em}$. The well known chiral Lagrangian describing the $ESB$ of the $SM$, can be written as an infinite expansion with terms of increasing number of gauge fields and derivatives of the Goldstone bosons ($GB$), with an infinite number of arbitrary parameters. This chiral Lagrangian can be seen as a low momentum expansion for the corresponding Green functions. At some given order in the number of $GB$ derivatives one can work only with a finite number of terms and parameters. In this case the model can only be applied to much smaller energies than $4\pi v$ which is the parameter controlling this expansion ($4\pi v
\simeq 3 TeV$ since $v\simeq 250 GeV$).

The first term of this effective Lagrangian ($O(p^2)$) is given by:

$${\cal L}^{(2)}=\frac{v^2}{4}tr[D_{\mu} U(D^{\mu}U)^{\dagger}]
\label{eq:lag1}$$

and by the Yang-Mills Lagrangian:

$${\cal L}_{YM}=-\frac{1}{4} B_{\mu\nu} B^{\mu\nu} - \frac{1}{2}
Tr[F_{\mu\nu}F^{\mu\nu}]
\label{eq:YM}$$

We choose the GB parametrization as an unitary matrix $U$ belonging to the quotient space $SU(2)_L \times SU(2)_R / SU(2)_{L+R}$.

$$U = {\rm exp} (i \frac{\vec{\tau}.\vec{\pi}}{v})$$

where $\vec{\tau} \equiv ({\tau}_1, {\tau}_2,{\tau}_3)$ are the Pauli matrices and $\vec{\pi} \equiv ({\pi}_1(x), {\pi}_2(x), {\pi}_3(x))$ represents the triplet of $GB$.

The covariant derivative $D_{\mu}U$ is defined as:

$$D_{\mu}U = {\partial}_{\mu} U + ig W_{\mu} U 