"Background Field Integration-by-Parts" and the Connection Between One-Loop and Two-Loop Heisenberg-Euler Effective Actions

TL;DR

This paper introduces integration-by-parts rules for Feynman diagrams in scalar QED with background fields, revealing a simple diagrammatic interpretation of mass renormalization at two loops, simplifying calculations without explicit integrals.

Contribution

It provides a novel diagrammatic approach to mass renormalization in two-loop scalar QED effective actions, generalizing previous results and simplifying computations.

Findings

Diagrammatic interpretation of mass renormalization

Simplified computation of two-loop effective actions

Generalization beyond self-dual background fields

Abstract

We develop integration-by-parts rules for Feynman diagrams involving massive scalar propagators in a constant background electromagnetic field, and use these to show that there is a simple diagrammatic interpretation of mass renormalization in the two-loop scalar QED Heisenberg-Euler effective action for a general constant background field. This explains why the square of a one-loop term appears in the renormalized two-loop Heisenberg-Euler effective action. No integrals need be evaluated, and the explicit form of the background field propagators is not needed. This dramatically simplifies the computation of the renormalized two-loop effective action for scalar QED, and generalizes a previous result obtained for self-dual background fields.