 Solution

ID: past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2016/1/b/i/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

η = - g

s = (p_{1} + p_{2})_{η}^{2}, t = (p_{1} - k_{1})_{η}^{2}, u = (p_{1} - k_{2})_{η}^{2}, 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; k l} = \sum_{a} (T^{a})_{ji} (T^{a})_{k l}

. Up to an irrelevant common phase,

M = \frac{g _{s}^{2}}{s} C_{ji; k l} [\overset{v}{ˉ} (p_{2}) γ^{μ} u (p_{1})] [\overset{u}{ˉ} (k_{1}) γ_{μ} 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{∣ M ∣^{2}}_{spin} = \frac{g _{s}^{4} ∣ C _{ji; k l} ∣ ^{2}}{4 s ^{2}} tr (\neq p_{2} γ^{μ} \neq p_{1} γ^{ν}) tr (\neq k_{1} γ_{μ} \neq k_{2} γ_{ν}) .

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

- u /2

. Therefore, before the remaining colour sum and average,

\overline{∣ M ∣^{2}}_{spin} = 2 g_{s}^{4} ∣ C_{ji; k l} ∣^{2} \frac{t ^{2} + u ^{2}}{s ^{2}} .

Here the bar averages spins only; it does not silently include the colour average requested next.

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