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

Schwinger pair production rate and time in some space-dependent fields are studied analytically by using the worldline instantons formalism. The instanton path, action, production rate, vacuum-decay time and tunneling time are obtained, which depend on the Keldysh parameter significantly. It is found that under a given field the instanton paths with different Keldysh parameter are exactly in the same plane. Moreover, the normalized instanton actions are all bounded within the region from π to 2π . We also find the relationship between vacuum-decay time and tunneling time. Graphic abstract Schwinger pair production rate and time in some space-dependent fields are studied analytically by using the worldline instantons formalism. The instanton path, action, production rate, vacuum-decay time and tunneling time are obtained, which depend on the Keldysh parameter significantly. It is found that under a given field the instanton paths with different Keldysh parameter are exactly in the same plane. Moreover, the normalized instanton actions are all bounded within the region from π to 2π . We also find the relationship between vacuum-decay time and tunneling time.