Finite volume treatment of $\pi\pi$ scattering in the $\rho$ channel

Abstract

We make a theoretical study of $\pi\pi$ scattering with quantum numbers $J^{PC}=1^{--}$ in a finite box. To calculate physical observables for infinite volume from lattice QCD, the finite box dependence of the potentials is not usually considered. We quantify such effects by means of two different approaches for vector-isovector $\pi\pi$ scattering based on Unitarized Chiral Perturbation Theory results: the Inverse Amplitude Method and another one based on the $N/D$ method. We take into account finite box effects stemming from higher orders through loops in the crossed $t,u-$channels as well as from the renormalization of the coupling constants. The main conclusion is that for $\pi\pi$ phase shifts in the isovector channel one can safely apply L\"uscher based methods for finite box sizes of $L$ greater than $2 m_\pi^{-1}$.