Bounds on the Unitarity Triangle, $\sin 2\beta$ and $K\to \pi \nu \bar{\nu }$ Decays in Models with Minimal Flavour Violation

Abstract

We present a general discussion of the unitarity triangle from ${ϵ}_K$, $\Delta M_{d,s}$ and $K\to \pi \nu \overline{\nu }$ in models with minimal flavour violation (MFV), allowing for arbitrary signs of the generalized Inami--Lim functions $F_{tt}$ and $X$ relevant for $({ϵ}_K,\Delta M_{d,s})$ and $K\to \pi \nu \overline{\nu }$, respectively. In the models in which $F_{tt}$ has a sign opposite to the one in the Standard Model, i.e. $F_{tt}<0$, the data for $({ϵ}_K,\Delta M_{d,s})$ imply an absolute lower bound on the $B_d\to \psi K_S$ CP asymmetry $a_{\psi K_S}$ of 0.69, which is substantially stronger than 0.42 arising in the case of $F_{tt}>0$. An important finding of this paper is the observation that for given $Br(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ and $a_{\psi K_S}$ only two values for $Br(K_L\to {\pi }^0\nu \overline{\nu })$, corresponding to the two signs of $X$, are possible in the full class of MFV models, independently of any new parameters arising in these models. This provides a powerful test for this class of models. Moreover, we derive absolute lower and upper bounds on $Br(K_L\to {\pi }^0\nu \overline{\nu })$ as functions of $Br(K^{+}\to {\pi }^{+}\nu \overline{\nu })$. Using the present experimental upper bounds on $Br(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ and $|V_{ub}/V_{cb}|$, we obtain the absolute upper bound $Br(K_L\to {\pi }^0\nu \overline{\nu })<7.1\cdot 10^{-10}$ (90% C.L.).

Comments: 25 pages, LaTeX, 4 figures. Few comments and one figure in view of new B-factory results added, conclusions unchanged