The Rare Decays $K^{+}\to {\pi }^{+}\nu \bar{\nu }$\ and $K_L\to {\mu }^{+}{\mu }^{-}$\ Beyond Leading Logarithms

Abstract

We analyze the branching ratio for the FCNC mode $K^{+}\to {\pi }^{+}\nu \bar{\nu }$\ in the standard model with QCD effects taken into account consistently to next-to-leading order. This involves a two-loop renormalization group analysis for the charm contribution, presented in this paper, and the calculation of $O({\alpha }_s)$ corrections to all orders in $m_t/M_W$ for the top-quark case that we have described elsewhere. The inclusion of next-to-leading corrections reduces considerably the theoretical uncertainty due to the choice of the renormalization scales, inherent in any calculation to finite order in perturbation theory. For $K^{+}\to {\pi }^{+}\nu \bar{\nu }$\ this point has not been discussed previously. In particular, the related uncertainty in the determination of $|V_{td}|$ from $B(K^{+}\to {\pi }^{+}\nu \bar{\nu })$ is reduced from $\sim 30\%$ to $\sim 7\%$ for typical values of the parameters. Simultaneously also the dependence of $B(K^{+}\to {\pi }^{+}\nu \bar{\nu })$ on the choice of $m_c$ is considerably reduced. We also give the next-to-leading order expression for the short-distance part of $K_L\to {\mu }^{+}{\mu }^{-}$. Impacts of our calculations on the determination of the unitarity triangle are presented.

40, MPI-Ph/93-44

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