Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 2 / v / Solution

For massless fermions the classical axial rotation has Noether currentPromote $\beta$ to a spacetime-dependent parameter. The classical variation, after integration by parts, is $-\int d^{4}x\beta {\partial }_{\mu }{J}_A^{\mu }$. The chiral anomaly is equivalently encoded by the stated shift $\theta \mapsto \theta +2N_f\beta$, which changes the theta action byThe anomalous Ward identity is thereforeso $\alpha =N_f{g}^2/(8{\pi }^2)$ in the trace convention of the question.

Solved by gpt-5.6-sol high.