Self-duality, helicity and background field loopology

TL;DR

This paper demonstrates that helicity simplifies the calculation of higher-loop effective Lagrangians in quantum electrodynamics, especially for self-dual background fields, enabling algebraic derivations from one-loop quantities.

Contribution

It introduces a helicity-based approach to compute two-loop effective Lagrangians, generalizing integration-by-parts rules to background field propagators.

Findings

Two-loop Heisenberg-Euler Lagrangian is simpler with self-dual fields.

Algebraic derivation of two-loop results from one-loop quantities.

Helicity plays a key role in simplifying background field calculations.

Abstract

I show that helicity plays an important role in the development of rules for computing higher loop effective Lagrangians. Specifically, the two-loop Heisenberg-Euler effective Lagrangian in quantum electrodynamics is remarkably simple when the background field has definite helicity (i.e., is self-dual). Furthermore, the two-loop answer can be derived essentially algebraically, and is naturally expressed in terms of one-loop quantities. This represents a generalization of the familiar ``integration-by-parts'' rules for manipulating diagrams involving free propagators to the more complicated case where the propagators are those for scalars or spinors in the presence of a background field.