Weak charged-current quark scattering

For the massless contact interaction $-(G_F/\sqrt{2})[\bar{d}{\gamma }^{\alpha }(P+Q{\gamma }^5)u][\bar{c}{\gamma }_{\alpha }(P+Q{\gamma }^5)s]$ with real $P,Q$, spin and colour averaging givesThe gamma matrix trace identities give the averaged amplitude $4G_F^2[(C^2+4P^2{Q}^2)(p_2\cdot k_1)(p_1\cdot k_2)+(C^2-4P^2{Q}^2)(p_2\cdot k_2)(p_1\cdot k_1)]$, where $C=P^2+Q^2$. Substitute the massless Mandelstam variables and divide by $64{\pi }^{2}s$. The two singlet colour contractions give factor one after final summation and initial averaging. This is a low-energy Fermi interaction result, not its high-energy extrapolation.

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.

For massless momenta, define $T^{\alpha \beta }(a,b)=\mathrm{Tr}[/a{\gamma }^{\alpha }(1-{\gamma }^5)/b{\gamma }^{\beta }(1-{\gamma }^5)]$. The gamma matrix trace identities and the antisymmetric tensor contraction give $T^{\alpha \beta }(a,b)T_{\alpha \beta }(c,d)=256(a\cdot c)(b\cdot d)$. Reversing an antiquark's spinor order changes the momentum pairing and creates the $(1-y{)}^2$ angular factor in a weak charged-current quark scattering rate.

For the massless common vector-axial coupling at both vertices in weak charged-current quark scattering, subtracting opposite directions givesIt is undefined for $P=Q=0$. Overall rate normalization cancels. Pure vector or axial interactions have zero asymmetry, while $P=\pm Q$ gives its maximal coefficient. The inequality $4P^2{Q}^2\le (P^2+Q^2{)}^2$ implies $|A|\le 1$. This unpolarized observable does not distinguish purely left- from purely right-handed couplings applied at both vertices.