False vacuum decay in the Lorentzian path integral

False vacuum decay is a non-perturbative phenomenon in quantum field theory and important quantum process in cosmology. It has relied on the Euclidean formalism developed by Coleman, but there are several subtle issues in cosmological application as a negative mode problem or ambiguity in the definition of the decay rate in the presence of the gravity. Instead of the Euclidean path integral, we directly evaluate the Lorentzian path integral to discuss false vacuum decay and estimate the decay probability. To make the Lorentzian path integral convergent, the deformation of an integral contour is performed on the basis of the Picard-Lefschetz theory. We show that the nucleation probability of a critical bubble, for which the corresponding bounce action is extremized, has the same exponent as the Euclidean approach. We also extend our computation to the nucleation of a bubble larger or smaller than the critical one to which the Euclidean formalism is not applicable.