Closed-form weak-field expansion of two-loop Euler-Heisenberg Lagrangians

TL;DR

This paper derives closed-form expressions for the weak-field expansion coefficients of two-loop Euler-Heisenberg Lagrangians using Faulhaber numbers, providing a new mathematical approach to quantum field theory calculations.

Contribution

It introduces a novel method to express two-loop Euler-Heisenberg Lagrangians in closed form using Faulhaber numbers, enhancing analytical tools in quantum electrodynamics.

Findings

Closed-form expressions for expansion coefficients derived

Utilization of Faulhaber numbers in quantum field theory

Simplification of two-loop Euler-Heisenberg calculations

Abstract

We obtain closed-form expressions, in terms of the Faulhaber numbers, for the weak-field expansion coefficients of the two-loop Euler-Heisenberg effective Lagrangians in a magnetic or electric field. This follows from the observation that the magnetic worldline Green's function has a natural expansion in terms of the Faulhaber numbers.