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The strong CP problem is considered one of the major unsolved physics problems. A non-vanishing value of $\theta$ in QCD $\theta$-term violates CP-symmetry. A primary strong CP violating observable is the electric dipole moment of the neutron $d_n$. The current observed upper bound of $d_n$ is extremely small, which is commonly taken as an indication that $\theta$ is anomalously small, hence the strong CP problem.

However, a new preprint on arXiv by Schierholz, Absence of strong CP violation, published today basically claims that there is no strong CP problem: It turns out that the electric dipole moment of the neutron $d_n$ could not be used to detect the CP violating $\theta$. In other words, $d_n$ could be extremely small while $\theta$ is NOT small.

We know that there are several proposed solutions to solve the strong CP problem. The most well-known is Peccei–Quinn theory, involving new pseudoscalar particles called axions. If the above new paper is correct, all the efforts of "solving" CP problem would be a waste of time!

My question: Could the electric dipole moment of the neutron $d_n$ be used to effectively detect the CP violating $\theta$? If not, then is there really a strong CP problem?

Added note:

See more CP-related discussions in terms of the infinite volume limit of QCD:

and dissenting views:

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    $\begingroup$ Is your question basically to have us read the paper and tell you if it is correct or not? $\endgroup$
    – octonion
    Commented Mar 21 at 19:20
  • $\begingroup$ There is also the issue that the "strong CP problem" is basically an unwillingness of physicists to accept that the physical constants in the laws of the Universe are what they are. For the "strong CP problem" to be a problem, one has to accept that theta should have a value other than zero for some reason, and there exists no set of "meta" laws of physics about what the laws of physics or the values of its physical constants should be. Theta is zero because Nature says so is a perfectly valid answer to that problem. Anything else is presumptuous. $\endgroup$
    – ohwilleke
    Commented Jul 16 at 22:21
  • $\begingroup$ This is a mischaracterization of why particle physicists care about naturalness. The strong CP problem is sharp in a UV theory where theta bar is an output, rather than an input. For example if CP is an exact symmetry in the UV--the puzzle is generating large CP violation in e.g. CKM while keeping theta bar small, since it generically gets renormalized to O(1). We can understand the strong CP problem generally by spurion analysis of the SM which warns us of this issue. Of course if it was only the SM into the UV there's no strong CP problem, but we would like a UV theory which explains the SM. $\endgroup$
    – SethK
    Commented Jul 16 at 23:08
  • $\begingroup$ @ohwilleke Note that the baryon asymmetry of the universe suggests that there is more CP violation somewhere than we have currently observed in the standard model. If CP violation in the strong sector is too small, then we will have to get more creative to explain why there is enough matter in the universe to self-organize into physicists who wonder where the antimatter went. $\endgroup$
    – rob ♦
    Commented Jul 17 at 3:49
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    $\begingroup$ @rob The notion that there had to be baryon symmetry immediately after the Big Bang is also just an arbitrary assumption. The mass-energy of the universe was not zero then. Why should the aggregate baryon number of the universe be zero then? All available evidence to date indicates that it was not, and that the Standard Model doesn't require any new physics to be free of mathematical pathologies right up to the GUT scale. $\endgroup$
    – ohwilleke
    Commented Jul 17 at 20:45

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I'll restrict myself to answer

Blockquote My question: Could the electric dipole moment of the neutron $d_n$ be used to effectively detect the CP violating $\theta$? If not, then is there really a strong CP problem?

In these lectures on the CP Problem or this SE post you can see how you can relate the value of of the theta angle $\theta$ to the value of the neutron eDM. A priori it should be possible to obtain a similar result for the proton but, as far as I know, we mostly focus on the neutron eDM because it is easier to measure. So, even if there were no $d_n$, you could still have other similar magnitudes that are not null.

But OK, let's imagine that there is some mechanism that suppresses all QCD eDMs (and other CP-violating quantities) without that requiring a non-null $\bar{\theta}$. Could we still have a CP problem? Well, it could still be that QCD does not violate CP-symmetry, but we would not be able to measure it experimentally, so we would lose all experimental justification to search for reasons that drive $\bar{\theta}$ to be small! This doesn't mean that axions, or whatever other solution to the Strong CP problem, don't exist. We would have just lost the biggest reason to believe they exist (there are others, as we think axions might be cold dark matter, see this again).

Just as a side note, in the preprint you're mentioning, the author is only proposing another solution to the CP problem, albeit it involves only reducing the impact of the value of $\theta$ on the value of $d_n$, and thus proposing another explanation to why $d_n$ is so small. So he is not "removing" the CP problem, he is just (attempting to) solve it.

Note 2: There are other physical implications of the theta term, like a $\theta$-dependent vacuum energy (see Vafa-Witten theorem), but I am not sure whether they can be experimentally tested. Maybe someone else can add on this.

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