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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.
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
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":
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.
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
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.
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 $\theta$.
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.
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
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.
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