Two-loop self-dual Euler-Heisenberg Lagrangians (I): Real part and helicity amplitudes

TL;DR

This paper derives a simple closed-form expression for the two-loop Euler-Heisenberg Lagrangian in self-dual backgrounds for scalar and spinor QED, revealing deep connections to helicity and supersymmetry, and applies it to photon amplitude calculations.

Contribution

It provides a novel, simplified expression for the two-loop effective Lagrangian in self-dual backgrounds, linking self-duality, helicity, and supersymmetry, and computes related photon amplitudes.

Findings

Closed-form Lagrangian expressions in terms of digamma functions.

Scalar and spinor QED Lagrangians are remarkably similar.

Application to low energy limits of two-loop photon helicity amplitudes.

Abstract

We show that, for both scalar and spinor QED, the two-loop Euler-Heisenberg effective Lagrangian for a constant Euclidean self-dual background has an extremely simple closed-form expression in terms of the digamma function. Moreover, the scalar and spinor QED effective Lagrangians are very similar to one another. These results are dramatic simplifications compared to the results for other backgrounds. We apply them to a calculation of the low energy limits of the two-loop massive N-photon `all +' helicity amplitudes. The simplicity of our results can be related to the connection between self-duality, helicity and supersymmetry.