Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part VII, Chapter 1

Anomalies

When quantum mechanics breaks classical symmetries: the profound consequences of anomalies

What is an Anomaly?

An anomaly is a symmetry of the classical action that is violated by quantum effects. What appears to be a perfect symmetry at the classical level fails to survive the process of quantization.

This is not a bug—it's a fundamental feature of quantum field theory! Anomalies have profound physical consequences and place powerful constraints on consistent quantum theories.

1.1 Classical Chiral Symmetry

Consider the Lagrangian for a massless fermion (like a massless quark):

$$\mathcal{L} = \bar{\psi} i \gamma^\mu \partial_\mu \psi$$

This Lagrangian has a chiral symmetry. We can decompose any Dirac fermion into left-handed and right-handed components:

\begin{align*} \psi_L &= \frac{1}{2}(1 - \gamma^5)\psi \\ \psi_R &= \frac{1}{2}(1 + \gamma^5)\psi \end{align*}

The massless fermion Lagrangian can be written as:

$$\mathcal{L} = \bar{\psi}_L i \gamma^\mu \partial_\mu \psi_L + \bar{\psi}_R i \gamma^\mu \partial_\mu \psi_R$$

This is invariant under independent U(1) rotations of left and right fields:

\begin{align*} \psi_L &\to e^{i\alpha_L} \psi_L \\ \psi_R &\to e^{i\alpha_R} \psi_R \end{align*}

Vector and Axial Symmetries

We can rewrite these as:

  • Vector symmetry (V): αL = αR = α
    $$\psi \to e^{i\alpha}\psi$$
  • Axial symmetry (A): αL = -αR = α
    $$\psi \to e^{i\alpha\gamma^5}\psi$$

1.2 Noether Currents

By Noether's theorem, each symmetry gives a conserved current. For the axial symmetry:

$$j_5^\mu = \bar{\psi} \gamma^\mu \gamma^5 \psi$$

Classically, this current is conserved:

$$\partial_\mu j_5^\mu = 0 \quad \text{(classical)}$$

But in quantum theory, this conservation law is violated!

1.3 The Triangle Diagram

The anomaly arises from the triangle diagram (also called AVV diagram):

          γ (photon)
           ↗   ↖
          /     \
    fermion loop
          \     /
           ↘   ↙
          γ (photon)
             ↑
        j₅μ (axial current)

One-loop Feynman diagram with an axial current insertion and two photon vertices.

Computing this triangle diagram with careful regularization yields:

$$\boxed{\partial_\mu j_5^\mu = \frac{e^2}{16\pi^2} F^{\mu\nu} \tilde{F}_{\mu\nu}}$$

where Fμν is the electromagnetic field strength and F̃μν = (1/2)εμνρσFρσis its dual. This is the famous Adler-Bell-Jackiw (ABJ) anomaly.

Key Insight

The term FμνF̃μν is proportional to E⃗ · B⃗, which measures the helicity of the electromagnetic field. The anomaly shows that axial charge is not conserved in the presence of electromagnetic fields!

1.4 Physical Consequences: π⁰ → γγ Decay

The most famous consequence of the chiral anomaly is the decay of the neutral pion:

$$\pi^0 \to \gamma + \gamma$$

Without the anomaly, this decay would be forbidden by the conservation of axial current. But the anomaly allows it, and the predicted decay rate agrees beautifully with experiment!

$$\Gamma(\pi^0 \to \gamma\gamma) = \frac{\alpha^2 m_{\pi}^3}{64\pi^3 f_\pi^2}$$

where fπ ≈ 93 MeV is the pion decay constant and α ≈ 1/137 is the fine structure constant.

Experimental Verification

  • Theory prediction: Γ ≈ 7.7 eV
  • Experimental value: Γ = 7.8 ± 0.5 eV
  • Outstanding agreement! One of the great triumphs of QFT.

1.5 Non-Abelian Anomalies

In non-Abelian gauge theories (like QCD or the Standard Model), we can have anomalies in the gauge currents themselves. For a gauge symmetry with generators Ta:

$$\partial_\mu j_a^\mu = \frac{g^2}{16\pi^2} d_{abc} F_b^{\mu\nu} \tilde{F}_{c,\mu\nu}$$

where dabc is a group theory factor (related to Tr[Ta{Tb, Tc}]).

Disaster if Non-Zero!

If a gauge symmetry is anomalous, the theory is inconsistent! Gauge invariance is essential for renormalizability and unitarity. An anomalous gauge theory makes no sense as a quantum theory.

1.6 Anomaly Cancellation in the Standard Model

The Standard Model must have its gauge anomalies cancel. The anomaly coefficient is proportional to:

$$\mathcal{A} \propto \sum_{\text{fermions}} Q_f^3$$

where the sum is over all fermions and Qf are their charges.

The Miraculous Cancellation

For one generation of quarks and leptons:

ParticleQQ³×3 colors
u (up quark)+2/3+8/27+24/27
d (down quark)-1/3-1/27-3/27
e (electron)-1-1-27/27
ν (neutrino)000
Total:0 ✓

The anomaly cancels exactly! This requires the precise particle content of each generation. If we had different charges or numbers of particles, the Standard Model would be inconsistent.

1.7 Global Anomalies

While gauge anomalies must cancel, global anomalies (like the chiral anomaly) are perfectly acceptable and even physically interesting. They tell us about the structure of the vacuum and can have observable consequences.

Types of Anomalies

  • Gauge anomalies: MUST cancel (theory inconsistent otherwise)
  • Global anomalies: CAN exist (interesting physics!)
  • Mixed anomalies: Involve both gauge and global symmetries
  • Gravitational anomalies: Involve coupling to gravity

1.8 Atiyah-Singer Index Theorem

The mathematical foundation of anomalies is the Atiyah-Singer index theorem, which relates the anomaly to topological properties of the gauge field configuration.

$$\text{Index}(\mathcal{D}) = \int d^4x \frac{g^2}{32\pi^2} F_{\mu\nu} \tilde{F}^{\mu\nu}$$

where D̸ is the Dirac operator. The index counts:

$$\text{Index} = n_{\text{zero modes, +}} - n_{\text{zero modes, -}}$$

This connects anomalies to instantons and the topology of gauge field configurations!

1.9 Anomalies Beyond the Standard Model

Anomaly cancellation is a powerful constraint on extensions of the Standard Model:

Grand Unified Theories (GUTs)

Must ensure anomaly cancellation in the unified gauge group. This constrains possible representations of matter fields.

String Theory

Green-Schwarz mechanism cancels anomalies in 10-dimensional superstring theory, requiring gauge groups SO(32) or E₈×E₈.

Beyond Standard Model Searches

Any new fermions or gauge bosons must preserve anomaly cancellation. This guides model building!

Key Takeaways

  • Anomalies: classical symmetries broken by quantum effects
  • Chiral (axial) anomaly: ∂μj5μ = (e²/16π²)FμνF̃μν
  • Triangle diagram is the source of the anomaly
  • Explains π⁰ → γγ decay beautifully
  • Gauge anomalies MUST cancel (Standard Model does!)
  • Anomaly cancellation constrains particle content
  • Related to topology via Atiyah-Singer index theorem
  • Powerful tool for BSM physics and theory building
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