Gauge covariance and the fermion-photon vertex in three- and four- dimensional, massless quantum electrodynamics

TL;DR

This paper investigates the gauge covariance of fermion-photon vertices in three- and four-dimensional massless QED using Schwinger-Dyson equations, identifying conditions for gauge covariance and providing an analytic solution in 3D.

Contribution

It introduces a criterion for gauge covariance of vertex Ans"{a}tze and demonstrates that the Curtis and Pennington Ansatz satisfies this, with an analytic solution in 3D QED.

Findings

Curtis and Pennington Ansatz satisfies gauge covariance criterion.

Analytic solution obtained for quenched, massless 3D QED.

Spectral representation cannot support dynamical chiral symmetry breaking.

Abstract

In the quenched approximation, the gauge covariance properties of three vertex Ans\"{a}tze in the Schwinger-Dyson equation for the fermion self energy are analysed in three- and four- dimensional quantum electrodynamics. Based on the Cornwall-Jackiw-Tomboulis effective action, it is inferred that the spectral representation used for the vertex in the gauge technique cannot support dynamical chiral symmetry breaking. A criterion for establishing whether a given Ansatz can confer gauge covariance upon the Schwinger-Dyson equation is presented and the Curtis and Pennington Ansatz is shown to satisfy this constraint. We obtain an analytic solution of the Schwinger-Dyson equation for quenched, massless three-dimensional quantum electrodynamics for arbitrary values of the gauge parameter in the absence of dynamical chiral symmetry breaking.