Scalar scattering amplitude in Gaussian wave-packet formalism

We compute an $s$-channel $2\to 2$ scalar scattering $\varphi \varphi \to \mathrm{\Phi }\to \varphi \varphi$ 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 $\mathrm{\Phi }$, persist even when we do not take into account the time-boundary effect for $2\to 2$, proposed earlier. The result can be interpreted that a heavy scalar $1\to 2$ decay $\mathrm{\Phi }\to \varphi \varphi$, taking into account the production of $\mathrm{\Phi }$, does not exhibit the in-state time-boundary effect unless we further take into account in-boundary effects for the $2\to 2$ scattering. We also show various plane-wave limits.