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Phase shifts for $s$-wave $\pi\pi$ scattering in both the $I=0$ and $I=2$channels are determined from a lattice QCD calculation performed on 741 gaugeconfigurations obeying G-parity boundary conditions with a physical pion massand lattice size of $32^3\times 64$. These results support our recent study ofdirect CP violation in $K\to\pi\pi$ decay \cite{Abbott:2020hxn}, improving ourearlier 2015 calculation \cite{Bai:2015nea}. The phase shifts are determinedfor both stationary and moving $\pi\pi$ systems, at three ($I=0$) and four($I=2$) different total momenta. We implement several $\pi\pi$ interpolatingoperators including a scalar bilinear "$\sigma$" operator and pairedsingle-pion bilinear operators with the constituent pions carrying variousrelative momenta. Several techniques, including correlated fitting and abootstrap determination of p-values have been used to refine the results and acomparison with the generalized eigenvalue problem (GEVP) method is given. Adetailed systematic error analysis is performed which allows phase shiftresults to be presented at a fixed energy.

Lattice determination of $I= 0$ and 2 $\pi\pi$ scattering phase shifts with a physical pion mass