8 · quantum gauge theory and the standard model

classical non-abelian gauge theory

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. A principal $G$-bundle $P\to M$ carries connections $A$ whose curvature $$ F=dA+A\wedge A $$ transforms in the adjoint representation. The Yang-Mills action $$ S_{\mathrm{YM}}=-\frac14\int\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\,d^4x $$ is invariant under the infinite-dimensional group of gauge transformations $$ A\mapsto gAg^{-1}+g\,dg^{-1}. $$ Matter fields appear as sections of associated bundles and are minimally coupled through the covariant derivative $D_A$. The classical theory is thus completely determined by the geometry of $P$ and the choice of representation for the matter fields.

example (bianchi). the identity $D_A F=0$ is the geometric Bianchi identity for the curvature of a connection, recovered as an off-shell identity before any equation of motion is imposed.

the problem of gauge redundancy

In the path-integral formulation the naive functional integral $$ \int\mathcal{D}A\,e^{iS_{\mathrm{YM}}[A]} $$ diverges because of the infinite volume of the gauge group. Equivalently, in the canonical formalism the conjugate momentum to $A_0$ vanishes and the theory is constrained: the Gauss law generates gauge transformations and must annihilate physical states.

A consistent quantization must therefore eliminate (or otherwise control) the gauge redundancy while preserving Poincare invariance, unitarity, and locality.

remark (orbits). the classical configuration space is the space of connections modulo gauge transformations. quantization must be performed on that stack of orbits, not on the redundant space of all connections.

gauge fixing and the faddeev-popov procedure

One introduces a gauge-fixing condition $G(A)=0$ (for example the Lorenz gauge $\partial_\mu A^\mu=0$ or a Coulomb gauge). The Faddeev-Popov determinant $$ \det\Bigl(\frac{\delta G(A^g)}{\delta g}\Bigr) $$ compensates for the volume of the gauge orbit. Representing the determinant by an integral over anticommuting ghost fields $c$ and $\bar c$ yields an effective action $$ S_{\mathrm{eff}}=S_{\mathrm{YM}}+S_{\mathrm{gf}}+S_{\mathrm{ghost}} $$ that is no longer gauge invariant but is invariant under a new global symmetry: BRST symmetry.

example (abelian ghost). in QED the Faddeev-Popov operator is essentially $\square$ in Lorenz gauge, so the ghosts decouple from matter. in non-Abelian theories the ghost kinetic operator is the covariant Laplacian along the gauge orbit and ghosts circulate in loops.

brst symmetry and physical states

The BRST operator $Q$ is nilpotent, $Q^2=0$, and acts on the enlarged algebra of fields that now includes ghosts and auxiliary Nakanishi-Lautrup fields. Physical states are identified with the cohomology of $Q$: $$ Q|\mathrm{phys}\rangle=0,\qquad |\mathrm{phys}\rangle\sim|\mathrm{phys}\rangle+Q|\chi\rangle. $$ Equivalently, in the operator formalism the physical Hilbert space is the cohomology of the BRST charge at ghost number zero. Gauge-invariant operators descend to well-defined operators on this cohomology. The construction guarantees that the $S$-matrix is unitary on the physical subspace and independent of the choice of gauge-fixing condition.

example (qed as warmup). the Gupta-Bleuler condition of Chapter 6 is the abelian precursor of BRST cohomology: unphysical polarizations are $Q$-exact and drop from physical matrix elements.

yang-mills perturbation theory

With a gauge-fixed, BRST-invariant action one may expand about free fields and apply the Feynman rules. The quadratic part of the action supplies propagators for the gauge bosons, ghosts, and matter fields; the cubic and quartic terms generate interaction vertices.

Ultraviolet divergences appear at loop level and are absorbed by renormalization of the gauge coupling, wave-function factors, and, when present, matter masses and Yukawa couplings. The theory is renormalizable in four dimensions precisely because the gauge symmetry (or its BRST residue) severely restricts the form of counterterms.

example (three-gluon vertex). the non-Abelian cubic vertex carries structure constants $f^{abc}$ and a distinctive momentum factor antisymmetric in the legs. it has no abelian analogue and is the diagrammatic signature of $[A,A]$ in the curvature.

spontaneous symmetry breaking and the higgs mechanism

When the gauge symmetry is spontaneously broken by the vacuum expectation value of a scalar field transforming in a nontrivial representation of $G$, the spectrum changes qualitatively. Goldstone modes are absorbed by the gauge bosons, which acquire longitudinal polarizations and nonzero masses proportional to the vacuum expectation value. The residual unbroken subgroup remains massless.

The resulting theory is still renormalizable ('t Hooft) and describes massive vector bosons interacting with fermions and a physical Higgs scalar.

example (abelian higgs). a $U(1)$ scalar with $|\langle\phi\rangle|=v/\sqrt{2}$ gives the photon a mass $m_A=ev$ while eating the angular Goldstone mode, leaving one physical radial Higgs excitation.

anomalies

Not every classical gauge symmetry survives quantization. Chiral fermions can produce an anomalous divergence of the gauge current $$ \partial_\mu J^{a\mu}\propto\epsilon^{\mu\nu\rho\sigma}\operatorname{Tr}\bigl(T^a F_{\mu\nu}F_{\rho\sigma}\bigr). $$ Consistency of the quantum theory requires that the anomaly cancel when summed over all fermion species. In four dimensions the cancellation conditions are algebraic constraints on the representation content of the fermions.

Anomalous global symmetries, by contrast, are physically acceptable and give rise to important selection rules and nonperturbative effects (for example the $\mathrm{U}(1)_A$ anomaly of QCD).

remark (consistency). a gauged anomaly would destroy BRST nilpotency and unitarity. anomaly cancellation is therefore not optional phenomenology; it is a consistency condition on the quantum gauge theory.

the electroweak theory

The electroweak sector is based on the gauge group $\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$. Left-handed fermions transform as doublets under $\mathrm{SU}(2)_L$, right-handed fermions as singlets; hypercharges are assigned so that electric charge $Q=T_3+Y/2$ is anomaly-free.

A complex Higgs doublet acquires a vacuum expectation value that breaks the group to $\mathrm{U}(1)_{\mathrm{em}}$. The three broken generators produce the massive $W^\pm$ and $Z$ bosons; the unbroken generator produces the massless photon. Yukawa couplings of the Higgs to fermions generate fermion masses after symmetry breaking. The resulting theory, after inclusion of the strong interactions, is the Standard Model.

example (weak mixing). the neutral mass eigenstates are mixtures of $W^3$ and $B$ with Weinberg angle $\theta_W$, yielding $Z$ and the photon. charged currents are mediated by $W^\pm$ alone.

quantum chromodynamics

The strong interaction is described by an unbroken $\mathrm{SU}(3)_c$ gauge theory coupled to quarks transforming in the fundamental representation. Asymptotic freedom implies that the coupling decreases at short distances, justifying a perturbative treatment of high-energy processes. At long distances the coupling grows and the theory is believed to confine colour, producing colour-singlet hadrons as the physical asymptotic states.

Instantons, the $\theta$-vacuum, and the $\mathrm{U}(1)_A$ anomaly give rise to additional nonperturbative phenomena (the $\eta'$ mass, the strong CP problem, and related questions).

example (asymptotic freedom). the one-loop beta function of pure $\mathrm{SU}(N)$ Yang-Mills is negative. with $N_f$ light quark flavours the sign persists for $N_f$ below a definite bound, including the physical case of QCD.

the standard model as quantum gauge theory

The full Standard Model Lagrangian is the sum of the Yang-Mills terms for $\mathrm{SU}(3)_c\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$, the covariant kinetic terms for three generations of quarks and leptons, the Higgs kinetic term and potential, and the Yukawa interactions. After electroweak symmetry breaking the spectrum consists of massless gluons and the photon; massive $W^\pm$ and $Z$ bosons; a physical Higgs scalar; and massive quarks and leptons (except for the neutrinos in the minimal formulation).

All observed high-energy scattering processes among these particles are described by the corresponding BRST-cohomology classes of the quantized theory. Precision tests (electroweak observables, deep-inelastic scattering, jet physics, flavour-changing processes) confirm the structure to high accuracy.

summary

Classical gauge geometry is quantized by converting gauge redundancy into BRST cohomology. Gauge fixing and ghosts produce a perturbatively renormalizable theory whose physical states are the BRST-closed, gauge-invariant observables. Spontaneous symmetry breaking generates masses for vector bosons while preserving renormalizability; anomaly cancellation constrains the fermion content.

The resulting framework realises the non-Abelian connections of classical field theory as the gluons, $W$, $Z$, and photon of the Standard Model. Starting from a connection $A\in\Omega^1(P,\mathfrak{g})$, one passes to a gauge-fixed BRST complex with charge $Q$, and the physical content is the cohomology of gauge-invariant observables in $H^*(Q)$. That cohomological reduction is the passage from classical bundle geometry to experimentally verified quantum gauge theory.