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Grand Unification in Non-Commutative Geometry

arXiv:hep-ph/9209224v1 10 Sep 1992ZU-TH- 30/1992ETH/TH/92-413 September 1992Grand Unification in Non-Commutative GeometryA. H. Chamseddine1 * G. Felder2 and J. Fr¨ohlich31 Theoretische Physik, Universit¨at Z¨urich, CH 8001 Z¨urich Switzerland2 Department of Mathematics, ETH, CH 8092 Z¨urich Switzerland3 Theoretische Physik, ETH, CH 8093 Z¨urich SwitzerlandAbstractThe formalism of non-commutative geometry of A. Connes is used to construct modelsin particle physics.

The physical space-time is taken to be a product of a continuousfour-manifold by a discrete set of points. The treatment of Connes is modified in sucha way that the basic algebra is defined over the space of matrices, and the breakingmechanism is planted in the Dirac operator.

This mechanism is then applied to threeexamples. In the first example the discrete space consists of two points, and the twoalgebras are taken respectively to be those of 2×2 and 1×1 matrices.

With the Diracoperator containing the vacuum breaking SU(2)×U(1) to U(1), the model is shown tocorrespond to the standard model. In the second example the discrete space has threepoints, two of the algebras are identical and consist of 5×5 complex matrices, and thethird algebra consists of functions.

With an appropriate Dirac operator this modelis almost identical to the minimal SU(5) model of Georgi and Glashow. The thirdand final example is the left-right symmetric model SU(2)L × SU(2)R × U(1)B−L.

* Supported by the Swiss National Foundation (SNF)

1. IntroductionAt present energies the standard model of electroweak interactions has passed all ex-perimental tests.

One of the essential ingredients of this model is the Higgs field. Thepresence of the Higgs field is required to break the gauge symmetry spontaneously.From the four-dimensional point of view, there is no apparent geometrical reason forthe Higgs field.

Although there are some possible candidates, e.g. a Kaluza-Kleintheory or a compactified string model, there are no compelling models yet.

A newpicture was put forward recently by Connes [1-2], where the experimental validityof the standard model was taken as an indication for a non-commutative picture ofspace-time. Space-time is taken to be a product of a continuous Euclidean manifoldM4 by a discrete ”two-point” space.

The fibers in the two copies of space are takento be U(1) and SU(2) respectively. The vector potential defined in this space willhave as components U(1) and SU(2) gauge fields along the continuous directionsand the scalar Higgs field along the discrete directions.

Therefore, non-commutativegeometry offers a geometrical picture for the unification of the gauge and Higgs field.The advantage of this approach over the Kaluza-Klein approach is that there is notruncation of any physical modes, while, in the latter, an infinite number of massivemodes is truncated. In this formalism it was shown by Connes and Lott [3-4], andelaborated upon in great detail by Kastler [5], on how to construct the standardmodel.

Inclusion of the SU(3) strong interaction proved to be more difficult and wasonly achieved recently [3-5]. Other constructions were proposed by different authors,such as Coquereux et al , Dubois-Violette et al.

and Balakrishna et al. [6], but theylack a compelling geometrical structure and will not be followed here.It is usually expected that the standard model [7] will be replaced by a differenttheory at higher energies which one hopes to be more unified.

In particular, the grandunified theories (GUTs) seem to provide (in their supersymmetric forms) acceptablemodels. The problem that will be addressed in this paper is to find a way to buildGUTs models and other possible models at energies higher than the weak scale,within the non-commutative picture.The strategy adopted in references [1-5] is only appropriate in the case of aproduct symmetry such as SU(2) × U(1).

If one follows this strategy without modi-fication, many difficulties will be encountered and no phenomonologically successfulgrand-unified model can be built. This strategy also excludes a single gauge group.To explore other possibilities, we note that a typical GUT involves at least two scales:1

the grand unification scale, where the three coupling constants of SU(3), SU(2) andU(1) coincide, and the electroweak scale. In addition, there could be intermediatescales.

By choosing space-time to be a product of a continuous four-dimensional Rie-mannian manifold by a discrete set of points we immediately see that the simplestpossibility for the choice of a Dirac operator including more than one scale is to takethe discrete space to consist of three points. This is the situation we shall be mostlyinterested in, although the extension to a discrete space of N points is straightfor-ward.

By generalizing the algebra of functions to be given by a direct sum of algebrasof matrix-valued functions and by planting the symmetry breaking mechanism in theDirac operator, it will turn out to be possible to construct unified models.The plan of this paper is as follows: In section 2, we modify the prescription ofConnes [1-2] in such a way that the discrete space consists of three points generalizableto N points. We introduce the idea of planting the symmetry breaking in the Diracoperator and prove that this does not break gauge invariance.In section 3, andas a warm up, we apply this prescription to construct the standard model of theelectorweak interactions.

In section 4, we construct the SU(5) model and obtainthe minimal model (apart from an extra Higgs singlet) of Georgi an Glashow [8]. Insection 5, we construct the model SU(2)L×SU(2)R×U(1)B−L of Pati and Mohapatra[9] by taking the discrete space to consist of four points.

Section 6, contains ourconclusions.2. A new prescription for model buildingConsider a model of non-commutative geometry consisting of the triple (A, h, D),where h is a Hilbert space, A is an involutive algebra of operators on h, and D is anunbounded self-adjoint operator on h. An example important for our construction isthe following one: Let X be a compact Riemannian spin-manifold, A1 the algebra offunctions on X, and (h1, D1, Γ1) the Dirac-K cycle with h1 ≡L2(X, √gddx) on A1.Let (A2, h2, D2) be given by A2 = Mn(C) ⊕Mp(C) ⊕Mq(C), where Mn(C) is theset of all n × n matrices and h2 = h2,1 ⊕h2,2 ⊕h2,3 where h2,1 h2,2 and h2,3 are theHilbert spaces Cn, Cp and Cq, respectively.

Then A and D are taken to beA = A1 ⊗A2D = D1 ⊗1 + Γ1 ⊗D2(2.1)To every fǫA we associate a triplet (f1, f2, f3) of matrix-valued functions onX, where f1, f2, and f3 are n × n, p × p, and q × q matrices, respectively. Thedecomposition of h2 corresponds to the decomposition h = h1 ⊕h2 ⊕h3 for whichthe action of f is block-diagonalf →diag(f1, f2, f3).

(2.2)2

In this decomposition, the operator D becomesD =npqn/∂⊗1γ5 ⊗M12γ5 ⊗M13pγ5 ⊗M21/∂⊗1γ5 ⊗M23qγ5 ⊗M31γ5 ⊗M32/∂⊗1(2.3)where M ∗mn = Mnm and m, n = 1, 2, 3, m ̸= n . The gamma matrices we use satisfy:γ∗a = −γa, {γa, γb} = −2δab,γ5 = γ1γ2γ3γ4, γ∗5 = γ5, and gab = −δab is the Euclideanmetric.An important difference between our approach and the prescription given byConnes et al.

is that they choose all the matrices Mmn to be of the same size, i.e.n = p = q, and proportional to the identity matrix. In our approach they can begeneral matrices and do not commute with elements of A.

The novel idea that wewill advance is that the matrices Mmn of the model determine the tree level vacuum-expectation values of Higgs fields and the desired symmetry breaking scheme. Thismodification allows us, first, to simplify the construction of the standard model andthen go beyond this model to grand unification models.Let E be a vector bundle characterized by the vector space E of its sections.

Weshall consider the example where E = A. Let ρ be a self-adjoint element in the space,Ω1(A), of one formsρ =Xiaidbi,(d1 = 0)(2.4)where Ω∗(A) = ⊕∞n=0Ωn(A) is the universal differential algebra, with Ω0(A) = A;See [1].

(The space Ωn(A) plays the role of n-forms in non-commutative geometry. )An involutive representation of Ω∗(A) is provided by the map π : Ω∗(A) →B(h)defined byπ(a0da1...dan) = a0[D, a1][D, a2]...[D, an](2.5)where B(h) is the algebra of bounded operators on h. The image of the one-form ρisπ(ρ) =Xiai[D, bi],(2.6)where the elements ai and bi are represented byai →diag(ai1, ai2, ai3)bi →diag(bi1, bi2, bi3)(2.7)interpreted as bounded operators on h. The product ai[D, bi] is defined in termsof standard multiplication.

Using the expression of eq. (2.3) for D, the commutator3

[D, b] can be easily evaluated and is given by[D, b] =/∂b1γ5 ⊗(M12b2 −b1M12)γ5 ⊗(M13b3 −b1M13)γ5 ⊗(M21b1 −b2M21)/∂b2γ5 ⊗(M23b3 −b2M23)γ5 ⊗(M31b1 −b3M31)γ5 ⊗(M32b2 −b3M32)/∂b3(2.8)Inserting eq. (2.8) in eq.

(2.6), we obtainπ(ρ) =A1γ5 ⊗φ12γ5 ⊗φ13γ5 ⊗φ21A2γ5 ⊗φ23γ5 ⊗φ31γ5 ⊗φ32A3(2.9).where the new variables A and φ are functions of the a′s and the b′s given byAm =Xiaim/∂bim,m = 1, 2, 3,φmn =Xiaim(Mmnbin −bimMmn),m ̸= n,(2.10)and satisfy A∗m = Am and φ∗mn = φnm.The two-form dρ is:dρ =Xidaidbi(2.11)and its immage under the involutive representation π is given byπ(dρ) =Xi[D, ai][D, bi](2.12)At this point we can address the question of gauge invariance. If one wishes for theaction of a spinor field< Ψ, (D + π(ρ))Ψ >(2.13)to be invariant under the transformation Ψ →gΨ = gΨ, where gǫA satisfiesgǫU(A) = {gǫA|g∗g = 1} is unitary, then ρ must transform inhomogeneously ac-cording togρ = gρg∗+ gdg∗(2.14)This is consistent with the definition of ρ in eq.

(2.4):gρ =Xi(gai)d(big∗) −g(Xiaibi) −1dg∗(2.15)where the second term could be included in the first term by enlarging the set of theai’s and bi’s. It is possible to define gauge transformations explicitly on the elementsai and bi :ai →gai = gaibi →gbi = big∗(2.16)4

provided one imposes the constraintXiaibi = 1(2.17)This is no loss in generality, as the field Pi aibi is independent. We shall use theconstraint (2.17) and the transformations (2.16) when convenient.Similarly, thetransformation of dρ could be easily derived to bedρ →g(dρ) = dgρg∗+ dgdg∗+ gdρg∗−gρdg∗(2.18)Working in the representation π, we see from eq.

(2.15) thatπ( gρ) = gπ(ρ)g∗+ g[D, g∗](2.19)and this can be written in the formπ( gρ) =Xigai[D, gbi](2.20)As expected, the Dirac operator is not acted up on by the gauge transformations (i.e.gD ≡D). The curvature θ, defined byθ = dρ + ρ2(2.21)is easily seen to be covariant under the gauge transformationsθ →gθ = gθg∗(2.22)To see how gauge transformations act on the components of π(ρ), we first give therepresentation of g:g →diag(g1, g2, g3)(2.23)where g1, g2 and g3 are n × n, p × p and q × q unitary matrix-valued functionsrespectively.

A simple computation, using the commutator [D, g] in eq. (2.8), givesthe component form of eq.

(2.19):gAm = gmAmg∗m + gm/∂g∗m,m = 1, 2, 3g(φmn + Mmn) = gm(φmn + Mmn)g∗n,m ̸= n(2.24)In this form it becomes manifest that the Am are the usual gauge fields, while thecombinations φmn + Mmn are scalar fields transforming covariantly under the mixedgauge transformations gm and gn. (We use that gMmn = Mmn in D .) The fields φmn5

are the physical fields, and the Mmn are the vacuum expectation values of the Higgsfields. In other words, the Higgs potential will turn out to have its minimun whenφmn = 0, indicating that the scalar fields appearing are the fluctuations aroundthe vacuum state, and that we are in the spontaneously broken phase.To passto the symmetric phase, we must reexpress all the scalar fields in the combinationφmn + Mmn.It was noted by Connes and Lott [4] that the representation π is ambiguous, afact that will explain the appearence of auxiliary fields.

This can be seen from the factthat if π(ρ) is set to zero π(dρ) is not necessarily zero, and the correct space of formsto work on isΩ∗(A)Kerπ+dKerπ, where Kerπ is the kernel of the map π. Thus the auxiliaryfields can be either quotiented out or eliminated through their equations of motion asthey are non-dynamical.

We choose to keep the auxiliary fields explicitly in our calcu-lations (rather than modding them out) since the step of identifying which fields aregenuinely independent is complicated and model-dependent. However, Proposition 4in [4] shows that, for the Yang-Mills functional, the two procedures are equivalent.Next we proceede to compute π(dρ) which is a lengthy calculation.

The elements ofthis matrix are functions of the ai′s and the bi′s and must be reexpressed in termsof the fields Am, φmn and possibly new independent fields. We first considerπ(dρ)11 =Xi/∂ai1/∂bi1 +Xi(M12ai2 −ai1M12)(M21bi1 −bi2M21)+Xi(M13ai3 −ai1M13)(M31bi1 −bi3M31)= /∂A1 + M12φ21 + φ12M21 + M13φ31 + φ13M31 −X11(2.25)where the auxiliary field X11 is given byX11 =Xiai1/∂2bi1 + [M12M21 + M13M31, bi1].

(2.26)Before continuing our calculations, we would like to point out the following problemand the necessary modifications needed to remedy it. The partXiai1/∂2bi1of the auxiliary field X11 is an n × n matrix whose elements are arbitrary functions.Thus the terms of the scalar Higgs potential could be absorbed in it.

This, of course,would be undesirable for any model (since all the scalar fields would remain masslessat the classical level). What saves the potential from disappearing alltogether is toinclude the information about the mixing between the three generations of quarks6

and leptons in the Dirac operator.This mixing is related to the fermionic massmatrix. Therefore the Dirac operator used in eq.

(2.3) should be modified toD =/∂⊗I ⊗Iγ5 ⊗M12 ⊗K12γ5 ⊗M13 ⊗K13γ5 ⊗M21 ⊗K21/∂⊗I ⊗Iγ5 ⊗M23 ⊗K23γ5 ⊗M31 ⊗K31γ5 ⊗M32 ⊗K32/∂⊗I ⊗I(2.27)where Kmn = K∗nm. The matrix K commutes with the ai and bi.

This modificationimplies that π(ρ) is obtained by substitutingφmn →φmn ⊗Kmn(2.28)and π(dρ)11 given in eq. (2.25), now becomes *π(dρ)11 = /∂A1+|K12|2(M12φ21+φ12M21)+|K13|2(M13φ31+φ13M31)−X11 (2.28)where the new field X11 is given byX11 =Xiai1/∂2bi1 + [|K12|2M12M21 + |K13|2M13M31, bi1](2.29)where |Kij|2 = K∗ijKij.

The other elements of π(dρ) can be found easily and expressedin the compact and generalizable formπ(dρ)mm = /∂Am +Xn̸=m|Kmn|2(Mmnφnm + φmnMnm) −Xmm(2.30)where the Xmm fields are defined byXmm =Xiaim/∂2bim + [Xn̸=m|Kmn|2MmnMnm, bim]. (2.31)The non-diagonal element π(dρ)12 is given byπ(dρ)12 = γ5K12−Xi/∂ai1(M12bi2 −bi1M12) + K13K32Xi(M12ai2 −ai1M12)/∂bi2+Xi(M13ai3 −ai1M13)(M32bi2 −bi3M32).

(2.32)and can be rewritten in terms of the fields Am and φmn and a new field X12π(dρ)12 = −γ5K12/∂φ12 + A1M12 −M12A2+ K13K32M13φ32 + φ13M32 −X12(2.33)* We omit the tensor product signs to simplify notation. Thus, e.g.

Kmn means1 ⊗1 ⊗Kmn and Mmn means 1 ⊗Mmn ⊗1.7

where the new field X12 is given byX12 =Xiai1(M13M32bi2 −bi1M13M32)(2.34)Similarly the other non-diagonal elements may be written in a compact and general-izable form:π(dρ)mn = −γ5Kmn/∂φmn + AmMmn −MmnAn+Xp̸=m,nKmpKpnMmpφpn + φmpMpn−Xmn,m ̸= n,(2.35)where the fields Xmn are defined byXmn =XiaimXp̸=m,nKmpKpnMmpMpnbin −bimMmpMpn,m ̸= n,(2.36)The elements π(dρ)mn are self adjoint,π(dρ)∗mn = π(dρ)nm(2.37)Collecting all these results, the representation of the curvature π(θ) can be writtenin terms of components. First, the diagonal elements are givenπ(θ)mm = 12γµνF mµν+ Xp̸=m(|Kmp|2|φmp+Mmp|2−Ym−X′mmm = 1, 2, 3 (2.38)where we have definedX′mm =Xiaim/∂2bim + (∂µAmµ + AµmAmµ )Fµν = ∂µAmν −∂νAmµ + [Amµ , Amν ]Ym =Xp̸=mXiaim|Kmp|2|Mmp|2bim(2.39).The non-diagonal elements of π(θ) are given by (m ̸= n):π(θ)mn = −γ5Kmn/∂φmn + Am(φmn + Mmn) −(φmn + Mmn)An−Xmn+Xp̸=m,nKmpKpn(φmp + Mmp)(φpn + Mpn) −MmpMpn(2.40)where we have used the notation |φmp|2 = φmpφpm.

The curvature is self-adjoint :π(θ)∗mn = π(θ)nm.8

The fields Ym and Xmn are not all independent. Depending on the structure ofthe mass matrices Mmn,some of them could be expressed in terms of φmn.

If it sohappens that all the X-fields are independent then after eliminating all the auxiliaryfields, the scalar potential will disappear. This does not happen if the mass matricesare chosen in such a way as to correspond to a possible vacuum with symmetrybreaking.

In the examples that we consider here, the potential will survive.The Yang-Mills action is given by the positive-definite expressionI = 18Trωθ2|D|−4(2.41)where Trω is the Dixmier trace. It is defined byTrω(|T|) = limω1log NNXi=0µi(T)(2.42))where T is a compact opreator, and µi are the eigenvalues of |T|.

This trace effectivelypicks out the coefficient of the logarithmic divergences.For the Dirac operatorswe shall consider the Dixmier trace to be equivalently replaced with a heat kernelexpression, using the identity|D|−4 =Z ∞0dǫǫe−ǫ|D|2. (2.43)and the expansiontrfe−ǫ|D|2=Zd4x√gf(x)(a0ǫ2 + a1ǫ + .

. .

)(2.44)where a0 = 1, g is the metric, and a1 = R is the curvature scalar. This can be usedto show that the action (2.41) is equal toI = 18Zd4x√gTrtr(π2(θ))(2.45)where tr is taken over the Clifford algebra, and Tr is taken over the matrix structure.Using eqs (2.38) and(2.45) the action takes the familiar form (in Euclidean space):I = −3Xm=1Tr14F mµνF µνm −12Xp̸=m|Kmp|2|φmp + Mmp|2 −Ym−X′mm2−12Xp̸=m|Kmp|2∂µ(φmp + Mmp) + Aµm(φmp + Mmp) −(φmp + Mmp)Aµp2+ 12Xn̸=mXp̸=m,n|Kmp|2(φmp + Mmp)(φpn + Mpn) −MmpMpn−Xmn2(2.46)9

where we have used the notation |Dµφ|2 ≡DµφDνφgµν, and when we analyticallycontinue to Minkowski space by the change x4 →it the action changes by IE →−IM.This action contains the Yang-Mills action for the gauge fields Aµm , kinetic energiesfor the scalar fields φmn, m ̸= n, and a potential for the scalar fields. In the laststep, the independent fields from the set X′mm, Xmn, and Ym must be eliminated.The result depends on the particular choices of Mmn and is model-dependent.

If thepotential survives, it is positive definite being a sum of squares and it is minimizedfor φmn = 0. Now we are ready to apply this construction to model building.3.

The SU(2) × U(1) standard modelTo clarify the general formalism developed in the last section, we consider thesimple example where the Riemannian manifold is extended by two points. In all theformulas of the last section we now setai3 = bi3 = 0,M13 = M23 = 0(3.1)We take the elements a1ǫM2(A), and a2ǫM1(A) to be 2 × 2 and 1 × 1 matrices,respectively.

The matrix M12 is then a 2 × 1 matrix and will be chosen to beM12 ≡µS = µ01(3.2)The choice of M12 dictates the breaking mechanism. With these choices π(ρ) takesthe formπ(ρ) =(A1)JIHJH∗IA2(3.3)where HI is a 2×1 doublet.

We shall also impose the graded tracelessness condition,Tr(Γ1π(ρ)) = 0, where Γ1 is the grading matrix Γ1 = diag(1, −1). This impliesTrA1 = A2(3.4)In terms of the elements ai and bi the Higgs field H takes the formH = µXiai1(Sbi2 −bi1S)(3.5)while the fields Xmn and Ym are given byX12 = 0 = X21Y1 = µ2 Xiai1Tbi1Y2 = µ2(3.6)10

where T is the matrix0001. This implies that the only auxilirary fields are X′11,X′22, and Y1, and these should be eliminated.

This step can be immediately takenand results in the disappearence of the terms involving these fields. The final actionthen takes the form (in Minkowski space ):I =14(F 1µν)JI (F 1µν)IJ + (F 2µν)(F 2µν)+ 12TrKK∗∂µ(HI + HI0) + (Aµ1)IJ(HJ + HJ0 ) −(HI + HI0)Aµ22−12Tr(KK∗)2 −(TrKK∗)2(HI + HI0)(H∗I + H∗0I) −µ22(3.7)where we have normalized the trace such that Tr1 = 1.

Note that, for this nor-malization of the trace, Tr(KK∗)2 −(TrKK∗)2 is positive (non-negative), by theSchwarz inequality for Tr. Thus the coupling constant of the quartic term in theHiggs potential is non-negative which guarantees stability of the theory at tree level.We also have that Tr(KK∗)2−(TrKK∗)2 ≤(n−1)(TrKK∗)2, where n is the numberof rows and columns of K. Therefore, the order of magnitude of the bare quarticHiggs coupling constant is the same as the order of magnitude of the square of thebare gauge coupling constant.

The potential is minimised when HI = 0. The fieldsare already expanded around the vacuum state, and the minimum corresponds tothe broken phase.

To display gauge invariance explicitely, the action could be easilyexpressed in terms of the shifted field H + H0.The gauge fields are in the familiar form of the standard model [7], but in thebroken phase. By writing(A1)JI = iA0 + A3A1 −iA2A1 + iA2A0 −A3A2 = 2iA0(3.8)one finds that A1µ −iA2µ = Wµ and A0µ +A3µ = Zµ are the W and Z gauge fields.

Theleptons fit naturally in this scheme, and can be included by introducing the spinorsL subject to the chirality conditionγ5 ⊗Γ1L = L(3.9)where this condition will only be imposed after we have performed the Wick rotationfrom Euclidean to Minkowski space. The spinors L then take the formL =lLe−R(3.10)11

where the left-handed electron and neutrino are in the first copy and form a doubletof SU(2): lL =νee−L, while the right-handed electron is in the second copy andis a singlet as can be deduced from the form of the elements ai and bi. The leptonicaction is then given byIl =< L, (D + π(ρ))L >=Zd4xLD + π(ρ)L(3.11a)In terms of components this becomesIl =Zd4xhlL(D + π(A))lL + eR(/∂+ A)eR+ lL(H + H0)eRK + eR(H∗+ H∗0)lLK∗i(3.11b)Thus, as required, the electron becomes massive, while the neutrino remains massless.Introducing SU(3) and the quarks is more complicated in this approach.

Thereason is that SU(3) is not broken, and no Higgs fields are necessary.It can beintroduced in an essentially commutative way. The solution adopted in [3-4] wasto introduce a bimodule.

One must introduce, in addition, a new algebra B whichmust be taken to be M1(C) ⊕M3(C).The mass matrices in the Dirac operatoralong these directions are taken to be zero, forcing the vanishing of the Higgs fieldsalong the same directions. Because the hypercharge assignments of the quarks aredelicate, the different U(1) factors must be related.

This is achieved with the fol-lowing assignments: On the algebra A we must set TrA1 = 0, A2 = −Y and on thealgebra B we must set B1 = −Y = −TrB2. This prescription guarantees the correcthypercharge assignments for the quarks and leptons.

The following point is in order.Although the relation between the different U(1) factors could be obtained from themathematical condition of the unimodularity of the algebras considered, it is clearthat this condition is not natural, especially since the main motivation behind thenon-commutative picture is to explain the geometric origin of the Higgs fields, andof the phenomena of symmetry breaking. Introducing SU(3) in a commutative wayand decoupling it from the rest is not very convincing.

However we shall proceed inour construction for illustration and to show that it is perfectly possible to obtainthe standard model using this method.The quarks are taken to be in the representationQ =1√2qLdR(3.12)12

subject to the chirality condition γ5 ⊗Γ1(Q) = Q, and qL =uLdLis a left-handedSU(2) doublet. Unfortunately, an action similar to that of the leptons will leavethe up quarks massless.

To avoid this one must also introduce the ”dual” quarkrepresentation˜Q =1√2 ˜qLuR(3.13)where ˜q =dL−uL= iτ2q is also a doublet of SU(2). By taking the matrix K to beK = diag(heαβ, hdαβ, huαβ)(3.14)where the hαβ’s are matrices in generation space.

By defining the spinorψα =LαQα˜Qα(3.15)where α = 1, 2, 3 refer to the three families, the full fermionic action can then bewritten asIf =< ψ, (D + π(ρ))ψ >=Zd4xψ(D + π(ρ))ψ(3.16)and when this action is expanded in terms of components, it gives exactly thefermionic action of the standard model. This shows that the standard model canbe obtained within the non-commutative setting.But as mentioned earlier, theSU(2) × U(1) sector fits more naturaly into this formalism than the SU(3) sector.4.

The SU(5) unified theoryThe way the strong interactions were introduced in the standard model suggeststhat a unified picture is more desirable from the geometrical point of view. This wasalso one of the reasons why model builders constructed unified theories.

Anotherreason is that it appears to be natural to assume that, at higher energies, the standardmodel is replaced by a more unified picture. The simplest example of such a schemeis the SU(5) gauge theory [8], which is the lowest rank group containing SU(3) ×SU(2) × U(1) as a subgroup.The SU(5) theory is spontaneously broken at twoscales.

At the grand unification scale M, SU(5) is broken to SU(3) × SU(2) × U(1)and this, in turn, is broken at the weak scale µ to U(1)em. The role of scales innon-commutative geometry is to measure the distance between the different copies13

of the space. Thus to reproduce the SU(5) theory we need to take the space to bethe product of the Reimannian manifold times three points.

In the minimal SU(5)theory the first stage of breaking is achieved through the use of the adjoint Higgsrepresentation 24 and the second stage through the fundamental 5 representation.The vacuum expectation value of the adjoint Higgs is taken to beΣ0 = Mdiag (2, 2, 2, −3, −3)(4.1)which breaks the symmetry from SU(5) to SU(3) × SU(2) × U(1). This is furtherbroken to U(1)em when the fundamental Higgs acquires the vacuum expectation valueH0 = µ00001≡µS(4.2)If the Riemannian manifold is extended by three points, the simplest possibilityto obtain two scales and not three, as well as a Higgs field belonging to the adjointrepresentaion and not to a product representation, is to identify two copies.

In otherwords we must have a permutation symmetry under the exchange 1 ↔2. Thereforewe must identifyai1 = ai2,bi1 = bi2(4.3)The operators in the first and second copies must be taken to be 5 × 5 matrices.

Toobtain the fundamental Higgs fields, the third copy must correspond to 1×1 matrices,and, to avoid an extra U(1) factor we take these elements to be real. Therefore wemust consider the algebraA = M5(C) ⊕M5(C) ⊕M1(R)(4.4)With these choices the vector potential π(ρ) becomesπ(ρ) =AΣHΣAHH∗H∗0(4.5)where A1 = A2 = A = γµ(Aµ)JI is a self-adjoint 5×5 gauge vector, ΣJI is a self-adjoint5 × 5 scalar field, and HI is a complex scalar field.

The reason that Σ is self-adjointlies in the permutation symmetry, as Σ21 = Σ12 = Σ∗12. This is also the reason whythe H′s in the first and second row are equal.

The vector A3 vanishes, because the14

self-adjointness condition implies that Aµ3 = −A∗µ3, but as Aµ3 = Pi ai3∂µbi3 is real,it must vanish. The tracelessness conditionTr(Γ1π(ρ)) = 0(4.6)where Γ1 = diag(1, 1, −1) impliesTrA = 0(4.7)reducing U(5) to SU(5).

In our method, the symmetry breaking pattern is specifiedby choosing the mass matrices Mmn in the Dirac operator to correspond to the desiredvacuum. We shall then takeM12 = M21 = Σ0M13 = M23 = H0(4.8)Writing the fields explicitly we find (see eq.

(2.27)):AIJ =Xi(ai1/∂bi1)IJ,Xiai1bi1 = 1φ12 ≡ΣIJ =Xiai1[Σ0, bi1]IJφ13 ≡HI = µXiai1(Sbi3 −bi1S)I(4.9)To determine the potential by eliminating the auxiliary fields, we must determinewhether the new functions Xmn, m ̸= n, and Ym are independent. First we findX12 = µ2 Xiai1[SS∗, bi1](4.10)which is equal to X21.

Clearly this is a new field which cannot be expressed in termsof the Σ and the H and thus must be eliminated. Next we calculateX13 =Xiai1(M13M32bi3 −bi1M13M32)= −3MHI(4.11)which is equal to X23, and where we have used the property that Σ0H0 = −3MH0.Obviously these fields are not auxiliary and contribute to the potential.

The otherfields X31, X32, are the conjugate of X13. The Y ′s are more subtle.

First, we havethatY1 =Xiai1|K12|2Σ20 + |K13|2H0H∗0bi1(4.12)15

which appears to be independent. However, because of the property of Σ0Σ20 = 15TrΣ20 −MΣ0(4.13)and eq.

(4.10), and the definition of Σ: Σ = Pi ai1[Σ0, bi1], we can rewrite it asY1 = |K12|2(−MΣ + Σ20) + |K13|2(H0H∗0 + X12)(4.14)It thus must be kept, as it will contribute when X12 is eliminated. Finally Y2 = Y1and Y3 = 2µ2|K31|2.

Collecting all these results, the action takes the formI = Tr12FµνF µν + |K12|2∂µ(Σ + Σ0) + [Aµ, Σ + Σ0]IJ2+ |K132|(∂µ + Aµ)(H + H0)I2−V (H, Σ)(4.15)where the potential V (H, Σ) is the scalar potential and is given byV = Tr|K12|2(Σ + Σ0)2 + |K13|2(H + H0)(H + H0)∗−Y1−X′112+K413||H + H0|2 −H0H∗0 −X12IJ2+ |K12K232|(Σ + Σ0 + 3M)(H + H0)I2+ 2K31|4(H + H0)∗(H + H0) −µ2−X′332(4.16)We now eliminate the auxiliary field X12 from the first two terms of the potential.This is then followed by eliminating X′11 and X′33 to get the manifestly gauge invariantpotentialV (Σ, H) =Tr|K12|4 −(Tr|K12|2)2(Σ + Σ0)2 + M(Σ + Σ0) −15TrΣ20IJ2+ Tr|K12K23|2(Σ + Σ0 + 3M)(H + H0)I2+ 2Tr|K31|4 −(Tr|K31|2)2(H + H0)∗(H + H0) −µ22(4.17)Clearly the potential is positive-definite, and the minimum occurs when Σ = 0 = H.Also, in order not to loose the Σ potential, the matrix K12 should be different fromthe identity matrix. So the picture we have is that of a space-time consisting of threecopies where two of the copies are identical and seperated by a distance of orderM −1.

These in turn are seperated from the third copy by a distance of order µ−1.At the next stage, we introduce the fermions into the picture. As is known, thefermions fit neatly as left-handed chiral spinors, in 5 + 10 representations of SU(5)16

denoted by ψI + ψIJ [8]. The fermionic action contains, besides the kinetic energiesinteracting with the SU(5) gauge fields, the Yukawa couplingsIy =Zd4xfαβψcIαH∗JψIJβ + f ′αβǫIJKLMψcαIJψKLβHM + h.c(4.18)where fαβ and f ′αβ are matrices in the family space, α, β = 1, 2, 3, and ψc = Cψ ,is the charge conjugate spinor having the same chirality as ψ, (C being the chargeconjugation matrix).In the present formulation of non-commutative geometry, the full fermionic ac-tion including both the kinetic terms and the Yukawa couplings must be obtainedfrom an expression of the form < Ψ, (D + π(ρ))Ψ >, where Ψ is some appropriaterepresentation for the spinors.

We wish to incorporate the (5 + 10)L into one spinorwhere we shall use the equivalence 5L = 5R. We define the spinor Ψ which transformsasΨ →gΨ = g ⊗gΨ(4.19)under the antisymmetric tensor product representation of the group U(A) of unitaryelements of the algebra A (acting on h ∧h).

A useful representation for this spinoris ΨAB, where the indices A and B take the values A = I1, I2, 1 along the directionsof the three spaces. It must then satisfyΨAB = −ΨBA.

(4.20)This together with the permutation symmetry 1 ↔2 implies that ΨAB has thefolowing components:ΨI1J1 = ΨI2J2 =1√6ψIJΨI1J2 = ΨI2J1 = 0ΨI11 = −Ψ1I1 = ΨI21 =1√2ψI(4.21)By further imposing the chirality conditionγ5Γ1 ⊗Γ1Ψ = Ψ, which can be writtenin the formγ5(Γ1)A′A (Γ1)B′B ΨA′B′ = ΨAB(4.22)one finds that ψIJ is left-handed and ψI is right-handed:ψIJ = ψIJ (L)ψI = ψI (R)(4.23)17

To put it differently, the fermions fit neatly in one spinor in a representationtransforming under the antisymmetric product of U(A).The fermionic action isthenI1f =< Ψ, (D + π(ρ) ⊗1 + 1 ⊗π(ρ))Ψ >=Zd4xΨABDΨAB + π(ρ)CAΨCB + π(ρ)CBΨAC=Zd4x13ψIJ (L)(/∂ψIJ (L) + AKI ψKJ (L) + AKJ ψIK (L)) + ψI (R)(/∂+ A)JI ψJ (R)+ 1√3(K13ψI (R)(H + H0)JψIJ (L) + h.c)(4.24)and where the particle assignments are taken to beψIJ (L) =0uc3−uc2u1d1−uc30uc1u2d2uc2−uc10u3d3−u1−u2−u30e+−d1−d2−d3−e+0L(4.25)for the 10 representation, andψcI =d1d2d3ecνcR(4.26)for the 5 representation.It is clear that this interaction provides masses to theleptons and down quarks but not to the up quarks. (The neutrinos will be alwaysmassless since they have no right-handed partners).

This situation is identical tothe one we faced for the Yukawa couplings of the quarks in the standard model. Interms of the SU(5) couplings such Yukawa couplings come from the second termin eq(4.18) and require the introduction of the epsilon tensor of SU(5).

We thenintroduce the spinor in the completely antisymmetric tensor product U(A), (actingon h ∧h ∧h). It can be represented by a spinor χABC completely antisymmetric inA, B, C. Because of the permutation symmetry 1 ↔2 the non-vanishing componentsare χI1J1K1 = χI2J2K2 ≡χIJK and χI1J11 = χI2J21 ≡χIJ.The chirality condition onχ isγ5Γ1 ⊗Γ1 ⊗Γ1χ = χ.

(4.27)This implies thatχIJK = χIJK(L)χIJ = χIJ(R)(4.28)18

However, since we do not wish to introduce more particles, these spinors will berelated to ΨAB by the identificationχIJK(L) =16√2ǫIJKMNψMN (L)χIJ(R) =1√6CψIJ (L)(4.29)The action corresponding to the χ-spinor is thenI2f =< χ, (D + π(ρ) ⊗1 ⊗1 + 1 ⊗π(ρ) ⊗1 + 1 ⊗1 ⊗π(ρ))χ >=Zd4xχABCDχABC + 3π(ρ)AEχEBC(4.30)The component form of this action then takes the formI2f =Zd4x23ψIJ (L)(/∂ψIJ (L) + 2(A)KI ψKJ (L))+16√3K31ǫIJKMNψcIJ(H∗K + H∗0K)ψMN + h.c(4.31)Notice that we have defined the spinors in such a way that their kinetic energies areproperly normalized. Finally, in order to give different masses to the up and downquarks, we must introduce the spinorλ =Ψχ(4.32)and take the matrix K13 to be of the formK = diag(fαβ, f′αβ)(4.33)where fαβ and f′αβ are matrices in generation space.

In this way the fermionic actioncould be written compactly in the form< λ, (D + π(ρ))λ >(4.34)Note that the fermionic mass matrix is proportional to K13, while K12 is necessaryfor the survival of the Σ self couplings in the potential.To summarize, the present picture looks very attractive: The discrete structureof space becomes apparent, first, at the weak scale, and the Higgs field H is associatedwith the mediation between the third copy and the two identical copies which, atthat scale, would appear to coincide. As we climb up in energy, probing the smallerdistance scale, we encounter the Higgs field Σ associated with the mediation between19

the two identical copies. The fermions fit into one representation (and its conjugate)and their action takes a very simple form.

Of course, from the phenomenologicalpoint of view, the gauge group SU(5) has a serious drawback connected with protondecay. The rate predicted by this model is ruled out experimentally, and only in morecomplicated models this problem is avoided.

The analysis of such models, however, isbeyond the scope of this paper. The construction of a phenomenologically successfulmodel will be left to the future.

Here we content ourselves with the construction ofsome prototype models, in order to master and illustrate the new techniques advancedhere.5. Left-right SU(2)L × SU(2)R × U(1)B−L symmetric modelAnother class of attractive models which are of phenomenological interest is theleft-right symmetric models.

The simplest one of which is the SU(2)L × SU(2)R ×U(1)B−L theory [9]. The non-commutative geometry setting is perfectly appropriatefor product groups.

The Higgs fields used in the breaking are usually taken to bea (2, 2) and (3, 1) + (1, 3) with respect to SU(2)L × SU(2)R. It is also possible toreplace the (3, 1)+(1, 3) by doublets (2, 1)+(1, 2). But this choice is less prefered forphenomenological reasons [9].

As we have learned from the SU(5) theory in order toget an adjoint Higgs representation, two copies must be identified, i.e. interchangedby permutation symmetry.

Thus, for each SU(2), the Riemannian manifold should beextended by two points. The total extension is by four points.

One can immediatelysee that the algebra must be taken to beA2 = M2(C) ⊕M2(C) ⊕M2(C) ⊕M2(C)(5.1)The elements a1, b1, a2, b2, a3, b3, a4, b4, are 2 × 2 matrices.We must require thepermutation symmetries 1 ↔2 and 3 ↔4. Thena1 = a2b1 = b2a3 = a4b3 = b4(5.2)The vector potential π(ρ) takes the formπ(ρ) =A1∆1ΦΦ∆1A1ΦΦΦ∗Φ∗A2∆2Φ∗Φ∗∆2A2(5.3)where A1 and A2 are U(2)L and U(2)R gauge fields, ∆1 and ∆2 are triplets in theadjoint representaions of the respective groups, and Φ is (2, 2) with respect to theproduct groups.20

The mass matrices entering the Dirac operator are taken to beM12 = M21 =00v10≡v1SM13 = M14 = M23 = M24 =u100u2M34 = M43 =00v20(5.4)where u1, v1, v2 are taken to be real, and u2 is taken to be complex, with the phaseof u2 related to CP violation. To reduce the gauge group from U(2)L × U(2)R toSU(2)L × SU(2)R × U(1)B−L we impose the tracelessness conditionTr(Γ1π(ρ)) = 0(5.5)where Γ1 = diag(1, 1, −1, −1).

The scalar fields in π(ρ) are given in terms of the aiand bi by∆1 = v1Xiai1Sbi1∆2 = v2Xiai3Sbi3Φ =Xiai1(M13bi3 −bi1M13)(5.6)These expressions are important in determining which of the auxiliary fields areindependent. The X- and Y - fields are given byX12 = 2(|u2|2 −|u1|2)Xiai1[T, bi1]X13 = u1Xiai1(v1Sbi3 −bi1v∗2S∗)X34 = (|u2|2 −|u1|2)Xiai3[T, bi3]Y1 = TrKK∗2|u1|2 + (2|u2|2 −2|u1|2 + |v1|2)Xiai1Tbi1Y3 =2|u1|2 + (2|u2|2 −2|u1|2 + |u1|2)Xiai3Tbi3(5.7)where the other functions could be determined from the above using the permutationsymmetry, and the matrix T is given by: T =0001.

For simplicity we haveassumed that K12 = K34 = K13 = K.21

From the form of the action in eq. (2.46) it is clear that the field X13 is auxiliaryand eliminating it will remove the whole term that appears in θ13.

The remainingpotential is given byV = V1(Φ, ∆1) + V2(Φ, ∆2)(5.8)where the two parts areV1(Φ, ∆1) = 2Tr(KK∗)2 −(TrKK∗)22|Φ + M13|2 + |∆1 + M12|2−2αZ1 −|M12|2 −2|M13|22+ 8Tr(KK∗)2|Φ + M13|2 −|M13|2 −βZ12(5.9a)where α = 2|u2|2 −2|u1|2 + |v1|2 and β = |u2|2 −|u1|2 and Z1 = Pi ai1Tbi1 is anauxiliary field. The second part V2 has a similar structure and can be obtained bythe substitutionsV2(Φ, ∆2) = V1(Φ →Φ∗, ∆1 →∆2, v1 →v2, Z1 →Z2)(5.9b)where Z2 = Pi ai3Tbi3.

Elimination of Z1 and Z2 will yield a potential of the desiredform.The leptons have the formΨ =ψ1ψ1ψ2ψ2(5.10)where ψ1 and ψ2 are doublets under the two SU(2) groups.After imposing thechirality condition(γ5 ⊗Γ1)Ψ = Ψ(5.11)one getsψ1 = ψ1 (L)ψ2 = ψ2 (R)(5.12)By writing ψ =νee−one finds that the usual leptons (with neutrinos acquiringMajorana masses ) emerge. The required coupling is< Ψ, (D + π(ρ))Ψ >=Zd4xΨ(D + π(ρ))Ψ(5.13)However, in order to make the right fermions heavy, one must introduce the conjugatefermionsχ =χ1χ1χ2χ2(5.14)22

required to also satisfy the chirality condition. We shall make the identifications,χ1 = iτ2ψ1 and χ2 = iτ2ψ2.

The other term needed in the action is< χ, (D + π(ρ))χ >=Zd4xχ(D + π(ρ))χ(5.15)and provides, in addition to the kinetic terms which appear again, the coupling of theconjugate Higgs. The quarks can be introduced in a similar manner.

However, thecorrect coupling to U(1) can only be achieved after introducing SU(3). This can bedone in a way identical to that in the standard model, where the U(3) × U(1) groupis coupled through the bimodule structure with both U(1) and TrU(3) related tothe U(1)B−L in order to provide the correct hypercharge assignments for the quarks.The phenomenological details of this model will be treated elsewhere.6.Summary and conclusionWe have achieved the main objective set for this paper: The construction of aformalism using the framework of non-commutative geometry of Alain Connes [1-2].

We have modified one particular point in that we choose the basic algebra tobe a direct sum of matrix algebras. This simplifies the computations and makes itpossible to consider large groups, while in the original setting this becomes a verycomplicated task, since all the elements in the curvature have to be computed one byone.

Another improvement is choosing the vacuum state of the potential to appear inthe Dirac operator. Such a choice might appear to break gauge invariance.

But thisis not so, as the Dirac operator does not transform under gauge transformations, andthe actions constructed are shown to be gauge invariant. This is also seen in detail inthe component form of the actions.

We have derived the formulas for the Yang-Millsaction corresponding to a continuous space-time multiplied by three points, but havewritten the formulas in a way applicable to a space extended by N points. We havestudied three examples in great detail, and showed, in a step by step calculation,how to extract the bosonic action by eliminating the auxiliary fields, and how tointroduce fermions in a realistic way.

Of course, this formalism does not pretend tosolve the fundamental problem of explaining the fermionic mass matrices, and thusdoes not reduce the number of parameters associated with the fermion masses. Ithowever specifies the Higgs sector and reduces the number of possible terms at thetree level, since the potential takes a very specific form.

At the classical level, thecosmological constant always naturally comes out to be zero. All this provides a verygood motivation to investigate some of the problems arising in this formalism.

Thefirst question that one may ask is on how restrictive this new formalism is. Obviouslyit is somewhat restrictive, but not to the point that only few models remain.

Onecommon feature is that the models that one can construct favor the minimal Higgs23

representations. Indeed, if in the SU(5) example one wanted to introduce the Higgsrepresentation 45 one finds that this can only be done by taking it as an externalscalar field not associated to any vector.Of course, this would be self-defeatingand cannot be considered to be natural.

In this respect, an SO(10) model whichis acceptable phenomenologically is not easy to construct, since it would requirecomplicated Higgs’s such as 120 or 16s.The second question that require further study is the question of whether space-time supersymmetry can be embedded in non-commutative geometry. This questiondoes not appear to have an obvious answer.

The difficulty is that the basic buildingblock in non-commutative geometry is the Dirac operator, while in supersymmetry,what appears to be more fundamental is the supersymmetric covariant derivative,Dα,which satisfies{Dα, Dβ} = (/∂)βαIt is like a square root of the Dirac operator.It would be extremely interestingif the second question could be answered in the affirmative and will have positiveconsequences for constructing models which are acceptable phenomenologically.Most fundamental, however, is the question of quantization of theories in non-commutative geometry.At present we only have information about the classicalaction, and any quantum effects can only be dealt with by starting from the actionextracted in the classical limit. The non-commutative geometry setting advocatedin this paper (if preserved after quantization ) imposes certain constraints on thecounterterms admissible in the renormalization of the quantum theories.

It is likelythat these constraints yield relations between the square of gauge coupling constantsand certain quartic Higgs coupling constants. We hope to report on some of thesequestions in future projects.AcknowledgmentsWe would like to thank D. Wyler for very useful discussions.24

References[1] A. Connes, Publ. Math.

IHES 62 44 (1983). [2] A. Connes, in the interface of mathematics and particle physics , Clarendonpress, Oxford 1990, Eds D. Quillen, G. Segal and S. Tsou[3] A. Connes and J. Lott,Nucl.Phys.B Proc.Supp.

18B 29 (1990), North-Holland,Amsterdam. [4] A. Connes and J. Lott, to appear in Proceedings of 1991 Summer Cargese con-ference.

[5] D. Kastler , Marseille preprints[6] R. Coquereaux, G. Esposito-Far´ese, G. Vaillant, Nucl. Phys.B353 689 (1991);M. Dubois-Violette, R. Kerner, J. Madore, J.

Math. Phys.31 (1990) 316;B. Balakrishna, F. G¨ursey and K. C. Wali, Phys.

Lett. 254B (1991) 430.

[7] S. Glashow Nucl. Phys.22,579 (1961);A. Salam and J.

Ward Phys. Lett13,168 (1964);S. Weinberg, Phys.

Rev. Lett.19, 1264 (1967);A. Salam, in Elementary Particle Theory (editor N. Svartholm), Almquist andForlag, Stockholm.

[8] H. Georgi and S. Glashow, Phys. Rev.

Lett32 438 (1976);For a review of unified theories see the book by G. Ross Grand Unified Theories,Frontiers in Physics Series, vol 60, Benjamin Publishing. [9] R. Mohapatra and J. Pati, Phys.

Rev.D11 566 (1975);R. Mohapatra and G. Senjanovich Phys. Rev.D21165 (1981);For a review of left-right symmettric models see the book by R. MohapatraUnification and Supersymmetry Springer-Verlag, Berlin.25

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