Scalar scattering amplitude in the Gaussian wave-packet formalism

We compute an s-channel 2 -> 2 scalar scattering phi phi -> Phi -> phi phi in the Gaussian wavepacket formalism at the tree level. We find that wave-packet effects, including shifts of the pole and the width of the propagator of Phi, persist even when we do not take into account the time boundary effect for 2 -> 2 proposed earlier. An interpretation of the result is that a heavy scalar 1 -> 2 decay Phi -> phi phi, taking into account the production of Phi, does not exhibit the in-state time boundary effect unless we further take into account in-boundary effects for the 2 -> 2 scattering. We also show various plane-wave limits.