The Kaon bag parameter \( {\hat{B}}_K \) 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 \( {\hat{B}}_K \) 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 \( \overline{\textrm{MS}} \) schemes. Our world averages combine all available lattice data, including conversion between the 3- and 4-flavour theories as appropriate. We obtain the result \( {\hat{B}}_K^{\left(f=3\right)}=0.7627(60) \) , 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 |ϵK| = 2.171(65)pert.(71)non-pert.(153)param. × 10−3. We briefly discuss applications of our results to D-meson mixing.
