An Addendum to the Heisenberg-Euler effective action beyond one loop

TL;DR

This paper investigates two-loop order quantum electrodynamics effects on the Heisenberg-Euler effective action, revealing a finite one-particle reducible contribution in constant fields, which impacts high-precision vacuum nonlinearity studies.

Contribution

It provides the first explicit calculation of two-loop contributions to the Heisenberg-Euler effective action, highlighting a finite one-particle reducible term previously thought to vanish.

Findings

Finite one-particle reducible contribution at two-loop order.

Clarification of the distinction between the Heisenberg-Euler action and one-particle irreducible actions.

Relevance for high-precision tests of quantum vacuum nonlinearity.

Abstract

We study the effective interactions of external electromagnetic fields induced by fluctuations of virtual particles in the vacuum of quantum electrodynamics. Our main focus is on these interactions at two-loop order. We discuss in detail the emergence of the renowned Heisenberg-Euler effective action from the underlying microscopic theory of quantum electrodynamics, emphasizing its distinction from a standard one-particle irreducible effective action. In our explicit calculations we limit ourselves to constant and slowly varying external fields, allowing us to adopt a locally constant field approximation. One of our main findings is that at two-loop order there is a finite one-particle reducible contribution to the Heisenberg-Euler effective action in constant fields, which was previously assumed to vanish. In addition to their conceptual significance, our results are relevant for high-precision probes of quantum vacuum nonlinearity in strong electromagnetic fields.