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 current is the baryon current and 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.


Relation between Chiral anomaly, Skyrme model, Baryon current and WZ-term

The “topological”-part B topB_{top} of the baryon current (the piece that is not generally conserved, reflecting the chiral anomaly), is the Wess-Zumino-Witten term of the exponentiated pion field:

(1)e iπ/f π: 0,1×( 3) cpt= 0,1×S 3S 3SU(2) e^{i \vec \pi/f_\pi} \;\colon\; \mathbb{R}^{0,1} \times (\mathbb{R}^3)^{cpt} \;=\; \mathbb{R}^{0,1} \times S^3 \longrightarrow S^3 \simeq SU(2)
B top Tr((e iπ/f πde iπ/f π)(e iπ/f πde iπ/f π)(e iπ/f πde iπ/f π)) =(e iπ/f π) *(θ)(e iπ/f π) *(θ)(e iπ/f π) *(θ)Ω 3( 3,1) \begin{aligned} B_{top} & \coloneqq \; Tr \big( ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} ) \wedge ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} ) \wedge ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} ) \big) \\ & =\; \big\langle (e^{i \vec \pi/f_\pi})^\ast(\theta) \wedge (e^{i \vec \pi/f_\pi})^\ast(\theta) \wedge (e^{i \vec \pi/f_\pi})^\ast(\theta) \big\rangle \;\;\in\; \Omega^3(\mathbb{R}^{3,1}) \end{aligned}

Here the expression in the first line uses the fact that SU(2) is a matrix group, while the second line exporesses the same via pullback of the Maurer-Cartan form θ\theta from the group manifold.

The homotopy class of the exponentiated pion field (1), as a continuous function, is an element of the (co-)homotopy group of spheres π 3(S 3)π 3(S 3)\pi_3(S^3) \simeq \pi^3(S^3) \simeq \mathbb{Z}, is the Skyrmion-number, or, in fact, the baryon-number, encoded in the knotted stucture of the pion field.

See also


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 Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D14:3432-3450 (1976).

Last revised on March 25, 2020 at 10:28:46. See the history of this page for a list of all contributions to it.