$\sin 2\beta$ from $K\to\pi\nu\bar\nu$

Abstract

We point out that the measurement of just the two branching fractions \linebreak $B(K^+\to\pi^+\nu\bar\nu)$ and $B(K_L\to\pi^0\nu\bar\nu)$ can in a theoretically clean manner determine $\sin 2\beta$ almost independently of $m_t$ and $V_{cb}$. This allows to obtain an interesting relation between the CP asymmetry $A_{CP}(\psi K_S)$ in B physics and the branching ratios for these two rare K decays. The recently calculated next-to-leading order QCD corrections improve the accuracy of this analysis. We find typically $\Delta\sin 2\beta=\pm 0.11$ provided $B(\kpn)$ and $B(\klpn)$ are measured within $\pm 10\%$ accuracy. With decreasing uncertainty in $\Lambda_{\overline{MS}}$ and $m_c$ this error could be reduced to $\Delta\sin 2\beta < \pm 0.10$. The determination of $\sin 2\alpha$ and $\sin 2\gamma$ on the other hand is rather poor. However respectable determinations of the Wolfenstein parameter $\eta$ and of $\mid V_{td}\mid$ can be obtained.