Dharwadker and Khachatryan's prediction of the Higgs boson mass

I foresee that I will be making a series of posts about "Higgs Boson Mass predicted by the Four Color Theorem", which has to be the world champion among crackpot physics papers right now, because they may have predicted the Higgs mass correctly! The formula for the Higgs mass that they offer is very simple ... m_H = 1/2 (m_W+ + m_W- + m_Z) ... and for anyone impressed by the result, it might be tempting to appropriate the formula, but try to justify it on some other basis.

However, that's not enough for snarxiv blog. I don't want to leave this stone unturned. So I'm going to be drilling down into the "logic" of the paper, trying to unearth the quasi-deductive process whereby this formula is supposed to follow from Ashay Dharwadker's rather unusual construction. Let's start with what must be the final step in the logic - found on pages 55 and 56:

Since the Higgs particle/antiparticle will be identified (as a Cooper pair), their combined energy would then be the sum of the masses of all other bosons defined on the particle frame. We can have all types of bosons superposed on a single particle frame, and the single Cooper pair of the Higgs particle/antiparticle must be able to attribute energy/rest mass to all types of bosons on this particle frame, by the Higgs-Kibble mechanism. The particle frames of the bosons can be superposed at a point in space-time because they follow the Bose-Einstein statistics. Hence, this Cooper pair must have at least enough energy to attribute the sum of the rest masses of all types of bosons defined on the particle frame. On the other hand, the most important property of Bose condensation is that the Cooper pair of the Higgs particle/antiparticle must have minimum energy, so it can have at most the energy required to attribute the sum of the rest masses of all types of bosons defined on the particle frame. This must be the lowest energy state possible for the Higgs boson when it undergoes Bose condensation.

The "particle frame" is a type of disk structure, illustrated many times in the paper, and all the standard model particles are associated with regions of this disk.

It appears that the logic is as follows: The Higgs boson provides the mass for everything. For some reason, we will suppose that it delivers this mass in the form of a "Cooper pair" made of a Higgs particle and a Higgs antiparticle. The energy must be enough to provide the masses of all the massive bosons. But the energy of the Cooper pair will be a minimum. Therefore "masses of all the massive bosons" = "mass of Higgs particle + mass of Higgs antiparticle" and you get the formula.

The logic is illogical because Cooper pairs don't play a role in the Higgs mechanism, Bose condensation is irrelevant when we are comparing different species of boson, and probably for other reasons too. And that's not even addressing the rest of the theoretical framework, starting with Dharwadker's almost certainly wrong proof of the four-color theorem, which employs the Steiner system S(5,8,24). The symmetry group of S(5,8,24) is the exceptional group M₂₄, a subgroup of the permutation group S₂₄, and it appears that types of particle are associated with permutations, possibly elements of M₂₄. But M₂₄ has about a quarter of a billion elements, so either we're talking about certain special elements, or some very large equivalence classes...

Anyway, I don't know when I'll return to this. Mostly I just wanted to begin to understand how the paper is supposed to work. Usually the theme of this blog is about trying to find more sense than expected in a crazy idea. Here I'm instead analyzing the logic of a paper, ultimately in order to show its flaws - but also just to bring into the open how it's supposed to work. So perhaps this post will help other readers of the paper who want to understand where the prediction comes from, but who are lost among its peculiarities.

I should add that I was driven to look again at the paper by the always interesting Marni Sheppeard, who is taking it seriously.

Eight years of Higgs observations in one graph

Faster than nothing

"Mr Nothing", whose ideas were first discussed on this blog back in July, has released a paper on OPERA's superluminal neutrinos. He actually has an equation which fits OPERA, MINOS, and SN1987A, something which only a few other theorists can do. :-)

(The equation is on page 5, number 6.1. All the energy dependence is in the factor "ln(E/Eμ)".)

More amazing neutrino data

The question no-one is asking

What are the implications for physics if neutrinos are slower than light?

Inspiration strikes!

What is the situation? We have a functioning field theory, the standard model, in which there are over a dozen numbers that are just input parameters. We have a supposed derivation of a few of those parameters, in terms of quantities associated with the icosahedron. How are we to give these derivations causal significance in a more fundamental theory?

The basic idea is as follows. We have an icosahedron associated with each point in space. (The exact nature of the association doesn't matter at this stage.) Then, we have a field whose expectation value has a specific functional dependence on the properties of the icosahedron. Or rather, we have several such fields, each with its special functional relation. Then, we combine those fields so as to give rise to the standard model parameters. And we're done!

For example, consider the problematic relationship: dihedral angle equals sum of three physical parameters (let's overlook the peculiarity of mr nothing's "GUT angle" for now), plus another quantity. If you were "measuring" an icosahedron through, say, a TQFT, there's simply no reason why the TQFT would directly detect the existence of that decomposition.

So instead, we suppose there is a "dihedral angle field", with a VEV equal to the icosahedron's dihedral angle, and a "Weinberg angle field", which gets its VEV from, say, that DFQ angle (we may suppose that it's a slightly non-Euclidean icosahedron). And then we have a "Cabibbo angle field", whose VEV equals "d.a.f. VEV - W.a.f. VEV - 'GUT angle' - pi/3" - with this relationship being enforced, not by any property of the icosahedron, but simply by algebraic fiat. Voila, we now have the Cabibbo angle as an output from our "icosahedral theory of nothing", ready in turn to be an input to the set of relationships which reproduces the standard model.

Analysis of nothing, part 2

So, let's recapitulate. The dihedral angle of the icosahedron is supposed to be the sum of three angles of physical significance, plus 60 degrees. The exponential of the surface area of the icosahedron, divided by the mass of all leptons expressed in units of electron mass, allegedly equals ... another whimsical expression relying on the Weinberg angle. And, counting all the fermions and gauge bosons, we get 24, and this has something to do with SU(5) and the icosahedron.

Also, there are a bunch of alleged identities involving the Weinberg angle, which I have not yet bothered to check numerically, nor have I checked whether a simultaneous solution to these equations is even possible. Also, I suspect that the closeness of the "DFQ angle" mentioned in the previous post, and the experimentally measured value of the Weinberg angle, may subliminally be at work here. (Let me note in passing that if you embedded an icosahedron in a slightly non-Euclidean geometry, it should be possible to make the DFQ angle exactly equal to the measured Weinberg angle - at one's preferred energy scale, that is, since the value of the Weinberg angle flows.)

The attentive reader may recall that the premise of this blog was, that abstracts from the snarxiv could - if used in moderation - serve as genuine inspiration for physics. The current experiment is to see whether an authentic bit of numerological physics, found in the wild (the physics blogosphere), can similarly provide inspiration. So rather than proceed with the numerological analysis, for a moment I want to switch tracks and ask, in what sort of physical theory could the relationships listed at the start of this post actually exist and actually play a role in physical causation and explanation?

Since Weinberg angle, Cabibbo angle, Higgs VEV, etc., all acquire physical significance on account of the roles they play in a particular quantum field theory (the standard model), we are presumably looking for a beyond-standard-model theory which reduces to the standard model in some limit, and in which icosahedra matter. Perhaps there are compact dimensions shaped like icosahedra; perhaps there are icosahedral branes. Perhaps there are interaction vertices dual to icosahedra, or perhaps we calculate certain amplitudes by integrating over icosahedra.

See, this is the fun part: take the work of mr nothing, and try to hybridize it with the conventional apparatus of physical theory. But I must say that the "sum of angles" worries me. We are supposed to be using the icosahedron to explain the standard model, not vice versa. And while the dihedral angle is certainly a natural property of the icosahedron, the peculiar decomposition into a sum of four angles does not appear to be natural. Even supposing that an icosahedral structure appears at some level of our theory, why would the theory be sensitive to the existence of that decomposition of the dihedral angle? Unfortunately, the similarity of the DFQ angle and the Weinberg angle doesn't seem to help, because the DFQ angle isn't a natural part of the dihedral angle, so far as I can see. I shall need to meditate on the geometry of the icosahedron for a little while...