We report on the status and prospects of the RBC-UKQCD collaborations' computations of $\Delta m_K$, the $K_L$-$K_S$ mass difference. This calculation is an important element of our wider programme of computations of the long-distance contributions to flavour changing neutral current processes in kaon physics, which also includes $\epsilon_K$ and the amplitudes for $K\to\pi\ell^+\ell^-$ and $K^+\to\pi^+\nu\bar{\nu}$ rare decays. The computation of $\Delta m_K$ is performed on a $64^3\times 128\times 12$ lattice with the Iwasaki gauge action and Moebius domain wall fermions at an inverse lattice spacing of 2.359(7)\,GeV, with a physical pion mass ($m_\pi\simeq 136$ MeV) and with four values of the charm-quark mass spanning its physical value (giving us control of the corresponding discretisation effects). Techniques such the use of random volume sources, all-to-all propagators and a new approach to all-mode-averaging are being used to reduce the statistical uncertainty. Results will be presented from about 60 configurations (separated by 20 configurations) which represents a fraction of the planned 160 measurements for the project.

Weak Decays and Matrix Elements