Spurious poles in a finite volume

TL;DR

This paper examines how spurious poles from effective-range expansions affect three-body energy levels in finite volumes, demonstrating a method to remove these poles and improve the accuracy of finite-volume energy predictions.

Contribution

It adapts a recent infinite-volume pole removal procedure to finite-volume calculations, improving the accuracy of three-body energy level predictions.

Findings

Spurious poles cause peculiar volume dependence of energy levels.

A pole removal method can be adapted for finite-volume calculations.

Exact energy levels are systematically approximated with improved EFT order.

Abstract

Using effective-range expansion for the two-body amplitudes may generate spurious sub-threshold poles outside of the convergence range of the expansion. In the infinite volume, the emergence of such poles leads to the inconsistencies in the three-body equations, e.g., to the breakdown of unitarity. We investigate the effect of the spurious poles on the three-body quantization condition in a finite volume and show that it leads to a peculiar dependence of the energy levels on the box size $L$. Furthermore, within a simple model, it is demonstrated that the procedure for the removal of these poles, which was recently proposed in Ref.[1] in the infinite volume, can be adapted to the finite-volume calculations. The structure of the exact energy levels is reproduced with an accuracy that systematically improves order by order in the EFT expansion.