Propagator Dyson-Schwinger Equations of Coulomb Gauge Yang-Mills Theory Within the First Order Formalism

TL;DR

This paper derives and analyzes the propagator Dyson-Schwinger equations for Coulomb gauge Yang-Mills theory using the first order formalism, highlighting symmetries, invariances, and key properties of Green's functions.

Contribution

It introduces a detailed derivation of Dyson-Schwinger equations in Coulomb gauge within the first order formalism, emphasizing BRS invariance and symmetry relations.

Findings

Energy independence of ghost Green's functions

Cancellation of energy divergences

Explicit form of propagator Dyson-Schwinger equations

Abstract

Coulomb gauge Yang-Mills theory within the first order formalism is considered with a view to deriving the propagator Dyson-Schwinger equations. The first order formalism is studied with special emphasis on the BRS invariance and it is found that there exists two forms of invariance - invariance under the standard BRS transform and under a second, non-standard 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 non-standard 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.