Bilinear R-parity Violation and Small Neutrino Masses: a Self-consistent   Framework
  We study extensions of supersymmetric models without R-parity which include an anomalous U(1)_H horizontal symmetry. Bilinear R-parity violating terms induce a neutrino mass at tree level of approximately $(\theta^2)^\delta$ eV where $\theta\approx 0.22$ is the U(1)_H breaking parameter and $\delta$ is an integer number that depends on the horizontal charges of the leptons. For $\delta=1$ a unique self-consistent model arises in which i) all the superpotential trilinear R-parity violating couplings are forbidden by holomorphy; ii) the tree level neutrino mass falls in the range suggested by the atmospheric neutrino problem; iii) radiative contributions to neutrino masses are strongly suppressed resulting in a squared solar mass difference of few 10^{-8} eV^2 which only allows for the LOW (or quasi-vacuum) solution to the solar neutrino problem; iv) the neutrino mixing angles are not suppressed by powers of $\theta$ and can naturally be large. 
J. M. Mira, E. Nardi, D. A. Restrepo, and J. W. F. Valle