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Scalar scattering amplitude in Gaussian wave-packet formalism

We compute an $s$-channel $2\to2$ scalar scattering $ϕϕ\toΦ\toϕϕ$ in the Gaussian wave-packet formalism at the tree-level. We find that wave-packet effects, including shifts of the pole and width of the propagator of $Φ$, persist even when we do not take into account the time-boundary effect for $2\to2$, proposed earlier. The result can be interpreted that a heavy scalar $1\to2$ decay $Φ\toϕϕ$, taking into account the production of $Φ$, does not exhibit the in-state time-boundary effect unless we further take into account in-boundary effects for the $2\to2$ scattering. We also show various plane-wave limits.