Analytic expansions of multi-hadron finite-volume energies: I. Two-particle states

TL;DR

This paper derives analytic formulas for the finite-volume energies of weakly-interacting two-particle systems, applicable to various states and momenta, aiding understanding of energy spectra and setting groundwork for three-particle studies.

Contribution

It provides the first detailed analytic expansions for two-particle finite-volume energies, including excited states and non-zero total momentum, based on scattering amplitude relations.

Findings

Analytic expansions for ground and excited states derived.

Role of degeneracies and power-counting schemes clarified.

Framework established for future three-particle energy analyses.

Abstract

We derive analytic expansions for the finite-volume energies of weakly-interacting two-particle systems, using the general relations between scattering amplitudes and energies derived by L\"uscher and others. The relations hold for ground and excited states with both zero and non-zero total momentum in the finite-volume frame. A number of instructive aspects arise in the derivation, including the role of accidental degeneracies and the importance of defining a power-counting scheme in the expansions. The results give intuition concerning the imprint of weakly-interacting systems on the energy spectrum, while also providing a useful basis for the analogous results concerning three-particle excited states, to appear.