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.

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