Rare decay K⁺ →π⁺ ν ν̄ at the next-to-next-to-leading order m QCD

We calculate the charm quark contribution to the rare decay K⁺ →π⁺ ν ν̄ in the next-to-next-to-leading order of QCD. This new contribution reduces the theoretical uncertainty in the relevant parameter P _c from ± 10.1% down to ± 2.4%, corresponding to scale uncertainties of ± 1.3%, ± 1.0%, ± 0.006, and ± 1.2° in B (K⁺ →π⁺ ν ν̄ and in |V_td|, sin2^β extracted from the K →π⁺ ν ν̄ system. The error P_c0. 37±0.04 is now fully dominated by the current uncertainty of ±3.8% in the charm quark mass m_c. We find B (K⁺ →π⁺ ν ν̄)=(8.0±1.1)×10^-11, where the quoted error stems almost entirely from the present uncertainties in m_c and the Cabibbo-Kobayashi-Maskawa elements.