The two Higgs doublet type-II seesaw model: Naturalness and $\bar{B}\to X_s\gamma$ versus heavy Higgs masses

We extend the work \cite{Ouazghour:2018mld} to a more general and detailed analysis through studying the naturalness problem and $B$ physics constraints within the context of two higgs doublets model augmented with a complex scalar triplet field ($2HDMcT$). We first derive the modified Vetman conditions and show that naturalness problem might be evaded at the electroweak scale. Then, we evaluate the branching ratio of $\bar{B}\to X_s\gamma$ radiative decay at the Next to leading order. Besides unitarity, boundedness from below constraints and the present collider data from Higgs signal rate measurements, the parameter space of $2HDMcT$ is then revisited to see how the modified Veltman conditions and $\bar{B}\to X_s\gamma$ experimental bounds induce significant delimitations. Our analysis shows that the naturalness affects drastically the masses of heavy Higgs $h_3$, $A_2$, $H_2^\pm$ and $H^{\pm\pm}$. More specifically, we considered three benchmak scenarios corresponding to $\mu_2 > 20, 40, 70$ GeV. For $\mu_2 > 40$, we find stringent upper limits on $m_{h_2}$, $m_{A_1}$ and $m_{H_1^\pm}$ at $880$, $920$ and $918$ GeV respectively. If, in addition, the measurements of $\bar{B}\to X_s\gamma$ decay rate are also considered, the lower masses limits of nonstandard Higgs bosons ($h_2$, $h_3$, $A_1$, $A_2$, $H_1^\pm$, $H_2^\pm$, $H^{\pm\pm}$) are pushed up to higher values located between $490$ and $750$ GeV. Besides, we also demonstrate that the $\bar{B}\to X_s\gamma$ experimental limits can only be accommodated within $2HDMcT$ if two conditions are fullfilled : $\alpha_1$ parameter positive and the Higgs mass hierarchy $m_{h_{3}}>m_{H_2^\pm}>m_{H^{\pm\pm}}$.