nLab
chiral anomaly

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

renormalization

Theorems

States and observables

Operator algebra

Local QFT

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: 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 physics.stackexchange.com/a/306242/5603

References

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.