### 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 10^{13}and 10^{17 }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 ofm_{H}is not much sensible on the choice of the uniﬁcation scale...

The recent discovery of the Higgs boson with a mass m(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,_{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 stablethe fact thatm_{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 10^{12}and 10^{17}Gev, there exists a realistic value r≃1 which brings back the computed value ofm_{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 massk_{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.

**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 ofC^{∞}(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⊕M_{3}(C) ...

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

A_{G}:= M_{4}(H)⊕M_{8}(C) ...

one generates the ﬁeld σ by a inner ﬂuctuation which respects the ﬁrst-order condition imposed by the partD_{v}of the Dirac operator that contains the Majorana massk_{R}[20]. The breaking toA_{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]_{ρ}JbJ^{-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 [thetwisted 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 algebraA_{G}containing Asm together with an automorphism ρ ofC^{∞}(M)⊗B such that

*i) (**C*^{∞}*(M)**⊗B*,H,*\(\begin{equation} D\!\!\!/ \end{equation}\)**+D*; ρ) is a twisted spectral triple satisfying the twisted 1st-order condition (1.8);_{v}*ii) a twisted ﬂuctuation of**\(\begin{equation} D\!\!\!/ \end{equation}\)**+D*by B generates an extra scalar ﬁeld σ, together with an additional vector ﬁeld Xµ;_{v}*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*, neglecting the interaction term between σ and Xµ. ...*\(\begin{equation} D\!\!\!/ \end{equation}\)**+D*_{v}

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 mentiona very recent work of Chamseddine, Connes and Mukhanov [8] where the algebraA_{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.

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