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

A bstract The Kaon bag parameter B̂K B ̂ K 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 B̂K B ̂ K to next-to-next-to-leading order (NNLO) and provide world averages for both 3- and 4-flavour theories. In the course of this, as our main technical development, we carry out the two-loop matching between the RI-(S)MOM and MS̄ MS ¯ schemes. Our world averages combine all available lattice data, including conversion between the 3- and 4-flavour theories as appropriate. We obtain the result B̂K⁽ᶠ⁼³⁾=0.7627(60) B ̂ K f = 3 = 0.7627 60 , which comprises the complete set of 3- and 4-flavour 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 reflecting a mild tension among the lattice inputs after inclusion of NNLO corrections in the scheme conversion and matching across flavour thresholds. Our averages imply an updated value | ϵ K | = 2 . 171(65) pert. (71) non-pert. (153) param. × 10 − 3 . We briefly discuss applications of our results to D -meson mixing.