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.