Anomaly

Remarks

Example: in quantum electrodynamics with one massless Dirac fermion, the axial transformation $\psi \mapsto e^{i\alpha\gamma^5}\psi$ leaves the classical action invariant. Its current $j^{\mu 5} = \bar\psi\gamma^\mu\gamma^5\psi$ satisfies, with $\epsilon^{0123} = +1$, $$\partial_\mu j^{\mu 5} = \frac{e^2}{16\pi^2}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$ The right side is not removable by a local counterterm compatible with gauge invariance, so the axial symmetry is anomalous.

Non-example: the vector current $j^\mu = \bar\psi\gamma^\mu\psi$ of the same theory stays conserved with a gauge-invariant regularization. So the phase symmetry $\psi \mapsto e^{i\alpha}\psi$ is not anomalous.

In the path integral, Fujikawa's view is that the anomaly is the Jacobian of the transformation of the fermion measure. In the presence of several currents one can often move the anomaly from one current to another, but not remove it from all of them.