Show that (13.51) follows from (13.50). The invariance under reordering of the spinors is an important property of the $V-A$ interaction. The effect of reordering in the scalar product of bilinear covariants is, in general, much more involved. The answer is the Fierz theorem.
To obtain the amplitude $9 \pi\left(\nu_{e} \mathrm{e}^{-} \rightarrow \nu_{e} \mathrm{e}^{-}\right)$, we add the amplitudes ( $T^{N C}$ and $9 \pi^{C C}$ ) for the two diagrams of Fig. 13.4. If we take $\rho=1$, then $G_{N}=G$ [see $(12.88)]$, and we find $\mathscr{R}=9{K}^{N C}+9 \pi^{C C}$ is given by $(13.46)$ with
$$
c_{V} \rightarrow c_{V}+1, \quad c_{A} \rightarrow c_{A}+1 .
$$
Thus, the $\nu_{e} \mathrm{e}^{-}$and $\bar{\nu}_{e} \mathrm{e}^{-}$elastic scattering cross sections are in turn given by (13.48) and (13.49) with these replacements.
It is customary to present the results of a given neutrino-electron cross section measurement as an ellipse of possible values of $c_{V}$ and $c_{A}$ in the $c_{V}, c_{A}$ plane. Recent results are shown in Fig. 13.5. The three "experimental" ellipses mutually intersect to give two possible solutions. The $c_{A}$ dominant solution is
$$
\begin{aligned}
&c_{A}^{e}=-0.52 \pm 0.06 \\
&c_{V}^{e}=0.06 \pm 0.08
\end{aligned}
$$
in excellent agreement with the standard model and $\sin ^{2} \theta_{W} \approx \frac{1}{4}$ (see Table 13.2).