We present an explicit formula for the exact rate of pair production of oppositely charged scalar particles by a smooth potential step $\varphi (\mathbf{x})\propto \tanh kz$ in three dimensions. As a check we recover from this the known results for an infinitely sharp step as well as for a uniform electric field.

• Received 9 June 2009

DOI:https://doi.org/10.1103/PhysRevD.80.065010