RI-(S)MOM to MS conversion for BK at two-loop order

Abstract

The Kaon bag parameter BˆK plays a critical role in constraining the parameters of the CKM matrix and in probing physics beyond the Standard Model. In this work, we improve the precision of BˆK to next-to-next-to-leading order (NNLO) and provide world averages for both 3- and 4-favour theories. In the course of this, as our main technical development, we carry out the two-loop matching between the RI-(S)MOM and MS schemes. Our world averages combine all available lattice data, including conversion between the 3- and 4-favour theories as appropriate. We obtain the result Bˆ (f=3) K = 0.7627(60), which comprises the complete set of 3- and 4-favour lattice results and can be used directly in phenomenological applications. The error is dominated by lattice uncertainties and missing higher-order corrections (residual scale dependence). Our averages include a PDG rescaling factor of 1.28 refecting a mild tension among the lattice inputs after inclusion of NNLO corrections in the scheme conversion and matching across favour thresholds. Our averages imply an updated value |ϵK| = 2.171(65)pert.(71)non-pert.(153)param. ×10−3 . We briefy discuss applications of our results to D-meson mixing.