A Dark Sector Extension of the Almost-Commutative Standard Model

A Dark Sector Extension of the Almost-Commutative Standard Model

Christoph A. Stephan 111Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, Potsdam, Germany

We consider an extension of the Standard Model within the frame work of Noncommutative Geometry. The model is based on an older model [St09] which extends the Standard Model by new fermions, a new -gauge group and, crucially, a new scalar field which couples to the Higgs field. This new scalar field allows to lower the mass of the Higgs mass from GeV, as predicted by the Spectral Action for the Standard Model, to a value of GeV. The short-coming of the previous model lay in its inability to meet all the constraints on the gauge couplings implied by the Spectral Action. These shortcomings are cured in the present model which also features a “dark sector” containing fermions and scalar particles.

1 Introduction

Noncommutative Geometry has in the last two decades proved to be of considerable interest for particle physics. The construction of the Standard Model of particle physics in terms of spectral triples [C94, CM08] provides deep insights into the geometric nature of high energy physics. In conjunction with the Spectral Action [CC97] on obtains a highly predictive, mathematically sound foundation of particle physics. For readers interested in an introduction to the field we recommend [Sc05a] and [DS12].

The geometrical basis for the construction of particle models is provided by almost-commutative geometries consisting of a spectral triple on a compact Riemannian manifold to model space(time) and an internal space constructed from a matrix algebra. The geometry and the dynamics of the models are encoded in a generalised Dirac operator which comprises the Dirac operator on the manifold, covariant derivatives with respect to the gauge group of the model and scalar fields (and Dirac or Majorana masses) with their Yukawa matrices. The particle content of the model and the interactions are fixed by the matrix algebra and its representation on the spinor space. The dynamical part of the theory is given by the Dirac action and the Spectral Action. Interpreting the Dirac Action and the Spectral Action as an effective action, valid at some cut-off energy, imposes further constraints on the model’s coupling constants. These constraints make the Spectral Action highly predictive, namely for the pure Standard Model one predicts the mass of the Higgs boson to be GeV, [CC97]. This value has recently been shown to be too high [ATLAS12, CMS12], the Higgs has a mass of GeV.

Although the Standard Model takes a prominent place [ISS04, JS08, CC08] within the possible almost-commutative geometries one can go further and construct models beyond the Standard Model. Finding extension within the almost-commutative framework of the classical algebra and with respect to the classical axioms [C96] is dating back to the beginnings of the field, [PS97, PSS99, SZ01]. In these extensions the Standard Model algebra or the Dirac operator were only mildly enlarged to include for example lepto-quarks. More recently the techniques from the classification scheme developed in [ISS04] were used to enlarge the models further, [St06a, St07, SS07, St09]. Here the model in [St09] will be of central interest, since it predicted approximately the correct Higgs mass.

In the case of finite spectral triples of KO-dimension six [B07, C06, CCM07] another classification leads to more general versions of the Standard Model algebra [CC08, CC10] under some extra assumptions. Considering the first order axiom, see Appendix B, as being dynamically imposed on the spectral triple one finds a Pati-Salam type model [CCS13]. From the same geometrical basis one can promote the Majorana mass of the neutrinos to a scalar field [CC12, DLM13] which allows to lower the Higgs mass to its experimental value. Another recent line of research which is aiming at supersymmetric extensions of the Standard Model has been started in [BS11, BS12]. The inclusion of grand unified theories seems to necessitate non-associative spectral triples [W97, FB13] and has recently gained interest again. One can furthermore include torsion [HPS10, PS12] and also impose scale invariance introducing dilaton fields [CC06, AKL11].

For the present paper we concentrate on a variation of the model in [St09]. The original model contains a new scalar field mixing with the Standard Model Higgs. Unfortunately it does not allow to meet all the constraints imposed by the Spectral Action. It was a long held believe [LS01] that fluctuations of the Dirac operator by centrally extended lifts allow to eliminate such constraints for the gauge couplings of abelian gauge groups. This claim is not true and so the problem of the non-matching triangle for the high energy values of the gauge couplings of the Standard Models prevails. We will show in this paper how to construct a model that overcomes these short comings by enlarging the particle content while respecting the axioms of noncommutative geometry. The model is compatible with the measured value of the Higgs mass, has a matching of the running gauge couplings consistent with the constraints from the Spectral Action and ensures that the potential of the scalar fields remains stable up to the Planck mass. Furthermore the new particles may have interesting phenomenological implications, perhaps as candidates for dark matter.

In section 2 we will give a short introduction into the differential geometric setting of generalised Dirac operators, in particular those of Chamseddine-Connes type, and the Spectral Action. Section 3 gives details of the model covering the gauge group, its representation on the fermionic Hilbert space and the Lagrangian of the model. Furthermore the high energy boundary conditions of the Spectral action are computed. The numerical analysis for a point in the parameter space of the model is carried out in section 4. Appendix A provides details on the normalisation conventions of the Yang-Mills sector and in Appendix B the details of the underlying spectral triple are discussed. The reader mainly interested in the physical model can concentrate on the sections 3 and 4.

2 The Spectral Action

From the differential geometric point of view one can consider the Dirac operator of an almost-commutative spectral triple as a generalised twisted Dirac operator on a Riemannian manifold . So any particle model for which the fermionic action is expressible in terms of such a generalised Dirac operator qualifies to be investigated from the spectral point of view, i.e. it may be worthwhile to extend the Spectral Action principle to particle models that are not necessarily based in noncommutative geometry. In these models of particle physics the matter content is encoded in a Hermitian vector bundle equipped with a connection . The Levi-Civita connection induces a connection on the spinor bundle , which we denote again by . A symmetric Dirac operator on the twisted bundle is defined in terms of the Levi-Civita connection and :


for any positively oriented orthonormal frame , any section of and any section of . For the twisted Dirac operator we get


where is the Laplacian associated to the connection

is the scalar curvature of the manifold and is the curvature of . The curvatures of , and are related as:


The fluctuated Dirac operator can then be expressed in terms of a selfadjoint endomorphism on added to the twisted Dirac operator,


where denotes the volume element. The Higgs endomorphism encodes the scalar fields, the Yukawa masses or couplings and Dirac mass terms. For one has the Lichnerowicz formula


where the potential is given by


The Spectral Action for is defined as the number of eigenvalues in an interval for some cut-off energy . The Spectral Action is a counting function and can be expressed as with a suitable cut-off function which has support in the interval and is constant near the origin. Performing a Laplace transformation and a heat kernel expansion [CC97, NVW02] one finds the asymptotic expression


as . Here , and are the first moments of the cut-off function . We will consider them as free parameters to be determined by experiment. Using Gilkey’s general formulas [Gi95] one obtains for the following Seeley-DeWitt coefficients [CC97, IKS97, CCM07, CC10]


with the abbreviations and .

Assume now that the principle fibre bundle of the particle model has the structure group with subgroups either equal to or with connection one-forms and curvature two-forms . Then the Yang-Mills action is given by the normalisation


where denotes the respective gauge coupling. The details for the choice of the Lie algebra basis are given in Appendix A. Furthermore assume that encodes complex scalar (multiplet) fields . Then the scalar part of the action is normalised to


where the mass terms can be absorbed into the cosmological term which is identified with . The scalar fields obtained from the fluctuations of the Dirac operator have mass dimension zero while the Yukawa mass matrices have mass dimension one. The dynamical term in (12) allows to determine the normalisation for the physical scalar fields with mass dimension one in terms of the Yukawa mass matrices and scalar fields . Imposing the standard normalisation of scalar-spinor interaction terms in the Dirac action at eliminates the Yukawa mass matrices in favour of the Yukawa coupling matrices with mass dimension zero:


If the Higgs endomorphism of the Dirac operator is constructed directly in terms of Yukawa couplings and scalar fields with mass dimension one it is not necessary to invoke the normalisation relation (13).

The standard normalisation of the Einstein-Hilbert action and for the coupling of the scalar curvature to the scalar fields implies


where the are real parameters.

The physical normalisation relations (11), (12), (13) and (14) imply relations among the gauge couplings, the quartic scalar couplings and the Yukawa couplings [CC97, T03]. Since these relations are not stable under the renormalisation group flow one interprets the Spectral Action as an effective action valid at a cut-off scale . Ideally this cut-off scale can be chosen such that all relations among the couplings are fulfilled at while the low energy values, say at the Z-boson mass, agree with experimental data.

3 The Model

The model we are investigating here can be either formulated directly in terms of the generalised twisted Dirac operator (4) or it can be constructed from a spectral triple, see Appendix B. The internal Hilbert space of the model extends the Standard Model Hilbert space [CC97] by generations of vector-like -particles, chiral - and -particles and vectorlike -particles. The structure group of the Standard Model is enlarged by an extra subgroup, so the total group is . The Standard Model particles and the -particles are neutral with respect to the subgroup while the -particles are neutral with respect to the Standard Model subgroup . Furthermore the model contains two scalar fields: a scalar field in the Standard Model Higgs representation and a new scalar field carrying only a charge.

The Hilbert spaces of the fermions and the scalar fields expressed in terms of their detailed representations of are written out in detail in the following list. The subscripts indicate the particle species in question, e.g. , for quarks and leptons, , for up- or down-type quarks and , for electron- or neutrino-type leptons. The superscript indicates that we restrict to the particle sector for brevity.


Here we chose the standard normalisations of [MV83, MV84, MV85] for the representation, i.e. the charges, of the abelian subgroups. The details for the Higgs endomorphism of the Dirac operator are given in Appendix B.

Let us now summarise the terms that will comprise the Lagrangian (21) of the model. Readers not interested in the computational details of the Spectral Action or the construction of the spectral triple can take the Lagrangian in this subsection as a definition and as a starting point for phenomenological investigations. Together with the boundary conditions at the cut-off scale, (26), (30), (32), (34) and (36), the Lagrangian (21) constitutes the main physical content of the model as an effective field theory.

Taking the Dirac inner product as defined in [CCM07] we can decompose it into the fermionic subspaces in (15). In this way we get the fermionic Lagrangian of the Standard Model in its euclidean formulation, the fermionic Lagrangian for the -particles,


and the fermionic Lagrangian of the - and the - particles


The Dirac operators of the form and in (17) and (18) are the twisted Dirac operators with gauge covariant derivatives according to the representations in (15). Observe that the mass terms of the -particles and the -particles are given by Dirac mass matrices since their charges are left-right symmetric. The mass terms of the -particles are induced by the new scalar field , where we have taken the liberty to normalise the scalar field to mass dimension one and the Yukawa couplings to mass dimension zero. If the Dirac operator is obtained from fluctuations, see Appendix B, then the mass dimension of the scalar fields is zero and we have Yukawa mass matrices. The formal justification for passing from one normalisation to the other will be given in the next section.

The Yang-Mills Lagrangian for the Standard Model subgroup takes again its usual form and for the subgroup we add


where is the field strength tensor for the -covariant derivative. For details on the normalisation see Appendix A. Finally the Lagrangian of the scalar fields


contains the dynamical terms as well as the the symmetry breaking potential with interaction term of the -field and the -field. The normalisation is again chosen according to [MV83, MV84, MV85] in order to obtain Lagrangians of the form for the real valued fields. Putting everything together the Dirac inner product and the Spectral Action (7) provide the Lagrangian


3.1 Dimensionless parameters

The Spectral Action (7) does not only supply the bosonic part of the Lagrangian (21) but also relations among the free parameters of the model. These relations serve as boundary conditions at the cut-off scale for the renormalisation group flow. Let us start with the computation of the Yang-Mills terms in (7) which are encoded in the -term in the fourth Seeley-DeWitt coefficient in (10). Here the trace is taken over the whole inner Hilbert space of the fermions.

-coupling relation

We first calculate the total sum of the squares of the hypercharges , taking into account their multiplicities according to the dimension of the respective fermionic subspaces consisting of the Standard Model particles and the -particles.

The trace of the squared Yang-Mills curvature terms can then be computed for each subgroup of separately, see Appendix A. Inserting the relevant term into the Spectral Action (7) yields after normalising

the relation


between the gauge coupling , the fourth moment of the cut-off function and .

-coupling relation

Proceeding as in the previous paragraph we calculate the trace of the curvature terms in :

The normalisation


yields a relation between the gauge coupling and the fourth moment of the cut-off function .

-coupling relation

Repeating the steps from the curvature terms in for the curvature terms,

we find the relation


Here we would like to note the deviations from the relations that appear in the Standard Model case. We see the additional factors proportional to in (22), (23) and (24) which have their origin in the -particles and the -particles. Similar variations of such relations have already been considered in [St07, SS07, St09].

-coupling relation

For the total sum of the squared charges of the -particles we find

This allows us to write the trace of the curvatures squared in as

and we obtain the relation


Gauge coupling relations

All four relations (22), (23), (24) and (25) depend linearly on the fourth moment of the cut-off function . Therefore we can eliminate to obtain the final relations among , , and at the cut-off scale :


We notice again the deviation of these relations compared to the case of the Standard Model, see [CC97]. Yet the relation remains unchanged, although the actual value of may still deviate from the Standard Model result, since the are charged under the Standard Model subgroup and can therefore change the running of the couplings under renormalisation group flow.


We would like to point out that the relations for the abelian subgroups cannot be normalised away by choosing different numerical values of the -charges in the central extension, see Appendix B. This possibility of reducing the number of boundary conditions was implied in [LS01]. It was used in many succeeding publications, including the publications of the author, see [St07, St09], etc. Yet, the conclusion that the free choice of the -charges implies that the boundary condition of the respective gauge coupling is empty does not hold, since only the product of the (squared) -charges and the gauge coupling is a measurable physical quantity. But this product can be fixed, for example by a measurement of the numerical value or by boundary conditions such as (26). Normalising the value of the -charges forces automatically a reciprocal normalisation of the gauge coupling resulting in the same value for the product. Therefore the boundary conditions of all gauge couplings need to be taken into account.

Yukawa coupling relations

Next we will find relations similar to (26) for the traces of the squared Yukawa coupling matrices and , for the definitions see (76). We compute the normalisation of the kinetic part of for the Higgs endomorphism in in terms of the scalar fields and with mass dimension zero. Inserting this into the Spectral Action (7) and equating this part of the Lagrangian with the standard kinetic Lagrangian for the mass dimension one fields and ,


yields the normalisation in terms of the traces of the squared Yukawa mass matrices (75):


If the Higgs endomorphism had already been given in terms of scalar fields of mass dimension one and Yukawa coupling matrices, then the relations (28) would immediately imply the desired boundary conditions (30) as was first noted in [T03]. In the present case we need a second relation to eliminate the spurious mass scale. This relation is provided by the Dirac action. We write the Higgs endomorphism as a function of the Yukawa mass matrices and the scalar fields of mass dimension one, , and impose equality of the Dirac action with the equivalent normalisation with respect to the Yukawa couplings and mass dimension zero scalar fields. Assuming the endomorphism is linear in the Yukawa matrices and the scalar fields we find that


This implies the boundary conditions


where the last equality follows from (23).

Scalar quartic coupling relations

To obtain the boundary conditions for the quartic couplings , and of the scalar fields we need to compute the quartic part of for the mass dimension one scalar fields and . Inserting again into the Spectral Action (7) and equating with the standard normalisation yields the desired boundary conditions. To simplify matters we split the computation into three parts for each of quartic couplings , and .

Restricting the Spectral Action to the term proportional to we find


where the traces of the fourth powers of the Yukawa mass matrices and the Yukawa coupling matrices are defined in Appendix B, see (77) and (78). The mass scale has been divided out in order to obtain the standard normalisation of the Yukawa couplings [MV83, MV84, MV85]. The boundary condition of which is associated to the quartic potential of the scalar field


follows then with (23). Restricting to the term proportional to gives


which yields, again with (23) the boundary condition of :


Restricting the Spectral Action to the term proportional to


we obtain the boundary condition for the coupling of the interaction term for the two scalar fields: