Upper and lower bounds of the lightest CP-even Higgs boson in the two-Higgs-doublet model

By imposing validity of the perturbation and stability of vacuum up to an energy scale $\mathrm{\Lambda }$ ($\le {10}^{19}$ GeV), we evaluate mass bounds of the lightest CP-even Higgs-boson mass ($m_h$) in the two-Higgs-doublet model (2HDM) with a softly-broken discrete symmetry. The most general bounds are obtained as a function of $\mathrm{\Lambda }$ vering all the free parameters under the experimental constraint from the $rho$ parameter and the b \to s \gamma$ results. We find that, while the upper bound is almost the same as in the SM, the lower bound is significantly reduced. In the decoupling regime where the model behaves like the SM at low energy, the lower bound is given, for example, by about 100 GeV for $\mathrm{\Lambda }={10}^{19}$ GeV and $m_t=175$ GeV, which is smaller by about 40 GeV than the corresponding lower bound in the SM. In general case, the $m_h$ is no longer bounded from below by these conditions. If we consider the experimental $b\to s\gamma$ constraint, small $m_h$ are excluded in Model II of the 2HDM.