Over two years ago, a series of posts on tHWZ relations was launched here, starting with the observation that
<ϕ> ~ √2 mt, mt ~ √2 mH
which I was prompted to record, when Andrew Oh-Willeke remarked that
<ϕ> ~ 2 mH
(Where <ϕ>, also often written as v, is the Higgs field "vacuum expectation value".)
Recently a paper appeared on arxiv, noting that first relation, in the form
4 mH2 = 2 mt2 = v2
and Andrew commented that
"I think it is more likely that the observed relationship is really an approximation of the relationships
sum((Fi(^2)=v^2/2 and sum((Bj)^2)=v^2/2 for all fundamental fermion rest masses Fi and fundamental boson rest masses Bj"
which is an aspect of the LC&P sum rule, of which he also says that it is
"quite a bit more profound than the fact that the heaviest fermion by
itself accounts for about half of the Higgs vev squared, or that the
Higgs mass square accounts for about a quarter of the Higgs vev squared."
I agree that the LC&P sum rule looks to be the fundamental thing here. But there is an interesting final twist which he didn't note.
1. The sum of the squares of all the fundamental particle masses, is approximately the square of the Higgs VEV.
2. The contributions to this total from bosons and fermions are approximately equal. (Given the love of supersymmetry in the particle physics community, it really is remarkable that this isn't visibly being talked about.)
3. The top quark is responsible for the great majority of the fermion contribution, and thus about half of the total.
4. The Higgs boson is responsible for about half the bosonic contribution, and thus about a quarter of the total.
So where does the rest of the bosonic contribution come from? It comes from the W and Z bosons. So we have a fifth fact:
5. The W and Z bosons are responsible for the other half of the bosonic contribution, and thus for the remaining quarter of the total.
If we write this up as an equation, we get
mH2 ~ mW2 + mZ2 ~ 1/2 mt2
The first part of this equation appeared as a blog comment by S. Vik, who is apparently a retired physicist from Wilfrid Laurier University in Canada. At the time I gave it a low probability of being meaningful, but I did record it. It would be ironic if it is yet another genuine clue to what lies beneath the standard model.