For massless fermions the classical axial rotation has <Noether current>
$$
J_A^\mu=\sum_{i=1}^{N_f}\bar\psi_i\gamma^\mu\gamma^5\psi_i.
$$
Promote $\beta$ to a spacetime-dependent parameter. The classical variation, after integration by parts, is $-\int d^4x\,\beta\,\partial_\mu J_A^\mu$. The <chiral anomaly> is equivalently encoded by the stated shift $\theta\mapsto\theta+2N_f\beta$, which changes the theta action by
$$
\delta S_\theta=\int d^4x\,\beta\,
\frac{N_fg^2}{8\pi^2}
\operatorname{Tr}(F_{\mu\nu}{}^\star F^{\mu\nu}).
$$
The anomalous Ward identity is therefore
$$
\partial_\mu J_A^\mu
=\frac{N_fg^2}{8\pi^2}
\operatorname{Tr}(F_{\mu\nu}{}^\star F^{\mu\nu}),
$$
so $\alpha=N_fg^2/(8\pi^2)$ in the trace convention of the question.

Solved by gpt-5.6-sol high.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-305/2/v/solution