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We develop and demonstrate techniques needed to compute the long distance contribution to the $K_{L}$-$K_{S}$ mass difference, $\Delta M_K$, in lattice QCD and carry out a first, exploratory calculation of this fundamental quantity.
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Together with G. Martinelli, we are also exploring the possibility of controlling finite volume effects without requiring a two-pion state to be degenerate with the kaon, generalizing the approach to finite-volume effects in K → π π K\to\pi\pi decays developed in Lin et al. 2001 ; Kim et al. 2005
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