Standard Model correlations between K -> \pi \nu \bar\nu decays and observables in B Physics

Including the recent preliminary results of BaBar and BELLE experiments, we update the currently allowed intervals for various CKM parameters: $\bar\rho$, $\bar\eta$, $\sin2\beta$, $\sin2\alpha$, $\sin^2\gamma$. We also update the SM prediction for the rates of the $K_L \to \pi^0 \nu \bar\nu$ and $K^+ \to \pi^+ \nu \bar\nu$ decays, their ratio $a_{\pi \nu \bar\nu}$, as well as for certain observables related to $B$ Physics like the CP asymmetries $a_{\psi K_S}$ in $B^0_d \to J/\psi K_S$ and $a_{\psi \phi}$ in $B^0_s \to J/\psi \phi$ or the mass differences $\Delta m_q$ ($q=d,s$) in the $B^0_q-\bar B^0_q$ systems. We investigate the correlations between them. The strongest correlations are between i) $a_{\pi \nu \bar\nu}$ and $a_{\psi K_S}$, ii) $\BR(K^+ \to \pi^+ \nu \bar\nu)$ and $\Delta m_d/\Delta m_s$ and iii) $\BR(K_L \to \pi^0 \nu \bar\nu)$ and $a_{\psi \phi}$. These correlations are likely to be violated in the presence of New Physics and therefore provide stringent tests of the Standard Model.