### (Why) Saving Pati-Salam (?)

**Seven empirically driven reasons for a partial (or grand) unification model**

The symmetry G(224) = SU(2)_{L}×SU(2)_{R}×SU(4)_{C}... supplemented by Left–Right discrete symmetry which is natural to G(224), brings a host of desirable features.Including some of those mentioned above which served as motivations for grand uniﬁcation,they are:

(i)Uniﬁcation of all sixteen members of a family within one left-right self-conjugate multiplet with a neat explanation of their quantum numbers;

(ii)Quantization of electric charge;

(iii)Q_{e−}/Q_{p}= −1;

(iv)Quark-lepton uniﬁcation through SU(4)-color;

(v)Conservation of parity at a fundamental level...(It appears aesthetically attractive to assume that symmetries like Parity (P), Charge Conjugation (C), CP and Time Reversal (T) break only spontaneously like the gauge symmetries. While such a preference a priori is clearly subjective... observations of neutrino oscillations and the likely need for leptogenesis, suggesting the existence of ν_{R}’s à la SU(4)-color and SU(2)L× SU(2)R, seem to go well with the notion of exact conservation of parity at a fundamental level);

(vi)RH neutrino as a compelling member of each family that is now needed for seesaw and leptogenesis;

(vii)B–L as a local symmetry. It has been realized eventually thatthis is needed to protect νR’s from acquiring Planck scale massesand to set (for reasons noted above) M(ν^{i}_{R}) ∝ M_{B−L}∼ M_{GUT}, bothcrucial to seesaw and leptogenesis; ...These... features constitute the hallmark of G(224). Historically, all the ingredients underlying these features, and explicitly (i)–(vii), including the RH ν’s, B–L and SU(4)-color, were introduced into the literature only through the symmetry G(224) [6]; this was well before SO(10) or (even) SU(5) appeared. Any simple or semi-simple group that contains G(224) would of course naturally possess these features. So does therefore SO(10), which is the smallest simple group containing G(224). In fact, all the advantages of SO(10), which distinguish it from SU(5) and are now needed to understand neutrino oscillations as well as baryogenesis via leptogenesis, arise entirely through the symmetry G(224).SO(10) being the smallest extension preserves even the family-multiplet structure of G(224) without needing additional fermion...

I have... implicitly assumed the existence of an underlying uniﬁed theory including gravity—be it string/M theory or something yet unknown—that would describe nature in a predictive manner and explain some of its presently unexplainable features, of the type mentioned above. Such a theory inevitably would operate at very short distances... and very likely in higher dimensions.It then becomes imperative, for reasons stated above,that such a theory, as and when it evolves to be predictive, should lead to an effective grand uniﬁcation-like symmetry (possessing SU(4)-color) in 4Dnear the string/GUT-scale, rather than the SM symmetry. If such a symmetry does emerge from the underlying theory as a preferred solution in 4D... it would explain observations in the real world, beyond those encompassed by grand uniﬁcation.

The picture depicted above is of course clearly a wish and a goal, yet to be realized. Entertaining such a wish amounts to hoping that the current difficulties of string/M theory as regards the large multiplicity of string vacua... and lack of selectivity... would eventually be overcomepossibly through a better understanding and/or formulation of the theory, and mostlikely through the introduction of some radically new ingredients(Perhaps as radical as Bohr’s quantization rule that selected out a discrete set of orbits from an unstable continuum, which in turn found its proper interpretation within quantum mechanics).Entertaining such a hope no doubt runs counter to the recently evolved view of landscape..., combined with anthropism... Such a hope is nevertheless inspired... by the striking successes we have hadover the last 400 yearsin our attempts at an understanding of nature at a fundamental level.To mention only a few that occurred in the last 100 years,they include ﬁrst and foremost the insights provided by the two theories of relativity and quantum mechanics.In the present contextthey include also the successes of the ideas of the standard model, grand uniﬁcation and inﬂation.Each of these have aided in varying degrees to our understanding of nature.

Jogesh C. Pati, June 7 2006

**Reading more extensively Pati's article, one will notice his emphasis on another symmetry not quoted in this excerpt: SUSY of course! It is important to underline nevertheless that - as far as the blogger can understand - SUSY is important for consistency of string theory and it gives a spectrum compatible with gauge coupling unification but it is not necessary for the part of the Pati's argumentation shown above.**

**A spectral noncommutative inspired heuristic incentive for non-SUSY GUTs**

The assumption that space-time is a noncommutative space formed as a product of a continuousfour dimensional manifold times a ﬁnite space predicts, almost uniquely, the Standard Modelwith all its fermions, gauge ﬁelds, Higgs ﬁeld and their representations.A strong restriction on thenoncommutative space results from the ﬁrst order condition which came from the requirement thatthe Dirac operator is a diﬀerential operator of order one. Without this restriction, invariance underinner automorphisms requires the inner ﬂuctuations of the Dirac operator to contain a quadraticpiece expressed in terms of the linear part.We apply the classiﬁcation of product noncommutativespaces without the ﬁrst order condition and show that this leads immediately to a Pati-SalamSU(2)_{R}×SU(2)_{L}×SU(4)_{C}type model which uniﬁes leptons and quarks in four colors.Besides the gauge ﬁelds, there are 16 fermions in the (2,2,4) representation, fundamental Higgs ﬁeldsin the (2,2,1), (2,1,4) and (1,1,1+15) representations. Depending on the precise form of theinitial Dirac operator there are additional Higgs ﬁelds which are either composite depending onthe fundamental Higgs ﬁelds listed above, or are fundamental themselves [in the (2,2,1+15) and (3,1,10) and (1,1,6) representations]. These additional Higgsﬁelds break spontaneously the Pati-Salam symmetries at high energies to those of the StandardModel...

Remarkably, we note thata very close model to the [case with a generic initial Dirac operator]... is the one considered by Marshak and Mohapatra where the U (1) of the left-right model is identiﬁed with the B−L symmetry... 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) toSU(2)_{R}×SU(2)_{L}×SU(4)_{C}and these have been extensively studied [1].

It remains tominimize the potential to determine all possible minima as well as studyingthe uniﬁed model andcheck whether it allows for uniﬁcation of coupling constantsg_{R}=g_{L}=gin addition to determining the top quark mass and Higgs mass.Obviously, this modeldeserves careful analysis, which will be the subject of future work.

One has learnt then that - beyond the Standard Model - only a non-SUSY partial unification scenario with a Pati-Salam model and possibly a Marshak and Mohapatra version of an SO(10) grand unification theory could fit in the noncommutative framework. One can then hope that the spectral paradigm and the almost commutative fine structure of spacetime at the attoscale are good and radical enough ingredients to help to grant the wish and pursue the goal of Pati: building partial or grand unification models which symmetries prove to be preferred solutions in 4D of an underlying theory!

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