Bounds on the Unitarity Triangle, $\sin 2β$ and $K\toπν\barν$ Decays in Models with Minimal Flavour Violation

We present a general discussion of the unitarity triangle from $\epsilon_K$, $\Delta M_{d,s}$ and $K\to\pi\nu\bar\nu$ in models with minimal flavour violation (MFV), allowing for arbitrary signs of the generalized Inami-Lim functions $F_{tt}$ and $X$ relevant for $(\epsilon_K,\Delta M_{d,s})$ and $K\to\pi\nu\bar\nu$, respectively. The models in which $F_{tt}$ has a sign opposite to the one in the Standard Model, i.e. $F_{tt}0$. We find absolute lower and upper bounds on the branching ratio $Br(K_L\to\pi^0\nu\bar\nu)$ as functions of $Br(K^+\to\pi^+\nu\bar\nu)$. Moreover, we point out that for given $Br(K^+\to\pi^+\nu\bar\nu)$ and $a_{\psi K_S}$ only two values for $Br(K_L\to\pi^0\nu\bar\nu)$, 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. The present upper bounds on $Br(K^+\to\pi^+\nu\bar\nu)$ and $|V_{ub}/V_{cb}|$ imply the absolute upper bound $Br(K_L\to\pi^0\nu\bar\nu)< 7.1 \cdot 10^{-10}$ (90% C.L.).