$ΔM_s/ΔM_d$, $\sin 2β$ and the angle $γ$ in the Presence of New $ΔF=2$ Operators

We present formulae for the mass differences $\Delta M_d$ and $\Delta M_s$ in the $\bar B_{d,s}B_{d,s}$ 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 (they are equal only if the SM $(V-A)\otimes(V-A)$ operators dominate). We present simple strategies involving the ratio $\Delta M_s/\Delta M_d$, $\sin 2\beta$ 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 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. As in this scenario the lower bound on $\sin2\beta$ following from $\epsilon$ and $\Delta M_d$ is identical to the SM one ($\sin2\beta>0.5 $), the future low values of the CP asymmetry $a_{\psi K_S}$ may further constrain the stop mixing or rule out this scenario altogether.