Improved Approximations for Fermion Pair Production in Inhomogeneous Electric Fields

TL;DR

This paper develops improved approximation methods for calculating fermion pair production rates in inhomogeneous electric fields, using complex plane instantons and WKB techniques, with applications to localized electric and magnetic fields.

Contribution

It introduces refined instanton-based approximations for fermion pair production, demonstrating the equivalence of scalar and spinor QED instanton actions and extending results to combined electric and magnetic fields.

Findings

Derived pair-production rates for localized electric fields.

Showed the equivalence of scalar and spinor QED instanton actions.

Extended calculations to include magnetic fields.

Abstract

Reformulating the instantons in a complex plane for tunneling or transmitting states, we calculate the pair-production rate of charged fermions in a spatially localized electric field, illustrated by the Sauter electric field E_0 sech^2 (z/L), and in a temporally localized electric field such as E_0 sech^2 (t/T). The integration of the quadratic part of WKB instanton actions over the frequency and transverse momentum leads to the pair-production rate obtained by the worldline instanton method, including the prefactor, of Phys. Rev. D72, 105004 (2005) and D73, 065028 (2006). It is further shown that the WKB instanton action plus the next-to-leading order contribution in spinor QED equals the WKB instanton action in scalar QED, thus justifying why the WKB instanton in scalar QED can work for the pair production of fermions. Finally we obtain the pair-production rate in a spatially localized electric field together with a constant magnetic field in the same direction.