# Three-loop Euler-Heisenberg Lagrangian in 1+1 QED, part 1: single fermion-loop part

## TL;DR

This paper calculates the three-loop Euler-Heisenberg Lagrangian in 1+1 QED, focusing on the one-fermion-loop contribution, and develops algorithms to analyze the weak-field expansion coefficients.

## Contribution

It introduces new computational methods for weak-field expansion coefficients and combines diagrammatic and worldline formalisms for the first time in this context.

## Key findings

- Coefficients are of the form r1 + r2 * zeta(3) with rational r1, r2.
- First two coefficients are computed analytically.
- Four additional coefficients are obtained through numerical integration.

## Abstract

We study the three-loop Euler-Heisenberg Lagrangian in spinor quantum electrodynamics in 1+1 dimensions. In this first part we calculate the one-fermion-loop contribution, applying both standard Feynman diagrams and the worldline formalism which leads to two different representations in terms of fourfold Schwinger-parameter integrals. Unlike the diagram calculation, the worldline approach allows one to combine the planar and the non-planar contributions to the Lagrangian. Our main interest is in the asymptotic behaviour of the weak-field expansion coefficients of this Lagrangian, for which a non-perturbative prediction has been obtained in previous work using worldline instantons and Borel analysis. We develop algorithms for the calculation of the weak-field expansions coefficients that, in principle, allow their calculation to arbitrary order. Here for the non-planar contribution we make essential use of the polynomial invariants of the dihedral group D4 in Schwinger parameter space to keep the expressions manageable. As expected on general grounds, the coefficients are of the form r1+r2*zeta(3) with rational numbers r1, r2. We compute the first two coefficients analytically, and four more by numerical integration.