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Heisenberg-Euler Effective Lagrangians : Basics and Extensions

Gerald V. Dunne

TL;DR

The Heisenberg–Euler framework computes the one-loop QED effective action in a constant background field, revealing vacuum nonlinearities, light–light scattering, and pair production. This survey surveys extensions to inhomogeneous backgrounds, nonabelian and supersymmetric theories, and two-loop corrections, highlighting powerful methods such as zeta-function regularization and worldline formalisms. A key theme is the deep link between strong-field asymptotics and renormalization-group data (beta functions), exemplified in constant-field and self-dual backgrounds, where two-loop expressions simplify dramatically and reveal connections to helicity and supersymmetry. The work underscores HE Lagrangians as a fundamental tool for low-energy EFTs, vacuum structure probes, and the interplay between perturbative and nonperturbative phenomena in gauge theories. Overall, the results provide a versatile, technically rich toolkit for exploring quantum vacuum effects across abelian and nonabelian settings, with implications for precision QED and beyond.

Abstract

I present a pedagogical review of Heisenberg-Euler effective Lagrangians, beginning with the original work of Heisenberg and Euler, and Weisskopf, for the one loop effective action of quantum electrodynamics in a constant electromagnetic background field, and then summarizing some of the important applications and generalizations to inhomogeneous background fields, nonabelian backgrounds, and higher loop effective Lagrangians.

Heisenberg-Euler Effective Lagrangians : Basics and Extensions

TL;DR

The Heisenberg–Euler framework computes the one-loop QED effective action in a constant background field, revealing vacuum nonlinearities, light–light scattering, and pair production. This survey surveys extensions to inhomogeneous backgrounds, nonabelian and supersymmetric theories, and two-loop corrections, highlighting powerful methods such as zeta-function regularization and worldline formalisms. A key theme is the deep link between strong-field asymptotics and renormalization-group data (beta functions), exemplified in constant-field and self-dual backgrounds, where two-loop expressions simplify dramatically and reveal connections to helicity and supersymmetry. The work underscores HE Lagrangians as a fundamental tool for low-energy EFTs, vacuum structure probes, and the interplay between perturbative and nonperturbative phenomena in gauge theories. Overall, the results provide a versatile, technically rich toolkit for exploring quantum vacuum effects across abelian and nonabelian settings, with implications for precision QED and beyond.

Abstract

I present a pedagogical review of Heisenberg-Euler effective Lagrangians, beginning with the original work of Heisenberg and Euler, and Weisskopf, for the one loop effective action of quantum electrodynamics in a constant electromagnetic background field, and then summarizing some of the important applications and generalizations to inhomogeneous background fields, nonabelian backgrounds, and higher loop effective Lagrangians.

Paper Structure

This paper contains 51 sections, 228 equations, 3 figures.

Table of Contents

  1. Introduction: the Heisenberg-Euler Effective Lagrangian
  2. The Spinor and Scalar QED one loop results
  3. Physical Applications
  4. Nonlinear QED Processes : Light-Light scattering
  5. Pair-production from vacuum in an electric field.
  6. Charge renormalization, $\beta$-functions and the strong-field limit.
  7. Low-energy effective field theory
  8. Weak-field expansions of Heisenberg-Euler
  9. Spinor QED case
  10. Scalar QED case
  11. Strong-field expansions of Heisenberg-Euler
  12. Spinor QED case
  13. Scalar QED case
  14. Inhomogeneous backgrounds : beyond the constant field strength limit
  15. Solvable inhomogeneous backgrounds
  16. ...and 36 more sections

Figures (3)

  • Figure 1: The diagrammatic perturbative expansion of the one loop effective action (\ref{['action']}).
  • Figure 2: A static electric field can tear apart a virtual $e^+ e^-$ pair from the vacuum, producing an asymptotic electron and positron, as shown on the left. On the other hand, a static magnetic field does not break this virtual dipole apart, as shown on the right for a magnetic field directed out of the page.
  • Figure 3: The two loop diagram for the two loop effective Lagrangian. The double line refers to a propagator in the presence of the constant background field, while the wavy line represents the internal virtual photon.