Schwinger pair production rate and time for some space-dependent electromagnetic fields via worldline instantons formalism

Schwinger pair production in some of space-dependent electromagnetic fields is studied analytically by using worldline instantons formalism for scalar quantum electrodynamics. With the increase of the modified Keldysh parameter $\gamma_{keb}$, the pair production rate decreases, the region of instanton paths expands regardless of the electromagnetic field form. We also find that all of the paths of the instantons for various electromagnetic fields are exactly in the same plane while with the different elliptical shapes. Moreover, the magnitudes of the normalized instantons action with different external electromagnetic fields are all bounded within the region from $\pi$ to $2\pi$ in spatial inhomogeneity fields. We further analytically identify and obtain two kinds of the pair-production time associated to the worldline instantons action.