ǫK at NNLO: The Charm-Top-Quark Contribution

We perform a next-to-next-to-leading order (NNLO) QCD analysis of the charmtop-quark contribution ηct to the effective |∆S| = 2 Hamiltonian in the Standard Model. ηct represents an important part of the short distance contribution to the parameter ǫK . We calculate the three-loop anomalous dimension of the leading operator Q̃S2, the three-loop mixing of the current-current and penguin operators into Q̃S2, 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 ηct = 0.494±0.046, which is roughly 7% larger than the next-to-leading-order (NLO) value ηNLO ct = 0.455±0.069. This results in a prediction for |ǫK | = (1.99±0.25)×10−3 , which corresponds to an enhancement of approximately 3.3% with respect to the value obtained using ηNLO ct .