mardi 29 avril 2014

3 solutions orthodoxes + 1 réponse "évidente" à un problème particulier de naturalité du boson de Higgs

Un cas particulier de problème de naturalité lié au boson de Higgs
Dans un texte intitulé Naturalness and the Higgs Boson Proliferation Instability Problem écrit pour une très récente conférence-atelier sur la philosophie du LHC, le physicien James Wells discute du problème de l'instabilité du secteur électrofaible du Modèle Standard vis à vis d'une prolifération de bosons scalaires. Voici comment il situe cette problématique par rapport à la question de la naturalité déjà abordée dans ce blog:
... the scalar proliferation instability problem is perhaps the least controversial, most generic and most important subcategory of the general naturalness problem. It is not unexpected that solutions to the general naturalness problem can solve or help solve the proliferation problem. Discussions of the general naturalness problem, however, have been plagued often with quasi-mystical arguments involving quantum gravity and arguments involving cancellations of bare mass terms with regulator-dependent cutoff scales, which we have no access to and are mere non-physical intermediate book-keeping devices for calculation. Considering the requirements for stability in the presence of a large number of heavy or condensing scalars in nature – a generic and motivated consideration – is a more concrete problem to address ... the requirement of proliferation stability puts significant restrictions on theory, and implies that there is more to discover beyond the Standard Model to complete our understanding of the weak scale.
Il n'est pas le premier à la soulever comme il le rappelle lui même en citant un extrait d'un article important de Martinus Veltman, Reflections on the Higgs system :
The introduction of higher Higgs multiplets, or of more than one doublet has the obvious disadvantage that in general no zero mass vector boson survives. In other words, the observed zero photon mass is then an ‘accident.’ For this reason alone these schemes are very unattractive.

A propos de la pertinence et du caractère générique d'éventuels bosons scalaires non chargés 
Alors que le célèbre physicien nobelisé qui vient d'être cité envisageait des bosons scalaires chargés sous une symétrie de jauge du Modèle Standard, Wells pour sa part préfère concentrer sa discussion sur des scalaires qui seraient non chargés sous ce dernier. Pour comprendre que cette dernière hypothèse n'a rien de gratuit pour Wells et pour apprécier sa pertinence dans le cadre de sa réflexion sur le secteur du boson de Higgs il faut je pense se plonger dans son cours Lectures on Higgs Boson Physics in the Standard Model and Beyond :
... the Higgs boson operator is the only bosonic one with dimension less than four that is both gauge invariant and Lorentz invariant. In my mind, this is one of the most important realizations that one can make about the Standard Model and about the Higgs boson in particular. As a consequence, hidden worlds have a much better chance of communicating with the Higgs boson than any other particle in the Standard Model. This is one reason why I think the Higgs boson is particularly susceptible to deviations from expected phenomenology at the LHC.

On peut aussi revenir au premier article de Wells évoqué au premier paragraphe pour trouver des arguments qui justifient le caractère générique des bosons scalaires dans les extensions usuelles du Modèle Standard :
It is often the case that more fundamental theories that try to explain dark matter and inflation (Baumann , "TASI Lectures on Inflation" 2009), explain flavor (Babu "TASI Lectures on Flavor Physics" 2009), or which strive to be compatible with a theory of quantum gravity, such as string theory, generically predict that there should be many more particles and much more dynamics than just what is described by the Standard Model. Regarding this last category, one should expect dozens, or perhaps even thousands of more Higgs bosons of exotic sectors that condense and break symmetries (Dijkstra et al. "Supersymmetric Standard Model Spectra from RCFT orientifolds 2005).

Trois solutions orthodoxes
Voyons maintenant les trois solutions potentielles au problème d'instabilité telles que les décrit J. Wells (je les qualifie d'orthodoxes car il s'agit essentiellement des solutions usuelles au problème de la hiérarchie des masses des particules scalaires dans le cadre des théories quantiques des champs renormalisables):
... there are several solutions to the proliferation instability problem that we have been describing above. Operationally, any solution theory has the burden of enforcing stability in the presence of a large number of massive or condensing scalars. The prospective solutions include banishing fundamental scalars as in technicolor and composite Higgs theories, banishing high-scale hierarchies as in large extra dimensions or warped extra dimensions, or invoking supersymmetry (Pomarol "Beyond the Standard Model" 2012). Not surprisingly, given our discussion above pointing out the important connection between the proliferation instability problem and naturalness, this triumvirate of general approaches also can potentially solve the proliferation instability problem. 
The first solution, to banish the entire category of scalars from the theory, clearly would take care of any problem systemic to scalars. However, there are well-known challenges to matching data with this approach (Pomarol 2012), not to mention that the recent discovery of a weakly interacting Higgs boson consistent with being elementary puts strain on this idea. 
The second solution is banishing the existence of high scales through extra dimensions. The idea is to reinterpret the singlet number of the very large mass Planck scale of gravity as the ratio of two numbers involving the weak scale and a very large extra dimensional volume or warp factor, in the case of warped extra dimensions (Csaki "Extra-dimensions and Branes" 2004). This approach may not work well to solve the proliferation problem. If we indeed have dozens or more condensing scalars in nature – let’s call the number NH – at scales not too far away from the Higgs mass scale, there is still the potential of destabilizing the hierarchy from large NH. For the Higgs mass to be stable the sum of contributions to the Higgs mass-squared operator would have to be of order the Higgs boson mass, m2H ∼ NHξ2, where ξ is the typical vacuum expectation value of the exotic condensing scalar. Thus, lowering the high-scale nearer to the weak scale through large or warped extra dimensions would soften the destabilization problem some, but may not eradicate it. 
The third solution, supersymmetry, is next to consider. It is a remarkable feat of supersymmetry that the Higgs sector is generically completely stable to a large number of extra condensing Higgs bosons, in stark contrast to non-supersymmetric field theories. The key is a special property of supersymmetry invariance that requires interactions to be analytic in their fields ...  The only requirements are that there are no pure singlet scalar states in nature under all possible symmetries (Bagger & Poppitz "Destabilizing Divergences in Supergravity-Coupled Supersymmetric theories" 1993), and the technically natural µ term that connects µHu·Hd together is near the weak scale. This is a restriction on two narrow and technical criteria compared to the admittance of a very large number of possible exotic Higgs states charged under many different exotic symmetries.

Une autre réponse "évidente" au problème (... derrière laquelle se cache peut-être une solution non orthodoxe ?)
On peut évidemment supposer que Wells en ne discutant que trois solutions en néglige d'autres. Voilà ce qu'il écrit à ce propos:
Another response to the proliferation problem is to assume that there simply is no proliferation of Higgs bosons in nature, and so no proliferation instability problem arises. The difficulty with this position is that we would be required to believe that the Higgs boson is very special and that unlike any other representations in nature, the spin 1/2 fermions and spin 1 vector bosons, there is just one Higgs boson and not another. This position fails a modern day Copernican test of making sure our theories do not require us to believe we are particularly special. 

Et si le paradigme géométrique non commutatif était justement le cadre idéal pour développer des modèles où il n'est pas nécessaire de supposer mais où on peut s'assurer qu'il n'y a pas de prolifération de scalaires potentiellement dangereux pour la stabilité du vide électrofaible? La question me parait légitime dans la mesure où contrairement aux modèles les plus discutés par les physiciens - où les bosons scalaires sont ajoutés "à la main" et les paramètres nécessaires aux brisures spontanées de symétrie introduits de manière ad-hoc - ceux construits dans le cadre du paradigme spectral non commutatif fournissent au contraire un secteur scalaire "clé en main", parcimonieux en nouvelles particules dans les extensions au Modèle Standard et dont les mécanismes de brisure de symétrie découlent d'un principe "dynamique" d'action spectral. Le mot dynamique est entre guillemets car l'espacetemps de la géométrie noncommutative n'est pas exactement l'espace-temps de la théorie quantique relativiste ordinaire.
(à suivre ...)


//ajout du 24/06/14
Au delà de l'intérêt heuristique du programme non commutatif dans la recherche de nouveaux modèles pour la physique des très hautes énergies il faut aussi signaler son rôle épistémologique il me semble pour la compréhension de la place singulière du modèle standard dans le paysage des théories quantiques des champs. En effet si la supersymétrie peut apparaître comme  une solution générique au problème de naturalité de l'échelle électrofaible dans le vaste paysage des théories d'unification les plus étudiées par les physiciens, on se doit à l'inverse de souligner que le modèle standard et certaines extensions minimales non supersymétriques (avec neutrinos droits de Majorana) sont parmi les rares constructions théoriques qui passent sous les fourches caudines de la géométrie spectrale non commutative. Dit autrement, et en opposition avec la dernière affirmation de Wells, il me semble que le modèle standard et particulièrement le secteur de Higgs ont nécessairement une structure très particulière (avec un réglage fin de certains paramètres) aux yeux des théories quantiques qui se basent sur des espaces-temps qui, quelles que soient le nombre de leur dimensions spatiales supplémentaires (et les raffinements de leur compactifications) restent commutatifs!

Pour finir et puisque l'exposé de Wells s'inscrivait dans un séminaire sur la philosophie du LHC je ne peux m'empêcher de rêver à ce qu'aurait pensé de la dernière phrase de James D. Wells, le physicien Yuri Ne'eman, lui qui écrivit un jour un texte intitulé The Hamlet principle in physical science - Copernican assumptions restricting scientific exploration... Si j'ignore le contenu de ce texte, la deuxième partie de son titre me paraît suffisamment explicite comme mise en garde sur la pertinence heuristique de la dernière remarque de Wells (dont j'admire par ailleurs l'analyse épistémologique des théories effectives ici et). Je pense donc que la critique de la présence de réglages fins dans le modèle spectral non commutatif repose sur une hypothèse copernicienne qui non seulement manque de pertinence épistémologique mais se dresse comme un obstacle à l'approfondissement du potentiel heuristique du programme spectral non commutatif. Pour ce qui est de la référence non explicite à Hamlet, il me semble que le passage suivant, tiré d'un livre de Ne'eman écrit avec Yuval Kirsch, peut raisonnablement nous mettre sur la bonne voie:
We think that Hamlet’s words ‘There are more things in Heaven and Earth, Horatio, than are dreamt of in our Philosophy’ (I, v. 167) are closer to the lessons of the past.
Yuri Ne'eman et Yovla Kirsch, The Particles Hunter, 1996

Remarques
Cet article fait suite à la lecture du dernier billet de Peter Woit qui m'a permis de découvrir le nouvel article de Wells. 
Il semble que depuis le 30/04/14 les documents concernant la conférence (et donc le dernier texte de Wells) ne soient malheureusement plus en accès libre.
La dernière citation quant-à elle est tirée d'un commentaire à cet autre billet de Woit.

lundi 28 avril 2014

Vers une extension non associative du formalisme spectral non commutatif pour aborder la physique de plus haute énergie?

Voici quelques "morceaux choisis" de deux récents articles de Latham Boyle et Shane Farnsworth déjà évoqués dans ce blog...

De l'espace-temps riemannien à l'espacetemps non commutatif
Often, noncommutative geometry is taken to mean the noncommutativity of the 4-dimensional spacetime coordinates themselves; it is regarded as a property of quantum gravity that presumably becomes manifest at the Planck energy scale (i.e. the exceedingly high energy scale of 1019 GeV). By contrast, from the perspective of the spectral reformulation of the standard model, all of the non-gravitational fields in nature at low energies are reinterpreted as the direct manifestations of noncommutative geometry, right in front of our nose, staring us in the face! What is the motivation for reformulating the familiar action for the standard model (coupled to gravity) in the unfamiliar langage of spectral triples and spectral action? ... The spectral action ... packages all of the complexity of gravity and the standard model of particle physics into two simple and elegant terms which, in turn, follow from a simple principle (the spectral action principle described above). The compactness and tautness of this formulation suggest that it may be a step in the right direction. To give a provocative analogy: much as Minkowski “discovered” that the rather cumbersome Lorentz transformations (which formed the basis of Einstein’s original formulation of special relativity) could be elegantly re-interpreted as the geometrical statement that we live in a 4-dimensional Minkowski spacetime, Chamseddine and Connes seem to have discovered that the rather cumbersome action for the standard model coupled to gravity can be elegantly re-interpreted as the geometric statement that we live in a certain type of noncommutative geometry. 

Des défauts des théories de Kaluza-Klein aux avantages des théories spectrales non commutatives
To see what is compelling about this picture, let us contrast it with Kaluza-Klein (KK) theory. To see the contrast clearly, it is enough to consider the original and simplest KK model. ... one starts from the simple and purely gravitational action for Einstein gravity [on a 5D manifold ... the product of a 4D manifold and a circle ( M4 × S1)] and obtains something tantalizingly close to 4D Einstein gravity plus 4D gauge theory – but it also captures what is unappealing about KK theory. For one thing, one typically obtains extra, unwanted fields with unwanted couplings ... , and one must explain why these extra fields and couplings are not observed in nature. For another thing, the reduction from the initial 5D action, which has the huge symmetry group Diff(M4 × S1), to the final 4D action, which has [a] much smaller symmetry group ..., fundamentally relies on the assumption that the 5D metric ... only depends on the 4D coordinates ... This assumption is supposed to be justified, in turn, by the fact that the compactified direction is so small; but this justification assumes that one has stabilized the extra dimension – i.e. found a way to make it small and keep it small, without letting it shrink down to a singularity or blow up to macroscopic size. The problem of stabilizing extra dimensions in KK theory is a famously thorny one and, furthermore, is ultimately at the root of the so-called landscape problem in string theory. Thus the spectral and KK approaches share a similar spirit: in both cases the goal is to reinterpret the action describing ordinary 4-dimensional physics as arising from a simpler action formulated on an “extension” of 4-dimensional spacetime. But the spectral action seems to achieve this goal more elegantly and directly. In KK theory, the starting point is an action with too many fields and too much symmetry, and one must then jump through many hoops to explain why these extra fields and symmetries are unobserved in nature. By contrast, in the spectral approach, the field content and symmetries of the standard model are obtained directly.


Des groupes non abéliens aux algèbres non associatives
The fundamental point is that, in the ordinary approach to physics, the basic input is a symmetry group: this is the starting point for specifying a model (like the standard model of particle physics). By contrast, in the spectral approach, the fundamental input is an algebra, and the symmetry group then emerges as the automorphism group of that algebra. Symmetry groups are associative by nature, but algebras are not. Just as some of the most beautiful and important groups are noncommutative, some of the most beautiful and important algebras (including Lie algebras, Jordan algebras and the Octonions) are nonassociative. Just as it would be unnatural to restrict our attention to commutative groups, it is unnatural to restrict our attention to associative algebras. In either case, imposing such an unnatural restriction amounts to blinding ourselves to something essential that the formalism is trying to tell us. From this standpoint, our task is to formulate the spectral approach to physics in such a way that the incorporation of nonassociative algebras becomes obvious and natural. 

De l'invariance par difféomorphismes et symétrie de jauges internes à la covariance par ∗-automorphismes 
Usually, the automorphism invariance of the spectral action is presented as a consequence of the “spectral action principle” (the principle that the action only depends on the spectrum of D which is, itself, invariant under automorphism). But in extending to the nonassociative case, we have found it important and clarifying to make a change of perspective, in which we elevate the ∗-automorphism covariance to a fundamental underlying principle which then provides guidance about all subsequent steps... The idea is that, in the spectral approach, the principle of ∗-automorphism covariance subsumes and replaces the traditional covariance principles of physics: diffeomorphism covariance (in Einstein gravity) and gauge covariance (in gauge theory).  


Du Modèle Standard aux modèles de grande unification 
We would like to reformulate the most successful Grand Unified Theories (GUTs) – e.g. those based on SU(5), SO(10) ... – in terms of the spectral action, but in order to do this, we are forced to use nonassociative input algebras. To appreciate this point, first note that the represen- tation theory of associative ∗-algebras is much more restricted than the representation theory of Lie groups: Lie groups (like SU(5)) have an infinite number of irreducible representations, but associative algebras (like the corresponding ∗-algebra M5(C) of 5×5 complex matrices, whose automorphism group is SU(5)) only have a finite number. In particular, if we ask whether key fermionic representations needed in GUT model building – such as the 10 of SU(5), the 16 of SO(10) ... – are available as the irreps of algebras with the correct corresponding automorphism groups, the answer is “no” for associative algebras, and “yes” for nonassociative algebras.  

De l'ancien modèle spectral presque commutatif aux récents modèles spectraux non commutatifs: des modèles déjà tous non associatifs!
We have seen that the finite geometry K that encodes the standard model corresponds to an algebra B that is associative. But, to evaluate the spectral action, one then tensors this finite geometry with a continuous geometry to form a new geometry K′, and one can check that the corresponding algebra B′ is not associative ... In this sense, non-associativity already appears in the traditional NCG embedding of the standard model. It is interesting to consider whether this non-associativity might be connected to the generalized inner fluctuations considered in [25, 26], which bear a striking resemblance to the inner derivations of a non-associative (and in particular, an alternative) algebra [13].

Des outils mathématiques plutôt originaux pour une phénoménologie physique qui n'est pas exotique pour autant
It is also worth stressing again that, although the input algebra is nonassociative, the resulting physical model is not: at the end of the day, the resulting action functional is an ordinary gauge theory, build from ordinary (associative) scalar, spinor, gauge and metric fields. 

//Ajout du 18 octobre 2014 
Voici deux vidéos de Latham Boyle et Shane Farnsworth présentant les idées précédentes lors d'un cycle de conférences intitulé "Quantum physics and non-commutative geometry" organisé par Alan L. Carey, Victor Gayral, Matthias Lesch, Walter van Suijlekom et Raimar Wulkenhaar à l'institut Hausdorff de recherche mathématique de Bonn.

Latham Boyle:Non commutative geometry,non associative geometry,and the standard model of particle physics, 23 September 2014


Shane Farnsworth: Rethinking Connes' Approach to the Standard Model of Particle Physics via NCG
23 September

vendredi 18 avril 2014

Bossons sur le prochain boson scalaire

Dernières avancées sur le modèle spectral...
Voici le résumé d'un séminaire du physicien français Pierre Martinetti sur les dernières avancées d'un modèle spectral non commutatif de la physique fondamentale intégrant un nouveau boson scalaire :
Afin d’éviter d’éventuelles instabilités due à la "faible" masse du boson de Higgs, divers groupes de phénoménologues ont proposé d’introduire un nouveau champ scalaire, couplé au Higgs et stabilisateur du vide électrofaible. Récemment Connes et Chamseddine ont montré comment - dans les modèles de géométrie non-commutative - la prise en compte de ce nouveau champ permettait de ramener la prédiction initiale de la masse du Higgs (170 Gev) à la masse observée (126 Gev). La question est donc de comprendre dans quelle mesure ce nouveau champ peut-être obtenu de manière naturelle dans le formalisme non-commutatif, ou s’il s’agit d’un ’’artefact commode’’. On montrera comment la classification des triplets spectraux de dimension finie de Chamseddine-Connes-Marcolli permet de considérer un modèle "grand symétrique", dont la réduction vers le modèle standard génère de manière naturelle le nouveau champ scalaire à partir du terme de Majorana de masse pour le neutrino. On soulignera l’interprétation de cette réduction en terme "d’émergence de la géométrie".  On mentionnera également d’éventuels implications en cosmologie.
Pierre Martinetti, Grande symétrie, action spectrale et masse du Higgs, 23 octobre 2013 

.... et sur une possible super grande symétrie ...
Pour avoir une petite idée des éventuelles implications cosmologiques précédemment évoquées et surtout pour en savoir plus sur la façon dont ce nouveau boson scalaire sigma s'inscrit dans le cadre spectral non commutatif, on dispose depuis quelques jours du dernier article du même auteur coécrit avec deux physiciens italiens et qui résume bien les avancées de leur modèle "grand symétrique" ainsi que les perspectives fascinantes qu'il ouvre sur une vision renouvelée de la fusion entre espace-temps et espace interne des quarks et des leptons via le degré de liberté de spin. Ce dernier article permet de mieux voir les hypothèses (extrapolations contrôlables du modèle presque commutatif passé) et les limites (hypothèses non contrôlées) de leur modèle. Se trouve ainsi explicité ce en quoi le modèle "grand symétrique" ne s'intègre pas (encore?) complètement dans le cadre axiomatique actuel de la géométrie non commutative.

The point is thus to understand how to obtain σ intrinsically within the NCG framework... 

Under natural assumptions (irreducibility of the representation, existence of a cyclic vector), technical requirements of the NCG model (there is a representation of the opposite algebra that commutes with the action of the algebra and is implemented by an operator that commutes with the chirality) and a hypothesis on the role of quartenion, one has that the most general finite dimensional algebra satisfying the axiom of noncommutative manifolds is of the form [4]: M
a(H)⊕M2a(C), a∈N, and acts on an Hilbert space of dimension d=2×(2×a)2. The case a=1 is too small to get the gauge group of the SM as the group of unitaries of M(H)⊕M2(C). The next choice a = 2 yields d = 32, that is the number of fermions per generation ... of the Standard Model. Interestingly, a=4 yields d=128, which is 4 times the number of particles/ generation. Viewing 4 as the number of components of a Dirac spinor on a 4-dimensional manifold, one can thus decompose the total Hilbert space (for 1 generation), using the fermion doubling of the model [18], as 
H = L2(M,S)⊗HF  = L2(M)⊗HF where  HF = C4HF  = C4⊗ C32= C128. (9)
In writing (9) we ignore global obstruction and assume that the r.h.s. equality of the first equation above holds on a local trivialization of the spin bundle. The idea we want to promote is that by mixing the spin degrees of freedom ... with the internal degrees of freedom ...then the Hilbert space H of the standard model is big enough to represent the grand algebra (a=4) AG= M4 (H)⊕M8(C) ...
... starting from the grand algebra one can generate the field σ by a fluctuation of the Majorana mass term which respects the first order condition imposed by the Majorana mass term. The further reduction to the standard model ... is obtained by the first order condition on the non-Majorana part of the Dirac operator.
... one may imagine a cosmological scenario beginning with a “pre-geometric phase”, described by the grand algebra and the finite dimensional Dirac operator γ5⊗DR. The right neutrino would then play the role of a “primary elementary particle”, that generates the field σ. Then usual geometry (encoded within the free Dirac operator...) emerges at a later stage, and provokes the reduction to the SM. This makes and interesting echo to a recent inflationary interpretation of the field σ [2]. Moreover very recent data [1] seem to indicate an inflationary scale at a scale of 1016GeV, a scale in broad agreement with the unification of the coupling constant required by this approach. 
A.Devastato, F.Lizzi et P.MartinettiHiggs mass in Noncommutative Geometry, 2 avril 2014 

... à replacer dans un cadre non commutatif élargi ?
L'article en question relaie au passage deux avancées récentes qui visent à étendre le cadre conceptuel de la géométrie non commutative : 

  • l'une repose sur un formalisme algébrique non associatif proposé par Latham Boyle et Shane Farnsworth dans lequel la géométrie non commutative développé par Connes peut s'inscrire. Cette reformulation pour ainsi dire a l'intérêt de suggérer des contraintes logiques nouvelles sur le modèle physique spectral qui coïncident exactement avec certaines hypothèses ad-hoc passées :
Connes has developed a notion of non-commutative geometry (NCG) that generalizes Riemannian geometry, and provides a framework in which the standard model of particle physics, coupled to Einstein gravity, may be concisely and elegantly reformulated. We point out that his formalism may be recast in a way that generalizes immediately from non-commutative to non-associative geometry. In the process, several of the standard axioms and formulae are conceptually reinterpreted. This reformulation also suggests a new constraint on the finite NCG corresponding to the standard model of particle physics. Remarkably, this new condition resolves a long-standing puzzle about the NCG embedding of the standard model, by precisely eliminating from the action the collection of 7 unwanted terms that previously had to be removed by an extra (empirically-motivated) assumption
Latham Boyle and Shane Farnsworth, Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics01/2014

  • l'autre avancée est la mise en évidence de la nature "dynamique" (via le principe d'action spectrale et grâce à une vision élargie du concept de "fluctuations internes") d'une contrainte mathématique (condition d'ordre un) qui permettait de déterminer les champs de jauge d'un modèle physique spectral. 
A strong restriction on the noncommutative space results from the fi rst order condition which came from the requirement that the Dirac operator is a di erential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece expressed in terms of the linear part. We apply the classi cation of product noncommutative spaces without the first order condition and show that this leads immediately to a Pati-Salam SU(2)R SU(2)L SU(4) 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 fields in the (2; 2; 1), (2; 1; 4) and (1; 1; 1 + 15) representations. Depending on the precise form of the initial Dirac operator there are additional Higgs fields which are either composite depending on the fundamental Higgs fields listed above, or are fundamental themselves. These additional Higgs fields break spontaneously the Pati-Salam symmetries at high energies to those of the Standard Model.
Ali H. Chamseddine , Alain Connes et Walter D. van Suijlekom, Beyond the Spectral Standard Model: Emergence of Pati-Salam Uni fication 2013