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.

Fostering an idea with experience

In the previous entry I wrote how hard it is to establish a new idea, if the only existing option to get experimental confirmation is to become very, very precise. Fortunately, this is not the only option we have. Besides experimental confirmation, we can also attempt to test an idea theoretically. How is this done?

The best possibility is to set up a situation, in which the new idea creates a most spectacular outcome. In addition, it should be a situation in which older ideas yield a drastically different outcome. This sounds actually easier than it is. There are three issues to be taken care of.

The first two have something to do with a very important distinction. That of a theory and that of an observation. An observation is something we measure in an experiment or calculate if we play around with models. An observation is always the outcome if we set up something initially, and then look at it some time later. The theory should give a description of how the initial and the final stuff are related. This means that we look for every observation for a corresponding theory to give it an explanation. To this comes the additional modern idea of physics that there should not be an own theory for every observation. Rather, we would like to have a unified theory, i.e. one theory which explains all observations. This is not yet the case. But at least we have reduced it to a handful of theories. In fact, for anything going on inside our solar system we need so far just two: The standard-model of particle physics and general relativity.

Coming back to our idea, we have now the following problem. Since we do a gedankenexperiment, we are allowed to chose any theory we like. But since we are just a bunch of people with a bunch of computers we are not able to calculate all the possible observations a theory can describe. Not to mention all possible observations of all theories. And it is here, where the problem starts. The older ideas still exist, because they are not bad, but rather explain a huge amount of stuff. Hence, for many observations in any theory they will be still more than good enough. Thus, to find spectacular disagreement, we do not only need to find a suitable theory. We also need to find a suitable observation to show disagreement.

And now enters the third problem: We actually have to do the calculation to check whether our suspicion is correct. This is usually not a simple exercise. In fact, the effort needed can make such a calculation a complete master thesis. And sometimes even much more. Only after the calculation is complete we know whether the observation and theory we have chosen was a good choice. Because only then we know whether the anticipated disagreement is really there. And it may be that our choice was not good, and we have to restart the process.

Sounds pretty hopeless? Well, this is actually one of the reasons why physicists are famed for their tolerance to frustration. Because such experiences are indeed inevitable. But fortunately it is not as bad as it sounds. And that has something to do with how we chose the observation (and the theory). This I did not specify yet. And just guessing would indeed lead to a lot of frustration.

The thing which helps us to hit more often than not the right theory and observation is insight and, especially, experience. The ideas we have tell us about how theories function. I.e., our insights give us the ability to estimate what will come out of a calculation even without actually doing it. Of course, this will be a qualitative statement, i.e. one without exact numbers. And it will not always be right. But if our ideas are correct, it will work out usually. In fact, if we would regularly not estimate correctly, this should require us to reevaluate our ideas. And it is our experience which helps us to get from insights to estimates.

This defines our process to test our ideas. And this process can actually be well traced out in our research. E.g. in a paper from last year we collected many of such qualitative estimates. They were based on some much older, much more crude estimates published several years back. In fact, the newer paper already included some quite involved semi-quantitative statements. We then used massive computer simulations to test our predictions. They were indeed as good confirmed as possible with the amount of computers we had. This we reported in another paper. This gives us hope to be on the right track.

So, the next step is to enlarge our testbed. For this, we already came up with some new first ideas. However, these will be even more challenging to test. But it is possible. And so we continue the cycle.

Getting better

One of our main tools in our research are numerical simulations. E.g. the research of the previous entry would have been impossible without.

Numerical simulations require computers to run them. And even though computers become continuously more powerful, they are limited in the end. Not to mention that they cost money to buy and to use. Yes, also using them is expensive. Think of the electricity bill or even having space available for them.

So, to reduce the costs, we need to use them efficiently. That is good for us, because we can do more research in the same time. And that means that we as a society can make scientific progress faster. But it also reduces financial costs, which in fundamental research almost always means the taxpayer's money. And it reduces the environmental stress which we exercise by having and running the computers. That is also something which should not be forgotten.

So what does efficiently mean?

Well, we need to write our own computer programs. What we do nobody did before us. Most of what we do is really the edge of what we understand. So nobody was here before us and could have provided us with computer programs. We do them ourselves.

For that to be efficient, we need three important ingredients.

The first seems to be quite obvious. The programs should be correct before we use them to make a large scale computation. It would be very wasteful to run on a hundred computers for several months, just to figure out it was all for naught, because there was an error. Of course, we need to test them somewhere, but this can be done with much less effort. But this takes actually quite some time. And is very annoying. But it needs to be done.

The next two issues seems to be the same, but are actually subtly different. We need to have fast and optimized algorithms. The important difference is: The quality of the algorithm decides how fast it can be in principle. The actual optimization decides to which extent it uses this potential.

The latter point is something which requires a substantial amount of experience with programming. It is not something which can be learned theoretically. And it is more of a craftsmanship than anything else. Being good in optimization can make a program a thousand times faster. So, this is one reason why we try to teach students programming early, so that they can acquire the necessary experience before they enter research in their thesis work. Though there is still today research work which can be done without computers, it has become markedly less over the decades. It will never completely vanish, though. But it may well become a comparatively small fraction.

But whatever optimization can do, it can do only so much without good algorithms. And now we enter the main topic of this entry.

It is not only the code which we develop by ourselves. It is also the algorithms. Because again, they are new. Nobody did this before. So it is also up to us to make them efficient. But to really write a good algorithm requires knowledge about its background. This is called domain-specific knowledge. Knowing the scientific background. One reason more why you cannot get it off-the-shelf. Thus, if you want to calculate something new in research using computer simulations that means usually sitting down and writing a new algorithm.

But even once an algorithm is written down this does not mean that it is necessarily already the fastest possible one. Also this requires on the one hand experience, but even more so it is something new. And it is thus research as well to make it fast. So they can, and need to be, made better.

Right now I am supervising two bachelor theses where exactly this is done. The algorithms are indeed directly those which are involved with the research mentioned in the beginning. While both are working on the same algorithm, they do it with quite different emphasis.

The aim in one project is to make the algorithm faster, without changing its results. It is a classical case of improving an algorithm. If successful, it will make it possible to push the boundaries of what projects can be done. Thus, it makes computer simulations more efficient, and thus satisfies allows to do more research. One goal reached. Unfortunately the 'if' already tells that, as always with research, there is never a guarantee that it is possible. But if this kind of research should continue, it is necessary. The only alternative is waiting for a decade for the computers to become faster, and doing something different in the time in between. Not a very interesting option.

The other one is a little bit different. Here, the algorithm should be modified to serve a slightly different goal. It is not a fundamentally different goal, but subtly different so. Thus, while it does not create a fundamentally new algorithm, it still does create something new. Something, which will make a different kind of research possible. Without the modification, the other kind of research may not be possible for some time to come. But just as it is not possible to guarantee that an algorithm can be made more efficient, it is also not always possible that an algorithm with any reasonable amount of potential can be created at all. So this is also true research.

Thus, it remains exciting of what both theses will ultimately lead to.

So, as you see, behind the scenes research is quite full of the small things which make the big things possible. Both of these projects are probably closer to our everyday work than most of the things I have been posting before. The everyday work in research is quite often grinding. But, as always, this is what makes the big things ultimately possible. Without such projects as these two theses, our progress would be slowed down to a snail's speed.

Tackling ambiguities

I have recently published a paper with a rather lengthy and abstract title. I wanted to enlighten in this entry a little bit what is going on.

The paper is actually on a problem which occupies me by now since more than a decade. And this is the problem how to really define what we mean when we talk about gluons. The reason for this problem is a certain ambiguity. This ambiguity arises because it is often much more convenient to have auxiliary additional stuff around to make calculations simple. But then you have to deal with this additional stuff. In a paper last year I noted that the amount of stuff is much larger than originally anticipated. So you have to deal with more stuff.

The aim of the research leading to the paper was to make progress with that.

So what did I do? To understand this, it is first necessary to say a few words about how we describe gluons. We describe them by mathematical functions. The simplest such mathematical functions makes, loosely speaking, a statement about how probable it is that a gluon moves from one point to another. Since a fancy word for moving is propagating, this function is called a propagator.

So the first question I posed was whether the ambiguity in dealing with the stuff affects this. You may ask whether this should happen at all. Is a gluon not a particle? Should this not be free of ambiguities? Well, yes and no. A particle which we actually detect should be free of ambiguities. But gluons are not detected. Gluons are, in fact, never seen directly. They are confined. This is a very peculiar feature of the strong force. And one which is not satisfactorily fully understood. But it is experimentally well established.

Since therefore something happens to gluons before we can observe them, there is now a way out. If the gluon is ambiguous, then this ambiguity has to be canceled by whatever happens to it. Then whatever we detect is not ambiguous. But cancellations are fickle things. If you are not careful in your calculations, something is left uncanceled. And then your results become ambiguous. This has to be avoided. Of course, this is purely a problem for us theoreticians. The experimentalists never have this problem. A long time ago I actually already wrote together with a few other people a paper on this, showing how it may proceed.

So, the natural first step is to figure out what you have to cancel. And therefore to map the ambiguity in its full extent. The possibilities discussed since decades look roughly like this:

As you see, at short distances there is (essentially) no ambiguity. This is actually quite well understood. It is a feature very deeply embedded in the strong interaction. It has to do with the fact that, despite its name, the strong interaction makes itself less known the shorter the distance. But for weak effects we have very precise tools, and we therefore understand it.

On the other hand at long distances - well, there we knew for a long time not even qualitatively what is going on for sure. But, finally, over the decades, we were able to constrain the behavior at least partly. Now, I tested a large part of the remaining range of ambiguities. In the end, it indeed mattered little. There is almost no effect left of the ambiguity on the behavior of the gluon. So, it seems we have this under control.

Or do we? One of the important things in research is that it is never sufficient to confirm your result just by looking at a single thing. Either your explanation fits everything we see and measure, or it cannot be the full story. Or may even be wrong and the agreement with part of the observations is just a lucky coincidence. Well, actually not lucky. Rather terrible, since this misguides you.

Of course, doing all in one go is a horrendous amount of work, and so you work on a few at the time. Preferably, you first work on those where the most problems are expected. It is just ultimately that you need to have covered everything. But you cannot stop and claim victory before you did.

So I did, and looked in the paper at a handful of other quantities. And indeed, in some of them there remain effects. Especially, if you look at how strong the strong interaction is, depending on the distance where you measure it, something remains:

The effects of the ambiguity are thus not qualitative. So it does not change our qualitative understanding of how the strong force works. But there remains some quantitative effect, which we need to take into account.

There is one more important side effect. When I calculated the effects of the ambiguity, I learned also to control how the ambiguity manifests. This does not alter that there is an ambiguity, nor that it has consequences. But it allows others to reproduce how I controlled the ambiguity. This is important because now two results from different sources can be put together, and when using the same control they will fit such that for experimental observables the ambiguity cancels. And thus we have achieved the goal.

To be fair, however, this is currently at the level of an operative control. It is not yet a mathematically well-defined and proven procedure. As with so many cases, this still needs to be developed. But having operative control allows to develop the rigorous control easier than starting without it. So, progress has been made.

Using evolution for particle physics

(I will start to illustrate the entries with some simple sketches. I am not very experienced with it, and thus, they will be quite basic. But with making more of them I should gain experience, and they should become better eventually)

This entry will be on the recently started bachelor thesis of Raphael Wagner.

He is addressing the following problem. One of the mainstays of our research are computer simulations. But our computer simulations are not exact. They work by simulating a physical system many times with different starts. The final result is then an average over all the simulations. There is an (almost) infinite number of starts. Thus, we cannot include them all. As a consequence, our average is not the exact value we are looking for. Rather, it is an estimate. We can also estimate in which range around the real result should be.

This is sketched in the following picture

The black line is our estimate and the red lines give the range were the true value should be. From left to right some parameter runs. In the case of the thesis, the parameter is the time. The value is roughly the probability for a particle to survive this time. So we have an estimate for the survivability probability.

Fortunately, we know a little more. From quite basic principles we know that this survivability cannot depend in an arbitrary way on the time. Rather, it has a particular mathematical form. This function depends only on a very small set of numbers. The most important one is the mass of the particle.

What we then do is to start with some theory. We simulate it. And then we extract from such a survival probability the masses of the particles. Yes, we do not know them beforehand. This is because the masses of particles are changed in a quantum theory by quantum effects. These are which we simulate, to get a final value of the masses.

Up to now, we try to determine the mass in a very simple-minded way: We determined them by just looking for numbers for the mathematical functions which are closest to the data. That seems reasonable. Unfortunately, the function is not so simple. Thus, you can mathematically show that this does not give necessarily the best result. You can imagine this in the following way: Imagine you want to find the deepest valley in area. Surely, walking down hill will get you in a valley. But only walking down hill this will usually not be the deepest one:

But this is the way we determine the numbers so far. So there may be other options.

There is a different possibility. In the picture of the hills, you could rather deploy a number of ants, of which some prefer to walk up, some down, and some sometimes so and otherwise opposite. The ants live, die, and reproduce. Now, if you give the ants more to eat if they live in a deeper valley, at some time evolution will bring the population to live in the deepest valley:

And then you have what you want.

This is called a genetic algorithm. It is used in many areas of engineering. The processor of the computer or smartphone you use to read this has likely been optimized using such algorithms.

The bachelor thesis is now to apply the same idea to find better estimates for the masses of the particles in our simulations. This requires to understand what would be the equivalent to the deepness of the valley and the food for the ants. And how long we let evolution run its course. Then, we have only to monitor the (virtual) ants to find our prize.

Searching for structure

This time I want to report on a new bachelor thesis, which I supervise. In this project we try to understand a little better the foundations of so-called gauge symmetries. In particular we address some of the ground work we have to lay for understanding our theories.

Let me briefly outline the problem: Most of the theories in particle physics include some kind of redundancy I.e., there are more things in it then we actually see in experiments. The surplus stuff is actually not real. It is just a kind of mathematical device to make calculations simpler. It is like a ladder, which we bring to climb a wall. We come, use the ladder, and are on top. The ladder we take again with us, and the wall remains as it was. The ladder made live simpler. Of course, we could have climbed the wall without it. But it would have been more painful.

Unfortunately, theories are more complicated than wall climbing.

One of the problems is that we usually cannot solve problems exactly. And as noted before, this can mess up the removal of the surplus stuff.

The project the bachelor student and I am working on has the following basic idea: If we can account for all of the surplus stuff, we should be able to know whether our approximations did something wrong. It is like preparing an engine. If something is left afterwards it is usually not a good sign. Unfortunately, things are again more complicated. For the engine, we just have to look through our workspace to see whether anything is left. But how to do so for our theories? And this is precisely the project.

So, the project is essentially about listing stuff. We start out with something we know is real and important. For this, we take the most simplest thing imaginable: Nothing. Nothing means in this case just an empty universe, no particles, no reactions, no nothing. That is certainly a real thing, and one we want to include in our calculations.

Of this nothing, there are also versions where some of the surplus stuff appears. Like some ghost image of particles. We actually know how to add small amounts of ghost stuff. Like a single particle in a whole universe. But these situations are not so very interesting, as we know how to deal with them. No, the really interesting stuff happens if well fill the whole universe with ghost images. With surplus stuff which we add just to make life simpler. At least originally. And the question is now: How can we add this stuff systematically? As the ghost stuff is not real, we know it must fulfill special mathematical equations.

Now we do something, which is very often done in theoretical physics: We use an analogy. The equations in question are not unique to the problem at hand, but appear also in quite different circumstances, although with a completely different meaning. In fact, the same equations describe how in quantum physics one particle is bound to each other. In quantum physics, depending on the system at hand, there may be one or more different ways how this binding occurs. You can count the number, and there is a set which one can label by whole numbers. Incidentally, this feature is where the name quantum originates from.

Returning to our original problem, we do the following analogy: Enumerating the ghost stuff can be cast into the same form as enumerating the possibilities of binding two particles together in quantum mechanics. The actual problem is only to find the correct quantum system which is the precise analogous one to our original problem. Finding this is still a complicated mathematical problem. Finding only one solution for one example is the aim of this bachelor thesis. But already finding one would be a huge step forward, as so far we do not have one at all. Having it will probably be like having a first stepping stone for crossing a river. From understanding it, we should be able to understand how to generate more. Hopefully, we will eventually understand how to create arbitrary such examples. And thus solve our enumeration problem. But this is still in the future. For the moment, we do the first step.

Playing indirectly

One of my research areas are neutron stars. To understand them requires to understand how the strong interactions behave when the matter is enormously densely packed. A new PhD student of mine has now started to work on this topic, and I would like to describe a little bit what we will be looking at.

I have already written in the past that this type of situation is very hard to deal with, because we cannot just do simulations. This is unfortunate, since simulations have been very successful in uncovering what happened in the early universe. In that case, the system is hot rather than dense. Though the reason for the problem is 'just' technical, chances are not too bright to resolve it in the near future.

Hence, I had already quite some time ago decided that a possibility is to play indirectly. The basic idea is that there are other methods, which would work. The price we have to pay is that we need to make approximations in these methods. But we would like to check these approximations, ideally against simulations. But we cannot, because there are no. So how to break the circle?

To escape this problem, we can again use a detour. We did this once, because we hoped that we will learn more about the qualitative features. That we can get some insight into this type of physics. Now, we have a much more quantitative approach. We use theories, which are very similar to the strong interactions (also called QCD), but are not QCD, but which can be simulated. And these we will use to break the circle.

Why is it possible to perform simulations for this type of theories? Well, the main reason is the difference between particles and anti-particles. In QCD, a quark and an anti-quark are fundamentally very different objects. Hence, a large density can mean two things. A large density could be having many more quarks than anti-quarks, but still have plenty of both. Or it could be just to have many of one type. For a neutron-star both situations are relevant. And thus, there may be actually many more particles present then we would think, just many of them anti-particles. This is at the heart of the problem, that there is so much more than just the superficial number of particles.

This problem is evaded by using a theory instead where there are no anti-quarks. To be more precise, a theory in which anti-quarks are the same as quarks. There exists a number of such theories. However, such a change is very drastic. It thus may happen that the so changed theory is so radically different from QCD that any comparison becomes meaningless. Thus, it is necessary to ensure that the theory is close enough to the original.

Two candidates for such theories have been identified so far. One is the so-called G2QCD, of which I talked about previously. Another one is very close to QCD, but instead of three color charges is has just two different ones. Both cases have their own merits. The first is closer to QCD. In this theory there are protons and neutrons. The latter does not have these, but it is very cheap to simulate. Both theories are hence quite different, but actually share both also many other traits with QCD.

It therefore stands to reason that whatever approximation describes both well will also work for QCD. Thus, we will now use the simulations of both theories to test the approximations made in the other methods. Especially, we will look at the properties of the quarks and gluons. We will then use the insights gained to improve the approximations. Until we describe both theories well enough. Then we will translate the approximations back to QCD. And if everything works out, we will have then an acceptable description of a piece of neutron star matter.

A model for a model

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

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

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

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

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

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

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

Precision may matter

The latest paper I have produced is an example of an often overlooked part of scientific research: It is not enough to get a qualitative picture. Sometimes the quantitative details modify or even alter the picture. Or, put more bluntly, sometimes precision matters.

When we encounter a new problem, we usually first try to get a rough idea of what is going on. It starts with a first rough calculation. Such an approach is often not very precise. Still, this creates a first qualitative picture of what is going on. This may be rough around the edges, and often does not perfectly fit the bill. But it usually gets the basic features right. Performing such a first estimate is often not a too serious challenge.

But once this rough picture is there, the real work begins. Almost fitting is not quite the same as fitting. This is the time where we need to get quantitative. This implies that we need to use more precise, probably different, but almost certainly more tedious methods. These calculations are usually not as simple, and a lot of work gets involved. Furthermore, we usually cannot solve the problem perfectly in the first round of improvement. We get things a bit rounder at the edges, and the picture normally starts to fit better. Still not everywhere, but better. Often, a second, and sometimes many more, rounds are necessary.

Fine, you may say. If things are improving, why bother doing even better? Is not almost fitting as good as fitting? But this is not quite the same. The best known examples we find in history. At the beginning of the 20th century, the picture of physics seem to fit the real world almost perfectly. There were just some small corners, where it seems to still require a bit of polishing. These small problems actually led to one of the greatest change in our understanding of the world, giving birth to both quantum physics and the theory of relativity. Actually, today we are again in a similar situation. Most of what we know, especially the standard model, fits the bill very nicely. But we still have some rough patches. This time, we have learned our lesson, and keep digging into these rough patches. Our secret hope is, of course, that a similar disruption will occur, and that our view of the world will be fundamentally changed. Whether this will be the case, or we just have to slightly augment things, we do not yet know. But it will be surely a great experience to figure it out.

Returning to my own research, it is precisely this situation which I am looking at. However, rather than looking at the whole world, I have been just looking at a very simplified theory. One that involves only the gluons. This is a much simpler theory than the standard model. Still, it is so complicated that we were not (yet) able to solve it completely. We made great progress, though, and it seems that we almost got it right. Still, also here, some rough edges remain. In this paper, I am looking precisely at these edges, and just check how rough they really are. I am not even trying to round them further. I am not the first to do it, and many other people have looked at them in one or the other way. However, doing it more than once, and especially from slightly different angles, is important. It is part of a system of check and balances, to avoid any error. Tt is also in science true: Nobody is perfect. And though there are many calculations, which are correct, even the greatest mind may fail sometime. And therefore it is very important to cross check any result.

In this particular case, everything is correct. But, by looking more precisely, I found some slight deviations. These were previously not found, as precision is almost always also a question of the amount of resources invested. In this case, the resources are mostly computing time, and I have just poured a lot of it into it. These slight deviations do not require a completely new view of the whole theory. But it changes some slight aspects. This may sound like not much. But if they should be confirmed, they provide closure in the following sense: Previously, some conclusions remained dangling, and seemed to be not at ease with each other. There were some ways out, but the previously known results rather suggested a more fundamental problem. My new contribution shifts these old results slightly, and makes them more precise. The new interpretation fits now much better with the suspected ways out rather than with a fundamental problem. Hence, looking closer has in this case improved our understanding.

Hence, theoretical physics has often more in common with a detective's work. We start with a suspicion. But then tedious work on the details is required to uncover more and more of the whole picture, until either the original suspicion is confirmed, or it shifts to a different suspect, which may have even been completely overlooked in the beginning. However, at least normally nobody tries to kill us if we come too close to the truth.

For each yes and no there is a perhaps

The last time, I was writing about my research on the Higgs. Especially, I was writing how we tested perturbation theory using numerical simulations. I was quite optimistic back then to have results by now, which could be of either of two types. Either perturbation theory is a good description, or it is not.

By now we have finished this project, and you can download the results from the arxiv. The arxiv is a server where you can find essentially every published result in particle physics of the last twenty years, legally and free of charge. But lets get back to our results. As I should have expected, things turned out to differently. Instead of a clear yes or no answer I got a perhaps.

The original question was, under which circumstances can perturbation theory be applied. It appears to be a simple enough question. Originally, it looked like this would depend on the relative sizes of the Higgs mass to the W and Z masses. And yes, it does. But. We found more.

We found different regimes. One is where the Higgs is lighter than the W and Z. Of course, this is not a situation we encounter in nature, where it is about 50% heavier. But as theoreticians, we are allowed to play this kind of games. Anyway, in this case, we confirmed what was already indirectly known from other investigations: Perturbation theory seems not to work. Always. While the first statement is not too surprising, the second statement is. Naively, one expected that if the interactions between the Higgs and the W and Z are of certain relative sizes, perturbation theory would still work. We did not find any hint of that. Is this already then a no? Unfortunately not. As I have described earlier, it is not so easy to relate a simulation to reality. Even if it is only a fantasy version of reality, as in this case. Hence, we cannot be sure that we have exhausted all possibilities. The only thing we can say for sure is that there are cases, where perturbation theory does not work. Perhaps there is something more, some other cases. And thus there is the first perhaps.

The situation gets even more interesting, when the Higgs is heavier than the W and Z, but lighter than twice their mass. In this regime, perturbation theory is expected to be pretty good. At least here, we find a rather clear answer: Perturbation theory does indeed well. Wherever we looked, we did not find anything to the contrary. Of course, again we cannot exclude that there is somewhere else a different case. But so far, everything seems to be fine.

When the Higgs finally hits the magic limit of twice the W and Z masses, something unexpected happens. This limit is particularly interesting, because above it, the Higgs can decay into W and Z. The expectation was that perturbation theory is still valid. At least until reaching several times the W and Z mass. But here, we found something odd. We found both possibilities, depending on the relative interaction strengths. In the one case, perturbation theory still works for a long time. In the other already a little bit above this critical mass perturbation theory starts to fail. We do not yet really understand, what is going on there, and what really characterizes the two different cases. We are working on this right now. But whatever it is, it is different than we expected. And this once more teaches to always expect that your naive expectations are not fulfilled. Things remain full of surprises, even if you think you understood them.

Knowing the limits

Some time ago, I have presented one of the methods I am using: The so-called perturbation theory. This fancy name signifies the following idea: If we know something, and we add just a little disturbance (a perturbation) to it, then this will not change things too much. If this is the case, then we can systematically give the consequences of the perturbation. Mathematically, this is done by first calculating the direct impact of the perturbation (the leading order). Then we look at the first indirection, which involves not only the direct effect, but also the simplest indirect effect, and so on.

Back then, I already wrote that, nice as the idea sounds, it is not possible to describe everything by it. Although it works in many cases very beautifully. But this leaves us with the question when does it not work. We cannot know this exactly. This would require to know the theory perfectly, and then there would be no need in the first place to do perturbation theory. So how can we then know what we are doing?

The second problem is that in many cases anything but perturbation theory is technically extremely demanding. Thus the first thing one checks is the simplest one: Whether perturbation theory makes itself sense. Indeed, it turns out that usually perturbation theory starts to produce nonsense if we increase the strength of the perturbation too far. This indicates clearly the breakdown of our assumptions, and thus the breakdown of perturbation theory. However, this is a best-case scenario. Hence, one wonders whether this approach could be fooling us. Indeed, it could be that this approximation breaks down long before it gets critical. So that it first produces bad (or even wrong) answers before it produces nonsensical ones.

This seems like serious trouble. What can be done to avoid it? There is no way inside perturbation theory to deal with it. One way is, of course, to compare to experiment. However, this is not always the best choice. On the one hand it is always possible that our underlying theory actually fails. Then we would misinterpret the failure of our ideas of nature as the failure of our methods. One would therefore like to have a more controllable way. In addition, we often reduce complex problems to simpler ones, to make them tractable. But the simpler problems often do not have a direct realization in nature, and thus we have no experimental access to them. Then this way is also not possible.

Currently, I find myself in such a situation. I want to understand, in the context of my Higgs research, to which extent perturbation theory can be used. In this context, the perturbation is usually the mass of the Higgs. The question then becomes: Up to which Higgs mass is perturbation theory still reliable? Perturbation theory itself predicts its failure at not more than eight time the mass of the observed Higgs particle. The question is, whether this is adequate, or whether this is too optimistic.

How can I answer this question? Well, here enters my approach not to rely only on a single method. It is true that we are not able to calculate as much with different methods than perturbation theory, just because anything else is too complicated. But if we concentrate on a few questions, enough resources are available to calculate things otherwise. The important task is then to make a wise choice. I.e. a choice from which one can read off the desired answer, in the present case whether perturbation theory applies or not. And at the same time to do something one can afford to calculate.

My present choice is to look at the relation of the W boson mass and the Higgs mass. If perturbation theory works, there is a close relation between both, if everything else is adjusted in a suitable way. The perturbative result can be found already in textbooks for physic students. To check it, I am using numerical simulations of both particles and their interactions. Even this simple question is an expensive endeavor, and several ten-thousand days of computing time (we always calculate how much time it would take a single computer to do all the work all by itself) have been invested. The results I found so far are intriguing, but not yet conclusive. However, in just a few weeks more time, it seems, that the fog will finally lift, and at least something can be said. I am looking with great anticipation to this date. Since either of two things will happen: Something unexpected, or something reassuring.

How to mix ingredients

I have written earlier that one particular powerful way to do calculations is to combine different methods. In this entry, I will be a bit more specific. The reason is that we just published a proceeding in which we describe our progress to prepare for such a combination. A proceeding is, by the way, just a fancy name for a write-up of a talk given at a conference.

How to combine different methods always depends on the methods. In this case, I would like to combine simulations and the so-called equations of motion. The latter describe how particles move and how they interact. Because you have usually an infinite number of them - a particle can interact with another one, or two, or three, or..., you can usually not solve them exactly. That is unfortunate. You may then ask, how one can be sure that the results after any approximation is still useful. The answer is that this is not always clear. It is here where the combination comes in.

The solution to the equations of motion are no numbers, but mathematical functions. These functions describe, for example, how a particle moves from one place to another. Since this travel depends on how far these places are apart, this must be described by a function. Similarly, there are results which describe how two, three, four... particles interact with each other depending on how far they are apart from each other. Since all of these functions then describe how two or more things are related to each other, a thing which is called a correlation in theoretical physics, these functions are correlation functions.

If we would be able to solve the equations of motions correctly, we would get the exact answer for all these correlation functions. We are not, and thus we will not get the exact ones, but different ones. The important question is how different they are from the correct ones. One possibility is, of course, to compare to experiment. But it is in all cases a very long way from the start of the calculation to results, which can be compared to experiment. There is therefore a great risk that a lot of effort is invested in vain. In addition, it is often not easy to identify then where the problem is, if there is a large disagreement.

It is therefore much better if a possibility exists to check this much earlier. This is the point where the numerical simulations come in. Numerical simulations are a very powerful tool. Up to some subtle, but likely practically irrelevant, fundamental questions, they essentially simulate a system exactly. However, the closer one moves to reality, the more expensive the calculations become. Most expensive is to have very different values in a simulation, e. g. large distances and small distances simultaneously, or large and small masses. There are also some properties of the standard model, like parity violation, which can even not be simulated up to now in any reasonable way at all. But within these limitations, it is possible to calculate pretty accurate correlation functions.

And this is then how the methods are combined. The simulations provide the correlation functions. They can then be used with the equations of motions in two ways. Either they can be used as a benchmark, to check whether the approximations make sense. Or they can be fed into the equations as a starting point to solve them. Of course, the aim is then not to reproduce them, as otherwise nothing would have been gained. Both possibilities have been used very successfully in the past, especially for hadrons made from quarks and to understand how the strong force has influenced the early universe or plays a role in neutron stars.

Our aim is to use this combination for Higgs physics. What we did, and have shown in the proceedings, are calculating the correlation functions of a theory including only the Higgs and the W and Z. This will now form the starting point for getting also the leptons and quarks into the game using the equations of motions. And we do this to avoid the problem with parity, and to include the very different masses of the particles. This will be the next step.

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.

(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.

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.

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.

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.

Methods. United, they are strong.

In the last few postings, I have collected a number of methods: perturbation theory, simulations, and the abstract equations of motion. I have furthermore gave you a bit of a taste of one of our most important strategies: divide and conquer. Or, more bluntly, if the original problem is too complicated, first try a simpler one, which resembles it. This lead us to a stack of models, which bit by bit always included more details of the world.

This list is by no means complete. Over the years, decades, and centuries, physicists have developed many methods. I could probably fill a blog all by its own just by giving a brief introduction to each of them. I will not do this here. Since the man purpose of this blog is to write about my own research, I will just contend myself with this list of methods. These are, right now, those which I use myself.

You may now ask, why do I use more than one method? What is the advantage in this? To answer this, lets have a look at my work-flow. Well, actually this is similar to what many people in theoretical particle physics do, but with some variations on the choice of methods and topics.

The ultimate goal of my work is to understand the physics encoded in the standard model of particle physics, and to get a glimpse of what else may be out there. Not an easy task at all. One, which many people work on, many hundreds, probably even thousands nowadays. And not something to be done in an afternoon, not at all. We know the standard model, more or less, since about forty years at the time of this writing. We think essentially as long as it exists about what else there might be in particle physics.

Thus, the first thing I do is to make the things more manageable. I do this, by making a simpler model of particles. I will give some examples of these simpler models in the next few entries. For now, lets say, I just keep a few of the particles, and one or two of their interactions, not more. This looks much more like something I can deal with. Ok, so now I have to treat this chunk of particles happily playing around with each other.

To get a first idea of what I am facing, I usually start off with perturbation theory, if no one else did this before me. This gives me an idea of what is going on, when the interactions are weak. This hides much of the interesting stuff, but it gives me a starting point. Also, very many insights of perturbation theory can be gained with a sheet of paper an a pencil (and many erasers), and probably a good table of mathematical formulas. Thus, I can be reasonably sure that what I do is right. Thus, whatever I will do next, it has to reduce to what I just did now when the interactions become weak.

Now I turn to the things, which really interest me. What happens, when the interactions are not weak? When they are strong? To get an idea of this, the next step is to perform some simulations of the theory. This will give me a rough idea, of what is going on. How the theory behaves. What kind of interesting phenomena will occur. Armed with this knowledge, I have already gained quite a lot of understanding of the model. I usually know then what are the typical way the particles arrange themselves. How their interaction changes, when looking at it from different directions. What the fate of the symmetries is. And lot more of details.

With this, I stand at a crossroad. I can either go on, and deepen my understanding by improving my simulations. Or, I can make use of the equations of motion to understand the internal workings a bit better. What usually decides for the latter is then that many questions about how a theory works can be best answered when going to extremes. Going to very long or very short distances when poking the particles. Looking a very light or very heavy particles. Simulations cannot do this with an affordable amount of computing time. So I formulate my equations. Then I have to make approximations, as they are usually too complicated. For this, I use the knowledge gained from the simulations. And then I solve the equations, thereby learning more about how the model works.

When I am done to my satisfaction, then I can either enlarge the model somewhat, by adding some more particles or interactions, or go a different model. Hopefully, at the end I arrive at the standard model.

What sounds so very nice and straightforward up to here is not. The process I describe is an ideal. Even if it should work out like this, I am talking about the several years of work. But usually it does not. I run across all kind of difficulties. It could turn out that my approximations for the equations of motion have been too bold, and I can get no sensible solution. Then I have to do more simulations, to improve the approximations. Or the calculations with the equations of motion tell me that I was looking at the wrong thing in my simulations. That the thing I was looking at was deceiving me, and gave me a wrong idea about what is going on. Or it can turn out that the model cannot be simulated efficiently enough, and I would have to wait a couple of decades to get a result. Then, I have to learn more about my model. Possibly, I even have to change it, and start from a different model. This often requires quite a detour to get back to the original model. This may even take many years of work. And then, it may happen that the different method give different results, and I have to figure out, what is going on, and what to improve.

You see, working on a problem means for me to go over the problem many times, comparing the different results. Eventually, it is the fact that the different methods have to agree in the end what guides my progress. Thus, a combination of different methods, each with their specific strengths and weaknesses, is what permits me to make progress. In the end, reliability is what counts. And with this nothing cuts it like a set of methods all pointing to the same answer.

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.