# Is the 125 GeV Higgs boson some kind of a “almost-commutative graviton” at the electroweak scale?

The clumsy "almost-commutative graviton" is provocative. I use it on purpose, to ask two questions in one :

• Is the observation of only one Higgs and no supersymmetric particle below 8 TeV (up to now) a sufficient evidence to substantiate the almost commutative spectral model?

• Can physicists consider now this kind of models pioneered by Connes and Chamseddine to be an effective (physical) and not only formal tentative unification of gravitation and Yang-Mill-Higgs interactions?

Recent developments of the almost-commutative spectral model regarding the Higgs boson and its mass:

(motives for "graviton" as a metaphore and "almost commutative" as a pedagogical reminder)

• I know that graviton is a spin 2 gauge boson associated to the gravitational field in a tentative quantification of general relativity. In the framework of Quantum Field Theory it is thus an object independant a priori from the Higgs that is a scalar boson responsible for masses of elementary particles from the Standard Model. Nevertheless I remind that Higgs interaction can be considered as derivated from gravitation in the noncommutative geometric setting (following Thomas Schücker).
• The adjective almost-commutative has a precise technical meaning but I use it also in my question to underline the fact that in any theoretical framework non-commutativity is a necessary but not sufficient tool to describe quantum phenomena, therefore it is clear that gravitation has not been quantized yet!
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It is hard figuring out what your actual question is. –  Peter Shor May 20 at 9:56
@PeterShor the question is clear, it is in the title of the question ;-). –  Dilaton May 20 at 10:13
Dear M. Shor, I apologize for the question not clear enough. I have tried to edit it in a better way but it keeps a fuzzy epistemological (rather than a sharp technical) character. Anyway, thank you to take part in physics.stackexchange! Au plaisir de vous lire et de découvrir vos contributions à la quantique. –  laboussoleestmonpays May 23 at 9:48
So your proposition is that the lack of beyond-standard-model physics at the LHC, is evidence for Connes et al's program of noncommutative physics? –  Mitchell Porter May 26 at 8:38
Thank you for this rephrasing Mitchell, it fits pretty well with what I have in mind and helped me to edit my question. –  laboussoleestmonpays May 28 at 8:05

Is the observation of only one Higgs and no supersymmetric particle below 8 TeV (up to now) a sufficient evidence to substantiate the almost commutative spectral model?

No, absolutely not. The Chamseddine-Connes model assumes the existence of a desert, with no new field excitations or strong coupling phenomena between the 1 TeV scale and the GUT scale, roughly $10^{16}$ TeV. This is a very strong assumption (although certainly not unknown in particle physics): They are assuming that there is no new physics across 16 orders of magnitude. (For reference, this is roughly the same separation of scales that separates the 1 cm scale of a large drop of water and the current limits of particle accelerators.)

Frankly, I think the desert hypothesis is a far stronger assumption than any of the other assumptions (what fields, what couplings, what Calabi-Yau, what non-commutative geometry) people make when speculating about physics beyond the Standard Model.

None of this is meant to discourage work on these NC models. I personally quite like the smell of them. But it should be remembered that the entire history of particle physics (from Newtonian mechanics to fluid dynamics to radio waves to molecular chemistry to QED to nuclear physics to quarks and gluons and the electroweak scale) covers a smaller gap of scales.

Can physicists consider now this kind of models pioneered by Connes and Chamseddine to be an effective (physical) and not only formal tentative unification of gravitation and Yang-Mill-Higgs interactions?

Again, no. To make the 'almost commutative' model a real model of gravity (instead of an intriguing way of expressing a short distance classical action for an effective field theory of the non-gravitational degrees of freedom), one must explain how to carry out path integral computations over the space of Dirac operators. This will entail explaining how to integrate over the gravitational degrees of freedom, and isn't likely to be much easier than any other approaches to quantum gravity.

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Thank you for this useful answer. I want to stress the recent developments in the Chamseddine-Connes model that I refer to in my question do not rely on the big desert hypothesis any more. To get a correct postdiction of the Higgs mass the model relies on the existence of a coupling with one new neutral singlet scalar field. And quite interestingly this new scalar field is not completely ad hoc : it could also be responsible for the existence of heavy Majorana right-handed neutrinos responsible for neutrino masses and mixing and fit well in some new scenarios of grand unification ... –  laboussoleestmonpays Jun 28 at 9:13
I complete the last comment with a quote from (arxiv.org/pdf/1304.8050v2.pdf) "One indication that there must be a new higher scale that effects the low energy sector is the small mass of the neutrinos which is explained through the see-saw mechanism with a Majorana mass of at least of the order of 10^11Gev. In addition and as noted above, a scalar field which acquires a vev generating that mass scale can stabilize the Higgs coupling and prevent it from becoming negative at higher energies and thus make it consistent with the low Higgs mass of 126 Gev." –  laboussoleestmonpays Jun 28 at 9:17
@laboussoleestmonpays I'm familiar with that paper. It's misleading to claim that they are no longer making a desert hypothesis. The only difference is that they are now considering a scenario where there is an oasis on the far side of the desert. –  user1504 Jun 28 at 13:50

2 remarks :

1) The original theory of Connes made a prediction for the mass of the Higgs boson which was $170$ Gev, if I'm not mistaken. And it has been ruled out.

Then there was a new 2012 article, and it is said that the experimental value of the mass of the Higgs is now compatible with the model.

I do not understand a word in the maths of the model, but at least, it does not appear like extraordinary trustable, to change so quickly (but maybe I am wrong).

2) Higgs boson is a scalar particle, and graviton is a spin-2 particle, so they are very different.

Apart from this, the vaccuum expectation of the Higgs scalar gives the mass to the $W,Z$ bosons, but also to the other particles.

But there is a difference between giving a mass, and being a (like-gauge) particle representing gravitational interaction between masses (in fact, stress-energy tensor). This is conceptually different.

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Dear @Trimok I understand the Higgs from the Standard Model is different from the graviton as a like-gauge particle carrying gravitational interaction in a Minkowski space-time. This is why I have added in my question the comment "Higgs interaction can be considered as derivated from gravitation in the noncommutative geometric setting" (BTW the article by Thomas Schücker I quote is pretty pedagogical, at least it helped me to understand a few words in the physics of the model ... ;-) – –  laboussoleestmonpays Jul 3 at 20:49

Let's focus first on the almost-commutative aspect. One could argue that a possibly better formulation for the 126 GeV Higgs scalar could be almost-commutative gauge boson to underline the fact that its phenomenology at the LHC agrees up to now quite well with the coupling terms and quantum numbers postulated by the Standard Model but mostly calculated in the almost-commutative spectral model from first principles. Among them are these two, quoted by Robert Brout in some Notes on Connes' Construction of the Standard Model from 1997:

I emphasize strongly the gauge principle and its handmaiden, anomaly freedom. One of the more remarkable things that has come up as the uncanny consistency of Connes’ approach with the constraints of anomaly freedom.

To say something about the gravitational aspect, one also has to keep in mind that the spectral action principle is a conjectural generalization of the equivalence principle, said it differently, it is General Relativity formulated with noncommutative geometry thus there could be a genuine connection of the scalar (spin 0) Higgs with gravitation in the noncommutative framework, even if it has nothing to do with the (spin 2) graviton envisioned by the non perturbatively renormalizable quantum field theory of gravitation.

This last speculation is also inspired by reading this article(1995) by Fedele Lizzi et al that dwells on a possible dual role played by the nondiagonal elements of the matrix algebra responsible for the noncommutative character of the fine structure of spacetime in the Connes vision. I quote :

Loosely speaking, this formulation is based on a doubling of space-time, considered as a two sheeted manifold... The nontrivial feature of the theory is that the Dirac operator, as well as the gauge potential (connection), have some nondiagonal elements, which couple the two sheets of space-time. These are classical scalar fields: one is related to the component of the metric in the discrete direction, and thus to the distance between the two sheets of space-time, and the others are Higgs fields, responsible for the breaking of the symmetry.

Remark #1 The question formulated by Mitchell Porter in a comment:

the lack of beyond-standard-model physics at the LHC, is evidence for Connes et al's program of noncommutative physics?

and the possible existence of the two-sheeted space-time are addressed more specifically here.

Remark #2 A possible continuation of my own question, in an attempt to deepen the understanding of the dual aspect of the noncommutative Higgs sector, making connection with more recent work, has been proposed here.

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