Identification of shallow two-body bound states in finite volume

TL;DR

This paper explores how finite volume effects influence the detection of shallow two-body bound states using Luscher's method, proposing a signature change in scattering length and analyzing the bound-state pole condition.

Contribution

It introduces a finite volume signature for shallow bound states based on phase-shift analysis and examines the finite size corrections to the bound-state pole condition.

Findings

Sign of the scattering length changes with bound-state formation in finite volume.

Bound-state pole condition is only strictly fulfilled in infinite volume, with finite size effects exponentially suppressed.

Numerical lattice simulations confirm theoretical predictions using positronium spectrum in scalar QED.

Abstract

We discuss signatures of bound-state formation in finite volume via the Luscher finite size method. Assuming that the phase-shift formula in this method inherits all aspects of the quantum scattering theory, we may expect that the bound-state formation induces the sign of the scattering length to be changed. If it were true, this fact provides us a distinctive identification of a shallow bound state even in finite volume through determination of whether the second lowest energy state appears just above the threshold. We also consider the bound-state pole condition in finite volume, based on Luscher's phase-shift formula and then find that the condition is fulfilled only in the infinite volume limit, but its modification by finite size corrections is exponentially suppressed by the spatial lattice size L. These theoretical considerations are also numerically checked through lattice simulations to calculate the positronium spectrum in compact scalar QED, where the short-range interaction between an electron and a positron is realized in the Higgs phase.