Electroweak corrections to the charm quark contribution toK+→π+νν¯

We compute the leading-log QED, the next-to-leading-log QED-QCD, and the electroweak corrections to the charm quark contribution relevant for the rare decay $K^{+}\to {\pi }^{+}\nu \overline{\nu }$. The corresponding parameter $P_c(X)$ is increased by up to 2% with respect to the pure QCD estimate to $P_c(X)=0.372\pm 0.015$ for $m_c(m_c)=(1.286\pm 0.013)\mathrm{GeV}$, ${\alpha }_s(M_Z)=0.1176\pm 0.0020$, and $|V_{us}|=0.2255$. For the branching ratio we find $B(K^{+}\to {\pi }^{+}\nu \overline{\nu })=(8.5\pm 0.7)\times {10}^{-11}$, where the quoted uncertainty is dominated by the Cabibbo-Kobayashi-Maskawa elements.