Lattice determination of $I=0$ and 2 $\ddot{I}€\ddot{I}€$ scattering phase shifts with a physical pion mass

Abstract

Phase shifts for $s$-wave $\ddot{I}€\ddot{I}€$ scattering in both the $I=0$ and $I=2$ 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 $32^3\times 64$. These results support our recent study of direct CP violation in $K\to \ddot{I}€\ddot{I}€$ decay \cite{Abbott:2020hxn}, improving our earlier 2015 calculation \cite{Bai:2015nea}. The phase shifts are determined for both stationary and moving $\ddot{I}€\ddot{I}€$ systems, at three ($I=0$) and four ($I=2$) different total momenta. We implement several $\ddot{I}€\ddot{I}€$ interpolating operators including a scalar bilinear "$\ddot{I}ƒ$" 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 (GEVP) method is given. A detailed systematic error analysis is performed which allows phase shift results to be presented at a fixed energy.

v3: Add a subsection "Higher partial wave correction", and correct the unit of scattering length. 88 pages and 14 figures v2: 1). Add reference 29 as an example of pipi scattering calculation above 4mpi threshold. 2). Modify the wording on page 3 for the footage. 3). Correct the σoperator on page 17