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We use a locally constant field approximation (LCFA) to study the one-loopHeisenberg-Euler effective action in a particular class of slowly varyinginhomogeneous electric fields of Lorentzian shape with $0\leq d\leq 4$inhomogeneous directions. We show that for these fields, the LCFA of theHeisenberg-Euler effective action can be represented in terms of a singleparameter integral, with the constant field effective Lagrangian with rescaledargument as integration kernel. The imaginary part of the Heisenberg-Eulereffective action contains information about the instability of the quantumvacuum towards the formation of a state with real electrons and positrons.Here, we in particular focus on the dependence of the instantaneous vacuumdecay rate on the dimension $d$ of the field inhomogeneity. Specifically forweak 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 theinhomogeneity and $E_{\rm cr}$ the critical electric field strength.

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