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

Phase shifts for <math display="inline"><mi>s</mi></math>-wave <math display="inline"><mi>π</mi><mi>π</mi></math> scattering in both the <math display="inline"><mi>I</mi><mo>=</mo><mn>0</mn></math> and <math display="inline"><mi>I</mi><mo>=</mo><mn>2</mn></math> channels are determined from a lattice QCD calculation performed on 741 gauge configurations obeying G-parity boundary conditions with a physical pion mass and lattice size of <math display="inline"><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math>. These results support our recent 2021 study of direct <math display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> violation in <math display="inline"><mi>K</mi><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi></math> decay, improving our earlier 2015 calculation. The phase shifts are determined for both stationary and moving <math display="inline"><mi>π</mi><mi>π</mi></math> systems, at three (<math display="inline"><mi>I</mi><mo>=</mo><mn>0</mn></math>) and four (<math display="inline"><mi>I</mi><mo>=</mo><mn>2</mn></math>) different total momenta. We implement several <math display="inline"><mi>π</mi><mi>π</mi></math> interpolating operators including a scalar bilinear “<math display="inline"><mi>σ</mi></math>” operator and paired single-pion bilinear operators with the constituent pions carrying various relative momenta. Several techniques, including correlated fitting and a bootstrap determination of p-values have been used to refine the results and a comparison with the generalized eigenvalue problem method is given. A detailed systematic error analysis is performed which allows phase shift results to be presented at a fixed energy.