We present a way to evaluate the scattering of unstable particles quantized in a finite volume with the aim of extracting physical observables for infinite volume from lattice data. We illustrate the method with the $\pi \rho$ scattering which generates dynamically the axial-vector $a_1(1260)$ resonance. Energy levels in a finite box are evaluated both considering the $\rho$ as a stable and unstable resonance and we find significant differences between both cases. We discuss how to solve the problem to get the physical scattering amplitudes in the infinite volume, and hence phase shifts, from possible lattice results on energy levels quantized inside a finite box.

• Received 9 January 2012

DOI:https://doi.org/10.1103/PhysRevD.85.054507