  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.
