New effect in wave-packet scatterings of quantum fields: Saddle points, Lefschetz thimbles, and Stokes phenomenon

We find a new contribution in wave-packet scatterings, which has been overlooked in the standard formulation of S-matrix. As a concrete example, we consider a two-to-two scattering of light scalars $\phi$ by another intermediate heavy scalar $\Phi$, in the Gaussian wave-packet formalism: $\phi\phi\to\Phi\to\phi\phi$. This contribution can be interpreted as an "in-time-boundary effect" of $\Phi$ for the corresponding $\Phi\to\phi\phi$ decay, proposed by Ishikawa et al., with a newly found modification that would cure the previously observed ultraviolet divergence. We show that such an effect can be understood as a Stokes phenomenon in an integral over complex energy plane: The number of relevant saddle points and Lefschetz thimbles (steepest descent paths) discretely changes depending on the configurations of initial and final states in the scattering.