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Study of vacuum behavior for inert models with discrete $Z_{2}$-like and abelian $U(1)$ symmetries

We study the vacuum behavior at one loop level in extended Higgs sectors with two doublets (2HDM), where $U(1)$ and $Z_{2}$ symmetries are considered to protect the $CP$ symmetry in the Higgs potential and to avoid Flavor Changing Neutral Currents at tree level in the Yukawa sector. In the Inert Higgs Model case, a detailed comparison is made between both models by using the energy evolution of couplings, which should satisfy energy scale dependent relations deduced for minima and stationary points of the Higgs potential at tree level. Besides, perturbative unitarity constraints at tree level are considered to generate the allowed parameter space compatible with perturbativity (absence of Landau poles). Our studies illustrate exclusion regions for Higgs masses and other combinations of couplings in the scalar sector, in particular for splittings of mass square for neutral scalars $A^{0}$ and $H^{0}$, as well as the difference between the sum of these and the charged Higgs mass square. From the vacuum stability for inert-2HDM at the tree and one loop levels, analyses lead us to find out new hierarchical structures for scalar masses. To complete vacuum studies on the Inert model, and based on reparameterization invariance of the Higgs potential, we compute original discriminants that allow ensuring the presence of a global electroweak minimum at tree level. Moreover, the behavior in high energy scales drives out analyzing criticality phenomena for the additional parameters of extended Higgs sectors. Finally, and using the consistency with the electroweak precision analyses of oblique parameters, we describe several implications from different regimes of the inert model on charged and pseudoscalar Higgs searches.