There are two notions of "naturalness" used in particle physics today.

  • Dirac naturalness: all dimensionless parameters $g$ in a theory should be order $1$.
  • Technical naturalness: an observed coupling constant $g_{\text{eff}}$ can be much smaller than $1$ if a symmetry is restored when it is set to zero.

The distinction is a bit confusing and often blurred even by practitioners, but technical naturalness is used overwhelmingly more often than Dirac naturalness. For example, the focus on the hierarchy problem is because the Higgs mass is one of the only SM parameters that is not protected by a symmetry. From the standpoint of Dirac naturalness, almost every SM parameter is unnatural.

Model builders will often establish technical naturalness and declare victory, but I'm not sure what the exact motivation is. The reason technical naturalness is good is that quantum corrections mean $$g_{\text{eff}} = g + f(g)$$ where $g$ is a parameter in the Lagrangian. Since quantum corrections (usually) preserve symmetries, we must have $f(0) = 0$, which means $f(g)$ is linear plus subleading terms. That means that $g_{\text{eff}}$ is of the same order of magnitude as $g$.

In other words: suppose we measure $g_{\text{eff}} < 10^{-5}$. If the smallness is technically natural, then we have $g \lesssim 10^{-5}$, while if it is not, we must have, say $$34.37692 < g < 34.37693$$ Without technical naturalness, we would have to explain why $g$ is some incredibly specific value ("a pencil standing on its tip"). With technical naturalness, we merely have to explain why it is small.

That does seem like a bit of progress, but from the standpoint of Dirac naturalness it's just kicking the can down the road. What do model builders typically imagine would justify small coupling constants in the fundamental Lagrangian? Is there a motivation for this from string theory?

  • $\begingroup$ Please do read arXiv:1801.02176. $\endgroup$ – AccidentalFourierTransform Aug 9 '18 at 22:44
  • $\begingroup$ @AccidentalFourierTransform I've read plenty of Sabine's stuff, though I do think some of the criticisms are a bit simplistic. For instance, if you say that naturalness implies low-scale SUSY, then not observing SUSY invalidates naturalness. But there are tons and tons of ways to make natural theories without SUSY. I think the focus on SUSY alone is a bit myopic. $\endgroup$ – knzhou Aug 10 '18 at 11:26
  • $\begingroup$ Related: physics.stackexchange.com/q/278137/2451 $\endgroup$ – Qmechanic Aug 10 '18 at 13:31

The way I understand this both are just constraints for making a theory more appealing.

Dirac Naturalness: Dirac naturalness basically describes the fact that we only keep the strongest couplings and then rescale the action by a choice of units such that the strongest coupling is unity. If the other couplings are much smaller we can drop them since they are not important. So all remaining couplings also need to be of order unity.

Technical naturalness: Broken symmetries are the only small effect that can change things drastically since even a small term in the Lagrangian can lead to a completely different quantum phase. Therefore keeping such terms even if they are small may be necessary.

Maybe an analogy to condensed matter theory is instructive: If we try to find a description of many-body systems in condensed matter theory then starting from the plain description we have many strongly interacting electrons, which is nasty to deal with. A more convenient description is afforded by making use of Landau Liquid theory type concepts and finding an effective free particle description, which amounts to identifying quasi particles. Additional effects from the interaction are then moved into small interaction corrections. Notice in this example: The theory was made simpler by changing our idea of what the fundamental building blocks of our theory are.

To make the connection to high energy physics: Unlike Condensed matter theory here the is no preconceived notion of what fundamental building blocks are. Therefore what is fundamental is up to definition. One looks for fields that satisfy as many symmetries as possible or almost satisfy them because this can be expected to make the mathematical treatment easier. An almost satisfied symmetry means that we can only expect small corrections, which is consistent with Technical Naturalness.

  • $\begingroup$ Interesting, I’ve never heard it put that way! I’m going to hold out for a more traditional “particle physics” answer, but I like this one. $\endgroup$ – knzhou Aug 9 '18 at 22:40
  • $\begingroup$ It's neat that this is almost the exact opposite of what we do in particle physics. You're looking for a description with approximate symmetries broken only by small numbers. We're looking to run the procedure in reverse to get rid of the small numbers. From your perspective, the SM, with its Yukawa couplings all on the order of $0.001$ or $0.000001$, is basically as good as it gets! $\endgroup$ – knzhou Aug 10 '18 at 11:22
  • $\begingroup$ You must agree that part of the reason the standard model works so well is just for that reason. If the couplings were larger predictions would be difficult. To make the argument more fun if you wanted to you could redefine fields in the standard model in an ill-mannered way by means of some deviously chosen unitary transformations that involve multiple fields such that you end up with the same theory just with all couplings close to 1 but a Hamiltonian that is very far from being diagonal. This theory then, however, is very hard to solve and hence not very useful. $\endgroup$ – Michael Aug 11 '18 at 19:15

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