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: $phiphitoPhitophiphi$. This contribution can be interpreted as an in-time-boundary effect of $Phi$ for the corresponding $Phitophiphi$ 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.
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