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We consider the quark sector of theories containing three scalar SU(2)_Ldoublets in the triplet representation of A4 (or of S4) and three generationsof quarks in arbitrary A4 (or S4) representations. We show that, for allpossible choices of quark field representations and for all possible alignmentsof the Higgs vacuum expectation values that can constitute global minima of thescalar potential, it is not possible to obtain simultaneously non-vanishingquark masses and a non-vanishing CP-violating phase in the CKM quark mixingmatrix. As a result, in this minimal form, models with three scalar fields inthe triplet representation of A4 or of S4 cannot be extended to the quarksector in a way consistent with experiment.

Models with three Higgs doublets in the triplet representations of A4 or S4