& 0 & -2
\end{pmatrix}
+\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & W_\mu^+ & Y_\mu^0 \\
W_\mu^- & 0 & X_\mu^- \\
Y_\mu^{0\dagger} & X_\mu^+ & 0
\end{pmatrix}\end{aligned}$$ The gauge kinetic terms for $\Phi_1,\Phi_2$ are $$\mL_{gk}=(D_\mu\Phi_1)^\dagger(D^\mu\Phi_1)+
(D_\mu\Phi_2)^\dagger(D^\mu\Phi_2)$$ The first order (in $\frac{v}{f}$) gauge boson mixing for $A^3,A^8,B_x$ takes the form $$\begin{aligned}
\begin{pmatrix}
A^3 \\ A^8 \\ B_x
\end{pmatrix}
=
\begin{pmatrix}
0 & c_W & -s_W \\
\sqrt{1-\frac{t_W^2}{3}} & \frac{s_W t_W}{\sqrt{3}} & \frac{s_W}{\sqrt{3}} \\
-\frac{t_W}{\sqrt{3}} & s_W\sqrt{1-\frac{t_W^2}{3}} & c_W\sqrt{1-\frac{t_W^2}{3}}
\end{pmatrix}
\begin{pmatrix}
Z' \\ Z \\ A
\end{pmatrix}\end{aligned}$$ We note that $Z',Z$ are not the ultimate mass eigenstate fields. For future convenience we split the $Y^0$ field into real and imaginary parts $$Y_\mu^0\equiv\frac{1}{\sqrt{2}}(Y_{R\mu}+iY_{I\mu}),\quad
Y_\mu^{0\dagger}\equiv\frac{1}{\sqrt{2}}(Y_{R\mu}-iY_{I\mu})$$ In this paper we intend to focus on the neutral sector, in which there are six scalar degrees of freedom: $\eta,\zeta,H,\chi,\sigma,\omega$. Four degrees of freedom will be eaten to give mass to massive neutral gauge bosons and are unphysical. The remaining two are physical and need to play the role of the observed Higgs-like boson and the pseudo-axion which has been discussed a lot in the literature. The pseudo-axion actually corresponds to the Goldstone boson of a spontaneously broken global $U(1)$ symmetry in the SLH. To give it a mass, the 