have taken $Y_\nu = 0.3$ for $(3,1)$ case and for $(3,3)$ case we have taken $Y^{ai}_\nu = 0.3$; $a=i=1,2,3$ while all the off-diagonal entries are taken to be zero. The rest of the parameters are kept same as described previously for the $(3,1)$ Majoron seesaw case. Thus, Majoron models phenomenologically consistent with the neutrino oscillation data, can also have a completely stable vacuum all the way up to Planck scale.

Before concluding we should note the cosmological advantages of the Majoron completion. The first is that it can also provide a dark matter candidate, namely the Majoron [@Berezinsky:1993fm], providing an alternative to the $\Lambda$CDM paradigm. The Majoron is assumed to get mass from gravitational effects that explicitly violate the global lepton number [@coleman:1988tj]. Assuming that its mass lies in the keV range one can show that it can be a viable warm dark matter candidate, decaying to neutrinos, with a strength proportional to their tiny mass [@Schechter:1981cv]. Hence, it is naturally long-lived on a cosmological scale, as required, with lifetime $\tau_J$ larger than the age of the Universe $t_0 =13.8\,\mathrm{Gyr}\simeq 4\times10^{17}\,\mathrm{s}$. Indeed, the massive Majoron dark matter scenario has been shown to be consistent with cosmic microwave background data for 