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Axial anomaly and broken Lorentz invariance

  1. Apr 4, 2012 #1

    DrDu

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    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?
     
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  3. Apr 4, 2012 #2

    Physics Monkey

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    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 [itex] \epsilon^{abcd} n_a A_b \partial_c A_d [/itex] with [itex] n_a [/itex] 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.
     
  4. Apr 4, 2012 #3

    DrDu

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    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?
     
  5. Apr 4, 2012 #4
    U(1) symmetries can indeed be anomalous.
     
  6. Apr 4, 2012 #5
    The axial anomaly of nonabelian gauge theories actually comes from a broken U(1) symmetry.
    In QCD, you have [itex]U(N_f)_L\times U(N_f)_R[/itex] chiral symmetry, with both a conserved left-handed current [itex]L^\mu[/itex] and a right-handed one, [itex]R^\mu[/itex]. It is now possible to make a linear combination of those two currents, leading to a vector current [itex]V^\mu=(L^\mu+R^\mu)/2[/itex] and an axial current [itex]A^\mu=(L^\mu-R^\mu)/2[/itex] (note: [itex]A^\mu[/itex] is not the gauge field). The symmetry group is now [itex]U(N_f)_V\times U(N_f)_A[/itex], which is isomorphic to the original one. It now decomposes as

    [itex]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.[/itex]

    You can now analyze each part separately, and see what it does. You get the following results:
    [itex]U(1)_V[/itex] remains unbroken, it stands for baryon-number conservation.
    [itex]SU(N_f)_V[/itex] is broken in the case when quarks have different masses.
    [itex]SU(N_f)_A[/itex] is broken when quarks have nonzero mass.
    [itex]U(1)_A[/itex] is broken at quantum level for non-vanishing quark masses, that's what is referred to as chiral/axial anomaly.
     
    Last edited: Apr 4, 2012
  7. Apr 4, 2012 #6
    In the last line, did you mean U(1)A?
     
  8. Apr 4, 2012 #7
    Yes, I did, thank you! :) Corrected it!
     
  9. Apr 5, 2012 #8

    DrDu

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    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.
     
    Last edited: Apr 5, 2012
  10. Apr 6, 2012 #9

    tom.stoer

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    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
     
  11. Apr 6, 2012 #10

    DrDu

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    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.
     
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