dominate the RG behavior of $\lambda$ as shown in Fig. \[RG-31-Running\].

![ []{data-label="RG-31-Running"}](plots/HighScale03.pdf "fig:"){width="49.00000%"} ![ []{data-label="RG-31-Running"}](plots/HighScale05.pdf "fig:"){width="49.00000%"}

Fig. \[RG-31-Running\] illustrates the effect of the new neutrino Yukawa coupling $Y_{\nu}$ on various other couplings. One sees that the RG running of $\lambda$ can be completely dominated by the Yukawa coupling $Y_{\nu}$, particularly for $Y_{\nu} \approx \mathcal{O}(1)$. For illustration we have taken two representative values of Yukawa couplings $Y_{\nu}=0.3,\,0.5$. One sees from Fig. \[RG-31-Running\] that for $Y_{\nu} = 0.3,\,0.5$, the Higgs quartic coupling $\lambda$ becomes negative at $\mu=10^{9},\,10^{8}\,\text{GeV}$ respectively i.e. faster than the [[Standard Model ]{}]{}case, Fig. \[RG-SM-Running\].

Some important conclusions can be drawn from this simple (3,1) seesaw scenario.

-   The problem of Higgs vacuum instability becomes more acute in a type-I seesaw completion of the Standard Model. As seen from the example of the minimal (3,1) seesaw mechanism illustrated in Fig. \[RG-31-Running\], the quartic coupling $\lambda$ runs faster to negative values. This was expected, since the addition of new fermions tends to destabilize the Higgs vacuum.

-   More importantly, the quartic coupling becomes negative much before the mass scale characterizing the right-handed neutrinos, making the theory inconsistent. Indeed, for $Y_\nu \gtrsim \mathcal{O} (10^{-2})$ and $m_\nu \sim 0.1$ eV, one has 