\gamma_a \gamma_b] \Rightarrow {\rm Tr} [\gamma_a \gamma_b] = 4 g_{a b}\\
{\rm Tr} [\gamma_a \gamma_b \gamma_c \gamma_d] &= 4 \left( g_{ab} g_{cd} - g_{ac} g_{bd} + g_{ad} g_{bc} \right)\\
{\rm Tr} [\gamma_5 \gamma_a \gamma_b \gamma_c \gamma_d] &= 4 i \epsilon_{abcd}\\\end{aligned}$$ $A$, $B$, and $C$ denote square matrices of the same dimension. The traces of products of odd numbers of $\gamma$ matrices, as well as those of odd numbers of $\gamma$ matrices with the $\gamma_5$ matrix, vanish.

\[sec:Derivation\]Derivation of the various weak-interaction differential cross sections
========================================================================================

In this section, the scattering amplitudes, as well as the DCSs and TCSs derived thereof, of the various scattering processes of the neutrini with the (anti)neutrini, as well as with electrons and positrons, will be extracted. Sections \[sec:D1\]-\[sec:D5\] deal with the channels in which only (anti)neutrini appear in the initial and final states. Sections \[sec:D6\] and \[sec:D7\] deal with the scattering of neutrini off electrons and positrons, respectively. Finally, Section \[sec:D8\] pertains to the neutrino-antineutrino annihilation to an $e^+ e^-$ pair.

\[sec:D1\]Scattering of neutrini of different flavour
-----------------------------------------------------

The scattering amplitude $\mathscr{T}$ for the interaction of neutrini of different flavour was obtained in Section \[sec:Preliminary\], see Eq. (\[eq:EQ100\]). For an electron neutrino as projectile, the 