I=3/2 and I=1/2 channels of K Ц decay at the physical point with periodic boundary conditions

We present a lattice calculation of the K -> pi pi matrix elements and amplitudes with both the Delta I = 3/2 and 1/2 channels and epsilon', the measure of direct CP violation. We use periodic boundary conditions (PBC), where the correct kinematics of K -> pi pi can be achieved via an excited two-pion final state. To overcome the difficulty associated with the extraction of excited states, our previous work [Z. Bai et al., Standard Model prediction for direct CP violation in K -> pi pi decay, Phys. Rev. Lett. 115, 212001 (2015); R. Abbott et al., Direct CP violation and the Delta I = 1/2 rule in K -> pi pi decay from the Standard Model, Phys. Rev. D 102, 054509 (2020).] successfully employed G-parity boundary conditions, where pions are forced to have nonzero momentum enabling the I = 0 two-pion ground state to express the on shell kinematics of the K -> pi pi decay. Here instead we overcome the problem using the variational method which allows us to resolve the two-pion spectrum and matrix elements up to the relevant energy where the decay amplitude is on shell. In this paper we report an exploratory calculation of K -> pi pi decay amplitudes and epsilon 0 using PBC on a coarser lattice size of 24(3) x 64 with inverse lattice spacing a(-1) = 1.023 GeV and the physical pion and kaon masses. The results are promising enough to motivate us to continue our measurements on finer lattice ensembles in order to improve the precision in the near future.