One possibility is provided by the fact that the state with $\theta=0$ is the bottom of the zone and thus has the lowest energy. This means if $\theta$ becomes a dynamical variable, the vacuum can relax to the $\theta=0$ state (just like electrons can drop to the bottom of the conduction band by emitting phonons). This is the basis of the axion mechanism [190-192]. The axion is a hypothetical pseudo-scalar particle, which couples to $G \tilde G$. The equations of motion for the axion field automatically remove the effective $\theta$ term, which is now a combination of $\theta_{QCD}$ and the axion expectation value. Experimental limits on the axion coupling are very severe, but an "invisible axion" might still exist.
page 138 in The QCD Vacuum, Hadrons and Superdense Matter by E. V. Shuryak
Technically this new global U(1) symmetry is not quite an exact symmetry. Like the strong CP symmetry itself, it is a pseudo-symmetry, broken only by non-perturbative or instanton (tunneling) effects. This is exactly why it works as desired. The trick is to make the Higgs field energy depend on the θ value in such a way that, for any initial value of θ, the Higgs field will choose a vacuum value such that the resulting physical parameter θeffective is zero. The Higgs vacuum expectation values aquire phases such that the phase of the quark mass matrix cancels against the initial θ.
http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-8964.pdf
Axions are proposed as a solution to the strong CP problem. In some sense, it's the simplest solution, because the QCD vacuum energy has a minimum at $\bar{\theta}=0$ and the axion allows the vacuum to relax to this ground state. Without the axion the QCD vacuum is frozen and $\bar{\theta}$ is therefore fixed at some not-necessarily minimum value.
For a great discussion of the history of axion models, see Axions: Past, Present, and Future by Mark Srednicki
By the time the papers were written Weinberg and Wilczek both used the name “axion” for this particle. I rather liked the alternate name “higglet” which was floating around for a while.
It may strike the reader (and has struck the author) that enormous theoretical superstructures are here being erected upon a very narrow foundation. The whole superstructure of axions could be made obsolete if a good alternative approach to the problem of strong CP invariance were found. Even if this does happen, I am confident that techniques for dealing with approximate Nambu- Goldstone bosons and their phenomenological (including cosmological) implications will be of enduring interest – so I won't be completely wasting your time.THE U(1) PROBLEM: INSTANTONS, AXIONS, AND FAMILONS by Frank Wilczek
An obvious question about the axion hypothesis is how natural it really is. Why introduce a global PQ “symmetry” if it is not actually a symmetry? What is the sense in constraining a theory so that the classical Lagrangian possesses a certain symmetry if the symmetry is actually anomalous? It could be argued that the best evidence that PQ “symmetries” are natural comes from string theory, which produces them without any contrivance.
Axions In String Theory by Peter Svrcek, Edward Witten
"My favorite solution is that the up quark mass becomes very small by evolving near to a chirally symmetric point. We know that if one of the masses is strictly zero, $\theta$ becomes invisible. The mechanism that I have in mind is that at a very high energy scale, where the weak interactions really communicate a lot with QCD, so that the $\theta$ angle at that scale would be influenced a lot by the weak interactions, one might put one of the quark masses, preferably the up quark mass, to zero. This would correspond to the $U(1)$ chiral symmetry which we know is broken by instantons. So if you scale down from that very high energy scale to low energy scales, the fact that the other quark masses (say the d quark) is unequal to zero will shift the up quark mass to the value it has in the real world. Now to me this appears to be a possible scenario, but it depends very much on numerical analysis whether this works out or not. Usually people who do more detailed calculations tend to disagree with the statement that this is a viable scenario for the $\theta$ angle problem. I am not so totally convinced yet, and I believe that, because of the uncertainties in QCD, this is a possible explanation. It could be for instance, that even if $\theta$ is large at high energies, the renormalization group would rotate it to small values just because the quark masses are so small. The renormalization group could get huge effects from instantons which align according to $\theta$. I have some hesitation to accept the bold, daring assumptions that e.g. Peccei, Quinn made. See: ’t Hooft
By taking this standard road, one needs to implement the axion in form of a hypothetical degree of freedom from beyond the SM. This requires an introduction of a singlet scalar field with a very large VEV plus either a hypothetical heavy quark [8] or an additional Higgs doublet [9] (for a review see, e.g., [10]).
Domestic Axion Gia Dvali
It is possible that an additional gravitational $\theta$ term:
$$ \theta_G R \tilde R, $$ where $R$ is the Riemann tensor, exists. If this is the case, the axion solution does not work.
there is an old believe that quantum gravity effects can generate an additional breaking of the axion shift symmetry (2) and therefore ruin the axion solution to the strong CP problem (see, e.g., [13]). […] The existence of a nonvanishing topological vacuum susceptibility in pure gravity is currently an open question. […] [I]n the absence of an axion (or a massless fermion), there exist two physically observable theta parameters, one from QCD (1) and one from gravity (9). Consequently, after the axion is introduced, it can only cancel a single combination of the two ϑ-terms, whereas the other combination remains physically observable. Hence, the strong CP problem is not solved.
Domestic Axion Gia Dvali
It is well known that a phase transition with PQ symmetry breaking can form axionic cosmic strings. These strings later become boundaries of domain walls [26] and decay producing axions.
Domestic Axion Gia Dvali
The notion of Peccei–Quinn (PQ) symmetry may seem contrived. Why should there be a U(1) symmetry which is broken at the quantum level but which is exact at the classical level? However, the reason for PQ symmetry may be deeper than we know at present. String theory contains many examples of symmetries which are exact classically but which are broken by quantum anomalies, including PQ symmetry [17–19]. Within field theory, there are examples of theories with automatic PQ symmetry, i.e. where PQ symmetry is a consequence of just the particle content of the theory without adjustment of parameters to special values. The strong CP problem by Pierre Sikivie
There, a global U(1) symmetry (the PQ symmetry) which is almost exact but broken by the axial anomaly of QCD plays a crucial role. After spontaneous breaking, the effective θ-angle of QCD is cancelled by the vacuum expectation value (VEV) of the associated pseudo Nambu-Goldstone boson, the axion a. The origin of such a convenient global symmetry is, however, quite puzzling from the theoretical point of view in many aspects. By definition, the PQ symmetry is not an exact symmetry. Besides, the postulation of global symmetries is not comfortable in the sense of general relativity. It is also argued that all global symmetries are broken by quantum gravity effects [5–10]. If we could regard the PQ symmetry as a U(1) gauge symmetry, there would be no suspicion about the exactness and the consistency with quantum gravity. The PQ symmetry is, however, broken by the QCD anomaly, and hence, it cannot be a consistent gauge symmetry as it is. https://arxiv.org/abs/1703.01112