  We study the 1-loop renormalization group equation running in the simplest
singlet Majoron model constructed by us earlier to accommodate the dark
radiation and dark matter content in the universe. A comprehensive numerical
study was performed to explore the whole model parameter space. A smaller
effective number of neutrinos $\triangle N_{eff}\sim 0.05$, or a Majoron
decoupling temperature higher than the charm quark mass, is preferred. We found
that a heavy scalar dark matter, $\rho$, of mass $1.5-4$ TeV is required by the
stability of the scalar potential and an operational type-I see-saw mechanism
for neutrino masses. A neutral scalar, $S$, of mass in the $10-100$ GeV range
and its mixing with the standard model Higgs as large as $0.1$ is also
predicted. The dominant decay modes are $S$ into $b\bar{b}$ and/or
$\omega\omega$. A sensitive search will come from rare $Z$ decays via the chain
$Z\rightarrow S+ f\bar{f}$, where $f$ is a Standard Model fermion, followed by
$S$ into a pair of Majoron and/or b-quarks. The interesting consequences of
dark matter bound state due to the sizable $S\rho \rho$-coupling are discussed
as well. In particular, shower-like events with an apparent neutrino energy at
$M_\rho$ could contribute to the observed effective neutrino flux in
underground neutrino detectors such as IceCube.
