Higgs potential and hidden light Higgs scenario in two Higgs doublet models

In two Higgs doublet models (2HDM), there exists an interesting possibility, the hidden light Higgs scenario, that the discovered SM-like Higgs boson is the heavier \textit{CP}-even Higgs boson $H^0$ and the lighter \textit{CP}-even $h^0$ has not been observed yet in any experiment. We study the current status of this scenario in Types I, II, X, and Y, through the scans of the 2HDM parameters with all relevant theoretical and experimental constraints. We employ not only the most up-to-date Higgs signal strength measurements with the feed-down effects, but also all the available LHC exclusion limits from heavy Higgs searches. Adjusting the heavier $H^0$ to the 125 GeV state while hiding the lighter $h^0$ from the LEP Higgs search prohibits the extreme decoupling limit: there exist upper bounds on the masses of the pseudoscalar $A^0$ and the charged Higgs $H^\pm$ below about 600 GeV. In addition, the $Z_2$ symmetry is shown to be a good approximate symmetry since the soft $Z_2$ symmetry breaking parameter $m_{12}^2$ should be less than about (45 GeV)$^2$. Most interestingly, a few parameters in the Higgs potential and the related Higgs triple and quartic couplings are shown to be meaningfully constrained by the current data. The double Higgs-strahlung process at an $e^+ e^-$ collider is also studied.