Implementing the three-particle quantization condition for $\pi^+\pi^+K^+$ and related systems

TL;DR

This paper advances the implementation of the three-particle quantization condition for nondegenerate particles, providing theoretical checks, code, and predictions for systems like nd K+ using chiral perturbation theory.

Contribution

It introduces practical modifications, theoretical validations, and numerical tools for applying the three-particle quantization formalism to systems with two identical and one different particle.

Findings

Derived the threshold expansion up to /L^5 for the lightest three-particle state.

Provided leading-order chiral perturbation theory predictions for nd K+ systems.

Validated the formalism with toy model numerical checks.

Abstract

Recently, the formalism needed to relate the finite-volume spectrum of systems of nondegenerate spinless particles has been derived. In this work we discuss a range of issues that arise when implementing this formalism in practice, provide further theoretical results that can be used to check the implementation, and make available codes for implementing the three-particle quantization condition. Specifically, we discuss the need to modify the upper limit of the cutoff function due to the fact that the left-hand cut in the scattering amplitudes for two nondegenerate particles moves closer to threshold; we describe the decomposition of the three-particle amplitude $\mathcal{K}_\text{df,3}$ into the matrix basis used in the quantization condition, including both $s$ and $p$ waves, with the latter arising in the amplitude for two nondegenerate particles; we derive the threshold expansion for the lightest three-particle state in the rest frame up to $\mathcal{O}(1/L^5)$; and we calculate the leading-order predictions in chiral perturbation theory for $\mathcal{K}_\text{df,3}$ in the $\pi^+\pi^+K^+$ and $\pi^+K^+K^+$ systems. We focus mainly on systems with two identical particles plus a third that is different ("2+1" systems). We describe the formalism in full detail, and present numerical explorations in toy models, in particular checking that the results agree with the threshold expansion, and making a prediction for the spectrum of $\pi^+\pi^+K^+$ levels using the two- and three-particle interactions predicted by chiral perturbation theory.