RI-(S)MOM to $$ \overline{\textrm{MS}} $$ conversion for BK at two-loop order

A bstract The Kaon bag parameter $$ {\hat{B}}_K $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mi>K</mml:mi> </mml:msub> </mml:math> 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 $$ {\hat{B}}_K $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mi>K</mml:mi> </mml:msub> </mml:math> 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 $$ \overline{\textrm{MS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>MS</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> schemes. Our world averages combine all available lattice data, including conversion between the 3- and 4-flavour theories as appropriate. We obtain the result $$ {\hat{B}}_K^{\left(f=3\right)}=0.7627(60) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mi>K</mml:mi> <mml:mfenced> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:mfenced> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>0.7627</mml:mn> <mml:mfenced> <mml:mn>60</mml:mn> </mml:mfenced> </mml:math> , 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.