Understanding naturalness

24 January 2019

The last few years have seen an explosion of original ideas concerning whether the universe is “natural” or not, and the LHC has brought the issue into sharp focus. But we’re only at the beginning of our understanding, says theorist Nathaniel Craig.

Nathaniel Craig

What is “naturalness”?

Colloquially, a theory is natural if its underlying parameters are all of the same size in appropriate units. A more precise definition involves the notion of an effective field theory – the idea that a given quantum field theory might only describe nature at energies below a certain scale, or cutoff. The Standard Model (SM) is an effective field theory because it cannot be valid up to arbitrarily high energies even in the absence of gravity. An effective field theory is natural if all of its parameters are of order unity in units of the cutoff. Without fine-tuning, a parameter can only be much smaller than this if setting it to zero increases the symmetry of the theory. All couplings and scales in a quantum theory are connected by quantum effects unless symmetries distinguish them, making it generic for them to coincide.

When did naturalness become a guiding force in particle physics?

We typically trace it back to Eddington and Dirac, though it had precedents in the cosmologies of the Ancient Greeks. Dirac’s discomfort with large dimensionless ratios in observed parameters – among others, the ratio of the gravitational and electromagnetic forces between protons and electrons, which amounts to the smallness of the proton mass in units of the Planck scale – led him to propose a radical cosmology in which Newton’s constant varied with the age of the universe. Dirac’s proposed solutions were readily falsified, but this was a predecessor of the more refined notion of naturalness that evolved with the development of quantum field theory, which drew on observations by Gell-Mann, ’t Hooft, Veltman, Wilson, Weinberg, Susskind and other greats.

Does the concept appear in other disciplines?

There are notions of naturalness in essentially every scientific discipline, but physics, and particle physics in particular, is somewhat unique. This is perhaps not surprising, since one of the primary goals of particle physics is to infer the laws of nature at increasingly higher energies and shorter distances.

Isn’t naturalness a matter of personal judgement?

One can certainly come up with frameworks in which naturalness is mathematically defined – for example, quantifying the sensitivity of some parameter in the theory to variations of the other parameters. However, what one does with that information is a matter of personal judgement: we don’t know how nature computes fine-tuning (i.e. departure from naturalness), or what amount of fine-tuning is reasonable to expect. This is highlighted by the occasional abandonment of mathematically defined naturalness criteria in favour of the so-called Potter Stewart measure: “I know it when I see it.” The element of judgement makes it unproductive to obsess over minor differences in fine-tuning, but large fine-tunings potentially signal that something is amiss. Also, one can’t help but notice that the degree of fine-tuning that is considered acceptable has changed over time.

What evidence is there that nature is natural?

Dirac’s puzzle, the smallness of the proton mass, is a great example: we understand it now as a consequence of the asymptotic freedom of the strong interaction. A natural (of order-unity) value of the QCD gauge coupling at high energies gives rise to an exponentially smaller mass scale on account of the logarithmic evolution of the gauge coupling. Another excellent example, relevant to the electroweak hierarchy problem, is the mass splitting of the charged and neutral pions. From the perspective of an effective field theorist working at the energies of these pions, their mass splitting is only natural if the cutoff of the theory is around 800 MeV. Lo and behold, going up in energy from the pions, the rho meson appears at 770 MeV, revealing the composite nature of the pions and changing the picture in precisely the right way to render the mass splitting natural.

Which is the most troublesome observation for naturalness today?

The cosmological-constant (CC) problem, which is the disagreement by 120 orders of magnitude between the observed and expected value of the vacuum energy density. We understand the SM to be a valid effective field theory for many decades above the energy scale of the observed CC, which makes it very hard to believe that the problem is solved in a conventional way without considerable fine-tuning. Contrast that with the SM hierarchy problem, which is a statement about the naturalness of the mass of the Higgs boson. Data so far show that the cutoff of the SM as an effective field theory might not be too far above the Higgs mass, bringing naturalness within reach of experiment. On the other hand, the CC is only a problem in the context of the SM coupled to gravity, so perhaps its resolution lies in yet-to-be-understood features of quantum gravity.

What about the tiny values of the neutrino masses?

Neutrino masses are not remotely troublesome for naturalness. A parameter can be much smaller than the natural expectation if setting it to zero increases the symmetry of the theory (we call such parameters “technically natural”). For the neutrino, as for any SM fermion, there is an enhanced symmetry when neutrino masses are set to zero. This means that your natural expectation for the neutrino masses is zero, and if they are non-zero, quantum corrections to neutrino masses are proportional to the masses themselves. Although the SM features many numerical hierarchies, the majority of them are technically-natural ones that could be explained by physics at inaccessibly high energies. The most urgent problems are the hierarchies that aren’t technically natural, like the CC problem and the electroweak hierarchy problem.

Has applying the naturalness principle led directly to a discovery?

It’s fair to say that Gaillard and Lee predicted the charm-quark mass by applying naturalness arguments to the mass-splitting of neutral kaons. Of course, the same arguments were also used to (incorrectly) predict a wildly different value of the weak scale! This is a reminder that naturalness principles can point to a problem in the existing theory, and a scale at which the theory should change, but they don’t tell you precisely how the problem is resolved. The naturalness of the neutral kaon mass splitting, or the charged-neutral pion mass splitting, suggests to me that it is more useful to refer to naturalness as a strategy, rather than as a principle.


A slightly more flippant example is the observation of neutrinos from Supernova 1987A. This marked the beginning of neutrino astronomy and opened the door to unrelated surprises, yet the large water-Cherenkov detectors that detected these neutrinos were originally constructed to look for proton decay predicted by grand unified theories (which were themselves motivated by naturalness arguments).

While it would be great if naturalness-based arguments successfully predict new physics, it’s also worthwhile if they ultimately serve only to draw experimental attention to new places.

What has been the impact of the LHC results so far on naturalness?

There have been two huge developments at the LHC. The first is the discovery of the Higgs boson, which sharpens the electroweak hierarchy problem: we seem to have found precisely the sort of particle whose mass, if natural, points to a significant departure from the SM around the TeV scale. The second is the non-observation of new particles predicted by the most popular solutions to the electroweak hierarchy problem, such as supersymmetry. While evidence for these solutions could lie right around the corner, its absence thus far has inspired both a great deal of uncertainty about the naturalness of the weak scale and a lively exploration of new approaches to the problem. The LHC null results teach us only about specific (and historically popular) models that were inspired by naturalness. It is therefore an ideal time to explore naturalness arguments more deeply. The last few years have seen an explosion of original ideas, but we’re really only at the beginning of the process.

The situation is analogous to the search for dark matter, where gravitational evidence is accumulating at an impressive rate despite numerous null results in direct-detection experiments. These null results haven’t ruled out dark matter itself; they’ve only disfavoured certain specific and historically popular models.

How can we settle the naturalness issue once and for all?

The discovery of new particles around the TeV scale whose properties suggest they are related to the top quark would very strongly suggest that nature is more or less natural. In the event of non-discovery, the question becomes thornier – it could be that the SM is unnatural; it could be that naturalness arguments are irrelevant; or it could be that there are signatures of naturalness that we haven’t recognised yet. Kepler’s symmetry-based explanation of the naturalness of planetary orbits in terms of platonic solids ultimately turned out to be a red herring, but only because we came to realise that the features of specific planetary orbits are not deeply related to fundamental laws.

Without naturalness as a guide, how do theorists go beyond the SM?

Naturalness is but one of many hints at physics beyond the SM. There are some incredibly robust hints based on data – dark matter and neutrino masses, for example. There are also suggestive hints, such as the hierarchical structure of fermion masses, the preponderance of baryons over antibaryons and the apparent unification of gauge couplings. There is also a compelling argument for constructing new-physics models purely motivated by anomalous data. This sort of “ambulance chasing” does not have a stellar reputation, but it’s an honest approach which recognises that the discovery of new physics may well come as another case of “Who ordered that?” rather than the answer to a theoretical problem.

What sociological or psychological aspects are at work?

If theoretical considerations are primarily shaping the advancement of a field, then sociology inevitably plays a central role in deciding what questions are most pressing. The good news is that the scales often tip, and data either clarify the situation or pose new questions. As a field we need to focus on lucidly articulating the case for (and against) naturalness as a guiding principle, and let the newer generations make up their minds for themselves.

bright-rec iop pub iop-science physcis connect