ϵK 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 ${\ensuremath{\eta}}_{ct}$ to the effective $|\ensuremath{\Delta}S|=2$ Hamiltonian in the standard model. ${\ensuremath{\eta}}_{ct}$ represents an important part of the short distance contribution to the parameter ${ϵ}_{K}$. We calculate the three-loop anomalous dimension of the leading operator ${\stackrel{\texttildelow{}}{Q}}_{S2}$, the three-loop mixing of the current-current and penguin operators into ${\stackrel{\texttildelow{}}{Q}}_{S2}$, 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 ${\ensuremath{\eta}}_{ct}=0.496\ifmmode\pm\else\textpm\fi{}0.047$, which is roughly 7% larger than the next-to-leading-order (NLO) value ${\ensuremath{\eta}}_{ct}^{\mathrm{NLO}}=0.457\ifmmode\pm\else\textpm\fi{}0.073$. This results in a prediction for $|{ϵ}_{K}|=(1.90\ifmmode\pm\else\textpm\fi{}0.26)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, which corresponds to an enhancement of approximately 3% with respect to the value obtained using ${\ensuremath{\eta}}_{ct}^{\mathrm{NLO}}$.