Circumnavigating the Great Loop of Scales or a Grand Tour of Physics (a hypothetical narrative)

Today I have decided to celebrate the third anniversary of the Higgs boson discovery and the 30 (35?) years of noncommutative geometry by musing on the famous "Glashow's snake" which symbolizes the eventual merging of empirical high energy particle physics with observational cosmology and following the Ariane thread woven with insights from the spectral noncommutative geometrization of physics. 

Summer heat oblige I will indulge myself with a watery metaphor.

The original Glashow's Snake: 

"The unity of the forces on the inside and 

the characteristic structure sizes on the snake, 

with the succession of scale size on the outside. 

The snake devours its tail, where the physics 

of very smallest (the Planck scale, 10^-30cm) 

is visible by peering back in time to

the outer reaches of the universe (10²⁵cm)."

Driven by the wind of spectral action...

Let's begin our navigation starting in the gravity sector, given a heading from macro to micro scales thus sailing clockwise on the Glashow's snake. Our caravel will be christened "the MC² theory" which is my nickname for the spectral noncommutative geometrization of physics (more explanation below). Here is an already old but still relevant and above all pedagogical review of this endeavour that fits nicely with the nautic metaphor chosen for this post:

Einstein was a passionate sailor. We speculate that this was no accident. The subtle harmony between geometries and forces becomes palpable to the sailor, he sees the curvature of the sail and feels the force that it produces. Before Einstein, it was generally admitted that forces are vector fields in an Euclidean space, R³, the scalar product being necessary to define work and energy. Einstein generalized Euclidean to Minkowskian and Riemannian geometry and we have two dreisätze or règles de trois. Take Coulomb’s static law for the electric field with coupling constant ε₀ and add Minkowskian geometry with its scale c, the speed of light: you obtain Maxwell’s theory. In particular, there appears the magnetic field with feeble coupling constant µ₀=1/(c²ε₀). Maxwell’s theory is celebrated today as Abelian or should we say, commutative Yang-Mills theory. The second dreisatz starts from Newton’s (static) universal law of gravitation, adds Riemannian geometry to obtain general relativity with new feeble, gravito-magnetic forces. 

Connes proposes two more dreisätze. Take a certain Yang-Mills theory with coupling constant g, coupled to a Dirac spinor of mass m. Add noncommutative geometry [1] with an energy scale Λ: you obtain a Yang-Mills-Higgs theory [2],[3]. The symmetry breaking scalar becomes a magnetic field of the Yang-Mills field and its mass and self-coupling λ are constrained in terms of g, m and Λ. His second dreisatz starts from general relativity, adds noncommutative geometry to obtain the Einstein-Hilbert action plus the Yang-Mills-Higgs action [4][5]. Now the Yang-Mills and the Higgs fields are magnetic fields of the gravitational field. Again there are constraints on λ, but they are different. 

Let us call noncommutative Yang-Mills the third and noncommutative relativity the fourth dreisatz. Note however that — unlike with supersymmetry — you cannot take any Yang-Mills theory and put ’noncommutative’ in front [6][7][8]. Note also that, behind noncommutative relativity, there stands a genuine noncommutative extension of Einstein’s principle of general relativity, the spectral {action} principle. One of the attractive features of noncommutative geometry is to unify gauge couplings with scalar self-couplings and Yukawa couplings. 

Noncommutative Yang-Mills and Noncommutative Relativity: A Bridge Over Trouble Water, Lionel Carminati, Bruno Iochum, Thomas Schucker (Marseille)

(Submitted on 13 Jun 1997)

Since the writing of this paper, some progress has been made in the noncommutative geometrization of the standard model or put it differently the improved spectral modelisation of spacetime at the very small scale. The most important one is probably the tentative quantization of spacetime through a new kind of Heisenberg-like relation due to A. Connes and Ali Chamseddine (it would probably deserve to be called a third dreisatz). As time is short and as a more thorough discussion of the possible physical consequences is still awaited, I will not dwell further on this subject. To tease the reader I will just add that the noncommutative Yang-Mills and relativity theories could be a bridge indeed over less dark matter thanks to the collaboration of Sacha Mukhanov who recently joined Chamseddine and Connes.

...tracking a new Higgs σ scalar

The standard model of particle physics with its now confirmed Brout-Englert-Higgs mechanism that has been checked up to the TeV scale can be completely computed from the spectral action principle which is a generalisation of the equivalence principle applied on an ordinary 4D Manifold dressed with a fine structure coordinatized by two Clifford algebras (it's hard then to resist the temptation to name this work by Connes Chamseddine and Mukhanov the MC² theory). Conceptually this new model of spacetime with "discrete" dimensions so to speak lead us from the electroweak sector to the Grand Unified Theory one. It should be more appropriate to talk about partial instead of grand unification as the most recent quantitative guess from noncommutative geometry posits the existence of a left-right symmetric Pati-Salam type model at an energy scale around 10¹⁴ GeV if one want to fit the mass of the Higgs boson. Such an extension of gauge symmetries requires a new Higgs-like scalar that naturally lives at the same high scale. Here is what can be said about it:

Connes’ non-commutative geometry (NCG) [1,2] is a generalization of Riemannian geometry which also provides a particularly apt framework for expressing and geometrically reinterpreting the action for the standard model of particle physics, coupled to Einstein gravity [3– 12] (for an introduction, see [13,14]). In a recent paper [15], we suggested a simple reformulation of the NCG framework, and pointed out three key advantages of this reformulation: (i) it unifies many of the traditional NCG axioms into a single, simpler axiom; (ii) it immediately yields a further generalization, from non-commutative to non-associative geometry [16]; and (iii) it resolves a key problem with the traditional NCG construction of the standard model, thereby making the NCG construction tighter and more explanatory than the traditional one based on effective field theory [17]. 

Here we report the discovery of three crucial and unexpected consequences of the reformulation in [15]. (i) First, it yields a new notion of the natural symmetry associated to any non-commutative space, and the action functional that lives on that space. (ii) Second, when we work out the realization of this symmetry for the non-commutative geometry used to describe the standard model of particle physics we find that the usual SU(3)_C×SU(2)_LU(1)_Y gauge symmetry is augmented by an extra U(1)_B-L factor. (iii) Third, as a consequence of this additional gauge symmetry, we find the standard model field content must be augmented by the following two fields: a U(1)_B-L gauge boson Cµ, and a single complex scalar field σ which is {a standard model} singlet and has charge B−L=2. 

The scalar field σ has important phenomenological implications. (i) First, although the traditional NCG construction of the standard model predicted an incorrect Higgs mass   (m_h≈170 GeV), several recent works [18–21] have explained that an additional real singlet scalar field σ can resolve this problem, and also restore the stability of the Higgs vacuum. Our σ field, although somewhat different (since it is complex, and charged under B−L), solves these same two problems for exactly the same reasons (as may be seen in the U(1)_B-L gauge where σ is real). (ii) Furthermore, precisely this field content (the standard model, extended by a right-handed neutrino in each generation of fermions, plus a U(1)_B-L  gauge boson Cµ, and a complex scalar field σ that is a singlet under SU(3)_C×SU(2)_LU(1)_Y but carries B−L=2) has been previously considered [22,23] because it provides a minimal extension of the standard model that can account for several cosmological phenomena that may not be accounted for by the standard model alone: namely, the existence of dark matter, the cosmological matter-antimatter asymmetry, and the scale invariant spectrum of primordial curvature perturbations.

Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry

Shane Farnsworth, Latham Boyle

(Submitted on 22 Aug 2014 (v1), last revised 14 Jan 2015 (this version, v2))

At any rate, we have here a clear advantage over grand unified theories which suffers of having arbitrary and complicated Higgs representations. In the noncommutative geometric setting, this problem is now solved by having minimal representations of the Higgs fields. Remarkably, we note that a very close model to the one deduced here is the one considered by Marshak and Mohapatra where the U (1) of the left-right model is identified with the B−L symmetry. They proposed the same Higgs fields that would result starting with a generic initial Dirac operator not satisfying the first order condition. Although the broken generators of the SU (4) gauge fields can mediate lepto-quark interactions leading to proton decay, it was shown that in all such types of models with partial unification, the proton is stable. In addition this type of model arises in the first phase of breaking of SO(10) to SU(2)_R×SU(2)_L×SU(4) and these have been extensively studied [1].

Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification

Ali H. Chamseddine, Alain Connes, Walter D. van Suijlekom

(Submitted on 30 Apr 2013 (v1), last revised 25 Sep 2014 (this version, v4))

We can now envision a comprehensive picture of physics from macroscale to microscale. Beyond the electroweak scale one would expect to recover a left-right symmetry of interactions between chiral subatomic particles. A minimal amount of new particles is required consisting of right-handed W_R and Z_R gauge bosons and three Majorana N_R plus a new Higgs-like scalar boson σ. The theory does not precisely predict their masses and it could be that all of them would remain inaccessible to any man-made collider experiment. But now we can take advantage of the seesaw mechanism - a theoretical mechanism naturally implemented in the partial unification model - that helps to check experimentally but indirectly the physics of inaccessibly small scales with accessible left-handed neutrino physics for instance. More on this another day, now let's consider another indirect observational test of the ultra small (10^-25  to 10^-30 cm): namely cosmology. 

... course to steer : seesaw inflation?

We are focusing now on the last and most important part of the Glashow's snake where tail and head literally meet each other. It is quite enjoyable a theoretical mechanism called seesaw might help to merge ultra small scale speculations with possible observational phenomena at the cosmological scale. 

... a new scalar field can play the role of inflaton in the presence of a non-minimal gravitational coupling. For example, one may introduce a SM singlet scalar to drive inflation and yield the inflationary predictions consistent with the observations [12a,12b,13], with a lower bound r>0.002 for n_s≥0.96 when possible quantum corrections are taken into account [13]. This scalar may be identified as a B-L Higgs field in the minimal B-L model [14]. Furthermore, the Higgs portal scalar dark matter can play the role of inflaton, leading to a unification of inflaton and dark matter particle [15a,15b]. For a scenario relating inflation, seesaw physics and Majoron dark matter, see Ref. [16].

Higgs Inflation, Seesaw Physics and Fermion Dark Matter

Nobuchika Okada, Qaisar Shafi

(Submitted on 22 Jan 2015)

A recent twist along these lines was the proposal that inflation and dark matter have a common origin (similar idea was suggested by Smoot in arXiv:1405.2776 [astro-ph]), with the inflaton identified to the real part of the complex singlet containing the majoron and breaking lepton number through its vev [30]. The resulting inflationary scenario is consistent with the recent CMB observations, including the B-mode observation by the BICEP2 experiment re-analized jointly with the Planck data, as illustrated in the {figure below}. The upper (red) contours correspond to the BICEP2 results, while the lower ones (green) follow from the new analysis released jointly with PLANCK [34]. The lines correspond to 68 and 95% CL contours. Further restrictions on the majoron dark matter scenario should follow from structure formation considerations

Status and implications of neutrino masses: a brief panorama

J. W. F. Valle

(Submitted on 8 Apr 2015)

We have a natural See-Saw Inflationary scenario based upon having heavy (10¹⁴ GeV) right-handed neutrinos to explain the observed light left-handed neutrinos. There needs to be a scalar field to produce the heavy right-handed neutrino mass and it is a natural source for inflation - See-Saw Inflation. 

This leads to plausible and generally physically possible though perhaps finetuned mechanisms to tie the neutrino sector to the four major fundamental issues in cosmology: Inflation, Dark Matter, Dark Energy, and Baryogenesis.  

The weakest argument presented here is for Dark Energy. It is not unreasonable to find that this inflation gets back into action when it is perturbed by the later symmetry breaking allowing the left-handed neutrinos to gain a very low Majorana mass, and then make a very slow roll and reasonably rapid decay to the new minimum producing the apparent Dark Energy accelerating the universe. Thus the seesaw mechanism completes its work. 

In one simple incarnation there are three right-handed neutrinos and related fields that correspond to energy levels of two GUT symmetry breakings, e.g. SO(10), and the last big Inflation. The lightest right-handed neutrino acting with its field produces the last high-scale inflation period and then the lightest left-handed neutrino acting with its field produces the late-time accelerating universe.  

It is interesting to note that the lowly left-handed neutrino and its high-borne right-handed neutrino partner appear to have a big role in the destiny of the universe and that measurements of the neutrino properties reflect on parameters both at the GUT scale and at the lowest energy scale. In particular, it is important to determine: 

• 1) Are these Majorana neutrinos? So I say to my Cuore colleagues go to it, as I know have a personal interest beyond being involved via (former) graduate student Michele Dolinski, post doc Tom Gutier in Cuoricino and my colleagues in Berkeley. Let’s see some good neutrinoless double-beta decay. 

• 2) Measuring the neutrino mass spectrum as these feed directly into the fits for the right-handed neutrino mass and the inflation potential. There will undoubtedly be joint fits between the neutrino data and the large scale structure and CMB data to make a global fit to the right-hand neutrino mass and the inflation self coupling. 

The large scale structure observations - e.g. galaxy and quasar-Lyman alpha surveys - may have interesting things to say not only about levels and coefficients but also about any structure in the potential or any splitting or right-hand neutrino masses. Theorists have their work cut out in continuing the resurrection of SO(10) and making a more coherent and exhaustive treatment of the neutrino and scalar sectors and the links to observables.

See Saw Inflation / Dark Matter / Dark Energy / Baryogenesis

George F. Smoot

(Submitted on 12 May 2014 (v1), last revised 19 May 2014 (this version, v2))

Even if this last article by the Nobel prize laureate astrophysicist G. Smoot is yet a bricolage as he says it himself it provides fascinating expectations sketching a physical landscape not so different from the noncommutative geometric perspective...

I leave the mimetic dark matter sector discussion for another day...

 The Great Loop of Scales or Grand Tour of Physics  

a possible extension of the Glashow's snake or Ouroboros

(L-R SYM stands for left-right symmetry and NCG for NonCommutative Geometry)

//last edit 25 August 2015 19 May 2019 : "grand Tour" added to make an explicit connection to the historical grand unification theories from the past inspired by Glashow in particular and a wink to the Grand Tour of Europe from the past XVIII & XIX centuries.