Equation (13.47) is valid if $\mathrm{m}^2 / \mathrm{s} \& 1$. If the electron mass $m$ is not ignored, show that the extra contribution to $(13.47)$ is
$$
-G^2 m^2 y\left(c_V^2-c_A^2\right) / 2 \pi
$$
The process $v, \mathrm{e}^{-} \rightarrow \nu, \mathrm{e}^{-}$offers the intriguing possibility of studying charged current and neutral current interference, see Fig. 13.4. The amplitude for diagram (a) is $\Re^{N C}$ of (13.46) with $\nu=\nu_e$. For diagram (b); we have
$$
\Re^{C C}=-\frac{G}{\sqrt{2}}\left[\bar{e} \gamma^\mu\left(1-\gamma^5\right) \nu_e\right]\left[\overline{\nu_e} \gamma_\mu\left(1-\gamma^5\right) e\right],
$$
where the minus sign relative to (13.46) arises from interchange of the outgoing leptons [see (6.9)]. We may use the Fierz reordering theorem, see, for example, Bailin (1982), to rewrite (13.50) as
$$
\mathscr{9 k}^{C C}=\frac{G}{\sqrt{2}}\left(\overline{\nu_e} \gamma^\mu\left(1-\gamma^5\right) \nu_e\right)\left(\bar{e} \gamma_\mu\left(1-\gamma^5\right) e\right) \text {. }
$$