Bounds on the unitarity triangle,sin2βandK→πνν¯decays in models with minimal flavor violation

We present a general discussion of the unitarity triangle from ${\varepsilon }_K,$ $\Delta M_{d,s}$ and $\overrightarrow{K}\pi \nu \overline{\nu }$ in models with minimal flavor violation (MFV), allowing for arbitrary signs of the generalized Inami-Lim functions $F_{\mathrm{tt}}$ and X relevant for $({\varepsilon }_K,\Delta M_{d,s})$ and $\overrightarrow{K}\pi \nu \overline{\nu },$ respectively. In the models in which $F_{\mathrm{tt}}$ has a sign opposite to the one in the standard model, i.e. $F_{\mathrm{tt}}<0,$ the data for $({\varepsilon }_K,\Delta M_{d,s})$ imply an absolute lower bound on the $B_d\to \psi K_S$ $\mathrm{CP}$ asymmetry $a_{\psi K_S}$ of 0.69, which is substantially stronger than 0.42 arising in the case of $F_{\mathrm{tt}}>0.\mathrm{}$ 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 ${Br(K}_{\mathrm{L}}\to {\pi }^0\nu \overline{\nu }),$ 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 ${Br(K}_{\mathrm{L}}\to {\pi }^0\nu \overline{\nu })$ as functions of ${Br(K}^{+}\to {\pi }^{+}\nu \overline{\nu }).\mathrm{}$ Using the present experimental upper bounds on ${Br(K}^{+}\to {\pi }^{+}\nu \overline{\nu })$ and $|V_{\mathrm{ub}}{/V}_{\mathrm{cb}}|,$ we obtain the absolute upper bound