Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2016 / 1 / b

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 areThis 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,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 givesThe gamma-matrix trace identity makes the contractionFor 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,Here the bar averages spins only; it does not silently include the colour average requested next.

Use the fundamental SU(3) normalization $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, with Hermitian generators. The average over the $3\times 3$ incoming colours and sum over the outgoing colours givesThe complete spin and colour averaged result is consequentlyFor massless two-body centre-of-momentum kinematics, $t=-s(1-\cos \theta )/2$ and $u=-s(1+\cos \theta )/2$, so the same answer is $2g_s^4(1+{\cos }^2\theta )/9$. This checks positivity, the symmetry under exchanging the outgoing directions, and the absence of a forward $t$-channel singularity in this particular process.