## Monday, April 23, 2012

### t, H, W, Z

Andrew Oh-Willeke mentions (in a blog post whose headline observation is the same as that made by  Dharwadker and Khachatryan) that the Higgs VEV is approximately twice the mass of the LHC's maybe-Higgs. This is one of a set of numerological connections between t, H, W and Z that have been on my list of items to ponder. Obviously I need to rush into print with those observations now, or else Andrew and others will get all the credit (when they figure it out for themselves)...

So, first item: top mass is approximately sqrt(2) times Higgs mass, Higgs VEV is approximately sqrt(2) times top mass. This is an extra twist on the basic observation that Higgs VEV is approximately two times Higgs mass.

Second item... This one isn't an observation so much as a conjunction of observations. I've already blogged Malcolm Mac Gregor's observation that m_top =approx m_W + m_Z. (He has a new paper today with lots of hadron numerology.) One then needs to consider this alongside the Dharwadker-Khachatryan observation (prediction, actually) that m_Higgs = m_W + 1/2 m_Z, and finally alongside the observation from the first item that m_top =approx sqrt(2) m_Higgs.

If you set the two expressions for m_top equal to each other, you get that m_W + m_Z "equals" sqrt(2) x (m_W + 1/2 m_Z), which would be true if m_W = 1/sqrt(2) m_Z, which isn't true. But maybe it's true "to zeroth order"? ... in the same unknown (and possibly nonexistent) theoretical framework where all these relationships aren't just coincidences.

2. Alejandro Rivero points out a discovery of Hans de Vries in which the eigenvalues of a certain operator are close to m_W, m_Z, m_t, and m_H. But the eigenvalues are supposed to be associated with values of "spin" which are wrong.

3. Mitchell, you say:
"… If you set the two expressions for m_top equal to each other,
you get that m_W + m_Z "equals" sqrt(2) x (m_W + 1/2 m_Z),
which would be true if m_W = 1/sqrt(2) m_Z,
which isn't true. …".

As you say since mW = 80 is not equal to (1/sqrt(2))mZ = 64
exact algebra relations imply
that the two expressions are not consistent
and
it looks like the discrepancy is very large
like 16/80 = 20 per cent.

If you look at the error bars for the two expressions
and use the values
mH = 125 and mW = 80 and mZ = 91 GeV and mT = 173 GeV

then you get for each expression separately:

(1/2) x (mW + mZ + Mw) = (1/2)( 251 ) =(rounding)= 125 GeV = mH
error = 0 per cent

mW + mZ = 171 GeV = 2 GeV lower than mT
error = 2/173 = 1 per cent

sqrt(2) x mH = 177 GeV = 4 GeV higher than mT
error = 4/173 = 2 percent

So:

sqrt(2) x (1/2) x (2 mW + mZ) - 4 = mW + mZ + 2
and
sqrt(2) x (1/2) x (2 mW + mZ) - mW - mZ = 6
and
(sqrt(2) - 1)mW = (1 - (1/sqrt(2))mZ + 6
and
0.414 mW = 0.293 mZ + 6
and
33 = 27 + 6

and

in reality the total error is about 6/173 = 3 per cent
which is really pretty good
and
certainly is not horrible like the 20 per cent error
that seems to me to be an artifact of doing exact algebra
without taking error bars into account.

Tony

4. It's a good observation. Still, "t=W+Z" is a strange relationship, so I wouldn't be surprised if that one is a coincidence and the other ones are the meaningful relations. The masses are all about the same order of magnitude, so it wouldn't be surprising to have one or two spurious relations mixed in with the relations which genuinely have a cause.

5. Matt Reece (in comments): m_H =approx sqrt(m_Z) sqrt(m_t). "Just a coincidence. There are lots of them if you look. They mean nothing."

6. In search of an explanation for Reece's "coincidence".

7. Latest tHWZ mass fits. Contains a M_W prediction.

8. Equation 10 in this paper is rather odd - why is m_q the constituent quark mass, rather than the current mass? - but since they derive the Higgs mass later, using m_top as an input, it goes on the list.

9. S.Vik says "Mass of H^2 = W^2 + Z^2 give or take a quark." The coincidence is unimpressive, but this goes on the list because sum rules often have this form.

10. Tony Smith says the sum of the squares of the W and Z masses in Planck units is about the order of the monster group. That has to be the least likely relation of any listed here :-) but the monster is good to think about, on account of its role in pure gravity in AdS3.

1. I have to correct this comment, I don't know how I could have read Tony's paper as saying that. Anyway, the relevant passage is halfway through version 1, and the actual formula is ((Mpl/Mep)(Mw/Mep))^2 = 9 x 10^53, where Mpl is the Planck mass and Mep ~ 1 MeV is "the positronium mass".

Also, later he considers ((Mpl/Mep)(Higgsvev/Mep))^2 = 9 x 10^54, and we have the more recent formula that HiggsVEV^2 = mt^2+mH^2+mW^2+mZ^2, so this is a little more like what I said he said. But basically, I completely misread him.

11. Volovik and Zubkov have updated their paper (mentioned in a comment on September 3), so I will mention Moffat's paper that also proposes a "second Higgs" at 325 GeV.

12. A model from Christopher Hill (inventor of "topcolor") in which m_t = m_H (to a first approximation).

13. The Higgs quartic coupling, lambda, needs to be -1/4 to make the Higgs VEV twice the Higgs mass.

Note that in Hill's paper, if we approximate the top yukawa as 1 at high energies (which Rodejohann and Zhang say is possible, if we take into account neutrinos, I think), we get that lambda = 1/2 at high energies. And meanwhile, the current discussion of a tuned Higgs mass revolves around the idea that lambda = 0 at high energies.

1. "The Higgs quartic coupling, lambda, needs to be -1/4 to make the Higgs VEV twice the Higgs mass."

That was wrong, the true value is 1/8, see edit #2 here

14. The noncommutative or spectral approach to the standard model generates "numerology" in the form of relations between standard model parameters.

The "Connes-Lott model" was an early one and I have learned (from hep-th/9412185 page 2, hep-th/9501142 page 23) that it produced the relations m_t = 2 m_W, m_H = 3.14 m_W. Which are not very good. But like some old musings of mine which aimed at the wrong value for m_H, the overall framework may still be of interest.

15. Section 5 of 1307.3536 discusses numerous possible interpretations of the RG behavior of the Higgs couplings and the top yukawa.

1. Also see remarks on page 31 about mass ratios m_H/m_t and m_H/m_W.

16. More RG mysteries: I haven't even parsed what this paper is saying, except that pole mass and running mass of the top quark have ... some special relationship ... at an energy between m_Z and 2 m_W.

17. A ten-year old paper which predicts a Higgs mass over three times too big. The prediction comes from requiring that "physical cosmological constant is renormalization scale independent", which is said to imply a relation among fourth powers of masses. It might be relevant for explaining this "vev-squared sum rule".

18. Andrew Oh-Willeke examines the predictive value of some of the more prominent formulas in comments here.

19. This one is really crazy: ten years ago, A. Rivero pointed out a coincidence between mass scales of tHWZ and of "doubly magic" nuclei. I mention it now because when he wrote, the hip value for the Higgs boson was 115 GeV; but the coincidence has improved - sharply! - now that we have the experimental value of 125 GeV.

20. F. Himpsel has a theoretical framework which he says can produce, as a first approximation, the relation that m_H = 1/2 v. But at first glance, I doubt it makes sense; see comments here.