Massless neutrino-electron contact cross-section (${\sigma }_{\nu }=G_F^{2}s(c_V^2+c_V{c}_A+c_A^2)/(3\pi )$)

For a left-handed incoming neutrino, unpolarized electrons, and electron vector/axial couplings $c_V,c_A$, the four-fermion interaction gives $\overline{|\mathcal{M}{|}^2}=4G_F^2[(c_V+c_A{)}^2{s}^2+(c_V-c_A{)}^2{u}^2]$. Integrating massless two-body Lorentz-invariant phase space yields $\sigma =G_F^{2}s(c_V^2+c_V{c}_A+c_A^2)/(3\pi )$. The electron spin average is included, while no sterile neutrino helicity is averaged in.

Crossing a massless neutrino to an antineutrino reverses the antisymmetric part of its spin tensor. The $s^2$ and $u^2$ weights exchange, giving ${\sigma }_{\bar{\nu }}=G_F^{2}s[(c_V-c_A{)}^2+(c_V+c_A{)}^2/3]/(4\pi )$ for the same electron couplings. This is a low-energy contact statement rather than a replacement for a full weak-boson propagator.

Electron-neutrino scattering has a charged-current contribution absent for muon neutrinos. Fierz rearrangement combined with the external fermionic sign adds a left-electron current: $c_V\to c_V+1$, $c_A\to c_A+1$. The right-electron coupling stays unchanged; the left-electron coupling increases by two.