Rare DecayK+→π+νν¯at the Next-to-Next-to-Leading Order in QCD

We calculate the charm quark contribution to the rare decay $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ in the next-to-next-to-leading order of QCD. This new contribution reduces the theoretical uncertainty in the relevant parameter $P_c$ from $\pm 10.1\%$ down to $\pm 2.4\%$, corresponding to scale uncertainties of $\pm 1.3\%$, $\pm 1.0\%$, $\pm 0.006$, and $\pm 1.2°$ in $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ and in $|V_{td}|$, $\sin 2\beta$, and $\gamma$ extracted from the $K\to \pi \nu \overline{\nu }$ system. The error in $P_c=0.37\pm 0.04$ is now fully dominated by the current uncertainty of $\pm 3.8\%$ in the charm quark mass $m_c$. We find $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu })=(8.0\pm 1.1)\times {10}^{-11}$, where the quoted error stems almost entirely from the present uncertainties in $m_c$ and the Cabibbo-Kobayashi-Maskawa elements.