Two Particle States and the $S$-matrix Elements in Multi-channel Scattering

TL;DR

This paper derives a generalized relation between finite-volume energy levels and multi-channel S-matrix elements in quantum scattering, extending L"uscher's formula to inelastic cases.

Contribution

It provides an exact relation between energy eigenvalues in a finite cubic box and multi-channel S-matrix elements, generalizing L"uscher's formula to inelastic scattering.

Findings

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

Extended L"uscher's formula to include inelastic scattering channels.

Provides a framework for analyzing multi-channel scattering in lattice QCD.

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 in all three directions. 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.