Euler-Heisenberg action for fermions coupled to gauge and axial vectors: Hessian diagonalization, sector classification, and applications

TL;DR

This paper derives a comprehensive one-loop effective action for Dirac fermions coupled to electromagnetic and axial vector fields, classifies stability sectors, and explores applications in physics including vacuum stability, anomaly structures, and cosmological models.

Contribution

It provides the first complete diagonalization of the Hessian for such systems, extending previous results to include propagating axial fields and revealing new stability and instability sectors.

Findings

Identified stability and instability sectors in the parameter space.

Derived nonperturbative pair-production rates showing vacuum stabilization.

Connected electromagnetic coupling to instanton-like axial configurations and applied to cosmological baryogenesis.

Abstract

We derive the closed-form one-loop Euler--Heisenberg effective actions for Dirac fermions coupled simultaneously to classical electromagnetic vector and massive pseudo-vector backgrounds within a controlled quasi-static approximation. Through complete diagonalization of the functional Hessian, we systematically delineate the parameter space into distinct sectors characterized by stability properties and spectral structure. We identify subspaces that encompass and extend results from previous studies into a broader class, admitting propagating axial fields as physically viable regimes; strikingly, we note a sector presenting chirality-asymmetric instability. This addresses long-standing questions regarding the well-defined nature, diagonalizability, and stability of the model. From the effective action, we derive novel nonperturbative pair-production rates for simultaneously propagating electromagnetic and axial vector backgrounds; remarkably, we find pronounced vacuum stabilization compared to previous results. Furthermore, we demonstrate that this framework allows for a unified derivation of the chiral anomaly structures in the general case and show that the electromagnetic coupling induces instanton-like configurations for the axial field, even when it is not a fundamental gauge field. As a proof-of-concept, we analyze a cosmological toy model of baryogenesis driven by an axial vector, providing numerical estimates that support the viability of this hypothesis. Additionally, we outline qualitative predictions for Weyl/Dirac semi-metals and briefly discuss potential applications in related phenomena, such as the Strong-CP problem.