Grand Unification in the Spectral Pati-Salam Model

We analyze the running at one-loop of the gauge couplings in the spectral Pati-Salam model that was derived in the framework of noncommutative geometry. There are a few different scenario's for the scalar particle content which are determined by the precise form of the Dirac operator for the finite noncommutative space. We consider these different scenarios and establish for all of them unification of the Pati-Salam gauge couplings. The boundary conditions are set by the usual RG flow for the Standard Model couplings at an intermediate mass scale at which the Pati-Salam symmetry is broken.


I. INTRODUCTION
This paper builds on two recent discoveries in the noncommutative geometry approach to particle physics: we showed in [9] how to obtain inner fluctuations of the metric without having to assume the order one condition on the Dirac operator. Moreover the original argument by classification [4] of finite geometries F that can provide the fine structure of Euclidean space-time as a product M × F (where M is a usual 4-dimensional Riemannian space) has now been replaced by a much stronger uniqueness statement [7,8]. This new result shows that the algebra where H are the quaternions, appears uniquely when writing the higher analogue of the Heisenberg commutation relations. This analogue is written in terms of the basic ingredients of noncommutative geometry where one takes a spectral point of view, encoding geometry in terms of operators on a Hilbert space H. In this way, the inverse line element is an unbounded self-adjoint operator D. The operator D is the tensor sum of the usual Dirac operator on M and a 'finite Dirac operator' on F , which is simply a hermitian matrix D F . The usual Dirac operator involves γ matrices which allow one to combine the momenta into a single operator. The higher analogue of the Heisenberg relations puts the spatial variables on similar footing by combining them into a single operator Y using another set of γ matrices and it is in this process that the algebra (1) appears canonically and uniquely in dimension 4. We refer to [7,8] for a detailed account. What matters for the present paper is that the above process leads without arbitrariness to the Pati-Salam [19] gauge group SU (2) R × SU (2) L × SU (4), together with the corresponding gauge fields and a scalar sector, all derived as inner perturbations of D [9]. Note that the scalar sector can not be chosen freely, in contrast to the early work on Pati-Salam unification [1, 10,12,13]. In fact, there are only a few possibilities for the precise scalar content, depending on the assumptions made on the finite Dirac operator. From the spectral action principle, the dynamics and interactions are described by the spectral action [2,3], where Λ is a cutoff scale and f an even and positive function. In the present case, it can be expanded using heat kernel methods, where F 4 , F 2 , F 0 are coefficients related to the function f and a k are Seeley deWitt coefficients, expressed in terms of the curvature of M and (derivatives of) the gauge and scalar fields. This action is interpreted as an effective field theory for energies lower than Λ.
One important feature of the spectral action is that it gives the usual Pati-Salam action with unification of the gauge couplings [9] (cf. Eq. (36) below). This is very similar to the case of the spectral Standard Model [6] where there is unification of gauge couplings. Since it is well known that the SM gauge couplings do not meet exactly, it is crucial to investigate the running of the Pati-Salam gauge couplings beyond the Standard Model and to find a scale Λ where there is grand unification: This would then be the scale at which the spectral action (3) is valid as an effective theory. There is a hierarchy of three energy scales: SM, an intermediate mass scale m R where symmetry breaking occurs and which is related to the neutrino Majorana masses (10 11 − 10 13 Gev), and the GUT scale Λ. For simplicity, we restrict our analysis to the running of the gauge couplings at one-loop. Indeed, at two loops, the gauge and scalar couplings are mixed and influence each other. Moreover, the running of the scalar mass terms can not be trusted at all because of quadratic divergences.
Thus, we analyze the running of the gauge couplings according to the usual (one-loop) RG equation where each takes the form The coefficient b is determined by the particle content and their representation theory [11,[14][15][16] for which we use [17] as well as the program PyR@TE. As mentioned before, depending on the assumptions on D F , one may vary to a limited extent the scalar particle content, consisting of either composite or fundamental scalar fields. We will not limit ourselves to a specific model but consider all cases separately. In fact, we establish grand unification for all of them, thus confirming validity of the spectral action at the corresponding scale, independent of the specific form of D F .

II. SPECTRAL PATI-SALAM AND GRAND UNIFICATION
One of the pressing questions at present is whether there is new physics beyond the Standard Model. The success of the spectral construction of the Standard Model, predicting its particle content, including gauge fields, Higgs fields as well as a singlet whose vev gives Majorana mass to the right handed neutrino, is a strong signal that we are on the right track. The route that led to this conclusion starts with classifying the algebras of the finite space. The results show that the only algebras which solve the fermion doubling problem are of the form M 2a (C) ⊕ M 2a (C) where a is an even integer. An arbitrary symplectic constraint is imposed on the first algebra restricting it from M 2a (C) to M a (H). The first non-trivial algebra one can consider is for a = 2 with the algebra [4] Coincidentally, and as explained in the introduction, the above algebra comes out as a solution of the two-sided Heisenberg quantization relation between the Dirac operator D and the two maps from the four spin-manifold and the two four spheres S 4 × S 4 [7,8]. This removes the arbitrary symplectic constraint and replaces it with a relation that quantize the four-volume in terms of two quanta of geometry and have far reaching consequences on the structure of space-time.
The existence of the chirality operator γ that commutes with the algebra breaks the quaternionic matrices M 2 (H) to the diagonal subalgebra and leads us to consider the finite algebra The Pati-Salam gauge group SU (2) R ×SU (2) L ×SU (4) is obtained as the inner automorphism group of A = C ∞ (M )⊗ A F , and the corresponding gauge bosons appear as inner perturbations of the (spacetime) Dirac operator [9]. Next, an element of the Hilbert space Ψ ∈ H is represented by where ψ c A is the conjugate spinor to ψ A . Thus all primed indices A correspond to the Hilbert space of conjugate spinors. It is acted on by both the left algebra M 2 (H) and the right algebra M 4 (C). Therefore the index A can take 16 values and is represented by where the index α is acted on by quaternionic matrices and the index I by M 4 (C) matrices. Moreover, when the grading breaks 1, .
2 (dotted index) is acted on by the first quaternionic algebra H R and a = 1, 2 is acted on by the second quaternionic algebra H L . When M 4 (C) breaks into C ⊕ M 3 (C) (due to symmetry breaking or through the use of the order one condition as in [4]) the index I is decomposed into I = 1, i and thus distinguishing leptons and quarks, where the 1 is acted on by the C and the i by M 3 (C) . Therefore the various components of the spinor ψ A are 1, This is a general prediction of the spectral construction that there is 16 fundamental Weyl fermions per family, 4 leptons and 12 quarks. The (finite) Dirac operator can be written in matrix form and must satisfy the properties where J 2 F = 1. A matrix realization of γ F and J F are given by where cc stands for complex conjugation. These relations, together with the hermiticity of D imply the relations and have the following zero components [5] ( aI arbitrary. These restrictions lead to important constraints on the structure of the connection that appears in the inner fluctuations of the Dirac operator. In particular the operator D of the full noncommutative space given by gets modified to where We have shown in [9] that components of the connection A which are tensored with the Clifford gamma matrices  aI whose values are given by those that were derived for the Standard Model, then the Higgs fields Σ . bJ aI , H aIbJ and H . aI . bJ will become composite and expressible in terms of more fundamental fields Σ J I , ∆ . aJ and φ b . a . We refer to this as the composite model. Depending on the precise particle content we determine the coefficients b R , b L , b in (5) that control the RG flow of the Pati-Salam gauge couplings g R , g L , g. We run them to look for unification of the coupling g R = g L = g. 1 in terms of the Standard Model gauge couplings g 1 , g 2 , g 3 . At the mass scale m R the Pati-Salam symmetry is broken to that of the Standard Model, and we take it to be the same scale that is present in the see-saw mechanism. It should thus be of the order 10 11 − 10 13 Gev. We now discuss the three models, in order of complexity.

A. Pati-Salam with composite Higgs fields
We first consider the case of a finite Dirac operator for which the Standard Model subalgebra C⊕H L ⊕M 3 (C) ⊂ A F satisfies the first-order condition [4]. This condition is extremely constraining and forces the couplings of the righthanded neutrino to be with a singlet. In this case, the off-diagonal term in (11) becomes and the diagonal structure of D F is determined by the following sub-matrices [5] D β1 where The Yukawa couplings k ν , k e , k u , k d are 3 × 3 matrices in generation space. Notice that this structure gives Dirac masses to all the fermions, but Majorana masses only for the right-handed neutrinos. One can also consider the special case of lepton and quark unification by equating k ν = k u , k e = k d which imply some simplifications. The inner perturbations of the finite Dirac operator of the above type were determined in [9] to be composite fields Σ bJ . aI and H .
aI . bJ , depending on fundamental Higgs fields φ b . a , Σ J I and ∆ . aJ in the following way: The field φ b . a is not an independent field and is given by We have listed the fundamental Higgs fields and their representations in Table I. We first assume that there is lepton quark unification, so that the Σ J I is decoupled. The β-functions for the Pati-Salam couplings g R , g L , g with the above particle content are found to be The solutions of the RG-equations are easily found to be We impose the boundary conditions (24) at the mass scale µ = m R . As a first approximation, we adopt the usual running of the SM gauge couplings, leaving a full analysis of the effect of the scalar fields after Pati-Salam symmetry breaking to future work. Also, we ignore the presence of non-renormalizable terms in the spectral action for the composite model. Then, after experimenting with different values of m R , we find a unification scale Λ ≈ 2.5×10 15 Gev if we set m R = 4.25 × 10 13 Gev (Figure 1). If the scalar field Σ J I is not decoupled -in other words, if there is no lepton-quark coupling unification-then there is an additional scalar (1 R , 1 L , 15) irreducible representation contributing to the β-function, giving a slightly different 3, 9 . This in turn gives a unification scale Λ ≈ 6.3 × 10 15 Gev for m R = 4.1 × 10 13 Gev.  bJ are now themselves fundamental Higgs fields [9,Sect. 5] and their representations are listed in Table II. The β-functions are computed to be Note that the SU (2) R and SU (2) L -sectors are not asymptotically free, due to the large scalar sector. Nevertheless, we can still run the gauge couplings with the boundary values set by (24). Adopting the same approximation as in the previous section, this results in Figure 2. The unification scale is Λ ≈ 6.3 × 10 16 Gev if we set m R = 1.5 × 10 11 Gev.

C. Left-right symmetric Pati-Salam with fundamental Higgs fields
As a final possibility we consider the most general case for D F which gives in addition to the fundamental Higgs fields in Table II the field H aIbJ in the (1 R , 3 L , 10) + (1 R , 1 L , 6) representation. The β-functions become Adopting once more the approximation that we made use of in the previous sections, we run the Pati-Salam gauge couplings from m R , resulting in Figure 3. We find the unification scale to be Λ ≈ 2.7 × 10 15 Gev if we set m R = 5.1 × 10 13 Gev.

III. CONCLUSIONS
We have analyzed the running of the Pati-Salam gauge couplings for the spectral model, considering different scalar field contents corresponding to the assumptions made on the finite Dirac operator. We stress that the number of possible models is quite restrictive and that one can not freely choose the particle content. We have identified the three main models, although there exists small variations on them. The different possibilities correspond to restrictions on the geometry of the finite space F . In all the models considered here, we establish unification of the gauge couplings, with boundary conditions set by the usual Standard Model gauge couplings at an intermediate mass scale.
Besides the direct physical interest of such grand unification, it also determines the scale at which the asymptotic expansion of Equation (3) is actually valid as an effective theory. In order to see this, note that the scale-invariant term F 0 a 4 in (3) for the spectral Pati-Salam model contains the terms [9]: Normalizing this to give the Yang-Mills Lagrangian demands which requires gauge coupling unification, g R = g L = g. Note that the similar result for the Standard Model gauge couplings does not hold (at least at the one-loop level) because the three couplings actually do not meet, even though they are required to be unified in the spectral action [2]. We consider this to be strong evidence for the spectral Pati-Salam model as a realistic possibility to go beyond the Standard Model. To summarize, the spectral construction of particle physics models based on a spectral triple with a noncommutative space with metric dimension four and whose finite part has KO dimension 6 leads directly to a family of Pati-Salam models with gauge symmetry SU (2) R × SU (2) L × SU (4) and well defined Higgs structure. Breaking of SU (2) R × SU (4) to U (1) × SU (3) occurs at some scale m R ∼ 10 11 − 10 13 Gev with a unification scale where the three coupling constants meet of the order of 10 16 Gev. All these breakings will have the Standard Model as an effective theory at low energies.