The QED photon-fermion vertex from its Dyson-Schwinger equation in 4D: the full vertex, the transverse form factors and the perturbative solution

TL;DR

This paper derives and analyzes the Dyson-Schwinger equations for the photon-fermion vertex in QED, providing exact integral equations, approximations, and perturbative solutions for the vertex's transverse form factors across gauges.

Contribution

It formulates the exact non-linear integral equations for the transverse vertex in QED and computes their perturbative solutions, extending previous one-loop results to a more comprehensive non-perturbative framework.

Findings

Derived exact integral equations for the transverse vertex

Provided perturbative asymptotic expressions for form factors

Analyzed various kinematical configurations

Abstract

We investigate the Dyson-Schwinger equation for the photon-fermion one-particle irreducible vertex in QED in linear covariant gauges. The longitudinal component of this vertex is described using the Ball-Chiu basis, while its transverse part is expressed with the K{\i}z{\i}lersu-Reenders-Pennington basis. Combining the vertex Ward-Takahashi identity with the vertex equation, we derive a set of exact, non-linear integral equations governing the transverse vertex. These equations hold for any linear covariant gauge and must be solved self-consistently. We discuss several approximations to the exact equations, generalizing results previously obtained at the perturbative one-loop level. Various kinematical configurations are also examined. Furthermore, we compute the perturbative solution of the transverse vertex Dyson-Schwinger equations for all transverse form factors and derive their perturbative asymptotic expressions.