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

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 ≤ d ≤ 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 cr ) d/2 , where E 0 is the peak field strength of the inhomogeneity and E cr the critical electric field strength.

Pair production from the vacuum by a weakly inhomogeneous space-dependent electric potential

“…Our result for the probability P v (67) coincides with the one obtained in Ref. [31] for the particular case of d = 4.…”

“…The representations (57) and (59) coincide with the vacuum-to-vacuum transition probabilities obtained from the imaginary part of a locally constant field approximation (LCFA) for the one-loop effective action in d = 4 dimensions [22,31]. In this approximation, the effective action S is expanded about the constant field case, in terms of derivatives of the background field strength F µν ,…”

“…However, it should be stressed that unlike the representations obtained in Refs. [22,31], we derive Eqs. (57) and (59) in the framework of the general formulation of strong-field QED in the presence of x-electric potential steps [24], where P v are defined by Eqs.…”

“…It is a particular case of an inhomogeneous field that was used to study the LCFA for the probability P v in Ref. [31]. Using the Eqs.…”

Reducible contributions to quantum electrodynamics in external fields

“…These have been used to study low energy photon amplitudes [24,25] and the structure of the quantum vacuum, see [26] for a recent review. Aside from this, the particle propagator can also be constructed exactly (non-perturbatively) in the presence of constant fields, plane waves, and other symmetric fields, allowing the calculation of a variety of electron-seeded and photon-seeded processes, see [26][27][28][29] for reviews.Recently, however, it was found that historical calculations had overlooked the possibility of one particle reducible (1PR) contributions to processes in constant background fields [30][31][32]. These contributions involve a tadpole, displayed in figure 1, attached somewhere in the corresponding Feynman diagram describing the process.…”