= Weak charged-current quark scattering
For the massless contact interaction $-(G_F/\sqrt2)[\bar d\gamma^\alpha(P+Q\gamma^5)u][\bar c\gamma_\alpha(P+Q\gamma^5)s]$ with real $P,Q$, spin and colour averaging gives
$$
\frac{d\sigma}{d\Omega}=\frac{G_F^2s}{128\pi^2}\left[(P^2+Q^2)^2(1+\cos^2\theta)+8P^2Q^2\cos\theta\right].
$$
The <gamma matrix trace identities> give the averaged amplitude $4G_F^2[(C^2+4P^2Q^2)(p_2\cdot k_1)(p_1\cdot k_2)+(C^2-4P^2Q^2)(p_2\cdot k_2)(p_1\cdot k_1)]$, where $C=P^2+Q^2$. Substitute the massless <Mandelstam variables> and divide by $64\pi^2s$. 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.
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