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.

What could the Higgs be made of?

One of the topics I am working on is how the standard model of particle physics can be extended. The reason is that it is, intrinsically, but not practically, flawed. Therefore, we know that there must be more. However, right now we have only very vague hints from experiments and astronomical observations how we have to improve our theories. Therefore, many possibilities are right now explored. The one I am working on is called technicolor.

A few weeks ago, my master student and I have published a preprint. By the way, a preprint is a paper which is in the process of being reviewed by the scientific community, whether it is sound. They play an important role in science, as they contain the most recent results. Anyway, in this preprint, we have worked on technicolor. I will not rehearse too much about technicolor here, this can be found in an earlier blog entry. The only important ingredient is that in a technicolor scenario one assumes that the Higgs particle is not an elementary particle. Instead, just like an atom, it is made from other particles. In analogy to quarks, which build up the protons and other hadrons, these parts of the Higgs are called techniquarks. Of course, something has to hold them together. This must be a new, unknown force, called techniforce. It is imagined to be again similar, in a very rough way, to the strong force. Consequently, the carrier of this fore are called technigluons, in analogy to the gluons of the strong force.

In our research we wanted to understand the properties of these techniquarks. Since we do not yet know if there is really technicolor, we can also not be sure of how it would eventually look like. In fact, there are many possibilities how technicolor could look like. So many that it is not even simple to enumerate them all, much less to calculate for all of them simultaneously. But since we are anyhow not sure, which is the right one, we are not yet in a position where it makes sense to be overly precise. In fact, what we wanted to understand is how techniquarks work in principle. Therefore, we just selected out of the many possibilities just one.

Now, as I said, techniquarks are imagined to be similar to quarks. But they cannot be the same, because we know that the Higgs behaves very different from, say, a proton or a pion. It is not possible to get this effect without making the techniquarks profoundly different from the quarks. One of the possibilities to do so is by making them a thing in between a gluon and a quark, which is called an adjoint quark. The term 'adjoint' is referring to some mathematical property, but these are not so important details. So that is what we did: We assumed our techniquarks should be adjoint quarks.

The major difference is now what happens if we make these techniquarks light and lighter. For the strong force, we know what happens: We cannot make them arbitrarily light, because they gain mass from the strong force. This appears to be different for the theory we studied. There you can make them arbitrarily light. This has been suspected since a long time from indirect observations. What we did was, for the first time, to directly investigate the techniquarks. What we saw was that when they are rather heavy, we have a similar effect like for the strong force: The techniquarks gain mass from the force. But once they got light enough, this effect ceases. Thus, it should be possible to make them massless. This possibility is necessary to make a Higgs out of them.

Unfortunately, because we used computer simulations, we could not really go to massless techniquarks. This is far too expensive in terms of the time needed to do computer simulations (and actually, already part of the simulations were provided by other people, for which we are very grateful). Thus, we could not make sure that it is the case. But our results point strongly in this direction.

So is this a viable new theory? Well, we have shown that a necessary condition is fulfilled. But there is a strong difference between necessary and sufficient. For a technicolor theory to be useful it should not only have a Higgs made from techniquarks, and no mass generation from the techniforce. It must also have more properties, to be ok with what we know from experiment. The major requirement is how strong the techniforce is over how long distances. There existed some indirect earlier evidence that for this theory the techniforce is not quite strong enough for sufficiently far distances to be good enough. Our calculations have again a more direct way of determining this strength. And unfortunately, it appears that we have to agree with this earlier calculations.

Is this the end of technicolor? Certainly not. As I said above, technicolor is foremost an idea. There are many possibilities how to implement this idea, and we have just checked one. Is it then the end of this version? We have to agree with the earlier investigations that it appears so in this pure form. But, in fact, in this purest form we have neglected a lot, like the rest of the standard model. There is still a significant chance that a more complete version could work. After all, the qualitative features are there, it is just that the numbers are not perfectly right. Or perhaps just a minor alteration may already do the job. And this is something where people are continuing working on.

(Un-)Dead stars and particles

I have already written about some aspects of my research on neutron stars. But what is the problem with them? And what do I want to understand?

First: What are they? If you have a sufficiently massive star, it will not die in a fizzle, like our sun, but it will end violently. It will explode in a supernova. In this process, a lot of its mass gets compressed at its core. If this core is not too heavy, a remainder, a corpse of the star, will remain: A neutron star. If it is too heavy, the remainder will, however, collapse further into a black hole. But this is not the interesting case for me.

Such a neutron star is actually far from dead. It continues its life after its death. But it no longer emits light and warmth, but usually x-rays, neutrinos, and occasionally so-called gravitational waves.

Second: What do I want to understand about them? Neutron stars are enigmatic objects. Their size is about ten kilometers, not more than a larger city. At the same time they have about one to two times the mass of our sun. Thus, they are incredible dense. In fact, they are so dense that there is no place for atoms, but they consist out of the atomic nuclei. That is the case in the outer layers of the neutron star, perhaps the first kilometer or so. Going further inward, the density increases. Then everything gets so tight, that it is no longer possible to separate the nuclei, and they start to overlap. In addition, whatever electrons there still were have already after the first few meters been soaked up to change almost all of the protons into neutrons. In a certain sense, it is just one big atomic nucleus.

And even further in? Well, nobody knows. However, there are many speculations. Do we have there a different kind of matter, so-called strange matter? Such matter is obtained when one starts to replace the up and down quarks in the neutrons with strange quarks. Or does also the neutrons dissolve, and we just have a bunch of quarks? And if yes, how would such quark matter behave? Would it be a fluid, a superconducting metal, or possibly even a crystal?

And that is, where my research starts. What I want to understand is, which form this matter takes. And I am not alone with this. I have just organized a workshop, which partly focused on this subject, and we have worked hard on getting a better understanding, of what goes on in there. That becomes even more interesting as more and more results come in from astronomical observations on neutron stars. They provide us with a lot of indirect evidence on how the matter inside the neutron star's core must behave. But if we understand the strong force correctly, we should be able to calculate this.

The central problem involved in these calculations is the density. The standard approach to particle physics (and to physics in general) is to attempt to simplify the problem, and study its parts in isolation. That is quite well working for many cases, like the Higgs. However, the properties of the neutron star is determined not by the individual neutrons, but in how they interact with each other when there are many of them. Thus, by breaking the system apart you destroy what you want to study. Thus, you have to study the neutrons - or more appropriately the quarks - all together. This enlarges the complexity severely, and it is what stops us in our tracks. Particularly, because it is hard to find efficient ways to calculate anything for a real neutron star.

One way around this is to attempt to indirectly understand it, by studying a simpler system. The alternative is to simplify the system itself. This can be done by making things a bit more fuzzy. This fuzziness is achieved by not tracking each and every quark and what it precisely does. Instead, groups of quarks are tracked, and their activities is averaged. This can be a very simple step. For example, one can treat a neutron instead of being made from three quarks as made from one quark and the rest. And then approximate the rest by a single particle with simple properties. Such an approximation already gives a rough estimate of how things work. Of course, if one wants to get the last bit of precision out of the theory, then one has to return to the original three quarks.

But the problem is complicated, and thus one follows this strategy: Creating less and simpler objects first, an then refine them again. This simpler objects are often called 'effective degrees of freedom', because they effectively mimic many complicated objects. And then we solve the simpler theory describing them, the so-called effective theory. Afterwards, we go back. We refine the effective theory and the simple particles again, introducing the problems bit by bit. And solving them on the way. And that is, where we are currently. Still far away from understanding a neutron stars as a set of elementary particles, as quarks and gluons, but closing in, step by step.

(Almost) Nothing is forever: Decays

A truly remarkable fact is that most elementary particles do not live forever. Most of them exist only for a very brief moment in time. The only particles, for which we are reasonable sure that they live forever are photons. Then there are a number of particles, which live for almost forever. This means their life expectancy is much, much longer than the current age of the universe. The electron, the proton, and at least one neutrino belong to this group. And so do many nuclei. That is good, because almost all matter around us is made of these particles. It is rather reassuring that it will not disintegrate spontaneously anytime soon.

However, the vast majority of elementary particles has not a very long life expectancy. For some, their lifetime is still such that we can observe them directly. But for particles like the infamous Higgs, this is not the case.

If we want to learn something about such particles, we have to cope with this problem. A very important help is that a particle cannot just vanish without a trace. If a particle's life ends, it decays into other, lighter particles. For example, the Higgs can decay into two photons. Or into two light quarks. Or in some other particles, as long as the sum of their masses is lighter than the Higgs' mass. And these decays follow very specific rules. Take again the Higgs. If it is lying somewhere around, and then decays into an electron and a positron, these two particles are not completely free in their actions. Very basic rules of physics require them then to move away from the position of the Higgs in opposite directions, with the same speed. And this speed is also fixed.

These laws are what helps us. If we detect the electron and the positron, and measure their speed, we can reconstruct that they come from a Higgs. This is a bit indirect, and one has to measure things rather precisely. But it is possible. And it was exactly this approach with which we have detected (very likely) the Higgs last year.

Of course, there are a lot of subtleties involved. And not every decay can happen, which seems to be permitted at a first, superficial look. And so on. But all this follows very precise rules. And the experimental physicists have become very good at using these rules to their advantage.

And here enters my own research. One of my projects is about some particles which are build from Higgs particles and Ws and Zs. If I want to tell my experimental colleagues to check, whether my theory is right, I have to tell them for what to look. Since those complicated particles are not expected to live very long, they will decay. Hence, I should be able to tell how they decay, and into which particles with what speed. This will give what we call a signature: The traces a unstable particle leaves at the end of its life. Doing this is somewhat complicated, as you not only have to understand the structure of the particles, but also their dynamics. Fortunately, people have developed very sophisticated methods. I use them now in simulations. With them I start to obtain results how my complicated particles decay into two Ws. In addition, I learn how often and how effectively this will happen, and how long the original particle lived.

Of course, as always, my first results are not the final answer yet. But things look encouraging. And, what is best, I start to find hints that the complicated states may even live long enough that, just maybe, they could be seen by my experimental colleagues. This is important, because a particle which lives to short requires a precision to observe higher than what we have currently. But it is still a long road before anything is certain. As always.

Taking a detour helps

Almost all relevant physical systems are pretty complicated. One I am working on is how the interior of so-called neutron stars look like. Neutron stars is what is left of stars somewhat heavier than our sun, but not too heavy, after they became a supernova. In a neutron star the atoms collapse due to the strong gravitation. Only the atomic nuclei remain, and are packed very densely. Neutron stars have roughly one to two times the mass of our sun, but have a radius of only about ten kilometers, barely larger than a small city. These star remnants are very interesting for astronomy and astrophysics. But I am more interested what happens in their most inner core.

Deep inside the neutron star, everything is even more packed. In fact, even the atomic nuclei are no longer separated, but are mashed into a big mess. Because their are so densely packed, even the nucleons are overlapping. Thus, the substructure of them, the quarks may become the most important players.

But this nobody knows yet for sure. It has been a challenge to understand such matter since more than thirty years. It is a joint effort of theoreticians, like me, people smashing atoms on each other in accelerators, so-called heavy-ion experiments, and people observing actual neutron stars with telescopes of many kinds.

In general, if quarks come into play, very often simulations have been very helpful. But it turns out that we are not (yet) clever enough to simulate a neutron star's interior. The algorithms, which we have developed to deal with single nuclei are just too inefficient to deal with so many nuclei. For technical reasons, this is called the sign problem, denoting the particular technical problem involved. This obstruction is also known since decades, without us being able so far to remove it.

An alternative have been other methods and models, but we would like to have a combination, to be more sure of our results.

One possibility has been to circumvent the problem. We have looked at theories which are similar to the strong nuclear force, but slightly modified. The modification were such that numerical simulations were possible. We made this detour for two reasons. We hoped that we could learn something in general. And we wanted to use these results to provide us with tests for our models and other methods. In a way we cheated: We evaded the problem by doing a simpler problem. And hoped that we would learn enough by this to solve the original problem or get a new insight.

However, so far our detours had serious drawbacks. The replacement theories were only able to solve some problems, but never all at the same time. Some had the problem that the mass creation by the strong force did not work in the right way. This would yield wrong answers for size and mass of a neutron stars. Or the nucleons were not repellent enough, so that all neutron stars would collapse further to so-called quark stars, much smaller than neutron stars and made from quarks.

And here comes my own research into play. Just recently we found another theory, which we call G2-QCD for very technical reasons. Irrespective of the name, it has neither of these problems. However, it is still not QCD. E. g., it has besides the nucleons further exotic objects flying around. But it is anyway the theory closest to the original one so far investigated. And we can actually simulate it. That is something we just done very recently. The results are very encouraging, though we are yet far from a final answer for neutron stars. Nonetheless, we have now an even stronger test for all the models and results from other methods available. This should provide even more constraints on our understanding of neutron stars, though still an enormous amount of work has to be done. But this is research: Mostly progress by small steps. And we thus continue on with this theory.

And this is just one example in my research where it is worthwhile to take a detour, and this is true for physics in general: Often the study of a simpler problem helps to reveal the solution of the original one. Even if we did not (again yet) succeeded, we made progress.

Enthusiasm vs. Statistics

Being a theoretician permits one to speculate and make predictions what should be seen in experiments. How reality could, in fact, be. However, this is not always as simple as it looks. Not so much because of how to arrive at a prediction, but when it comes to judge whether a prediction is true.

For example, I am working on interesting new phenomena in connection with the Higgs. Recently, I found some rather interesting results, which lead to some predictions. And now the ATLAS experiment  at the LHC continues to find that the Higgs properties are not exactly what they are expected to be in the standard model. What is actually not completely off from what I would expect because of my own calculations.

Should I now get excited, and cry that I have the explanation? I better not. Why? Why should I not be enthusiastic about these results?

Well, here comes the detective part in particle physics.

First, no calculation we can make today in any theory attempting to describe nature is exact. We always have to make some kind of approximations. For some of them we can make a firm statement how large the error, we are making, is at most. But in many cases, we cannot even reliably estimate the maximum size of the error. Do not get me wrong. It is not that most things are completely uncontrolled. In many cases we just cannot proof how large the error can be at most. But we have experience, experiments, and other kinds of approximations to which we can compare. This gives us a rather good idea of the size of the errors. But still, we cannot be absolutely sure about the true size. We then prefer to be better safe than sorry.

This is one of the reasons why I am not immediately enthusiastic. The approximations are yet too crude to be sure that what ATLAS sees must be unequivocally what my calculations give.

But there is more. It is not only theory which has errors. Experiments have errors as well. The reason is that nature is not strictly consequential. Because of quantum physics we cannot make the firm prediction that if A happens then B has to happen. We can just make a statement how probable B happens if A happens. As a consequence, modern particle physics experiments have, even if the machine itself is perfectly understood and perfectly build, an intrinsic error. Like any error for a probability it becomes smaller when we make more measurements.

Right now, this error for the consistency problems found by ATLAS is large. Not large in the sense of huge, but so large that there is a fair chance that the inconsistencies will go away, and we just see a random glitch of nature.

That sounds a bit odd at first. What should a glitch of nature be? Take a dice. If you throw it often enough, and just note the number of times a number comes up, then for very many throws every number will be there the same number of times. Try it. You will see that this will take a large number of throws before it happens, but it will happen eventually.

However, it may happen that you throw it ten times, and you will never get a one. Would you now conclude that there is no one on the dice? No, you would know that there is a one, just by looking at it. But you may need to throw some more times to get it at last once. But what if you get just told the numbers and never are allowed to look at the dice? Would you know that there is a one? Or could just somebody use a non-standard dice? What you would not expect is that nature just avoids ones, right?

In a particle physics experiment, it is like this. We cannot see the dice. We just get the counts. And like the case without ones, the current results of ATLAS could be a similar glitch. It just came up like this, and we have to go on, and count more.

Fortunately, you can make a statement how improbable it is not to get a one, if you throw the dice often enough. That is a number which quickly becomes small the larger the number of throws is.

Is particle physics, we can do the same thing. For the ATLAS experiment right now, there is a very good chance that things will turn out to be what they should be, and it is jut the good, old Higgs. What do I mean by good? Well, that is something like a one in a hundred chance, or so. That seems to be a far cry. But we physicists made the bitter experience that a one-in-a-hundred chance will turn against you in some cases. We make so many hundred measurements that at least some will turn out against the chances. That led in the past to false claims of discoveries, and nowadays we have become very careful, rather waiting long to reduce it to a one-in-a-million chance then to be premature.

Thus, I currently also think that the glitches seen by ATLAS are more likely not more than just such an effect. And I stay my enthusiasm for other occasions. But if the results of ATLAS should stay even with more data, well, then there may be finally the point reached to be enthusiastic. In spite of the potential problems lurking in my own calculations. Because then there is something new to be explained. And this may still be my own solution.

A Higgs and a Higgs make what?

Recently, I have mostly written about increasingly technical details of my work. Though these are absolutely necessary foundations for what I do, they are of limited use when taken out of context. I will try to be a bit more up-to-date and less technical, and will try to write more about what motivates me at a current time. What am I really working on?

So let me start with what I am doing right now, just before I started writing this entry. You will probably all have heard about the Higgs discovery. Or, more appropriately, of a particle of which we strongly suspect, but do not yet know, that it is the Higgs. But let me assume for now that it is. As exciting and important as this discovery is in itself, there is much more to this. One of the fascinating things about modern theories is that they do not only describe one or two phenomena, but have an enormous richness.

Concerning the standard model, and in particular the Higgs sector, there are quite some subtle phenomena going on. Why subtle? Well, the Higgs is actually quite a seclusive type. It does not play very much with the rest of the standard model, perhaps except for the top quark. In our language, we say that it is weakly coupled. The Higgs is not the only such particle in the standard model. Everything which has to do with electromagnetism is also not very strongly coupled.

Nonetheless, electrically charged particles play along very well. They like to group together in what we call atoms. These are so-called bound states of electrically charged particles, like the proton and the electron.

Now, for the Higgs actually something similar applies. It has been already suspected in the early 1980ies that Higgs particles could form bound states. In fact, there are very strong theoretical arguments for it, as soon as you include enough of the standard model. The crucial question was (and still is), how long do such Higgs atoms live? Of course, normal atoms live essentially forever, if no physicist comes by and smashes them or some chemist tries to let them react with each other. This is, because the things making up a normal atom are stable themselves. Electrons and protons are, to the best of our knowledge, very, very stable. Even the neutrons, once packed into a nucleus, remain stable. Well, at least if do not select too exotic a nucleus.

Anyway, this is different in the Higgs case. Here the constituents of these 'atoms', just two Higgses, really, are unstable themselves. Thus, it is at all not clear whether you can ever observe such a thing. But since rather deep theoretical arguments say that this could be, I want to know what the answer is. Even more, I am not satisfied with whether they could exist in principle, but if we can see them in an experiment, say the LHC.

To get an answer to this question, I have to invest everything I know. I first have to gather the basic foundation of the theory describing the Higgs. Then, I use simulations to determine the properties of such Higgs atoms. To be able to do this, I have to simplify quite a lot, because otherwise the simulations would be unbearable slow. Once I have these properties, I put them into a model. This step is necessary, because the simulations are not very efficient to give experimental predictions. Thus, I have to take a detour to get an answer. Such a model can be obtained using the appropriate equations to link both worlds.

Then, finally, I have a description of these atoms, and how they interact. With this, I have finally reached the point to use different types of simulations to make an experimental prediction.

That may sound like an afternoons work. But, unfortunately, it is not. Determining the properties of the atoms, even very roughly, has already required something like eighteen months. Constructing the model took another month in its simplest version. And right now, I just get acquainted with the basic simulations for an experiment. I just finished 'rediscovering' the well-known Z, as a first exercise. I hope that I will be able to present a first result at a small workshop in January, almost two years after I started thinking about this question. This result will be extremely simplified, and will be at best a motivation to go on. I have estimated that to get a real quantitative result, which is correct within about 20-30 percent, will require probably another ten-twenty man-years, and likely a couple of thousand core-years of computing time. But well, if we can find them, that would be really something. But even if not, then we have learned a lot about the theory we are working with. And that would be something in itself. So stay tuned, what will happen next.

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.