### A 40 year-old complex quest looking for a principle of minimality

**The past of SO(10) grand unification quest**

Blue: number of papers per year with the keyword "SO(10)" in the title as a function of the years. Red: subset of papers with the keyword "supersymmetry" either in the title or in the abstract. (Source: inSPIRE)

By looking at the plot above it is possible to reconstruct the following historical phases:• 1974 ÷ 1986: Golden age of grand uniﬁcation. These are the years of the foundation in which the fundamental aspects of the theory are worked out. The ﬁrst estimate of the proton lifetime yields τp ∼ 10^{31}yr [37], amazingly close to the experimental bound τp > 10^{30}yr [38]. Hence the great hope that proton decay is behind the corner.

• 1987 ÷ 1990: Great depression. Neither proton decay nor magnetic monopoles are observed so far. Emblematically the last workshop on grand uniﬁcation is held in 1989 [39].

•> 1991: SUSY-GUTs. The new data of the Large Electron-Positron collider (LEP) seem to favor low-energy supersymmetry as a candidate for gauge coupling uniﬁcation. From now on almost all the attention is caught by supersymmetry

•> 1998: Neutrino revolution. Starting from 1998 experiments begin to show that atmospheric [40] and solar [41] neutrinos change ﬂavor. SO(10) comes back with a rationale for the origin of the sub-eV neutrino mass scale.

•> 2010: LHC era. Has supersymmetry something to do with the electroweak scale? The lack of evidence for supersymmetry at the LHC would undermine SUSY-GUT scenarios. Back to nonsupersymmetric GUTs?

•> 2019: Next generation of proton decay experiments sensitive to τp∼10^{34-35}yr [42]. The future of grand uniﬁcation relies heavily on that.

Luca Di Luzio (Submitted on 14 Oct 2011)

**A stumbling block for future development**

Despite the huge amount of work done so far, the situation does not seem very clear at the moment. Especially from a theoretical point of view no model of grand uniﬁcation emerged as "the" theory. The reason can be clearly attributed to the lack of experimental evidence on proton decay.

In such a situation a good guiding principle in order to discriminate among models and eventually falsify them is given by minimality, where minimality deals interchangeably with simplicity, tractability and predictivity. It goes without saying that minimality could have nothing to do with our world, but it is anyway the best we can do at the moment.It is enough to say that if one wants to have under control all the aspects of the theory the degree of complexity of some minimal GUT is already at the edge of the tractability.Quite surprisinglyafter 37 years there is still no consensus on which is the minimal theory. Maybe the reason is also that minimality is not a universal and uniquely deﬁned concept, admitting a number of interpretations. For instance it can be understood as a mere simplicity related to the minimum rank of the gauge group... if we stick to the SO(10) case,minimality is closely related to the complexity of the symmetry breaking sector. Usually this is the most challenging and arbitrary aspect of grand uniﬁed models.While the SM matter nicely ﬁt in three SO(10) spinorial families, this synthetic feature has no counterpart in the Higgs sector where higher dimensional representations are usually needed in order to spontaneously break the enhanced gauge symmetry down to the SM.Establishing the minimal Higgs content needed for the GUT breaking is a basic question which has been addressed since the early days of the GUT program. Let us stress that the quest for the simplest Higgs sector is driven not only by aesthetic criteria but it is also a phenomenologically relevant issue related to the tractability and the predictivity of the models.Indeed, the details of the symmetry breaking pattern, sometimes overlooked in the phenomenological analysis, give further constraints on the low-energy observables such as the proton decay and the effective SM ﬂavor structure. For instance in order to assess quantitatively the constraints imposed by gauge coupling uniﬁcation on the mass of the lepto-quarks responsible for proton decay it is crucial to have the scalar spectrum under control. Even in that case some degree of arbitrariness can still persist due to the fact that the spectrum can never be ﬁxed completely but lives on a manifold deﬁned by the vacuum conditions. This also means thatif we aim to a falsiﬁable (predictive) GUT scenario, better we start by considering a minimal Higgs sector.

**A possible noncommutative breakthrough with geometric unification**

The freedom in the choice of the gauge group and the fermionic representations have led to any attempts to unify all the gauge interactions in one group, and the fermions in one irreducible representation. The most notable among the uniﬁcation schemes are models based on the SO (10) gauge group and groups containing it such as E6, E7 and E8. The most attractive feature of SO (10) is that all the fermions in one family ﬁt into the 16 spinor representation and the above delicate hypercharge assignments result naturally after the breakdown of symmetry. However, what is gained in the simplicity of the spinor representation and the uniﬁcation of the three gauge coupling constants into one SO (10) gauge coupling is lost in the complexity of the Higgs sector. To break the SO (10) symmetry into SU(3)c×U(1)em one needs to employ many Higgs ﬁelds in representations such as 10, 120, 126 [1]. The arbitrariness in the Higgs sector reduces the predictivity of all these models and introduces many arbitrary parameters, in addition to the unobserved proton decay...

[The noncommutative geometric] approach predicts a unique fermionic representation of dimension 16[with gauge couplings uniﬁcation in a Pati-Salam type model at a high energy somewhere between 6.5×10^{12}−1.4×10^{17}GeV]...The main advantage of our approach over the grand uniﬁcation approach is that the reduction to the Standard Model gauge group is not due to plethora of Higgs ﬁelds[but to a minimal scalar spectrum (the Higgs fields are fixed andbelong to the 16 × 16 and 16 × 16bar products with respect to the Pati-Salamgroup) with a specific dynamics which are outputs from noncommutative geometry axioms and the spectral action principle with the input of the fermion spectrum]... The spectral action is the pure gravitational sector of the noncommutative space. This is similar in spirit to the Kaluza-Klein approach, but with the advantage of having a ﬁnite spectrum, and not the inﬁnite tower of states. Thus the noncommutative geometric approach manages to combine the advantages of both grand uniﬁcation and Kaluza Klein without paying the price of introducing many unwanted states.

We still have few delicate points which require further understanding... [one] is to determine the number of generations. From the physics point, because of CP violation, we know that we need to take N ≥ 3, but there is no corresponding convincing mathematical principle.

We would like to stress that the spectral actionof the standard model comes out almost uniquely, predicting the number of fermions, their representations and the Higgs breaking mechanisms, with very little input.[It]predicts a real singlet scalar field beyond the minimal Standard Model whose large vev sets the scale of right handed Majorana neutrino mass providing a natural see-saw mechanism. This new scalar also gets mixed in a non-trivial way with the Higgs ﬁeld of the Standard Model so that the renormalization group equations of the combined Higgs-singlet system solves the stabilization problem faced with a light Higgs ﬁeld of the order of 125 GeV avoiding making the Higgs quartic coupling negative at very high energies. Remarkably, the form of the Higgs-singlet potential derived before from the spectral action[2]agrees with the one proposed recently by different groups[3],[4],[5],[6]. The quartic couplings are determined at uniﬁcation scale in terms of the gauge and Yukawa couplings. Running these relations down with the scale, give values consistent with the present data for the Higgs and top quark mass.

Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model, Ali H. Chamseddine, Alain Connes, 2007[+ update adapted fromResilience of the Spectral Standard Model, 2012andNoncommutative Geometry in Physics2014 ]

Last editing work on 25th July 2014

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