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 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.
Highlights
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We study the three-loop Euler-Heisenberg Lagrangian in spinor quantum electrodynamics in 1+1 dimensions
The calculation has been performed in parallel using standard Feynman diagrams and the worldline formalism, treating the constant external field non-perturbatively in both cases
Summary
2.1 Worldline representation of 2-photon and 4-photon amplitudes in a constant field. The representations (2.8), (2.10) hold mutatis mutandis in 4D QED, in the 2D case they are useful, because here all matrices appearing in the above expressions (the Fi’s and all worldline Green’s functions) commute with each other. Heisenberg Lagrangians directly using the concept of multi-loop worldline Green’s functions [53, 56, 57]; this would obscure the decomposition (2.11), which we will find very useful in the following This is because, first, it will allow us to substantially reduce the number of terms in the integrand using the symmetries of the problem; and second, because the gauge invariance term-by-term of this decomposition implies that for each term we can use a different gauge parameter ξ in the sewing procedure (2.15)
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