Anomalies

A classical symmetry can be destroyed by quantum corrections — this is an anomaly. The most famous is the chiral (ABJ) anomaly: the axial current is classically conserved for massless fermions but receives a non-zero divergence at one loop from triangle diagrams with two photon legs. Anomalies in global symmetries are physically real (explaining π⁰→2γ); anomalies in gauge symmetries would destroy the theory's consistency. In the Standard Model, gauge anomalies cancel precisely — a non-trivial constraint that requires exactly 3 quark colors.

Key Concepts

Classical chiral symmetry: ψ → e^{iαγ⁵}ψ, j^{5μ} = ψ̄γμγ⁵ψ, classically ∂μj^{5μ}=0 for m=0
ABJ anomaly: ∂μj^{5μ} = (e²/16π²)εμνρσFμνFρσ = (e²/4π²)E·B — from triangle diagram
Triangle diagram: axial current vertex + two photon vertices; UV finite but anomalous by regularization
π⁰→2γ: explained by the chiral anomaly — the amplitude is entirely determined by the anomaly coefficient
Gauge anomaly: non-conservation of gauge current → non-unitary S-matrix → theory inconsistent
SM anomaly cancellation: Σ_L Y³ − Σ_R Y³ = 0 and others hold exactly per generation — requires N_c = 3

Key Equations

ABJ anomaly

\partial_\mu j^{5\mu}=\frac{e^2}{16\pi^2}\varepsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}=\frac{e^2}{4\pi^2}\mathbf{E}\cdot\mathbf{B}

π⁰→γγ amplitude

\mathcal{M}(\pi^0\to\gamma\gamma)=\frac{\alpha N_c}{\pi f_\pi}\varepsilon^{\mu\nu\rho\sigma}\varepsilon^*_\mu k_{1\nu}\varepsilon^{\prime*}_\rho k_{2\sigma}

Hypercharge anomaly

\sum_{\rm left}Y^3-\sum_{\rm right}Y^3=0\quad\Rightarrow\quad N_c=3

Index theorem

n_+-n_-=\frac{g^2}{16\pi^2}\int F^a\tilde F^a\,d^4x\quad(\text{Atiyah-Singer})

Worked Example

Example Problem

Problem

The π⁰→2γ decay rate is Γ = α²m³_π/(64π³f²_π). Using mπ = 135 MeV and fπ = 93 MeV, estimate Γ and compare to the measured value ~7.7 eV.

Solution

Γ = (1/137)²×(0.135)³/(64π³×0.093²) GeV = (5.33×10⁻⁵)×(2.46×10⁻³)/(64×31.0×8.65×10⁻³) = 1.31×10⁻⁷/(17.18) ≈ 7.6×10⁻⁹ GeV = 7.6 eV ✓.

Key Takeaways

The ABJ anomaly breaks the classical chiral symmetry at the quantum level: ∂μj^{5μ} ≠ 0 due to one-loop triangle diagrams
Anomalies in global symmetries are physical (π⁰→2γ, strong CP problem); anomalies in gauge symmetries destroy consistency (non-unitary S-matrix)
In the SM, gauge anomalies cancel within each generation — the precise cancellation requires exactly 3 quark colors (N_c=3)
The Atiyah-Singer index theorem gives a topological interpretation: the anomaly coefficient equals the topological index of the Dirac operator