Effective Euler-Heisenberg Lagrangians in models of QED

TL;DR

This paper presents a direct diagrammatic calculation of the effective four-photon Euler-Heisenberg Lagrangian in massive charged vector boson QED, embedded in the electroweak theory, using a novel unitary gauge approach.

Contribution

It provides the first direct diagrammatic evaluation of the Euler-Heisenberg Lagrangian for vector bosons in QED using the unitary gauge, confirming previous results obtained by different methods.

Findings

Recovered the known Euler-Heisenberg Lagrangian for vector bosons

Validated the diagrammatic approach in the unitary gauge

Extended calculations to scalar and spinor QED for comparison

Abstract

The main goal of this paper is a direct diagrammatic evaluation of the effective four-photon Lagrangian of the Euler-Heisenberg type for the quantum electrodynamics of massive charged vector bosons. This QED model is naturally embedded in the standard electroweak theory, and we have carried out the corresponding one-loop calculation in the unitary gauge. As far as we know, such a work has not been published before since usually the $R$-gauge techniques are preferred for the computation of loop Feynman diagrams. We have recovered the result obtained many years ago by Vanyashin and Terent'ev, who used a specific functional method. For completeness and for validation of our techniques, we have also redone the analogous calculations for scalar and spinor QED.