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.