Neutrino mixing : a small step beyond the Standard Model and a giant leap for geometry of spacetime?
The present paper shows that the modification of the standard model required by the phenomenon of neutrino mixing in fact resulted in several improvements on the previous descriptions of the standard model via noncommutative geometry. In summary we have shown that the intricate Lagrangian of the standard model coupled with gravity can be obtained from a very simple modification of space-time geometry provided one uses the formalism of noncommutative geometry. The model contains several predictions and the corresponding section 5 of the paper can be read directly, skipping the previous sections.
(Submitted on 23 Oct 2006)
In our first approach  to the understanding of the Lagrangian of the Standard Model coupled to gravity, we used the above new paradigm of spectral geometry to model space-time as a product of an ordinary 4-manifold (we work after Wick rotation in the Euclidean signature) by a finite geometry F. This finite geometry was taken from the phenomenology i.e. put by hand to obtain the Standard Model Lagrangian using the spectral action. The algebra AF , the Hilbert space HF and the operator DF for the finite geometry F were all taken from the experimental data. The algebra comes from the gauge group, the Hilbert space has as a basis the list of elementary fermions and the operator is the Yukawa coupling matrix. This worked fine for the minimal Standard Model, but there was a problem  of doubling the number of Fermions, and also the Kamiokande experiments on solar neutrinos showed around 1998 that, because of neutrino oscillations, one needed a modification of the Standard Model incorporating in the leptonic sector of the model the same type of mixing matrix already present in the quark sector. One further needed to incorporate a subtle mechanism, called the see-saw mechanism, that could explain why the observed masses of the neutrinos would be so small. At first our reaction to this modification of the Standard Model was that it would certainly not fit with the noncommutative geometry framework and hence that the previous agreement with noncommutative geometry was a mere coincidence. After about 8 years it was shown in  and  that the only needed change (besides incorporating a right handed neutrino per generation) was to make a very simple change of sign in the grading for the anti-particle sector of the model (this was also done independently in ). This not only delivered naturally the neutrino mixing, but also gave the see-saw mechanism and settled the above Fermion doubling problem. The main new feature that emerges is that when looking at the above table of signs giving the KO-dimension, one finds that the finite noncommutative geometry F is now of dimension 6 modulo 8. Of course the space F being finite, its metric dimension is 0 and its inverse line-element is bounded. In fact this is not the first time that spaces of this nature— i.e. whose metric dimension is not the same as the KO-dimension— appear in noncommutative geometry and this phenomenon had already appeared for quantum groups and related homogeneous spaces .
... since we want the finite geometry F to be of KO dimension 6, we are left only with the second case and we obtain among the very few choices of lowest dimension the case AF = M2(ℍ)⊕M4(ℂ) where H is the skew field of quaternions...
We can now describe the predictions obtained by comparing the spectral model with the standard model coupled to gravity. The status of “predictions” in the above spectral model is based on two hypothesis:
(1) The model holds at unification scale
(2) One neglects the new physics up to unification scale.
The spectrum of the fermionic particles, which is the number of states in the Hilbert space per family is predicted to be 42 = 16 which is a consequence of the algebra of the discrete space being M2(ℍ)⊕M4(ℂ). In addition the surviving algebra consistent with the axioms of noncommutative geometry, in particular the order one condition, is given by ℂ⊕ℍ⊕M3(ℂ) which gives rise to the gauge group of the standard model. A consequence of this is that the 16 spinors get the correct quantum number with respect to the standard model gauge group which follows the decomposition:
(4, 4) → (1R+1'R+2L, 1+3) = (1R, 1) + (1'R, 1) + (2L, 1) + (1R, 3) + (1'R, 3) + (2L, 3)
These spinors correspond to νR, eR, lL, uR, dL, qL respectively, where lL is the left-handed neutrino-electron doublet and qL is the left-handed up-down quark doublet. In addition to the gauge bosons of SU(3)×SU(2)×U(1) which are the inner fluctuations of the metric along continuous directions, we also have a Higgs doublet which correspond to the inner fluctuations of the metric along the discrete directions. What is peculiar about this Higgs doublet, is that its mass term as determined from the spectral action comes with a negative sign and a quartic term with a plus sign, thus predicting the phenomena of spontaneous breakdown of the electroweak symmetry
If you think K-theory is useless for physics think again and have a look at the recent developments in the field of topological insulators and superconductors and their periodic table (the Physics Nobel Prize 2016 for David Thouless is my guess for next year).
As far as the phenomenologists are concerned I guess the following skeptical blog post gives an idea of the consensus:
Is it possible that [Connes'] approach will provide new insights into the Standard Model and beyond? Not likely. As far as I understood, the fine structure of space-time has no implications that could be observed at the LHC or in other experiments in foreseeable future. Next, the Standard Model is not a unique system that allows for such a geometrical embedding.
Alain Connes' Standard Model, Jester Sunday 11 February 2007
Echo at the electroweak scale?
... the Higgs-Brout-Englert boson has been discovered with a mass around 125 GeV. This mass is problematic, or at least intriguing, because it lies just below the threshold of stability, meaning that electroweak vacuum is a metastable state rather than a stable one. One solution to stabilize the electroweak vacuum is to postulate there exists another scalar field, called σ, suitably coupled to the Higgs. Chamseddine and Connes have noticed in  that taking into account this new scalar field in the spectral action, by promoting the Yukawa coupling of the right neutrino (which is one of the constant component of the matrix DF ) to a field,
kR → kR σ, (13)
then one obtains the correct coupling to the Higgs as well as a way to pull back the mass of the Higgs from 170 to 126GeV. In [10, 15], ... its is shown how the substitution (13) can be obtained as a fluctuation of the Dirac operator, but in a slightly modified version inspired by the notion of twisted spectral triple introduced previously by Connes and Moscovici. The field σ thus appears as a Higgs-like field associated to a spontaneous symmetry breaking to the standard model of a “grand symmetry” model where the spin degrees of freedom (C∞(M) acting on the space of spinors) are mixed with the internal degrees of freedom (AF acting on the space of particles)
Beyond the Standard Model with noncommutative geometry, strolling towards quantum gravity Pierre Martinetti 2015