Scalar sector properties of two-Higgs-doublet models with a global U(1) symmetry

We analyze the scalar sector properties of a general class of two-Higgs-doublet models which has a global U(1) symmetry in the quartic terms. We find constraints on the parameters of the potential from the considerations of unitarity of scattering amplitudes, the global stability of the potential and the $\rho$-parameter. We concentrate on the spectrum of the non-standard scalar masses in the decoupling limit which is preferred by the Higgs data at the LHC. We exhibit charged-Higgs induced contributions to the diphoton decay width of the 125\,GeV Higgs boson and its correlation with the corresponding $Z\gamma$ width.