A model for a model

One of the more disturbing facts of modern theoretical particle physics is complexity. We can formulate the standard model on, more or less, two pages of paper. But to calculate most interesting quantities is so seriously challenging that even an approximate result takes many person-years, and often even much, much more.

Fortunately, the standard model is in one respect kind: For many interesting questions only a small part of it is relevant. This does not mean that the parts are really independent. But the influence of the other parts on the subject in question is so minor that the consequences are often much smaller than any reasonable theoretical or experimental accuracy could resolve. For example, many features of the strong interactions can be determined without ever considering the Higgs explicitly. In fact, it is even possible to learn much about the strong interactions just from looking at the gluons alone, neglecting the quarks. This reduced theory is called Yang-Mills theory. It is a very reduced model for the core features of the strong interactions.

Unfortunately, even this theory, which contains only a single of the particles of the standard model, is very complex. One of our lines of research is dealing with the resulting problems. One of these problems has to do with the properties of the local symmetry of this model, the so-called gauge symmetry. This feature leads to certain, technically necessary, redundancies. But when doing calculations, we need to do approximations. This may mess up the classification of what is redundant and what is not. Getting this straight is important, and this is the research topic I write about today.

And it is here where the title comes into play. Even if the theory of only gluons is much simpler than the original theory, it is still so complicated that the redundancies are pretty messed up. Therefore, we decided by now that it would be better to understand first a different case. A case, in which the same redundancies appear, but all the rest is simpler. A (simpler) model for a (more complicated) model.

This strategy creates the bridge to my previous entry on supersymmetry.

Theories which have this supersymmetry are, in almost all cases, much simpler than theories without. As I wrote, there are different levels of supersymmetry. In its simplest form, supersymmetry relates the kinds of possible particles, and constrains a few interactions. In the maximum version, essentially the whole structure of the theory, and almost all details, are constrained. These constrains are so rigid and powerful that we can solve the theory almost exactly. Nonetheless, this theory has the same kind of redundancies as Yang-Mills theory, and even the full standard model. Thus, we can study what approximations do to these redundancies. Especially, using the exact knowledge, we can reverse engineer essentially everything we want.

In fact, we make a kind of theoretical experiment: We take the theory. We treat it with a method - in our case we use the so-called equations-of-motion. We know the results. Now, we perform the same type of approximations we do in the more complicated models, or even the full theory. We see how this modifies the results. Well, actually we will see, since we are still working on this bit. From the change of the results, we will learn a lot of things. One is which kind of approximations make a qualitative change. Since any qualitative difference compared to the exact result will be a wrong result, we should not do such approximations. Not in this theory, and especially not in more complicated theories. Just small quantitative changes are probably fine, though there is no guarantee. And we can explicitly see if the approximations start to mix redundant parts such that they are treated wrongly. From this we will (hopefully) learn more about how to correctly treat the redundancies in the more complicated models.