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.