Truncating first-order Dyson-Schwinger equations in Coulomb-Gauge Yang-Mills theory

TL;DR

This paper explores a truncation scheme for Dyson-Schwinger equations in Coulomb-gauge Yang-Mills theory, revealing limitations in capturing the expected linear Coulomb potential and proposing a novel BRST-type operator for non-perturbative analysis.

Contribution

It introduces a new BRST-type operator and systematically analyzes a truncation scheme, highlighting its insufficiency for reproducing the linear Coulomb potential in Coulomb gauge QCD.

Findings

Truncation scheme fails to produce linear Coulomb potential

Vertex dressings do not alter the main result

Novel BRST operator aids in non-perturbative analysis

Abstract

The non-perturbative domain of QCD contains confinement, chiral symmetry breaking, and the bound state spectrum. For the calculation of the latter, the Coulomb gauge is particularly well-suited. Access to these non-perturbative properties should be possible by means of the Green's functions. However, Coulomb gauge is also very involved, and thus hard to tackle. We introduce a novel BRST-type operator r, and show that the left-hand side of Gauss' law is r-exact. We investigate a possible truncation scheme of the Dyson-Schwinger equations in first-order formalism for the propagators based on an instantaneous approximation. We demonstrate that this is insufficient to obtain solutions with the expected property of a linear-rising Coulomb potential. We also show systematically that a class of possible vertex dressings does not change this result.