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

DefineThe gauge-field transformation is precisely the one for which the gauge covariant derivative $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$ transforms by $D_{\mu }^{\prime }=U{D}_{\mu }{U}^{†}$. Since $[D_{\mu },D_{\nu }]=-ig{F}_{\mu \nu }$, it follows immediately thatFor $U=1+i{\alpha }^c{T}_c+O({\alpha }^2)$,Define the structure constant of a Lie algebra by $[T_b,T_c]=i{f}_{bc}{}^a{T}_a$. Antisymmetry then gives

Writing the source interaction as $\mathrm{Tr}(J^{\mu }{A}_{\mu })$, gauge invariance requires the matrix current to transform in the Adjoint representation,The fermion bilinear with this transformation law isup to a convention-dependent overall coupling or generator normalization. It is the Noether current that appears when the free derivative is replaced by $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$.

Write $\Psi ={\Psi }^a{T}_a$. The infinitesimal form of $\Psi \mapsto U\Psi U^{†}$ isThe adjoint gauge covariant derivative iswhich transforms as $D_{\mu }\Psi \mapsto U(D_{\mu }\Psi )U^{†}$. A gauge-invariant Lagrangian is thereforeThe trace and cyclicity make each term invariant under conjugation.

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.