in Open Questions

Demystifying the Hierarchy Problem

There is no hierarchy problem in the Standard Model. The Standard Model has only one scale: the electroweak scale. Therefore, there can’t be any hierarchy problem because there is no hierarchy. But, of course, there are good reasons to believe that the Standard Model is incomplete and almost inevitably if you introduce new physics at a higher scale, you get problems with hierarchies. Formulated more technically, only when the cutoff $\Lambda$ is physical, we have a real hierarchy problem.

However, whenever someone starts talking about the hierarchy problem, you should ask: which one?

  • There is a tree-level hierarchy problem which you get in many extensions of the Standard Model. As an example, let’s consider GUT models. Here the Standard Model gauge symmetry is embedded into some larger symmetry group. This group breaks at an extremely high scale and the remnant symmetry is what we call the Standard Model gauge symmetry. If you now write down the Higgs potential for GUT model, the standard assumption is that all parameters in this potential are of the order of the GUT scale because, well there isn’t any other scale and we need to produce a GUT scale vacuum expectation value. The mystery is now how the, in comparison, tiny vacuum expectation value of the electroweak Higgs comes about. In a GUT this Standard Model Higgs lives in the same representation as several superheavy scalars. The superheavy masses of these scalars are no problem if we assume that all parameters in the GUT Higgs potential are extremely large numbers. But somehow these parameters must cancel to yield the tiny mass of the Standard Model Higgs. If you write down two random large numbers it’s extremely unlikely that they cancel so exactly that you get a tiny result. Such a cancelation needs an explanation and this is what people call the tree-level hierarchy problem. The prefix “tree-level” refers to the fact that no loops are involved here. The problem arises solely by investigating the tree-level Higgs potential.
  • But there is also a hierarchy problem which has to do with loops, i.e. higher orders in perturbation theory. The main observation is that our bare Higgs mass $m$ (the parameter in the Lagrangian) gets modified if we move beyond the tree-level. While this happens for all particles, it leads to a puzzle for scalar particles like the Higgs boson because here the loop corrections are directly proportional to the cutoff scale $\Lambda$. Concretely, the physical Higgs mass we can measure is given by $$ m^2_P = m^2 + \sigma (m^2) +\ldots , $$ where $m$ is the bare mass, $m_P$ the physical mass and $\sigma (m^2) $ the one-loop corrections. The puzzle is now that if we want to get a light Higgs mass $m_P^2 \ll \Lambda^2$, we need to fine-tune the bare parameter $m^2$: $$ m^2 \approx \Lambda^2 +m_P^2. $$ For example, for a physical Higgs mass $m_P \approx 125$ GeV and a cutoff scale around the Planck scale $\Lambda \approx  10^19$ GeV, we find that $$ m^2 = (1+10^{-34}) \Lambda^2 .$$ This means that our bare mass $m$ must be tuned extremely precisely to yield the light Higgs mass that we observe. This is automatically the case if we have a large cutoff scale. If we include higher order in perturbation theory, the situation gets even worse. At each order of perturbation theory, we must repeat the procedure and fine-tune the bare Higgs mass even further. This is what people usually call the hierarchy problem because the core of the problem is that the cutoff scale $\Lambda$ is so far above the electroweak scale.

Now, here’s the catch. Nature doesn’t know anything about loops. Each loop represents a term in our perturbation series. Perturbation theory is a tool we physicists invented to describe nature. But nature knows nothing about the bare mass and loop corrections. She only knows the whole thing $m_P$, which is what we can measure in experiments. In other words, we can’t measure the bare mass $m$ or, say, the one-loop correction $\sigma (m^2)$. Therefore, these parameters only exist within our description and we can simply adjust them to yield the value for the physical Higgs mass that we need.

The situation could be very different. If we could measure $m$ or $\sigma (m^2)$, for example, because they can be calculated using other measurable parameters, there would be a real problem. If two measurable parameters would cancel so precisely, we would have every right to wonder. But as long as the bare mass is only something which exists in our description and isn’t measurable, there is not really a deep problem because we can simply adjust these unphysical parameters at will.

Similar arguments are true for the tree-level hierarchy problem. As long as we haven’t measured the GUT scale Higgs potential parameters, there is nothing to really wonder about. Maybe the large symmetry gets broken differently, was never a good symmetry in the first place or maybe the parameters happen to cancel exactly.

Two great papers which discuss this point of view in more technical terms are

So … there isn’t really a hierarchy problem?

There is one. But to understand it we need to look at the whole situation a bit differently.

Criticality

The main idea of this alternative perspective is to borrow intuition from condensed matter physics. Here, we can also use field theory because we can excite the atoms our system consists of to yield waves. In addition, there can be particle-like excitations which are usually called phonons. For our problem here, the most important observation is that here we also have a cutoff scale $\Lambda$ which represents the inverse atomic spacing/the lattice spacing. Beyond this scale, our description doesn’t make sense.

With this in mind, we can understand what a hierarchy problem really is from a completely new perspective. Naively, we expect that if we excite our condensed matter system we only get small excitations. In technical terms, we generically only expect correlation lengths of the order of the lattice spacing. All longer correlation lengths need an explanation. The correlation length is inversely proportional to the mass associated with the excitation. Hence, in particle physics jargon we would say that for a system with cutoff $\Lambda$, we only expect particles with a mass of order $\Lambda$, i.e. superheavy particles.

Now the mystery is that we know that there are light elementary particles although the cutoff scale is presumably extremely high. In condensed matter jargon this means that we know that there are excitations with an extremely long correlation length compared with the fundamental lattice spacing.

This is the mystery we call the hierarchy problem.

But this is not only a helpful alternative perspective. It also allows us to think about possible solutions in completely different terms. We can now ask: under what circumstances do we get excitations with extremely long correlation length compared to the lattice spacing?

The answer is: whenever the system is close to a critical point. (The most famous example is the liquid-vapor critical point.)

A solution of the hierarchy problem, therefore, requires an explanation why nature seems so close to a critical point.

There are, as far as I know, two types of possible answers.

  • Either, someone/something tuned the fundamental parameters externally (whatever that means in this context). Condensed matter systems can be brought to a critical point by adjusting the temperature and pressure.
  • Or, there is a dynamical reason why nature evolved towards a critical point. This is known as self-organized criticality.

In the first category, we have multiverse-anthropic-principle type explanation.

If you are, like me, not a fan of these types of arguments, there is good news: self-organized criticality is a thing in nature. There are many known systems which evolve automatically towards a critical point. The most famous one is a sandpile.

For a brilliant discussion of self-organized criticality in general, see

A defining feature of systems close to a critical point is that we get complexity at all scales. Under normal circumstances, interesting phenomena only happen on scales comparable to the lattice spacing (which in particle physics possibly means the Planck scale).  But luckily, there are complex phenomena at all scales in nature, not just at extremely small scales. Otherwise, humans wouldn’t exist. This, I think, hints beautifully towards the interpretation that nature is fundamentally close to a critical point.

PS: As far as I know, Christof Wetterich was the first one who noticed the connection between criticality and the hierarchy problem in the paper mentioned above. Combining it with self-organized criticality was proposed in Self-organizing criticality, large anomalous mass dimension and the gauge hierarchy problem by Stefan Bornholdt and Christof Wetterich. Recently, the connection between self-organized criticality and solutions of the hierarchy problem was emphasized by Gian Francesco Giudice in The Dawn of the Post-Naturalness Era.)

PS: Please let me know if you know any paper in which the self-organized criticality idea is applied to the hierarchy problem in fundamental physics.

P.S. I wrote a textbook which is in some sense the book I wished had existed when I started my journey in physics. It's called "Physics from Symmetry" and you can buy it, for example, at Amazon. And I'm now on Twitter too if you'd like to get updates about what I'm recently up to.

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