Axial-Vector Vertex in Spinor Electrodynamics

1969/01/25 by Stephen L. Adler · 2 voices · 4,316 citations
Mathematics · Physics and Astronomy · #Atomic and Subatomic Physics Research #Black Holes and Theoretical Physics #Combinatorics #Graph #Massless particle #Mathematical physics #Mathematics #Meson #Perturbation theory (quantum mechanics) #Physics #Pseudovector #Quantum Chromodynamics and Particle Interactions #Quantum electrodynamics #Quantum mechanics #Spinor #Vector field #Vertex (graph theory) #Vertex function

paper · doi:10.1103/physrev.177.2426

published in Physical Review 177(5), 2426-2438 (American Institute of Physics)

openalex publication_date 1969/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Working within the framework of perturbation theory, we show that the axial-vector vertex in spinor electrodynamics has anomalous properties which disagree with those found by the formal manipulation of field equations. Specifically, because of the presence of closed-loop "triangle diagrams," the divergence of axial-vector current is not the usual expression calculated from the field equations, and the axial-vector current does not satisfy the usual Ward identity. One consequence is that, even after the external-line wave-function renormalizations are made, the axial-vector vertex is still divergent in fourth- (and higher-) order perturbation theory. A corollary is that the radiative corrections to \ensuremathν_ll elastic scattering in the local current-current theory diverge in fourth (and higher) order. A second consequence is that, in massless electrodynamics, despite the fact that the theory is invariant under \ensuremathγ₅ tranformations, the axial-vector current is not conserved. In an Appendix we demonstrate the uniqueness of the triangle diagrams, and discuss a possible connection between our results and the \ensuremathπ⁰\ensuremath→2\ensuremathγ and \ensuremathη\ensuremath→2\ensuremathγ decays. In particular, we argue that as a result of triangle diagrams, the equations expressing partial conservation of axial-vector current (PCAC) for the neutral members of the axial-vector-current octet must be modified in a well-defined manner, which completely alters the PCAC predictions for the \ensuremathπ⁰ and the \ensuremathη two-photon decays.

Citations

• The axial vector current in beta decay

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Discussions

• The initial valence quark assignment for the neutral pion came from the quark model, but the best direct experimental constraint on the quark content of the $\pi^0$ probably comes from its measured tw [stackexchange, 1 points]
• Continuing the saga! I've now read S. Adler (1969) "Axial-Vector Vertex in Spinor Electrodynamics" doi.org/10.1103/Phys... and THIS clarifies things tremendously! (Students: read Adler first, then Be [bsky, 0 points, 1 comments]

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• Anomalous Ward Identities in Spinor Field Theories
• Perturbative results of matrix elements of the axial current and their relation with the axial anomaly
• Partially conserved axial vector current and applications
• Axial vector current anomaly problem without regularization in Dyson scheme
• Effective field theory for radiative corrections to charged-current processes II: Axial-vector coupling