Pion-pion Scattering with Elongated Boxes

Pion-pion scattering offers an important benchmark for a lattice QCD study of hadron-hadron interactions. Scattering can happen in one of three isospin channels, each having distinct properties. The attractive I $=$ 0 and I $=$ 1 channels are dominated by the broad $\sigma $ and narrow $\rho $ resonances, respectively, while the I $=$ 2 channel has no low energy resonance. Our group has calculated the $\sigma $ and $\rho $ resonance properties using elongated boxes to scan the relevant kinematic region at two pion masses. Here we present new results for the isospin-2 channel, thus completing the full study of $\pi \pi $ scattering. In addition, we establish a link to the physical point of all three channels simultaneously using the Inverse Amplitude Method.~~