Dyson-Schwinger Equations For Propagators Research Articles

A machine learning pipeline for autonomous numerical analytic continuation of Dyson-Schwinger equations

Dyson-Schwinger equations (DSEs) are a non-perturbative way to express n-point functions in quantum field theory. Working in Euclidean space and in Landau gauge, for example, one can study the quark propagator Dyson-Schwinger equation in the real and complex domain, given that a suitable and tractable truncation has been found. When aiming for solving these equations in the complex domain, that is, for complex external momenta, one has to deform the integration contour of the radial component in the complex plane of the loop momentum expressed in hyper-spherical coordinates. This has to be done in order to avoid poles and branch cuts in the integrand of the self-energy loop. Since the nature of Dyson-Schwinger equations is such, that they have to be solved in a self-consistent way, one cannot analyze the analytic properties of the integrand after every iteration step, as this would not be feasible. In these proceedings, we suggest a machine learning pipeline based on deep learning (DL) approaches to computer vision (CV), as well as deep reinforcement learning (DRL), that could solve this problem autonomously by detecting poles and branch cuts in the numerical integrand after every iteration step and by suggesting suitable integration contour deformations that avoid these obstructions. We sketch out a proof of principle for both of these tasks, that is, the pole and branch cut detection, as well as the contour deformation.

Linking continuum and lattice quark mass functions via an effective charge

The quark mass function is computed both by solving the quark propagator Dyson-Schwinger equation and from lattice simulations implementing overlap and Domain-Wall fermion actions for valence and sea quarks, respectively. The results are confronted and seen to produce a very congruent picture, showing a remarkable agreement for the explored range of current-quark masses. The effective running-interaction is based on a process-independent charge rooted on a particular truncation of the Dyson-Schwinger equations in the gauge sector, establishing thus a link from there to the quark sector and inspiring a correlation between the emergence of gluon and hadron masses.

Locating the critical endpoint of QCD: Mesonic backcoupling effects

We study the effects of pion and sigma meson backcoupling on the chiral order parameters and the QCD phase diagram and determine their effect on the location of the chiral critical endpoint. To this end, we solve a coupled set of truncated Dyson-Schwinger equations for Landau gauge quark and gluon propagators with ${N}_{\mathrm{f}}=2+1$ dynamical quark flavors and explicitly backcoupled mesons. The corresponding meson bound-state properties and the quark-meson Bethe-Salpeter vertices are obtained from their homogeneous Bethe-Salpeter equation. We find chiral-restoration effects of the pion and/or sigma meson backcoupling and observe a (small) shift of the critical endpoint towards smaller chemical potentials. The curvature of the chiral crossover line decreases. Our results indicate that the location of the critical endpoint in the phase diagram is mainly determined by the microscopic degrees of freedom of QCD (in contrast to its critical properties).

Correlation functions of Landau gauge Yang-Mills theory

Correlation functions of Yang-Mills theory in the Landau gauge are calculated from their equations of motion. The employed setup is completely parameter free and leads, within errors, to good quantitative agreement with corresponding lattice results for the ghost and gluon propagators as well as the ghost-gluon and three-gluon vertices. Also the four-gluon vertex is calculated. The present setup allows for the first time for a unique subtraction of quadratic divergences in the gluon propagator Dyson-Schwinger equation. Thus, there are no ambiguities which can arise due to the use of models or auxiliary workarounds. In addition, several self-tests of the results are described that allow assessing the truncation error in a self-consistent way. This enables a new perspective on how to identify limitations of the present setup and develop future improvements.

Deep reinforcement learning for complex evaluation of one-loop diagrams in quantum field theory.

In this paper we present a technique based on deep reinforcement learning that allows for numerical analytic continuation of integrals that are often encountered in one-loop diagrams in quantum field theory. To extract certain quantities of two-point functions, such as spectral densities, mass poles or multiparticle thresholds, it is necessary to perform an analytic continuation of the correlator in question. At one-loop level in Euclidean space, this results in the necessity to deform the integration contour of the loop integral in the complex plane of the square of the loop momentum, to avoid nonanalyticities in the integration plane. Using a toy model for which an exact solution is known, we train a reinforcement learning agent to perform the required contour deformations. Our study shows great promise for an agent to be deployed in iterative numerical approaches used to compute nonperturbative two-point functions, such as the quark propagator Dyson-Schwinger equation, or more generally, Fredholm equations of the second kind, in the complex domain.

Analytic properties of the quark propagator from an effective infrared interaction model

In this paper, I investigate the analytic properties of the quark propagator Dyson-Schwinger equation (DSE) in the Landau gauge. In the quark self-energy, the combined gluon propagator and quark-gluon vertex is modeled by an effective interaction (the so-called Maris-Tandy interaction), where the ultraviolet term is neglected. This renders the loop integrand of the quark self-energy analytic on the cut plane $\ensuremath{-}\ensuremath{\pi}<arg(x)<\ensuremath{\pi}$ of the square of the external momentum. Exploiting the simplicity of the truncation, I study solutions of the quark propagator in the domain $x\ensuremath{\in}[\ensuremath{-}5.1,0]\phantom{\rule{0.16em}{0ex}}{\text{GeV}}^{2}\ifmmode\times\else\texttimes\fi{}i[0,10.2]\phantom{\rule{0.16em}{0ex}}{\text{GeV}}^{2}$. Because of a complex conjugation symmetry, this region fully covers the parabolic integration domain for Bethe-Salpeter equations (BSEs) for bound state masses of up to 4.5 GeV. Employing a novel numerical technique that is based on highly parallel computation on graphics processing units (GPUs), I extract more than 6500 poles in this region, which arise as the bare quark mass is varied over a wide range of closely spaced values. The poles are grouped in 23 individual trajectories that capture the movement of the poles in the complex region as the bare mass is varied. The raw data of the pole locations and residues is provided as Supplemental Material, which can be used to parametrize solutions of the complex quark propagator for a wide range of bare mass values and for large bound-state masses. This study is a first step towards an extension of previous work on the analytic continuation of perturbative one-loop integrals, with the long-term goal of establishing a framework that allows for the numerical extraction of the analytic properties of the quark propagator with a truncation that extends beyond the rainbow by making adequate adjustments in the contour of the radial integration of the quark self-energy.

The Quark Propagator in the Complex Domain: A Status Report

In these proceedings I review the status of an ongoing project that aims at solving the quark propagator Dyson-Schwinger equation in the complex domain. The novel aspect of the approach is that the non-analyticities arising throughout the iteration of the equation are to be taken into account in an appropriate way through dynamic contour deformation. Because of the complexity of the approach, these studies are undertaken in a heavily truncated scenario that serves as a toy model for the development of the numerical techniques that are required to treat the system in a mathematically sound way.

The quark propagator in QCD andG2QCD

QCD-like theories provide testing grounds for truncations of functional equations at non-zero density, since comparisons with lattice results are possible due to the absence of the sign problem. As a first step towards such a comparison, we determine for QCD and $G_2$ QCD the chiral and confinement/deconfinement transitions from the quark propagator Dyson-Schwinger equation at zero chemical potential by calculating the chiral and dual chiral condensates, respectively.

Different critical points of chiral and deconfinement phase transitions in (2 + 1)-dimensional fermion–gauge interacting model

Based on the truncated Dyson-Schwinger equations for fermion and massive boson propagators in QED(3), the fermion chiral condensate and the mass singularities of the fermion propagator via the Schwinger function are investigated. It is shown that the critical point of the chiral phase transition is apparently different from that of the deconfinement phase transition and in Nambu phase the fermion is confined only for small gauge-boson mass.

Running coupling in the conformal window of large-Nf QCD

Quantum Chromodynamics with a relatively large number of fundamentally charged quark flavours in the chiral limit is considered. A self-consistent solution of the quark, gluon and ghost propagator Dyson-Schwinger equations in Landau gauge exhibits a phase transition. Above the critical number of fermion flavours the non-perturbative running coupling develops a plateau over a wide momentum range, and the propagators follow a power law behaviour for these momenta. Hereby, the critical number of quark flavours depends crucially on the beyond-tree-level tensor structures of the quark-gluon vertex.

Infrared Self-consistent Solutions of Bispinor QED<sub><span class="cmr-7">3</span></sub>

Quantum electrodynamics in three dimensions in the bispinor formulation is considered. It is shown that the Dyson-Schwinger equations for fermion and boson propagators may be self-consistently solved in the infrared domain if on uses the Salam's vertex function. The parameters defining the behavior of the propagators are found numerically for different values of coupling constant and gauge parameter. For weak coupling the approximated analytical solutions are obtained. The renormalized gauge boson propagator (transverse part) is shown in the infrared domain to be practically gauge independent.

Brief remarks on the similarities of the infrared solutions for the ghost propagator Dyson-Schwinger equation in Landau and Coulomb gauges

This brief note is devoted to reconcile the conclusions from a recent analysis of the IR solutions for the ghost propagator Dyson-Schwinger equations in Coulomb gauge with previous studies in Landau gauge.

Low-momentum ghost dressing function and the gluon mass

We study the low-momentum ghost propagator Dyson-Schwinger equation (DSE) in Landau gauge, assuming for the truncation a constant ghost-gluon vertex, as it is extensively done, and a simple model for a massive gluon propagator. Then, regular DSE solutions (the zero-momentum ghost dressing function not diverging) appear to emerge and we show the ghost propagator to be described by an asymptotic expression reliable up to the order ${\cal O}(q^2)$. That expression, depending on the gluon mass and the zero-momentum Taylor-scheme effective charge, is proven to fit pretty well the low-momentum ghost propagator obtained through big-volume lattice simulations.

On the IR behaviour of the Landau-gauge ghost propagator

We examine analytically the ghost propagator Dyson-Schwinger Equation (DSE) in the deep IR regime and prove that a finite ghost dressing function at vanishing momentum is an alternative solution (solution II) to the usually assumed divergent one (solution I). We furthermore find that the Slavnov-Taylor identities discriminate between these two classes of solutions and strongly support the solution II. The latter turns out to be also preferred by lattice simulations within numerical uncertainties.

Propagator Dyson-Schwinger equations of Coulomb gauge Yang-Mills theory within the first order formalism

Coulomb gauge Yang-Mills theory within the first order formalism is considered with a view of deriving the propagator Dyson-Schwinger equations. The first order formalism is studied with special emphasis on the Becchi-Rouet-Stora (BRS) invariance and it is found that there exists two forms of invariance---invariance under the standard BRS transform and under a second, nonstandard transform. The field equations of motion and symmetries are derived explicitly and certain exact relations that simplify the formalism are presented. It is shown that the Ward-Takahashi identity arising from invariance under the nonstandard part of the BRS transform is guaranteed by the functional equations of motion. The Feynman rules and the general decomposition of the two-point Green's functions are derived. The propagator Dyson-Schwinger equations are derived and certain aspects (energy independence of ghost Green's functions and the cancellation of energy divergences) are discussed.

Solution of coupled vertex and propagator Dyson-Schwinger equations in the scalar Munczek-Nemirovsky model

In a scalar $\phi^2\chi$ model, we exactly solve the vertex and $\phi$ propagator Dyson-Schwinger equations under the assumption of a spatially constant (Munczek-Nemirovsky) propagator for the $\chi$ field. Various truncation schemes are also considered.

Study of relativistic bound states for scalar theories in the Bethe-Salpeter and Dyson-Schwinger formalism

The Bethe-Salpeter equation for Wick-Cutkosky like models is solved in dressed ladder approximation. The bare vertex truncation of the Dyson-Schwinger equations for propagators is combined with the dressed ladder Bethe-Salpeter equation for the scalar S-wave bound state amplitudes. With the help of spectral representation the results are obtained directly in Minkowski space. We give a new analytic formula for the resulting equation simplifying the numerical treatment. The bare ladder approximation of Bethe-Salpeter equation is compared with the one with dressed ladder. The elastic electromagnetic form factors is calculated within the relativistic impulse approximation.

General axial gauge for the light-front Schwinger model

Abstract The canonical light-front (LF) quantization of the massive Schwinger model and its perturbation theory are presented. The closed fermion loops contribution is calculated via the Pauli-Villars regularization and the massless limit is safely taken. The Dyson-Schwinger equations for exact propagators are solved for the gauge and the fermion fields, showing their dependence on the axial gauge fixing parameter α. The four-fermion Green function is found and for coinciding pairs of end-points it becomes explicitly a gauge invariant expression.

Dimensionally regularized study of nonperturbative quenched QED

We study the dimensionally regularized fermion propagator Dyson-Schwinger equation in quenched nonperturbative QED in an arbitrary covariant gauge. The nonperturbative fermion propagator is solved in $D\ensuremath{\equiv}4\ensuremath{-}2\ensuremath{\epsilon}<4$ dimensional Euclidean space for a large number of values of $\ensuremath{\epsilon}.$ Results for $D=4$ are then obtained by the extrapolation $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}0.$ The nonperturbative renormalization is performed numerically, yielding finite results for all renormalized quantities. This demonstrates, apparently for the first time, that it is possible to successfully implement nonperturbative renormalization of Dyson-Schwinger equations within a gauge invariant regularization scheme such as dimensional regularization. Here we present results using the Curtis-Penningon fermion-photon proper vertex for two values of the coupling, namely $\ensuremath{\alpha}=0.6$ and $\ensuremath{\alpha}=1.5$, and compare these to previous studies employing a modified ultraviolet cut-off regularization. The results using the two different regularizations are found to agree to within the numerical precision of the present calculations.

Renormalization and chiral symmetry breaking in quenched QED in arbitrary covariant gauges.

We extend a previous Landau-gauge study of subtractive renormalization of the fermion propagator Dyson-Schwinger equation (DSE) in strong-coupling, quenched QED_4 to arbitrary covariant gauges. We use the fermion-photon proper vertex proposed by Curtis and Pennington with an additional correction term included to compensate for the small gauge-dependence induced by the ultraviolet regulator. We discuss the chiral limit and the onset of dynamical chiral symmetry breaking in the presence of nonperturbative renormalization. We extract the critical coupling in several different gauges and find evidence of a small residual gauge-dependence in this quantity.