&= \frac{g^2_N}{16} \bar{u}(\nu_a) \gamma_\mu (1 - \gamma_5) u(\nu_a) \frac{-g^{\mu \nu} + q^\mu q^\nu / M_Z^2}{q^2 - M_Z^2} \bar{u}(\nu_b) \gamma_\nu (1 - \gamma_5) u(\nu_b) \, \, \, ,\end{aligned}$$ which, for $q^2 \ll M_Z^2$, results in $$\begin{aligned}
 \label{eq:EQ100}
\mathscr{T} &= \frac{g^2_N}{16 M_Z^2} \bar{u}(\nu_a) \gamma^\mu (1 - \gamma_5) u(\nu_a) \bar{u}(\nu_b) \gamma_\mu (1 - \gamma_5) u(\nu_b)\nonumber\\
&= \frac{G_F}{2\sqrt{2}} \bar{u}(\nu_a) \gamma^\mu (1 - \gamma_5) u(\nu_a) \bar{u}(\nu_b) \gamma_\mu (1 - \gamma_5) u(\nu_b) \, \, \, .\end{aligned}$$

The square of the scattering amplitude $\lvert \mathscr{T} \rvert^2$, summed over the final states and properly averaged (for unpolarised cross sections) over the spin orientations in the initial state, is the backbone of the evaluation of the DCS of a process, complemented by the flux of the incident beam and the permissible Lorentz invariant phase space.

For massless projectiles, the DCS is obtained via the expression $$\frac{d\sigma}{d\tau} = \frac{\lvert \mathscr{T} \rvert^2}{2(s-m_t^2)} \, \, \, ,$$ where $d\tau$ denotes the infinitesimal Lorentz invariant phase space and $m_t$ stands for the rest mass of the target, i.e., $0$ for neutrini (Sections \[sec:D1\]-\[sec:D5\] and \[sec:D8\]), and $m_e$ for electrons and positrons (Sections \[sec:D6\] and \[sec:D7\]). The infinitesimal Lorentz invariant phase space may be obtained via the relation $$d\tau = \frac{p_f d\Omega}{(4 \pi)^2 \sqrt{s}} 