In the $m_q \rightarrow 0$ limit the QCD lagrangian has the symmetry $U(N)_V \times U(N)_A$. Including just the two lightest quarks, $N=2$, and looking at the $U(2)_A=SU(2)_A \times U(1)_A$ part, we have upon SSB 8 PNGB's from the $SU(2)_A$. However, no goldstone boson is observed that matches the $U(1)_A$-part ("The $U(1)_A$ problem"). The way I see this explained is that $U(1)_A$ was never a real symmetry (the associated axial current is anomalous), and thus there is no SSB or GB either.
In the Peccei-Quinn solution to the strong CP problem one adds a new anomalous U(1) symmetry to cancel the CP-violating theta-term. How come we can have axions as a PNGB of this anomalous symmetry, when in the case of the $U(1)_A$ problem anomalousness was used to explain why there should not be a PNGB?
This question has been answered here: Spontaneous symmetry breaking of anomalous global abelian symmetries and $U(1)$ of QCD
The answer (I think) seems to say that for $U(1)_{PQ}$ the symmetry is exact in the $m_a \ll \lambda_{QCD}$ limit (the anomaly is small) and thus we get PNGBs from SSB, while for $U(1)_{A}$ the anomaly is not small so we get no goldstone bosons.
- Are anomalies then merely the reason for explicit symmetry breaking, and if this breaking is small enough (so that we can have SSB) we will have (pseudo-) goldstone bosons?
- What happens to the goldstone bosons between this limit where the anomaly is small and a limit where it isn't (maybe for example with time dependent axion mass around when the axion mass is "turned on" at $T=T_{QCD}$)?
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1@BrendanDarrer please stop inserting links where the OP did not see fit to insert any. – ZeroTheHero Jul 12 '22 at 13:42
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@ZeroTheHero: Ok, no problem! – Brendan Darrer Jul 12 '22 at 13:45
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2@ZeroTheHero: Speaking in general rather than this specific case, hyperlinks might be helpful and clarifying for some readers, and improve the question. – Qmechanic Jul 12 '22 at 13:55
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1@Qmechanic sure, adding links can be helpful if the links are to some relevant papers or sources but systematically editing questions by adding links to wikipedia articles is distracting. After all, the authors did not think that such links would be helpful. – ZeroTheHero Jul 12 '22 at 14:18
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@ZeroTheHero: Thanks for your feedback. I only meant to clarify the question with the links. But if you find it distracting, I will limit it my use of them, only to questions that would benefit from it. – Brendan Darrer Jul 12 '22 at 14:34
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1One assumes you are cool and unconflicted about pions as pseudogoldstone bosons of chiral flavor symmetry? – Cosmas Zachos Jul 12 '22 at 15:14
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1@CosmasZachos Yes – sb59 Jul 12 '22 at 15:36
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1So the added remark in the answer by @Thomas to the linked question settles the issue, then. – Cosmas Zachos Jul 12 '22 at 15:47
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@CosmasZachos Okay, so is the answer to the first question of mine in the last section "yes"? What about the second? Also, in Thomas' argument for $U(1){PQ}$ being exact, what happens to the $(\partial{\mu} a)^2$? Can't the second term dominate even in this $m_a \ll \lambda_{QCD}$ provided the kinetic part is very small? – sb59 Jul 12 '22 at 16:10
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@CosmasZachos The first "question" is really just a restatement of the linked post (although I realize it looks like thats the main question I have by how I wrote it, but in that case it would be a repost). The two questions at the end hardly qualify as a list question imo as one of them is a yes/no question. As to the question in my previous comment, I came up with it after posting the question. I'll edit the question – sb59 Jul 12 '22 at 16:24
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1@BrendanDarrer Links are sometimes necessary, but I honestly fail to see how adding links to wikipedia or other trivial sources fall in the necessary category and clarify much. Moreover, I personally would not substitute my judgment on what sources are needed to understand a question for the judgment of the OP and would ask for additional clarifications in a comment before editing a question by adding wiki links. – ZeroTheHero Jul 12 '22 at 17:53
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@ZeroTheHero: Thanks for the advice. I take your point! – Brendan Darrer Jul 12 '22 at 18:46
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Yes, both anomalies and quark masses provide the explicit breaking in the process, and $m_a\sim m_\pi f_\pi /f_a$, a tiny number. Normal texts absorb the huge $f_a$ into the normalization of a, in which your kinetic term question collapses: it deconstructs itself! *All* of your questions have been answered adequately unambiguously in the linked question. – Cosmas Zachos Jul 13 '22 at 13:25