Neutrino masses and mixing with non-anomalous abelian flavor symmetries

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The experimental data on atmospheric and solar neutrinos are used to test the framework of non-anomalous Abelian horizontal gauge symmetries with only three light active neutrinos. We assume that the hierarchy in mass-squared splittings is not accidental and that the small breaking parameters are not considerably larger than 0.2. We find that the small angle MSW solution of the solar neutrino problem can only be accommodated if the $\nu_\mu-\nu_\tau$ mass hierarchy depends on the charges of at least three sterile neutrinos. The large angle MSW solution can be accommodated in simpler models if $\nu_e$ and $\nu_\mu$ form a pseudo-Dirac neutrino, but it is difficult to induce large enough deviation from maximal mixing. The vacuum oscillation solution can be accommodated rather simply. We conclude that it is possible to accommodate the neutrino parameters in the framework of Abelian horizontal symmetries, but it seems that these parameters by themselves will not provide convincing evidence for this framework.