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We perform a comprehensive study of The Higgs potential of the two Higgsdoublet model extended by a real triplet scalar field $\Delta$. This model,dubbed $2\mathcal{HDM+T}$, has a rich Higgs spectrum consisting of threeCP-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 aset of non trivial conditions for the boundedness from below (BFB). Then wederive the Veltman conditions by considering the quadratic divergencies ofHiggs boson self energies in $2\mathcal{HDM+T}$. We find that the parameterspace is severely delimited by these theoretical constraints, as well asexperimental exclusion limits and Higgs signal rate measurements at LEP andLHC. Using HiggsBounds-5.3.2beta and HiggSignals-2.2.3beta public codes anexclusion test at $2\sigma$ is then performed on the physical scalars of$2\mathcal{HDM+T}$. Our analysis provides a clear insight on the nonstandardscalar masses, showing that the allowed ranges are strongly sensitive to thesign of mixing angle $\alpha_1$, essentially when naturalness is involved. For$\alpha_1 < 0$ scenario, our results place higher limits on the bounds of allscalar 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 thatconsistency with theoretical constraints and current LHC data, essentially onthe diphoton decay channel, favors Higgs masses varying within wide allowedranges: $[153\,,\,973]$~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 generallycorrelated.

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