The Higgs Potential in 2HDM extended with a Real Triplet Scalar: A roadmap

We perform a comprehensive study of The Higgs potential of the two Higgs doublet model extended by a real triplet scalar field $\Delta$. This model, dubbed $2\mathcal{HDM+T}$, has a rich Higgs spectrum consisting of three CP-even Higgs $h_{1,2,3}$, one CP-odd $A_0$ and two pairs of charged Higgs $H^\pm_{1,2}$. First, we determine the perturbative unitarity constraints and a set of non trivial conditions for the boundedness from below (BFB). Then we derive the Veltman conditions by considering the quadratic divergencies of Higgs boson self energies in $2\mathcal{HDM+T}$. We find that the parameter space is severely delimited by these theoretical constraints, as well as experimental exclusion limits and Higgs signal rate measurements at LEP and LHC. Using HiggsBounds-5.3.2beta and HiggSignals-2.2.3beta public codes an exclusion test at $2\sigma$ is then performed on the physical scalars of $2\mathcal{HDM+T}$. Our analysis provides a clear insight on the nonstandard scalar masses, showing that the allowed ranges are strongly sensitive to the sign of mixing angle $\alpha_1$, essentially when naturalness is involved. For $\alpha_1 < 0$ scenario, our results place higher limits on the bounds of all scalar masses, and show that the pairs $(h_2, H_1^\pm)$ and $(h_3, H_2^\pm)$ are nearly mass degenerate varying within the intervals $[130\,,\,246]$~GeV and $[160\,,\,335]$~GeV respectively. When $\alpha_1$ turns positive, we show that consistency with theoretical constraints and current LHC data, essentially on the diphoton decay channel, favors Higgs masses varying within wide allowed ranges: $[153\,,\,973]$~GeV for $m_{A_0}$; $[151\,,\,928]$~GeV for ($m_{h_2}$, $m_{H_1^\pm}$) and $[186\,,\,979]$~GeV for ($m_{h_3}$, $m_{H_2^\pm}$). Finally, we find that the $\gamma \gamma$ and $Z\gamma$ Higgs decay modes are generally correlated.