Derivative Expansion of the Effective Action for QED in 2+1 and 3+1 dimensions

TL;DR

This paper develops a derivative expansion method for the one-loop effective action in QED in 2+1 and 3+1 dimensions, providing explicit formulas for terms involving derivatives of the electromagnetic field.

Contribution

It introduces an explicit expression for the next term in the derivative expansion of the effective action, valid for arbitrary field strengths, for both fermion and scalar QED.

Findings

Derived explicit form for the second term in the derivative expansion.

Presented Feynman rules for perturbative expansion in derivatives.

Analyzed cases of pure electric and magnetic fields.

Abstract

The derivative expansion of the one-loop effective action in QED$_3$ and QED$_4$ is considered. The first term in such an expansion is the effective action for a constant electromagnetic field. An explicit expression for the next term containing two derivatives of the field strength $F_{\mu\nu}$, but exact in the magnitude of the field strength, is obtained. The general results for fermion and scalar electrodynamics are presented. The cases of pure electric and pure magnetic external fields are considered in details. The Feynman rules for the perturbative expansion of the one-loop effective action in the number of derivatives is developed.