Abstract
Abstract We compute an $s$-channel $2\to2$ scalar scattering $\phi\phi\to\Phi\to\phi\phi$ in the Gaussian wave-packet 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 $2\to2$ proposed earlier. An interpretation of the result is that a heavy scalar $1\to2$ decay $\Phi\to\phi\phi$, 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 $2\to2$ scattering. We also show various plane-wave limits.
Highlights
Introduction and summaryIt is well-known that a plane-wave S-matrix is ill-defined when taken literally because its matrix element is proportional to the energy-momentum delta function, which always gives either zero or infinity when squared to compute a probability
This paper is organized as follows: In Sec. 2, we present basic setup of the Gaussian formalism, and compute the Gaussian S-matrix for the s-channel 2 → 2 scattering: φφ → Φ → φφ
We have computed the Gaussian S-matrix for the s-channel 2 → 2 scalar scattering: φφ → Φ → φφ
Summary
It is well-known that a plane-wave S-matrix is ill-defined when taken literally because its matrix element is proportional to the energy-momentum delta function, which always gives either zero or infinity when squared to compute a probability. It has been claimed that the Gaussian formalism gives a deviation from the Fermi’s golden rule [3, 4], in which the probability is suppressed only by a power of the deviation from the energy-momentum conservation rather than the conventional exponential suppression; see Refs. [2], a scalar decay Φ → φφ has been computed in the Gaussian formalism, and the previously-claimed power-law deviation from the Fermi’s golden rule has been identified to come from the configuration in which the decay interaction is placed near a time-boundary. This paper is organized as follows: In Sec. 2, we present basic setup of the Gaussian formalism, and compute the Gaussian S-matrix for the s-channel 2 → 2 scattering: φφ → Φ → φφ. In Appendix A, we compare with the φφ → φφ scattering in the φ4 theory
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