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 Incoming links: Photon-gluon colour orthogonality

Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2021 / 2 / a / iv / Solution 2026-10-25

There are two tree-level Feynman diagrams, with intermediate photon and gluon respectively. Their color charge tensors are

δ_{ji} δ_{k l}

and

\sum_{A} T_{ji}^{A} T_{k l}^{A}

. The pure photon factor is

C_{γ} = \frac{1}{N _{c}^{2}} \sum_{i, j, k, l} ∣ δ_{ji} δ_{k l} ∣^{2} = 1.

Their interference vanishes after the color charge sum because of photon-gluon colour orthogonality:

C_{γ g} = \frac{1}{N _{c}^{2}} \sum_{A} tr (T^{A}) tr (T^{A}) = 0.

Thus there is no term proportional to

e^{2} g_{s}^{2}

, and with the

F_{u}

of part (iii),

\overline{∣ M_{γ} + M_{g} ∣^{2}} = \frac{2 F _{u}}{s ^{2}} [e^{4} Q_{u}^{2} Q_{t}^{2} + \frac{2}{9} g_{s}^{4}] = \frac{F _{u}}{s ^{2}} [\frac{32}{81} e^{4} + \frac{4}{9} g_{s}^{4}] .

Here

Q_{u} = Q_{t} = 2/3

. The absence of interference concerns the specified averaged and summed result; a fixed choice of external color charge need not have zero interference.

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