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 + → π + ν ν 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( α 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 + → π + ν ν this point has not been discussed previously. In particular, the related uncertainty in the determination of | V td | from B( K + → π + ν ν ) is reduced from ∼ 30% to ∼ 7% for typical values of the parameters. Simultaneously also the dependence of B( K + → π + ν ν ) 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 → μ + μ − . Impacts of our calculations on the determination of the unitarity triangle are presented.