Assuming that the lepton, quark and Higgs fields belong to the three-dimensional reducible representation of the permutation group S_3, we suggest a minimal S_3 invariant extension of the standard model. We find that in the leptonic sector, the exact S_3 x Z_2 symmetry, which allows 6 real independent parameters with two CP-violating phases, is consistent with experimental data and predicts a maximal mixing of the atmospheric neutrinos, and that the third neutrino is the lightest neutrino. With the exact S_3 only, there are 10 real parameters and five phases in the quark sector. A set of values of these parameters that are consistent with experimental observations is given.Comment: 15 pages, LaTeX, 1 figure, erros correcte

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The idea of Nelson and Strassler to obtain a power law suppression of parameters by a superconformal force is applied to understand the smallness of the parameter and neutrino masses in R-parity violating supersymmetric standard models. We find that the low-energy sector should contain at least another pair of Higgs doublets, and that a suppression of ՇO(10 Ϫ13 ) for the parameter and neutrino masses can be achieved generically. The superpotential of the low-energy sector happens to possess an anomaly-free discrete R symmetry, either R 3 or R 6 , which naturally suppresses certain lepton-flavor violating processes, the neutrinoless double beta decays and also the electron electric dipole moment. We expect that the escape energy of the superconformal sector is ՇO(10) TeV so that this sector will be observable at the CERN Large Hadron Collider ͑LHC͒. Our models can accommodate a large mixing among neutrinos and give the same upper bound of the lightest Higgs boson mass as the minimal supersymmetric standard model.

We consider finite groups of small order for family symmetry. It is found that the binary dihedral group Q 6 , along with the assumption that the Higgs sector is of type II, predicts mass matrix of a nearest neighbor interaction type for quarks and leptons. We present a supersymmetric model based on Q 6 with spontaneously induced CP phases. The quark sector contains 8 real parameters with one independent phase to describe the quark masses and their mixing. Predictions in the jV ub j ÿ , jV ub j ÿ sin2 1 , and jV ub j ÿ jV td =V ts j planes are given. The lepton sector contains also 9 parameters. A normal as well as an inverted spectrum of neutrino masses is possible, and we compute V e3 . We find that jV e3 j 2 > 10 ÿ4 in the case of a normal spectrum, and jV e3 j 2 > 8 10 ÿ4 in the case of an inverted spectrum. It is also found that Q 6 symmetry forbids all Baryon number violating terms of d 4, and the contributions to EDMs from the A terms vanish in this model.

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