= Solution
The <different-flavour quark annihilation into a massive pair> has only an $s$-channel <gluon> graph at tree level when the two <Quark flavour>[quark flavours] differ. Its <Scattering amplitude>[amplitude] contains one incoming and one outgoing <Color charge>[colour current], with common magnitude factor $g_s^2/s$. <On shell>[On-shell] current conservation removes the gauge-parameter-dependent longitudinal part of the <gluon> propagator. To display the <Dirac field>[Dirac] signs transparently, evaluate <Gamma-matrix trace>[spin traces] with the auxiliary <Minkowski metric>[mostly-minus] <Minkowski metric>[metric] $\eta=-g$ and $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}$; the invariants $s,t$ are then the usual positive-energy <Mandelstam variables>[Mandelstam invariants], numerically identical to those defined in the paper.
The incoming <Color factor>[colour average] and outgoing <Color factor>[colour sum] give
$$
C_{\mathrm{col}}=\frac1{3^2}\sum_{a,b}\operatorname{tr}(T^aT^b)\operatorname{tr}(T^aT^b)
=\frac19\frac14\delta^{ab}\delta^{ab}=\frac29.
$$
Averaging over the two <spins> of each incident <particle> supplies an additional $1/4$. The physical <spin sums> are $\sum u\bar u=\not p+M$ and $\sum v\bar v=\not p-M$. Therefore
$$
\boxed{\overline{|T|^2}=\frac{g_s^4}{18s^2}
\operatorname{tr}[\gamma_\mu(\not p_2-M)\gamma_\alpha(\not p_1+M)]\,
\operatorname{tr}[\gamma^\mu\not k_1\gamma^\alpha\not k_2].}
$$
The coefficient is equivalently $8\pi^2\alpha_s^2/(9s^2)$, with $\alpha_s=g_s^2/(4\pi)$.
\b[The two top projectors cannot both contain $-M$.] The printed expression has that error. It is not repaired by changing the <Minkowski metric>[metric] convention consistently: a <particle> and <antiparticle> still require the appropriate distinct projectors. In the auxiliary convention, its outgoing <tensor> would obey $q_\mu H_{\mathrm{printed}}^{\mu\alpha}=8M^2q^\alpha$, rather than zero for $q=p_1+p_2$. At <Particle-production threshold>[threshold] it gives a zero contracted <trace>, while the correctly averaged vector-current production <Scattering amplitude>[amplitude] squared has a nonzero limit $4g_s^4/9$. The <Lorentz-invariant phase space>[phase-space] <cross section> still vanishes at <Particle-production threshold>[threshold] because the final <velocity> tends to zero. The <Color charge>[colour] and <spin> prefactor above is consequently the prefactor for the corrected <trace>; no overall constant can make the misprinted massive <trace> the physical answer for all kinematics.
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