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

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- Solution v

Solution

For massless fermions the classical axial rotation has Noether current

J_{A}^{μ} = \sum_{i = 1}^{N_{f}} \overset{ˉ}{ψ}_{i} γ^{μ} γ^{5} ψ_{i} .

Promote

β

to a spacetime-dependent parameter. The classical variation, after integration by parts, is

- \int d^{4} x β \partial_{μ} J_{A}^{μ}

. The chiral anomaly is equivalently encoded by the stated shift

θ \mapsto θ + 2 N_{f} β

, which changes the theta action by

δ S_{θ} = \int d^{4} x β \frac{N _{f} g ^{2}}{8 π ^{2}} Tr (F_{μν}^{⋆} F^{μν}) .

The anomalous Ward identity is therefore

\partial_{μ} J_{A}^{μ} = \frac{N _{f} g ^{2}}{8 π ^{2}} Tr (F_{μν}^{⋆} F^{μν}),

so

α = N_{f} g^{2} / (8 π^{2})

in the trace convention of the question.

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