{\xi}_B} [f_0(B_{\mu})]^2$$

$$\begin{aligned}
{\cal L}_{FPG} & = & \int d^4 y {c^{\dag}}_m(x) \frac{\delta
f_m(x)}{\delta {\theta}_n (y)} c_n(y)  \;\;\;\;\;\;  m,n= 0,1,2,3\end{aligned}$$

Therefore, the effective action and the effective Lagrangian (non-linear $SM$ Lagrangian, ${\cal L}_{NLSM}$) take the form:

$$S_{eff} =  \int d^4x [ {\cal L}(x) + {\cal L}_{GF} + {\cal L}_{FPG}]$$

$${\cal L}_{NLSM} = {\cal L}^{(2)}+{\cal L'}_1+{\cal
L}^{(4)}+{\cal L}_{YM}+{\cal L}_{GF}
+{\cal L}_{FPG}+{\cal L}_{\Psi}
\label{eq:MENL}$$

In the following we will to work in the Landau gauge (${\xi}_W={\xi}_B=0$). In this gauge the perturbative $\pi$ propagator remains massless and the Faddeev-Popov ghosts only couple to the gauge field. There is not direct coupling of the $\pi$ field to the ghosts. Moreover, the counterterms necessary to cancel divergences with ${\cal L}^{(2)}$ at one loop are gauge invariant functions of the gauge fields and the $GB$ alone (the ${\cal L}'_1$, ${\cal L}_1$, ${\cal L}_2$, ${\cal L}_3$, ${\cal L}_4$ and ${\cal L}_5$ terms mentioned above). In other gauges, or other parametrizations of the coset space, the counterterms could also be functions of the ghost fields, and their structure would be determined using the Becchi-Rouet-Stora ($BRS$) invariance.

Thus, the gauge fixing and the Faddeev-Popov term in the Landau gauge read:

$$\begin{aligned}
{\cal L}_{GF} & = & - \frac{1}{2 {\xi}} 