A comment on generalized Schwinger effect

A spatially homogeneous, time-dependent, electric field can produce charged particle pairs from the vacuum. When the electric field is constant, the mean number of pairs which are produced depends on the electric field and the coupling constant in a non-analytic manner, showing that this result cannot be obtained from the standard perturbation theory of quantum electrodynamics. When the electric field varies with time and vanishes asymptotically, the result may depend on the coupling constant either analytically or non-analytically. We investigate the nature of this dependence in several specific contexts. We show that the dependence of particle production on coupling constant is non-analytic for a class of time-dependent electric fields, with the leading order non-analytic behaviour being controlled by a specific parameter which can be identified. We also demonstrate that, for another class of electric fields, which vary rapidly, the dependence of particle production on coupling constant is analytic. Finally, we describe what happens to these results when we go beyond the leading order, using some specific examples