Three-loop Euler-Heisenberg Lagrangian in 1+1 QED. Part I. Single fermion-loop part

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 expansion 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 + r₂ζ₃ with rational numbers r₁, r₂. We compute the first two coefficients analytically, and four more by numerical integration.