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 θ_ij ≳ m(l_i^-)/m(l_j^-) (i < j) and unusual order of magnitude predictions for neutrino mass ratios: Hi(ν_e)/m(ν_μ) ∼ sin² θ₁₂; m(ν_i)/m(ν_τ) ∼ 10^-7 sin² θ_i3 (i = 1, 2). Bounds from laboratory experiments exclude m_ντ ≳ 3 MeV and cosmological constraints exclude m_ντ ≳ 100 eV. Neither the solar nor the atmospheric neutrino problems are likely to be solved by ν_μ - ν_e oscillations. These conclusions can be evaded if holomorphy plays an important role in the lepton Yukawa couplings.