Fate of the scalar quartic couplings in the inert models

In this article we consider three inert models, namely the inert singlet model (ISM), inert triplet model (ITM), and inert doublet model (IDM) as beyond Standard Model scenarios, and look into the running of the scalar quartic couplings at one- and two-loop levels. Interestingly, the Landau poles at one-loop and Fixed points at two-loop have been observed for these scenarios, very similar to the $\phi^4$ theory, which is absent in the Standard Model. The role of Higgs portal couplings in attaining these features has been examined. For inert singlet and triplet models, the portal couplings give rise to terms without any residual phases, enhancing this Fixed point behaviour. In the case of the inert doublet model, $\lambda_{4,5}$ terms which have the residual phases, spoil this behaviour. Finally, we consider perturbative unitarity constraints to put limits on the scalar quartic couplings for various perturbativity scales in both one- and two-loop. Larger portal couplings, i.e. $\gsim 2$, get strong perturbative bounds as low as $10^{2-3}$ GeV.