Higgs potential is given by V&=&m\_1\^2[Tr]{}(\^)+m\_2\^2[Tr]{}(\^)+\_1[Tr]{}(\^)\^2+\_2[Tr]{}(\^)\^2+\_3[Tr]{}(\^)[Tr]{}(\^)\
&&+\_4[Tr]{}(\^\^)+\_5[Tr]{}(\^)[Tr]{}(\^T\_iT\_j)\
&&+\_1[Tr]{}(\^)\_P\^[ij]{}+\_2[Tr]{}(\^T\_i T\_j)\_P\^[ij]{} , \[potential\] where $\tau_i$ are the $2\times 2$ Pauli matrices and \_P=P\^P,         P=(

[ccc]{}-1/&i/&0\
0&0&1\
1/&i/&0

) .

The neutral components of the doublet and the real and the complex triplets have the VEVs, $v_\phi$, $v_\xi$, and $v_\chi$, respectively, which are defined as \^0&=& ,\
\^0&=&v\_+\_r\^0 ,\
\^0&=&v\_+  . After electroweak symmetry breaking, the custodial $SU(2)_{\rm V}$ symmetry remains in the Higgs sector $(SU(2)_L\times SU(2)_R\to SU(2)_{\rm V})$, by which the real and complex triplets have the same VEV, $v_\Delta\equiv v_\xi=v_\chi$. Consequently this leads to $\rho=1$ at the tree level [@Chanowitz:1985ug]. In this case the VEVs are constrained as $v^2=v_\phi^2 + 8v_\Delta^2$, where $v=(\sqrt{2}
G_F)^{-\frac{1}{2}} \simeq 246$ GeV. Therefore, differently from usual triplet models, $v_\Delta$ can be of order 100 GeV in this model without explicit inconsistency with the experimental value of the rho parameter. It is convenient to introduce the doublet-triplet mixing angle $\theta_H$, \_H=  . The experimental constraint on $\theta_H$ is discussed in Ref. [@Haber:1999zh]. In the potential Eq.(\[potential\]), the last two terms with the coupling constants $\mu_1$ and $\mu_2$ explicitly violate the discrete $Z_2$ symmetry under the transformation of $\Phi\to \Phi$ and $\Delta \to -\Delta$. Without the $Z_2$ symmetry, 