Matt Strassler [June 13, 2012]
Our world may consist of the obvious three spatial dimensions that we can see and feel and move around in, but it may in principle have other dimensions of space that for one reason or another we cannot see and move around in, at least not easily. But principle is one thing, practice is another. Do we have any evidence in favor of or against extra dimensions, and if so, for dimensions of what sizes and types?
If you’ve read about what extra dimensions of space are (see also here, here and here) and if you’ve understood why we expect Kaluza-Klein (KK) partner particles for any particles which can move in an extra dimension, you’re ready to understand what we know experimentally about extra dimensions.
One of the key observations made in my article on “what is an extra dimension?” is that there are two ways that a dimension can be “extra” — undetectable to your senses and brain, and perhaps to all but the most clever scientific experiments that you can perform. I illustrated this with figures that are reproduced here. The first is that an extra dimension can be so small that you, and everything that you and your scientific experiments are made of, basically fill it from side to side, like a large ship filling a ship canal. Only very small objects (and experiments that are sensitive to them) can show signs of the extra dimension. (See Figures 1 and 2.)
A second is that you, and everything that you and your scientific experiments are made of, can be tethered or trapped in some way so that, like a ship tethered to and towed along one side of a ship canal (Figure 3), you are insensitive to the presence of the extra dimension.
These ideas are not limited to a single extra dimension; there may be multiple extra dimensions, see Figure 4.
Now, what can we do as scientists to rule out various possibilities? KK partner particles are very helpful, though we can make some progress using more intuitive arguments that I’ll give you first.
How Big Could a Fully Accessible Extra Dimension Be?
If all known particles were able to move in an extra dimension, then the atoms out of which we are made, and the light that we see, could move around in more than three dimensions. Clearly these dimensions can’t be infinite in size, or macroscopic in size, or we’d see them, literally; we could turn our heads in the direction of that extra dimension, just as we can turn our heads to the right or look down out our feet, and our eyes could absorb photons coming from that direction, just as they can see light coming from the horizon or from overhead. Since we don’t see them, any such extra dimensions have to be finite in extent and microscopic.
In this case, the analogy of a two-dimensional strip or tube that appears one-dimensional from the point of view of something large — the example of a canal viewed by a large freighter was given in Figure 2 above — applies by analogy to us. The world might have four or five or eight or twelve space dimensions, for all we know, as long as all but the three dimensions we can see easily are finite and small in extent. We may just be too big to move around within them, just as a big freighter that completely fills a canal from side to side will be unable to move right or left.
However, suppose that the extra dimensions that our atoms can move around in are significantly larger than the atoms themselves. Well, then we’d know that already, for many reasons. For one thing (for those of you who took freshman-year physics) all of the calculations of the heat capacity of gases presuppose that molecules in the gas are moving only in three spatial dimensions; if they could really move in more dimensions, the heat capacity calculations would disagree with data, by a lot. For another, all of our calculations of atomic physics — the energy levels of hydrogen, for instance — presuppose that electrons can move only in the three spatial dimensions that we can see. If there were more spatial dimensions, then electrons could move around in ways that we are not expecting them to, and that would completely change how they orbit within atoms, ruining our predictions. Since our predictions work very well indeed, we conclude that if there are any extra dimensions like the direction across a canal, the distance across those dimensions must be smaller than 1/10,000,000,000 meters (an “Angstrom”) or so.
The same general logic applies to atomic nuclei. If there were extra dimensions as large as atomic nuclei (1/100,000,000,000,000 meters, a millionth of a billionth of a meter) then nuclear physics would be in many ways drastically altered compared to what we observe.
These are just a few among many similar considerations. But we can go further. There’s a very powerful argument that gets us down another factor of about 100 or more. It relies on those KK partner particles, and it applies to a wider class of extra dimensions.
Implications of the Absence of KK Partner Particles
In the previous few paragraphs I asked what we’d observe if all known particles moved in extra dimensions larger than a certain size. But in the next few we’ll see that we can rule out, via a simple argument, the possibility that almost any of the known particles moves in extra dimensions that are larger than about 1/100,000,000,000,000 meters (ten billionths of a billionth of a meter.) So this will also cover cases where some of the known particles move around in an extra dimension (like the small boat in Figure 3) while others don’t (like the tethered boat in Figure 3.)
It turns out to be efficient to start the argument with electrically charged particles, and then move on to the electrically neutral ones. Any particle and its KK partner particles must have the same electric charge — because a KK partner particle is secretly just the original particle moving in the direction of an extra dimension. So, for example, if the electron has KK partner particles, those particles must be heavier than the electron and carry the same electric charge as the electron. [There's a special case where the KK partners might have different (but still non-zero) charges; since that case will also be ruled out by the coming argument, I won't dwell on it here.]
Any electrically charged particle (let’s call it E, and call its anti-particle E*) can always be produced in the process
- electron + positron → E + E*,
in particle-antiparticle annihilation via a virtual photon. This is just the same type of process as electron + positron → muon + anti-muon. The only thing that is required for it to occur occasionally is that the collision of the electron and positron must have energy that is more than twice the mass of E times c2.
Well, at the LEP collider, which operated in the 1990s, the collision energy of electrons and positrons slightly exceeded 200 GeV. So if there were any E particle that we didn’t already know about and that had a mass less than 100 GeV/c2, it would have been produced in this way, and in abundance. Would it have been observed, though, by the four experiments that were studying LEP’s collisions? Well, exactly how it would have been observed would depend on exactly how it decays, but you can make an exhaustive list of the possibilities and come to the conclusion that it is virtually impossible for any sort of E to have gone unnoticed. [For example; if E were a KK partner of the electron it could form a new stable charged particle (easily observed) or decay to an electron and a photon (easily observed) or decay to an electron plus undetectable particles such as neutrinos (easily observed); even if it weirdly decayed entirely to hadrons, this would have been observed in the total rate for hadron production.]
So right away, this tells you that there are no KK partners of any electrically charged particle with mass below about 100 GeV/c2, which in turn means that electrons, muons, taus, up quarks, down quarks, strange quarks, charm quarks and bottom quarks cannot be moving in any extra dimensions whose extent is larger than about 1/100,000,000,000,000 meters (ten billionths of a billionth of a meter). Results from the Large Hadron Collider [LHC] (which has proton-proton collisions in which the process quark + anti-quark → E + E* may occur) have probably improved these limits by a small factor, of about 2 to about 10 depending on the type of particle.
[The experiments at the Tevatron, predecessor to the LHC, and other experiments that were predecessors to those at LEP, also made nice measurements that constrained the possibility of extra dimensions, but for brevity, and because their results are largely superseded, I will not refer to them; apologies to those who did these pioneering measurements.]
Now you might ask: wait, couldn’t the muon and the tau particle be KK partners of the electron, and the bottom and strange quarks be KK partners of the down quark? The answer is not obvious, but it turns out to be no. The way the weak nuclear force works on these particles assures that all of them get their masses from the Higgs field (or fields), and that none of them get masses directly from the effect of being KK partners of some other particle. [For instance, in electron-positron collisions, the muon and tau are produced with the same angular distribution, whereas KK partners of the electron would have a different angular distribution.] A couple of other reasons are given at the end of this article.
What about the other known particles? Well, KK partner particles (call them `N’) for the three known neutrinos would also be easily observed, because the process
- electron+positron → N + N*
will also occur, through the weak nuclear force, i.e. through a Z virtual particle. Now N might be detectable (if it decays to something detectable) and then, just as for KK partners of charged particles, the LEP collider experiments put a similar bound on any extra dimensions that neutrinos travel in. And even if N were as difficult to detect as a neutrino itself, the process
- electron+positron → N + N* + photon ,
giving a photon and nothing else detectable, can be used to look for neutrinos and for any other undetectable particles. No excess of collisions of this type were observed at LEP. Similar techniques can be used at the LHC, starting with a quark and anti-quark. In the end, limits on the sizes of extra dimensions for the known types of neutrinos are obtained, and they are not much different from the limits for charged particles.
Photons and gluons are also neutral and lightweight; what about them? Similar arguments apply, though with a bit more subtlety. Processes such as
- electron + positron → KK photon + ordinary photon
- electron + positron → 2 KK photons
would be expected to occur, but none were observed. One might even have observed, in analogy to the “resonant” process electron + positron → Z particle, the process in which a single KK photon is created (and subsequently decays to a known particle and its anti-particle, such as a muon/anti-muon pair.) Nothing like this was observed at LEP; nor has anything like it been observed at the LHC. As before, this tells us that any extra dimensions accessible to photons are no larger than about ten billionths of a billionth of a meter.
(Again, the possibility that the Z particle is a KK partner of a photon might occur to you, but again, for a not obvious reason, the pattern of the interactions of the Z particle and photon with matter make clear that this is not the case. For instance, the Z particle interacts directly with neutrinos even though they carry no electric charge, something which would be impossible for photons and their KK partners.)
As for gluons, one may follow similar logic, just replacing the electron and positron at LEP with a quark and anti-quark at the LHC, and replacing the photons and their KK partners at LEP with gluons and their partners at the LHC. The absence of any surprises in the data (so far) puts similar constraints on extra dimensions for gluons.
What about the heavy known particles, the W+, W- and Z particles, and the top quark? The limits are similar, but to see this requires one more observation. As noted in my article on KK partners, the KK partners of a heavy particle of mass m in a rather large extra dimension of length L will have masses M that are not that different from the heavy particle itself. With a single simple extra dimension the formula for the first KK partner will be
- M2 = m2 + (h/c L)2
(where h is Planck’s constant and c is the speed of light) or something like this (depending on the shape of the extra dimensions.) If m is very small and L is very small, then M will be proportional to 1/L, but if m is large and L is large, M will be just slightly larger than m. For W, Z and top, with masses of 80, 91 and 173 GeV/c2, an extra dimension with a radius of 10 billionths of a billionth of a meter would produce a KK partner particle whose mass is not much higher than that of the W, Z or top itself. No such particles (which would be easy to detect) have been observed at LEP or the LHC, so yet again, the limits on extra dimensions are of the same size as for the other particles.
To summarize: for all these particles, any extra dimensions in which they can move must be smaller than a few billionths of a billionth of a meter — something like ten million times shorter than the distance across an atom, and a hundred times shorter than the distance across a proton. In short, small. The search continues at the LHC for any sign of even smaller extra dimensions — by looking for the corresponding KK partner particles with masses in the range of a few hundred GeV/2 to a few TeV/c2. One looks for KK partners of electrons, muons and taus; of the six quarks, of the three neutrinos; of photons, gluons, W and Z particles — all and any of them. None have yet been found, so the constraints keep tightening on how big an extra dimension accessible to these particles can be.
We left some loopholes: Higgs particles; dark matter; new types of neutrinos; and gravitons, the particles that are (as yet undetected, since they are extremely hard to produce) ripples in the gravitational field.
Since knowledge about the Higgs particle is spotty at best, we can say rather little about it as yet. The one thing we can say is that a Standard Model Higgs (the simplest form of Higgs particle) has the same constraints on extra dimensions, roughly, as the W or Z, because it interacts very directly with them. The constraints could be loosened for a more unusual set of Higgs particles, but this is a long story, about which we’ll know more much more experimentally by the end of 2012.
For dark matter particles, or any new type of neutrino, we can say even less. Nothing is known; these particles may not even exist, so there’s not a lot we can say about what their properties are.
Gravitons, however, are a very interesting possibility. They interact so weakly with ordinary matter that the LEP collider would not have had such an easy time making them; the process
- electron + positron → photon + KK partner of graviton
is certainly allowed, but the rate is very tiny. However, this is precisely where the story gets very interesting… because although, as we saw, extra dimensions accessible to any of the known particles must be 100 times or more smaller than the distance across a proton, extra dimensions inaccessible to all known particles but accessible to gravity could be quite a bit larger!
Extra Dimensions Inaccessible to All Known Particles But Not to Gravitons
This is a big subject that I’m going to address in a few steps.
First, let’s see that we can already learn something from Newton’s law of gravity.
Next, we’ll see what can be learned from the fact that Kaluza-Klein partners of the graviton would have effects that we have not observed. [coming soon]