is given by [@Joshipura:1992hp] $$\begin{aligned}
  V(\sigma,H)=-\mu_{H}^{2}H^{\dagger}H-\mu_{\sigma}^{2}\sigma^{\dagger}\sigma+\lambda (H^{\dagger}H)^{2}+\lambda_{\sigma}(\sigma^{\dagger}\sigma)^{2}
  +\lambda_{H\sigma}(H^{\dagger}H)(\sigma^{\dagger}\sigma).
  \label{eq:maj-pot}\end{aligned}$$

This potential is bounded from below if $\lambda_{\sigma}$, $\lambda_{H}$ and $\lambda_{H\sigma}+2\sqrt{\lambda_{\sigma}\lambda_{H}}$ are all positive. In addition to the standard gauge invariance, the theory is also invariant under lepton number. The above potential can develop a minimum for non-zero vacuum expectation values of both H and $\sigma$ if $\lambda$, $\lambda_{\sigma}$ and $4\lambda\lambda_{\sigma}-\lambda_{H\sigma}^{2}$ are all positive. The vevs break both the electroweak and lepton number symmetries, three of the degrees of freedom in $H$ being eaten by the massive gauge bosons, while the imaginary part of the $\sigma$ corresponds to the Majoron $J$. The real parts of $H$ and $\sigma$ will mix with each other to give two CP-even mass eigenstates $h_{1}$ and $h_{2}$. The lighter of these is identified with [[Standard Model ]{}]{}Higgs boson.

Vacuum stability in type-I seesaw with Majoron {#sec:result-maj .unnumbered}
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![ []{data-label="RG-31aJ-Running"}](plots/HighScaleMajoron01.pdf "fig:"){width="49.00000%"} ![ []{data-label="RG-31aJ-Running"}](plots/HighScaleMajoron03.pdf "fig:"){width="49.00000%"} ![ []{data-label="RG-31aJ-Running"}](plots/HighScaleMajoron04.pdf "fig:"){width="49.00000%"} ![ []{data-label="RG-31aJ-Running"}](plots/HighScaleMajoron05.pdf "fig:"){width="49.00000%"}

.

For simplicity we take again the simplest dark extension of the type-I seesaw mechanism based on the Majoron extension of the (3,1) scheme considered above. From Eq. \[seesaw formula in type-I seesaw with Majoron\] one finds that $$\begin{aligned}
 Y_{R}\approx\frac{10^{14}\,\text{GeV}}{v_{\sigma}}.\end{aligned}$$ for a Yukawa coupling $Y_{\nu}$ of 