chiral anomaly



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In quantum field theory with chiral fermions (spinor fields) ψ\psi with chiral version of the Dirac current J=iΨ¯ΓΨJ = i \bar \Psi \Gamma \Psi, a chiral anomaly is a non-conservation of this current

divJ=Anomaly. div J = Anomaly \,.

(See at Ward identity.)

In the standard model of particle physics this happens and plays a role for pion decay? and for baryogenesis. Here the anomaly term is the Pontryagin 4-form Anomaly=F F Anomaly = \langle F_\nabla \wedge F_\nabla\rangle of the gauge field \nabla, hence the curvature 4-form of the corresponding Chern-Simons line 3-bundle.

If there are instantons, i.e. if the gauge field principal connection \nabla has a nontrivial underlying principal bundle, then also the Chern-Simons line 3-bundle is topologically nontrivial the anomaly term F F \langle F_\nabla \wedge F_\nabla\rangle is a non-exact integral form, hence the above equation is to be read as the local expression identifying j\star j with the local 3-connection on the CS 3-bundle.


The orginal observation is due to

  • Stephen Adler. Axial-Vector Vertex in Spinor Electrodynamics Physical Review 177 (5): 2426. (1969)

  • John Bell, Roman Jackiw, A PCAC puzzle: π 0γγ\pi^0 \to \gamma \gamma in the σ-model“. Il Nuovo Cimento A 60: 47. (1969)

A detailed mathematical derivation is in

Detailed argument for the theta vacuum (Yang-Mills instanton vacuum) from chiral symmetry breaking is offered in

Review includes

Discussion in the rigorous context of causal perturbation theory/perturbative AQFT is (for m>0m \gt 0) in

and (for m=0m = 0) in

and reviewed in the context the master Ward identity in

Application to baryogenesis is due to

  • Gerard 't Hooft, Symmetry Breaking through Bell-Jackiw Anomalies Phys. Rev. Lett. 37 (1976) (pdf)

  • Gerard 't HooftComputation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D14:3432-3450 (1976).

Last revised on March 24, 2019 at 09:57:03. See the history of this page for a list of all contributions to it.