We discuss formation of an $S$-wave bound state in finite volume on the basis of Lüscher’s phase-shift formula. It is found that although a bound-state pole condition is fulfilled only in the infinite-volume limit, its modification by the finite-size corrections is exponentially suppressed by the spatial extent $L$ in a finite box $L^3$. We also confirm that the appearance of the $S$-wave bound state is accompanied by an abrupt sign change of the $S$-wave scattering length even in finite volume through numerical simulations. This distinctive behavior may help us to distinguish the loosely bound state from the lowest energy level of the scattering state in finite-volume simulations.

• Received 16 October 2006

DOI:https://doi.org/10.1103/PhysRevD.74.114507