Atmospheric and solar neutrino masses from horizontalU(1) symmetry

We study the neutrino mass matrix in supersymmetric models in which the quark and charged lepton mass hierarchies and also the suppression of baryon or lepton number violating, i.e., R-parity violating, couplings are all explained by horizontal ${U\left(1\right)\mathrm{}}_X$ symmetry. It is found that the neutrino masses and mixing angles suggested by recent atmospheric and solar neutrino experiments arise naturally in this framework which fits in best with gauge-mediated supersymmetry breaking with large $\tan \beta .$ This framework highly favors the small angle Mikheyev-Smirnov-Wolfenstein oscillation of solar neutrinos, and determines the order of magnitude of all the neutrino mixing angles and mass hierarchies.