The Chiral (ABJ) Anomaly

Every symmetry argument so far — Noether's theorem, the Ward–Takahashi identities — assumed that a classical symmetry survives quantization. Anomalies are the exceptions: a symmetry of the classical action that is unavoidably broken by quantum effects. The prototype is the chiral (Adler–Bell–Jackiw) anomaly, the quantum non-conservation of the axial current. It is not a failure of the theory but a calculable, physical effect — it explains the pion's decay to two photons and constrains which gauge theories are consistent.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$; ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$.

Two currents, classically both conserved

A massless Dirac fermion coupled to electromagnetism has two $U(1)$ symmetries and hence two Noether currents:

$$j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi (\text{vector}),j^{\mu 5}=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi (\text{axial}).$$

The vector current counts particles minus antiparticles; the axial current counts right-handed minus left-handed. Classically, for massless fermions, both are conserved: ${\partial }_{\mu }{j}^{\mu }=0$ and ${\partial }_{\mu }{j}^{\mu 5}=0$ (the mass term is the only thing that would violate axial conservation, and it is absent).

The anomaly (Theorem)

Quantum mechanically, the two conservation laws cannot both hold. Regularizing the theory in a way that preserves the vector (gauge) current — mandatory, since gauge invariance is non-negotiable for consistency — forces the axial current to be non-conserved:

$$\boxed{{\partial }_{\mu }{j}^{\mu }=0,{\partial }_{\mu }{j}^{\mu 5}=\frac{e^2}{16{\pi }^2}{ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }{F}_{\rho \sigma }=\frac{e^2}{2{\pi }^2}\mathbf{E}\cdot \mathbf{B}.}$$

The right-hand side is a specific, finite, scheme-independent number — the anomaly. It is exact at one loop and receives no higher-order corrections (the Adler–Bardeen theorem). The axial charge is not conserved in a background of parallel $\mathbf{E}$ and $\mathbf{B}$ fields: chirality is created out of the vacuum.

Derivation 1: the triangle diagram

The anomaly first appeared in the one-loop triangle diagram with one axial and two vector vertices — the amplitude for $j^{\mu 5}\to \gamma \gamma$:

The loop integral is linearly divergent, so it must be regularized. The decisive fact: no regulator can preserve both the vector and axial Ward identities simultaneously. Imposing $q_{\mu }{\Gamma }^{\mu \nu \rho }=0$ for the two vector currents (gauge invariance) leaves the axial Ward identity violated by exactly the coefficient above. The anomaly is the unavoidable "leftover" of the triangle's ambiguity, pinned down by demanding gauge invariance.

Derivation 2: the path-integral measure (Fujikawa)

The cleaner, all-orders derivation is Fujikawa's: the anomaly lives in the path-integral measure. Under a chiral rotation $\psi \to e^{i\alpha {\gamma }^5}\psi$, the classical action of a massless fermion is invariant — but the Grassmann measure $\mathcal{D}\bar{\psi }\mathcal{D}\psi$ is not. Its Jacobian is a determinant that, carefully regularized, equals exactly the anomaly:

$$\mathcal{D}\bar{\psi }\mathcal{D}\psi ⟶\mathcal{D}\bar{\psi }\mathcal{D}\psi \exp \left(-i\int d^{4}x\alpha (x)\frac{e^2}{16{\pi }^2}{ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }{F}_{\rho \sigma }\right).$$

This is the failure case of the Ward-identity derivation: the identity ${\partial }_{\mu }⟨j^{\mu 5}\cdots ⟩=0$ assumed an invariant measure, and here that assumption is false. The Fujikawa method makes the anomaly's topological character manifest — the right-hand side is a total derivative, $ϵFF={\partial }_{\mu }{K}^{\mu }$, with $K^{\mu }$ the Chern–Simons current, tying the anomaly to the topology of the gauge field.

Physical consequence: ${\pi }^0\to \gamma \gamma$

The anomaly is not academic — it quantitatively predicts the decay of the neutral pion. Treating the pion as the pseudo-Goldstone of chiral symmetry, its coupling to two photons is fixed entirely by the anomaly, giving

$$\Gamma ({\pi }^0\to \gamma \gamma )=\frac{{\alpha }^2{m}_{\pi }^3}{64{\pi }^3{f}_{\pi }^2}\approx 7.7\text{eV},$$

in striking agreement with experiment. Historically this was decisive evidence for three colors of quarks: the rate is proportional to $N_c^2$, and only $N_c=3$ matches. The anomaly thus provided an early, clean measurement of the color quantum number of QCD.

Types of anomaly: when is it fatal vs. useful?

A global anomaly is a genuine prediction. A gauge anomaly is a catastrophe: it would break the gauge invariance that the BRST proof of unitarity relies on, rendering the theory inconsistent. Any would-be gauge theory must therefore have its gauge anomalies cancel — a powerful constraint developed on the next page.

Summary

• An anomaly is a classical symmetry broken unavoidably by quantization.
• The chiral anomaly: ${\partial }_{\mu }{j}^{\mu 5}=\frac{e^2}{16{\pi }^2}ϵFF\ne 0$, exact at one loop.
• Two derivations: the triangle diagram (no regulator saves both Ward identities) and the Fujikawa non-invariant measure (all-orders, topological).
• Global anomalies are physical (${\pi }^0\to \gamma \gamma$, measures $N_c=3$); gauge anomalies are fatal and must cancel.

Where this leads

• Gauge-anomaly cancellation in the Standard Model: anomaly cancellation.
• The topological side ($ϵFF$, $\theta$-vacuum): solitons, instantons and topology.
• The index-theorem viewpoint: the anomaly is the analytical index of the Dirac operator, $n_{+}-n_{-}=\frac{1}{8{\pi }^2}\int \mathrm{tr}(F\wedge F)$ — see math/differential-geometry: the Atiyah–Singer index theorem.
• The measure derivation: Ward–Takahashi identities.

References

• Adler, Phys. Rev. 177, 2426 (1969); Bell & Jackiw, Nuovo Cim. A 60, 47 (1969).
• Fujikawa, Phys. Rev. Lett. 42, 1195 (1979).
• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 19.
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 22.