We show that in theories that reduce, at the Fermi scale, to an extension of the standard model with two doublets, there can be additional dimension five operators giving rise to neutrino masses. In particular there exists a singlet operator which can not generate neutrino masses at tree level but generates them through operator mixing. Under the assumption that only this operator appears at tree level we calculate the neutrino mass matrix. It has the Zee mass matrix structure and leads naturally to bimaximal mixing. However, the maximal mixing prediction for solar neutrinos is very sharp even when higher order corrections are considered. To allow for deviations from maximal mixing a fine tuning is needed in the neutrino mass matrix parameters. However, this fine tuning relates the departure from maximal mixing in solar neutrino oscillations with the neutrinoless double beta decay rate.

11 pages, 1 figure, revtex