obtained by the authors in [@Long]. They have the same form that ${\cal L}'_1$, ${\cal L}_1$, ${\cal L}_2$, ${\cal L}_3$, ${\cal L}_4$ and ${\cal L}_5$, whereas the rest of the terms in eq.(\[eq:lag2\]) are not needed for renormalization.

We will describe the interactions between gauge bosons and fermions by the same Lagrangian than that of the $SM$, which can also describe the couplings between fermions and scalars by means of the usual Yukawa terms. However the latter are not used in our calculations since we work all the time with massless fermions.

Once we have set the Lagrangian to be used, the next step is to define the quantum theory. This can be done in a standard way using the Faddeev-Popov method and choosing some appropriate $R_{\xi}$ covariant gauge. The gauge fixing functions are constructed by requiring that the terms of the form $W^{i}_{\mu}
{\partial}^{\mu} {\pi}_i(x)$ and $B_{\mu} {\partial}^{\mu} {\pi}_3$ cancel out; thus yielding:

$$\begin{aligned}
f_i(W_{i\mu})  & =  & {\partial}^{\mu} W_{i\mu} - \frac{v}{2} {\xi}_W
g{\pi}_i,
\;\;\;\;\;  i=1,2,3     \nonumber  \\
f_0(B_{\mu})  & =  & {\partial}^{\mu} B_{\mu}  + \frac{v}{2} {\xi}_B g'
{\pi}_3\end{aligned}$$

Now we can write the gauge-fixing and Faddeev-Popov terms as:

$${\cal L}_{GF} = - \frac{1}{2 {\xi}_W} \sum_{i=1}^{3} [f_i(W_{i\mu})]^2 -
\frac{1}{2 