The limits of the standard model II

The last entry focused on the low-energy, or long-distance limit, of the standard model. This time, lets have a look at the opposite limit, the one of very short distances, or, as discussed previously, the one of very high energies.

If we go to smaller and smaller distances, we try to look deeper and deeper into something. Just like with the ocean: First, we just see the essentially plain waters. When we go nearer, we see the large movements of very large waves. When we go closer, we see that on the large waves there are small waves, and even deeper, we see ripples on all of the smaller waves. However, if we would go even closer, we would see that the water is made up of water molecules, out of discrete things. Wow. We just made a jump from one description - a continuous amount of water - to another one - that of water molecules. This means, when we look at shorter and shorter distances, we can learn how the things work in the interior, and in detail. Therefore, by looking at smaller and smaller distances, or higher and higher energies, we learn something about the nature of things.

In terms of theories, physicists like to speak of the description in terms of the water molecules as the 'underlying theory'. The description in terms of water as a fluid is called the 'low-energy effective theory', i. e., a theory which describes the relevant features of the underlying theory if we are looking on distances where we cannot distinguish the individual constituents of the underlying theory anymore.

In doing so, we actually notice something: As we have discussed, molecules are made up of atoms and the atoms are made up out of even smaller particles, and these smaller particles are described by another theory, the standard model. Hence, the theory of water molecules is out of a sudden no longer an underlying theory, but a low-energy effective theory for the standard model. Thus, a theory can be both, an underlying theory, and a low-energy effective theory. It is just a question of whether we look at it from larger or smaller distances than the characteristic scale of it.

The characteristic scale, which I have introduced here without warning, is actually a rather sloppy term: It means essentially distances in which the typical behavior of the objects in a theory show themselves. In the case of the waves, this scale is of the order of kilometers down to micrometers, the water-theory with water as molecules then takes over until one reaches the domain of femtometers, where the standard model comes into play. To not give a scale range, one usually uses one intermediate scale to indicate such a characteristic scale. For the theory of water molecules this is typically some Angstrom (about 0.0000000001 meters). For the standard model, it depends on the sector: about 0.000000000000001 meter for the strong interactions, and about 0.000000000000000001 meter for the weak sector, about one-thousand times smaller.

But wait: How can we be sure that the standard model is the underlying theory? The answer is we cannot. In fact, we firmly believe that it is not. The reason for thinking so is the following: On the one hand, it lacks gravity. We think, that the last theory in such a hierarchy of effective theories should include both quantum physics with the standard model as well as gravity. We just do not see right now any logical possibility that these two should be and remain separate.

On the other hand, there is a technical problem, which shows that the standard model is incomplete. If you do calculations in the standard model, then it turns out that for doing everything mathematically consistently you have to consider arbitrarily large energies. However, if you do this, the results are infinite, and thus at first glance meaningless. If you, however, assume that the standard model is just a low-energy effective theory, it is possible to remove these infinities by defining a small number of parameters appropriately. This is called renormalization, and the proof that this is possible for the standard model, at least to some extent, has been awarded with a noble prize. In a way, the standard model is telling us: "Hey, I m not the final answer, but you can parametrize your ignorance such that I still make sense, if you just do not poke me with too large energies."

Ok, all well and fine. But what is the underlying theory to the standard model? We do not know right now. And to figure this out, we have to look at physics a ever shorter distance scales and thus ever higher energies to get an answer to this. That is the reason we built and use the LHC and its predecessors. There is also the possibility to indirectly interfere the very high energy behavior by making very precise measurements. You could imagine this in the following way: You would also figure out that water is made out of molecules when you would weigh water very carefully. Then you would notice that it is not possible to have an arbitrary weight of water, but only discrete portions. And similarly we try to infer the high-energy behavior by very precise measurements.

Anyway, this is the current goal: To see what is the underlying theory of the standard model. This process of identifying the next underlying theory has been driven physics since centuries. Will it ever terminate? That is a good question,a and one we cannot answer (yet). The only thing sure right now is that it did not terminate with the standard model. And that we do not even yet fully understand the standard model, though this is necessary to answer whether something we observe is genuinely a signal of the underlying theory, or just a feature of the standard model. A difficult question indeed.