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

Abstract

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.

Tadpole diagrams in constant electromagnetic fields

“…In this section, we briefly recall the determination of the 1PR contribution to the Heisenberg-Euler effective Lagrangian at two loops [1], specializing to a constant backgrounds field from the outset. In d space-time dimensions and in constant electromagnetic fields F ≡ F µν , we have Γ HE = V (d) L HE , where V (d) denotes the space-time volume and L HE ≡ L HE (F ) is the Heisenberg-Euler effective Lagrangian evaluated in the prescribed field.…”

“…[1] it has recently been emphasized that by definition the Heisenberg-Euler effective action Γ HE [2,3] is not one-particle irreducible (1PI) if relying on Schwinger's definition [4]. One-particle reducible (1PR) tadpole diagrams, i.e., Feynman subdiagrams consisting of quantum fluctuations of charged particles that couple to the rest of the diagram only via a single photon line, start contributing at two loops.…”

“…[1], who evaluated the 1PR contribution to Γ HE in constant fields. Similar contributions are expected to become relevant in the context of higher-loop calculations of essentially all physical quantities to be studied in external electromagnetic fields.…”

Photon polarization tensor in pulsed Hermite- and Laguerre-Gaussian beams

“…[4] and references therein. Correspondingly, our quantum particle degrees of freedom are electrons, positions and photons.…”

“…More specifically, the latter derivation makes use of the fact that for probe photons and background fields 2 with frequencies and momenta delimited from above by υ ≪ m e , the dominant contribution to the photon polarization tensor can be inferred straightforwardly from the constant-field result in two steps [4,33,34]: First, the Heisenberg-Euler Lagrangian in constant fields is adopted to inhomogeneous electromagnetic fields as…”