# Implementing the three-particle quantization condition including higher partial waves

## TL;DR

This paper develops a comprehensive framework for three-particle quantization in finite volume, including higher partial waves like d-waves, and demonstrates its application to systems such as 3π+ relevant for lattice QCD.

## Contribution

It introduces a systematic expansion for the three-particle divergence-free K matrix including higher partial waves, extending previous approaches and enabling more accurate finite-volume spectrum analysis.

## Key findings

- Higher partial waves significantly affect the three-particle spectrum.
- The framework can identify potential unphysical solutions in the quantization condition.
- Application to 3π+ system shows sensitivity to d-wave interactions.

## Abstract

We present an implementation of the relativistic three-particle quantization condition including both $s$- and $d$-wave two-particle channels. For this, we develop a systematic expansion about threshold of the three-particle divergence-free K matrix, $\mathcal{K}_{\mathrm{df,3}}$, which is a generalization of the effective range expansion of the two-particle K matrix, $\mathcal{K}_2$. Relativistic invariance plays an important role in this expansion. We find that $d$-wave two-particle channels enter first at quadratic order. We explain how to implement the resulting multichannel quantization condition, and present several examples of its application. We derive the leading dependence of the threshold three-particle state on the two-particle $d$-wave scattering amplitude, and use this to test our implementation. We show how strong two-particle $d$-wave interactions can lead to significant effects on the finite-volume three-particle spectrum, including the possibility of a generalized three-particle Efimov-like bound state. We also explore the application to the $3\pi^+$ system, which is accessible to lattice QCD simulations, where we study the sensitivity of the spectrum to the components of $\mathcal{K}_{\mathrm{df,3}}$. Finally, we investigate the circumstances under which the quantization condition has unphysical solutions.