annihilation represents the main source of $Z^0Z^0$ pairs in $pp$ colliders like the $LHC$. As in the case of gluon fusion, there is no dependence on the chiral parameters in the calculation, at our level of approximation. The cross section at tree level, which only receives contribution from the $t$ and $u$ channels (Figure \[fig:qq\]), has the same well-known expression as in the $MSM$.

The $Z^0Z^0$ fusion calculated using the chiral Lagrangian, only receives contribution from the vertex $Z^0Z^0Z^0Z^0$ (Figures \[fig:GF4\] and \[fig:ZZ\]). The only dependence on the effective Lagrangian parameters is through the following combination eq.(\[eq:GF4\]):

$${\alpha}_4 + {\alpha}_5 + 2({\alpha}_6 + {\alpha}_7 + {\alpha}_{10})$$

Again we can use the relations derived from the $C$, $P$ and $T$ to relate the helicity amplitudes and thus to check our results. These relations are:

$$\begin{aligned}
|M_{{{\lambda}_1}{{\lambda}_2}{{\lambda}_3}{{\lambda}_4}}| &
\stackrel{C}{=}  &
|M_{{{\lambda}_2}{{\lambda}_1}{{\lambda}_4}{{\lambda}_3}}|        \\
|M_{{{\lambda}_1}{{\lambda}_2}{{\lambda}_3}{{\lambda}_4}}| &
\stackrel{P}{=}  &
|M_{{-{\lambda}_1}{-{\lambda}_2}{-{\lambda}_3}{-{\lambda}_4}}|    \\
|M_{{{\lambda}_1}{{\lambda}_2}{{\lambda}_3}{{\lambda}_4}}| &
\stackrel{T}{=}  &
|M_{{{\lambda}_3}{{\lambda}_4}{{\lambda}_1}{{\lambda}_2}}|\end{aligned}$$

In order to relate some helicity amplitudes we have also taken into account whether the scattering particles are identical or not., There are still $15$ independent amplitudes remaining after applying these symmetry relations.

The $W^+W^- \rightarrow Z^0Z^0$ reaction represents another source of $Z^0Z^0$ pairs from $pp$ beams. Every helicity 