Making big plans

 Occasionally, you have an idea, and you can do the required research within a couple of weeks. But this is the rare exception. Most research requires months, and often years, to complete. In particle physics, with its huge experiments running for decades, this is probably even more aware to people than in many other cases. This requires plans. A very recent example of such a plan is the European Strategy on Particle Physics (Update), in which all of Europe came together to make a plan. I have contributed to this as coordinating the theory input for the national Austrian roadmap. It is a huge effort to get everyone agreeing on what to do next - and what to do in the next half-a-century. Because this is how long you have to plan in advance for the big experiments.

Aside from these big plans, there are also smaller ones. Even for me as a theoretician. Occasionally, I have to sit down, and formulate a research plan for a couple of years into the future. The reason is often that I write a so-called grant proposal to get a considerable amount of money to hire postdocs and PhD students. Such a large proposal requires you to formulate what you want to do with all these people, usually for about five years. Last year, we got one already, for dark matter.

This year, I write another one. Why again, if we just got one? Well, on the one hand each would roughly take up half my time. So, I can manage both, and thereby do more. But putting this up front is cheating. The main reason is that it is unlikely I will get it in the first attempt. As there are currently many more people wanting to do particle physics than resources are allocated for this purpose at the national and international level, these resources need to be distributed. Thus, you write a proposal to get some of these. Then some panel judges the submitted proposals, and decides, who will get resources. And thus where efforts in particle physics will be concentrated. Usually, the are many more proposals the panel would like to fund than there are resources available, and so small points tip the scale to one or the other proposals, and the others are rejected. One can then try again. On average, less than one in five proposals is successful. Thus, you often need to try again, with an optimized proposal. And thus, I already submit another one.

Coming back to the original topic: For such a proposal I need to make a five-years plan. Of course, its research. Nobody can guarantee me that I (or, more likely, someone else) will not discover something which requires a fundamental change of plans. This is always allowed. But you are still required to make a plan what you want to do, if nothing unexpected happens. Usually, in my personal experience, about half what is planned will be done, and the rest of the resources is spent on unexpected stuff. Which is as well.

Still, you need to make a plan, if everything happens as you would expect it now. And that is what I did.

The first thing you need to decide is to what part of your research you would like to base it on. If you read my blog since a while, you may have seen that I actually do quite a lot of different topics, ranging from neutron stars to quantum gravity. But not all of this research is something I would like to extend at this level. The neutron star physics is something I currently do not work too much on. It is very interesting. But I would need to focus much more efforts on it, and needed to mainly concentrate on technical details. That is not what I currently want. The quantum gravity part is very exciting, and we develop quickly new ideas. There is much more to come. But currently it is too much at an exploratory stage as I would be able to formulate a large-scale five-years program. This will have to cook for a little time longer before it warrants this kind of attention.

So, I am down to my Higgs physics and beyond-the-standard-model research. For the latter, we are currently having enough people to work on. Also, it is a bit more speculative, as we did not yet see anything new in experiments. It is thus somewhat less easy to identify where to concentrates ones efforts on. The combination of our current research and what the next few years of experiments, especially LHC Run 3, will bring, will make this clearer.

So I concentrate this time on our attempts to find some new, subtle effects from theory in experiments: That there is an additional Higgs contribution inside the proton.

Right now, what we did was making a good guess, and looked, whether experiment told us we are right. Iterating this would be a time-honed approach to identifying a new effect. But for this plan, I wanted to be more ambitious. I wanted to have some prediction that rather just guess and iterate. This is very demanding. As a suitable tool, I choose simulations. While I will not be able to really simulate an actual proton and its Higgs content, the effort made possible by such a big grant should be enough to get a decent proxy for it. Something, which is close enough to the real thing that from a guess I can move to something which only requires a few more numbers, which I can get from experiments. That would be a huge success. We then use slightly different methods to fix the numbers.

But this is not easy. Based on what we learned so far, this is a big endeavour. At least for a theoretician. I estimated that I will need about four people with PhD, plus myself, and five more doing a PhD to get there. Not to mention that many master students and bachelor students will be able to work on this as well. This also means that especially several of the PhD students will work on this project, but will complete their PhD only on a part of it, and be done before the whole project is done. This required me to break the project down into smaller workpackages, 17 in total. Each of them is a milestone in itself, and provides intermediate (and eventually final) results. Each requires several of the people, and each at least half a year of time, and some even a year. I needed to make a plan, how each of them intersect with the other, and how they depend on each other. If you are interested in how such a thing looks in the end (it has a lot of tech babble in it), contact me. But it is actually not that different from any other large scale project, even in industry, like building a house. Thus, you also need some project management skills to do research. Even as a theoretician.

I am quite pleased with how it turned out in the end. It really has a good flow, and a succession of reasonable and manageable steps. In the end, it holds the promise of a guaranteed discovery - i.e. we will see a new physics effect, as long as we just keep on with the experiments, it will happen. Likely by the end of the runtime of the LHC in about 15 years. Or with the next generation of machines latest, which are part of the Strategy mentioned in the beginning. By this, I come full circle: My small research project ties in into the big ones. And together, we push the boundaries of human knowledge just a bit further.

Talking about the same thing

In this blog entry I will try to explain my most recent paper. The theme of the paper is rather simply put: You should not compare apple with oranges. The subtlety comes from knowing whether you have an apple or an orange in your hand. This is far less simple than it sounds.

The origin of the problem are once more gauge theories. In gauge theories, we have introduced additional degrees of freedom. And, in fact, we have a choice of how we do this. Of course, our final results will not depend on the choice. However, getting to the final result is not always easy. Thus, ensuring that the intermediate steps are right would be good. But they depend on the choice. But then they are only comparable between two different calculations, if in both calculations the same choice is made.

Now it seems simple at first to make the same choice. Ultimately, it is our choice, right? But this is actually not that easy in such theories, due to their mathematical complexity. Thus, rather than making the choice explicit, the choice is made implicitly. The way how this is done is, again for technical reasons, different for methods. And because of all of these technicalities and the fact that we need to do approximations, figuring out whether the implicit conditions yield the same explicit choice is difficult. This is especially important as the choice modifies the equations describing our auxiliary quantities.

In the paper I test this. If everything is consistent between two particular methods, then the solutions obtained in one method should be a solution to the equations obtained in the other method. Seems a simple enough idea. There had been various arguments in the past which suggested that this should be he case. But there had been more and more pieces of evidence over the last couple of years that led me to think that there was something amiss. So I made this test, and did not rely on the arguments.

And indeed, what I find in the article is that the solution of one method does not solve the equation from the other method. The way how this happens strongly suggests that the implicit choices made are not equivalent. Hence, the intermediate results are different. This does not mean that they are wrong. They are just not comparable. Either method can still yield in itself consistent results. But since neither of the methods are exact, the comparison between both would help reassure that the approximations made make sense. And this is now hindered.

So, what to do now? We would very much like to have the possibility to compare between different methods at the level of the auxiliary quantities. So this needs to be fixed. This can only be achieved if the same choice is made in all the methods. The though question is, in which method we should work on the choice. Should we try to make the same choice as in some fixed of the methods? Should we try to find a new choice in all methods? This is though, because everything is so implicit, and affected by approximations.

At the moment, I think the best way is to get one of the existing choices to work in all methods. Creating an entirely different one for all methods appears to me far too much additional work. And I, admittedly, have no idea what a better starting point would be than the existing ones. But in which method should we start trying to alter the choice? In neither method this seems to be simple. In both cases, fundamental obstructions are there, which need to be resolved. I therefore would currently like to start poking around in both methods. Hoping that there maybe a point in between where the choices of the methods could meet, which is easier than to push all all the way. I have a few ideas, but they will take time. Probably also a lot more than just me.

This investigation also amazes me as the theory where this happens is nothing new. Far from it, it is more than half a century old, older than I am. And it is not something obscure, but rather part of the standard model of particle physics. So a very essential element in our description of nature. It never ceases to baffle me, how little we still know about it. And how unbelievable complex it is at a technical level.

Being formal

One of the topics I am working on is about basic properties of so-called gauge symmetries. I just have published a new paper on it, and here I want to describe what it is about.

A gauge symmetry is, very roughly stated, a useful tool for which we pay the price of a very redundant description. Pictorially speaking, we can say the same thing with very many different words. This may sound awful. However, in practice, it seems to work just like a charm. So what is there still to investigate?

Well, knowing how to use something in one way, and understanding it fully are two very different things. And actually, we are not, on a very strict and formal level, absolutely sure that we know how to use gauge symmetry. Though this is likely the case. But the situation is nonetheless for two reasons not really satisfactory.

The first is more a question of approach. When we use something, we would really like to know what we are actually doing. The second is that if we would understand it better, there may very well be ways to use it much better than we currently do. So there are reasons for understanding it better.

But what is it what I actually want to do?

It all starts when going back to the meaning of symmetries. Symmetries introduce redundant directions, meaning that when you have a symmetry you have more directions to point then there are actually. That is in general very helpful on a technical level.

But here enters the problem. If we have directions, we should be able to say 'go in this direction'. To do so, we introduce coordinate systems. Now comes the catch: For a local symmetry this is easier said than done, especially when it comes to the gauge symmetries of the standard model.

The problem is somewhat abstract. When you think about directions, then usually you think about left, right, up or down, and so on. This is true if you think about our usual space around us. But not everything has the same geometry. Especially, symmetries can also have a direction of bending. This is still not a major issue. But, there are some symmetries where the bending of directions becomes so strong, that some directions bend back on themselves or meet others, when going too far. And this is a problem. If they bend back, or even worse, bend on a different direction, what is direction anyway? I can start walking in one direction, and then I am in another direction. Sounds like a catastrophe, right?

Well, the reason it sounds like that is that we insisted to define directions once and for all. This is what we are used to. A direction is a direction is a direction. Unfortunately, not everything is so straight and something, especially gauge symmetries, have additional directions which are warped, and can intersect each other. The problem then arises how to orient oneself, if directions change. The answer to this is that it is necessary to give up directions which are always the same. Rather, you need to define directions only in some area around you, and when you move, you may need to change them.

The aforementioned paper now investigates this bending of directions. In a sense, it tries to map how far it is possible to go in a fixed direction, before this direction changes. Finally, it attempts to draw a map of where these directional changes occur. That sounds now pretty graphical, but the reality is once more mathematically involved. But in the end, this map hopefully will help to setup useful collections of coordinate systems, and a dictionary telling you where to use which coordinate system to get your directions.

The details are pretty involved. But the rough outline of what I did was to put myself at many points, create there coordinate systems, follow the directions they give and check when they started to make no more sense - when they hit other directions or themselves. Then I got a list of collisions, and where they occurred. And from this I could get a map of collisions. What I did not yet do is to make something useful out of the map. That comes next.

Big blobs

One of the things I have discussed in my blog is how particles arise in quantum theories. Putting it into one (hand-waving) sentence, then particles are just isolated peaks in the quantum fields which fill up the universe. But is this all that there is possible?

The answer is no, and I had to do with the alternatives several times in my own research. But what are these alternatives?

Particles, I said, are isolated peaks. They are, what we call localized - existing at a single place. A single, and very slender, peak on a background of (nearly) nothing else. Of course, there are also bound states, like the hydrogen atom, and other such objects. These are two, or more particles, being close to each other, and which move in the same direction. However, in this case the individual particles are still, more or less, distinct.

Here, I want to introduce another concept. It arises, when one takes many particles, and puts them very close together. Then the peaks start to overlap, until it is impossible to say where one starts, and where another ends. In many cases such a bunch of particles is just unstable, and the particles fly apart pretty quickly. But several theories, most notably the strong interactions, provide another option. When carefully balancing how the particles are together, they form a super-particle, and the whole bunch behaves almost like one big particle. This is different from the bound states, because the particles are no longer individually detectable inside, it is just one big blob. Of course, it is possible to disassemble this blob, and the original particles come out. Hence, such blobs are not called particles, but pseudo-particles. A more fancy name for them is 'topological excitations'. This name has been given to them because of certain properties linked to the mathematical field of 'topology'. One of the particularly important features of these blobs is that they are, without external disturbance, extremely stable. The reason is, pictorially speaking, the way the particles are interwoven makes knots, which do not open.

So aside from the fascinating fact that these things exist, what is their use for physics? They play especially a role in theories where everything interacts strongly with each other, like the strong force. It is hypothesized that in such theories blobs emerge easily, and may even play the most important role. This would mean that effectively not the original particles, but the blobs are the usually encountered objects. And how they interact makes up the phenomena we see in experiments. Single particles are then just some minor disturbance to the game of the big blobs. The blobs become what physicists call the 'effective degrees of freedom', meaning the important players.

Is this true, especially in the strong interactions? It depends. We do not have an equivalent formulation of the theory in terms of blobs instead of particles, so we do not know for sure. We do know that several features, like mass generation, can be very simply explained just by using the blobs. There, it helped us a lot in understanding what is going on. Other features, like the famous confinement, turn out to be a much tougher cookie. We still are not sure, whether it is really possible.

Finally, what are my stakes in the blobs? One of the questions to be posed is, whether the properties of remaining individual particles are determined by the what the blobs do. Is their movement constrained by them? Are their interactions mainly with a blob involved, rather then directly between the particles? I am trying to answer these questions by simulations. Some preliminary findings are already available, but there will be more to come.

Triviality is not trivial

OK, starting with a pun is probably not the wisest course of action, but there is truth in it as well.

When you followed the various public discussiosn on the Higgs then you will probably have noticed the following: Though finding it, most physicists are not really satisfied with it. Some are even repelled by it. In fact, most of us are convinced that the Higgs is only a first step towards something bigger. Why is this so? Well, there are a number of reasons, from purely aesthetic ones to deeply troubling ones. As the latter also affect my own research, I will write about a particular annoying nuisance: The triviality referred to in the title.

To really understand this problem, I have to paint a somewhat bigger picture, before coming back to the Higgs. Let me start: As a theoretician, I can (artificially) distinguish between something I call classical physics, and something I call quantum physics.

Classical physics is any kind of physics which is fully predictive: If I know the start conditions with sufficient precision, I can predict the outcome as precisely as desired. Newton's law of gravity, and even the famous general theory of relativity belong to this class of classical physics.

Quantum physics is different. Quantum phenomena introduce a fundamental element of chance into physics. We do not know why this is so, but it is very well established experimentally. In fact, the computer you use to read this would not work without it. As a consequence, in quantum physics we cannot predict what will happen, even if we know the start as good as possible. The only thing we can do is make very reliable statements of how probable a certain outcome is.

All kinds of known particle physics are quantum physics, and have this element of chance. This is also experimentally very well established.

The connection between classical physics and quantum physics is the following: I can turn any kind of classical system into a quantum system by adding the element of chance, which we also call quantum fluctuations. This does not necessarily go the other way around. We know theories where quantum effects are so deeply ingrained that we cannot remove them without destroying the theory entirely.

Let me return to the Higgs. For the Higgs part in the standard model, we can write down a classical system. When we then want to analyze what happens at a particle physics experiment, we have to add the quantum fluctuations. And here enters the concept of triviality.

Adding quantum fluctuations is not necessarily a small effect. Indeed, quantum fluctuations can profoundly and completely alter the nature of a theory. One possible outcome of adding quantum fluctuations is that the theory becomes trivial. This technical term means the following: If I add quantum fluctuations to a theory, the resulting theory will describe particles which do not interact, no matter how complicated they do in the classical version. Hence, a trivial quantum theory describes nothing interesting. What is really driving this phenomena depends on the theory at hand. The important thing is that it can happen.

For the Higgs part of the standard model, there is the strong suspicion that it is trivial, though we do not have a full proof for (or against) it. Since we cannot solve the theory entirely, we cannot (yet) be sure. The only thing we can say is that if we add only a part of the quantum fluctuations, only a part of the so-called radiative corrections, the theory makes still sense. Hence it is not trivial to decide whether the theory is trivial, to reiterate the pun.

Assuming that the theory is trivial, can we escape it? Yes, this is possible: Adding something to a trivial theory can always make a theory non-trivial. So, if we knew for sure that the Higgs theory is trivial, we would know for sure that there is something else. On the other hand, trivial theories are annoying for a theoretician, because you either have nothing or have to remove artificially part of the quantum fluctuations. This is what annoys me right now with the Higgs. Especially as I have to deal with it in my own research.

Thus, this is one out of the many reasons people would prefer to discover soon more than 'just' the Higgs.

News about the state of the art

Right now, I am at workshop in Benasque, Spain. This workshop is called 'After the Discovery: Hunting for a non-standard Higgs Sector'. The topic is essentially this: We now have a Higgs. How can we find what else is out there? Or at least assure that it is currently out of our reach? That there is something more is beyond doubt. We know too many cases where our current knowledge is certainly limited.

I will not go on with describing all what is presented on this workshop. This is too much. And there are certainly other places on the web, where this is done. In this entry I will therefore just describe how what is discussed at the workshop relates to my own research.

One point is certainly what experiments find. At such specialized workshops, you can get much more details of what they actually do. Since any theoretical investigation is to some extent approximative, it is always good to know, what is known experimentally. Hence, if I get a result in disagreement with the experiment, I know that there is something wrong. Usually, it is the theory, or the calculations performed. Some assumption being too optimistic, some approximation being too drastic.

Fortunately, so far nothing is at odds with what I have. That is encouraging. Though no reason for becoming overly confident.

The second aspect is to see what other peoples do. To see, which other ideas still hold up against experiment, and which failed. Since different people do different things, combining the knowledge, successes and failures of the different approaches helps you. It helps not only in avoiding too optimistic assumptions or other errors. But other people's successes provide new input.

One particular example at this workshop is for me the so-called 2-Higgs-Doublet models. Such models assume that there exists besides the known Higgs another set of Higgs particles. Though this is not obvious, the doublet in the name indicates that they have four more Higgs particles, one of them being just a heavier copy of the one we know. I have recently considered to look also into such models, though for quite different reasons. Here, I learned how they can be motivated for entirely different reasons, and especially why there are so interesting for ongoing experiments. I also learned much about their properties, and what is known (and not known) about them. This gives me quite some new perspectives, and some new ideas.

Ideas, I will certainly realize, once being back.

Finally, collecting all the talks together, they draw the big picture. They tell me, where we are now. What we know about the Higgs, what we do not know, and where there is room (left) for much more than just the 'ordinary' Higgs. It is an update for my own knowledge about particle physics. And it finally delivers the list, of what will become looked at in the next couple of months and years. I now know better where to look for the next result relevant for my research, and relevant for the big picture.

Blessing and bane: Redundancy

We have just recently published a new paper. It is part of my research on the foundations of theoretical particle physics. To fully appreciate its topic, it is necessary to say a few words on an important technical tool: Redundancy.

Most people have heard the term already when it comes to technology. If you have a redundant system, you have two or more times the same system. If the first one fails, the second takes over, and you have time to do repairs. Redundancy in theoretical physics is a little bit different. But it serves the same ends: To make life easier.

When one thinks about a theory in particle physics, one thinks about the particles it describes. But if we would write down a theory only using the particles which we can observe in experiment, these theories would become very quickly very complicated. Too complicated, in fact, in most cases. Thus people have very early on found a trick. If you add artificially something more to the theory, it becomes simpler. Of course, we cannot just simply add it really, because otherwise we would have a different theory. What we really do is, we start with the original theory. Then we add something additional. We make our calculations. And from the final result we remove then what we added. In this sense, we added a redundancy to our theory. It is a mathematical trick, nothing more. We imagine a theory with more particles, and by removing in the end everything too much, we end up with the result for our original problem.

Modern particle physics would not be imaginable without such tricks. It is one of the first things we learn when we start particle physics, the power of redundancy. A particular powerful case is to add additional particles. Another one is to add something external to the system. Like opening a door. It is the latter kind with which we had to deal.

Now, what has this to do with our work? Well, redundancies are a powerful tool. But one has to be careful with them nonetheless. As I have written, we remove at the end everything we added too much. The question is, can this be done? Or becomes everything so entwined that this is no longer possible? We have looked at especially was such a question.

To do this, we regarded a theory of only gluons, the carrier of the strong force. There has been a rather long debate in the scientific community how such gluons move from one place to another. A consensus has only recently started to emerge. One of the puzzling things were that you could prove mathematical certain properties of their movement. Surprisingly, numerical simulations did not agree with this proof. So what was wrong?

It was an example of reading the fine-print carefully enough. The proof made some assumptions. Making assumptions is not bad. It is often the only way of making progress: make an assumption, and see whether everything fits together. Here it did not. When studying the assumptions, it turned out that one had to do with such redundancies.

What was done, was essentially adding an artificial sea of such gluons to the theory. At the end, this sea was made to vanish, to get the original result. The assumption was that the sea could be removed without affecting how the gluons move. What we found in our research was that this is not correct. When removing the sea, the gluons cling to it in a way that for any sea, no matter how small, they still moved differently. Thus, removing the sea little by little is not the same as starting without the sea in the first place. Thus, the introduction of the sea was not permissible, and hence we found the discrepancy. There have been a number of further results along the way, where we learned a lot more about the theory, and about gluons, but this was the essential result.

This may seem a bit strange. Why should an extremely tiny sea have such a strong influence? I am talking here about a difference of principle, not just a number.

The reason for this can be found in a very strange property of the strong force, which is called confinement: A gluon cannot be observed individually. When the sea is introduced, it offers the gluons the possibility to escape into the sea, a loophole of confinement. It is then a question of principle: Any sea, no matter how small, provides such a loophole. Thus, there is always an escape for the gluons, and they can therefore move differently. At the same time, if there is no sea to begin with, the gluons remain confined. Unfortunately, this loophole was buried deep into the mathematical formalism, and we had to first find it.

This taught us an important lesson that, while redundancies are a great tool, one has to be careful with them. If you do not introduce your redundancies carefully enough, you may alter the system in a way too substantial to be undone. We now know what to avoid, and can go on, making further progress.

Picking the right theory

One of the more annoying facts of modern particle physics is that our theories cannot stand purely alone. All of them have some parameters, which we are not able to predict (yet). The standard-model of particle physics has some thirty-odd parameters, which we cannot fathom by theoretical investigations alone. An example are the masses of the particles. We need to measure them in an experiment.

Well, you may say that this is not too bad. If we just have to make a couple of measurements, and then can predict all the rest, this is acceptable. We just need to find as many independent quantities as there are parameters, and we are done. In principle, this is correct. But, once more, it is the gap between experiment and theory which makes live complicated. Usually, what we can easily measure is something which is very hard to compute theoretically. And, of course, something we ca calculate easily is hard to measure, if at all possible.

The only thing left is therefore to do our best, and get as close to each other as possible. As a consequence, we usually do not know the parameters exactly, but only within a certain error. This may still not seem too bad. But here enters something which we call theory space. This is an imaginary space in which every point corresponds to one particular set of parameters for a given theory.

In principle, when we change the parameters of our theory, we do not just change some quantitative values. We are really changing our theory. Even a very small change may produce a large effect. This is not only a hypothetical situation - we know explicit examples where this happens. Why is this so?

There are two main reasons for this.

One is that the theory depends very sensitively on its parameters. Thus, a slight change of the parameters will induce a significant change in everything we can measure. Finding then precisely the parameters which reproduce a certain set of measurements requires a very precise determination of the parameters, possibly with many digits of precision. This is a so-called fine-tuning problem. An example of this is the standard model itself. The mass of the Higgs is fine-tuned in the standard model, it depends strongly on the parameters. This was the reason why we were not able to fully predict its mass before we measured it, although we had many, many more measurements than the standard model has parameters. We could just not pinpoint it precisely enough, but only within a rough factor of ten.

A related problem is then the so-called hierarchy problem of modern particle physics. It is the question why the parameters are fine-tuned just to the value we measure in the experiment, and not slightly off and giving a completely different result. Especially this takes today the form of "Why is the Higgs particle rather light?"

But the standard model is not as worse as it can get. In so-called chaotic models the situation is far worse, and any de-tuning of the parameters leads to an exponential effect. That is the reason why these models are called chaotic - because everything is so sensitive, there is no structure at first sight. Of course, one can, at least in principle, calculate this dependency exactly. But any ever so slight imprecision is punished by an exponentially large effect. This makes chaotic models a persistent challenge to theoreticians. Fortunately, particle physics is not of this kind, but such models are known to be realized in nature nonetheless. E.g. some fluids behave like this, under certain conditions.

The other reason is that theory space can have separate regions. Inside these regions, so-called phases, the theory shows quite distinct behaviors. A simple example is water. If you take the temperature and pressure as parameters (though they are not really fundamental parameters of the theory of water, but this is just for the sake of the argument) then two such phases could be solid and liquid. In a similar manner theories in particle physics can show different phases.

If the parameters of a theory are very close by to the boundary between two phases, a slight change will push the theory from one phase to another. Thus, also here it is necessary to determine the parameters very precisely.

In the case of my research on Higgs physics, I am currently facing such a challenge. The problem is that the set of basic parameters are not unique - you can always exchange one set against another set, as log as the number is kept, and there is no trivial connection between two new parameters. Especially, depending on the methods used, particular set are more convenient. However, this implies that for every method it is necessary to redo the comparison to experiment. Since I work on Higgs physics, the dependence is strong, as I said above. It is thus highly challenging to make the right pick. And currently some for my results significantly depend on this determination.

So what can I do? Instead of pressing on, I have to make a step back, and improve my connection to experiment. Such a cycle is quite normal. First you make your calculation, getting some new and exciting result, but with large errors. To improve your errors, you have to go back, and make some groundwork, to improve the basis. Then you return back to the original result, and get an improved one. Then you do the next step, and repeat.

Right now, I am not only for the Higgs physics in such a process. Also our project on a neutron star's interior currently underoes such an improvement. This work is rarely very exciting or yields unexpected results. But it is very necessary, and one cannot get away without, once one wants to get reliable results. Also this is part of a researcher's (everyday) life.

Hiding our ignorance

The radiative corrections discussed last time have another important aspect. For this, it is useful to recall the entry on Einstein's famous relation E=m*c*c. This relation told us that you can convert energy to mass, and thus to particles.

Now, quantum physics is a cheater. Always was, always will be. One of the most basic things it cheats about is knowledge. It tells you that certain pairs exist of which you cannot know both at the same time with certainty. If you know one very precisely, you can have only little knowledge about the other. The most important and fundamental such pair is position and speed. If you know the position of a particle well, you cannot know its speed very well. And the other way around. This is an observation of nature, which has been confirmed in numerous experiments. We cannot yet really explain why this is so, and have to accept it for the time being as an experimental fact. What we can do is derive an enormous amount of knowledge from this fact.

Among this is that a very similar relation holds for energy and time. If we know time very precisely, we do not know the energy very precisely. If you combine this with Einstein's formula, you get a very interesting consequence: For very short periods of time, energy is not very well defined, and may be much larger than assumed. Since this energy is equivalent to mass, this means that for very short periods of time you can have particles pop out of nowhere and vanish again. To be precise, you can have a pair of a particle and an anti-particle for very brief moments in time.

This seem to be almost unbelievable: Something hops into and out of existence, just like this. However, you can measure actually this effect, and it has been experimental confirmed very well. Also, it should not be taken too literally. What really happens is that quantum physics does something, and in our mathematical description it appears like you would have these pairs.

So what does this have to do with the radiative corrections? Radiative corrections are quantum corrections. As such they involve precisely this type of process: Something hoping out of the vacuum. It then briefly interacts with whatever you are actually looking at. Then it vanishes again. Therefore, radiative corrections include all the possible interactions of some particle with all other possible particles. Now comes the real boon of this: In reality this happens with all particles, not only those we know of. This has been used in the past to predict new particles, like the top quark, some of the neutrinos, and, yes, also the Higgs.

Great, so I can get everything from it! you may say. Unfortunately, it is not that simple. The heavier the particles, the less their contributions to radiative corrections, and thus the more precise an experiment has to be to detect their influence. As a consequence, the Higgs was the last particle for which we had strong such indirect evidence. And this was already experimentally challenging.

But it is much more troublesome for theory. Since we do not actually know what is there, our calculations have a problem. We create at very short times a lot of energy, but we do not know where to put it, since we do not know all the particles. Our theories thus lack something. And this something haunts us as failures of our theories, when we try to calculate radiative corrections. This was a very big problem for theories for a while, but we finally managed it. The key concept was named 'renormalization', which is again somewhat of a misnomer. Anyway, it gives a name to the process of hiding our ignorance. In fact, what we do is that we introduce in our theories placeholders for all these unknown particles. These placeholders are designed on purpose to remove all the problems we have. The way we designed them they can never described something of nature, but they absorb all the problems we encounter with our ignorance.

Since we know that we have these problems, it also tells us that the standard model cannot be the end - or for that matter any theory having such problems. They only describe our world at (relatively) low energies: The standard model is a low-energy effective theory, as was briefly indicated before. Here, you now have a better view of what the reason for the infinities encountered back then is: That we do not know what particles may appear in our radiative corrections, and thus that we do not know where to direct our energy to. And that the parameters used back then just mock up the unknown particles.

You may wonder whether this is a generic sickness of quantum theories. This is very hard to tell for a realistic theory. Of course, we assume that if we would know the theory of everything, it should not have these problems. We can indeed construct toy theories of toy worlds, which do not have these problems, so we think it is possible. Whether this is true in the end or not, we cannot say yet - perhaps we will need in the end a whole new theoretical concept to deal with the real world. For now, renormalization prevents us from the need to know everything already. This permits us to discover nature step by step.

What means 'radiative correction'?

A term, which comes up very often when one reads about the Higgs, are radiative corrections. The thing hiding behind this name is also very essential in both my own work, and in particle physics in general. So what is it?

Again, the name is historic. There are two parts in it, referring to radiation and to correction. It describes something one comes across when one wants to calculate very precisely something in quantum physics.

When we sit down to calculate something in theoretical quantum physics, we have many methods available. A prominent one is perturbation theory. The basic idea of perturbation theory is to first solve a simpler problem, and then add the real problem in small pieces, until one has the full answer.

Usually, when you starts to calculate something with perturbation theory in quantum physics, you assume that the quantum effects are, in a certain sense, small. A nice starting point is then to neglect quantum physics completely, and do just the ordinary non-quantum, often called classical, part. To represent such a calculation, we have developed a very nice way using pictures. I will talk about this soon. Here, it is only necessary to say that the picture of this level of calculation looks like a (very, very symbolic) tree. Therefore, this simplest approximation is also known as tree-level.

Of course, neglecting quantum effects is not a very good description of nature. Indeed, we would not be able to build the computer on which I write this blog entry, if we would not take quantum effects into account. Or have the Internet, which transports it to you. In perturbation theory we add these quantum contributions now piece by piece, in order of increasing 'quantumness'. This can be mathematically very well formulated what this means, but this is not so important here.

If the quantum contributions are small, these pieces are just small corrections to the tree-level result. So, here comes the first part of the topic, the correction.

When people did this in the early days of quantum mechanics, in the 1920ies, the major challenge was to describe atoms. In atoms, most quantum corrections involve that the electron of an atom radiates a photon or captures a photon radiated from somewhere else. Thus, the quantum corrections where due to radiation, and hence the name radiative corrections, even if quantum corrections would be more precise. But, as always, not the best name sticks, and hence we are stuck with radiative corrections for quantum corrections.

Today, our problems have become quite different from atoms. But still, if we calculate a quantum correction in perturbation theory, we call it a radiative correction. In fact, by now we have adapted the term even when what we calculate is no small correction at all, but may be the most important part. Even if we use other methods than perturbation theory. Then, the name radiative correction is just the difference between the classical result and the quantum result. You see, there is no limit to the abuse of notation by physicists.

Indeed, calculating radiative corrections for different particles is a central part of my research. More or less every day, I either compute such radiative corrections, or develop new techniques to do so. When I finally arrive at an expression for the radiative correction, I can do two things with them. Either I can try to understand from the mathematical structure of the radiative corrections what are the properties of the particles. For example, what is its mass. Or how strongly does it interact with other particles. Or I can combine the radiative corrections for several particles or interactions to determine a new quantity. These can be quite complicated. Recently, one of the things I have done was to use the radiative corrections of gluons to calculate the temperature of the phase transition of QCD. There, I have seen that at a certain temperature the radiative correction to the behavior of gluons change drastically. From this, I could infer that a phase transition happened.

So you see, this term, being used so imprecisely, is actually an everyday thing in my life as a theoretician.

Using E=m*c*c

The last time I gave you a first, brief glimpse of special relativity. Special relativity has one property on which all modern experiments at accelerators like the LHC are based on. It is encoded in Einstein's most famous equation E=m*c*c, where E stands for energy, m for mass, and c is the speed of light. But what does this equation, which is already part of pop culture, really mean?

Let us have a look at its part. The symbol c denotes the speed of light. As discussed last time, the speed of light is always and everywhere constant. It is thus a constant of proportionality, without any dynamical meaning. In fact, its value is no longer measured anymore. In the international most used system of physical units it is defined to have a certain value, roughly 300000 km/s. Since it is so devoid of meaning, most particle physicist have decided that you do not need it, really, and replaced it with one. Ok, this may sound pretty strange to you, since one is not even a speed, it is just a number. But all such things like physical units are man-made. Nature knows only what a distance is, but not what a kilometer is. Thus, you must be able to formulate all laws of nature without such man-made things like a second or a kilometer.

Indeed, this is possible. However, we are just human, and thus working only with numbers turned out to be inconvenient for the mind. Thus, we usually set only so much irrelevant constants to one until we are left with just one single physical unit. Depending on the circumstances, for a particle physicist this is either the so called femtometer (short fm) or fermi, 0.000000000000001 meter, what is roughly the size of a proton. Or we use energy, measured in giga-electron volt (short GeV), or 100000000 electron volt. An electron volt is the amount of kinetic energy an electron gains when it is accelerated by one volt of voltage. That is roughly the voltage of an ordinary battery. Both units are very convenient when you do particle physics. If you are an astrophysicist, this would not be the case. They measure distances, e.g., in megaparsec, which is roughly 3261567 light years.

Anyway, lets get back to the equation. If we set c to one, it reads E=m. Much simpler. The left hand side now denotes an energy E, and the right-hand side a mass m. This is actually not what you can read off a scale. This is called weight, and depends on the planet you are on. Mass is a unique property of a body, form which one can derive the weight, once you chose a planet.

Since the left-hand side is an energy, measured in GeV, so is the right-hand side. Thus, we measure mass not in kilogram, but in energy. A proton has then roughly the mass of 1 GeV, while an electron has a mass of about 0.005 GeV.

But this equation is not just about units. It has a much deeper meaning. As it stands, it says that mass is equal energy. What does this mean? You know that you have to invest energy to get an object moving, again the kinetic energy. But the right-hand side does not contain a speed, so the energy on the left-hand side seems not to be a kinetic energy. This is correct. The reason is that this formula is actually a special case of a more general one, which only applies if you consider something which does not move. It makes the explicit statement that a body with a mass m at rest has an energy E. Thus, the energy has nothing to do with moving, and is therefore called a rest energy. If the particle should start to move, this energy is increased by the kinetic energy, but never decreased. This means that every body has a minimum energy equal to its rest energy, which in turn is equal to its mass.

Why is this so? The first answer is that it necessarily comes out of the mathematics, once you set up special relativity. That is a bit unsatisfactory. In quantum field theory, mass comes out as an arbitrary label that every body has, and which can take on any value. Only by experiment we can decide what particles of which mass do exist. We cannot yet predict the mass a particle has. That is one of the unsolved mysteries of physics. Note that the Higgs effect or the strong force seem to create mass. Thus, it seems we can predict mass. But this is a bit imprecise. Both of them do not really create mass, but add more to Einstein's equation. This makes particles behave as if they have a certain mass. But it is not quite the same.

Let me get back to where I started. Why is this equation so important? Well, as I said, the energy gets only bigger by moving. Now, think of a single particle, which moves very fast. Thus, it has a lot of energy. At the LHC, the protons have currently 4000 times more energy than they have at rest. If you stop the proton by a hard wall, than most of this energy will go on and move the wall. But since a wall is usually pretty heavy, and even 4000 times the proton rest mass is not much on the scale of such walls, they do not move in a way that we would notice.

But now, let us collide this proton with another such proton. What will happen? We have a lot of energy and a head-on collision. One thing Einstein's equation permits, if you formulate it for more than one particle, is the following: You are permitted to convert all of this energy into new particles. At least, as long as the sum of kinetic and rest energy does not exceed the total energy of the two protons before-hand. By this, you can create new particles. And this is what makes this equation so important for modern experiments. You can create new particles, and observe them, if you just put enough energy into the system. And that is, why we use big accelerators like the LHC: To make new particles by converting the energy of the protons.

Unfortunately, we cannot predict to what the energy is converted, as already noted earlier. But well, at least we can create particles.

Oh, and there is a subtlety with the wall: If we are good, and hit a single particle inside the wall, then the same happens as when we collide just two protons. But in most cases, we do not hit a single particle, but it is more like the first shot of billiard, giving just a bit of energy to every particle in the wall. And then the wall as a whole is affected, and not a single particle. Just when you wonder if you ever hear of fixed-target experiment, instead of a collider. This it, how this works: Shot at a wall, and hope to hit just a single particle.

The speed of light - and its consquences

So far, I did not say anything about gravity. This will remain so. However, I will have to say something about special relativity. Somehow, special relativity is often associated with gravity. This is actual not the case. Einstein's theory of special relativity does not make any reference to gravity. Only the theory of general relativity, of which special relativity is just a small subset, does so.

If special relativity is not about gravity, what is it about? Well, it is about the fact that our universe is a bit more weird than one expects.

What do you expect of a law of nature? One property is likely that it is always valid. This simple requirement has quite profound consequences. Assume for a second that you and your experiment are alone in the universe. This means ta you have no point of reference. If now the experiment moves, you could not say whether it is moving, or you. Still, you would expect that it gives the same results, irrespective of whether it or you are moving. We made experiments to test this idea, and it was confirmed beautifully. This fact that experiments are independent of relative motion, is one of the basic observations leading to special relativity.

The next ingredient is much more harder to believe. Take a light beam. What speed does it have? Well, the speed of light, of course. Now, if you move the thing creating the light at a fixed speed, how fast should the light move? Naively, one would expect that the light would now move faster. Unfortunately, our universe does not tick that way: The light still moves with the same speed. Actually, the light is actually made up out of massless photons, which travel at the speed of light. And this observation is the same for anything which is massless: All massless particles move at the speed of light. And the speed of light is always the same, no matter how fast the light source moves.

This is nothing we can really explain. It is an experimental fact. Our universe is like this. But this observation is the second basic fact underlying special relativity.

If we cannot explain why this second fact comes about, can we at least describe it? We can, and that is what leads to special relativity.

Now, how do we describe this? Well, this is a bit more involved. Take the universe. Then at each instance you have three directions in space. Distances you measure do not depend on the direction in space. You can also measure time elapsing. Works out also nicely. But now, try to measure a space-time distance. You do this by measuring a distance in the space direction. Then you take the elapsed time, and calculate what distance a light ray would have moved during this time. By this, you can talk about a distance in time direction.

The speed of an object is given (if the speed does not change) by the ratio of distance over time required to move over this distance. If you now want that the speed of light is independent of whether the light source moves or not, something peculiar is found: To get this, a distance in space-time direction is obtained by subtracting the distance in the time direction and in the space direction, when you do your Phytagorean geometry. That is completely different than what you have in the three time directions, but the only way to get the light speed to agree with experiment. The geometry of space-time is hence quite different from the one we know just from space.

As a theoretician, I say that the way we measure the distances is not like the ordinary one in three space directions (a so-called Euclidean measure), but we have rather a Lorentz measure. This is in honor to the first one who has described it. Again, we cannot yet explain why this is the case, it just is an experimental fact.

You may wonder why you never noticed this in real lifer. The answer is that when the light source moves very slowly compared to the light ray, the effect is negligible. This becomes only relevant, if you move at a considerable fraction of the speed of light. But then all the nice effects result of which you may have heard in Science Fiction movies or novels: Things like the twin paradox, time dilatation, length contraction, and so on. All of these result from these two basic observations. And are described by the theory of special relativity.

This all sound pretty weird, and it is. It is nothing we have a real handle on with our everyday experience. Just like the quantum effects. The universe is just this way.

As you see, gravity enters here nowhere. And also no quantum stuff. If you add gravity, you get to the theory of general relativity. If you add quantum stuff, you end up with quantum field theory. The standard model is of the latter kind. And this combination leads to very interesting effects, which I will discuss in more detail next time.

Two worlds: Theory and experiment

You will probably have heard that we have found the Higgs boson - or something similar to it. We are not quite sure yet. You may also have heard that we found it in an experiment, and that this was a triumph for theory, which predicted it long ago. This seems to be a wonderful combination, theory and experiment. But, as always, nothing is just as simple as it seems.

Let us undertake the journey and accompany a theoretical idea from its inception until its experimental test, to see what is going on.

Having an idea of how physics beyond the standard model could look like is essentially simple. Though, of course, many ideas have already found by some of the people thinking about it since the early 1970ies. The interesting question after having an idea is, how to check, whether it is actually describing nature, or is just an interesting mathematical toy.

To do this, two things are necessary. The first is to check whether the idea is compatible with what we know so far about nature. The second is to use the idea to predict something which is different from the standard model. That is necessary, so that we can distinguish both, and decide how nature can be described. To do both we have to to somehow compare to an experiment.

Unfortunately, experiments cannot directly work with the mathematical stuff a theoretician writes down. Modern particle experiments work in the following way: You send something into a box and then detect what comes out of the box. In case of the suspected Higgs, we send in protons. The box is an empty space where these protons hit each other. Because the encounter is violent enough, everything comes apart, and out of the box come a lot of other (known) particles. These are then detected. Actually, we can pretty well by now not only say that there is a particle, but also what particle it is, and where it is headed with which speed. The set of detected particles is what we call an event. We then do many collisions and collect many events. The reason for this is that quantum physics forbids us to know precisely what is going on, but only what happens on the average. And to get an average, we have to average over many events.

At any rate, we end up with such information. That is what modern experiments do.

Now, the theoretician has to somehow convert his idea to something which can be compared to this experimental outcome.

In most cases, things roughly proceed as follows:

What we actually collide are not protons, but the quarks and gluons inside the proton. Thus, the theoretician first computes how quarks and gluons become converted into a new particle. Unfortunately, the experiment can only talk about the protons going into the box. So we have to first compute how we find quarks inside the proton. This is actually very complicated, and so far only partially solved. Nonetheless, we can do it sufficiently well for our purpose, though it is a challenging calculation.

The next problem is that the new particles lives usually only for a very short time. Too short to escape the box. It will decay into other particles before it can leave the box. In fact, it will often decay into particles, which in turn still do not live long enough to escape the box, but also decay first. So you have to calculate the whole chain of decays, until you reach particles, which are so stable that they will escape the box, and can be detected in the detector.

Once you have this, you have what we call a cross section. This is a number, which tells you how often two colliding protons will end up being a certain set of particles, which come from the decay chain of the new particle. Usually, you also know how often these particles go with which speed into which direction.

Unfortunately, we cannot yet compare to experiment, for two reasons.

The first is that the detector is not perfect. For example, the detector has to have a hole where the protons enter. Also, we cannot suspend the detector in thin air, and the holding devices produce blind spots. In addition, we are actually not able to measure all the speeds and directions perfectly. And it can happen that we mistake one particle for a particle of a different species. All of this is part of the so-called detector efficiency. An experimentalist can determine this with a great amount of work for a given detector. As a theoretician, we have to combine our prediction with this detector efficiency, to make a reliable prediction. Just think what would happen if our idea produces a signal which would escape preferentially along the way the protons came in. If we would not take the detector efficiency into account, we would just see nothing, and would decide our idea is wrong. But knowing the detector efficiency, we can figure out what is going on.

The second problem is what we call background. This background has two origins.

One is that the remainder quarks and gluons of the protons usually do not go away nicely, but will produce many other particles in other collisions. At LHC we even have that usually more than two protons collide. This produces a lot of debris in the detector. To find the new particle then means to separate all the debris from the particles into which the new particle has decayed.

The second problem is that other processes may mimic the searched for particle to some extent. For example, by having similar decay products. Then we have to distinguish both cases.

Because of the detector efficiency, we are not able to resole both types of background perfectly. And neither can we resolve the signal perfectly. We just get a big pileup, and have to find the signal in it. To do this, theoreticians have to calculate all this background. By comparing than what one gets from background alone and from background plus the desired signal, we have reached our goal: We know, how our searched for new particle appears in the detector. And how the experiment will look like, if we were incorrect, and just the known bunch of things is there.

All of this is quite laborious, and a lot of groundwork. And all too often it comes out that for a given detector efficiency we will not be able to get the signal out of the background. Or that the number of times we have to try is so large that we cannot afford it - we would just have to get too many events to get a reasonable reliable average.

But well, this is life. Your options are then either to wait for a better experiment (which will usually take a couple of decades to build), or to go back to the drawing board. And find a new signal, which the present experiment can find. And then it may still happen that they find nothing, and this means you idea was incorrect from the very beginning. And then you can go back to step one. In case of the Higgs, it appears likely that it turned out be correct. But there the process was so complicated that it took 48 years to have good enough experiments and reliably enough theory. Physics beyond the standard model may require even more, if we are unlucky. If we are lucky, we may have something in a few months.

Where physicists go to

You may wonder, why it is so quite here. The reason is that I do what all physicists do from time to time: We go to conferences, present and discuss our ideas and listen to other people's ideas. This exchange is incredible important to make progress, because only in discussions we can remove the flaws we do not see ourselves, and get the inspiration to find new ideas.

I am right now at two such conferences in Australia - since the people are spread all over the world, it does not really matter where we meet. Someone has always to travel far. Thus, the next entry on topic will take some time, likely until August, but that is just how it is. But until then, I will post everything interesting I hear directly on Twitter, so stay tuned.

Above and beyond

Before I can introduce my last research topic, I have first to go above and beyond the standard model. Given that I am calling this blog a tourist guide to the standard model, this is a bit like trespassing. But that is how my research went over the last years. So let me keep the name, but nonetheless venture beyond.

With keeping the name, I am using a very important concept in scientific naming: For historic reasons. For historic reasons, this blog was named, but then reality overtook it. This is something very often happening in physics. In the beginning, we encounter a phenomena. To not always have to describe it, we give it a name, based on what we have encountered so far. Very often, when we really understand what is going on, this name would no longer be appropriate. But at this time, the name stuck, and thus we stick with it. You should keep this in mind, when you think that the name of something seems to have nothing to do with the thing. Then the name is just there for historic reasons. And that is something one will also encounter beyond the standard model.

So let us go beyond the standard model. As I told you, the standard model is just that: A model. It has its limits. We know today, mostly for reasons of mathematical consistency and for astronomical observations that the standard model cannot be the end. We know that with what we have we cannot explain what happens at very high energies. We also cannot explain with the standard model why the outer rims of the galaxies rotate faster than they should. We do not know why the universe is expanding faster and faster. We do not know why the particles have precisely the masses they have. We cannot tell, why nature is such that the sun can shine: Given the standard model, we can explain how the sun shines. But we cannot explain why the standard model is such that this is possible. And there are many other things.

However, irritatingly, all our experiments here on earth have been in accordance with the standard model.

This is very unsatisfactory, to say the least. As a consequence, almost since the birth of the standard model in its present form people have started to consider extensions of it. This is commonly known as beyond-the-standard-model models, or BSM for short. Unfortunately, the observations listed above are not pointing all in the same direction. Actually, most just point somewhere, for the amount of knowledge we have of them. Thus, we are, right now, very much in the dark when it comes to figure out how we should extend the standard model.

This lead to very many proposals. Even listing all conceptual proposals would easily fill a couple of months worth of blog entries. But they can be broadly distinguished by their approach. Some have a top-down approach. They try to envisage the underlying theory, which will solve all (or many) of the known problems in one strike. You may have heard of (super)string theory. This is one example of this approach. The other is bottom-up. Here, one just tries to resolve a single problem. The ones I am working on belong to the latter category.

Now, what is the state of affairs? Top-down approaches are often rather complicated theories, and it is often very complicated to calculate anything at all. Thus, progress on this side is naturally slow. The bottom-up approaches are often more tractable. It is often not too complicate to design a theory, which solves the problem one was setting out to solve. However, in doing so one usually gets for a completely different thing a result which disagrees with the known experiments. Thus, one is forced to modify the model to sate this problem. But then the next springs up, and one finds oneself adding bits and pieces to the model, until it becomes rather baroque.

You may say now: Well, if things are either too complicated or too much mingling, maybe you are on the wrong track. And I would not really disagree with you. Of course, nothing prevents nature from being really complicated or baroque. Just because both versions do not look aesthetically pleasing to us does not mean it cannot be. But this way of thinking has never been right in the history of physics. When something got complicated, and any amendment made it worse, then we were on the wrong track.

So, why do we not abandon everything we did, take the standard model as the basis, and start over from scratch? This has actually been done many times, and so far was not successful. On the other hand, going back to the complicated theories, one hope is that by making the theory more complex by making more and more things agree with experiment, we can hope that at some point a pattern emerges. This has occurred in the past, and is thus a real possibility.

Thus, today, all of these possibilities are followed. We try to imagine solution to the problems, test them against experiments, both old and new, and reiterate. A spectacular new observation at any experiment would greatly help. That is why any ever so slight deviation from the standard model expectation is greeted with great enthusiasm by the theorist. Even if we know that in most cases this will be a coincidental fluke, which will go away when we keep looking more carefully.

Where will this lead us to? We do not know yet. That is the really exciting part of particle physics: We try to push the boundaries of knowledge, and we can only speculate what we will find there.

Modelling reality

Ever wondered why it is called the standard model of particle physics? And what a physicist has in mind, when she talks about models?

Models are the basic ingredient of what a theoretical physicist is doing. The problem is that we do not know the answer, we do not know the fundamental theory of everything. Thus, the best we can do is take what we know, and make a guess. The result of such a guess is a model. Such a model should describe what we see. Thus, the standard model of particle physics is the one model what we know about particle physics right now, as incomplete as it may be. It is called the standard one, because it is our best effort to describe nature so far, to model nature in terms of mathematics. There are also other standard models. We have one for how a sun functions, the standard model of the sun, or how the universe evolved, the standard model of cosmology.

Now, when I say, it is our best guess this implies that it is not necessarily right. Well, actually it is, in a sense. It was made the standard model, because it describes (or, if you read this in a couple of years, perhaps has described) our experiments as good as we can wish for. That means, we have found no substantial evidence against this model within the domain accessible in the experiment. This sentence has two important warning signs attached.

The one is about the domain. We do not know what is the final theory. But what we do know is the models. And any decent model will tell us, what it can describe, and what not. This also applies to the standard model. It tells us: 'Sorry guys, I cannot tell what is happening at very large energies, and on the matter of gravitation, well I stay away from this entirely.' This means that this standard model will only remain the standard model until we have figured out what is going on elsewhere. At higher energies, or what is up with gravitation. However, this does not mean that the standard model will be completely useless once we managed that. As with many standard models in the past, it likely will just become part of the large picture, and remain a well-trusted companion, at least in some area of physics. Happened to Newton's law, which was superseded by special relativity, and later by general relativity. Happened to Maxwell's theory of electromagnetism, which was superseded by Quantumelectrodynamics, and later by the standard model. Of course, there is once more no guarantee, and it may happen that we have to replace the standard model entirely, once we see the bigger picture. But this seems right now unlikely.

The other thing was about the experiment. Models are created to describe experiments (or observations, when we think about the universe). Their justification rests on describing experiments. We can have some experimental result, and cook up a model to explain it. Then we do a prediction, and make an experiment to test it. Either it works, and we go on. Or it does not, and then we discard the model. While people developed the standard model, this was a long, painful process during which many models have been developed, proposed, checked, and finally discarded. Only one winner remained, the model which we now call the standard model.

Ok, nice and cozy, and that how science works. But I was talking about methods the last couple of times, so what has this to do with it? Well, this should just prepare you for an entirely different type of models, to avoid confusion. Hopefully. Now the standard model is the model of particle physics. But, honestly, it is a monster. Just writing it down during a lecture requires something like fifteen minutes, two blackboards, and two months of preparation to explain all the symbols, abbreviations and notions involved to write it in such a brief version. I know, I have done it. If you want to solve it, things go often from bad to worse. That is where models come in once more.

Think of the following: You want to describe how electric current flows inside a block of, say, Aluminum. In principle, this is explained by the standard model. The nuclei of Aluminum come from the strong force, and the electrons from the electromagnetic one, and both are decorated with some weak interaction effects. If you really wanted to try describing this phenomena using the standard model, you would be very brave indeed. No physicist has yet tried to undertake such an endeavor. The reason is that the description using the standard model is very, very complicated, and actually most of it turns out to be completely irrelevant for the electric current in Aluminum. To manage complexity, therefore, physicists investigating aluminum do not use the standard model of particle physics in its full glory, but reduce it very, very much, and end up with a much simpler theory. This models Aluminum, but has forgotten essentially everything about particle physics. This is then a model of Aluminum. And it works nice and well for Aluminum. Applying it to, say, copper, will not work, as Aluminum nuclei have been put into it as elementary entities, to avoid the strong interactions. You would need a different model for copper then, or at least different parameters.

So, we threw away almost all of the power of the standard model. For what? Actually, for a price worth the loss: The final model of Aluminum is sufficiently simple to solve it. Most of our understanding of materials, technology, chemistry, biology (all described by the standard model of particle physics, in principle) rests on such simplified models. With only the standard model, we would not be able to accomplish anything useful for these topics, even knowing so much about particles. In fact, historically, the development was even the other way around. We started with simple models, describing few things, and generalized bit by bit.

Ok, you may say. You see the worth of simplified models for practical applications. But, you may ask, you surely do not simplify in particle physics? Well, unfortunately, we have to, yes. Even when only describing particles, the standard model is so complicated that we are not really able to solve it. So we very often make models only describing part of it. Most what we know about the strong interactions has been learned by throwing away most of the weak interactions, to have a simpler model. When talking about nuclear physics, we even reduce further. Also, when we talk about physics beyond the standard model, we often first create very simple-minded models, and in fact neglect the standard model part. Only, if we start to do experiments, we start to incorporate some parts of the standard model.

Again, we do this for the sake of manageability. Only by first solving simpler models, we understand how to deal with the big picture. In particle physics the careful selection of simplified models was what drove our insight since decades. And it will continue to do so. This strategy is called divide and conquer. It is a central concept in physics, but also in many other areas where you have to solve complicated problems.

Of course, there is always a risk. The risk is that we simplify the model too much. That we loose something important on the way. We try to avoid that, but it has happened, and will happen again. Therefore, one has to be careful with such simplifications, and double-check. Often, it turns out that a model makes very reliable predictions for some quantities, but fails utterly for others. Often, our intuition and experience tells us ahead what is a sensible question for a given model. But sometimes, we are wrong. Then experiment is one of the things which puts us back on track. Or that we are actually able to calculate something in the full standard model, and find a discrepancy compared to the simple model.

In the past, such simplified models were created by very general intuition, and including some of the symmetries of the original theory. Over time, we have also learned how to construct, more or less systematically, models. This systematic approach is referred to as effective field theory. This name comes about as it creates a (field) theory which is an effective (thus manageable) version of a more complicated field theory in a certain special case, e.g. low energies.

Thus, you see that models are in fact a versatile part of our tool kit. But they are only to some extent a method - we have still to specify how we perform calculations in them. And that will lead us then to the important concept of combining methods next time.

Fermions

You thought bosons were strange? Well, wait, now comes the really strange quantum stuff - fermions.

At first sight, fermions are innocently looking and differing from bosons by the fact that they have half-integer spin. In the standard model, all quarks and leptons are fermions, and have spin one half. No elementary particle is known (though some hypothesized) which are fermions and have a larger spin than one half. But again, some particles made up from several elementary particles may look from afar like having a larger half-integer spin. E.g. the Delta, a heavier cousin of the proton and made up also from three quarks, has spin three halves.

In contrast to bosons, fermions dislike being at the same place. In fact, they can never take the same position, much like the classical balls. But there is a difference to the classical balls. For fermions, this not only applies to position, but also to all quantum numbers and energies. As a consequence, there can never be two fermions being having the same energy. This is the famous Pauli exclusion principle.

This principle has very fundamental consequences: It is responsible for the stability of all matter. If your desk would be made out of bosons, only electromagnetic repulsion would prevent it from collapsing to a pile of bosons. But because it is made out of fermions - all the quarks and electrons - it could never collapse to a single point. Because the fermions can just not get so near to each other. That is the fundamental reason which prevents a white dwarf or a neutron star from collapsing.

Very similar, it also prevents the electrons in an atom, which are attracted by the nucleus by electric forces, from collapsing into the lowest energy level, or into the nucleus outright. All of chemistry works the way it works because the electrons, since they are fermions, cannot all go into the lowest energy level. Otherwise, our chemistry, and thus our biology, would be very different, indeed.

But this is not the only strange thing about fermions. Fermions are also very strange in many other respects. As a consequence of the Pauli principle they obey again a different statistics, the so-called Fermi-Dirac statistics. The consequence of this are at the heart of why there are electric insulators.

But fermions are also strange in the sense that when you turn your coordinate system by 360 degrees, i.e. once fully around, everything is unchanged. Only the fermions do not play along: You have to turn your coordinate system twice around so that they look again the same (or, more precisely, their wave-function explained next time, looks the same). That is so mind-boggling that it is hard to believe it is true, and one cannot really intuitively understand this. It is a very deep combination of our space-time structure and quantum physics. There is no classical objects which behaves like this.

The mathematical consequences of these properties are little less strange. Fermions are the only objects which we cannot describe by ordinary numbers. Theoretical physicists had to invent a whole new type of numbers (well, actually borrow them from your friendly mathematician next door) to describe fermions - so-called Grassmann numbers. These are really strange. If you multiply an ordinary number with itself, you get a new number. If you multiply a Grassmann number with itself, you always get zero. That is the mathematical realization of the Pauli principle. This feature makes fermions very hard to handle in actual calculations, and they have been a bane especially to numerical simulations.

Nonetheless, they are there, and we are bound to live with them, as we are bound to live with bosons. Though - you can always combine two fermions to make something which looks from afar like a boson. But you can never combine two bosons such that they look from afar like a fermion. This fact has been found to be exploited very often by nature, as already described last time. And it lies at the heart of some ideas, so-called technicolor scenarios, to get rid of the Higgs with all its annoying properties: In such proposed extensions of the standard model, the Higgs is just a combinations of two new particles, so-called techniquarks.

Bosons

The first type of particles are bosons bosons. Those are these having integer spin. In the standard model, there is the Higgs particle, which has spin zero, and the photons, the W and Z bosons, and the gluons, which all have spin one.

Particles with spin zero are also called scalar particles. Since their spin is zero, the properties of such particles are the simplest when changing to a different coordinate system: They just look the same.

Particles with spin one are also called vector particles. Such vector particles are described like photons. The name vector stems from the fact that under a coordinate transformation the fields describing a vector particle changes in the same way as a line which connects the origin of a coordinate system and an event. The latter line is also called a vector, and hence the name for particles of spin one.

There is actually also a hypothetical particle with spin two, the graviton. Such a particle is also called a tensor particle. Tensors are generalizations of vectors when it comes to coordinate transformations, and fields of spin two particles transform in the same way as such tensors. In general, tensors are rectangular collections of numbers, where the columns transform like a vector under coordinate transformation.

Elementary particles with higher spin are not known. However, particles made up from elementary particles add their spin together (though not necessarily in the sense 1+1=2 - it can also be subtracted, 1-1=0, and everything in between), and can thus have higher spins.

Furthermore, to each such type of bosons, there exists a so-called pseudo bosons, i. e. a pseudo scalar, a pseudo vector (sometimes for historical reasons also called an axial vector), and a pseudo tensor. The difference between a boson and a pseudo boson is what happenes if you reflect the world in a mirror (a parity transformation). Ordinary bosons just become bosons once more. In contrast, the fields of pseudo bosons are multiplied by minus one.

Ok, after all this classification and name stuff, what is special about bosons? The most striking feature is that you can pile them upon each other. That is different from the small balls one often uses to imagine elementary particles: We can stack such balls next to each other, but never ever can two of these balls be at the same place. But bosons can. That is very hard to get in line with our ideas of how things work, and it shows just how quantum bosons are: they behave in a way which is just unexpected.

This is, of course, only true, if the bosons do not repel each other by some force. For example, if you would have two electrically same-name charged bosons, you would have a hard time to bring them together. But if they have oppositely named charges then they would just love to sit at exactly the same place.

In fact, if bosons do not repel each other because of a force acting between them, they have a tendency to lump together - two bosons rather prefer to be at the same place than being apart. This phenomenon is again a pure quantum effect: If you would have two balls, which are not talking to each other, they ignore each other very consequently. The reason for this different behavior is encoded in what physicists call statistics. In case of the boson this statistics is called Bose-Einstein statistics, in contrast to the classical statistics of the balls. Statistics describes how particles distribute themselves. Classical statistics is essentially randomly distributed, but bosons with Bose-Einstein statistics are not entirely randomly distributed but tend to get together.

This property also pertains to a different thing: The energies the particles have. While classical particles have just their energy, independent of every other particle, as long as they do not interact, bosons tend to have the same energy.

The extreme case of getting together is occurring when a sizable fraction of all available bosons are involved, and all of them have the lowest possible energy. That is what is called a Bose-Einstein condensate. This type of stuff is a state of matter similar to being liquid or being solid. But it only occurs under rather extreme conditions, in particular at very low temperatures. On Earth, there is no naturally occurring case of such a condensate. But it was possible to create such condensates in the laboratory using atoms.

In particle physics, such condensates play a central role. The Higgs effect was associated with a condensate of Higgs particles: It is just such a Bose-Einstein condensate. The same applies to the mass generation from the strong force, though in this case it is not the quarks that form a condensate. Since they are fermions, as will be discussed next, this is not directly possible. But states made up from two quarks (or a quark and an anti-quark) can condense. Since spin adds, such states have either spin zero or one, and thus behave like a boson, if one is not looking too closely. And these effective bosons are, loosely speaking, condensing to a Bose-Einstein condensate in this case.

These are only some examples, but such condensates play very often a role, from superconductors to the interiors of neutron stars. Thus bosons, with their strange properties, are very important to physics, and especially particle physics.

Spin

One of the most intriguing and most important properties of an elementary particle is its spin. At the same time, spin is one of the conceptually most problematic quantities, and has led to an enormous amount of misunderstandings.

The reason for this is that there is something in classical physics, which is very closely related to the concept of spin. But this relation is in spirit, rather than literally, and this has led to a lot of confusion. This analogue is angular momentum.

So, first, what is angular momentum? Angular momentum is connected with any kind of rotation of a particle around some center. Formally, it is a product involving the radius of the rotation and the speed along the path of the object. In classical physics, without friction, it is conserved, and it is what keeps the planets' orbits in their respective plane. It is likely also responsible for the fact that all the orbits are more or less in the same plane, or that the milky way has the over-all form of a discus (neglecting the spiral arms). In essence, it is just a reformulation of the ordinary speed, mixed with the mass of a particle. Essentially a kinematic quantity, despite its importance.

If an object just rotates, e.g. a ball, then each of the elements of the ball rotates. This can be described by giving the ball as such an angular momentum. Since the geometry of the ball is known and fixed, it is possible to defer from this total angular momentum the angular momentum of every piece of the ball.

In the world of particles, this angular momentum is reappearing whenever there is something having some kind of relative motion. E. g. in an atom, it is possible to assign the electrons an angular momentum, which is then often called orbital angular momentum (a somewhat complicated name). However, the electrons are not actually small spheres orbiting around the nucleus, bur rather smeared out over the whole of the atom. What this precisely means, I will discuss later. The important thing is that this smeared out something has a kind of orbital movement (the whole object 'rotates' in a certain sense), and can therefore be assigned such an orbital angular momentum.

It is a remarkable observation in quantum physics that angular momentum cannot take any value it likes. It is quantized. The reason for this quantization is the inherent relation between angular momentum and speed, and then speed and energy. Because energy is quantized this implies that angular momentum is quantized.

As orbital angular momentum depends on the momentum, and thus on the speed, its numerical value changes when we as the observer are changing our movement. This does not change the path of the rotating objects, just our perception of it, of course. Therefore, this change of values is closely tied to our change of our coordinate system, when we move.

Now enter spin: It was very early on recognized in quantum physics that elementary particles have a property which changes in the same way as the angular momentum of the ball when we change our coordinate system. This was an intrinsic property of the particles, unchangeable. However, the elementary particles are point-like, at least to the extent we can resolve them. Thus, they cannot rotate in any way, as they do not have any extension. In fact, if this would be an ordinary angular momentum, and the elementary particles would have a small extension, then within our experimental knowledge about the upper limit of this extension, their surface would need to rotate much faster than the speed of light.

Thus, this property got its own name: Spin. This is still inspired by the similarity to (orbital) angular momentum under a change of coordinate system, but by keeping strictly the difference in name, it can always be distinguished from it. However, from time to time its is useful to refer to them both together, and in this case they are called total angular momentum, which is in principle somewhat a misnomer.

Now spin is also quantized, and there exist both half-integer and integer values for it (when choosing appropriate units). This is different from ordinary angular momentum for two reasons. First, there is no simple explanation for the quantization like for angular momentum. There is indeed a complicated explanation, which shows that for the space-time structure which we have, these are the only two possibilities consistent with this type of change under a change of the coordinate system. Second, angular momentum, when measured in the same units, can have only integer values.

The latter is an intriguing difference. It has a very important consequence: Particles having integer spin behave very different from those having a half-integer spin. Therefore, these two types of particles received different names: The former are called bosons, and the latter are called fermions. This distinction is of fundamental importance to particle physics, and therefore the next two entries will discuss both types of particles in more detail. Also, none of these types behave in the same way as an ordinary small ball. But it turns out that if one takes the classical (long-distance) limit, both behave in the same way, and like small balls: Classically fermions and bosons can not be distinguished, their existence is a pure quantum effect, which is intricately linked to the structure of space and time. That is one of the reasons why some people believe that the quantum effect of spin and gravity may be related at a deeper level, but we are very far from understanding whether this suspicion is correct.