Closed-form two-loop Euler-Heisenberg Lagrangian in a self-dual background

TL;DR

This paper derives a simple closed-form expression for the two-loop Euler-Heisenberg Lagrangian in scalar QED within a self-dual background, facilitating analysis of its expansions and properties.

Contribution

It provides the first explicit closed-form expression for the two-loop scalar QED effective Lagrangian in a self-dual background using the digamma function.

Findings

Explicit closed-form in terms of digamma function

Simplified analysis of weak- and strong-field expansions

Insights into the two-loop beta function and analytic continuation

Abstract

We show that the two-loop Euler-Heisenberg effective Lagrangian for scalar QED in a constant Euclidean self-dual background has a simple explicit closed form expression in terms of the digamma function. This result leads to a simple analysis of the weak- and strong-field expansions, the two-loop scalar QED beta function, and the analytic continuation properties of the effective Lagrangian and its imaginary part.