Long distance contribution to $K_{L}$-$K_{S}$ mass difference

We present a method to non-perturbatively determine the long-distance contribution to the $K_{L}$-$K_{S}$ mass difference. The calculation is performed on 2+1 flavor, domain wall fermion, $16^3\times32$ configurations with a 421 MeV pion mass and a kaon mass of 559 MeV . We include only current-current operators and drop all disconnected diagrams in the calculation. The largest contribution comes from quadratically divergent, short distance lattice artifacts. This quadratic divergence is eliminated through the GIM mechanism by introducing a valence charm quark. The remaining short distance effects are then removed by using RI/MOM technique which allows their exact replacement by the physical short distance part.

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