In this work, we report the lattice calculation of two important quantities which emerge from second order, K0 -- K¯0 mixing : DeltaMK and epsilonK. The RBC-UKQCD collaboration has performed the first calculation of DeltaMK with unphysical kinematics [1]. We now extend this calculation to near-physical and physical ensembles. In these physical or near-physical calculations, the two-pion energies are below the kaon threshold, and we have to examine the two-pion intermediate states contribution to DeltaMK, as well as the enhanced finite volume corrections arising from these two-pion intermediate states. We also report the ?rst lattice calculation of the long-distance contribution to the indirect CP violation parameter, the epsilonK. This calculation involves the treatment of a short-distance, ultra-violet divergence that is absent in the calculation of DeltaMK, and we will report our techniques for correcting this divergence on the lattice. In this calculation, we used unphysical quark masses on the same ensemble that we used in [1]. Therefore, rather than providing a physical result, this calculation demonstrates the technique for calculating epsilonK, and provides an approximate understanding the size of the long-distance contributions.;Various new techniques are employed in this work, such as the use of All-Mode-Averaging (AMA), the All-to-All (A2A) propagators and the use of super-jackknife method in analyzing the data.
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