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 flow 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 $m_\nu$. In large part of the parameter space this effect dominates over the standard one-loop contributions to $m_\nu$; we estimate 1 MeV $\lsim m_\nu \lsim 1 GeV$. Laboratory, cosmological and astrophysical constraints imply $m_\nu \lsim 100 eV$. To be phenomenologically viable, these models must be supplemented with some additional mechanism to ensure approximate alignment and to suppress $m_\nu$.