= 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 is
$$
\operatorname{Tr}_R(T_R^aT_R^b)=T(R)\delta^{ab}.
$$
With 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.
Back to article page
