nLab 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

See also:

  • Valentin Benedetti, Horacio Casini, Javier M. Magan, ABJ anomaly as a U(1)U(1) symmetry and Noether’s theorem [arXiv:2309.03264]

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

Textbook account:

Further review:

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).

Via holographic QCD:

The WZW term of QCD chiral perturbation theory

The gauged WZW term of chiral perturbation theory/quantum hadrodynamics which reproduces the chiral anomaly of QCD in the effective field theory of mesons and Skyrmions:


The original articles:

See also:

  • O. Kaymakcalan, S. Rajeev, J. Schechter, Nonabelian Anomaly and Vector Meson Decays, Phys. Rev. D 30 (1984) 594 (spire:194756)

Corrections and streamlining of the computations:

  • Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (doi:10.1016/0370-2693(84)90986-9))

  • H. Kawai, S.-H. H. Tye, Chiral anomalies, effective lagrangians and differential geometry, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (doi:10.1016/0370-2693(84)90780-9)

  • J. L. Mañes, Differential geometric construction of the gauged Wess-Zumino action, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (doi:10.1016/0550-3213(85)90487-0)

  • Tomáš Brauner, Helena Kolešová, Gauged Wess-Zumino terms for a general coset space, Nuclear Physics B Volume 945, August 2019, 114676 (doi:10.1016/j.nuclphysb.2019.114676)

See also

Interpretation as Skyrmion/baryon current:

Concrete form for NN-flavor quantum hadrodynamics in 2d:

  • C. R. Lee, H. C. Yen, A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators, Chinese Journal of Physics, Vol 23 No. 1 (1985) (spire:16389, pdf)

Concrete form for 2 flavors in 4d:

  • Masashi Wakamatsu, On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action, (arXiv:1108.1236, spire:922302)

Including light vector mesons

Concrete form for 2-flavor quantum hadrodynamics in 4d with light vector mesons included (omega-meson and rho-meson):

Including heavy scalar mesons

Including heavy scalar mesons:

specifically kaons:

specifically D-mesons:


specifically B-mesons:

  • Mannque Rho, D. O. Riska, N. N. Scoccola, above (2.1) in: The energy levels of the heavy flavour baryons in the topological soliton model, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) (doi:10.1007/BF01283544)

Including heavy vector mesons

Inclusion of heavy vector mesons:

specifically K*-mesons:

Including electroweak interactions

Including electroweak fields:

Discussion for the full standard model of particle physics:

  • Jeffrey Harvey, Christopher T. Hill, Richard J. Hill, Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions, Phys. Rev. D77:085017, 2008 (arXiv:0712.1230)

Last revised on June 13, 2024 at 07:29:56. See the history of this page for a list of all contributions to it.