t, H, W, Z: noncommutative edition

In May, I blogged about a Higgs-VEV-squared sum rule due to Lopez Castro and Pestieau (LC & P) - and which, I would like to repeat, was closely anticipated by A. Garces Doz, in a pseudonymous comment at Lubos Motl's blog, at a time when the Higgs mass was not yet known. (The comment seems to have been lost from Lubos's blog, perhaps when the old "JS-Kit" comment system was shut down, but fortunately I made a copy here.)

In their paper (second link above, page 2), LC&P express their equation in terms of standard model couplings. Rewritten slightly, it is:

2 λ + 1/4 g² + 1/4 (g²+g'²) + 1/2 Σ y_f² = 1

This may seem less enlightening than the original.

But consider this discussion between Urs Schreiber and Jacques Distler, dating from 2006, regarding the mysteries of the Chamseddine-Connes-Marcolli noncommutative standard model. From his personal notes, Urs reproduces the equation

g₃² = g₂² = 3 λ = 1/4 Σ g_Y²

I do not actually see how to obtain this from the CCM paper. Perhaps it's implied by something in section 5.4. It can at least be verified that there are odd formulae in which squares of yukawas appear, e.g. equation 5.25 - and that may be "enough". As I have previously noted, the LC&P sum rule is still true if you only use the top yukawa, i.e. if you replace the fourth term on the left-hand side with just "y_t²".

The important observation here is that these noncommutative models have a tendency to produce, in Jacques Distler's words, "relations among the coupling constants over and above those guaranteed by gauge invariance and renormalizability"; and that these can include squares of couplings, such as appear in the rewritten version of the LC&P sum rule with which I began this post. So perhaps these "noncommutative" or "spectral" models have at least a fighting chance of explaining it.

P.S.: While I'm here, I'll also observe that it would be interesting to see whether the derivation of the Higgs mass via asymptotic safety, can be extended or modified to also produce a Higgs VEV that is roughly twice the mass. Out of all the t, H, W, Z numerology that I've collected so far, that ought to be the simplest relation to add to the A.S. scenario.