The rare decays K⁺ → π⁺νν and K_L → μ⁺μ^- beyond leading logarithms

We analyze the branching ratio for the FCNC mode K⁺ → π⁺νν 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⁺νν this point has not been discussed previously. In particular, the related uncertainty in the determination of |V_td| from B(K⁺ → π⁺νν) is reduced from ∼ 30% to ∼ 7% for typical values of the parameters. Simultaneously also the dependence of B(K⁺ → π⁺νν) 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.