= Solution
In pure <QCD>, different incoming and outgoing <quark flavours> allow only the tree-level $s$-channel <gluon> graph. A <quark-gluon vertex> preserves flavour, so no flavour-changing $t$-channel exchange is available. Label the incoming momenta $p_1,p_2$ and outgoing momenta $k_1,k_2$. For transparent <spin> <traces>, use the auxiliary metric $\eta=-g$ and its usual <gamma matrices>; the positive-energy <Mandelstam variables> are
$$
s=(p_1+p_2)_\eta^2,\qquad t=(p_1-k_1)_\eta^2,\qquad u=(p_1-k_2)_\eta^2,\qquad s+t+u=0.
$$
This is a consistent change of metric notation, not a change of the physical invariants. Define the fixed-index colour tensor $C_{ji;kl}=\sum_a(T^a)_{ji}(T^a)_{kl}$. Up to an irrelevant common phase,
$$
\mathcal M=\frac{g_s^2}{s}C_{ji;kl}
[\bar v(p_2)\gamma^\mu u(p_1)]
[\bar u(k_1)\gamma_\mu v(k_2)].
$$
The indices $i,j$ label incoming colours and $k,l$ outgoing colours. Averaging over the two <spins> of each incoming <particle> and summing the outgoing <spins> gives
$$
\overline{|\mathcal M|^2}_{\rm spin}
=\frac{g_s^4|C_{ji;kl}|^2}{4s^2}
\operatorname{tr}(\not p_2\gamma^\mu\not p_1\gamma^\nu)
\operatorname{tr}(\not k_1\gamma_\mu\not k_2\gamma_\nu).
$$
The <gamma-matrix trace> identity makes the contraction
$$
32[(p_1\cdot k_1)(p_2\cdot k_2)+(p_1\cdot k_2)(p_2\cdot k_1)]
=8(t^2+u^2).
$$
For instance $p_1\cdot k_1=p_2\cdot k_2=-t/2$ and the crossed products are $-u/2$. Therefore, before the remaining colour sum and average,
$$
\boxed{\overline{|\mathcal M|^2}_{\rm spin}
=2g_s^4|C_{ji;kl}|^2\frac{t^2+u^2}{s^2}.}
$$
Here the bar averages <spins> only; it does not silently include the colour average requested next.
\Image[/past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2016-qed-self-energy-and-quark-annihilation.png]
{title=One-loop QED fermion self-energy and different-flavour quark annihilation}
{description=The left panel shows the internal fermion and photon in the self-energy insertion. The right panel is the single pure-QCD tree topology for the specified flavour-changing pair final state; arrows show fermion-number flow.}
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