CP-odd invariants in models with several Higgs doublets

We present CP-odd Higgs-basis invariants, which can be used to signal explicit CP violation in a multi-Higgs system, written in an arbitrary Higgs basis. These invariants are constructed in terms of the parameters of the Lagrangian and are independent of the gauge symmetry breaking. In particular, they do not vanish at high energies and can thus be relevant for baryogenesis. It is shown through specific examples how these CP-odd invariants can also be useful to determine the character of CP breaking (i.e., whether it is hard or soft CP breaking) in a given Higgs Lagrangian. We analyse in detail the cases of two and three Higgs doublets.