Coulomb field in a constant electromagnetic background

TL;DR

This paper derives the nonlinear Maxwell equations up to third order in a constant electromagnetic background, analyzing the modified Coulomb field and charge distribution in quantum electrodynamics with the Euler-Heisenberg Lagrangian.

Contribution

It provides a detailed formulation of nonlinear Maxwell equations in a constant background and calculates the linear electric response and Coulomb field corrections within QED.

Findings

Modified Coulomb field in nonlinear electrodynamics

Corrections to total charge and charge density

Explicit formulas for Euler-Heisenberg Lagrangian case

Abstract

Nonlinear Maxwell equations are written up to the third-power deviations from a constant-field background, valid within any local nonlinear electrodynamics including QED with a Euler-Heisenberg (EH) effective Lagrangian. The linear electric response to an imposed static finite-sized charge is found in the vacuum filled by an arbitrary combination of constant and homogeneous electric and magnetic fields. The modified Coulomb field and corrections to the total charge and to the charge density are given in terms of derivatives of the effective Lagrangian with respect to the field invariants. These are specialized for the EH Lagrangian.