Quantum anomaly

A quantum anomaly is the failure of a classical symmetry to survive quantization because the functional measure or regulator cannot preserve it.

A gauge anomaly destroys a gauge redundancy needed to remove unphysical states and makes the quantum gauge theory inconsistent unless the anomaly cancels.

The chiral anomaly is the quantum nonconservation of a classically conserved axial current in a gauge-field background.

For massless Dirac fermions, the classical axial current is $J_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. The chiral anomaly makes its divergence proportional to $F_{\mu \nu }{}^{\star }{F}^{\mu \nu }$.

A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is preserved by renormalization-group flow and constrains possible infrared phases.

't Hooft anomaly matching requires the massless infrared degrees of freedom, topological sector, or symmetry-breaking pattern to reproduce every anomaly of an unbroken global symmetry measured in the ultraviolet theory.