Neutrino scattering on an antiquark ($d{\sigma }_{\bar{u}}/dy=G_F^{2}s(1-y{)}^2/\pi$)

For a massless neutrino scattering on an unpolarized up antiquark through the Fermi interaction, $d\sigma /dy=G_F^{2}s(1-y{)}^2/\pi$. The corresponding down-quark channel has $d\sigma /dy=G_F^{2}s/\pi$. Their difference follows from the chiral weak-current trace contraction. The parton distribution functions weight these rates in a hadron.