mercredi 1 mai 2013

Rêve (algébrique) de grande unification, brisures (dynamiques) de symétries élargies et oasis noncommutatif (en extension)

Un quantum d'obstination (11)

Et si un modèle d'espace-temps presque-commutatif et un principe d'action spectrale étaient les outils vraiment efficaces pour prolonger les succès du Modèle Standard par de nouveaux moyens (une théorie quantique des champs augmentée ... d'une théorie des chants quantiques ;-)? 
The Standard Model (SM) continues to conform to all experimental data. The question remains whether this model will continue to hold at much higher energies, or whether there is a unified theory whose low-energy limit is the SM. One indication that there must be a new higher scale that effects the low energy sector is the small mass of the neutrinos which is explained through the see-saw mechanism with a Majorana mass of at least of the order of 10^11Gev. In addition and as noted above, a scalar field which acquires a vev generating that mass scale can stabilize the Higgs coupling and prevent it from becoming negative at higher energies and thus make it consistent with the low Higgs mass of 126 Gev [11]. Another indication of the need to modify the SM at high energies is the failure (by few percent) of the three gauge couplings to be unified at some high scale which indicates that it may be necessary to add other matter couplings to change the slopes of the running of the RG equations. 
This leads us to address the issue of the breaking from the natural algebra A which results from the classfication of irreducible finite geometries of KO-dimension 6 (modulo 8) performed in [9], to the algebra corresponding to the SM. This breaking was effected in [9], [8] using the requirement of the first order condition on the Dirac operator...  
In this work we shall examine the hypothesis that the first order condition on the subalgebra (2) arises as the vacuum solution of the spectral action. In this way the first order condition becomes a dynamical requirement instead of being imposed from outside... 
The important point to notice is the novel phenomena of the appearance of composite Higgs field ...The importance of this point should not be underestimated. The reason is that the main disadvantage of grand unified theories is the need for complicated Higgs representations with arbitrary potentials. In the noncommutative geometric setting, this problem is now solved by having minimal representations of the Higgs fields allowing for (quadratic) products of these representations. We also note that a very close model to the one deduced here is the one considered by Marshak and Mohapatra ...  
We conclude that the study of noncommutative spaces based on a product of a continuous four dimensional manifold times a finite space of KO-dimension 6; without the first order condition gives rise to almost unique possibility in the form of a Pati-Salam type model. This provides a setting for unification avoiding the desert and which goes beyond the SM. In addition one of the vacua of the Higgs fields gives rise at low energies to a Dirac operator satisfying the first order condition. 
 A. H. Chamseddine, A. Connes et W. D. van Suijlekom, Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification 30/04/13

// Et si la géométrie spectrale noncommutative était un outil mathématique naturellement quantique idéal pour "résoudre" un problème physique de réglage fin?
The aim of this paper is to determine the interactions of the dilaton field φ with all other fields present in the spectral action formulation of the standard model. Because of the spectral character of the action, it is completely determined from the form of [a generalized dirac Operator] and there is no room for fine tuning the results. It is then very reassuring to find that the resulting interactions are identical to those constructed in the literature by postulating a hidden scale invariance of the matter interactions ... These are also equivalent to the interactions of the radion field in the Randall Sundrum model ... All of these results now support the conclusion that space-time at high-energies reveals its discrete structure, and is governed by noncommutative geometry.  
The Dirac operator being a differential operator has the dimensions of mass. The spectral action in noncommutative geometry is defined as a function of a dimensionless operator which is taken to be the Dirac operator divided by some arbitrary large mass scale. The arbitrariness of the mass scale naturally suggests to make this scale dynamical by introducing a dilaton field in the Dirac operator of the noncommutative space defined by the standard model. To understand the appearance of the mass scales of the spectral action, we evaluated all interactions of the dilaton with the matter sector in the standard model. We found the remarkable result that the low-energy action, when evaluated in the Einstein frame, is scale invariant except for the Einstein-Hilbert term and the dilaton kinetic term. The resulting model is almost identical to the one proposed in the literature ... The main motivation in these works is the observation that the standard model is classically almost scale invariant, with the symmetry only broken by the mass term in the Higgs potential. The symmetry is restored by the use of a dilaton field. When coupled to gravity, neither the dilaton kinetic energy nor the scalar curvature are scale invariant, leading to a Jordan-Brans-Dicke theory of gravity. The vacuum expectation value of the Higgs field is then dependent on the dilaton and is classically undetermined. Quantum corrections break the scale invariance of the scalar potential and change the vacuum expectation value of the Higgs field. The dilaton acquires a large negative expectation value given by −m and a small mass. The hierarchy in mass scales is due to the large Yukawa coupling of the top quark. The dilaton expectation value can range between the GUT scale of 10^15 Gev to the Planck scale of 2.4·10^18 Gev. The hierarchy in mass scales is not possible if the dilaton kinetic energy and the gravitational action were scale invariant. It is remarkable that all the essential features of building a scale invariant standard model interactions to generate a mass hierarchy and predict the Higgs mass are naturally included in the spectral action without any fine tuning. It is worth mentioning that the scalar potential of exactly the same model considered here was shown to admit extended inflation and a metastable ground state. It also evades the problems of the original version of extended inflation.
 A. H. Chamseddine, A. Connes, Scale Invariance in the Spectral Action 16/03/06

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