Neutrino Masses and Mixing in Supersymmetric Models without $R$ Parity

We study neutrino masses and mixing in Supersymmetric Models without $R$ parity and with generic soft Supersymmetry breaking terms. Neutrinos acquire mass from various sources: tree level neutrino--neutralino mixing, loop effects and non--renormalizable operators. Abelian horizontal symmetries (invoked to explain the smallness and hierarchy in quark parameters) replace $R$ parity in suppressing neutrino masses. We find lower bounds on the mixing angles: $\sin\theta_{ij} \gsim m(\ell_i^-)/m(\ell_j^-)$ ($i<j$) and unusual order of magnitude predictions for neutrino mass ratios: $m(\nu_e)/m(\nu_\mu)\sim\sin^2\theta_{12}$; $m(\nu_i)/m(\nu_\tau)\sim 10^{-7} \sin^2\theta_{i3}$ ($i=1,2$). Bounds from laboratory experiments exclude $m_{\nu_\tau} \gsim 3\ MeV$ and the cosmological constraint that excludes $m_{\nu_\tau} \gsim 100\ eV$ is not evaded. Neither the solar nor the atmospheric neutrino problems are likely to be solved by $\nu_\mu-\nu_e$ oscillations. These conclusions can be evaded if holomorphy plays an important role in the lepton Yukawa couplings.