Show that $|\vartheta(\nu \mathrm{q} \rightarrow \nu \mathrm{q})|^2$ behaves like $s^2, s^2(1-y)^2$, $s^2 y^2$ for pure $V-A$, pure $V+A$, and $S, P$ neutral couplings of the quark, respectively. Pure $V \pm A$ denote $\gamma^{\bar{\mu}}\left(1 \pm \gamma^5\right)$ couplings, and $S, P$ stands for the scalar, pseudoscalar interaction amplitude
$$
\Re=\frac{G_N}{\sqrt{2}}\left(\bar{u}_p\left(1-\gamma^5\right) u_p\right)\left(\bar{u}_q\left(g_S-g_P \gamma^5\right) u_q\right)
$$
The parton model predictions for the neutral current (NC) processes $\nu \mathrm{N} \rightarrow \nu \mathrm{X}$ and $\bar{\nu} \mathrm{N} \rightarrow \bar{\nu} \mathrm{X}$ are obtained by following the calculation of the $\mathrm{CC}$ processes $\nu \mathrm{N} \rightarrow \mu^{-} \mathrm{X}$ and $\bar{\nu} \mathrm{N} \rightarrow \mu^{+} \mathrm{X}$ of Section 12.8. For an isoscalar target, we find that the cross section per nucleon is
$$
\begin{aligned}
\frac{d \sigma(\nu \mathrm{N} \rightarrow \nu \mathrm{X})}{d x d y}= & \frac{G_N^2 x s}{2 \pi}\left[g_L^2\left(Q(x)+(1-y)^2 \bar{Q}(x)\right)\right. \\
& \left.+g_R^2\left(\bar{Q}(x)+(1-y)^2 Q(x)\right)\right],
\end{aligned}
$$
where, if we assume only $\mathrm{u}, \mathrm{d}, \overline{\mathrm{u}}, \overline{\mathrm{d}}$ quarks within the nucleon,
$$
g_L^2 \equiv\left(g_L^u\right)^2+\left(g_L^d\right)^2
$$
and similarly for $g_R^2$. We may integrate over $x$ and define
$$
Q \equiv \int x Q(x) d x=\int x[u(x)+d(x)] d x
$$
see (12.75). Cross section (12.92) and that for $\bar{\nu} \mathrm{N} \rightarrow \bar{\nu} \mathrm{X}$ become
$$
\begin{aligned}
& \frac{d \sigma^{N C}(\nu)}{d y}=\frac{G_N^2 s}{2 \pi}\left\{g_L^2\left(Q+(1-y)^2 \bar{Q}\right)+g_R^2\left(\bar{Q}+\left(1-y^2\right) Q\right)\right\}, \\
& \frac{d \sigma^{N C}(\bar{\nu})}{d y}=\frac{G_N^2 s}{2 \pi}\left\{g_I^2\left(\bar{Q}+(1-y)^2 Q\right)+g_R^2\left(Q+\left(1-y^2\right) \bar{Q}\right)\right\},
\end{aligned}
$$
which are to be contrasted with the charged current expressions (12.76) and $(12.77)$
$$
\begin{aligned}
& \frac{d \sigma^{C C}(\nu)}{d y}=\frac{G^2 s}{2 \pi}\left(Q+(1-y)^2 \bar{Q}\right) . \\
& {\frac{d \sigma^{C C}}{d y}}^{(\bar{\nu})}=\frac{G^2 s}{2 \pi}\left(\bar{Q}+(1-y)^2 Q\right) .
\end{aligned}
$$
Correcting (12.95) and (12.96) for the neutron excess in an iron target and for an s quark contribution, the present data give
$$
g_L^2=0.300 \pm 0.015, \quad g_R^2=0.024 \pm 0.008
$$
The experimental verdict is that the weak neutral current is predominantly $V-A$ (i.e., left-handed) but, since $g_R \neq 0$, not pure $V-A$. The NC and the CC have a tantalizingly similar structure, but the $\mathrm{CC}$ is believed to have a pure $V-A$ form. Chapter 13 takes up this point, but first we must look more carefully at the quark sector.