Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 2 / b / Solution

For a massless Dirac fermion,The local gauge symmetry gives the vector current $J^{\mu }=e\bar{\psi }{\gamma }^{\mu }\psi$. The global chiral transformation $\psi \mapsto e^{i\beta {\gamma }^5}\psi$ gives the classical axial currentUsing the massless Dirac equation and $\{{\gamma }^5,{\gamma }^{\mu }\}=0$ gives ${\partial }_{\mu }{J}_{\mathrm{ax}}^{\mu }=0$ classically.

Quantum mechanically, a gauge-invariant regulator for the fermion measure is not invariant under the axial rotation. In the Fujikawa form, its infinitesimal Jacobian containsThe first nonzero term in the heat kernel expansion is quadratic in $F_{\mu \nu }$. Usingand the Gaussian momentum integral gives the chiral anomalyThe overall sign follows the gamma-matrix, charge, and Levi-Civita conventions stated in the question. The same coefficient is obtained from the one-loop axial-vector-vector triangle diagram.