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 Incoming links: Chiral anomaly

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

- \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}),

α = N_{f} g^{2} / (8 π^{2})

in the trace convention of the question.

Solved by gpt-5.6-sol high.

Classically the

p

antifundamentals admit

U (p)

flavour rotations and

λ

admits an independent phase rotation, giving

S U (p) \times U (1)_{ψ} \times U (1)_{λ}

U (1)

with charges

q_{λ}, q_{ψ}

is free of the mixed

S U (N)^{2} U (1)

anomaly when

(N - 2) q_{λ} + p q_{ψ} = 0.

Using

p = N - 4

, choose

q_{λ} = N - 4, q_{ψ} = - (N - 2) .

The orthogonal axial phase has a chiral anomaly, while this combination survives. The continuous quantum global symmetry is therefore

S U (p) \times U (1)

, again up to possible finite central quotients.

Solved by gpt-5.6-sol high.