$\Delta M_s/\Delta M_d$, $\sin 2\beta$ and the angle $\gamma$ in the Presence of New $\Delta F=2$ Operators

We present formulae for the mass differences $\Delta M_d$ and $\Delta M_s$ in the \BBds systems and for the CP violation parameter $\epsilon$ which are valid in minimal flavour violation models giving rise to new four-fermion $\Delta F=2$ operators. Short distance contributions to $\Delta M_s$, $\Delta M_d$ and $\epsilon$ are parameterized by three {\it real} functions $F^s_{tt}$, $F^d_{tt}$ and $F^\epsilon_{tt}$, respectively ($F^s_{tt} = F^d_{tt} = F^\epsilon_{tt}$ holds only if the Standard Model $(V-A) \otimes (V-A)$ operators dominate). We present simple strategies involving the ratio $\Delta M_s/\Delta M_d$, $\sin2\beta$ and $\gamma$ that allow to search for the effects of the new operators. We point out that their sizable contributions to the ratio $\Delta M_s/\Delta M_d$ would in principle allow $\gamma$ to be larger than $90^\circ$. Constraints on the functions $F^i_{tt}$ imposed by the present (and future) experimental data are also discussed. As an example we show that for large $\tan\bar\beta\equiv v_2/v_1$ and $H^+$ not too heavy, $F^s_{tt}$ in the MSSM with heavy sparticles can be substantially smaller than in the SM due the charged Higgs box contributions and in particular due to the growing like $\tan^4\bar\beta$ contribution of the double penguin diagrams involving neutral Higgs boson exchanges. As a result the bounds on the function $F^s_{tt}$ can be violated which allows to exclude large mixing of stops. In this scenario the range of $\sin2\beta$ following from $\epsilon$ and $\Delta M_d$ is identical to the SM ones ($0.5<\sin 2\beta<0.8$). On the other hand $\gamma$ following from $\Delta M_s/\Delta M_d$ is lower.