More similar than expected

Some while ago I have written about a project a master student and myself have embarked upon: Using a so-called supersymmetric theory - or SUSY theory for short - to better understand ordinary theories.

Well, this work has come to fruition, both in the form of the completion of the master project as well as new insights written up in a paper. This time I would like to present these results a little bit.

To start, let me briefly rehearse what we did, and why. One of the aims of our research is to better understand how the theories work we are using to describe nature. A particular feature of these theories is redundancy. This redundancy makes many calculations possible, but at the same time introduces new problems, mainly about how to get unique results.

Now, in theories like the standard model, we have all problems at the same time: All the physics, all the formalism, and especially all the redundancy. But this is a tedious mess. It is therefore best to reduce the complexity and solve one problem at a time. This is done by taking a simpler theory, which has only one of the problems. This is what we did.

We took a (maximal) SUSY theory. In such a theory, the supersymmetry is very constraining, and a lot of features are exactly known. But the implications of redundancy are not. So we hoped that by applying the same procedures we use to deal with the redundancy in ordinary theories to this theory, we could check whether our approach is valid.

Of course, the first, and expected, finding was that even a very constraining theory is not simple. When it comes to technical details, anything interesting becomes a hard problem. So it required a lot of grinding work before we got results. I will not bore you with the details. If you want them, you can find them in the paper. No, here I want to discuss the final result.

The first finding was a rather comforting one. Doing the same things to this theory that we do to ordinary theories did not do too much damage. Using these approximations, the final results were still in agreement with what we do know exactly about this theory. This was a relief, because this lends a tiny amount of support more to what we are usually doing.

The real surprise was, however, a very different one. We knew that this theory shows a very different kind of behavior than all the theories we are usually dealing with. So we did expect that, even if our methods work, the results will still be drastically different from the other cases we are dealing with. But this was not so.

To understand better what we have found, it is necessary to know that this theory is similar in structure to a conventional theory. This conventional one is a theory of gluons, but without quarks to make the strong interactions complete. In the SUSY theory, we also have gluons. In addition, we have further new particles, which are needed to get.

The first surprise was that the gluons behaved unexpectedly similar to their counterparts in the original theory. Of course, there are differences, but these differences were expected. They came from the differences of both theories. But where they could be similar, they were. And not roughly so, but surprisingly precisely so. We have an idea why this could be the case, because there is one structural property, which is very restricting, and which appears in both theories. But we know that this is not enough, as we now other theories where this still is different, despite also having this one constraining structure. Since the way how the gluons are very similar is strongly influenced by the redundancy features of both theories, we can hope that this means we are treating the redundancy in a reasonable way.

The second surprise was that the new particles mirror the behavior of the gluons. Even though these particles are connected by supersymmetry to the gluons, the connection would have allowed many possible shapes of relations. But no, the relation is an almost exact mirror. And this time, there is no constraining structure which gives us a clue why, out of all possible relations, this one is picked. However, this is again related to redundancy, and perhaps, just speculating here, this could indicate more about how this redundancy works.

In total, we have learned quite a lot. We have more support for what we doing in ordinary theories. We have seen that some structures might be more universal than expected. And we may even have a clue in which direction we could learn more about how to deal with the redundancy in more immediately relevant theories.

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.

A partner for every particle?

A master student has started with me a thesis on a new topic, one on which I have not been working before. Therefore, before going into details about the thesis' topic itself, I would like to introduce the basic physics underlying it.

The topic is the rather famous concept of supersymmetry. What this means I will explain in a minute. Supersymmetry is related to two general topics we are working on. One is the quest for what comes after the standard model. It is with this respect that it has become famous. There are many quite excellent introductions to why it is relevant, and why it could be within the LHC's reach to discover it. I will not just point to any of these, but write nonetheless here a new text on it. Why? Because of the relation to the second research area involved in the master thesis, the ground work about theory. This gives our investigation a quite different perspective on the topic, and requires a different kind of introduction.

So what is supersymmetry all about? I have written about the fact that there are two very different types of particles we know of: Bosons and fermions. Both types have very distinct features. Any particle we know belong to either of these two types. E.g. the famous Higgs is a boson, while the electron is a fermion.

One question to pose is, whether these two categories are really distinct, or if there are just two sides of a single coin. Supersymmetry is what you get if you try to realize the latter option. Supersymmetry - or SUSY for short - introduces a relation between bosons and fermions. A consequence of SUSY is that for every boson there is a fermion partner, and for every fermion there is a boson partner.

A quick counting in the standard model shows that it cannot be supersymmetric. Moreover, SUSY also dictates that all other properties of a boson and a fermion partner must be the same. This includes the mass and the electric charge. Hence, if SUSY would be real, there should be a boson which acts otherwise like an electron. Experiments tell us that this is not the case. So is SUSY doomed? Well, not necessarily. There is a weaker version of SUSY where it only approximately true - a so-called broken symmetry. This allows to make the partners differently massive, and then they can escape detection. For now.

SUSY, even in its approximate form, has many neat features. It is therefore a possibility desired by many to be true. But only experiment (and nature) will tell eventually.

But the reason why we are interested in SUSY is quite different.

As you see, SUSY puts tight constraints on what kind of particles are in a theory. But it does even more. It also restricts the way how these particles can interact. The constraints on the interactions are a little bit more flexible than on the kind of particles. You can realize different amounts of SUSY by relaxing or enforcing relations between the interactions. What does 'more or less' SUSY mean? The details are somewhat subtle, but a hand-waving statement is that more SUSY not only relates bosons and fermions, but in addition also partner particles of different particles more and more. There is an ultimate limit to the amount of SUSY you can have, essentially when everything and everyone is related and every interaction is essentially of the same strength. That is what is called a maximal SUSY theory. A fancy name is N=4 SUSY for technical reason, if you should come across it somewhere on the web.

And it is this theory which is interesting to us. Having such very tight constraints enforces a very predetermined behavior. Many things are fixed. Thus, calculations are more simple. At the same time, many of there more subtle questions we are working on are nonetheless still there. Using the additional constraints, we hope to understand this stuff better. With these insights, we may have a better chance to understand the same stuff in a less rigid theory, like the standard model.