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 θij > ∼ m(l − i )/m(l − j ) (i < j) and unusual order of magnitude predictions for neutrino mass ratios: m(νe)/m(νμ) ∼ sin 2 θ12; m(νi)/m(ντ ) ∼ 10 −7 sin θi3 (i = 1, 2). Bounds from laboratory experiments exclude mντ > ∼ 3 MeV and the cosmological constraint that excludes mντ > ∼ 100 eV is not evaded. 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.