Two Particle States in an Asymmetric Box and the Elastic Scattering Phases

TL;DR

This paper derives formulas relating two-particle energy levels in an asymmetric box to elastic scattering phases, generalizing previous cubic box results and enabling better finite-volume analysis of scattering data.

Contribution

It provides a generalization of L"uscher's formulae to asymmetric rectangular boxes, connecting finite-volume energy eigenstates with continuum scattering phases.

Findings

Derived relations between energy eigenstates and scattering phases for asymmetric boxes

Extended L"uscher's formulae to non-cubic geometries

Discussed applications to finite-volume scattering analysis

Abstract

The exact two-particle energy eigenstates in a generic asymmetric rectangular box with periodic boundary conditions in all three directions are studied. Their relation with the elastic scattering phases of the two particles in the continuum are obtained for both $D_4$ and $D_2$ symmetry. These results can be viewed as a generalization of the corresponding formulae in a cubic box obtained by L\"uscher before. In particular, the s-wave scattering length is related to the energy shift in the finite box. Possible applications of these formulae are also discussed.