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We develop and demonstrate techniques needed to compute the long distancecontribution to the $K_{L}$-$K_{S}$ mass difference, $\Delta M_K$, in latticeQCD and carry out a first, exploratory calculation of this fundamentalquantity. The calculation is performed on 2+1 flavor, domain wall fermion,$16^3\times32$ configurations with a 421 MeV pion mass and an inverse latticespacing $1/a=1.73$ GeV. We include only current-current operators and drop alldisconnected and double penguin diagrams. The short distance part of the massdifference in a 2+1 flavor calculation contains a quadratic divergence cut offby the lattice spacing. Here, this quadratic divergence is eliminated throughthe GIM mechanism by introducing a valence charm quark. The inclusion of thecharm quark makes the complete calculation accessible to lattice methodsprovided the discretization errors associated with the charm quark can becontrolled. The long distance effects are discussed for each parity channelseparately. While we can see a clear signal in the parity odd channel, thesignal to noise ratio in the parity even channel is exponentially decreasing asthe separation between the two weak operators increases. We obtain a massdifference $\Delta M_K$ which ranges from $6.58(30)\times 10^{-12}$ MeV to$11.89(81)\times 10^{-12}$ MeV for kaon masses varying from 563 MeV to 839 MeV.Extensions of these methods are proposed which promise accurate results forboth $\Delta M_K$ and $\epsilon_K$, including long distance effects.

Long distance contribution to the $K_L-K_S$ mass difference