Constraints from Heavy Higgs boson masses in the two Higgs doublet model

Upon the absence of signals of new physics at the LHC, a reasonable strategy is to assume that new particles are very heavy and the other model parameters are unknown yet. In the aligned two Higgs doublet model, however, heavy Higgs boson masses above 500 GeV enhance some couplings in the scalar potential, which causes a breakdown of the perturbative unitariry in general. Some tuning among model parameters is required. We find that one information on the heavy Higgs boson mass, say $M_H$, has significant theoretical implications: (i) the other heavy Higgs bosons should have similar masses to $M_H$ within $\pm \mathcal{O}(10)\%$; (ii) the inequalities from the theoretical constraints are practically reduced to an equation such that $m_{12}^2 \tan\beta$ is constant, where $m_{12}^2$ is the soft $Z_2$ breaking parameter and $\tan\beta$ is the ratio of two vacuum expectation values; (iii) the triple Higgs coupling $\lambda_{HHh}$ is constant over $\tan\beta$ while $\lambda_{HHH}$ and $\lambda_{AAH}$ are linearly proportional to $\tan\beta$. The double Higgs-strahlung process of $e^+ e^- \to ZHH$ is also studied, of which the total cross section is almost constant with the given $M_H$.