Two-Higgs-Doublet Models with CP violation

We consider the Two-Higgs-Doublet Model (2HDM) and determine the range of parameters for which CP violation and Flavor Changing Neutral Current effects are naturally small. It corresponds to small values of the mass parameter m_{12}^2, describing soft (\phi_1,\phi_2) mixing in the potential. We discuss how, in this approach, some Higgs bosons can be heavy, with mass of the order of 1 TeV. The possibility that at the Tevatron, LHC and an e^+e^- Linear Collider, only one Higgs boson will be found, with properties indistinguishable from those in the Standard Model (SM), we define as the SM-like scenario. While this scenario can be obtained with large \mu^2 \sim \Re m_{12}^2 parameter, in which case there is decoupling, we here discuss the opposite case of small \mu^2, without decoupling.