CP-symmetry of order 4 and its consequences

Extended Higgs sectors offer rich opportunities for various forms of CP-violation. Here, we describe a new form of CP-conservation and discuss its consequences. We give a concrete example of a three-Higgs-doublet model dubbed CP4-3HDM with a CP-symmetry of order 4 and no other other accidental symmetries. If the vacuum conserves this symmetry, the model is CP-conserving with pairwise mass-degenerate extra neutral Higgs bosons. These fields cannot be classified as CP-even or CP-odd but they can be combined into complex physical fields which are CP-half-odd, that is, they pick up the $i$ factor upon CP transformation. These CP-half-odd scalars can be Yukawa-coupled to the fermion bilinears in a CP-conserving way. We discuss fundamental and phenomenological features of the model, and stress a peculiar clash between the CP-symmetry and any convention for the particle-antiparticle assignment.