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Flip Tanedo | USLHC | USA

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The Hierarchy Problem: why the Higgs has a snowball’s chance in hell

The Higgs boson plays a key role in the Standard Model: it is related to the unification of the electromagnetic and weak forces, explains the origin of elementary particle masses, and provides a weakly coupled way to unitarize longitudinal vector boson scattering.

As particle physics community eagerly awaits CERN’s special seminar on a possible Higgs discovery (see Aidan’s liveblog), it’s a good time to review why Higgs—the last piece of the Standard Model—is also one of the big reasons why we expect even more exciting physics beyond the Standard Model.

The main reason is called the Hierarchy problem. This is often ‘explained’ by saying that quantum corrections want to make the Higgs much heavier than we need it to be… say, 125-ish GeV. Before explaining what that means, let me put it in plain language:

The Higgs has a snowball’s chance in hell of having a mass in that ballpark.

This statement works as an analogy, not just an idiom. (This analogy is adapted from one originally by R. Rattazzi involving a low energy particle passing through a thermal bath. Edit: I’m told this analogy was by G. Giudice, thanks Duccio.)

If you put a glass of water in a really hot place—you expect it to also become really hot, maybe even to off into steam.  It would be really surprising if we put an ice cube in a hot oven and 10 minutes later it had not melted. This is because the ambient thermal energy is expected to be transferred to the ice cube by the energetic air molecules bouncing off it. Sure, it is possible that the air molecules just happen to bounce in a way that doesn’t impart much thermal energy—but that would be ridiculously improbable, as we learn in thermodynamics.

The Higgs is very similar: we expect its mass to be around 125 GeV (not too far from W and Z masses), but ambient quantum energy wants to make its mass much larger through interactions with virtual particles. While it is possible that the Higgs stays light without any additional help, it’s ridiculously improbable, as we learn from quantum physics.

Remark: the relation between thermal/statistical uncertainty and quantum uncertainty is actually one that is deeply woven into their mathematical descriptions and is the reason why quantum (or statistical) field theory is the common language of both particle physics and condensed matter physics.

Quantum corrections: the analogy of the point electron

The phrase “quantum corrections” is somewhat daunting, so let’s appeal to a slightly more familiar problem (from H. Murayama) and draw some pictures. The analog of the Hierarchy problem in classical physics is the question of the electron self energy:

The electron has charge but is nearly point-like. It must have a very large charge density and thus have a very large self-energy (mass).

Self-energy here just means the contribution to the electron mass coming from repulsive electrostatic energy of one part of the electron from another. The problem thus reduces to: how can the electron mass be so small when we expect it to be large due to electrostatic energy? Yet another way to pose the question is to say that the electron mass has contributions from some ‘inherent mass’ (or ‘bare mass’) m0 and the electrostatic energy, ΔE:

mmeasured = m0ΔE

Since mmeasured is small while ΔE is large, then it seems that m0 must be very specifically chosen to cancel out most of ΔE but still leave the correct tiny leftover value for the electron mass. In other words, the ‘bare mass’ m0 must be chosen to uncomfortably high precision.

I walk through the numbers in a previous post (see also the last few pages of these lectures to undergraduates [pdf] from here), but here’s the main idea: the reason why there isn’t a huge electrostatic contribution to the electron mass is that virtual electron–positron pairs smear out the electric charge over a radius larger than the size of the electron:

In other words: current experimental bounds tell me that the electron is smaller than 10-17 cm and the “electron hierarchy problem” arises when I calculate the energy associated with packing in one unit of electric charge into that radius. The resolution is that even though the electron may be tiny, at a certain length scale quantum mechanics becomes relevant and you start seeing virtual electron–positrion pairs which interact with the physical electron to smear out the charge over a larger distance (this is called vacuum polarization).

The distance at which this smearing takes place is predicted by quantum mechanics—it’s the distance where the virtual particles have enough energy to become real—and when you plug in the numbers, it’s precisely where it needs to be to prevent a large electrostatic contribution to the electron mass. Since we’re now experts with Feynman diagrams, here’s what such a process looks like in that language:

Higgs: the petulant child of the Standard Model

The Hierarchy problem for the Higgs is the quantum version of the above problem. “Classically” the Higgs has a mass that comes from the following diagram (note the Higgs vev):

This diagram is perfectly well behaved. The problem occurs from contributions that include loops of virtual particles—these play the role of the electrostatic contribution to the electron mass in the above analogy:

As an exercise, use the Higgs Feynman rules to draw other contributions to the Higgs mass which contain a single loop; for our present purposes the one above is sufficient. Recall, further, that  one of our rules for drawing diagrams was that momentum is conserved. In the above diagram, the incoming Higgs has some momentum (which has to be the same as the outgoing Higgs), but the virtual particle momenta (k) can be anything. What this means is that we have to sum over an infinite number of diagrams, each with a different momentum k running through the loop.

We’ll ignore the mathematical expression that’s actually being summed, but suffice it to say that it is divergent—infinity. This is a good place for you to say, what?! the Higgs mass isn’t infinity… that doesn’t even make sense! That’s right—so instead of summing up to diagrams with infinite loop momentum, we should stop where we expect our model to break down. But without any yet undiscovered physics, the only energy scale at which we know our description must break down is the gravitational scale: MPlanck ~ 1018 GeV. And thus, as a rough estimate, these loop diagrams want to push the Higgs mass up to 1018 GeV… which is way heavier than we could ever hope to discover from a 14 TeV (= 14,000 GeV) LHC. (Recall that these virtual contributions to the Higgs mass are what were analogous to thermal energy in our “snowball in Hell” analogy.)

But here’s the real problem: the Standard Model really, really wants the Higgs to be around the 100 GeV scale. This is because it needs something to “unitarize longitudinal vector boson scattering.” It needs to have some Higgs-like state accessible at low energies to explain why certain observed particle interactions are well behaved. Thus if the Higgs indeed has a mass around 125 GeV, then the only way to make sense of the 1018 GeV mass contribution from the loop diagram above is if the “classical” (or “tree”) diagram has a value which precisely cancels that huge number to leave only a 125 GeV mass. This is the analog of choosing m0 in the electron analogy above.

Unlike the electron analogy above, we don’t know what kind of physics can explain this 1016 ‘fine-tuning’ of our Standard Model parameters. For this reason, we expect there to be some kind of new physics accessible at TeV energies to explain why the Higgs should be right around that scale rather than being at the Planck mass.

Outlook on the Hierarchy

The Hierarchy problem has been the main motivation for new physics at the TeV scale for over two decades. There are a few obvious questions that you may ask.

1. Is it really a problem? Maybe some number just has to be specified very precisely.

Indeed—it is possible that the Higgs mass is 125 GeV due to some miraculous almost-cancellation that set it to be in just the right ballpark to unitarize longitudinal vector boson scattering. But such miracles are rare in physics without any a priori explanation. The electron mass is an excellent example. There are some apparent (and somewhat controversial) counter-examples: the cosmological constant problem is a much more severe ‘fine-tuning’ problem which may be explained anthropically rather than through more fundamental principles.

2. I can draw loop diagrams for all of the Standard Model particles… why don’t they all have Hierarchy problems?

If you’ve asked this question, then you get an A+. Indeed, based on the arguments in this post, it seems like any diagram with a loop gives a divergence when you sum over the possible intermediate momenta so that we would expect all Standard Model particles to have Planck-scale masses due to quantum corrections. However, the important point was that we never wrote out the mathematical form of the thing that we’re summing.

It turns out that the Hierarchy problem is unique for scalar particles like the Higgs. Loop contributions to fermion masses are not so sensitive to the ‘cutoff’ scale where the theory breaks down. This is manifested in the mathematical expression for the fermion mass and is ultimately due to the chiral structure of fermions in four dimensions. Gauge boson masses are also protected, but from a different mechanism: gauge invariance. More generally, particles that carry spin are very picky about whether they’re massive or massless, whereas scalar particles like the Higgs are not, which makes the Higgs susceptible to large quantum corrections to its mass.

3. What are the possible ways to solve the Hierarchy problem?

There are two main directions that most people consider:

  1. Supersymmetry. Recall in our electron analogy that the solution to the ‘electron mass hierarchy problem’ was that quantum mechanics doubled the number of particles: in addition to the electron, there was also a positron. The virtual electron–positron contributions solved the problem by smearing out the electric charge. Supersymmetry is an analogous idea where once again the set of particles is doubled, and in doing so the loop contributions of one particle to the Higgs are cancelled by the loop contributions of its super-partner. Supersymmetry has deep connections to an extension of space-time symmetry since it relates matter particles to force particles.
  2. Compositeness/extra dimensions. The other solution is that maybe our description of physics breaks down much sooner than the Planck scale. In particular, maybe at the TeV scale the Higgs no longer behaves like a scalar particles, but rather as a bound state of two fermions. This is precisely what happens with the mesons: even though the pion is a scalar, there is no pion ‘hierarchy problem’ because as you probe smaller distances, you realize the pion is actually a bound state of two quarks and it starts behaving as such. One of the beautiful developments of theoretical physics in the 1990s and early 2000s was the realization that this is precisely what is being described by theories of extra dimensions through the so-called holographic principle.

So there you have it—while you’re celebrating the [anticipated] Higgs discovery with fireworks on July 4th, also take a moment to appreciate that this isn’t the end of a journey culminating in the Standard Model, but the beginning of an expedition for exciting new physics at the terascale.

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