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

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. The calculation is performed on 2+1 flavor, domain wall fermion, $16^3\times32$ configurations with a 421 MeV pion mass and an inverse lattice spacing $1/a=1.73$ GeV. We include only current-current operators and drop all disconnected and double penguin diagrams. The short distance part of the mass difference in a 2+1 flavor calculation contains a quadratic divergence cut off by the lattice spacing. Here, this quadratic divergence is eliminated through the GIM mechanism by introducing a valence charm quark. The inclusion of the charm quark makes the complete calculation accessible to lattice methods provided the discretization errors associated with the charm quark can be controlled. The long distance effects are discussed for each parity channel separately. While we can see a clear signal in the parity odd channel, the signal to noise ratio in the parity even channel is exponentially decreasing as the separation between the two weak operators increases. We obtain a mass difference $\Delta M_K$ 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 for both $\Delta M_K$ and $\epsilon_K$, including long distance effects.