Light states in real 3HDMs with spontaneous CP violation and softly broken symmetries

Abstract Scalar sectors with several Higgs doublets, a CP invariant potential, and a CP violating vacuum, possess a mass spectrum in which, unexpectedly, new scalars cannot have masses much larger than the electroweak scale once perturbativity requirements are imposed on the quartic couplings of the Higgs potential and despite the presence of free quadratic (mass) terms that can be arbitrarily large. The minimal model in which this behavior is manifest involves 3 Higgs doublets. We analyze and illustrate in detail that kind of scenario including an additional simplifying assumption: the quartic part of the potential is shaped by some discrete symmetry. Besides analytic results, a numerical analysis is presented together with some phenomenological consequences derived only from the properties of the scalar sector alone.