Stringent tests of constrained Minimal Flavor Violation through ΔF=2 transitions

New Physics contributions to ΔF=2 transitions in the simplest extensions of the Standard Model (SM), the models with constrained Minimal Flavor Violation (CMFV), are parametrized by a single variable S(v), the value of the real box diagram function that in CMFV is bounded from below by its SM value S ₀(x _t). With already very precise experimental values of ε _K, ΔM _d, ΔM _s and precise values of the CP-asymmetry S_ψK_S and of B̂_K entering the evaluation of ε _K, the future of CMFV in the ΔF=2 sector depends crucially on the values of {pipe}V _cb{pipe}, {pipe}V _ub{pipe}, γ, F_Bs √̂B_Bs and F_Bd √̂B_Bd. The ratio ξ of the latter two non-perturbative parameters, already rather precisely determined from lattice calculations, allows then together with ΔM _s/ΔM _d and S_ψ K_S to determine the range of the angle γ in the unitarity triangle independently of the value of S(v). Imposing in addition the constraints from {pipe}ε _K{pipe} and ΔM _d allows to determine the favorite CMFV values of {pipe}V _cb{pipe}, {pipe}V _ub{pipe}, F_Bs √̂B_Bs and F_Bd √̂B_Bd as functions of S(v) and γ. The {pipe}V _cb{pipe}⁴ dependence of ε _K allows to determine {pipe}V _cb{pipe} for a given S(v) and γ with a higher precision than it is presently possible using tree-level decays. The same applies to {pipe}V _ub{pipe}, {pipe}V _td{pipe} and {pipe}V _ts{pipe} that are automatically determined as functions of S(v) and γ. We derive correlations between F_Bs √̂B_Bs and F_Bd √̂B_Bd, {pipe}V _cb{pipe}, {pipe}V _ub{pipe} and γ that should be tested in the coming years. Typically F_Bs √̂B_Bs and F_Bd √̂B_Bd have to be lower than their present lattice values, while {pipe}V _cb{pipe} has to be higher than its tree-level determination, with a significance depending on the evolution of their errors. Within the SM this would imply values for B(K⁺ → π⁺νν̄ and B(K_L → π⁰νν̄) that are typically larger by (15-20) % than those presently quoted in the literature. The region in the space of these three parameters allowed by CMFV indicates tensions in this class of models and hints for the presence of new sources of flavor violation and/or new local operators in ΔF=2 data that are strongly suppressed in these models. As a byproduct we propose to reduce the present uncertainty in the charm contribution to ε _K by using the experimental value of ΔM _K.