Fermilab Today on naturalness

Today, Fermilab Today published a nice short article about naturalness written by Jim Pivarski of CMS.



The article contains an unlikely arrangement of stones which seems to depend – much like the Higgs boson mass – on fine, unlikely cancellations. Pivarski also writes that it's unlikely for many parts of the same car to break at the same moment.

I chose another example of an unnatural arrangement of matter – one that was being offered to me by several Czech discount servers today. ;-) While the "Wine Bottle Chain Holder – Holds Bottles In the Air" seems even more extreme than the rocks above because the forces don't even seem to balance (not even in an unstable way), you may actually buy it. And just for six or ten bucks!

See the amazon.com link at the bottom. You simply can't afford not to possess this miraculous wine bottle holder! ;-)




An important point to notice – a point that Pivarski sensibly points out, too – is that it may just "look" unlikely for the chain holder to keep the bottle floating in the air. In reality, there could be an underlying explanation that makes the arrangement likely – we have just been overlooking the explanation. I don't actually know "exactly" what the explanation is because I don't have the chain holder and the pictures aren't quite sufficient. It still does look like a miracle to me! ;-)

But I ultimately believe that Nature contains no miracles or contradictions.




A question is what sort of explanations of the "apparently unlikely cancellations" are kosher.

To answer this question, we must understand the reason why the arrangement (or the lightness of the Higgs boson) seems unlikely to us in the first place. It's because we may argue that the numerical parameters like the squared Higgs mass are a priori "almost uniformly distributed in an interval, like the interval\[

0\leq m_h^2 \leq m_{GUT}^2,

\] which means that the probability that \(m_h^2\) is equal to or smaller than \(10^{-30}m_{GUT}^2\) is very small, pretty much \(10^{-30}\). So we don't expect such a cancellation to occur in Nature – much like we don't expect the weird rocks or (even worse) the wine bottle chain holder to exist.

However, the LHC is showing us that the Higgs boson does exist and its being is unbearably light. That means that we either admit that \(10^{-30}\) is a high enough probability – which is pretty bad (this standard would lead us to say it's OK and not evidence of crime for someone to win $100 million in a lottery five times in a row) – or we must conclude that something in the calculation of the probability was wrong. What was really wrong was the assumption of the "uniformity" of the distribution. In the real word, it must be much more likely than \(10^{-30}\) that the Higgs mass is close to zero. In other words, the distribution must be much more non-uniform.

At the end, when we know everything about the dynamical mechanisms that determine the Higgs mass (including the choice of the right string/M-theoretical vacuum), the actual probability distribution is\[

\rho(m_h^2) = \delta (m_h^2 - 10^{-30}m_{GUT}^2),

\] a simple delta-function positioned at the correct value of the Higgs mass (which is unnaturally small). Clearly, we don't have the "full explanation" yet. But we expect to be "somewhat closer" to a partial understanding that doesn't spit out the exact value of the mass but at least some rough estimate – some probability distribution for the Higgs mass that doesn't make the observed low value of the mass insanely unlikely.

Supersymmetry makes the low figures much more likely – it guarantees some cancellations. But to do so, the SUSY breaking scale must be low enough, not too much above the Higgs mass. Well, this is a rough description people would be satisfied with years ago. As the illusion was increasingly suggesting that the superpartners are heavier or substantially heavier than the Higgs mass, people began to be much more careful about the statements that the superpartners shouldn't be "too much heavier" than the Higgs. How much heavier? Which superpartners?

Yup, the $200 lasso holder may be bought for $8.49.

It turns out that only the top squarks and perhaps higgsinos (and, due to some additional interactions, also gluinos) are really important for the naturalness argument. They shouldn't be "orders of magnitude" heavier than the Higgs. The amount of fine-tuning – how crazy arrangement of stones we find tolerable in Nature – is a matter of subjective preferences. I personally find (and I have always found) a fine-tuning at "one part in a thousand" tolerable. Several years ago, many phenomenologists would proudly reduce their tolerance and would sometimes declare even "one in ten" fine-tuning unacceptable. Well, I think that Nature has already proved them wrong. There don't seem be any particles that are important for the stabilization of the Higgs mass and whose mass is "really close" to the Higgs mass. So some fine-tuning is bound to be required.

Once we accept the anthropic reasoning as a factor influencing the probability distributions, many of the worries about the unnaturalness evaporate. The reason is that the anthropic reasoning allows us to assume the existence of intelligent life – a rough aspect of the observed experimental data – and the existence of intelligent life heavily favors the existence of a light Higgs boson. A light Higgs boson (and light quarks and electrons, which sort of require a light Higgs boson although the Yukawa couplings may hide the smallness, too) is needed for the stars to live for a long time (relatively to the nuclear scale) and to contain parameterically many atoms. The number of atoms in a star is pretty much a power of the Planck-to-proton mass ratio. It is only large if the Higgs boson is much lighter than the Planck mass! And a planet orbiting a 15-atom star wouldn't have too much (or enough) potential to produce moderately and occasionally intelligent animals like us.

So if you say that the probability distributions above should be modified by the knowledge that intelligent life exists, the distributions are peaked near the small values of the Higgs mass – and they similarly reproduce many hierarchies. But the existence of intelligent life, while spiritually "important", is just a vaguely defined aspect of observations. Of course that if we would be using all the observations, we could just conclude that the right distribution is the delta-function I have already mentioned:\[

\rho(m_h^2) = \delta (m_h^2 - 10^{-30}m_{GUT}^2)

\] That would give us the right results but all of them would be extracted from the experiments. The theoretical reasoning would be completely useless. We know that over the history of science, science was capable of calculating many things without measuring them. So many physicists still think about the working hypothesis that the lightness of the Higgs boson can be justified by a logic or mechanism that doesn't require the experimental data – not even their rough aspects such as the existence of intelligent beings – to be assumed. Of course that to a certain extent, it is just a hope. Even the best theories we have ever had need to assume certain things that we had to extract from the experiments. There's no guarantee that the dependence on the observable data may be eliminated entirely.

Many such potential explanations of the Higgs boson's lightness may be found in the literature. Many of them, especially if you take a strong version of them (no tolerance to even modest fine-tuning), have already been ruled out because the Higgs boson is low and the particle seems to be somewhat isolated.

It's not clear whether some physics at a near-electroweak scale has to be found. The anthropic principle may be the reason why the Higgs mass seems so unnaturally light. But even without the anthropic principle, it's plausible that there exists an overlooked explanation of the low Higgs mass that however doesn't create any new physics near the electroweak scale. Such an "intelligent targeting done from a distance" violates the lore of effective field theories but I think that string/M-theory ultimately may violate this lore. It may produce patterns and relationships that are "unexplainable" by field-theoretical reasoning itself but that may be established using stringy arguments. In fact, string theory has already produced many examples of such extra constraints. The UV-IR connections may be viewed as a place to start.

So I have always been ready to encounter the major possibility that the Higgs boson is an island of a sort. I just don't have a fantastic problem with it and I don't think it would be a revolution in physics of any kind. It's just the damn ordinary Standard Model. Just some vague arguments about naturalness, arguments that have to exclude the anthropic bias and make several other additional assumptions, would be shown illegitimate. Well, not a big deal.

I want to emphasize that this does not mean that I consider the existence of supersymmetry questionable. For theoretical reasons, SUSY is pretty much inevitable in any consistent theory of quantum gravity with fermions and at least some stability. But whether many/most superpartners must be really close to the Higgs mass is an entirely open question, as far as I can say.