as follows $$\begin{aligned}
\Phi_1=\exp\left(\frac{i\Theta'}{f}\right)
\exp\left(\frac{it_\beta\Theta}{f}\right)
\begin{pmatrix}
0 \\ 0 \\ fc_\beta
\end{pmatrix} \label{eq:phi1} \\
\Phi_2=\exp\left(\frac{i\Theta'}{f}\right)
\exp\left(-\frac{i\Theta}{ft_\beta}\right)
\begin{pmatrix}
0 \\ 0 \\ fs_\beta
\end{pmatrix} \label{eq:phi2}\end{aligned}$$ Here we introduced the shorthand notation $s_\beta\equiv\sin\beta,c_\beta\equiv\cos\beta,t_\beta\equiv\tan\beta$. $f$ is the Goldstone decay constant which is supposed to be at least a few$\TeV$. $\Theta$ and $\Theta'$ are $3\times 3$ matrix fields, defined by $$\begin{aligned}
\Theta=\frac{\eta}{\sqrt{2}}+
\begin{pmatrix}
\textbf{0}_{2\times 2} & h \\
h^\dagger & 0
\end{pmatrix},\quad
\Theta'=\frac{\zeta}{\sqrt{2}}+
\begin{pmatrix}
\textbf{0}_{2\times 2} & k \\
k^\dagger & 0
\end{pmatrix}\end{aligned}$$ where $h$ and $k$ are parameterized as ($v\approx 246\GeV$ denotes the vacuum expectation value of the Higgs doublet) $$\begin{aligned}
h & =\begin{pmatrix} h^0 \\ h^- \end{pmatrix},\quad
h^0=\frac{1}{\sqrt{2}}(v+H-i\chi) \\
k & =\begin{pmatrix} k^0 \\ k^- \end{pmatrix},\quad
k^0=\frac{1}{\sqrt{2}}(\sigma-i\omega)\end{aligned}$$ The covariant derivative in the electroweak sector can be written as $$D_\mu=\partial_\mu-igA_\mu^a T^a+ig_xQ_xB_\mu^x,\quad
g_x=\frac{gt_W}{\sqrt{1-t_W^2/3}}$$ Here $t_W\equiv\tan\theta_W$.$A_\mu^a$ and $B_\mu^x$ denote the $SU(3)_L$ and $U(1)_X$ gauge fields, respectively. The $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge quantum number of $\Phi_1,\Phi_2$ is $(\textbf{1},\textbf{3})_{-\frac{1}{3}}$, therefore for $\Phi_1,\Phi_2$, $Q_x=-\frac{1}{3}$, and $A_\mu^a T^a$ can be written as $$\begin{aligned}
A_\mu^a T^a=\frac{A_\mu^3}{2}
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{pmatrix}
+\frac{A_\mu^8}{2\sqrt{3}}
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 