ig [W_{\mu}(x), O(x)]\end{aligned}$$

Then the complete electroweak chiral Lagrangian with the whole set of $SU(2)
\times U(1)_Y$, Lorentz, $C$, $P$ and $T$ invariant operators up to dimension four has the form:

$${\cal L}={\cal L}^{(2)}+{\cal L}_{YM}+{\cal L'}_1+ \sum_{i=1}^{13} {\cal
L}_i
\label{eq:lag3}$$

In this equation, the different terms ${\cal L}_i$ are the $SU(2)_L
\times U(1)_Y$ invariant functions of the gauge vector bosons and the $GB$ fields, containing four derivatives, whereas ${\cal L}'_1$ has dimension two. They have the following expressions:

$$\begin{aligned}
{\cal L'}_1  & = & \frac{1}{4} g^2 {\alpha}_0 v^2 [Tr(TV_{\mu})]^2
\nonumber   \\
{\cal L}_1  & = & \frac{1}{2} g^2 {\alpha}_1 B_{\mu\nu} Tr(TF^{\mu\nu})
\nonumber   \\
{\cal L}_2  & = & \frac{1}{2} i g {\alpha}_2 B_{\mu\nu}
Tr(T[V^{\mu},V^{\nu}])  \nonumber   \\
{\cal L}_3  & = & i g {\alpha}_3 Tr(F_{\mu\nu}[V^{\mu},V^{\nu}])
\nonumber   \\
{\cal L}_4  & = & {\alpha}_4 [Tr(V_{\mu}V_{\nu})]^2   \nonumber  \\
{\cal L}_5  & = & {\alpha}_5 [Tr(V_{\mu}V^{\mu})]^2   \nonumber  \\
{\cal L}_6  & = & {\alpha}_6 Tr[(V_{\mu}V_{\nu})]
Tr(TV^{\mu}) Tr(TV^{\nu})  \nonumber    \\
{\cal L}_7  & = & {\alpha}_7 Tr[(V_{\mu}V^{\mu})] [Tr(TV^{\nu})]^2
\nonumber   \\
{\cal L}_8  & = & \frac{1}{4} g^2 {\alpha}_8 [Tr(TF_{\mu\nu})]^2
\nonumber   \\
{\cal L}_9  & = & \frac{1}{2} i g {\alpha}_9 Tr(TF_{\mu\nu})
Tr(T[V^{\mu},V^{\nu}])  \nonumber   \\
{\cal L}_{10}  & = & \frac{1}{2} {\alpha}_{10}
[Tr(TV_{\mu})Tr(TV_{\nu})]^2     \nonumber  \\
