Adler–Bardeen Chiral Anomaly

Triangle (AVV): One axial vertex (A) + two vector vertices (V). The diagram has a linear divergence — momentum routing ambiguity (shift p→p+aΔ changes the answer).
Bardeen (1969): Cannot simultaneously preserve both vector AND axial Ward identities. Choice: preserve vector (gauge symmetry) → axial anomaly: ∂_μj^μ_5 = e²/(16π²) ε^{μνρσ}F_{μν}F_{ρσ} = e²/(8π²) E·B
Adler-Bardeen theorem: The anomaly is exact — receives contributions only at one loop; all higher loops cancel (non-renormalization theorem). This makes anomaly cancellation in the Standard Model exact.