stability it is quite desirable to further extend or embed the type-I seesaw. A natural way to do this is to include a candidate for dark matter intimately connected with the neutrino mass generation mechanism. In the next two sections, we consider two such scenarios and analyze the vacuum stability properties for both cases.

The Majoron completion of the Type-I Seesaw  {#sec:maj}
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We now consider the type-I seesaw extensions of the Standard Model, in which lepton number is promoted to a spontaneously broken symmetry within the gauge framework [@Chikashige:1980ui; @Schechter:1981cv]. In addition to the right-handed neutrinos $\nu_{R}$ we add a complex singlet $\sigma$ carrying two units of lepton number.

The relevant Lagrangian is given by $$\begin{aligned}
 \mathcal{L}=-\sum_{a,i} Y_{\nu}^{ai}\bar{\ell}_{L}^{a}\tilde{H}\nu_{R_i} - \frac{1}{2}\sum_{i,j} Y^{ij}_{R}\sigma\overline{\nu_{R_i}^{c}}\nu_{R_j}+\text{H.c.}\end{aligned}$$ The resulting neutrino mass matrices in $\nu_{L}$ and $\nu_{R}$ basis is given by $$\begin{aligned}
  \mathcal{M}_{\nu}=
 \begin{pmatrix}
 0 & \frac{Y_{\nu}v_{H}}{\sqrt{2}} \\
 \frac{Y_{\nu}^{T}v_{H}}{\sqrt{2}} & \frac{Y_{R}v_{\sigma}}{\sqrt{2}} \\
 \end{pmatrix}\end{aligned}$$ The effective light neutrino mass obtained by perturbative diagonalization of the above mass matrix is of the form $$\begin{aligned}
 m_{\nu}\simeq Y_{\nu}Y_{R}^{-1}Y_{\nu}^{T}\frac{v_{H}^{2}}{\sqrt{2}v_{\sigma}}
 \label{seesaw formula in type-I seesaw with Majoron}\end{aligned}$$

In the presence of the complex scalar singlet $\sigma$, the most general Higgs potential which is capable of driving electroweak and lepton number symmetry breaking 