Anomaly Cancellation and Consistency

The chiral anomaly is harmless — indeed useful — when it afflicts a global symmetry. When it afflicts a gauge symmetry it is fatal: it destroys the gauge invariance that underwrites unitarity and renormalizability. A consistent chiral gauge theory must therefore have its gauge anomalies cancel. This requirement is not a technicality — it constrains the particle content so tightly that it essentially predicts the structure of a Standard Model generation.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Why gauge anomalies must cancel

A gauge anomaly means the gauge current is not conserved at the quantum level, ${\partial }_{\mu }{j}^{\mu a}\ne 0$. This is catastrophic because the entire consistency of a gauge theory rests on gauge invariance:

• The Ward/Slavnov–Taylor identities that make the unphysical polarizations decouple fail, so unitarity is lost (negative-norm states leak into physical amplitudes).
• The BRST cohomology no longer defines a sensible physical Hilbert space.
• Renormalizability ('t Hooft's proof) breaks down.

So a chiral gauge theory is consistent only if the total gauge anomaly vanishes.

The anomaly coefficient

The gauge anomaly comes from the triangle diagram with three gauge currents. Its coefficient is a group-theory trace over all chiral fermions running in the loop:

$$\boxed{{\mathcal{A}}^{abc}=\mathrm{Tr}[T^a\{T^b,T^c\}{]}_{\text{L}}-\mathrm{Tr}[T^a\{T^b,T^c\}{]}_{\text{R}}=0\text{(required)}.}$$

The theory is anomaly-free iff this symmetric trace vanishes, summed over the representation content, with left- and right-handed fermions contributing with opposite sign. Two ways to satisfy it:

• Non-chiral (vector-like) theories — left and right transform identically, so the two traces cancel term by term. QED and QCD are automatically safe.
• Chiral theories — left and right differ (the electroweak sector); cancellation is a non-trivial constraint on the charges.

Groups with no symmetric invariant $d^{abc}=\mathrm{Tr}[T^a\{T^b,T^c\}]$ — such as $SU(2)$ — are automatically anomaly-free; only $SU(N\ge 3)$ and $U(1)$ factors can be dangerous.

The Standard Model miracle

The electroweak theory is chiral — only left-handed fermions form $SU(2{)}_L$ doublets — so anomaly cancellation is a real constraint. Remarkably, it holds exactly, but only when quarks and leptons are combined and summed over color. The gauge-anomaly conditions ($SU(2{)}^{2}U(1)$, $U(1{)}^3$, and the mixed gauge–gravitational $U(1)$) all reduce to sums of hypercharges that vanish generation by generation. The cleanest example is the $U(1{)}_Y$-gravitational anomaly, ${\sum }_f{Y}_f=0$ over a generation:

$$\underset{\text{quark doublet}\times N_c}{\underbrace{3\times 2\times \frac{1}{6}}}+\underset{\text{quark singlets}}{\underbrace{3\times (-\frac{2}{3}-\frac{1}{3}\times \cdots )}}+\underset{\text{lepton doublet}}{\underbrace{2\times (-\frac{1}{2})}}+\underset{e_R}{\underbrace{(+1)}}=0.$$

The factor of 3 from color is essential: leptons alone do not cancel, quarks alone do not cancel, but quarks + leptons together do. Anomaly cancellation thus demands that quarks and leptons come in matched sets — a deep hint of quark–lepton unification and part of why the number of colors and the fractional quark charges are what they are.

't Hooft anomaly matching

Anomalies of global symmetries, while not fatal, obey a powerful constraint: 't Hooft anomaly matching. Because the global anomaly coefficient is a renormalization-group invariant (it cannot change under the RG flow), the anomaly computed with the UV degrees of freedom must equal the anomaly computed with the IR degrees of freedom. This is a nonperturbative consistency condition linking short- and long-distance physics:

• In QCD, the chiral anomaly computed with quarks (UV) must match that computed with the pion and baryons (IR) — a check on the pattern of chiral symmetry breaking and confinement.
• It constrains proposals for composite or new-physics models: any IR theory must reproduce the UV anomalies.

Global vs. gauge anomalies, and Witten's $SU(2)$ anomaly

Beyond the perturbative (triangle) anomaly, there is a subtler global (non-perturbative) gauge anomaly — Witten's $SU(2)$ anomaly — arising from the topology of the gauge group (${\pi }_4(SU(2))={\mathbb{Z}}_2$). A theory with an odd number of $SU(2{)}_L$ fermion doublets is inconsistent regardless of the perturbative cancellation. The Standard Model has an even number per generation (thanks, again, to color $\times$ generations), so it passes this test too.

Summary

• Gauge anomalies must cancel or the theory loses unitarity and renormalizability.
• Cancellation condition: the symmetric trace $\mathrm{Tr}[T^a\{T^b,T^c\}{]}_L-(L\to R)=0$; vector-like theories are automatic.
• In the Standard Model the anomalies cancel only when quarks and leptons (with the factor of 3 from color) are combined — a hint of unification.
• 't Hooft matching constrains global anomalies across the RG; Witten's anomaly is a topological consistency check.

Where this leads

• The theory whose consistency this guarantees: the Standard Model.
• The topological origin of anomalies: solitons, instantons and topology.
• The anomaly itself: the chiral anomaly.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 20.2.
• 't Hooft, in Recent Developments in Gauge Theories (1980).
• Witten, Phys. Lett. B 117, 324 (1982).
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 22.4.