Lattice Calculation of the $K_L$-$K_S$ mass difference

We report progress on calculating the $K_L$-$K_S$ mass difference in lattice QCD. T he calculation is performed on a 2+1 flavor, domain wall fermion, $16^3\times 32$ en semble with a 421 MeV pion mass. We include only current-current operators and drop all disconnected and double penguin diagrams in the calculation. The calculation is made finite through the GIM mechanism by introducing a valence charm quark. The long distance effects are discussed separately for each of the two parity channels. Whil e we find a clear long distance contribution from the parity odd channel, the signal to noise ratio in the parity even channel is exponentially decreasing and the two-p ion state can be seen in only a subclass of amplitudes. We obtain the mass differenc e $\Delta M_K$ in a range from $5.12(24)\times 10^{-12}$ to $9.31(66) \times 10^{-12}$ MeV for kaon masses between 563 and 839 MeV.