I was thinking again, about whether there is some way to explain the Higgs mass as a multiple of the nucleon mass. Not only is the fact that 53=125 surely a red herring, 125 or 126 wouldn't even be the dimensionless quantity of interest. The Higgs is about 126 GeV, but the nucleon mass is somewhat less than 1 GeV...
And then I remembered that other number beloved of physics numerologists, 137. Specifically, I vaguely recalled that there is some instability for a nucleus with atomic number of 137, precisely because the fine structure constant is about 1/137. I was reminded of the recently discovered fact that the value of the Higgs mass (along with the specific value of the top quark mass, and a few other parameters) places the standard model vacuum on the brink of instability.
It is tempting to suppose that some unknown physics has forced the Higgs to a critical value. In the previous post I speculated that "the Higgs field could be a QCD meson condensate weighed down by virtual nucleons". Could it be that the density of these virtual nucleons is bounded by an analogue of this 137-instability?! A crackpot idea, yes; but the first thing is to check what the actual ratio of the masses is.
Let us say that the Higgs mass is 125-126 GeV. The nucleon mass is about 939 MeV. This gives a ratio between 133 and 134. To my mind, this is close enough to 137 that one should persist a while longer with the idea. So what is the mechanism that destabilizes element 137 - which is jocularly known as "feynmanium", because Feynman was the one who noted the instability?
It turns out that the problem emerges first in the Bohr model - the innermost electron would orbit the nucleus faster than light ... and then in a more sophisticated version when using the Dirac equation - a ground-state instability ... and that even more sophisticated analyses push the problem out to atomic number 173, or entirely abolish it. The fact that the "137 instability" appears in different formalisms is mildly encouraging, since it suggests a phenomenon at work that might still exist, even in a wildly different theoretical context.
The next step was to see whether anyone else has had thoughts along these lines. The numerology mentioned by "fzero" in the previous post is getting there, but it's a little back-to-front: it uses the running of the fine-structure constant, to reach a scale where it is approximately 1/125. But as I have already mentioned, 1 GeV is only a ballpark number; 939 MeV is the objectively interesting quantity, and that suggests that we should go with the low-energy value, ~ 1/137.
A search for "feynmanium higgs" turned up a blog comment by "Juan F." which is halfway there. Feynmanium is mentioned, but Juan F. is still using something more like fzero's relation, with the number 126 mooted as significant because it is a "magic number" in nuclear physics.