  We report calculations of a wave-packet amplitude of the two-body scattering
$\phi \phi \to \Phi \to \phi \phi$, which leads to the measured probability in
realistic experiments. We elucidate the decay amplitude of $ \Phi \rightarrow
\phi \phi$ from this. In such an amplitude of wave packets, there are in and
out time boundaries for the initial $\Phi$ and final $\phi\phi$ configurations,
respectively. In this paper, we prove that the effect of the in time boundary
of $\Phi\to\phi\phi$ emerges from $\phi\phi\to\Phi\to\phi\phi$ without assuming
any time boundary \emph{a priori}. This effect has been overlooked in the
standard plane-wave formulation and can exhibit distinct phenomena in wide
areas of science. We confirm the result in different integration orders. The
result is also interpreted as a Stokes phenomenon in the Lefschetz-thimble
decomposition.
