Anisotropic Lattice Calculation of Pion Scattering Using an Asymmetric Box

Pion-pion elastic scattering phase shifts in the I=2, J=0 channel are calculated within quenched approximation using improved gauge and Wilson fermion actions on anisotropic lattices in an asymmetric box. Using the modified L\"{u}scher's formula applicable in an asymmetric volume, we are able to access more low-momentum modes than a cubic box with similar volume. The calculation is performed at three lattice spacings at $\beta=2.080$, $\beta=2.215$ and $\beta=2.492$. For each lattice spacing, several valence quark mass values are taken in the quenched approximation which enable us to extrapolate the results to the chiral limit. Continuum limit extrapolation is then performed and the extrapolated results are compared with the known results from both theory and experiment. We find that our lattice results for the scattering phase shifts in this channel are consistent with the experimental data when the three-momentum of the pion is below 300MeV. The results are also consistent with previous lattice calculations and other theoretical calculations using Chiral Perturbation Theory or dispersion relations. We have also given the pion-pion scattering length in the chiral and continuum limit