Intuitive

The strong CP puzzle is the observation that in the standard model nothing forbids that strong interactions violate CP symmetry but so far such a CP violation by strong interactions has never been observed.

The puzzle is regarded as deep and interesting since upon closer inspection there are possibly two sources of how strong interactions could violate CP symmetry. These two sources come from completely different sectors and thus it's somewhat a miracle that they cancel exactly.

One possible source comes from the nontrivial structure of the ground state of the theory of strong interactions (QCD).

The other possible source comes from the chiral anomaly. Thus contribution to the total CP violation of strong interactions depends on the couplings of the quark fields to the Higgs field. The Higgs sector and the ground state of the theory of strong interactions are, in the standard model, completely different parts of the theory and there is no connection between them besides that they both are possible sources for CP violation in strong interactions.

Concrete

Nothing forbids that we add a new term to the Lagrangian

$$\mathcal{L}_{QCD} = \ldots + \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu}, $$

$$\bar\theta=\theta_{\rm QCD}-\theta_{\rm F}<10^{-10} ,$$

where

• $\theta_{\rm QCD}$ is the coefficient of the term $\alpha_s^2/ 8 \pi \, G\tilde G $ in the Lagrangian that we get when we consider QCD alone.
• $\theta_{\rm F} = \arg \det M_u M_d$ is an additional contribution to the effective complete theta parameter $\theta$ that results when we consider QCD in the presence of fermions. The term enters since we have to diagonalize the mass matrices to switch to the mass basis and this diagonalization process necessarily involves a chiral rotation.
• The experimental bound $\bar\theta<10^{-10}$ comes from the observation that the dipole moment of the neutron is tiny: $|d_n| \le 3.6 \times 10^{-26} e \, {\rm cm}$ (Source).

Take note that $$ \tilde{F}^{\mu\nu} F_{\mu\nu} \propto \vec E \cdot \vec B \propto \partial_t \vec A\cdot \vec B $$

For a gentle introduction see the series "Demystifying the QCD vacuum":

1. http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-1/
2. http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-2/
3. http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-3-the-untold-story/
4. http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-4-physical-implications-of-theta/
5. http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-5-anomalies-and-the-strong-cp-problem/

• A great introduction to and review of the strong CP problem is The Strong CP Problem Revisited by Hai-Yang Cheng
• Another good summary can be found in chapter 4 of GAUGE THEORIES IN THREE DIMENSIONS (= AT FINITE TEMPERATURE) by R. Jackiw. Especially he discusses the two possible ways to see how $\theta$ emerges. Either via the functional approach or via the Hamiltonian approach.

Since one can show that no physical operator can connect states with different values of 6), it follows that $\theta$ labels completely disjoint sectors of the theory. In particular, a complete physical theory may be built on each $\theta$ vacuum (Callan, Dashen, and Gross, 1976; Jackiw and Rebbi, 1976). It will be shown below that different $\theta$ worlds have different physical properties. Therefore, in order to calculate the expectation values of observables in a pure state, not in a mixture, one must project onto a single $\theta$ sector.QCD and instantons at finite temperature by David J. Gross et. al.

Abstract

Unfortunately the relation between $d_n$, $\bar{\theta}$ is not known to better than an order of magnitude (see e.g. [5][6] and references therein), so the constraint on $\bar{\theta}$ quoted in (1.1) — obtained using naive dimensional analysis (NDA) — suffers from a large uncertainty.https://arxiv.org/pdf/1412.3805.pdf

Why is it interesting?

Nothing in QCD forbids the so-called $\theta$-term, which violates CP-symmetry. The strength of the CP violation is proportional to some parameter, called $\theta$. However, so far no CP-violation was ever observed in strong interactions. Hence, the strong CP problem is, why this $\theta$ parameter is so small, or even zero.

FAQ

Why does a massless quark solve the problem?

If at least one quark were massless, the Lagrangian would have a chiral symmetry $u \to e^{i\alpha \gamma_5} \Psi$. Such a chiral rotation shifts the $\theta$ parameter: $\theta \to \theta + c \alpha$. When at least one quark is massless, we can perform one completely arbitrary chiral rotation and hence eliminate $\theta \to 0$.

However, when all quarks do have mass, we can't choose the phase $\alpha$ arbitrarily. Instead, it is fixed by the requirement that we must get a real mass for the quarks. Hence $\alpha$ is not an arbitrary parameter and $\theta$ can not be shifted to zero. Of course, it is possible that $\alpha$ happens to be exactly the right value to set $\theta$ to zero. However, there is no reason why this should be the case and this is the strong CP problem. Why should this seemingly unrelated, but fixed parameter $\alpha$ exactly cancel the bare parameter $\theta$?

A chiral rotation shifts $\theta$, because a chiral rotation is basically a change of basis, i.e. a relabelling of the quark fields. Such changes of basis are accompanied by a Jacobian. In this case, the Jacobian is proportional to $F \tilde F$ and thus shifts $\theta$.

Why is there no $\theta$ angle in the weak sector?

Completely analogous we could at a $\theta$ term to the Lagrangian that involves the $SU(2)_L$ gauge bosons. However, in contrast to the $\theta$ angle in the strong sector, it can be set to zero through the "mechanism" described in the answer to the question above.

The crucial difference between strong and weak interactions is that all quarks interact strongly, whereas only the left-chiral fermions interact weakly. This means that a transformation of the right-chiral fermions does not change the integral measure and hence there is no shift of $\theta$ associated with these rotations.

Thus, we have the freedom to rotate both, the left-chiral and right-chiral fields with the same arbitrary angle $\alpha$ and thus set $\theta_{weak} \to 0$. It is necessary to have this freedom to rotate both, the left-chiral and right-chiral fermions with the same angle, because otherwise the mass terms are change under such a rotation and hence $\alpha$ is not arbitrary and $\theta$ can not be set to zero.

Gluons couple to right-chiral and left-chiral quarks and thus transforming these leads to separate shifts of $\theta_{strong}$. However, we cannot perform arbitrary rotations, because the mass terms must be real.

To summarize: We have two places where the rotations make a difference: for the mass terms and for the $\theta$ term. In the absence of massless fermions, we can't perform arbitrary chiral rotations, because the mass terms must be real. This means immediately that we can't rotate $\theta_{QCD}$ to zero. (Of course, it could miraculously be that the rotation that makes the mass terms real, cancels at the same time $\theta_{QCD}$. However there is no reason why this should be the case and this is the strong CP puzzle).

In the weak sector the situation is almost the same. However, the weak bosons do not care about the right-chiral fermions and hence there is no shift of $\theta_{weak}$ when we rotate them. This extra freedom allows us to make the mass terms real (or leave them real) while rotating at the same time $\theta_{weak}$ to zero.

Source: https://physics.stackexchange.com/questions/27462/why-is-there-no-theta-angle-topological-term-for-the-weak-interactions/27463

See also https://physics.stackexchange.com/questions/176424/why-is-the-strong-cp-term-theta-fracg232-pi2-g-mu-nua-tildeg?noredirect=1&lq=1

What about $\theta$ in unified theories?

First, above the unification scale the vacuum angle is determined by a single parameter. When we descend down in a controllable way this parameter does not split in two independent parameters [10], rather all surviving gauge groups G1,2,.. inherit it from G. Thus, if a mechanism could be found to screen θ at short distances, then this mechanism would eliminate θ in all G1,2,.. subgroups simultaneouslyhttps://arxiv.org/pdf/1701.00467.pdf

't Hooft's favorite Solution and Opinion on the Peccei-Quinn Solution?

"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 i sstricly 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 wheter this works out or not. Usually people who do more detailed calculations thend to disagree with the statement tha this is a viable scenario for the $\theta$ angle problem. I am not so totally convinced yet, and I believe that, because of the uncertainities 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 mases are so small. The renormalization group could get hughe effects from instantons which align according to $\theta$. I have some hesitation to accept the wold, daring assumptions that e.g. Peccei, Quinn made. See: https://books.google.de/books?id=bvSxDgAAQBAJ&lpg=PA232&ots=ceJzZ6xXeG&dq=Renormalization%20Scheme%20Dependence%20beta%20function%20hooft&hl=de&pg=PA234#v=onepage&q&f=false

Is the $\theta$-term gauge invariant?

No, see https://physics.stackexchange.com/questions/367514/gauge-invariance-of-theta-term-in-qcd . The $\theta$-term is proportional to the winding number, which changes under large gauge transformations.

Why can we compactify space from to a sphere?

See: Gauge vacuums and the conformal group by A. P. Balachandran, A. M. Din, J. S. Nilsson, and H. Rupertsberger

Are other boundary conditions possible?

Yes:

p. We have used the fact that π3(S3) = Z to characterize the number of components of the gauge group SU(2). In the argument it is crucial that the spatial topology is S3. If this is not the case, the treatment should be refined. For instance, in noncompact three-dimensional Euclidean space the type of gauge transformations described by the elements of G = {g : S3 → G} are those approaching the identity at infinity fast enough and in a way that does not depend on angles. An equivalent way to describe them is to consider those gauge transformations that, outside a compact set surrounding the origin of coordinates, go to the identity very fast. The classes generated by these gauge transformations can be characterized by an integer number, but this may not exhaust the topological characterization of all possible nontrivial transformations. Working on a three-dimensional box with periodic boundary conditions results in a spatial topology that is that of a three-dimensional torus T 3, and the topological structure of the mappings G = {g : T 3 → G} is in general richer than the one described by the single winding number appearing in S3. In this case we also have other gauge transformations not included inG0 and associated to the fact that the space is not simply connected. These additional transformations are physically relevant, and play an important role in ’t Hooft’s theory of confinement in nonabelian gauge theories. From this point of view, the topology of the space of gauge transformation often depends on the type of physical questions asked. Hence, apart from the θ angle, there may be other angles or quantum number characterizing the physical states (or the vacuum) of the theory (see for instance [15] and references therein)."An Invitation to Quantum Field Theory", by Alvarez-Gaume et. al.