$K^+\to\pi^+\nu\bar\nu$ Beyond Leading Logs and $V_{td}$

The calculation of QCD corrections beyond the leading logarithmic approximation for the short-distance dominated rare decay $K^+\to\pi^+\nu\bar\nu$ is summarized. This analysis requires the complete ${\cal O}(\alpha_s)$ corrections to the top contribution to all orders in the top-quark mass and a two-loop renormalization group (RG) calculation for the charm contribution. The inclusion of these QCD corrections reduces considerably the theoretical uncertainty in calculating $B(K^+\to\pi^+\nu\bar\nu)$. Implications for the determination of $V_{td}$ from this decay are discussed.