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It has been known since decades that imposing a symmetry group G on thescalar sector of multi-Higgs-doublet models has consequences for CP-violation.In all examples of two- and three-Higgs-doublet models equipped withsymmetries, one observes the following intriguing property: if G preventsexplicit CP-violation (CPV), at least in the neutral Higgs sector, then it alsoprevents spontaneous CPV, and if G allows explicit CPV, then it allows forspontaneous CPV. One is led to conjecture that this is a general phenomenon. Inthis paper, we prove this conjecture for any rephasing symmetry group G and anynumber of doublets.

Group-theoretic restrictions on generation of CP-violation in multi-Higgs-doublet models