On the equivalence of left-hand-cut and three-body formalisms

Abstract

The "left-hand cut problem" in lattice QCD arises when exchange singularities invalidate standard relations used to extract scattering amplitudes from the finite-volume spectrum. In recent years, several approaches were proposed to resolve this problem. We establish analytically and verify numerically the equivalence between the three-body approach proposed in [JHEP 06 (2024) 051] and the two-body left-hand cut formalism developed in [JHEP 08 (2024) 075]. We test the equivalence in a scalar model describing a particle scattering from an $S$-wave two-particle bound state through one-particle exchange. Both approaches reproduce the same infinite-volume particle--bound-state amplitude and correctly encode the associated left-hand cut in the finite-volume spectrum. Crucially, we identify that the exponentially suppressed finite-volume effects neglected in both formalisms can be numerically large, compromising amplitude extraction at small lattice volumes.