Two-Loop Vacuum Diagrams in Background Field and Heisenberg-Euler Effective Action

TL;DR

This paper derives a simplified expression for the two-loop scalar QED Heisenberg-Euler effective action in even dimensions under specific background field conditions, revealing new recursion relations and explicit forms in two dimensions.

Contribution

It introduces algebraic methods to reduce two-loop effective actions to one-loop quantities in specific background fields, providing new recursion relations and explicit results.

Findings

Reduction of two-loop to one-loop quantities in scalar QED

Derivation of recursion relations between loop diagrams

Explicit two-loop effective action in two dimensions

Abstract

We show that in arbitrary even dimension, the two-loop scalar QED Heisenberg-Euler effective action can be reduced to simple one-loop quantities, using just algebraic manipulations, when the constant background field satisfies F^2 = -f^2 I, which in four dimensions coincides with the condition for self-duality, or definite helicity. This result relies on new recursion relations between two-loop and one-loop diagrams, with background field propagators. It also yields an explicit form of the renormalized two-loop effective action in a general constant background field in two dimensions.