Renormalization-group-induced neutrino mass in supersymmetry without R parity

We study supersymmetric models without R parity and with universal soft supersymmetry-breaking terms. We show that as a result of the renormalization group Row of the parameters, a misalignment between the directions in field space of the down-type Higgs vacuum expectation value v(d) and of the mu term is always generated. This misalignment induces a mixing between the neutrinos and the neutralinos, resulting in one massive neutrino. By means of a simple approximate analytical expression, we study the dependence on the different parameters that contribute to the misalignment and to m(v). In large part of the parameter space this effect dominates over the standard one-loop contributions to m(v); we estimate 1 MeV less than or similar to m(v) less than or similar to 1 GeV. Laboratory, cosmological, and astrophysical constraints imply m(v) less than or similar to 100 eV. To be phenomenologically viable, these models must be supplemented with some additional mechanism to ensure approximate alignment and to suppress m(v).