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

Abstract

We present formulae for the mass differences $\Delta M_d$ and $\Delta M_s$ in the \BBds systems and for the CP violation parameter $ϵ$ 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 $ϵ$ are parameterized by three {\it real} functions $F_{tt}^s$, $F_{tt}^d$ and $F_{tt}^{ϵ}$, respectively ($F_{tt}^s=F_{tt}^d=F_{tt}^{ϵ}$ holds only if the Standard Model $(V-A)\otimes (V-A)$ operators dominate). We present simple strategies involving the ratio $\Delta M_s/\Delta M_d$, $\sin 2\beta$ 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 $F_{tt}^i$ 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, $F_{tt}^s$ 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 $F_{tt}^s$ can be violated which allows to exclude large mixing of stops. In this scenario the range of $\sin 2\beta$ following from $ϵ$ 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.

Comments: 39 pages, 13 figures, uses axodraw.sty. V2: new BaBar and Belle results on \sin2\beta incorporated; some explanations of the details of calculation of \tan^4\bar\beta terms added