What is a gauge anomaly?

A gauge anomaly refers to a situation in theoretical physics, particularly in quantum field theory, where a gauge symmetry that is expected to be preserved during quantization is actually violated. This occurs when the quantum version of the theory does not respect the gauge invariance that is present in the classical Lagrangian. In more detail, gauge theories are built on the principle of gauge invariance, meaning that the physical predictions should not depend on certain local transformations of the fields. An anomaly can arise if the calculations involving quantum corrections—like loops in Feynman diagrams—introduce terms that break this symmetry. The presence of a gauge anomaly can have significant implications: 1. **Inconsistency of the Theory**: If a gauge anomaly is present, it can render the quantum field theory inconsistent, suggesting that the quantum version of the theory cannot be defined consistently. 2. **Implications for Particle Physics**: Gauge anomalies play a crucial role in determining viable theories of particle physics. For example, the Standard Model is constructed to avoid gauge anomalies, ensuring that it is mathematically self-consistent. 3. **Anomalies and Topology**: Some anomalies have deep connections with the topology of the field configurations and can provide insights into the global properties of the gauge fields involved. A classic example of a gauge anomaly is the axial anomaly in quantum electrodynamics (QED) and quantum chromodynamics (QCD), where the conservation of a certain current is violated due to the effects of quantum fluctuations. Addressing gauge anomalies is crucial for building consistent models in particle physics and string theory.