Contents

# Contents

## Idea

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

$div J = Anomaly \,.$

(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 $Anomaly = \langle F_\nabla \wedge F_\nabla\rangle$ 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 $\langle F_\nabla \wedge F_\nabla\rangle$ is a non-exact integral form, hence the above equation is to be read as the local expression identifying $\star j$ 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 $B_{top}$ 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)$e^{i \vec \pi/f_\pi} \;\colon\; \mathbb{R}^{0,1} \times (\mathbb{R}^3)^{cpt} \;=\; \mathbb{R}^{0,1} \times S^3 \longrightarrow S^3 \simeq SU(2)$
\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 $\theta$ 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 $\pi_3(S^3) \simeq \pi^3(S^3) \simeq \mathbb{Z}$, is the Skyrmion-number, or, in fact, the baryon-number, encoded in the knotted stucture of the pion field.

## 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: $\pi^0 \to \gamma \gamma$ 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 $m \gt 0$) in

and (for $m = 0$) 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.