Electroweak corrections to the charm quark contribution to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>K</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mi>ν</mml:mi><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math>

Abstract

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 ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}}$. The corresponding parameter ${P}_{c}(X)$ is increased by up to 2% with respect to the pure QCD estimate to ${P}_{c}(X)=0.372\ifmmode\pm\else\textpm\fi{}0.015$ for ${m}_{c}({m}_{c})=(1.286\ifmmode\pm\else\textpm\fi{}0.013)\mathrm{GeV}$, ${\ensuremath{\alpha}}_{s}({M}_{Z})=0.1176\ifmmode\pm\else\textpm\fi{}0.0020$, and $|{V}_{us}|=0.2255$. For the branching ratio we find $B({K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}})=(8.5\ifmmode\pm\else\textpm\fi{}0.7)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$, where the quoted uncertainty is dominated by the Cabibbo-Kobayashi-Maskawa elements.