$\sin 2\beta$ 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 $B(\text{kpn})$ and $B(\text{klpn})$ 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.

Comments: 9 pages, MPI-PhT/94-19