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We will report on the first full calculation of the K L − K S mass difference in lattice QCD. The calculation is performed on a 2+1 flavor, domain wall fermion, 243ÃŮ64 ensemble with a 329 MeV pion mass and a 575 MeV kaon mass. Both double penguin diagrams and disconnected diagrams are included in this calculation. The calculation is made finite through the GIM mechanism by introducing a 949 MeV valence charm quark. While the double penguin diagrams contribute a very small fraction to the mass difference, there is a large cancellation between disconnected diagrams and other types of digrams. We obatain the mass difference ∆M K =3.30(34) × 10 −12 MeV for these unphysical kinematics.

In Ref. [1] we have presented the results of an exploratory lattice QCD computation of the long-distance contribution to the K þ → π þ νν decay amplitude. In the present paper we describe the details of this calculation, which includes the implementation of a number of novel techniques. The K þ → π þ νν decay amplitude is dominated by short-distance contributions which can be computed in perturbation theory with the only required nonperturbative input being the relatively well-known form factors of semileptonic kaon decays. The long-distance contributions, which are the target of this work, are expected to be of Oð5%Þ in the branching ratio. Our study demonstrates the feasibility of lattice QCD computations of the K þ → π þ νν decay amplitude, and in particular of the long-distance component. Though this calculation is performed on a small lattice (16 3 × 32) and at unphysical pion, kaon and charm quark masses, m π ¼ 420 MeV, m K ¼ 563 MeV and m MS c ð2 GeVÞ ¼ 863 MeV, the techniques presented in this work can readily be applied to a future realistic calculation.

We report a first, complete lattice QCD calculation of the long-distance contribution to the K + → π + νν decay within the standard model. This is a second-order weak process involving two four-Fermi operators that is highly sensitive to new physics and being studied by the NA62 experiment at CERN. While much of this decay comes from perturbative, short-distance physics there is a long-distance part, perhaps as large as the planned experimental error, which involves nonperturbative phenomena. The calculation presented here, with unphysical quark masses, demonstrates that this contribution can be computed using lattice methods by overcoming three technical difficulties: 1) a short-distance divergence that results when the two weak operators approach each other, 2) exponentially-growing, unphysical terms which appear in Euclidean, second-order perturbation theory and 3) potentially large finite-volume effects. A follow-on calculation with physical quark masses and controlled systematic errors will be possible with the next generation of computers.Keywords: rare kaon decay, lattice QCD Introduction -An important objective of experimental high-energy physics is the search for direct and indirect signs of new physics. Complementary to the direct search for new particles and forces at high energy, is the search for subtle deviations from standard model predictions at lower energies. The rare kaon decays K → πνν are such examples. As flavor-changing-neutral-current processes, the K → πνν decay amplitudes arise as oneloop, electroweak effects. The small size of one-loop, standard model effects makes these decays particularly sensitive to new phenomena.

We review the status of the RBC-UKQCD collaborations’ computations of the KL-KS mass difference. After a brief discussion of the theoretical framework which had been developed previously by the collaboration, we describe our latest computation, performed at physical quark masses, and present our preliminary result mKL - mKS = (5.5 ± 1.70) × 10-12 MeV.