Shedding Light on the "Dark Side" of $B^0_d$--$\bar{B^{0}_{d}}$ Mixing through $B_{d} \to \pi^{+}\pi^{-}$, $K \to \pi\nu\bar{\nu}$ and $B_{d,s} \to \mu^{+}\mu^{-}$

In a wide class of NP models, which can be motivated through generic arguments and within SUSY, we obtain large contributions to $B^0_d$--$\bar B^0_d$ mixing, but not to $\Delta B=1$ processes. If we assume such a scenario, the solutions $\phi_d\sim 47^\circ\lor 133^\circ$ for the $B^0_d$--$\bar B^0_d$ mixing phase implied by $A_{CP}^{mix}(B_d\to J/\psi K_S)$ cannot be converted directly into a constraint in the $\rho$--$\eta$ plane. However, we may complement $\phi_d$ with $|V_{ub}/V_{cb}|$ and the recently measured CP asymmetries in $B_d\to\pi^+ \pi^-$ to determine the unitarity triangle, with its angles $\alpha$, $\beta$ and $\gamma$. To this end, we have also to control penguin effects, which we do by means of the $B_d\to\pi^\mp K^\pm$ branching ratio. Interestingly, the present data show a perfectly consistent picture not only for the ``standard'' solution of $\phi_d\sim 47^\circ$, but also for $\phi_d\sim 133^\circ$. In the latter case, the preferred region for the apex of the unitarity triangle is in the second quadrant, allowing us to accommodate conveniently $\gamma>90^\circ$, which is also favoured by other non-leptonic B decays such as $B\to\pi K$. Moreover, also the prediction for BR$(K^+\to\pi^+\nu\bar\nu})$ can be brought to better agreement with experiment. Further strategies to explore this scenario with the help of $B_{d,s}\to\mu^+\mu^-$ decays are discussed as well.