Why is technical naturalness enough? There are two notions of "naturalness" used in particle physics today.


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*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? 
 A: The previous answer is not correct, so I thought I'd correct it. While we often phrase naturalness in terms of the size of coupling constants, as these are what we usually work with mathematical, we could rephrase everything in terms of physical measurable quantities and the same considerations would still arise. So the issues surrounding naturalness have nothing to do with our choice of how to describe the physical system, contrary to the previous answer.
The underlying rationale behind Dirac naturalness is the observation that if you have a physical system with some length scale $L$, then we should expect all physical quantities, once suitably made dimensionless, to be $O(1)$ with respect to that length scale. What $O(1)$ actually means isn't that clear, but I think most model builders would consider the gauge couplings and the Yukawas of the third generation of matter (which are $\sim 10^{-2}$ or larger) to be Dirac natural in this sense. Whether we should consider the Yukawas for the second and first generation of matter (which can be as low as $\sim 10^{-5}$) to be Dirac natural is less clear. The Higgs mass ($\sim 10^{-17}$) and the cosmological constant ($\sim 10^{-30}$) are definitely not Dirac natural, and so considered problematic by most (but not all) model builders.
Technical naturalness is a broader concept than Dirac naturalness. A quantity is technically natural if it does not suffer from large quantum corrections. The reason we might care about technical naturalness is that it allows us to punt our explanation for why a quantity is small to higher energy scales. A simple example is the neutrino mass. This is technically natural, because the neutrino has a chiral symmetry when it is massless. But one cannot tell if the neutrino mass is Dirac natural just by studying physical process with energies comparable to the neutrino mass.  Instead, we have to appeal to energy scales much greater than the neutrino mass itself. In the Standard Model this is realised through the seesaw mechanism, which is Dirac natural. (Actually, in the Standard Model this isn't Dirac natural because it relies on the Higgs mass being light. However, it doesn't introduce any new hierachies beyond the usual Higgs mass problem. Hypothetically, you could realize light fermions through a strongly coupled gauge theory breaking a chiral symmetry, and in this case you would have a Dirac natural explanation.)
Of course, it is possible that there are technically natural explanations that can never be realized through a Dirac natural explanation at much higher energy scales. If this is the case then we should rule out these technically natural explanations. But in practice it is hard to know for sure whether this is the case (maybe we have simply not found the Dirac natural explanation yet), so I think this is why model builders focus on technical rather than Dirac naturalness.
This also explains why model builders are much more interested in the Higgs mass than the Yukawas in the Standard Model. Aside from the much greater fine-tuning required by the former compared to the latter, the Yukawas in the Standard Model are technically natural, and so maybe GUT or Planck scale physics provides a Dirac natural explanation for their small size. The Higgs mass is not technically natural, and so if we want our physics to be Dirac natural we must invoked new physics at the weak scale.
To summarize: The absence of technical naturalness implies that the lack of naturalness needs to be solved at roughly the energy scales the problem arises if we want the theory to be Dirac natural. But if a quantity is technically natural, we may not be able to tell if it is Dirac natural without understanding physics at much higher energy scales.
A: 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.
