Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 4 / d / Solution

The gluon two-point function through one loop contains the tree propagator and these one-particle-irreducible insertions: a gluon loop with two three-gluon vertices, a gluon tadpole with one four-gluon vertex, a Faddeev-Popov ghost field loop, a fermion loop, and the gluon two-point counterterm. Gauge fixing is required before these Feynman diagrams and the propagator are defined.

Only the fermion loop changes when the fermion representation changes. Its two gauge vertices contain the representation matrices $T_R^a$, and its color factor isWith the conventional normalization, $T(F)=1/2$ for the fundamental representation and $T(\mathrm{adj})=C_A=N$ for the adjoint representation of $SU(N)$. The momentum and spinor integral is otherwise the same, apart from the number and type of fermion species; the pure-gluon and ghost diagrams are unchanged.