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

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 Lüscher 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 perturbative interactions on the energy spectrum, while also providing a useful basis for the analogous results concerning three-particle excited states, to appear. We have also developed a Mathematica notebook that automates the expansions described in this work.

Multi-hadron interactions from lattice QCD

“…The Lüscher formalism provides an indirect connection between the energy levels obtained from Euclidean correlation functions and the two-particle scattering amplitude . Following the original works [24,25], several theoretical extensions have achieved a formalism that enables the study of generic two-to-two systems below the first inelastic threshold with more than two particles [29][30][31][32][33][34][35][36][37][38]. The two-particle formalism comes in the form of a determinant equation, the so-called two-particle quantization condition:…”

Higher partial wave contamination in finite-volume formulae for 1-to-2 transitions

“…As is well known, the A + 1 also couples to higher angular-momentum states. After = 0, the next lowest angular-momentum state that contributes is = 4 [19][20][21][22][23][24]. Naively, if we were to truncate at = 4, our angular momentum indices would run over 2×4+1 = 9 additional values.…”