# Energy shift of the three-particle system in a finite volume

## TL;DR

This paper derives finite-volume energy shifts for the two lowest three-particle states using a recent quantization condition, extending previous results and confirming known formulas up to specific order corrections.

## Contribution

It provides new derivations of energy shifts in finite volume for three-particle systems, including particle-dimer states, up to order L^{-6} and L^{-3} respectively, using the particle-dimer framework.

## Key findings

- Energy shift formulas agree with existing literature.
- Lowest three-particle state results match previous findings.
- Particle-dimer state shifts reproduce Luescher equation results.

## Abstract

Using the three-particle quantization condition recently obtained in the particle-dimer framework, the finite-volume energy shift of the two lowest three-particle scattering states is derived up to and including order $L^{-6}$. Furthermore, assuming that a stable dimer exists in the infinite volume, the shift for the lowest particle-dimer scattering state is obtained up to and including order $L^{-3}$. The result for the lowest three-particle state agrees with the results from the literature, and the result for the lowest particle-dimer state reproduces the one obtained by using the Luescher equation.