Two Particle States in a Box and the $s$-Matrix in Multi-Channel Scattering

TL;DR

This paper derives a generalized relation between energy eigenstates in a finite cubic box and the multi-channel S-matrix elements, extending Lüscher's formula to inelastic scattering scenarios.

Contribution

It provides an exact relation connecting finite-volume energy levels to multi-channel S-matrix elements, generalizing Lüscher's formula for inelastic scattering.

Findings

Derived a relation between energy eigenvalues and S-matrix in multi-channel scattering.

Extended Lüscher's formula to inelastic, multi-channel cases.

Provides a theoretical framework for analyzing two-particle states in a finite volume.

Abstract

Using a quantum mechanical model, the exact energy eigenstates for two-particle two-channel scattering are studied in a cubic box with periodic boundary conditions. A relation between the exact energy eigenvalue in the box and the two-channel $S$-matrix elements in the continuum is obtained. This result can be viewed as a generalization of the well-known L\"uscher's formula which establishes a similar relation in elastic scattering.