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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 $ε_K$, $ΔM_{d,s}$ and $K \to πν\overlineν$ in models with minimal flavour violation (MFV), allowing for arbitrary signs of the generalized Inami--Lim functions $F_{tt}$ and $X$ relevant for $(ε_K,ΔM_{d,s})$ and $K \to πν\overlineν$, 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 $(ε_K, ΔM_{d,s})$ imply an absolute lower bound on the $B_d\toψK_{S}$ CP asymmetry $a_{ψK_S}$ 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 $Br(K^+\toπ^+ν\overlineν)$ and $a_{ψK_S}$ only two values for $Br(K_{L}\toπ^0ν\overlineν)$, 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 $Br(K_{L}\toπ^0ν\overlineν)$ as functions of $Br(K^+\toπ^+ν\overlineν)$. Using the present experimental upper bounds on $Br(K^+\toπ^+ν\overlineν)$ and $|V_{ub}/V_{cb}|$, we obtain the absolute upper bound $Br(K_{L}\toπ^0ν\overlineν)< 7.1 \cdot 10^{-10}$ (90% C.L.).