THE RARE DECAYS K+-]PI+NU(NU)OVER-BAR AND K(L)-]MU+MU- BEYOND LEADING LOGARITHMS

We analyze the branching ratio for the FCNC mode K+ --> pi nunuBAR 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(as) 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+ --> pi+ nunuBAR this point has not been discussed previously. In particular, the related uncertainty in the determination of \V(td)\ from B(K+ --> pi+ nunuBAR) is reduced from approximately 30% to approximately 7% for typical values of the parameters. Simultaneously also the dependence of B(K+ --> pi+ nunuBAR) on the choice of mc is considerably reduced. We also give the next-to-leading order expression for the short-distance part of K(L) --> nu+nu-. Impacts of our calculations on the determination of the unitarity triangle are presented.