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In my (undergraduate) advanced elementary particles class last semester, we learnt that for a 2 quark (u/d) model the symmetry of the Lagrangian is (and breaks as)

$$ U(2)_L \otimes U(2)_R = SU(2)_L \otimes SU(2)_R \otimes U(1)_L \otimes U(1)_R \rightarrow SU(2)_V \otimes U(1)_B $$

where $SU(2)_V$ is the isospin symmetry and $U(1)_B$ is the baryon number conserving symemtry. On a first glance, according to the Goldstone theorem, we should be getting 4 pseudo-Goldstone bosons, but that doesn't seem to be the case, as the only pseudo-Goldstone bosons that I can find seem to be the three pions that are related to the $SU(2)$ chiral symmetry breaking. Upon doing some research, I found that this is related to an anomaly on the so called $U(1)_A$ group.

Similarly, for a 3 quark (u/d/s) model we have a symmetry that breaks as: $$ U(3)_L \otimes U(3)_R = SU(3)_L \otimes SU(3)_R \otimes U(1)_L \otimes U(1)_R \rightarrow SU(3)_V \otimes U(1)_B $$ According to the Goldstone theorem, we expect 9 pseudo-Goldstone bosons, and we can easily see that we have the three pions, the four Kaons, the eta meson and the eta prime meson. Except that the eta prime meson is much heavier than expected, and that's apparently due to (again) an anomaly in the breaking of the $U(1)_A$ group, which (apparently) also means that the eta prime is not a pseudo-Goldstone boson.

All of this leads me to ask the following four questions:

  1. What is the $U(1)_A$ group and how is it related to $U(1)_L \otimes U(1)_R$?
  2. What is this so called anomaly that's affecting $U(1)_A$?
  3. Why is there no 4th meson in the case of the 2 quark flavor model, but there is a 9th meson in the case of the 3 quark flavor model?
  4. If eta prime is not a pseudo-Goldstone boson, how would it be categorized/characterized?

I have tried to research and have found several other questions here, here, here, and here, but none seem to give a clear answer to any of my questions and I can't put the pieces together.

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    $\begingroup$ The linked questions answer all of your questions, one way or another. You should be able to put your pieces together; your list does not cohere, anyway. Review chiral anomalies. $\endgroup$
    – Cosmas Zachos
    Commented Jul 14, 2023 at 0:40
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    $\begingroup$ Further link. $\endgroup$
    – Cosmas Zachos
    Commented Jul 14, 2023 at 11:15
  • $\begingroup$ Do you understand that the η is, roughly, an SU(2) singlet, and an SU(3) octet, but the η' is both an SU(2) and SU(3) singlet? What kind of categorization for the η' are you asking for? It is a heavier-than-usual pseudoscalar meson... $\endgroup$
    – Cosmas Zachos
    Commented Jul 14, 2023 at 13:43
  • $\begingroup$ @CosmasZachos Thank you for the comments. I think my confusion stems from the fact that, while in SU(3) theory we point to the eta prime and say that it is a "would-be" Goldstone boson, as it is a pseudoscalar meson. There is no such analogue in SU(2) flavor symmetry as far as I can see. There is no pseudoscalar "would-be" goldstone meson composed out of u and d quarks. Why is that? $\endgroup$
    – Yaezir
    Commented Jul 14, 2023 at 15:05
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    $\begingroup$ I see, I think this clears up things a little. Thank you! $\endgroup$
    – Yaezir
    Commented Jul 14, 2023 at 16:44

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