What is so important (to me) about the Higgs?

In a few weeks, I will give my inaugural lecture at the University of Graz. I have entitled it "The Higgs as a touchstone of theoretical particle physics". Of course, with this title I could easily give a lecture on many of the problems the Higgs present us with which are begging for new physics. I could write about how we have no idea why the Higgs has a mass of the size it has. How we do not understand why it does what it does. And so many other things.

But this is not what I want to write about. Nor is it what i want to talk about. It is rather something more mundane but also much more subtle. It is something for which I do not need any new physics to worry about. I can very well worry just about the known physics.

It has something to do with redundancy. In theoretical physics it is often very convenient - well, often indispensable - to add something redundant in our description. A redundancy is like writing instead of 2 2+0. Both statements have the same meaning. But adding zero to two is redundant. In this example, the zero is just redundant, but not useful. In particle physics, what we add is both redundant and useful.

It is this redundancy which helps us disentangling complicated problems. Of course, we are not allowed to change something by introducing the redundancy. But as long as we respect this requirement, we are pretty free what kind of redundancy we add. In the previous example, we could just write 2+0+0, and have another version of redundancy.

OK, so what has this to do with the Higgs? Well, if we measure something, it is of course independent of these redundancies - after all, they are man-made. And nothing made by us should influence what we measure. But if we look at the ordinary version of how we describe the Higgs, than there is a slight mismatch. In our theoretical description of the Higgs, there is some remainder of the redundancy still lingering. It is, like 2+0.001 pops up. Nonetheless, our theoretical description of the Higgs is spot on the experimental results. But how can this be if there is still redundancy polluting our result? It is this where the Higgs becomes a touchstone of understanding theoretical particle physics: In explaining why this is not correct and correct at the same time.

As always, the answer appears to be in the fine-print. In the standard way how we approach the Higgs, we are not doing it exactly. Well, to be honest, we could not do it exactly. We make some approximations. The consequences of these approximations is the appearance of the residual redundancy. Since our calculations are so spot on, these approximations appear to be good. However, after performing these approximations, we have now way short of experiment to confirm our approximations. That is highly unsatisfactory. We must be able to do it such that we can predict whether the approximations work. And understand why they work. It is in this sense that the Higgs is a touchstone. If we are not even able to answer these questions, how can we expect to solve the many outstanding questions?

This question has bugged people already 35 years ago, and some understanding of why it works was achieved. But not of when it works, at least not in the form of numbers. We have made some progress with this, especially recently. The amazing result was that it appears to work only in a very limited range of masses of the Higgs - with the observed Higgs mass essentially right in the middle of the possible range. This is even more surprising as already slight modifications of how many Higgs particles there are seem to change this. So, why is this so? Why is the Higgs mass just there? And what would happen otherwise? Understanding these questions will be very important to go beyond what is known. Without understanding, we may easily be fooled by our approximations, if we are not that lucky next time. This is the reason, why I think the Higgs is a touchstone for theoretical particle physics.