\label{eq:EQ220}
\mathscr{T} &= \frac{G_F}{\sqrt{2}} \bar{u}(\nu_e) \gamma^\mu (1 - \gamma_5) u(\nu_e) \bar{u}(e^-) \gamma_\mu \left( g_V^l + 1 - (g_A^l +1) \gamma_5 \right) u(e^-)\nonumber\\
&= \frac{G_F}{\sqrt{2}} \bar{u}(\nu_e) \gamma^\mu (1 - \gamma_5) u(\nu_e) \bar{u}(e^-) \gamma_\mu (C_V - C_A \gamma_5) u(e^-) \, \, \, .\end{aligned}$$ where $C_V \coloneqq g_V^l + 1$ and $C_A \coloneqq g_A^l +1$ for the scattering of an electron neutrino off an electron. Equation (\[eq:EQ220\]) provides an explanation for the differences in the interactions between electron and muon/$\tau$-lepton neutrini with the electron. As, below the temperature $T_1$, the CM energy is not sufficient for the creation of muons and $\tau$-leptons in the final state, the CC graph contributes to the scattering amplitude only in case of an incident electron neutrino. To be able to apply the results of the calculation also in the case of incident muon and $\tau$-lepton neutrini, I will retain the constants $C_V$ and $C_A$ (applicable for the scattering of an electron neutrino off an electron) and bear in mind that, in the case of an incident $\nu_{\mu,\tau}$, they must be replaced by $\widetilde{C}_V = C_V-1 = g_V^l$ and $\widetilde{C}_A = C_A-1 = g_A^l$. Evidently, $$\begin{aligned}
\lvert \mathscr{T} \rvert^2 = &\frac{G_F^2}{2} \, {\rm Tr} [{\ensuremath{\mathrlap{\!\not{\phantom{k}}}k}}^\prime \gamma^\mu (1-\gamma_5) {\ensuremath{\mathrlap{\!\not{\phantom{k}}}k}} 