One or two things I know about a (spectral noncommutative) dynamical symmetry breaking of a Pati-Salam symmetry down to the Standard Model one

A nice recap

In a recent article submitted on 5 Nov 2014 on arxiv and entitled : Twisted spectral triple for the Standard Model and spontaneous breaking of the Grand Symmetry, Agostino Devastato and Pierre Martinetti have written a nice (clear,  explicit) summary of the state of the art of the spectral non-commutative geometric based physics beyond the Standard Model in general and report also on important progress in their own work :

Noncommutative geometry [NCG] provides a description of the standard model of elementary particles [SM] in which the mass of the Higgs −at unification scale Λ−is a function of the other parameters of the theory, especially the Yukawa coupling of fermions [7]. Assuming there is no new physics between the electroweak and the unification scales (the “big desert hypothesis”), the flow of this mass under the renormalization group yields a prediction for the Higgs observable mass m_H. It is well known that in the absence of new physics the three constants of interaction fail to meet at a single unification scale, but form a triangle which lays between 10¹³ and 10¹⁷ GeV. The situation can be improved by taking into account higher order term in the NCG action [19], or gravitational effects [18]. Nevertheless, the prediction of m_H is not much sensible on the choice of the unification scale... 

The recent discovery of the Higgs boson with a mass m_H≃126 Gev suggests the big desert hypothesis should be questioned. There is indeed an instability in the electroweak vacuum which is meta-stable rather than stable (see [3] for the most recent update). There does not seem to be a consensus in the community whether this is an important problem or not: on the one hand the mean time of this meta-stable state is longer than the age of the universe, on the other hand in some cosmological scenario the meta-stabililty may be problematic [23, 24]. Still, the fact that m_H is almost at the boundary value between the stable and meta-stable phases of the electroweak vacuum suggests that “something may be going on”. In particular, particle physicists have shown how a new scalar field suitably coupled to the Higgs - usually denoted σ - can cure the instability (e.g. [11, 22]) 

Taking into account this extra field in the NCG description of the SM induces a modification of the flow of the Higgs mass, governed by the parameter r=k_ν/k_t , which is the ratio of the Dirac mass of the neutrino and of the Yukawa coupling of the quark top. Remarkably, for any value of Λ between 10¹² and 10¹⁷ Gev, there exists a realistic value r≃1 which brings back the computed value of m_H to 126 Gev [6]. 

The question is then to generate the extra field σ in agreement with the tools of noncommutative geometry. Early attempts in this direction have been done in [29], but they require the adjunction of new fermions (see [30] for a recent state of the art). In [6], a scalar σ correctly coupled to the Higgs is obtained without touching the fermionic content of the model, simply by turning the Majorana mass k_R of the neutrino into a field k_R → k_R σ... Usually the bosonic fields in NCG are generated by inner fluctuations of the geometry. However this does not work for the field σ because of the first-order condition [[D,a],JbJ ^-1] = 0 ∀a,b ∈A ... where A and D are the algebra and the Dirac operator of the spectral triple of the standard model, and J the real structure. 

In [9, 10] it was shown how to obtain σ by a inner fluctuation that does not satisfy the first-order condition, but in such a way that the latter is retrieved dynamically, as a minimum of the spectral action. The field σ is then interpreted as an excitation around this minimum. 

A new twist(ted spectral triple) 

... in [20] another way had been investigated to generate σ in agreement with the first-order condition, taking advantage of the fermion doubling in the Hilbert space H of the spectral triple of the SM [26, 27, 28].  

More specifically, under natural assumptions on the representation of the algebra and an ad-hoc symplectic hypothesis, it is shown in [5] that the algebra in the spectral triple of the SM should be a sub-algebra of C^∞(M)⊗A_F, where M is a Riemannian compact spin manifold (usually of dimension 4) while 

 A_F=M_a(H)⊕M_2a(C)    a∈N  ... 

The algebra of the standard model  

A_sm:=C⊕H⊕M₃(C)  ... 

is obtained from A_F for a=2, by the grading and the first-order conditions. Starting instead with the “grand algebra” (a=4) 

 A_G:= M₄(H)⊕M₈(C)  ... 

one generates the field σ by a inner fluctuation which respects the first-order condition imposed by the part D_v of the Dirac operator that contains the Majorana mass k_R [20].  The breaking to A_sm is then obtained by the first-order condition imposed by the free Dirac operator \(\begin{equation} D\!\!\!/ \end{equation}\):= \(\begin{equation} \partial\!\!\!/ \end{equation}\)⊗ I.  

Unfortunately, before this breaking not only is the first-order condition not satisfied, but the commutator 

[\(\begin{equation} D\!\!\!/ \end{equation}\),A] A∈C^∞(M)⊗A_G  ... 

is never bounded. This is problematic both for physics, because the connection 1-form containing the gauge bosons is unbounded; and from a mathematical point of view, because the construction of a Fredholm module over A and Hochschild character cocycle depends on the boundedness of the [later] commutator...  

In this paper, we solve this problem by using instead a twisted spectral triple (A,H,D,ρ) [14]. Rather than requiring the boundedness of the commutator, one asks that there exists a automorphism ρ of A such that the twisted commutator 

[D,a]_ρ  := Da−ρ(a)D  ... 

is bounded for any a ∈A. Accordingly, we introduce... a twisted first-order condition 

[[D,a]ρ,JbJ^-1]_ρ :=[D,a]_ρJbJ^-1 −Jρ(b)J^-1[D,a]_ρ=0 ∀a,b ∈A  ... 

We then show that a for a suitable choice of a subalgebra B of C^∞(M)⊗A_G, a twisted fluctuation of \(\begin{equation} D\!\!\!/ \end{equation}\)+D_v that satisfies [the twisted first-order condition] generates a field σ - slightly different from the one of [6] - together with an additional vector field X_µ. 

Furthermore, the breaking to the standard model is now spontaneous, as conjectured by Lizzi in [20]. Namely the reduction of the grand algebra A_G to A_sm is obtained dynamically, as a minimum of the spectral action. The scalar and the vector fields then play a role similar as the one of the Higgs in the electroweak symmetry breaking.  

Mathematically, twists make sense as explained in [14], for the Chern character of finitely summable spectral triples extends to the twisted case, and lands in ordinary (untwisted) cyclic cohomology. Twisted spectral triples have been introduced to deal with type III examples, such as those arising from transverse geometry of codimension one foliation. It is quite surprising that the same tool allows a rigorous implementation in NCG of the idea of a “bigger symmetry beyond the SM”. 

The main results of the paper are summarized in the following theorem. 

Theorem 1.1. Let H be the Hilbert space of the standard model described in §2.1. There exists a sub-algebra B of the grand algebra A_G containing Asm together with an automorphism ρ of C^∞(M)⊗B such that 

• i) (C^∞(M)⊗B,H, \(\begin{equation} D\!\!\!/ \end{equation}\)+D_v; ρ) is a twisted spectral triple satisfying the twisted 1st-order condition (1.8); 
• ii) a twisted fluctuation of \(\begin{equation} D\!\!\!/ \end{equation}\)+D_v by B generates an extra scalar field σ, together with an additional vector field Xµ; 
• iii) the spectral triple of the standard model is obtained as the minimum of the spectral action induced by a twisted fluctuation of / D. The same result is obtained from a twisted fluctuation of \(\begin{equation} D\!\!\!/ \end{equation}\)+D_v, neglecting the interaction term between σ and Xµ. ...

Starting with the “not so grand algebra” B, one builds a twisted spectral triple whose fluctuations generate both an extra scalar field σ and an additional vector field X_µ. This is a Pati-Salam like model - the unitary of B yields both an SU(2)_R and an SU(2)_L - but in a pre-geometric phase since the Lorentz symmetry (in our case: the Euclidean SO(n) symmetry) is not explicit. The spectral action spontaneously breaks this model to the standard model, with both a scalar and a vector field playing a role similar as the one of Higgs field. We thus have a dynamical model of emergent geometry.

The last word

Finally, let us mention a very recent work of Chamseddine, Connes and Mukhanov [8] where the algebra A_F for a=2 is obtained without the ad-hoc symplectic hypothesis, but from an higher degree Heisenberg relation for the space-time coordinates. It would be interesting to understand whether the case a=4 enters this framework.