Spectrum of three-body bound states in a finite volume

Abstract

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 $\Delta E=c({\kappa }^2/m)(\kappa L{)}^{-3/2}|A{|}^2\exp (-2\kappa L/\sqrt{3})$, where $\kappa$ is the bound-state momentum, $L$ 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 $\kappa L\gg 1$.

Comments: An error in the calculation of the overlap integral in Eq. (11) is corrected. Our main result given in Eq. (26) remains the same, except the numerical value of the constant c, which changes approximately by 10%