Three-body Unitarity in the Finite Volume

TL;DR

This paper develops a method to extrapolate three-particle scattering amplitudes from finite-volume lattice QCD simulations to infinite volume, ensuring unitarity and accurately identifying genuine three-body states.

Contribution

It introduces a finite-volume quantization condition for three-body systems based on a relativistic amplitude that respects unitarity, with a demonstration for identical scalar particles.

Findings

Finite-volume poles cancel, isolating genuine three-body eigenvalues.

Derived quantization condition for three identical scalar particles.

Numerical implementation confirms the method's validity.

Abstract

The physical interpretation of lattice QCD simulations, performed in a small volume, requires an extrapolation to the infinite volume. A method is proposed to perform such an extrapolation for three interacting particles at energies above threshold. For this, a recently formulated relativistic $3\to 3$ amplitude based on the isobar formulation is adapted to the finite volume. The guiding principle is two- and three-body unitarity that imposes the imaginary parts of the amplitude in the infinite volume. In turn, these imaginary parts dictate the leading power-law finite-volume effects. It is demonstrated that finite-volume poles arising from the singular interaction, from the external two-body sub-amplitudes, and from the disconnected topology cancel exactly leaving only the genuine three-body eigenvalues. The corresponding quantization condition is derived for the case of three identical scalar-isoscalar particles and its numerical implementation is demonstrated.