Axial anomaly and broken Lorentz invariance

[DrDu]

I had a look at Jackiws article on axial anomaly in scholarpedia:
http://www.scholarpedia.org/article/Axial_anomaly
Apparently, axial anomaly also breaks Lorentz invariance. Even if this effect would be very weak, doesn't this pull the plug on relativity?

 

[Physics Monkey]

Jackiw is not claiming that the axial anomaly breaks Lorentz invariance by itself. Instead, he is saying that the Chern-Simons term (closely associated to the axial anomaly), when added to a four dimensional Lagragian in the form $\epsilon^{abcd} n_a A_b \partial_c A_d$ with $n_a$ a fixed vector of your choice, leads to a Lorentz non-invariant theory. Basically you have to pick the vector n. This theory also breaks CPT if memory serves.

A much more natural place to add the Chern-Simons term is in 2+1 dimensions where it preserves Lorentz invariance and is topological.

 

[DrDu]

Ok, I understand.
Another question: Is it necessary to have a non-abelian gauge field to obtain an anomaly? I mean in general and not necessarily the axial anomaly. Would a U(1) not do?

 

[chrispb]

U(1) symmetries can indeed be anomalous.

 

[Polyrhythmic]

Ok, I understand.
Another question: Is it necessary to have a non-abelian gauge field to obtain an anomaly? I mean in general and not necessarily the axial anomaly. Would a U(1) not do?

U(1) symmetries can indeed be anomalous.

The axial anomaly of nonabelian gauge theories actually comes from a broken U(1) symmetry.
In QCD, you have $U(N_f)_L\times U(N_f)_R$ chiral symmetry, with both a conserved left-handed current $L^\mu$ and a right-handed one, $R^\mu$. It is now possible to make a linear combination of those two currents, leading to a vector current $V^\mu=(L^\mu+R^\mu)/2$ and an axial current $A^\mu=(L^\mu-R^\mu)/2$ (note: $A^\mu$ is not the gauge field). The symmetry group is now $U(N_f)_V\times U(N_f)_A$, which is isomorphic to the original one. It now decomposes as

$U(N_f)_V\times U(N_f)_A\equiv U(1)_V\times SU(N_f)_V\times U(1)_A\times SU(N_f)_A.$

You can now analyze each part separately, and see what it does. You get the following results:
$U(1)_V$ remains unbroken, it stands for baryon-number conservation.
$SU(N_f)_V$ is broken in the case when quarks have different masses.
$SU(N_f)_A$ is broken when quarks have nonzero mass.
$U(1)_A$ is broken at quantum level for non-vanishing quark masses, that's what is referred to as chiral/axial anomaly.

 

[AdrianTheRock]

In the last line, did you mean U(1)_A?

 

[Polyrhythmic]

In the last line, did you mean U(1)_A?

Yes, I did, thank you! :) Corrected it!

 

[DrDu]

Thank you, my question maybe aims at equation 19 in the scholarpedia article. As far as I can see, it is not stated to which field the potential A or the field strength F belongs. From the section "Mathematical Connections to Axial Symmetry Anomalies" I got the impression that it has to be a non-abelian gauge field. Which one and why does it have to be non-abelian?

Ok, I just learned that there are different anomalies depending on the particle type concerned (leptons or baryons) with F being either the electromagnetic or the gluon field (or both).
Apparently, either field will give rise to an anomaly. However, only in the case of non-abelian fields, the volume integral over the divergence of the axial current is non-vanishing when there are instantons.

 

[tom.stoer]

the basic ingredient are chiral structures; the vanishing of the sum of all chiral gauge-anomalies in the electro-weak sector of the standard model causes several constraints among the lepton couplings (electroweak isospin and hypercharge => electric charge).

Anyway - I don't get to the point why chiral anomalies shall have anything to do with broken Lorentz invariance

 

[DrDu]

Anyway - I don't get to the point why chiral anomalies shall have anything to do with broken Lorentz invariance

Dear Tom, I misunderstood what Jackiw was saying. PhysicsMonkey (post no. 2) cleared up this already. Jackiw had written an article (I don't have the reference at hand) about the cosequences of adding a Chern Simons term to the Lagrangian. That's not directly related to anomalies.