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: sinθij ≳ m(li−)m(lj−) (i < j) and unusual order of magnitude predictions for netrino mass ratios: m(νe)m(νμ) ∼ sin2 θ12; m(νi)m(ντ) ∼ 10−7 sin2 θi3 (i = 1, 2). Bounds from laboratory experiments exclude mντ ≳ 3 MeV and cosmological constraints exclude mντ ≳ 100 eV. Neither the solar nor the atmospheric neutrino problems are likely to be solved by νμ − νe oscillations. These conclusions can be evaded if holomorphy plays an important role in the lepton Yukawa couplings.