dropped using $C$ invariance that relates the magnitude of the helicity amplitudes $M_{{\lambda}_1{\lambda}_2{\lambda}_3{\lambda}_4}$ in the process with $W^+$ and with $W^-$. If we apply the relations between amplitudes derived from $P$ and $T$ invariance, there will be only $25$ remaining amplitudes. These subprocesses cross sections are affected by all the ${\alpha}_k$, but ${\alpha}_{10}$, as we can deduce from the Feynman rules eqs.(\[eq:GF3\] and \[eq:GF4\]).

The number of helicity amplitudes corresponding to the processes $W^{\pm}\gamma  \rightarrow  W^{\pm}Z^0$ is $2 \times (2
\times 3^3)$. The $2$ factor disappears when we take into account $C$ invariance as in the previous case. The number of remaining amplitudes is reduced again, by another $2$ factor after applying $P$ invariance. Therefore in these processes we have $3^3=27$ independent helicity amplitudes that we calculate using the Feynman rules shown in eqs.(\[eq:GF3\] and \[eq:GF4\]). Here we can observe that the set of the chiral lagrangian parameters affecting the $W^{\pm}\gamma \rightarrow
W^{\pm}Z^0$ cross sections is the following: ${\alpha}_0$, ${\alpha}_1$, ${\alpha}_2$, ${\alpha}_3$, ${\alpha}_8$, ${\alpha}_9$, ${\alpha}_{11}$, ${\alpha}_{12}$ and ${\alpha}_{13}$.

In order to see the relative importance of the different channels we have evaluated the contribution to the total cross section to obtain $Z^0Z^0$ and $W^{\pm}Z^0$ pairs in $pp$ colliders from the 