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 2HDM+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 2HDM+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 2HDM+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 (\sin \alpha_1 < 0).