Three particles in a finite volume

TL;DR

This paper derives an analytical expression for how the energy levels of a shallow three-particle bound state in a finite cubic volume depend on the volume size, focusing on the unitary limit and short-range interactions.

Contribution

It provides a new formula for the volume dependence of three-particle bound states in a finite box, highlighting the role of the three-body asymptotic normalization constant.

Findings

Derived the volume dependence of energy levels in the unitary limit.

Identified the exponential decay factor related to the bound-state momentum.

Connected short-range interactions to finite-volume energy shifts.

Abstract

The volume-dependence of a shallow three-particle bound state in the cubic box with a size $L$ is studied. It is shown that, in the unitary limit, the energy-level shift from the infinite-volume position is given by $\Delta E=c (\kappa^2/m)\,(\kappa L)^{-3/2}|A|^2 \exp(-2\kappa L/\sqrt{3})$, where $\kappa$ is the bound-state momentum and $|A|^2$ denotes the three-body analog of the asymptotic normalization constant, which encodes the information about the short-range interactions in the three-body system.