However, like many theories, supersymmetry looks fine in the abstract but leaves many questions unresolved when you get down to the concrete details of how it connects to the world we actually see. At some energy, supersymmetry must break down, because we haven't yet seen any "superpartners." This means that the two particle partners—for example, the electron and the selectron—cannot have exactly the same mass; if they did, we would see both. The unseen partner must have a bigger mass if it has so far eluded detection. We want to know how this could happen in a way consistent with all known properties of elementary particles. The problem for most theories incorporating supersymmetry-breaking is that all sorts of other interactions and decays are predicted which experiment has already ruled out. The most obvious candidates for breaking supersymmetry permit the various kinds of quarks to mix together, and particles would have a poorly defined identity. The absence of this mixing and the retention of the various quark identities is a stringent constraint on the content of the physical theory associated with supersymmetry-breaking, and is one important reason that people were not completely satisfied with supersymmetry as an explanation of the TeV scale. To find a consistent theory of supersymmetry requires introducing physics that gives masses to the supersymmetric partners of all the particles we know to exist, without introducing interactions we don't want. So it's reasonable to look around for other theories that might explain why particle masses are associated with the TeV energy scale and not one that is sixteen orders of magnitude higher.

There was a lot of excitement when it was first suggested that extra dimensions provide alternative ways to address the origin of the TeV energy scale. Additional spatial dimensions may seem like a wild and crazy idea at first, but there are powerful reasons to believe that there really are extra dimensions of space. One reason resides in string theory, in which it is postulated that the particles are not themselves fundamental but are oscillation modes of a fundamental string. The consistent incorporation of quantum gravity is the major victory of string theory. But string theory also requires nine spatial dimensions, which, in our observable universe, is obviously six too many. The question of what happened to the six unseen dimensions is an important issue in string theory. But if you're coming at it from the point of view of the relatively low-energy questions, you can also ask whether extra dimensions could have interesting implications in our observable particle physics or in the particle physics that should be observable in the near future. Can extra dimensions help answer some of the unsolved problems of three-dimensional particle physics?

People entertained the idea of extra dimensions before string theory came along, although such speculations were soon forgotten or ignored. It's natural to ask what would happen if there were different dimensions of space; after all, the fact that we see only three spatial dimensions doesn't necessarily mean that only three exist, and Einstein's general relativity doesn't treat a three-dimensional universe preferentially. There could be many unseen ingredients to the universe. However, it was first believed that if additional dimensions existed they would have to be very small in order to have escaped our notice. The standard supposition in string theory was that the extra dimensions were curled up into incredibly tiny scales—10 33 centimeters, the so-called Planck length and the scale associated with quantum effects becoming relevant. In that sense, this scale is the obvious candidate: If there are extra dimensions, which are obviously important to gravitational structure, they'd be characterized by this particular distance scale. But if so, there would be very few implications for our world. Such dimensions would have no impact whatsoever on anything we see or experience.

From an experimental point of view, though, you can ask whether extra dimensions really must be this ridiculously small. How large could they be and still have escaped our notice? Without any new assumptions, it turns out that extra dimensions could be about seventeen orders of magnitude larger than 10-33 cm. To understand this limit requires more fully understanding the implications of extra dimensions for particle physics.

If there are extra dimensions, the messengers that potentially herald their existence are particles known as Kaluza-Klein modes. These KK particles have the same charges as the particles we know, but they have momentum in the extra dimensions. They would thus appear to us as heavy particles with a characteristic mass spectrum determined by the extra dimensions' size and shape. Each particle we know of would have these KK partners, and we would expect to find them if the extra dimensions were large. The fact that we have not yet seen KK particles in the energy regimes we have explored experimentally puts a bound on the extra dimensions' size. As I mentioned, the TeV energy scale of 10-16 cm has been explored experimentally. Since we haven't yet seen KK modes and 10-16 cm would yield KK particles of about a TeV in mass, that means all sizes up to 10-16 are permissible for the possible extra dimensions. That's significantly larger than 10 33 cm, but it's still too small to be significant.