  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.
