$\theta_H\simeq \pi/2$. When all masses other than $m_{H_3}^{}$ are zero, $m_{H_3}^{}$ is bounded by $m_{H_3}<\sqrt{2\pi/3}\, v$ ($\simeq 356$ GeV) from $|x_1^\pm| < 8\pi$. However numerical analysis shows that the actual upper bound is a few decade GeV greater. This excess comes from some delicate cancellation in Eq.(\[x1\]). When we impose the experimental data from $Z\to b\bar b$ which give the constraint on the combination of $m_{H_3}^{}$ and $\theta_H$ [@Haber:1999zh], the allowed region is further limited in the dark shadowed regions. The upper bounds on $m_{H_3}^{}$ and $m_{H_5}^{}$ do not change, but the remained allowed region is in the vicinity of $m_{H_5}^{}\simeq \sqrt{3} m_{H_3}^{}$.

In Fig. 1(b) we can see that the upper bound on $m_{\tilde H_1^0}$ is about 710 GeV, which is almost the same as that on the mass of the SM Higgs boson [@Lee:1977yc]. Larger values of $m_{\tilde H_1^0}$ are allowed for smaller values of $\alpha$ and $\theta$ as well as smaller $m_{H_5}^{}$ and $m_{\tilde
H_1^{0'}}$ values. For instance, taking the limit $\alpha \to 0,~ \theta_H \to 0,~m_{H_5}^{}\to 0$ and $m_{\tilde H_1^{0'}}\to 0$, we obtain $m_{\tilde H_1^0} < \sqrt{8\pi/3}\, v$ ($\simeq 712$ GeV) in the condition $|x_1^+|<8\pi$. The mass bound for another singlet $\tilde H_1^{0'}$ can be obtained 