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Why exactly does the observed smallness of the $\theta$-angle in QCD constitute a fine-tuning problem?

I thought that the content of 't Hooft's technical naturalness was exactly that a parameter is allowed to be small if, when taken to zero, it leads to an increased symmetry?

In this case, taking $\theta$ to zero leads to CP-invariance, so what is the problem?

Michael Angelo
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3 Answers3

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The $\theta$-angle poses a naturalness problem in the context of the Standard Model (SM).

In fact, as you correctly remark, when you consider QCD in isolation, sending $\theta \to 0$ restores $CP$-symmetry; that is, a small $\theta$ is technically natural according to 't Hooft. Much more strongly, if QCD was the whole thing, the experimental bounds on $\theta$ would simply suggest $\theta \equiv 0$.

However, in the SM there are other potential sources of CP violation: sending $\theta \to 0$ alone is not enough in order to restore CP; in order to restore CP, you must also to send to zero the CP violating phases in the CKM matrix.

These phases are, however, experimentally known to be non zero, which means that (from a naturalness point of view) you would expect an equally big $\theta$ angle.

This is why the strong experimental bounds on $\theta$ pose a naturalness problem within the SM.

pppqqq
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This is a somewhat subjective thing and I think you'll get about five different answers if you ask ten different people. My impression of the situation is this:

  • All that technical naturalness does is say that if some parameter is small at a low scale, it's also small at a high scale. It doesn't explain why it's small at a high scale. If one is worried about the parameters in your theory taking seemingly unlikely values, then technical naturalness only means that you can delay resolving the question to the next layer in the EFT hierarchy.
  • One can always argue that we don't know what the probability distribution of parameters is, so naturalness is meaningless. But the $\theta$-angle is a very clean example, because it's literally an angle. Absent any more structure, you would expect some UV theory to pick $\theta$ randomly in $[0, 2 \pi)$, which makes the observed smallness very unlikely.
  • If you say, "well, can't there just be some unknown structure at high energies that prefers small $\theta$?", then of course the answer is yes, but identifying such structure is precisely what we mean by doing model building to solve the strong CP problem.

For some more perspectives on this, see here and here.

knzhou
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The theta_QCD=0 discussion is reminiscent of the talk about neutrino masses. Yes, setting them to zero preserved a symmetry and didn’t add extra noise to the SM. But they are not zero. I suppose theta is similar.

misha
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