Improved approximations for fermion pair production in inhomogeneous electric fields

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 E₀sech²(z/L), and in a temporally localized electric field such as E₀sech²(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. D 72 105004 (2005) and 73 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.