Naturalness is dead. Long live naturalness.

I was elated when I saw that Gian Francesco Giudice announced the “Dawn of the Post-Naturalness Era,” as the title of his recent paper promises. The craze in particle physics, I thought, might finally come to an end; data brought reason back to Earth after all.

But disillusionment followed swiftly when I read the paper.

Gian Francesco Giudice is a theoretical physicist at CERN. He is maybe not the most prominent member of his species, but he has been extremely influential in establishing “naturalness” as a criterion to select worthwhile theories of particle physics. Together with Riccardo Barbieri, Giudice wrote one of the pioneering papers on how to quantify naturalness, thereby significantly contributing to the belief that it is a scientific criterion. To date the paper has been cited more than 1000 times.

Giudice was also the first person I interviewed for my upcoming book about the relevance of arguments from beauty in particle physics. It became clear to me quickly, however, that he does not think naturalness is an argument from beauty. Instead, Giudice, like many in the field, believes the criterion is mathematically well-defined. When I saw his new paper, I hoped he’d come around to see the mistake. But I was overly optimistic.

As Giudice makes pretty clear in the paper, he still thinks that “naturalness is a well-defined concept.” I have previously explained why that is wrong, or rather why, if you make naturalness well-defined, it becomes meaningless. A quick walk through the argument goes as follows.

Naturalness in quantum field theories – ie, theories of the type of the standard model of particle physics – means that a theory at low energies does not sensitively depend on the choice of parameters at high energies. I often hear people say this means that “the high-energy physics decouples.” But note that changing the parameters of a theory is not a physical process. The parameters are whatever they are.

The processes that are physically possible at high energies decouple whenever effective field theories work, pretty much by definition of what it means to have an effective theory. But this is not the decoupling that naturalness relies on. To quantify naturalness you move around between theories in an abstract theory space. This is very similar to moving around in the landscape of the multiverse. Indeed, it is probably not a coincidence that both ideas became popular around the same time, in the mid 1990s.

If you now want to quantify how sensitively a theory at low energy depends on the choice of parameters at high energies, you first have to define the probability for making such choices. This means you need a probability distribution on theory space. Yes, it’s the exact same problem you also have for inflation and in the multiverse.

In most papers on naturalness, however, the probability distribution is left unspecified which implicitly means one chooses a uniform distribution over an interval of about length 1. The typical justification for this is that once you factor out all dimensionful parameters, you should only have numbers of order 1 left. It is with this assumption that naturalness becomes meaningless because you have now simply postulated that numbers of order 1 are better than other numbers.

You wanted to avoid arbitrary choices, but in the end you had to make an arbitrary choice. This turns the whole idea ad absurdum.

That you have to hand-select a probability distribution to make naturalness well-defined used to be well-known. One of the early papers on the topic clearly states

“The “theoretical license” at one’s discretion when making this choice [for the probability distribution] necessarily introduces an element of arbitrariness to the construction.” 

Anderson and Castano, Phys. Lett. B 347:300-308 (1995)

Giudice too mentions “statistical comparisons” on theory space, so I am sure he is aware of the need to define the distribution. He also writes, however, that “naturalness is an inescapable consequence of the ingredients generally used to construct effective field theories.” But of course it is not. If it was, why make it an additional requirement?

(At this point usually someone starts quoting the decoupling theorem. In case you are that person let me say that a) no one has used mass-dependent regularization schemes since the 1980s for good reasons, and b) not only is it questionable to assume perturbative renormalizability, we actually know that gravity isn’t perturbatively renormalizable. In other words, it’s an irrelevant objection, so please let me go on.)

In his paper, Giudice further claims that “naturalness has been a good guiding principle” which is a strange thing to say about a principle that has led to merely one successful prediction but at least three failed predictions, more if you count other numerical coincidences that physicists obsess about like the WIMP miracle or gauge coupling unification. The tale of the “good guiding principle” is one of the peculiar myths that gets passed around in communities until everyone believes it.

Having said that, Giudice’s paper also contains some good points. He suggests, for example, that the use of symmetry principles in the foundations of physics might have outlasted its use. Symmetries might just be emergent at low energies. This is a fairly old idea which goes back at least to the 1980s, but it’s still considered outlandish by most particle physicists. (I discuss it in my book, too.)

Giudice furthermore points out that in case your high energy physics mixes with the low energy physics (commonly referred to as “UV/IR mixing”) it’s not clear what naturalness even means. Since this mixing is believed to be a common feature of non-commutative geometries and quite possibly quantum gravity in general, I have picked people’s brains on this for some years. But I only got shoulder shrugs, and I am none the wiser today. Giudice in his paper also doesn’t have much to say about the consequences other than that it is “a big source of confusion,” on which I totally agree.

But the conclusion that Giudice comes to at the end of his paper seems to be the exact opposite of mine.

I believe what is needed for progress in the foundations of physics is more mathematical rigor. Obsessing about ill-defined criteria like naturalness that don’t even make good working hypotheses isn’t helpful. And it would serve particle physicists well to identify their previous mistakes in order to avoid repeating them. I dearly hope they will not just replace one beauty-criterion by another.

Giudice on the other hand thinks that “we need pure unbridled speculation, driven by imagination and vision.” Which sounds great, except that theoretical particle physics has not exactly suffered from a dearth of speculation. Instead, it has suffered from a lack of sound logic.

Be that as it may, I found the paper insightful in many regards. I certainly agree that this is a time of crisis but that this is also an opportunity for change to the better. Giudice’s paper is very timely. It is also merely moderately technical, so I encourage you to give it a read yourself.