm [*Proof*]{} of Proposition \[proposition1\]. It is sufficient to prove the following statement. Suppose $X$ and $Y$ are (non-commutative) polynomials in $E_{ij}$ with $i,j\leq n$. Then the product of $\widetilde{\hspace{.05cm}X\hspace{.05cm}}$ and $\widetilde{\hspace{.05cm}Y\hspace{.05cm}}$ in $\Z_{n+1}$ coincides with the image in $\Z_{n+1}$ of $X\, P_{n} Y$, where $P_{n}$ is the projector for $\gl_{n}$, modulo the left ideal in $\Z_{n+1}$, generated by all $\s_{i,n+1}$, $i\leq n$. Again we note that due to the structure of the projector for any $\lambda=\sum_k\lambda_k\ve_k$ with $\lambda_{n+1}=0$, the product $X\mult_\lambda Y$ related to $\gl_n$ coincides with product $X\mult_\lambda Y$ related to $\gl_{n+1}$. Thus it remains to prove that for any $X$ and $Y$ as above the element $\widetilde{\hspace{.05cm}X\hspace{.05cm}}\mult_\lambda \widetilde{\hspace{.05cm}Y
\hspace{.05cm}}$ belongs to the ideal in $\Z_{n+1}$, generated by all $\s_{i,n+1}$, $i\leq n$, once $\lambda_{n+1}\not=0$. But for $\lambda$ with $\lambda_{n+1}\not=0$ we see, by weight arguments, that $\widetilde{\hspace{.05cm}X\hspace{.05cm}}\mult_\lambda \widetilde{\hspace{.05cm}Y
\hspace{.05cm}}$ can be presented as an image in $\Z_{n+1}$ of the sum $\sum X_i Y_i$, such that the $(n+1)$-st component of the weight of each $Y_i$ is not zero. Thus each $Y_i$ necessarily belongs to the left ideal generated by $E_{j,n+1}$, $j=1,\dots ,n$. Finally we apply Lemma \[lemma1\] to complete the proof.
.2cm The statement of Proposition \[proposition1\] concerning the ideal $V_{n+1}'$ is proved similarly. ${\relax\ifmmode\else\unskip\quad\fi{\hbox{\rlap{$\sqcap$}$\sqcup$}}}$
. The coefficients in the relations – , , – are stable with respect to the above inclusions of $\Z_n$ to $\Z_{n+1}$.
The stability of the coefficients is understood in the following sense. Let $\cal{R}$ be a relation for $\Z_{n+1}$ from our defining list $\mathfrak{R}$, see Subsection \[section3.3\].
Assume that ${\cal{R}} $ does not contain any term with $\s_{i,n+1},\ i=1,\ldots n,$ as a left factor. Then if we suppress in $\cal{R}$ terms which contain $\s_{i,n+1}$, $i=1,\ldots n$, as a right factor (such term automatically contains $\s_{n+1,j}$, $j=1,\dots, n$, as a left factor), we get a relation in $\Z_{n}$.
.2cm Call “cut” the result of this procedure of getting the relations in $\Z_{n}$ from the relations in $\Z_{n+1}$ (under the formulated conditions). Then all relations in $\Z_{n}$ can be obtained by cutting appropriate relations in $\Z_{n+1}$.
.2cm Moreover, each relation in $\Z_{n}$ extends uniquely to a relation in $\Z_{n+1}$ from which it can be obtained by the cut procedure; in other words, there is a bijection between the set of relations in $\Z_{n}$ and the set of those relations in $\Z_{n+1}$ which do not contain any term with $\s_{i,n+1}$, $i=1,\ldots n$, as a left factor.
.2cm The stabilization rule is certainly not an isolated $\gl$ phenomenon; it can be generalized to certain other quadruplets of algebras replacing those which participate in the diagram $${{\gl_n}^{\textstyle{\nearrow}}_{\textstyle{\searrow}}}\begin{array}{c}\gl_n\oplus\gl_n\\[1em]\gl_{n+1}\end{array}
\! \phantom{}^{\textstyle{\searrow}}_{\textstyle{\nearrow}} \gl_{n+1}\oplus\gl_{n+1}\ .$$
0
Completeness of relations {#sectionproofs}
=========================
#### 1.
We first give general arguments, proving the weakened version, in which $\Uh$ is enlarged to $\UUh$, of Theorem \[mthe\].
.2cm As before, denote by $\mathfrak{R}$ the system , , , , and of relations. We shall see that it is equivalent to the system of the ordering rules. The system $\mathfrak{R}$ follows from since is the set of defining relations for the weight generators; we have to verify the opposite implication. For a moment denote the generators from the set $\{\hs_{ij},\tt_i\}$ by symbols $\pp_L$, labeled by a single index $L$, $L=1,2,\dots ,n^2$. The number of ordering rules for $n^2$ variables $\pp_L$ is $n^2(n^2-1)/2$. So, to prove the completeness, it is sufficient to show that the dimension of the subspace (over $\UUh$) spanned by $\mathfrak{R}$ is at least $n^2(n^2-1)/2$. Any relation from $\mathfrak{R}$ is a sum of products $\pp_L\mult\pp_M$ with coefficients in $\Uh$ plus, possibly, a term of zero degree in $\pp$’s. Denote by $\mathfrak{R}_0$ the system $\mathfrak{R}$ with degree zero terms dropped. It suffices to show that \_0 /[2]{} .\[fafa\]Once the coefficients from $\Uh$ in all relations from $\mathfrak{R}_0$ are placed on the same side, say, on the right, from the monomials $\pp_L\mult\pp_M$, one can give arbitrary numerical values to the variables $\th_{ij}$ (respecting linear dependencies between them). To check the assertion it is enough to find a set of values for which the corresponding system with numerical coefficients has $n^2(n^2-1)/2$ linearly independent relations . But when all $\th_{ij}$ tend to $\infty$ (in the following way: $\th_{i,i+1}=c_{i,i+1}h$, $h\rightarrow\infty$ and $c_{i,i+1}$ are constants), we directly observe that the system $\mathfrak{R}_0$ becomes simply $\pp_L\mult\pp_M=\pp_M\mult\pp_L$, $M>L$. The proof of the completeness over $\UUh$ is finished. ${\relax\ifmmode\else\unskip\quad\fi{\hbox{\rlap{$\sqcap$}$\sqcup$}}}$
Note that we did not use in the above arguments the compatibility of the ordering $\prec$ with the partial order $<$ on $\h^*$.
#### 2.
Given an order, let $X$ be a formal vector of all unordered products $\pp_L\mult\pp_K$ and $Y$ a formal vector of all ordered products. To rewrite $\mathfrak{R}$ in the form of ordering relations, one has to solve for $X$ a linear system of equations X=Y+C ,\[caode\]where $C$ is a vector of degree 0 terms; ${\cal{A}}$ and ${\cal{B}}$ are certain matrices with coefficients in $\Uh$ (by the above proof, ${\cal{A}}$ is a square matrix). The solution of this system may cause an appearance of coefficients from $\UUh$ (not from $\Uh$) in the ordering relations. This happens, for example, for the lexicographical order for the generators $z_{ij}$ and $t_i$ (with $z_{ii}=t_i$) of $\Z_n$ for $n>2$ (we don’t give details; it is an explicit calculation). It follows from (and statement (e) of Section \[section2new\]) that for the order - the solution of the system is defined over the ring $\Uh$. However, this shows only that for this order possible terms from $\UUh$ in the determinant of ${\cal{A}}$ simplify in the combinations ${\cal{A}}^{-1}{\cal{B}}$ and ${\cal{A}}^{-1}C$; the systems $\mathfrak{R}$ and $\mathfrak{R}^\prec$ may still be not equivalent over $\Uh$, in the sense that the elements of the matrix ${\cal{A}}^{-1}$ may not belong to $\Uh$ and we cannot transform the system $\mathfrak R$ to the system $\mathfrak{R}^\prec$ by composing linear over $\Uh$ combinations of relations from $\mathfrak R$.
#### 3.
We now pass to the proof of Theorem \[mthe\]. Let ${\cal{F}}$ be the free algebra with the weight generators $\s_{ij}$ and $t_i$ over $\Uh$. Let $\mathfrak{R}^\prec$ be the set of ordering relations . Both $\mathfrak{R}$ and $\mathfrak{R}^\prec$ are defined over $\Uh$ and we have the homomorphism $\varpi :\, {\cal{F}}/\mathfrak{R}\rightarrow {\cal{F}}/\mathfrak{R}^\prec$. According to the weak form of theorem \[mthe\], see paragraph 1 of this subsection, the homomorphism $\varpi$ becomes the isomorphism after taking the tensor product with $\UUh$ (over $\Uh$). We shall now prove that $\varpi$ itself is the isomorphism.
The proof is done by induction in $n$, with the help of the stabilization law and an explicit calculation of certain determinants (and one can follow the precise structure of appearing denominators at each step). The induction base is $n=1$, there is nothing to prove for $\Z_1$.
.2cm All we have to show in general case is that the numerator of the determinant of the matrix ${\cal{A}}$, figuring in , is a product of linear factors of the form . The relations are weighted so the matrix ${\cal{A}}$ has a block structure, blocks ${\cal{A}}_\lambda$ are labeled by the relation weights. The determinant of ${\cal{A}}$ is the product of the determinants of the blocks ${\cal{A}}_\lambda$.
Consider $\Z_{n-1}$ as a subspace in $\Z_n$ as in section \[subsection3.4\]. Fix a weight $\lambda$ for $\Z_{n-1}$. Call ${\cal{L}}_\lambda^{(n)}$ the linear subsystem ${\cal{A}}_\lambda X_\lambda ={\cal{B}}_\lambda Y_\lambda +C_\lambda$ of , corresponding to the weight $\lambda$ for $\Z_n$. The system ${\cal{L}}_\lambda^{(n)}$ contains the subsystem $_{\phantom{\lambda}}^{(n)}
\! {\cal{L}}_\lambda^{(n-1)}$, corresponding to the generators from $\Z_{n-1}$ (recall that the relations are labeled by pairs of generators, so the subsystem $_{\phantom{\lambda}}^{(n)}\! {\cal{L}}_\lambda^{(n-1)}$ is well defined). Compare $_{\phantom{\lambda}}^{(n)}\! {\cal{L}}_\lambda^{(n-1)}$ with the corresponding system ${\cal{L}}_\lambda^{(n-1)}$ for $\Z_{n-1}$. By the stabilization principle, the system ${\cal{L}}_\lambda^{(n-1)}$ is the cut of the system $_{\phantom{\lambda}}^{(n)}\! {\cal{L}}_\lambda^{(n-1)}$ in the sense of section \[subsection3.4\]: there is a bijection between the two systems and the relations from $_{\phantom{\lambda}}^{(n)}\! {\cal{L}}_\lambda^{(n-1)}$ have, compared to the corresponding relations from ${\cal{L}}_\lambda^{(n-1)}$, extra terms with $\s_{ni}\mult \s_{jn}$ for certain $i,j<n$. By induction, ${\cal{L}}_\lambda^{(n-1)}$ is equivalent, over its own $\Uh$, to the system of ordering relations. Making the same transformation with the system $_{\phantom{\lambda}}^{(n)}
\! {\cal{L}}_\lambda^{(n-1)}$ preserves the ordered form since the terms $\s_{ni}\mult \s_{jn}$ are ordered. This argument shows that we need to consider only the subset of relations labeled by those pairs of generators $(\pp_L,\pp_M)$ for which $\pp_L$ or $\pp_M$ do not belong to $\Z_{n-1}$, (\_L,\_M) : \_L\_[n-1]{} \_M\_[n-1]{} .\[indrests\]Applying the just constructed ordering rules (equivalent to the system $_{\phantom{\lambda}}^{(n)}\! {\cal{L}}_\lambda^{(n-1)}$) to these remaining relations, we leave in them only ordered terms $\pp_{L'}\pp_{M'}$, $L'<M'$, with two generators from $\Z_{n-1}$, $\pp_{L'},\pp_{M'}\in \Z_{n-1}$. .2cm
We shall now consider separately each type of weight relations listed in the beginning of Section \[section3.3\]. The relations of types 1 and 2 do not cause any difficulty.
.2cm The number of relations of the types 3 or 4 grows with $n$. The change of variables , as well as the renormalization and its inverse, have allowed denominators, so we can work with the generators $\tt_i$ and $\hs_{ij}$ instead of $t_i$ and $\z_{ij}$.
#### 4. Relations of type 4.
For the relations and , the restriction shows that we have to consider only the subsystem, corresponding to the pairs $(\z_{ni},\z_{in})$, $i<n$, of the relations . By the arguments from the paragraph above, we assume that the only unordered quadratic monomials in this subsystem are $\hs_{in}\mult\hs_{ni}$, $i<n$. Rewrite this subsystem in the form : \_[in]{}\_[ni]{}+\_[a:a<n,ai]{} \_[an]{}\_[na]{}=… ,where dots stand for ordered terms. Therefore, the matrix $\mathring{\mathfrak{A}}$, whose determinant we need to calculate, is simply \_[ij]{}:= , \[bacd\]where, we recall, $\th_{ij}=\th_{i}-\th_{j}$; in particular, $\th_{ii}=0$. The determinant of such matrix is well known. The matrix $\mathring{\mathfrak{A}}$ is the specialization of the matrix \_[ij]{}:= \[cma\]at $x_i=\th_i$ and $y_j=-\th_j+1$. The determinant of the matrix $ {\mathfrak{A}}$, calculated in [@Ca], is $\det {\mathfrak{A}}=\prod_{i,j:i<j}\,
\Bigl( (x_i-x_j)(y_i-y_j)\Bigr)/\prod_{i,j}\, (x_i+y_j)$. It follows that $\det \mathring{\mathfrak{A}}=\prod_{i,j:i<j}\, \th_{ij}^2/(\th_{ij}^2-1)$. The inverse of $\mathring{\mathfrak{A}}$ has thus allowed denominators.
#### 5. Relations of type 3.
For the relations and of the type 3, the restriction shows that we have to consider only the relations of the weights $\ve_i-\ve_n$ and $\ve_n-\ve_i$, $i<n$.
.2cm We start with the weight $\ve_i-\ve_n$ with a fixed $i$, $i<n$. The unordered quadratic monomials of the weight $\ve_i-\ve_n$ are $$\begin{aligned}
&\hs_{an}\mult\hs_{ia}&\ \ \ \ {\mathrm{with}}\ \ a:\, 2a<i+n\ ,\ a\neq i\ ,\label{fcf31}\\[.5em]
&\hs_{ij}\mult\hs_{jn} &\ \ \ \ {\mathrm{with}}\ \ j:\, i+n\leq 2j\ ,\label{fcf32}\\[.5em]
&\hs_{in}\mult\tt_b\ .&\label{fcf33}\end{aligned}$$ All relations and participate in our system. However, the system is block-triangular and can be analyzed.
.2cm Denote by ${\mathfrak{r}}_{ikl}$ the relation from the list whose left hand side starts with $\hs_{ik}\mult\hs_{kl}$. Let $\boldsymbol{\kappa}_{ikl}$ be the coefficient of the term $\hs_{ik}\mult\hs_{kl}$ in ${\mathfrak{r}}_{ikl}$. The relations ${\mathfrak{r}}_{ijn}$ can be rewritten in the form (since $\th_{jj}=0$) \_[ij]{}\_[jn]{}\_[ijn]{}=\_[a:ai,a<n]{}\_[an]{}\_[ia]{}+… .Here dots stand for ordered terms with $\tt_b\mult\hs_{in}$ (the term with $\hs_{jn}\mult \hs_{ij}$ is absorbed into the sum). Among the unordered monomials - only the monomials enter the relations ${\mathfrak{r}}_{ijn}$ with $j$ such that $2j<i+n$ and $j\neq i$. Thus the subsystem $\{ {\mathfrak{r}}_{ijn}\, |\, j:\, 2j<i+n\, ,\, j\neq i\}$ contains as many relations as unordered monomials. The matrix, whose determinant we have to calculate in order to express, using this subsystem, the unordered monomials in terms of ordered monomials is $\mathring{\mathfrak{A}}_{aj}':=
B_{ai}/(\th_{ja}+1)$; the $a$-th row contains $B_{ai}$ as the common factor, so the determinant of the matrix $\mathring{\mathfrak{A}}'$ is the product of $B_{ai}$ (over $a$ such that $\, 2a<i+n$ and $a\neq i$) times the determinant of the matrix of the same form as before. Thus the inverse of the determinant of the matrix $\mathring{\mathfrak{A}}'$ belongs to $\Uh$. We use this subsystem to order the monomials .
.2cm After the monomials are ordered, the rest of the relations ${\mathfrak{r}}_{ijn}$ (with $j:\, i+n\leq 2j$) turns into the set of the ordering relations for the monomials ; each relation contains exactly one unordered monomial of the form with the coefficient $\boldsymbol{\kappa}_{ijn}$ whose inverse has allowed determinants.
.2cm The set of relations provides the ordering rules for the monomials $\hs_{in}\mult\tt_k$ once one knows the ordered expressions for all monomials $\hs_{an}\mult\hs_{ia}$.
#### 6. Relations of type 3, weight $\ve_n-\ve_i$.
The considerations of Section \[section2new\] show that for any two orders on the weight basis of $\p$, compatible with the partial order $<$ on $\h^*$, the ordering relations for them are equivalent over $\Uh$. Define, instead of - , the order $\grave{\prec}$ by $$\label{not4p}\z_{ij}\grave{\prec}\z_{kl}\quad \text{if}
\quad i-j>k-l\quad\text{or}\quad \left\{\begin{array}{ll}i>k\quad &{\mathrm{if}}
\quad i-j=k-l>0\ ,\\[.3em]
i<k\quad &{\mathrm{if}}\quad i-j=k-l<0\ ,\\[.3em] {\mathrm{arbitrarily}}
\quad &{\mathrm{if}}\quad i-j=k-l=0\ .\end{array}\right.$$ The peculiarity of the order $\grave{\prec}$ is that the anti-involution $\epsilon$, see , transforms the set of quadratic ordered monomials of any non-zero weight $\lambda$, $\lambda\neq 0$, into the set of quadratic ordered monomials of the weight $(-\lambda)$.
It is proved in [@KO2], that the system $\mathfrak{R}$ is closed under the anti-involution $\epsilon$ (that is, $\mathfrak{R}$ and $\epsilon (\mathfrak{R})$ are equivalent over $\Uh$). For the order $\grave{\prec}$, the application of the anti-involution $\epsilon$ reduces the question about the equivalence over $\Uh$ of $\mathfrak{R}$ and the set of the ordering relations for the weight $\ve_n-\ve_i$ to the same question for the weight $\ve_i-\ve_n$. By the preceding paragraph, the equivalence assertion follows for the order $\grave{\prec}$ and therefore for any other order, compatible with the partial order $<$ on $\h^*$, for example, the order $\prec$.
The proof of the theorem \[mthe\] is completed. ${\relax\ifmmode\else\unskip\quad\fi{\hbox{\rlap{$\sqcap$}$\sqcup$}}}$
.2cm The set $\mathfrak{R}^\prec$ of ordering relations is, by construction, closed over $\Uh$ under the involution $\omega$, see . As a by-product of the equivalence of $\mathfrak{R}$ and $\mathfrak{R}^\prec$ over $\Uh$ we observe that $\mathfrak{R}$ is closed over $\Uh$ under the involution $\omega$ as well.
.2cm Note that all denominators, which appeared in the proof, are of the form $\th_{ij}\pm \varsigma$, $i<j$, where $\varsigma =0,1$ or 2.
#### 7.
As the proof shows, essentially the only matrix we have to invert is of the form . The matrix, inverse to reads (\^[-1]{})\_[ij]{}=-\_[a:ai]{} \_[b:bj]{} .\[bacdinv\]The verification of in the form $\sum_j (\mathring{\mathfrak{A}}^{-1})_{ij}
\mathring{\mathfrak{A}}_{jk}=\delta_{jk}$, where $\delta_{jk}$ is the Kronecker delta, reduces to the identity \_[b:bi]{} -\_[j:ji]{} \_[b:bi,j]{} =\_[ik]{}\_[b:bi]{} ,which is checked by an evaluation of residues and the values at infinity of both sides as functions of $\th_i$.
.2cm The inverse of the more general matrix reads (\^[-1]{})\_[ij]{}=(x\_j+y\_i)\_[a:aj]{} \_[b:bi]{} .\[cmainv\]It is demonstrated similarly to , by an appropriate evaluation of residues and the values at infinity.
.2cm The formula is equivalent (not directly equal) to the specialization of at $x_i=\th_i$ and $y_j=-\th_j+1$.
.2cm The formula provides a recursive way to transform the system $\mathfrak{R}$ into the set of ordering relations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Loïc Poulain d’Andecy for an independent partial confirmation of correctness of the relations from subsection \[section3.3\] by a computer aided check of the Poincaré–Birkhoff–Witt theorem for the algebra $\Z_4$ defined by the generators and relations. We are indebted to Elena Ogievetskaya for valuable help in preparation of the manuscript.
.2cm A part of the present work was done during visits of S. K. to CPT and CIRM in Marseille. The authors thank the staff of the Institutes for providing excellent working conditions during these visits. S. K. was supported by the RFBR grant 08-01-00667, joint CNRS-RFBR grant 09-01-93106 and the grant for Support of Scientific Schools 3036-2008-2. Both authors were supported by the ANR project GIMP No. ANR-05-BLAN-0029-01.
[\[EHW\]]{}
R. M. Asherova, Yu. F. Smirnov and V. N. Tolstoy, [*Projection operators for simple Lie groups. II. General scheme for construction of lowering operators. The groups SU(n)*]{} (Russian), Teoret. Mat. Fiz. [**15**]{} (1973) 107–119. R. M. Asherova, Yu. F. Smirnov and V. N. Tolstoy, [*Description of a certain class of projection operators for complex semi-simple Lie algebras*]{} (Russian), Matem. Zametki [**26**]{} (1979) 15–25. A. Cauchy, [*Mémoire sur les fonctions alternées et sur les sommes alternées*]{}, in: “Exercices d’analyse et de physique mathématique”, Volume 2, Paris, Bachelier (1841) 151–159. S. Khoroshkin and O. Ogievetsky, [*Mickelsson algebras and Zhelobenko operators*]{}, J. Algebra [**319**]{} (2008) 2113–2165. S. Khoroshkin and O. Ogievetsky, [*Structure constants of diagonal reduction algebras of $\gl$ type*]{}, to appear. S. Khoroshkin and O. Ogievetsky, [*Examples of diagonal reduction algebras*]{}, to appear. J. Mickelsson, [*Step algebras of semisimple subalgebras of Lie algebras*]{}, Rep. Math. Phys. [**4**]{}:4 (1973) 303–318. D. Zhelobenko, [*Representations of reductive Lie algebras*]{}, Nauka, Moscow (1994).
[^1]: Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud Toulon – Var; laboratoire affilié à la FRUMAM (FR 2291)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Strongly repelling bosons in two-dimensional harmonic traps are described through breaking of rotational symmetry at the Hartree-Fock level and subsequent symmetry restoration via projection techniques, thus incorporating correlations beyond the Gross-Pitaevskii (GP) solution. The bosons localize and form polygonal-ring-like crystalline patterns, both for a repulsive contact potential and a Coulomb interaction, as revealed via conditional-probability-distribution analysis. For neutral bosons, the total energy of the crystalline phase saturates in contrast to the GP solution, and its spatial extent becomes smaller than that of the GP condensate. For charged bosons, the total energy and dimensions approach the values of classical point-like charges in their equilibrium configuration.'
author:
- Igor Romanovsky
- Constantine Yannouleas
- Uzi Landman
date: '25 June 2004; Physical Review Letters [**93**]{}, 230405 (2004)'
title: |
Crystalline boson phases in harmonic traps:\
Beyond the Gross-Pitaevskii mean field
---
Bose-Einstein condensates (BEC’s) in harmonic traps [@corn] are normally associated with weakly interacting neutral atoms, and their physics is described adequately by the Gross-Pitaevskii (GP) mean-field theory [@dal]. Lately, however, experimental advances in controlling the interaction strength [@cor; @grei; @par; @wei] permit the production of novel bosonic states in the regime of strong interparticle repulsions. Theoretical efforts motivated by this capability include studies of the Bose-Hubbard model [@jak; @mot], and investigations about the “fermionization” limit of an one-dimensional (1D) gas of trapped impenetrable bosons [@gir2; @dunj; @ast], often referred to as the Tonks-Girardeau (TG) regime [@gir2; @gir]. In this Letter, we address the still open problem of strongly repelling (impenetrable) bosons in higher dimensions, and in particular in two dimensions (2D).
We describe the strongly repelling bosons through symmetry breaking at the Hartree-Fock (HF) mean-field level followed by post-Hartree-Fock symmetry restoration, thus incorporating correlations beyond the GP solution. This two-step method, which has not been applied yet to the bosonic many-body problem, has been shown to successfully describe strongly correlated electrons in 2D semiconductor quantum dots [@yl]. We focus here on results for 2D interacting bosons in a harmonic trap, with the extension to 3D systems being straightforward.
To illustrate our method, we consider systems with a few bosons. The method describes the transition from a BEC state to a crystalline phase, in which the trapped localized bosons form crystalline patterns. In 2D, these patterns are ring-like, both for a repulsive contact and a Coulomb interaction. At the mean-field level, these crystallites are static and are portrayed directly in the single-particle densities. After restoration of symmetry, the single-particle densities are rotationally symmetric, and thus the crystalline symmetry becomes “hidden”; however, it can be revealed in the conditional probability distribution (CPD, anisotropic pair correlation), $P({\bf r},{\bf r}_0)$, which expresses the probability of finding a particle at ${\bf r}$ given that the “observer” (i.e., reference point) is riding on another particle at ${\bf r}_0$ [@yl4].
Mean-field symmetry breaking for bosonic systems has been discussed earlier in the context of two-component condensates, where each species is associated with a different space orbital [@esr]. We consider here one species of bosons, but allow each particle to occupy a different space orbital $\phi_i({\bf r}_i)$. The permanent $|\Phi_N \rangle = {\it Perm}[\phi_1({\bf r}_1), ..., \phi_N({\bf r}_N)]$ serves as the many-body wave function of the [*unrestricted*]{} Bose-Hartree-Fock (UBHF) approximation. This wave function reduces to the Gross-Pitaevskii form with the [*restriction*]{} that all bosons occupy the same orbital $\phi_0({\bf r})$, i.e., $|\Phi^{\text{GP}}_N \rangle =\prod_{i=1}^N \phi_0({\bf r}_i)$, and $\phi_0({\bf r})$ is determined self-consistently at the restricted Bose-Hartree-Fock (RBHF) level [@note1] via the equation [@esr2] $ [ H_0({\bf r}_1) + (N-1) \int d{\bf r}_2 \phi^*_0({\bf r}_2)
V({\bf r}_1,{\bf r}_2) \phi_0({\bf r}_2)]
\phi_0({\bf r}_1) = \varepsilon_0 \phi_0({\bf r}_1)$. Here $V({\bf r}_1,{\bf r}_2)$ is the two-body repulsive interaction, which can be either a long-range Coulomb force, $V_C=Z^2e^2/(\kappa |{\bf r}_1 -{\bf r}_2|)$, for charged bosons or a contact potential, $V_{\delta}= g\delta({\bf r}_1 -{\bf r}_2)$, for neutral bosons. The single-particle hamiltonian is given by $H_0({\bf r}) =
-\hbar^2 \nabla^2 /(2m) + m \omega_0^2 {\bf r}^2/2$, where $\omega_0$ characterizes the harmonic confinement.
[*First step: Symmetry breaking.*]{} Going beyond the GP approach to the [*unrestricted*]{} Hartree-Fock level (i.e., using the permanent $|\Phi_N \rangle$) results in a set of UBHF equations with a higher complexity than that encountered in electronic structure problems \[13(a)\]. Consequently, we simplify the UBHF problem by considering explicit analytic expressions for the space orbitals $\phi_i({\bf r}_i)$. In particular, since [*the bosons must avoid occupying the same position in space in order to minimize their mutual repulsion*]{}, we take all the orbitals to be of the form of displaced Gaussians [@note2], namely, $\phi_i({\bf r}_i) = \pi^{-1/2} \sigma^{-1}
\exp[-({\bf r}_i - {\bf a}_i)^2/(2 \sigma^2)]$. The positions ${\bf a}_i$ describe the vertices of concentric regular polygons, with both the width $\sigma$ and the radius $a=|{\bf a}_i|$ of the regular polygons determined variationally through minimization of the total energy $E_{\text{UBHF}} = \langle \Phi_N | H | \Phi_N \rangle$ /$\langle \Phi_N | \Phi_N \rangle$, where $H = \sum_{i=1}^N H_0({\bf r}_i) + \sum_{i < j}^{N}
V( {\bf r}_i,{\bf r}_j)$ is the many-body hamiltonian.
With the above choice of localized orbitals, the unrestricted permanent $|\Phi_N \rangle$ breaks the continuous rotational symmetry. However, for both the cases of a contact potential and a Coulomb force, the resulting energy gain becomes substantial for stronger repulsion. Controlling this energy gain (the strength of correlations) is the ratio $R$ between the strength of the repulsive potential and the zero-point kinetic energy. Specifically, for a 2D trap, one has $R_{\delta} = gm/(2\pi\hbar^2)$ for a contact potential and $R_W=Z^2e^2/(\hbar \omega_0 l_0)$ for a Coulomb force, with $l_0=\sqrt{\hbar/(m\omega_0)}$ being the characteristic harmonic-oscillator length. (The subscript $W$ in the case of a Coulomb force stands for “Wigner”, since the Coulomb crystallites in harmonic traps are finite-size analogs of the bulk Wigner crystal [@wig].)
{width="6.2cm"}
In Fig. 1, we display as a function of the parameters $R_{\delta}$ (a) and $R_W$ (b), respectively, the total energies for $N=6$ bosons calculated at several levels of approximation. In both cases the lowest UBHF energies correspond to a (1,5) crystalline configuration, namely one boson is at the center and the rest form a regular pentagon of radius $a$. Observe that the GP total energies are slightly lower than the $E_{\text{RBHF}}^{\text{Gauss}}$ ones; however, both exhibit an unphysical behavior since they diverge as $R_{\delta} \rightarrow \infty$. This behavior contrasts sharply with that of the unrestricted Hartree-Fock energies, $E_{\text{UBHF}}$, and those of the projected (PRJ) states (see below), which saturate as $R_{\delta} \rightarrow \infty$; in fact, a value close to saturation is achieved already for $R_{\delta}$ ($R_W$) $\sim$ 10. We have checked that for all cases with $N=2 - 7$, the total energies exhibit a similar behavior. For a repulsive contact potential, the saturation of the UBHF energies is associated with the ability of the trapped bosons (independent of $N$) to minimize their mutual repulsion by occupying different positions in space, and this is one of our central results. For $N=2$, the two bosons localize at a distance $2a$ apart to form an antipodal dimer. For $N \leq 5$ the preferred UBHF crystalline arrangement is a single ring with no boson at the center \[usually denoted as $(0,N)$\]. $N=6$ is the first case having one boson at the center \[designated as $(1,N-1)$\], and the (0,6) arrangement is a higher energy isomer.
The saturation found here for 2D trapped bosons interacting through strong repelling contact potentials is an illustration of the “fermionization” analogies that appear in strongly correlated systems in all three dimensionalities. Indeed such energy saturation has been shown for the TG 1D gas [@gir; @gir2], and has also been discussed for certain 3D systems (i.e., three trapped bosons [@blum] and an infinite boson gas [@heis]). Saturation of the energy and the length of the trapped atom cloud (and thus of the interparticle distance) has been measured recently for the 1D TG gas (see in particular Fig. 3 and Fig. 4 in Ref. [@wei] and compare to the similar trends predicted here for the 2D case in Fig. 1 and Fig. 2).
For the Coulomb potential \[see Fig. 1(b)\], the displayed total energies have been referenced to the classical energy $E^{\text{cl}}_C$ [@yl3] (plus the zero-point energy) of six trapped point charges in their (1,5) equilibrium configuration, since the total energy of a Wigner crystallite (independently of whether it consists of bosons or fermions) is expected to approach $E^{\text{cl}}_C$ as $R_W \rightarrow \infty$. We see again that the $E_{\text{RBHF}}^{\text{Gauss}}$ energies (one common Gaussian orbital) diverge as $R_W \rightarrow \infty$. In contrast, the unrestricted HF energies $E_{\text{UBHF}}$ remain finite and approach slowly $E^{\text{cl}}_C$ as $R_W \rightarrow \infty$. A similar behavior is exhibited by the total energies for all $N=2 - 7$ cases of charged bosons.
![ Variationally determined widths ($\sigma$) and ring radii ($a$) for $N=6$ harmonically confined 2D bosons as a function of (a) $R_\delta$ and (b) $R_W$, obtained according to the various approximations (as marked in the figure, see also caption of Fig. 1 ). The saturation values of $a$ of the lowest-energy configuration for $2 \leq N \leq 7$ are on the right in (a). Lengths in units of $l_0$. For the UBHF case \[displaying an (1,5) crystallite\] the interparticle distance on the pentagonal shell is $d= ((5-5^{1/2})/2)^{1/2}
a \approx 1.176 a$, showing the same saturation trend as the radius $a$. ](bos_crph_fig2.ps){width="7.2cm"}
In Fig. 2, we display for the $N=6$ bosons the radii of the polygonal rings $a$ and widths $\sigma$ of the Gaussian orbitals obtained in various approximations, as a function of $R_{\delta}$ (a) and $R_W$ (b). For the contact potential, in the RBHF/G approximation we find that $a=0$ and the width (marked as RBHF/G in Fig. 2) keeps increasing continuously as $R_{\delta} \rightarrow \infty$ (this reflects the unsuccessful attempt of the common orbital to minimize the mutual repulsion between the bosons by spreading out as far as possible). In contrast, the unrestricted widths $\sigma_{\text{UBHF}}$ associated with the displaced Gaussian orbitals (that correspond to a lower total energy, see Fig. 1) saturate to a constant value. Similar behaviors are also exhibited by $\sigma_{\text{RBHF}}^{\text{Gauss}}$ and $\sigma_{\text{UBHF}}$ in the case of a Coulomb force \[see of Fig. 2(b)\].
The radii $a$ associated with the pentagonal ring of localized orbitals, however, exhibit a different behavior depending on whether the repulsive potential is a contact or a Coulomb one. Indeed, in the Coulomb case, the radii $a_{\text{UBHF}}$ keep increasing with $R_W$, approaching the equilibrium radius $a^{\text{cl}}_C= 1.334 l_0 R_W^{1/3}$ of six $Ze$ classical point-charges in a harmonic trap in the (1,5) configuration [@yl3]. In contrast, for a repulsive contact potential, the radii $a_{\text{UBHF}}$ saturate to a constant value $\approx 2 l_0$. The dependence of the saturation values of $a$ on $N$ (for $3 \leq N \leq 7$) for the lowest-energy configurations is shown on the right in Fig. 2(a). The different behavior of the boson positions in the UHBF crystallite is a natural consequence of the long-range character of the Coulomb potential versus the short-range contact potential.
[*Second step: Restoration of broken symmetry.*]{} Although the optimized UBHF permanent $|\Phi_N \rangle$ performs exceptionally well regarding the total energies of the trapped bosons, in particular in comparison to the resctricted wave functions (e.g., the GP anzatz), it is still incomplete. Indeed, due to its localized orbitals, $|\Phi_N \rangle$ does not preserve the circular (rotational) symmetry of the 2D many-body hamiltonian $H$. Instead, it exhibits a lower point-group symmetry, i.e., a $C_2$ symmetry for $N=2$ and a $C_5$ one for $N=6$ (see below). As a result, $|\Phi_N \rangle$ does not have a good total angular momentum. This is resolved through a post-Hartree-Fock step of [*restoration*]{} of broken symmetries via projection techniques \[13(b)\],[@yl3], yielding a new wave function $|\Psi_{N,L}^{\text{PRJ}} \rangle$ [@note45] with a definite angular momentum $L$. Here, we focus on the properties of the ground state, i.e., $L=0$; the corresponding energy is $E_0^{\text{PRJ}}$.
For $N=6$ 2D bosons, Fig. 1 shows that the $E_0^{\text{PRJ}}$ energies share with the UBHF ones the saturation property for the case of a contact-potential repulsion, as well as the property of converging to $E^{\text{cl}}_C$ as $R_W \rightarrow \infty$ for the case of a Coulomb repulsion. In both cases, however, the projections bring further lowering [@note43] of the total energies compared to the UBHF ones. Thus, for strong interactions (large values of $R_\delta$ or $R_W$) the restoration-of-broken-symmetry step yields an excellent approximation of both the exact many-body wave function and the exact total energy [@note44].
![ (a-c): Single-particle densities for $N=6$ 2D harmonically trapped neutral bosons with a contact interaction and $R_\delta=25$. (a) The single-orbital self-consistent GP case. (b) The symmetry broken UBHF case (static crystallite). (c) The projected (symmetry-restored wave function, see Ref. [@note45]) case (collectively fluctuating crystallite). The crystalline structure of the outer ring in this last case is “hidden”, but it is revealed in the conditional probability distribution [@yl4] displayed in (d), where the observation point is denoted by a black dot (on the right). Lengths in units of $l_0$. ](bos_crph_fig3.eps){width="8.0cm"}
The transformations of the single-particle densities (displayed in Fig. 3 for $N=6$ neutral bosons interacting via a contact potential and $R_\delta=25$) obtained from application of the successive approximations provide an illustration of the two-step method of symmetry breaking with subsequent symmetry restoration. Indeed, the GP single-particle density \[Fig. 3(a)\] is circularly symmetric, but the UBHF one \[Fig. 3(b)\] explicitly exhibits a (1,5) crystalline configuration. After symmetry restoration \[Fig. 3(c)\], the circular symmetry is re-established, but the single-particle density is radially modulated unlike the GP density. In addition, the crystalline structure in the projected wave function is now hidden; however, it can be revealed through the use of the CPD [@yl4] \[see Fig. 3(d)\], which resembles the (crystalline) UBHF single-particle density, but with one of the humps on the outer ring missing (where the observer is located). In particular, $P({\bf r}_0, {\bf r}_0) \approx 0$ and the boson associated with the observer is surrounded by a “hole” similar to the exchange-correlation hole in electronic systems. This is another manifestation of the “fermionization” of the strongly repelling 2D bosons. However, here as in the 1D TG case [@gir; @gir2], the vanishing of $P({\bf r}_0,{\bf r}_0)$ results from the impenetrability of the bosons. For the GP condensate, the CPD is independent of ${\bf r}_0$, i.e., $P_{\text{GP}}({\bf r}, {\bf r}_0) \propto |\phi_0({\bf r})|^2$, reflecting the absence of any space correlations.
It is of importance to observe that the radius of the BEC \[GP case, Fig. 3(a)\] is significantly larger than the actual radius of the strongly-interacting crystalline phase \[projected wave function, Fig. 3(c)\]. This is because the extent of the crystalline phase saturates, while that of the GP condensate grows with no bounds as $R_\delta
\rightarrow \infty$. Such dissimilarity in size (between the condensate and the strongly-interacting phase) has been also predicted [@dunj] for the trapped 1D Tonks-Girardeau gas and indeed observed experimentally [@wei]. In addition, the 2D single-particle momentum distributions for neutral bosons have a one-hump shape with a maximum at the origin (a behavior exhibited also by the trapped 1D TG gas). The width of these momentum distributions versus $R_\delta$ increases and saturates to a finite value, while that of the GP solution vanishes as $R_\delta \rightarrow \infty$.
In conclusion, we provided a solution to strongly repelling bosons in 2D harmonic traps using a two-step method of breaking of rotational symmetry at the unrestricted Bose-Hartree-Fock level and of subsequent symmetry restoration. This method yields substantially lower total energies compared to the GP solution, through the inclusion of correlations beyond the single-orbital Bose-Einstein condensate. We find that the bosons become localized and form crystalline patterns made of concentric polygonal rings, both for a repulsive contact and a Coulomb interaction. For neutral bosons the total energy of the crystalline phase saturates with increasing strength of the repulsion, in contrast to the GP condensate whose energy diverges. Furthermore, the spatial extent saturates and becomes smaller than that of the GP condensate, which grows without limit. For charged bosons, the total energy and spatial extent of the crystalline phase approach the classical values of point-like charges in their equilibrium configuration as $R_W \rightarrow \infty$. In light of the above, we trust that our predictions will provide the impetus for experimental efforts to access the regime of strongly repelling bosons in two dimensions. To this end we anticipate that extensions of methodologies developed for the recent realization of the Tonks-Girardeau regime in 1D (using a finite small number of trapped $^{87}$Rb and optical lattices, with a demonstrated wide variation of $R_{\delta}$ from 5 to 200 [@par] and from 1 to 5 [@wei]) will prove most promising. Control of the interaction strength via the use of the Feshbach resonance may also be considered [@cor].
This research is supported by the U.S. D.O.E. (Grant No. FG05-86ER45234 ) and by the NSF.
[99]{} E.A. Cornell and C.E. Wieman, Rev. Mod. Phys. [**74**]{}, 875 (2002); W. Ketterle, Rev. Mod. Phys. [**74**]{}, 1131 (2002). F. Dalfovo [*et al.*]{}, Rev. Mod. Phys. [**71**]{}, 463 (1999). S.L. Cornish [*et al.*]{}, Phys. Rev. Lett. [**85**]{}, 1795 (2000). M. Greiner [*et al.*]{}, Nature (London) [**415**]{}, 39 (2002). B. Paredes [*et al.*]{}, Nature [**429**]{}, 277 (2004). G.T. Kinoshita, T. Wenger, and D.S. Weiss, Science [**305**]{}, 1125 (2004), published after the submission of this Letter. D. Jaksch [*et al.*]{}, Phys. Rev. Lett. [**81**]{}, 3108 (1998). See, e.g., O.I. Motrunich and T. Senthil, Phys. Rev. Lett. [**89**]{}, 277004 (2002). M.D. Girardeau and E.M. Wright, Laser Phys. [**12**]{}, 8 (2002). V. Dunjko [*et al.*]{}, Phys. Rev. Lett. [**86**]{}, 5413 (2001). See, e.g., G.E. Astrakharchik and S. Giorgini, Phys. Rev. A [**66**]{}, 053614 (2002). M. Girardeau, J. Math. Phys. [**1**]{}, 516 (1960). (a) C. Yannouleas and U. Landman, Phys. Rev. Lett. [**82**]{}, 5325 (1999); (E) [**85**]{}, 2220 (2000); Phys. Rev. B [**68**]{}, 035325 (2003); (b) J. Phys.: Condens. Matter [**14**]{}, L591 (2002); Phys. Rev. B [**68**]{}, 035326 (2003). C. Yannouleas and U. Landman, Phys. Rev. Lett. [**85**]{}, 1726 (2000); cond-mat/0401610. P. Öhberg and S. Stenholm, Phys. Rev. A [**57**]{}, 1272 (1998); B.D. Esry , and C.H. Greene, Phys. Rev. A [**59**]{}, 1457 (1999). For the corresponding Hartree-Fock terminology for fermions, see Ref. \[13(a)\]. B.D. Esry, Phys. Rev. A [**55**]{}, 1147 (1997). For strongly correlated electrons in parabolic quantum dots the Gaussian form is adequate in most cases, leading to formation of electron crystallites [@yl] (Wigner molecules). E. Wigner, Phys. Rev. [**46**]{}, 1002 (1934). D. Blume and C.H. Greene, Phys. Rev. A [**66**]{}, 013601 (2002). H. Heiselberg, J. Phys. B: At. Mol. Opt. Phys. [**37**]{}, S141 (2004). C. Yannouleas and U. Landman, Phys. Rev. B [**69**]{}, 113306 (2004). The projected multi-permanent wave function can be written as $ 2 \pi |\Psi_{N,L}^{\text{PRJ}} \rangle = \int^{2\pi}_0 d\gamma
|\Phi_{N}(\gamma) \rangle \exp(i\gamma L)$, where $|\Phi_{N}(\gamma) \rangle$ is the original UBHF permanent having each localized orbital rotated by an azimuthal angle $\gamma$, with $L$ being the total angular momentum. The projection yields wave functions for a whole rotational band. The projected ground-state ($L=0$) energy is given by $E_0^{\text{PRJ}} = \langle \Psi_{N,0}^{\text{PRJ}} | H |
\Psi_{N,0}^{\text{PRJ}} \rangle /
\langle \Psi_{N,0}^{\text{PRJ}} | \Psi_{N,0}^{\text{PRJ}}
\rangle$. The projected ground state is always lower in energy than the original broken-symmetry one \[P.-O. Löwdin, Rev. Mod. Phys. [**34**]{}, 520 (1962), in particular sect. 3\]. Energies calculated from the [*symmetry-breaking*]{} mean field approach (i.e., the UBHF energies and projections thereof) improve relative to the exact energies for larger $N$; see P. Ring and P. Schuck, [*The Nuclear Many-body Problem*]{} (Springer-Verlag, New York, 1980), Ch. 11.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In 2014, Voronov introduced the notion of thick morphisms of (super)manifolds as a tool for constructing $L_{\infty}$-morphisms of homotopy Poisson algebras. Thick morphisms generalise ordinary smooth maps, but are not maps themselves. Nevertheless, they induce pull-backs on $C^{\infty}$ functions. These pull-backs are in general non-linear maps between the algebras of functions which are so-called “non-linear homomorphisms”. By definition, this means that their differentials are algebra homomorphisms in the usual sense. The following conjecture was formulated: an arbitrary non-linear homomorphism of algebras of smooth functions is generated by some thick morphism. We prove here this conjecture in the class of formal functionals. In this way, we extend the well-known result for smooth maps of manifolds and algebra homomorphisms of $C^{\infty}$ functions and, more generally, provide an analog of classical “functional-algebraic duality” in the non-linear setting.'
address:
- 'Department of Mathematics, University of Manchester, Manchester, UK'
-
author:
- 'Hovhannes M. Khudaverdian'
title: 'Non-linear homomorphisms of algebras of functions are induced by thick morphisms'
---
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Introduction
============
A map $\varphi\colon\,\, M\to N$ defines the linear map $$\label{first}
\varphi^*\colon\,\,
C^{\infty}(N)\to C^{\infty}(M)\,,$$ which is homomorphism of algebras of functions. In 2014 Ted Voronov introduced the notion of a [*thick morphism*]{} (see [@thickfirst], [@thicktheor]) of manifolds, which generalises ordinary maps. A thick morphism defines a non-linear map $\Phi^*\colon\,\, C^{\infty}(N)\to C^{\infty}(M)$. This notion provides a natural way to construct $L_\infty$ morphisms for homotopy Poisson algebras (see [@thickfirst],[@thicktheor], and [@thickkoszul] and also Appendix A). The notion of thick morphisms turns out to be also related with quantum mechanis and the construction of spinor representation (see [@tvoscil] and [@thickaction]). The pull-back $\Phi^* \colon\,\,
C^{\infty}(N)\to C^{\infty}(M)$ corresponding to a thick morphism is not in general a homomorphism of algebras (just because it is non-linear). However as it was proved by Voronov, the differential of this non-linear map is a usual pull-back. This motivated him to define so called [*non-linear homomorphisms*]{}.
(Th.Voronov, see [@thicktheor]) \[defofnonlinearhom1\] Let $\A,\B$ be two algebras. A map $L$ from an algebra $\A$ to an algebra $\B$ is called [*a non-linear homomorphism*]{} if at an arbitrary element of algebra $\A$ its derivative is a homomorphism of the algebra $\A$ to the algebra $\B$.
One can say that a thick morphism induces a non-linear homomorphism of algebras of functions in the same way as a usual morphism $\varphi$ induces usual (linear) homomorphism . A natural question was formulated in [@thicktheor]: is it true that every non-linear algebra homomorphism between algebras of smooth functions arises from a thick morphism as the pull–back? Note that the pull-backs by thick morphisms are formal mappings of algebras. Hence in the above definition of non-linear homomorphisms one can consider formal maps only. We prove here this conjecture for formal maps (”formal functionals”).
The structure of the paper is as follows. We recall the construction of thick morphisms, and we define a class of [*formal functionals*]{} which are induced by thick morphisms. We recall the proof of Voronov’s result that the functional induced by a thick morphism is a non-linear homomorphism (see [@thickfirst] and [@thicktheor] for detail). Then we show that the converse implication also holds. In Appendix A we briefly discuss the relation of thick morphisms with $L_\infty$ morphisms of homotopy Poisson algebras. In appendix B we recall some useful polarisation formulae.
[**Acknowledgment.**]{} I am grateful to Th.Th.Voronov not only for continuous help during the work on the paper, but also for possibility to learn thick morphisms first-hand. I am deeply grateful to A.S.Schwarz. He expressed firm belief that thick morphisms can be formulated in terms of functionals which are non-linear homomorphisms. This encouraged me to prove Theorem \[2\], the main result of this paper. I am also grateful to A.Verbovetsky for many useful comments.
Part of this work was done during my visit to Lyon in autumn 2019. I thank O.Kravchenko, C.Roger and Th.Stroble for hospitality. This work was partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR).
Thick morphisms and non-linear functionals
==========================================
Consider two manifolds $M$ and $N$. We denote by $x^i$ local coordinates on $M$, and we denote $y^a$ local coordinates on $N$. To define the thick morphism $\Phi\colon\,\,M\Rrightarrow N$ we consider a function, $S=S(x,q)$, where $x$ is the point on $M$ and $q$ is covector in $T^*N$. We supposee that $S=S(x,q)$ is a formal function, power series over $q$: $$S=S(x,q)=S_0(x)+S_1^a(x)q_a+
S_2^{ab}(x)q_bq_a+
S_3^{cba}(x)q_cq_bq_a+\ldots=$$ $$\label{action1}
S_0(x)+S_1^a(x)q_a+S_+(x,q)
\,, {\rm where}\,\,
S_+(x,q)=
\sum_{k=2}^\infty S^{a_1\dots a_k}q_{a_1}\dots q_{a_k}\,,$$ coefficients $S_{k}^{a_1\dots a_k}(x)$ are usual smooth functions on $x$.
A formal function $S(x,q)$ is called [*generating function of thick morphism*]{}.
In fact $S(x,q)$ is geometrical object which transforms non- trivially under changing of local coordinates (see for detail [@thicktheor]). Here and below we consider only local coordinates $x^i$ on $M$ and $y^a$ on $N$.
To generating function $S(x,q)$ corresponds thick morphism $\Phi=\Phi_S\, \colon\,\,M\Rrightarrow N$ which is defined in the following way: it defines pull-back $\Phi_S^*$ such that to every smooth function $g(y)\in C^\infty(M)$ corresponds a function
$$\label{defofthickmorphism1}
f(x) =\Phi^*(g)=g(y)+S(x,q)-y^aq_a\,$$
where $y^a=y^a(x),q_b=q_b(x)$ are chosen in a way that $$\label{defofthickmorphism2}
y^a={\p S(x,q)\over \p q_a}\,,\quad
q_b={\p g(y)\over \p y^b}\,.$$
Conditions imply that left hand side of equation does not depend on $y^a$ and $q_b$: $${\p \over \p y^a}
\left(
g(y)+S(x,q)-y^aq_a
\right)=0\,,\,
{\p \over \p q_b}
\left(
g(y)+S(x,q)-y^aq_a
\right)=0\,.$$
In the special case if $S(x,q)=S^a(x)q_a$ $y^a=S^a(x)$ and $\Phi^*g$ is the usual pull-back corresponding to the map $y^a=S^a(x)$: $$\label{usualpullback1}
f(x) =\Phi^*(g)=g(y)+S(x,q)-y^aq_a=g\left(S^a(x)\right)\,,$$ and this pull-back corresponds to the usual morphism $y^a=S^a(x)$.
In the general case (if action $S(x,q)$ is not linear over $q$) maps and become formal maps. They become formal power series in $g$ (see for details also equation below). Namely equation defines the formal functional $L(x,g)$ on $C^\infty(N)$ such that $$\label{formalfunctional1}
L(x,g)=
L_0(x,g)+
L_1(x,g)+
L_2(x,g)+\dots=
\sum L_k(x,g)\,,\,
\left(g\in C^\infty(N)\right)$$ where every summand $L_k(x,g)$ takes values in smooth functions on $M$ and it has an order $k$ in $g$: $L_k(x,\lambda g)=\lambda^kL_k(x,g)$. We suppose that $$\label{generalisedfunctions}
L_k(g)=\int L(x,y_1,\dots,y_k)
g(y_1)\dots g(y_k)
dy_1\dots dy_k\,,$$ the kernel $L(x,y_1,\dots,y_n)$ of the functional $L_k(x,g)$ can be generalised functions.
\[defofformalfunctionals\] We denote by $\A$ the space of all formal functionals which have appearance . We denote by $\A_k$ the subspace of functionals which have order $k$ on $g$, ($k=0,1,2,\dots $).
For arbitrary functional $L(x,g)\in \A$ (see equation ) functional $L_k(x,g)$ is projection of functional $L(x,g)$ on subspace $\A_k$. We sometimes denote this projection by $[L(x,g)]_k$ $$\label{formalfunctional1a}
L(x,g)=
\sum L_k(x,g)\,,\quad
L_k(x,g)=[L(x,g)]_k\,.$$ It is useful to denote by $\A_{\geq k}$ ($\A_{\leq k}$) the subspace of functionals which have order bigger or equal than $k$ (less or equal than $k$), $$\label{lessthank}
\A_{\geq k}=\oplus_{i\geq k}A_a\,,\quad
\A_{\leq k}=\oplus_{0\leq i\leq k}A_i\,,$$ and we say that two functionals $L_1, L_2\in\A$ coincide up to the order $k$ if $L_1-L_2\in \A_{\geq k+1}$. We will write in this case that $$\label{equalityoffunctionals}
L_1(g)=L_2(g) ({\rm mod}\, \A_{k+1})$$ Explain how every formal generating function $S(x,q)$, (see equation ) defines thick morphism $\Phi_S$, i.e. how $S(x,q)$ defines a map $\Phi_S^*(g)$ which is a formal functional in $\A$. Functional $\Phi_{S(x,q)}^*(g)$ defines non-linear pull-back, assigning to every smooth function $g\in C^\infty(N)$ a formal sum of smooth functions $\left[\Phi_{S(x,q)}^*(g)\right]_k$, ($k=0,1,2,\dots$). $$\label{defofthickmorphism1a}
\Phi_{S(x,q)}^*(g)=
\sum
\left[\Phi_{S(x,q)}^*(g)\right]_k=
\left[\Phi_{S(x,q)}^*(g)\right]_0+
\left[\Phi_{S(x,q)}^*(g)\right]_1+\dots\,,$$ where $\left[\Phi_{S(x,q)}^*(g)\right]_k$ is component of the functional $\Phi_{S(x,q)}^*(g)$ which has order $k$ in $g$ (see equation ). We will explain how to calculate this map recurrently step by step and we will write explicitly the results of calculations of its first components. (See Propositions \[mapforthickmorphisms\] and \[expressionforthickmorphism\])
As it was mentioned above a map $y^a=y^a(x)$ in equation has to be viewed as a formal sum of smooth maps depending on $g$: $$\label{defofthickmorphism2a}
y^a(x)=y^a(x,g)=\sum y_k(x,g)=
y^a_0(x)+y^a_1(x,g)+\dots=$$ Here every term $y^a_k(x)=y^a_k(x,g)$ is a smooth map of order $k$ in $g$: $$y^a_k(x,\lambda g)=\lambda^k
(x,g)\,.$$ We will show how to calculate map step by step recurrently, and we will write the expressions for calculating first few components of this formal map (see Proposition \[mapforthickmorphisms\] below).
One can see from equations and that initial term $y^a_0(x)$ in equation is equal to $$\label{formalmapforinitialterm}
y_0^a(x)=\left[
{\p S(x,q)\over \p q_a}
\right]_{q=0}=S^a_1(x)\,,$$ and every next term $y^a_{k+1}(x)=y^a_{k+1}(x,g)$ in is expressed recurrently via previous terms $\{y^a_0(x),\dots, y^a_k(x)\}$: $$\label{formalmapforarbitraryterm}
y^a_{k+1}=
\left[
{\p S(x,q)\over \p q_a}\big\vert_
{q_a={\p g(y)\over \p y^a}\big\vert_
{y^a=y^a_{\leq k}(x)}}
\right]_{k+1}\,.$$ Here $y^a_{\leq k}(x)=\sum_{i\leq k} y_i(x)$ according to equation , and $[\,\,]_{r}$ means $r$-th component of the map (see expansion ).
We have already expression for initial component $y_0(x)$ of map $y^a(x)$ in equation . Write down expression for next components $y_1^a(x)$ and $y^a_2(x)$ of this map. We have $$y^a_{1}=
\left[
{\p S(x,q)\over \p q_a}\big\vert_
{q_a={\p g(y)\over \p y^a}\big\vert_{y^a=y^a_{\leq 0}(x)}}
\right]_{1}=$$ $$\label{formalmapforfirstterm}
\left[
\left(
S^a(x)+2S^{ab}(x){\p g(y)\over \p y^a}
\right)
_{y^a=S^a(x)}
\right]_1=2S^{ab}(x)g_b^*(x) \,,$$ and $$y^a_{2}=
\left[
{\p S(x,q)\over \p q_a}\big\vert_
{q_a={\p g(y)\over \p y^a}\big\vert_{y^a=y^a_{\leq 1}(x)}}
\right]_{2}=$$ $$\left[
\left(
S^a(x)+
\sum_{j\geq 1}(j+1)S^{a{b_1\dots b_j}}(x)
{\p g(y)\over \p y^{b_1}}\dots
{\p g(y)\over \p y^{b_j}}
\right)
\big\vert_{y^a=y^a_0(x)+y^a_1(x)}
\right]_{2}=$$ $$= \left[
\left(
S^a(x)+
2S^{ab}(x)
{\p g(y)\over \p y^{b}}
+
3S^{abc}(x)
{\p g(y)\over \p y^{b}}
{\p g(y)\over \p y^{c}}
\right)
\big\vert_{y^a=S^a_0(x)+2S^{ab}g_b^*(x)}
\right]_{2}=$$ $$\label{formalmapforsecondterm}
=3S^{abc}(x)g^*_b(x)g^*_c(x)
+4S^{ab}(x)S^{cd}(x)g^*_{bc}(x)g_d^*(x)$$ where in equations and we used notations $$\label{usefulnotation1}
g^*(x)=g\left(y^a\right)
\big\vert_{y^a=S_1^a(x)}\,,\quad
g_a^*(x)={\p g(y)\over \p y^a}
\big\vert_{y^a=S^a_1(x)}\,,\quad
g_{ab}^*(x)={\p^2 g(y)\over \p y^b \p y^a}
\big\vert_{y^a=S_1^a(x)}\,.$$
Thus collecting the answers in equations , and we come to
\[mapforthickmorphisms\] For thick morphism $\Phi_{S(x,q)}$ formal map $y^a(x)=y^a(x,g)$ in can be calculated recurrently by the equations , . In particular up to order $k\leq 2$ it is defined by the following expression: for arbitrary $g\in C^\infty(N)$, $$y^a(x)=y^a(x,g)=
\underbrace {S_1^a(x)}_
{
\hbox
{\footnotesize term of
order $0$ in $g$}}+$$ $$\label{answerforformalmap1}
=
+\underbrace{2S_2^{ab}(x)g_b^*(x)}
_
{\hbox
{\footnotesize term of
order $1$ in $g$}}+
\underbrace
{
3S^{abc}(x)g^*_b(x)g^*_c(x)+
4S^{ab}(x)S^{cd}(x)g^*_{bc}(x)g_d^*(x)
}_ {\hbox
{\footnotesize term of
order $2$ in $g$}}
\, \,({\rm mod \A_3})\,.$$
Use this Proposition to calculate components $[\Phi_S^*(g)]_k$ of functional $\Phi_S^*(g)$.
Due to definition we have that $$\Phi^*_S(g)=
\left(
g(y^a)+S(x,q)-y^aq_a
\right)
\big\vert_{y^a={\p S(x,q)\over \p q_a}\,,
q_a={\p g(y)\over \p y^a}
}
=$$ $$\label
{calculationinthirdorder1}
\left(
g(y^a)+S_0(x)-\sum_{k\geq 2} (k-1)
S_k^{a_1\dots a_k}(x)
{\p g(y)\over \p y^{a_1}}
\dots
{\p g(y)\over \p y^{a_k}}
\right)
\big\vert_{y^a=y^a_0(x)+y^1_1(x)+\dots}\,,$$ where $y^a=y^a_0(x)+y^1_1(x)+\dots$ is a formal map . Here we used the fact that according to equations , and $$S(x,q)-y^aq_a=S(x,q)-{\p S(x,q)\over \p q_a}q_a=
\sum_k S_k^{a_1\dots a_k}(x)q_{a_1}\dots q_{a_k}-
\sum_k kS_k^{a_1\dots a_k}(x)q_{a_1}\dots q_{a_k}=$$ $$\sum_k (1-k)S_k^{a_1\dots a_k}(x)q_{a_1}\dots q_{a_k}\,.$$ Now using equation and equation in Proposition \[mapforthickmorphisms\] write down first few components $\left[\Phi_S^*(g)\right]_k$ of non-linear functional $\Phi_S^*(g)$ $$\left[ \left( g(y)\right)_{y^a(x)}\right]_{\leq 3}=
g\left(y^a_{\leq 2}(x)\right)=
g\left(y_0(x)+y^a_1(x)+y^a_2(x)\right)=$$ $$g\left(S_1^a(x)+2S^{ab}(x)g_b^*(x)+
4S^{ab}(x)S^{cd}(x)g^*_{bc}(x)g_d^*(x)
\right)=$$ $$g^*(x)+2S^{ab}(x)g_a^*(x)g_b^*(x)+
3S^{abc}(x)g_c^*(x)g_b^*(x)g_a^*(x)+
2S^{ab}(x)S^{cd}(x)g_{ab}^*(x)g^*_a(x)g^*_d(x)\,,$$ where we denoted by $\left[ g\left(y\right)_{y^a(x)}\right]_{\leq 3}$ projection of functional $g\mapsto g\left(y^a\left(x\right)\right)$ on $\A_3$. Hence it follows from equation that $$\left[\Phi_S^*(g)\right]_{\leq 3}=
\left[\Phi_S^*(g)\right]_0+
\left[\Phi_S^*(g)\right]_1+
\left[\Phi_S^*(g)\right]_2+
\left[\Phi_S^*(g)\right]_3=$$ $$S_0(x)+g\left(y^a_{\leq 2}(x)\right)
-S_2^{ab}(x)
{\p g(y)\over \p y^{a}}
{\p g(y)\over \p y^{b}}
\big\vert_{y^a=S^a_0(x)+2S^{ab}(x)g_b^*(x)}-$$ $$-2S_3^{abc}(x)
{\p g(y)\over \p y^{c}}
{\p g(y)\over \p y^{b}}
{\p g(y)\over \p y^{a}}
\big\vert_{y^a=S^a_0(x)}$$
Collecting together the terms we come to formal power sums we come to
\[expressionforthickmorphism\] Formal functional $\Phi_S^*(g)$ corresponding to thick morphism $\Phi_{S(x,q)}$ can be calculated recurrently by equations .
In particular up to the order $\leq 3$ it is defined by the following expression $$\Phi_S^*(g)=
\underbrace{S_0(x)}_
{\footnotesize \hbox{term of order $0$ in $g$}}+
\underbrace
{
g\left(S^a(x)\right)
}_
{\footnotesize \hbox{term of order $1$ in $g$}}+
\underbrace
{
S^{ab}(x)g_b^*(x)g_b^*(x)
}_
{\footnotesize \hbox{terms of order $2$ in $g$}}+$$ $$\label{expansion2}
\underbrace
{
S^{abc}(x)
g_c^*(x)
g_b^*(x)
g_a^*(x)
+
2 S^{ac}
S^{bd}(x)g_{ab}^*(x)
g_d^*(x)
g_c^*(x)
}_
{{\footnotesize \hbox{terms of order $3$ in $g$}}}
\, ({\rm mod} \A_4)\,.$$
Thick morphisms define in general non-linear functionals $\Phi_S^*(g)$ belonging to space of formal functionals $\A$ (see definition of formal functionals in \[defofformalfunctionals\]). As it was mentioned in introduction these non-linear functionals are non-linear homomorphisms. Return to definition \[defofnonlinearhom1\] of non-linear homomorphisms formulating it for formal functionals.
\[defoflocal\] Let $L=L(x,g)$ be formal functional in $\A$ (see definition \[defofformalfunctionals\]). According to definition \[defofnonlinearhom1\] this formal functional is [*non-linear homomorphism*]{} if its differential is usual homomorphism, i.e. for every function $g$ there exists a map $$\label{formalmap}
y^a(x)=K^a(x,g)\,,$$ such that for an arbitrary function $h$ $$\label{differentialislocal}
L(g+\vare h)-L(g)=
\vare h \left(y^a(x,g)\right)\,,\quad(\vare^2=0)\,.$$
The map $y^a(x,g)=K^a(x,g)$ in is in general a formal map: $$y^a(x,g)=K^a_0(x)+
K^a_1(x,g)+
K^a_2(x,g)+\dots=$$ $$\label{formalmap1}
K^a_0(x)+
\int K^a_1(x,y)g(y)dy+
\int K^a_1(x,y_1,y_2)g(y_1)g(y_2)dy_1dy_2+\dots$$
Now we formulate
\[1\] Let $\Phi=\Phi_S\colon M\Rrightarrow N$ be an arbitrary thick morphism. Then formal functional $\Phi_S^*(g)$ is non-linear homomorphism, i.e. for arbitrary functions $g$ there exists a map $y^a(x)=y^a(x,g)$ such that for an arbitrary function $h$, ( $h\in C^\infty {N}$) $$\label{theorem1}
\Phi_S^*(g+\vare h)-
\Phi_S^*(g)=\vare h\left(y^a(x,g)\right)\,,\quad
\vare^2=0\,.$$
This very important observation was made by Voronov in his pioneer work [@thickfirst] on thick morphisms.
\[newexample\] For example consider pull-back $$\label{newexample1}
L(g)= \Phi_S^*(g)\,.$$ According to Theorem \[1\] this is non-linear homomorphism. One can show that the map $y^a=y^a(x,g)$ in equation which we constructed above (see equations , and equation in Proposition \[mapforthickmorphisms\]) is just formal map $K^a(x,g)$ for this functional. (See the proof of Theorem \[1\] in the next section.)
For non-linear homomorphisms we will use the notion of so called [*support map*]{}.
\[defofsupport\] If $L(g)$ is a functional which is non-linear homomorphism then a map $K_0^a(x)$ corresponding to the functional $L(g)$, which is the zeroth part of the formal map $K^a(x)$ (see equations and ) will be called [support map corresponding to functional $L(g)$]{}.
Consider functional $L(x,g)$ corresponding to thick morphism (see equation in example \[newexample\]). If $
S(x,q)=S_0(x)+S_1^a(x)q_a+\dots
$ is generating function which defines this thick morphism, then it follows from equations and that support map is equal to $
K_0^a(x)=S^a_1(x)
$ (see also equation .)
\[associate\] Let $L$ be an arbitrary functional in $\A$, $$L(x,g)=\sum_k L_k(x,g)\,, \, {\rm where}\,\,
L_k(x,g)=[L(x,g)]_k\in \A_k$$ (see equations and ). Taking the values of this functional on linear functions $y=y^al_a$ we assign to this functional, [*formal function*]{} $$\label{actionassociated1}
S_L(x,q)=L(x,g)\big\vert_{g=y^aq_a}=
S_0(x)+
\sum_k
S_k^{a_1\dots a_k}(x)
q_{a_1}\dots q_{a_k}\,,$$ where tensors $\{S_k^{a_1\dots a_k}(x)\}$ can be expressed through polarised form of functionals $L_k$ (see equations and in Appendix B): $$S_k^{a_1\dots a_k}(x)=
L_k^{\rm polaris.}
\left(
x,y^{a_1},\dots, y^{a_k}\right)\,,$$ where $\{y^a\}$ are coordinates on $N$. E.g. $$S_L^{ab}(x)=L_2^{\rm polaris.}(x,y^a,y^b)=
{1\over 2}\left(L_2\left(y^a+y^b\right)-
L_2\left(y^a\right)-L_2\left(y^a\right)\right)\,.$$ We say that $S_L(x,q)$ is [*formal function associated with functional $L$*]{}.
Let $S=S(x,q)$ be an arbitrary formal generating function . Let $\Phi_S$ be a thick morphism defined by this generating function, and let $L(x,g)$ be a formal functional, $L(x,g)\in \A$, which defines pull-back of functions produced by this thick morphism: $L(x,g)=\Phi^*_{S(x,q)}(g)$. Then one can see that formal generating function associated with functional $ L(x,g)=\Phi^*_{S(x,q)}(g)$ coincides with formal generating function $S(x,q)$: $$\label{actionisthesame1}
L(x,g)=\Phi^*_{S(x,q)}(g)\Rightarrow
S_L(x,q)\equiv S(x,q)\,.$$ Indeed in the case if function $g=y^al_a$ is linear then calculations of pull-back $\Phi_S^*(g)$ by formulae and become evident. Indeed in this case we immediately come to equation since according to equations and $$f(x)=g(y)+S(x,q)-y^aq_a=S(x,l)$$ because for linear function $g(y)=y^aq_a$.
It turns out that converse implication is also valid for non-linear homomorphisms.
\[2\] Let $L=L(x,g)\in\A$ be an arbitrary non-linear homomorphism, and let $S(x,q)$ be an action associated to it. Then $$L(g)=\Phi_S^*(g)\,.$$
This is main result of this paper.
Proof of the Theorems
=====================
We recall here the proof of Theorem \[1\] and give a proof of Theorem \[2\].
Proof of Theorem \[1\]
----------------------
Check straightforwardly that a formal map $y^a(x,g)$ constructed in Proposition \[mapforthickmorphisms\] (see equations , and equation in Proposition \[mapforthickmorphisms\]) is just a map corresponding to function $g$ i.e. equation $$\label{checkstraightforwardly}
\Phi_S^*(g+\vare h)-
\Phi_S^*(g)=\vare h(y(x,g))\,, (\vare^2=0)$$ is satisfied. (See also example \[newexample\].)
Using definition we see that in $$\Phi_S^*(g+\vare h)-
\Phi_S^*(g)=$$ $$\left[
\left(g(y)+\vare h(y)\right)\big\vert_{y^a=y^a(x,g+\vare h)}+
S(x,q)\big\vert_{q_a=q_a(x,g+\vare h)}-
y^aq_a\big\vert_{y^a=y^a(x,g+\vare h),q_a=q_a(x,g+\vare h)}
\right]-$$ $$\label{checkstraightforwardly2}
\left[
\left(g(y)\right)\big\vert_{y^a=y^a(x,g)}+
S(x,q)\big\vert_{q_a=q_a(x,g)}-
y^aq_a\big\vert_{y^a=y^a(x,g),q_a=q_a(x,g)}
\right]$$ Here we introduced notation $$q_a(x,g)={\p g(y)\over \p y^a}\big\vert_{y^a=y^a(x,g)}\,.$$ To see that right hand sides of equations and coincide we note that in equation the following relations hold $$\label{A}
\left(g(y)+\vare h(y)\right)\big\vert_{y^a=y^a(x,g+\vare h)}
-
g(y)\big\vert_{y^a=y^a(x,g)}=
\vare{\p g(y)\over \p y^a}\big\vert_{y^a=y^a(x,g)}t^a=
\vare q_a(x,g)t^a\,,$$ $$\label{B}
S(x,q)\big\vert_{q_a=q_a(x,g+\vare h)}-
S(x,q)\big\vert_{q_a=q_a(x,g)}=
\vare {\p S(x,q)\over \p q_a}\big\vert_{y^a=y^a(x,g)}r_a=
\vare y^a(x,g)r_a(x,g;h)\,,$$ and $$\label{C}
y^aq_a\big\vert_{y^a=y^a(x,g+\vare h),q_a=q_a(x,g)+\vare h)}-
y^aq_a\big\vert_{y^a=y^a(x,g,q_a=q_a(x,g)}=
\vare t^a q_a(x,g)+\vare y^a(x,g)r_a$$ In equations , and we used notations $t^a,r_b$ such that $$y^a(x,g+\vare h)-
y^a(x,g)=\vare t^a\,
{\rm and}\,\,
q_a(x,g+\vare h)-
q_a(x,g)=\vare r_a\,.$$ Comparing right hand sides of equations , and we come to conclusion that equation is obeyed.
Proof of Theorem \[2\]
----------------------
To prove Theorem \[2\] we will formulate two lemmas.
\[lemma1\]
Let $L=L(x,g)=\sum_{k\geq 0} L_k(x,g)$ be an arbitrary functional in $\A$ which is non-linear homomorphism (see definition\[defoflocal\]). Let $S_0(x)$ be a function which is equal to value of this functional on function $g=0$ $$\label{affinecomponent}
S_0(x)=L(x,g)\big\vert_{g=0}\,,$$ we will call sometimes this function [*an affine component of functional $L$*]{}.
Let a map $K_0^a(x)$ be a support map corresponding to this functional (see definition \[defoflocal\]). Then $$L(g)=S_0(x)+g(K^a_0(x))\,\, ({\rm mod} \A_2)\$$
\[lemma2\] Let $L(x,g)$ and $\widetilde L(x,g)$ be two functionals on $\A$ which both are non-linear homomorphisms, and which coincide up to the order $k-1$ ($k\geq 2$): $$\begin{matrix}
\widetilde L(g)=
\sum_i\widetilde L_i(x,g)\,,
\quad \widetilde L_i(x,g)\in A_i\cr
L(g)=
\sum_i L_i(x,g)\,,
\quad L_i(x,g)\in A_i\cr
\widetilde L_j=L_j \,\, {\rm for}\,\,j\leq k-1\cr
\end{matrix}$$ Then the difference of these functionals in the order $k$ is given by $k$-linear functional $T_k(x,\p g)\in A_k$: $$\widetilde L_k(x,g)-L_k(x,g) =T_k(\p g)$$ where $$\label{lemma2func}
\A_k\ni T_k(\p g)=T^{a_1\dots a_k}(x)
g^*_{a_1}(x)\dots g^*_{a_k}
\quad {\rm and}\,\,
g_a^*(x)={\p g(y)\over \p y}\big\vert_
{y^a=K^a(x)}\,,$$ $K_0^a(x)$ is a support map \[defofsupport\] which is the same for both these functionals, and tensor $T^{a_1\dots a_k}$ is defined by equation $$\label{formulaforaction1}
T^{a_1\dots a_k}(x)=
\widetilde L_k^{\rm polaris.}
\left(x,y^{a_1},\dots, y^{a_k}
\right)
-L^{\rm polaris.}_k
\left(x,y^{a_1},\dots, y^{a_k}\right)$$ where $\widetilde L_k(x,g)$ and $L_k(x,g)$ are the terms of order $k$ in the expansion of functionals $\widetilde L(x,g)$ and $L(x,g)$), and respectively $\widetilde L^{\rm polaris.}_k(x,g_1,\dots,g_k)$ is polarised form of functional $\widetilde L_k(x,g)$, and $L^{\rm polaris.}_k(x,g_1,\dots,g_k)$ is polarised form of functional $L_k(x,g)$ (see equation in definition \[defofpolar\] in Appendix B).
Prove Theorem \[2\] using these lemmas.
Let $L=L(g)$ be a functional in $\A$ which is non-linear homomorphism, i.e, condition (see definition \[defoflocal\]) holds for this functional, and $$L(x,g)=L_0(x,g)+L_1(x,g)+\dots+L_k(x,g)+\dots\,,$$ where every functional $L_r(x,g)$ has order $r$ in $g$: $L_r\in A_r$.
Consider an action $S(x,q)$ associated with this functional (see equation in definition \[associate\]).
Consider the sequence of thick morphisms $\{\Phi_k\}$ ($k=0,1,2,\dots$) such that the thick morphism $\Phi_k$ is generated by the action $$\bS_k(x,q)=
S_0(x)+S_1^a(x)q_a+S_2^{ab}(x)q_aq_b+\dots+
S_k^{a_1\dots a_k}(x)q_{a_1}\dots q_{a_k}\,,$$ and respectively the sequence $\{\Phi_k^*(g)=\Phi^*_{\bS_k}(g)\}$ of functionals, generated by these thick morphisms.
Prove that for every $k$, non-linear homomorphism $L(g)$ coincides up to terms of order $k$ in $g$ with functional $\Phi^*_{\bS_k}$: $$\label{inductivestatement}
L(g)=\Phi_k^*(g)
({\rm mod }\, \A_{k+1})\,.$$ This will be the proof of Theorem \[2\].
Thick morphisms $\{\Phi_k\}$ can be viewed as a sequense of morphisms tending to morphims $\Phi_S$.
We prove equation by induction. If $k=1$ then ${\bf S}_1(x)=S_0(x)+S_1^a(x)q_a$ and $$\Phi_1^*(g)=
S_0(x)+g(S_1^a(x))=L(g)
({\rm mod }\, \A_{2})\,.$$ due to Lemma \[lemma1\]. Thus equation is obeyed if $k=1$. Now suppose that equation is obeyed for $k=m$, $m\geq 1$. Prove it for $k=m+1$. Denote by $$\label{temporarynotation}
\widetilde L(g)=\Phi_m^*(g)\,.$$ Due to Theorem \[1\] this functional is also non-linear homomorphism. Both functionals are non-linear homomorphisms and by inductive hypothesis functionals $L(g)$ and $\tilde L(g)$ coincide up to the order $m$. Hence lemma \[lemma2\] implies that there exists tensor $T^{a_1\dots a_{m+1}}(x)$ such that $$\label{comparing}
L(g)=\widetilde L(g)+T_{m+1}(\p g)=
\Phi^*_{\bS_m}(g)+T_{m+1}(\p g)
\,({\rm mod}\A_{m+2}),$$ where $$T_{m+1}(\p g)=T^{a_1\dots a_{m+1}}(x)g_{a_1}^*(x)\dots g_{a_{m+1}}^*(x)\,,
\left(g_a^*(x)={\p g(y)\over \p y^a}\big\vert_{y^a=S^a_1(x)}\right)\,,$$ and tensor $T^{a_1\dots a_{m+1}}(x)$ according to equation is defined by equation $$\label{defofT}
T^{a_1,\dots,a_{m+1}}_{m+1}=
L_{m+1}^{\rm polaris.}\left(x,y^{a_1},\dots,y^{a_{m+1}}\right)-
\widetilde
L^{\rm polaris.}_{m+1}\left(x,y^{a_1},
\dots,y^{a_{m+1}}\right)\,,$$ where $L^{\rm polaris.}_{m+1}$ is polarised form of functional $L_{m+1}(g)$ which contains terms of order $m+1$ of functional $L(g)$. Respectively functional $\widetilde L^{\rm polaris.}_{m+1}$ is polarised form of functional $\widetilde L_{m+1}(g)$ which contains terms of order $m+1$ of functional $\widetilde L(g)=\Phi_{\bS_m}(g)$. It is easy to see that functional $\widetilde L^{\rm polaris.}_{m+1}$ is vanished on arbitrary linear functions: $$\label{vanishesonlinear}
\widetilde L\left(x,l_1,\dots,l_{m+1}\right)=0\,,\quad
\hbox{if functions $l_i$ are linear:
$l_i=y^al_{ai}$, $i=1,\dots,m+1$}\,.§:$$ Indeed functional $\widetilde L(g)=\Phi^*_{\bS_m}(g)$ is assigned to the action $\bS_m(x,q)$ which is a polynomial of order $\leq m$, hence due to equation it vanishes for arbitrary linear function $g=y^al_a$, hence polarised form vanishes also on linear functions ( see equation in Appendix B). Thus we come to condition . This condition means that in particular $$\widetilde
L^{\rm polaris.}_{m+1}\left(x,y^{a_1},
\dots,y^{a_{m+1}}\right)=0\,,\quad{\rm for}\,\,
\widetilde L(g)=\Phi_{m+1}^*(g)\,,$$ hence we come to conclusion that tensor $T^{a_1\dots a_{m+1}}(x)$ in equation is equal to $S^{a_1\dots a_{m+1}}(x)$.
We see that $$\label{m+2before}
L(g)=\Phi^*_m(g)+S_{m+1}(\p g)\,\,
({\rm mod} \A_{m+2})\,.$$ On the other hand up to the terms of order $m+1$, right hand sight of this equation is equal to $\Phi^*_{m+1}$: $$\label{m+2after}
\Phi^*_{m+1}(g)=\Phi^*_m(g)+S_{m+1}(\p g)
({\rm mod} \A_{m+2})\,.$$ One can see it straightforwardly using equation or it is much easier to check equation taking differential of this equation. Namely taking differential of equation and using equations and we come to equation $$h\left(y^a_{m+1}(x,g)\right)
=h\left(y^a_{m}(x,g)\right)+S_{m+1}^{aa_1\dots a_m}
g_{a_1}^*\dots
g_{a_m}^*
({\rm mod} \A_{m+1})\,,$$ where $y^a_{\bS_{k}}(x,g$ is a map $y^a(x,g)$ corresponding to thick morphism $\Phi_{\bS_k}$ ($\Phi^*_{\bS_k}(g+\vare h)-\Phi^*_{\bS_k}(g)=
h\left(y^a_{\bS_{m+1}}(x,g)\right) h$). Comparing left hand sides of equations and we see that equation holds for $k=m+1$. This ends the proof.
.
It remians to prove lemmas.
Proofs of lemmas
================
Proof of the Lemma \[lemma1\]
-----------------------------
Let $L=L(x,g)$ be a functional in $\A$ which is non-linear homomorphism. $$\label{lemma11}
L(x,g)=
L_0(x)+L_1(x,g)+\dots=
L_0(x)+L_1(x,g)({\rm mod\,} \A_2)$$ If we put $g=0$ we come to $L_0(x)=S_0(x)=L(g)\big\vert_{g=0}$.
Differentiate equation . Using equation we come to $$L(x,g+\vare h)-L(x,g)=
\vare
h\left(
y^a(x,g)
\right)=
\vare h
\left(
K_0^a(x)+K_1^a(x,g)+\dots
\right)
=\vare h\left(
K_0^a(x)
\right) \, ({\rm mod} \A_1)$$ This is true for arbitrary smooth function $h$. This implies that $L_1(x,g)=g\left(K_0^a(x)\right) $. Hence $$L(g)=L_0(g)+L_1(g) \, ({\rm mod} \A_2)=
S_0(x)+g\left(K_0^a(x)\right)\,({\rm mod} \A_2)=$$ $$S_0(x)+\int K(x,y)g(y)dy+\hbox{\footnotesize terms of order
$\geq 2$ in $g$}\,,\quad {\rm with}
\,\, K(x,y)=\delta(y^a-K^a_0(x))\,.$$
First lemma is proved.
Proof of lemma \[lemma2\]
-------------------------
Let functionals $L(g)$ and $\widetilde L(g)$ both be functionals which are non-linear homomorphisms (see definition \[defoflocal\]). Suppose these functionals coincide up to the order $k-1$ ($k=2,3,\dots$). According to expansion this means that difference of these functionals is a functional $T_k(g)$ of order $k$ $$\label{lemma21}
\widetilde L(g)-L(g)=T_k(x,g)\in \A_{k+1}\, \quad
{\rm i.e. }\,\,
\widetilde L(g)-L(g)-T_k(g)=0 ({\rm mod} \A_{k+1})\,,$$ where $$\label{lemma22}
T_k(x,g)=\int T(x,y_1,\dots,y_k)
g(y_1)\dots g(y_k)dy_1\dots dy_k\,.$$ Take the differential of equation . We come to $$\left(\widetilde L(g+\vare h)-
\widetilde L(g)\right)-
\left(
L(g+\vare h)-
L(g)\right)=
\vare h\left(\widetilde {y^a}(x,g)\right)
-\vare h\left( {y^a}(x,g)\right)=$$ $$=T_k(x,g+\vare h)-T_k(g)+\hbox {terms of order $\geq k$ in $g$}=$$ $$\label{lemma23}=
\vare kT^{\rm polaris.}_k
\left(h,\underbrace {g,\dots,g}_{\hbox {$k-1$ times}}\right)+
\hbox {terms of order $\geq k$ in $g$}=$$ Here $T^{\rm polaris.}_k=T_k(x,g_1,g_2,\dots,g_k)$ is the polarisation of the form $T_k(x,g)$ (see equation in definition \[defofpolar\]). Recall that if function $T(x,y_1,y_2,\dots,y_k)$ which correspond to functional $T_k(g)$ in equation is symmetric function on variables $y_1,\dots y_k$ then (see equation ) $$T^{\rm polaris.}_k(x,g_1,\dots,g_k)=
\int T(x,y_1,y_2,\dots,y_k)
g_1(y_1)g_2(y_2)\dots g_k(y_k)dy_1 dy_2\dots dy_k\,.$$
Formal maps $y^a(x,g)$ corresponding to differential $dL(g)=L(g+\vare h)-L(g)$ of functional $L(g)$ and $\widetilde y^a(x,g)$ corresponding to differential $d\widetilde L(g)=
\widetilde L(g+\vare h)-\widetilde L(g)$ of functional $\widetilde L(g)$ according to equation are given by formal power series $$y^a(x,g)=
K^a_0(x)+
K^a_1(x,g)+\dots+
K^a_{k-2}(x,g)+K^a_{k-1}(x,g)+
{\footnotesize \hbox {terms of order $\geq k$ in $g$}}$$ and $$\label{lemma25}
\widetilde y^a(x,g)=
\widetilde K^a_0(x)+
\widetilde K^a_1(x,g)+\dots+
\widetilde K^a_{k-2}(x,g)+
\widetilde K^a_{k-1}(x,g)+
{\footnotesize \hbox {terms of order $\geq k$ in $g$}}\,.$$ Recall that here $K^a_r(x,g)$ and $\widetilde K^a_r(x,g)$ are maps of order $r$ in $g$: $$K^a_r(x,g)=\int K(x,y_1,\dots, y_r)g(y_1)\dots g(y_r)
dy_1\dots dy_r\,.$$ Since functionals $L(g)$ and $\widetilde L(g)$ coincide up to the order $k-1$, their differentials coincide up to the order $k-2$. Hence it follows from equation that in equation all the maps $K^a_r$ coincide with maps $\widetilde K^a_r$ for $r=0,1,2,\dots, k-2$ $$K_0^a(x)
=\widetilde K_0^a(x)\,,
\dots\,,
K_{k-2}^a(x,g)
=\widetilde K_{k-2}^a(x,g)\,,$$ and it is the difference between maps $\widetilde K_{k-1}$ and $K_{k-1}$ which produces the functional $T_k(x,g)$.
Rewrite equation projecting all terms on subspace $A_{k-1}$. We come to $$\left[\left(\widetilde L(g+\vare h)-
\widetilde L(g)\right)\right]_{k-1}-
\left[
L(g+\vare h)-
L(g)\right]_{k-1}=
\vare
\left[h\left(\widetilde {y^a}(x,g)\right)
-\vare h\left( {y^a}(x,g)\right)
\right]_{k-1}=$$ $${\p h\over \p y^a}\big\vert_{y^a=K^a_0(x)}
\left[ \widetilde K_{k-1}^a(x,g)-
K_{k-1}^a(x,g)\right]
=
{\p h\over \p y^a}\big\vert_{y^a=K^a_0(x)}
P^a_{k-1}(x,g)=$$ $$$$ $$=T_k(x,g+\vare h)-T_k(g)=$$ $$\label{lemma26}=
\vare kT^{\rm polaris.}_k
\left(x,h,\underbrace {g,\dots,g}_{\hbox {$k-1$ times}}\right)
\,.$$ where we denote by $P^a_{k-1}(x,g)$ the difference between maps $\widetilde K_{k-1}^a(x,g)$ and $ K_{k-1}^a(x,g)$ $$\label{lemma26}
P^a_{k-1}(x,g)=\widetilde K_{k-1}^a(x,g)-
K_{k-1}^a(x,g)=\int P^a_{k-1}(x,y_1,\dots,y_{k-1})
g(y_1)\dots g(y_{k-1})dy_1\dots dy_k\,.$$ The map $P^a_{k-1}(x,g)$ has order $n-1$ over $g$. Consider polarisation $P^{a\, {\rm polaris.}}_{k-1}(x,_1,\dots,g_{k-1})$ of this map. Equation implies $${\p h\over \p y^a}\big\vert_{y^a=K^a_0(x)}
P^{a\, {\rm polaris.}}_{k-1}(x_1,\dots,g_{k-1})
= \vare kT^{\rm polaris.}_k
\left(x,h,g_1,\dots,g_{k-1}\right)
\big\vert_{g_1=\dots=g_{k-1}=g}
\,.$$ Thus we come to equation $$\label{lemma27}
T^{\rm polaris.}_k
\left(x,g_1,\dots,g_{k}\right)=
{1\over k}{\p g_1\over \p y^a}\big\vert_{y^a=K^a_0(x)}
P^{a\,{\rm polaris.}}_{k-1}(x,g_2,\dots,g_k)\,,$$ where $g_1,\dots,g_k$ are arbitrary functions and left hand side of this equation is symmetric with respect to transposition of functions $\{g_1,\dots,g_k\}$. It follows from equation that $$P^{a\,{\rm polaris.}}_{k-1}(x,g_2,\dots,g_k)\,,
=
kT^{\rm polaris.}_k
\left(x,y^a,g_2,\dots,g_{k}\right)$$ hence $$\label{lemma2key}
T^{\rm polaris.}_k
\left(x,g_1,\dots,g_{k}\right)=
{\p g_1\over \p y^a}\big\vert_{y^a=K^a_0(x)}
T^{\rm polaris.}_k
\left(x,y^a,g_2,\dots,g_{k}\right)\,.$$ Equation and symmetricity of functional $T_k(x,g_1,\dots,g_k)$ imply that $$T^{\rm polaris.}_k
\left(x,g_1,g_2,\dots,g_{k}\right)=
{\p g_1\over \p y^a}\big\vert_{y^a=K^a_0(x)}
T^{\rm polaris.}_k
\left(x,y^a,g_2,\dots,g_{k}\right)=$$ $$T^{\rm polaris.}_k
\left(x,g_2,g_1,\dots,g_{k}\right)=
{\p g_2\over \p y^a}\big\vert_{y^a=K^a_0(x)}
T^{\rm polaris.}_k
\left(x,y^a,g_1,\dots,g_{k}\right)=\dots=$$ $${\p g_1\over \p y^{a_1}}\big\vert_{y^{a_1}=K^{a_1}_0(x)}
\dots
{\p g_k\over \p y^{a_k}}\big\vert_{y^{a_k}=K^{a_k}_0(x)}
T^{\rm polaris.}_k
\left(x,y^{a_1},\dots,y^{a_k}\right)=$$ $$\label{mainstatement}
=g_{a_1}^*(x)
\dots g_{a_k}^*(x)
T^{a_1\dots a_k}(x)\,,$$ where $$T^{a_1\dots a_k}(x)=T_k\left(x,y^{a_1},\dots,y^{a_k}\right)=
{\rm and}\,\,
g_a^*(x)=
{\p g\over \p y^{a}}\big\vert_{y^{a}=K^{a}_0(x)}\,.$$ Now returning to equation and comparing it with formulation of lemma \[lemma2\] we come to proof of lemma \[lemma2\]: $$\widetilde L_k(g)-L_k(g)=T_k(x,g_1,\dots,g_k)
\big\vert_{g_1=\dots=g_k=g}=T_k(\p g)\,.$$
Appendix A. Thick morphisms and $L_\infty$ maps
================================================
We briefly here discuss why thick morphisms is an adequate tool to describe $L_\infty$-morphisms of homotopy Poisson algebras (see [@thickfirst] and [@thicktheor] for detail). For this purpose we need to consider thick morphisms of supermanifolds. However we can catch some improtant features considering just usual manifolds. We first consider thick morphisms for usual manifolds, and show that in this case thick morphisms describe morphisms of algebras of functions on these manifolds which are provided with multilinear symmetric brackets. It turns out that if we consider supermanifold, then under some assumptions these algebras become homotopy Poisson algebras.
Let $M$ be an arbitrary manifold, and $H=H(x,p)$ be a function (Hamiltonian) on cotangent bundle $T^*M$. This Hamiltonian $H$ defines the series of symmetric brackets on $M$ via canonical symplectic structure on $T^*M$ $$\langle \emptyset \rangle_H\,,
\langle f_1 \rangle_H\,,
\langle f_1, f_2 \rangle_H\,,
\langle f_1, f_2,f_3 \rangle_H\,,
\dots
\langle f_1, f_2,\dots,f_k \rangle_H\,,$$
where $$\langle \emptyset \rangle_H=H(x,p)\big\vert_{p=0}=H_0(x)$$ $$\langle f_1 \rangle_H=\left(H,f_1\right)\big\vert_{p=0}=
H_1^a(x){\p f_1(x)\over \p x^a}\,,$$ $$\langle f_1,f_2 \rangle_H=
\left(\left(H,f_1\right),f_2\right)\big\vert_{p=0}=
H_1^{ab}(x)
{\p f_1(x)\over \p x^a}
{\p f_2(x)\over \p x^a}\,,$$ and so on: $$\label{homotopybracketsformanifoldsusual}
\langle f_1,f_2,\dots,f_k \rangle_H=
\underbrace
{(\dots (}
_{\hbox {$k$ times}}
H,f_1),f_2 )\dots f_k)\big\vert_{p=0}=
H_k^{a_1\dots a_k}(x)
{\p f_1(x)\over \p x^{a_1}}
\dots
{\p f_k(x)\over \p x^{a_k}}\,.$$ Here $(\_,\_)$ is Poisson bracket on $T^*M$ corresponding to canonical symplectic structure: $$\label{canonicalpoissonbracket}
\left(f(x,p),g(x,p)\right)=
{\p f(x,p)\over \p p_a}
{\p g(x,p)\over \p x^a}
-
{\p g(x,p)\over \p p_a}
{\p f(x,p)\over \p x^a}\,.$$ We suppose that Hamiltonian $H=H(x,p)$ is a [*formal Hamiltonian*]{}, i.e. formal function, power series over $p$: $$\label{formalhamiltonian}
H=H(x,p)=H_0(x)+H_1^a(x)p_a+H_2^{ab}(x)p_bp_a+
H_3^{abc}(x)p_cp_bp_a+\dots$$ where all coefficients are smooth functions on $x$.
\[canonicalcoordinatesremark\] All these formulae are written in local coordinates $(x^a,p_b)$ in $T^*M$ corresponding to local coordinates $x^a$ on $M$ (if $x^{a'}$ are new local coordinates on $M$, then new local coordinates $(x^{a'}, p_{b'})$) are $$\label{canonicalcoordinatesonbundle}
x^{a'}=x^{a'}(x)\,, p_{b'}={\p x^b(x')\over \p x_{b'}}p_b\,.$$
Notice that every Hamiltonian $H(x,p)$ defines vector field $$\label{vectorfieldonfunctions1}
X_H=\int
H\left(
f\left(x\right),
{\p f\left(x\right)\over \p x}
\right)dx$$ on the space of function. Vector field $X_H$ assigns to every function $f\in C^\infty(M)$ infinitesimal curve $$\label{vectorfieldonfunctions2}
f+\vare X_H=
f(x)+
\vare H\left(
f\left(x\right),
{\p f\left(x\right)\over \p x}\right)\,,
\quad (\vare^2=0)\,.$$
Now consider two manifolds $M$ and $N$. Let $H_M(x,p)$ be formal Hamiltonian on $M$, and let $H_N(y,q)$ be formal Hamiltonian on $N$. Hamiltonian $H_M(x,p)$ induces on $M$ the sequence of multilienar symmetric brackets $\left\{\langle f_1,\dots,f_p\rangle_M\right\}$ on functions on $M$, and respectively Hamiltonian $H_N(y,q)$ induces on $N$ the sequence of multilinear symmetric brackets $\left\{\langle g_1,\dots,g_q\rangle_M\right\}$ on functions on $N$ ($p,q=0,1,2,3,\dots$).
We say that formal functional $L(g)$ is morphism of multilinear symmetric brackets on $N$ to multilinear symmetric brackets on $M$ if vector fields $X_{H_M}$ and $X_{H_N}$ are connected by functional $L(g)$, i.e. according to formulae $$\label{bracketsconnected}
L\left(g+\vare X_N\right)=
L\left(g\right)+\vare X_M\,.$$ Consider thick morphism $\Phi_S\colon M\Rrightarrow N$ generated by $S(x,q)$ and consider formal functional $\Phi_S^*(g)$ on $C^\infty(N)$ defined by this thick morphism (see equations — and remark \[infactmanifolds\]).
We say that Hamiltonians $H_M$ and $H_N$ are $S$-related if $$\label{hamiltoniansrelated}
H_M\left(x,{\p S\left(x,q\right)\over \p x}\right)
\equiv
H_N\left({\p S\left(x,q\right)\over \p q},q\right)$$ The following remarkable theorem takes place:
\[Voronovtoy\](Voronov, 2014) If Hamiltonians $H_M$ and $H_N$ are $S$-related, then formal functional $L(g)$ defined by thick morphism $\Phi_S$, $L(g)=\Phi_S^*(g)$ defines morphisms of multilinear brackets
$\left\{\langle f_1,\dots,f_p\rangle_M\right\}$ and $\left\{\langle g_1,\dots,g_q\rangle_M\right\}$ $\left\{\langle g_1,\dots,g_q\rangle_M\right\}$. In other words thick morphism connects these brackets.
Now consider the case of supermanifolds.
In this case all the constructions above will remain the same, just in some formulae will appear a sign factor. (See [@thickfirst] and [@thicktheor] for detail). In particular arbitrary Hamiltonian $H=H(x,p)$ which is a function on cotangent bundle $T^*M$ to supermanifold $M$ will define the collection of symmetric brackets like in the case . On the other hand if Hamiltonian $H_M$ is [*odd*]{} and Hamiltonian $H_M$ obeys condition $$\label{master1}
\left(H_M,H_M\right)\equiv 0\,,$$ then these brackets will become [*homotopy Poisson brackets*]{}. This is famous construction of homotopy Poisson brackets derived by odd Hamiltonian $H_M$ which obeys so called master-equation (see for detail [@thickkoszul]).
Appendix B. Polarisation of functionals
=======================================
It is useful to consider polarised form of formal functionals.
\[defofpolar\] Let $L_k(x,g)$ be formal functional of order $k$, $L_k(x,g)\in \A_k$ (See for definition \[defofformalfunctionals\].) Polarisation of functional $L_k(x,g)$ is the functional $L_k^{\rm polaris.}(x,g_1,\dots,g_k)$ which linearly depends on $k$ functions $g_1,\dots,g_k$ such that for every function $g$ $$\label{polarisation1}
L_k(x,g) =
L_k^{\rm polaris.}(x,g_1,\dots, g_k)
\big\vert_{g_1=g_2=\dots=g_k=g}\,.$$ Using elementary combinatoric one can express polarised form $L_k^{\rm polaris.}(x,g_1,\dots, g_k)$ explicitly in terms of functional $L_k(x,g)$, ($L_k\in A_k$): $$\label{polarisation2}
L_k^{\rm polaris.}(x,g_1,\dots,g_k)=
{1\over k!} \sum(-1)^{k-n}
L_k\left(
x,g_{i_1}+\dots+g_{i_n}
\right)\,,$$ where summation goes over all non-empty subsets of the set $\{g_1,\dots,g_k\}$. E.g. if $L=L_3$ then $$L^{\rm polaris.}(x,g_1,g_2, g_r)=
{1\over 6}
\left(
L_3\left(x,g_1+g_2+g_3\right)
-L_3\left(x,g_1+g_2\right)
-L_3\left(x,g_1+g_3\right)
-L_3\left(x,g_2+g_3\right)\right.$$ $$\left.
+L_3\left(x,g_1\right)
+L_3\left(x,g_2\right)
+L_3\left(x,g_3\right)
\right)\,.$$ If functional $L_r(x,g)$ is expressed through (generalised) functions $L(x,y_1,\dots,y_r)$ (see equation ) such that it is symmetric with respect to coordinates $y_1,\dots,y_r$ then $$\label{polarisation}
L^{\rm polaris.}(g_1,\dots,g_r)=
\int L(x,y_1,\dots,y_r)g_1(y_1)\dots g(y_r)dy_1\dots dy_r\,.$$
It is useful also to note that if $L(x,g)=L_0(x)+L_1(x,g)+\dots+L_n(x,g)$ then for every $k\colon\, k=0,1,\dots,n$ $$\label{polarisation3}
L_k^{\rm polaris.}(x,g_1,\dots,g_k)=
{1\over k!} \sum(-1)^{k-n}
L\left(
x,g_{i_1}+\dots+g_{i_n}
\right)\,,$$ where summation goes over all subsets of the set $\{g_1,\dots,g_k\}$ including empty subset. (For empty subset $L(x,\emptyset)=L_0(x)$.)
[10]{}
Th.Th. Voronov “Nonlinear pullbacks” of functions and $L_\infty$-morphisms for homotopy Poisson structures. J. Geom. Phys. 111 (2017), 94-110. arXiv:1409.6475
Th.Th. Voronov Microformal geometry and homotopy algebras. Proc. Steklov Inst. Math. 302 (2018), 88-129. arXiv:1411.6720
Th. Th. Voronov. Thick morphisms of supermanifolds and oscillatory integral operators. , 71(4):784–786, 2016.
H.M.Khudaverdian, Th.Th.Voronov. Thick morphisms, higher Koszul brackets, and $L_\infty$-algebroids. arXiv:1808.10049
H.M.Khudaverdian, Th,Th,Voronov. Thick morphisms of supermanifolds, quantum mechanics and spinor representation. J. Geom. Phys. 113 (2019), DOI: 10.1016/j.geomphys.2019.103540, arXiv:1909.00290 (with H. Khudaverdian)
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'David Walnut, Götz E. Pfander, Thomas Kailath'
title: Cornerstones of Sampling of Operator Theory
---
Introduction
============
The problem of identification of a time-variant communication channel arose in the 1950s as the problem of secure long-range wireless communications became increasingly important due to the geopolitical situation at the time. Some of the theoretical and practical advances made then are described in this paper, and more recent advances extending the theory to more general operators, and onto a more rigorous mathematical footing, known as [*sampling of operators*]{} are developed here as well.
The launching point for the theory of operator sampling is the early work of the third-named author in his Master’s thesis at MIT, entitled “Sampling models for linear time-variant filters” [@Kai59], see also [@Kai62; @Kai63], and [@Kai61] in which he reviews the identification problem for time-variant channels. The third named author as well as Bello in subsequent work [@Bel69] were attempting to understand and describe the theoretical limits of identifiability of time-variant communication channels. Section \[section:historical\] of this paper describes in some detail their work and explores some of the important mathematical challenges they faced. In Section \[section:operatorsampling\], we describe the more recently developed framework of operator sampling. Results addressing the problem considered by Bello are based on insights on finite dimensional Gabor systems which are presented in Section \[section:finiteGabor\]. Malikiosis’s recent result [@M13] allows for the generalization of those results to a higher-dimensional setting, these are stated and proven in Section \[section:higherdimensional\]. We conclude the paper in Section \[section:outlook\] with a short summary of the sampling of operators literature, that is, of results presented in detail elsewhere.
Historical Remarks. {#section:historical}
===================
The Cold War Origins of the Rake System.
----------------------------------------
In 1958, Price and Green published [*A Communication Technique for Multi-path Channels*]{} in Proc. IRE [@PG58], in which they describe a communication system called Rake, designed to solve the [*multi-path problem*]{}. When a wireless transmitter does not have line-of-sight with the receiver, the transmitted signal is reflected possibly multiple times before reaching the receiver. Reflection by stationary objects such as the ground or buildings introduces random time delays to the signal, and reflection or refraction by moving objects such as clouds, the troposphere, ionosphere, or a moving vehicle produce random frequency or Doppler shifts in the signal as well. Due to scattering and absorption, the reflected signals are randomly amplitude-attenuated too. The problem is to recover the transmitted signal as accurately as possible from the superposition of time-frequency-shifted and randomly amplitude-attenuated versions of it. Since the location and velocities of the reflecting objects change with time, the effects of the unknown, time-variant channel must be estimated and compensated for.
Price and Green’s paper [@PG58] was the disclosure in the literature of a long-distance system of wide-band or spread-spectrum communications that had been developed in response to strategic needs related to the Cold War. This fascinating story has been described in several articles by those directly involved ([@Sch82; @Sch83; @P83; @G08]). We present a summary of those remarks and of the Rake system below. The goal is to motivate the original work of the third-named author on which the theory of operator sampling is based.
In the years following World War II, the Soviet Union was exercising its power in Eastern Europe with a major point of contention being Berlin, which the Soviets blockaded in the late 1940s. This made secure communication with Berlin a top priority. As Paul Green describes it,
> \[T\]he Battle of Berlin was raging, the Russians having isolated the city physically on land, so that the Berlin Airlift was resorted to, and nobody knew when all the communication links would begin to be subjected to heavy Soviet jamming. [@G08]
By 1950, with a shooting war in Korea about to begin, the Army Signal Corps approached researchers at MIT to develop secure, and reliable wireless communication with the opposite ends of the world. According to Green,
> It is difficult today to recall the fearful excitement of those times. The Russians were thought to be 12 feet high in anything having to do with applying mathematics to communication problems (“all Russians were Komogorovs or Kotelnikovs”)....\[T\]here was a huge backlog of unexploited theory lying around, and people were beginning to build digital equipment with the unheard of complexity of a hundred or so vacuum tube-based bits (!). And the money flowed. [@G08]
The effort was called Project Lincoln (precursor to Lincoln Laboratory). The researchers were confronted by two main problems: 1) making a communications system robust to noise and deliberate jamming, and 2) enabling good signal recovery from multiple paths.
Spread Spectrum communications and NOMAC
----------------------------------------
The technique chosen to address the first problem is an application of the notion, already well-understood and used by that time, that combatting distortions from noise and jamming can be achieved by spreading the signal over a wide frequency band. The idea of spreading the spectrum had been around for a long time [@P83; @T80; @PSM82] and can be found even in a now famous Hedy Lamarr-George Antheil patent of 1942 [@MA42; @P83], which introduced the concept later called “frequency hopping”. The system called NOMAC (Noise Modulation and Correlation) was developed in the early 1950s and used noise like (pseudo-noise or PN) signals to achieve spectrum spreading. Detailed discussion of its history can be found in [@P83; @G08; @WW92].
The huge backlog of “unexploited theory” mentioned above included the recent work of Claude Shannon on communication theory [@Sha49], of Norbert Wiener on correlation functions and least mean squares prediction and filtering [@W49], and recent applications of statistical decision theory to detection problems in radar and communications.
The communication problem addressed by NOMAC was to encode data represented by a string of ones and zeros into analog signals that could be electromagnetically transmitted over a noisy communication channel in a way that foiled “jamming” by enemies. The analog signals $x_1(\cdot)$ and $x_0(\cdot)$, commonly called Mark and Space, associated with the data digits 1 and 0, were chosen to be waveforms of approximate bandwidth B, and with small cross correlation. The target application was $60$wpm teletype, with $22$ msec per digit (called a baud), which corresponds to a transmission rate of $1/0.022 \,\text{sec} =45$Hz. The transmitted signals were chosen to have a bandwidth of $10$KHz, which was therefore expected to yield a “jamming suppression ration” of $10,000/45=220$, or $23$db [@G08; @WW92]. The jamming ratio is often called the “correlation gain”, because the receiver structure involves cross correlation of the received signal with each of the possible transmitted signals. If the correlation with the signal $x_1(\cdot)$ is larger than the one with the signal $x_0(\cdot)$, then it is decided that the transmitted signal corresponded to the digit 1. This scheme can be shown to be optimum in the sense of minimum probability of error provided that the transmitted signals are not distorted by the communications channel and that the receiver noise is white Gaussian noise (see, for example, [@Hel60]). The protection against jamming is because unless the jammer has good knowledge of the noise like transmitted signals, any jamming signals would just appear as additional noise at the output of the crosscorrelations.
More details on the nontrivial ideas required for building a practical system can be found in the references. We may mention that the key ideas arose from three classified MIT dissertations by Basore [@B52], Pankowski [@P52], and Green [@G53], in fact, documents on NOMAC remained classified until 1961 [@G08].
A transcontinental experiment was run on a NOMAC system, but was found to have very poor performance because of the presence of multiple paths; the signals arriving at the receiver by these different paths sometimes interfere destructively. This is the phenomenon of “fading”, which causes self jamming of the system. Some improvement was achieved by adding additional circuitry and the receiver to separately identify and track the two strongest signals and combine them after phase correction; this use of time and space diversity enabled a correlation gain of $17$db, $6$db short of the expected performance. It was determined that this loss was because of the neglected weaker paths, of which there could be as many as 20 or 30. So attention turned to a system that would allow the use of all the different paths.
The Rake system
---------------
One conceptual basis for this new system was provided by the doctoral thesis of Robert Price [@P53], the main results of which were published in 1956 [@P56]. In a channel with severe multi-path the signal at the receiver is composed of large number of signals of different amplitudes and phases and so Price made the assumption that the received “signal” was a Gaussian random process. He studied the problem of choosing between the hypothesis $$H_i: w(\cdot)=Ax_i(\cdot)+n(\cdot), \quad i=0,1,$$ where the random time variant linear communication channel $A$ is such that the $\{Ax_i(\cdot)\}$ are Gaussian processes. In this case, the earlier cross correlation detection scheme makes no sense, because the “signal” arriving at the receiver is not deterministic but is a sample function of a random process, which is not available to the receiver because it is corrupted by the additive noise. Price worked out the optimum detection scheme and then ingeniously interpreted the mathematical formulas to conclude that the new receiver forms least mean-square estimates of the $\{Ax_i(\cdot)\}$ and then crosscorrelates the $w(\cdot)$ against these estimates. In practice of course, one does not have enough statistical information to form these estimates and therefore more heuristic estimates are used and this was done in the actual system that was built. The main heuristic, from Wiener’s least mean-square smoothing filter solution and earlier insights, is that one should give greater weight to paths with higher signal-to-noise ratio.
So Price and Green devised a new receiver structure comprised of a delay line of length $3$ms intervals (the maximum expected time spread in their channel), with 30 taps spaced every $1 / 10$Khz, or $100\, \mu$s. This would enable the capture of all the multi-path signals in the channel. Then the tap gains were made proportional to the strength of the signal received at that tap. Since a Mark/Space decision was only needed every $22$ms (for the transmission rate of $60$wpm), and since the fading rate of the channel was slow enough that the channel characteristics remain constant over even longer than $22$ms, tap gains could be averaged over several $3$ms intervals. The new system was called “Rake”, because the delay line structure resembled that in a typical garden rake! Trials showed that this scheme worked well enough to recover the $6$db loss experienced by the NOMAC system. The system was put into production and was successfully used for jam-proof communications between Washington DC and Berlin during the “Berlin crisis” in the early 60s.
HF communications is no longer very significant, but the Rake receiver has found application in a variety of problems such as sonar, the detection of underground nuclear explosions, and planetary radar astronomy (pioneered by Price and Green, [@G68; @P68]) and currently it is much used in mobile wireless communications. It is interesting to note that the eight racks of equipment needed to build the Rake system in the 1960s is now captured in a small integrated circuit chip in a smart phone!
However the fact that the Rake system did not perform satisfactorily when the fading rates of the communication channel were not very slow led MIT professor John Wozencraft, (who had been part of the Rake project team at Lincoln Lab) to suggest in 1957 (even before the open 1958 publication of the Rake system) to his new graduate student Thomas Kailath a fundamental study of linear time-variant communication channels and their identifiability for his Masters thesis. While linear time-variant linear systems had begun to be studied at least as early as 1950 (notably by Zadeh [@Z50]), in communication systems there are certain additional constraints, notably limits on the bandwidths of the input signal and the duration of the channel memory. So a more detailed study was deemed to be worthwhile.
Kailath’s Time-Variant Channel Identification Condition {#section:KailathSufficient}
-------------------------------------------------------
In the paper [@Kai59], the author considers the problem of measuring a channel whose characteristics vary rapidly with time. He considers the dependence of any theoretical channel estimation scheme on how rapidly the channel characteristics change and concludes that there are theoretical limits on the ability to identify a rapidly changing channel. He models the channel $A$ as a linear time-variant filter and defines
> $A(\lambda,t)=$ response of $A$, measured at time $t$ to a unit impulse input at time $t-\lambda$.
$A(\lambda,t)$ is one form of the time-variant impulse response of the linear channel that emphasizes the role of the “age” variable $\lambda$. The channel response to an input signal $x(\cdot)$ is $$Ax(t) = \int A(\lambda,t)\,x(t-\lambda)\,d\lambda.$$ An impulse response $A(\lambda,t)=A(\lambda)$ represents a time-invariant filter. Further, the author states
> Therefore the rate of variation of $A(\lambda,t)$ with $t$, for fixed $\lambda$, is a measure of the rate of variation of the filter. It is convenient to measure this variation in the frequency domain by defining a function $\mathcal A$ $$\mathcal A(\lambda, f)=\int_{-\infty}^\infty A(\lambda,t) e^{-2\pi i f t } dt \quad
> $$
Then he defines $$B=\max_{\lambda} [ b-a, \text{ where } \mathcal A (\lambda,f)=0 \text{ for } f\notin [a,b] \,].$$ While symmetric support is assumed in the paper, this definition makes clear that non-rectangular regions of support are already in view. Additionally, he defines the memory as the maximum time-delay spread in response to an impulse of the channel as $$L=\max_{t} [\min_{\lambda'}\text{ such that } A (\lambda,t)=0, \ \lambda\geq \lambda' ].$$ In short, the assumption in the continuation of the paper is that $${\operatorname{supp}}\mathcal A (\lambda,f) \subseteq [0,L]\times[-W,W]$$ where $W=B/2$. The function $\mathcal A (\lambda,f)$ is often called the *spreading function* of the channel. He then asks under what assumptions on $L$ and $B=2W$ can such a channel be measured? In the context of the Rake system, this translates to the question of whether there are limits on the rate of variation of the filter that can assure that the measurement filter can be presumed to be effective.
The author’s assertion is that as long as $BL\le 1$, then a “simple measurement scheme” is sufficient.
> We have assumed that the bandwidth of any “tap function”, $A_\lambda(\cdot)\,[=A(\lambda,\cdot)]$ , is limited to a frequency region of width $B$, say a low-pass region $(-W,W)$ for which $B=2W$. Such band-limited taps are determined according to the Sampling theorem, by their values at the instants $i/2W$, $i=0,\pm 1, \pm 2, \ldots $.
>
> If the memory, $L$, of the filter, $A(\lambda,t)$ is less than $1/2W$ these values are easily determined: we put in unit impulses to $A(\lambda,t)$ at instants $0,\ 1/2W,\ 2/2W, \ldots, T$, and read off from the responses the desired values of the impulse response $A(\lambda,t)$. \[...\] If $L\leq 1/2W$, the responses to the different input impulses do not interfere with one another and the above values can be unambiguously determined.
In other words, sufficiently dense samples of the tap functions can be obtained by sending an impulse train $\sum_n \delta_{n/2W}$ through the channel. Indeed, $$A\big(\sum_n \delta_{n/2W}\big)(t) = \sum_n \int A(\lambda,t)\,\delta_{n/2W}(t-\lambda)\,d\lambda = \sum_n A(t-n/2W,t).$$ Evaluating the operator response at $t=\lambda_0+n_0/2W$, $n_0\in{\mathbb{Z}}$, we obtain $$\begin{aligned}
A\big(\sum_n \delta_{n/2W}\big)(\lambda_0+n_0/2W)
& = & \sum_n A(\lambda_0+(n_0-n)/2W,\lambda_0+n_0/2W) \\
& = & A(\lambda_0,\lambda_0+n_0/2W)\end{aligned}$$ since $L\le 1/2W$ implies that $A(\lambda_0+(n_0-n)/2W,\lambda_0+n_0/2W)=0$ if $n\neq n_0$. In short, for each $\lambda$, the samples $A(\lambda,\lambda+n/2W)$ for $n\in{\mathbb{Z}}$ can be recovered.
The described Kailath sounding procedure is depicted in Figure \[fig:KailathSounding\]. In this visualization, we plot the kernel $\kappa(s,t)=A(t-s,t)$ of the operator $A$, that is, $$Ax(t) = \int A(\lambda,t)\,x(t-\lambda)\,d\lambda = \int A(t-s,t)\,x(s)\,ds = \int \kappa (t,s)\,x(s)\,ds.$$
\[fig:KailathSounding\]
at (0,1)[ ![Kailath sounding of $A$ with ${\operatorname{supp}}\mathcal A (\lambda,f) \subseteq [0,L]\times[-W,W]$ and $L=1/2W$. The kernel $\kappa(t,s)$ is displayed on the $(t,s)$ plane, the impulse train $\sum_n \delta_{n/2W}(s)$ on the $s$-axis, and the output signal $Ax(t)=A\big(\sum_n \delta_{n/2W}\big)(t)
=\sum_n A(t-n/2W,t)=\sum_n\kappa (t,n/2W)$. The sample values of the tab functions $A_\lambda(t)=A(\lambda,t)=\kappa(t,t-\lambda)$ can be read off $Ax(t)$.[]{data-label="fig:KailathSounding"}](KailathSampling1 "fig:"){width="11cm"} ]{}; at (10.5,1.5) [$t$-axis]{}; at (-.6,4)[$s$-axis]{}; at (2.5,4.1)[$\kappa(t,s)$]{}; at (1.2,.8)[$0$]{}; at (-0.4,2.1)[$1/2W{=}L$]{}; at (0.05,3.3)[$2L$]{}; at (-.3,4.5)[$3L$]{}; at (3.8,.9)[$L$]{}; at (6.1,1.1)[$2L$]{}; at (8.4,1.3)[$3L$]{};
Necessity of Kailath’s Condition for Channel Identification. {#section:kailathnecessity}
------------------------------------------------------------
For the “simple measurement scheme” to work, $BL\le 1$ is sufficient but could be restrictive.
> We need, therefore, to devise more sophisticated measurement schemes. However, we have not pursued this question very far because for a certain class of channels we can show that the condition $$L\leq 1/2W, \text{ i.e. }, BL\leq 1$$ is necessary as well as sufficient for unambiguous measurement of $A(\lambda,t)$. The class of channels is obtained as follows: We first assume that there is a bandwidth constraint on the possible input signals to $A(\lambda,t)$, in that the signals are restricted to $(-W_i,W_i)$ in frequency. We can now determine a filter $A_{W_i}(\lambda,t)$ that is equivalent to $A(\lambda,t)$ over the bandwidth $(-W_i,W_i)$, and find necessary and sufficient conditions for unambiguous measurement of $A_{W_i}(\lambda,t)$. If we now let $W_i\to \infty$, this condition reduces to condition (1), viz: $L\leq 1/2W$. Therefore, condition (1) is valid for all filters $A(\lambda,t)$ that may be obtained as the limit of band-limited channels. This class includes almost all filters of physical interest. The argument is worked out in detail in Ref. 6 [^1] but we give a brief outline here.
The class of operators in view here can be described as limits (in some unspecified sense) of operators whose impulse response $A(\lambda,t)$ is bandlimited to $[-W_i,W_i]$ in $\lambda$ for each $t$ and periodic with period $T>0$ in $t$ for each $\lambda$. Here, $T$ is assumed to have some value larger than the maximum time over which the channel will be operated. We could take it as the duration of the input signal to the channel.
The restriction to input signals bandlimited $(-W_i,W_i)$ indicates that it suffices to know the values of $A(\lambda,t)$ or ${\cal A}(\lambda, f)$ for a finite set of values of $\lambda$: $\lambda=0$, $1/2W_i$, $2/2W_i$, $\ldots$, $L$, assuming for simplicity that $L$ is a multiple of $1/2W_i$. Therefore, we can write $$\begin{aligned}
A(\lambda,t) = \sum_n A(n/2W_i,t)\,{\operatorname{sinc}}_{W_i}(\lambda-n/2W_i),\end{aligned}$$ where ${\operatorname{sinc}}_{W_i}(t) = \sin(2\pi W_i t)/(2\pi W_i t)$ so that as $W_i\to\infty$, ${\operatorname{sinc}}_{W_i}(t)$ becomes more concentrated at the origin.
Also, $T$-periodicity in $t$ allows us to write $$A(\lambda,t) = \sum_k A(\lambda,k/T)\,e^{2\pi ikt/T},$$ so that combining gives $$A(\lambda,t) = \sum_n \sum_k A(n/2W_i,k/T)\,{\operatorname{sinc}}_{W_i}(\lambda-n/2W_i)\,e^{2\pi ikt/T}.$$
Based on the restriction to bandlimited input signals which are $T$ periodic, we have obtained a representation of $A$ which is neither compactly supported in $\lambda$ nor bandlimited in $t$. However, the original restriction that $${\operatorname{supp}}\mathcal A (\lambda,f) \subseteq [0,L]\times[-W,W]$$ motivates the assumption that we are working with finite sums, viz. $$A(\lambda,t) = \sum_{n/2W_i\in[0,L]} \sum_{k/t\in[-W,W]} A(n/2W_i,k/T)\,{\operatorname{sinc}}_{W_i}(\lambda-n/2W_i)\,e^{2\pi ikt/T}.$$ This is how the author obtains the estimate that there are at most $(2W_iL+1)(2WT+1)$ degrees of freedom in any impulse response $A$ in the given class. For any input signal $x(t)$ bandlimited to $[-W_i,W_i]$, the output will be bandlimited to $[-W-W_i,W+W_i]$. Specifically, $$\begin{aligned}
Ax(t) & = & \int A(\lambda,t)\,x(t-\lambda)\,d\lambda \\
& = & \sum_{n/2W_i\in[0,L]} \sum_{k/t\in[-W,W]} A(n/2W_i,k/T)\,e^{2\pi ikt/T} \\
& & \qquad\qquad\qquad\int x(t-\lambda)\,{\operatorname{sinc}}_{W_i}(\lambda-n/2W_i)\,d\lambda \\
& = & \sum_{n/2W_i\in[0,L]} \sum_{k/t\in[-W,W]} A(n/2W_i,k/T)\,e^{2\pi ikt/T} \\
& & \qquad\qquad\qquad(x\ast {\operatorname{sinc}}_{W_i})(t-n/2W_i).\end{aligned}$$ Since $e^{2\pi ikt/T}\,(x\ast {\operatorname{sinc}}_{W_i})(t-n/2W_i)$ is bandlimited to $[-W_i,W_i]+(k/T)$ for $k/T\in[-W,W]$, it follows that $Ax(t)$ is bandlimited to $[-W-W_i,W+W_i]$.
If we restrict our attention to signals $x(t)$ time-limited to $[0,T]$, the output signal $Ax(t)$ will have duration $T+L$, and $Ax(\cdot)$ will be completely determined by its samples at $\frac{n}{2(W+W_i)}\in[0,T+L]$, from which we can identify $2(T+L)(W+W_i)+1$ degrees of freedom.
In order for identification to be possible, the number of degrees of freedom of the output signal must be at least as large as the number of degrees of freedom of the operator, i.e. $$\begin{aligned}
2W_iT+2W_i L+2W T+2W L+1&=\\2(T+L)(W_i+W)+1 &\geq (2 W T + 1 ) (2 W_i L + 1) \\ &=2WT+2W_i L+1+4W_i W TL\end{aligned}$$ which reduces ultimately to $$\begin{aligned}
\frac{1}{1-1/(2W_iT)} \geq 2W L = B L.\end{aligned}$$ That is, $BL$ needs to be strictly smaller than $1$ in the approximation while $BL=1$ may work in the limiting case $W_i \to \infty$ (and/or $T \to \infty$).
This result got a lot of attention because it corresponded with experimental evidence that Rake did not function well when the condition $BL<1$ was violated. It led to the designation of “underspread” and “overspread” channels for which $BL$ was less than or greater than 1.
Some Remarks on Kailath’s Results
---------------------------------
This simple argument is surprising, particularly in light of the fact that the author obtained a deep result in time-frequency analysis with none of the tools of modern time-frequency analysis at his disposal. He very deftly uses the extremely useful engineering “fiction” that the dimension of the space of signals essentially bandlimited to $[-W,W]$ and time-limited to $[0,T]$ is approximately $2WT$. The then recent papers of Landau, Slepian and Pollak [@SP61; @LSP61], which are mentioned explicitly in [@Kai59], provided a rigorous mathematical framework for understanding the phenomenon of essentially simultaneous band- and time-limiting. While the existence of these results lent considerable mathematical heft to the argument, they were not incorporated into a fully airtight mathematical proof of his theorem.
> In the proof we have used a degrees-of-freedom argument based on the sampling theorem which assumes strictly bandlimited functions. This is an unrealistic assumption for physical processes. It is more reasonable to call a process band (or time) limited if some large fraction of its energy, say 95%, is contained within a finite frequency (or time) region. Recent work by Landau and Slepian has shown the concept of approximately $2TW$ degrees of freedom holds even in such cases. This leads us to believe that our proof of the necessity of the $BL\leq 1$ condition is not merely a consequence of the special properties of strictly band-limited functions. It would be valuable to find an alternative method of proof.
While Kailath’s Theorem is stated for channel operators whose spreading functions are supported in a rectangle, it is clear that the later work of Bello [@Bel69] was anticipated and more general regions were in view. This is stated explicitly.
> We have not discussed how the bandwidth, B is to be defined. There are several possibilities: we might take the nonzero $f$-region of $\mathcal A(\lambda,f)$; or use a“counting" argument. We could proceed similarly for the definition of $L$. As a result of these several possibilities, the value 1, of the threshold in the condition $BL\leq 1$ should be considered only as an order of magnitude value.
>
> ...constant and predictable variations in $B$ and $L$, due for example to known Doppler shifts or time displacements, would yield large values for the absolute values of the time and frequency spreadings. However such predictable variations should be subtracted out before the $BL$ product is computed; [*what appears to be important is the area covered in the time- and frequency-spreading plane rather than the absolute values of $B$ and $L$.*]{} (emphasis added)
The reference to “counting” as a definition of bandwidth clearly indicates that essentially arbitrary regions of support for the operator spreading function were in view here, and that a necessity argument relying on degrees of freedom and not the shape of the spreading region was anticipated. The third-named author did not pursue the measurement problem studied in his MS thesis because he went on in his PhD dissertation to study the optimum (in the sense of minimum probability of error) detector scheme of which Rake is an intelligent engineering approximation. See [@Kai60; @Kai61; @Kai63].
The mathematical limitations of the necessity proof in [@Kai59] can be removed by addressing the identification problem directly as a problem on infinite-dimensional space rather than relying on finite-dimensional approximations to the channel. This approach also avoids the problem of dealing with simultaneously time and frequency-limited functions. In this way, the proof can be made completely mathematically rigorous. This approach is described in Section \[section:pfandernecessity\].
Bello’s time-variant Channel Identification Condition
-----------------------------------------------------
Kailath’s Theorem was generalized by Bello in [@Bel69] along the lines anticipated in [@Kai59]. Bello’s argument follows that of [@Kai59] in its broad outlines but with some significant differences. Bello clearly anticipates some of the technical difficulties that have been solved more recently by the authors and others and which have led to the general theory of operator sampling.
Continuing with the notation of this section, Bello considers channels with spreading function $\mathcal A (\lambda,f)$ supported in a rectangle $[0,L]\times[-W,W]$. If $L$ and $W$ are all that is known about the channel, then Kailath’s criterion for measurability requires that $2WL\le 1$. Bello considers channels for which $2WL$ may be greater than $1$ but for which $$S_A = |{\operatorname{supp}}\mathcal A (\lambda,f)| \le 1$$ and argues that this is the most appropriate criterion to assess measurability of the channel modeled by $A$.
In order to describe Bello’s proof we will fix parameters $T\gg L$ and $W_i\gg W$ and following the assumptions earlier in this section, assume that inputs to the channel are time-limited to $[0,T]$ and (approximately) bandlimited to $[-W_i,W_i]$. Under this assumption, Bello considers the spreading function of the channel to be approximated by a superposition of point scatterers, viz. $$\mathcal A (\lambda,f) = \sum_n\sum_k A_{n,k}\,\delta(f-(k/T))\,\delta(\lambda-(n/2W_i)).$$ Hence the response of the channel to an input $x(\cdot)$ is given by $$\begin{aligned}
\label{eqn:belloresponse}
Ax(t) & = & \int\!\!\!\int x(t-\lambda)\,e^{2\pi i f(t-\lambda)}\mathcal A (\lambda,f)\,d\lambda\,df \\ \notag
& = & \sum_n\sum_k A_{n,k}\,x(t-(n/2W_i))\,e^{2\pi i(k/T)(t-(n/2W_i))}.\end{aligned}$$ Note that this is a continuous-time Gabor expansion with window function $x(\cdot)$ (see, e.g., [@Gro01]). By standard density results in Gabor theory, the collection of functions $\set{x(t-(n/2W_i))\,e^{2\pi i(k/T)(t-(n/2W_i))}}$ is overcomplete as soon as $2TW_i>1$. Consequently, without further discretization, the coefficients $A_{n,k}$ are in principle unrecoverable. Taking into consideration support constraints on $\mathcal A$, we assume that the sums are finite, viz. $$\bigg(\frac{n}{2W_i},\frac{k}{T}\bigg) \in {\operatorname{supp}}\mathcal A.$$ Hence determining the channel characteristics amounts to finding $A_{n,k}$ for those pairs $(n,k)$. It should be noted that for a given spreading function $\mathcal A (\lambda,f)$ for which ${\operatorname{supp}}\mathcal A$ is a Lebesgue measurable set, given $\epsilon>0$, there exist $T$ and $W_i$ sufficiently large that the number of such $(n,k)$ is no more than $2 S_A W_i T(1+\epsilon)$. On the other hand, for a given $T$ and $W_i$, there exist spreading functions $\mathcal A (\lambda,f)$ with arbitrarily small non-convex $S_A$ for which the number of nonzero coefficients $A_{n,k}$ can be large. For example, given $T$ and $W_i$, $S_A$ could consist of rectangles centered on the points $(n/(2W_i), k/T)$ with arbitrarily small total area.
By sampling, (\[eqn:belloresponse\]) reduces to a discrete, bi-infinite linear system, viz. $$\label{eqn:belloresponsediscrete}
Ax\bigg(\frac{p}{2W_i}\bigg) = \sum_n \sum_k A_{n,k}\,x\bigg(\frac{p-n}{2W_i}\bigg)\,e^{2\pi i\frac{k}{T}(\frac{p-n}{2W_i})}$$ for $p\in{\mathbb{Z}}$. Note that (\[eqn:belloresponsediscrete\]) is the expansion of a vector in a discrete Gabor system on $\ell^2({\mathbb{Z}})$, a fact not mentioned by Bello, and of which he was apparently unaware. Specifically, defining the translation operator $\mathcal T$ and the modulation operator $\mathcal M$ on $\ell^2$ by $$\label{eqn:translationandmodulation}
\mathcal Tx(n) = x(n-1),\qquad{\mbox{\rm and}}\qquad \mathcal Mx(n) = e^{\pi i n/(TW_i)}x(n),$$ (\[eqn:belloresponsediscrete\]) can be rewritten as $$\label{eqn:belloresponsediscretegabor}
Ax\bigg(\frac{p}{2W_i}\bigg) = \sum_n \sum_k (\mathcal T^n\,\mathcal M^k x)(p)\,A_{n,k}.$$ Since there are only finitely many nonzero unknowns in this system, Bello’s analysis proceeds by looking at finite sections of (\[eqn:belloresponsediscretegabor\]) and counting degrees of freedom.
[*Necessity.*]{} Following the lines of the necessity argument in [@Kai59], we note that there are at least $2(T+L)(W+W_i)$ degrees of freedom in the output vector $Ax(t)$, that is, at least that many independent samples of the form $Ax(p/2W_i)$, and as observed above, no more than $2 S_A W_i T(1+\epsilon)$ nonzero unknowns $A_{n,k}$. Therefore, in order for the $A_{n,k}$ to be determined in principle, it must be true that $$2 W_i T(1+\epsilon) S_A \le 2(T+L)(W+W_i)$$ or $$S_A \le \frac{(T+L)(W+W_i)}{W_i T(1+\epsilon)}.$$ Letting $T,\,W_i\to\infty$ and $\epsilon\to 0$, we arrive at $S_A\le 1$.
[*Sufficiency.*]{} Considering a section of the system (\[eqn:belloresponsediscretegabor\]) based on the assumption that ${\operatorname{supp}}\mathcal A \subseteq [0,L]\times[-W,W]$, the system has approximately $2W_i(T+L)$ equations in $(2W_iT)(2WL)$ unknowns. Since $L$ and $2W$ are simply the dimensions of a rectangle that encloses the support of $\mathcal A$, $2WL$ may be quite large and independent of $S_A$. Hence the system will not in general be solvable. However by assuming that $S_A<1$, only approximately $S_A(2W_iT)$ of the $A_{n,k}$ do not vanish and the system reduces to one in which the number of equations is roughly equal to the number of unknowns. In this case it would be possible to solve (\[eqn:belloresponsediscretegabor\]) as long as the collection of appropriately truncated vectors $\{\mathcal T^n \mathcal M^k x\colon A_{n,k}\ne 0\}$ forms a linearly independent set for some vector $x$.
In his paper, Bello was dealing with independence properties of discrete Gabor systems apparently without realizing it, or at least without stating it explicitly. Indeed, he argues in several different ways that a vector $x$ that produces a linearly independent set should exist, and intriguingly suggests that a vector consisting of $\pm 1$ should exist with the property that the Grammian of the Gabor matrix corresponding to the section of (\[eqn:belloresponsediscretegabor\]) being considered is diagonally dominant. The setup chosen below to prove Bello’s assertion leads to the consideration of a matrix whose columns stem from a Gabor system on a finite-dimensional space, not on a sequence space.
Operator Sampling {#section:operatorsampling}
=================
The first key contribution of operator sampling is the use of frame theory and time-frequency analysis to remove assumptions of simultaneous band- and time-limiting, and also to deal with the infinite number of degrees of freedom in a functional analytic setting (Section \[section:Operator classes and operator identification\]). A second key insight is the development of a “simple measurement scheme” of the type used by the third-named author but that allows for the difficulties identified by Bello to be resolved. This insight is the use of periodically-weighted delta-trains as measurement functions for a channel. Such measurement functions have three distinct advantages.
First, they allow for the channel model to be essentially arbitrary and clarify the reduction of the operator identification problem to a finite-dimensional setting without imposing a finite dimensional model that approximates the channel. Second, it combines the naturalness of the simple measurement scheme described earlier with the flexibility of Bello’s idea for measuring channels with arbitrary spreading support. Third, it establishes a connection between identification of channels and finite-dimensional Gabor systems and allows us to determine windowing vectors with appropriate independence properties.
In Section \[section:Operator classes and operator identification\], we introduce some operator-theoretic descriptions of some of the operator classes that we are able to identify, and discuss briefly different ways of representing such operators. Such a discussion is beneficial in several ways. First, it contains a precise definition of identifiability, which comes into play when considering the generalization of the necessity condition for so-called overspread channels (Section \[section:pfandernecessity\]). Second, we can extend the necessity condition to a very large class of inputs. In other words, we can assert that in a very general sense, no input can identify an overspread channel. Third, it allows us to include both convolution operators and multiplication operators (for which the spreading functions are distributions) in the operator sampling theory. The identification of multiplication operators via operator sampling reduces to the classical sampling formula, thereby showing that classical sampling is a special case of operator sampling. In Section \[section:pfandernecessity\] we present a natural formalization of the original necessity proof of [@Kai59] (Section \[section:kailathnecessity\]) to the infinite-dimensional setting, which involves an interpretation of the notion of an under-determined system to that setting. Finally, in Section \[section:mainidentification\] we present the scheme given first in [@PW06b; @PW13] for the identification of operator classes using periodically-weighted delta trains and techniques from modern time-frequency analysis.
Operator classes and operator identification {#section:Operator classes and operator identification}
--------------------------------------------
We formally consider an arbitrary operator as a [*pseudodifferential operator*]{} represented by $$\begin{aligned}
\label{eqn:operator1}
Hf(x) = \int \sigma_H(x,\xi)\widehat{f}(\xi)\,e^{2\pi ix\xi}\,d\xi,\end{aligned}$$ where $\sigma_H(x,\xi)\in L^2({\mathbb{R}}^2)$ is the [*Kohn-Nirenberg*]{} (KN) symbol of $H$. The [*spreading function*]{} $\eta_H(t,\nu)$ of the operator $H$ is the [*symplectic Fourier transform*]{} of the KN symbol, viz. $$\begin{aligned}
\label{eqn:operator2}\eta_H(t,\nu) = \int\!\!\!\!\int \sigma_H(x,\xi)\,e^{-2\pi i(\nu x - \xi t)}\,dx\,d\xi\end{aligned}$$ and we have the representation $$\begin{aligned}
\label{eqn:operator3}Hf(x) = \int\!\!\!\!\int \eta_H(t,\nu)\,\mathcal{T}_t\,\mathcal{M}_\nu f(x)\,d\nu\,dt\end{aligned}$$ where $\mathcal{T}_tf(x) = f(x-t)$ is the [*time-shift operator*]{} and $\mathcal{M}_\nu f(x) = e^{2\pi i \nu x}\,f(x)$ is the [*frequency-shift operator*]{}.
This is identical to the representation given in [@Kai59] where $\eta_H(t,\nu) = \mathcal A(\nu,t)$, see Section \[section:KailathSufficient\].
To see more clearly where the spreading function arises in the context of communication theory, we can define the [*impulse response*]{} of the channel modeled by $H$, denoted $h_H(x,t)$, by $$Hf(x) = \int h_H(x,t)\,f(x-t)\,dt.$$ Note that if $h_H$ were independent of $x$, then $H$ would be a convolution operator and hence a model for a time-invariant channel. In fact, with $\kappa_H(x,t)$ being the [*kernel*]{} of the operator $H$, $$\begin{aligned}
Hf(x) &= \int \kappa_H(x,t)\,f(t)\,dt \\
&= \int h_H(x,t)\,f(x-t)\,dt \\
&= \iint \eta_H(t,\nu)\,e^{2\pi i\nu (x-t)}\,f(x-t)\,d\nu\,dt\label{eqn:operatorrepresentations1} \\
&= \int \sigma_H(x,\xi)\, \widehat f(\xi)\,e^{2\pi i x \xi} d\xi,\label{eqn:operatorrepresentations2}\end{aligned}$$ where $$\begin{aligned}
h_H(x,t) &= \kappa_H(x,x-t) \nonumber \\
&= \int \sigma_H (x,\xi)\, e^{2\pi i \xi t}\, d\xi, \nonumber \\
&= \int \eta_H(t,\nu)\, e^{2\pi i \nu (x-t)}\, d\nu. \label{eqn:symbolrelations}\end{aligned}$$ With this interpretation, the maximum support of $\eta_H(t,\nu)$ in the first variable corresponds to the maximum spread of a delta impulse sent through the channel and the maximum support of $\eta_H(t,\nu)$ in the second variable corresponds to the maximum spread of a pure frequency sent through the channel.
Since we are interested in operators whose spreading functions have small support, it is natural to define the following operator classes, called [*operator Paley-Wiener spaces*]{} (see [@Pfa10]).
\[generaloperatorpaleywienerspaces\] For $S\subseteq {\mathbb{R}}^2$, we define the operator Paley-Wiener spaces $OPW(S)$ by $$\begin{aligned}
OPW(S) & = \{H\in \mathcal L (L^2({\mathbb{R}}), L^2({\mathbb{R}})) \colon \ {\operatorname{supp}}\eta_H\subseteq S,\,\norm{\sigma_H}_{L^{2}}<\infty\}.\end{aligned}$$
\[rem:generaloperatorpaleywienerspaces\] In [@Pfa10; @PW06], the spaces $OPW^{p,q}(S)$, $1\le p,\,q<\infty$, were considered, where $L^2$-membership of $\sigma_H$ is replaced $$\norm{\sigma_H}_{L^{p,q}} = \Big(\int\Big(\int\abs{\sigma_H (x,\xi)}^q d\xi\Big)^{p/q}\,dx\Big)^{1/p}$$ with the usual adjustments made when either $p=\infty$ or $q=\infty$. $OPW^{p,q}(S)$ is a Banach space with respect to the norm $\norm{H}_{OPW^{p,q}} = \norm{\sigma_H}_{L^{p,q}}$. Note that if $S$ is bounded, then $OPW^{\infty,\infty}(S)$ consists of all bounded operators whose spreading function is supported on $S$. In fact, the operator norm is then equivalent to the $OPW^{\infty,\infty}(S)$ norm, where the constants depend on $S$ [@KP12].
The general definition is beneficial since it also allows the inclusion of convolution operators with kernels whose Fourier transforms lie in $L^q({\mathbb{R}})$ ($OPW^{\infty,q}({\mathbb{R}})$) and multiplication operators whose multiplier is in $L^p({\mathbb{R}})$ ($OPW^{p,\infty}({\mathbb{R}})$).
The goal of operator identification is to find an input signal $g$ such that each operator $H$ in a given class is completely and stably determined by $Hg$. In other words, we ask that the operator $H\mapsto Hg$ be continuous and bounded below on its domain. In our setting, this translates to the existence of $c_1,\,c_2 >0$ such that $$\label{eqn:boundedbelow}
c_1\,\|\sigma_H\|_{L^2} \le \|Hg\|_{L^2}\le c_2\,\|\sigma_H\|_{L^2},\quad H\in OPW(S).$$ This definition of identifiability of operators originated in [@KP06]. Note that (\[eqn:boundedbelow\]) implies that the mapping $H\mapsto Hg$ is [*injective*]{}, that is, that $Hg=0$ implies that $H\equiv 0$, but is not equivalent to it. The inequality (\[eqn:boundedbelow\]) adds to injectivity the assertion that $H$ is also stably determined by $Hg$ in the sense that a small change in the output $Hg$ would correspond to a small change in the operator $H$. Such stability is also necessary for the existence of an algorithm that will reliably recover $H$ from $Hg$. In this scheme, $g$ is referred to as an [*identifier*]{} for the operator class $OPW(S)$ and if (\[eqn:boundedbelow\]) holds, we say that [*operator identification*]{} is possible.
In trying to find an explicit expression for an identifier, we use as a starting point the “simple measurement scheme” of [@Kai59], in which $g$ is a delta train, viz. $g = \sum_n\delta_{nT}$ for some $T>0$. In the framework of operator identification the channel measurement criterion in [@Kai59] takes the following form [@KP06; @PW06b; @Pfa10].
\[thm:main-simple\] For $H\in OPW \big([0, T] {\times}[- \Omega / 2, \Omega / 2]\big)$ with $T\Omega{\leq} 1$, we have $$\begin{aligned}
\|H\sum_{k\in{\mathbb{Z}}}\delta_{kT}\|_{L^2({\mathbb{R}})}=T\|\sigma_H\|_{L^2},\notag \end{aligned}$$ and $H$ can be reconstructed by means of $$\begin{aligned}
\kappa_H(x+t,x)=\chi_{[0,T]}(t)\sum_{n\in{\mathbb{Z}}} \big(H\sum_{k\in{\mathbb{Z}}}\delta_{kT}\big)(t+nT)\, \frac{\sin(\pi T (x-n))}{\pi T (x-n)} \label{eqn:operatorreconstruction-simple}
\end{aligned}$$ where $\chi_{[0,T]}(t)=1$ for $t\in[0,T]$ and $0$ elsewhere and with convergence in the $L^2$ norm and uniformly in $x$ for every $t$.
As was observed earlier, the key feature of this scheme is that the spacing of the deltas in the identifier is sufficiently large so as to allow the response of the channel to a given delta to “die out” before the next delta is sent. In other words, the parameter $T$ must exceed the time-spread of the channel. On the other hand, the rate of change of the channel, as measured by its bandwidth $\Omega$, must be small enough that its impulse response can be recovered from “samples” of the channel taken $T$ time units apart. In particular, the samples of the impulse response $T$ units apart can be easily determined from the output. In the general case considered by Bello, in which the spreading support of the operator is not contained in a rectangle of unit area, this intuition breaks down.
Specifically, suppose that we consider the operator class $OPW(S)$ where $S\subseteq[0,T_0]\times[-\Omega_0/2,\Omega_0/2]$ and $T_0\Omega_0\gg 1$ but where $|S|<1$. Then sounding the channel with a delta train of the form $g=\sum_n \delta_{nT_0}$ would severely [*undersample*]{} the impulse response function. Simply increasing the sampling rate, however, would produce overlap in the responses of the channel to deltas close to each other. An approach to the undersampling problem in the literature of classical sampling theory is to sample at the low rate transformed versions of the function, chosen so that the interference of the several undersampled functions can be dealt with. This idea has its most classical expression in the Generalized Sampling scheme of Papoulis [@Pa77]. Choosing shifts and constant multiples of our delta train results in an identifier of the form $g=\sum_n c_n\,\delta_{nT}$ where the weights $(c_n)$ have period $P$ (for some $P\in{\mathbb{N}}$) and $T>0$ satsifies $PT>T_0$.
If $g$ is discretely supported (for example, a periodically-weighted delta-train), then we refer to operator identification as [*operator sampling*]{}. The utility of periodically-weighted delta trains for operator identification is a cornerstone of operator sampling and has far-reaching implications culminating in the developments outlined in Sections \[section:higherdimensional\] and \[section:outlook\].
Kailath’s necessity proof and operator identification {#section:pfandernecessity}
-----------------------------------------------------
In Section \[section:kailathnecessity\] we presented the proof of the necessity of the condition $BL\le 1$ for channel identification as given in [@Kai59]. The argument consisted of finding a finite-dimensional approximation of the channel $H$, and then showing that, given any putative identifier $g$, the number of degrees of freedom present in the output $Hg$ must be at least as large as the number of degrees of freedom in the channel itself. For this to be true in any finite-dimensional setting, we must have $BL<1$ and so in the limit we require $BL\le 1$. In essence, if $BL>1$, we have a linear system with fewer equations than unknowns which necessarily has a nontrivial nullspace. The generalization of this notion to the infinite-dimensional setting is the basis of the necessity proof that appears in [@KP06]. In this section, we present an outline of that proof, and show how the natural tool for this purpose once again comes from time-frequency analysis.
To see the idea of the proof, assume that $BL>1$ and for simplicity let $S=[-\frac L 2,\frac L 2]\times[-\frac B 2,\frac B 2]$. The goal is to show that for any sounding signal $s$ in an appropriately large space of distributions[^2], the operator $\Phi_s\colon OPW(S)\longrightarrow L^2({\mathbb{R}})$, $H\mapsto Hs$, is not stable, that is, it does not possess a lower bound in the inequality (\[eqn:boundedbelow\]).
First, define the operator $E\colon l_0({\mathbb{Z}}^2)\longrightarrow OPW(S)$, where $l_0({\mathbb{Z}}^2)$ is the space of finite sequences equipped with the $l^2$ norm, by $$E(\sigma) = E(\set{\sigma_{k,l}})
= \sum_{k,l}\sigma_{k,l} \mathcal M_{k\lambda/L}\mathcal T_{l\lambda/B}\,P\,\mathcal T_{-l\lambda/B}\mathcal M_{-k\lambda/L}$$ where $1<\lambda$ is chosen so that $1<\lambda^4<BL$ and where $P$ is a time-frequency localization operator whose spreading function $\eta_P(t,\nu)$ is infinitely differentiable, supported in $S$, and identically one on $[-\frac L {2\lambda},\frac L {2\lambda}]\times[-\frac B {2\lambda},\frac B {2\lambda}]$. It is easily seen that the operator $E$ is well-defined and has spreading function $$\eta_{E(\sigma)}(t,\nu) = \eta_P(t,\nu)\,\sum_{k,l}\sigma_{k,l}\,e^{2\pi i(k\lambda t/L - l\lambda\nu/B)}.$$ By construction, it follows that for some constant $c_1$, $\norm{E(\sigma)}_{OPW(S)}\ge c_1\norm{\sigma}_{l^2({\mathbb{Z}}^2)}$, for all $\sigma$, and that for any distribution $s$, $Ps$ decays rapidly in time and in frequency.
Next define the Gabor analysis operator $C_g\colon L^2({\mathbb{R}})\longrightarrow l^2({\mathbb{Z}}^2)$ by $$C_g(s) = \set{\ip{s}{\mathcal M_{k\lambda^2/L}\mathcal T_{l\lambda^2/B}g}}_{k,l\in{\mathbb{Z}}}$$ where $g(x)=e^{-\pi x^2}$. A well-known theorem in Gabor theory asserts that $\set{\mathcal M_{k\alpha}\mathcal T_{l\beta}g}_{k,l\in{\mathbb{Z}}}$ is a Gabor frame for $L^2({\mathbb{R}})$ for every $\alpha\beta<1$ ([@Lyu92; @SW92; @Sei92b]). Consequently $C_g$ satisfies, for some $c_2>0$, $\norm{C_g(s)}_{l^2({\mathbb{Z}}^2)}\ge c_2\,\norm{s}_{L^2({\mathbb{R}})}$ for all $s$, since $\lambda^2/L\,\cdot\lambda^2/B = \lambda^4/BL < 1$.
For any $s$, consider the composition operator $$C_g\circ\Phi_s\circ E \colon l_0({\mathbb{Z}}^2) \longrightarrow l^2({\mathbb{Z}}^2).$$ The crux of the proof lies in showing that this composition operator is not stable, that is, it does not have a lower bound. Since $C_g$ and $E$ are both bounded below, it follows that $\Phi_s$ cannot be stable. Since $s\in S'_0({\mathbb{R}})$ was arbitrary, this completes the proof.
To complete this final step we examine the canonical bi-infinite matrix representation of the above defined composition of operators, that is, the matrix $M=(m_{k',l',k,l})$ that satisfies $$(C_g\circ\Phi_s\circ E(\sigma))_{k',l'} = \sum_{k,l} m_{k',l',k,l}\,\sigma_{k,l}.$$ It can be shown that $M$ has the property that for some rapidly decreasing function $w(x)$, $$\label{eqn:matrixdecay}
\abs{m_{k',l',k,l}} \le w(\max\set{\abs{\lambda k'-k},\abs{\lambda l'-l}}).$$ The proof is completed by the following Lemma. Its proof can be found in [@KP06] and generalizations can be found in [@Pfa05].
\[fig:skewmatrix\]
in [1,...,22]{}[ ]{} in [1,...,12]{}[ ]{} (m3)\[matrix anchor= north west, matrix of math nodes,left delimiter=, right delimiter=, nodes in empty cells, row sep=0.2em, minimum size=1.3em, ampersand replacement=&\] [ ]{}; [\[loosely dotted, shorten >=15pt, shorten <=15pt, thick\] (m3-2-1.north west) – (m3-4-5.north west); (m3-5-7.north west) – (m3-7-11.north west); (m3-8-13.north west) – (m3-10-17.north west); (m3-10-18.south east) – (m3-12-22.south east); ]{}
[myback]{} plot\[ybar\] (, [ 4\*pow(abs()+1.2, -2]{} ); plot\[ybar\] (, [ 4\*pow(abs()+1.2, -2]{} ); plot\[ybar\] (, [ 4\*pow(abs()+1.2, -2]{} );
\[lem:instability\] Given $M=(m_{j',j})_{j',j\in{\mathbb{Z}}^2}$. If there exists a monotonically decreasing function $w\colon R^+_0\longrightarrow R^+_0$ with $w=O(x^{-2-\delta})$, $\delta>0$, and constants $\lambda>1$ and $K_0>0$ with $\abs{m_{j',j}}<w(\norm{\lambda j'-j}_\infty)$ for $\norm{\lambda j'-j}_\infty>K_0$, then $M$ is not stable.
Intuitively, this result asserts that a bi-infinite matrix whose entries decay rapidly away from a skew diagonal behaves like a finite matrix with more rows than columns (see Figure \[fig:skewmatrix\]). Such a matrix will always have a nontrivial nullspace. In the case of an infinite matrix what can be shown is that at best its inverse will be unbounded.
We can make a more direct connection from this proof to the original necessity argument in [@Kai59] in the following way. If we restrict our attention to sequences $\{\sigma_{k,l}\}$ with a fixed finite support of size say $N$, then the image of this subspace of sequence space under the mapping $E$ is an $N$-dimensional subspace of $OPW(S)$. The operator $P$ is essentially a time-frequency localization operator. This fact is established in [@KP06] and follows from the rapid decay of the Fourier transform of $\eta_P$. Since $\eta_P$ itself is concentrated on a rectangle of area $BL/\lambda^2$, its Fourier transform will be concentrated on a rectangle of area $\lambda^2/BL$. From this it follows that for $\sigma$ as described above, the operator $E(\sigma)$ essentially localizes a function to a region in the time-frequency plane of area $N(\lambda^2/BL)$.
Considering now the Gabor analysis operator $C_g$, we observe that the Gaussian $g(x)$ essentially occupies a time-frequency cell of area $1$, and that this function is shifted in the time-frequency plane by integer multiples of $(\lambda^2/B, \lambda^2/L)$. Hence to “cover” a region in the time-frequency plane of area $N(\lambda^2/BL)$ would require only about $$\frac{N(\lambda^2/BL)}{\lambda^4/BL} = \frac{N}{\lambda^2}$$ time-frequency shifts. So roughly speaking, in order to resolve $N$ degrees of freedom in the operator $E({\sigma_{k,l}})$, we have only $N/\lambda^2 < N$ degrees of freedom in the output of the operator $E({\sigma_{k,l}})s$.
Identification of operator Paley-Wiener spaces by periodically weighted delta-trains {#section:mainidentification}
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Theorem \[thm:main-simple\] is based on arguments outlined in Section \[section:KailathSufficient\] and applies only to $OPW(S)$ if $S$ is contained in a rectangle of area less than or equal to one. In the following, we will develop the tools that allow us to identify $OPW(S)$ for any compact set $S$ of Lebesgue measure less than one.
In our approach we discretize the channel by covering the spreading support $S$ with small rectangles of fixed sidelength, which we refer to as a [*rectification*]{} of $S$. As long as the measure of $S$ is less than one, it is possible to do this in such a way that the total area of the rectangles is also less than one. This idea seems to bear some similarity to Bello’s philosophy of sampling the spreading function on a fixed grid but with one fundamental difference. Bello’s approach is based on replacing $t$ and $x$ by samples, thereby approximating the channel. For a better approximation, sampling on a finer grid is necessary, which results in a larger system of equations that must be solved. In our approach, as soon as the total area of the rectification is less than one, the operator modeling the channel is completely determined by the discrete model. Once this is achieved, identification of the channel reduces to solving a single linear system of equations at each point.
{height="4cm"}{height="4cm"}
Given parameters $T>0$ and $P\in{\mathbb{N}}$, we assume that $S$ is rectified by rectangles of size $T\times\Omega$, where $\Omega = 1/(TP)$, such that the total area of the rectangles is less than one. Given a period-$P$ sequence $c=(c_n)_{n\in{\mathbb{Z}}}$, we then define the [*periodically weighted delta-train*]{} $g$ by $g = \sum_{n\in{\mathbb{Z}}} c_n\,\delta_{nT}$. The goal of this subsection is to describe the scheme by which a linear system of $P$ equations in a priori $P^2$ unknowns can be derived by which an operator $H\in OPW(S)$ can be completely determined by $Hg(x)$. In this sense, the “degrees of freedom” in the operator class $OPW(S)$, and that of the output function $Hg(x)$ are precisely defined and can be effectively compared.
The basic tool of time-frequency analysis that makes this possible is the [*Zak transform*]{} (see [@Gro01]).
\[def:zaktransform\] The non-normalized Zak Transform is defined for $f\in{\cal S}({\mathbb{R}})$[^3], and $a>0$ by $$\displaystyle{Z_a f(t,\nu) = \sum_{n\in{\mathbb{Z}}} f(t-an)\,e^{2\pi ia n\nu}}.$$
$Z_af(t,\nu)$ satisfies the quasi-periodicity relations $$\displaystyle{Z_af(t+a,\nu) = e^{2\pi ia\nu}\,Z_af(t,\nu)}$$ and $$\displaystyle{Z_af(t,\nu+1/a) = Z_af(t,\nu)}.$$ $\sqrt{a}\,Z_a$ can be extended to a unitary operator from $L^2({\mathbb{R}})$ onto $L^2([0,a]{\times}[0,1/a])$.
A somewhat involved but elementary calculation yields the following (see [@PW14] and Section \[section:proof of lemma\]).
\[lem:matrixequationquasiperiodic\] Let $T>0$, $P\in{\mathbb{N}}$, $c=(c_n)$, and $g$ be given as above. Then for all $(t,\nu)\in{\mathbb{R}}^2$, and $p=0,\,1,\,\dots,\,P{-}1$, $$\begin{aligned}
\label{eqn:matrixequationquasiperiodic}
& e^{-2\pi i\nu T p}\,(Z_{TP}\circ H)g(t + T p,\nu) \nonumber\\
& = \Omega\,\sum_{q,\,m=0}^{P-1} (T^q\,M^m c)_p\,
e^{-2\pi i\nu T q}\,\eta^{Q}_H(t + T q,\nu + m/TP).
\end{aligned}$$
Here $\mathcal T$ and $\mathcal M$ are the translation and modulation operators given in Definition \[def:translationmodulation\], and $\eta^{Q}_H(t,\nu)$ is the [*quasiperiodization*]{} of $\eta_H$, $$\label{eqn:quasiperiodization}
\eta^{Q}_H(t,\nu) = \sum_k\sum_\ell \eta_H(t+kTP,\nu+\ell/T)\,e^{-2\pi i\nu kTP}$$ whenever the sum is defined.
\[fig:PWSounding\]
at (0,1)[ ![Channel sounding of $OPW([0,2/3]{\times}[-1/4,1/4]\, \cup\, [4/3,2]{\times}[-1/2,1/2])$ using a $P$-periodically weighted delta train $g$. The kernel $\kappa(x,y)$ takes values on the $(x,y)$-plane, the sounding signal $g$, a weighted impulse train, is defined on the $y$-axis, and the output signal $Hg(x)=\int \kappa(x,y)g(y)dy$ is displayed on the $x$-axis. Here, the sample values of the tab functions $h(x,t)=\kappa(x,t-x)$ are not easily read of the response $Hg(x)$ as, for example for $x\in [2T,3T]=[4/3,2]$ we have $Hg(x)=0.7\kappa(x,0)+0.6\kappa(x,2T)=0.7h(x,x)+.6h(x,2T-x)$. In detail, we have $g = \ldots +0.7\delta_{-2}+0.5\delta_{-4/3}+ 0.6\delta_{-2/3} +0.7\delta_0+0.5\delta_{2/3}+ 0.6\delta_{4/3}+0.7\delta_{2}+ 0.5\delta_{8/3}+\ldots$, so $P=3$, $T=2/3$, $\Omega=1/PT=1/2$, $c_n=0.7$ if $n \!\! \mod 3=0$, $c_n=0.5$ if $n\!\!\mod 3=1$, $c_n=0.6$ if $n\!\! \mod 3=2$. .](PWSampling.png "fig:"){width="11cm"} ]{}; at (10,1.5) [$x$-axis]{}; at (-.5,4.1)[$y$-axis]{}; at (3.8,4.4)[$\kappa(x,y)$]{}; at (1.5,.8)[$0$]{}; at (-0.1,1.8)[$1/P\Omega{=}T$]{}; at (.45,2.75)[$2T$]{}; at (0.13,3.7)[$3T$]{}; at (-.2,4.6)[$4T$]{}; at (3,.85)[$T$]{}; at (4.5,1)[$2T$]{}; at (6.05,1.15)[$3T$]{}; at (7.6,1.28)[$4T$]{}; at (9.15,1.43)[$5T$]{};
Under the additional simplifying assumption that the spreading function $\eta_H(t,\nu)$ is supported in the large rectangle $[0,TP]\times [0,1/T]$, and by restricting (\[eqn:matrixequationquasiperiodic\]) to the rectangle $[0,T]\times[0,1/(TP)]$, we arrive at the $P\times P^2$ linear system $$\label{eqn:basiclinearsystem}
{\bf Z}_{Hg}(t,\nu)_p = \sum_{q,m=0}^{P-1} G(c)_{p,(q,m)}\,\boldsymbol\eta_H(t,\nu)_{(q,m)}$$ where $$\label{eqn:boldZvector}
{\bf Z}_{Hg}(t,\nu)_p = (Z_{TP}\circ H)g(t+pT,\nu)\,e^{-2\pi i \nu pT},$$ $$\label{eqn:boldetavector}
\boldsymbol\eta_H(t,\nu)_{(q,m)} = \Omega\,\eta_H(t+qT,\nu+m/TP)\,e^{-2\pi i \nu qT}\,e^{-2\pi iqm/P},$$ and where $G(c)$ is a finite Gabor system matrix (\[def:fullgabormatrix\]). If (\[eqn:basiclinearsystem\]) can be solved for each $(t,\nu)\in [0,T]\times[0,1/(TP)]$, then the spreading function for an operator $H$ can be completely determined by its response to the periodically-weighted delta-train $g$.
As anticipated by Bello, two issues now become relevant. (1) We require that ${\operatorname{supp}}\eta_H$ occupy no more than $P$ of the shifted rectangles $[0,T]\times[0,1/(TP)]+(qT,k/(TP))$, so that (\[eqn:basiclinearsystem\]) has at least as many equations as unknowns. This forces $|{\operatorname{supp}}\eta_H|\le 1$. (2) We require that $c$ be chosen in such a way that the $P\times P$ system formed by removing the columns of $G(c)$ corresponding to vanishing components of $\boldsymbol\eta_H$ is invertible. That such $c$ exist is a fundamental cornerstone of operator sampling and is the subject of the next section.
Based on the existence of $c$ such that any set of $P$ columns of $G(c)$ form a linearly independent set, we can prove the following [@PW13].
\[thm:reconstruction\] For $S\subseteq (0,\infty){\times}{\mathbb{R}}$ compact with $\abs{S}<1$, there exists $T>0$ and $P\in{\mathbb{N}}$ , and a period-$P$ sequence $c=(c_n)$ such that $g=\sum_n c_n\,\delta_{nT}$ identifies $OPW(S)$. In particular, there exist period-$P$ sequences $b_j=(b_{j,k})$, and integers $0\le q_j,\,m_j\le P{-}1$, for $0\le j\le P{-}1$ such that $$\begin{aligned}
h(x,t) &= e^{-\pi it/T}\sum_k\sum_{j=0}^{P-1} \big[b_{j,k}\,Hg(t - (q_j-k)T) \nonumber \\
& \hskip-.25in e^{2\pi i m_j(x-t)/PT}\,\phi((x-t)+(q_j-k)T)\, r(t-q_j T)\big] \label{eqn:reconstructionformula}\end{aligned}$$ where $r,\phi\in {\cal S}({\mathbb{R}})$ satisfy $$\label{eq:r_phi_2}
\sum_{k\in{\mathbb{Z}}} r(t + kT) = 1 = \sum_{n\in{\mathbb{Z}}} \widehat{\phi}(\gamma + n/PT),$$ where $r(t)\widehat{\phi}(\gamma)$ is supported in a neighborhood of $[0,T]{\times}[0,1/PT]$, and where the sum in converges unconditionally in $L^{2}$ and for each $t$ uniformly in $x$.
Equation is a generalization of which is easily seen by choosing $\phi(x)=\sin(\pi PTx)/(\pi PTx)$ and $r(t)$ to be the characteristic function of $[0,T)$.
Linear Independence Properties of Gabor Frames {#section:finiteGabor}
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Finite Gabor Frames
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\[def:translationmodulation\] Given $P\in{\mathbb{N}}$, let $\omega=e^{2\pi i/P}$ and define the [*translation operator $\mathcal T$*]{} on $(x_0,\,\dots,\,x_{P-1})\in\mathbb C^P$ by $$\mathcal{T}x=(x_{P-1},x_0,\,x_1,\,\ldots,x_{P-2}),$$ and the [*modulation operator $\mathcal M$*]{} on $\mathbb C^P$ by $$\mathcal{M}x=(\omega^0 x_0, \omega^1 x_1,\,\dots,\, \omega^{P-1} x_{P-1}).$$ Given a vector $c\in\mathbb C^P$ the [*finite Gabor system with window $c$*]{} is the collection $\set{\mathcal{T}^q \mathcal{M}^p c}_{q,p=0}^{P-1}$. Define the [*full Gabor system matrix*]{} $G(c)$ to be the $P\times P^2$ matrix $$\label{def:fullgabormatrix}
G(c) = \left[\,\, D_0\,W_P\,\,\vrule\,\, D_1\,W_P\,\, \vrule \,\,\cdots \,\, \vrule\,\, D_{P-1}\,W_P \,\,\right]$$ where $D_k$ is the diagonal matrix with diagonal $$\mathcal{T}^kc = (c_{P-k},\,\dots,\,c_{P-1},\,c_0,\,\dots,\,c_{P-k-1}),$$ and $W_P$ is the $P\times P$ Fourier matrix $W_P = (e^{2\pi inm/P})_{n,m=0}^{P-1}$.
\[rem:translationmodulation\] (1) For $0\le q,\,p\le P-1$, the $(q+1)$st column of the submatrix $D_pW_P$ is the vector $\mathcal{M}^p\mathcal{T}^qc$ where the operators $\mathcal{M}$ and $\mathcal{T}$ are as in Definition \[def:translationmodulation\]. This means that each column of the matrix $G(c)$ is a unimodular constant multiple of an element of the finite Gabor system with window $c$, namely $\set{e^{-2\pi ipq/P}\,\mathcal{T}^q \mathcal{M}^pc}_{q,p=0}^{P-1}$.
\(2) Note that the finite Gabor system defined above consists of $P^2$ vectors in $\mathbb C^P$ which form an overcomplete tight frame for $\mathbb C^P$ [@LPW05]. For details on Gabor frames in finite dimensions, see [@LPW05; @KPR08; @FKL09] and the overview article [@Pfa12].
\(3) Note that we are abusing notation slightly by identifying a vector $c\in\mathbb C^P$ with an $P$-periodic sequence $c=(c_n)$ in the obvious way.
[@DE03] The [*Spark*]{} of an $M\times N$ matrix F is the size of the smallest linearly dependent subset of columns, i.e., $$Spark(F) = \min\set{\norm{x}_0\colon Fx=0,\ \ x\ne 0}$$ where $\norm{x}_0$ is the number of nonzero components of the vector $x$. If $Spark(F)=M+1$, then $F$ is said to have [*full Spark*]{}. $Spark(F)=k$ implies that any collection of fewer than $k$ columns of $F$ is linearly independent.
Finite Gabor frames are generically full Spark
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The existence of Gabor matrices with full Spark has been addressed in [@LPW05; @M13]. The results in these two papers are as follows.
\[thm:genericfullsparkprime\][@LPW05] If $P\in{\mathbb{N}}$ is prime then there exists a dense, open subset of $c\in\mathbb C^P$ such that every minor of the Gabor system matrix $G(c)$ is nonzero. In particular, for such $c$, $G(c)$ has full Spark.
\[thm:genericfullspark\][@M13] For every $P\in{\mathbb{N}}$, there exists a dense, open subset of $c\in\mathbb C^P$ such that the Gabor system matrix $G(c)$ has full Spark.
The goal of this subsection is to outline the proof of Theorems \[thm:genericfullsparkprime\] and \[thm:genericfullspark\]. We will adopt some of the following notation and terminology of [@M13].
Let $P\in{\mathbb{N}}$ and let $M$ be an $P\times P$ submatrix of $G(c)$. For $0\le \kappa<P$ let $\ell_\kappa$ be the number of columns of $M$ chosen from the submatrix $D_\kappa W_P$ of (\[def:fullgabormatrix\]). While the vector $\ell=(\ell_\kappa)_{\kappa=0}^{P-1}$ does not determine $M$ uniquely, it describes the matrix $M$ sufficiently well for our purposes. Define $M_\kappa$ to be the $P\times\ell_\kappa$ matrix consisting of those columns of $M$ chosen from $D_\kappa W_P$. Given the [*ordered partition*]{} $B=(B_0,\,B_1,\,\dots,\,B_{P-1})$ where $\set{B_0,\,B_1,\,\dots,\,B_{P-1}}$ forms a partition of $\set{0,\,\dots,\,P-1}$, and where for each $0\le\kappa<P$, $|B_\kappa|=\ell_\kappa$, let $M_\kappa(B_\kappa)$ be the $\ell_\kappa\times\ell_\kappa$ submatrix of $M_\kappa$ whose rows belong to $B_\kappa$. Then $\det(M) = \prod \det(M_\kappa(B_\kappa))$ where the product is taken over all such ordered partitions $B$. This formula is called the [*Lagrange expansion*]{} of the determinant.
Each ordered partition $B$ corresponds to a permutation on ${\mathbb{Z}}_P$ as follows. Define the [*trivial partition*]{} $A=(A_0,\,A_1,\,\dots,\,A_{P-1})$ by $$A_j = \set{\sum_{i=0}^{j-1}\ell_i, \big(\sum_{i=0}^{j-1}\ell_i\big)+1,\,\dots,\,\big(\sum_{i=0}^j\ell_i\big)-1}$$ so that $A_0=[0,\ell_0-1]$, $A_1=[\ell_0, \ell_0+\ell_1+1]$, $\dots$, $A_{P-1}=[\ell_0+\,\cdots\,+\ell_{P-2},P-1]$. Then given $B=(B_0,\,B_1,\,\dots,\,B_{P-1})$ there is a permutation $\sigma\in S_P$ such that $\sigma(A_\kappa)=B_\kappa$ for all $\kappa$. This $\sigma$ is unique up to permutations that preserve $A$, that is, up to $\tau\in S_P$ such that $\tau(A_\kappa)=A_\kappa$ for all $\kappa$. Call such a permutation [*trivial*]{} and denote by $\Gamma$ the subgroup of $S_P$ consisting of all trivial permutations. Then the ordered partitions $B$ of ${\mathbb{Z}}_P$ can be indexed by equivalence classes of permutations $\sigma\in S_P/\Gamma$.
The key observation is that $\det(M)$ is a homogeneous polynomial in the $P$ variables $c_0,\,c_1,\,\dots,\,c_{P-1}$ and we can write $$\label{eqn:determinant}
\det(M) = \sum_{\sigma\in S_P/\Gamma} a_\sigma\,C^\sigma$$ where the monomial $C^\sigma$ is given by $$C^\sigma = \prod_{\kappa=0}^{P-1}\,\prod_{j\in \sigma(A_\kappa)} c_{(j-\kappa)(mod\ P)}.$$ If it can be shown that this polynomial does not vanish identically then we can choose a dense, open subset of $c\in\mathbb C^P$ for which $\det(M)\ne 0$. Since there are only finitely many $P\times P$ submatrices of $G(c)$ it follows that there is a dense, open subset of $c$ for which $\det(M)\ne 0$ for all $M$, and we conclude that, for these $c$, $G(c)$ has full Spark.
Following [@M13], we say that a monomial $C^{\sigma_0}$ [*appears uniquely*]{} in (\[eqn:determinant\]) if for every $\sigma\in S_P/\Gamma$ such that $\sigma\ne\sigma_0$, $C^{\sigma}\ne C^{\sigma_0}$. Therefore, in order to show that the polynomial (\[eqn:determinant\]) does not vanish identically, it is sufficient to show that (1) there is a monomial $C^\sigma$ that appears uniquely in (\[eqn:determinant\]) and (2) the coefficient $a_\sigma$ of this monomial does not vanish.
Obviously, whether or not (\[eqn:determinant\]) vanishes identically does not depend on how the variables $c_i$ are labelled. More specifically, if the variables are renamed by a cyclical shift of the indices, viz., $c_i \mapsto c_{(i+\gamma)mod\ P}$ for some $0\le\gamma<P$, then $$\det(M)(c_{\gamma+1},\,\dots,\,c_{P-1},\,c_0,\,\dots,\,c_\gamma) = \pm\,\det(M')(c_0,\,\dots,\,c_{P-1})$$ where $M'$ is an $P\times P$ submatrix described by the vector $$\ell'=(\ell_{\gamma+1},\,\dots,\,\ell_{P-1},\,\ell_0,\,\dots,\,\ell_\gamma).$$
### The lowest index monomial
In [@LPW05], a monomial referred to in [@M13] as the [*lowest index (LI) monomial*]{} is defined that has the required properties when $P$ is prime. In order to see this, note first that each coefficient $a_\sigma$ in the sum (\[eqn:determinant\]) is the product of minors of the Fourier matrix $W_P$ and since $P$ is prime, Chebotarev’s Theorem says that such minors do not vanish [@SL96]. More specifically, $$a_\sigma\,C^\sigma = \pm\,\prod_{\kappa=0}^{P-1} \det(M_\kappa(\sigma(A_\kappa)))$$ and for each $\kappa$, the columns of $M_\kappa$ are columns of $W_P$ where each row has been multiplied by the same variable $c_j$ and $M_\kappa(\sigma(A_\kappa))$ is a square matrix formed by choosing $\ell_\kappa$ rows of $M_\kappa$. Hence for each $\kappa$, $\det(M_\kappa(\sigma(A_\kappa)))$ is a monomial in $c$ with coefficients a constant multiple of a minor of $W_P$. Since $a_\sigma$ is the product of those minors, it does not vanish.
Note moreover that each submatrix $M_\kappa(\sigma(A_\kappa))$ is an $\ell_\kappa\times\ell_\kappa$ matrix, so that $\det(M_\kappa(\sigma(A_\kappa)))$ is the sum of a multiple of the product of $\ell_\kappa!$ diagonals of $M_\kappa(\sigma(A_\kappa))$. Hence $a_\sigma\,C^\sigma$ is the sum of multiples of the product of $\prod_{\kappa=0}^{P-1} \ell_\kappa!$ generalized diagonals of $M$.
We define the LI monomial as in [@LPW05] as follows. If $M$ is $1\times 1$, then $\det(M)$ is a multiple of a single variable $c_j$ and we define the LI monomial, $p_M$ by $p_M=c_j$. If $M$ is $d\times d$, let $c_j$ be the variable of lowest index appearing in $M$. Choose any entry of $M$ in which $c_j$ appears, eliminate the row and column containing that entry, and call the remaining $(d-1)\times(d-1)$ matrix $M'$. Define $p_M = c_j\,p_{M'}$. It is easy to see that the monomial $p_M$ is independent of the entry of $M$ chosen at each step. In order to show that the LI monomial appears uniquely in (\[eqn:determinant\]), we observe as in [@LPW05] that the number of diagonals in $\det(M)$ that correspond to the LI monomial is $\prod_{\kappa=0}^{P-1}\ell_\kappa!$. Since this is also the number of generalized diagonals appearing in the calculation of each $\det(M_\kappa(\sigma(A_\kappa)))$, it follows that this monomial appears only once. The details of the argument can be found in Section \[section:prime\]. Note that because $P$ is prime, this argument is valid no matter how large the matrix $M$ is. In other words, $M$ does not have to be an $P\times P$ submatrix in order for the result to hold. Consequently, given $k<P$ and $M$ an arbitrary $P\times k$ submatrix of $G(c)$, we can form the $k\times k$ matrix $M_0$ by choosing $k$ rows of $M$ in such a way that the LI monomial of $M_0$ contains at most only the variables $c_0,\,\dots,\,c_{k-1}$. This observation leads to the following theorem for matrices with arbitrary Spark.
\[thm:smallsparksampta13\][@PW14] If $P\in{\mathbb{N}}$ is prime, and $0<k< P$, there exists an open, dense subset of $c\in{\mathcal{C}}^k\times\set{0}\subseteq \mathbb C^P$ with the property that $Spark(G(c))=k+1$.
This result has implications for relating the capacity of a time-variant communication channel to the area of the spreading support, see [@PW14].
### The consecutive index monomial
In [@M13], a monomial referred to as the [*consecutive index (CI) monomial*]{} is defined that has the required properties for any $P\in{\mathbb{N}}$. The CI monomial, $C^I$, is defined as the monomial corresponding to the identity permutation in $S_P/\Gamma$, that is, to the equivalence class of the trivial partition $A=(A_0,\,A_1,\,\dots,\,A_{P-1})$. Hence $$C^I = \prod_{\kappa=0}^{P-1}\,\prod_{j\in A_\kappa} c_{(j-\kappa)mod\ P}.$$ For each $\kappa$, the monomial appearing in $\det(M_\kappa(A_\kappa))$, $\prod_{j\in A_\kappa} c_{(j-\kappa)mod\ P}$, consists of a product of $\ell_k$ variables $c_j$ with consecutive indices modulo $P$.
That $a_I\ne 0$ follows from the observation that for each $\kappa$, $\det(M_\kappa(A_\kappa))$ is a monomial whose coefficient is a nonzero multiple of a Vandermonde determinant and hence does not vanish (for details, see [@M13]). The proof that $C^I$ appears uniquely in (\[eqn:determinant\]) amounts to showing that, with respect to an appropriate cyclical renaming of the variables $c_i$, the $CI$ monomial uniquely minimizes the quantity $\Lambda(C^\sigma)=\sum_{i=0}^{P-1} i^2\,\alpha_i$, where $\alpha_i$ is the exponent of the variable $c_i$ in $C^\sigma$. An abbreviated version of the proof of this result as it appears in [@M13] is given in Section \[sec:malikiosis\].
As a final observation, we quote the following corollary that provides an explicit construction of a unimodular vector $c$ such that $G(c)$ has full Spark.
[@M13] Let $\zeta=e^{2\pi i/(P-1)^4}$ or any other primitive root of unity of order $(P-1)^4$ where $P\ge 4$. Then the vector $$c=(1,\,\zeta,\,\zeta^4,\,\zeta^9,\,\dots,\,\zeta^{(P-1)^2})$$ generates a Gabor frame for which $G(c)$ has full Spark.
Generalizations of operator sampling to higher dimensions {#section:higherdimensional}
=========================================================
The operator representations , , and hold verbatim for higher dimensional variables $x,\xi,t,\nu\in {\mathbb{R}}^d$. In this section, we address the identifiability of $$\begin{aligned}
OPW (S) & = \{H\in \mathcal L (L^2({\mathbb{R}}^d), L^2({\mathbb{R}}^d)) \colon
{\operatorname{supp}}\mathcal F_s \sigma_H\subseteq S,\,\norm{\sigma_H}_{L^2}<\infty\}\end{aligned}$$ where $S\subseteq {\mathbb{R}}^{2d}$.
Looking at the components of the multidimensional variables separately, Theorem \[thm:main-simple\] easily generalizes as follows.
\[thm:main-simple-higher\] For $H\in OPW \big(\prod_{\ell=1}^d [0, T_\ell] {\times}\prod_{\ell=1}^d[- \Omega_\ell / 2, \Omega_\ell / 2]\big)$ with $T_\ell\Omega_\ell{\leq} 1$, $\ell=1,\ldots,d$, we have $$\begin{aligned}
\|H\sum_{k_1\in{\mathbb{Z}}}\ldots \sum_{k_d\in{\mathbb{Z}}}\delta_{(k_1T_1,\ldots, k_dT_d)}\|_{L^2({\mathbb{R}})}=T_1\ldots T_d \|\sigma_H\|_{L^2},\notag \end{aligned}$$ and $H$ can be reconstructed by means of $$\begin{aligned}
\kappa_H(x+t,x)&=\chi_{\prod_{\ell=1}^d [0, T_\ell]}(t)\sum_{n_1\in{\mathbb{Z}}}\ldots \sum_{n_d\in{\mathbb{Z}}} \\ \quad &\big(H \sum_{k_1\in{\mathbb{Z}}}\ldots \sum_{k_d\in{\mathbb{Z}}}\delta_{(k_1T_1,\ldots, k_dT_d)} \big)(t+(n_1T_1,\ldots,n_d T_d) \\ &\quad
\frac{\sin(\pi T_1 (x_1-n_1))}{\pi T_1 (x_1-n_1)} \ldots \frac{\sin(\pi T_d (x_d-n_d))}{\pi T_d (x_d-n_d)} \notag \end{aligned}$$ with convergence in the $L^2$ norm.
In the following, we address the situation where $S$ is not contained in a set $\prod_{\ell=1}^d [0, T_\ell] {\times}\prod_{\ell=1}^d[- \Omega_\ell / 2, \Omega_\ell / 2]\big)$ with $T_\ell\Omega_\ell{\leq} 1$, $\ell=1,\ldots, d$. For example, $S=[0,1]\times [0,2]\times[0,\frac 1 4]\times[0,1]\subseteq {\mathbb{R}}^4$ of volume $\frac 1 2$ is not covered by Theorem \[thm:main-simple-higher\].
To give a higher dimensional variant of Theorem \[thm:reconstruction\], we shall denote pointwise products of finite and infinite length vectors $k$ and $T$ by $k{ {\star} }T$, that is, $k{ {\star} }T=(k_1T_1,\ldots,k_d T_d)$ for $k,T\in \mathbb C^d$. Similarly, $k/ T=(k_1/T_1,\ldots,k_d /T_d)$.
\[thm:reconstruction-higher\] If $S\subseteq (0,\infty)^d {\times}{\mathbb{R}}^d$ is compact with $\abs{S}<1$ then $OPW (S)$ is identifiable. Specifically, there exist $T_1,\ldots,T_d>0$ and pairwise relatively prime natural numbers $P_1,\ldots, P_d$ such that $$S\subseteq \prod_{\ell=1}^d[0,P_\ell T_\ell]{\times}\prod_{\ell=1}^d [-1/(2T_\ell), 1/(2T_\ell)],$$ and a sequence $c=(c_n) \in \ell^\infty({\mathbb{Z}}^d)$ which is $P_\ell$ periodic in the $\ell$-th component $n_\ell$ such that $g=\sum_{n\in{\mathbb{Z}}^d} c_n\,\delta_{n { {\star} }T}$ identifies $OPW^{2}(S)$. In fact, for such $g$ there exists for each $j\in J=\prod_{\ell =1}^d \{0,1,\ldots,P_\ell{-}1\}$ a sequences $b_j=(b_{j,k})$ which is $P_\ell$ periodic in $k_\ell$ and $2d$-tuples $(q_j,m_j) \in J\times J$ with $$\begin{aligned}
h(x,t) &= e^{-\pi i \sum_{\ell=1}^d t_\ell/T_\ell}\sum_{k\in{\mathbb{Z}}^d}\sum_{j\in J} \big[b_{j,k}\,Hg(t - (q_j-k){ {\star} }T) \nonumber \\
& \hskip-.25in e^{2\pi i m_j\cdot ( (x-t)/P{ {\star} }T)}\,\phi((x-t)+(q_j-k){ {\star} }T)\, r(t-q_j { {\star} }T)\big]. \label{eqn:reconstruction-higher}\end{aligned}$$ The functions $r,\phi\in {\cal S}({\mathbb{R}}^d)$ are assumed to satisfy $$\label{eq:r_phi_2}
\sum_{k\in{\mathbb{Z}}^d} r(t + k { {\star} }T) = 1 = \sum_{n\in{\mathbb{Z}}^d} \widehat{\phi}(\gamma + (n/P{ {\star} }T),$$ and $r(t)\widehat{\phi}(\gamma)$ is supported in a neighborhood of $\prod_{\ell=1}^d [0,T_\ell]{\times}\prod_{\ell=1}^d[0,1/P_\ell T_\ell]$. The sum in converges unconditionally in $L^{2}$ and for each $t$ uniformly in $x$.
This result follows from adjusting the proof of Theorem \[thm:reconstruction-higher\] to the higher dimensional setting. For example, it will employ the Zak transform $$\displaystyle{Z_{T { {\star} }P} f(t,\nu) = \sum_{n\in{\mathbb{Z}}^d} f(t-n{ {\star} }P { {\star} }T)\,e^{2\pi i \nu \cdot (P{ {\star} }T)}},$$ where $P=(P_1,\ldots, P_d)$. We are then led again to a system of linear equations of the form $$\label{eqn:basiclinearsystemhigher}
{\bf Z}_{Hg}(t,\nu)_p = \sum_{q\in J} \sum_{m \in J} G(c)_{p,(q,m)}\,\boldsymbol\eta_H(t,\nu)_{(q,m)}$$ where as before $${\bf Z}_{Hg}(t,\nu)_p = (Z_{T { {\star} }P}\circ H)g(t+p{ {\star} }T,\nu)\,e^{-2\pi i \nu p{ {\star} }T},$$ $$\begin{aligned}
\boldsymbol\eta_H(t,\nu)_{(q,m)} = &(T_1 P_1 \ldots T_d P_d )^{-1}\,\eta_H(t+q{ {\star} }T, \nu+(m/T{ {\star} }P )\, \\ & e^{-2\pi i \nu \cdot( q{ {\star} }T)}\,e^{-2\pi i q \cdot (m/P)},\end{aligned}$$ and where $G(c)$ is now a multidimensional finite Gabor system matrix similar to (\[def:fullgabormatrix\]).
In order to show that the spreading function for operator $H$ can be completely determined by its response to the periodically-weighted $d$-dimensional delta-train $g$, we need to show that (\[eqn:basiclinearsystemhigher\]) can be solved for each $(t,\nu)\in \prod_{\ell=1}^d [0,T_\ell]{\times}\prod_{\ell=1}^d[0,1/(T_\ell P_\ell)]$ if $c\in \mathbb C^{P_1\times \ldots \times P_d}$ is chosen appropriately.
To see that a choice of $c$ is possible, observe that the product group ${\mathbb{Z}}_{P_1}\times \ldots \times {\mathbb{Z}}_{P_d}$ is isomorphic to the cyclic group ${\mathbb{Z}}_{P_1\cdot \ldots \cdot P_d}$ since the $P_\ell$ are chosen pairwise relatively prime. Theorem \[thm:genericfullspark\] applied to the cyclic group ${\mathbb{Z}}_{P_1\cdot \ldots \cdot P_d}$ guarantees the existence of $\widetilde c \in \mathbb C^{P_1\cdot \ldots \cdot P_d}$ so that the Gabor system matrix $G(\widetilde c)$ is full spark. We can now define $c\in \mathbb C^{P_1\times \ldots \times P_d}$ by setting $$c_{n_1,\ldots,n_d}=\widetilde{c}_{n_1+n_2\, P_1+n_3\,P_1P_2+\ldots + n_d\,P_1\ldots P_{d-1}}, \quad n=(n_1,\ldots,n_d)\in J$$ and observe that $G(c)$ is simply a rearrangement of $G(\widetilde c)$, hence, $G(c)$ is full spark.
Further results on operator sampling {#section:outlook}
====================================
The results discussed in this paper are discussed in detail in [@Kai62; @Bel69; @KP06; @PW06b; @Pfa10] and [@PW14]. The last listed article contains the most extensive collection of operator reconstruction formulas, including extensions to some $OPW(S)$ with $S$ unbounded. Moreover, some hints on how to use parallelograms to rectify a set $S$ for operator sampling efficiently are given.
A central result in [@PW14] is the classification of all spaces $OPW(S)$ that are identifiable for a given $g=\sum_{n\in{\mathbb{Z}}} {c_n}\delta_{nT}$ for $c_n$ being $P$-periodic.
The papers [@PW06; @Pfa10] address some functional analytic challenges in operator sampling, and [@KP12] focuses on the question of operator identification if we are restricted to using more realizable identifiers, for example, truncated and modified versions of $g$, namely, $\widetilde {g} (t)=\sum_{n=0}^N {c_n} \varphi( t-nT)$. The problem of recovering parametric classes of operators in $OPW(S)$ is discussed in [@BGE11; @BGE11b].
In the following, we briefly review literature that address some other directions in operator sampling.
Multiple Input Multiple Output
------------------------------
A Multiple Input Multiput Output (MIMO) channel $\bf H$ with $N$ transmitters and $M$ receivers can be modeled by an $N\times M$ matrix whose entries are time-varying channel operators $H_{mn}\in OPW(S_{mn})$. For simplicity, we write ${\bf H} \in OPW(\bf S)$. Assuming that the operators $H_{mn}$ are independent, a sufficient criterion for identifiability is given by $\sum_{n=1}^N |S_{mn}| \leq 1$ for $m=1,\ldots, M$. Conversely, if for a single $m$, $\sum_{n=1}^N |S_{mn}| > 1$, then $OPW(\bf S)$ is not identifiable by any collection $s_1,\ldots, s_N$ of input signals [@PW07; @Pfa08].
A somewhat dual setup was considered in [@HP10]. Namely, a Single Input Single Output (SISO) channel with $S$ being large, say $S=[0,M]\times [-N/2,N/2]$ with $N,M\geq 2$. As illustrated above, $OPW([0,M]\times [-N/2,N/2])$ is not identifiable, but if we are allowed to use $MN$ (infinite duration) input signals $g_1,\ldots, g_{MN}$, then $H\in OPW([0,M]\times [-N/2,N/2])$ can be recovered from the $MN$ outputs $Hg_1,\ldots, Hg_{MN}$.
Irregular Sampling of Operators
-------------------------------
The identifier $g=\sum_{n\in{\mathbb{Z}}} c_n \delta_{nT}$ is supported on the lattice $T{\mathbb{Z}}$ in ${\mathbb{R}}$. In general, for stable operator identification, choosing a discretely supported identifier is reasonable, indeed, in [@KP12] it is shown that identification for $OPW(S)$ in full requires the use of identifiers that neither decay in time nor in frequency. (Recovery of the characteristics of $H$ during a fixed transmission band and a fixed transmission interval can be indeed recovered when using Schwartz class identifiers [@KP12].)
In irregular operator sampling, we consider identifiers of the form $g=\sum_{n\in{\mathbb{Z}}} c_n \delta_{\lambda_n}$ for nodes $\lambda_n$ that are not necessarily contained in a lattice. If such $g$ identifies $OPW(S)$, then we refer to ${\operatorname{supp}}g=\{\lambda_n\}$ as a sampling set for $OPW(S)$, and similarly, the [*sampling rate*]{} of $g$ is defined to be $$D(g)=D({\operatorname{supp}}g)=D(\Lambda) = \lim_{r\to \infty} \frac{n^-(r)}{r}$$ where $$n^-(r) = \inf_{x\in{\mathbb{R}}}\# \{ \Lambda \cap [x,x+r]\}$$ assuming that the limit exists [@HP10; @PW14].
To illustrate a striking difference between irregular sampling of functions and operators, note that ${\mathbb{Z}}$ is a sampling set for $OPW([0,1]\times [-\frac 1 2, \frac 1 2])$ as well as for the Paley Wiener space $PW([-\frac 1 2, \frac 1 2])$, but the distribution $
g=c_0\delta_{\lambda_0}+\sum_{n\in{\mathbb{Z}}\setminus\{0\}} c_n \delta_{n}$ does not identify $OPW([0,1]\times [-\frac 1 2, \frac 1 2])$, regardless of our choice of $c_n$ and $\lambda_0\neq 0$. This shows that, for example, Kadec’s $\frac 1 4$th theorem does not generalize to the operator setting [@HP09].
In [@PW14] we give with $D(g)=D(\Lambda)\geq B(S)$ a necessary condition on the (operator) sampling rate based on the bandwidth $B(S)$ of $OPW(S)$ which is defined as $$\begin{aligned}
\label{eqn:bandwidth}
B(S) = \sup_{t\in{\mathbb{R}}}|{\operatorname{supp}}\eta(t,\nu)| = \Big\|\int_{\mathbb{R}}\chi_{S}(\cdot,\nu)\,d\nu\Big\|_\infty.\end{aligned}$$ Here, $\chi_S$ denotes the characteristic function of $S$. This quantity can be interpreted as the maximum vertical extent of $S$ and takes into account gaps in $S$. Moreover, in $\cite{PW14}$ we discuss the goal of constructing $\{\lambda_n\}$ of small density, and/or large gaps in order to reserve time-slots for information transmission. Results in this direction can be interpreted as giving bounds on the capacity of a time-variant channel in $OPW(S)$ in terms of $|S|$ [@PW14].
Finally, we give in [@PW14] an example of an operator class $OPW(S)$ that cannot be identified by any identifier of the form $g=\sum_{n\in{\mathbb{Z}}} c_n \delta_{nT}$ with $T>0$ and periodic $c_n$, but requires coefficients that form a bounded but non-periodic sequence. In this case, $S$ is a parallelogram and $B(S)=D(g)$ (see Figure \[fig:nonperiodic\])
Ł[9]{} (0,0) – (4,2.828) – (8,6.657)– (4,3.828) – (0,0); (0,0) grid (Ł-0.4, Ł+0.4); (-0.2,0) – (Ł+0.4,0) node\[below\] [$t$]{}; (0,-0.2) – (0,Ł+0.4) node\[above\] [$\nu$]{}; at (2,0) [$1$]{}; at (4,0) [$2$]{}; at (6,0) [$3$]{}; at (8,0) [$4$]{}; at (0,2) [$1$]{}; at (0,4) [$2$]{}; at (0,6) [$3$]{}; at (0,8) [$4$]{};
Sampling of $OPW(S)$ with unknown $S$.
--------------------------------------
In some applications, it is justified to assume that the set $S$ has small area, but its shape and location are unknown. If further $S$ satisfies some basic geometric assumptions that guarantee that $S$ is contained in $[0,TP]\times[-1/2T, 1 / 2T]$ and only meets few rectangles of the rectification grid $[kT,(k+1)T]\times [q /TP,(q+1)/TP]$, then recovery of $S$ and, hence, an operator in $OPW(S)$ is possible [@PW14; @HB13].
\[figure:colors\]
Ł[9]{} (0,0) – (1,1) – (2,0) – (0,0); (2,0) – (3,1) – (4,0) – (2,0); (4,6) – (5,7) – (6,6) – (4,6);
(0,0) – (1,1) – (0,2) – (0,0); (2,2) – (3,3) – (2,4) – (2,2); (6,2) – (7,3) – (6,4) – (6,2);
(6,4) – (4,6) – (6,6) – (6,4); (4,2) – (2,4) – (4,4) – (4,2); (6,2) – (4,4) – (6,4) – (6,2);
(0,0) grid (Ł-0.4, Ł+0.4); (-0.2,0) – (Ł+0.4,0) node\[below\] [$t$]{}; (0,-0.2) – (0,Ł+0.4) node\[above\] [$\nu$]{}; at (2,0) [$T$]{}; at (0,2) [$\Omega$]{}; at (6,0) [$LT$]{}; at (0,6) [$L\Omega$]{};
The independently obtained results in [@PW14; @HB13] employ the same identifiers $g=\sum_{n\in{\mathbb{Z}}} c_n \delta_{\lambda_n}$ as introduced above. Operator identification is therefore again reduced to solving , that is, the system of $P$ linear equations $$\label{eqn:basiclinearsystemsimple}
{\bf Z}(t,\nu) = G(c)\boldsymbol\eta (t,\nu)$$ for the vector $\boldsymbol \eta (t,\nu)\in\mathbb C^{P^2}$ for $(t,\nu)\in [0,T]\times[-1/2TP, 1/2TP]$. While the zero components of $\boldsymbol \eta (t,\nu)$ are not known, the vector is known to be very sparse. Hence, for fixed $(t,\nu)$, we can use the fact that $G(c)$ is full spark and recover $\boldsymbol \eta (t,\nu)$ if it has at most $P/2$ nonzero entries. Indeed, assume ${\boldsymbol{\eta}}(t,\nu)$ and $\widetilde{\boldsymbol{\eta}}(t,\nu)$ solve and both have at most $P/2$ nonzero entries. Then $\boldsymbol {\eta} (t,\nu)-\widetilde{\boldsymbol{\eta}} (t,\nu)$ has at most $P$ nonzero entries and the fact that $G(c)$ is full spark indicates that $G(c)( \boldsymbol {\eta} (t,\nu)-\widetilde{\boldsymbol{\eta}} (t,\nu))=0$ implies $\boldsymbol {\eta} (t,\nu)-\widetilde{\boldsymbol{\eta}} (t,\nu)=0$.
Clearly, under the geometric assumptions alluded to above, the criterion that at most $P/2$ rectangles in the grid are met can be translated to the unknown area of $S$ has measure less than or equal to 1/2.
In [@HB13], this area 1/2 criterion is improved by showing that $H$ can be identified whenever at most $P-1$ rectangles in the rectification grid are met by $S$. This result is achieved by using a joint sparsity argument, based on the assumption that for all $(t,\nu)$, the same cells are active.
Alternatively, the “area 1/2” result can be strengthened by not assuming that for all $(t,\nu)$, the same cells are active. This requires solving , for $\boldsymbol{\eta} (t,\nu)$ sparse, for each considered $(t,\nu)$ independently, see Figure \[fig:rectification\] and [@PW14].
It must be added though, that solving for $\boldsymbol{\eta} (t,\nu)$ being $P/2$ sparse is not possible for moderately sized $P$, for example for $P > 15$. If we further reduce the number of active boxes, then compressive sensing algorithms such as Basis Pursuit and Orthogonal Matching Pursuit become available, as is discussed in the following section.
Finite dimensional operator identification and compressive sensing
------------------------------------------------------------------
Operator sampling in in the finite dimensional setting translates into the following matrix probing problem [@PRT08; @CD12; @BD14]. For a class of matrices $\boldsymbol{\mathcal M}\in\mathbb C^{P\times P}$, find $c\in \mathbb C^P$ so that we can recover $M\in\boldsymbol{\mathcal M}$ from $Mc$.
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The classes of operator considered here are of the form $M_{\boldsymbol \eta}=\sum_\lambda \boldsymbol{\eta}_\lambda B_\lambda$ with $B_{\lambda}=B_{p,q}=\mathcal T^p \mathcal M^q$, and the matrix identification problem is reduced to solving $$\label{eqn:basiclinearsystemsimpleno}\boldsymbol Z=M_{\boldsymbol{\eta}} c=\sum_{p,q=0}^{P-1} {\boldsymbol{\eta}}_{p,q} \big(\ \mathcal T^p \mathcal M^q c\big) =G(c)\boldsymbol{\eta},$$ where $c$ is chosen appropriately; this is just with the dependence on $(t,\nu)$ removed.
If $\boldsymbol\eta$ is assumed to be $k$-sparse, we arrive at the classical compressive sensing problem with measurement matrix $G(c)\in \mathbb C^{P \times P^2}$ which depends on $c=(c_0,c_1,\ldots,c_{P-1})$. To achieve recovery guarantees for Basis Pursuit and Orthogonal Matching Pursuit, averaging arguments have to be used that yield results on the expected qualities of $G(c)$. This problem was discussed in [@PRT08; @PR09; @PRT13] as well as, in slightly different terms, in [@AHS08; @HS09]. The strongest results were achieved in [@KMR14] by estimating Restricted Isometry Constants for $c$ being a Steinhaus sequence. These results show that with high probability, $G(c)$ has the property that Basis Pursuit recovers $\boldsymbol{\eta}$ from $G(c)\boldsymbol{\eta}$ for every $k$ sparse $\boldsymbol{\eta}$ as long as $k \leq C \, P / \log^2 P$. for some universal constant $C$.
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Stochastic operators and channel estimation
-------------------------------------------
It is common that models of wireless channels and radar environments take the stochastic nature of the medium into account. In such models, the spreading function $\eta(t,\nu)$ (and therefore the operator’s kernel and Kohn–Nirenberg symbol) are random processes, and the operator is split into the sum of its deterministic portion, representing the mean behavior of the channel, and its zero-mean stochastic portion that represents the noise and the environment.
; ; ;
The detailed analysis of the stochastic case was carried out in [@PZ14a; @PZ14b]. One of the foci of these works lies in the goal of determining the second-order statistics of the (zero mean) stochastic process $\eta(\tau,\nu)$, that is, its so called covariance function $R(\tau,\nu,\tau',\nu') = \mathbb E \{\eta(\tau,\nu)\, \overline{\eta(\tau',\nu')}\}$. In [@PZ14a; @PZ14b], it was shown that a necessary but not sufficient condition for the identifiability of $R\eta(\tau,\nu,\tau',\nu')$ from the output covariance $
A(t,t') = \mathbb E \{ H g(t) \, \overline{H g(t')}\}
$ is that $R(\tau,\nu,\tau',\nu')$ is supported on a bounded set of 4-dimensional volume less than or equal to one. Unfortunately, for some sets $S\subseteq {\mathbb{R}}^4$ of arbitrary small measure, the respective stochastic operator Paley–Wiener space $StOPW(S)$ of operators with $R\eta$ supported on $S$ is not identifiable; this is a striking difference to the deterministic setup where the geometry of $S$ does not play a role at all.
In [@OPZ; @PZ14c] the special case of *wide-sense stationary operators with uncorrelated scattering*, or WSSUS operators is considered. These operators are characterized by the property that $$\label{eq:scattering}
R\eta(t,\nu,t',\nu') = C_\eta (t,\nu) \, \delta(t-t') \, \delta(\nu-\nu').$$ The function $C_\eta(t,\nu)$ is then called *scattering function* of $H$. Our results on the identifiability of stochastic operator classes allowed for the construction of two estimators for scattering functions [@OPZ; @PZ14c]. The estimator given in [@PZ14c] is applicable, whenever the scattering function of $H$ has bounded support. Note that the autocorrelation of a WSSUS operator is supported on a two dimensional plane in ${\mathbb{R}}^4$ which therefore has 4D volume 0, a fact that allows us to lift commonly assumed restrictions on the size of the 2D area of the support of the scattering function.
For details, formal definitions of identifiability and detailed statements of results we refer to the papers [@OPZ; @PZ14a; @PZ14b; @PZ14c].
Appendix: Proofs of Theorems.
=============================
Proof of Lemma \[lem:matrixequationquasiperiodic\] {#section:proof of lemma}
--------------------------------------------------
In order to see how the time-frequency shifts of $c$ arise, we will briefly outline the calculation that leads to (\[eqn:matrixequationquasiperiodic\]). It can be seen by direct calculation using the representation given by (\[eqn:operator3\]), that if $g = \sum_{n}\delta_{nTP}$ then $\ip{Hg}{s} = \ip{\eta_H}{Z_{TP} s}$ for all $s\in{\cal S}({\mathbb{R}})$ where the bracket on the left is the $L^2$ inner product on ${\mathbb{R}}$ and that on the right the $L^2$ inner product on the rectangle $[0,TP]{\times}[0,1/(TP)]$. Periodizing the integral on the left gives $$\begin{aligned}
\ip{\eta_H}{Z_{TP} s} = & \int_0^{1/(TP)}\int_0^{TP} \sum_{k}\sum_{m} \eta_H(t+kTP,\nu+m/(TP)) \\
& \hskip.5in e^{-2\pi i\nu kTP}\overline{Z_{TP} s(t,\nu)}\,dt\,d\nu.\end{aligned}$$ Since this holds for every $s\in{\cal S}({\mathbb{R}})$, we conclude that $$\begin{aligned}
(Z_{TP}\circ H)g(t,\nu) & \\
& \hskip-.5in = 1/(TP)\,\sum_{k}\sum_{m} \eta_H(t+kTP,\nu+m/(TP))\,e^{-2\pi i\nu kTP}.\end{aligned}$$
Given $g = \sum_{n\in{\mathbb{Z}}} c_n\,\delta_{nT}$, for a period-$P$ sequence $c=(c_n)$, and letting $n=mP-q$ for $m\in{\mathbb{Z}}$ and $0\le q<P$, we obtain $$\begin{aligned}
g & = \sum c_n\,\delta_{nT} = \sum_{q=0}^{P-1} \sum_{m\in{\mathbb{Z}}} c_{mP-q}\,\delta_{mPT - qT} \\
& = \sum_{q=0}^{P-1} c_{-q} \mathcal T_{-qT}\,\bigg(\sum_{m\in{\mathbb{Z}}}\,\delta_{mPT}\bigg).\end{aligned}$$ Since for $\alpha\in{\mathbb{R}}$, the spreading function of $H\circ \mathcal T_\alpha$ is $\eta_H(t-\alpha,\nu)\,e^{2\pi i\nu\alpha}$, we arrive at $$\begin{aligned}
(Z_{TP}\circ H)g(t,\nu) & \nonumber \\
& \hskip-.75in = 1/(TP)\,\sum_{q=0}^{P-1} c_{-q}\,\sum_{k}\sum_{m} \eta_H(t+kTP+qT,\nu+m/(TP)) \nonumber \\
& \hskip.75in e^{-2\pi i(\nu+m/(TP))qT}\,e^{-2\pi i\nu kTP}. \label{eqn:weighteddeltatrain}\end{aligned}$$
Letting $m=jP+\ell$ for $j\in{\mathbb{Z}}$ and $0\le \ell<P$, we obtain $$\begin{aligned}
(Z_{TP}\circ H)g(t,\nu) & \nonumber \\
& \hskip-.75in = 1/(TP)\,\sum_{q=0}^{P-1} c_{-q}\,\sum_{k}\sum_{j}\sum_{\ell=0}^{P-1} \eta_H(t+kTP+qT,\nu+j/T+\ell/(TP)) \nonumber \\
& \hskip.75in e^{-2\pi i\nu qT}\,e^{-2\pi i\ell q/P}\,e^{-2\pi i\nu kTP} \nonumber \\
& \hskip-.75in = 1/(TP)\,\sum_{q=0}^{P-1}\sum_{\ell=0}^{P-1} \big(c_{-q}\,e^{-2\pi i\ell q/P}\big)\,
e^{-2\pi i\nu qT}\,\eta^{Q}_H(t + T q,\nu + \ell/TP).\end{aligned}$$
Finally, replacing $t$ by $t+pT$ for $p=0,\,1,\,\dots,\,P{-}1$, and changing indices by replacing $q$ by $q-p$, we obtain $$\begin{aligned}
(Z_{TP}\circ H)g(t+pT,\nu) & \\
& \hskip-1.25in = 1/(TP)\,\sum_{q=0}^{P-1}\sum_{\ell=0}^{P-1} \big(c_{-q}\,e^{-2\pi i\ell q/P}\big)
\,e^{-2\pi i\nu qT}\,\eta^{Q}_H(t + (q+p) T,\nu + \ell/TP) \\
& \hskip-1.25in = 1/(TP)\,\sum_{q=0}^{P-1}\sum_{\ell=0}^{P-1} \big(c_{-(q-p)}\,e^{-2\pi i\ell (q-p)/P}\big) \nonumber \\
& \hskip.75in e^{-2\pi i\nu (q-p)T}\,\eta^{Q}_H(t + q T,\nu + \ell/TP).\end{aligned}$$ The observation that $(\mathcal T^q\,\mathcal M^m c)_p = c_{p-q}\,e^{2\pi i m (p-q)/P}$ completes the proof.
Proof of Theorem \[thm:genericfullsparkprime\] {#section:prime}
----------------------------------------------
To see why this is true, define $\mu(M)$ to be the number of diagonals of $M$ whose product is a multiple of $p_M$, and proceed by induction on the size of the matrix $M$. If $M$ is $1\times 1$ then the result is obvious. Suppose that $M$ is $n\times n$ and that it is described by the vector $\ell=(\ell_0,\,\dots,\,\ell_{P-1})$. Assuming without loss of generality that the variable of smallest index in $p_M$ with a nonzero exponent is $c_0$, there is a row of $M$ in which the variable $c_0$ appears $ \ell_j$ times for some index $j$. Choose one of these terms and delete the row and column in which it appears. Call the remaining matrix $M'$. The vector $\ell$ describing $M'$ is $(\ell_0,\,\dots,\,\ell_{j-1},\,\ell_j-1,\,\ell_{j+1},\,\dots,\,\ell_{P-1})$, and is independent of which term was chosen from the given row to form $M'$. By the construction of the LI monomial, $p_M=c_0\,p_{M'}$ and by the induction hypothesis $$\mu(M') = \ell_0!\,\cdots\,\ell_{j-1}!\,(\ell_j-1)!\,\ell_{j+1}!\,\cdots\,\ell_{P-1}!.$$ Since there are $\ell_j$ ways to choose a term from the given row to produce $M'$ we have that $$\mu(M) = \ell_j\,\mu(M') = \ell_0!\,\cdots\,\ell_{j-1}!\,\ell_j( l_j-1)!\,\ell_{j+1}!\,\cdots\,\ell_{P-1}! = \prod_{\kappa=0}^{P-1} \ell_\kappa!$$ which was to be proved.
Since each term $a_\sigma\,C^\sigma$ in (\[eqn:determinant\]) is made up of a sum of precisely this many terms, it follows that exactly one of these terms is a multiple of the LI monomial. Alternatively, we can think of the LI monomial as the one corresponding to the $\sigma\in S_P/\Gamma$ that minimizes the functional $\Lambda_0(C^\sigma) = \sum_{i=0}^{L-1} i^2\,H(\alpha_i)$ where $\alpha_i$ is the exponent of $c_i$ in $C^\sigma$ and where $H(\alpha_i)=0$ if $\alpha_i=0$ and $1$ otherwise.
Because by Chebotarev’s Theorem, $a_\sigma\ne 0$ for all $\sigma$ the proof works for any square submatrix $M$, no matter what size. This gives us Theorem \[thm:genericfullsparkprime\].
Proof of Theorem \[thm:genericfullspark\] {#sec:malikiosis}
-----------------------------------------
We first need to assert the existence of a cyclical renumbering of the variables such that with respect to the new trivial partition $A'=(A_\kappa')_{\kappa=0}^{P-1}$, the CI monomial is given by $$C^I = \prod_{\kappa=0}^{P-1}\,\prod_{j\in A_\kappa'} c_{j-\kappa}$$ in other words, if $j\in A_\kappa'$ then $0\le j-\kappa <P$. Note first that since $\min(A_\kappa')=\sum_{i=0}^{\kappa-1}\ell_i'$ for all $\kappa$, $j\in A_\kappa'$ implies that $j\ge\sum_{i=0}^{\kappa-1}\ell_i'$. Therefore, it will suffice to find a $0\le\gamma<P$ such that for all $\kappa$, $\sum_{i=0}^{\kappa-1}\ell_i' - \kappa\ge 0$ so that $j-\kappa \ge \sum_{i=0}^{\kappa-1}\ell_i' - \kappa\ge 0$.
Let $0\le \gamma<P$ be such that the quantity $\sum_{i=0}^{\gamma-1} \ell_i - \gamma$ is minimized, let $$\ell' = (\ell_i')_{i=0}^{L-1} = (\ell_{(i+\gamma)mod\ P})_{i=0}^{P-1},$$ and let $A' = (A_\kappa')_{\kappa=0}^{P-1}$ be the corresponding trivial partition. Now fix $\kappa$ and assume that $\kappa+\gamma \le P$. Then $$\begin{aligned}
\sum_{i=0}^{\kappa-1}\ell_i' - \kappa
& = & \sum_{i=0}^{\kappa-1}\ell_{(i+\gamma)} - \kappa \\
& = & \bigg(\sum_{i=0}^{\kappa+\gamma-1}\ell_i - (\kappa+\gamma)\bigg) - \bigg(\sum_{i=0}^{\gamma-1}\ell_i - \gamma\bigg) \\
& \ge & 0\end{aligned}$$ since the second term in the difference is minimal. If $\kappa+\gamma \ge P+1$ then remembering that $\sum_{i=0}^{P-1}\ell_i = L$ $$\begin{aligned}
\sum_{i=0}^{\kappa-1}\ell_i' - \kappa
& = & \sum_{i=0}^{\kappa-1}\ell_{(i+\gamma)mod\ P} - \kappa \\
& = & \sum_{i=\gamma}^{P-1} \ell_i + \sum_{i=0}^{\kappa+\gamma-P-1}\ell_i - \kappa \\
& = & \sum_{i=0}^{P-1} \ell_i - \sum_{i=0}^{\gamma-1} \ell_i + \sum_{i=0}^{\kappa+\gamma-P-1}\ell_i - \kappa \\
& = & \bigg(\sum_{i=0}^{(\kappa+\gamma-P)-1}\ell_i - (\kappa+\gamma-P)\bigg) - \bigg(\sum_{i=0}^{\gamma-1}\ell_i - \gamma\bigg) \\
& \ge & 0.\end{aligned}$$
In order to complete the proof, we must show that $\Lambda(C^\sigma)\ge\Lambda(C^I)$ for all $\sigma\in S_P/\Gamma$ with equality holding if and only if $\sigma$ is trivial. This will follow by direct calculation together with the following lemma which follows from a classical result on rearrangements of series ([@HLP52], Theorems 368, 369). This result is Lemma 3.3 in [@M13].
First, however, we adopt the following notation. For $0\le n<P$, let $b_n = \kappa$ if $n\in A_\kappa$. With this notation, given $\sigma\in S_P/\Gamma$, $$C^\sigma = \prod_{n=0}^{P-1} c_{(\sigma(n)-b_n)\ mod\ P}$$ and under the above assumptions, $$C^I = \prod_{n=0}^{P-1} c_{(n-b_n)}.$$ Moreover, $$\begin{aligned}
\Lambda(C^\sigma)
& = & \sum_{i=0}^{P-1} i^2\,\alpha_i \\
& = & \sum_{i=0}^{P-1} i^2\,(\#\{n\colon (\sigma(n)-b_n)\ mod\ P = i\}) \\
& = & \sum_{i=0}^{P-1} \big( (\sigma(n)-b_n)\ mod\ P \big)^2.\end{aligned}$$
\[lem:rearrangement\] Given two finite sequences of real numbers $(\alpha_n)$ and $(\beta_n)$ defined up to rearrangement, the sum $$\sum_n \alpha_n\,\beta_n$$ is maximized when $\alpha$ and $\beta$ are both monotonically increasing or monotonically decreasing. Moreover, if for every rearrangement $\alpha'$ of $\alpha$, $$\sum_n \alpha_n'\,\beta_n \le \sum_n \alpha_n\beta_n$$ then $\alpha$ and $\beta$ are [*similarly ordered*]{}, that is, for every $j,\,k$, $$(\alpha_j-\alpha_k)(\beta_j-\beta_k)\ge 0.$$
In particular, for every $\sigma\in S_P$, $$\sum_{n=0}^{P-1} n\,b_n \ge \sum_{n=0}^{P-1} \sigma(n)\,b_n$$ with equality holding if and only if $\sigma$ is trivial.
The first part of the lemma is simply a restatement of Theorems 368 and 369 of [@HLP52]. To prove the second part, note first that $b_n$ is a non-decreasing sequence and in particular is constant on each $A_\kappa$. Theorem 368 in [@HLP52] states that a sum of the form $\sum_{n=0}^{P-1} \sigma(n)\,b_n$ is maximized when $\sigma(n)$ is monotonically increasing, which proves the given inequality. Since $b_n$ is constant on each $A_\kappa$, it follows that if $\sigma$ is trivial, then we have equality.
If $\sigma$ is not trivial then we will show that the sequences $\sigma(n)$ and $b_n$ are not similarly ordered. Letting $\kappa$ be the minimal index such that $A_\kappa$ is not left invariant by $\sigma$, there exists $m\in A_\kappa$ such that $\sigma(m)\in A_\mu$ for some $\mu>\kappa$, and for some $\lambda>\kappa$ there exists $k\in A_\lambda$ such that $\sigma(k)\in A_\kappa$. Therefore, $b_m=\kappa < \lambda = b_k$ but since $\mu>\kappa$, $\sigma(m) > \sigma(k)$, and so $\sigma(n)$ and $b_n$ are not similarly ordered.
In order to complete the proof, define ${\mathcal{C}}_1,\,{\mathcal{C}}_2\subseteq\{0,\,\dots,\,P-1\}$ by $n\in{\mathcal{C}}_1$ if $0\le \sigma(n)-b_n <P$, and $n\in{\mathcal{C}}_2$ if $-P+1\ge \sigma(n)-b_n <0$ (note that always $\abs{\sigma(n)-b_n} <P$) so that when $n\in C_2$, $(\sigma(n)-b_n)\ mod\ P = \sigma(n)-b_n + P$. Let $\sigma'(n)=\sigma(n)$ if $n\in{\mathcal{C}}_1$ and $\sigma(n)+P$ if $n\in{\mathcal{C}}_2$, and let $(a_n)_{n=0}^{P-1}$ be an increasing sequence enumerating the set $\sigma({\mathcal{C}}_1)\cup(\sigma({\mathcal{C}}_2)+P)$. Therefore, $$\begin{aligned}
\Lambda(C^\sigma) - \Lambda(C^I)
& = \sum_{n=0}^{P-1} (\sigma'(n)-b_n)^2 - \sum_{n=0}^{P-1} (n-b_n)^2 \\
& = \bigg[\sum_{n=0}^{P-1} (\sigma'(n)-b_n)^2 - \sum_{n=0}^{P-1} (a_n-b_n)^2\bigg]
\\ & \qquad \qquad+ \bigg[\sum_{n=0}^{P-1} (a_n-b_n)^2 - \sum_{n=0}^{P-1} (n-b_n)^2\bigg] \\
& = 2\,\bigg[\sum_{n=0}^{P-1} a_nb_n - \sigma'(n)b_n\bigg] + \bigg[\sum_{n=0}^{P-1} (a_n-b_n)^2 - (n-b_n)^2\bigg] \\
& = I + II.\end{aligned}$$ Since $a_n$ is increasing, $I\ge 0$ by Lemma \[lem:rearrangement\], and since $a_n\ge n$ for all $n$, $(a_n-b_n)\ge(n-b_n)\ge 0$ so that $(a_n-b_n)^2\ge (n-b_n)^2$ and hence $II\ge 0$. It remains to show that equality holds only if $\sigma$ is trivial. If $\Lambda(C^\sigma)=\Lambda(C^I)$ then $I = II = 0$. Since $II=0$, ${\mathcal{C}}_2=\emptyset$ for if $a_n\in\sigma({\mathcal{C}}_2)+P$ then $a_n>n$ and we would have $II > 0$. Since ${\mathcal{C}}_2=\emptyset$, $\sigma'(n)=\sigma(n)$ so that $$\begin{aligned}
0 & = & \Lambda(C^\sigma) - \Lambda(C^I) \\
& = & \sum_{n=0}^{P-1} (\sigma(n)-b_n)^2 - \sum_{n=0}^{P-1} (n-b_n)^2 \\
& = & 2\,\sum_{n=0}^{P-1} (n\,b_n - \sigma(n)\,b_n)\end{aligned}$$ which by Lemma \[lem:rearrangement\] implies that $\sigma$ is trivial. The proof is complete.
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[^1]: Ref. 6 is [@Kai59].
[^2]: $S'_0({\mathbb{R}})$, the dual space of the Feichtinger algebra $S_0({\mathbb{R}})$ [@Gro01], or ${\cal S}'({\mathbb{R}})$, the space of tempered distributions [@PW06]. These spaces are large enough to contain weighted infinite sums of delta distributions.
[^3]: ${\cal S}({\mathbb{R}})$ denotes the Schwartz class of infinitely-differentiable, rapidly-decreasing functions.
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
Cayetano Di Bartolo\
Departamento de Física, Universidad Simón Bolívar, Apartado 89000,\
Caracas 1080-A, Venezuela.
date:
title: THE GAUSS CONSTRAINT IN THE EXTENDED LOOP REPRESENTATION
---
6.8in 8.8in
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[ 11.15.-q, 04.60.-m ]{}
The Ashtekar variables [@ash1; @ash2] present general relativity as a gauge theory. As a consequence, it is possible to quantize the theory in the loop space [@rovsmo88]. In this representation, the wave functions are loop dependent. All the information relevant to the loop is found in its ’multitangents fields’ which can be thought of as loop coordinates [@extended]. The loop representation poses some advantages over the connection representation: the Gauss constraint is solved in the former, and furthermore, using knot invariant wave functions, the diffeomorphism constraint is solved.
Recently, the gravity extended loop representation was introduced [@arena; @gravext] as a generalization of the loop representation. In this new representation, the wave function domain is the extended loop space. This vector space of infinite dimension includes the loop coordinates as a particular subset. The extended loop representation has several interesting features: it allows to deal with regularization problems of wave functions which are inherent to the loop representation; it supplies [@remarks] a new context in which to deal with constraint regularization and renormalization, and it supplies new calculation tools to handle loop dependent objects.
In this work we will discuss the problem of the gauge covariance of the extended representation, some aspects of this problem were pointed out and discussed in references [@remarks; @troy]. We shall show here that there exists a sector of the state space which is gauge invariant. This sector contains most of the wave functions know in the extended loop representation.
In order to deal with the extended loop representation, it is useful to define a number of vector spaces. We define the vector space $\cal E$ of infinite dimension as the space of all the linear combinations of a linearly independent base of vectors $\b\DG_{\aa\nu}$ (the subindex ${\aa\nu}$ labels the elements of the base),
$${\b E} \in {\cal E} \leftrightarrow {\b E}=
{\b\DG}_{\aa\nu}E^{\aa\nu} \; .
\label{Espacio1}$$
The dotted Greek indexes correspond to an ordered set of Greek indexes $ {\aa \nu } = \nu_1 \va \nu 2n $ where $n=n({\aa\nu })$ is the number of elements in the set. A Greek index has a discrete part and a continuous part, $ \mu = (\, a,\,x\,)$ with $a\,\in\, \{1,2,3 \} $ and $ x\, \in\, R^3 $. We are adopting the generalized Einstein convention for repeated indexes
$$\varphi_{\mu}E^{\mu} = \sum^3_{i=1} \int d^3x \varphi_{ix} E^{ix}
\;\;\; \mbox{ and } \;\;\;
\varphi_{\aa\mu }E^{\aa\mu } = \varphi E + \sum_{n=1} \varphi_{\va \mu
1n}
E^{\va \mu 1n} \; .$$
In (\[Espacio1\]), $E^{\va \nu 1n}$ is the component of range n of ${\b E}$. The components of the vector ${\b\DG}_{\aa\nu}$ are:
$$\DG^{\aa\mu}_{\aa\nu}\,=\,
\left\{
\begin{array}{ll}
0 & \mbox{if $\, n(\aa\mu)\,\neq \,n(\aa\nu)$} \\
1 & \mbox{if $ \,n(\aa\mu)\,= \,n(\aa\nu)=0$} \\
\mbox{$ \DG^{\mu_1}_{\nu_1}\cdots\DG^{\mu_n}_{\nu_n}$} &
\mbox{if $ n=n(\aa\mu)\,= \,n(\aa\nu)\geq 1$}
\end{array}
\right.$$
where
$$\DG^{ax}_{by} \, \equiv \delta^a_b \, \delta(x-y) \, .$$
In the space ${\cal E}$, we introduce the vector product:
$${\b E_1} \times {\b E_2} \equiv
{\b\DG}_{\aa\mu\,\aa\nu}E^{\aa\mu}_1 E^{\aa\nu}_2 \;.
\label{producto}$$
The extended loop space [@extended] ${\cal E}_y $ is the vector subspace of elements of ${\cal E}$ that satisfy the following differential constraint:
$$\frac{\partial }{\partial x^a} E^{{\aa\alpha}ax{\aa\beta}}=
\DG^{\aa\alpha}_{\aa\nu\, ax}E^{\aa\nu ax \aa\beta } -
\DG^{\aa\beta}_{ax \,\aa\nu}E^{\aa\alpha ax \aa\nu }+
\delta_{x,y} \,\lbrack \DG^{\aa\alpha} E^{\aa\beta } -
\DG^{\aa\beta}E^{\aa\alpha}\rbrack
\label{VD}$$
where there is a sum over “a” and “y” is the origin of the extended loop. This equation can be written as follows (without explicitly indicating the $\DG^{\aa\mu}_{\aa\nu}$ quantities):
$$\frac{\partial }{\partial x^a} E^{\vb ax1{i} ax \vb ax{i+1}n}=
\lbrack \delta (x_{i}-x) - \delta (x_{i+1}-x) \rbrack
E^{ \vb ax1n }$$
where $0\leq i \leq n$ and $x_0\equiv x_{n+1}\equiv y$. The differential constraint relates the components of range n+1 and range n of ${\bf E}$, and basically, indicates that the completely transverse parts of ${\bf E}$ are free, i.e., to solve the constraint means to establish an isomorphism between ${\cal E}_y$ and the transverse vector space [@extended]. It can be shown that the product(\[producto\]) is closed in ${\cal E}_y$. The vectors in ${\cal E}_y$ with the positive null range component together with the product (\[producto\]) define the extended loop group.
We define ${\cal N}$ as the space constituted by the linear applications $\varphi$, $\varphi: \cal E \to C\llap{/}$, that satisfy
$$\exists \, \mbox{ m integer } / \,
\varphi({\b E})= \varphi_{\aa\mu} E^{\aa\mu} = \sum_{n=0}^{m}
{\varphi}_{\va \mu 1n} E^{\va \mu 1n} \;\;.$$
The extended loop representation for a gauge theory or for quantum gravity is dual to the connection representation, and it is formally obtained from the extended loop transform,
$$\varphi({\b E})= \int DA \, \hat{\varphi}(A) \, W_{\b E}[A]
\label{transformada}$$
where
$$W_{\b E}[A]= Tr[A_{\aa \mu}]E^{\aa \mu}
\label{wilson}$$
is the Wilson extended functional and $A_{\aa \mu}$ denotes the product
$$A_{\vb ax1n} = A_{a_1}(x_1)\cdots A_{a_n}(x_n) \, .$$
The ${\b E}$ coordinate in (\[wilson\]) is a vector in ${\cal E}_y$. However, the origin of extended coordinate is not relevant. Due to the trace in (\[wilson\]), the cyclic part of ${\b E}$ is the only one that contributes, this part loses every notion of the origin; in effect, it inherits from (\[VD\]) the following differential constraint without origin:
$$V^{\aa\alpha}_{x\,\,\aa\mu}E^{\aa\mu}=
-\frac{\partial }{\partial x^a} E^{(ax{\aa\alpha})_c}+
\lbrack {\b\DG}_{\aa\nu} \, , \,
{\b\DG}_{ax} \rbrack^{\aa\alpha} \; E^{(ax {\aa\nu})_c}=0
\label{VDC}$$
with
$$V^{\aa\alpha}_{x\,\,\aa\mu}\equiv
-\frac{\partial }{\partial x^a} {\b\DG}^{(ax{\aa\alpha})_c}_{\aa\mu}+
\lbrack {\b\DG}_{\aa\nu} \, , \,
{\b\DG}_{ax} \rbrack^{\aa\alpha} \; {\b\DG}^{(ax {\aa\nu})_c}_{\aa\mu}
\label{VDC2}$$
where there is a sum in ’a’ and no integration in ’x’, the notation $(\,)_c$ meaning the sum of all the cyclic permutations of indices and the commutator is formed with the product ’$\times$’. Formally, the differential constraint guarantees the gauge invariance of $W_{\b E}[A]$ and of the theory. However, the functional (\[wilson\]) is not well defined for all the elements of the extended loop space. There exist pairs of connections and vectors of the extended loop space for which the infinite sum in (\[wilson\]) does not converge; but we shall see that there is a sector in the state space where this problem is not important.
Under the infinitesimal gauge transformation,
$$(\delta A_{ax})= \Lambda_{x,a} + A_{ax}\Lambda_x-\Lambda_x A_{ax}
\label{calibre}$$
the quantity $A_{\aa\mu}$ transforms according to
$$\delta A_{\va \mu 1n}= \sum^n_{h=1}
A_{\va\mu 1{h-1}} \, (\delta A_{\mu_h}) \, A_{\va\mu {h+1}n }
=
\delta_{\va \mu 1n}^{\aa\alpha\,ax\,\aa\beta}
A_{\aa\alpha}(\delta A_{ax})A_{\aa\beta} \,.$$
Its trace satisfies
$$\delta Tr[A_{\aa\mu}] = \delta^{\aa\alpha ax \aa\beta}_{\aa\mu}
Tr\lbrack A_{\aa\beta \aa\alpha} (\delta A_{ax}) \rbrack =
\delta^{\aa\alpha ax \aa\beta}_{\aa\mu}
\delta^{\aa\nu}_{\aa\beta \aa\alpha}
Tr\lbrack A_{\aa\nu} (\delta A_{ax}) \rbrack \, .$$
Using in this expression, the identity
$$\delta^{{\aa\alpha}\, ax \, {\aa\beta}}_{\aa\mu}
\delta^{\aa\nu}_{\aa\beta\,\aa\alpha}
= \delta^{ (ax \, {\aa\nu})_c}_{\aa\mu}$$
and replacing $\delta A_{ax}$ from (\[calibre\]), we get
$$\delta Tr[A_{\aa\mu}] =
Tr\lbrack -\Lambda_x A_{\aa\nu} \frac{\partial }{\partial x^a}+
\Lambda_x A_{\aa\alpha} (\delta^{\aa\alpha}_{\aa\nu ax}-
\delta^{\aa\alpha}_{ax \aa\nu})\rbrack
\delta^{(ax \, \aa\nu)_c}_{\aa\mu}
=
Tr\lbrack \Lambda_x A_{\aa\alpha}\rbrack\, V^{\aa\alpha}_{x\aa\mu}
\;.
\label{deltatraza}$$
We introduce the gauge dependent quantities
$$\varphi^{\va I1n}_{\va \mu 1n} \equiv \int DA \, \hat{\varphi}(A) \,
A^{I_1}_{\mu_1}\cdots A^{I_n}_{\mu_n}$$
and
$$\varphi_{\va \mu 1n} \equiv \varphi^{\va I1n}_{\va \mu 1n}
Tr(T^{I_1} \cdots T^{I_n}) =
\int DA \, \hat{\varphi}(A) \, Tr(A_{\mu_1}\cdots A_{\mu_n})$$
where the $T^I$ are the gauge group generators. Because of (\[deltatraza\]), $\varphi_{\aa\mu}$ transforms according to
$$\delta\varphi_{\aa\mu}= \int DA \,
Tr\lbrack \Lambda_x A_{\aa\alpha}\rbrack\, V^{\aa\alpha}_{x\aa\mu}
\label{transfo1}$$
under the gauge change (\[calibre\]). We shall suppose now that $\hat{\varphi}(A)$ is such that in some gauge G the following “cut condition” is satisfied:
$$\exists \;M\;\; / \; \varphi^{\va I1n}_{\va \mu 1n}=0
\;\; \forall n \, > \, M \, .
\label{corte}$$
This implies that in the gauge G the function
$$\varphi({\b E})= \int DA \, \hat{\varphi}(A) \,
Tr[A_{\aa\mu}] E^{\aa\mu}
= \varphi_{\aa\mu} E^{\aa\mu} \; ,$$
is not dependent on the components of range $n(\aa\mu)>M$ in ${\b E}$. Next, let us show that $\varphi({\b E})$ is invariant under infinitesimal gauge changes. We shall only take gauge transformations for which $\Lambda_x$ can be expanded in a ’power series’ in the connection,
$$\Lambda_x=\lambda_x + \lambda^{ J,\mu}_x
A^J_\mu + \cdots \; .
\label{serie}$$
From (\[transfo1\]) we have
$$\delta \varphi({\b E})=
\int DA \,
Tr\lbrack \Lambda_x A_{\aa\alpha}\rbrack\, V^{\aa\alpha}_{x\aa\mu}
E^{\aa\mu}
\label{transfo2}$$
In this expression, as a consequence of the “cut condition” the sum in $\aa\alpha$ is finite, ( $Max\,\,n(\aa\alpha)=M$ ), and from the definition of $V^{\aa\alpha}_{x\aa\mu}$, we have that $Max\,\,n(\aa\mu)=M+1$, i.e., the sum in (\[transfo2\]) involves a finite number of ranges of ${\b E}$, and because (\[VDC\]), we have
$$\delta \varphi({\b E})=0$$
The functionals $\varphi({\b E})=\varphi_{\aa\mu} E^{\aa\mu}$, which satisfy the “cut condition”, are gauge invariant, and belong to ${\cal N}$. These functionals satisfy Mandelstam identities which reflect the specific structure of the gauge group [@gravext]. In effect, for SU(2), the Mandelstam identities -which relates group generator product traces- lead to the following relations:
$$\begin{aligned}
\varphi_{\aa\mu\aa\nu} =&& \varphi_{\aa\nu \aa\mu} \nonumber \\
\varphi_{\aa \mu } =&& \varphi_{ \overline{\aa\mu}}
\label{mandelstam}\\
\varphi_{\aa\mu \aa\nu \aa\gamma} +
\varphi_{\aa\mu \aa\nu \overline{\aa\gamma}}
=&&
\varphi_{\aa\nu \aa\mu \aa\gamma} +
\varphi_{\aa\nu \aa\mu \overline{\aa\gamma}}
\nonumber\end{aligned}$$
where to overline a set of Greek indexes means to invert the order and the multiplication by a sign, as follows:
$$\delta^{\overline{\va\mu 1n}}_{\aa \nu}\equiv
\delta^{\va\mu 1n}_{\overline{\aa \nu}}\equiv
(-1)^n \delta^{\va\mu n1}_{\aa \nu} \, .$$
The set of functionals that belong to ${\cal N}$ and that satisfy the identities (\[mandelstam\]) contains a sector, gauge invariant, of the state space of the quantum gravity extended loop representation.
In quantum gravity, the wave functions, $\varphi({\b E})= \varphi_{\aa\mu} E^{\aa\mu}$, must be invariant under diffeomorphism. This implies that by evaluating $\varphi({\b E})$ in the coordinate of a loop $\Gamma$, ${\b E} = {\b X}(\Gamma)$, a knot invariant is obtained. Reciprocally, if we have a knot invariant of the form
$$\Psi(\Gamma) = \Psi'({\b X(\Gamma)}) = \,
\sum^M_{n=0} \Psi_{\va\mu 1n} {\b X(\Gamma)}^{\va\mu 1n}
\, ,
\label{nudo}$$
the function $\Psi'({\b E})$ belongs to the state space in the extended loop representation. If the perturbative expansion of the expectation value of the Wilson functional in the Chern-Simons theory is considered, every one of orders of the expansion has the form [@a3] (\[nudo\]). An example of a knot invariant which is not a finite linear combination of the form (\[nudo\]) is the exponential of the Gauss invariant. However, there exist an infinite sequence of functionals in ${\cal N}$ which tend to the exponential of the Gauss invariant.
The following is a possible generalization of the exponential of the Gauss invariant in the extended loop representation:
$$\exp^{*}({\b E}) \equiv \lim_{M\to\infty} F_M({\b E})
\label{ex}$$
where
$$F_M({\b E}) \equiv \sum^M_{n=0} \frac{a^n }{2^n n!}
g_{\mu_1\,\nu_1} \cdots g_{\mu_n\,\nu_n}
E^{(\mu_1\,\nu_1 \cdots \mu_n\,\nu_n)_S}
\label{FM}$$
and $g_{ax \, by} \equiv - \varepsilon_{abc}
\partial_c \nabla^{-2} \delta (x-y)$ is the propagator of the Chern-Simons theory. In (\[FM\]), the subscript ’S’ indicates a sum over all the permutations of the Greek indexes. The functions $F_M(\b E)$ belong to ${\cal N}$, satisfy the Mandelstam identities (\[mandelstam\]) and, it can be shown that, they are diffeomorphism invariant.[^1] By evaluating (\[ex\]) in loop coordinates, we get
$$\exp^*[{\b X(\Gamma)}] = \exp [a \,\rho(\Gamma)]
\label{gauss}$$
where $\rho(\Gamma)$ is the Gauss self-linking number of the loop $\Gamma$. In order to obtain (\[gauss\]), the following algebraic constraint, satisfied by the loop coordinate, was used,
$$[\b X(\Gamma)]^{{\aa\beta}_1\nu_1 \cdots {\aa\beta}_n\nu_n
{\aa\beta}_{n+1} }
\delta^{\va \mu 1m}_{\va {\aa\beta} 1{n+1}}
=
[\b X(\Gamma)]^{\va \mu 1m}\, [\b X(\Gamma)]^{\va \nu 1n}
\;.
\label{VA}$$
As shown in reference [@GaPu], $\exp [-3\Lambda \,\rho(\Gamma)/2 ]$ is a formal solution of gravity with cosmological constant in the loop representation. It is expected that its extension (\[ex\]) is the corresponding solution in the extended loop representation. However, $\exp^*$ does not converge for every vector $\b E$. A possible solution to this problem would be to limit the extended loop space to those vectors that satisfy an algebraic constraint equal to (\[VA\]). In this case, (\[ex\]) converges to
$$\exp^*[{\b E}] =
\exp [\frac{a}{2} g_{\mu_1\mu_2}\, E^{\mu_1}\,E^{\mu_2}] \,.$$
The problem of the appropriate definition of $\exp^*$ and the action of gravity constraints on this functional is still under study. It is interesting to point out that there are also problems in the lattice, where it has not been possible to define an exponential of the Gauss number which simultaneously satisfies the Hamilton and diffeomorphism constraint [@privada].
We wish especially to thank Rodolfo Gambini for his critical comments.
[99]{}
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R. Gambini, private communication.
[^1]: For each $n$, $E^{\va \nu 1n}$ is a vector density of weight one in each argument $\nu=(ax)$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Software repositories contain a plethora of useful information that can be used to enhance software projects. Prior work has leveraged repository data to improve many aspects of the software development process, such as, help extract requirement decisions, identify potentially defective code and improve maintenance and evolution. However, in many cases, project stakeholders are not able to fully benefit from their software repositories due to the fact that they need special expertise to mine their repositories. Also, extracting and linking data from different types of repositories (e.g., source code control and bug repositories) requires dedicated effort and time, even if the stakeholder has the expertise to perform such a task.
Therefore, in this paper, we use bots to automate and ease the process of extracting useful information from software repositories. Particularly, we lay out an approach of how bots, layered on top of software repositories, can be used to answer some of the most common software development/maintenance questions facing developers. We perform a preliminary study with 12 participants to validate the effectiveness of the bot. Our findings indicate that using bots achieves very promising results compared to not using the bot (baseline). Most of the participants (90.0%) find the bot to be either useful or very useful. Also, they completed 90.8% of the tasks correctly using the bot with a median time of 40 seconds per task. On the other hand, without the bot, the participants completed 25.2% of the tasks with a median time of 240 seconds per task. Our work has the potential to transform the MSR field by significantly lowering the barrier to entry, making the extraction of useful information from software repositories as easy as chatting with a bot.
author:
- Ahmad Abdellatif
- Khaled Badran
- Emad Shihab
bibliography:
- 'bots\_bib.bib'
title: 'MSRBot: Using Bots to Answer Questions from Software Repositories'
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose a data-to-text generation model with two modules, one for tracking and the other for text generation. Our tracking module selects and keeps track of salient information and memorizes which record has been mentioned. Our generation module generates a summary conditioned on the state of tracking module. Our model is considered to simulate the human-like writing process that gradually selects the information by determining the intermediate variables while writing the summary. In addition, we also explore the effectiveness of the writer information for generation. Experimental results show that our model outperforms existing models in all evaluation metrics even without writer information. Incorporating writer information further improves the performance, contributing to content planning and surface realization.'
author:
- |
Hayate Iso[${}^{\thanks{~~Work was done during the internship at Artificial Intelligence Research Center, AIST}\dag}$]{} Yui Uehara[${}^{\ddag}$]{} Tatsuya Ishigaki[${}^{\natural\ddag}$]{} Hiroshi Noji[${}^{\ddag}$]{} \
**[ Eiji Aramaki[${}^{\dag\ddag}$]{} Ichiro Kobayashi[${}^{\flat\ddag}$]{} Yusuke Miyao[${}^{\sharp\ddag}$]{} Naoaki Okazaki[${}^{\natural\ddag}$]{} Hiroya Takamura[${}^{\natural\ddag}$]{}]{}\
Nara Institute of Science and Technology[${}^{\ddag}$]{}Artificial Intelligence Research Center, AIST\
[${}^\natural$]{}Tokyo Institute of Technology[${}^\flat$]{}Ochanomizu University[${}^\sharp$]{}The University of Tokyo\
[ {iso.hayate.id3,aramaki}@is.naist.jp [email protected]]{}\
[ {yui.uehara,ishigaki.t,hiroshi.noji,takamura.hiroya}@aist.go.jp ]{}\
[ [email protected] [email protected] ]{}**
bibliography:
- 'acl2019.bib'
title: 'Learning to Select, Track, and Generate for Data-to-Text'
---
Introduction
============
Advances in sensor and data storage technologies have rapidly increased the amount of data produced in various fields such as weather, finance, and sports. In order to address the information overload caused by the massive data, data-to-text generation technology, which expresses the contents of data in natural language, becomes more important [@barzilay2005collective]. Recently, neural methods can generate high-quality short summaries especially from small pieces of data [@liu2018table].
Despite this success, it remains challenging to generate a high-quality long summary from data [@wiseman2017challenges]. One reason for the difficulty is because the input data is too large for a naive model to find its salient part, i.e., to determine which part of the data should be mentioned. In addition, the salient part moves as the summary explains the data. For example, when generating a summary of a basketball game (Table \[tab:example\] (b)) from the box score (Table \[tab:example\] (a)), the input contains numerous data records about the game: e.g., *Jordan Clarkson scored 18 points*. Existing models often refer to the same data record multiple times [@puduppully2019data]. The models may mention an incorrect data record, e.g., *Kawhi Leonard added 19 points*: the summary should mention *LaMarcus Aldridge*, who scored 19 points. Thus, we need a model that finds salient parts, tracks transitions of salient parts, and expresses information faithful to the input.
In this paper, we propose a novel data-to-text generation model with two modules, one for saliency tracking and another for text generation. The tracking module keeps track of saliency in the input data: when the module detects a saliency transition, the tracking module selects a new data record[^1] and updates the state of the tracking module. The text generation module generates a document conditioned on the current tracking state. Our model is considered to imitate the human-like writing process that gradually selects and tracks the data while generating the summary. In addition, we note some writer-specific patterns and characteristics: how data records are selected to be mentioned; and how data records are expressed as text, e.g., the order of data records and the word usages. We also incorporate writer information into our model.
The experimental results demonstrate that, even without writer information, our model achieves the best performance among the previous models in all evaluation metrics: 94.38% precision of relation generation, 42.40% F1 score of content selection, 19.38% normalized Damerau-Levenshtein Distance (DLD) of content ordering, and 16.15% of <span style="font-variant:small-caps;">BLEU</span> score. We also confirm that adding writer information further improves the performance.
[0.52]{}
<span style="font-variant:small-caps;">Team</span> <span style="font-variant:small-caps;">H/V</span> <span style="font-variant:small-caps;">Win</span> <span style="font-variant:small-caps;">Loss</span> <span style="font-variant:small-caps;">Pts</span> <span style="font-variant:small-caps;">Reb</span> <span style="font-variant:small-caps;">Ast</span> <span style="font-variant:small-caps;">Fg\_Pct</span> <span style="font-variant:small-caps;">Fg3\_Pct</span> <span style="font-variant:small-caps;">$\dots$</span>
------------------------------------------------------ --------------------------------------------------- --------------------------------------------------- ---------------------------------------------------- --------------------------------------------------- --------------------------------------------------- --------------------------------------------------- ------------------------------------------------------- -------------------------------------------------------- -------------------------------------------------------
<span style="font-variant:small-caps;">Knicks</span> H [**16**]{} [**19**]{} [**104**]{} <span style="font-variant:small-caps;">46</span> <span style="font-variant:small-caps;">26</span> <span style="font-variant:small-caps;">45</span> <span style="font-variant:small-caps;">46</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">Bucks</span> V [**18**]{} [**16**]{} [**105**]{} <span style="font-variant:small-caps;">42</span> <span style="font-variant:small-caps;">20</span> <span style="font-variant:small-caps;">47</span> <span style="font-variant:small-caps;">32</span> <span style="font-variant:small-caps;">$\dots$</span>
\
<span style="font-variant:small-caps;">Player</span> <span style="font-variant:small-caps;">H/V</span> <span style="font-variant:small-caps;">Pts</span> <span style="font-variant:small-caps;">Reb</span> <span style="font-variant:small-caps;">Ast</span> <span style="font-variant:small-caps;">Blk</span> <span style="font-variant:small-caps;">Stl</span> <span style="font-variant:small-caps;">Min</span> <span style="font-variant:small-caps;">City</span> <span style="font-variant:small-caps;">$\dots$</span>
--------------------------------------------------------------------- ------------------------------------------------------- ------------------------------------------------------- ------------------------------------------------------- ------------------------------------------------------- ------------------------------------------------------- ------------------------------------------------------- --------------------------------------------------- --------------------------------------------------------- -------------------------------------------------------
<span style="font-variant:small-caps;">Carmelo Anthony</span> <span style="font-variant:small-caps;">H</span> [**30**]{} [**11**]{} [**7**]{} <span style="font-variant:small-caps;">0</span> [**2**]{} <span style="font-variant:small-caps;">37</span> <span style="font-variant:small-caps;">New York</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">Derrick Rose</span> <span style="font-variant:small-caps;">H</span> [**15**]{} [**3**]{} [**4**]{} <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">1</span> <span style="font-variant:small-caps;">33</span> <span style="font-variant:small-caps;">New York</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">Courtney Lee</span> <span style="font-variant:small-caps;">H</span> [**11**]{} [**2**]{} [**3**]{} <span style="font-variant:small-caps;">1</span> <span style="font-variant:small-caps;">1</span> <span style="font-variant:small-caps;">38</span> <span style="font-variant:small-caps;">New York</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">Giannis Antetokounmpo</span> <span style="font-variant:small-caps;">V</span> [**27**]{} [**13**]{} [**4**]{} [**3**]{} <span style="font-variant:small-caps;">1</span> <span style="font-variant:small-caps;">39</span> <span style="font-variant:small-caps;">Milwaukee</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">Greg Monroe</span> <span style="font-variant:small-caps;">V</span> [**18**]{} [**9**]{} [**4**]{} <span style="font-variant:small-caps;">1</span> [**3**]{} <span style="font-variant:small-caps;">31</span> <span style="font-variant:small-caps;">Milwaukee</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">Jabari Parker</span> <span style="font-variant:small-caps;">V</span> [**15**]{} [**4**]{} [**3**]{} <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">1</span> <span style="font-variant:small-caps;">37</span> <span style="font-variant:small-caps;">Milwaukee</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">Malcolm Brogdon</span> <span style="font-variant:small-caps;">V</span> [**12**]{} [**6**]{} [**8**]{} <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">38</span> <span style="font-variant:small-caps;">Milwaukee</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">Mirza Teletovic</span> <span style="font-variant:small-caps;">V</span> [**13**]{} <span style="font-variant:small-caps;">1</span> <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">21</span> <span style="font-variant:small-caps;">Milwaukee</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">John Henson</span> <span style="font-variant:small-caps;">V</span> <span style="font-variant:small-caps;">2</span> <span style="font-variant:small-caps;">2</span> <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">0</span> <span style="font-variant:small-caps;">14</span> <span style="font-variant:small-caps;">Milwaukee</span> <span style="font-variant:small-caps;">$\dots$</span>
<span style="font-variant:small-caps;">$\dots$</span> <span style="font-variant:small-caps;">$\dots$</span> <span style="font-variant:small-caps;">$\dots$</span> <span style="font-variant:small-caps;">$\dots$</span> <span style="font-variant:small-caps;">$\dots$</span> <span style="font-variant:small-caps;">$\dots$</span> <span style="font-variant:small-caps;">$\dots$</span>
[0.43]{}
;
Related Work
============
Data-to-Text Generation
-----------------------
Data-to-text generation is a task for generating descriptions from structured or non-structured data including sports commentary [@tanaka1998reactive; @chen2008learning; @taniguchi2019generating], weather forecast [@liang2009learning; @mei2016talk], biographical text from infobox in Wikipedia [@lebret2016neural; @sha2018order; @liu2018table] and market comments from stock prices [@murakami2017learning; @aoki2018generating].
Neural generation methods have become the mainstream approach for data-to-text generation. The encoder-decoder framework [@cho2014learning; @sutskever2014sequence] with the attention [@bahdanau2015neural; @luong2015effective] and copy mechanism [@gu2016incorporating; @gulcehre2016pointing] has successfully applied to data-to-text tasks. However, neural generation methods sometimes yield fluent but inadequate descriptions [@tu2017context]. In data-to-text generation, descriptions inconsistent to the input data are problematic.
Recently, @wiseman2017challenges introduced the <span style="font-variant:small-caps;">RotoWire</span> dataset, which contains multi-sentence summaries of basketball games with box-score (Table \[tab:example\]). This dataset requires the selection of a salient subset of data records for generating descriptions. They also proposed automatic evaluation metrics for measuring the informativeness of generated summaries. @puduppully2019data proposed a two-stage method that first predicts the sequence of data records to be mentioned and then generates a summary conditioned on the predicted sequences. Their idea is similar to ours in that the both consider a sequence of data records as content planning. However, our proposal differs from theirs in that ours uses a recurrent neural network for saliency tracking, and that our decoder dynamically chooses a data record to be mentioned without fixing a sequence of data records.
Memory modules
--------------
The memory network can be used to maintain and update representations of the salient information [@weston2015memory; @sukhbaatar2015end; @graves2016hybrid]. This module is often used in natural language understanding to keep track of the entity state [@kobayashi2016dynamic; @hoang2018entity; @bosselut2018simulating].
Recently, entity tracking has been popular for generating coherent text [@kiddon2016globally; @ji2017dynamic; @yang2017reference; @clark2018neural]. @kiddon2016globally proposed a neural checklist model that updates predefined item states. @ji2017dynamic proposed an entity representation for the language model. Updating entity tracking states when the entity is introduced, their method selects the salient entity state.
Our model extends this entity tracking module for data-to-text generation tasks. The entity tracking module selects the salient entity and appropriate attribute in each timestep, updates their states, and generates coherent summaries from the selected data record.
$t$ 199 200 201 202 203 204 205 206 207 208 209
--------- ---------------------------------------------------------- --------------------------------------------------------- ------------- ---------------------------------------------------------- -------- ----- ---------------------------------------------------------- ---------- ----- ---------------------------------------------------------- --------- -- -- -- -- -- --
$Y_{t}$ Jabari Parker contributed 15 points , four rebounds , three assists
$Z_{t}$ 1 1 0 1 0 0 1 0 0 1 0
<span style="font-variant:small-caps;">Jabari</span> <span style="font-variant:small-caps;">Jabari</span> <span style="font-variant:small-caps;">Jabari</span> <span style="font-variant:small-caps;">Jabari</span> <span style="font-variant:small-caps;">Jabari</span>
<span style="font-variant:small-caps;">Parker</span> <span style="font-variant:small-caps;">Parker</span> <span style="font-variant:small-caps;">Parker</span> <span style="font-variant:small-caps;">Parker</span> <span style="font-variant:small-caps;">Parker</span>
$A_{t}$ <span style="font-variant:small-caps;">First Name</span> <span style="font-variant:small-caps;">Last Name</span> - <span style="font-variant:small-caps;">Player Pts</span> - - <span style="font-variant:small-caps;">Player Reb</span> - - <span style="font-variant:small-caps;">Player Ast</span> -
$N_{t}$ - - - 0 - - 1 - - 1 -
Data {#section:data}
====
Through careful examination, we found that in the original dataset <span style="font-variant:small-caps;">RotoWire</span>, some NBA games have two documents, one of which is sometimes in the training data and the other is in the test or validation data. Such documents are similar to each other, though not identical. To make this dataset more reliable as an experimental dataset, we created a new version.
We ran the script provided by @wiseman2017challenges, which is for crawling the <span style="font-variant:small-caps;">RotoWire</span> website for NBA game summaries. The script collected approximately 78% of the documents in the original dataset; the remaining documents disappeared. We also collected the box-scores associated with the collected documents. We observed that some of the box-scores were modified compared with the original <span style="font-variant:small-caps;">RotoWire</span> dataset.
The collected dataset contains 3,752 instances (i.e., pairs of a document and box-scores). However, the four shortest documents were not summaries; they were, for example, an announcement about the postponement of a match. We thus deleted these 4 instances and were left with 3,748 instances. We followed the dataset split by @wiseman2017challenges to split our dataset into training, development, and test data. We found 14 instances that didn’t have corresponding instances in the original data. We randomly classified 9, 2, and 3 of those 14 instances respectively into training, development, and test data. Finally, the sizes of our training, development, test dataset are respectively 2,714, 534, and 500. On average, each summary has 384 tokens and 644 data records. Each match has only one summary in our dataset, as far as we checked. We also collected the writer of each document. Our dataset contains 32 different writers. The most prolific writer in our dataset wrote 607 documents. There are also writers who wrote less than ten documents. On average, each writer wrote 117 documents. We call our new dataset <span style="font-variant:small-caps;">RotoWire-Modified</span>.[^2]
Saliency-Aware Text Generation
==============================
At the core of our model is a neural language model with a memory state $\boldsymbol{h}^{\textsc{LM}}$ to generate a summary $y_{1:T} = (y_1, \dots, y_T)$ given a set of data records $\boldsymbol{x}$. Our model has another memory state $\boldsymbol{h}^{\textsc{Ent}}$, which is used to remember the data records that have been referred to. $\boldsymbol{h}^{\textsc{Ent}}$ is also used to update $\boldsymbol{h}^{\textsc{LM}}$, meaning that the referred data records affect the text generation.
Our model decides whether to refer to $\boldsymbol{x}$, which data record $r\in \boldsymbol{x}$ to be mentioned, and how to express a number. The selected data record is used to update $\boldsymbol{h}^{\textsc{Ent}}$. Formally, we use the four variables:
1. $ Z_t $: binary variable that determines whether the model refers to input $\boldsymbol{x}$ at time step $t$ ($ Z_t = 1 $).
2. $ E_t $: At each time step $t$, this variable indicates the salient entity (e.g., <span style="font-variant:small-caps;">Hawks</span>, <span style="font-variant:small-caps;">LeBron James</span>).
3. $ A_t $: At each time step $t$, this variable indicates the salient attribute to be mentioned (e.g., <span style="font-variant:small-caps;">Pts</span>).
4. $ N_t $: If attribute $A_t$ of the salient entity $E_t$ is a numeric attribute, this variable determines if a value in the data records should be output in Arabic numerals (e.g., 50) or in English words (e.g., five).
To keep track of the salient entity, our model predicts these random variables at each time step $t$ through its summary generation process. Running example of our model is shown in Table \[tab:annotate\] and full algorithm is described in Appendix \[sec:algorithm\]. In the following subsections, we explain how to initialize the model, predict these random variables, and generate a summary. Due to space limitations, bias vectors are omitted.
Before explaining our method, we describe our notation. Let $\mathcal{E}$ and $\mathcal{A}$ denote the sets of entities and attributes, respectively. Each record $r \in \boldsymbol{x}$ consists of entity $e\in\mathcal{E}$, attribute $a\in\mathcal{A}$, and its value $\boldsymbol{x}[e, a]$, and is therefore represented as $r = (e, a, \boldsymbol{x}[e, a])$. For example, the box-score in Table \[tab:example\] has a record $r$ such that $e = \textsc{Anthony Davis}, a = \textsc{Pts},$ and $\boldsymbol{x}[e, a] = 20$.
Initialization {#sec:init}
--------------
Let $\boldsymbol{r}$ denote the embedding of data record $r \in \boldsymbol{x}$. Let $\bar{\boldsymbol{e}}$ denote the embedding of entity $e$. Note that $\bar{\boldsymbol{e}}$ depends on the set of data records, i.e., it depends on the game. We also use $\boldsymbol{e}$ for static embedding of entity $e$, which, on the other hand, does not depend on the game.
Given the embedding of entity $\boldsymbol{e}$, attribute $\boldsymbol{a}$, and its value $\boldsymbol{v}$, we use the concatenation layer to combine the information from these vectors to produce the embedding of each data record $(e,a,v)$, denoted as $\boldsymbol{r}_{e,a,v}$ as follows: $$\begin{aligned}
\boldsymbol{r}_{e,a,v} = \tanh\left(\boldsymbol{W}^{\textsc{R}}(\boldsymbol{e} \oplus \boldsymbol{a} \oplus \boldsymbol{v})\right),
\label{init}\end{aligned}$$ where $\oplus$ indicates the concatenation of vectors, and $\boldsymbol{W}^{\textsc{R}}$ denotes a weight matrix.[^3]
We obtain $\bar{\boldsymbol{e}}$ in the set of data records $\boldsymbol{x}$, by summing all the data-record embeddings transformed by a matrix: $$\begin{aligned}
\bar{\boldsymbol{e}} = \tanh\left(\sum_{a\in \mathcal{A}} \boldsymbol{W}^{\textsc{A}}_{a}\boldsymbol{r}_{e, a, \boldsymbol{x}[e,a]}\right),\end{aligned}$$ where $\boldsymbol{W}^{\textsc{A}}_{a}$ is a weight matrix for attribute $a$. Since $\bar{\boldsymbol{e}}$ depends on the game as above, $\bar{\boldsymbol{e}}$ is supposed to represent how entity $e$ played in the game.
To initialize the hidden state of each module, we use embeddings of $<$<span style="font-variant:small-caps;">SoD</span>$>$ for $\boldsymbol{h}^{\textsc{LM}}$ and averaged embeddings of $\bar{\boldsymbol{e}}$ for $\boldsymbol{h}^{\textsc{ENT}}$.
Saliency transition {#subsec:saliency}
-------------------
Generally, the saliency of text changes during text generation. In our work, we suppose that the saliency is represented as the entity and its attribute being talked about. We therefore propose a model that refers to a data record at each timepoint, and transitions to another as text goes.
To determine whether to transition to another data record or not at time $t$, the model calculates the following probability: $$\begin{aligned}
p(Z_t = 1 \mid \boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t-1}^{\textsc{Ent}}) = \sigma(\boldsymbol{W}_{z}(\boldsymbol{h}_{t-1}^{\textsc{LM}} \oplus \boldsymbol{h}_{t-1}^{\textsc{Ent}})),\end{aligned}$$ where $\sigma (\cdot) $ is the sigmoid function. If $p(Z_t = 1 \mid \boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t-1}^{\textsc{Ent}})$ is high, the model transitions to another data record.
When the model decides to transition to another, the model then determines which entity and attribute to refer to, and generates the next word (Section \[sec:ent\]). On the other hand, if the model decides not transition to another, the model generates the next word without updating the tracking states $\boldsymbol{h}^{\textsc{Ent}}_t = \boldsymbol{h}^{\textsc{Ent}}_{t-1}$ (Section \[sec:out\]).
Selection and tracking {#sec:ent}
----------------------
When the model refers to a new data record ($Z_t = 1$), it selects an entity and its attribute. It also tracks the saliency by putting the information about the selected entity and attribute into the memory vector $\boldsymbol{h}^{\textsc{Ent}}$. The model begins to select the subject entity and update the memory states if the subject entity will change.
Specifically, the model first calculates the probability of selecting an entity: $$\begin{aligned}
&p(E_t = e \mid \boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t-1}^{\textsc{Ent}})\nonumber\\
\propto
&\begin{cases}
\exp\left({\boldsymbol{h}^{\textsc{Ent}}_s\boldsymbol{W}^{\textsc{Old}}\boldsymbol{h}_{t-1}^{\textsc{LM}}}\right) & \text{if } e \in \mathcal{E}_{t-1}\\
\exp\left({\bar{\boldsymbol{e}}\boldsymbol{W}^{\textsc{New}}\boldsymbol{h}_{t-1}^{\textsc{LM}}}\right) & \text{otherwise}
\end{cases},\end{aligned}$$ where $\mathcal{E}_{t-1}$ is the set of entities that have already been referred to by time step $t$, and $s$ is defined as $ s = {\max \{s: s \leq t - 1, e = e_s \}} $, which indicates the time step when this entity was last mentioned.
The model selects the most probable entity as the next salient entity and updates the set of entities that appeared ($\mathcal{E}_t = \mathcal{E}_{t - 1} \cup \{e_t\}$).
If the salient entity changes $(e_t \not= e_{t - 1})$, the model updates the hidden state of the tracking model $\boldsymbol{h}^{\textsc{Ent}}$ with a recurrent neural network with a gated recurrent unit [<span style="font-variant:small-caps;">Gru</span>; @chung2014empirical]: $$\begin{aligned}
\boldsymbol{h}_{t}^{\textsc{Ent}'}
=
\begin{cases}
\boldsymbol{h}_{t-1}^{\textsc{Ent}} \hspace{2.8cm}\text{ if } e_t = e_{t-1}\\
\textsc{Gru}^{\textsc{E}}(\bar{\boldsymbol{e}}, \boldsymbol{h}_{t-1}^{\textsc{Ent}}) \hspace{0.4cm} \text{ else if } e_t \not \in \mathcal{E}_{t-1}\\
\textsc{Gru}^{\textsc{E}}(\boldsymbol{W}^\textsc{S}_s\boldsymbol{h}_s^{\textsc{Ent}}, \boldsymbol{h}_{t-1}^{\textsc{Ent}}) \hspace{0.2cm} \text{ otherwise.}
\end{cases}\end{aligned}$$ Note that if the selected entity at time step $t$, $e_t$, is identical to the previously selected entity $e_{t-1}$, the hidden state of the tracking model is not updated.
If the selected entity $e_t$ is new ($e_t \not \in \mathcal{E}_{t-1}$), the hidden state of the tracking model is updated with the embedding $\bar{\boldsymbol{e}}$ of entity $e_t$ as input. In contrast, if entity $e_t$ has already appeared in the past ($e_t \in \mathcal{E}_{t-1}$) but is not identical to the previous one $(e_t \not= e_{t-1})$, we use $\boldsymbol{h}_s^{\textsc{Ent}}$ (i.e., the memory state when this entity last appeared) to fully exploit the local history of this entity.
Given the updated hidden state of the tracking model $\boldsymbol{h}_{t}^{\textsc{Ent}}$, we next select the attribute of the salient entity by the following probability: $$\begin{aligned}
&p(A_t = a \mid e_t,\boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t}^{\textsc{Ent}'}) \\
\propto &\exp\left(\boldsymbol{r}_{e_t, a, \boldsymbol{x}[e_t,a]}\boldsymbol{W}^{\textsc{Attr}} (\boldsymbol{h}_{t-1}^{\textsc{LM}} \oplus \boldsymbol{h}_{t}^{\textsc{Ent}'})\right).\nonumber\end{aligned}$$ After selecting $a_t$, i.e., the most probable attribute of the salient entity, the tracking model updates the memory state $\boldsymbol{h}_{t}^{\textsc{Ent}}$ with the embedding of the data record $\boldsymbol{r}_{e_t, a_t, \boldsymbol{x}[e_t, a_t]}$ introduced in Section \[sec:init\]: $$\begin{aligned}
\boldsymbol{h}_{t}^{\textsc{Ent}} = \textsc{Gru}^{\textsc{A}}(\boldsymbol{r}_{e_t, a_t, \boldsymbol{x}[e_t,a_t]}, \boldsymbol{h}_{t}^{\textsc{Ent}'}).\end{aligned}$$
Summary generation {#sec:out}
------------------
Given two hidden states, one for language model $\boldsymbol{h}_{t-1}^{\textsc{LM}}$ and the other for tracking model $\boldsymbol{h}_{t}^{\textsc{Ent}}$, the model generates the next word $y_t$. We also incorporate a copy mechanism that copies the value of the salient data record $\boldsymbol{x}[e_t, a_t]$.
If the model refers to a new data record ($Z_t = 1$), it directly copies the value of the data record $\boldsymbol{x}[e_t, a_t]$. However, the values of numerical attributes can be expressed in at least two different manners: Arabic numerals (e.g., [*14*]{}) and English words (e.g., [*fourteen*]{}). We decide which one to use by the following probability: $$\begin{aligned}
p(N_t = 1 \mid \boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t}^{\textsc{Ent}}) = \sigma(\boldsymbol{W}^{\textsc{N}}(\boldsymbol{h}_{t-1}^{\textsc{LM}}\oplus \boldsymbol{h}_{t}^{\textsc{Ent}})),\end{aligned}$$ where $\boldsymbol{W}^{\textsc{N}}$ is a weight matrix. The model then updates the hidden states of the language model: $$\begin{aligned}
\boldsymbol{h}_t'&= \tanh\left(\boldsymbol{W}^{\textsc{H}}(\boldsymbol{h}_{t-1}^{\textsc{LM}} \oplus\boldsymbol{h}_{t}^{\textsc{Ent}})\right),\label{eq:hidden}\end{aligned}$$ where $\boldsymbol{W}^{\textsc{H}}$ is a weight matrix.
If the salient data record is the same as the previous one ($Z_t = 0$), it predicts the next word $y_t$ via a probability over words conditioned on the context vector $\boldsymbol{h}_t'$: $$\begin{aligned}
p(Y_t \mid \boldsymbol{h}_t') = \text{softmax}(\boldsymbol{W}^{\textsc{Y}} \boldsymbol{h}_t').\label{eq:prob}\end{aligned}$$ Subsequently, the hidden state of language model $\boldsymbol{h}^{\textsc{LM}}$ is updated: $$\begin{aligned}
\boldsymbol{h}_t^{\textsc{LM}} &= \textsc{LSTM}(\boldsymbol{y}_{t}\oplus\boldsymbol{h}_t', \boldsymbol{h}_{t-1}^{\textsc{LM}}),\end{aligned}$$ where $\boldsymbol{y}_t$ is the embedding of the word generated at time step $t$.[^4]
Incorporating writer information {#subsec:writer}
--------------------------------
We also incorporate the information about the writer of the summaries into our model. Specifically, instead of using Equation (\[eq:hidden\]), we concatenate the embedding $\boldsymbol{w}$ of a writer to $\boldsymbol{h}_{t-1}^{\textsc{LM}}\oplus\boldsymbol{h}_{t}^{\textsc{Ent}}$ to construct context vector $\boldsymbol{h}_t'$: $$\begin{aligned}
\boldsymbol{h}_t'&= \tanh\left(\boldsymbol{W}'^{\textsc{H}}(\boldsymbol{h}_{t-1}^{\textsc{LM}} \oplus\boldsymbol{h}_{t}^{\textsc{Ent}}\oplus \boldsymbol{w})\right),
\end{aligned}$$ where $\boldsymbol{W}'^{\textsc{H}}$ is a new weight matrix. Since this new context vector $\boldsymbol{h}_t'$ is used for calculating the probability over words in Equation (\[eq:prob\]), the writer information will directly affect word generation, which is regarded as surface realization in terms of traditional text generation. Simultaneously, context vector $\boldsymbol{h}_t'$ enhanced with the writer information is used to obtain $\boldsymbol{h}_t^{\textsc{LM}}$, which is the hidden state of the language model and is further used to select the salient entity and attribute, as mentioned in Sections \[subsec:saliency\] and \[sec:ent\]. Therefore, in our model, the writer information affects both surface realization and content planning.
Learning objective
------------------
We apply fully supervised training that maximizes the following log-likelihood: $$\begin{aligned}
&\log p(Y_{1:T}, Z_{1:T}, E_{1:T}, A_{1:T}, N_{1:T} \mid \boldsymbol{x})\\
=&\sum_{t=1}^T\log p(Z_t = z_t \mid \boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t-1}^{\textsc{Ent}})~\\
+ &\sum_{t:Z_t = 1}\log p(E_t = e_t \mid \boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t-1}^{\textsc{Ent}})\\
+ &\sum_{t:Z_t = 1}\log p(A_t = a_t \mid e_t, \boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t}^{\textsc{Ent}'})\\
+&\sum_{t:Z_t = 1, a_t \text{is num\_attr}} \log p(N_t = n_t \mid \boldsymbol{h}_{t-1}^{\textsc{LM}}, \boldsymbol{h}_{t}^{\textsc{Ent}})\\
+ & \sum_{t: Z_t = 0}\log p(Y_{t} = y_t \mid \boldsymbol{h}_t')\end{aligned}$$
--------------------------------------------------------- ------- ----------- -------------------------------------------------- ----------- ----------- ----------- -----------
<span style="font-variant:small-caps;">CO</span>
\# P% P% R% F1% DLD%
<span style="font-variant:small-caps;">Gold</span> 27.36 93.42 100. 100. 100. 100. 100.
<span style="font-variant:small-caps;">Templates</span> 54.63 100. 31.01 58.85 40.61 17.50 8.43
@wiseman2017challenges 22.93 60.14 24.24 31.20 27.29 14.70 14.73
@puduppully2019data 33.06 83.17 33.06 43.59 37.60 16.97 13.96
<span style="font-variant:small-caps;">Proposed</span> 39.05 **94.43** **35.77** **52.05** **42.40** **19.38** **16.15**
--------------------------------------------------------- ------- ----------- -------------------------------------------------- ----------- ----------- ----------- -----------
Experiments
===========
Experimental settings
---------------------
We used <span style="font-variant:small-caps;">RotoWire-Modified</span> as the dataset for our experiments, which we explained in Section \[section:data\]. The training, development, and test data respectively contained 2,714, 534, and 500 games.
Since we take a supervised training approach, we need the annotations of the random variables (i.e., $Z_t$, $E_t$, $A_t$, and $N_t$) in the training data, as shown in Table \[tab:annotate\]. Instead of simple lexical matching with $r \in \boldsymbol{x}$, which is prone to errors in the annotation, we use the information extraction system provided by @wiseman2017challenges. Although this system is trained on noisy rule-based annotations, we conjecture that it is more robust to errors because it is trained to minimize the marginalized loss function for ambiguous relations. All training details are described in Appendix \[sec:settings\].
Models to be compared
---------------------
We compare our model[^5] against two baseline models. One is the model used by @wiseman2017challenges, which generates a summary with an attention-based encoder-decoder model. The other baseline model is the one proposed by @puduppully2019data, which first predicts the sequence of data records and then generates a summary conditioned on the predicted sequences. @wiseman2017challenges’s model refers to all data records every timestep, while @puduppully2019data’s model refers to a subset of all data records, which is predicted in the first stage. Unlike these models, our model uses one memory vector $\boldsymbol{h}^{\textsc{Ent}}_{t}$ that tracks the history of the data records, during generation. We retrained the baselines on our new dataset. We also present the performance of the <span style="font-variant:small-caps;">Gold</span> and <span style="font-variant:small-caps;">Templates</span> summaries. The <span style="font-variant:small-caps;">Gold</span> summary is exactly identical with the reference summary and each <span style="font-variant:small-caps;">Templates</span> summary is generated in the same manner as @wiseman2017challenges.
In the latter half of our experiments, we examine the effect of adding information about writers. In addition to our model enhanced with writer information, we also add writer information to the model by @puduppully2019data. Their method consists of two stages corresponding to content planning and surface realization. Therefore, by incorporating writer information to each of the two stages, we can clearly see which part of the model to which the writer information contributes to. For @puduppully2019data model, we attach the writer information in the following three ways:
1. concatenating writer embedding $\boldsymbol{w}$ with the input vector for LSTM in the content planning decoder (stage 1);
2. concatenating writer embedding $\boldsymbol{w}$ with the input vector for LSTM in the text generator (stage 2);
3. using both 1 and 2 above.
For more details about each decoding stage, readers can refer to @puduppully2019data.
Evaluation metrics
------------------
As evaluation metrics, we use BLEU score [@papineni2002bleu] and the extractive metrics proposed by @wiseman2017challenges, i.e., relation generation (RG), content selection (CS), and content ordering (CO) as evaluation metrics. The extractive metrics measure how well the relations extracted from the generated summary match the correct relations[^6]:
- RG: the ratio of the correct relations out of all the extracted relations, where correct relations are relations found in the input data records $\boldsymbol{x}$. The average number of extracted relations is also reported.
- CS: precision and recall of the relations extracted from the generated summary against those from the reference summary.
- CO: edit distance measured with normalized Damerau-Levenshtein Distance (DLD) between the sequences of relations extracted from the generated and reference summary.
Results and Discussions
=======================
We first focus on the quality of tracking model and entity representation in Sections \[subsec:result\_tracking\] to \[subsec:result\_output\], where we use the model without writer information. We examine the effect of writer information in Section \[subsec:result\_writer\].
Saliency tracking-based model {#subsec:result_tracking}
-----------------------------
As shown in Table \[tab:result\], our model outperforms all baselines across all evaluation metrics.[^7] One of the noticeable results is that our model achieves slightly higher RG precision than the gold summary. Owing to the extractive evaluation nature, the generated summary of the precision of the relation generation could beat the gold summary performance. In fact, the template model achieves 100% precision of the relation generations.
The other is that only our model exceeds the template model regarding F1 score of the content selection and obtains the highest performance of content ordering. This imply that the tracking model encourages to select salient input records in the correct order.
![Illustrations of static entity embeddings $\boldsymbol{e}$. Players with colored letters are listed in the ranking top 100 players for the 2016-17 NBA season at <https://www.washingtonpost.com/graphics/sports/nba-top-100-players-2016/>. Only [*LeBron James*]{} is in [red]{} and the other players in top 100 are in [blue]{}. Top-ranked players have similar representations of $\boldsymbol{e}$.[]{data-label="fig:static_ent"}](figs/static.pdf){width="\linewidth"}
{width="\linewidth"}
Qualitative analysis of entity embedding {#subsec:result_entity}
----------------------------------------
Our model has the entity embedding $\bar{\boldsymbol{e}}$, which depends on the box score for each game in addition to static entity embedding $\boldsymbol{e}$. Now we analyze the difference of these two types of embeddings.
We present a two-dimensional visualizations of both embeddings produced using PCA [@pearson1901liii]. As shown in Figure \[fig:static\_ent\], which is the visualization of static entity embedding $\boldsymbol{e}$, the top-ranked players are closely located.
We also present the visualizations of dynamic entity embeddings $\bar{\boldsymbol{e}}$ in Figure \[fig:ebar\]. Although we did not carry out feature engineering specific to the NBA (e.g., whether a player scored double digits or not)[^8] for representing the dynamic entity embedding $\bar{\boldsymbol{e}}$, the embeddings of the players who performed well for each game have similar representations. In addition, the change in embeddings of the same player was observed depending on the box-scores for each game. For instance, *LeBron James* recorded a double-double in a game on April 22, 2016. For this game, his embedding is located close to the embedding of *Kevin Love*, who also scored a double-double. However, he did not participate in the game on December 26, 2016. His embedding for this game became closer to those of other players who also did not participate.
Duplicate ratios of extracted relations {#subsec:result_duplicate}
---------------------------------------
As @puduppully2019data pointed out, a generated summary may mention the same relation multiple times. Such duplicated relations are not favorable in terms of the brevity of text.
Figure \[fig:ratio\] shows the ratios of the generated summaries with duplicate mentions of relations in the development data. While the models by @wiseman2017challenges and @puduppully2019data respectively showed 36.0% and 15.8% as duplicate ratios, our model exhibited 4.2%. This suggests that our model dramatically suppressed generation of redundant relations. We speculate that the tracking model successfully memorized which input records have been selected in $\boldsymbol{h}_s^{\textsc{Ent}}$.
![Ratios of generated summaries with duplicate mention of relations. Each label represents number of duplicated relations within each document. While @wiseman2017challenges’s model exhibited 36.0% duplication and @puduppully2019data’s model exhibited 15.8%, our model exhibited only 4.2%.[]{data-label="fig:ratio"}](figs/relations.pdf){width="\linewidth"}
Qualitative analysis of output examples {#subsec:result_output}
---------------------------------------
Figure \[fig:example\] shows the generated examples from validation inputs with @puduppully2019data’s model and our model. Whereas both generations seem to be fluent, the summary of @puduppully2019data’s model includes erroneous relations colored in [ orange]{}.
Specifically, the description about <span style="font-variant:small-caps;">Derrick Rose</span>’s relations, “15 points, four assists, three rounds and one steal in 33 minutes.”, is also used for other entities (e.g., <span style="font-variant:small-caps;">John Henson</span> and <span style="font-variant:small-caps;">Willy Hernagomez</span>). This is because @puduppully2019data’s model has no tracking module unlike our model, which mitigates redundant references and therefore rarely contains erroneous relations.
However, when complicated expressions such as parallel structures are used our model also generates erroneous relations as illustrated by the underlined sentences describing the two players who scored the same points. For example, “11-point efforts” is correct for <span style="font-variant:small-caps;">Courtney Lee</span> but not for <span style="font-variant:small-caps;">Derrick Rose</span>. As a future study, it is necessary to develop a method that can handle such complicated relations.
Use of writer information {#subsec:result_writer}
-------------------------
-------------------------------------------------------- ------- ----------- -------------------------------------------------- ----------- ----------- ----------- -----------
<span style="font-variant:small-caps;">CO</span>
\# P% P% R% F1% DLD%
@puduppully2019data 33.06 83.17 33.06 43.59 37.60 16.97 13.96
+ $\boldsymbol{w}$ in stage 1 28.43 **84.75** **45.00** **49.73** **47.25** 22.16 18.18
+ $\boldsymbol{w}$ in stage 2 35.06 80.51 31.10 45.28 36.87 16.38 17.81
+ $\boldsymbol{w}$ in stage 1 & 2 28.00 82.27 44.37 48.71 46.44 **22.41** **18.90**
<span style="font-variant:small-caps;">Proposed</span> 39.05 **94.38** 35.77 52.05 42.40 19.38 16.15
+ $\boldsymbol{w}$ 30.25 92.00 **50.75** **59.03** **54.58** **25.75** **20.84**
-------------------------------------------------------- ------- ----------- -------------------------------------------------- ----------- ----------- ----------- -----------
[0.446]{}
;
[0.5]{}
;
We first look at the results of an extension of @puduppully2019data’s model with writer information $\boldsymbol{w}$ in Table \[tab:author\]. By adding $\boldsymbol{w}$ to content planning (stage 1), the method obtained improvements in CS (37.60 to 47.25), CO (16.97 to 22.16), and BLEU score (13.96 to 18.18). By adding $\boldsymbol{w}$ to the component for surface realization (stage 2), the method obtained an improvement in BLEU score (13.96 to 17.81), while the effects on the other metrics were not very significant. By adding $\boldsymbol{w}$ to both stages, the method scored the highest BLEU, while the other metrics were not very different from those obtained by adding $\boldsymbol{w}$ to stage 1. This result suggests that writer information contributes to both content planning and surface realization when it is properly used, and improvements of content planning lead to much better performance in surface realization.
Our model showed improvements in most metrics and showed the best performance by incorporating writer information $\boldsymbol{w}$. As discussed in Section \[subsec:writer\], $\boldsymbol{w}$ is supposed to affect both content planning and surface realization. Our experimental result is consistent with the discussion.
Conclusion
==========
In this research, we proposed a new data-to-text model that produces a summary text while tracking the salient information that imitates a human-writing process. As a result, our model outperformed the existing models in all evaluation measures. We also explored the effects of incorporating writer information to data-to-text models. With writer information, our model successfully generated highest quality summaries that scored 20.84 points of <span style="font-variant:small-caps;">BLEU</span> score.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the anonymous reviewers for their helpful suggestions. This paper is based on results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO), JST PRESTO (Grant Number JPMJPR1655), and AIST-Tokyo Tech Real World Big-Data Computation Open Innovation Laboratory (RWBC-OIL).
Algorithm {#sec:algorithm}
=========
The generation process of our model is shown in Algorithm \[alg\]. For a concise description, we omit the condition for each probability notation. $<$<span style="font-variant:small-caps;">SoD</span>$>$ and $<$<span style="font-variant:small-caps;">EoD</span>$>$ represent “start of the document” and “end of the document”, respectively.
Initialize $\{\boldsymbol{r}_{e, a, v}\}_{r \in \boldsymbol{x}}$, $\{\bar{\boldsymbol{e}}\}_{e\in\mathcal{E}}$, $\boldsymbol{h}_0^{\textsc{LM}}$, $\boldsymbol{h}_0^{\textsc{Ent}}$\
$t \leftarrow 0$\
$e_t, y_t \leftarrow \textsc{None}, <\textsc{SoD}>$\
$y_{1:t-1}$;
Experimental settings {#sec:settings}
=====================
We set the dimensions of the embeddings to 128, and those of the hidden state of RNN to 512 and all of parameters are initialized with the Xavier initialization [@glorot2010understanding]. We set the maximum number of epochs to 30, and choose the model with the highest <span style="font-variant:small-caps;">Bleu</span> score on the development data. The initial learning rate is 2e-3 and AMSGrad is also used for automatically adjusting the learning rate [@reddi2018convergence]. Our implementation uses DyNet [@neubig2017dynet].
[^1]: We use ‘data record’ and ‘relation’ interchangeably.
[^2]: For information about the dataset, please follow this link: <https://github.com/aistairc/rotowire-modified>
[^3]: We also concatenate the embedding vectors that represents whether the entity is in home or away team.
[^4]: In our initial experiment, we observed a word repetition problem when the tracking model is not updated during generating each sentence. To avoid this problem, we also update the tracking model with special trainable vectors $\boldsymbol{v}_{\textsc{REFRESH}}$ to refresh these states after our model generates a period: $\boldsymbol{h}_t^{\textsc{Ent}} = \textsc{Gru}^{A}(\boldsymbol{v}_{\textsc{Refresh}}, \boldsymbol{h}_t^{\textsc{Ent}})$
[^5]: Our code is available from <https://github.com/aistairc/sports-reporter>
[^6]: The model for extracting relation tuples was trained on tuples made from the entity (e.g., team name, city name, player name) and attribute value (e.g., “Lakers”, “92”) extracted from the summaries, and the corresponding attributes (e.g., “<span style="font-variant:small-caps;">Team Name</span>”, “<span style="font-variant:small-caps;">Pts</span>”) found in the box- or line-score. The precision and the recall of this extraction model are respectively 93.4% and 75.0% in the test data.
[^7]: The scores of @puduppully2019data’s model significantly dropped from what they reported, especially on BLEU metric. We speculate this is mainly due to the reduced amount of our training data (Section \[section:data\]). That is, their model might be more data-hungry than other models.
[^8]: In the NBA, a player who accumulates a *double*-digit score in one of five categories (points, rebounds, assists, steals, and blocked shots) in a game, is regarded as a good player. If a player had a double in two of those five categories, it is referred to as *double-double*.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We call a group $G$ nilpotently Jordan of class at most $c$ $(c\in\mathbb{N})$ if there exists a constant $J\in\mathbb{Z}^+$ such that every finite subgroup $H\leqq G$ contains a nilpotent subgroup $K\leqq H$ of class at most $c$ and index at most $J$.\
We show that the birational automorphism group of a $d$ dimensional variety over a field of characteristic zero is nilpotently Jordan of class at most $d$.
address: |
Rényi Alfréd Matematikai Kutatóintézet\
Reáltanoda utca 13-15.\
Budapest, H1053\
Hungary
author:
- Attila Guld
title: 'Finite subgroups of the birational automorphism group are ‘almost’ nilpotent'
---
[^1]
Introduction
============
\[nilpJord\] A group $G$ is called Jordan, solvably Jordan or nilpotently Jordan of class at most $c$ ($c\in\mathbb{N}$) if there exists a constant $J=J(G)\in\mathbb{Z}^+$, only depending on $G$, such that every finite subgroup $H\leqq G$ has a subgroup $K\leqq H$ such that $|H:K|\leqq J$ and $K$ is Abelian, solvable or nilpotent of class at most $c$, respectively.
The notion of Jordan groups and solvably Jordan groups was introduced by V. L. Popov (Definition 2.1 in [@Po11]) and Yu. Prokhorov and C. Shramov (Definition 8.1 in [@PS14]), respectively.
\[main\] The birational automorphism group of a $d$ dimensional variety over a field of characteristic zero is nilpotently Jordan of class at most $d$.
\[C\] It is enough to prove the theorem over the field of the complex numbers. Indeed, let $K$ be a field of characteristic zero and $X$ be a variety over $K$. We can fix a finitely generated field extension $L_0|\mathbb{Q}$ and an $L_0$-variety $X_0$ such that $X\cong X_0\times_{L_0}\operatorname{Spec}K$. Fix a field embedding $L_0\hookrightarrow\mathbb{C}$ and let $X^*\cong X_0\times_{L_0}\operatorname{Spec}\mathbb{C}$. For an arbitrary finite subgroup $G\leqq\operatorname{Bir}(X)$ we can find a finitely generated field extension $L_1|L_0$ such that the elements of $G$ can be defined as birational transformations over the field $L_1$. Hence $G\leqq \operatorname{Bir}(X_1)$, where $X_1\cong X_0\times_{L_0}\operatorname{Spec}L_1$. We can extend the fixed field embedding $L_0\hookrightarrow\mathbb{C}$ to a field embedding $L_1\hookrightarrow\mathbb{C}$. Therefore $X^*\cong X_0\times_{L_0}\operatorname{Spec}\mathbb{C}\cong X_1\times_{L_1}\operatorname{Spec}\mathbb{C}$, and we can embed $G$ to the birational automorphism group of the complex variety $X^*$. As the birational class of the complex variety $X^*$ only depends on the birational class of the variety $X$, it is enough to examine complex varieties.
In the following discussion we shortly sketch the history of Jordan type properties in birational geometry over fields of *characteristic zero*. Research about investigating the Jordan property of the birational automorphism group of a variety was initiated by J.-P. Serre ([@Se09]) and V. L. Popov ([@Po11]). In [@Se09] J.-P. Serre settled the problem for the Cremona group of rank two (by showing that it enjoys the Jordan property), while in the articles [@Po11], [@Za15] V. L. Popov and Yu. G. Zarhin solved the question for one and two dimensional varieties. They found that the birational automorphism group of a curve or a surface is Jordan, save when the variety is birational to a direct product of an elliptic curve and the projective line. This later case was examined in [@Za15], where -based on calculations of D. Mumford- the author was able to conclude that the birational automorphism group contains Heisenberg $p$-groups for arbitrarily large prime numbers $p$. Hence it does not enjoy the Jordan property.\
In [@PS14] and [@PS16] Yu. Prokhorov and C. Shramov made important contributions to the subject using the arsenal of the Minimal Model Program and assuming the Borisov-Alexeev-Borisov (BAB) conjecture (which has later been verified in the celebrated article [@Bi16] of C. Birkar; for a survey paper on the work of C. Birkar and its connection to the Jordan property, the interested reader can consult with [@Ke19]). Amongst many highly interesting results, Yu. Prokhorov and C. Shramov proved that the birational automorphism group of a rationally connected variety and the birational automorphism group of a non-uniruled variety is Jordan. To answer a question of D. Allcock, they also introduced the concept of solvably Jordan groups, and showed that the birational automorphism group of an arbitrary variety is solvably Jordan.\
The landscape is strikingly similar in differential geometry. The techniques are fairly different, still the results converge to similar directions. In the following we briefly review the history of the question of Jordan type properties of diffeomorphism groups of smooth compact real manifolds. (We note that there are many other interesting setups which were considered by differential geometers; for a very detailed account see the Introduction of [@MR18].) As mentioned in [@MR18], during the mid-nineties É. Ghys conjectured that the diffeomorphism group of a smooth compact real manifold is Jordan, and he proposed this problem in many of his talks ([@Gh97]). The case of surfaces follows from the Riemann-Hurwitz formula (see [@MR10]), the case of 3-folds are more involved. In [@Zi14] B. P. Zimmermann proved the conjecture for them using the geometrization of compact 3-folds (which follows from the work of W. P. Thurston and G. Perelman). I. Mundet i Riera also verified the conjecture for several interesting cases, like tori, projective spaces, homology spheres and manifolds with non-zero Euler characteristic ([@MR10],[@MR16], [@MR18]).\
However, in 2014, B. Csikós, L. Pyber and E. Szabó found a counterexample ([@CPS14]). Their construction was remarkably analogous to the one of Yu. G. Zarhin. They showed that if the manifold $M$ is diffeomorphic to the direct product of the two-sphere and the two-torus or to the total space of any other smooth orientable two-sphere bundle over the two-torus, then the diffeomorphism group contains Heisenberg $p$-groups for arbitrary large prime numbers $p$. Hence $\operatorname{Diff}(M)$ cannot be Jordan. As a consequence, É. Ghys improved on his previous conjecture, and proposed the problem of showing that the diffeomorphism group of a compact real manifold is nilpotently Jordan ([@Gh15]). As the first trace of evidence, I. Mundet i Riera and C. Saéz-Calvo showed that the diffeomorphism group of a 4-fold is nilpotently Jordan of class at most 2 ([@MRSC19]). Their proof uses results from the classification theorem of finite simple groups.\
Motivated by these antecedents, in this article we investigate the nilpotently Jordan property for birational automorphism groups of varieties.\
The idea of the proof stems from the following picture. Let $X$ be a $d$ dimensional complex variety. We can assume that $X$ is smooth and projective. Let $G\leqq\operatorname{Bir}(X)$ be an arbitrary finite subgroup. Consider the MRC (maximally rationally connected) fibration $\phi:X\dashrightarrow Z$ (Theorem \[MRC\]). Because of the functoriality of the MRC fibration, a birational $G$-action is induced on $Z$, making $\phi$ $G$-equivariant. After a smooth regularization (Lemma \[reg\]) we can assume that both $X$ and $Z$ are smooth and projective, $G$ acts on them by regular automorphisms and $\phi$ is a $G$-equivariant morphism. Since the general fibres of $\phi$ are rationally connected, we can run a $G$-equivariant relative Minimal Model Program over $Z$ on $X$ (Theorem \[MMP\]). It results a $G$-equivariant Mori fibre space $\varrho:W\to Y$ over $Z$. $$\xymatrix{
X\ar@ {-->} [r]^{\cong} \ar[rd]_{\phi} & W \ar[r]^\varrho \ar[d] & Y \ar[ld]^\psi\\
& Z
}$$ We can understand the $G$-action on $X$ by analyzing the $G$-actions on $\psi:Y\to Z$ and on $\varrho:W\to Y$. We will apply induction on the relative dimension $e=\dim X-\dim Z$ to achieve this (Theorem \[AlmostMain\]). Actually, we will prove a slightly stronger theorem then Theorem \[main\] and will show that $\operatorname{Bir}(X)$ is nilpotently Jordan of class at most $(e+1)$. The base of the induction is when $e=0$. Then $X$ is non-uniruled and a theorem of Yu. Prokhorov and C. Shramov (Theorem 1.8 in [@PS14]) shows us that the birational automorphism group of $X$ is Jordan.\
Otherwise, the inductive hypothesis will show us that $H=\operatorname{Im}(G\to\operatorname{Aut}_{\mathbb{C}}(Y))$ has a bounded index nilpotent subgroup of class at most $e$. To perform the inductive step, we will take a closer look at the $G$-action on the generic fibre $W_\eta\to\operatorname{Spec}K(Y)$. We will use two key ingredients. The first one is based on the boundedness of Fano varieties, and will allow us to embed $G$ into the semilinear group $\operatorname{GL}(n, K(Y))\rtimes\operatorname{Aut}_{\mathbb{C}}(K(Y))$, where $n$ is bounded in terms of $e$ (Proposition \[Fano\]). The second one is a Jordan type theorem on certain finite subgroups of a semilinear group (Theorem \[groupmain\]). Putting these together will finish the proof.\
The article is organized in the following way. In Section \[P\] we recall the definition and some basic facts about nilpotent groups, we also recall the concept of the MRC fibration. In Section \[FGV\] we collect results about finite birational group actions on varieties. In particular, it contains the theorem of Yu. Prokhorov and C. Shramov about the Jordan property of the birational automorphism group of non-uniruled and rationally connected varieties (Theorem \[nu\]), the regularization lemma (Lemma \[reg\]), the theorem on the $G$-equivariant MMP (Theorem \[MMP\]) and the proposition about certain finite group actions on Fano varieties (Proposition \[Fano\]). At the end of the section we investigate some questions about bounds on the number of generators of finite subgroups of the birational automorphism group. The boundedness of the generating set helps us to give a more accurate bound on the nilpotency class (Remark \[NoB\]). Section \[gp\] deals with the proof of the Jordan type theorem on semilinear groups (Theorem \[groupmain\]). Finally, in Section \[PMT\] we prove our main theorem.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author is very grateful to E. Szabó for many helpful discussions.
Preliminaries {#P}
=============
Nilpotent groups
----------------
We recall the definition of nilpotent groups and some of their basic properties.
Let $G$ be a group. Let $\operatorname{Z}_0(G)=1$ and define $\operatorname{Z}_{i+1}(G)$ as the preimage of $\operatorname{Z}(G/\operatorname{Z}_i(G))$ under the natural quotient group homomorphism $G\to G/\operatorname{Z}_i(G)$ $(i\in\mathbb{N})$. The series of groups $1=\operatorname{Z}_0(G)\leqq\operatorname{Z}_1(G)\leqq\operatorname{Z}_2(G)\leqq...$ is called the upper central series of $G$.\
Let $\gamma_0(G)=G$ and let $\gamma_{i+1}(G)=[\gamma_i(G),G]$ ($i\in\mathbb{N}$, and $[,]$ denotes the commutator operation). The series of groups $G=\gamma_0(G)\geqq\gamma_1(G)\geqq\gamma_2(G)\geqq...$ is called the lower central series of $G$.\
$G$ is called nilpotent if one (hence both) of the following equivalent conditions hold:
- There exists $n\in\mathbb{N}$ such that $\operatorname{Z}_n(G)=G$.
- There exists $n\in\mathbb{N}$ such that $\gamma_n(G)=1$.
If $G$ is a nontrivial nilpotent group, then there exists a natural number $c$ for which $\operatorname{Z}_c(G)=G$, $\operatorname{Z}_{c-1}(G)\neq G$ and $\gamma_c(G)=1$, $\gamma_{c-1}(G)\neq 1$ holds. $c$ is called the nilpotency class of $G$. (If $G$ is trivial, then its nilpotency class is zero.)
Note that $\operatorname{Z}_1(G)$ is the centre of the group $G$, while $\gamma_1(G)$ is the commutator subgroup. A non-trivial group $G$ is nilpotent of class one if and only if it is Abelian.\
Nilpotency is the property between the Abelian and the solvable properties. The Abelian property implies nilpotency, while nilpotency implies solvability.
The following proposition describes one of the key features of nilpotent groups. They can be built up by successive central extensions.
\[CE\] Let $G$ be a group and $A\leqq\operatorname{Z}(G)$ be a central subgroup of $G$. If $G/A$ is nilpotent of class at most $c$, then $G$ is nilpotent of class at most $(c+1)$.
We will use also the two properties below about nilpotent groups.
\[ICmap\] Let $G$ be a nilpotent group of class at most $n$. Fix $n-1$ arbitrary elements in $G$, denote them by $g_1,g_2,...g_{n-1}$, and let $1\leqq j \leqq n$ be an arbitrary integer. The map $\varphi_j$ defined by the help of iterated commutators of length $(n-1)$ $$\begin{gathered}
\varphi_j:G\to\gamma_{n-1}(G)\\
g\mapsto [[...[[[[...[[g_1,g_2],g_3]...],g_{j-1}],g],g_j]...],g_{n-1}]\end{gathered}$$ gives a group homomorphism.
\[IC\] Let $G$ be a group. $G$ is nilpotent of class at most $n$ if and only if $\forall g_1,g_2,...,g_{n+1}\in G$: $[[...[[g_1,g_2],g_3]...],g_{n+1}]=1$.
Typical examples of nilpotent groups are finite $p$-groups (where $p$ is a prime number). If we restrict our attention to finite nilpotent groups, even more can be said. (Recall that a $p$-Sylow subgroup of a finite group is the largest $p$-group contained in the group.) A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups (Theorem 6.12 in [@CR62]).
The maximally rationally connected fibration
--------------------------------------------
We recall the concept of the maximally rationally connected fibration. For a detailed treatment see Chapter $4$ of [@Ko96], for the non-uniruledness of the basis see Corollary 1.4 in [@GHS03].
\[MRC\] Let $X$ be a smooth proper complex variety. The pair $(Z,\phi)$ is called the maximally rationally connected (MRC) fibration if
- $Z$ is a complex variety,
- $\phi:X\dashrightarrow Z$ is a dominant rational map,
- there exist open subvarieties $X_0$ of $X$ and $Z_0$ of $Z$ such that $\phi$ descends to a proper morphism between them $\phi_0:X_0\to Z_0$ with rationally connected fibres,
- if $(W,\psi)$ is another pair satisfying the three properties above, then $\phi$ can be factorized through $\psi$. More precisely, there exists a rational map $\tau: W\dashrightarrow Z$ such that $\phi=\tau\circ\psi$.
The MRC fibration exists and is unique up to birational equivalence. Moreover the basis $Z$ is non-uniruled.
Finite group actions on varieties {#FGV}
=================================
In this section we introduce techniques which help us to solve partial cases of our problem and help us to build up the full solution from the special cases.\
Jordan property
---------------
Yu. Prokhorov and C. Shramov proved the following theorem (Theorem 1.8 in [@PS14] and Theorem 1.8 in [@PS16]). It will serve us as a starting point of an inductive argument in the proof of our main theorem and will be an important ingredient when we look for bounds on the number of generators of finite subgroups of the birational automorphism group (Theorem \[bfsg\]).
\[nu\] Let $X$ be variety over a field of characteristic zero. Assume that $X$ is either non-uniruled or rationally connected. Then the birational automorphism group of $X$ is Jordan (in other words, it is nilpotently Jordan of class at most 1).
Smooth regularization
---------------------
The next lemma is a slight extension of the well-known (smooth) regularization of finite group actions on varieties (Lemma-Definition 3.1. in[@PS14]).
\[reg\] Let $X$ and $Z$ be complex varieties and $\phi:X\dashrightarrow Z$ be a dominant rational map between them. Let $G$ be a finite group which acts by birational automorphisms on $X$ and $Z$ in such a way that $\phi$ is $G$-equivariant. There exist smooth projective varieties $X^*$ and $Z^*$ with regular $G$-actions on them and a $G$-equivariant projective morphism $\phi^*: X^*\to Z^*$ such that $X^*$ is $G$-equivariantly birational to $X$, $Z^*$ is $G$-equivariantly birational to $Z$ and $\phi^*$ is $G$-equivariantly birational to $\phi$. In other words, we have a $G$-equivariant commutative diagram. $$\xymatrix{
X \ar@{-->}[r]^\cong \ar@{-->}[d]^\phi & X^* \ar[d]^{\phi^*}\\
Z \ar@{-->}[r]^\cong & Z^*
}$$
Let $K(Z)\leqq K(X)$ be the field extension corresponding to the function fields of $Z$ and $X$, induced by $\phi$. Take the induced $G$-action on this field extension and let $K(Z)^G\leqq K(X)^G$ be the field extension of the $G$-invariant elements. Consider a projective model of it, i.e. let $\varrho_1: X_1\to Z_1$ be a (projective) morphism, where $X_1$ and $Z_1$ are projective varieties such that $K(X_1)\cong K(X)^G$ and $K(Z_1)\cong K(Z)^G$, and $\varrho_1: X_1\to Z_1$ induces the field extension $K(Z_1)\cong K(Z)^G\leqq K(X)^G\cong K(X_1)$. By normalizing $X_1$ in the function field $K(X)$ and $Z_1$ in the function field $K(Z)$ we get projective varieties $X_2$ and $Z_2$, moreover $\varrho_1$ induces a $G$-equivariant morphism $\varrho_2:X_2\to Z_2$ between them.\
As the next step, we can take a $G$-equivariant resolution of singularities $\widetilde{Z_2}\to Z_2$. After replacing $Z_2$ by $\widetilde{Z_2}$ and $X_2$ by the irreducible component of $X_2\times_{Z_2}\widetilde{Z_2}$ which dominates $\widetilde{Z_2}$, we can assume that $Z_2$ is smooth. Hence $G$-equivarianlty resolving the singularities of $X_2$ finishes the proof.
Minimal Model Program and boundedness of Fano varieties
-------------------------------------------------------
Applying the results of the famous article by C. Birkar, P. Cascini, C. D. Hacon and J. McKernan ([@BCHM10]) enables us to use the arsenal of the Minimal Model Program. As a consequence, we can examine rationally connected varieties (fibres) with the help of Fano varieties (fibres). For the later we can use boundedness results because of yet another famous theorem by C. Birkar ([@Bi16]). (This theorem was previously known as the BAB Conjecture).
\[MMP\] Let $X$ and $Z$ be smooth projective complex varieties such that $\dim Z<\dim X$. Let $\phi:X\to Z$ be a dominant morphism between them with rationally connected general fibres. Let $G$ be a finite group which acts by regular automorphisms on $X$ and $Z$ in such a way that $\phi$ is $G$-equivariant. We can run a $G$-equivariant Minimal Model Program (MMP) on $X$ relative to $Z$ which results a Mori fibre space. In particular, the Minimal Model Program gives a $G$-equivariant commutative diagram $$\xymatrix{
X\ar@ {-->} [r]^{\cong} \ar[rd]^{\phi} & W \ar[r] \ar[d] & Y \ar[ld]\\
& Z
}$$ where $W$ is $G$-equivariantly birational to $X$, $\dim Y< \dim X$ and the generic fibre of the morphism between $W$ and $Y$ is a Fano variety with (at worst) terminal singularities.
By Corollary 1.3.3 of [@BCHM10], we can run a relative MMP on $\phi:X\to Z$ (which results a Mori fibre space) if the canonical divisor of $X$ is not $\phi$-pseudo-effective. It can be done equivariantly if we have finite group actions. (See Section 2.2 in [@KM98] and Section 4 of [@PS14] for further discussions on the topic.) So, it remains to show that the canonical divisor of $X$ is not $\phi$-pseudo-effective.\
By generic smoothness, a general fibre of $\phi$ is a smooth rationally connected projective complex variety. Therefore if $x$ is a general closed point of a general fibre $F$, then there exists a free rational curve $C_x$ running through $x$, lying entirely in the fibre $F$ (Theorem 1.9 of Chapter 4 in [@Ko96]). Since $C_x$ is a free rational curve, $C_x.K_X\leqq-2$. Since the inequality holds for every general closed point of every general fibre, $K_X$ cannot be $\phi$-pseudo-effective.
The lemmas and the theorems above open the door for us to use induction on the relative dimension of the MRC fibration while proving Theorem \[main\]. So we only need to deal with Fano varieties of bounded dimensions.
\[Fano\] Let $e$ be a natural number. There exists a constant $n=n(e)\in\mathbb{N}$, only depending on $e$, with the following property. If
- $K$ is a field of characteristic zero,
- $F$ is a Fano variety over $K$ of dimension at most $e$, with terminal singula/-rities,
- $G$ is a finite group which acts faithfully on $F$ by regular automorphisms of the $\mathbb{Q}$-scheme $F$, and acts on $\operatorname{Spec}K$ by regular automorphisms of the $\mathbb{Q}$-scheme $\operatorname{Spec}K$, in such a way that the structure morphism $F\to\operatorname{Spec}K$ is $G$-equivariant,
then $G$ can be embedded into the semilinear group $\operatorname{\Gamma L}(n, K)\cong \operatorname{GL}(n, K)\rtimes \operatorname{Aut}K$ in such a way that $G\hookrightarrow \operatorname{\Gamma L}(n,K)\twoheadrightarrow\operatorname{Aut}K$ corresponds to the $G$-action on $\operatorname{Spec}K$.
Fix $K$, $F$ and $G$ with the properties described by the theorem. There exists a finitely generated field extension $L_0|\mathbb{Q}$ and a Fano variety $F_0$ over $L_0$ such that $F\cong F_0\times_{L_0}\operatorname{Spec}K$. Consider an embedding of fields $L_0\hookrightarrow\mathbb{C}$, and let $F_1\cong F_0\times_{L_0}\operatorname{Spec}\mathbb{C}$. Since complex Fano varieties with terminal singularities of bounded dimension form a bounded family (Theorem1.1 in[@Bi16]), there exist constants $P=P(e),M=M(e)\in\mathbb{N}$, only depending on $e$, such that $P$-th power of the anticanonical divisor embeds $F_1$ to the $M_1$-dimensional complex projective space, where $M_1\leqq M$. Since the $P$-th power of the anticanonical divisor is defined over any field, this embedding is defined over any field, in particularly over $K$. So we have a closed embedding of the form $F\hookrightarrow \mathbb{P}_K^{M_1}\cong\mathbb{P}(\operatorname{H^0}(X,-K_F^P)^*)$.\
By the functorial property of a (fixed) power of the anticanonical divisor, an equivariant $G$-action is induced on the commutative diagram below. $$\xymatrix{
F\ar@{^{(}->}[r] \ar[d] & \mathbb{P}(\operatorname{H^0}(X,-K_F^P)^*) \ar[ld] \\
\operatorname{Spec}K
}$$ Since $F\hookrightarrow\mathbb{P}(\operatorname{H^0}(X,-K_F^P)^*)$ is a closed embedding, the semilinear action of $G$ on the vector space $\operatorname{H^0}(X,-K_F^P)$ is faithful. Hence $G$ embeds to $\operatorname{\Gamma L}(\operatorname{H^0}(X,-K_F^P))$. Clearly $G\to \operatorname{Aut}K$ corresponds to the $G$-action on $\operatorname{Spec}K$. As $\dim\operatorname{H^0}(X,-K_F^P)\leqq M(e)+1$, we finished the proof.
Bound on the number of generating elements of finite subgroups of the birational automorphism groups
----------------------------------------------------------------------------------------------------
Now we turn our attention on finding bounds on the number of generating elements of finite subgroups of the birational automorphism group of varieties. It will be important for as when we will investigate commutator relations (Lemma \[DN\]), and it will be crucial to have a bound on the number of the elements of a generating set of the group.\
The next theorem and its proof are essentially due to Y. Prokhorov and C. Shramov. (We use the world essentially as they only considered the case of finite Abelian subgroups (Remark 6.9 of [@PS14]).) It is also important to note that the proof of Remark 6.9 of [@PS14] uses the result of C. Birkar about the boundedness of Fano varieties (Theorem 1.1 in [@Bi16]).
\[bfsg\] Let $X$ be a variety over a field of characteristic zero. There exists a constant $m=m(X)\in\mathbb{Z}^+$, only depending on the birational class of $X$, such that if $G\leqq\operatorname{Bir}(X)$ is an arbitrary finite subgroup of the birational automorphism group, then $G$ can be generated by $m$ elements.
First we show the theorem in the special cases when $X$ is either non-uniruled or rationally connected. By Remark 6.9 of [@PS14] and Theorem 1.1 of [@Bi16], there exists a constant $m=m(X)\in\mathbb{Z}^+$, only depending on the birational class of $X$, such that if $A\leqq\operatorname{Bir}(X)$ is an arbitrary finite Abelian subgroup of the birational automorphism group, then $A$ can be generated by $m$ elements. Since $\operatorname{Bir}(X)$ is Jordan when $X$ is non-uniruled or rationally connected (Theorem \[nu\]), the result on the finite Abelian groups implies the claim of the theorem in both of these special cases.\
Now let $X$ be arbitrary. Arguing as in Remark \[C\] we can assume that $X$ is a complex variety. Consider the MRC fibration $\phi:X\dashrightarrow Z$. By Lemma \[reg\] we can assume that both $X$ and $Z$ are smooth projective varieties, and $G$ acts on them by regular automorphisms. Let $\rho$ be the generic point of $Z$, and let $X_\rho$ be the generic fibre of $\phi$. $X_\rho$ is a rationally connected variety over the function field $k(Z)$.\
Let $G_\rho\leqq G$ be the maximal subgroup of $G$ acting fibrewise. $G_\rho$ has a natural faithful action on $X_\rho$, while $G/G_\rho=G_Z$ has a natural faithful action on $Z$. This gives a short exact sequence of groups $$1\to G_\rho\to G\to G_Z\to 1.$$ By the rationally connected case there exists a constant $m_1(X_{\rho})$, only depending on the birational class of $X_{\rho}$, such that $G_\rho$ can be generated by $m_1(X_{\rho})$ elements. By the non-uniruled case there exists a constant $m_2(Z)$, only depending on the birational class of $Z$, such that $G_Z$ can be generated by $m_2(Z)$ elements. So $G$ can be generated by $m(X_{\rho}, Z)=m_1(X_{\rho})+m_2(Z)$ elements. Since $m(X_{\rho},Z)$ only depends on the birational classes of $X_{\rho}$ and $Z$, and both of the birational classes of $X_{\rho}$ and $Z$ only depend on the birational class of $X$, this finishes the proof.
In case of rationally connected varieties we will use a slightly stronger version of the theorem. To prove it, we need a theorem about fixed points of rationally connected varieties. It is due to Yu. Prokhorov and C. Shramov (Theorem 4.2 of [@PS14]).
\[afp\] Let $e$ be a natural number. There exits a constant $R=R(e)\in\mathbb{Z}^+$, only depending on $e$, with the following property. If $X$ is a rationally connected complex projective variety of dimension at most $e$, and $G\leqq\operatorname{Aut}(X)$ is an arbitrary finite subgroup of its automorphism group, then there exists a subgroup $H\leqq G\leqq \operatorname{Aut}(X)$ such that $H$ has a fixed point in $X$, and the index of $H$ in $G$ is bounded by $R$.
\[bgrc\] Let $e$ be a natural number. There exits a constant $m=m(e)\in\mathbb{Z}^+$, only depending on $e$, with the following property. If $K$ is an arbitrary field of characteristic zero, $X$ is a rationally connected variety over $K$ of dimension at most $e$, and $G\leqq\operatorname{Bir}(X)$ is an arbitrary finite subgroup of the birational automorphism group, then $G$ can be generated by $m$ elements.
Fix $K$, $X$ and $G$ with the properties described by the theorem. Arguing as in the case of Remark \[C\], we can assume that $K$ is the field of the complex numbers.\
Using Lemma \[reg\], we can assume that $X$ is smooth and projective and $G$ is a finite subgroup of the biregular automorphism group $\operatorname{Aut}(X)$.\
By Theorem \[afp\], we can assume that $G$ has a fixed point in $X$. Denote it by $P$.\
By Lemma 4 of [@Po14] $G$ acts faithfully on the tangent space of the fixed point $P$. So $G$ can be embedded to $\operatorname{GL}(\operatorname{T}_PX)$, whence $G$ can be embedded to $\operatorname{GL}(e,\mathbb{C})$. Therefore the claim of the theorem follows from Lemma \[bg\]. This finishes the proof.
Calculations in the general semilinear group {#gp}
============================================
This section contains the group theoretic ingredient of the proof of the main theorem.
\[groupmain\] Let $c,n$ and $m$ be positive integers. Let $F$ be the family of those finite groups $G$ which have the following properties.
- There exists a field $K$ of characteristic zero containing all roots of unity such that $G$ is a subgroup of the semilinear group $\operatorname{\Gamma L}(n,K)\cong \operatorname{GL}(n)\rtimes \operatorname{Aut}K$.
- Every subgroup of $G$ can be generated by $m$ elements.
- The image of the composite group homomorphism $G\hookrightarrow \operatorname{\Gamma L}(n,K)\twoheadrightarrow\operatorname{Aut}K$, denoted by $\Gamma$, is nilpotent of class at most $c$ ($c \in\mathbb{N}$) and fixes all roots of unity.
There exists a constant $C=C(c,n,m)\in\mathbb{Z}^+$, only depending on $c,n$ and $m$, such that every finite group $G$ belonging to $F$ contains a nilpotent subgroup $H\leqq G$ with nilpotency class at most $(c+1)$ and with index at most $C$.
First, we recall a slightly strengthened version of Jordan’s theorem.
\[Jor\] Let $n$ be a positive integer. There exists a constant $J=J(n)\in\mathbb{Z}^+$, only depending on $n$, such that if a finite group $G$ is a subgroup of a general linear group $\operatorname{GL}(n, K)$, where $K$ is a field of characteristic zero, then $G$ contains a characteristic Abelian subgroup $A\leqq G$ of index at most $J$.
The only claim of the above theorem which does not follow immediately from Theorem 2.3 in [@Br11] is that we require the Abelian subgroup of bounded index $A\leqq G$ to be characteristic (i.e. invariant under all automorphisms of $G$) instead of being normal (i.e. invariant under the inner automorphisms of $G$). In the following we will prove some lemmas which help us to deduce the above variant of the theorem from the one which can be found in [@Br11].
\[bg\] Let $n$ be a positive integer. There exists a constant $r=r(n)\in\mathbb{Z}^+$, only depending on $n$, such that if a finite group $G$ is a subgroup of a general linear group $\operatorname{GL}(n, K)$, where $K$ is a field of characteristic zero, then $G$ can be generated by $r$ elements.
It is enough to prove the lemma when $K$ is algebraically closed, so we can assume it. By Theorem 2.3 in [@Br11], $G$ contains a diagonalizable subgroup of bounded index. Since finite diagonal groups of $\operatorname{GL}(n,K)$ can be generated by $n$ elements, the lemma follows.
\[ind\] Let $J$ and $r$ be positive integers. There exists a constant $L=L(J,r)\in\mathbb{N}$, only depending on $r$ and $J$, such that if $G$ is a finite group which can be generated by $r$ elements, then $G$ has at most $L$ many subgroups of index $J$.
Fix an arbitrary finite group $G$ which can be generated by $r$ elements. We can construct an injective map of sets from the set of index $J$ subgroups of $G$ to the set of group homomorphisms from $G$ to the symmetric group of degree $J$. Since $G$ can be generated by $r$ elements the later set has boundedly many elements, hence the former set has boundedly many elements as well. So we only left with the task of constructing such an injective map.\
Let $S$ be a set with $J$ elements. We can identify the symmetric group of degree $J$, denoted by $\operatorname{Sym}_J$, with the symmetry group of the set $S$. Fix an arbitrary element $x\in S$. For every index $J$ subgroup $K\leqq G$, fix a bijection $\mu_K$ between the set of the left cosets of $K$ and the set $S$, subject to the following condition, $K$ is mapped to the fixed element $x$, i.e. $\mu_K(K)=x$. Let $H\leqq G$ be an arbitrary subgroup of index $J$. $G$ acts on the set of the left cosets of $H$ by left multiplication. Using the bijection $\mu_H$, this induces a group homomorphism $\phi_H: G\to \operatorname{Sym}_J$ . The constructed assignment is injective as the stabilizator subgroup of $x$ in the image group $\operatorname{Im}\phi_H$ uniquely determines $H$.
Let $K$ be an arbitrary field of characteristic zero, and let $G$ be an arbitrary finite subgroup of $\operatorname{GL}(n,K)$. By Theorem 2.3 in [@Br11] $G$ contains an Abelian subgroup $A\leqq G$ of index bounded by $J_0=J_0(n)$. Consider the set $S$ of the smallest index Abelian subgroups of $G$. By Lemma \[bg\] and Lemma \[ind\] there exists a constant $L=L(n)$, only depending on $n$, such that $S$ has at most $L$ many elements. Take the intersection of the subgroups contained in $S$, it gives a characteristic Abelian subgroup of index at most $J_0^L$.
Next we prove a lemma about nilpotent groups.
\[DN\] Let $c,J$ and $m$ be positive integers. There exists a constant $C=C(c,J,m)\in \mathbb{N}$, only depending on $c,J$ and $m$, such that if
- $G$ is a nilpotent group of class at most $(c+1)$,
- $G$ can be generated by $m$ elements,
- the cardinality of $\gamma_{c}(G)$ is at most $J$,
then $G$ has a nilpotent subgroup $H\leqq G$ of class at most $c$ whose index is bounded by $C$.
Fix a generating system $g_1,...,g_m\in G$. Consider the group homomorphisms (Proposition \[ICmap\]) $$\begin{gathered}
\varphi_{i_1,i_2,...,i_{c}}:G\to\gamma_{c}(G)\\
g\mapsto [[[...[[g_{i_1},g_{i_2}],g_{i_3}]...],g_{i_{c}}],g],\end{gathered}$$ where $1\leqq i_1,i_2,...,i_{c}\leqq m$, i.e. for every ordered length $c$ sequence of the generators we assign a group homomorphism using the iterated commutators. Let $H$ be the intersection of the kernels. $$H=\bigcap\limits_{1\leqq i_1,i_2,...,.i_{c} \leqq m} \operatorname{Ker}\varphi_{i_1,i_2,...,i_{c}}$$ Using the fact that the length $c$ iterated commutators give group homomorphisms in every variable if we fix the other variables (Proposition \[ICmap\]), one can show that all the length $c$ iterated commutators of $H$ vanish. Hence $H$ is nilpotent of class at most $c$ (Proposition \[IC\]).\
On the other hand $H$ is the intersection of $m^c$ many subgroups of index at most $|\gamma_{c}(G)|\leqq J$. Hence the index of $H$ is bounded in terms of $c,J$ and $m$. This finishes the proof.
Now we are ready to prove the main theorem of the section.
Let $K$ be an arbitrary field of characteristic zero containing all roots of unity, and let $G$ be an arbitrary finite subgroup of $\operatorname{\Gamma L}(n,K)$ belonging to $F$. Consider the short exact sequence of groups given by $$1\to N\to G\to\Gamma\to 1$$ where $N=\operatorname{GL}(n,K)\cap G$ and $\Gamma=\operatorname{Im}(G\to\operatorname{Aut}K)$. By Theorem \[Jor\], $N$ contains a characteristic Abelian subgroup of index bounded by $J=J(n)\in\mathbb{Z}^+$. Since $A$ is characteristic in $N$ and $N$ is normal in $G$, $A$ is a normal subgroup of $G$.\
Consider the natural action of $G$ on the vector space $V=K^n$. Since $A$ is a finite Abelian subgroup of $\operatorname{GL}(V)$ and the ground field $K$ contains all roots of unity, $A$ decomposes $V$ into common eigenspaces of its elements: $V=V_1\oplus V_2\oplus...\oplus V_r$ $(r\leqq n)$. As $A$ is normal in $G$, $G$ respects this decomposition, i.e. $G$ acts on the set of linear subspaces $\{V_1,V_2,...,V_r\}$ by permutations. The kernel of this group action, denoted by $G_1$, is a bounded index subgroup of $G$ (indeed $|G:G_1|\leqq r!\leqq n!$). Furthermore, $A$ is central in $G_1$, i.e. $A\leqq \operatorname{Z}(G_1)$. To see this, notice that on an arbitrary fixed eigenspace $V_i$ $(1\leqq i\leqq r)$ $A$ acts by scalar matrices in such a way that all scalars are drawn from the set of the roots of unity. Since $G_1$ leaves $V_i$ invariant by definition and $\operatorname{Im}(G_1\to\operatorname{Aut}K)$ fixes all roots of unity, our claim follows. After replacing $G$ with the bounded index subgroup $G_1$, we can assume that $A\leqq \operatorname{Z}(G)$.\
As $A$ is a central subgroup of $G$, we can consider the quotient group $\overline{G}=G/A$. By Proposition \[CE\], we only need to prove that $\overline{G}$ has a bounded index nilpotent subgroup of class at most $c$. Our strategy will be that, first we prove that $\overline{G}$ has a bounded index nilpotent subgroup of class at most $(c+1)$, then we will apply Lemma \[DN\].\
Let $\overline{N}=N/A$, and consider the short exact sequence of groups $$1\to \overline{N}\to\overline{G}\to\Gamma\to 1.$$ The number of elements of $\overline{N}$ is bounded by $J(n)$, by the definition of $A$, and $\Gamma$ is nilpotent of class at most $c$, by the definition of $G$.\
$\overline{G}$ acts on $\overline{N}$ by conjugation, and the kernel of this action is the centralizer group $\operatorname{C}_{\overline{G}}(\overline{N})=\{g\in\overline{G}|\; ng=gn\;\forall n\in\overline{N}\}$. Therefore $\overline{G}/\operatorname{C}_{\overline{G}}(\overline{N})$ embeds into the automorphism group of $\overline{N}$ which has cardinality at most $J!$. Hence $\operatorname{C}_{\overline{G}}(\overline{N})$ has bounded index in $\overline{G}$. Hence, after replacing $\overline{G}$ with $\operatorname{C}_{\overline{G}}(\overline{N})$, $\overline{N}$ with $\overline{N}\cap \operatorname{C}_{\overline{G}}(\overline{N})$ and $\Gamma$ with the image group $\operatorname{Im}(\operatorname{C}_{\overline{G}}(\overline{N})\to \Gamma)$, we can assume that $\overline{G}$ is the central extension of the Abelian group $\overline{N}$ and nilpotent group $\Gamma$ whose nilpotency class is at most $c$. Therefore we can assume that $\overline{G}$ is nilpotent of class at most $(c+1)$ (Proposition \[CE\]).\
Notice that $\gamma_c(\overline{G})$ maps to $\gamma_c(\Gamma)=1$, which implies that the former group is contained in $\overline{N}$. So $|\gamma_c(\overline{G})|\leqq|\overline{N}|\leqq J$. Hence we are in the position to apply Lemma \[DN\], which finishes the proof.
\[NoB\] In the above proof we only used the assumption that $G$ can be generated by $m$ elements via Lemma \[DN\]. So if we omit this condition from Theorem \[groupmain\], we can still prove that there exists a constant $D=D(n)\in\mathbb{Z}^+$, only depending on $n$ (not even on $c$), such that if $G$ belongs to the corresponding family of groups, then $G$ contains a nilpotent subgroup $H\leqq G$ with nilpotency class at most $(c+2)$ and with index at most $D$.
Proof of the Main Theorem {#PMT}
=========================
Using the techniques developed in the previous sections, we will prove our main theorem.
\[AlmostMain\] Fix a non-uniruled complex variety $Z_0$. Let $F_{Z_0}$ be the collection of 5-tuples $(X, Z,\phi, G, e)$, where
- $X$ is a complex variety,
- $Z$ is a complex variety, which is birational to $Z_0$,
- $\phi: X\dashrightarrow Z$ is a dominant rational map such that there exist open subvarieties $X_1$ of $X$ and $Z_1$ of $Z$ such that $\phi$ descends to a morphism between them $\phi_1:X_1\to Z_1$ with rationally connected fibres,
- $G\leqq \operatorname{Bir}(X)$ is a finite group of the birational automorphism group of $X$, which also acts by birational automorphisms on $Z$ in such a way that $\phi$ is $G$-equivariant,
- $e\in\mathbb{N}$ is the relative dimension $e=\dim X-\dim Z_0$.
Then the following claims hold.
- There exist constants $\{m_{Z_0}(e)\in\mathbb{Z}^+|\,e\in\mathbb{N}\}$, only depending on the birational class of $Z_0$, such that if the 5-tuple $(X,Z,\phi, G, e)$ belongs to $F_{Z_0}$, then $G$ can be generated by $m_{Z_0}(e)$ elements.
- There exist constants $\{J_{Z_0}(e)\in\mathbb{Z}^+|\,e\in\mathbb{N}\}$, only depending on the birational class of $Z_0$, such that if the 5-tuple $(X,Z,\phi, G, e)$ belongs to $F_{Z_0}$, then $G$ has a nilpotent subgroup $H\leqq G$ of nilpotency class at most $(e+1)$ and index at most $J_{Z_0}(e)$.
(Proof of the First Claim) Let $(X, Z,\phi, G, e)$ be an arbitrary 5-tuple belonging to $F_{Z_0}$. By Lemma \[reg\] we can assume that both $X$ and $Z$ are smooth projective varieties, and $G$ acts on them by regular automorphisms. Let $\rho$ be the generic point of $Z$, and let $X_\rho$ be the generic fibre of $\phi$. $X_\rho$ is a rationally connected variety of dimension $e$ over the function field $K(Z)$.\
Let $G_\rho\leqq G$ be the maximal subgroup of $G$ acting fibrewise. $G_\rho$ has a natural faithful action on $X_\rho$, while $G/G_\rho=G_Z$ has a natural faithful action on $Z$. This gives a short exact sequence of groups $$1\to G_\rho\to G\to G_Z\to 1.$$ By Theorem \[bgrc\] there exists a constant $m_1(e)$, only depending on $e$, such that $G_\rho$ can be generated by $m_1(e)$ elements. By Theorem \[bfsg\] there exists a constant $m_2(Z)$, only depending on the birational class of $Z$, such that $G_Z$ can be generated by $m_2(Z)$ elements. So $G$ can be generated by $m_{Z_0}(e)=m_1(e)+m_2(Z)$ elements. Since $m_{Z_0}(e)$ only depends on $e$ and the birational class of $Z_0$, this finishes the proof of the first claim.\
(Proof the Second Claim) We will apply induction on $e$. If $e=0$, then $X$ and $Z_0$ are birational, hence $G\leqq \operatorname{Bir}(Z_0)$ and the claim of the theorem follows from Theorem \[nu\]. So we can assume that $e>0$ and the claim of the theorem holds if the relative dimension is strictly smaller than $e$.\
Let $(X, Z,\phi, G, e)$ be a 5-tuple belonging to $F_{Z_0}$. After regularizing $\phi$ in the sense of Lemma \[reg\], we may assume that $X$ and $Z$ are smooth projective varieties, $G$ acts on them by regular automorphisms and $\phi$ is a $G$-equivariant (projective) morphism.\
Hence by Theorem \[MMP\], we can run a relative $G$-equivariant MMP on $\phi: X\to Z$. It results a $G$-equivariant commutative diagram $$\xymatrix{
X \ar@{-->}[r]^\cong \ar[rd]_{\phi} & W \ar[r]^{\varrho}\ar[d] & Y \ar[ld]^{\psi}\\
& Z
}$$ where $\varrho:W\to Y$ is a Mori fibre space and $\psi: Y\to Z$ is a dominant morphism with rationally connected general fibres (as so does $\phi$). Let $H$ be the image of $G\to\operatorname{Aut}_{\mathbb{C}}(Y)$, and let $f$ be the relative dimension $f=\dim Y-\dim Z$. The 5-tuple $(Y, Z,\psi, H, f)$ clearly belongs to $F_{Z_0}$. Moreover, since $f<e$, we can use the inductive hypothesis. Let $H_1\leqq H$ be the nilpotent subgroup of nilpotency class at most $(f+1)$ and index at most $J_{Z_0}(f)$. After replacing $H$ with its bounded index subgroup $H_1$ (and $G$ with the preimage of $H_1$), we can assume that $H$ is nilpotent of class at most $e$.\
Let $\eta\cong\operatorname{Spec}K(Y)$ be the generic point of $Y$, and let $W_\eta$ be the generic fibre of $\varrho$. Since $\varrho:W\to Y$ is a Mori fibre space, $W_\eta$ is a Fano variety over $K(Y)$ with (at worst) terminal singularities. Furthermore, $G$ acts on the structure morphism $W_\eta\to\operatorname{Spec}K(Y)$ equivariantly by scheme automorphisms. Hence we can apply Proposition \[Fano\], and we can embed $G$ to $\operatorname{\Gamma L}(n,K(Y))\cong\operatorname{GL}(n,K(Y))\rtimes \operatorname{Aut}K(Y)$ where $n=n(e)$ only depends on $e$ (since $\dim W_\eta\leqq e$). Moreover, the image group $\Gamma=\operatorname{Im}(G\hookrightarrow\operatorname{\Gamma L}(n,K(Y))\twoheadrightarrow \operatorname{Aut}K(Y))$ corresponds to the $G$-action on $\operatorname{Spec}K(Y)$, therefore it corresponds to the $H$-action on $Y$. Hence $\Gamma$ fixes all roots of unity, as $Y$ is a complex variety, and $\Gamma$ is nilpotent of class at most $e$, as so does $H$. Furthermore, by the first claim of the theorem, every subgroup of $G$ can be generated by $m=m_{Z_0}(e)$ elements (where $m$ only depends on $e$ and the birational class of $Z_0$). So we are in the position to apply Theorem \[groupmain\] to the group $G$, which finishes the proof.
\[NoB2\] In accordance with Remark \[NoB\], we need to consider bounds on the number of generators of finite subgroups of the birational automorphism group to give a more accurate bound on the nilpotency class.
To close our article, we prove our main theorem.
Let $X$ be a $d$ dimensional complex variety. We can assume that $X$ is smooth and projective. We can also assume that $X$ is non-uniruled by Theorem \[nu\]. Let $G\leqq\operatorname{Bir}(X)$ be an arbitrary finite subgroup of the birational automorphism group of $X$. Let $\phi: X\dashrightarrow Z$ be the MRC fibration, and let $e=\dim X-\dim Z$ be the relative dimension. By the functoriality of the MRC fibration (Theorem $5.5$ of Chapter $4$ in [@Ko96]), $G$ acts on the base $Z$ by birational automorphisms making the rational map $\phi$ $G$-equivariant. Hence the 5-tuple $(X,Z,\phi,G, e)$ belongs to the collection $F_Z$ defined in the previous theorem. Therefore $G$ has a nilpotent subgroup of class at most $(e+1)$ and index at most $J_Z(e)$. Since $e<d$ (as $X$ is non-uniruled), moreover the relative dimension $e$ and the birational class of the base $Z$ only depends on the birational class of $X$, the theorem follows.
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[^1]: The research was partly supported by the National Research, Development and Innovation Office (NKFIH) Grant No. K120697. The project leading to this application has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 741420).
|
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Experimental evaluation\[sec:experiment\]
=========================================
Extending the DBpedia ontology\[sec:dbpedia\]
---------------------------------------------
This section presents all the steps that were undertaken in order to prepare and conduct an experiment on a crowdsourcing platform [^1]. Our aim was to answer the following research question: can SLDM mine new, meaningful axioms, that can be added to the ontology. To answer the question, we used 2015-04 with the ontology, and we followed the experimental protocol described:
1. We conducted exploratory data analysis to select a set of classes.
2. For the selected classes, we used SLDM to generate superclass expressions, and used them to obtain a set of axioms for a selected class, with the class in the left-hand side, and an expression in the right-hand side of an axiom.
3. We translated the generated axioms into natural language sentences.
4. We generated test questions to ensure that participants of the experiment are paying attention to their tasks.
5. These sentences were then posed to for verification by the contributors.
6. We collected and analyzed the results of the verification.
In the following sections, the details of the experimental protocol are explained.
### Exploratory data analysis
To select a set of classes from the ontology, what would allow us to conduct a high quality, statistically reliable experimental evaluation, we performed exploratory data analysis. For every class in the ontology, we computed the following characteristics using 2015-04:
1. the number of class instances,
2. the number of different triples, for which the subject belongs to the class,
3. the number of different predicates, for which there exists a triple in the dataset with a given predicate, and the subject belonging to the class,
4. the depth of the class in the subsumption hierarchy in the ontology (the shortest path from the root of the hierarchy `owl:Thing` to the class).
Histograms of the obtained values are presented in Figures \[fig:hist1\]–\[fig:hist4\]. On the basis of the histograms, we chose a set of criteria that the selected classes should fulfill.
To provide enough statistical support, we chose classes with more that 1000 instances and every instance occurring as a subject on average in more than 50 triples. To avoid generating a very large number of axioms, which would increase the costs of the crowdsourcing verification, we decided to keep the number of different predicates in range from 20 to 35. Finally, we decided on selecting classes with the depth of at least 3, and in such a manner that all selected classes should have pairwise different parents and at least three different grandparents.
We selected 5 classes, which meet all the aforementioned criteria: `Journalist, ProgrammingLanguage, Book, MusicGenre, Crater` together with their ancestors: `Agent, Person, Work, Software, WrittenWork, TopicalConcept, Genre, Place, NaturalPlace`. Full hierarchy is presented in Figure \[fig:hierarchy1\]. For each of these 14 classes, we used SLDM to generate two sets of axioms: the first one using the minimal support threshold $\theta_\sigma=0.5$ and the second one with $\theta_\sigma=0.8$. The obtained axioms are available in the repository <https://bitbucket.org/jpotoniec/sldm>, in the subfolder `CF_source`.
To get some insights into the novelty of the mined axioms, we used HermiT reasoner[^2] [@hermit] and for each of the mined set of axioms, we calculated how many of them were already logically entailed by the *DBpedia* ontology and how many of them were logically entailed by the *DBpedia* ontology enriched with the mined axioms for the superclass. The detailed statistics are presented in Table \[tab:entailment\_stats\]. For example, for the class Book and the minimal support threshold $\theta_\sigma=0.8$, 35 axioms were mined by SLDM, out of which 7 were logically entailed by the ontology and 15 were logically entailed by the ontology with the mined axioms for the class WrittenWork asserted. It must be noted that in all the cases, SLDM was able to discover more than it was already present in the ontology.
[.48]{}
coordinates [ (0,15) (10, 40) (20, 128) (30, 168) (40, 78) (50, 18) (60, 2) ]{};
[.48]{}
coordinates [ (0,6) (10, 10) (20, 74) (30, 175) (40, 104) (50, 41) (60, 12) (70, 8) (80, 4) (90, 7) (100, 2) ]{};
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coordinates [ (0,57) (10, 233) (20, 89) (30, 30) (40, 11) (50, 12) (60, 8) (70, 4) ]{};
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coordinates [ (0,32) (10, 82) (20, 106) (30, 188) (40, 36) (50, 5) ]{};
=\[draw=black,thick,anchor=west\]
child [ node [`dbo:Agent`]{} child [ node [`dbo:Person`]{} child [node [`dbo:Journalist`]{}]{} ]{}]{} child \[missing\] child [ node [`dbo:Work`]{} child [ node [`dbo:Software`]{} child [node [`dbo:Programminglanguage`]{}]{}]{} child \[missing\] child [ node [`dbo:WrittenWork`]{} child [node [`dbo:Book`]{}]{}]{} ]{} child \[missing\] child \[missing\] child \[missing\] child \[missing\] child [ node [`dbo:TopicalConcept`]{} child [ node [`dbo:Genre`]{} child [node [`dbo:MusicGenre`]{}]{}]{} ]{} child \[missing\] child [ node [`dbo:Place`]{} child [ node [`dbo:NaturalPlace`]{} child [node [`dbo:Crater`]{}]{}]{} ]{};
[l|>p[1.4cm]{}>p[1.4cm]{}>p[1.4cm]{}|>p[1.4cm]{}>p[1.4cm]{}>p[1.4cm]{}]{} Class & &\
& Overall & In the ontology & In the superclass & Overall & In the ontology & In the superclass\
Agent & 9 & 3 & - & 22 & 3 & -\
Person & 9 & 7 & 9 & 22 & 7 & 20\
Journalist & 109 & 8 & 10 & 131 & 8 & 21\
Work & 11 & 3 & - & 14 & 3 & -\
Software & 17 & 4 & 12 & 45 & 4 & 14\
ProgrammingLanguage & 16 & 6 & 12 & 28 & 6 & 13\
WrittenWork & 14 & 4 & 12 & 63 & 4 & 15\
Book & 35 & 7 & 15 & 108 & 7 & 29\
TopicalConcept & 10 & 3 & - & 13 & 3 & -\
Genre & 13 & 4 & 11 & 27 & 4 & 13\
MusicGenre & 13 & 5 & 13 & 27 & 5 & 27\
Place & 21 & 3 & - & 186 & 5 & -\
NaturalPlace & 13 & 4 & 12 & 24 & 4 & 24\
Crater & 12 & 5 & 11 & 25 & 5 & 12\
### Translation of the axioms to natural language
Ontological axioms expressed in OWL (e.g., using Turtle) cannot be easily understood by English speakers that are not familiar with the Semantic Web technologies. Therefore, we proposed a procedure of translation of OWL axioms to English. We wanted to use a simple variant of the language, so we decided to choose Attempto Controlled English [@kaljurand2007verbalizing]. It is a controlled version of normal English language that involves advantages of formal representation (well defined syntax, possibility of automatic processing) and natural language (expressiveness and ease of understanding) [@fuchs2008attempto]. Each person that knows basics of English should be able to understand a translated sentence without any knowledge about its formal representation. Usage of controlled language has one other, very important feature: the translation is fully reversible, so we do not lose any information.
We decided to create a whole translator on our own. The reason for that was the fact that existing tools (e.g., [^3] [@owlverb]) are restricted and work only for some of our examples.
In the OWL axioms generated by SLDM, we identified a set of structural templates and for every template, we provided a corresponding template in English. The URIs in the axioms were replaced by their corresponding labels during the translation. The core idea of the translation tool is to analyze an axiom level by level and match it to the templates. The output is a set of simple sentences, that represent more and more specific parts of constrains. Sample axiom, that pertains `dbo:Journalist` class
`dbo:Journalist`` ``dbo:nationality`` `\
` (``dbo:governmentType`` ``owl:Thing`\
` ``dbo:leader`` ``owl:Thing``)`
can be translated into sentences *Every journalist has nationality. Nationality has government type. Nationality has leader.* During the translation, we also performed some pruning to make the final sentences more readable and limit the costs of the experiment. We removed axioms that contained concepts that are characteristic for internal structure of or act as metadata, for example, predicate `dbp:hasPhotoCollection`. We reason that such axioms are very hard to understand for a non-expert, and thus cannot be efficiently verified by the contributors of a crowdsourcing platform. We also removed axioms containing namespaces from other Linked Data sets, for example, namespace, in order to decrease number of axioms to verify, and avoid displaying numerical URIs to the users, for example, we removed the axiom `dbo:Book` `wikidata:Q1930187`[^4].
After application of the translation tool to the axioms generated by SLDM, we obtained a set of sentences. Each of these sentences was then used as a base to form a question. A question consists of: a sentence, which is to be verified; a set of three allowed answers, from which only one is to be selected: *Yes*, *No*, *I don’t know*; an optional field to explain why a particular answer was selected.
### Test questions
The quality of the results achieved from crowdsourcing experiment can be significantly improved by introducing test questions [@mortensen2013crowdsourcing]. The right answer to these questions is known before the experiment. They are used to check reliability of crowdsourcing platform contributors. They can be used in two ways:
1. One prepares a quiz for the contributors, that contains only the test questions. If a contributor passes the test, she is allowed to answer payable questions.
2. For each set of questions, that are presented to a contributor, one question is a test question. If the contributor does not answer the test question correctly, the other answers from her are discarded, and she does not get paid for them.
During the experiment, we used the second solution, because it requires contributor attention for every set of questions. A good practice recommends having 10-20% test questions in the input dataset[^5]. Some of our test questions were correct (i.e., required an answer *Yes*) and some were incorrect (i.e., required an answer *No*), in order to ensure that a contributor cannot select always the same answer and ignore the questions completely.
To obtain the test questions, we used reasoner *Pellet* [@pellet] to find in the set of axioms generated by SLDM axioms that logically follows from the ontology. The questions corresponding to these axioms were then used as the test questions with a known correct answer *Yes*.
To obtain test questions with a correct *No* answer, we selected some of the axioms generated by SLDM containing only a named class in the right-hand side and replaced the class by some other, unrelated class, obtaining, for example, an axiom `dbo:Journalist` `dbo:Book`. We also generated some false axioms by adding to the left-hand side of an axiom inferred from the ontology, obtaining, for example, a sentence *Not every software is software*.
### CrowdFlower experiment setup
The last activity to do before starting a crowdsourcing experiment is to setup the settings of the experiment and create an instruction for the contributors. Both of these steps are crucial with respect to ensuring quality of the experimental results.
We set up the settings in the following way:[^6] As our questions are quite simple, we requested for contributors of the lowest level, as this allowed us to obtain the results faster. We decided on presenting 10 questions at once (i.e., on a single page) to a single contributor, as it should not take more than a few minutes to answer all of them. We chose to pay 0.03 USD for answering one page of questions. This is a typical payment on *CrowdFlower* for the contributors of the lowest level, and should maintain their commitment. We chose to request answers from 20 distinct contributors to one question. In the preliminary experiments, we requested only 3 answers, but in such a case, a single disagreement (e.g., when a contributor does not understand a question) makes the result unreliable. However, we did not want to increase the number too much, to keep the costs under control.
An instruction for a crowdsourcing experiment should be as simple as possible, yet answer all questions a contributor can ask. Moreover, it must contain examples of real questions, both positive and negative, and all steps that should be undertaken to solve them. For our experiment, we inform the contributors, that axioms are represented as sentences and their task is to decide whether a given sentence is true, false, or is not clear. In the instructions, we also mentioned one true sentence, one false sentence, and one not clear sentence with an explanation in each case. The full text of the instruction is also available in the repository.
### Experimental results
For the crowdsourcing experiment, we generated two sets of axioms: one with the minimal support threshold $\theta_\sigma=0.5$ and the other with the threshold $\theta_\sigma=0.8$. From each of the sets, we removed axioms that were logically entailed by the ontology or by the mined axioms for a superclass. The first set was translated to 425 payable questions and 61 test questions. Each of the payable questions was asked to 20 distinct contributors, leading to 8500 trusted answers. The second set was translated to 168 payable questions and 56 test questions; the payable questions yielded $20\cdot 168=3360$ trusted answers. Reliability of all contributors was checked with the test questions and when it dropped below 70% (more than 30% of the presented test questions had wrong answers), they were refused to continue answering the questions.
A summary of the results is presented in Table \[tab:cf-summary\]. In the first set, $69.41\%$ (resp. $38.69\%$ in the second set) of the axioms were accepted by at least $80\%$ of the contributors and $97.17\%$ (resp. $89.88\%$) by at least $55\%$ of the contributors. Our aim was to verify whether SLDM can mine new, meaningful axioms that can be added to the ontology. We found both results to answer positively to the question. All the verified knowledge was new, because the axioms that could be inferred from the ontology were used as the test questions.
[rr>p[1cm]{}|rr|rr]{} *Yes* & *No* & *I don’t know* & &\
16–20 & 0–5 & 0–5 & 295 & $69.41\%$ & 65 & $38.69\%$\
11–15 & 6–10 & 0–5 & 40 & $9.41\%$ & 41 & $24.40\%$\
11–15 & 0–5 & 0–5 & 78 & $18.35\%$ & 45 & $26.79\%$\
6–10 & 11–15 & 0–5 & 3 & $0.71\%$ & 3 & $1.79\%$\
6–10 & 6–10 & 0–5 & 5 & $1.18\%$ & 13 & $7.74\%$\
0–5 & 16–20 & 0–5 & 2 & $0.47\%$ & 0 & $0.00\%$\
0–5 & 11–15 & 0–5 & 2 & $0.47\%$ & 1 & $0.60\%$\
& 425 & $100.00\%$ & 168 & $100.00\%$\
To measure the level of disagreement for each question, we treated the numbers of *Yes*, *No*, and *I don’t know* answers as coordinates in a space and measured the Euclidean distance from all three crisp answers. We treated the minimal distance as the disagreement measure, where higher value means higher disagreement. For example, a question with 7 *Yes*, 8 *No*, and 5 *I don’t know* has coordinates $(7, 8, 5)$ and the nearest crisp answer is all *No* with coordinates $(0, 20, 0)$ (distance: $\sqrt{218}$). We present top 5 questions with the highest disagreement, selected from the axioms mined with the minimal support threshold $\theta_\sigma=0.5$ as given in Table \[tab:cf-disagreement\].
Question 1 probably refers to a geographical position of a crater, but the name of the predicate is very vague and, as it is in the `dbp` namespace, it lacks a description. Question 2 was mined, because $31,172$ out of $51,019$ instances, that is, over $50\%$, of `dbo:WrittenWork` is asserted to the class `http://purl.org/ontology/bibo/Book`. Questions 3 and 4 display a similar problem with a complex structure and verbalization requiring knowledge about knowledge representation (KR). Finally, question 5, on top of requiring knowledge about KR, requires also expert knowledge from the domain of law.
[r|rr>p[1cm]{}|p[.75]{}]{} \# & *Yes* & *No* & *I don’t know* & A question and the axiom it originated from\
1 & 7 & 8 & 5 & “Every crater has E or W. E or W is Literal.”\
& & & & `dbo:Crater dbp:eOrW rdf:PlainLiteral`\
2 & 10 & 9 & 1 & “Every written work (Written work is any text written to read it (e.g. - books, newspaper, articles)) is Book \*.”\
& & & & `dbo:WrittenWork <http://purl.org/ontology/bibo/Book>`\
3 & 10 & 8 & 2 & “Every genre has instrument. instrument has is Primary Topic Of \*. is Primary Topic Of \* is Thing. instrument has label. label is Literal.”\
& & & & `dbo:Genre dbo:instrument ((foaf:isPrimaryTopicOf owl:Thing) (rdfs:label rdf:PlainLiteral))`\
4 & 10 & 8 & 2 & “Every genre has instrument. instrument has is Primary Topic Of \*. is Primary Topic Of \* is Document \*. is Primary Topic Of \* has Language (A language of the resource.). Language (A language of the resource.) is string. is Primary Topic Of \* has Language (A language of the resource.) which value is en.”\
& & & & `dbo:Genre dbo:instrument foaf:isPrimaryTopicOf (foaf:Document dc:language xsd:string dc:language enxsd:string)`\
5 & 10 & 6 & 4 & “Every journalist has nationality. nationality has Legislature. Legislature has see Also. see Also is Thing. Legislature has type that is Bicameralism \*.”\
& & & & `dbo:Journalist dbo:nationality (dbp:legislature ((rdfs:seeAlso owl:Thing) (dbo:type dbr:Bicameralism)))`
[^1]: <https://www.crowdflower.com/>
[^2]: <http://www.hermit-reasoner.com/>
[^3]: <https://github.com/Kaljurand/owl-verbalizer>
[^4]: The prefix `wikidata:` corresponds to <http://www.wikidata.org/entity/>. The complete list of removed axioms is available in the *Git* repository, in the file `CF_source/removed_axioms.txt`.
[^5]: <http://www.success.crowdflower.com>
[^6]: See <https://success.crowdflower.com/hc/en-us/articles/201855719-Guide-to-Basic-Job-Settings-Page> for additional explanation of the settings
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Building on the recent coded-caching breakthrough by Maddah-Ali and Niesen, the work here considers the $K$-user cache-aided wireless multi-antenna (MISO) symmetric broadcast channel (BC) with random fading and imperfect feedback, and analyzes the throughput performance as a function of feedback statistics and cache size. In this setting, our work identifies the optimal cache-aided degrees-of-freedom (DoF) within a factor of 4, by identifying near-optimal schemes that exploit the new synergy between coded caching and delayed CSIT, as well as by exploiting the unexplored interplay between caching and feedback-quality. The derived limits interestingly reveal that — the combination of imperfect quality current CSIT, delayed CSIT, and coded caching, guarantees that — the DoF gains have an initial offset defined by the quality of current CSIT, and then that the additional gains attributed to coded caching are exponential, in the sense that any linear decrease in the required DoF performance, allows for an exponential reduction in the required cache size.'
author:
- 'Jingjing Zhang and Petros Elia [^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'final\_refs.bib'
nocite:
- '[@KPR:99; @BR:05; @BK:06; @BGW:10]'
- '[@ZE:15]'
title: 'Fundamental Limits of Cache-Aided Wireless BC: Interplay of Coded-Caching and CSIT Feedback'
---
Introduction\[sec:intro\]
=========================
Recent work by [@MN14] explored — for the single-stream broadcast setting — how careful caching of content at the receivers, and proper encoding across different users’ requested data, can allow for higher communication rates. The key idea was to use coding in order to create multicast opportunities, even if the different users requested different data content. This *coded caching* approach — which went beyond storing popular content closer to the user — involved two phases; the placement phase (during off peak hours) and the delivery phase (during peak hours). During the placement phase, content that was predicted to be popular (a library of commonly requested files), was coded and placed across user’s caches. During the delivery phase — which started when users requested specific files from the predicted library of files — the transmitter encoded across different users’ requested data content, taking into consideration the requests and the existing cache contents. This approach — which translated to efficient interference removal gains that were termed as ‘coded-caching gains’ — was shown in [@MN14] to provide substantial performance improvement that far exceeded the ‘local’ caching gains from the aforementioned traditional ‘data push’ methods that only pre-store content at local caches.
Our interest here is to explore coded caching, not in the original single-stream setting in [@MN14], but rather in the feedback-aided multi-antenna wireless BC. This wireless and multi-antenna element now automatically brings to the fore a largely unexplored and involved relationship between coded caching and CSIT-type feedback quality. This relationship carries particular importance because both CSIT and coded caching are powerful and crucial ingredients in handling interference, because they are both hard to implement individually, and because their utility is affected by one another (often adversely, as we will see). Our work tries to understand how CSIT and caching resources jointly improve performance, as well as tries to shed some light on the interplay between coded caching and feedback.
### Motivation for the current work
A main motivation in [@MN14] and in subsequent works, was to employ coded caching to remove interference. Naturally, in wireless networks, the ability to remove interference is very much linked to the quality and timeliness of the available feedback, and thus any attempt to further our understanding of the role of coded caching in these networks, stands to benefit from understanding the interplay between coded caching and (variable quality) feedback. This joint exposition becomes even more meaningful when we consider the connections that exist between feedback-usefulness and cached side-information at receivers, where principally the more side information receivers have, the less feedback information the transmitter might need.
This approach is also motivated by the fact that feedback is hard to get in a timely manner, and hence is typically far from ideal and perfect. Thus, given the underlying links between the two, perhaps the strongest reason to jointly consider coded caching and feedback, comes from the prospect of using coded caching to alleviate the constant need to gather and distribute CSIT, which — given typical coherence durations — is an intensive task that may have to be repeated hundreds of times per second during the transmission of content. This suggests that content prediction of a predetermined library of files during the night (off peak hours), and a subsequent caching of parts of this library content again during the night, may go beyond boosting performance, and may in fact offer the additional benefit of alleviating the need for prediction, estimation, and communication of CSIT during the day, whenever requested files are from the library. Our idea of exploring the interplay between feedback (timeliness and quality) and coded caching, hence draws directly from this attractive promise that content prediction, once a day, can offer repeated and prolonged savings in CSIT.
Cache-aided broadcast channel model
-----------------------------------
### $K$-user BC with pre-filled caching
In the symmetric $K$-user multiple-input single-output (MISO) broadcast channel of interest here, the $K$-antenna transmitter, communicates to $K$ single-antenna receiving users. The transmitter has access to a library of $N\geq K$ distinct files $W_1,W_2, \dots, W_N$, each of size $|W_n| = f$ bits. Each user $k \in \{1,2,\dots,K\}$ has a cache $Z_k$, of size $|Z_k| = Mf$ bits, where naturally $M \leq N$. Communication consists of the aforementioned *content placement phase* and the *delivery phase*. During the placement phase — which usually corresponds to communication during off-peak hours — the caches $Z_1, Z_2, \dots, Z_K$ are pre-filled with content from the $N$ files $\{W_n\}_{n=1}^{N}$. The delivery phase commences when each user $k$ requests from the transmitter, any *one* file $W_{R_k}\in \{W_n\}_{n=1}^{N}$, out of the $N$ library files. Each file can be requested with equal probability. Upon notification of the users’ requests, the transmitter aims to deliver the (remaining of the) requested files, each to their intended receiver, and the challenge is to do so over a limited (delivery phase) duration $T$.
For each transmission, the received signals at each user $k$, will be modeled as $$\begin{aligned}
y_{k}=\hv_{k}^{T} \xv + z_{k}, ~~ k = 1, \dots, K\end{aligned}$$ where $\xv\in\mathbb{C}^{K\times 1}$ denotes the transmitted vector satisfying a power constraint $\E(||\xv||^2)\leq P$, where $\hv_{k}\in\mathbb{C}^{K\times 1}$ denotes the channel of user $k$ in the form of the random vector of fading coefficients that can change in time and space, and where $z_{k}$ represents unit-power AWGN noise at receiver $k$.
![Cache-aided $K$-user MISO BC.[]{data-label="fig:model"}](chanmodel){width="0.8\columnwidth"}
At the end of the delivery phase, each receiving user $k$ combines the received signal observations $y_{k}$ — accumulated during the delivery phase — with the fixed information in their respective cache $Z_k$, to reconstruct their desired file $W_{R_k}$.
Coded caching and CSIT-type feedback
------------------------------------
Communication also takes place in the presence of channel state information at the transmitter. CSIT-type feedback is crucial in handling interference, and can thus substantially reduce the resulting duration $T$ of the delivery phase. This CSIT is typically of imperfect-quality as it is hard to obtain in a timely and reliable manner. In the high-SNR (high $P$) regime of interest, this current-CSIT quality is concisely represented in the form of the normalized quality exponent [@YKGY:12d][@CE:13it] $$\begin{aligned}
\alpha & := -\lim_{P \rightarrow \infty} \frac{\log \E[||{\hv_{k}}-{\hat \hv_{k}}||^2]}{\log P}, ~k\in \{1,\dots,K\}\end{aligned}$$ where ${\hv_{k}}-{\hat \hv_{k}}$ denotes the estimation error between the current CSIT estimate ${\hat \hv_{k}}$ and the estimated channel ${\hv_{k}}$. The range of interest[^3] is $\alpha\in[0,1]$. We also assume availability of delayed CSIT (as in for example [@MAT:11c], as well as in a variety of subsequent works [@YKGY:12d; @CE:13it; @GJ:12o; @CE:12d; @KYG:13; @CYE:13isit; @VV:09; @TJSP:12; @LH:12; @HC:13], see also [@VV:11t; @AGK:11o; @Lee2012; @Tandon2012b] as well as [@TAV:2015; @BW:2015; @LTA:2015]) where now the delayed estimates of any channel, can be received without error but with arbitrary delay, even if this delay renders this CSIT completely obsolete. As it is argued in [@YKGY:12d], this mixed CSI model (partial current CSIT, and delayed CSIT) nicely captures different realistic settings that might involve channel correlations and an ability to improve CSI as time progresses. This same CSI model is particularly well suited for our caching-related setting here, because it explicitly reflects two key ingredients that are directly intertwined with coded caching; namely, feedback timeliness and feedback quality.
In terms of caching, we will consider the normalized $$\begin{aligned}
\label{eq:gamma1}
\gamma := \frac{M}{N}\end{aligned}$$ as well as the cumulative $$\begin{aligned}
\Gamma := \frac{KM}{N} = K\gamma.\end{aligned}$$ The latter simply means that the sum of the sizes of the caches across all users, is $\Gamma$ times the volume of the $N$-file library. As in [@MN14], we will consider the case where $\Gamma = \{1,2,\cdots K\}$.
#### Intuitive links between $\alpha$ and $\gamma$
As we will see, $\alpha$ is not only linked to the performance — where a higher $\alpha$ allows for better interference management and higher performance over the wireless delivery link — but is also linked to caching; after all, the bigger the $\gamma$, the more side information the receivers have, the less interference one needs to handle (at least in symmetric systems), and the smaller the $\alpha$ that is potentially needed to steer interference. This means that principally, a higher $\gamma$ implies that more common information needs to be transmitted, which may (in some cases) diminish the utility of feedback which primarily aims to facilitate the opposite which is the transmission of private information. It is for example easy to see (we will see this later) that in the presence of $\Gamma = K-1$, there is no need for CSIT in order to achieve the optimal performance.
Measures of performance in current work
---------------------------------------
As in [@MN14], the measure of performance here is the duration $T$ — in time slots, per file served per user — needed to complete the delivery process, *for any request*. The wireless link capabilities, and the time scale, are normalized such that one time slot corresponds to the optimal amount of time it would take to communicate a single file to a single receiver, had there been no caching and no interference. As a result, in the high $P$ setting of interest — where the capacity of a single-user MISO channel scales as $\log_2(P)$ — we proceed to set $$\begin{aligned}
\label{eq:f}
f = \log_2(P)\end{aligned}$$ which guarantees that the two measures of performance, here and in [@MN14], are the same and can thus be directly compared[^4].
A simple inversion leads to the equivalent measure of the per-user DoF $$\begin{aligned}
\label{eq:TtoDoF}
d(\gamma,\alpha)=\frac{1-\gamma}{T}\end{aligned}$$ which captures the joint effect of coded caching and feedback[^5].
Notation and assumptions
------------------------
### Notation
We will use the notation $H_n := \sum_{i=1}^{n} \frac{1}{i}$, to represent the $n$-th harmonic number, and we will use $\epsilon_n := H_n-\log (n)$ to represent its logarithmic approximation error, for some integer $n$. We remind the reader that $\epsilon_n$ decreases with $n$, and that $\epsilon_\infty :=\lim \limits_{n \rightarrow \infty} H_n - \log (n) $ is approximately $0.5772$. $\mathbb{Z}$ will represent the integers, $\mathbb{Z}^{+}$ the positive integers, $\mathbb{R}$ the real numbers, $\binom{n}{k}$ the $n$-choose-$k$ operator, and $\oplus$ the bitwise XOR operation. We will use $[K]:= \{1,2,\cdots,K\}$. If $\psi$ is a set, then $|\psi|$ will denote its cardinality. For sets $A$ and $B$, then $A \backslash B$ denotes the difference set. Complex vectors will be denoted by lower-case bold font. We will use $||\xv||^2$ to denote the magnitude of a vector $\xv$ of complex numbers. For a transmitted vector $\xv$, we will use $\text{dur}(\xv)$ to denote the transmission duration of that vector. For example, having $\text{dur}(\xv) = \frac{1}{10}T$ would simply mean that the transmission of vector $\xv$ lasts one tenth of the delivery phase. In our high-$P$ setting of interest, we will also use $\doteq$ to denote *exponential equality*, i.e., we will write $g(P)\doteq P^{B}$ to denote $\displaystyle\lim_{P\to\infty}\frac{\log g(P)}{\log P}=B$. Similarly $\dotgeq$ and $\dotleq$ will denote exponential inequalities. Logarithms are of base $e$, unless we use $\log_2(\cdot)$ which will represent a logarithm of base 2.
### Main assumptions
Throughout this work, we assume availability of current CSIT with some quality $\alpha$, of delayed CSIT (D-CSIT), as well as ask that each receiver knows their own channel perfectly. We also adhere to the common convention (see for example [@MAT:11c]) of assuming perfect and global knowledge of delayed channel state information at the receivers (delayed global CSIR), where each receiver must know (with delay) the CSIR of (some of the) other receivers. We will assume that the entries of *each specific* estimation error vector are i.i.d. Gaussian. For the outer (lower) bound to hold, we will make the common assumption that the current channel state must be independent of the previous channel-estimates and estimation errors, *conditioned on the current estimate* (there is no need for the channel to be i.i.d. in time). We will make the assumption that the channel is drawn from a continuous ergodic distribution such that all the channel matrices and all their sub-matrices are full rank almost surely. We also make the soft assumption that the transmitter *during the delivery phase* is aware of the feedback statistics. We note though that, while our main scheme assumes knowledge of $\alpha$ during the caching phase, most results will be the outcome of a simpler scheme that does not require knowledge of $\alpha$ during this caching phase. Removing this assumption entails, for $\alpha>0$, a performance penalty which is small.
Prior work
----------
The benefits of coded caching on reducing interference and improving performance, were revealed in the seminal work by Maddah-Ali and Niesen in [@MN14] who considered a caching system where a server is connected to multiple users through a shared link, and designed a novel caching and delivery method that jointly offers a multicast gain that helps mitigate the link load, and which was proven to have a gap from optimal that is at most 12. This work was subsequently generalized in different settings, which included the setting of different cache sizes for which Wang et al. in [@WLTL:15] developed a variant of the algorithm in [@MN14] which achieves a gap of at most 12 from the information theoretic optimal. Other extensions included the work in [@MND13] by Maddah-Ali and Niesen who considered the setting of decentralized caching where the achieved performance was shown to be comparable to that of the centralized case [@MN14], despite the lack of coordination in content placement. For the same original single-stream setting of [@MN14], the work of Ji et al. in [@JTLC:14] considered a scenario where users make multiple requests each, and proposed a scheme that has a gap to optimal that is less than 18. Again for the setting in [@MN14], the work of Ghasemi and Ramamoorthy in [@HA:2015], derived tighter outer (lower) bounds that improve upon existing bounds, and did so by recasting the bound problem as one of optimally labeling the leaves of a directed tree. Further work can be found in [@WLG:15] where Wang et al. explored the interesting link between caching and distributed source coding with side information. Interesting conclusions are also drawn in the work of Ajaykrishnan et al. in [@APPV:15], which revealed that the effectiveness of caching in the single stream case, is diminished when $N$ approaches and exceeds $K^2$.
Deviating from single-stream error free links, different works have considered the use of coded caching in different wireless networks, without though particular consideration for CSIT feedback quality. For example, work by Huang et al. in [@HuangWDY015], considered a cache-aided wireless fading BC where each user experiences a different link quality, and proposed a suboptimal communication scheme that is based on time- and frequency-division and power- and bandwidth-allocation, and which was evaluated using numerical simulations to eventually show that the produced throughput decreases as the number of users increases. Further work by Timo and Wigger in [@TW:15] considered an erasure broadcast channel and explored how the cache-aided system efficiency can improve by employing unequal cache sizes that are functions of the different channel qualities. Another work can be found in [@MN:15isit] where Maddah-Ali and Niesen studied the wireless interference channel where each transmitter has a local cache, and showed distinct benefits of coded caching that stem from the fact that content-overlap at the transmitters allows effective interference cancellation.
Different work has also considered the effects of caching in different non-classical channel paradigms. One of the earlier such works that focused on practical wireless network settings, includes the work by Golrezaei et al. in [@GSDMC:12], which considered a downlink cellular setting where the base station is assisted by helper nodes that jointly form a wireless distributed caching network (no coded caching) where popular files are cached, resulting in a substantial increase to the allowable number of users by as much as $400 - 500\%$. In a somewhat related setting, the work in [@PBKD:15] by Perabathini et al. accentuated the energy efficiency gains from caching. Further work by Ji et al. in [@JWTLCEL:15] derived the limits of so-called combination caching networks in which a source is connected to multiple user nodes through a layer of relay nodes, such that each user node with caching is connected to a distinct subset of the relay nodes. Additional work can also be found in [@NSW:12] where Niesen et al. considered a cache-aided network where each node is randomly located inside a square, and it requests a message that is available in different caches distributed around the square. Further related work on caching can be found in [@BBD:15; @MCOFBJ:14; @HKD:14; @HKS:15; @SJTLD:15; @JTLC:14].
Work that combines caching and feedback considerations in wireless networks, has only just recently started. A reference that combines these, can be found in [@DBAD:15] where Deghel et al. considered a MIMO interference channel (IC) with caches at the transmitters. In this setting, whenever the requested data resides within the pre-filled caches, the data-transfer load of the backhaul link is alleviated, thus allowing for these links to be instead used for exchanging CSIT that supports interference alignment. An even more recent concurrent work can be found in [@GKY:15] where Ghorbel et al. studied the capacity of the cache-enabled broadcast packet erasure channel with ACK/NACK feedback. In this setting, Ghorbel et al. cleverly showed — interestingly also using a retrospective type algorithm, this time by Gatzianas et al. in [@GGT:13] — how feedback can improve performance by informing the transmitter when to resend the packets that are not received by the intended user and which are received by unintended users, thus allowing for multicast opportunities. The first work that considers the actual interplay between coded caching and CSIT quality, can be found in [@ZFE:15] which considered the easier problem of how the optimal cache-aided performance (with coded caching), can be achieved with reduced quality CSIT.
Outline and contributions
-------------------------
In Section \[sec:mainResults\], Lemma \[lem:outer\], we offer a lower bound for the optimal $T^*(\gamma,\alpha)$. Then in Theorem \[thm:bigGamma\] we calculate the achievable $T(\gamma,\alpha)$, for $\Gamma \in \{1, 2, \cdots , K\}$, $\alpha \in[0,1]$, and prove it to be less than four times the optimal, thus identifying the optimal $T^*(\gamma,\alpha)$ within a factor of 4. A simpler expression for $T$ (again within a factor of $4$ from optimal), and its corresponding per-user DoF, are derived in Theorem \[thm:bigGamma\], while a simple approximation of these is derived in Corollary \[cor:LargeGammaLogApprox\], where we see that the per-user DoF takes the form $d(\gamma,\alpha) = \alpha + (1-\alpha) \frac{1-\gamma}{\log{\frac{1}{\gamma}}}$, revealing that even a very small $\gamma = e^{-G}$ can offer a substantial DoF boost $
d(\gamma = e^{-G},\alpha) - d(\gamma = 0,\alpha) \approx (1-\alpha)\frac{1}{G}.$
In Section \[sec:CacheAidedCSIT\] we discuss practical implications. In Corollary \[cor:alphaGainTot1\] we describe the savings in current CSIT that we can have due to coded caching, while in Corollary \[cor:alphaThreshold2GtotLarge\] we quantify the intuition that, in the presence of coded-caching, there is no reason to improve CSIT beyond a certain threshold quality. Furthermore in Section \[sec:vanishingFractionCSIT\] we show how cache-aided communications can utilize a vanishingly-small portion of D-CSIT compared to traditional D-CSIT schemes, simply because caching helps ‘skip’ the parts of the schemes that require the highest D-CSIT load. In Section \[sec:schemeAlphaBigGamma\] we present the caching-and-delivery schemes, which build on the interesting connections between MAT-type retrospective transmission schemes (cf. [@MAT:11c]) and coded caching. The caching part is modified from [@MN14] to essentially *‘fold’* (linearly combine) the different users’ data into multi-layered blocks, in a way such that the subsequent transmission algorithm (which employs parts of the QMAT algorithm in [@KGZE:16]) is suited to efficiently unfold these. The caching and transmission algorithms are calibrated so that the caching algorithm — which is modified from that in [@MN14] to adapt the caching redundancy to $\alpha$ — creates the same multi-destination delivery problem that is efficiently solved by the last stages of the QMAT scheme. Section \[sec:additionalProofs\] in the Appendix presents the outer bound proof, and the proof for the gap to optimal.
Throughput of cache-aided BC as a function of CSIT quality and caching resources\[sec:mainResults\]
===================================================================================================
The following results hold for the $(K,M,N,\alpha)$ cache-aided $K$-user wireless MISO BC with random fading, $\alpha\in [0,1]$ and $N\geq K$, where $\gamma = \frac{M}{N}$ and $\Gamma = K\gamma$. We begin with an outer bound (lower bound) on the optimal $T^*$.
\[lem:outer\] The optimal $T^*$ for the $(K,M,N,\alpha)$ cache-aided $K$-user MISO BC, is lower bounded as $$\begin{aligned}
T^*(\gamma,\alpha) \geq \mathop {\text{max}}\limits_{s\in \{1, \dots, \lfloor \frac{N}{M} \rfloor \}} \frac{1}{(H_s \alpha+1-\alpha)} (H_s -\frac{Ms}{\lfloor \frac{N}{s} \rfloor}).\end{aligned}$$
The proof is presented in Section \[sec:lower\] and it uses the bound from Lemma \[lem:lowerSecond\] whose proof can be found in Section \[sec:lowerSecond\].
Achievable throughput of the cache-aided BC
-------------------------------------------
The following identifies, up to a factor of 4, the optimal $T^*$, for all $\Gamma \in \{1, 2, \cdots , K\}$ (i.e., $M\in \frac{N}{K}\{1,\cdots,K\}$). The result uses the expression $$\begin{aligned}
\label{eq:alphaBreak}
\alpha_{b,\eta} = \frac{\eta-\Gamma}{\Gamma(H_K-H_\eta-1)+\eta}, \ \eta = \ceil{\Gamma},\dots,K-1. \end{aligned}$$ Note that the above does not hold for $\Gamma = K$, as this would imply no need for delivery.
\[thm:bigGammaBest\] In the $(K,M,N,\alpha)$ cache-aided MISO BC with $N$ files, $K\leq N$ users, $\Gamma \in \{1, 2, \cdots , K\}$, and for $\eta = \arg\max_{\eta{'}\in [\Gamma,K-1]\cap \mathbb{Z}} \{\eta{'} \ : \ \alpha_{b,\eta'}\leq \alpha\}$, then $$\begin{aligned}
\label{eq:gammabigBest}
T = \max\{1-\gamma, \frac{(K-\Gamma)(H_K-H_\eta)}{(K-\eta)+\alpha(\eta+K(H_K-H_\eta-1))}\}\end{aligned}$$ is achievable and always has a gap-to-optimal that is less than 4, for all $\alpha,K$. For $\alpha \geq \frac{K(1-\gamma)-1}{(K-1)(1-\gamma)} $, $T$ is optimal.
The caching and delivery scheme that achieves the above performance is presented in Section \[sec:schemeAlphaBigGamma\], while the corresponding gap to optimal is bounded in Section \[sec:gapCalculation\].
The above is achieved with a general scheme whose caching phase is a function of $\alpha$. We will henceforth consider a special case ($\eta =\Gamma$) of this scheme, which provides similar performance (it again has a gap to optimal that is bounded by 4), simpler expressions, and has the practical advantage that the caching phase need not depend on the CSIT statistics $\alpha$ of the delivery phase. For this case, we can achieve the following performance.
\[thm:bigGamma\] In the $(K,M,N,\alpha)$ cache-aided MISO BC with $\Gamma \in \{1, 2, \cdots , K\}$, $$\begin{aligned}
\label{eq:gammabig}
T = \frac{(1-\gamma)(H_K-H_{\Gamma})}{\alpha(H_K-H_{\Gamma})+(1-\alpha)(1-\gamma)}\end{aligned}$$ is achievable and has a gap from optimal $$\begin{aligned}
\label{eq:gap2}
\frac{T}{T^*}<4\end{aligned}$$ that is less than 4, for all $\alpha,K$. Thus the corresponding per-user DoF takes the form $$\begin{aligned}
\label{eq:gammabigDoF}
d(\gamma,\alpha) = \alpha + (1-\alpha)\frac{1-\gamma}{H_K-H_\Gamma}.\end{aligned}$$
The scheme that achieves the above performance will be described later on as a special (simpler) case of the scheme corresponding to Theorem \[thm:bigGammaBest\]. The corresponding gap to optimal is bounded in Section \[sec:gapCalculation\].
The following corollary describes the above achievable $T$, under the logarithmic approximation $H_n\approx\log (n)$. The presented expression is exact in the large $K$ setting[^6] where $\frac{H_K-H_{\Gamma}}{\log(\frac{1}{\gamma})} = 1$.
\[cor:LargeGammaLogApprox\] Under the logarithmic approximation $H_n\approx\log (n)$, the derived $T$ takes the form $$\begin{aligned}
\label{eq:gammabigApprox}
T(\gamma,\alpha) = \frac{(1-\gamma)\log(\frac{1}{\gamma})}{\alpha\log(\frac{1}{\gamma})+(1-\alpha)(1-\gamma)}\end{aligned}$$ and the derived DoF takes the form $$\begin{aligned}
\label{eq:gammabigApproxDoFLog}
d(\gamma,\alpha) = \alpha + (1-\alpha) \frac{1-\gamma}{\log{\frac{1}{\gamma}}}.\end{aligned}$$
For the large $K$ setting, what the above suggests is that current CSIT offers an initial DoF boost of $d^*(\gamma=0,\alpha) = \alpha$ (cf. [@KGZE:16]), which is then supplemented by a DoF gain $$d(\gamma,\alpha) - d^*(\gamma=0,\alpha) \rightarrow (1-\alpha)\frac{1-\gamma}{\log(\frac{1}{\gamma})}$$ attributed to the synergy between delayed CSIT and caching [^7]. These synergistic gains (see also [@ZEsynergy:16]) are accentuated for smaller values of $\gamma$, where we see an exponential effect of coded caching, in the sense that now a microscopic $\gamma = e^{-G}$ can offer a substantial DoF boost $$\begin{aligned}
d(\gamma = e^{-G},\alpha) - d(\gamma = 0,\alpha) \approx (1-\alpha)\frac{1}{G}.\end{aligned}$$
\[ex:GapToOptimal\] In a MISO BC system with $\alpha = 0$, $K$ antennas and $K$ users, in the absence of caching, the optimal per-user DoF is $d^*(\gamma=0,\alpha=0) = 1/H_K$ (cf. [@MAT:11c]) which vanishes to zero as $K$ increases. A DoF of $1/4$ can be guaranteed with $\gamma \approx \frac{1}{50}$ for all $K$, a DoF of $1/7$ with $\gamma \approx \frac{1}{1000}$, and a DoF of $1/11.7$ can be achieved with $\gamma \approx 10^{-5}$, again for all $K$.
#### Interplay between CSIT quality and coded caching in the symmetric MISO BC
The derived form in (and its approximation in ) nicely capture the synergistic as well as competing nature of feedback and coded caching. It is easy to see for example that the effect from coded-caching, reduces with $\alpha$ and is proportional to $1-\alpha$. This reflects the fact that in the symmetric MISO BC, feedback supports broadcasting by separating data streams, thus diminishing multi-casting by reducing the number of common streams. In the extreme case when $\alpha = 1$, we see — again for the symmetric MISO BC — that the caching gains are limited to local caching gains[^8].
Cache-aided CSIT reductions\[sec:CacheAidedCSIT\]
=================================================
We proceed to explore how coded caching can alleviate the need for CSIT.
Cache-aided CSIT gains
----------------------
To capture the cache-aided reductions on the CSIT load, let us consider $$\begin{gathered}
\label{eq:alphaGainCode2}
\bar{\alpha}(\gamma,\alpha) := \arg\min_{\alpha'}\{\alpha': (1-\gamma) T^*(\gamma=0,\alpha') \leq T(\gamma,\alpha)\}\end{gathered}$$ which is derived below in the form $$\bar{\alpha}(\gamma,\alpha) = \alpha + \delta_\alpha(\gamma,\alpha)$$ for some $\delta_\alpha(\gamma,\alpha)$ that can be seen as the *CSIT reduction due to caching* (from $\bar{\alpha}(\gamma,\alpha)$ to the operational $\alpha$).
\[cor:alphaGainTot1\] In the $(K,M,N,\alpha)$ cache-aided MISO BC, then $$\begin{aligned}
\label{eq:alphaGainTotalGeneral}
\bar{\alpha}(\gamma,\alpha) = \alpha + \frac{(1-\alpha)(H_{K\gamma}-\gamma H_K)}{(H_K-1)(H_K-H_{K\gamma})}\end{aligned}$$ is achievable, and implies a cache-aided CSIT reduction $$\delta_\alpha(\gamma,\alpha) = \frac{(1-\alpha)(H_{K\gamma}-\gamma H_K)}{(H_K-1)(H_K-H_{K\gamma})}.$$
The proof is direct from Theorem \[thm:bigGamma\].
The above is made more insightful in the large $K$ regime, for which we have the following.
\[cor:alphaGainTot1Asymptotic1\] In the $(K,M,N,\alpha)$ cache-aided MISO BC, then $$\begin{aligned}
\label{eq:alphaGainCodingAsymptotic1b}
\bar{\alpha}(\gamma,\alpha) = \alpha+(1-\alpha) \frac{1-\gamma}{\log(\frac{1}{\gamma})}\end{aligned}$$ which implies CSIT reductions of $$\delta_\alpha(\gamma,\alpha) = (1-\alpha)d(\gamma,\alpha = 0) = (1-\alpha) \frac{1-\gamma}{\log(\frac{1}{\gamma})}.$$
The proof is direct from the definition of $\bar{\alpha}(\gamma,\alpha)$ and from Theorem \[thm:bigGamma\].
Furthermore we have the following which quantifies the intuition that, in the presence of coded-caching, there is no reason to improve CSIT beyond a certain threshold quality. The following uses the definition in , and it holds for all $K$.
\[cor:alphaThreshold2GtotLarge\] For any $\Gamma\in \{1,\dots,K\}$, then $$\begin{aligned}
\label{eq:alphaThreshold1}
T^*(\gamma,\alpha) = T^*(\gamma,\alpha = 1) = 1-\gamma\end{aligned}$$ holds for any $$\begin{aligned}
\alpha \geq \alpha_{b,K-1} = \frac{K(1-\gamma)-1}{(K-1)(1-\gamma)}\end{aligned}$$ which reveals that CSIT quality $\alpha = \alpha_{b,K-1}$ is the maximum needed, as it already offers the same optimal performance $T^*(\gamma,\alpha = 1)$ that would be achieved if CSIT was perfect.
This is seen directly from Theorem \[thm:bigGammaBest\] after noting that the achievable $T$ matches $T^*(\gamma,\alpha = 1) = 1-\gamma$.
#### How much caching is needed to partially substitute current CSIT with delayed CSIT (using coded caching to ‘buffer’ CSI)
As we have seen, in addition to offering substantial DoF gains, the synergy between feedback and caching can also be applied to reduce the burden of acquiring current CSIT. What the above results suggest is that a modest $\gamma$ can allow a BC system with D-CSIT to approach the performance attributed to current CSIT, thus allowing us to partially substitute current with delayed CSIT, which can be interpreted as an ability to buffer CSI. A simple calculation — for the large-$K$ regime — can tell us that $$\gamma^{'}_{\alpha}:= \arg\min_{\gamma^{'}}\{\gamma^{'}: d(\gamma^{'},\alpha = 0) \geq d^*(\gamma = 0,\alpha)\} = e^{-1/\alpha}$$ which means that $
\gamma^{'}_{\alpha}= e^{-1/\alpha}$ suffices to achieve — in conjunction with delayed CSIT — the optimal DoF performance $d^*(\gamma = 0,\alpha)$ associated to a system with delayed CSIT and $\alpha$-quality current CSIT.
Let $K$ be very large, and consider a BC system with delayed CSIT and $\alpha$-quality current CSIT, where $\alpha = 1/5$. Then $\gamma^{'}_{\alpha = 1/5}= e^{-5} = 0.0067 \approx 1/150$ which means that $$d^*(\gamma = 0.0067,\alpha=0) \geq d^*(\gamma = 0,\alpha=1/5)$$ which says that the same high-$K$ per-user DoF performance $d^*(\gamma = 0,\alpha=1/5)$, can be achieved by substituting all current CSIT with coded caching employing $\gamma\approx 1/150$.
Vanishing fraction of delayed CSIT\[sec:vanishingFractionCSIT\]
---------------------------------------------------------------
In the following we briefly explore how caching allows for a reduced D-CSIT load. We do so for the case of $\alpha = 0$. When $\alpha = 0$, the delivery scheme which we describe in Section \[sec:schemeAlphaBigGamma\], draws directly from the MAT scheme [@MAT:11c]. This scheme can have up to $K$ phases which are of decreasing time duration and which use a decreasing number of transmit antennas. Essentially each phase is lighter than the previous one, in terms of implementation difficulty. What we will see is that caching will allow us to bypass the first $\Gamma$ phases, which are the longest and most intensive, leaving us with the remaining $K-\Gamma$ communication phases that are easier to support with delayed feedback because they involve fewer transmissions, with fewer transmit antennas and to fewer users, and thus involve fewer D-CSIT scalars that must be communicated. In brief — after normalization to account for the condition that each user receives a total of $\log_2(P)$ bits of data — each phase $j = \Gamma+1,\Gamma+2,\dots,K$ will have a *normalized* duration $T_j = \frac{1}{j}$. During each phase $j$, we will need to send D-CSIT that describes the channel vectors for $K-j$ users, and during this same phase the transmitted vectors will have support $K-j+1$ because only $K-j+1$ transmit antennas are active. Thus during phase $j$, there will be a need to send $T_j(K-j+1)(K-j) = \frac{1}{j}(K-j+1)(K-j)$ D-CSIT scalars, and thus a need to send D-CSIT for up to a total of $$\begin{aligned}
& L(\Gamma) = \sum_{j=\Gamma+1}^K \frac{1}{j}(K-j+1)(K-j) \nonumber \\ &=
(K^2+K)(H_K-H_{\Gamma}) -\frac{K(1-\gamma)(3K-K\gamma-1)}{2} \nonumber \end{aligned}$$ channel scalars, while in the absence of caching (corresponding to $\Gamma = 0$), we will have to send D-CSIT on $$\begin{aligned}
L(\Gamma = 0) & = \sum_{j=1}^K \frac{1}{j}(K-j+1)(K-j) \nonumber \\ & = (K^2+K)H_K -\frac{3K^2}{2} + \frac{K}{2}\nonumber \end{aligned}$$ channel scalars.
To reflect the frequency of having to gather D-CSIT, and to provide a fair comparison between different schemes of different performance that manage to convey different amounts of actual data to the users, we consider the measure $Q(\Gamma)$ that normalizes the above number $L(\Gamma)$ of full D-CSIT scalars, by the coherence period $T_c$ and by the total number of full data symbols sent. In our case, under the assumption that each user receives a total of $\log_2(P)$ bits, the total number of full data symbols sent is $K$, and thus we have $$\begin{aligned}
Q(\Gamma) & = \frac{L(\Gamma)}{T_c K} \nonumber \\
&= \frac{(K^2+K)(H_K-H_{\Gamma}) -\frac{K(1-\gamma)(3K-K\gamma-1)}{2}}{T_c K}\nonumber\end{aligned}$$ while without caching, we have $$\begin{aligned}
Q(\Gamma = 0) & = \frac{L(\Gamma)}{T_c K}= \frac{(K+1)H_K - \frac{3}{2}K +\frac{1}{2}}{T_c}.\nonumber\end{aligned}$$ Consequently we see that in the large $K$ limit, $$\begin{aligned}
Q(\Gamma) \rightarrow \frac{K \bigl(\log(\frac{1}{\gamma})-\frac{3}{2}+2\gamma-2\gamma^2\bigr)}{T_c}\nonumber\end{aligned}$$ $$\begin{aligned}
Q(\Gamma = 0) \rightarrow \frac{1}{T_c}K\log(K)\nonumber \end{aligned}$$ which implies that $$\begin{aligned}
\lim_{K\rightarrow \infty} \frac{Q(\Gamma) }{Q(\Gamma = 0)} = 0 \nonumber\end{aligned}$$ which in turn tells us that as $K$ increases, for any fixed $\gamma$, caching allows for a substantial reduction (down to a vanishingly small portion) from the original cost of D-CSIT. This is illustrated in Figure \[fig:vanishingDelayedCSIT2\].
![Illustration of the vanishing fraction of D-CSIT cost, due to caching.[]{data-label="fig:vanishingDelayedCSIT2"}](vanishing)
This reduction is important because retrospective delayed-feedback methods suffer from an increased cost of supporting their CSIT requirements (cf. [@KC:12]) (albeit at the benefit of allowing substantial delays in the feedback mechanisms); after all, in the presence of perfect CSIT and zero forcing (no caching), the same cost is $$Q_{ZF} = \frac{K^2}{T_c K} = \frac{K}{T_c}$$ which gives that $$\begin{aligned}
\lim_{K\rightarrow \infty} \frac{Q(\Gamma = 0) }{Q_{ZF}} = \infty\nonumber\end{aligned}$$ which in turn verifies the above claim, and shows that the increase in the cost of supporting the D-CSIT (without caching) can be unbounded compared to ZF methods. On the other hand, we see that $$\begin{aligned}
\lim_{K\rightarrow \infty} \frac{Q(\Gamma) }{Q_{ZF}} = \log(\frac{1}{\gamma} - \frac{3}{2}+2\gamma-2\gamma^2)\nonumber\end{aligned}$$ which means that $$\lim_{K\rightarrow \infty} \frac{Q(\Gamma) }{Q_{ZF}} <1, \ \gamma\geq \frac{1}{10}.$$ One interesting conclusion that comes out of this, is that caching can allow for full substitution of current CSIT (as we have seen above), with a very substantial reduction of the cost of D-CSIT as well, where for $\gamma\geq \frac{1}{10}$ this cost is even less than that of the very efficient ZF, which has to additionally deal though with harder-to-obtain current CSIT. This cost reduction is also translated into a reduction in the cost of disseminating global channel state information at the receivers (global CSIR), where each receiver must now know (again with delay that is allowed to be large) the CSIR of only a fraction of the other receivers.
Cache-aided retrospective communications {#sec:schemeAlphaBigGamma}
========================================
We proceed to describe the communication scheme, and in particular the process of placement, folding-and-delivery, and decoding. In the end we calculate the achievable duration $T$.
The caching part is modified from [@MN14] to *‘fold’* (linearly combine) the different users’ data into multi-layered blocks, in a way such that the subsequent Q-MAT transmission algorithm (cf. [@KGZE:16]) (specifically the last $K-\eta_\alpha$ ($\eta_\alpha\in\{\Gamma,\dots,K-1\}$) phases of the QMAT algorithm) can efficiently deliver these blocks. Equivalently the algorithms are calibrated so that the caching algorithm creates a multi-destination delivery problem that is the same as that which is efficiently solved by the last stages of the QMAT-type communication scheme. We henceforth remove the subscript in $\eta_\alpha$ and simply use $\eta$, where now the dependence on $\alpha$ is implied.
Placement phase
---------------
We proceed with the placement phase which modifies on the work of [@MN14] such that when the CSIT quality $\alpha$ increases, the algorithm caches a decreasing portion from each file, but does so with increasing redundancy. The idea is that the higher the $\alpha$, the more private messages one can deliver directly without the need to multicast, thus allowing for some of the data to remain entirely uncached, which in turn allows for more copies of the same information across different users’ caches.
Here each of the $N$ files $W_n, n = 1, 2, \ldots, N$ ($|W_n| = f$ bits) in the library, is split into two parts $$\begin{aligned}
\label{eq:splitCachedUncached}
W_n = (W_n^c, W_n^{\overline{c}})\end{aligned}$$ where $W_n^c$ ($c$ for ‘cached’) will be placed into one or more caches, while the content of $W_n^{\overline{c}}$ ($\overline{c}$ for ‘non-cached’) will never be cached anywhere, but will instead be communicated — using CSIT — in a manner that causes manageable interference and hence does not necessarily benefit from coded caching. The split is such that $$\begin{aligned}
\label{eq:WNcSize}
|W_n^c| = \frac{KMf}{N\eta}\end{aligned}$$ where $\eta\in\{\Gamma,\dots,K-1\}$ is a positive integer, the value of which will be decided later on such that it properly regulates how much to cache from each $W_n$. Now for any specific $\eta$, we equally divide $W_n^c$ into $\binom{K}{\eta}$ subfiles $\{W^c_{n,\tau}\}_{\tau \in \Psi_{\eta}} $, $$\begin{aligned}
\label{eq:WnTau}
W_n^c = \{W^c_{n,\tau}\}_{\tau \in \Psi_{\eta}}\end{aligned}$$ where[^9] $$\begin{aligned}
\label{eq:PsiEta} \Psi_{\eta}:= \{\tau \subset [K] \ : \ |\tau| = \eta\}\end{aligned}$$ where each subfile has size $$\begin{aligned}
\label{eq:WnTauSize}
|W^c_{n,\tau}| = \frac{KMf}{N\eta\binom{K}{\eta}} = \frac{Mf}{N\binom{K-1}{\eta-1}} \ \text{bits}.\end{aligned}$$
Now drawing from [@MN14], the caches are filled as follows $$\begin{aligned}
\label{eq:ZkFill1} Z_k=\{W^c_{n,\tau}\}_{n \in [N], \tau\in \Psi_{\eta}^{(k)}}\end{aligned}$$ where $$\begin{aligned}
\label{eq:PsiEta_k}
\Psi_{\eta}^{(k)} := \{\tau \in \Psi_{\eta} \ : \ k\in \tau\}.\end{aligned}$$ Hence each subfile $W^c_{n, \tau}$ is stored in $Z_k$ as long as $k\in\tau$, which means that each $W^c_{n, \tau}$ (and thus each part of $W_n^c$) is repeated $\eta$ times in the caches. As $\eta$ increases with $\alpha$, this means that CSIT allows for a higher redundancy in the caches; instead of content appearing in $\Gamma$ different caches, it appears in $\eta\geq \Gamma$ caches instead, which will translate into multicast messages that are intended for more receivers.
Data folding
------------
At this point, the transmitter becomes aware of the file requests $R_k, k=1,\dots,K$, and must now deliver each requested file $W_{R_k}$, by delivering the constituent subfiles $\{W^c_{R_k,\tau}\}_{\tau\in\Psi_{\eta} \backslash \Psi_{\eta}^{(k)}}$ as well as $W_{R_k}^{\overline{c}}$, all to the corresponding receiver $k$. We quickly recall that:
1. subfiles $\{W^c_{R_k,\tau}\}_{\tau \in \Psi_{\eta}^{(k)}} $ are already in $Z_k$;
2. subfiles $\{W^c_{R_k,\tau}\}_{\tau\in\Psi_{\eta} \backslash \Psi_{\eta}^{(k)}}$ are directly requested by user $k$, but are not cached in $Z_k$;
3. subfiles $ Z_k \backslash \{W^c_{R_k,\tau}\}_{\tau \in \Psi_{\eta}^{(k)}} = Z_k \backslash W^c_{R_k}$ are cached in $Z_k$, are not directly requested by user $k$, but will be useful in removing interference.
We assume the communication here has duration $T$. Thus for each $k$ and a chosen $\eta$, we split each subfile $W^c_{R_k,\tau}, \ \tau\in\Psi_{\eta} \backslash \Psi_{\eta}^{(k)}$ (each of size $|W^c_{R_k,\tau}| = \frac{Mf}{N\binom{K-1}{\eta-1}}$ as we saw in ) into $$\begin{aligned}
\label{eq:WRktauSplit}
W^c_{R_k,\tau} = [ W^{c,f}_{R_k,\tau} \ \ W^{c,\overline{f}}_{R_k,\tau} ]\end{aligned}$$ where $W^{c,f}_{R_k,\tau}$ corresponds to information that appears in a cache somewhere and that will be eventually ‘folded’ (XORed) with other information, whereas $W_{R_k,\tau}^{c,\overline{f}}$ corresponds to information that is cached somewhere but that will not be folded with other information. The split yields $$\begin{aligned}
\label{eq:WRktauSplitSizes}
|W_{R_k,\tau}^{c,\overline{f}}| = \frac{f \alpha T-f(1-\frac{KM}{N\eta})}{\binom{K-1}{\eta}}\end{aligned}$$ where in the above, $f \alpha T$ represents the load for each user without causing interference during the delivery phase, where $f(1-\frac{KM}{N\eta})$ is the amount of uncached information, and where $|W_{R_k,\tau}^{c,f}|=|W^c_{R_k,\tau}|-|W_{R_k,\tau}^{c,\overline{f}}|$.
We proceed to fold cached content, by creating linear combinations (XORs) from $\{W_{R_k,\tau}^{c,f}\}_{\tau\in\Psi_{\eta} \backslash \Psi_{\eta}^{(k)}}, \forall k$. We will use $P_{k,k'}(\tau)$ to be the function that replaces inside $\tau$, the entry $k'\in \tau$, with the entry $k$. As in [@MN14], the idea is that if we deliver $$\begin{aligned}
\label{eq:WRktau1}
W_{R_k,\tau}^{c,f} \oplus (\oplus_{k'\in \tau}\underbrace{W^{c,f}_{R_{k'},P_{k,k'}(\tau)}}_{\in Z_k})\end{aligned}$$ the fact that $W^{c,f}_{R_{k'}, P_{k,k'}(\tau)} \in Z_k$, guarantees that receiver $k$ can recover $W_{R_k,\tau}^{c,f}$, while at the same time guarantees that each other user $k'\in \tau$ can recover its own desired subfile $W^{c,f}_{R_k',P_{k,k'}(\tau)} \notin Z_{k'}, \forall k' \in \tau$.
Hence delivery of each $W_{R_k,\tau}^{c,f} \oplus (\oplus_{k'\in \tau}W^{c,f}_{R_{k'},P_{k,k'}(\tau)})$ of size $|W_{R_k,\tau}^{c,f} \oplus (\oplus_{k'\in \tau}W^{c,f}_{R_{k'},P_{k,k'}(\tau)})| = |W_{R_k,\tau}^{c,f}|$ (cf. ), automatically guarantees delivery of $W^{c,f}_{R_{k'},P_{k,k'}(\tau)}$ to each user $k'\in \tau$, i.e., simultaneously delivers a total of $\eta+1$ distinct subfiles (each again of size $|W^{c,f}_{R_{k'},P_{k,k'}(\tau)}| = |W_{R_k,\tau}^{c,f}|$ bits) to $\eta+1$ distinct users. Hence *any* $$\begin{aligned}
\label{eq:XpsiDef} X_{\psi} := \oplus_{k \in \psi} W^{c,f}_{R_k,\psi \backslash \{k\}}, \psi \in \Psi_{\eta+1}\end{aligned}$$ — which is of the same form as in , and which is referred to here as an *order-($\eta+1$) folded message* — can similarly deliver to user $k\in \psi$, her requested file $W^{c,f}_{R_k,\psi \backslash k}$, which in turn means that each order-($\eta+1$) folded message $X_{\psi}$ can deliver — with the assistance of the side information in the caches — a distinct, individually requested subfile, to each of the $\eta + 1$ users $k\in \psi$ ($\psi\in \Psi_{\eta+1}$).
Thus to satisfy all requests $\{W_{R_k} \backslash Z_k \}_{k=1}^K$, the transmitter must deliver
uncached messages $W_{R_k}^{\overline{c}}, \ k=1,\dots,K$
cached but unfolded messages $\{W^{c,\overline{f}}_{R_k,\psi \backslash \{k\}}\}_{\psi \in \Psi_{\eta+1}}, \ k=1,\dots,K$
and the entire set $$\begin{aligned}
\label{eq:foldedMessages}
\mathcal{X}_\Psi := \{ X_{\psi} = \oplus_{k \in \psi} W^{c,f}_{R_k,\psi \backslash \{k\}}\}_{\psi \in \Psi_{\eta+1}}\end{aligned}$$ consisting of $$\begin{aligned}
\label{eq:cardinalityMathcalXPsi}
|\mathcal{X}_\Psi|=\binom{K}{\eta+1}\end{aligned}$$ folded messages of order-$(\eta+1)$, each of size (cf. ,) $$\begin{aligned}
\label{eq:XpsiSize}
|X_{\psi}| & = |W^{c,f}_{R_k,\tau}| = |W^c_{R_k,\tau}| - |W_{R_k,\tau}^{c,\overline{f}}| \nonumber\\
& = \frac{f(1-\gamma-{\alpha T)}}{\binom{K-1}{\eta}} \ \text{(bits)}.\end{aligned}$$
Transmission
------------
The transmission scheme is taken from [@KGZE:16], and each transmission takes the form $$\begin{aligned}
\label{txformperfect}
\xv_{t} = \textbf{G}_{c,t} \xv_{c,t}+ \sum_{k\in \bar{\psi}}\gv_{k,t} a_{k,t}^{*} +\sum_{k=1}^{K} \gv_{k,t} a_{k,t}\end{aligned}$$ where $t\in[0, T]$, where $\xv_{c,t}$ is a $K$-length vector for MAT-type symbols, where $a_{k,t}^{*}$ is the additional auxiliary symbols that carry residual interference (here we ‘load’ this round with additional requests from the users, for the very first round, $a_{k,t}^{*} =0$) (in the above, $\bar{\psi}$ is a set of ‘undesired’ users). In the above, each unit-norm precoder $\gv_{k,t}$ for user $k=1,2,\dots,K$, is simultaneously orthogonal to the CSI of all other channels, i.e., $$\begin{aligned}
\hat{\hv}_{k',t}^{T} \gv_{k,t} = 0, \ \ \forall k' \in [K] \backslash k.\end{aligned}$$ Each precoder $\textbf{G}_{c,t}$ is defined as $\textbf{G}_{c,t} = [\gv_{c,t}, \textbf{U}_{c,t}]$, where $\gv_{c,t}$ is simultaneously orthogonal to the channel estimates of the undesired receivers, and $\textbf{U}_{c,t} \in \C^{K\times(K-1)}$ is a randomly chosen, isotropically distributed unitary matrix[^10].
Throughout communication
we will allocate power such that $$\begin{aligned}
\E\{|\xv_{c,t}|_1^2\} & \doteq \E\{|a_{k,t}^{*}|^2\} \doteq P , \\ \E\{|\xv_{c,t}|_{i\neq 1}^2\} & \doteq P^{1-\alpha}, \ \E\{|a_{k,t}|^2\} \doteq P^{\alpha} \notag\end{aligned}$$ where $|\xv_{c,t}|_i, i=1,2,\cdots,K,$ denotes scalar $i$ in vector $\xv_{c,t}$, and we will allocate rate such that
each $\xv_{c,t}$ carries $f(1-\alpha)$ bits per unit time,
each $a_{k,t}^{*}$ carries $f (1-\alpha)$ bits per unit time.
and each $a_{k,t}$ carries $f \alpha $ bits per unit time.
Recall that instead of employing matrix notation, after normalization, we use the concept of signal duration $\text{dur}(\xv)$ required for the transmission of some vector $\xv $. We also note that due to time normalization, the time index $t\in [0, T]$, need not be an integer.
For any $\alpha$, our scheme will be defined by an integer $\eta\in [\Gamma,K-1]\cap \mathbb{Z}$, which will be chosen as $$\begin{aligned}
\label{eq:etaAlpha}
\eta = \arg\max_{\eta{'}\in [\Gamma,K-1]\cap \mathbb{Z}} \{\eta{'} \ : \ \alpha_{b,\eta'}\leq \alpha\}\end{aligned}$$ for $$\begin{aligned}
\label{eq:alphaBreak2}
{\alpha}_{b,\eta} = \frac{\eta-\Gamma}{\Gamma(H_K-H_\eta-1)+\eta} .\end{aligned}$$ $\eta$ will define the amount of cached information that will be folded ($\{W^{c,f}_{R_k,\tau}\}_{\tau\in \Psi_{\eta} \backslash \Psi_{\eta}^{(k)}}$ ), and thus also the amount of cached information that will not be folded ($\{W^{c,\overline{f}}_{R_k,\tau}\}_{\tau\in \Psi_{\eta}\backslash \Psi_{\eta}^{(k)}}$ ) and which will be exclusively carried by the different $a_{k,t}$. In all cases,
all of $\{X_{\psi}\}_{\psi \in \Psi_{\eta+1}}$ (which are functions of the cached-and-to-be-folded $\{W^{c,f}_{R_k,\tau}\}_{\tau\in \Psi_{\eta} \backslash \Psi_{\eta}^{(k)}}$) will be exclusively carried by $\xv_{c,t}, \ t\in[0, T]$, while
all of the uncached $W^{\overline{c}}_{R_k}$ (for each $k=1,\dots,K$) and all of the cached but unfolded $\{W^{c,\overline{f}}_{R_k,\tau}\}_{\tau\in\Psi_{\eta} \backslash \Psi_{\eta}^{(k)}}$ will be exclusively carried by $a_{k,t}, t\in[0,T]$.
*Transmission of $\{X_{\psi}\}_{\psi \in \Psi_{\eta+1}}$:* From [@KGZE:16], we know that the transmission relating to $\xv_{c,t}$ can be treated independently from that of $a_{k,t}$, simply because — as we will further clarify later on — the $a_{k,t}$ do not actually interfere with decoding of $\xv_{c,t}$, as a result of the scheme, and as a result of the chosen power and rate allocations which jointly adapt to the CSIT quality $\alpha$. For this reason, we can treat the transmission of $\xv_{c,t}$ separately.
Hence we first focus on the transmission of $\{X_{\psi}\}_{\psi \in \Psi_{\eta+1}}$, which will be sent using $\xv_{c,t}, \ t\in[0,T]$ using the last $K-\eta$ phases of the QMAT algorithm in [@KGZE:16] corresponding to having the ZF symbols $a_{k,t}$ set to zero. For ease of notation, we will label these phases starting from phase $\eta+1$ and terminating in phase $K$. The total duration is the desired $T$. Each phase $j = \eta+1, \dots,K$ aims to deliver order-$j$ folded messages (cf. ), and will do so gradually: phase $j$ will try to deliver (in addition to other information) $N_j := (K-j+1)\binom{K}{j}$ order-$j$ messages which carry information that has been requested by $j$ users, and in doing so, it will generate $N_{j+1} := j\binom{K}{j+1}$ signals that are linear combinations of received signals from $j+1$ different users, and where these $N_{j+1}$ signals will be conveyed in the next phase $j+1$. During the last phase $j=K$, the transmitter will send fully common symbols that are useful and decoded by all users, thus allowing each user to go back and retroactively decode the information of phase $j=K-1$, which will then be used to decode the information in phase $j=K-2$ and so on, until they reach phase $j = \eta+1$ (first transmission phase) which will complete the task. We proceed to describe these phases. We will use $T_j$ to denote the duration of phase $j$.
*Phase $\eta+1$:* In this first phase of duration $T_{\eta+1}$, the information in $\{X_{\psi}\}_{\psi \in \Psi_{\eta+1}}$ is delivered by $\xv_{c,t}, \ t\in[0,T_{\eta+1}] $, which can also be rewritten in the form of a sequential transmission of shorter-duration $K$-length vectors $$\begin{aligned}
\label{eq:vectorfirstphase}
\xv_{\psi} = [x_{\psi,1}, \dots, x_{\psi,K-\eta}, 0, \dots, 0]^{T}\end{aligned}$$ for different $\psi$, where each vector $\xv_{\psi}$ carries exclusively the information from each $X_{\psi}$, and where this information is uniformly split among the $K-\eta$ independent scalar entries $x_{\psi,i}, \ i=1,\dots,K-\eta$, each carrying $$\begin{aligned}
\label{eq:SizeScalar_x_psi}
\frac{|X_{\psi}|}{(K-\eta)} = \frac{f(1-\gamma-\alpha T)}{\binom{K-1}{\eta}(K-\eta)}\end{aligned}$$ bits (cf. ). Hence, given that the allocated rate for $\xv_{c,t}$ (and thus the allocated rate for each $\xv_{\psi}$) is $(1-\alpha)f$, we have that the duration of each $\xv_{\psi}$ is $$\begin{aligned}
\label{eq:durationVector_XPsi}
\text{dur}(\xv_{\psi}) = \frac{|X_{\psi}|}{(K-\eta)(1-\alpha)f}.\end{aligned}$$ Given that $|\mathcal{X}_\Psi|=\binom{K}{\eta+1}$, then $$\begin{aligned}
\label{eq:durationcal1}
T_{\eta+1} = \binom{K}{\eta+1} \text{dur}(\xv_{\psi}) =\frac{ \binom{K}{\eta+1} |X_{\psi}|}{(K-\eta)(1-\alpha)f}.\end{aligned}$$
After each transmission of $\xv_{\psi}$, each user $k\in[K]$ receives (in addition to information originating from $a_{k,t}$ which will be treated as noise and thus neglected for now), a linear combination $L_{\psi,k}$ of the transmitted $K-\eta$ symbols $x_{\psi,1}, x_{\psi,2}, \dots, x_{\psi,K-\eta}$. Next the transmitter will send an additional $K-\eta-1$ signals $L_{\psi,k'}, \ k'\in[K]\backslash \psi$ (linear combinations of $x_{\psi,1}, x_{\psi,2}, \dots, x_{\psi,K-\eta}$ as received — up to noise level — at each user $k'\in [K] \backslash \psi$) which will help each user $k\in \psi$ resolve the already sent $x_{\psi,1}, x_{\psi,2}, \dots, x_{\psi,K-\eta}$. This will be done in the next phase $j=\eta+2$.
*Phase $\eta+2$:* The challenge now is for signals $\xv_{c,t}, \ t \in (T_{\eta+1},T_{\eta+1}+T_{\eta+2}] $ to convey all the messages of the form $$L_{\psi,k'}, \ \forall k'\in[K]\backslash \psi, \ \forall \psi\in \Psi_{\eta+1}$$ to each receiver $k\in \psi$. Note that each of the above linear combinations, is now — during this phase — available (up to noise level) at the transmitter. Let $$\begin{aligned}
\Psi_{\eta+2} = \{\psi\in [K] \ : \ |\psi|=\eta+2 \}\end{aligned}$$ and consider for each $\psi\in \Psi_{\eta+2}$, a transmitted vector $$\xv_\psi = [x_{\psi,1}, \dots, x_{\psi,K-\eta-1}, 0, \dots, 0]^{T}$$ which carries the contents of $\eta+1$ different linear combinations $f_i(\{L_{\psi\backslash\{k\},k}\}_{k\in \psi}), i=1,\dots,\eta+1$ of the $\eta+2$ elements $\{L_{\psi\backslash\{k\},k}\}_{\forall k\in \psi}$ created by the transmitter. The linear combination coefficients defining each linear-combination function $f_i$, are predetermined and known at each receiver. The transmission of $\{\xv_{\psi}\}_{\forall \psi \in \Psi_{\eta+2}}$ is sequential.
It is easy to see that there is a total of $(\eta+1)\binom{K}{\eta+2}$ symbols of the form $f_i (\{L_{\psi\backslash\{k\},k}\}_{k\in \psi}), i=1,\dots,\eta+1, \ \psi\in \Psi_{\eta+2}$, each of which can be considered as an order-$(\eta+2)$ signal intended for $\eta+2$ receivers in $\psi$. Using this, and following the same steps used in phase $\eta+1$, we calculate that $$\begin{aligned}
\label{eq:durationPhase2}
T_{\eta+2} = \binom{K}{\eta+2}\text{dur}(\xv_{\psi}) = T_{\eta+1} \frac{\eta+1}{\eta+2}.\end{aligned}$$
We now see that for each $\psi$, each receiver $k \in \psi$ recalls their own observation $L_{\psi \backslash \{k\}, k}$ from the previous phase, and removes it from all the linear combinations $\{ f_i (\{L_{\psi\backslash\{k\},k}\}_{\forall k\in \psi})\}_{i=1,\dots,\eta+1}$, thus now being able to acquire the $\eta+1$ independent linear combinations $\{L_{\psi \backslash \{k'\}, k'}\}_{\forall k' \in \psi \backslash \{k\}}$. The same holds for each other user $k'\in \psi$.
After this phase, we use $L_{\psi,k}, \psi \in \Psi_{\eta+2}$ to denote the received signal at receiver $k$. Like before, each receiver $k, k\in \psi$ needs $K-\eta-2$ extra observations of $x_{\psi,1}, \dots, x_{\psi,K-\eta-1}$ which will be seen from $L_{\psi,k'},\forall k'\notin \psi$, which will come from order-$(\eta+3)$ messages that are created by the transmitter and which will be sent in the next phase.
*Phase $j$ $(\eta+3 \leq j \leq K)$:* Generalizing the described approach to any phase $j\in[\eta+3,\dots,K]$, we will use $\xv_{c,t}, \ t \in [\sum_{i=\eta+1}^{j-1}T_{i},\sum_{i=\eta+1}^{j}T_{i} ] $ to convey all the messages of the form $$L_{\psi,k'}, \ \forall k'\in[K]\backslash \psi, \ \forall \psi\in \Psi_{j-1}$$ to each receiver $k\in \psi$. For each $$\begin{aligned}
\psi\in \Psi_{j} := \{\psi\in [K] \ : ~ |\psi|=j \}\end{aligned}$$ each transmitted vector $$\xv_\psi = [x_{\psi,1}, \dots, x_{\psi,K-j-1}, 0, \dots, 0]^{T}$$ will carry the contents of $j-1$ different linear combinations $f_i(\{L_{\psi\backslash\{k\},k}\}_{k\in \psi}), i=1,\dots,j-1$ of the $j$ elements $\{L_{\psi\backslash\{k\},k}\}_{\forall k\in \psi}$ created by the transmitter. After the sequential transmission of $\{\xv_{\psi}\}_{\forall \psi \in \Psi_{j}}$, each receiver $k$ can obtain the $j-1$ independent linear combinations $\{L_{\psi \backslash \{k'\}, k'}\}_{\forall k' \in \psi \backslash \{k\}}$. The same holds for each other user $k'\in \psi$. As with the previous phases, we can see that $$\begin{aligned}
\label{eq:durationPhaseJ}
T_j = T_{\eta+1} \frac{\eta+1}{j}, \ j= \eta+3,\dots,K.\end{aligned}$$ This process terminates with phase $j = K$, during which each $$\xv_\psi = [x_{\psi,1}, 0 , 0, \dots, 0]^{T}$$ carries a single scalar that is decoded easily by all. Based on this, backwards decoding will allow for users to retrieve $\{ X_{\psi}\}_{\psi \in \Psi_{\eta+1}}$. This is described immediately afterwards. In treating the decoding part, we briefly recall that each $a_{k,t}, \ k=1,\dots,K$ carries (during period $t\in[0,T]$), all of the uncached $W^{\overline{c}}_{R_k}$ and all of the unfolded $\{W^{c,\overline{f}}_{R_k,\tau}\}_{\tau\in\Psi_{\eta} \backslash \Psi_{\eta}^{(k)}}$.
Decoding
--------
The whole transmission lasts $K-\eta$ phases. For each phase $j, j=\eta+1,\cdots,K$ and the corresponding $\psi$, the received signal $y_{k,t}, \ t \in [\sum_{i=\eta+1}^{j-1}T_{i},\sum_{i=\eta+1}^{j}T_{i} ]$ of desired user $k~(k \in \psi)$ takes the form $$\begin{aligned}
y_{k,t} = \underbrace{\overbrace{\hv_{k,t}^{T} \textbf{G}_{c,t} \xv_{c,t}}^{\text{rate} \ 1-\alpha} }_{L_{\psi,k}, \ \text{power} \ \doteq \ P}+ \underbrace{\overbrace{\hv_{k,t}^{T} \sum^K_{k \in \bar{\psi}}\gv_{k,t} a_{k,t}^{*}}^{\text{rate} \ 1-\alpha} }_{\doteq \ P^{1-\alpha}}+ \underbrace{\overbrace{\hv_{k,t}^{T} \gv_{k,t} a_{k,t}}^{\text{rate} \ \alpha }}_{P^{\alpha}}
$$ The received signal $y_{k',t}$ of undesired user $k'~(k' \in [K]\backslash \psi)$ takes the form $$\begin{aligned}
y_{k',t} = \underbrace{\overbrace{\hv_{k',t}^{T} \sum^K_{k' \in \bar{\psi}}\gv_{k',t} a_{k',t}^{*}}^{\text{rate} \ 1-\alpha} }_{\ \text{power} \ \doteq \ P} + \underbrace{\overbrace{\hv_{k',t}^{T} \textbf{G}_{c,t} \xv_{c,t}}^{\text{rate} \ 1-\alpha} }_{L_{\psi,k'}, \ \ \doteq \ P^{1-\alpha}}+ \underbrace{\overbrace{\hv_{k',t}^{T} \gv_{k',t} a_{k',t}}^{\text{rate} \ \alpha }}_{P^{\alpha}}\end{aligned}$$ where in both cases, we ignore the Gaussian noise and the ZF-related noise up to $P^{0}$. In addition to somehow send $L_{\psi,k'}$ to the next MAT phase, as we see in [@KGZE:16], after each phase, $L_{\psi,k'}$ is first quantized with $(1-2\alpha) \log P$ bits, which results in a residual quantizaton noise $n_{\psi,k'}$ with power scaling as $P^{\alpha}$. Then, the transmitter quantizes the quantization noise $n_{\psi,k'}$ with an additional $\alpha \log P$ bits, which will be carried by the auxiliary data symbols $a_{k',t}^{*}$ in the corresponding phase in the next round (note: additional requests from the users). In this way, we can see that the ‘common’ signal $\xv_{c,t}$ can be decoded with the assistance of an auxiliary data symbol from the next round. After this, each user $k$ will remove $\hv_{k,t}^{T} \textbf{G}_{c,t} \xv_{c,t}$ from their received signals, and readily decode their private symbols $a_{k,t}, \ t\in[0,T]$, thus allowing for retrieval of their own unfolded $\{W^{c,\overline{f}}_{R_k,\psi \backslash \{k\}}\}_{\psi \in \Psi_{\eta+1}}$ and uncached $W_{R_k}^{\overline{c}}$. In terms of decoding the common information, as discussed above, each receiver $k$ will perform a backwards reconstruction of the sets of overheard equations $$\ba{c}
\{L_{\psi,k'}, \ \forall k'\in[K]\backslash \psi\}_{\forall \psi\in \Psi_{K}} \\ \downarrow \\ \{L_{\psi,k'}, \ \forall k'\in[K]\backslash \psi\}_{\forall \psi\in \Psi_{K-1}} \\ \vdots \\ \downarrow \\ \{L_{\psi,k'}, \ \forall k'\in[K]\backslash \psi\}_{\forall \psi\in \Psi_{\eta+2}} \ea$$ until phase $\eta+2$. At this point, each user $k$ has enough observations to recover the original $K-\eta$ symbols $x_{\psi,1}, x_{\psi,2}, \dots, x_{\psi,K-\eta}$ that fully convey $X_{\psi}$, hence each user $k$ can reconstruct their own set $\{W^{c,f}_{R_k,\psi \backslash \{k\}}\}_{\psi \in \Psi_{\eta+1}}$ which, combined with the information from the $a_{k,t}, \ t=[0,T]$ allow for each user $k$ to reconstruct $\{W^c_{R_k,\psi \backslash \{k\}}\}_{\psi \in \Psi_{\eta+1}}$ which is then combined with $Z_{k}$ to allow for reconstruction of the requested file $W_{R_k}$.
Calculation of $T$
------------------
To calculate $T$, we recall from that $$\begin{aligned}
T &= \sum \limits_{j=\eta+1}^{K} T_j = T_{\eta+1}\sum \limits_{j=\eta+1}^{K} \frac{\eta+1}{j} \nonumber \\ &= (\eta+1)(H_K-H_\eta)T_{\eta+1} \label{eq:durationcal2}\end{aligned}$$ which combines with and to give $$\begin{aligned}
\label{eq:proofT1}
T = \frac{(K-\Gamma)(H_K-H_\eta)}{(K-\eta)+\alpha(\eta+K(H_K-H_\eta-1))}\end{aligned}$$ as stated in Theorem \[thm:bigGammaBest\]. The bound by $T=1-\gamma$ seen in the theorem, corresponds to the fact that the above expression applies, as is, only when $\alpha\leq \alpha_{b,K-1} = \frac{K(1-\gamma)-1}{(K-1)(1-\gamma)}$ which corresponds to $\eta = K-1$ (where $X_\psi$ are fully common messages, directly desired by all), for which we already get the best possible $T=1-\gamma$, and hence there is no need to go beyond $\alpha=\frac{K(1-\gamma)-1}{(K-1)(1-\gamma)}$.
Conclusions \[sec:conclusions\]
===============================
This work studied the previously unexplored interplay between coded-caching and CSIT feedback quality and timeliness. This is motivated by the fact that CSIT and coded caching are two powerful ingredients that are hard to obtain, and by the fact that these ingredients are intertwined in a synergistic and competing manner. In addition to the substantial cache-aided DoF gains revealed here, the results suggest the interesting practical ramification that distributing predicted content ‘during the night’, can offer continuous amelioration of the load of predicting and disseminating CSIT during the day.
Appendix\[sec:additionalProofs\]
================================
Lower bound on $T^*$ (proof of Lemma \[lem:outer\])\[sec:lower\]
----------------------------------------------------------------
This part draws from the bound in [@MN14], and it is similar to that in [@ZEsynergy:16] which deals with the case of $\alpha = 0$. To lower bound $T$, we consider the easier problem where we want to serve $s\leq K $ different files to $s$ users, each with access to all caches. We also consider that we repeat this (easier) last experiment $\lfloor \frac{N}{s} \rfloor$ times, thus spanning a total duration of $T \lfloor \frac{N}{s} \rfloor$ (and up to $\lfloor \frac{N}{s} \rfloor s$ files delivered). At this point, we transfer to the equivalent setting of the $s$-user MISO BC with delayed CSIT and imperfect current CSIT, and a side-information multicasting link to the receivers, of capacity $d_m$ (files per time slot). Under the assumption that in this latter setting, decoding happens at the end of communication, and once we set $$\begin{aligned}
\label{eq:fromdmToCaching} d_m T \lfloor \frac{N}{s} \rfloor = sM \end{aligned}$$ (which guarantees that the side information from the side link, throughout the communication process, matches the maximum amount of information in the caches), we have that $$\begin{aligned}
T \lfloor \frac{N}{s} \rfloor d^{'}_{\Sigma}(d_m) \geq \lfloor \frac{N}{s} \rfloor s\end{aligned}$$ where $d^{'}_{\Sigma}(d_m)$ is any sum-DoF upper bound on the above $s$-user MISO BC channel with delayed CSIT and the aforementioned side link. Using the bound $$d^{'}_{\Sigma}(d_m) = s \alpha + \frac{s}{H_s} (1-\alpha+d_m)$$ from Lemma \[lem:lowerSecond\], and applying , we get $$\begin{aligned}
T \lfloor \frac{N}{s} \rfloor \bigl( s\alpha + \frac{s}{H_s}(1-\alpha + \frac{sM}{T \lfloor \frac{N}{s} \rfloor }) \bigr)\geq \lfloor \frac{N}{s} \rfloor s\end{aligned}$$ and thus that $$\begin{aligned}
T \geq \frac{1}{(H_s \alpha+1-\alpha)} (H_s -\frac{Ms}{\lfloor \frac{N}{s}\rfloor })\end{aligned}$$ which implies a lower bound on the original $s$-user problem. Maximization over all $s$, gives the desired bound on the optimal $T^*$ $$\begin{aligned}
T^* \geq \mathop {\text{max}}\limits_{s\in \{1, \dots, \lfloor \frac{N}{M} \rfloor \}} \frac{1}{(H_s \alpha+1-\alpha)} (H_s -\frac{Ms}{\lfloor \frac{N}{s}\rfloor})\end{aligned}$$ required for the original $K$-user problem. This concludes the proof.
Bounding the sum-DoF of the $s$-user MISO BC, with delayed CSIT, $\alpha$-quality current CSIT, and additional side information\[sec:lowerSecond\]
--------------------------------------------------------------------------------------------------------------------------------------------------
We begin with the statement of the lemma, which we prove immediately below.
\[lem:lowerSecond\] For the $s$-user MISO BC, with delayed CSIT, $\alpha$-quality current CSIT, and an additional parallel side-link of capacity that scales as $d_m \log P$, the sum-DoF is upper bounded as $$\begin{aligned}
\label{eq:lowerSecond}
d_{\Sigma}(d_m) \leq s \alpha + \frac{s}{H_s} (1-\alpha+d_m).\end{aligned}$$
Our proof traces the proof of [@CYOG:14], adapting for the additional $\alpha$-quality current CSIT.
Consider a permutation $\pi$ of the set $\mathcal{E}= \{1,2,\cdots,s\}$. For any user $k , k \in \mathcal{E}$, we provide the received signals $y_k^{[n]}$ as well as the message $W_k$ of user $k$ to user $k+1, k+2, \cdots, s$. We define the following notation $$\Omega^{[n]} := \{\hv_k^{[n]}\}_{k=1}^{s},~~ \hat{\Omega}^{[n]} := \{\hat\hv_k^{[n]}\}_{k=1}^{s}, ~~\mathcal{U}^{[n]} := \{\Omega^{[n]}, \hat{\Omega}^{[n]}\},$$ $$\hv_k^{[t]} := \{\hv_k^{(i)}\}_{i=1}^{t} ,~~ y_k^{[t]}:= \{y_k^{(i)}\}_{i=1}^{t}, t=1,2,\cdots,n,$$ $$W_{[k]} := \{W_1,W_2,\cdots,W_k\},~~y_{[k]}^{[n]} := \{y_1^{[n]},y_2^{[n]},\cdots,y_k^{[n]}\}.$$ Then for $k =1,2,\cdots,s$, we have $$\begin{aligned}
& n (R_k-\epsilon_n) \notag \\
&\leq I(W_k; y_{[k]}^{[n]},y_0^{[n]}, W_{[k-1]} | \mathcal{U}^{[n]}) \label{eq:DoFbound1} \\
& = I(W_k; y_{[k]}^{[n]},y_0^{[n]}| W_{[k-1]}, \mathcal{U}^{[n]}) \label{eq:DoFbound2} \\
& = I(W_k; y_{[k]}^{[n]}| W_{[k-1]}, \mathcal{U}^{[n]}) + I(W_k; y_0^{[n]}| y_{[k]}^{[n]}, W_{[k-1]}, \mathcal{U}^{[n]}) \notag \\
& = h(y_{[k]}^{[n]}|W_{[k-1]}, \mathcal{U}^{[n]}) - h(y_{[k]}^{[n]}| W_{[k]}, \mathcal{U}^{[n]}) \notag \\
& ~~ + h(y_0^{[n]}| y_{[k]}^{[n]}, W_{[k-1]}, \mathcal{U}^{[n]}) - h(y_0^{[n]}|y_{[k]}^{[n]}, W_{[k]}, \mathcal{U}^{[n]}) \label{eq:DoFbound7}\end{aligned}$$ where follows from Fano’s inequality, where holds due to the fact that the messages are independent, and where the last two steps use the basic chain rule. Note that $W_0 = 0$. $$\begin{aligned}
&\sum_{k=1}^{s-1} \big( \frac{h(y_{[k+1]}^{[n]}|W_{[k]}, \mathcal{U}^{[n]})}{k+1} - \frac{h(y_{[k]}^{[n]}| W_{[k]}, \mathcal{U}^{[n]})}{k} \big) \notag \\
&=\sum_{t=1}^{[n]} \sum_{k=1}^{s-1} \big( \frac{h( y_{1}^{(t)},\cdots,y_{k+1}^{(t)}|y_{1}^{[t-1]},\cdots,y_{k+1}^{[t-1]}, W_{[k]}, \mathcal{U}^{[n]})}{k+1} \notag \\
&~~- \frac{h(y_{1}^{(t)},\cdots,y_{k}^{(t)}|y_{1}^{[t-1]},\cdots,y_{k}^{[t-1]}, W_{[k]}, \mathcal{U}^{[n]})}{k} \big) \label{eq:DoFbound3} \\
&=\sum_{t=1}^{[n]} \sum_{k=1}^{s-1} \big( \frac{h( y_{1}^{(t)},\cdots,y_{k+1}^{(t)}|y_{1}^{[t-1]},\cdots,y_{k+1}^{[t-1]}, W_{[k]}, \mathcal{U}^{[t]})}{k+1} \notag \\
&~~- \frac{h(y_{1}^{(t)},\cdots,y_{k}^{(t)}|y_{1}^{[t-1]},\cdots,y_{k}^{[t-1]}, W_{[k]}, \mathcal{U}^{[t]})}{k} \big) \label{eq:DoFbound4} \\
&\leq \sum_{t=1}^{[n]} \sum_{k=1}^{s-1} \big( \frac{h( y_{1}^{(t)},\cdots,y_{k+1}^{(t)}|y_{1}^{[t-1]},\cdots,y_{k+1}^{[t-1]}, W_{[k]}, \mathcal{U}^{[t]})}{k+1} \notag \\
&~~- \frac{h(y_{1}^{(t)},\cdots,y_{k}^{(t)}|y_{1}^{[t-1]},\cdots,y_{k+1}^{[t-1]}, W_{[k]}, \mathcal{U}^{[t]})}{k} \big) \label{eq:DoFbound5} \\
&\leq \sum_{t=1}^{[n]} \sum_{k=1}^{s-1} \frac{1}{k+1} \alpha \log P + n \cdot o(\log P) \label{eq:DoFbound6} \\
& \leq n (H_s-1) \alpha \log P + n \cdot o(\log P) \end{aligned}$$ where follows from the linearity of the summation, where holds since the received signal is independent of the future channel state information, where uses the fact that conditioning reduces entropy, and where is from the fact that Gaussian distribution maximizes differential entropy under the covariance constraint and from [@KYG:13 Lemma 2]. From , we then have $$\begin{aligned}
&\sum_{k=1}^{s} \frac{n (R_k-\epsilon_n) }{k} \notag \\
&\leq \sum_{k=1}^{s-1} \big( \frac{h(y_{[k+1]}^{[n]}|W_{[k]}, \mathcal{U}^{[n]})}{k+1} - \frac{h(y_{[k]}^{[n]}| W_{[k]}, \mathcal{U}^{[n]})}{k} \big) \notag \\
&~~+ h(y_1^{[n]}|\mathcal{U}^{[n]})-\frac{1}{s} h(y_{[s]}^{[n]}|W_{[s]}, \mathcal{U}^{[n]}) \notag \\
&~~+ \!\! \sum_{k=1}^{s-1} \big( \frac{H(y_0^{[n]}|y_{[k+1]}^{[n]}, W_{[k]}, \mathcal{U}^{[n]})}{k+1} \! - \! \frac{H(y_0^{[n]}|y_{[k]}^{[n]}, W_{[k]}, \mathcal{U}^{[n]})}{k} \big) \notag \\
&~~+ H(y_0^{[n]}|y_1^{[n]}, \mathcal{U}^{[n]})-\frac{1}{s} H (y_0^{[n]}|y_{[s]}^{[n]},W_{[s]}, \mathcal{U}^{[n]}) \notag \\
& \leq n (H_s-1) \alpha \log P + \underbrace{h(y_1^{[n]}|\mathcal{U}^{[n]})}_{\leq n \log P} + \underbrace{H(y_0^{[n]}|y_1^{[n]}, \mathcal{U}^{[n]})}_{\leq n \cdot d_m \log P} \notag \\
&~~+ \sum_{k=1}^{s-1} \big( (\frac{1}{k+1}-\frac{1}{k}) H(y_0^{[n]}|y_{[k]}^{[n]}, W_{[k]}, \mathcal{U}^{[n]} \big) + n \cdot o(\log P) \notag \\
&\leq n (H_s-1) \alpha \log P + n \log P +n \cdot d_m \log P + n \cdot o(\log P).\end{aligned}$$ Dividing by $n \log P$ and letting $P \rightarrow \infty$ gives $$\begin{aligned}
\sum_{k=1}^{s} \frac{d_k}{k} \leq (H_s-1) \alpha + 1 + d_m\end{aligned}$$ which implies that $$\begin{aligned}
d_{\Sigma}(d_m) \leq s \alpha + \frac{s}{H_s} (1-\alpha+d_m)\end{aligned}$$ which completes the proof of Lemma \[lem:lowerSecond\].
Bounding the gap between the achievable $T$ and the optimal $T^*$ \[sec:gapCalculation\]
----------------------------------------------------------------------------------------
Our aim here is to show that $$\frac{T(\gamma,\alpha > 0)}{T^*(\gamma,\alpha > 0)}<4$$ and we will do so by showing that the above gap is smaller than the gap we calculated in [@ZEsynergy:16] for $\alpha = 0$, which was again bounded above by 4. For this, we will use the expression[^11] $$\begin{aligned}
\label{eq:gapProofT1}
T(\gamma,\alpha > 0) = \frac{(1-\gamma)(H_K-H_{\Gamma})}{\alpha(H_K-H_{\Gamma})+(1-\alpha)(1-\gamma)}\end{aligned}$$ from Theorem \[thm:bigGamma\], and the expression $$\begin{aligned}
T^*(\gamma,\alpha > 0) \geq \mathop {\text{max}}\limits_{s\in \{1, \dots, \lfloor \frac{N}{M} \rfloor \}} \frac{1}{(H_s \alpha+1-\alpha)} (H_s -\frac{Ms}{\lfloor \frac{N}{s} \rfloor})\end{aligned}$$ from Lemma \[lem:outer\]. Hence we have $$\begin{aligned}
\frac{T}{T^*} & \leq \frac{\frac{(1-\gamma)(H_K-H_{K \gamma})}{\alpha(H_K-H_{K \gamma})+(1-\alpha)(1-\gamma)}}{\max\limits_{s\in \{1, \dots, \lfloor \frac{N}{M} \rfloor\}} \frac{1}{(H_s \alpha+1-\alpha)} (H_s -\frac{Ms}{\lfloor \frac{N}{s} \rfloor})} \label{eq:bound20} \\
& \leq \underbrace{\frac{\frac{(1-\gamma)(H_K-H_{K \gamma})}{\alpha(H_K-H_{K \gamma})+(1-\alpha)(1-\gamma)}}{\frac{1}{(H_{s_c} \alpha+1-\alpha)}(H_{s_c} -\frac{Ms_c}{\lfloor \frac{N}{s_c} \rfloor})}}_{g(s_c,\gamma)}\label{eq:bound20b}\end{aligned}$$ where $s=s_c\in \{1, \dots, \lfloor \frac{N}{M} \rfloor\}$, but where this $s_c$ will be chosen here to be exactly the same as in the case of $\alpha = 0$. This will be useful because, for that case of $\alpha = 0$, we have already proved that the same specific $s_c$ guarantees that $$\begin{aligned}
\frac{H_{K}-H_{K\gamma}}{H_{s_c} -\frac{Ms_c}{\lfloor \frac{N}{s_c} \rfloor}} < 4 \label{bound21}\end{aligned}$$ for the appropriate ranges of $\gamma$. This will apply towards bounding .
The proof is broken in two cases, corresponding to $\gamma\in[\frac{1}{36} ,\frac{K-1}{K}]$, and $\gamma\in[0 ,\frac{1}{36}]$.
### Case 1 ($\alpha > 0,\gamma\in[\frac{1}{36} ,\frac{K-1}{K}]$)
As when $\alpha = 0$ (cf. [@ZEsynergy:16]), we again set $s=1$, which reduces to $$\frac{T(\alpha>0,\gamma)}{T^*(\alpha>0,\gamma)} \leq \frac{\frac{(1-\gamma)(H_K-H_{K \gamma})}{\alpha(H_K-H_{K \gamma})+(1-\alpha)(1-\gamma)}}{1-\gamma}.$$ For this case — when $\alpha$ was zero, and when we chose the same $s = 1$ — we have already proved that $$\frac{T(\alpha=0,\gamma)}{1-\gamma}<4.$$ As a result, since $T(\alpha>0,\gamma)<T(\alpha=0,\gamma)$, and since $1-\gamma\leq T^*$, we conclude that $$\frac{T(\alpha>0,\gamma)}{T^*}<4, \ \gamma\in[\frac{1}{36} ,\frac{K-1}{K}]$$ which completes this part of the proof.
### Case 2 ($\alpha > 0, \gamma\in[0,\frac{1}{36}]$)
Going back to , we now aim to bound $$\begin{aligned}
\label{eq:def_g1}
g(s_c,\gamma):=\frac{\frac{(1-\gamma)(H_K-H_{K \gamma})}{\alpha(H_K-H_{K \gamma})+(1-\alpha)(1-\gamma)}}{\frac{1}{(H_{s_c} \alpha+1-\alpha)}(H_{s_c} -\frac{Ms_c}{\lfloor \frac{N}{s_c} \rfloor})} < 4.\end{aligned}$$ We already know from the case of $\alpha = 0$ (cf. ) that $$\begin{aligned}
\frac{H_K-H_{K\gamma}}{H_{s_c} -\frac{Ms_c}{\lfloor \frac{N}{s_c} \rfloor}} <4 \label{bound22-aa}\end{aligned}$$ holds. Hence we will prove that $$\begin{aligned}
\label{bound22-bb}
g(s_c,\gamma) \leq \frac{H_K-H_{K\gamma}}{H_{s_c} -\frac{Ms_c}{\lfloor \frac{N}{s_c} \rfloor}}\end{aligned}$$ to guarantee the bound. We note that is implied by $$\begin{aligned}
H_{s_c} \leq \frac{H_K-H_{K \gamma}}{1-\gamma} \label{bound22}\end{aligned}$$ which is implied by $$\begin{aligned}
\log(s_c) \leq \frac{\log(1/ \gamma)}{1-\gamma}-\epsilon_6, \ \epsilon_6=H_6-\log(6) \label{bound23}\end{aligned}$$ because $H_{s_c} \leq \log(s_c)+\epsilon_6, \forall s_c \geq 6,\forall \gamma \in [0, \frac{1}{36}], \forall K$. Furthermore is implied by $$\begin{aligned}
\frac{1}{2} \log(\frac{1}{\gamma}) \leq \frac{\log(1/ \gamma)}{1-\gamma}-\epsilon_6 \label{bound24}\end{aligned}$$ because $\gamma \in [\frac{1}{(s_c+1)^2}, \frac{1}{s_c^2}]$ means that $s_c \leq \sqrt{\frac{1}{\gamma}}$. Since $\frac{1}{1-\gamma} \geq 1$, then is implied by $$\begin{aligned}
\frac{1}{2} \log(\frac{1}{\gamma}) \leq \log(\frac{1}{\gamma})-\epsilon_6 \label{bound25}.\end{aligned}$$ It is obvious that holds since $\gamma \leq \frac{1}{36}$. Towards this, by proving , we guarantee and the desired bound. This completes the proof.
[^1]: The authors are with the Mobile Communications Department at EURECOM, Sophia Antipolis, 06410, France (email: [email protected], [email protected]). The work of Petros Elia was supported by the European Community’s Seventh Framework Programme (FP7/2007-2013) / grant agreement no.318306 (NEWCOM\#), and from the ANR Jeunes Chercheurs project ECOLOGICAL-BITS-AND-FLOPS.
[^2]: An initial version of this paper has been reported as Research Report No. RR-15-307 at EURECOM, August 25, 2015, http://www.eurecom.fr/publication/4723 as well as was uploaded on arxiv in November 2015.
[^3]: In the high SNR regime of interest here, $\alpha=0$ corresponds to having essentially no current CSIT (cf. [@DJ:14]), while having $\alpha = 1$ corresponds (again in the high SNR regime) to perfect and immediately available CSIT (cf. [@Caire+:10m]).
[^4]: We note that setting $f = \log_2(P)$ is simply a normalization of choice, and does not carry a ‘forced’ relationship between SNR and file sizes. The essence of the derived results would remain the same for any other non-trivial normalization.
[^5]: The DoF measure is designed to exclude the benefits of having some content already available at the receivers (local caching gain), and thus to limit the DoF between 0, and the interference free optimal DoF of 1.
[^6]: For large $K$, this approximation $\frac{H_K-H_{\Gamma}}{\log(\frac{1}{\gamma})} = 1$ is tight for any *fixed* $\gamma$.
[^7]: We note that these interference-removal gains, particularly in the large $K$ regime, are not a result of extra performance boost directly from D-CSIT, because in the large $K$ setting, this latter performance boost is negligible (vanishes to zero) without caching.
[^8]: This conclusion is general (and not dependent on the specific schemes), because the used schemes are optimal for $\alpha = 1$. The statement holds because we can simply uniformly cache a fraction $\gamma$ of each file in each cache, and upon request, use perfect-CSIT to zero-force the remaining requested information, to achieve the optimal $T^*(\gamma,\alpha = 1) = 1-\gamma$, which leaves us with local (data push) caching gains only.
[^9]: We recall that in the above, $\tau$ and $W^c_{n,\tau}$ are sets, thus $|\tau|,|W^c_{n,\tau}|$ denote cardinalities; $|\tau| = \eta$ means that $\tau$ has $\eta$ different elements from $[K]$, while $|W^c_{n,\tau}|$ describes the size of $W^c_{n,\tau}$ in bits.
[^10]: Whenever possible, we will henceforth avoid going into the details of the Q-MAT scheme. Some aspects of this scheme are similar to MAT, and a main new element is that Q-MAT applies digital transmission of interference, and a double-quantization method that collects and distributes residual interference across different rounds (this is here carried by $a_{k,t}^{*}$), in a manner that allows for ZF and MAT to coexist at maximal rates. Some of the details of this scheme are ‘hidden’ behind the choice of $\textbf{G}_{c,t}$ and behind the loading of the MAT-type symbols $\xv_{c,t}$ and additional auxiliary symbols $a_{k,t}^{*}$. The important element for the decoding part later on, will be how to load the symbols, the rate of each symbol, and the corresponding allocated power. An additional element that is hidden from the presentation here is that, while the Q-MAT scheme has many rounds, and while decoding spans more than one round, we will — in a slight abuse of notation — focus on describing just one round, which we believe is sufficient for the purposes of this paper here.
[^11]: We note that the here derived upper bound on the gap corresponding to the $T$ in Theorem \[thm:bigGamma\], automatically applies as an upper bound to the gap corresponding to the $T$ from Theorem \[thm:bigGammaBest\], because the latter $T$ is smaller than the former.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is well-known that an undirected graph has no odd cycle if and only if it is bipartite. A less obvious, but similar result holds for directed graphs: a strongly connected digraph has no odd cycle if and only if it is bipartite. Can this result be further generalized to more general graphs such as edge-colored graphs? In this paper, we study this problem and show how to decide if there exists an odd properly colored cycle in a given edge-colored graph. As a by-product, we show how to detect if there is a perfect matching in a graph with even (or odd) number of edges in a given edge set.'
author:
- Gregory Gutin
- Bin Sheng
- 'Magnus Wahlstr[ö]{}m'
title: 'Odd Properly Colored Cycles in Edge-Colored Graphs'
---
Introduction
============
A graph $G$ is [*edge-colored*]{} if each edge of $G$ is assigned a color. (Edge-colorings can be arbitrary, not necessarily proper.) A cycle $C$ in an edge-colored graph is [*properly colored (PC)*]{} if no pair of adjacent edges of $C$ have the same color. It is not hard to see that PC cycles in edge-colored graphs generalize directed cycles (dicycles) in digraphs: in a digraph $D$ replace every arc $uv$ by an undirected path $ux_{uv}v$, where $x_{uv}$ is a new vertex, and edges $ux_{uv}$ and $x_{uv}v$ are of colors 1 and 2, respectively. Similarly, PC cycles generalize cycles in undirected graphs, e.g., by assigning every edge a distinct color.
One of the central topics in graph theory is the existence of certain kinds of cycles in graphs. In digraphs, it is not hard to decide the existence of any dicycle by simply checking whether a given digraph is acyclic [@GG]. The problem of existence of PC cycles in edge-colored graphs is less trivial. To solve the problem, we may use Yeo’s theorem [@AY]: if an edge-colored graph $G$ has no PC cycle then $G$ contains a vertex $z$ such that no connected component of $G-z$ is joined to $z$ with edges of more than one color. Thus, we can recursively find such vertices $z$ and delete them from $G$; if we end up with a trivial graph (containing just one vertex) then $G$ has no PC cycles; otherwise $G$ has a PC cycle. Clearly, the recursive algorithm runs in polynomial time.
One of the next natural questions is to decide whether a digraph has an odd (even, respectively) dicycle, i.e. a dicycle of odd (even) length, respectively. For odd dicycles we can employ the following well-known result (see, e.g., [@GG; @HNC]): A strongly connected digraph is bipartite if and only if it has no odd dicycle. (Note that the result does not hold for non-strongly connected digraphs.) Thus, to decide whether a digraph $D$ has an odd cycle, we can find strongly connected components of $D$ and check whether all components are bipartite. This leads to a simple polynomial-time algorithm. The question of whether we can decide in polynomial time whether a digraph has an even dicycle, is much harder and was an open problem for quite some time till it was solved, in affirmative, independently by McCuaig, and Robertson, Seymour and Thomas (see [@RST]) who found highly non-trivial proofs.
In this paper we consider the problem of deciding the existence of an odd PC cycle (and of finding one, if it exists) in an edge-colored graph in polynomial time. We show that while a natural extension of the odd dicycle solution does not work, an algebraic approach using Tutte matrices and the Schwartz-Zippel lemma allows us to prove that there is a randomized polynomial-time algorithm for solving the problem. The existence of a deterministic polynomial-time algorithm for the odd PC cycle problem remains an open question, as does the existence of a polynomial-time algorithm for the even PC cycle problem.
In this paper, we allow our graphs to have multiple edges (but no loops) and call them, for clarity, [*multigraphs*]{}. In edge-colored multigraphs, we allow parallel edges of different colors (there is no need to consider parallel edges of the same color). For an edge $xy$ and a vertex $v$, we use $\chi(xy)$ and $\chi(v)$ to denote the color of $xy$ and the set of colors of edges incident to $v$, respectively. For any other terminology and notation not provided here, we refer the readers to [@GG]. There is an extensive literature on PC paths and cycles: for a detailed survey of pre-2009 publications, see Chapter 16 of [@GG]; more recent papers include [@AD+; @FM; @Lo1; @Lo2; @Lo3].
Graph-Theoretical Approaches
============================
Recall that to solve the odd dicycle problem, in the previous section, we used the following result: A strongly connected digraph is bipartite if and only if it has no odd dicycle. It is not straightforward to generalize strong connectivity to edge-colored multigraphs. Indeed, color-connectivity[^1], introduced by Saad [@Sa] under another name, does not appear to be useful to us as, in general, it does not partition vertices into components. Thus, we will use cyclic connectivity introduced by Bang-Jensen and Gutin [@BJG] as follows. Let $P=\{H_1, \dots, H_p\}$ be a set of subgraphs of an edge colored multigraph $G$. The [*intersection graph*]{} $\Omega(P)$ of $P$ has the vertex set $P$ and the edge set $\{H_iH_j: V(H_i)\cap V(H_j)\neq \emptyset, 1\leq i < j\leq p\}$. A pair $x,y$ of vertices in an edge-colored multigraph $H$ is *cyclic connected* if $H$ has a collection of PC cycles $P=\{C_1, \dots, C_p\}$ such that $x$ and $y$ belong to some cycles in $P$ and $\Omega(P)$ is a connected graph. A maximum cyclic connected induced subgraph of $G$ is called a [*cyclic connected component*]{} of $G$. Note that cyclic connected components partition the vertices of $G$. Also note that cyclic connectivity for digraphs, where dicycles are considered instead of PC cycles, coincides with strong connectivity. One could wonder whether every non-bipartite cyclic connected edge-colored graph has an odd PC cycle. Unfortunately, it is not true, see a graph $H$ in Fig. \[fig:counterExample\]. It is not hard to check that $H$ is not bipartite and cyclic connected. It has even PC cycles, such as $v_1v_2v_5v_3v_1$, but no odd PC cycles.
\[figure1\]
(5,5)\[fill\]circle \[radius=0.07\]–(5,7)\[fill\]circle \[radius=0.07\]node \[left\][$v_3$]{}; (5,5) node \[below\][$v_1$]{}–(7,5)\[fill\]circle \[radius=0.07\] node \[below\][$v_2$]{}; (5,7)–(6,6)\[fill\]circle \[radius=0.07\]node \[below\][$v_4$]{}; (6,6)–(7,7)\[fill\]circle \[radius=0.07\]node \[right\][$v_5$]{}; (5,7)–(6,8)\[fill\]circle \[radius=0.07\]node \[above\][$v_6$]{}; (6,8)–(7,7); (7,7)–(7,5); (5,7)–(7,7);
(10,5)node\[left\][colour 1]{} –(11,5) ; (10,6)node\[left\][colour 2]{} –(11,6) ; (10,7) node\[left\][colour 3]{} –(11,7) ;
Another natural idea is to find some odd PC closed walk first, and hope to find an odd PC cycle in it. Unfortunately, we cannot generate all possible PC closed walks in polynomial time, and moreover a PC closed walk does not necessarily contain an odd PC cycle, see the graph in Figure \[fig:oddWalk\]. It contains an odd PC walk, but not an odd PC cycle.
\[figure2\]
(0,0) circle \[radius=0.07\]node\[left\][$v_1$]{}–(1,0)circle \[radius=0.07\]node\[below\][$v_2$]{}; (1,0) –(2,1)\[fill\]circle \[radius=0.07\] node \[below\][$v_5$]{}; (2,1) –(3,0) node \[below\][$v_3$]{}; (3,0)\[fill\]circle \[radius=0.07\] –(4,0)\[fill\]circle \[radius=0.07\] node \[below\][$v_4$]{}; (0,0) –(0,1)\[fill\]circle \[radius=0.07\] node \[left\][$v_0$]{}; (0,1) –(0,2)\[fill\]circle \[radius=0.07\] node \[left\][$v_6$]{}; (0,2) –(1,2)\[fill\]circle \[radius=0.07\] node \[above\][$v_7$]{}; (1,2) –(2,1) ; (2,1) –(3,2)\[fill\]circle \[radius=0.07\] node \[above\][$v_8$]{}; (3,2) –(4,2)\[fill\]circle \[radius=0.07\] node \[right\][$v_9$]{}; (4,0) –(4,2) ; (6,0)node\[left\][colour 1]{} –(7,0) ; (6,1)node\[left\][colour 2]{} –(7,1) ; (6,2) node\[left\][colour 3]{} –(7,2) ;
Algebraic Approach {#sec3}
==================
In an edge-colored multigraph $G$, a vertex $v$ is [*monochromatic*]{} if $|\chi(v)|=1$. Let $G'$ be the multigraph obtained from $G$ by recursively deleting monochromatic vertices such that $G'$ has no monochromatic vertex. Following Szeider [@Sz], let $G_x$, $x\in V(G')$ denote a graph with vertex set $$\{x_i,x'_i:\ i\in \chi(x)\}\cup \{x''_a,x''_b\} \mbox{ and edge set }$$ $$\{x''_ax''_b,x'_ix''_a,x'_ix''_b:\ i\in \chi(x)\}\cup \{x_ix'_i:\ i\in \chi(x)\}.$$ Let $G^*$ denote a graph with vertex set $\bigcup_{x\in V(G')}V(G_x)$ and edge set $E_1\cup E_2$, where $E_1=\bigcup_{x\in V(G')}E(G_x)$ and $E_2=\{y_qz_q:\ yz\in E(G'), \chi(yz)=q\}.$ Let $c=\max\{\chi(x):\ x\in V(G)\}$. Note that $$\label{eq0} |V(G^*)| = O(c|V(G)|)$$ A subgraph of an edge-colored multigraph is called a *PC cycle subgraph* if it consists of several vertex-disjoint PC cycles. We will use the following result of [@GE].
\[thm2\] Let $G$ be a connected edge-colored multigraph such that $G'$ is non-empty and $G^*$ constructed as above. Then $G$ has a PC cycle subgraph with $r$ edges if and only if $G^{*}$ has a perfect matching with exactly $r$ edges in $E_2$.
Using Theorem \[thm2\], the problem of deciding if there exists an odd PC cycle in $G$ reduces to that of deciding if there is a perfect matching with an odd number of edges from $E_2$ in the graph $G^{*}$ (indeed, $G^*$ has an odd PC cycle subgraph if and only if it has an odd PC cycle). We use the properties of Tutte matrices to solve the reduced problem. For a graph $G=(V,E)$ with $V=\{v_1,v_2,..., v_n\}$, the [*Tutte matrix*]{} $A_{G}$ is the $n \times n$ multivariate polynomial matrix with entries $$\label{eq1}
A_{G}(i,j) =
\left\{
\begin{array}{rl}
x_{ij} & \text{if } v_{i}v_{j}\in E ~~\text{and}~~i<j \\
-x_{ji} & \text{if } v_{i}v_{j}\in E ~~\text{and}~~i>j \\
0 & \text{otherwise,}
\end{array}
\right.$$ where $x_{ij}$ are distinct variables. Tutte [@tt] proved that a graph $G$ has a perfect matching if and only if $\det A_{G}$ is not identically 0.
We say that a matrix $A$ is *skew symmetric* if $A + A^{T}=0$. Note that the Tutte matrix is skew symmetric. In our argument, we will use the notion of Pfaffian of a skew symmetric matrix. Let $A = [a_{ij}]$ be a $2n \times 2n$ skew symmetric matrix. The [*Pfaffian*]{} of $A$ is defined as follows. $$\label{eq:pf}
\operatorname{pf}A = \sum_{\sigma} sgn(\sigma) \prod_{i=1}^n a_{\sigma(2i-1),\sigma(2i)},$$ where $sgn(\sigma)$ is the signature of $\sigma$ and the summation is over all permutations $\sigma$ such that $\sigma(2i-1)<\sigma(2i)$ for each $1 \leq i \leq n$, and $\sigma(2i)<\sigma(2i+2)$ for each $1 \leq i < n$ (i.e., each partition of the set $\{1,\ldots,2n\}$ into pairs is included in the sum exactly once). Note in particular that for a graph $G$, this formula for $\operatorname{pf}A_G$ enumerates every perfect matching of $G$ exactly once.
Observe that, if we regard $a_{ij}$ as indeterminate, $\operatorname{pf}A$ is a multi-linear polynomial. For an odd skew symmetric matrix $A$, the Pfaffian is defined to be zero. We will use the following well-known relation between Pfaffian and determinants of skew symmetric matrices (see, e.g., [@LM]).
\[thm3\] If $A$ is a skew symmetric matrix, then $\det A = (\operatorname{pf}A)^2$.
Given a graph $G=(V,E)$ and a subset of edges $E_0\subseteq E(G)$, we now define another skew symmetric matrix $A_{G,E_0}$ whose entries are $$\label{eq2}
A_{G, E_0}(i,j) =
\left\{
\begin{array}{rl}
-A_{G}(i,j) & \text{if } v_{i}v_{j}\in E_0 \\
A_{G}(i,j) & \text{if } v_{i}v_{j}\notin E_0 \\
\end{array}
\right.$$ It is easy to see that $A_{G, E_0}$ is also a skew symmetric matrix, thus by Theorem \[thm3\], $\det A_{G,E_0}= (\operatorname{pf}A_{G,E_0})^2$. Note that $A_G$ and $A_{G,E_0}$ only differ at entries corresponding to edges in $E_0$. We call a perfect matching $M$ in a graph $G$ [*$E_0$-odd*]{} ([*$E_0$-even*]{}, respectively) if $|M\cap E_0|$ is odd (even, respectively). Here is our key result.
\[lemma2\] Given a graph $G$ with even number of vertices and an edge subset $E_0\subseteq E(G)$, let $A_{G}$ and $A_{G,E_0}$ be defined as in (\[eq1\]) and (\[eq2\]). Then $\det A_{G,E_0}=\det A_{G}$ if and only if all the perfect matchings of $G$ are of same $E_0$-parity.
As both $A_{G}$ and $A_{G,E_0}$ are skew symmetric, by Theorem \[thm3\], $$\det A_G = (\operatorname{pf}A_G)^2 \mbox{ and } \det A_{G,E_0}
=(\operatorname{pf}A_{G,E_0})^2.$$ Thus $\det A_{G,E_0}=\det A_{G}$ if and only if $\operatorname{pf}A_{G,E_0}=\operatorname{pf}A_{G}$ or $\operatorname{pf}A_{G,E_0}=-\operatorname{pf}A_{G}$. By (\[eq:pf\]), $\operatorname{pf}A_G$ and $\operatorname{pf}A_{G,E_0}$ both enumerate perfect matchings of $G$, and by the definitions of $A_G$ and $A_{G,E_0}$, for each such matching $M$ its contributions to $\operatorname{pf}A_G$ and $\operatorname{pf}A_{G,E_0}$ differ (by a sign term) if and only if $M$ is $E_0$-odd. Hence $\operatorname{pf}A_{G,E_0}=\operatorname{pf}A_{G}$ ($\operatorname{pf}A_{G,E_0}=-\operatorname{pf}A_{G}$, respectively) if and only if each perfect matching in $G$ is $E_0$-even ($E_0$-odd, respectively).
For $G=G^*$ and $E_0=E_2\subseteq E(G^*)$, by Lemma \[lemma2\], if $\det {A^{*}}_{G,E_2}\neq \det A^{*}_{G}$, then $G^*$ has a $E_2$-odd perfect matching and a $E_2$-even perfect matching. If $\det {A^{*}}_{G,E_2}=\det A^{*}_{G}$, then every perfect matching of $G^*$ is either $E_0$-even or $E_0$-odd. In such a case, we can find a perfect matching $M$ of the graph $G^*$ in polynomial time, and decide the parity of $M\cap E_2$. So we have an algorithm for deciding if $G^*$ has a perfect matching with even or odd number of edges in $E_2$. Unfortunately, we do not know whether this algorithm is polynomial or not as there is no polynomial algorithm to decide whether a multivariate polynomial is identically zero. Fortunately, we can have a polynomial randomized algorithm due to the following well-known lemma, called the Schwartz-Zippel lemma.[^2]
\[thm6\] Let $P(x_{1},x_{2},\dots,x_{n})$ be a multivariate polynomial of total degree at most $d$ over a field $F$, and assume that $P$ is not identically zero. Pick $r_{1},r_{2},\dots,r_{n}$ uniformly at random from a finite set $S$ of values where $S\subset F$. Then the probability $\mathbb{P}(P(r_{1},r_{2},\dots,r_{n})=0)\leq \frac{d}{|S|}$.
Now it is not hard to prove the following:
\[thm7\] Let $G$ be an edge-colored multigraph with $n$ vertices and let $c=\max\{\chi(v):\ v\in V(G)\}$. There is a randomized algorithm running in time $O((cn)^{\omega})$, where $\omega < 2.3729$, that decides if there is an odd PC cycle in $G$, with false negative probability less than 1/4.
As $f(\textbf{x})=\det A^{*}_{G,E_2} -\det A^{*}_{G}$ is a multivariate polynomial of degree at most $n$, we may choose a set $S$ of real values, such that $|S|> 4n$, and use Lemma \[thm6\] to decide if $\det {A^{*}}_{G,E_2}\neq \det A^{*}_{G}$ in polynomial time, with false negative less than 1/4. To see that the running time is $O((cn)^{\omega})$, recall (\[eq0\]) and observe that computing the determinants of $A^{*}_{G,E_2}$ and $A^{*}_G$ will take time $O((cn)^\omega)$, where $\omega < 2.3729$, by the algorithm in [@VVW]. Finally, for the case that $f(\textbf{x}) \equiv 0$, we can use the algorithm of Mucha and Sankowski [@MS] to find a perfect matching in time $O((cn)^\omega)$, and then decide its $E_0$-parity.
Open Problems
=============
We have proved that the odd PC cycle problem can be solved in randomized polynomial time. A natural question is whether this problem can be solved in (deterministic) polynomial time. It was proved in [@GJSWY] that if an edge-colored graph $G $ has no PC closed walk then $G$ has a monochromatic vertex. This can be viewed as a characterization of edge-colored graphs with no PC closed walk and it implies that deciding whether $G$ has a PC closed walk is polynomial-time solvable. We leave open the question of characterizing edge-colored graphs with no odd PC closed walk, as this differs from graphs with no odd PC cycle as we saw in Section \[sec3\].
[**Acknowledgment.**]{} Research of BS was partially supported by China Scholarship Council.
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A. Yeo. A note on alternating cycles in edge-coloured graphs, J. Combin. Theory Ser. B 69 (1997), 222–225.
R. Zippel. Probabilistic algorithms for sparse polynomials. EUROSAM’79, Proceedings of the International Symposium on Symbolic and Algebraic Computation, 216–226, 1979.
[^1]: We will not define color-connectivity; an interested reader can find its definition in Sec. 16.6 of [@GG].
[^2]: It was independently discovered by several authors: Schwartz [@S], Zippel [@Z], DeMillo and Lipton [@DL].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The complex effect of genetic algorithm’s (GA) operators and parameters to its performance has been studied extensively by researchers in the past but none studied their interactive effects while the GA is under different problem sizes. In this paper, We present the use of experimental model (1) to investigate whether the genetic operators and their parameters interact to affect the offline performance of GA, (2) to find what combination of genetic operators and parameter settings will provide the optimum performance for GA, and (3) to investigate whether these operator-parameter combination is dependent on the problem size. We designed a GA to optimize a family of traveling salesman problems (TSP), with their optimal solutions known for convenient benchmarking. Our GA was set to use different algorithms in simulating selection ($\Omega_s$), different algorithms ($\Omega_c$) and parameters ($p_c$) in simulating crossover, and different parameters ($p_m$) in simulating mutation. We used several $n$-city TSPs ($n=\{5, 7, 10, 100, 1000\}$) to represent the different problem sizes (i.e., size of the resulting search space as represented by GA schemata). Using analysis of variance of 3-factor factorial experiments, we found out that GA performance is affected by $\Omega_s$ at small problem size (5-city TSP) where the algorithm Partially Matched Crossover significantly outperforms Cycle Crossover at $95\%$ confidence level. Under intermediate problem sizes (7-city and 10-city TSPs), we found out that the mean GA performance is affected by the $\Omega_s \times \Omega_c$ interaction where the average performance of GA across $p_c$ and $p_m$ varies at different $\Omega_s$-$\Omega_c$ combinations. At big problem sizes (100-city and 1000-city TSPs), we observed that a 3-way interaction among $\Omega_s$, $\Omega_c$, and $p_m$ exist to affect the GA performance averaged across different $p_c$. Similarly, we also observed that the 3-way interaction among $\Omega_s$, $p_c$ and $p_m$ affects the GA performance averaged across all $\Omega_c$. To explain these three-way interactions, we used the Duncan’s Multiple Range Test at $5\%$ probability level to perform pairwise comparison of means of GA performance.'
author:
- |
Jaderick P. Pabico and Elizer A. Albacea\
\
\
bibliography:
- 'ga-em.bib'
title: The Interactive Effects of Operators and Parameters to GA Performance Under Different Problem Sizes
---
Introduction
============
Genetic Algorithms (GAs) are probabilistic search techniques suited for solving large, complex, multidimensional, multimodal, discontinuous, and/or noisy search and optimization problems. Applied to such problems, GAs outperformed several tested search and optimization procedures such as the gradient techniques and some various forms of random search [@Beaty93; @Davis87b; @Davis91; @Goldberg89; @Grefenstette89; @Holland92]. In the past years, the GA algorithms for selection, crossover, and mutation and the GA parameters population size, crossover probability, and mutation rate have received much attention in research [@Lee94; @Back92; @Schraudolph92]. These studies show that depending on the operators used and the parameter setting, the behavior of the GA can range from that of random search to hill climbing [@Lee94]. Thus, designing a GA that would meet a specific problem domain’s resource constraints would require a significant effort in trying to find out the right GA operator-parameter combination.
Many researchers have attempted to find a set of genetic operators and parameters for GAs to perform optimally for solving a given problem domain [@DeJong75; @Grefenstette86; @Schaffer89; @Davis91; @Back92; @Schraudolph92; @Lee94]. These researchers have used techniques such as hand optimization, a meta-GA, brute force search, and adapting parameters which are costly and time consuming [@DeJong75; @Grefenstette86; @Schaffer89; @Davis91]. The techniques’ results can only give parameter settings that are robust on a particular problem (such as the Traveling Salesman Problem (TSP)), but not on all other problems in a particular domain (such as the combinatorial problem domain where TSP is classified) [@Davis91]. Furthermore, the parameters found in any of these techniques become a liability for GA when the GA structure is modified, such as using another crossover algorithm. Thus, the optimal parameters that resulted from any of the techniques described above may not be good for any GA solving another problem, even to those belonging to the same domain. On the other hand, experimental models can be used to answer the following questions which can not be answered by the techniques used by other researchers:
1. Are these genetic operators and their parameters act independently or dependently on GA performance?
2. If they act independently, how these operators and their parameters affect GA performance? What trend (i.e, linear, quadratic, etc.) these parameters give on GA performance?
3. If they act dependently, which of these operators and their parameters interactively affect GA performance and how?
Results of past studies [@Pabico96a; @Pabico96d] have shown that experimental models can be a standardization technique for GAs. In these studies, an optimal set of genetic operators and parameters for GAs solving problems under the parametric optimization domain was found. The interactive effects of crossover probability, mutation rate, and population size on GA convergence velocity in parameterizing a multiple objective model were determined [@Pabico96d]. The convergence velocity was measured using the offline metric proposed by de Jong [@DeJong75] while the interaction was measured using a three-factor factorial analysis on the variance of the GA operator-parameter combinations. A GA that uses the combination of 0.60 one-point crossover probability, mutation rate varied over generation and gene representation, and a population density of 30 was found efficient under this problem domain [@Pabico96d]. No explanation, however, was given on how these operators and parameters affect GA performance. In our current effort, we aim to find the same optimal set of genetic operators and parameters for a GA solving problems under the combinatorial optimization domain. In addition, we will attempt to explain how these operators and parameters affect GA performance and investigates whether problem size is also a factor.
In this paper, we report the results of applying experimental models in measuring the interactive effects of operators and parameters on GA performance. Measuring the effects follows that the specific operators and parameters can be determined to give GA its best performance. Specifically, we used the $n$-factor ANOVA on the interactive effects of operators and parameters to GA convergence. An $n$-factor ANOVA, depending upon a certain probability level, tells how $n$ factors interactively affect a certain response measure (i.e., GA performance) via the goodness-of-fit of the data to the $n$-factor linear model. Although only a few researches have been reported to have used experimental models to compute for and compare different algorithms’ performance [@Alviar86; @Alviar87; @Pabico96a; @Pabico96d], this method offers flexibility and ease of use compared to mathematical analyses or analyses of algorithms.
Our main objective in this study is to show that experimental models can be a standardization method for GA. Specifically, we aim (1) to investigate the relationship between the problem size and the GA operators and their parameters, (2) to investigate whether the selection, crossover and mutation operators act independently on GA performance using $n$-factor ANOVA, and (3) to suggest genetic operators and their parameters for GA in solving optimization problems under the combinatorial domain. With the promise of GA’s general applicability to solve problems, many optimization and search studies can be conducted to try and use this technique. Knowing the relationships between problem size and the genetic operators and parameters that would give GAs an optimal performance, researchers can save time fine tuning their GAs. Further, having known that experimental model can be a standardization technique for GAs, more genetic operators can be devised that can give efficient GAs.
Review of Related Literature
============================
Refinements on Traditional Parameters
-------------------------------------
The operators of a traditional GA are selection ($\Omega_s$), crossover ($\Omega_c$), and mutation ($\Omega_m$). The GAs parameter settings are population size ($\lambda$), crossover probability ($p_c$), and mutation rate ($p_m$). A traditional GA uses the roulette wheel selection, one-point crossover with $p_c=0.6$, and bit-mutation with $p_m=0.033$. The population size, set according to the user’s discretion, is an important factor because the population of individuals serves as a mechanism with distributed knowledge. This knowledge is being represented by all the genes in the entire population [@Lee94]. Other parameter settings reported in the literature are $p_c=0.6$, $p_m=0.001$, $50\leq\lambda\leq 100$ [@DeJong75], $p_c\in[0.75, 0.95]$, $p_m\in[0.005,0.01]$, $20\leq\lambda\leq 30$ [@Schaffer89], and $p_c=0.95$, $p_m=0.01$, $\lambda=30$ [@Grefenstette86].
GA has been used in parametric optimization and much effort has been put into refining the GA to improve its convergence speed. Researchers [@DeJong75; @Grefenstette86; @Schaffer89; @Davis91] have used four techniques to find good parameter seetings for GA. These techniques are (1) hand optimization, (2) using a meta-GA, (3) brute force search, and (4) parameters that adapt. de Jong [@DeJong75] carried out hand optimization to find parameter values for the traditional GA which were good across a set of numerical function optimization problems. The parameter values for single-point crossover and bit mutation were worked out by hand while holding the population size constant.
Using a meta-GA, the same parameters were optimized by the use of another GA [@Grefenstette86]. With the same set of problems, the GA-optimized GA improved slightly over the GA with hand-optimized parameters. However, a robust parameter setting that would perform well across the range of problems considered was not found.
Davis [@Davis91] proposed a method that would make the operators evolve or adapt to the problem as the GA iterates. The adapting parameters can be used to study new operators and evaluate its performance. This could be an effective technique for separating the valuable operators from those that are not. Schaffer, et al. [@Schaffer89] sampled the possible parameter settings across a range of values using the same set of problems that Grefenstette [@Grefenstette86] and de Jong [@DeJong75] used. It was concluded that a GA’s optimal parameter setting vary from one problem to another.
Measures of GA Performance
--------------------------
de Jong [@DeJong75] designed two measures to quantify GA’s search technique’s performance. These are online performance and offline performance. The online performance measures the ongoing performance of the GA and is the running average of all evaluations performed. Mathematically, the online performance is given as $${\rm Online}=\frac{1}{\Lambda}\sum\limits_{i=1}^\Lambda f_i$$ where $\Lambda$ is the current number of evaluations and $f_i$ is the $i$th value of the objective function. This measure is appropriate in situations where the cost of evaluating an individual is related in a monotonically increasing way to its fitness value. The offline performance measures convergence and is the running average of the best performance value. The offline performance is computed as $${\rm Offline}=\frac{1}{G}\sum\limits_{i=1}^G f_{{\rm max},i}$$ where $G$ is the current generation and $f_{{\rm max},i}={\rm max}\{f_{i,j}:1\leq j\leq\lambda\} $ is the best function value obtained from the $i$th generation. This measure can be used when there is no additional cost for evaluating less-fitted individuals.
Methodology
===========
GA Architectures for TSP
------------------------
To solve for TSP, we considered different GA architecture designs. In designing these architectures, the choice for genetic operators is important. Our reasons for choosing the specific genetic operators considered in this study are discussed in the following subsections and are summarized in Table \[Factors\].
1. [**Selection algorithms**]{}. We considered two selection algorithms in this study: Remainder Stochastic Independent Sampling (RSIS) and Stochastic Universal Sampling (SUS). We selected these two algorithms over the usual roullete–wheel method because they are known to have reduced selection bias [@Goldberg89], giving us assurance that the highly fit individual found at each generation will not be lost by chance in the succeeding generations [@Baker87].
2. [**Crossover algorithms and probabilities**]{}. We considered two crossover algorithms specifically designed for solving combinatorial problems: Partially Matched Crossover (PMX) and Cycle Crossover (CX). For each algorithm, five crossover probabilities were used, 0.60, 0.65, 0.70, 0.75, and 0.80, which gave us 10 algorithm–probability combinations.
3. [**Mutation algorithms**]{}. We decided to use the inversion algorithm to simulate mutation because this method was designed solely for combinatorial problems. We considered five levels of mutation rates as a parameter for this algorithm: 0.02, 0.04, 0.06, 0.08, and 0.10.
[ c p[0.15cm]{} c p[0.15cm]{} c]{} &
------------------------------------------------------------------------
& [**Algorithm**]{} &
------------------------------------------------------------------------
& [**Parameter Setting**]{}\
Selection & &RSIS & &\
& &SUS & &\
Crossover & &PMX & & 0.60, 0.65, 0.70, 0.75, 0.80\
& &CX & & 0.60, 0.65, 0.70, 0.75, 0.80\
Mutation & &Inversion & & 0.02, 0.04, 0.06, 0.08, 0.10\
To determine whether these GA architectures are dependent or independent on the problem size, we considered five different $n$-city TSPs, where $n = \{5, 7, 10, 100, 1000\}$. Varying the size of the problem is important to see whether it will have an effect on the operators and parameters found by ANOVA (i.e., will ANOVA give the same operators and parameters regardless of the size of the problem?). Each $n$-city TSP corresponds to a search space whose size is $n!=\Pi^{n}_{k=1}k = 1 \times 2 \times \cdots \times n$.
We have utilized a total of 100 GA architecures solving TSP under five different problem sizes. We run all GAs until the optimum value for the TSP was reached. For each GA run, we recorded the corresponding offline performance. We performed all GA runs under a multi-programming operating system that is why we only measured the offline performance instead of the actual wall-clock running time.
Fitness Function for TSP
------------------------
We transformed the TSP into a maximization problem (i.e., the closed-route that will give the maximum profit) and built the problem around a profit matrix, ${\rm \bf PR}$, of known optimum. ${\rm\bf PR}$ is similar to a graph’s weighted adjacency matrix, encoding the profit of going from one node to the connecting node. Thus, adjacency and profit between the $i$th and the $j$th nodes is defined if ${\rm \bf PR}_{ij}>0$. If all off-diagonal elements in the matrix are positive, then the graph is fully-connected. In TSP, the value of the elements along the diagonal of the matrix does not matter.
We constructed ${\rm\bf PR}$ creating an $n\times n$ diagonally symmetric positive sparse matrix, ${\rm \bf SMat}$, of random elements and by creating a vector, ${\rm \bf Rt}$, of length $n+1$ whose first $n$ elements are the random permutation of the first $n$ integers and ${\rm\bf Rt}_{n+1}={\rm\bf Rt}_1$. ${\rm \bf Rt}$ is the closed route where the maximum profit can be obtained. For example, if $n=5$, ${\rm \bf SMat}$ and ${\rm \bf Rt}$ might be: $$\begin{aligned}
{\rm\bf SMat} &=& \left[\begin{array}{ccccc}
17 & 22 & 27 & 15 & 17\\
22 & 16 & 18 & 20 & 15\\
27 & 18 & 18 & 16 & 17\\
15 & 20 & 16 & 13 & 16\\
17 & 15 & 17 & 16 & 10
\end{array}\right]\nonumber\\
{\rm\bf Rt} &=& \left[\begin{array}{cccccc}
4 & 3 & 5 & 1 & 2 & 4\end{array}\right]\label{RouteEg}\end{aligned}$$ By taking notice of the maximum element of ${\rm\bf SMat}$, $\max({\rm\bf SMat})=27$, and adding it by a constant, say ${\rm MAd}=1$, ${\rm\bf PR}$ can be computed using: $${\rm \bf PR}_{i,j} = \left\{ \begin{array}{ll}
{\rm\bf SMat}_{i,j}, & {\rm if\ }i\neq {\rm \bf Rt}_y\\
& {\rm and\ }j \neq {\rm \bf Rt_{y+1}}\\
& \forall 1\leq y\leq n\\
{\rm\bf PR}_{j,i}=\max({\rm\bf SMat})+{\rm MAd}, & {\rm otherwise.}
\end{array}\right.
\label{Profit}$$ The second case, ${\rm\bf PR}_{i,j}={\rm\bf PR}_{j,i}$, in equation \[Profit\] is necessary so that the same closed route but of different direction (example, in equation \[RouteEg\], ${\rm\bf Rt}^*=[4\quad 2\quad 1\quad 5\quad 3\quad 4]$) will have the same maximum profit. The above equation makes sure that the maximum profit TSP will have a maximum profit of $n\times (\max({\rm\bf SMat})+{\rm MAd})$. With respect to our example, the profit of traversing the optimum route is $5 \times ( 27 + 1) = 168$.
The fitness, $f_i$, of the $i$th randomly generated closed-route can be computed by traversing the route using the profit matrix: $$f=\sum_{y=1}^n{\rm\bf PR}_{{\rm\bf Rt}_y,{\rm\bf Rt}_{y+1}}.$$
Experimental Model
------------------
To provide basis for comparison of GA performance as affected by four factors, we used a four-factor ANOVA model. The factors known to have an effect on GA performance are (1) the algorithm used in simulating selection, (2) the algorithm and (3) parameter used in simulating crossover, and (4) the algorithm and parameter used in simulating mutation. If two selection algorithms produce the same relative GA efficiencies with two crossover and mutation algorithms, then either selection algorithms can be used to evaluate GA efficiencies for any combination of crossover and mutation algorithms. If the results are dependent of selection algorithm, then any one or all combinations of the crossover and mutation algorithms may not be adequate for discriminating among the selection-crossover-mutation algorithm combinations.
The factorial treatment design was used to evaluate whether the four factors act independently on GA performance. The factors that we specifically considered in this study are :
1. the selection algorithms ($\Omega_s$) assumed to be discrete with two levels, RSIS and SUS;
2. the crossover algorithms ($\Omega_c$) assumed to be discrete with two levels, PMX and CX;
3. the crossover probabilities ($p_c$) assumed to be continuous with five levels from 0.60 to 0.80 on 0.05 intervals; and
4. the mutation rate ($p_m$) with five continuous levels from 0.02 to 0.10 via 0.02 intervals.
By determining whether $\Omega_s$, $\Omega_c$, $p_c$, and $p_m$ in combination interact to influence the offline performance of the GA, we can find the combinations of GA operators and parameters that would give the best GA offline performance.
The performance ($P$) of the GA is a function of selection algorithm used ($\Omega_s$), crossover algorithm used ($\Omega_c$), crossover probability used ($p_c$), mutation rate ($p_m$) used, the random error ($\epsilon$[^1]) inherrent to the experiments used which can not be accounted for by $\Omega_s$, $\Omega_c$, $p_c$, and $p_m$, and the interactive effects of $\Omega_s$, $\Omega_c$, $p_c$, and $p_m$. The ANOVA model is therefore $$\begin{array}{rcl}
P&=&\epsilon+\alpha_1\Omega_s+\alpha_2\Omega_c + \alpha_3 p_c+\alpha_4 p_m+\\
& &\quad\alpha_5\Omega_s\Omega_c+\alpha_6\Omega_sp_c+\alpha_7\Omega_s p_m+
\alpha_8\Omega_cp_c+\\
& &\quad\alpha_9\Omega_c p_m +\alpha_{10}p_cp_m+\alpha_{11}\Omega_s\Omega_cp_c+\\
& &\quad\alpha_{12}\Omega_s\Omega_cp_m+
\alpha_{13}\Omega_sp_cp_m+\alpha_{14}\Omega_cp_cp_m +\\
& &\quad\alpha_{15}\Omega_s\Omega_cp_cp_m.
\end{array}$$ We replicated each GA run four times, each replicate using different random seeds but starting with the same initial population. The analysis of variance tests the hypothesis that $\alpha_i=0,\;\forall\;i$, with a probability of $5\%$.
### Varying the Problem Size
To represent varying problem size, we used different TSP sizes. These sizes are the family of $n$-city TSPs where $n=\{5, 7, 10, 100, 1000\}$. Interestingly, we note here that when solutions are encoded into GA chromosomes using the permutation form, the size of the problem space becomes $n!$. Increasing the search space from $(n-1)!$ is not disadvantageous to GA but rather advantageous because each chromosome can provide $n$ more schemes, a desirable characteristics according to GA’s schema theorem [@Goldberg89]. Thus, problem sizes were grouped in terms of the size of the search space brought about by the normal encoding of the solutions to chromosomes. Both $n=7$ and $n=10$ (with search spaces of $6!$ and $9!$, respectively) belong to the intermediate problem size while both $n=100$ and $n=1000$ (with search spaces of $99!$ and $999!$, respectively) belong to the big problem size. $n=5$ represent the small problem size with 120 search points. Because of the extensive computing resources required for performing the experiment involving the bigger problem sizes (i.e, $n=100$ and $n=1000$), only the following levels of genetic parameters were used:
1. the crossover probabilities ($p_c$) with three levels 0.60, 0.70, and 0.80 ; and
2. the mutation rate ($p_m$) with three levels 0.001, 0.010, and 0.100.
### Comparing the Mean GA Performance
To analyze the factors with continuous levels (i.e., $p_c$ and $p_m$), we partitioned their of sum of squares using trend contrasts. Based on the result of the trend comparison, we performed a regression analysis to model the effect of the factors on GA performance. However, we did not perform the regression when the number of points for regression is less than four. Instead, we performed pairwise comparison on the means of the factors involved. For other factors such as $\Omega_s$ and $\Omega_c$, we conducted a pairwise comparison of means using the Duncan’s Multiple range Test (DMRT) at 5% probability level to explain the significant effect of these factors to GA performance.
Results and Discussion
======================
Optimum GA Operators for 5-City TSP
-----------------------------------
The ANOVA result for the 5-city TSP shows that there is no $z$-way interaction present, where $z\geq 2$. Table \[RESULT05\] shows that only $\Omega_c$ has a significant effect on the average GA performance. All other factors have no effect. A simple comparison of means shows that PMX is a better crossover scheme than CX.
2.0em
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
[**Source of**]{} [**Degree of**]{} [**Sum of**]{} [**Mean**]{} $F$-[**Value**]{} Pr$>F$
[**Variation**]{} [**Freedom**]{} [**Squares**]{} [**Square**]{}
Replication 3 86682.58 28894.19 1389.71 0.0001
$\Omega_s$ 1 23.22 23.22 1.12 0.2914
$\Omega_c$ 1 1817.18 1817.18 87.40 0.0001
$p_c$ 4 32.72 8.18 0.39 0.8133
$p_m$ 4 31.88 7.97 0.38 0.8204
$\Omega_s\times\Omega_c$ 1 3.97 3.97 0.19 0.6623
$\Omega_s\times p_c$ 4 58.95 14.73 0.71 0.5864
$\Omega_s\times p_m$ 4 158.94 39.73 1.91 0.1085
$\Omega_c\times p_c$ 4 61.31 15.32 0.74 0.5672
$\Omega_c\times p_m$ 4 45.87 11.46 0.55 0.6980
$p_c\times p_m$ 16 8.19 0.51 0.02 1.0000
$\Omega_c\times\Omega_c\times p_c$ 4 42.79 10.69 0.51 0.7251
$\Omega_s\times\Omega_c\times p_m$ 4 28.68 7.17 0.34 0.8475
$\Omega_s\times p_c\times p_m$ 16 19.81 1.23 0.06 1.0000
$\Omega_c\times p_c\times p_m$ 16 37.54 2.34 0.11 1.0000
$\Omega_s\times\Omega_c\times p_c\times p_m$ 16 25.81 1.61 0.08 1.0000
Error 297 6175.07 20.79
Total 399 95254.58
CV=2.07
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
The difference of mean offline performance between PMX and CX can be explained by how these two crossover algorithms behave for some inputs. Given two strings $C_A$ and $C_B$, $C_A\neq C_B$, that encode the solutions to the 5-city TSP, PMX will always create two new strings $C_A^\prime$ and $C_B^\prime$ where $C_i\neq C_i^\prime$ and $f(C_i)\neq f(C_i^\prime)$. However, in CX, for some $C_A$ and $C_B$, the created strings might be the same as the parents strings, $C_A^\prime=C_B$ and $C_B^\prime=C_A$. This defeats the purpose of creating new solutions by crossing-over the parent strings. Take for instance $C_A=\{6, 2, 0, 3, 4, 7, 9, 1, 8, 5\}$ and $C_B=\{7, 0, 5, 2, 8, 1, 3, 4, 9, 6\}$. Applying CX on these two solutions gives $C_A^\prime=\{7, 0, 5, 2, 8, 1, 3, 4, 9, 6\}$ and $C_B^\prime=\{6, 2, 0, 3, 4, 7, 9, 1, 8, 5\}$. Inputs of this type make CX unable to create new solutions. Table \[CX\_PMX\] shows the relative performance of PMX over CX in terms of new solutions found for all $\Omega_s$–$p_c$–$p_m$ combinations.
[c c c c c c c c c c ]{} & & & &
------------------------------------------------------------------------
&\
$\Omega_s$ & $p_c$ & $p_m$ & Actual & Expected & % & & Actual & Expected & %\
& & & Count & Count & & & Count & Count &\
RSIS & 0.6 & 0.001 & 2988 & 2988 & 100 & & 1872 & 2992 & 62.57\
RSIS & 0.6 & 0.010 & 2981 & 2981 & 100 & & 1871 & 3004 & 62.28\
RSIS & 0.6 & 0.100 & 2945 & 2945 & 100 & & 1895 & 3014 & 62.87\
RSIS & 0.7 & 0.001 & 3520 & 3520 & 100 & & 2264 & 3504 & 64.61\
RSIS & 0.7 & 0.010 & 3499 & 3499 & 100 & & 2158 & 3504 & 61.82\
RSIS & 0.7 & 0.100 & 3508 & 3508 & 100 & & 2202 & 3516 & 62.63\
RSIS & 0.8 & 0.001 & 4000 & 4000 & 100 & & 2481 & 3981 & 62.32\
RSIS & 0.8 & 0.010 & 3992 & 3992 & 100 & & 2495 & 3986 & 62.09\
RSIS & 0.8 & 0.100 & 4001 & 4001 & 100 & & 2583 & 4055 & 63.70\
SUS & 0.6 & 0.001 & 2962 & 2962 & 100 & & 1842 & 2989 & 61.63\
SUS & 0.6 & 0.010 & 2955 & 2955 & 100 & & 1827 & 2975 & 61.41\
SUS & 0.6 & 0.100 & 2975 & 2975 & 100 & & 1908 & 2943 & 64.83\
SUS & 0.7 & 0.001 & 2497 & 2497 & 100 & & 2143 & 3488 & 61.44\
SUS & 0.7 & 0.010 & 3490 & 3490 & 100 & & 2138 & 3474 & 61.54\
SUS & 0.7 & 0.100 & 3463 & 3463 & 100 & & 2240 & 3461 & 64.72\
SUS & 0.8 & 0.001 & 3957 & 3957 & 100 & & 2472 & 4001 & 61.78\
SUS & 0.8 & 0.010 & 3955 & 3955 & 100 & & 2468 & 3991 & 61.84\
SUS & 0.8 & 0.100 & 3985 & 3985 & 100 & & 2555 & 3965 & 64.44\
Optimum GA Operators for 7-City and 10-City TSPs
------------------------------------------------
A $z$-way interaction is present when simple interaction effects of $z-1$ control variables are not the same at different levels of the $z$th control control variable. As shown in the analysis of variance tables (Tables \[RESULT07\] and \[RESULT10\]) a four-way interaction is not present among $\Omega_s$, $\Omega_c$, $p_c$, and $p_m$. However, a two-way interaction is present between $\Omega_s$, and $\Omega_c$. The offline performance of the GA behave differently at different $\Omega_s$–$\Omega_c$ combinations (averaged across $p_c$ and $p_m$) which means that varying the values of $p_c$ and $p_m$ will not affect the average offline performance of the GA. The DMRT groupings explain these interactions as shown in Table \[DMRT07\]. At 7-City TSP, RSIS–CX, RSIS–PMX, and SUS–PMX are not different from each other while SUS–CX and SUS–PMX have the same effect on GA performance. At 10-City TSP, RSIS–PMX, SUS–CX, and SUS–PMX have the same effect on GA performance and are different from RSIS–CX. The effect of replication (i.e, random seed) on mean GA performance is significant at 7-City TSP only. The presence of significant variability among replications at 7-City TSP suggests that the GA offline performance is dependent on the random number used. This confirms the earlier results of experiments conducted by Goldberg, et al. [@Goldberg92] that GA offline performance is dependent also on the initial population used.
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
[**Source of**]{} [**Degree of**]{} [**Sum of**]{} [**Mean**]{} $F$-[**Value**]{} Pr$>F$
[**Variation**]{} [**Freedom**]{} [**Squares**]{} [**Square**]{}
Replication 3 502.88 167.63 7.15 0.0001
$\Omega_s$ 1 443.50 443.50 18.92 0.0001
$\Omega_c$ 1 120.51 120.51 5.14 0.0241
$p_c$ 4 57.31 14.33 0.61 0.6550
$p_m$ 4 198.60 49.65 2.12 0.0786
$\Omega_s\times\Omega_c$ 1 256.91 256.91 10.96 0.0010
$\Omega_s\times p_c$ 4 99.53 24.88 1.06 0.3759
$\Omega_s\times p_m$ 4 123.41 30.85 1.32 0.2640
$\Omega_c\times p_c$ 4 66.55 16.64 0.71 0.5859
$\Omega_c\times p_m$ 4 122.37 30.59 1.30 0.2682
$p_c\times p_m$ 16 179.65 11.23 0.48 0.9562
$\Omega_c\times\Omega_c\times p_c$ 4 45.69 79.44 1.95 0.1024
$\Omega_s\times\Omega_c\times p_m$ 4 32.94 723.85 1.41 0.2322
$\Omega_s\times p_c\times p_m$ 16 8.98 314.55 0.38 0.9857
$\Omega_c\times p_c\times p_m$ 16 16.77 186.71 0.72 0.7782
$\Omega_s\times\Omega_c\times p_c\times p_m$ 16 13.90 106.09 0.59 0.8888
Error 297 6963.44 23.45
Corrected Total 399 10083.49
CV=1.54
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
[**Source of**]{} [**Degree of**]{} [**Sum of**]{} [**Mean**]{} $F$-[**Value**]{} Pr$>F$
[**Variation**]{} [**Freedom**]{} [**Squares**]{} [**Square**]{}
Replication 3 243.23 81.08 1.06 0.3683
$\Omega_s$ 1 1512.35 1512.35 19.69 0.0001
$\Omega_c$ 1 2461.45 2461.45 32.05 0.0001
$p_c$ 4 690.55 172.64 2.25 0.0640
$p_m$ 4 619.73 154.93 2.02 0.0920
$\Omega_s\times\Omega_c$ 1 735.17 735.17 9.57 0.0022
$\Omega_s\times p_c$ 4 331.62 82.90 1.08 0.3668
$\Omega_s\times p_m$ 4 273.36 68.34 0.89 0.4703
$\Omega_c\times p_c$ 4 585.87 146.47 1.91 0.1092
$\Omega_c\times p_m$ 4 122.16 30.54 0.40 0.8103
$p_c\times p_m$ 16 816.52 51.03 0.66 0.8282
$\Omega_c\times\Omega_c\times p_c$ 4 45.25 79.44 0.59 0.6708
$\Omega_s\times\Omega_c\times p_m$ 4 59.33 723.85 0.77 0.5438
$\Omega_s\times p_c\times p_m$ 16 47.43 314.55 0.62 0.8694
$\Omega_c\times p_c\times p_m$ 16 51.30 186.71 0.67 0.8249
$\Omega_s\times\Omega_c\times p_c\times p_m$ 16 1576.64 1.28 0.59 0.2065
Error 297 22811.24 76.81
Corrected Total 399 34778.01
CV=1.91
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
----------------------- -------------------- ---------------------
$\Omega_s$–$\Omega_c$
[**7-City TSP**]{} [**10-City TSP**]{}
RSIS–CX 316.26a 453.58b
RSIS–PMX 315.76a 461.25a
SUS–CX 312.55b 460.18a
SUS–PMX 315.25ab 462.43a
----------------------- -------------------- ---------------------
ANOVA Result for 100-City and 1000-City TSPs
--------------------------------------------
Tables \[RESULT100\] and \[RESULT1000\] show the ANOVA of GA offline performance for 100-city and 1000-city TSP, respectively. As both results show, two three-way interactions, $\Omega_s$–$\Omega_c$–$p_m$ and $\Omega_s$–$p_c$–$p_m$, exhibit significant differences among their factors.
DMRT explains the significant differences of these factors (Tables \[DMRT100\], \[DMRT100b\], \[DMRT100\], and \[DMRT1000b\]). Solving a 100-city TSP, the least $\Omega_s$–$\Omega_c$–$p_m$ combination for a GA is SUS, CX, and 0.001, respectively. No specific best combination can be recommended as several combinations can be bests as seen by the DMRT groupings (Table \[DMRT100\]). Three different groupings were identified by DMRT for the $\Omega_s$–$p_c$–$p_m$ combinations (Table \[DMRT100b\]). The least $\Omega_s$–$\Omega_c$–$p_m$ combination for a GA that solves 1000-city TSP has $\Omega_s={\rm SUS}$, $\Omega_c={\rm CX}$, and $p_m=0.001$ (Table \[DMRT1000\]). Two inferior $\Omega_s$–$p_c$–$p_m$ combinations were also identified , SUS–0.70–0.001 and SUS–0.80–0.001 (Table \[DMRT1000b\]). All other combinations are better.
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
[**Source of**]{} [**Degree of**]{} [**Sum of**]{} [**Mean**]{} $F$-[**Value**]{} Pr$>F$
[**Variation**]{} [**Freedom**]{} [**Squares**]{} [**Square**]{}
Replication 3 240337 80112 14.96 0.0001
$\Omega_s$ 1 28871 28871 5.39 0.0222
$\Omega_c$ 1 175147 175147 32.70 0.0001
$p_c$ 2 5659 2829 0.53 0.5912
$p_m$ 2 25907 12953 2.42 0.0940
$\Omega_s\times\Omega_c$ 1 18249 18249 3.41 0.0677
$\Omega_s\times p_c$ 2 11559 5779 1.08 0.3437
$\Omega_s\times p_m$ 2 88359 44179 8.25 0.0005
$\Omega_c\times p_c$ 2 31973 15986 2.98 0.0549
$\Omega_c\times p_m$ 2 51259 25629 4.78 0.0103
$p_c\times p_m$ 4 15830 3957 0.74 0.5676
$\Omega_c\times\Omega_c\times p_c$ 2 2684 1342 0.25 0.7788
$\Omega_s\times\Omega_c\times p_m$ 2 97260 48630 9.08 0.0002
$\Omega_s\times p_c\times p_m$ 4 67941 16985 3.17 0.0167
$\Omega_c\times p_c\times p_m$ 4 12972 3243 0.61 0.6596
$\Omega_s\times\Omega_c\times p_c\times p_m$ 4 37991 9497 1.77 0.1397
Error 105 562436 5356
Total 143 1474443
CV=2.33
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
----------------------- ----------- ---------- ----------
$\Omega_s$–$\Omega_c$ 0.001 0.010 0.100
RSIS, CX 4599.1a-c 4626.4ab 4535.2c
RSIS, PMX 4639.6a 4620.6ab 4643.7a
SUS, CX 4438.2d 4555.8bc 4616.3ab
SUS, PMX 4638.7a 4618.6ab 4634.9a
----------------------- ----------- ---------- ----------
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
[**Source of**]{} [**Degree of**]{} [**Sum of**]{} [**Mean**]{} $F$-[**Value**]{} Pr$>F$
[**Variation**]{} [**Freedom**]{} [**Squares**]{} [**Square**]{}
Replication 3 24256820 8085606 15.15 0.0001
$\Omega_s$ 1 2799316 2799316 5.24 0.0240
$\Omega_c$ 1 17317700 17317700 32.44 0.0001
$p_c$ 2 575886 287943 0.54 0.5847
$p_m$ 2 2576564 1288282 2.41 0.0945
$\Omega_s\times\Omega_c$ 1 1929517 1929517 3.61 0.0600
$\Omega_s\times p_c$ 2 1039831 519915 0.97 0.3810
$\Omega_s\times p_m$ 2 8608709 4304354 8.06 0.0006
$\Omega_c\times p_c$ 2 3224993 1612496 3.02 0.0530
$\Omega_c\times p_m$ 2 5050079 2525039 4.73 0.0108
$p_c\times p_m$ 4 1675328 418832 0.78 0.5377
$\Omega_c\times\Omega_c\times p_c$ 2 281360 140680 0.26 0.7688
$\Omega_s\times\Omega_c\times p_m$ 2 9601609 4800804 8.99 0.0002
$\Omega_s\times p_c\times p_m$ 4 6794845 1698711 3.18 0.0164
$\Omega_c\times p_c\times p_m$ 4 1407357 351839 0.66 0.6218
$\Omega_s\times\Omega_c\times p_c\times p_m$ 4 3891617 972904 1.82 0.1300
Error 105 56052384 533832
Total 143 147083926
CV=2.01
---------------------------------------------- ------------------- ----------------- ---------------- ------------------- --------
------------ ------- ----------- ----------- -----------
$\Omega_s$ $p_c$ 0.001 0.010 0.100
RSIS 0.60 4613.2a-c 4689.1a 4567.3bc
RSIS 0.70 4643.4ab 4564.4bc 4606.7a-c
RSIS 0.80 4601.8a-c 4617.1a-c 4597.8a-c
SUS 0.60 4551.9bc 4561.4bc 4621.7a-c
SUS 0.70 4528.7c 4623.6a-c 4648.4ab
SUS 0.80 4534.8c 4576.6bc 4606.7a-c
------------ ------- ----------- ----------- -----------
----------------------- ------------ ----------- -----------
$\Omega_s$–$\Omega_c$ 0.001 0.010 0.100
RSIS, CX 45960.6a-c 46185.2ab 45338.5c
RSIS, PMX 46348.5a 46149.5ab 46375.2a
SUS, CX 44308.1d 45517.6bc 46129.6ab
SUS, PMX 46311.4a 46143.2ab 46276.4a
----------------------- ------------ ----------- -----------
------------ ------- ------------ ------------ ------------
$\Omega_s$ $p_c$ 0.001 0.010 0.100
RSIS 0.60 46085.7a-d 46844.0a 45677.5b-d
RSIS 0.70 46393.8a-c 45602.0b-d 46005.5a-d
RSIS 0.80 45984.2a-d 46056.1a-d 45927.5a-d
SUS 0.60 45438.5cd 45574.8b-d 46751.1a-d
SUS 0.70 45226.2d 46184.9a-d 46431.5ab
SUS 0.80 45264.5d 45731.5b-d 46026.4a-d
------------ ------- ------------ ------------ ------------
Summary and Conclusion
======================
This study aimed to find the interactive effects of different genetic operators and their parameters on GA offline performance using 4-way ANOVA. Several $n$-city TSPs were considered as test beds, where $n=\{5, 7, 10, 100, 1000\}$. Problem size (i.e., search space) was hypothesized to have an effect on the optimum GA operators and parameter settings.
ANOVA shows that at a smaller problem size (i.e., 5-city TSP), only $\Omega_c$ has a significant effect on GA offline performance. All other operators and parameters do not affect GA offline performance when the problem size is small. This difference was explained by the way the two $\Omega_c$ algorithms behave. It was found out that PMX is better than CX. When the problem size is intermediate (i.e., 7-City and 10-City TSPs), $\Omega_s$ and $\Omega_c$ interact to affect the mean GA performance. No trend as to what $\Omega_s$–$\Omega_c$ combination is best for this problem size can be concluded as DMRT showed different groupings at different problem sizes.
At bigger problem sizes ($n$-city TSPs where $n=\{100, 1000\}$), the $\Omega_s$–$\Omega_c$–$p_m$ and $\Omega_s$–$p_c$–$p_m$ combinations affect the GA offline performance. No specific behavior on the continuous parameters (i.e, $p_m$ and $p_c$) were found by the regression analysis. Instead DMRT explains the significant three-way interaction among the factors ($\Omega_s$, $\Omega_c$, $p_c$, and $p_m$). Table \[SUMM\] summarizes the results of this study.
[l c p[6.5cm]{} ]{} & [**Significant**]{} & [**Best/Worst Setting**]{}\
& [**Factor**]{}\
5-city TSP & $\Omega_c$ & PMX is better than CX\
\
7-City TSP & $\Omega_s$–$\Omega_c$ & RSIS–CX, RSIS–PMX, and SUS–PMX behave the same while SUS–CX and SUS–PMX have the same effect\
\
10-city TSP & $\Omega_s$–$\Omega_c$ & RSIS–CX is an inferior combination than the other\
\
100-city TSP& $\Omega_s$–$\Omega_c$–$p_m$ & both SUS–CX–0.001 is worst\
& $\Omega_s$–$p_c$–$p_m$ & No recommendation\
\
1000-city TSP& $\Omega_s$–$\Omega_c$–$p_m$ & SUS–CX–0.001 is worst\
& $\Omega_s$–$p_c$–$p_m$ & both SUS–0.70–0.001 and SUS–0.80–0.001 are inferior\
It is now therefore concluded that at a smaller problem size, only $\Omega_c$ will have a significant effect on GA offline performance. Between the two $\Omega_c$ considered, PMX has a significantly higher mean GA offline performance than that of CX. When the problem size is intermediate, $\Omega_s$ and $\Omega_c$ interact to affect GA performance. No recommendation as to what combination is best can be given as different groupings were found by DMRT at different problem size within the intermediate range. At bigger problem sizes, the combination of $\Omega_s$–$\Omega_c$–$p_m$ and $\Omega_s$–$p_c$–$p_m$ significantly affect the mean GA offline performance. $\Omega_s={\rm SUS}$, $\Omega_c={\rm CX}$, $p_m=0.001$ is a worst setting for a GA that solves 100-city TSP. The combination of $\Omega_s={\rm SUS}$, $\Omega_c={\rm CX}$, $p_m=0.001$ is worst for a GA that solves a 1000-city TSP. Similarly, both $\Omega_s={\rm SUS}$, $p_c=0.70$, $p_m=0.001$ and $\Omega_s={\rm SUS}$, $p_c=0.80$, $p_m=0.001$ combinations are worst for the same problem.
[^1]: The random error effect for each test run is assumed to be N($0, \sigma^2$), where N is the normal distribution function with mean 0 and variance $\sigma^2$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this note we reprove a theorem of Gromov using Ricci flow. The theorem states that a, possibly non-constant, lower bound on the scalar curvature is stable under $C^0$-convergence of the metric.'
address: 'Department of Mathematics, UC Berkeley, CA 94720, USA'
author:
- Richard H Bamler
title: A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature
---
Introduction
============
In [@Gromov], Gromov proved the following theorem as an application of his theory of singular spaces with positive scalar curvature:
\[Thm:mainthm\] Let $M$ be a (possibly open) smooth manifold and $\kappa : M \to {\mathbb{R}}$ a continuous function. Consider a sequence of $C^2$ Riemannian metrics $g_i$ on $M$ that converges to a $C^2$ Riemannian metric $g$ in the local $C^0$-sense. Assume that for all $i = 1, 2, \ldots$ the scalar curvature of $g_i$ satisfies $R (g_i) \geq \kappa$ everywhere on $M$. Then $R (g) \geq \kappa$ everywhere on $M$.
In this note, we present an independent proof of this theorem using Ricci flow.
Let us first sketch the idea of our proof in the most elementary setting: Consider the case in which $M$ is closed and $\kappa$ is constant on $M$ and assume that all our metrics, $g, g_1, g_2, \ldots$, are smooth. We may then solve the Ricci flow equation $$\label{eq:RF}
\partial_t g_t = - 2 \operatorname{Ric}(g_t),$$ starting with the limiting metric $g_0 = g$ for some short time and obtain a smooth solution $(g_t)_{t \in [0,\tau^*)}$ for some $\tau^* > 0$. By a theorem of Koch and Lamm (cf [@Koch-Lamm-II sec 5.3]), there are constants ${\varepsilon}= {\varepsilon}( g ) > 0$ and $\tau^* > \tau = \tau (g) > 0$ such that any smooth metric $g'$ on $M$ that is $(1+{\varepsilon})$-bilipschitz to $g$ can be evolved into a smooth Ricci flow $(g')_{t \in [0, \tau)}$ on the uniform time-interval $[0, \tau)$. So the metrics $g_i$, for sufficiently large $i$, can be evolved into a Ricci flow $(g_{i,t})_{t \in [0, \tau)}$ on $[0, \tau)$. By the weak maximum principle applied to the evolution equation for the scalar curvature, $$\label{eq:scalarcurvevol}
\partial_t R (g_{i,t}) = \Delta R (g_{i,t}) + 2 |{\operatorname{Ric}} (g_{i,t}) |^2,$$ it follows that $R (g_{i,t}) \geq \kappa$ for all $t \in [0, \tau)$, for sufficiently large $i$. Using again the results of Koch and Lamm, one can see that the Ricci flows $(g_{i,t})_{t \in [0, \tau)}$ converge locally smoothly to $(g_t)_{t \in [0, \tau)}$ on $M \times (0,\tau)$, modulo diffeomorphisms. So we have $R (g_t) \geq \kappa$ for all $t \in (0, \tau)$. Letting $t$ go to zero yields $R(g) = R(g_0) \geq \kappa$.
The difficulties in the following proof arise from the general form of the theorem: $M$ need not be closed, $\kappa$ need not be constant and the metrics $g, g_1, g_2, \ldots $ have only regularity $C^2$. We will overcome the first two difficulties by localizing the argument presented in the previous paragraph. This localization will be the main challenge of this note. The key estimate for this localization can be found in Lemma \[Lem:scalestimate\]. The issue concerning the $C^2$-regularity of the metrics can be addressed by considering the Ricci DeTurck flow instead of the Ricci flow.
I would like to thank John Lott for pointing out Gromov’s theorem to me.
Ricci DeTurck flow {#sec:Preliminaries}
==================
The Ricci DeTurck flow, as introduced in [@DeTurck], differs from the Ricci flow by a family of diffeomorphisms. Its evolution equation is strongly parabolic, as opposed to the Ricci flow equation, which is only weakly parabolic. This parabolicity is a consequence of a gauge fixing, which we will recall in the following. Choose a background metric ${\overline{g}}$ on $M$ and define the Bianchi operator $$X_{{\overline{g}}}^i ( h ) = ({\overline{g}} + h)^{ij} ({\overline{g}} + h)^{pq} \big( - \nabla^{{\overline{g}}}_p h_{qj} + \tfrac12 \nabla^{{\overline{g}}}_j h_{pq} \big) ,$$ which assigns a vector field to every symmetric $2$-form $h$ on $M$. The evolution equation of the Ricci DeTurck flow now reads $$\label{eq:RdTflow}
\partial_t g_t = - 2 \operatorname{Ric}(g_t) - \mathcal{L}_{X (g_t)} g_t.$$ The evolution equation of the difference $h_t := g_t - {\overline{g}}$ takes the form $$\partial_t h_t = \triangle h_t + 2 \operatorname{Rm}* h_t + Q ( h_t, \nabla h_t, \nabla^2 h_t),$$ where $$\begin{gathered}
\label{eq:Q}
Q ( h_t, \nabla h_t, \nabla^2 h_t) = ({\overline{g}} + h_t) ^{-1} * ({\overline{g}} + h_t) ^{-1} * \nabla h_t * \nabla h_t \\
+ \big( ({\overline{g}} + h_t) ^{-1} - {\overline{g}}^{-1} \big) * \nabla^2 h_t . \end{gathered}$$ (Here all covariant derivatives are taken with respect to ${\overline{g}}$.) So if $h_t$ is $C^0$-close to ${\overline{g}}$, then (\[eq:RdTflow\]) is strongly parabolic.
Given a solution $(g_t)_{t \in I}$ of the Ricci DeTurck flow equation (\[eq:RdTflow\]), we can construct a solution $({\widetilde{g}}_t)_{t \in I}$ to the Ricci flow equation (\[eq:RF\]) by pulling back via a family of diffeomorphisms as follows: Let $(\Phi_t)_{t \in I}$ be a flow of the time-dependent family of vector fields $X_{{\overline{g}}} (g_t)$, meaning that $$\label{eq:Phi}
\partial_t \Phi_t = X_{{\overline{g}}} (g_t) \circ \Phi_t.$$ Then ${\widetilde{g}}_t := \Phi^*_t g_t$ satisfies the Ricci flow equation (\[eq:RF\]).
We will also need to use the heat kernel on a Ricci flow and Ricci DeTurck flow background. Consider first the heat kernel ${\widetilde{K}} (x,t; y,s)$, $s < t$, on a Ricci flow background $({\widetilde{g}}_t = \Phi^*_t g_t)_{t \in I}$, that is for fixed $(y,s) \in M \times I$ we have $$\partial_t {\widetilde{K}} (x, t; y, s) = \Delta_{{\widetilde{g}}_t, x} {\widetilde{K}} (x,t; y,s)$$ and ${\widetilde{K}} (\cdot, t; y,s)$ approaches a $\delta$-function centered at $y$ as $t \searrow s$. Then, for fixed $(x,t) \in M \times I$, the function ${\widetilde{K}} (x,t; \cdot, \cdot )$ is a kernel of the conjugate heat equation $$- \partial_s {\widetilde{K}} (x,t ; y, s) = \Delta_{{\widetilde{g}}_s, y} {\widetilde{K}} (x,t; y,s) - {\widetilde{R}}(y,s) {\widetilde{K}} (x,t; y,s).$$ Here ${\widetilde{R}}(y,s)$ denotes the scalar curvature of ${\widetilde{g}}_s$ in $y$. Note that this equation implies that for all $s < t$ $$\label{eq:KintegralRF}
\int_M {\widetilde{K}} (x,t; y,s) d{\widetilde{g}}_s (y) = 1$$
Consider now the push-forward $K(x,t; y,s)$ of ${\widetilde{K}}(x,t; y,s)$ under $\Phi_t$. That is $$K(x,t; y,s) := {\widetilde{K}} (\Phi^{-1}_t (x), t; \Phi^{-1}_s (y), s).$$ This kernel is the associated heat kernel on the Ricci DeTurck flow background $(g_t)_{t \in I}$ and it satisfies $$\label{eq:heateqRdT}
\partial_t K (x,t; y,s) = \Delta_{g_t, x} K (x,t; y,s) - \partial_{X_{{\overline{g}}} (g_t), x} K(x, t; y,s)$$ for fixed $(y,s) \in M \times I$, as well as $$- \partial_s K (x,t ; y, s) = \Delta_{{\widetilde{g}}_s, y} K (x,t; y,s) - R(y,s) K (x,t; y,s) + \partial_{X_{{\overline{g}}} (g_s), y} K(x, t; y,s).$$ for fixed $(x,t) \in M \times I$. As a direct consequence of (\[eq:KintegralRF\]), we also obtain $$\label{eq:KintegralRFdT}
\int_M K (x,t; y,s) dg_s (y) = 1.$$
Proof
=====
In the following, we will fix some dimension $n \geq 2$ of the manifold $M$ and we will not mention this dependence anymore. We will also frequently consider Euclidean space ${\mathbb{R}}^n$ with the standard Euclidean metric $g_{\operatorname{eucl}}$ and origin $o \in {\mathbb{R}}^n$.
Let us first establish and recall a short-time existence result for Ricci DeTurck flows, which is mainly a consequence of the work of Koch and Lamm (cf [@Koch-Lamm]) and which will become important for us. It states that metrics that are sufficiently close to the Euclidean metric in the $C^0$-sense can be evolved by the Ricci DeTurck flow on a uniform time-interval. This flow becomes instantly smooth and depends continuously on the initial data. Note that we have phrased the following lemma specifically such that it can be applied to our situation. In fact, with some additional work, the lemma can be strengthened in several aspects: For example, the Ricci DeTurck flow $(g_t)_{t \in [0,1)}$ can actually be extended to the time-interval $[0, \infty)$, the condition that $g - g_{\operatorname{eucl}}$ is compactly supported is not necessary and we have bounds on all higher derivatives of $g_t$.
\[Lem:KochLamm\] There are constants ${\varepsilon}> 0$, $C_1 < \infty$ such that the following is true:
Consider a Riemannian metric $g$ on ${\mathbb{R}}^n$ of regularity $C^2$ that is $(1+{\varepsilon})$-bilipschitz close to the standard Euclidean metric $g_{\operatorname{eucl}}$ and assume that $g - g_{\operatorname{eucl}}$ is compactly supported. Then there is a continuous family of Riemannian metrics $(g_t)_{t \in [0, 1)}$ on ${\mathbb{R}}^n$ such that the following holds:
1. For all $t \geq 0$, the metric $g_t$ is $1.1$-bilipschitz to $g_{\operatorname{eucl}}$.
2. $(g_t)$ is smooth on ${\mathbb{R}}^n \times (0,1)$ and the map $[0,1) \to C^2 ({\mathbb{R}}^n)$, $t \mapsto g_t$ is continuous.
3. $g_0= g$ and $(g_t)_{t \in (0, 1)}$ is a solution to the Ricci DeTurck equation $$\label{eq:RdTflow-eucl}
\partial_t g_t = - 2 \operatorname{Ric}(g_t) - \mathcal{L}_{X_{g_{\operatorname{eucl}}} (g_t)} g_t.$$
4. For any $t > 0$ and any $m = 0, \ldots, 10$ we have $$|\partial^m g_t| < \frac{C_1}{t^{m/2}}.$$
5. If $(g_{i,t})_{t \in [0,1)}$ is a sequence of solutions to (\[eq:RdTflow-eucl\]) that are continuous on ${\mathbb{R}}^n \times [0,1)$ and smooth on ${\mathbb{R}}^n \times (0,1)$ and if $g_{i,0}$ converges to some metric $g_{0}$ uniformly in the $C^0$-sense, then $(g_{i,t})$ converges to $(g_t)$ uniformly in the $C^0$-sense on ${\mathbb{R}}^n \times [0,1)$ and locally in the smooth sense on ${\mathbb{R}}^n \times (0,1)$.
The lemma essentially follows from the work of Koch and Lamm (cf [@Koch-Lamm sec 4]). In their paper, the authors analyze and solve (\[eq:RdTflow-eucl\]) by rewriting the term $Q$ in (\[eq:Q\]) as $$Q( h_t, \nabla h_t, \nabla^2 h_t) = R_1 ( h_t, \nabla h_t ) + \nabla^* R_2 ( h_t, \nabla h_t ),$$ where $$R_1 (h_t, \nabla h_t ) = (g_{\operatorname{eucl}} + h_t) ^{-1} * (g_{\operatorname{eucl}} + h_t) ^{-1} * \nabla h_t * \nabla h_t$$ and $R_2 (h_t, \nabla h_t)$ is a $(0,3)$-tensor, which has the form $$R_2 (h_t, \nabla h_t ) = \big( (g_{\operatorname{eucl}} + h_t) ^{-1} - g_{\operatorname{eucl}}^{-1} \big) * \nabla h_t.$$ (Here and in the rest of the proof, all covariant derivatives are taken with respect to $g_{\operatorname{eucl}}$.) So the evolution equation for $h_t = g_t - g_{\operatorname{eucl}}$ becomes $$\label{eq:generalform}
\partial_t h_t = \Delta h_t + R_1 (h_t, \nabla h_t ) + \nabla^* R_2 (h_t, \nabla h_t).$$
The existence of $(g_t)_{t \in [0, 1)}$ and assertions (a), (c) and (d) are consequences of [@Koch-Lamm Theorem 4.3], which follows from a general analysis of equations of the form (\[eq:generalform\]). This theorem also provides the bound $$\label{eq:C0bound}
\Vert h_t \Vert_{C^0({\mathbb{R}}^n \times [0,1))} \leq C \Vert h_0 \Vert_{C^0({\mathbb{R}}^n)}$$ as well as bounds on the derivatives of $h_t$ on ${\mathbb{R}}^n \times (0,1)$.
Likewise, one may look at two different solutions, $(g^1_t)$ and $(g^2_t)$, of (\[eq:RdTflow-eucl\]) and find that for any $a > 0$, the multi-valued function $(g^1_t - g_{\operatorname{eucl}}, a (g^1_t - g^2_t))$ satisfies an equation of a form similar to that of (\[eq:generalform\]). So if $a > 0$ is chosen small enough such that $$\Vert g^1_0 - g_{\operatorname{eucl}} \Vert_{C^0({\mathbb{R}}^n)} + a \Vert g^1_0 - g^2_0 \Vert_{C^0({\mathbb{R}}^n)} < {\varepsilon}',$$ for some universal ${\varepsilon}' > 0$, then one can derive the bound, which is similar to (\[eq:C0bound\]): $$\begin{gathered}
\Vert g^1_t - g_{\operatorname{eucl}} \Vert_{C^0({\mathbb{R}}^n \times [0,1))} + a \Vert g^1_t - g^2_t \Vert_{C^0({\mathbb{R}}^n \times [0,1))} \\
\leq C' \big( \Vert g^1_0 - g_{\operatorname{eucl}} \Vert_{C^0({\mathbb{R}}^n)} + a \Vert g^1_0 - g^2_0 \Vert_{C^0({\mathbb{R}}^n)} \big).\end{gathered}$$ So if $\Vert g^1_0 - g_{\operatorname{eucl}} \Vert_{C^0({\mathbb{R}}^n)} < \frac{{\varepsilon}}2$, then we may choose $a := \frac{{\varepsilon}}2 \Vert g^1_0 - g^2_0 \Vert_{C^0({\mathbb{R}}^n)}^{-1}$ and deduce $$\label{eq:lipschitz}
\Vert g^1_t - g^2_t \Vert_{C^0({\mathbb{R}}^n \times [0,1))} \leq 2 C' \Vert g^1_0 - g^2_0 \Vert_{C^0({\mathbb{R}}^n)}.$$ This implies assertion (d).
For assertion (b), observe that (\[eq:lipschitz\]) implies that difference quotients of $(g_t)$ are uniformly bounded. So we obtain a uniform bound on $\nabla g_t$. Similarly, we obtain a uniform bound on $\nabla^2 g_t$. So $$\partial_t \nabla^2 h_t = \Delta \nabla^2 h_t + R_{1,t} + \nabla^* R_{2,t},$$ where $$|R_{1,t}| < C'' |\nabla^3 h_t| + C'' \qquad \text{and} \qquad |R_{2,t}| < C'' |h_t| \cdot |\nabla^3 h_t| + C''.$$ The continuity of $\nabla^2 h_t$, then follows similarly as the continuity for $h_t$ in [@Koch-Lamm Theorem 4.3].
Next, we analyze the heat kernel $K(x,t; y,s)$ on a Ricci DeTurck flow background, as introduced in section \[sec:Preliminaries\]. Our main observation will be that, on a small time-interval, the kernel can be bounded from above by a standard Gaussian.
\[Lem:hkestimate\] For any $A < \infty$ there are constants $C_2 = C_2 (A), D = D(A) < \infty$ and $0 < \theta = \theta (A) < \frac12$ such that the following is true:
Let $(g_t)_{t \in [0,\theta]}$ be a smooth solution to the Ricci DeTurck equation (\[eq:RdTflow-eucl\]) on ${\mathbb{R}}^n$. Assume that $g_t$ is $1.1$-bilipschitz to $g_{\operatorname{eucl}}$ for all $t \in [0, \theta]$ and assume that $| \partial^m g_t | < A$ for all $m = 0, \ldots, 10$. Consider the heat kernel $K(x,t; y,s)$ on ${\mathbb{R}}^n \times [0,\theta]$ as discussed in section \[sec:Preliminaries\]. Then, for any $(x,t), (y,s) \in {\mathbb{R}}^n \times [0, \theta]$ with $s < t$, we have $$K(x,t; y,s ) < \frac{C_2}{(t-s)^{n/2}} \exp \Big( { - \frac{ d_{g_{\operatorname{eucl}}}^2 (x, y)}{D (t-s)}} \Big)$$ and for any $r > 0$ $$\int_{{\mathbb{R}}^n \setminus B (x, r)} K(x,t; y,s) dg_s (y) < C_2 \exp \Big( { - \frac{ r^2}{D (t-s)}} \Big).$$ Here $B(x,r)$ denotes the $r$-ball in ${\mathbb{R}}^n$ with respect to the Euclidean metric $g_{\operatorname{eucl}}$.
We remark that we can actually choose $D > 4$ arbitrarily as in the work of Cheng, Li and Yau (cf [@CLY]).
Consider the associated Ricci flow ${\widetilde{g}}_t = \Phi^*_t g_t$, where $\Phi_t$ is defined as in (\[eq:Phi\]) with $\Phi_0 = \operatorname{id}_{{\mathbb{R}}^n}$ and the heat kernel ${\widetilde{K}} (x,t; y,s) = K(\Phi_t^{-1} (x), t; \Phi^{-1}_s (y), s)$ on a Ricci flow background. Using the derivative bounds on $g_t$, we find that for small enough $\theta$, the metrics ${\widetilde{g}}_t$ stay $1.2$-bilipschitz close to $g_{\operatorname{eucl}}$. Moreover, we can find a constant $C' = C'(A) < \infty$ such that for all $x \in {\mathbb{R}}^n$ and $s, t \in [0, \theta]$ $$\operatorname{dist}_{g_{\operatorname{eucl}}} (\Phi_t(x),\Phi_s(x)) < C' |s-t|.$$
Using [@CetalIII Theorem 26.25] and the derivative bounds on $g_t$, we get that for sufficiently small $\theta$ $${\widetilde{K}} (x,t; y,s ) < \frac{C}{(t-s)^{n/2}} \exp \Big( { - \frac{ d_{g_{\operatorname{eucl}}}^2 (x, y)}{D (t-s)}} \Big).$$ Here $C = C(A), D= D(A) <\infty$ are some uniform constants. So $$\begin{aligned}
{1}
K(x,t; y,s ) &< \frac{C}{(t-s)^{n/2}} \exp \Big( { - \frac{ d_{g_{\operatorname{eucl}}}^2 (\Phi_t(x), \Phi_s(y))}{D (t-s)}} \Big) \\
& \leq \frac{C}{(t-s)^{n/2}} \exp \Big( { - \frac{ d_{g_{\operatorname{eucl}}}^2 (\Phi_t(x), \Phi_t(y)) - d_{g_{\operatorname{eucl}}}^2 (\Phi_t(y), \Phi_s(y)) }{2D (t-s)}} \Big) \\
& \leq \frac{C}{(t-s)^{n/2}} \exp \Big( { - \frac{ d_{{\widetilde{g}}_t}^2 (x,y) - C' (t-s)^2 }{2D (t-s)}} \Big) \\
& \leq \frac{C''}{(t-s)^{n/2}} \exp \Big( { - \frac{ d_{g_{\operatorname{eucl}}}^2 (x,y) }{2(1.2)^2 D (t-s)}} \Big)\end{aligned}$$ This proves the first assertion of the lemma, after adjusting $D$. The second assertion follows by integration and adjusting $D$ again.
We now state and prove the our key estimate:
\[Lem:scalestimate\] There is an ${\varepsilon}> 0$ and for every $\delta > 0$ there is a $\tau = \tau ( \delta) > 0$ such that the following is true:
Let $g_0$ be a $C^2$-Riemannian metric on ${\mathbb{R}}^n$ that is $(1+ {\varepsilon})$-bilipschitz close to the standard Euclidean metric $g_{\operatorname{eucl}}$ and assume that $g_0 - g_{\operatorname{eucl}}$ is compactly supported. Assume that $R (g_0) > a$ on $B (o, 1)$ for some $a \in {\mathbb{R}}$. Then there is a solution $(g_t)_{t \in [0,1)}$ to the Ricci DeTurck flow equation (\[eq:RdTflow-eucl\]) with initial metric $g_0$ such that $R (o, t) > a - \delta$ for all $t \in [0, \tau]$.
Choose ${\varepsilon}> 0$ from Lemma \[Lem:KochLamm\]. Then $g_0$ can be evolved to a solution $(g_t)_{t \in [0,1)}$ of the Ricci DeTurck flow (\[eq:RdTflow-eucl\]) such that $g_t$ is $1.1$-bilipschitz to $g_{\operatorname{eucl}}$ for all $t \in [0,1)$. By Lemma \[Lem:KochLamm\](d), we can find a uniform constant $C_3 < \infty$ such that for all $t \in (0,1)$ $$\label{eq:C3}
|{\operatorname{Rm}}| (\cdot, t), \; |R|(\cdot, t) < \frac{C_3}t.$$ Note that the scalar curvature bound follows already from the bound on the Riemannian curvature and is only mentioned for convenience. We even have the more precise lower bound $R (\cdot, t) > - \frac{n}{2t}$, which will, however, not be essential for us.
The scalar curvature $R$ of $g_t$ satisfies the equation $$\partial_t R = \Delta R - \partial_{X_{g_{\operatorname{eucl}}} (g_t)} R + 2 |{\operatorname{Ric}}|^2,$$ which is the analogue of (\[eq:scalarcurvevol\]) for Ricci DeTurck flow. So the scalar curvature is a supersolution for the associated heat equation on a Ricci DeTurck flow background (compare with (\[eq:heateqRdT\])). Hence it follows that for any $x \in {\mathbb{R}}^n$ and any $0 < s < t \leq 1$ $$\label{eq:Rxtlowerbound}
R(x,t) \geq \int_{{\mathbb{R}}^n} K(x,t; y,s) R(y,s) dg_s(y).$$ Alternatively, this equation can be derived from the corresponding equations involving the heat kernel ${\widetilde{K}} (x,t; y,s)$ on a Ricci flow background and then pulling back via the diffeomorphisms $\Phi_t$.
Let us now choose the constants that we will be using throughout the proof. Let $C_1 < \infty$ be the constant from Lemma \[Lem:KochLamm\] and let $\theta := \theta (2C_1) > 0$ and $D := D (2 C_1), C_2 := C_2 (2C_1) < \infty$ be the constants from Lemma \[Lem:hkestimate\]. Next choose $\beta > 0$ small enough such that $$1-\beta > \sqrt{1-\theta}$$ and set $$c:= \frac{\beta^2}{D\theta}.$$ Choose $\lambda > 0$ such that $$1+ \lambda < \frac{(1-\beta)^2}{1-\theta} .$$ Note that $$\sum_{k = 1}^\infty \frac{2C_2 C_3}{(1-\theta)^k} \exp \big( { - c (1+ \lambda)^k } \big) < \infty.$$ So for any $\delta > 0$, we can find a large number $N = N(\delta) \in {\mathbb{N}}$ such that $$\label{eq:lessthandelta}
\sum_{k = N}^\infty \frac{2C_2 C_3}{(1-\theta)^k} \exp \big( { - c (1+ \lambda)^k } \big) < \delta.$$ We can finally define $$\tau = \tau (\delta ) := (1- \theta)^N.$$
Next, we choose times and radii $$t_k := (1 - \theta)^k, \qquad \text{and} \qquad r_k := 1 - (1- \beta)^{k},$$ for $k = 1, 2, \ldots$ and set $$a_k := \inf_{B(o, r_k)} R(\cdot, t_k).$$ By (\[eq:C3\]), we have $$\label{eq:akC3}
|a_k| \leq \frac{C_3}{t_{k}}$$ and by Lemma \[Lem:KochLamm\](b), we find that $$\label{eq:liminf}
\liminf_{k \to \infty} a_k \geq a.$$
We will now estimate $a_k$ from below in terms of $a_{k+1}$. Let $x \in B (o, r_k)$. First note that by Lemma \[Lem:hkestimate\], we have $$\begin{gathered}
\int_{{\mathbb{R}}^n \setminus B( x, r_{k+1} - r_k ) } K(x,t_k ; y, t_{k+1} ) dg_{t_{k+1}} (y)
< C_2 \exp \Big({ - \frac{(r_{k+1} - r_k)^2}{D (t_k- t_{k+1})} }\Big) \\
< C_2 \exp \Big({ - \frac{ \beta^2 (1-\beta)^{2k} }{D \theta (1- \theta)^k} }\Big)
< C_2 \exp \big({ - c (1+\lambda)^k } \big).\end{gathered}$$ Then, by (\[eq:Rxtlowerbound\]), (\[eq:C3\]), (\[eq:KintegralRFdT\]) and (\[eq:akC3\]) $$\begin{aligned}
{1}
R(x, t_k) &\geq \int_{{\mathbb{R}}^n} K(x,t_k; y, t_{k+1} ) R(y, t_{k+1}) dg_{t_{k+1}} (y) \\
&\geq a_{k+1} \int_{B (x, r_{k+1} - r_k)} K(x,t_k; y, t_{k+1} ) dg_{t_{k+1}} (y) \\
&\qquad\qquad
- \frac{C_3}{t_{k+1}} \int_{{\mathbb{R}}^n \setminus B(x, r_{k+1} - r_k)} K(x,t_k; y, t_{k+1} ) dg_{t_{k+1}} (y) \\
&\geq a_{k + 1} - \Big( \frac{C_3}{t_{k+1}} + a_{k+1} \Big) \int_{{\mathbb{R}}^n \setminus B (x, r_{k+1} - r_k)} K(x,t_k; y, t_{k+1} ) dg_{t_{k+1}} (y) \\
&\geq a_{k+1} - \frac{2C_3}{t_{k+1}} \cdot C_2 \exp \big( { - c (1+ \lambda)^k } \big).\end{aligned}$$ So we conclude that $$a_k \geq a_{k+1} - \frac{2C_2 C_3}{(1-\theta)^k} \exp \big( { - c (1+ \lambda)^k } \big).$$ Together with (\[eq:liminf\]) and (\[eq:lessthandelta\]), this implies that for all $k \geq N$ $$R(o, t_k) \geq a_k > a - \delta.$$ In particular, this proves the claim for $t = \tau$ and for $t = t_k$ for all $k \geq N$. The lower bound on $R(o, t)$ for any $t \in [0, \tau]$ follows similarly, e.g. by perturbing the parameter $\theta$ slightly or by parabolic rescaling with a bounded factor. For our purposes, it is, however, enough to know the bound $R(o, t_k) > a - \delta$ for all $k \geq N$.
We can finally prove Theorem \[Thm:mainthm\].
Consider the constant ${\varepsilon}> 0$ from Lemma \[Lem:scalestimate\]. Assume that for some $x \in M$ and some $\kappa' \in {\mathbb{R}}$ we have $R (g, x) < \kappa' < \kappa (x)$. By restricting $M$ to a subset, we may assume that for some $\kappa'' \in {\mathbb{R}}$ $$R (g, x) < \kappa' < \kappa'' < \kappa (y) \qquad \text{for all} \qquad x, y \in M.$$ Next, we choose a chart $\Phi : U \subset M \to {\mathbb{R}}^n$ around $x$ such that $\Phi (x) = o$ and such that $\Phi_* g$ on $\Phi (U)$ is $(1+{\varepsilon})$-bilipschitz to $g_{\operatorname{eucl}}$. By rescaling, we may assume that $\Phi (U)$ contains the closure of $B(o, 1)$.
Let $\varphi : {\mathbb{R}}^n \to [0,1]$ be a smooth cutoff function that is constantly equal to $1$ on $B(o, 1)$ and whose support is contained in $\Phi(U)$. Consider the metric $$g_0 := \varphi \Phi_* g + (1- \varphi ) g_{\operatorname{eucl}}.$$ This metric is still $(1+{\varepsilon})$-bilipschitz to $g_{\operatorname{eucl}}$ and $C^2$ and satisfies $$\label{eq:lessthankappas}
R (g_0, o) < \kappa' .$$ Similarly, the metrics $$g_{i, 0} := \varphi \Phi_* g_i + (1- \varphi ) g_{\operatorname{eucl}}$$ are $(1+{\varepsilon})$-bilipschitz close to $g_{\operatorname{eucl}}$ and $C^2$ and satisfy $$R (g_{i, 0}) > \kappa'' \qquad \text{on} \qquad B(o, 1).$$ Moreover, $g_{i,0}$ converge to $g_0$ uniformly in the $C^0$-sense.
Apply Lemma \[Lem:scalestimate\] to each metric $g_{i, 0}$ with $a = \kappa''$ and $\delta = \frac12 (\kappa'' - \kappa')$. Then we get a sequence of Ricci DeTurck flows $(g_{i, t})_{t \in [0, 1)}$ such that for any $t \in [0, \tau (\delta)]$ we have $$R ( g_{i,t}, o) > \kappa'' - \delta .$$ By Lemma \[Lem:KochLamm\](e), these Ricci DeTurck flows converge to the Ricci DeTurck flow $(g_t)_{t \in [0,1)}$ starting from $g_0$. The convergence is uniformly $C^0$ on ${\mathbb{R}}^n \times [0,1)$ and locally smooth on ${\mathbb{R}}^n \times (0,1)$. So for all $t \in (0, \tau]$ $$R (g_t, o) \geq \kappa'' - \delta.$$ Using Lemma \[Lem:KochLamm\](b), it follows that $$R (g_0, o) = \lim_{t \to 0} R (g_{0,t}, o) \geq \kappa'' - \delta > \kappa',$$ in contradiction to (\[eq:lessthankappas\]).
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Wenjie He$^a$, Lingmin Zhang$^b$, Daniel W. Cranston$^c$, Yufa Shen$^b$, Guoping Zheng$^b$'
title: 'Choice number of complete multipartite graphs $K_{3*3,2*(k-5),1*2}$ and $K_{4,3*2,2*(k-6),1*3}$'
---
*$^a$Applied Mathematics Institute, Hebei University of Technology,\
Tianjin 300130, P.R. China\
*
*$^b$Dept. of Math. and Phys., Hebei Normal University of Science and Technology,\
Qinhuangdao 066004, P.R. China*
*$^c$DIMACS, Rutgers University,\
Piscataway, NJ 08854, USA*
*MSC*: 05C15
**Keywords:** List coloring, Complete multipartite graphs, Chromatic choosable graph, Ohba’s conjecture
Introduction {#intro}
============
*List colorings* of graphs were introduced independently by V.G. Vizing [@vizing] and [@erdos]. For a graph $G=(V,E)$ and each vertex $u\in V(G)$, let $L(u)$ denote a list of colors available for $u$. We call $L=\{L(u)|u\in V(G)\}$ a *list assignment* of $G$, and we call $L$ a *$k$-list assignment* if $|L(u)|\geq k$ for each $u\in V(G)$. An *$L$-coloring* of $G$ is a coloring in which each vertex receives a color from its own list such that adjacent vertices get different colors. A graph $G$ is called *$k$-choosable* if $G$ is $L$-colorable for every $k$-list assignment $L$. The *chromatic number* $\chi(G)$ of $G$ is the smallest integer $k$ such that $G$ is $k$-colorable, and the *choice number* $Ch(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ is $k$-choosable.
It is easy to see that every graph $G$ satisfies $Ch(G)\geq\chi(G)$. However, the equality does not necessarily hold. In fact, Erdős et al. [@erdos] showed that there are bipartite graphs with choice numbers that are arbitrarily large. If a graph $G$ satisfies $Ch(G)=\chi(G)$, then $G$ is called *chromatic-choosable*. Much of the work on choice number has studied graph classes in which every graph is chromatic-choosable. The most famous conjecture in this area is the *List Coloring Conjecture* (see [@haggkvist]), which states that every line graph is chromatic-choosable. Galvin [@galvin] proved the special case that every line graph of a bipartite graph is chromatic-choosable (for more information about chromatic-choosability, we refer the reader to a survey by Woodall [@woodall]). In this paper, we focus our attention on Ohba’s conjecture:
\[conj1\] If $|V(G)|\leq 2\chi(G)+1$, then $Ch(G)=\chi(G)$.
Because every $k$-chromatic graph is a subgraph of a complete $k$-partite graph, Ohba’s conjecture is true if and only if it is true for every complete multipartite graph. Thus, we can restate Ohba’s conjecture in the following way.
\[conj2\] If $G$ is a complete $k$-partite graph with $|V(G)|=2k+1$, then $Ch(G)=\chi(G)=k$.
As a result of the formulation in Conjecture \[conj2\], all of the work done on proving Ohba’s conjecture has focused on proving it for specific classes of complete multipartite graphs. We use the notation $K_{r*s}$ to denote a complete $s$-partite graph in which each part has $r$ vertices. Analagously, we use the notation $K_{r*s,t*u}$ to denote a complete $(s+u)$-partite graph, in which $s$ parts have $r$ vertices and $u$ parts have $t$ vertices. Here we list the complete multipartite graphs for which the choice number is known.
\[first\] $Ch(K_{2*k})=k.$
$Ch(K_{3*k})=\ceil{\frac{4k-1}{3}}.$
$Ch(K_{3*r,1*t})=\max(r+t,\ceil{\frac{4r+2t-1}{3}}.$
If $k\geq 3$, then $Ch(K_{3*2,2*(k-2)})=k.$
\[useful\] $Ch(K_{4,2*(k-1)})=
\begin{cases}
k, & \mbox{if $k$ is odd} \\
k+1, &\mbox{if $k+1$ is even.}
\end{cases}
$
If $m\leq 2s+1$, then $Ch(K_{m,2*(k-s-1),1*s})=k.$
\[last\] $Ch(K_{s+2,3,2*(k-s-2),1*s})=k$ for $s\in\{2,3,4\}$.
To obtain partial results for Ohba’s conjecture from Theorems \[first\] through \[last\], we consider subgraphs of the graphs studied in the seven theorems above that are $k$-partite and have $2k+1$ vertices. In particular, we conclude that the choice number is $k$ for every multipartite graph of the following form: $K_{3*2,2*(k-3),1}$, $K_{4,2*(k-2),1}$, $K_{s+3,2*(k-s-1),1*s}$ (for all $s$), and $K_{s+2,3,2*(k-s-2),1*s}$ (for $s\leq 4$). For $K_{4,2*(k-2),1}$, if $k$ is odd, then the result follows directly from Theorem \[useful\]. If $k$ is even, then we first color the vertex $v$ in the unique part of size 1. Since the remaining graph $G-v$ is $(k-1)$-choosable by Theorem \[useful\], we see that $G$ is $k$-choosable.
Preliminaries
=============
In this section we introduce three tools that significantly reduce the number of cases we must consider in each of our proofs.
For a graph $G=(V,E)$ and a subset $X\subseteq V$, let $G[X]$ denote the subgraph of $G$ induced by $X$. For a list assignment $L$ of $G$, let $L|_X$ denote $L$ restricted to $X$, and let $L(X)$ denote the union $\cup_{u\in X}L(u)$. If $A$ is a set of colors, let $L\backslash A$ denote the list assignment obtained from $L$ by deleting the colors in $A$ from each $L(u)$ with $u\in V(G)$. When $A$ consists of a single color $a$, we write $L-a$ instead of $L\backslash\{a\}$.
We say that a graph $G$ satisfies *Hall’s condition* for a list assignment $L$ if $|L(X)|\geq |X|$ for every subset $X\subseteq V(G)$. A result of Hall implies the following theorem (this is commonly called Hall’s Theorem):
\[hall\] If $G$ satisfies Hall’s condition, then there exists an $L$-coloring of $G$ in which each vertex receives a distinct color.
Kierstead [@kierstead] used Theorem \[hall\] to prove the following lemma. This result will be of great use to us.
Let $L$ be a list assignment for a graph $G=(V,E)$ \[lemma1\] and let $X\subseteq V(G)$ be a maximal non-empty subset such that $|L(X)|<|X|$. If $G[X]$ is $L|_X$-colorable, then graph $G$ is $L$-colorable.
Let $X$ be a maximal subset of $V$ such that either $X=\emptyset$ or $|L(X)|<|X|$. Let $C=L(X)$. By the maximality of $X$, every subset $Y\subset V\backslash X$ satisifes $|L(Y)\backslash C|\geq |Y|$. Let $L'(v)=L(v)\backslash C$ for every $v\in V\backslash X$. Note that $G\backslash V$ satisfies Hall’s condition for $L'$. Hence $G\backslash V$ is $L'$-colorable. By hypothesis, $X$ is $L|_X$-colorable. Since none of the colors used on $X$ are used on $V\backslash X$, we can combine the two colorings to give an $L$-coloring of $G$.
Kierstead used Lemma \[lemma1\] to prove the following lemma.
\[lemma2\] A graph $G=(V,E)$ is $k$-choosable if $G$ is $L$-colorable for every $k$-list assignment $L$ such that $|L(V)| < |V|$.
We would like to apply Lemma \[lemma2\] in the middle of constructing a coloring. However, at that point the number of colors available at one vertex may be different from the number of colors available at another vertex. Thus we will prove a more general version of Kierstead’s second lemma, which will apply even when different vertices may have lists of different sizes. We need the following definition. Let $L$ be a list assignment. We say $G$ is *$L$-size-choosable* if $G$ is $L_1$-colorable for every list assignment $L_1$ such that $|L_1(v)|=|L(v)|$ for each $v\in V(G)$. This is a generalization of $k$-choosability, since distinct vertices may have lists of different sizes.
Now we can state Lemma \[keylemma\], which is a generalization of Lemma \[lemma2\] to the case where distinct vertices may have lists of different sizes. Our proof of Lemma \[keylemma\] is essentially the same as Kierstead’s proof of Lemma \[lemma2\].
\[keylemma\] Let L be a list assignment such that $|L(v)| < |V|$ for each $v\in V$. A graph $G=(V,E)$ is $L$-size-choosable if $G$ is $L_1$-colorable for every list assignment $L_1$ such that $|L_1(V)| < |V|$ and $|L_1(v)|=|L(v)|$ for each $v\in V$.
Fix a list assignment $L_{0}$ such that $|L_{0}(v)|=|L(v)|$ for each $v\in V$. Suppose $G$ is $L_{1}$-colorable for every list assignment $L_{1}$ such that $|L_{1}(V)|<|V|$ and $|L_{1}(v)|=|L(v)|$ for each $v\in V$.
We show that the hypothesis of Lemma 2.1 holds for $G$ and $L_{0}$. Consider any maximal non-empty subset $X\subset V$ such that $|L_{0}(X)|<|X|$. Let $A=L_{0}(X)$. Choose $u\in V-X$ such that $|L_{0}(u)|\geq |L_{0}(w)|$ for all $w\in V-X$. We define a list assignment $L_{1}$. We consider two cases depending on whether $|L_{0}(u)|\leq|A|$ or not.
If $|L_{0}(u)|\leq|A|$, then we define $L_{1}$ as follows. If $v\in X$, then $L_{1}(v)=L_{0}(v)$. If $v\not\in X$, then $L_{2}(v)$ is a subset of $A$ of size $|L_{0}(v)|$.
If $|L_{0}(u)|>|A|$, then we define $L_{1}$ as follows. Let $B$ be a subset of $L_{0}(u)\backslash A$ of size $|L_{0}(u)|-|A|$. If $v\in X$, then $L_{1}(v)=L_{0}(v)$. If $v\not\in X$, then $L_{1}(v)$ is a subset of $A\cup B$ of size $|L_{0}(v)|$.
In the first case, $|L_{1}(V)|=|L_{0}(X)|<|X|\leq|V|$. In the second case, $|L_{1}(V)|=|A\cup B|=|L_{0}(u)|<|V|$. By hypothesis, $G$ is $L_{1}$-colorable. Since $L_{0}|_{X}=L_{1}|_{X}$, we see that $G[X]$ is $L_{0}|_{X}$-colorable. Thus, by Lemma $2.2$, $G$ is $L_{0}$-colorable.
In the process of constructing a coloring, we will repeatedly choose a color to use on 2 or 3 vertices, then delete that color from the lists of colors available at each uncolored vertex. We must then show that we can color the remaining uncolored vertices from their lists. Each time we choose a color to use on one or more vertices, Lemma \[keylemma\] enables us to assume that the total number of colors available on the uncolored vertices is smaller than the number of uncolored vertices. We use this technique frequently in the proofs in Sections \[sec1\] and \[sec2\].
Ohba’s conjecture is true for $K_{4,3*2,2*(k-6),1*3}$ {#sec1}
=====================================================
We are now ready to prove our first main theorem. In Sections \[sec1\] and \[sec2\], we will often conclude a case in the analysis by saying that we can finish by coloring greedily. By this we mean that we can color the uncolored vertices greedily in order of nondecreasing list size. Frequently we will use phrases like “there exists some vertex in $X$, say $x_1$, such that color $c_1\in L(x_1)$”; by this we mean that without loss of generality we may assume that the desired vertex is $x_1$.
\[theorem1\] If $G=K_{4,3*2,2*(k-6),1*3}$, then $Ch(G)=k$.
Let $G=K_{4,3*2,2*(k-6),1*3}$. We label the parts of sizes 4, 3, and 1 as follows: $X=\{x_1,x_2,x_3,x_4\}$, $Y=\{y_1,y_2,y_3\}$, $Z=\{z_1,z_2,z_3\}$, $W_1=\{w_1\}$, $W_2=\{w_2\}$, and $W_3=\{w_3\}$ (we do not label the parts of size 2 since they will be less important in the argument).
We begin by handling the case when all the vertices in a part of size 3 or 4 have a common color. Clearly, we should use this common color on all the vertices in the part. Intuitively, this case should be easier than the general case. In fact, the analysis is straightforward. However, for brevity, we observe that the remaining uncolored vertices will form a subgraph of $K_{4,3,2*(k-5),1*2}$ and recall that Shen et al. [@shen] proved that $Ch(K_{4,3,2*(k-5),1*2})=k-1$. So for the rest of this proof, we assume that no color appears on all the vertices in a part of size 3 or 4.
Similarly, if the 2 vertices in a part of size 2 have a common color, we will use the common color on both of them. To formalize this, we induct on the number of parts of size 2 in which the vertices have a common color. The induction step is easy. We use the common color on each vertex in the part (of size 2), remove that color from the lists of all other vertices, then recurse on the graph with both vertices of that part deleted. Hence, for our base case, we assume that no color appears on both vertices in a part of size 2.
We first consider the case when no color appears on more than 2 vertices in $X$ (later, we will consider the case when a color appears on 3 vertices in $X$ and show that that case reduces to the present case).
By Lemma \[keylemma\], we can assume that some color, say $c_1$, appears on two vertices in $Y$, say $y_1$ and $y_2$. Use $c_1$ on $y_1$ and $y_2$. Let $L(v)$ denote the list of available colors at each vertex $v$ after we have used color $c_1$ on $y_1$ and $y_2$. Now let $U$ be a maximal subset of uncolored vertices $U\subset V(G)$ such that $|L(U)|<|U|$. Note that $|L(x_1)|+|L(x_2)|+|L(x_3)|+|L(x_4)|\geq 4k-2$. Since no color appears on three vertices of $X$, we have $|L(x_1)\cup L(x_2)\cup L(x_3)\cup L(x_4)|\geq (4k-2)/2 = 2k-1\geq |U|$. Hence, $U$ contains at most 3 vertices from $X$; call these $x_1$, $x_2$, and $x_3$. Since each pair of vertices in the same part of size 2 have disjoint lists, each part of size 2 contains at most 1 vertex of $U$. Since each vertex in a part of size 2 has at least $k-1$ colors available, we can greedily color the vertices of $U$ in parts of size 2. Since there are only $k-6$ parts of size 2, each vertex loses at most $k-6$ colors. So we have reduced our problem to coloring the vertices of $U$ in parts of sizes 1, 3, and 4. Let $L'(v)$ denote the list of available colors at each uncolored vertex $v\in U$ after we have colored all the vertices of $U$ in parts of size 2. We have the following inequalities: $|L'(w_1)|\geq 5$, $|L'(w_2)|\geq 5$, $|L'(w_3)|\geq 5$ and $|L'(y_3)|\geq 6$. Wlog, we also have the inequalities: $|L'(x_1)|\geq 6$, $|L'(x_2)|\geq 5$, $|L'(x_3)|\geq 5$, $|L'(z_1)|\geq 6$, $|L'(z_2)|\geq 5$, and $|L'(z_3)|\geq 5$. We assume that each inequality holds with equality. Let $U'$ denote this set of 10 vertices. The set $U$ may not contain all of $U'$, but if we can color the graph $G[U']$, that will imply that $G$ is $L$-colorable.
At this point, we observe that the case when 3 vertices of $X$ have a common color reduces to the present case. In that case we use the common color on the three vertices on which it appears. By the same analysis as above, we again reduce the problem to coloring the vertices of $U$ that are in parts of sizes 1, 3, and 4. In that case $U$ contains at most 3 vertices of $Y$ and at most 1 vertex of $X$. By relabeling the vertices of $Y$ as $x_1$, $x_2$, and $x_3$ and relabeling vertex $x_1$ as $y_1$, we reach the present situation. Each of the inequalities given above still holds.
Let $A=L'(y_3)\cup L'(w_1)\cup L'(w_2)\cup L'(w_3)$. We consider two cases: $|A|\geq 7$ and $|A|=6$.
**Case 1:** $|A|\geq 7$.\
Since $|U'|=10$, by Lemma \[keylemma\], we may assume that $|L'(U')|\leq 9$. Since $|L'(x_1)|+|L'(x_2)|+|L'(x_3)|\geq 16$, at least $16-9=7$ colors each appear on two vertices in $X$ (since no color appears on all three vertices of $X$). So we can choose some color $c_2$ that appears on two vertices in $X$, say $x_1$ and $x_2$, such that $c_2\notin L'(z_1)$. Use color $c_2$ on vertices $x_1$ and $x_2$. Let $L''(v)=L'(v)-c_2$ for each uncolored vertex $v\in U'$. Since $|L''(z_2)|+|L''(z_3)|\geq 8$, by Lemma \[keylemma\] we may assume that vertices $z_2$ and $z_3$ must have a common color, call it $c_3$. Use color $c_3$ on vertices $z_2$ and $z_3$. Let $L'''(v)$ denote the lists of remaining colors for each vertex $v\in U'\backslash
\{z_2,z_3\}$. We have the following inequalities: $|L'''(x_3)|\geq 4$, $|L'''(y_3)|\geq 4$, $|L'''(z_1)|\geq 6$, $|L'''(w_1)|\geq 3$, $|L'''(w_2)|\geq 3$, $|L'''(w_3)|\geq 3$, and $|L''(y_3)\cup L''(w_1) \cup L''(w_2) \cup L''(w_3)|\geq 5$. It is easy to verify that Hall’s condition holds. Hence, $G$ is $L$-colorable.
**Case 2:** $|A|=6$.\
Since $|U'|=10$, by Lemma \[keylemma\], we may assume that $|L'(U')|\leq 9$. Since $|L'(x_1)|+|L'(x_2)|+|L'(x_3)|\geq 16$, at least $16-9=7$ colors appear on two vertices in $X$. So we can choose some color $c_2$ that appears on two vertices in $X$, say $x_1$ and $x_2$, such that $c_2\notin A$. Use $c_2$ on vertices $x_1$ and $x_2$. Let $U''=U'\backslash \{x_1,x_2\}$ and $L''(v)=L'(v)-c_2$ for each vertex $v\in U''$. By Lemma \[keylemma\], we may assume that $|L''(U'')| < |U''| = 8$. Since $|L''(z_1)|+|L''(z_2)|+|L''(z_3)|\geq 14$, we see that $14-7=7$ colors must each appear on two vertices in $Z$. So we can choose some color $c_3$ that appears on two vertices in $Z$, say $z_1$ and $z_2$, such that $c_3\notin A$. Use color $c_3$ on vertices $z_1$ and $z_2$.
Let $L'''(v)$ denote the lists of remaining colors for each uncolored vertex $v$. We have the inequalities: $|L'''(x_3)|\geq 4$, $|L'''(y_3)|\geq 6$, $|L'''(z_3)|\geq 4$, $|L'''(w_1)|\geq 5$, $|L'''(w_2)|\geq 5$, and $|L'''(w_3)|\geq 5$. We can finish by coloring greedily. Hence, $G$ is $L$-colorable.
Ohba’s conjecture is true for $K_{3*3,2*(k-5),1*2}$ {#sec2}
===================================================
We will now prove our second main theorem. The proof is similar to the proof of Theorem \[theorem1\]; however, the one fewer part of size 1 requires a more complex argument.
If $G=K_{3*3,2*(k-5),1*2}$, then $Ch(G)=k$. \[theorem2\]
It is easy to handle the case when all the vertices in a part of size 2 or size 3 have a common color. We will use that common color on all the vertices in that part. To formalize this, we use induction on the number of parts of size 2 or 3 in which all the vertices have a common color.
The induction step is easy. Let $S$ be a part (of size 2 or 3) in which the vertices have a common color. We use the common color on each vertex in $S$, remove that color from the lists of all other vertices, then recurse on $G-S$. If $S$ has size 2, then we recurse on a graph with one fewer part of size 2. If $S$ has size 3, then we recurse on a proper subgraph of the graph we consider when $S$ has size 2 (so the claim follows). Hence, for our base case, we assume that no color appears in the lists of all the vertices in a part of size 2 or 3.
We label the parts of sizes 3 and 1 as follows: $X=\{x_1,x_2,x_3\}$, $Y=\{y_1,y_2,y_3\}$, $Z=\{z_1,z_2,z_3\}$, $W_1=\{w_1\}$, and $W_2=\{w_2\}$ (we do not label the parts of size 2 because they will be less important in the argument). We would like to find 2 vertices in $X$, say $x_1$ and $x_2$, with a common color, say $c_1$, and 2 vertices in $Y$, say $y_1$ and $y_2$, with a common color, say $c_2\neq c_1$, such that there exists a vertex in $Z$, call it $z_1$, such that $\{c_1,c_2\}\cap L(z_1)=\emptyset$. It will also be fine if part $Z$ is interchanged with part $X$ or part $Y$ in these conditions. We now show that we can do this.
\[lemmaclaim\] We can find 2 vertices in $X$, say $x_1$ and $x_2$, with a common color, say $c_1$, and 2 vertices in $Y$, say $y_1$ and $y_2$, with a common color, say $c_2\neq c_1$, such that there exists a vertex in $Z$, call it $z_1$, such that $\{c_1,c_2\}\cap L(z_1)=\emptyset$. It will also be fine if part $Z$ is interchanged with part $X$ or part $Y$ in these conditions.
By Lemma \[keylemma\], we can assume that $|L(V)|<|V|=2k+1$. Note that $|L(x_1)|+|L(x_2)|+|L(x_3)|=3k$. Since $|L(x_1)\cup L(x_2)\cup L(x_3)|\leq |L(V)|\leq 2k$, there are at least $k$ colors that each show up on at least 2 vertices in $X$; the same is true for parts $Y$ and $Z$. Recall that $k\geq 5$. Note that if at least 4 colors each appear on 2 vertices in $X$ and also each appear on 2 vertices in $Y$, then the claim holds for the following reason. Each of these 4 colors does not appear on at least 1 vertex of $Z$. Since there are 3 vertices in $Z$, 2 of these 4 colors (call them $c_1$ and $c_2$) “miss” the same vertex in $Z$. So we can use color $c_1$ on 2 vertices of $X$ and use color $c_2$ on 2 vertices of $Y$. Hence, we may assume that some color that appears on 2 vertices of $X$ must appear on either 0 or 1 vertices of $Y$; we consider these two cases separately.
Suppose that color $c_1$ appears on 2 vertices in $X$, but that color $c_1$ does not appear on any vertex in $Y$. Now we can use color $c_1$ on 2 vertices of $X$, and use any choice of $c_2\neq c_1$ on 2 vertices in $Z$. Hence, we can choose colors $c_1$ and $c_2$ as desired.
Instead suppose that color $c_1$ appears on 2 vertices of $X$, but in $Y$ color $c_1$ only appears on one vertex, say $y_1$. Recall that at least $k$ colors appear on 2 vertices in $Z$. Consider the at least $k-1\geq 4$ colors other than $c_1$ that appear on 2 vertices in $Z$. If one of these colors does not appear at $y_2$ or $y_3$, then the claim holds. So we may assume that at least 4 of the colors that each appear on two vertices in $Z$ also appear on both $y_2$ and $y_3$. Again, the claim holds, as in the first paragraph of the proof.
Use color $c_1$ on vertices $x_1$ and $x_2$. Let $G'=G\backslash \{x_1,x_2\}$ and $L'=L\backslash c_1$. Let $U$ be a maximal nonempty subset $U\subseteq V(G')$ such that $|L(U)|<|U|$. By Lemma 2.1, $G'$ is $L'$-colorable if $G'[U]$ is $L'|_U$-colorable. Thus, the remainder of our argument will show that $G'[U]$ is $L'|_U$-colorable. Note that each part of size 2 has at most one vertex in $U$ (otherwise, $|L(U)|\geq 2k-1 \geq |U|$, since the lists of any two vertices in the same part of size 2 must be disjoint). Since each vertex in a part of size 2 has at least $k-1$ colors available, we can greedily color all the vertices of $U$ in parts of size 2 without using color $c_2$. (Note that the size of the list for each vertex decreases by at most $k-5$ since there are only $k-5$ parts of size 2.) So now we only need to color the vertices of $U$ in parts of sizes 3 and 1. In fact, we will color all the uncolored vertices (not just those in $U$) in parts of sizes 3 and 1. Let $U'$ denote the set of uncolored vertices in parts of sizes 3 and 1. Let $L''(v)$ denote the lists of colors available at each vertex $v\in U'$ after we have colored all the vertices of $U$ in parts of size 2. We have the following inequalities: $|L''(x_3)|\geq 5$, $|L''(w_1)|\geq 4$, $|L''(w_2)|\geq 4$, Without loss of generality, we have the additional inequalitites: $|L''(y_1)|\geq 5$, $|L''(y_2)|\geq 4$, $|L''(y_3)|\geq 4$, $|L''(z_1)|\geq 5$, $|L''(z_2)|\geq 4$, $|L''(z_3)|\geq 4$. We assume that each of the inequalities holds with equality. Let $A=L''(x_3)\cup L''(w_1)\cup L''(w_2)$. We consider two cases: $|A|\geq 6$ and $|A|=5$.
**Case 1:** $|A|\geq 6$.\
Use color $c_2$ on vertices $y_1$ and $y_2$. Note that $|L''(z_1)|\geq 5$ (recall that $c_1\notin L(z_1)$) and that vertex $z_1$ is only adjacent to 4 uncolored vertices in $G[U']$. Hence, any coloring of the other 6 uncolored vertices in $U'$ can be extended to $z_1$. So let $U''=U'\backslash\{y_1,y_2,z_1\}$. Now we need to show that $G[U'']$ is $L''|_{U''}$-colorable. By Lemma \[keylemma\], we may assume that $|L''(U'')| < |U''| = 6$. Since $|L''(z_2)-c_2|+|L''(z_3)-c_2|\geq 6$, there exists a color $c_3\in L''(z_2)\cap L''(z_3)$; use $c_3$ on vertices $z_2$ and $z_3$. Let $U'''=U''\backslash \{z_2,z_3\}$ and $L'''(v)=L''(v)-\{c_3\}$ for every vertex $v\in U'''$. Now we have $|L'''(w_1)|\geq 2$, $|L'''(w_2)|\geq 2$, $|L'''(y_3)|\geq 3$, $|L'''(x_3)|\geq 3$, and $|L'''(x_3)\cup L'''(w_1)\cup L'''(w_2)|\geq 4$. It is straightforward to verify that the four remaining uncolored vertices satisfy Hall’s condition. As a result, $G[U''']$ is $L'''|_{U'''}$-colorable. Thus, $G$ is $L$-colorable.
**Case 2:** $|A|=5$.\
We would like to find two vertices both in $Y$ (or both in $Z$), call them $y_1$ and $y_2$, such that there exists a color $c_3\in (L''(y_1)\cap L''(y_2))\backslash A$. (In Claim \[lemmaclaim\] we previously specified two vertices to be $y_1$ and $y_2$; now we relabel vertices if necessary.) We consider two subcases, depending on whether or not we can find such vertices.
**Subcase 2.1:** There exists $c_3\in(L''(y_1)\cap L''(y_2))\backslash A$.\
Use $c_3$ on vertices $y_1$ and $y_2$. Let $U'=U\backslash \{y_1,y_2\}$. and Let $L'''(v)=L''(v)- c_3$ for all $v\in U'$. Note that $|L'''(z_1)|+|L'''(z_2)|+|L'''(z_3)|\geq 11$. Since $|A|=5$ and no color in $A$ appears on all three vertices of $Z$, some vertex in $Z$ has a color available that is not in $A$. Wlog, say this is color $c_4$ on vertex $z_1$; use color $c_4$ on $z_1$. There are 6 remaining uncolored vertices. By Lemma \[keylemma\], we can assume that the union of the lists for these 6 remaining vertices has size at most 5. Since $|L'''(z_2)|+|L'''(z_3)|\geq 6$, there exists $c_5\in L(z_2)\cap L(z_3)$. After using $c_5$ on vertices $z_2$ and $z_3$, we can color the four remaining uncolored vertices greedily. Thus, $G$ is $L$-colorable.
**Subcase 2.2:** $(L''(y_i)\cap L''(y_j))-A=\emptyset$ for all $i\neq j\in \{1,2,3\}$.\
By symmetry, we can also assume $(L''(z_i)\cap L''(z_j))-A=\emptyset$ for all $i\neq j\in \{1,2,3\}$. Since $|L''(y_1)|+|L''(y_2)|+|L''(y_3)|\geq 13>2|A|=10$, there exists some vertex of $Y$, say $y_1$, with $c_4\in L(y_1)-A$. Use color $c_4$ on $y_1$ and let $U''=U'-y_1$. (Note that color $c_4$ is available on at most one vertex in $Z$.) By Lemma \[keylemma\], we may assume that $|L''(U'')-c_4| < |U''|=8$. Since $|L''(y_2)|+|L''(y_3)|\geq8$, there exists a color $c_5\in L''(y_2)\cap L''(y_3)$. Use color $c_5$ on $y_2$ and $y_3$.
Some vertex in $Z$, say $z_1$, has at least 4 available colors. Note that $z_1$ is only adjacent to 3 uncolored vertices in $U''$. Hence, any coloring of the other 6 uncolored vertices in $U''$ can be extended to $z_1$. Let $U'''=U''\backslash\{y_1,y_2,y_3,z_1\}$ and let $L'''(v)=L''(v)\backslash\{c_4,c_5\}$ for each vertex $v\in U'''$. By Lemma \[keylemma\], we may assume that $|L'''(U''')|<|U'''|=5$. Since $|L'''(z_2)|+|L'''(z_3)|\geq 5$, vertices $z_2$ and $z_3$ have a common color, call it $c_6$. Use color $c_6$ on $z_2$ and $z_3$, then color the remaining vertices greedily. Thus, $G$ is $L$-colorable.
Discussion
==========
We believe that our methods can be extended to prove Ohba’s conjecture for more multipartite graphs with three parts each of size at least 3. In particular, we suspect that our methods will be suitable to prove Ohba’s conjecture for $K_{4*2,3,2*(k-7),1*4}$ and $K_{4*3,2*(k-8),1*5}$. Further, we believe that our methods will be sufficient to prove Ohba’s conjecture for $K_{3*4,2*(k-7),1*3}$.
Acknowledgements
================
We would like to express our gratitude to the referees for their careful reading and helpful comments.
[12]{}
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{
"pile_set_name": "ArXiv"
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|
---
abstract: 'Bolometers are most often biased by Alternative Current (AC) in order to get rid of low frequency noises that plague Direct Current (DC) amplification systems. When stray capacitance is present, the responsivity of the bolometer differs significantly from the expectations of the classical theories. We develop an analytical model which facilitates the optimization of the AC readout electronics design and tuning. This model is applied to cases not far from the bolometers in the Planck space mission. We study how the responsivity and the NEP (Noise Equivalent Power) of an AC biased bolometer depend on the essential parameters: bias current, heat sink temperature and background power, modulation frequency of the bias, and stray capacitance. We show that the optimal AC bias current in the bolometer is significantly different from that of the DC case as soon as a stray capacitance is present due to the difference in the electro-thermal feedback. We also compare the performance of square and sine bias currents and show a slight theoretical advantage for the last one. This work resulted from the need to be able to predict the real behaviour of AC biased bolometers in an extended range of working parameters. It proved to be applicable to optimize the tuning of the Planck High Frequency Instrument (HFI) bolometers.'
address:
- '$^1$Laboratoire AstroParticule et Cosmologie (APC), Université Paris Diderot, CNRS/IN2P3, Observatoire de Paris, 10, rue Alice Domon et Léonie Duquet, 75205, Paris cedex 13, France'
- '$^2$ LERMA, Observatoire de Paris et CNRS, 61 Avenue de l’Observatoire, 75014 Paris, France'
- '$^*$Corresponding author: [email protected]'
author:
- 'Andrea Catalano,$^{1,2,*}$ Alain Coulais,$^2$ Jean-Michel Lamarre$^2$'
title: An analytical approach to optimize AC biasing of bolometers
---
Introduction
============
Bolometers are now the most sensitive receivers for astrophysical observations in the submillimetre spectral range. After decades of improvement, they are able to operate with a sensitivity limited by the photon noise of the observed source when operated outside of the atmosphere [@Bock2009]. The principle of a bolometer is that the heat deposited by the incoming radiation is measured by a thermometer. The theory of bolometers has been developed in founding papers [@Jones1953] and refined later [@Mather1982; @Mather1984a; @Mather1984b]. They have shown that their photometric responsivity strongly depends on its interaction with the readout electronics, through the variation of the electrical power deposited in the thermometer (the electro-thermal feedback). These theories have been developed for a semiconductor thermometer element biased by a Direct Current (DC) voltage through a load resistor. The readout electronics for bolometers experienced a radical change more than a decade ago. Most of them are now using a modulated bias current in order to get rid of low frequency noises that plague amplification systems [@Rieke1989; @Wilbanks1990; @Delvin1993; @Gaertner1997; @Kreysa2003].
The theory developed for a DC bias must be altered for Alternative Current (AC) biased bolometers in the presence of stray capacitance in the circuit. This was evidenced in the Planck-HFI instrument [@Lamarre2010] in spite of the fact that the readout electronics had been designed [@Gaertner1997] to mimic, as far as possible, the operation of a DC bias. Very significant differences were found in absolute responsivity and even in value of the optimal bias current for the Planck bolometers. This was shown to be mostly due to the effect of parasitic capacitances in the wiring, which cannot be neglected in many practical experimental setups. This effects have been studied in several papers dedicated to the characterization of bolometers and calorimeters by measuring their effective impedance (e.g. [@Vaillancourt2005]), but we are here essentially interested in effective tools able to predict the responsivity and optimise the tuning of bolometers in specific configurations. Brute force modelling based on numerical integration of thermal and electrical equations of the bolometers proved to be feasible but computationally too heavy to be applied on wide ranges of the many parameters of the models. To facilitate the computation, we have developed an analytical model of the responsivity of AC biased semiconductor bolometers. This model was used as an aid to predict the behaviour of the bolometer of Planck-HFI and to optimize their tuning. Its numerical application proved to be flexible and fast enough to study the effects of all variable parameters.
This paper describes this analytical model and its application with a set of parameters not too far from the realistic cases encountered in Planck-HFI. The next section is dedicated to the differential equations driving the thermal and the electrical behaviour of the electro-thermal system comprising the bolometer and its readout electronics. It focuses on the derivation of an analytical solution giving the responsivity for both the DC and AC biased cases. Section three addresses the various noises encountered and shows that the optimal bias currents are different in the two cases. In the fourth section the model is applied to analyze the effects of some essential parameters (cold stage temperature, modulation frequency, value of the stray capacitance, optical background). Section five deals with the shape of the periodic bias wave to cover the case of square bias current used in Planck-HFI.
The Theoretical Model
=====================
Bolometer Model
---------------
Let us consider a low temperature bolometer consisting of an absorber attached to a semiconductor thermometer. The bolometer is attached to a heat sink at temperature $T_0$ through a thermal link of thermal conductance $G_s$. The incoming optical power deposits energy in the absorber and heats the whole bolometer including the thermometer. The absorbed optical power will determine the equilibrium temperature $T_b$ of the bolometer:
$$\label{bolo}
G_s(T_b-T_0)=W_{tot}$$
Where $W_{tot}$ is the total power dissipated in the bolometer that is $W=P+Q$ where $Q$ is the absorbed radiant power and $P(t)=V(t)I(t)$ is the electrical power.
$G_s$ can be well represented in many cases by:
$$G_s=G_{s0}(T_b/T_0)^{\beta}$$
where $G_{s0}$ is the static thermal conductance at temperature $T_0$ ($100 mK$ in the case we investigate here).
The dominant electrical conduction mechanism in the thermometer is the variable range hopping between localised sites and the resistance of the device varies with both applied voltage and temperature.
The relation between the resistance and the temperature of the bolometer [@Piat2006] is set by : $$R(T,E)=R_* exp\left(\left(\frac{T_g}{T}\right)^n-\frac{eEL}{K_bT}\right)$$
where $T_g$ is a characteristic parameter of the material, $R_*$ is a parameter depending on the material and the geometry of the element, $L$ is related to the average hopping distance and $E$ is the electric field across the device. In absence of electrical non-linearities and other effects such as electron-phonon decoupling, the thermistor resistance depends only on temperature: $$\label{resi}
R(T)=R_* exp\left(\frac{T_g}{T}\right)^n$$
The impedance changes induced by the temperature variations can be measured by an appropriate readout circuit that we are going to detail and discuss hereafter.
Readout Electronics
-------------------
![Schemes of a DC (left) and an AC (right) bias circuits[]{data-label="fig:ACmod"}](Catalano_AO131102_Fig1a.pdf "fig:"){width="8cm"} ![Schemes of a DC (left) and an AC (right) bias circuits[]{data-label="fig:ACmod"}](Catalano_AO131102_Fig1b.pdf "fig:"){width="8cm"}
The Readout Electronics is designed to measure the impedance of the temperature sensitive element of the bolometer. This is done by injecting a current, and therefore depositing power in the bolometer, which changes its temperature. Consequently, the bolometer responsivity and performance strongly depend on the design of the readout electronics.
The Responsivity $ \Re$ is the derivative of the bolometer voltage with respect to the optical absorbed power $W$:
$$\Re=\frac{dV}{dW}$$
It is a strong indicator of the coupling efficiency achieved by readout electronics for a given bolometer.
We compare here after the responsivity obtained with a classical DC bias and a sine-shape AC bias.
**DC Responsivity :** the bolometer is biased with a DC bias voltage through a load resistance $R_L$, and the voltage $V_b$ is measured with an amplifier with an high input resistance (Fig \[fig:ACmod\] left). The general equation of a DC biased circuit is :
$$\label{equreu}
V_b^{DC}=\frac{R_b}{R_b+R_L}V_0$$
where $R_L$ and $R_b$ are the impedances of the load and the bolometer. $V_0$ is the total input voltage.
The electrical responsivity $\Re_{el}$ can be written using the Zwerling formalism [@Zwerdling1968]
$$\Re_{el}=\frac{\alpha \varphi_{DC} R_{b} I_{b}}{G_{e}}$$
where :
$$\varphi_{DC}=\frac{R_L}{R_b+R_L}$$
and $G_{e}$ is the equivalent thermal conductance :
$$G_{e}= G_{s0}-\alpha R_{b} I_{b}^{2}(2 \varphi_{DC} -1)$$
$\alpha$ is the temperature coefficient of resistance of the bolometer :
$$\label{eq2}
\frac{dR_b}{dT_b}=\alpha \cdot R_b$$
The advantage of this DC setup is the use of a well established theory [@Jones1953; @Mather1982]. The optical power $W_{opt}$ absorbed by the bolometer and the responsivity of the bolometer can be directly computed in the time domain. On the other hand, a DC bias current increases the level of low frequency noise, like Flicker noise, making the detection of a faint and slowly varying optical signal impossible. In addition, the Johnson noise produced by the load resistor forces us to put this element on the coldest cryogenic stage.
**AC Responsivity:** Let us consider now an AC bias circuit as presented in Fig. \[fig:ACmod\] right.
The voltage at the ends of bolometer is:
$$V_b^{AC}=V_0 \cdot \frac{R_b Z_c}{Z_L Z_c+R_b(Z_L+Z_c)}\label{aceq}$$
If we assume that the AC bias frequency $F_{mod}$ is much higher than the bolometer cut-off frequency ($F_{mod}>>\frac{G_e}{2 \pi C}$), then we can consider only the average electrical power and a steady state responsivity and neglect short term variations. We can derive a modified Zwerdling’s formula for the responsivity by following the method used in the previous section for a DC biased bolometer:
$$\Re_{el}=\frac{\alpha \varphi_{AC} R_b I_b}{G^{AC}_e}$$
where $G_{AC}$ the dynamic thermal conductance is equal to: $$G^{AC}_e=G_s+\frac{dG_s}{dT}(T-T_0)-\alpha R_b I_b^2 (2
\varphi_{AC}-1)$$
Here the $ \varphi_{AC}$ factor is: $$\varphi_{AC}=\frac{Z_L Z_c}{Z_L Z_c + R_b(Z_L + Z_c)}$$
where, if $Z_L$ is a resistor:
$$\displaystyle\lim_{Z_c\to\infty} \varphi_{AC} = \varphi_{DC}$$
If we consider the module of the $\varphi$ factor we can plot (Fig. \[fig:comp\]) a responsivity versus bias current for DC and AC currents (sine-shape) bolometer. We can conclude that in terms of responsivity a DC electronics is preferable. For an AC bias the maximum in responsivity is lower and is obtained with higher bias current in the bolometer. We show in the next section that for slowly varying signals, AC bias has a decisive advantage in sensitivity.
![Left: Simulation of responsivity versus bias current in the bolometer for DC (solid curve) and AC sine (dashed curve) bias currents in the case of a 3 mm bolometer (using parameters from Tab. \[tab:tab\]). For AC model we consider a sine wave bias with a stray capacitance of $C_p$ = 130 pF and plot the responsivity versus the r.m.s. value of the bias current. Right: Consistency of the AC model (dashed curve) with experimental measurements (stars points) taken from ground calibration of Planck HFI. The disagreement is of the order of 1 %.[]{data-label="fig:comp"}](Catalano_AO131102_Fig2a.pdf "fig:"){width="8cm"} ![Left: Simulation of responsivity versus bias current in the bolometer for DC (solid curve) and AC sine (dashed curve) bias currents in the case of a 3 mm bolometer (using parameters from Tab. \[tab:tab\]). For AC model we consider a sine wave bias with a stray capacitance of $C_p$ = 130 pF and plot the responsivity versus the r.m.s. value of the bias current. Right: Consistency of the AC model (dashed curve) with experimental measurements (stars points) taken from ground calibration of Planck HFI. The disagreement is of the order of 1 %.[]{data-label="fig:comp"}](Catalano_AO131102_Fig2b.pdf "fig:"){width="8cm"}
![Left: noise equivalent power versus bias current in the bolometer for different sources of noise in case of an AC readout electronics for the test bolometer $\lambda$ = 3 mm. Right: total NEP versus frequency for a DC and AC readout electronics at respective best bias currents.[]{data-label="fig:NEPs"}](Catalano_AO131102_Fig3a.pdf "fig:"){width="8cm"} ![Left: noise equivalent power versus bias current in the bolometer for different sources of noise in case of an AC readout electronics for the test bolometer $\lambda$ = 3 mm. Right: total NEP versus frequency for a DC and AC readout electronics at respective best bias currents.[]{data-label="fig:NEPs"}](Catalano_AO131102_Fig3b.pdf "fig:"){width="8cm"}
Noise
=====
The Noise Equivalent Power (NEP) is:
$$\label{nep}
NEP(f)=\frac{< \Delta S^2(f)>^{1/2}}{\Re(f)}$$
where $\Delta S^2$ is the power spectral density of the noise and $\Re$ is the responsivity of the detector. NEP is measured in $[W/ Hz^{1/2}]$.
In our model we take into account all the principal sources of noise in bolometric detection: Johnson noise, phonon noise, photon noise, Flicker noise and the preamplifier noise. The following is a review of the NEPs for the different sources of noise existing in literature (see for instance [@Mather1982; @Lamarre1986]) :
**Johnson noise:** Johnson noise is the electronic noise generated by the thermal agitation of electrons inside a bolometer at equilibrium. It has a white noise spectrum. The NEP for DC biased bolometers is [@Mather1982]: $$\label{johnnoise}
NEP_{john}=(4k_BT_bR_bI_b^2)^{1/2} \frac{|Z_b+R_b |}{|Z_b-R_b|}$$ where $R_b$ is the bolometer resistance and $Z_b$ is its dynamic impedance.
Let us notice with Mather [@Mather1982] that Johnson noise does not depend on load impedance. Let us assume here that it does not depend on stray capacitance in the case of AC biased bolometers. Hereafter, we shall use Eq. \[johnnoise\] indifferently with a DC model or an AC Model.
**Phonon noise:** The parameters of the bolometer are strongly dependent on the temperature, so small variations in temperature inside the bolometer produce a voltage variation at the ends of the detector.
It results [@Mather1982]: $$NEP_{phon}=(4k_BGT^2)^{1/2}$$
This result is independent of the readout electronics.
**Photon noise:** The Photon noise comes from the fluctuations of the incident radiation due to the Bose-Einstein distribution of the photon emission. The NEP is [@Lamarre1986]: $$NEP_{phot}=2 \int_{\Delta \nu} h \nu Q_{\nu} d \nu + (1+P^2)
\int_{\Delta \nu} \Delta(\nu) Q_{\nu}^2 d \nu$$ Where $Q_{\nu}$ is the absorbed optical power per unit of frequency, $\Delta(\nu)$ is the coherence spacial factor (equal to the inverse of the number of modes; $\Delta (\nu)=1$ if diffraction limited) and $P$ is the polarisation degree (0 non-polarised 1 polarised). This noise corresponds to the limitation in sensitivity of any instrument because it does not depends on performances of detectors and readout electronics.
**Flicker noise:** The Flicker noise depends on a distribution of time constants due to the recombination and generation phenomena appearing in semiconductors.
This noise shows a spectrum directly proportional to the bias current and inversly proportional to the frequency. To first order we have :
$$NEP_{fl} = const \frac{I_b}{\sqrt{freq}}$$
The Flicker noise is usually the dominant source of noise up to few Hertz. In the case of AC electronics, we can choose the working modulation frequency in order to keep the Flicker noise less then the photon noise (see Fig. \[fig:NEPs\] right).
**Preamplifier noise:** results from the impossibility to amplify a signal without adding noise, which is a consequence of the Heisenberg Uncertainty principle. It also depends on the available components and on the design of the amplifier. We assume that the power spectrum of signal fluctuation is constant and equal to: $$< \Delta S^2>_{pre}^{1/2}= const =\sigma_{PA} [V/Hz^{1/2}]$$ The $NEP_{pre}$ results from Eq. \[nep\] as: $$NEP_{pre}=\frac{\sigma_{PA}}{\Re}$$
The Total NEP of the instrument is:
$$NEP_{tot} = \left[ NEP_{john}^2 + NEP_{phon}^2 + NEP_{phot}^2 +
NEP_{fl}^2 + NEP_{pre}^2 \right]^{1/2}$$
Fig. \[fig:NEPs\] (left) presents the influence of the different contributions to the total NEP for an AC readout electronics.
Fig. \[fig:NEPs\] (right) shows the advantage of using of an AC system instead of a DC solution at low frequencies. It is clear that Flicker noise increases the DC total NEP at low frequencies making the AC solution mandatory for the measurement of low and very low frequency signals (less then few Hertz).
**Optimization of the Bias Current :** from Fig. \[fig:comp\] it is clear that in both cases (AC and DC), the responsivity strongly depends on the bias current in the bolometer. The optimisation of this parameter is therefore a key point. We want to present a general result, not depending on the preamplifier noise level. Since our practice and all the simulations (see Fig. \[fig:NEPs\]) show that the minimum NEP happens very near to the maximum responsivity (to better than 1% in practical cases), we have chosen to use the responsivity to illustrate this point. The amplitude and the position of the peak responsivity are different for the two types of bias current. In the case presented in Fig. \[fig:comp\], the bias currents corresponding to the maximum of the Responsivity are equal to $I^{DC}_{best}$ = 0.12 nA and $I^{AC}_{best}$ = 0.22 nA.
Variation of Responsivity with Readout Electronics and Environmental Parameters
===============================================================================
We are now interested in establishing the performance of AC readout electronics biased with a sine wave. We will derive the responsivity, the NEP and how the NEP depends on the main parameters (stray capacitance, modulation frequency, optical background and plate temperature) for three typical bolometers optimized to observe the sky between 0.3 and 3 mm cooled to a temperature of 100 mK. In order to obtain an analytical solution to this problem, we developed a model in the frequency domain using the Fourier formalism. The results could be also obtained in the case of a square AC model. The two methods are detailed in the appendices A and B.
![Relative variation of the total NEP versus stray capacitance from 0 to 300 pF for three typical bolometers with an AC sine bias.[]{data-label="fig:nepvscpsin"}](Catalano_AO131102_Fig4.pdf){width="10cm"}
![Relative variation of the total NEP versus modulation frequency of the AC sine from 85 Hz to 120 Hz for three typical bolometers[]{data-label="fig:nepvsfmodsin"}](Catalano_AO131102_Fig5.pdf){width="10cm"}
Stray Capacitance
-----------------
The first stage of preamplifiers for semiconductors bolometers are classically J-FETs giving optimal performance at 100 K or more. Rather long wiring is needed between the J-FETs and the bolometer to avoid an excessive thermal load on the sub-Kelvin stage supporting the bolometer. Stray capacitance of tens and even hundreds of picoFarads result from this design. In Fig. \[fig:nepvscpsin\], we plot for our three test bolometers the excess of NEP versus the value of the stray capacitance. For a typical value of 150 pF, the NEP excess is several percents (from 4 % to 8 %). Let us note here that the DC bias case is identical to an AC case without stray capacitance.
Modulation Frequency
--------------------
As we have seen in previous section, the use of an AC bias has the advantage of presenting a noise spectrum flat down to very low frequencies, while DC biased readouts show a large 1/f component at frequencies less than about 10 Hz. The modulation frequency of the electronics will be chosen therefore taking into account the requirement of keeping the Flicker noise less then the Johnson noise but also taking into account the scanning strategy of the instrument and the angular responsivity of the optics. In the case of Planck HFI for example [@Lamarre2010] the full width at half maximum $\delta$ ranges from 5 to 9 arcmin and the scanning speed is 6 degrees per second. So, in the limit of small angles, the maximum frequency of interest is given by the relation:
$$f \sim \frac{v_{ang}}{\delta}$$
where $f$ is the frequency of the optical modulation. In Fig. \[fig:nepvsfmodsin\] we consider the excess NEP with respect to a 85 Hz modulation frequency. In the worst case the excess NEP is 0.5 % per Hertz.
Optical Background
------------------
The background strongly affects the static performance of a bolometer by changing the operating point. With respect to others parameters, the background is the most uncertain and variable parameter during an observational campaign. A good understanding of the effect of the optical background on the static performances of a bolometers is therefore a key point during the calibration of the instrument. In Fig. \[fig:nepvswopthfi\] we present the relative variation of the total NEP versus the nominal background for our test bolometers. For the 3 mm bolometer, the nominal background is 0.3 pW; for 1 mm 0.6 pW and for 0.3 mm 3.6 pW. In the worst case the degradation in NEP is 2 % with respect the nominal background for a background increase of +16.5 %.
![Relative variation of the total NEP versus optical background. We consider for each bolometer a total range in background equal to 33 % of the nominal background.[]{data-label="fig:nepvswopthfi"}](Catalano_AO131102_Fig6.pdf){width="10cm"}
![Relative variation of the total NEP versus bolometer plate temperature from 95 mK to 105 mK for the three test bolometers.[]{data-label="fig:nepvst0hfi"}](Catalano_AO131102_Fig7.pdf){width="10cm"}
Bolometer Plate Temperature
---------------------------
Following the first order thermal model of a bolometer (Eq. \[bolo\]), we know that a change in temperature of the plate corresponds exactly to a change of background power on the bolometer. In this case the equivalent power generated from a change of the plate temperature is:
$$\Delta P_{plate}=G_s \Delta T_0$$
Our test bolometers were designed for a plate temperture of 100 mK. The NEP variation in the range 95 mK – 105 mK is reported in Fig. \[fig:nepvst0hfi\]. We find that a change of the plate temperature of 1 mK gives a change in NEP of 0.8 % in the worst case.
Comparison Between two Modulation Techniques : Sine AC Bias vs Square AC Bias
=============================================================================
We want now to compare the performances, in terms of responsivity and NEP, of a sine-wave and a square-wave AC electronics. The results are shown in Fig. \[fig:nepcomp\]. The sine case is better for both responsivity and NEP. This is more obvious for the responsivity (better by about 10 %) than for the total NEP (better by about 4 %). We conclude that in terms of NEP, a bolometer connected to a sine AC biased readout electronics would be more sensitive.
Let us remark that the difference in NEP is modest. Let us also notice that an AC sine bias would induce significant variations in the temperature of the fastest bolometers, bringing them into the non-linear regime. On the contrary, the square bias deposits a nearly constant power in the bolometers, that deviate from their mean temperature only by small amounts.
![Simulation of responsivity (left) and total NEP (right) of the three test bolometers in case of an AC sine bias (solid curves) and a square AC bias (dashed curves)[]{data-label="fig:nepcomp"}](Catalano_AO131102_Fig8a.pdf "fig:"){width="8cm"} ![Simulation of responsivity (left) and total NEP (right) of the three test bolometers in case of an AC sine bias (solid curves) and a square AC bias (dashed curves)[]{data-label="fig:nepcomp"}](Catalano_AO131102_Fig8b.pdf "fig:"){width="8cm"}
Conclusion
==========
wavelength $R_{*} [Ohm]$ $G_{so} [pW/K]$ $T_g [K]$ $n$ $\beta$
---------- ------------ --------------- ----------------- ----------- ----- ---------
Bolo \#1 3 mm 100 52 16 0.5 1.3
Bolo \#2 1 mm 94 70 16 0.5 1.3
Bolo \#3 0.3 mm 105 703 16.5 0.5 1.1
: *Parameters of the test bolometers used to illustrate the results of the analytical model.*[]{data-label="tab:tab"}
The analytical model presented in this paper has been developed for the HFI on board Planck satellite. It allowed us to predict the responsivity and the noise of semi-conductor bolometers cooled at 100 mK and biased by AC currents in a realistic environment. It sheds some light on the differences between AC and DC biased bolometer and on the different optimal bias currents for these two cases. Three test bolometers rather similar to Planck’s ones were used to illustrate our results. Our main conclusions are:
- The AC responsivity is always lower than the DC responsivity. This is due to a more effective electro-thermal feedback. The resulting excess of NEP depends on the relative part of the preamplifier noise in the total NEP. In our test cases the excess NEP ranges from 4 % to 10 %, which is more than compensated for by shifting of the low frequency noises out of the range of useful frequencies. Frequencies down to 1 mHz are measurable with a well designed AC readout electronics.
- The AC bias RMS current providing to the maximum of the responsivity is about twice larger than that obtained for a DC bias. This concerns the current through the bolometer and results from the different electro-thermal feedback.
- For a stray capacitance of $\sim$ 150 pF we obtain an excess NEP of 10 % in the worst case (3 mm bolometer) and 4% in the best case (0.3 mm bolometer).
- Around a modulation frequency of 90 Hz, the excess NEP ranges between 0.2 % and 0.5 % per Hz.
- The sensitivity of NEP to background is dlog(NEP)/dlog(Wbg) = 0.22 to 0.42
- The sensitivity of NEP to the plate temperature is dlog(NEP)/dlog(Tplate) = 0.3 to 0.8 around 100 mK, but is rather non-linear.
- The performances of a sine bias are better than the square bias. In our test cases, this result is more obvious in the responsivity (better by about 10 %) than in the total NEP (better by about 4 %). But non-linear effects may show up in the sine case for bolometers fast enough to respond to the modulation frequency.
Appendix A: Computing the Responsivity with a Sine Bias {#appendix-a-computing-the-responsivity-with-a-sine-bias .unnumbered}
=======================================================
Let us consider the bias circuit of Fig. \[fig:ACmod\] with a stray capacitance in parallel to the bolometer and a load capacitance in series. The value of the load capacitance is fixed to $C_b=4.7\cdot
10^{-12} F$ which is the typical value in HFI.
Let’s also consider a range of temperatures starting from the temperature of the plate (100mK for example) up to an arbitrary value. For each temperature we can calculate the impedance $R_b$ of the bolometer and its total power using Eqs. \[bolo\] and \[resi\]. In this simulation we assume that the parameters of the bolometers ($R_*$, $T_g$, $\beta$, etc......) are those of HFI.
If the optical background is constant in this run of simulations, the dissipated electrical power in the bolometer is: $$\label{wele}
W_{elec}=W_{tot}-W_{opt}$$ So, the r.m.s. Voltage at the ends of the bolometer is : $$\label{veff}
V_b=(R_b W_{elec})^{1/2}$$ and the r.m.s. bias current passing through the bolometer is: $$I_b=\frac{V_b}{R_b}$$
In general for a quadripole we have: $$F(V_b)=TF(\omega, R_b, C_p)\cdot F(V_0)$$ where $F$ indicate the Fourier transform and $TF(\omega, R_b, C_p)$ is the transfer function of the quadripole. Using the quadripole obtained from Eq. \[aceq\], the module of the transfer function is : $$|TF(\omega, R_b, C_p)|=\frac{R_b \omega C_b}{(1+\omega^2 R_b^2 (C_b+C_p)^2)^{1/2}}$$ So, the r.m.s. input voltage is: $$V_0=\frac{V_b}{|TF(\omega, R_b, C_p)|}$$
In order to calculate the optical responsivity let us consider a small step in temperature for each bolometer. If we keep $V_0$ unchanged, the step in temperature is due to a change of the optical background that can be computed:
$$W_{opt1}=W_{tot1}-W_{elec1}$$
where $W_{tot1}$ is calculated from the new temperature $T_{b1}$ and $W_{elec1}$ is derived from: $$W_{elec1}=\frac{V_{b1}^2}{R_{b1}}$$ $V_{b1}$ is equal to: $$V_{b1}=V_0 \cdot |TF(\omega, R_{b1}, C_p)|$$ assuming that $V_0$ is not varying, and using the $TF$ calculated from $R_{b1}$
The responsivity will be: $$\Re=| \frac{V_{b1}-V_b}{W_{opt1}-W_{opt}}|$$
with the responsivity and NEP equations from the previous section, it is possible to calculate the total NEP
Appendix B: Computing the Responsivity with a Square Bias {#appendix-b-computing-the-responsivity-with-a-square-bias .unnumbered}
=========================================================
With the same bias circuit (Fig. \[fig:ACmod\]), it is possible to derive the performances of a REU in which a square wave voltage applied to the bolometer, as in HFI. Let us assume that if the REU is *balanced*, a perfect square wave bias is passing through the bolometer even in presence of a stray capacitance. In HFI this is achieved by using a triangular wave plus a square wave.
A square wave can be decomposed as: $$V_b(\omega)=a cos (\omega t)+ \frac{a}{3} cos(3 \omega t)
+\frac{a}{5}cos(5 \omega t)+.......=$$ $$= \sum_{n=0}^{\bar{n}}\frac{a}{2n+1}cos((2n+1)\omega t)$$ The r.m.s. $V_{b}$ is equal to: $$\label{hfivb}
V_{b}=(\sum_{n=0}^{\bar{n}}(\frac{a}{\sqrt{2}(2n+1)})^2)^{1/2}$$
If the temperature of the bolometer is given and the optical power is constant we can calculate the r.m.s. $V_{b}$ as we did for the sine bias case (Eq. \[veff\] and Eq. \[wele\]). So we have: $$a=(W_{elec} R_b)^{1/2} \cdot \sum_{n=0}^{\bar{n}} 2(2n+1)^2$$ The r.m.s. bias current passing through the bolometer is: $$I_b=\frac{V_b}{R_b}$$
Now let’s derive the responsivity. As for the sine AC case, so we can calculate the $R_{b1}$, $W_{tot1}$, and the $TF$ starting using a small step in temperature due to an incoming optical signal.
On the other hand we cannot derive the electrical power following the same logic: if we keep the same set up of the REU, after a small step in temperature the bias passing through the bolometer is not a square wave anymore so, the Eq. \[hfivb\] is not applicable. We have to correct each term of the sum as follows: $$W_{elect1}=\frac{a^2}{R_{b1}}\sum_{n=0}^{\bar{n}} 2(2n+1)^2\cdot
\Upsilon((2n+1)\omega, R_b, R_{b1}, C_p)$$ where $$\Upsilon((2n+1)\omega, R_b, R_{b1}, C_p)=\frac{TF((2n+1)\cdot\omega,
R_b, C_p)}{TF((2n+1)\cdot \omega, R_{b1}, C_p)}$$
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|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The most efficient way to pack equally sized spheres isotropically in 3D is known as the random close packed state, which provides a starting point for many approximations in physics and engineering. However, the particle size distribution of a real granular material is never monodisperse. Here we present a simple but accurate approximation for the random close packing density of hard spheres of any size distribution, based upon a mapping onto a one-dimensional problem. To test this theory we performed extensive simulations for mixtures of elastic spheres with hydrodynamic friction. The simulations show a general (but weak) dependence of the final (essentially hard sphere) packing density on fluid viscosity and on particle size, but this can be eliminated by choosing a specific relation between mass and particle size, making the random close packed volume fraction well-defined. Our theory agrees well with the simulations for bidisperse, tridisperse and log-normal distributions, and correctly reproduces the exact limits for large size ratios.'
author:
- 'Robert S. Farr'
- 'Robert D. Groot'
title: Close packing density of polydisperse hard spheres
---
Introduction
============
Granular materials such as sediments and powders are widespread in nature and industrial contexts, and treating the grains as hard spheres is often a useful first approximation. In these systems, the manner in which the grains pack together has profound influence on properties. Of these, random close packing[@1] is most likely to be encountered in tapped and consolidated systems, although other possibilities, such as random loose packing[@2], and various crystalline arrangements (whose existence is very sensitive to the form of the size distribution and method of creation[@3; @4]) are also possible.
Even though the precise nature (and for monodisperse spheres, even the existence[@5]) of the random close packed state remains the subject of ongoing research[@6], it provides a starting point for many approximations in both physics[@7; @8] and engineering, and has great practical importance not only for the prediction of the density of granular materials, but also other properties. For example, the viscosity of dense dispersions will diverge at this point[@9] and it is related to the permeability in packed beds[@10].
One of the insights that have come forward from simulations[@6] is that the dense random packing density of hard spheres depends upon the (shear) friction coefficient, if the particles only lose energy by inelastic collisions. Further, the jamming density of a hard sphere system depends upon the initial state, and on the particular pathway chosen to cool down the system. In general, however, dissipative interactions play a role not just at contact. Granular particles suspended in a viscous medium also dissipate energy via long-range hydrodynamic interactions. Hence we anticipate that the dense random packing also depends upon solvent viscosity and on the range of the (hydrodynamic) friction. This is the first problem that we wish to address. To this end we developed a new simulation method that includes these effects. Using this method we not only find that the dense random packing depends on fluid viscosity, but – quite unexpectedly – also on particle size and mass. By analysing the various time scales in the problem we obtain a way to eliminate this dependence, which sheds new light upon the nature of the dense random packed state.
In practical cases the particle size distribution of a real granular material, or mixture of materials, is never monodisperse. Also for such polydisperse problems modelling techniques[@11; @12] have been used to calculate maximum packing fractions of spheres with a distribution of sizes. A typical example of a polydisperse system in a close packed state is shown in Fig. \[fig1\].
![\[fig1\] Close packed configuration of spheres from a log-normal distribution. The spread in the logarithm of radius is $\sigma = 0.6$. Only spheres with centres lying in one periodic image of the simulation cell are shown. ](figure1.eps){width="2.5in"}
The second problem that we wish to address is that these simulation methods are quite time consuming, and therefore their applicability is limited. It would be desirable to have an analytic expression for the close packing density, or a fast approximation, but progress in this direction has not been rapid. Ouchiyama and Tanaka[@13] have presented a theory based upon the volume occupied by a sphere in contact with other spheres of the mean diameter, but their results are at best qualitative, and the reasoning behind the method is not simple enough to suggest obvious improvements. Song [*et al*]{}.[@6] presented a theory for the packing of monodisperse spheres, but the generalisation to an arbitrary size distribution is not obvious. Recently Biazzo [*et al*]{}.[@14] presented a theory for binary mixtures, but like the Ouchiyama-Tanaka theory, it violates the exact upper limit for large fractions of big spheres that is given in Eq. (\[eq3\]) below. Thus, a comprehensive theory to predict the random close packing density of an arbitrary sphere mixture is still lacking.
We formulate such a theory, which is presented here in Section II. Next, we define our simulation method in Section III, and we present the combined theoretical and simulation results in Section IV. Conclusions are formulated in Section V.
Theory
======
Here we propose an approximate solution to the problem of polydisperse packing density, obtained by abstracting what we believe to be essential features of the physics and geometry of packing. The fundamental problem we wish to solve is as follows: suppose we have a normalised distribution $P_{3D}(D)$ of sphere diameters defined so that $P_{3D}(D){\rm d}D$ equals the number fraction of spheres with diameters in the range $(D, D+{\rm d}D)$ present in the system. Then we ask what is the functional ${\cal F}:P_{3D}(D)\mapsto\phi_{\max}$ that maps the size distribution onto the random close packed volume fraction?
In order to construct this functional, we begin by mapping the 3D sphere packing problem onto a packing problem of rods in 1D. The corresponding 1D distribution $P_{1D}(L){\rm d}L$ gives the number fraction of rods with rod length in the range $(L, L+dL)$ present in the system. To do this we imagine a large random, but non-overlapping, arrangement of spheres in 3D with size distribution $P_{3D}(D)$. This need not be close packed for the argument that follows. Now imagine drawing a straight line through this distribution, and counting each portion of the line which lies within a sphere as a rod (see Fig. \[fig2\]). The resulting distribution of rod lengths is then given by $$\label{eq1}
P_{1D}(L)=2L\frac{\int_{L}^{\infty}P_{3D}(D){\rm d}D}{
\int_{0}^{\infty}P_{3D}(D)D^{2}{\rm d}D}.$$
If rods of length $L_{i}$ are arranged on a line of length $\Lambda$ (with periodic boundary conditions), then the rod length fraction is clearly given by $\psi=\Lambda^{-1}\sum L_{i}$. This equals the volume fraction $\phi$ if there is a corresponding system of spheres in 3D.
![\[fig2\] How to map a 3D sphere distribution onto a rod distribution. A straight line through a random arrangement of spheres defines a set of rods. The probability that a sphere is hit by the line is proportional to its cross sectional area and so proportional to $D^{2}P_{3D}(D)$. The probability that such a hit produces a rod of length $L$ is $2L/D^{2}$ (for $L<D$), and Eq. (\[eq1\]) in the text follows. ](figure2.eps){width="3.0in"}
Let us assume that it is possible to map the closest random packing of spheres in 3D onto a problem of packing the above collection of rods on a line, where we search over all orderings of rods as well as their separations. This mapping will be achieved through an effective potential between the rods, which must have the following properties:
1\. It should lead to a maximum packing fraction $\psi_{max}$ which is unchanged if all the rods (or spheres) are magnified by an equal amount;
2\. The potential should be ‘hard’, in that it is either zero or infinite;
3\. The interaction potential between large rods should reach through small rods. This will allow very small rods to ‘rattle around’ in the gaps between the large rods, so that at high weight fractions of large rods the latter can form a stress-bearing network;
4\. The interaction range for mixtures of very unequal rods should be determined by the size of the smallest rod, so that small rods can form a dense randomly packed system in between the large rods, without leaving large gaps.
The true interaction potential between the rods will be both many-body and highly complicated, capturing topological aspects of 3D space. However, we suggest that the following pair potential is the simplest expression which honours the four requirements listed:
If two rods $i$ and $j$ have a gap $h$ between their nearest approaching ends, then we introduce the potential $$\label{eq2}
V(h)=\left\{
\begin{array}{lll}
\infty & {\rm if} & h < \min(fL_{i},fL_{j}) \\
0 & {\rm if} & h \ge \min(fL_{i},fL_{j})
\end{array}\right.$$ where $f>0$ is a ‘free volume’ parameter. If this potential applies between any pair of rods, regardless of intervening rods in the gap between them, then it will satisfy requirement 3. The other requirements follow naturally. Rather than taking the minimum of the two lengths one could introduce a more complicated function of $L_{i}$ and $L_{j}$, but the present choice appears to be the simplest to capture the physics. In the remainder of this article $f$ will be used as a fit parameter; it is the only parameter in our theory, and can be chosen by requiring that the theory reproduces the random close packing of monodisperse systems.
For each ordering of the rods on the line, there is a shortest line which can accommodate them without incurring an infinite potential energy, and this leads to a close packing fraction for this ordering, which is simply $\psi=\Lambda^{-1}\sum L_{i}$. The maximum packing fraction $\psi_{\max}$ is the maximum value attained by $\psi$ over all possible orderings of the rods. If all the $L$’s are equal, $\psi_{\max} = (1+f)^{-1}$, and any rod polydispersity (which will always be present if we use Eq. (\[eq1\])) will increase $\psi_{\max}$.
If we imagine inserting rods one at a time to form a packing, while increasing $\Lambda$ if necessary, then Eq. (\[eq2\]) constitutes a two-body potential between the inserted rod and all the rods currently in the packing. However, if rods are inserted in decreasing order of size, then the special choice of this potential means it only depends upon the newly inserted rod, and the packing further away is not disrupted by this process. To insert the new rod with a minimum increase of line length $\Lambda$ we need to identify the biggest gap. Therefore the following ‘greedy algorithm’[@15] may be used to find $\psi_{\max}$ for arbitrary $\{ L_{i}\}$:
\(a) The set of lengths $\{ L_{i}\}$ is labelled such that $L_{1} \ge L_{2} \ge \ldots \ge L_{N}$. These will be inserted in decreasing order of lengths into the growing optimal packing, starting with $L_{1}$.
\(b) Throughout the algorithm, we maintain a set of gaps $\{ g_{i}\}$, equal in number to the number of rods we have inserted into the packing. At the start, when we have only one rod, this set contains one element $g_{1} = fL_{1}$.
\(c) In order to insert rod $j$, we identify $g_{\max}$, the largest gap in the set of gaps, and we remove it from the set. We then add two new gaps to the set, namely $fL_{j}$ and $\max[g_{\max} - (1+f)L_{j}, fL_{j}]$.
This process implicitly increases the line length $\Lambda$, if that is needed to accommodate the new rod.
Our final approximation consists of choosing a large number of rods from the distribution $P_{1D}(L)$, and packing them by the greedy 1D packing algorithm to obtain our estimate $\psi_{\max}$ for $\phi_{\max}$, the close packed volume fraction of the sphere distribution $P_{3D}$. Since this is essentially a sorting routine, the predictions of this algorithm take much less time (ca 0.3 seconds on a modern desktop computer) than explicit 3D simulation (1 to 30 hours). When the rod lengths are chosen at equidistant values of the cumulative 1D distribution, 2000 rods are sufficient for 5 decimal places accuracy. Thus, for $N$ rods we choose rod length $L_i$ such that $\int_{L_i}^{\infty}P_{1D}(l){\rm d}l = (N-i+1/2)/N$. We use $N=20000$ rods.
One useful property of this procedure is that it correctly reproduces the exact solution for bidisperse spheres with infinite size ratio. This limit is given by $$\label{eq3}
\psi_{\max}=\min\left(
\frac{\phi_{RCP}}{1-w(1-\phi_{RCP})},\frac{\phi_{RCP}}{w}\right),$$ where $\phi_{RCP}$ is the maximum packing fraction for a [*monodisperse*]{} system, and $w$ is the mass fraction of large spheres on the total particle volume, so $w = \phi_{\rm large}/(\phi_{\rm large}+\phi_{\rm small})$. Numerical solution of the theory for size ratios down to 1:1000 shows minor ($\sim 1\%$) deviations from the exact limit, for $w$ values very close to the cusp, $w=1/(2-\phi_{RCP})$.
The description above is complete, except that we need to specify the parameter $f$, which should be chosen such that the predicted maximum packing for monodisperse spheres is the correct random close packing value $\phi_{RCP}$. For monodisperse spheres we have $P_{1D}(L) = 2LD_{0}^{-2}\theta(D_{0}-L)$ (where $\theta$ is the Heaviside step function), and we find that a value of $f = 0.7654$ leads to a packing fraction of approximately $0.6435$. This value agrees with the simulation result described below, and is our only fit parameter.
Simulation
==========
Several methods have been proposed in the literature to simulate the dense random packing of hard spheres. One method used frequently was introduced by Lubachevsky and Stillinger,[@21] who simulated hard spheres by Molecular Dynamics and slowly compress the system until it jams. There is a drawback to this method, namely that ever smaller time steps need to be taken as close packing is approached. Moreover, the physical relaxation time of the system diverges near the close packing density,[@26] and consequently long runs are necessary. To circumvent this problem O’Hern [*et al.*]{}[@16] used soft spheres, and located the minimum energy by a conjugate gradient (CG) algorithm. This method is much faster than a hard sphere simulation, but the disadvantage is that the CG algorithm only simulates the high friction limit.
In general the dense random packing dependends on friction,[@6] and dissipative interactions may play a role not just at hard sphere contact. Granular particles suspended in a viscous medium also dissipate energy via long-range hydrodynamic interactions. Hence we anticipate that the dense random packing also depends upon solvent viscosity and on the range of friction. Therefore we introduce a new simulation method to include these effects.
Following O’Hern [*et al.*]{}[@16] and Groot and Stoyanov[@17] we simulate repulsive elastic spheres in the limit $T \rightarrow 0$. The generalization of the repulsive force in this model to elastic spheres of unequal size is $$\label{eq4}
F_{ij}^{Rep}=2ER_{ij}(D_{ij}-r)\theta(D_{ij}-r).$$ Here, $r$ is the distance between particle centres, $D_{ij} = (D_{i}+D_{j})/2$ is the mean diameter and $R_{ij}$ is the harmonic mean radius given by $R_{ij} = \frac{1}{2}D_{i}D_{j}/D_{ij}$. The parameter $E$ is proportional to the linear elastic modulus of the particles[@17]. Henceforth we use $E = 1000$.
Instead of using a CG algorithm to search the energy minimum, or imposing energy dissipation at particle collision, we introduce a soft friction function of finite range that represents the hydrodynamic interaction between spheres. A soft friction has been introduced before to simulate hydrodynamics in fluids, in the context of Dissipative Particle Dynamics[@27; @19; @20]. However, application to particles of unequal size is new to our knowledge, and because the dense random packing depends on friction some care must be taken in defining the friction function.
The most general distance-dependent friction is $$\label{fric1}
{\rm\bf F}_{ij}^{frict}=-\gamma_{ij}{\rm\bf v}_{ij}^{r}g(r/r_c)$$ where $\gamma_{ij}$ is a friction factor that may depend on both particle sizes, ${\rm\bf v}_{ij}^{r}$ is the radial velocity difference, $g(r/r_c)$ is a distance dependent function, and $r_c$ is a cut-off distance that may again depend on particle size.
![\[newfig3\] Volume fraction obtained by extrapolating the $P-\phi$ curve to zero pressure. ](figure3.eps){width="2.5in"}
To demonstrate the importance of the friction function, we first concentrate on monodisperse systems and study the influence of friction range and strength, and of particle size. We use a periodic $10\times 10\times 10$ box, containing from 1222 up to 1290 particles. All particles have diameter 1 and mass 1, and interact with a repulsive force $F = 10^3(1-r)$ for $r < 1$. First we study the influence of the range of the friction interaction. To this end we use the friction interaction ${\rm\bf F}_{ij}^{frict} = -{\rm\bf v}_{ij}^r (1-r/r_c)^2/(1-1/r_c)^2 $; and the range is varied as $r_c =$ 1.5, 2, 3. With this choice for the friction force, the friction at particle contact is unity for all systems. All systems are evolved over $10^4$ steps or more, with $\delta t = 0.01$. For each system the pressure at T = 0 was averaged over 5 independent starting configurations.
Even though all systems have the same friction strength at $r = 1$, the mere range of friction appears to influence the pressure in the final state. As the friction range increases, so does the pressure at T = 0. In particular, the volume fraction to which the pressure extrapolates to zero – the dense random packing – varies systematically with the force range, see Fig. \[newfig3\]. Although the effect is not very large (about 1% variation), it is clear that the friction range does have an influence. The extrapolated value to $r_c = D = 1$, $\phi_{cp} = 0.6392\pm 0.0004$, compares well with the reported mean-field result for hard spheres with a friction interaction at contact,[@6] $\phi_{RCP} = 0.634$. Thus we conclude that, to eliminate the influence of the friction range, the range must be scaled relative to the particle size.
Next, to demonstrate the influence of particle size and friction strength, a series of simulations was done where the ratio of the friction range relative to the particle diameter was kept fixed at $r_c/D$ = 1.5. The friction force used in this study was ${\rm\bf F}_{ij}^{frict} = -\gamma_f {\rm\bf v}_{ij}^r (1-r/r_c)^2 $, where $\gamma_f$ is a fixed friction factor, to be specified as input variable. Two sizes were studied, $D = 1/2$ and $D = 1$, and two friction factors were used, $\gamma_f = 1$ and $\gamma_f = 4$. In these simulations all particle masses were put at $m = 1$. The box size was taken as $V = 8^3$, and the conservative force was taken as $F = 10^3D(D-r)$ for $r<D$, the same as in the previous simulations.
The results are shown in Fig. \[newfig4\]a. The red lines give the results of low viscosity ($\gamma_f = 1$) and the black lines give the results of higher viscosity ($\gamma_f = 4$). Results for small particles are denoted by open symbols and dashed curves, while results for big particles are denoted by closed symbols and full curves. This shows that both the friction factor and particle size influence the pressure in the glassy state. Consequently, the dense random packing density must depend on particle size. Even though the effect is small it is important, as it points at a reason why the dense random packing density is ill-defined. For practical reasons we wish to define a dense random packing density that does not depend on particle size, i.e. that is scale invariant. To obtain this, it is not sufficient to have a friction range that is proportional to the particle diameter; the packing density depends in a complicated way on particle size and on the strength and range of the friction force. Empirically there may seem to be some scaling when the friction is increased with the square of particle size (results for $D = 1/2$, $\gamma_f = 1$ in Fig. \[newfig4\]a partially overlap with $D = 1$, $\gamma_f = 4$), but the slopes of the curves are clearly different.
![\[newfig4\] Mean pressure at T = 0 as function of particle volume fraction; a (top), for big particles (full symbols) and small particles (open symbols) and for low (red) and high (black) viscosity; b (bottom) big particles (red dots) and small particles (black circles) using friction as in Eq. (\[eq5\]) and mass $m=D$. ](figure4a.eps "fig:"){width="2.4in"} ![\[newfig4\] Mean pressure at T = 0 as function of particle volume fraction; a (top), for big particles (full symbols) and small particles (open symbols) and for low (red) and high (black) viscosity; b (bottom) big particles (red dots) and small particles (black circles) using friction as in Eq. (\[eq5\]) and mass $m=D$. ](figure4b.eps "fig:"){width="2.4in"}
The above results show that we cannot just take any friction function. It must reflect the physical properties of the hydrodynamic interaction. One physical property is the scaling of the hydrodynamic force with particle size. On dimensional grounds the friction factor must be proportional to particle radius, as in Stokes’ law, $F=6\pi\eta av$. More generally, the (radial) squeeze mode of the hydrodynamic force between two rigid spheres [*at close contact*]{} behaves as[@18] $$\label{neweq5}
{\rm\bf F}_{ij}^{frict} = -\gamma {\rm\bf v}_{ij}^{r} R_{ij} g(h/R_{ij})
\approx -\gamma {\rm\bf v}_{ij}^{r} \frac {R_{ij}^2} h$$ where $\gamma$ is the friction factor that is proportional to fluid viscosity, and $h=r-D_{ij}$ is the distance of closest approach. The function $g(x)\sim(1/x)$ is a scaling function that represents the lubrication force.
There is a large body of evidence showing that correct long-range inertial hydrodynamics is generated even if the (divergent) lubrication force between particles is replaced by a finite distance-dependent friction[@19; @20]. It is important however to choose the harmonic mean radius as the scaling length (unlike the conjecture by Kim and Karilla[@18]), otherwise the friction between very unequal spheres would vanish if we remove the divergence of $g(x)$. Thus, we use the friction function $$\label{eq5}
{\rm\bf F}_{ij}^{frict}=-\gamma{\rm\bf v}_{ij}^{r}R_{ij}
(1-h/R_{ij})^{2}\theta(R_{ij}-h).$$ which captures the right physics regarding the scaling of the range and strength of the viscous interaction with particle size.
Now we can turn to the problem of defining a size invariant dense random packing density. Therefore we analyse the relevant time scales of the problem. For a monodisperse system of elastic particles, a first time scale is the oscillation time, $t_{el} = 2\pi(m/ED)^{1/2}$. This is the elastic time scale. The second time scale in the system is the drag relaxation time, $t_{d} = m/(\gamma D)$. The dense random packing can only be independent of particle size if we maintain a constant ratio between these two time scales, $t_{el}/t_{d} = 2\pi\gamma(D/Em)^{1/2}$. Thus, we have scale invariance only if the friction factor satisfies $\gamma\propto (m/D)^{1/2}$. Since for most systems mass scales as $m \propto D^{3}$, this implies that for (soft) elastic spheres with hydrodynamic interaction the dense random packing fraction is (weakly) particle size dependent.
To obtain a well-defined dense random packing we are forced to choose the particle mass $m$ proportional to $D$. When this choice is made, the above ratio of time scales becomes particle size independent, and consequently the dense random packing fraction is well-defined. This choice has been made henceforth. The predicted scaling was checked by simulation and holds exactly. The pressure as function of time for systems of particle diameter $D = 1/2$ and $D = 1$ fall on top of each other if we scale the friction range with particle size (as in Eq. (\[eq5\])) and simultaneously impose $m\propto D$. To demonstrate the improvement in system definition, two series of simulations were done, again for monodisperse systems of particle diameter $D = 1/2$ and $D = 1$, with repulsive force $F = 10^3 D(D-r)$ for $r < D$. For a realistic hydrodynamic scaling we used Eq. (\[eq5\]), with friction factor $\gamma = 0.74$. Because the systems are monodisperse the cut-off distance for the friction interaction is $r_c = 1.5D$, as in the previous case. To obtain scale invariance, we choose the masses as $m = D$. The systems had fixed volume $V = 10^3$ for $D = 1$ and $V = 5^3$ for $D = 1/2$. The systems were integrated over 5000 steps with time step $\delta t = 0.01$ and the pressure was averaged over 10 independent runs. The comparison between small and big particles is shown in Fig. \[newfig4\]b. The predicted scaling is followed excellently.
Now that the system is well-defined, we can define a fast algorithm to obtain the close packing density. We use a variation of the Lubachevsky-Stillinger algorithm[@21], where we make use of the relative softness of the interaction potential. We prepare the system in a random conformation (with particle overlaps) and then evolve it in an $(N, V, T)$ ensemble until we have a completely equilibrated state. For the parameters $\gamma = 1$ and $\delta t = 0.01$ that we used, this requires $3-50\times 10^3$ time steps. Then we switch to an $(N, P, T)$ ensemble, where the pressure is steered towards $P = 0.01$, which is close enough in practice to $P = 0$ (the error in $\phi_{\max}$ is of the order $10^{-5}$). If during a run the pressure falls below $0.001$ we switch to the L-S algorithm and compress the system in small steps until the pressure turns positive. The advantage of this method over the standard L-S algorithm is that the (high) pressure in the initial $(N, V, T)$ simulation quickly drives the system towards $P = 0$. The final $(N, P, T)$ simulation serves to run down the $P-\phi$ curve (see Fig. \[newfig4\]b) to locate the intercept at $P=0$. Some minor evolution can however still be observed at $P=0.01$. To gain further simulation speed we combined a linked cell neighbour search with a Verlet neighbour list[@22]. In the late stages of evolution, when particles hardly move, this leads to a large increase in simulation speed, particularly for systems of large particle size difference.
![\[fig3\] a (top): Close packing density as function of the friction factor $\gamma$. Each data point is an average over 10 independent runs; b (bottom) same data, plotted to $1/\sqrt{\gamma}$. ](figure5a.eps "fig:"){width="2.4in"} ![\[fig3\] a (top): Close packing density as function of the friction factor $\gamma$. Each data point is an average over 10 independent runs; b (bottom) same data, plotted to $1/\sqrt{\gamma}$. ](figure5b.eps "fig:"){width="2.4in"}
Even though the close packing density is now well-defined, it still depends on friction, or the cooling rate, see Fig. \[fig3\]. In fact this is the very source of the previously found dependence of the close packing density on particle size and mass. To study this relation, monodisperse systems of 6000 particles were used. All particles have diameter 1 and mass 1, and interact with a repulsive force with $E = 10^{3}$. We insert the particles in a box of size $16.83$ (or $\phi=0.66$), pre-equilibrate for $5\times 10^{4}$ to $2\times 10^{5}$ time steps of $\delta t = 0.01$ until the pressure has fully equilibrated. Then we run a constant pressure ensemble, steering the pressure to $P = 0.01$, until the volume fraction is stable over four decimal places ($1.5\times 10^{5}$ time steps). All results are averaged over 10 runs. Polycrystalline domains only start to occur for $\gamma<0.03$; all systems shown here are isotropic. Over about a decade we find a $\log(\gamma)$ dependence for $\gamma\rightarrow 0$ (see Fig. \[fig3\]a), and over two decades we find a power law decay $\phi \approx 0.64 + 0.0028/\sqrt{\gamma}$ for $\gamma>1/4$ (see Fig. \[fig3\]b). Therefore we have to make a choice for the friction factor, and only refer to the packing fraction at that value of $\gamma$. Our default value used in the next section is $\gamma =1$.
For comparison we also evolved a system from its initial conformation to equilibrium using a steepest descent method in an $(N, V, T)$ ensemble, which should compare well with the CG algorithm[@16]. The final pressure coincides with our result at $\gamma \rightarrow\infty$, which demonstrates that CG simulates the high viscosity limit. For truly hard spheres the modulus diverges, hence the ratio $t_{el}/t_{d} = 2\pi\gamma(D/Em)^{1/2}\rightarrow 0$. To simulate this with particles of finite modulus, low values of $\gamma$ would be preferred. This implies that the present method is closer to the physical case than the CG algorithm to generate hard spheres conformations, unless largely inelastic collisions are pertinent.
Results
=======
The simple approximation for ${\cal F}$ described in section II is now compared with the results from the sphere packing simulation method of section III. Consider first bidisperse spheres, as studied for example by Clarke and Wiley[@23] and by Yerazunis et al.[@24], where the larger spheres have $R$ times the radius of the smaller, so that $P_{3D}(D) \propto R^{3}(1-w) \delta(D-1/R) + w \delta(D-1)$. The simulation results for binary mixtures are shown in Table \[tab1\], and in Fig. \[fig4\], together with the theoretical prediction. The big particles have diameter $D_{1} =$ 1, and the small particle diameters are $D_{2} =$ 0.5, 0.3, 0.2 and 0.1. The dash-dot curve gives the exact upper limit of the volume fraction, Eq. (\[eq3\]). For diameter $D_{2}\ge 0.2$ we used $6000$ particles; for $D_{2} = 0.1$ we used up to $N = 49950$ particles (at $w = 0.8$) to prevent finite size effects. All runs were evolved over a minimum of 150 000 time steps, and convergence of the volume fraction was checked by extending the evolution of selected systems to 450 000 steps. All volume fraction results shown in Fig. \[fig4\] are stable up to four decimal places.
![\[fig4\] Maximum packing fraction for bidisperse spheres of different size. $R$ is the size ratio, and $w = \phi_{\rm large}/(\phi_{\rm large}+\phi_{\rm small})$ is the relative volume fraction of the large spheres. Symbols are simulation results, and solid lines are theoretical predictions, based on 20000 rods. The dashed curves give the upper limit for infinite size ratio, Eq. (\[eq3\]). ](figure6.eps){width="2.5in"}
$w$ $D_{2}/D_{1}=0.5$ $D_{2}/D_{1}=0.3$ $D_{2}/D_{1}=0.2$ $D_{2}/D_{1}=0.1$
------ ------------------- ------------------- ------------------- -------------------
0 0.6435 0.6435 0.6435 0.6435
0.2 0.6579 0.6695 0.6761
0.4 0.6690 0.6971 0.7152 0.7298
0.5 0.7557
0.6 0.6774 0.7236 0.7525 0.7835
0.7 0.6795 0.7324 0.7714 0.8150
0.75 0.8270
0.8 0.6749 0.7315 0.7769 0.7948
0.9 0.6650 0.6985 0.7111
0.95 0.6558 0.6690
1 0.6435 0.6435 0.6435 0.6435
: \[tab1\]Simulation results for the maximum packing fraction of bidisperse sphers, with diameters $D_{1}$ and $D_{2}$. The mass fraction present in the large spheres is given by $w = \phi_{\rm large}/(\phi_{\rm large}+\phi_{\rm small})$.
A bidisperse system, with moderate to large size ratios, has two distinct regimes (and a non-trivial crossover between them): When the proportion of large spheres is low, they are isolated from one another, like holes in a Swiss cheese; while the small spheres form a close-packed phase (the ‘cheese’) between them. On the other hand, when the proportion of large spheres is high, these form a close-packed structure, leaving the small spheres to ‘rattle around’ in the gaps between them. Recent theories of the close packed state of bidisperse spheres[@14; @25] do not address the ‘rattler’ regime adequately. In contrast, our theory captures both regimes (exactly, in the limit of infinite size ratio), and also the analogous regimes which are produced for larger numbers of size classes, such as tridiperse spheres.
![\[fig5\] Maximum packing fraction for a tridisperse distribution of spheres with size ratios 1:3:9. The composition diagram (a, top) is based on weight fractions; contour lines connect points of equal volume fraction, with bold lines at volume fractions of 0.65, 0.7, 0.75 and 0.8. The compositions used in the simulations are marked by the circles; b (bottom) shows the comparison between theory and simulation per sample point. ](figure7a.eps "fig:"){width="2.7in"} ![\[fig5\] Maximum packing fraction for a tridisperse distribution of spheres with size ratios 1:3:9. The composition diagram (a, top) is based on weight fractions; contour lines connect points of equal volume fraction, with bold lines at volume fractions of 0.65, 0.7, 0.75 and 0.8. The compositions used in the simulations are marked by the circles; b (bottom) shows the comparison between theory and simulation per sample point. ](figure7b.eps "fig:"){width="2.4in"}
Next, we consider a tridisperse distribution. In particular we consider the case where the sphere diameters are in the ratio 1:3:9. Fig. \[fig5\] shows the theoretical prediction compared with simulation results for selected points. The number of particles chosen varies from 6000 to 13400. Particular care was taken at large weight fractions of big particles, where these may form a stress bearing network. A finite size study showed that in that case (specifically sample point 1) the number of large particles in the system needs to be above 175 for reliable results. For sample point 1 we used 209 big particles from a total of 13300. Again, the theory compares very well with the simulation results. The differences may well be attributed to remaining (minor) finite size effects.
![\[fig6\] Maximum packing fraction for a log-normal distribution. The spread in log(radius) is denoted by $\sigma$. Simulation results are given by symbols; the solid line is the theoretical prediction, based on 20000 rods. ](figure8.eps){width="2.5in"}
Finally, we consider a log-normal distribution, which is defined as $P_{3D}(D) \propto \exp(- [\ln(D/D_{0})]^{2}/2\sigma^{2})/D$. Thus, from Eq. (\[eq2\]), we find the rod distribution in the theory as $P_{1D}(L) \propto L\ {\rm erfc}[\ln(L/D_{0})/\sigma\sqrt{2}]$. In these simulations we set an upper diameter to the particles $D = 1$, and choose the mean diameter such that less than 0.1% of the particles exceed this size. We used 6000 particles that were evolved over 50 000 to 200 000 steps, until the volume fraction had converged up to 4 decimal places. Fig. \[fig6\] shows the comparison between theory and simulation data. The system at $\sigma = 0.6$ is shown in Fig. \[fig1\]. Again, the theory compares very well with the simulation results.
Conclusions
===========
Concluding, we have introduced a theory for the close packing density of hard spheres of arbitrary size distribution, based on a mapping onto a one-dimensional problem. To test the theory we simulated the dense random packing of (soft) elastic spheres with hydrodynamic friction in 3D in the limit $T\rightarrow 0$, which approaches the hard sphere system. For the distributions studied we obtain excellent agreement between theory and simulation. The theory reproduces the infinite size ratio limit for bidisperse spheres. Hence we expect that this approximation will prove useful for more general size distributions. The simple structure of the approximation may also be amenable to further analysis, and open up new avenues for analytical approximations.
Application of this theory to other space dimensions than 3D is straightforward. However, a comparison between theory and simulations showed that, although the theory is qualitatively correct in 2D, it does not reproduce the simulations as accurately as in 3D. This may be related to the mean-field character of the theory, i.e. the explicit spatial correlation is lost in the theory. We therefore speculate that the theory will be accurate in 3D and in higher dimensions.
The simulations show a weak dependence of the dense random packing on fluid viscosity, if the friction force is of hydrodynamic origin. In general the dense random packing density also depends on particle size, mass and elastic modulus. For particles of diameter $D$, mass $m$ and elastic modulus $E$, suspended in a liquid of viscosity $\eta$, we infer that the dense random packing density should be a function of the dimensionless group $Q=\eta^2D/(Em)$. For systems of the same $Q$ value but different size, mass and viscosity we find excellent scaling of the pressure as function of time. Therefore we conclude that the size dependence of dense random packing is a kinetic effect that disappears when $m\propto D$. Although such scaling is artificial, it leads to a well-defined dense random packing, which is prudent to test theories that are based only on geometrical considerations.
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