Spectrum of three-body bound states in a finite volume

The spectrum of a bound state of three identical particles with a mass $m$ in a finite cubic box is studied. It is shown that in the unitary limit, the energy shift of a shallow bound state is given by $ΔE=c (κ^2/m)\,(κL)^{-3/2}|A|^2\exp(-2κL/\sqrt{3})$, where $κ$ is the bound-state momentum, $L$ is the box size, $|A|^2$ denotes the three-body analog of the asymptotic normalization coefficient of the bound state wave function and $c$ is a numerical constant. The formula is valid for $κL\gg 1$.

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