The Rare Decays K+→π+ννand Kₗ→μ+μ-Beyond Leading Logarithms

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We analyze the branching ratio for the FCNC mode K+→π+ννin 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(αₛ) corrections to all orders in mₜ/MW 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+ννthis point has not been discussed previously. In particular, the related uncertainty in the determination of |Vtd| 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 mc is considerably reduced. We also give the next-to-leading order expression for the short-distance part of Kₗ→μ+μ-. Impacts of our calculations on the determination of the unitarity triangle are presented.