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We consider the most general supersymmetric model with minimal particlecontent and an additional discrete Z_3 symmetry (instead of R-parity), whichallows lepton number violating terms and results in non-zero Majorana neutrinomasses. We investigate whether the currently measured values for lepton massesand mixing can be reproduced. We set up a framework in which Lagrangianparameters can be initialised without recourse to assumptions concerningtrilinear or bilinear superpotential terms, CP-conservation orintergenerational mixing and analyse in detail the one loop corrections to theneutrino masses. We present scenarios in which the experimental data arereproduced and show the effect varying lepton number violating couplings has onthe predicted atmospheric and solar mass^2 differences. We find that withbilinear lepton number violating couplings in the superpotential of the order 1MeV the atmospheric mass scale can be reproduced. Certain trilinearsuperpotential couplings, usually, of the order of the electron Yukawa couplingcan give rise to either atmospheric or solar mass scales and bilinearsupersymmetry breaking terms of the order 0.1 GeV^2 can set the solar massscale. Further details of our calculation, Lagrangian, Feynman rules andrelevant generic loop diagrams, are presented in three Appendices.Comment: 48 pages, 7 figures, v2 references added, typos corrected, published  versio

Neutrino masses in the lepton number violating MSSM