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In the references I read, it's always said that the Peccei-Quinn (PQ) symmetry is a $U(1)$ symmetry, and the PQ Lagrangian has a potential term that, for a certain temperature, assumes a mexican-hat shape that spontaneously breaks this symmetry. The Goldstone mode of this SSB is the axion, and the axion does many awesome things, and so on.

Now, in other SSBs (Higgs, chiral QCD...) the symmetry that is eventually broken has, indeed, some significance even when it's unbroken, in that it acts on the degrees of freedom of the theory in a way that does not change the theory (otherwise it wouldn't be a symmetry), and then the breaking creates Goldstone modes such as the pions, and so on. For me, understanding how the nonbroken symmetry behaves is crucial to understanding its breaking.

What is the Peccei-Quinn symmetry (apart from "something that, once broken, produces the axion")?

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    $\begingroup$ Understood eq (29-30) of this? Or the simpler model? The P-Q symmetry is any symmetry resulting in the requisite posited effective lagrangian. $\endgroup$ Dec 16, 2023 at 15:00
  • $\begingroup$ Near duplicate. $\endgroup$ Dec 16, 2023 at 17:36
  • $\begingroup$ @CosmasZachos your first comment is on point, thanks for the references (and for the answer!). The linked post, however, asks a very different thing, regarding a PQ transformation for fermions. $\endgroup$ Dec 16, 2023 at 18:20
  • $\begingroup$ Wilczek's paper details the hypercharge-orthogonal PQ chiral U(1) even more elegantly. $\endgroup$ Dec 16, 2023 at 20:31

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From your interaction in the comments with Cosmas Zachos it isn't entirely clear what you're looking for, but let me expand on the information they have given you.

Fundamentally, the Peccei-Quinn symmetry must be a symmetry acting on (some of the) colored fermions. This is crucial because the point of such a PQ symmetry is that it has a mixed anomaly with $SU(3)_C$, $\mathcal{A}\neq 0$ where $$ \mathcal{A} \delta^{ab} = \sum_i \text{Tr}\left(q_i T^a_{R_i} T^b_{R_i}\right) $$ is a sum over left-handed Weyl fermions $i$ with charges $q_i$ under the PQ symmetry which are in representations $R_i$ of $SU(3)_C$ which have generators $T_{R_i}^a$. This anomaly is roughly counting the fermionic zero-modes that an $SU(3)_C$ instanton has, weighted by their PQ charges, so it is crucial that the PQ symmetry is one which acts on fermions.

This non-vanishing mixed anomaly means that while the PQ symmetry is classically a good global symmetry, quantum mechanically it is violated by non-perturbative effects from $SU(3)_C$. That is, let us perform a PQ transformation by an angle $\alpha$, $\psi_i \rightarrow \psi_i e^{i \alpha q_i}$. While the Lagrangian is invariant under this transformation (since we said it was a good classical global symmetry), it modifies the path integral measure (c.f. Fujikawa) and so the partition function of the theory, effectively modifying the action as $$ S \rightarrow S + \alpha \mathcal{A} \int_\mathcal{M} \frac{\text{Tr} F \tilde{F}}{16 \pi^2} $$ where $F$ is the $SU(3)_C$ gauge field strength, $\tilde{F}$ the Hodge dual, and $\mathcal{M}$ whatever manifold we're working on.

This, then, is the unbroken PQ symmetry. It effects a change in the Lagrangian by the 'topological theta term' of the gauge theory, $\theta \text{Tr} F \tilde{F}$, which is a total derivative and so does not affect local dynamics. If this is the only PQ-violating effect, then one may use such a PQ transformation to modify the coefficient of this term as we wish, showing that the term is entirely unphysical.

If the PQ symmetry is spontaneously broken, as you say this results in an axion field which couples like the theta angle, $a(x) \text{Tr} F(x) \tilde{F}(x)$, and if the axion potential $V(a)$ has its minimum at $a=0$, this dynamically explains why we observe that the coefficient of $F(x) \tilde{F}(x)$ is extremely close to zero, which is the strong CP problem. This solves the strong CP problem using a spontaneously broken PQ symmetry which is 'high quality' (a good symmetry apart from the anomaly).

Now since you are particularly interested in unbroken PQ symmetry, let me tell you that it is also possible for a high quality PQ symmetry which is unbroken to solve the strong CP problem. This was the original hope of the 'massless up quark solution', which would have been the most beautiful way the problem could have been solved in the SM itself. Like we said above, if the only PQ-breaking effect is the anomaly, then the theta angle is unphysical. Is it possible that there is such a PQ symmetry acting on the SM fermions which is unbroken? A chiral rotation of only the up quark $\bar u \rightarrow e^{i \alpha} \bar u$ (where this is the 'right-handed' up quark written as a left-handed Weyl fermion) is such a symmetry.

Such a symmetry forbids a mass for the up quark, $H Q \bar u$. But we know we measure a nonzero up quark mass in the far infrared, which is a violation of this PQ symmetry. Could this nonzero up quark mass result entirely from the PQ-breaking effects of the QCD anomaly? See e.g. Georgi & McArthur or Choi, Kim, Sze for early discussions. This was a beautiful possibility which was on ice for some decades, because the largest PQ-breaking effects take place on scales at which the QCD gauge coupling has become large and we cannot calculate by hand.

Thankfully heroic efforts by lattice physicists have now resolved the issue for us (see e.g. this recent review) but the answer is in the negative

In view of the fact that a massless up quark would solve the strong CP problem, many authors have considered this an attractive possibility, but the results presented above exclude this possibility: the value of mu in Eq. (43) differs from zero by 26 standard deviations.

So the unbroken PQ solution to the strong CP problem, as originally envisioned in the 80s, is not realized in nature.

Now let me further mention that it is possible that an unbroken PQ solution may still be realized in physics Beyond the Standard Model which embeds the $SU(3)_C$ gauge group non-trivially in an ultraviolet gauge group. This influential paper by Agrawal & Howe has recently revived interest in this possibility, and it turns out this may also work naturally in certain theories of unification.

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