SMOM - $\overline{MS}$ Matching for $B_K$ at Two-loop Order

Abstract: The Kaon bag parameter, $B_K$, is a key non-perturbative ingredient in the search for new physics through CP-violation. It parameterizes the QCD hadronic matrix element of the effective weak $\Delta S=2$ four-quark operator which can only be computed non-perturbatively on the lattice. The perturbative matching of $B_K$ between the lattice renormalization schemes and $\overline{\mbox{MS}}$ scheme has been done before at one-loop order. In this proceedings we describe an ongoing calculation of the conversion factors for $B_K$ between the four non-exceptional RI-SMOM schemes and the $\overline{\mbox{MS}}$ scheme at two-loop order in perturbation theory.