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

  • Idrish Huet
  • Michel Rausch de Traubenberg
  • Christian SchubertEmail author
Open Access
Regular Article - Theoretical Physics


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.


Field Theories in Lower Dimensions Nonperturbative Effects Scattering Amplitudes 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2019

Authors and Affiliations

  • Idrish Huet
    • 1
    • 2
  • Michel Rausch de Traubenberg
    • 3
  • Christian Schubert
    • 4
    • 5
    Email author
  1. 1.Facultad de Ciencias en Física y MatemáticasUniversidad Autónoma de ChiapasTuxtla GutiérrezMexico
  2. 2.Theoretisch-Physikalisches InstitutFriedrich-Schiller-Universität JenaJenaGermany
  3. 3.Université de Strasbourg, CNRS, IPHC UMR7178Strasbourg CedexFrance
  4. 4.Instituto de Física y MatemáticasUniversidad Michoacana de San Nicolás de HidalgoMoreliaMexico
  5. 5.Kavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.

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