γK at next-to-next-to-leading order: The charm-top-quark contribution

We perform a next-to-next-to-leading order QCD analysis of the charm-top-quark contribution η_ct to the effective |ΔS|=2 Hamiltonian in the standard model. η_ct represents an important part of the short distance contribution to the parameter γ_K. We calculate the three-loop anomalous dimension of the leading operator Q∼_S2, the three-loop mixing of the current-current and penguin operators into Q∼_S2, and the corresponding two-loop matching conditions at the electroweak, the bottom-quark, and the charm-quark scale. As our final numerical result we obtain η_ct=0.496±0.047, which is roughly 7% larger than the next-to-leading-order (NLO) value η_ct^NLO=0.457±0.073. This results in a prediction for |γ_K|=(1.90±0.26)×10^-3, which corresponds to an enhancement of approximately 3% with respect to the value obtained using η_ct^NLO.