How gauge covariance of the fermion and boson propagators in QED constrain the effective fermion-boson vertex

TL;DR

This paper establishes the gauge covariance constraints on the fermion-photon vertex in QED, ensuring consistent solutions to the fermion propagator SDE across gauges and dimensions, and discusses implications for vacuum polarization and regularization methods.

Contribution

It derives the gauge covariance condition for the fermion-photon vertex in QED using spectral representations, linking it to the Landau-Khalatnikov-Fradkin transformation across dimensions.

Findings

The gauge covariance condition ensures consistent fermion propagator solutions in covariant gauges.

The vacuum polarization can be made gauge-independent under certain conditions.

The Gauge Technique in 4D fails to satisfy the covariance requirement.

Abstract

We derive the gauge covariance requirement imposed on the QED fermion-photon three-point function within the framework of a spectral representation for fermion propagators. When satisfied, such requirement ensures solutions to the fermion propagator Schwinger-Dyson equation (SDE) in any covariant gauge with arbitrary numbers of spacetime dimensions to be consistent with the Landau-Khalatnikov-Fradkin transformation (LKFT). The general result has been verified by the special cases of three and four dimensions. Additionally, we present the condition that ensures the vacuum polarization is independent of the gauge parameter. As an illustration, we show how the Gauge Technique dimensionally regularized in 4D does not satisfy the covariance requirement.