Two-pion scattering and $K \to \pi\pi$ decay calculations with periodic boundaries

Since RBC/UKQCD’s first effort to determine the $\Delta I = 1/2$ $K \to \pi\pi$ decay amplitudes with lattice simulation, which was published in 2015 and 2020, we have been revisiting this calculation with a different lattice setup with periodic boundary conditions and multiple lattice spacings to take the continuum limit and improve the precision. While there was an expectation that it could be difficult to extract physical kinematics of $K \to \pi\pi$ decay with periodic boundary conditions, we found it is possible through the variational method. Also periodic boundary conditions provide a relatively easy prescription to introduce electromagnetic and isospin breaking corrections, which is desired to be implemented in the near future. In this talk, I provide our recent status and discuss prospect of high-precision calculation of $K \to \pi\pi$ decay with periodic boundary conditions. As a related calculation, I also give some results for $\pi\pi$ scattering phase shifts and scattering lengths.