Generations of Higgs Bosons in Supersymmetric Models

We examine extensions of the MSSM with more than one generation of Higgs bosons. If one assumes that a symmetry eliminates the tree-level FCNC, then the extra scalar bosons do not acquire VEVs, do not couple to fermions and do not mix with the ordinary Higgs bosons; the lightest is absolutely stable. The two lightest neutral scalars, $\phi_S$, and $\phi_P$, are degenerate in mass, and the mass difference between those and the lightest charged scalar, $\phi_+$, is calculated. For most of the parameter space, the charged scalar is between a hundred MeV and a few GeV heavier than the neutral scalars. The $\phi_+$ will decay at the vertex; the signature for this decay will be like that of a chargino with a nearly degenerate undetected neutralino. Next, the possibility that the symmetry that eliminates FCNC is a flavor symmetry is discussed. In an example, the U(2) model, tree-level FCNC processes can be calculated in terms of quark masses. The strongest constraint on this model is from $D-\bar{D}$ mixing, which should be within an order of magnitude of the current bound.