# Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states

## TL;DR

This paper extends the relativistic three-particle quantization condition to include two-particle bound states and resonances, enabling detailed finite-volume spectrum analysis and scattering amplitude extraction for three identical scalar particles.

## Contribution

It introduces a method to lift the previous restrictions on two-particle states in the quantization condition, allowing for the study of bound states and resonances in three-particle systems.

## Key findings

- Extended the quantization condition to include bound states and resonances.
- Numerically explored spectra with dimers and trimers.
- Demonstrated extraction of scattering amplitudes below dimer breakup.

## Abstract

In this work, we use an extension of the quantization condition, given in Ref. [1], to numerically explore the finite-volume spectrum of three relativistic particles, in the case that two-particle subsets are either resonant or bound. The original form of the relativistic three-particle quantization condition was derived under a technical assumption on the two-particle K matrix that required the absence of two-particle bound states or narrow two-particle resonances. Here we describe how this restriction can be lifted in a simple way using the freedom in the definition of the K-matrix-like quantity that enters the quantization condition. With this in hand, we extend previous numerical studies of the quantization condition to explore the finite-volume signature for a variety of two- and three-particle interactions. We determine the spectrum for parameters such that the system contains both dimers (two-particle bound states) and one or more trimers (in which all three particles are bound), and also for cases where the two-particle subchannel is resonant. We also show how the quantization condition provides a tool for determining infinite-volume dimer-particle scattering amplitudes for energies below the dimer breakup. We illustrate this for a series of examples, including one that parallels physical deuteron-nucleon scattering. All calculations presented here are restricted to the case of three identical scalar particles.