# The Heisenberg-Euler effective action in slowly varying electric field inhomogeneities of Lorentzian shape

## TL;DR

This paper investigates the quantum vacuum instability in slowly varying Lorentzian-shaped electric fields using the LCFA, revealing how the decay rate depends on the inhomogeneity dimension and field strength.

## Contribution

It introduces a simplified integral representation of the Heisenberg-Euler effective action for Lorentzian inhomogeneous fields and analyzes vacuum decay dependence on inhomogeneity dimension.

## Key findings

- Effective action expressed as a single parameter integral.
- Vacuum decay rate suppressed by (E_0/E_cr)^{d/2} for weak fields.
- Decay rate varies with the dimension of inhomogeneity d.

## Abstract

We use a locally constant field approximation (LCFA) to study the one-loop Heisenberg-Euler effective action in a particular class of slowly varying inhomogeneous electric fields of Lorentzian shape with $0\leq d\leq 4$ inhomogeneous directions. We show that for these fields, the LCFA of the Heisenberg-Euler effective action can be represented in terms of a single parameter integral, with the constant field effective Lagrangian with rescaled argument as integration kernel. The imaginary part of the Heisenberg-Euler effective action contains information about the instability of the quantum vacuum towards the formation of a state with real electrons and positrons. Here, we in particular focus on the dependence of the instantaneous vacuum decay rate on the dimension $d$ of the field inhomogeneity. Specifically for weak fields, we find an overall parametric suppression of the effect with $(E_0/E_{\rm cr})^{d/2}$, where $E_0$ is the peak field strength of the inhomogeneity and $E_{\rm cr}$ the critical electric field strength.