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Multidimensional quantum tunneling in the Schwinger effect

We study the Schwinger effect, in which the external field having a spatiotemporal profile creates electron-positron pairs via multidimensional quantum tunneling. Our treatment is based on the trace formula for the QED effective action, whose imaginary part is represented by a sum over complex worldline solutions. The worldlines are multiperiodic, and the periods of motion collectively depend on the strength of spatial and temporal inhomogeneity. We argue that the classical action that leads to the correct tunneling amplitude must take into account both the full period, $\tilde{T}$ and the first fundamental period, $T_1$. In view of this argument we investigate pair production in an exponentially damped sinusoidal field and find that the initial momenta for multiperiodic trajectories lie on parabolic curves, such that on each curve the ratio $\tilde{T}/T_1$ stays uniform. Evaluation of the tunneling amplitude using these trajectories shows that vacuum decay rate is reduced by an order of magnitude, with respect to the purely time-dependent case, due to the presence of magnetic field.