FROM THE FERMI NATIONAL ACCELERATOR LABORATORY

What extra dimensions, you probably think, having just read the title. We know very well that the world around us is three-dimensional. We know East from West, North from South, up from down – what extra dimensions could there possibly be if we never see them?

Well, it turns out that we do not really know yet how many dimensions our world has. All that our current observations tell us is that the world around us is at least 3+1-dimensional. (The fourth dimension is time. While time is very different from the familiar spatial dimensions, Lorentz and Einstein showed at the beginning of the 20th century that space and time are intrinsically related.) The idea of additional spatial dimensions comes from string theory, the only self-consistent quantum theory of gravity so far. It turns out that for a consistent description of gravity, one needs more than 3+1 dimensions, and the world around us could have up to 11 spatial dimensions!

How could this be possible? The reason we do not feel these additional spatial dimensions in our everyday life (if they exist) is because they are very different from the three dimensions we are familiar with. It turns out that it is possible that our world is 'pinned' to a 3-dimensional sheet (a so-called 'brane') that is located in a higher dimensional space, To illustrate this, imagine an ant crawling on a sheet of paper in your hand. For the ant, the 'universe' is pretty much two-dimensional, as it cannot leave the surface of the paper. It only knows North from South and East from West, but up and down don't make any sense as long as it has to stay on the sheet of paper. In pretty much the same way, we could be restrained to a three-dimensional world, which is in fact a part of a more complicated multi-dimensional universe!

For an ant crawling on a sheet of paper the universe is pretty much two-dimensional.

These extra spatial dimensions, if they really exist, are thought to be curled-up, or "compactified". In the example with the ant, let's roll the sheet of paper so that it forms a cylinder. In this case, if the ant starts crawling in the direction of curvature, it will eventually come back to the same point it started from. This is an example of a compactified dimension. If the ant crawls in a direction parallel to the length of the cylinder, it would never come back to the same point (we are assuming that the paper cylinder is so long so that it never reaches the edge). This is an example of a "flat" dimension. According to string theory then, we live in a universe where our three familiar dimensions of space are "flat", but there are additional dimensions which are curled-up very tightly so that they have an extremely small radius: 10-30 cm or less.

These famous drawings by M.C. Escher illustrate the idea of a compactified spatial dimension.

So why would it matter to us if the universe has more than 3 spatial dimensions, if we can not feel them? Well, in fact we could "feel" these extra dimensions through their effect on gravity. While the forces that hold our world together (electromagnetic, weak, and strong interactions) are constrained to the 3+1-"flat" dimensions, the gravitational interaction always occupies the entire universe, thus allowing it to feel the effects of extra dimensions. Unfortunately, since gravity is a very weak force and since the radius of extra dimensions is tiny, it could be very hard to see any effects, unless there is some kind of mechanism that amplifies the gravitational interaction. Such a mechanism was recently proposed by Arkani-Hamed, Dimopoulos, and Dvali, who realized that the extra dimensions can be as large as one millimeter, and still we could have missed them in our quest for the understanding of how the universe works!

If the extra dimensions were indeed so large, the laws of gravity would be modified at distances comparable to the size of the extra dimensions. So, why don't we see this in experiments? In fact it turns out that we know very well how gravity works for large distances (Isaac Newton's famous law that says that gravitational force between two bodies falls off as the square of distance between them). However, no one has tested how well this works for distances less than about 1 mm. It is complicated to study gravitational interactions at small distances. Objects positioned so close to each other must be very small and very light, so their gravitational interactions are also small and hard to detect. While a new generation of gravitational experiments that should be capable of probing Newton's law at short distances (up to 1 micron) is under way, our current knowledge about gravity stops at distances of the order of 1 mm. We currently cannot say whether there are, or are not, possible extra dimensions smaller than 1 mm.

So far so interesting, but what does this have to do with particle physics and the D├ÿ experiment at Fermilab? Actually, there is a very direct connection. Since the particles that we accelerate at Fermilab are very energetic, we can easily probe distances as small as 10-19 cm by studying the products of their collisions. However, the particles involved in these collisions are very light, so the gravitational interaction between them is very weak. Fortunately, it turns out that in the theory proposed by Arkani-Hamed, Dimopoulos, and Dvali, the gravitational interaction is greatly enhanced if the colliding particles have sufficiently high energy. This enhancement is due to the so-called "winding modes" of the graviton – the gravitational force carrier – around the compactified extra dimensions. If the graviton is energetic enough, it could travel ┬¥ "wind" its way ┬¥ around the compactified dimensions many times. Each time it winds around, it gives rise to a small gravitational force between the colliding particles. If the number of revolutions that the graviton makes around the curled extra dimensions is large enough, the gravitational interaction is tremendously enhanced.

Two types of the extra-dimensional effects observable at collides. Left: a graviton escapes from our 3-dimensional world in extra dimensions (Megaverse), resulting in an apparent energy non-conservation in our three-dimensional world. Right: a graviton leaves our world for a short moment of time, just to come back and decay into a pair of photons (the D├ÿ physicists looked for that particular effect).

As the Fermilab Tevatron is the highest energy particle accelerator in the world, it is the perfect place to look for extra dimensions, since the higher the colliding particle energy is, the stronger enhancement of the gravitational interaction is expected. Physicists working at the D├ÿ experiment have looked for the effects of gravitational interactions between pairs of electrons or photons produced in high-energy collisions. If the gravitational interaction between the two electrons or two photons is large enough, the properties of such a final state system would be modified. There will be more pairs produced at high two-body masses, and also the angular distribution of these particles will be more uniform than one expects to see if gravity is weak enough to be ignored. When D├ÿ carefully analyzed the data they collected in 1992-1996, no such enhancements were found. The data agrees very well with the predictions from known physics processes, and the gravitational interaction does not seem to play any significant role at the energies that we are able to reach. So, no evidence for extra dimensions was found so far.

A display of the highest invariant mass (570 GeV) photon-pair event that D├ÿ has observed in Run I. Two colorful splashes of energy seen in the left and right "arms" of the detector mark the points where the two photons were intercepted by the D├ÿ calorimeter. Unfortunately, this event is unlikely to originate from extra dimensions, due to its angular properties. It is consistent with the rate that one would expect from known physics processes.

Although we have not seen extra dimensions, we were able to set rather strict limits on their size. These limits are stricter than those set by gravitational experiments, or accelerator experiments at lower energy machines, so far. These new limits also place significant constraints on Arkani-Hamed, Dimopoulos, and Dvali's theory.

Our search for extra dimensions is not over yet. In fact, it has only just started. We are also looking for the effects of extra dimensions in collisions that produce different types of particles, such as quarks. We are also looking for events where gravitons are produced in the collisions and then leave our three-dimensional world, travelling off into one of the other dimensions. This would cause an apparent non-conservation of energy from the point of view of our three dimensional world. With the next data-taking run scheduled to start in 2001, and likely to deliver twenty times the data presently accumulated, we will have a significantly extended sensitivity to large extra dimensions. We very well might see them!

If we are not so lucky, the next generation collider, LHC, that is being built at CERN (near Geneva, Switzerland) will allow us to ultimately probe the theory of large extra dimensions and either find them or show that the idea is actually wrong. But we will have to wait six more years or so, before we learn that.