All-Loop Result for the Strong Magnetic Field Limit of the Heisenberg-Euler Effective Lagrangian.

Felix Karbstein

doi:10.1103/physrevlett.122.211602

Abstract

We provide an explicit expression for the strong magnetic field limit of the Heisenberg-Euler effective Lagrangian for both scalar and spinor quantum electrodynamics. To this end, we show that the strong magnetic field behavior is fully determined by one-particle reducible contributions discovered only recently. The latter can efficiently be constructed in an essentially algebraic procedure from lower-order one-particle reducible diagrams. Remarkably, the leading strong magnetic field behavior of the all-loop Heisenberg-Euler effective Lagrangian only requires input from the one-loop Lagrangian. Our result revises previous findings based exclusively on one-particle irreducible contributions. In addition, we briefly discuss the strong electric field limit and comment on external field QED in the large N limit.

Highlights

Felix Karbstein*Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany and Theoretisch-Physikalisches Institut, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany (Received 19 March 2019; published 29 May 2019)
Introduction.—The Heisenberg-Euler effective Lagrangian [1,2,3] is a central quantity in the development of quantum field theory
It studies the effect of quantum fluctuations on the effective theory of prescribed electromagnetic fields in the vacuum, and allows for the systematic derivation of quantum corrections to Maxwell’s classical theory of electrodynamics

Summary

Felix Karbstein*

Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany and Theoretisch-Physikalisches Institut, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany (Received 19 March 2019; published 29 May 2019). We provide an explicit expression for the strong magnetic field limit of the Heisenberg-Euler effective Lagrangian for both scalar and spinor quantum electrodynamics To this end, we show that the strong magnetic field behavior is fully determined by one-particle reducible contributions discovered only recently. Introduction.—The Heisenberg-Euler effective Lagrangian [1,2,3] is a central quantity in the development of quantum field theory It studies the effect of quantum fluctuations on the effective theory of prescribed electromagnetic fields in the vacuum, and allows for the systematic derivation of quantum corrections to Maxwell’s classical theory of electrodynamics. It is convenient to formally treat the parameter α and the combination eFμν as independent The power of the former counts the number of internal photon lines nγ in a given diagram, and the power of the latter the number of couplings to the external fields. The Heisenberg-Euler effective Lagrangian admits a diagrammatic expansion in the number of loops l of the constituting Feynman diagrams,

Fμν is the classical

Maxwell term

The analogous expression for a purely electric field is LlHarEgeN ðEÞ