Long-distance contribution to ${ϵ}_K$ from lattice QCD

Abstract

A lattice QCD approach to the calculation of the long-distance contributions to ${ϵ}_K$ is presented. This parameter describes indirect CP violation in $K\to \pi \pi$ decay. While the short-distance contribution to ${ϵ}_K$ can be accurately calculated in terms of standard model parameters and a single hadronic matrix element, $B_K$, there is a long-distance part which is estimated to be approximately $5\%$ of the total and is more difficult to determine. A method for determining this small but phenomenologically important contribution to ${ϵ}_K$ using lattice QCD is proposed and a complete exploratory calculation of the contribution is presented. This exploratory calculation uses an unphysical light quark mass corresponding to a 339 MeV pion mass and an unphysical charm quark mass of 968 MeV, expressed in the $\overline{\mathrm{MS}}$ scheme at 2 GeV. This calculation demonstrates that future work should be able to determine this long-distance contribution from first principles with a controlled error of 10\% or less.