by replacing $m_{\tilde H_1^0}$ with $m_{\tilde H_1^{0'}}$ and $\alpha$ with $\alpha + \pi/2$, which can be seen from Eqs.(\[lambda1\])-(\[lambda3\]). Consequently the allowed regions in the $m_{H_3}^{}$ - $m_{\tilde H_1^{0'}}$ plane are given by the same as in Fig. 1(b). We find that contrary to the result in Fig. 1(a) there is only few difference in the case where we include the $Z\to b\bar b$ data.

The allowed region in the $m_{\tilde H_1^{0}}$ - $m_{\tilde H_1^{0'}}$ plane in Fig. 1(c) is symmetrical about the line of $m_{\tilde H_1^{0}} = m_{\tilde H_1^{0'}}(\equiv m)$. It is notable that at least one singlet receives very strict constraint from perturbative unitarity. The mass of lighter singlet, either $\tilde H_1^0$ or $\tilde H_1^{0'}$, is bounded from above by 322 GeV. In analytic calculation, this upper bound is obtained as $m < \sqrt{6\pi/11}\, v$ ($\simeq 322$ GeV) from $|x_1^\pm| < 8\pi$.

In the following, we evaluate the upper bound on $m_{\rm lightest}^{}$ $\left[\equiv {\rm Min}\left(m_{H_3}^{}, m_{H_5}^{}, m_{\tilde H_1^0}, m_{\tilde H_1^{0'}} \right)\right]$. We start from the case in which the constraints from the $Zb\bar b$ results are switched off. When the 3-plet is the lightest, we obtain the upper bound on $m_{\rm lightest} ( = m_{H_3}^{})$ as m\_[lightest]{} 