Improved Pion Correlation Functions Under Lattice Action

We present the first calculation of pion-pion ($\pi\pi$) scattering at physical quark mass from lattice QCD. We simulate QCD in a periodic $\sim 5$ fm box at inverse lattice spacing $a^{-1}=1.015,1.3784$ GeV using zM\ obius/M\ obius Domain Wall Fermions and Iwasaki Gauge Action. We form operators composed of localized hydrogen-like wave functions for scalar ($\overline{\psi}\psi$), pseudoscalar ($\overline{\psi}\gamma_5\psi$), and vector bilinears ($\overline{\psi}\gamma_\mu\psi$) We then calculate all-to-all (A2A) meson field propagators for up to three units of individual particle momenta as well as isospin $I=0,1,2$, with up to (in the $I=0,2$) three units of center of mass momentum. This allows us to project the resulting $O(10000)$ correlation functions onto definite isospin and irreducible representation (which allows us to project onto the lowest bose-symmetry allowed angular momenta of each of our operators). These projections form correlation function matrices of definite time separation, which then naturally define a generalized eigenvalue problem (GEVP) we can solve for the desired spectra. We then apply the L\ uscher method to obtain from our lattice energies continuum scattering phase shifts. After partially accounting for some of our systematic errors, we obtain fair to good agreement with phenomenological predictions for the phase shifts as derived from Roy equations and chiral perturbation theory. These results can help serve as a foundation on which to build a fully periodic calculation of a kaon decaying into two pions ($K\to\pi\pi)$, a very important decay related to the study of baryogenesis in the Standard Model. This work also contains algorithmic and mathematical developments which, while not used directly in the $\pi\pi$ scattering study, are likely to be useful in the study of $K\to\pi\pi$ (and might also be useful to interaction physics generally). These developments are highlighted briefly in the introduction