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

Abstract

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 $\overline{\mathrm{MS}}$ schemes. Our world averages combine all available lattice data, including conversion between the 3- and 4-flavour theories as appropriate. We obtain the result ${\hat{B}}_K^{(f=3)}=0.7627(60)$, 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 $|{\epsilon }_K|=2.171(65{)}_{\text{pert.}}(71{)}_{\text{non-pert.}}(153{)}_{\text{param.}}\times 10^{-3}$. We briefly discuss applications of our results to $D$-meson mixing.

33 pages, 11 figures, 9 tables, v2: match published version

References in corpus (14)

• Domain wall QCD with physical quark masses
• pySecDec: a toolbox for the numerical evaluation of multi-scale integrals
• On epsilon_K beyond lowest order in the Operator Product Expansion
• Correlations among new CP violating effects in Delta F = 2 observables
• Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point
• Epsilon_K at Next-to-Next-to-Leading Order: The Charm-Top-Quark Contribution
• Three loop anomalous dimension of non-singlet quark currents in the RI' scheme
• Precision computation of the kaon bag parameter
• Renormalization constants and beta functions for the gauge couplings of the Standard Model to three-loop order
• Neutral Kaon Mixing Beyond the Standard Model with $n_f=2+1$ Chiral Fermions Part 1: Bare Matrix Elements and Physical Results
• Standard-model prediction of ${\epsilon }_K$ with manifest CKM unitarity
• Precise MS-bar light-quark masses from lattice QCD in the RI/SMOM scheme
• Electroweak Corrections to the Charm-Top-Quark Contribution to ${\epsilon }_K$
• Power corrections to the CP-violation parameter ${\epsilon }_K$

Cited by in corpus (1)

• CP violation in the hyperon decays $\Sigma \to N\pi$