Show that a (charge-lowering) weak current of the form
$$
\bar{u}_e \gamma^\mu \frac{1}{2}\left(1-\gamma^5\right) u_v
$$
involves only left-handed electrons (or right-handed positrons). In the relativistic limit $(v \approx c)$, show that the electrons have negative helicity.
The $\frac{1}{2}\left(1-\gamma^5\right)$ in (12.9) automatically selects a left-handed neutrino (or a right-handed antineutrino). This $V-A$ (vector-axial vector) structure of the weak current can be directly exposed by scattering $\nu_e$ 's off electrons (see Section 12.7), just as the $\gamma^\mu$ structure of electromagnetism was verified by measurements of the angular distribution of $\mathrm{e}^{+} \mathrm{e}^{-}$scattering.
It is natural to hope that all weak interaction phenomena are described by a $V-A$ current-current interaction with a universal coupling $G$. For example, $\beta$-decay of Fig. 12.2 and $\mu$-decay of Fig. 12.4 can be described by the amplitudes
$$
\text { IR }\left(\mathrm{p} \rightarrow \mathrm{ne}^{+} \nu_e\right)=\frac{G}{\sqrt{2}}\left[\bar{u}_n \gamma^\mu\left(1-\gamma^5\right) u_p\right]\left[\bar{u}_{v_k} \gamma_\mu\left(1-\gamma^5\right) u_e\right]
$$
and
$$
\operatorname{\vartheta R}\left(\mu^{-} \rightarrow \mathrm{e}^{-} \bar{\nu}_e \nu_\mu\right)=\frac{G}{\sqrt{2}}\left[\bar{u}_{v_\mu} \gamma^\sigma\left(1-\gamma^5\right) u_\mu\right]\left[\bar{u}_e \gamma_\sigma\left(1-\gamma^5\right) u_{v_e}\right],
$$
respectively. The $1 / \sqrt{2}$ is pure convention (to keep the original definition of $G$ which did not include $\gamma^5$ ). We then proceed in analogy with the Feynman rules for QED. The calculations only involve particles, and the diagrams show only particle lines. Antiparticles do not appear. Thus, the outgoing $\bar{\nu}_e$ (of momentum $k$ ) in $\mu$-decay is shown in Fig. 12.4 as an ingoing $\nu_e$ (of momentum $-k$ ). As before, the spinor $u_{v_e}(-k)$ of (12.11) will be denoted $v_{v_e}(k)$, see (5.33). The same remarks apply to the outgoing $\mathrm{e}^{+}$of $(12.10)$.