Noncommutative Geometry and the Physics of the LHC Era

Abstract

Noncommutative geometry allows to unify the basic building blocks of particle physics, Yang-Mills-Higgs theory and general relativity, into a single geometrical framework. The resulting effective theory constrains the couplings of the standard model and reduces the number of degrees of freedom. After an introduction of the basic ideas of Noncommutative geometry, I will present its predictions for the standard model and the few known models beyond the standard model based on a classification scheme for finite spectral triples. Most of these models, including the Standard Model, are now ruled out by LHC data. But interesting extensions of the standard model which agree with the presumed Higgs mass, predict new particles (Fermions, Scalars and Bosons) and await further experimental data.

Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00

1 Introduction

Noncommutative geometry as an algebraic approach to geometry has surprisingly proved to be a useful tool to understand the geometry of the standard model of particle physics. Connes’ spectral triples allow to transfer geometrical notions from compact Riemannian spin manifolds to finite matrix spaces. The central object in this construction is a generalised Dirac operator which in such finite cases can be interpreted as a fermionic mass matrix of the particle model.

It came as a surprise that spectral triples, which were originally constructed to investigate for example singular foliations of manifolds, allow to incorporate the intricate structure of the standard model particle multiplets as well as its complicated representation theoretic structure. Product spaces built from spectral triples over manifolds and from finite spectral triples over matrix algebras proved to be very useful to construct models known from particle physics, in particular the standard model.

The Dirac operator is a central ingredient of a spectral triple and plays a multiple role. On the one hand it substitutes for the metric of the underlying space(-time), on the other hand it defines the dynamics of the fermions. If one constructs spectral triples for the standard model, the Higgs field becomes a natural part of the Dirac operator acting as an inter-twiner of representations. In the noncommutative context such scalar fields as the Higgs field are then interpreted as connections of the internal matrix space and are therefore considered as part of the metric.

In recent years the matrix geometries that underly the standard model have been classified in various ways. As another surprise it was found that the standard model not only fits into the noncommutative framework but is also in many ways a very minimal model and the chiral structure of the electro-weak sector appears to be quite natural in the setting of finite spectral triples.

But the classification schemes also allow to construct particle models beyond the standard model. These models generally have a rich phenomenology and may contribute interesting new insights to the dark matter problem which is probably one of the most important open question in today’s particle physics.

These conference proceedings were written as a supplement to a talk given at conference Quantum Mathematical Physics, A Bridge between Mathematics and Physics which took place in 2014 in Regensburg. The interested reader may profit from the talk and is referred to the web page [1].

The article is organised as follows. Section 2 aims to give an introduction into the notion of a spectral triple, its axioms, almost-commutative spectral triples which are central to the noncommutative approach to particle physics and some classification schemes for spectral triples. Furthermore it gives a short account of how gauge groups appear in the framework and how the relevant Dirac operators appear. In the third section we describe the spectral action which is a natural action functional on spectral triples. We also give an overview on alternative action functionals for Dirac operators. Some of those may naturally be applied in the Lorentzian setting. The fourth section deals with the predictions noncommutative geometry makes for the standard model and its short-comings in the light of recent experimental data, in particular the detection of a Higgs-like boson by the LHC experiments. The fifth section gives examples of models beyond the standard model with a particular focus on an extension which displays new fermions, gauge bosons and a new scalar field. This model seems to agree with experimental data and may provide interesting dark matter candidates. We end this article with some questions to noncommutative geometry and particle physics.

2 Spectral Triples

We try to give a short introduction into the basic concepts of noncommutative geometry as needed for particle physics. For more in-depth introductions we recommend the classical book of Connes [2] and the more recent general overview of Connes and Marcolli [3].

For a pedagogical introduction to spectral triples we recommend Khalkhali’s textbook [4] and for details on the particle physics side of the subject the excellent book by van Suijlekom [5] and the lecture notes of Schücker [6].

2.1 General Definitions

Let us give a brief account on the basic ingredients of noncommutative geometry in the formulation given by Connes [2, 7]. The basic geometric building blocks are spectral triples $(\mathcal{A},\mathcal{D},\mathcal{H})$ which, in the even, real case, are given by the following five components:

• a unital pre-C ^∗ algebra $\mathcal{A}$.

• a separable Hilbert space $\mathcal{H}$ on which the algebra is faithfully represented with the representation $\rho: \mathcal{A}\rightarrow End(\mathcal{H})$.

• an unbounded self-adjoint operator $\mathcal{D}: dom(\mathcal{D}) \rightarrow \mathcal{H}$ with compact resolvent. This operator will be referred to as the Dirac operator.

• an anti-unitary operator $J: \mathcal{H}\rightarrow \mathcal{H}$. In the mathematical literature J is referred to as the real structure while in physics literature it is usually called the charge conjugation operator.

• a unitary operator $\gamma: \mathcal{H}\rightarrow \mathcal{H}$, the abstract volume element or chirality operator.

Example

The standard example of a spectral triple is build on a compact n-dimensional Riemannian spin-manifold (M, g). In this case the algebra $\mathcal{A} = C^{\infty }(M)$ and the Hilbert space consists of the square-integrable spinors $\mathcal{H} = L^{2}(\Gamma (\Sigma ))$. The representation of the algebra on the Spinors is simply by point-wise multiplication. The Dirac operator is the standard Dirac operator associated to the Levi-Civita connection ∇ and is locally given by , where e _i , i = 1, . . , n form an orthonormal basis of the tangent space T _x (M) and X⋅ denotes the Clifford multiplication of a vector. The charge conjugation and the volume element are the standard operators from particle physics.

The five components of the spectral triple are required to fulfil the following set of axioms in order to constitute a spectral triple:

Axiom 1

This axiom assumes that there exists a classical (spectral) dimension n associated to the growth of the eigenvalues of the Dirac operator. It is also referred to as the axiom of finite summability. Since the resolvent of $\mathcal{D}$ is compact its eigenvalues form a decreasing sequence {α _i }. The axiom then states that there is a smallest $n \in \mathbb{N}$ such that the ith eigenvalue is asymptotically for $i \rightarrow \infty$ of the order of $\mathcal{O}(i^{-1/n})$. This smallest n is defined as the dimension of the spectral triple.

Remark 2.1

The dimension of a spectral triple can be zero. Indeed this happens for finite spectral triples build from matrix algebras which play a central rôle in noncommutative approach to particle physics.

Remark 2.2

The dimension of the spectral triple of an n-dimensional Riemannian spin manifold coincides by Weyl’s law for the growth of eigenvalues of with the dimension of the manifold.

Axiom 2

This axiom requires that the commutator of the Hilbert space representation of each element in the algebra with the Dirac operator is a bounded operator on $\mathcal{H}$. So we have for all $a \in \mathcal{A}$ that $[\mathcal{D},\rho (a)] \in \mathcal{B}(\mathcal{H})$.

From now on we will drop the explicit mentioning of the representation ρ when no confusion arises.

Remark 2.3

This axiom certainly holds for the spectral triple of a manifold since the commutator of the Dirac operator with a differentiable function equals Clifford multiplication with its differential, i.e. , $f \in C^{\infty }(M)$ and $\psi \in \Gamma (\Sigma ).$

Axiom 3

The Dirac operator is a first-order operator. In the algebraic setting of spectral triples the axiom states that for all $a,b \in \mathcal{A}$ we have $[[\mathcal{D},a],JbJ^{-1}] = 0$.

Remark 2.4

This axiom can again be checked easily for the spectral triple of a manifold since is a zero-order differential operator for all $f \in C^{\infty }(M)$ and therefore commutes with any function in the algebra.

Axiom 4

The axiom of strong regularity states that the elements of the algebra are differentiable in a suitable way. For any bounded operator $T \in \mathcal{B}(\mathcal{H})$ we define define $\delta (T):= [\vert \mathcal{D}\vert,T]$ where $dom\,\delta =\{ T \in \mathcal{B}(\mathcal{H})\vert T(dom\vert \mathcal{D}\vert ) \subset (dom\vert \mathcal{D}\vert )\,\mathrm{and}\,\delta (T) \in \mathcal{B}(\mathcal{H})\}$. Furthermore we define $\mathcal{H}^{\infty }:=\bigcap _{ j=1}^{\infty }dom\,\mathcal{D}^{j}$.

The axiom demands that $\mathcal{A}$, $[\mathcal{D},\mathcal{A}]$ and $End_{\mathcal{A}}(\mathcal{H}^{\infty })$ are in

$$\displaystyle{B^{\infty }(\mathcal{H}):=\bigcap _{ j=1}^{\infty }dom\,\delta \,^{j}.}$$

Axiom 5

The orientability axiom gives further conditions on the chirality operator. It demands that $\gamma ^{2} = id_{\mathcal{H}}$ and if the dimension n of the spectral triple is even that $\gamma \mathcal{D} = -\mathcal{D}\gamma$. In the odd-dimensional case $\gamma = id_{\mathcal{H}}$. Furthermore we require that γ can be represented by a Hochschild n-cycle. For further details of this axiom we refer to [7].

Remark 2.5

For the spectral triple of a manifold the abstract volume form γ coincides with the Clifford multiplication of the metric volume form dvol _g .

Axiom 6

The reality axiom demands a set of commutation relations for the real structure J with the other operators. In particular we have J ² = ε, $J\mathcal{D} =\epsilon '\mathcal{D}J$ and J γ = ε″γ J where the signs are given in the following table.

Table 1

Here the p is the so called KO-dimension. We require also that $[JaJ^{-1},b] = 0$ for all $a,b \in \mathcal{A}$, i.e. that JaJ ⁻¹ is in the opposite algebra $\mathcal{A}^{op}$.

For the spectral triple of a manifold the KO-dimension is usually taken to be equal to the dimension of the manifold itself. In the finite case of matrix algebras this requirement is dropped and it is precisely the fact that the KO-dimension and the dimension of the spectral triple need not coincide that allows Majorana mass terms for neutrinos.

That the abstract definition of a spectral triple gives for commutative $\mathcal{A}$ an equivalent definition for a compact Riemannian spin manifold is the central content of Connes’ reconstruction theorem. See Connes [7, 8] for an in depth proof of the theorem and Sanders [9] for a very readable introduction.

Note that the definition of a spectral triple does not require that the algebra $\mathcal{A}$ is commutative. It is precisely this possibility to pass to noncommutative algebras which allows to generalise the frame work to noncommutative spaces.

Example

To give an example of how to reconstruct geometric data from a spectral triple let us consider the spectral triple of a manifold

As we have mentioned above, the dimension of the manifold M can be recovered by Weyl’s law from the growth of the eigenvalues of . The metric distance d( p, q), p, q ∈ M, can be reconstructed from Connes’ distance formula where δ _p ( f) = f( p) are δ-distributions. This formula can be generalised to any spectral triple if we replace the δ-distributions by states on the algebra.

Remark 2.6

Suppose we are given the algebra $\mathcal{A}$, the Hilbert space $\mathcal{H}$, the representation of the algebra ρ, the real structure J and the chirality operator γ. Then these data in generally do not uniquely fix the Dirac operator $\mathcal{D}$. In the case of the spectral triple over a Riemannian manifold (M, g) one may for example replace the Dirac operator associated to the Levi-Civita connection by a Dirac operator associated to any connection compatible with the metric g, i.e. by a connection with torsion [10]. One can also add scalar potentials [11] or terms of higher form degrees, as long as they are compatible with the axioms. Here we will always assume that the Dirac operator is associated to the Levi-Civita connection.

Definition 2.7

Given all the components $(\mathcal{A},\mathcal{H},\rho,J,\gamma )$ of a real, even spectral triple save the Dirac operator. Then the configuration space of Dirac operators $\mathcal{C}(\mathcal{A})$ is defined to be

$$\displaystyle{ \mathcal{C}(\mathcal{A}):=\{ \mathcal{D}: dom(\mathcal{D}) \subset \mathcal{H}\rightarrow \mathcal{H}\vert (\mathcal{A},\mathcal{H},\mathcal{D})\;\mathrm{is}\,\mathrm{a}\,\mathrm{real},\,\mathrm{even},\,\mathrm{spectral}\,\mathrm{triple}\}}$$

The configuration space $\mathcal{C}(\mathcal{A})$ is in general too big for applications in physics. So one will usually choose a sub-space which has certain invariance or covariance properties. In the case of particle physics models this will in general be the space of generalised Dirac operator which can be interpreted as twisted Dirac operators associated to suitable principal fibre bundles.

It should be noted that spectral triples are tailored to describe Riemannian manifolds. But it is quite clear that to model physical space-time one would need a Lorentzian equivalent for a spectral triple. The endeavour to find a suitable replacement in the Lorentzian case proves to be an extremely difficult task. Some proposals have been made by Strohmaier [12], Paschke and Sitarz [13], Besnard [14], Franco and Eckstein [15] and others. But no conclusive definition has been achieved, yet. It seems that a major obstacle in the definition is the fact, that on a Lorentz manifold the Dirac equation has no solutions in a suitable Hilbert space. Solutions spaces of wave operators consist generally of tempered distributions and are therefore not square integrable.

2.2 Almost-Commutative Geometries

Spectral triples have the nice property that the tensor product of two spectral triples is again a spectral triple. This also holds for the even, real case with real structure and chirality operator discussed above. Given two spectral triples $(\mathcal{A}_{1},\mathcal{D}_{1},\mathcal{H}_{1},J_{1},\gamma _{1})$ and $(\mathcal{A}_{2},\mathcal{D}_{2},\mathcal{H}_{2},J_{2},\gamma _{2})$ one constructs a new spectral triple $(\mathcal{A},\mathcal{D},\mathcal{H},J,\gamma )$ with the data

• $\mathcal{A}:= \mathcal{A}_{1} \otimes \mathcal{A}_{2}$

• $\mathcal{H}:= \mathcal{H}_{1} \otimes \mathcal{H}_{2}$

• J: = J ₁ ⊗ J ₂

• γ: = γ ₁ ⊗γ ₂

• $\mathcal{D}:= \mathcal{D}_{1} \otimes id_{\mathcal{H}_{2}} +\gamma _{1} \otimes \mathcal{D}_{2}$

If we want to construct particle models from spectral triples it turns out that such products are very useful. In reminiscence to Kaluza-Klein theory one takes the spectral triple of a compact Riemannian spin manifold and tensorises with a finite spectral triple. A finite spectral triple is defined as follows.

Definition 2.8

A finite spectral triple is given by a matrix algebra with m summands $A_{f}:=\bigoplus _{ j=1}^{m}\mathrm{Mat}_{j}(n_{j}, \mathbb{K}_{j})$, where $n_{j} \in \mathbb{N}$, $\mathbb{K}_{j} = \mathbb{R}, \mathbb{C}, \mathbb{H}$, a finite Hilbert space $H_{f}:= \mathbb{C}^{N}$ and a Dirac operator $D_{f} \in \mathrm{ Mat}(N, \mathbb{C})$ such that (A _f , H _f , D _f ) with suitable real structure J _f and γ _f constitute a zero-dimensional spectral triple of KO-dimension p.

One of the reasons why such finite spectral triples are useful in particle physics is the fact that the group of unitaries $U(A_{f}):=\{ u \in A_{f}\vert u^{{\ast}}u = uu^{{\ast}} = id\}$ of the matrix algebra can be identified, when properly lifted to the automorphisms of the Hilbert space, with the structure group underlying the particle model. One can then interpret the tensor product of the L ²-spinors and the finite Hilbert space as a twisted spinor bundle associated to the fibre bundle product of the spin structure and a trivial vector bundle associated to a (trivial) principle fibre bundle associated to said structure group. There are at the moment two approaches to define the structure group of the particle model and we will come back to this point later.

In the context of particle physics the Hilbert space is the finite Hilbert space of particle multiplets. In this case its rank N is even and one has to count particles and anti-particles as well as left-handed and right-handed particles separately. Consequently the Hilbert space is four times too big and the construction of the Dirac action requires a projection on the physical Hilbert space. The Dirac operator plays the rôle of the fermionic mass matrix and encodes the Yukawa couplings as well as possible Dirac or Majorana mass terms. The real structure is then indeed the charge conjugation operator from particle physics and the volume form can be identified with the chirality operator that allows to project on left- and right-handed spinors. It turns out that the KO-dimension should be chosen to p = 6 so we will only consider this case. This choice of KO-dimension allows for Majorana masses for right-handed neutrinos in the standard model, Barrett [16] as well as Chamseddine, Connes and Marcolli [17]. But such Majorana masses are in conflict with the axiom of orientability [18]. KO-dimension six also implies that the number of summands in the matrix algebra A _f has to be even.

Definition 2.9

Given the spectral triple of a 4-dimensional Riemannian spin manifold and a finite spectral triple, we call their tensor product an almost-commutative spectral triple.

Example

The standard example of such an almost-commutative geometry is the spectral triple of the standard model. We will give the finite matrix algebra, its representation on the Hilbert space and the Dirac operator in some detail to illustrate the general structure of particle models in terms of spectral triples.

The finite algebra can be chosen to be

$$\displaystyle{A_{f} = \mathbb{C}\, \oplus \, \mathbb{H}\, \oplus \, M(3, \mathbb{C})\, \oplus \, \mathbb{C}.}$$

We see that the group of unitaries is U(A _f ) = U(1) × SU(2) × U(3) × U(1). The correct way to deal with the supplementary U(1) terms to obtain the standard model structure group G _SM  = U(1) × SU(2) × SU(3) will be explained later. The finite Hilbert space is taken from particle physics. It splits into four subspaces $H_{f}:= H_{L} \oplus H_{R} \oplus H_{L}^{c} \oplus H_{R}^{c} = \mathbb{C}^{96}$ which are given by

$$\displaystyle{H_{L} = (\mathbb{C}^{2} \otimes \mathbb{C}^{3} \otimes \mathbb{C}) \oplus (\mathbb{C}^{2} \otimes \mathbb{C}^{3}\mathbb{C}^{3})}$$

and

$$\displaystyle{H_{R} = (\mathbb{C} \otimes \mathbb{C}^{3} \otimes \mathbb{C}) \oplus (\mathbb{C} \otimes \mathbb{C}^{3} \otimes \mathbb{C}) \oplus (\mathbb{C} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3}) \oplus (\mathbb{C} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3}),}$$

where the superscript ⋅ ^c denotes the anti-particle spaces. One can choose as a basis for the Hilbert space the particle multiplets of the standard model, i.e.

$$\displaystyle{\left (\begin{array}{*{10}c} e\\ \nu _{e } \end{array} \right )_{L},\;\left (\begin{array}{*{10}c} \mu \\ \nu _{\mu } \end{array} \right )_{L},\;\left (\begin{array}{*{10}c} \tau \\ \nu _{\tau } \end{array} \right )_{L},\;\left (\begin{array}{*{10}c} u\\ d \end{array} \right )_{L},\;\left (\begin{array}{*{10}c} c\\ s \end{array} \right )_{L},\;\left (\begin{array}{*{10}c} t\\ b \end{array} \right )_{L},}$$

for the left-handed SU(2)-doublets and

$$\displaystyle{e_{R},\;\nu _{eR},\;\mu _{R},\;\nu _{\mu R},\;\tau _{R},\;\nu _{\tau R},\;u_{R},\,d_{R},\;c_{R},\;s_{R},\;t_{R},\;b_{R}}$$

for the right-handed SU(2)-singlets.

Furthermore we have the following representation ρ of algebra elements

$$\displaystyle{(a,b,c,d) \in \mathcal{A}_{f} = \mathbb{C} \oplus \mathbb{H} \oplus M_{3}(\mathbb{C}) \oplus \mathbb{C}}$$

which also decomposes into for direct summands ρ = ρ _L ⊕ρ _R ⊕ρ _L ^c ⊕ρ _R ^c that are given by

$$\displaystyle{\rho _{L}(a,b,c,d) = \left (\begin{array}{*{10}c} b \otimes 1_{3} \otimes 1_{3} & 0 \\ 0 &b \otimes 1_{3} \end{array} \right ),\quad }$$

$$\displaystyle{\rho _{R}(a,b,c,d) = \left (\begin{array}{*{10}c} a1_{3} \otimes 1_{3} & 0 & 0 & 0 \\ 0 &\bar{a}1_{3} \otimes 1_{3} & 0 & 0 \\ 0 & 0 &\bar{d}1_{3} & 0 \\ 0 & 0 & 0 &\bar{a}1_{3} \end{array} \right ),}$$

$$\displaystyle{\rho _{L}^{c}(a,b,c,d) = \left (\begin{array}{*{10}c} 1_{2} \otimes 1_{3} \otimes c& 0 \\ 0 &d \otimes 1_{3} \otimes 1_{2} \end{array} \right )\quad }$$

and

$$\displaystyle{\rho _{R}^{c}(a,b,c,d) = \left (\begin{array}{*{10}c} 1_{3} \otimes c& 0 & 0 & 0 \\ 0 &1_{3} \otimes c& 0 & 0 \\ 0 & 0 &d1_{3} & 0 \\ 0 & 0 & 0 &d1_{3} \end{array} \right ).}$$

The real structure is

$$\displaystyle{J_{f} = \left (\begin{array}{*{10}c} 0 &1_{48} \\ 1_{48} & 0 \end{array} \right )\circ \mathrm{complex}\;\mathrm{conjucation}}$$

and the chirality operator is the diagonal matrix $\gamma _{f} =\mathrm{ diag}(-1_{24},1_{24},1_{24},-1_{24})$. Note that the signs discriminating left-handed and right-handed particles change from the particle sector to the anti-particle sector. This is due to the fact that we construct a spectral triple of KO-dimension six.

A central object in the whole construction is the Dirac operator D _f . It maps left-handed particles to right-handed particles and vice versa but also contains Majorana mass terms which map right-handed neutrinos to right-handed anti-neutrinos. Therefore the configuration space of Dirac operators consists of matrices $D_{f} \in \mathcal{C}(A_{f}) \subset Mat(96, \mathbb{C})$ that split into four blocks

$$\displaystyle{D_{f} = \left (\begin{array}{*{10}c} \Delta &H\\ H^{{\ast} } & \bar{\Delta } \end{array} \right )}$$

where each block is a matrix, i.e. $\Delta,H \in Mat(48, \mathbb{C})$. The Dirac operator is selfadjoint with respect to the standard inner product on $H_{f} = \mathbb{C}^{96}$ and its block $\Delta$ decomposes into the following sub-matrices

$$\displaystyle{\Delta = \left (\begin{array}{*{10}c} 0 &M\\ M^{{\ast} } & 0 \end{array} \right )}$$

with

$$\displaystyle\begin{array}{rcl} M& =& \left (\left (\begin{array}{*{10}c} 1&0\\ 0 &0 \end{array} \right ) \otimes M_{e} + \left (\begin{array}{*{10}c} 0&0\\ 0 &1 \end{array} \right ) \otimes M_{\nu }\right ) {}\\ & & \oplus \left (\left (\begin{array}{*{10}c} 1&0\\ 0 &0 \end{array} \right ) \otimes M_{u} \otimes 1_{3} + \left (\begin{array}{*{10}c} 0&0\\ 0 &1 \end{array} \right ) \otimes M_{d} \otimes 1_{3}\right ).{}\\ \end{array}$$

Here we have the lepton mass matrices M _e  = diag(m _e , m _μ , m _τ ) and $M_{\nu } = C_{PMNS}\,\mathrm{diag}(m_{\nu _{e}},m_{\nu _{\mu }},m_{\nu _{\tau }})$ with the unitary PMNS-mixing matrix for the neutrinos as well as the quark mass matrices M _u  = diag(m _u , m _c , m _t ) and M _d  = C _CKM  diag(m _d , m _s , m _b ) with the unitary CKM-mixing matrix. Since the Majorana mass matrix H contains mainly entries equal to zero, apart from a 3 × 3 sub-matrix for the Majorana masses of the right-handed neutrinos, we will not give the details and refer to [17].

The final spectral triple of the standard model is then the tensor product of the finite spectral triple given above and the commutative spectral triple of a 4-dimensional compact Riemannian spin manifold . The almost commutative algebra $C^{\infty }(M) \otimes A_{f}$ can be seen as the algebra of smooth functions on M with values in the matrix algebra A _f . We notice that the (lifted) unitary group of $C^{\infty }(M) \otimes A_{f}$ is coordinate-dependent and therefore it can be identified with the gauge group of a (trivial) principal fibre bundle.

2.3 Classifications of Finite Spectral Triples

In consideration of the complexity of the standard model’s finite spectral triple one could (and should) ask the question how unique or at least how special this model is under certain extra assumptions. It therefore appears worthwhile to classify all possible finite spectral triples and, if the general classification turns out to yield too many spectral triples, find suitable conditions to single out spectral triple of interest to particle physics.

A complete and general classification has been performed independently by Paschke and Sitarz [19] and by Krajewski [20]. Although the possible particle models that can be built from finite geometries are strictly less than the models which can be constructed in the classical formalism based on fibre bundles, it is nevertheless a far too big set. We will briefly comment here on two proposals to cut down this set of all finite spectral triples to sub-sets of potential physical interest.

Let us first mention a recent classification scheme proposed by Connes and Chamseddine [21]. In this approach extra mathematical conditions are imposed on the finite algebra and its representation. One assumes that the action of the algebra has a separating vector and that the representation, i.e. the Hilbert space is irreducible. The second conditions demands that there are no nontrivial linear projections acting on the Hilbert space which commute with the representation of the algebra and the action of the real structure.

Applying these conditions to a finite matrix algebra one is essentially left with two summands over the complex numbers, i.e. $A_{f} = Mat(k, \mathbb{C}) \oplus Mat(k, \mathbb{C})$. If one imposes as a further condition that the algebra should be symmetric under a certain symplectic symmetry, one finds that only algebras of the type $A_{f} = Mat(r, \mathbb{H}) \oplus Mat(k, \mathbb{C})$ are allowed. The commutation relations with the chirality operator then require that r is an even number and the first realistic model turns out to have the algebra $A_{PS} = \mathbb{H} \oplus \mathbb{H} \oplus Mat(4, \mathbb{C})$.

We note that A _PS has a three-summand version of the standard model algebra as a sub-algebra, namely $\mathbb{C} \oplus \mathbb{H} \oplus Mat(3, \mathbb{C}) \subset \mathbb{H} \oplus \mathbb{H} \oplus Mat(4, \mathbb{C})$. It was shown by Connes, Chamseddine and van Suijlekom [22] that the particle model of the corresponding spectral triple is the well known Pati-Salam model.

An earlier classification scheme was proposed by Iochum, Jureit, Schücker and Stephan [23]. Here the idea was to put up a list of restrictions on the resulting particle models, that appear necessary from the point of view of particle physics. The models obtained from this classification proved to be very useful for a bottom-up approach in model building which has been exploited in [24, 25] and for the model [26] which will be discussed in more detail below.

The requirements are such that the resulting particle model should

• be irreducible i.e. to have the smallest possible internal Hilbert space (minimal approach),

• allow a non-degenerate Fermionic mass spectrum,

• be free of harmful gauge anomalies,

• have unbroken colour groups

• and possess no charged massless Fermions.

Carrying out the classification is a demanding combinatorial task [27] and has been done for up to six summands in the matrix algebra A _f [23] and can be summarised as follows in terms of A _f :

Table 2

We call spectral triples with the matrix algebra $A_{f} = \mathbb{C} \oplus Mat(2, \mathbb{C}) \oplus Mat(n, \mathbb{K}) \oplus \mathbb{C}$ of standard model type. The sub-algebra $Mat(n, \mathbb{K})$ is called the colour algebra. The corresponding Hilbert space, representation of the algebra, real structure, chirality operator and configuration space of Dirac operators coincide in their general form with those of the standard model, thus the denomination. Note that in this classification scheme neither the number n of colours nor the field $\mathbb{K}$ can be determined. They rest as an input from experiment. Spectral triples with matrix algebra $A_{f} = \mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C} \oplus Mat(n, \mathbb{K})$ are called of electro-strong type since the sub-algebra $Mat(n, \mathbb{K})$ is a colour algebra.

From the classification follows that the standard model is indeed one of the simpler models within the set of spectral triples. Furthermore the two types of spectral triples we found from the classification turned out to be very useful basic building blocks to construct models beyond the standard model.

2.4 Gauge Groups

Up to now we have defined spectral triples with their configuration spaces of Dirac operators and have classified certain interesting subsets. But the configuration spaces of Dirac operators is still far too big and we wish to constrain them by specifying a gauge or structure group. The Hilbert space can be considered as a space of sections of a vector (spinor) bundle associated to a principal fibre bundle. We will now define the necessary structure group for finite spectral triples with an almost-commutative spectral triple.

Following Lazzarini and Schücker [28] we define the unitary automorphism group of the Hilbert space of a spectral triple. It is a sub-group of the general automorphism group of the Hilbert space and contains those unitary automorphisms that are compatible with the structures of the spectral triple.

Definition 2.10

Let $(\mathcal{A},\mathcal{H},\mathcal{D})$ be a real, even spectral triple with real structure J and chirality operator γ. The automorphism group of the Hilbert space is defined as

$$\displaystyle\begin{array}{rcl} Aut_{\mathcal{A}}(\mathcal{H})&:=& \{U \in End(\mathcal{H})\vert UU^{{\ast}} = U^{{\ast}}U = id_{ \mathcal{H}},\,UJ = JU, {}\\ & & \;\;U\gamma =\gamma U,\,U\rho (\mathcal{A})U^{-1} \subset \rho (\mathcal{A})\} {}\\ \end{array}$$

where ρ is the representation of the algebra on the Hilbert space.

The last condition is called covariance condition. It ensures for example, that the leptons and the quarks are charged under the same weak SU(2)-sub-group in the case of the standard model.

Example

For the standard model spectral triple one finds the following automorphism group of the Hilbert space [28]

$$\displaystyle{Aut_{A_{f}}(H_{f}) = [U(2)_{w} \times U(3)_{c} \times \prod _{j=1}^{6}U(3)]/[U(1) \times U(1)]}$$

We notice that this group is larger than the standard model structure group U(1) × SU(2) × SU(3). It contains family mixing unitaries $\prod _{j=1}^{6}U(3)$ and can therefore only be considered as a receptacle group for the true structure group.

In almost-commutative geometry the true structure group is identified with the group of unitaries of the algebra lifted to the Hilbert space or with a certain subgroup which is then centrally extended. Let us first give the definition of a structure group following Connes and Chamseddine [29]. Assume that u ∈ U(A _f ). The lift of u to the automorphisms of the finite Hilbert space is then defined by

$$\displaystyle{L(u):=\rho (u)J\rho (u)J^{-1}.}$$

The structure group of the principal fibre bundle is then $G^{cc}:=\{ L(u)\vert u \in U(A_{f}),\,\det (u) = 1\}$. The last condition is called the unimodularity condition. So one can consider now the Hilbert space of the almost-commutative spectral triple as the space of sections associated to the principal fibre bundle $P = P_{Spin} \circ P_{G^{cc}}$ where P _Spin is the natural spin structure.

A second approach that eliminates the need of the unimodularity condition was developed by Lazzarini and Schücker [28]. It restricts the unitaries which are to be lifted to the non-commutative unitaries $U^{n}(A_{f})$, i.e. the unitaries of the non-abelian matrix summands in the finite algebra. Here one apparently loses all the U(1)-sub-groups which play an important part in the standard model. Yet, they can be reintroduced by centrally extending the lift w.r.t. the determinant of the the non-commutative unitaries. Let us denote this lift by $\mathbb{L}: U^{n}(A_{f}) \rightarrow Aut_{A_{f}}(H_{f})$ without giving the details (they are quite technical and we refer the interested reader to [28]). But we note that the lift is not unique. In particular the central charges of the central extension can be chosen freely. They are usually fixed on physical grounds by demanding that the resulting particle model be free of harmful U(1)-anomalies.

Definition 2.11

The structure group is in this case $G_{f}:=\{ \mathbb{L}(u)\vert u \in U^{n}(A_{f})\}$ with chosen central extension for the lift $\mathbb{L}$.

In the extensions of the standard model we will usually work with this definition as it proves to be more flexible than the original one by Chamseddine and Connes.

We notice that the almost-commutative Dirac operator is not gauge covariant and therefore not a Dirac operator induced by the associated principal fibre bundle $P = P_{Spin} \circ P_{G_{f}}$. So one defines the fluctuated Dirac operator $\mathcal{D}^{f}$. The finite Dirac operator D _f gets promoted to an inter-twiner $\Phi (D_{f}) \in End(L^{2}(\Gamma (\Sigma )) \otimes H_{f})$ adapted to the principal fibre bundle $P_{G_{f}}$. We will always assume that the fields that constitute the inter-twiner are minimal in the sense that for two sub-groups of G _f we only have one scalar multiplet of inter-twiners. This assumption avoids the possibility of different Higgs fields for leptons and quarks. We will also assume that the scalar fields that constitute the inter-twiner allow to recover the finite Dirac operator if replaced by the identity, i.e. $\Phi (D_{f}) \rightarrow D_{f}$.

The derivative part gets promoted to a twisted Dirac operator where ∇^f is a covariant derivative associated to $P_{G_{f}}$. One can define algorithms to obtain such fluctuated Dirac operators operationally and we refer to Connes and Chamseddine [29] for details.

Definition 2.12

For a given almost commutative spectral triple we define the set of covariant Dirac operators to be

This will be the final dynamical configuration space of Dirac operators for the particle model. Note that the covariant Dirac operators form a subset of the configuration space of Dirac operators of the almost-commutative spectral triple.

3 Action Principles

One of the central objects in the noncommutative approach to the standard model is the (covariant) almost-commutative Dirac operators. Therefore Chamseddine and Connes proposed an action principle on the space of Dirac operators called spectral action. This action has the disadvantage that it needs a Dirac operator with discrete eigenvalues and is therefore only well defined on compact Riemannian manifolds. We will discuss this point briefly when we consider alternative approaches to define an action for Dirac operators.

3.1 The Spectral Action

The Chamseddine-Connes spectral action [29] for a spectral triple is defined to be the number of eigenvalues of the $\mathcal{D}$ in an interval $[-\Lambda,\Lambda ]$ where $\Lambda$ is a positive real number. One usually considers the spectral action to be an effective action for the gauge bosons which is valid at $\Lambda$ and subject to renormalisation when one wishes to investigate different energy scales. For the fermions the action is given by the Dirac action for $\mathcal{D}$ with a projection on the physical Hilbert space.

Since it is not possible to calculate the spectral action exactly from the above definition, one smoothes the counting function and resorts to an alternative definition which allows for approximate calculations. We take this approximation as the actual definition of the spectral action.

Definition 3.1

Let f be a smooth cut-off function with support in [0, +1] which is constant near zero and let $\Lambda \in \mathbb{R}^{+}$. The bosonic spectral action for the Dirac operator $\mathcal{D}$ is

$$\displaystyle{S_{cc}(\mathcal{D}):=\mathrm{ Tr}\,f\left (\tfrac{\mathcal{D}^{2}} {\Lambda ^{2}} \right ).}$$

The trace Tr is the trace over the Hilbert space.

Performing a Laplace transform one gets from the heat trace asymptotics for $t \rightarrow 0$

$$\displaystyle{\mathrm{Tr}\left (e^{-t\,\mathcal{D}^{2} }\right ) \sim \sum _{n\geq 0}t^{n-2}a_{ 2n}(\mathcal{D}^{2})}$$

with Seeley-deWitt coefficients $a_{2n}(\mathcal{D}^{2})$. For the spectral action S _cc follows (by $t = \Lambda ^{-2}$) an asymptotic expansion

$$\displaystyle{S_{cc}(\mathcal{D}) \sim \Lambda ^{4}\,f_{ 4}\,a_{0}(\mathcal{D}^{2}) + \Lambda ^{2}\,f_{ 2}\,a_{2}(\mathcal{D}^{2}) + \Lambda ^{0}\,f_{ 0}\,a_{4}(\mathcal{D}^{2}) + O(\Lambda ^{-\infty })}$$

as $\Lambda \rightarrow \infty$ with f ₄, f ₂, f ₀ moments of cut-off function f.

Assume from now on that D is the covariant Dirac operator of an almost-commutative spectral triple, i.e. it is of the form . Assume further that the underlying manifold M is closed and dim(M) = 4.

The calculations of the Seeley-DeWitt coefficients are long but standard. So we give here the final results in a condensed notation. The a ₀-coefficient is essentially the volume of the manifold and will not concern us here. So we focus on the a ₂- and the a ₄-coefficient:

$$\displaystyle\begin{array}{rcl} a_{2}(D)& =& -\tfrac{dim(H_{f})} {96\pi ^{2}} \int _{M}R\,dvol - \tfrac{1} {48\pi ^{2}} \int _{M}tr(\Phi ^{2})dvol {}\\ a_{4}(D^{2})& =& \tfrac{11\,dim(H_{f})} {720} \,\chi (M) -\tfrac{dim(H_{f})} {320\pi ^{2}} \int _{M}\|W\|^{2}\,dvol {}\\ & +& \tfrac{1} {8\pi ^{2}} \int _{M}\,tr\big([\nabla ^{\mathcal{H}_{f} },\Phi ]\big) +\, tr(\Phi ^{4})dvol {}\\ & +& \tfrac{5} {96\pi ^{2}} \int _{M}\,tr\big(\Omega _{f}^{2}\big)dvol\; + \tfrac{1} {48\pi ^{2}} \int _{M}R\,tr(\Phi ^{2})dvol {}\\ \end{array}$$

where R is the scalar curvature of M, χ(M) is the Euler characteristic, W is the Weyl tensor of M and $tr(\Omega _{f}^{2})$ is the Yang-Mills Lagrangian of the covariant derivative ∇^f. We notice that the two terms

$$\displaystyle{- \tfrac{1} {48\pi ^{2}} \int _{M}tr(\Phi ^{2})dvol\quad }$$

and

$$\displaystyle{+ \tfrac{1} {8\pi ^{2}} \int _{M}\,tr\big([\nabla ^{\mathcal{H}_{f} },\Phi ]\big) +\, tr(\Phi ^{4})dvol}$$

provide us exactly with the Lagrangian of a scalar field that can act as a Higgs field. In particular they exhibit the “Mexican hat”-potential which can induce a symmetry breaking mechanism.

Note also that the quartic term for the scalar potentials as well as the Yang-Mills Lagrangian for the covariant derivative appear in the same Seeley-DeWitt coefficient a ₄. If we decompose into the Yang-Mills Lagrangians of the sub-groups of the finite structure group G _f and write it out in terms of the gauge coupling constants we obtain relations among the quartic couplings of the scalar fields in the inter-twiner $\Phi (D)$ and the gauge couplings [29]. These relations are not stable under renormalisation group flow and are therefore considered as boundary conditions for the flow, i.e. the spectral action is an effective action valid at the cut-off energy $\Lambda$. We notice furthermore that the trace of the square and the fourth power of the finite Dirac operator D _f also enter the relations among the quartic couplings and the gauge couplings. In this way also the Yukawa couplings get involved.

Further restrictions on the Yukawa couplings can be obtained by normalising the scalar fields to their proper mass dimension. In the present normalisation the fields have mass dimension 2. It was discovered by Tolksdorf and Thumstädter [30] that a proper normalisation of the bosonic and the fermionic action results in a new constraint on the trace of $D_{f}^{2}$, i.e. on the trace of the squared Yukawa matrix.

These constraints can be exploited to make the resulting particle models more predictive than the usual models based on the differential geometric Yang-Mills-Higgs approach. They provide extra boundary conditions at the cut-off energy which have to match the measured values at other energies via renormalisation group running.

3.2 Alternative Action Principles

Let give us give a short account on alternative approaches to action principles based on a Dirac operator. The first one is closely related to the spectral action. It is basically another way to regularise the counting function for the eigenvalues.

Kurkov, Lizzi, Sakellariadou and Watcharangkool [31] recently proposed to use a ζ-function regularisation to give a well defined value to the counting function of the eigenvalues. The value of the ζ-regularised spectral action can be given in terms of Seeley-DeWitt coefficients and turns out to be proportional to the a ₄-coefficient. This way of regularising the counting function is closely related to the definition of the spectral action as Weyl anomaly as suggested by Andrianov, Kurkov and Lizzi [32]. As the spectral action of Chamseddine and Connes this approach is only well defined on compact Riemannian manifolds. It was noted by Zahn [33] that the Weyl anomaly, if calculated on a globally hyperbolic Lorentzian manifold, leads to similar results and may serve as a Lorentzian substitute.

A second approach which builds on a wider class of Dirac operators was developed by Tolksdorf [34, 35]. Here the generalised Dirac operators contain also curvature terms of the gauge connection as well as higher powers of the inter-twiner scalar fields. Such Dirac-Yukawa operators can be decomposed naturally into two terms (subscript B for Bochner) in the same way as one can write the square of the Dirac operator in terms of the Bochner Laplace operator and a zero-order term $\mathcal{D}^{2} = \Delta _{B} + V _{D}$. Tolksdorf notices [36] that the action functional

$$\displaystyle{\mathcal{D}\mapsto \int _{M} {\ast} tr(V _{D})}$$

produces the bosonic action of the standard model (as well as the Einstein-Hilbert action) for a suitable generalised Dirac operator of Yukawa type. This construction does not depend on the signature of the metric on M. It is therefore perfectly suitable in the Lorentzian setting.

4 Physical Predictions

Let us now return to the almost-commutative geometry of the standard model . We wish to focus in this section on the constraints for the quartic couplings of the scalar fields, the gauge couplings and the Yukawa couplings.

4.1 Constraints in Parameter Space

Calculating the full spectral action for the standard model is a long and tedious task and we refer to the original publication of Chamseddine and Connes [29] for the details.

The standard model has only one scalar doublet, the Higgs doublet, and only one quartic coupling $\lambda$ in the Lagrangian. It has three gauge couplings g ₁, g ₂ and g ₃ corresponding to the sub-groups of its structure group G _sm  = U(1) × SU(2) × SU(3). In the almost-commutative setting this structure group arises from Lazzarini’s and Schücker’s lift of the noncommutative unitaries of the finite algebra $A_{f} = \mathbb{C}\, \oplus \, \mathbb{H}\, \oplus \, M(3, \mathbb{C})\, \oplus \, \mathbb{C}$ to the finite Hilbert space.

The finite Dirac operator D _f contains then the Yukawa couplings and the Majorana masses. Calculating the spectral action, normalising the fields and comparing the terms in the a ₄ Seeley-DeWitt coefficients leads to the following set of relations at cut-off $\Lambda$

$$\displaystyle{5\,g_{1}^{2} =\, 3\,g_{ 2}^{2} =\, 3\,g_{ 3}^{2} =\, 3\,\frac{Y _{2}^{2}} {H} \, \frac{\lambda } {24}\, =\, \frac{3} {4}\,Y _{2}}$$

where we Y ₂ denotes the trace of the Yukawa matrix squared and H the trace of the Yukawa matrix to the fourth power. The last constraint was first noticed in [30].

4.2 Consequences for the Standard Model

Let us analyse these constraints for the standard model. We need to make some extra assumptions. Namely we will assume the standard model particles constitute the whole particle spectrum, i.e. we have a big desert. Furthermore we take the experimental values of the gauge couplings at the mass of the Z-boson, i.e. g ₁(m _Z ) = 0. 3575, g ₂(m _Z ) = 0. 6514 and g ₃(m _Z ) = 1. 221 [37].

We also assume that the running of the couplings can be described by the standard (1-loop) renormalisation group equations and that the top-quark and the τ-neutrino Yukawa couplings dominate all other Yukawa couplings. We will take $t:=\ln (E/m_{Z}),\mathrm{d}g/\mathrm{d}t =:\beta _{g}$ and $\kappa:= (4\pi )^{-2}.$ Then the β-functions are [38, 39]:

$$\displaystyle\begin{array}{rcl} \beta _{g_{i}}& =& \kappa b_{i}g_{i}^{3},\quad b_{ i} = \left (\frac{20} {9} N + \frac{1} {6},-\frac{22} {3} + \frac{4} {3}N + \frac{1} {6},-11 + \frac{4} {3}N\right ), {}\\ & & {}\\ \beta _{t}& =& \kappa \left [-\sum _{i}c_{i}^{u}g_{ i}^{2} + Y _{ 2} +\, \frac{3} {2}\,g_{t}^{2}\,\right ]g_{ t}, {}\\ \beta _{\nu }& =& \kappa \left [-\sum _{i}c_{i}^{\nu }g_{ i}^{2} + Y _{ 2} +\, \frac{3} {2}\,g_{\nu }^{2}\,\right ]g_{\nu }, {}\\ \beta _{\lambda }& =& \kappa \left [\,\frac{9} {4}\,\left (g_{1}^{4} + 2g_{ 1}^{2}g_{ 2}^{2} + 3g_{ 2}^{4}\right ) -\left (3g_{ 1}^{2} + 9g_{ 2}^{2}\right )\lambda + 4Y _{ 2}\lambda - 12H + 4\lambda ^{2}\right ], {}\\ \end{array}$$

with

$$\displaystyle\begin{array}{rcl} c_{i}^{t} = \left (\frac{17} {12}, \frac{9} {4},8\right ),& c_{i}^{\nu } = \left (\frac{3} {4}, \frac{9} {4},0\right ), & {}\\ Y _{2} = 3g_{t}^{2} + g_{\nu }^{2},& H = 3g_{ t}^{4} + g_{\nu }^{4}.& {}\\ \end{array}$$

A numerical analysis then shows that g ₂ = g ₃ at $\Lambda = 1.1 \times 10^{17}$ GeV. The constraint discovered by Tolksdorf and Thumstäter results in an upper bound on the top-quark mass, m _top  < 190 GeV. This is in good agreement with experiment, the experimental value of the top-quark mass being m _top  = 171. 2 ± 2. 1 GeV. Furthermore no 4th generation of standard model particles is allowed as their masses were forced to be smaller than the top-quark mass and should therefore be detectable.

The numerical analysis can be sharpened using the experimental value of the top-quark mass as an extra input. This allows to calculate the value of the quartic Higgs coupling $\lambda$ at the cut-off energy and then use the renormalisation group equations to obtain a low energy value. This is the missing ingredient to calculate the Higgs boson mass and one finds m _H  = 168. 3 ± 2. 5 GeV. Unfortunately the Higgs mass has been measured by the LHC to be $m_{H} \sim 125$ GeV [40], so the almost-commutative standard model is experimentally excluded.

One can also argue that the almost-commutative standard model has excluded before since $\frac{5} {3}\,g_{1}(\Lambda )^{2}\neq \,g_{ 2}(\Lambda )^{2}$. But this discrepancy has generally been considered as less grim then the exact value of the Higgs mass.

5 Beyond the Standard Model

In view of the failure of the standard model within the setting of almost-commutative geometry as well as the seeming experimental necessity of candidates for dark matter particles it appears to make sense to search for models beyond the standard model.

5.1 Different Approaches

We will focus on some of the ways to construct models that enlarge the particle content of the standard model or change the gauge sector.

The classification scheme of finite spectral triples as proposed by Chamseddine and Connes [21] leads in general to extensions of the gauge sector w.r.t. the standard model gauge group. The simplest viable example is a Pati-Salam type model [22]. Here the symmetry breaking mechanism is far more involved compared to the standard model. This model produces several new scalar fields which may be used to obtain the correct Higgs mass [41].

Another approach with relations to the classification of Chamseddine and Connes was put forward by Devastato, Lizzi and Martinetti [42]. Here the gauge group is enlarged to incorporate also the Spin-group and a mixing of spinor degrees of freedom and the finite Hilbert space. This allows to obtain a new scalar field which replaces the Majorana masses of the right-handed neutrinos and also leads to a Higgs mass in accordance with experiment.

A different way to construct particle models was recently revived by Farnsworth and Boyle [43]. They take up an earlier idea of Wulkenhaar [44] which replaces the associative matrix algebras of almost-commutative geometries by non-associative ones. Whether this will lead to interesting models beyond the standard model is still under investigation.

Building on the classification scheme [23] for finite spectral triples we proposed a model building kit where one follows the following steps in order to obtain viable extensions of the standard model:

• find a finite geometry that has the standard model as sub-model (tricky)

•  = > particle content, gauge group and its representation

• make sure everything is anomaly free

• compute the spectral action = > constraints on parameters

• determine the cut-off scale $\Lambda$ with suitable sub-set of the constraints

• use renormalisation group equations to obtain low energy values of (hopefully) interesting parameters (Higgs couplings, Yukawa couplings)

• check with experiment! (and here we usually fail)

To find a finite spectral triple with the standard model as a sub-geometry, it proved successful to use the finite geometries of standard model type and electro strong type. These can be combined to build bigger finite spectral triples which contain in general new fermions and new gauge bosons. Whether the new model meets all the constraints imposed by further theoretical requirements, such as being free of gauge anomalies, or withstands the confrontation with experiment is often a non-trivial question.

5.2 An Interesting Example

To illustrate viable extensions of the standard model building on the classification [23] we will sketch the model prosed in [26]. The matrix algebra of the internal space is

$$\displaystyle{A_{f}:= \mathbb{C} \oplus M_{2}(\mathbb{C}) \oplus M_{3}(\mathbb{C}) \oplus \mathbb{C} \oplus _{i=1}^{6}\mathbb{C}_{ i}}$$

By a centrally extended lift in the sense of [28] one finds that the structure group of the model is simply the standard model group with an extra U(1) factor. So we have $G_{f} = U(1)_{Y } \times SU(2)_{w} \times SU(3)_{c} \times U(1)_{X}$

The finite Hilbert space is extended and has, additionally to the particles of the standard model, the following fermion content in each generation:

$$\displaystyle{X_{l}^{1} \oplus X_{ l}^{2} \oplus X_{ l}^{3}: (0,1,1,+1) \oplus (0,1,1,+1) \oplus (0,1,1,0)}$$

$$\displaystyle{X_{r}^{1} \oplus X_{ r}^{2} \oplus X_{ r}^{3}: (0,1,1,+1) \oplus (0,1,1,0) \oplus (0,1,1,+1)}$$

$$\displaystyle{V _{\ell}^{w},\,V _{ r}^{w}:\; (0,\bar{2},1,0)\quad V _{\ell}^{c},\,V _{ r}^{c}:\; (-1/6,1,\bar{3},0)}$$

where we have given the representations of the structure group and thus the dimension of the subspaces in a short hand notion. So for the V ^w particles are in the conjugate of the fundamental representation of the SU(2) _w sub-group and are neutral to all other sub-groups. They also couple vectorially to SU(2) _w (left-right-symmetric) and therefore form a $\mathbb{C}^{24}$ sub-Hilbert space since the anti-particles and three generations have also to be taken into account.

The model has also a new scalar field $\sigma:\, (0,1,1,+1)$. We see that all the X-type particles and the new scalar field do not couple to the standard model subgroup while the V -particle have vectorial couplings and therefore their masses should be of the order of the cut-off scale of the effective theory, i.e. the spectral action.

From the spectral action we find the Lagrangians that have to be added to the standard model Lagrangian. For the fermions and the gauge bosons we have

$$\displaystyle\begin{array}{rcl} \mathcal{L}_{ferm}& =& g_{\nu,X^{1}}\,\bar{\nu }_{r}\sigma X_{\ell}^{1} +\bar{ X}_{\ell}^{1}m_{ X}X_{r}^{1} + g_{ X^{2}}\,\bar{X}_{\ell}^{2}\sigma X_{ r}^{2} + g_{ X^{3}}\,\bar{X}_{\ell}^{3}\sigma X_{ r}^{3} {}\\ & & +\bar{V }_{\ell}^{c}m_{ c}V _{r}^{c} +\bar{ V }_{\ell}^{w}m_{ w}V _{r}^{w} +\ h.c. {}\\ \mathcal{L}_{gauge}& =& \frac{1} {g_{4}^{2}}F_{X}^{\mu \nu }F_{ X,\mu \nu }. {}\\ \end{array}$$

To keep the everything notationally short we only gave the fermionic Lagrangian for one generation. The full Lagrangian contains also CKM-type matrices for the X-particles.

For the scalar sector we write the full potential where we include the Higgs field H:

$$\displaystyle{\mathcal{L}_{scalar} = -\mu _{1}^{2}\vert H\vert ^{2} -\mu _{ 2}^{2}\vert \sigma \vert ^{2} + \frac{\lambda _{1}} {6}\,\vert H\vert ^{4} + \frac{\lambda _{2}} {6}\,\vert \sigma \vert ^{4} + \frac{\lambda _{3}} {3}\,\vert H\vert ^{2}\vert \sigma \vert ^{2}}$$

Then the symmetry breaking pattern

$$\displaystyle{U(1)_{Y } \times SU(2)_{w} \times SU(3)_{c} \times U(1)_{X} \rightarrow U(1)_{e\ell.} \times SU(3)_{c} \times \mathbb{Z}_{2}}$$

implies that the model has a new massive vector boson associated to the broken U(1) _X sub-group.

So the set of free parameters becomes larger with new Yukawa couplings for the X-particles new quartic couplings $\lambda _{2}$ and $\lambda _{3}$, a new gauge coupling g ₄ and Dirac mass terms for the X- and V -particles. But also the set of constraints from the spectral action becomes larger. We find the following boundary conditions on the couplings at the cut-off energy $\Lambda$:

$$\displaystyle\begin{array}{rcl} g_{2}(\Lambda )& =& g_{3}(\Lambda ) = \sqrt{\tfrac{7} {6}}\quad g_{1}(\Lambda ) = \sqrt{\tfrac{4} {3}}\;g_{4}(\Lambda ) {}\\ \lambda _{1}(\Lambda )& =& 36\; \frac{H} {Y _{2}}\,g_{2}(\Lambda )^{2}\;\lambda _{ 2}(\Lambda ) = 36\; \frac{tr(g_{\nu,X^{1}}^{4})} {tr(g_{\nu,X^{1}}^{2})^{2}}\,g_{2}(\Lambda )^{2} {}\\ \lambda _{3}(\Lambda )& =& 36\;\frac{tr(g_{\nu }^{2})} {Y _{2}} \,g_{2}(\Lambda )^{2} {}\\ Y _{2}(\Lambda )& =& tr(g_{\nu,X^{1}}^{2})(\Lambda ) + tr(g_{ X^{1}}^{2})(\Lambda ) + tr(g_{ X^{2}}^{2})(\Lambda ) = 6\;g_{ 2}(\Lambda )^{2}. {}\\ \end{array}$$

Here tr(g ²) indicates that we take the trace of the corresponding Yukawa matrices (three generations).

Since these constraints are quite difficult to analyse let us pick a convenient point in the parameter space

$$\displaystyle\begin{array}{rcl} Y _{2}& \approx & 3g_{top}^{2} + g_{\nu _{\tau }}^{2} {}\\ tr(g_{X^{1}}^{2})(\Lambda )& \approx & tr(g_{ X^{2}}^{2})(\Lambda ) \approx 0 {}\\ tr(g_{\nu,X^{1}}^{2})(\Lambda )& \approx & g_{\nu,X}(\Lambda )^{2} = 6\;g_{ 2}(\Lambda )^{2} {}\\ (m_{V ^{w}})_{ij}& \approx & \Lambda (m_{V ^{c}})_{ij} \approx 10^{15}GeV {}\\ \end{array}$$

where we again assumed that the top-quark and the τ-neutrino Yukawa coupling dominate the Yukawa couplings of the standard model particles.

From a renormalisation group analysis we find then that all of the constraints can be met at $\Lambda \sim 2 \times 10^{18}$ GeV. The masses of the scalar fields do now depend on the three couplings $\lambda _{1}$, $\lambda _{2}$ and $\lambda _{3}$. Their mass matrix is not diagonal anymore and the free parameter in the mass matrix for the mass eigenvalues is the vacuum expectation value of the new scalar field. See Fig. 1 for a plot of the eigenvalues w.r.t. the vacuum expectation value. The straight horizontal line is at 125 GeV, i.e. the mass of the standard model Higgs.

Fig. 1

Mass eigenvalues of the scalar fields

If we require the smaller mass eigenvalue to be the experimental value of the Higgs mass and also put in the experimental value of the top-quark mass with m _top  ≈ 172. 9 ± 1. 5 GeV we find the following values for the remaining parameters

• m _top  ≈ 172. 9 ± 1. 5 GeV

• $m_{\sigma _{1,SMS}} \approx 125 \pm 1.1$ GeV

• $m_{\sigma _{2}} \approx 445 \pm 139$ GeV

• $m_{Z_{X}} \approx 254 \pm 87$ GeV

• g ₄(m _Z ) ≈ 0. 36

• $m_{X_{2},X_{3}} \leq 50$ GeV

The quite substantial errors on these predictions originate in the very flat slope of the lower mass eigenvalue, i.e. small errors in the Higgs mass translate into large errors in the second mass eigenvalue and therefore in the other mass terms which depend on it. Whether this model is already excluded is an open question. But such types of models are referred to as dark sector models with a Higgs portal in the particle physics literature and they seem to be quite difficult to rule out.

6 Challenges for the Future

Let us just mention a couple of challenges and open questions that come into mind in the context of noncommutative geometry and models beyond the standard model.

One of the major open problems in particle physics at the time is probably the lack of mass in the universe. It appears that at least 60 % of the matter we observe astronomically via gravitational effects is unaccounted for by the standard model. So wether models as those discussed above contain viable dark matter candidates seems a pressing question.

On a more fundamental point it is necessary to mention one of the major short-comings of the almost-commutative approach to the standard model. The “space-time”, i.e. the manifold on which part of the spectral triple is built needs to be of Riemannian signature and has to be compact. This is in stark disagreement with the universe we observe and which we model by a non-compact Lorentzian manifold. For some proposals to solve this problem we refer to Strohmaier [12], Paschke and Sitarz [13], Besnard [14] and Franco and Eckstein [15].

A solution to either of these problems within the context of noncommutative geometry would certainly be a substantial breakthrough.