The Hierarchy Problem

What is the Hierarchy Problem?

An important feature of nature that puzzles scientists like myself is known as the hierarchy, meaning the vast discrepancy between aspects of the weak nuclear force and gravity. There are several different ways to describe this hierarchy, each emphasizing a different feature of it. Here is one:

  • The mass of the smallest possible black hole defines what is known as the Planck Mass. [Planck was the scientist who took the first step towards quantum mechanics.] (A more precise way to define this is as a combination of Newton’s gravitational constant G, Planck’s quantum constant h-bar, and the speed of light c: the Planck mass is the square root of h-bar times c divided by G.) The masses of the W and Z particles, the force carriers of the weak nuclear force, are about 10,000,000,000,000,000 times smaller than the Planck Mass. Thus there is a huge hierarchy in the mass scales of weak nuclear forces and gravity.

When faced with such a large number as 10,000,000,000,000,000, ten quadrillion, the question that physicists are naturally led to ask is: where did that number come from? It might have some sort of interesting explanation.

But while trying to figure out a possible explanation, physicists in the 1970s realized there was actually a serious problem, even a paradox, behind this number. The issue, now called the hierarchy problem, has to do with the size of the non-zero Higgs field, which in turn determines the mass of the W and Z particles.

The non-zero Higgs field has a size of about 250 GeV, and that gives us the W and Z particles with masses of about 100 GeV. But it turns out that quantum mechanics would lead us to expect that this size of a Higgs field is unstable, something like (warning: imperfect analogy ahead) a vase balanced precariously on the edge of a table. With the physics we know about so far, the tendency of quantum mechanics to jostle — those quantum fluctuations I’ve mentioned elsewhere — would seem to imply that there are two natural values for the Higgs field — in analogy to the two natural places for the vase, firmly placed on the table or smashed on the floor. Naively, the Higgs field should either be zero, or it should be as big as the Planck Energy, 10,000,000,000,000,000 times larger than it is observed to be. Why is it at a value that is non-zero and tiny, a value that seems, at least naively, so unnatural?

This is the hierarchy problem.

Many theoretical physicists have devoted significant fractions of their careers to trying to solve this problem. Some have argued that new particles and new forces are needed (and their theories go by names such as supersymmetry, technicolor , little Higgs, etc.) Some have argued that our understanding of gravity is mistaken and that there are new unknown dimensions (“extra dimensions”) of space that will become apparent to our experiments at the Large Hadron collider in the near future. Others have argued that there is nothing to explain, because of a selection effect: the universe is far larger and far more diverse than the part that we can see, and we live in an apparently unnatural part of the universe mainly because the rest of it is uninhabitable — much the way that although rocky planets are rare in the universe, we live on one because it’s the only place we could have evolved and survived. There may be other solutions to this problem that have not yet been invented.

Many of these solutions — certainly all the ones with new particles and forces or with new dimensions — predict that new phenomena should be visible at the Large Hadron Collider. Even as I write this, the Large Hadron Collider is slowly but surely excluding many of these possibilities. So far it has not seen any unexpected new phenomena. But these are still early days.

By the way, you will often read the hierarchy problem stated as a problem with the Higgs particle mass.  This is incorrect.  The problem is with how big the non-zero Higgs field is.  (For experts — quantum mechanics corrects not the Higgs particle mass but the Higgs mass-squared parameter, changing the Higgs field potential energy and thus the field’s value, making it zero or immense.  That’s a disaster because the W and Z masses are known.  The Higgs mass is unknown, and therefore it could be very large — if the W and Z masses were very large too.  So it is the W and Z masses — and the size of the non-zero Higgs field — that are the problem, both logically and scientifically.)


110 responses to “The Hierarchy Problem

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  3. This is an awesome explanation of the phenomena. Thank you.

  4. Marcel van Velzen

    Hello Prof. Strassler,

    At least part of the hierarchy problem seems more like a hierarchy gift to me because it makes the effective field theory picture of physics complete:

    The standard model is composed of A and B terms (the renormalizable and non-renormalizable terms). The A terms are the normal standard model terms, the B terms would spoil it. So the assumption is that the B terms are suppressed by some mass scale and are unobservable at the LHC. Now this seems like an empty statement, one could say: there are no B terms.
    It is here that gravity comes to our rescue. It can be shown that gravity cannot have A terms and only consists of B terms. So if the above picture is correct then gravity should be far weaker than the electroweak and strong forces (that contain A terms) and gravity is weak, thank goodness!
    Furthermore, we have indications from the past that when the B terms become important the problems with the B terms are solved by more substructure (for beta decay these were the W and Z).
    So it tells us: at really high energy (probably near the Planck scale) we will see more substructure which sounds not too bad.

    To me this is a reasonable picture of the current state of physics, it could have been much worse. Your view on the above would be greatly appreciated.

  5. All of the anthropic balances are fixed to produce the most energy efficient structure possible like an energy conservation law that maximizes gradient breakdown, (work), before energy becomes inert.

    You might like this, Matt:

    From here:

  6. Black holes can be less massive than the Planck mass. It is not the way it is defined. You can have TeV black holes if quantum gravity is stronger than its classical conterpart.

    • You are correct of course; I oversimplified. (Though this is not an issue of whether quantum gravity is stronger than its classical counterpart; black holes are already semi-classical. It is whether there are extra dimensions that make the true Planck mass much smaller than it appears at long distance.)

      I wanted a physical way of defining the Planck mass that does not involve throwing around concepts that people have never heard of, but I believe you are right that I could do a better job, perhaps by defining how heavy a quark would have to be before the strong nuclear force and gravity between a quark and an antiquark would be equal in strength.

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  9. You mention in this article that the size of the Higgs field is about 250 GeV.

    Now that the Higgs particle has been discovered, it’s notable that its mass is just about half that figure — 125 GeV. Is there any reason to think that this is significant, or is it just coincidence?

    (For that matter, how is the size of the Higgs field known? Calculated based on the W and Z masses? How precise is that 250 GeV figure?)

    • The value of the Higgs field is 246 GeV.

      There is no known reason to think there is a connection with the 125 GeV mass-energy (i.e. E=mc^2 energy) of the Higgs. A mass-energy of 123 GeV is just about ruled out. As far as anyone knows, that’s a coincidence.

      You can play lots of games with numerology; the top quark mass is closed to sqrt(2) * the Higgs mass, the Z particle mass is close to the Higgs mass / sqrt(2). But when you try to calculate these things in real theories, such simple ratios do not generally emerge for particle masses; quantum corrections move things around a lot.

      • Thanks!

        I figured it couldn’t be that easy — otherwise theorists could have predicted the Higgs mass based on the value of the Higgs field!

        I am still curious about the other question, though — how is the value of the Higgs field determined experimentally?

  10. A few times I’ve seen custodial symmetry mentioned in relation to the hierarchy problem. Is it something that has a simple explanation?

    • Not without math, no. The statement is simple enough: if you turn off all of the Higgs field’s interactions with matter and all of its interactions with hypercharge, the resulting theory of weak-isospin particles coupled to the Higgs field has an extra SU(2) global symmetry. With the interactions turned back on the symmetry is only approximate, but it is powerful enough to constrain the predictions of the theory in important ways. But to prove the extra SU(2) global symmetry is there is technical (not very hard, but not appropriate for this website.)

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  13. suppose for a moment we exist in a 4D spatial universe. With 2 notions of time. 1. homogenous and the other relativistic. The primary dimension defined as providing a unity time value across the entire universe which determines and provides space for anything to exist and move. Hence the velocity of light. This primary dimension could it not provide the opening space by virtue of its expansion which locally determines the value of light speed. Then if we consider Planks constant and change the value of C then all matter dissociates back into its primordial form. Light is currently constant ( 14 billion years apparently) but who is to say that it might be due for a change? Who knows what physical dynamics occurring at the periphery might suddenly affect the local expansion. Then all the matter which has been created hence gravity – just vanishes instantaneously. Max Planks theory can explain that and the cohesion in the SM. Change the value of C and all known physics changes with it.

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  16. Why is “gluon” effective only at very short distance even though it is massless like photon?

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  18. Do GUTs make this problem very much worse? E.g. if there exist X and Y bosons that couple quarks to leptons, they would have to be much more massive than the W and Z. The Higgs would have to couple to X and Y, right? Is this (yet more) bad news for GUTs, or is it assumed that anything that fixes the hierarchy problem for SM would also solve it for GUTs?

    • No, GUTs (grand unification of the three non-gravitational forces, which, if it occurs at all, probably does so for physics processes with energy above 10^16 GeV — compare to the LHC which has a collision energy of about 10^4 GeV) don’t make the problem hierarchy worse. The hierarchy problem at the weak scale is a problem of 1 part in 10^32: specifically, (M_Planck/m_W)^2. A GUT would only have a problem of (M_Planck/M_GUT)^2 ~ 10^6, totally negligible by comparison.

  19. On your “warning! bad analogy”, might a better one be to a coin on edge? We’d not be surprised if it landed heads (0 GeV) or tails (1000etc GeV), but to land on its edge (246 GeV), between heads and tails: very weird. Maybe?

    • I gave a similar example, but I think that Matt’s point is that the value is balanced much closer to zero, (the top of the table), than it is the floor, which is a unique and different, (perhaps better), way of describing it that I can’t for the life of me repeat with a similar example.

      • Yes, that’s the problem with the coin. But hey, there’s no perfect example.

          • A motorboat perched just over the edge of a waterfall that is kept in place by the counterbalancing force of the spinning propeller.

            I say that because the balance is fixed dynamically when it is observed in other facets of nature, e.g. the tendency for the boat to go over the falls if the water speeds up is quickly offset by an increase in the speed of the engine and the propeller.

            Maybe it doesn’t apply here, but … don’t bet on it… 😉

  20. could dark energy and dark matter be seeping into our universe,and thats why we dont know much about it,like an ocean that exists beyond,and were just a bubble that is slowly being filled,,,would this solve any part of the hierarchy problem,,,

  21. Hw can da weak nuclear force carrier be 10 quadrillion times smaller den planck’s mass? i thought that planck’s mass is the smallest a mass could be? im confused, can sum1 please explain it to me.

    • Planck mass in /not/ the smallest possible mass .. far from it. I think the confusion here is because planck time and planck distance ARE smallest possible measures. In contrast, Planck mass gets its name from the fact that it is the mass of a black hole whose Schwarzchild radius equals planck length. Actually, the question “Why is Planck mass so large?” can be rephrased as “Why is Gravity so weak?” They’re more or less equivalent questions.

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  24. is gravity really so weak at the planck length,what is holding mass together,maybe at the macro end gravity becomes weak,I believe this is refered to as the hierarchy problem,am I correct

    • What holds mass together at the subatomic scale are the electromagnetic force and the strong nuclear force.

      It’s not that gravity is weak at Planck length … it’s weak at ALL lengths (micro and macro), and constant over an infinite range. In contrast, the strong nuclear force has an extremely limited range. So I guess it would be acceptable to say that gravity at Planck length is RELATIVELY weak, since at larger range it doesn’t have to compete with powerful, yet short-range, forces. Note, however, that electormagnetism also has infinite range.

    • I should also have pointed out that the expression “holding mass together” doesn’t make sense. (Sorry that I carelessly used that expression myself.) “Holding matter together” or “holding stuff together” .. these would be acceptable. Mass is not “stuff”; mass is a property that stuff has.

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  38. at ten to the minus 33 cm,or planck length,gravity must be stronger than first realized,?

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  41. Amazing how different physicists give different explanations! Here is what one of your very well known colleagues has to say about the hierarchy problem: “Moving down the hierarchy, 10^-16 times the Planck scale is a TeV, the energy at which the unification of the weak and electromagnetic forces take place. This is called the weak interaction scale. This is the region in which we should see the Higgs boson, and it is also where many theorists expect to see supersymmetry. The LHC is being built to probe the physics at this scale(this is from a book published in 2006). A proton mass is 71,000 of that(!), another 71,000 brings us down to the electron, and perhaps 71,000,000 of that is the neutrino. Then, way down at the bottom, is the vacuum energy, which exists throughout space even in the absence of matter.”
    Now, Dr Strassler, I would be very grateful if you could explain where all those multiples of 71 came from….

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  89. I’ve heard the Hierarchy Problem stated as “…we don’t understand why the weak interaction is so much stronger than gravity…” Full disclosure, I’m a full amateur and not much more than an avid reader… but I’m having trouble understanding why the discrepancy between the strength of the weak interaction and gravity is of significance (assuming the statement above is even correct). Why do we expect gravity to be so much stronger than it is?

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