Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2019 / 1 / a / i

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- Solution i

Solution

There are four topologies for the one-loop gluon self-energy, shown in the upper row of the figure:

A closed quark loop with the two external gluons attached through quark-gluon Feynman vertices, summed over active quark flavors.
A loop of two internal gluons joining two three-gluon Feynman vertices, each carrying one external gluon.
A closed Faddeev-Popov ghost field loop with two ghost-gluon Feynman vertices.
A single four-gluon Feynman vertex with two external gluons and its remaining two legs contracted into a tadpole diagram.

The two internal-gluon graphs have symmetry factor

1/2

. The directed quark and ghost loops have symmetry factor one and a minus sign for a closed anticommuting loop. The four-gluon tadpole diagram must be included in the list even though its massless Feynman integral vanishes in dimensional regularization. Local counterterms are excluded here.

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