$\sin 2β$ from $K\toπν\barν$

Abstract

We point out that the measurement of just the two branching fractions \linebreak $B(K^+\toπ^+ν\barν)$ and $B(K_L\toπ^0ν\barν)$ can in a theoretically clean manner determine $\sin 2β$ almost independently of $m_t$ and $V_{cb}$. This allows to obtain an interesting relation between the CP asymmetry $A_{CP}(ψK_S)$ 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 $Δ\sin 2β=\pm 0.11$ provided $B(\kpn)$ and $B(\klpn)$ are measured within $\pm 10\%$ accuracy. With decreasing uncertainty in $Λ_{\overline{MS}}$ and $m_c$ this error could be reduced to $Δ\sin 2β< \pm 0.10$. The determination of $\sin 2α$ and $\sin 2γ$ on the other hand is rather poor. However respectable determinations of the Wolfenstein parameter $η$ and of $\mid V_{td}\mid$ can be obtained.