  A new idea for neutrino mass was proposed recently, where its smallness is
not due to the seesaw mechanism, i.e. not inversely proportional to some large
mass scale. It comes from a one-loop mechanism with dark matter in the loop
consisting of singlet Majorana fermions $N_i$ with masses of order 10 keV and
neutrino masses are scaled down from them by factors of about $10^{-5}$. We
discuss how this model may be implemented with the non-Abelian discrete
symmetry $A_4$ for neutrino mixing, and consider the phenomenology of $N_i$ as
well as the extra scalar doublet $(\eta^+,\eta^0)$.
