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chiral anomaly in nLab

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

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Operator algebra

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Perturbative QFT

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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.

Properties

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:ℝ 0,1×(ℝ 3) cpt=ℝ 0,1×S 3⟶S 3≃SU(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(ℝ 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)≃ℤ\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 physics.stackexchange.com/a/306242/5603

References

General

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 0→γγ\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:

General

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.