Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 3

For the convention in the question, a finite Yang-Mills gauge transformation acts covariantly on the field strength:or with $U$ and $U^{†}$ exchanged if the opposite convention is used for $D_{\mu }$. Cyclicity of the matrix trace givesso the Yang-Mills theory Lagrangian is gauge invariant.

The quadratic gauge-field operator has zero directions $A_{\mu }\sim A_{\mu }+D_{\mu }\alpha$ along each gauge orbit. It therefore has no inverse on the full field space. Gauge fixing removes this degeneracy and produces a propagator, while the Faddeev-Popov determinant accounts for the corresponding Jacobian.

Requiring $D_{\mu }\varphi$ to transform as $U^{†}(D_{\mu }\varphi )U$ and using the gauge-field transformation law givesIndeed, differentiating $U^{†}\varphi U$ produces two inhomogeneous derivative terms, and those cancel against the inhomogeneous part of the transformed connection.

For $U=1+i{\alpha }^a{T}_a+O({\alpha }^2)$,If $[T_a,T_b]=i{f}_{ab}{}^c{T}_c$, thenThis is the infinitesimal Adjoint representation of a Lie algebra.

At lowest derivative order, a general invariant effective Lagrangian through fourth order in the fields has the schematic formHere ${\lambda }_{abcd}$ is any invariant symmetric rank-four tensor, the ${\mathcal{I}}_A$ run over invariant contractions such as $(\mathrm{Tr}{\varphi }^2)\bar{\psi }\psi$ and $\bar{\psi }{\varphi }^2\psi$, and the ${\mathcal{J}}_B$ run over gauge- and Lorentz-invariant four-fermion contractions. The two sign symmetries forbid scalar cubic terms and Yukawa terms $\bar{\psi }\varphi \psi$. The covariant kinetic terms automatically contain the allowed cubic and quartic interactions involving $A_{\mu }$.

The canonical dimensions areand henceA coupling is relevant, marginal, or irrelevant when its dimension is positive, zero, or negative at the Gaussian fixed point. Thus gauge and scalar-quartic interactions are marginal in $d=4$, the mixed two-scalar fermion bilinear is marginal in $d=3$, and four-fermion interactions are marginal in $d=2$. Quantum corrections replace this engineering classification near an interacting fixed point by the eigenvalues of its RG stability matrix.

The one-loop four-scalar one-particle-irreducible diagrams, together with permutations of the external scalar labels, are:

The first four are forced by the scalar covariant derivative and scalar potential. There is no ordinary Yukawa fermion box because the imposed $\varphi \mapsto -\varphi$ symmetry forbids a one-scalar fermion vertex.