Table of Contents
- Introduction
- Classical Symmetries in Field Theories
- What Are Anomalies?
- Noether’s Theorem and Conservation Laws
- Path Integral and Anomalous Jacobians
- Chiral Anomalies
- Example: Axial Anomaly in QED
- Triangle Diagrams and Anomaly Calculations
- Anomaly Cancellation Conditions
- Gauge Anomalies vs Global Anomalies
- Gravitational Anomalies
- The Adler–Bell–Jackiw (ABJ) Anomaly
- The Role of γ₅ and Dimensional Regularization
- Anomalies in the Standard Model
- Witten’s Global SU(2) Anomaly
- Index Theorem and Topological Aspects
- The Atiyah–Singer Index Theorem
- Anomalies and Instantons
- Chern–Simons Terms and Anomalous Currents
- Gauge Invariance and Renormalizability
- The Green–Schwarz Mechanism
- Anomalies in String Theory
- Anomaly Inflow and Brane Dynamics
- Physical Consequences of Anomalies
- Conclusion
1. Introduction
Quantum anomalies occur when a symmetry of the classical action is broken upon quantization. Although the classical theory may have conserved currents corresponding to continuous symmetries, the quantum theory may fail to preserve them due to the regularization or renormalization process.
2. Classical Symmetries in Field Theories
In classical field theory, symmetries correspond to conserved currents via Noether’s theorem. These conservation laws typically survive quantization — except when an anomaly is present.
3. What Are Anomalies?
An anomaly is the breaking of a classical symmetry due to quantum effects. Anomalies usually emerge during the regularization of divergent integrals, especially in loop diagrams in perturbation theory.
4. Noether’s Theorem and Conservation Laws
Noether’s theorem relates continuous symmetries to conserved currents:
\[
\partial_\mu j^\mu = 0
\]
An anomaly manifests as:
\[
\partial_\mu j^\mu = \frac{e^2}{16\pi^2} F_{\mu\nu} \tilde{F}^{\mu\nu}
\]
5. Path Integral and Anomalous Jacobians
In the path integral formulation, anomalies arise from non-invariant measures. If the functional determinant changes under a symmetry transformation, it indicates an anomaly.
6. Chiral Anomalies
Chiral anomalies occur when left- and right-handed fermions transform differently under a gauge group. They appear in theories with massless fermions and chiral symmetry.
7. Example: Axial Anomaly in QED
The axial current \( j_5^\mu = \bar{\psi} \gamma^\mu \gamma^5 \psi \) is classically conserved in massless QED. However, loop calculations reveal:
\[
\partial_\mu j_5^\mu = \frac{e^2}{16\pi^2} F_{\mu\nu} \tilde{F}^{\mu\nu}
\]
This is the famous Adler–Bell–Jackiw (ABJ) anomaly.
8. Triangle Diagrams and Anomaly Calculations
Anomalies are calculated using triangle Feynman diagrams with one axial and two vector vertices. Regularization ambiguities cause current non-conservation.
9. Anomaly Cancellation Conditions
Gauge anomalies must cancel for consistency. In the Standard Model, anomalies cancel between different fermion generations due to specific charge assignments.
10. Gauge Anomalies vs Global Anomalies
- Gauge anomalies: break gauge invariance, render theory inconsistent.
- Global anomalies: affect global symmetries but don’t spoil consistency.
Both are important in building quantum field theories.
11. Gravitational Anomalies
Occur when fermionic currents couple to gravity. The non-conservation of energy-momentum tensor or gravitational currents can spoil diffeomorphism invariance.
12. The Adler–Bell–Jackiw (ABJ) Anomaly
One of the first anomalies discovered. It explains why the neutral pion \( \pi^0 \rightarrow \gamma\gamma \) decay occurs much faster than expected from symmetry arguments.
13. The Role of γ₅ and Dimensional Regularization
Dimensional regularization struggles with the treatment of \( \gamma^5 \) in \( d \neq 4 \). This technical issue makes calculating anomalies subtle and regularization-dependent.
14. Anomalies in the Standard Model
The SM is anomaly-free due to delicate cancellations:
- \( SU(2)^2 \times U(1) \)
- \( U(1)^3 \)
- \( \text{Gravitational} \times U(1) \)
These cancellations constrain possible fermion representations.
15. Witten’s Global SU(2) Anomaly
This is a global anomaly in SU(2) involving topological considerations. It occurs if a theory has an odd number of SU(2) doublets, which would spoil gauge invariance.
16. Index Theorem and Topological Aspects
The Atiyah–Singer Index Theorem relates the difference between zero modes of the Dirac operator and the topological charge of gauge fields, connecting anomalies to topology.
17. The Atiyah–Singer Index Theorem
\[
n_+ – n_- = \frac{1}{32\pi^2} \int d^4x\, F_{\mu\nu} \tilde{F}^{\mu\nu}
\]
This connects anomalies to instantons and topologically nontrivial configurations.
18. Anomalies and Instantons
Instantons are non-perturbative solutions that mediate tunneling between vacua. They contribute to anomalous processes that violate otherwise conserved symmetries (e.g., baryon/lepton number).
19. Chern–Simons Terms and Anomalous Currents
Chern–Simons terms can be added to cancel anomalies via boundary inflow. They are crucial in anomaly inflow mechanisms in string theory and condensed matter systems.
20. Gauge Invariance and Renormalizability
Uncancelled gauge anomalies destroy the renormalizability and consistency of a quantum field theory. Thus, all gauge anomalies must cancel for the theory to be viable.
21. The Green–Schwarz Mechanism
In string theory, anomalies are canceled via the Green–Schwarz mechanism, where a two-form field cancels anomalies through higher-dimensional interactions.
22. Anomalies in String Theory
Consistent string theories (like heterotic strings) must be anomaly-free. This imposes constraints on the allowed gauge groups and dimensions of spacetime.
23. Anomaly Inflow and Brane Dynamics
Anomalies on D-branes are canceled by bulk contributions flowing into the brane. This anomaly inflow ensures consistency of the combined bulk-brane system.
24. Physical Consequences of Anomalies
- Proton decay suppression
- \( \pi^0 \rightarrow \gamma\gamma \) decay
- Strong CP problem and axions
- Confinement and chiral symmetry breaking
Anomalies impact observable processes and constrain model building.
25. Conclusion
Quantum anomalies highlight the delicate interplay between symmetry and quantization. While often subtle, they carry profound implications — from the structure of the Standard Model to string theory and beyond. Understanding and managing anomalies is essential for constructing consistent quantum theories and probing the fundamental structure of nature.
