Chiral gauge theory
In quantum field theory, a chiral gauge theory is a quantum field theory with charged chiral fermions. For instance, the Standard Model, specifically the electroweak interaction, is a chiral gauge theory. For topological reasons, chiral charged fermions cannot be given a mass without breaking the gauge symmetry, which will lead to inconsistencies unlike a global symmetry. It is notoriously difficult to construct a chiral gauge theory from a theory which does not already contain chiral fields at the fundamental level. A consistent chiral gauge theory must have no gauge anomaly. Almost by necessity, regulators will have to break the gauge symmetry. This is responsible for gauge anomalies in the first place.