15 years ago large extra dimensions were on vogue.
The idea that our universe may have additional spatial dimensions so small we have not yet been able to observe them dates back more than a century. This idea received a tremendous boost by the realization that String Theory actually requires such additional dimensions for consistency, but they were normally assumed to be wrapped up to sizes about a Planck length, or 10
-35m. That’s so small you can forget about it. (Forgetting about them being the reason to wrap them up to begin with.)
Then in 1998/99 some smart physicists realized that if there are extra dimensions, they could be much larger
than the Planck length, and we wouldn’t have noticed. Better still, if these dimensions have the right size this would explain why gravity is so much weaker than the other interactions in the standard model, a problem called the “hierarchy problem” that causes physicists sleepless nights.
In these scenarios with large extra dimensions, the stuff that we are made of (quarks, electrons and so on) sits on a slice with three spatial dimensions, which is called a “brane”. This matter does not normally notice the additional dimension, but gravity does. This has the result that gravity is weak on long distances, but becomes much stronger on short distances, leading to a “lowered Planck scale” and quantum gravitational effects that are much larger than naively expected. Thus the excitement. (There are different models with different realizations of this, but the details won’t concern us in the following. For details read this earlier post.).
If one buys into this, one however has a new problem: The question why the extra dimensions have exactly this size. But sometimes finding a new way to formulate an old question can be a big step forward, so this should not deter us from exploring the idea.
Models with large extra dimensions also made predictions for the LHC due to the lowered Planck scale, most strikingly graviton and black hole production. In 2012, now that the end of the world is near, we know that nothing like this has been seen.
As I explained in this earlier post, it is quite rare that experiment can falsify a model, even if you might have heard so. Normally a model has free parameters that should be determined by experiment, or, if nothing is found, be constrained by experiment. That the LHC has not found evidence for large extra dimensions doesn’t falsify the idea, but it certainly “implausifies” it by constraining the parameters into an uninteresting range. Which is another way of saying, move on, there’s nothing to see here.
So you might think large extra dimensions are dead. But Cliff Burgess begs to differ. In two recent arXiv papers, he and his collaborators have put forward an extra dimensional model that offers an intriguing new perspective:
Accidental SUSY: Enhanced Bulk Supersymmetry from Brane Back-reaction
C. P. Burgess, L. van Nierop, S. Parameswaran, A. Salvio, M. Williams
arXiv:1210.5405
Running with Rugby Balls: Bulk Renormalization of Codimension-2 Branes
M. Williams, C.P. Burgess, L. van Nierop, A. Salvio
arXiv:1210.3753
The key point is that they are trying in the first place not to solve the hierarchy problem, but the cosmological constant problem: Why is the cosmological constant that we observe small and nonzero? The cosmological constant term in general relativity describes the energy of the vacuum and this energy should receive contributions all the way up to the Planck energy. This would be a huge contribution, leading to a very strong curvature of our universe, incompatible with observation. Why aren’t these Planck energetic quantum contributions there?
Burgess and his collaborators argue that a plausible reason is that space-time has additional dimensions, and the full space-time is not Lorentz-invariant. In other words, it’s a scenario with branes in higher dimensions. In such a situation, the troublesome quantum contributions, which normally, due to Lorentz-invariance, take on the form of a cosmological constant term, might not make themselves noticeable on the brane, which is where we live.
The example that they give is that of a cosmic string. If one calculates the metric that the string induces, one finds that space is flat but has a defect angle that depends on the string tension. The string itself however is unaffected by what it does to the background. The scenario that Burgess et al construct is basically a higher-dimensional version of this, where our universe plays the role of the string and creates a defect, but no curvature is induced in our universe itself.
Concretely, they have two additional large extra dimensions. (There might be more than that, but if they are much smaller their presence does not matter for the argument.) These additional dimensions have the topology of a sphere. On the two poles of the sphere, there are a brane each, one of which you can interpret as our universe. Like in the case with the cosmic string, the matter density on the branes induces a defect angle for the sphere, creating a manifold which they call a “rugby ball”. The radius of the sphere is flux-stabilized, which leaves one free parameter (a combination of the radius and the dilaton field).
These extra dimensions induce a vacuum energy on the brane, which is essentially the Casimir energy of this compact space, and this energy depends on the radius of the sphere. To use this scenario to get the right value of the cosmological constant, the radius should be of the order of about 5 μm, which is somewhat below current measurement precision (45 μm), but not so far below.
But what about the troublesome quantum corrections?
Supersymmetry must be broken on the brane (because we don’t see it) but is intact away from it. Supersymmetry solves the cosmological constant problem in the sense that it brings all the troublesome contributions in the bulk to zero. What remains to be shown though is that the cosmological constant on the brane does not receive large correction terms, which depends on the way the branes are coupled.
Cliff and his collaborators have shown that, in the scenario they constructed, the cosmological constant on the brane (read “in our universe”) does receive correction terms from high energies, but due to the way the branes are coupled these corrections are highly suppressed and do not ruin the smallness of the effective cosmological constant; they do not induce a large curvature. Think of the example with the cosmic string that stands in for higher dimensional branes. The geometry on the string (or brane) is flat regardless of the value of the tension. The large quantum corrections are there, but they contribute to the tension rather than inducing a curvature.
Now once you have fixed the radius of the “rugby ball” so that the cosmological constant matches with observation, you can use this to calculate the value of the lowered Planck scale. It turns out to be at least 10 TeV, so we wouldn’t see gravitons or black holes at the LHC. (Keep in mind that the LHC collides protons, which are composite particles. The average energy per individual collision of quarks or gluons is in most cases far below the total energy in the proton collision which is usually quoted. That’s why everybody wants a lepton collider.) However, since the string scale is somewhat below the Planck scale, one would expect to see string excitations at the LHC, though still at fairly high energies; we wouldn’t have seen them yet.
So to sum up, what this model achieves is the following: 1) It provides a setting in which there is a small cosmological constant whose small value is not ruined by large quantum corrections. 2) It makes the prediction that we should see corrections to Newton’s law not too far beyond present measurement precision. 3) It gives a plausible reason why we haven’t seen evidence for extra dimensions at the LHC so far but 4) predicts that we should see some glimpses of it in form of string excitations within the next years.
This doesn’t convince me to start working on large extra dimensions again, but it does convince me that large extra dimensions aren’t dead yet.
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Or, as I like to put it, you really have to be careful what secondary critera (publications in journals with high impact factor) you use to substitute for the primary goal (good science). If you use the wrong criteria you'll not only not reach an optimal configuration, but make it increasingly harder to ever get there because you're changing the background on which you're optimizing (selecting for people with non-optimal strategies). 
