given by this expression:

$$d {\hat{\sigma}} = \frac{1}{32 \pi {\hat{s}}} \frac{p'}{p} \sum {\mid M
\mid}^2 d{\rm cos} {\theta}
\label{eq:subpro}$$

where $M$ is the helicity amplitude, and the $\Sigma$ symbol refers to the sum of all the final polarizations and the average on the initial ones. In the case of electroweak gauge bosons scattering, where we separate the transversal and longitudinal polarizations in the initial state (as we will see in section 5), we take into account the different helicities contributing to $TT$, $TL$, $LT$ and $LL$ polarizations of the initial bosons, to do this average. We also include the corresponding branching ratios to the gold-plated events.

Let us consider now the processes corresponding to the elastic collision of gauge bosons (fusion processes). Symbolically this reaction can be written as:

$$V_1 (p_1,{\lambda}_1) V_2 (p_2,{\lambda}_2) \rightarrow
V_3 (p_3,{\lambda}_3) V_4 (p_4,{\lambda}_4)$$

We choose the following polarization vectors in the helicity base (${\lambda}_i=\pm 1,0$):

$$\begin{aligned}
{\epsilon}^{\pm}_1 & = & \frac{1}{\sqrt{2}} (0,{\mp}1, -{\rm i},0)   \\
{\epsilon}^0_1  & = & \frac{1}{m_{V_1}} (p,0,0,E_1)     \\
{\epsilon}^{\pm}_2 & = & \frac{1}{\sqrt{2}} (0,{\mp}1,  {\rm i},0)   \\
{\epsilon}^0_2  & = & \frac{1}{m_{V_2}} (p,0,0,-E_2)     \\
{\epsilon}^{\pm}_3 & = & \frac{1}{\sqrt{2}} (0, \mp {\rm cos} \theta,
 -{\rm i}, \pm {\rm sin}\theta) \\
{\epsilon}^0_3 