Super Gaussian enhancers in the Schwinger mechanism

We study the Schwinger mechanism in the presence of an additional uniformly oriented, weak super Gaussian of integer order $4N+2$. Using the worldline approach, we determine the relevant critical points to compute the leading order exponential factor analytically. We show that increasing the parameter $N$ gives rise to a strong dynamical enhancement. For $N=2$, this effect turns out to be larger compared to a weak contribution of Sauter type. For higher orders, specifically, for the rectangular barrier limit, i.e. $N \rightarrow \infty$, we approach the Lorentzian case as an upper bound. Although the mentioned backgrounds significantly differ in Minkowski spacetime, we show that the found coincidence applies due to identical reflection points in the Euclidean instanton plane. In addition, we also treat the background in perturbation theory following recent ideas. By doing so, we show that the parameter $N$ determines whether the weak contribution behaves perturbatively or nonperturbatively with respect to the field strength ratio and, hence, reveals an interesting dependence on the background shape. In particular, we show that for backgrounds, for which higher orders in the field strength ratio turn out to be relevant, a proposed integral condition is not fulfilled. In view of these findings, the latter may serve as an indicator for the necessity of higher order contributions.