Clifford Structures in Noncommutative Geometry and the Extended Scalar Sector

We consider aspects of the noncommutative approach to the standard model based on the spectral action principle. We show that as a consequence of the incorporation of the Clifford structures in the formalism, the spectral action contains an extended scalar sector, with respect to the minimal Standard Model. This may have interesting phenomenological consequences. Some of these new scalar fields carry both weak isospin and colour indexes. We calculate the new terms in spectral action due to the presence of these fields. Our analysis demonstrates that the fermionic doubling in the noncommutative geometry is not just a presence of spurious degrees of freedom, but it is an interesting and peculiar property of the formalism, which leads to physically valuable conclusions. Some of the new fields do not contribute to the physical fermionic action, but they appear in the bosonic spectral action. Their contributions to the Dirac operator correspond to couplings with the spurious fermions, which are projected out.


Introduction
The standard model of particle interactions can be efficiently described by a particular noncommutative geometry: an "almost commutative geometry". Over the years the model has been developing both in its mathematical and physical aspects. Its mathematical framework has its roots in a global view [1][2][3][4] of geometry based on the spectral properties of operators. The applications of this point of view to geometry are quite startling, the standard reference of the model in its modern version is [5], for a recent review see [6]. The model has predictive power, although it is premature to consider it a fully fledged theory to confront with experiment, with prediction with a significative number of digits. Its main success is in the description of the symmetries of the model, very few Yang-Mills models can be described by a noncommutative geometry (NCG), but the standard model and few more can. The Higgs field emerges naturally as an intermediate boson corresponding to the noncommutative part of the model, of a par with photons, W , Z and gluons. The actions for fermions and bosons are firmly based on the spectral properties of a generalized Dirac operator [7] 1 and the procedure is capable of obtaining numbers such as the mass of the Higgs. The numbers produced in [5], although encouraging, are not in agreement with present data, in particular the model requires the unification of all couplings at a single energy, and one calculates the Higgs boson mass around 170 GeV. Both these aspects are experimentally excluded, and the model can be fixed to allow the physical mass of the Higgs boson [12][13][14][15][16][17][18][19][20][21]. Efforts are also undertaken to use the model for other predictions, for example in [22].
The fact that the calculations made in the present model are encouraging, but not yet comparable with experiment, suggests that some improvement may happen also from the mathematical side. In [23] it was discussed a noncommutative version of the Clifford symmetry. One of the remarkable effects of the Clifford requirements is the appearance of scalar fields which are not present in the usual description.
The aim of this paper is to discuss in detail these new fields and their couplings. In particular we will calculate their contribution to the spectral action. The noncommutative model is by nature Euclidean and exhibits spurious degrees of freedom, known as "fermion doubling" [24], therefore for physical applications a Wick (anti)rotation accompanied by an elimination of these spuriuous degrees of freedom is necessary. We have described this procedure in detail in [25]. Here we find that not all of these extra bosons behave upon this procedure in a standard way: some of the new scalar fields present in the Euclidean Dirac operator are absent in the corresponding (Lorentzian) physical action for fermions.
The paper is organized as follows: in Sec. 2 we review the noncommutative geometric approach to the Standard Model, focusing on the modification of the formalism due to an introduction of the Clifford structures proposed in [23]. In Sec. 3 we introduce the new scalar fields, which come out from the fluctuations of the Dirac operator in the "Clifford-based" approach [23], and discuss their transformation properties upon the action of the gauge group. Sec. 4 is devoted to the bosonic spectral action: we compute the new terms with respect to the "standard" spectral approach [5]. In Sec. 5 we discuss the physical action derived from this model: we carry out the Wick rotation to the Lorentzian signature and get rid of the spurious degrees of freedom in the fermionic action. The last section contains our conclusions.

The standard model as a Noncommutative Geometry
In this section we sketch the main aspects of the model. We will be very brief, the reader familiar with this approach will need this section just to set the notations. First we outline the basic concepts of the spectral triples, which are common for both the "standard" approach [5] and the "Clifford-based" [23] approaches, whilst afterwords we discuss the peculiar features of the latter, which differ it from the former: the finite dimensional grading γ F and the finite dimensional Dirac operator D F .

The Standard Spectral Triple
In the spectral approach a geometry is described by a spectral triple [1][2][3], i.e. a * -algebra (possibly noncommutative) realized as bounded operators on a Hilbert space, and a self adjoint operator which generalizes the Dirac operator. The algebra describes the topology of the space, for the case at hand the Hilbert space describes the matter content and the Dirac operator gives a metric structure and enables the writing of action. Being based on operators all quantities are based on spectra, and in particular the actions for bosons and fermions can be written in purely spectral form. Also of fundamental importance are two more operators: the grading and the real structure, which generalize chirality (for the even dimensional case) and charge conjugation. The standard model emerges form this scheme. We will briefly describe this approach mainly to set notations, referring for details to the original literature [5,7] or the recent book [6]. We start choosing an algebra which is the product commutative infinite dimensional algebra of continuous functions on the manifold M, which represents the space-time times a noncommutative but finite dimensional matrix algebra For the standard model the finite algebra is where by H we indicate quaternions, and by Mat 3 (C) three by three complex matrices. Likewise the Hilbert space is the product of usual spinors times a finite dimensional Hilbert space, which contains all physical degrees of freedom: the generalized Dirac operator (which in the following we will simply call Dirac operator) is Where ∇ LC µ is the covariant derivative on the spinor bundle of M, which contains the Levi-Civita spin connection. Gravity in the action is considered background, and is not quantized. A curved background does not however play a major role in this paper, but is useful to retain it, as it enables some simplification in the calculations, as we will see in Sect. 5.
As we mentioned, there are two more operators which play an important role. They are the grading operator Γ and the antiunitary real structure J . The grading operator Γ is present in the even dimensional case, it satisfies Γ 2 = 1 and it is taken to be where γ 5 is the chirality matrix i.e. the usual product of all four Dirac's γ µ and γ F is an operator acting on H F . It is usually taken to have eigenvalue +1 on left handed states, and -1 on right handed one, but other choices are possible and we will discuss them later in the paper. The real structure operator J = J ⊗ J F , which is antiunitary in H, enables the definition of the opposite algebra The elements of the triple must satisfy several conditions, which render the space the noncommutative equivalent of a manifold [26]. There are conditions of compatibility between Γ, J and D 0 with signs which depend on the dimensions: The opposite algebra must commute with the algebra (order zero condition): and with one forms (the order one condition) The dimension of H F in (2.3) is 96. This number is obtained taking into account that there is a lepton left doublet plus two right handed singlets, and a doublet and two singlets for quarks times three colours. This makes 16 degrees of freedom, times 3 generations, and times two for particle/antiparticle, sums to 96. Since the spinor index has four degrees of freedom the element of the full Hilbert space H is described by 384 independent complex valued functions. Clearly there is some overcounting, called for historical reasons fermion doubling [24]. We will come back to this issue, as well as the fact that the model is at this stage Euclidean, in section 5.
We will label the elements of H F according the basis given by the elementary particles of the standard model (including right handed neutrinos): where Q L corresponds to 2 the quark doublet (u L , d L ) while L L corresponds to the lepton doublet (ν L , e L ), with the supercript c we indicate the elements of H F which correspond to the antiparticles and by boldface characters we indicate that the elements have to replicated by three generations, for example e = (e, µ, τ ) and so on. Quarks have an extra colour index, which we omit. Below we will use the following notation for matrices action on H F . We define the matrix unity E u R u R to be a matrix whose only nonzero element is an identity matrix in the u R location, likewise for E u R d R is an off diagonal matrix with nonvanishing entry in the u R d R , and so on. In the cases for which a singlet crosses a doublet then we assume that, for example, E u R ,L L is two identity matrices side by side, or vertically superimposed. The representation of the algebra is diagonal and with our notation, an element a = (λ, h, m) with λ ∈ C, h ∈ H and m ∈ Mat 3 (C) is represented by the matrix 3 : In our notations the real structure J F of the finite spectral triple reads: where cc is complex conjugation. So far we have been in the framework of [5]. From now on we focus on the peculiar properties of the construction of [23], which enables to incorporate the Clifford structures in the finite spectral triple. We refer to the original paper for all the details, and present here just the results.

Alternative Grading
The first novelty of the Clifford based construction is the grading γ F of the finite spectral triple, which has the following form: (2.12) which differs from the "standard grading" γ st F considered in [5]: The two are connected by the following formula where Q and L stand for the projectors of the "quark" and "leptonic" subspaces of H F respectively: (2.15)

The Dirac Operator
Another novelty of the Clifford based approach is the Dirac operator D F , which has the following form: and which is compatible with the new grading γ F and other requirements of the approach of [23]. The terms on the first on the third and on the last lines involve the usual Yukawa couplings and the Majorana mass terms, which are already present in [5]. The second and the fourth lines instead contain novel terms, which are the object of this paper: ∆ and K provide novel couplings of leptons and quarks, Ω couples leptons among themselves in the Euclidean action before the projection on the physical subspace. We will see later on that the projection to the physical subspace will eliminate some of these couplings. It is important that the selfconsistency the approach of [23] requires in particular that: • both entries ∆ D and ∆ L must differ from zero, • and at least two out of the three entries ∆ U , K and Ω must be different from zero.
In conclusion we present the explicit matrix form of the Dirac operator D F defined by (2.16): (2.17) Setting ∆ U,D,L = 0, Ω = 0 and K = 0, one obtains the standard D F of [5].

Fluctuations of the Dirac operator: Fields
The fluctuated Dirac operator is constructed in the following way: for generic elements a i , b i ∈ A. Both gauge and scalar fields in the spectral approach come out from these fluctuations. Presence of the new terms (with respect to [5]) in (2.17) indicates new scalar fields, not present in the Standard Model. Below we restrict ourselves to the following structures, where the dependence on the generation indexes is factorised: In these formulas the two component columns h ν,e,u,d (in the Weak isospin indexes) are chosen in the same way as it was done in [5] (hereafter v is an arbitrary complex constant of the dimension of the mass): and the three component columns d u,d,L (in the colour indexes) we choose as follows: The quantity s is the complex 3 by 2 matrix (in both colour and the weak isospin indexes): ω is the complex number, which we set to v, the dimensionful constant M R sets the Majorana mass scale for the right handed neutrinos, which is needed for the sea-saw mechanism. The quantitiesŶ u ,Ŷ d ,ŷ u ,ŷ d ,ŷ ∆u ,ŷ ∆ d ,ŷ ∆ L ,ŷ S andŷ M are arbitrary (dimensionless) complex 3 by 3 Yukawa matrices which act on the generation index. The tilde indicates charge conjugated weak isospin doublets e.g.h ν = σ 2 h * ν , where σ 2 stands for the second Pauli matrix. Considering the fluctuations (3.1) of the Dirac operator one can see, that in order to construct the fluctuated Dirac operator, D one has replace the constant matrices in (3.2) by the matrix valued functions according to the following rule: Note that upon the fluctuations of the Dirac operator M R remains a constant i.e. it does not transform into a field. By definition the gauge subgroups SU(2) and SU(3) are represented on the weak isospin fermionic doublets and colour fermionic triplets as a left multiplication by the unitary matrices U SU (2) and U SU (3) respectively 4 : while the gauge fields transform upon the adjoint representation of the gauge group. The transformation law of the scalar fields which is presented below maintains the gauge invariance of the fermionic action upon the simultaneous gauge transformation of the fermionic multiplets, gauge and scalar fields. In what follows Y stands for the abelian hypercharge of a given multiplet, which describes the action of the U(1) gauge subgroup.
The scalar doublet H is nothing but the Higgs field of the minimal Standard Model, which transforms as follows: For each of the three fields ∆ u , ∆ d and ∆ L the transformation law reads: (3.10) The field S carries both colour and weak isospin indexes and transforms in the following way: The last field Ω is the SU(2) × SU(3) singlet, and it transforms nontrivially just under the U(1) transformations: In the next section we compute the bosonic spectral action.

Bosonic Spectral Action
The aim of this section is to calculate the bosonic spectral action 5 where χ is some cutoff function, f 0 , f 2 , f 4 are the first three momenta of its Fuorier transform and a 0 , a 2 and a 4 are the first three nontrivial heat kernel coefficients on the manifold without boundary. The "fluctuated" (or covariant) Dirac operator is given by: where the covariant derivative ∇ µ involves the gauge and the Levi-Civita spin connections, whilst the 96 by 96 matrix M is nothing but the "fluctuated" version of D F , which is obtained from (2.17) via the prescription (3.6).
Comment: We notice that the asymptotic expansion (4.1) correctly describes the behaviour of the trace in the left hand side of (4.1) at the energies below the cutoff scale Λ, whilst the high momenta behaviour of the bosonic spectral action is drastically different [28]: high momenta bosons do not propagate, see also [29]. Physically it means that this model becomes strongly coupled at the energies above Λ in both U (1), SU (2) and SU (3) sectors. A similar high energy phase transition has been considered beyond the scope of the noncommutative geometry, see e.g. [30,31]. In what follows we do not discuss the high momenta regime and the mentioned above effects, so from now on the ansatz in the right hand side of (4.1) is identified with the definition of the bosonic spectral action.
We emphasise that the gauge content of these formalism is identical to the one of [5], therefore if one sets ∆ u,d,L = 0, Ω = 0 and S = 0 our operator D will coincide with the one of [5], hence it is sufficient to calculate the difference

Computational simplifications
The structure of the heat kernel coefficients on manifolds without boundaries is very well known (see e.g. [32]), and one can easily see that the scalar fields can contribute to a 2 through the combination: and to a 4 through the combination: and R stands for a scalar curvature. Note that the a 2 contribution can not contain covariant derivatives of the scalar field: the simplest scalar contribution which involves the scalar fields and their covariant derivatives has the the canonical dimension 3, whilst the integrand in (4.4) must have the canonical dimension 2. Therefore, to compute a contrib 2 is sufficient to neglect the dependence of scalars on coordinates. Now let us focus on the a 4 contribution. The computation of the scalar contribution to a 4 drastically simplifies, when the Dirac operator transforms in a homogeneous way upon the local Weyl transformation of the metric tensor and of the scalar fields. Even though the Dirac operator D does not exhibit this property (since it contains the constant Majorana mass terms for the right handed neutrinos) one can write: where the "intermediate" Dirac operatorD is obtained from D via the replacement of the constant M R by the scalar field σ. This field has no gauge indexes and it has already been considered in the context of the model to fix the Higgs mass in [12]. We emphasise that for the scope of the present article this field is needed at the intermediate step only, and by the end of the day it will be replaced by the constant M R . Upon the local Weyl transformation and one can easily check (using the method of conformal variations, see for example [32]) that the fourth heat kernel coefficient which is associated with/ D 2 is Weyl invariant. On the other side all heat kernel coefficients are gauge invariant. The only Weyl and gauge invariant combination of scalar fields of the dimension four which involves the derivatives is 6 : tr D µ (scalar field) † D µ (scalar field) − 1 6 R tr (scalar field) † (scalar field) , (4.10) thus it is sufficient to compute the coefficient in front of R(scalar field) † (scalar field), whilst the kinetic term, which contains all the covariant derivatives D µ , can be restored from (4.10). Note that for such a computation it is sufficient to consider constant scalar fields: ∂ µ (scalar) = 0 and set the gauge connection to zero. Since the same simplification is applicable for the a 2 contribution, let us assume it for a while. Using the well known Lichnerowics formula one can easily check that in our "simplified" regime the endomorphism E, which enters in (4.4) and (4.5) equals to: so the calculation of the new terms of the bosonic spectral action reduced to an algebraic exercise: one has to calculate tr M 2 and tr M 4 . We remind that all the terms which disappeared because of our simplification can be recovered via the Weyl and the gauge invariance of a 4 .

Relevant traces
One can check by a direct computation using e.g. Maple, the following formulas: and where the constants y 1 ,...,y 7 , z 1 ,...,z 25 depend on the Yukawa couplings as follows:

The full bosonic spectral action
Substituting (4.13), (4.14), (4.11) and (4.12) in (4.4) and (4.5), recovering the dependence on the derivatives and on the gauge fields according to (4.10) and setting σ = M R we arrive to the following answer for the new terms in the bosonic spectral action: This is the result of the spectral action computation with the new fields coming form the Clifford requirement grading.

Towards the physical action.
In this section we discuss how to make our spectral action applicable in a physical context. In order to do this one has to carry out two important steps: • get rid of the redundant fermionic degrees of freedom, • make the action Lorentzian.
The redundancy is usually solved projecting out the extra degrees of freedom [5,24,33], while the Euclidean vs. Lorentzian issue has several ramifications (see for example [34][35][36][37]), but the usual method is to perform a Wick rotation. We have shown in [25] that the two issues are intimately related, and given a prescription on how to deal with them.

General prescription: a review and discussion.
Now we briefly recall how the Wick rotation works following [25]. In order to pass from the Euclidean to a Lorentzian theory, each expression F which involves the vierbeins e a µ has to be transformed according to the following rule: As it was demonstrated in [25], upon the transformation (5.1) the Euclidean bosonic action S E bos , which comes out from the first three nonzero heat kernel coefficients, perfectly transforms into the "textbook" Lorentzian action S M bos , in particular where the metric tensors g E µν and g M µν have the signatures {+, +, +, +} and {+, −, −, −} respectively. We refer the reader to the quoted reference for the details.
A treatment of the fermionic action is more subtle, since the product space H contains extra degrees of freedom. Now we briefly recall what the problem is. The Hilbert space H of the almost commutative geometry has the following structure: variables with the index "c", which indicates the charge conjugated field: the charge conjugated spinor is obtained from the original one via the the charge conjugation operation (i.e. they are not independent variables, see (5.12) below). This other doubling is called in [25] the "charge conjugation doubling". In order to get rid of the mirror doubling one has to extract the particles with the correct chirality. The left ψ L and the right ψ R chiral spinors are by definition the eigenstates of the left and the right chiral projectors: In order to get rid of the redundant fermionic degrees of freedom with the wrong chirality one has to extract just left chiral fermions from H L and H c R and just right chiral fermions from H R and H c L , where we took into account the fact that for the physical fermions, which live in Lorentzian space-time, the antiparticles have the opposite chirality with respect to the original particles. So the subspace H + of H which contains just the fermions with correct chiralities has the following structure: In the original paper [5] such an extraction was presented in the form where the projector P + is defined via the grading as follows: In this formula γ st F stands for the "standard" grading introduced in [5]. Since we are working with the different grading γ F , in order to arrive to the correct subspace (5.5) the connection between the projector P + and the grading γ F takes a slightly different form: The Euclidean fermionic action introduced in [5], which is free of the mirror doubling reads: As [25] shows, after the Wick rotation of the vierbeins (5.1) one obtains: where the "intermediate" fermionic action S M doubled F is already Lorentz invariant. However, due to the charge conjugation doubling, it depends on twice more fermionic fields than it is needed, it is not real and therefore it is not suitable for the canonical quantisation. In order to complete a construction of the physical fermionic action one has to eliminate the charge conjugation doubling via the following identification of the variables in the action S M doubled F from the subspaces H c L and H c R with the variables from H L and H R : where the operation C M is the charge conjugation operation, which acts on the arbitrary spinor ψ as follows: Note that in contrast to the Euclidean charge conjugation J the operation C M changes a chirality. We emphasise that the identification ( Following [25] we remind that the last step must be performed before the quantisation: otherwise one will get an additional Pontrtyagin gauge action which comes out from the abelian axial anomaly. Below we apply these prescriptions to our model and we will find out a nontrivial outcome.

This model
Let us parametrise the elements of the Hilbert space H as follows: This is basically the parametrisation (2.10) of the elements of H F , the change of typeface indicates that the elements of H are spinors, no longer complex numbers. In these notations u R is a collection of 4-component spinors which transforms upon the action of the gauge group as the right handed quarks, u c R is an independent collection of 4-component spinors which transforms upon the action of the gauge group as the charge conjugated right handed quark field and so on. The typical element of H + , which is constructed according to (5.5), then becomes: 16) Composing the fermionic action (5.9), applying the Wick rotation procedure (5.1), removing the charge conjugation doubling according to the general prescription (5.11) and finally carrying out the axial transformation (5.13) we see that: where S M F is given by: In this formula for an arbitrary spinor ψ the bar stands for the Dirac conjugation:ψ ≡ ψ † γ 0 . Note that after the Wick rotation accompanied by the elimination of the fermionic quadrupling just the multiplets of the sructures [ψ R ] R and [ψ L ] L remain in the result, therefore we simplified the notations replacing them by ψ R and ψ L respectively. We now come to an important point of this noncommutative geometric construction. Note that the fields ∆ u , ∆ d , ∆ L and S, which are present in the Dirac operator and hence in the bosonic spectral action are absent in the fermionic action (5.18)! Let us clarify what has happened. If one looks carefully at the structure (J Ψ) † DΨ, Ψ ∈ H one immediately finds out that the mentioned fields always appear in interactions terms in the action (vertices) which involve spinors with unphysical chiralities. Therefore, when one restricts the fermions just to the "good-chirality" subspace H + , all these terms vanish. Indeed, the S field enters in (J Ψ) † DΨ in particular via the combination: When one restricts Ψ ∈ H + this expression turns into where we took into account the fact that the chiral projectors P L = 1 2 (1 4 − γ 5 ) and P R = Upon the restriction Ψ ∈ H + this expression turns into One can easily check that all other terms, which involve the ∆-fields vanish as well in the similar way.
6 Conclusions and Outlook.
Based on the purely algebraic idea to incorporate the Clifford structure in the finite dimensional spectral triple, proposed in [23], we arrive to a set of new scalar fields in the minimal version of the noncommutative standard model. Some of the new scalar fields (viz ∆ u,d,L and S) carry both colour and the weak isospin indexes. The fields of such a kind are of interest of recent phenomenological research, in particular the scalar lepto-quarks are the case [38].
We computed the new terms in the bosonic spectral action which come out from these fields. The equation (4.16) is one of the main results of this article. The scalar-scalar couplings between the new fields and the Higgs field may improve the minimal noncommutative standard model from the phenomenological point of view: they give positive contributions to the beta function of the Higgs self-interaction quartic constant at the level of the one loop [39], what is needed to avoid the vacuum instability problem [40,41].
We did not discuss in detail in this paper the possible phenomenological consequences of these new terms. The whole approach to the standard model based on noncommutative geometry is now reaching the level to be confronted with phenomenology, and of course the scalar sector seems to be of paramount importance. The new fields discussed here may possibly be part of this, but more work is necessary in this direction.
The approach is interesting from the mathematical point of view as well. It turns out that some of the new fields (viz ∆ u,d,L and S) are coupled to the spurious fermionic degrees of freedom, whose presence is due to the "product-based" construction of the almost commutative spectral triple. Therefore this model exhibits a very peculiar property, which is the another important result of this article. On the one side these fields do not enter in the physical (Minkowskian) fermionic Lagrangian (5.18), even though they appear in the Euclidean NCG Dirac operator. On the other side the physical ("Wick rotated" to the Lorentzian signature) bosonic spectral action keeps memory about these extra degrees of freedom: it depends on ∆ u,d,L and S. Therefore the fermionic quadrupling in the spectral approach is not just the presence of the extra fermions to be projected out: by the end of the day it effects nontrivially the bosonic action of the model, without altering the fermionic action.
In this article we considered an evolution of the spectral approach from an algebraic point of view. There are other interesting mathematical directions which can be taken. In particular we consider the manifold M without boundary, manifolds with boundaries have been considered within the spectral action formalism as well [42][43][44]. Recently another purely spectral feature has been discovered: parity anomaly on four dimensional manifolds with boundaries [45,46]. It would be interesting to understand the role played by the parity anomaly in the context of the spectral action approach, and the issue will deserve further scrutiny.