$B$-decay anomalies and scalar leptoquarks in unified Pati-Salam models from noncommutative geometry

Motivated by possible scalar-leptoquark explanations of the recently reported $B$-decay anomalies, we investigate whether the required leptoquarks can be accommodated within models based on noncommutative geometry (NCG). The models considered have the gauge structure of Pati-Salam models, $SU(4)\times SU(2)_L\times SU(2)_R$, with gauge coupling unification at a single scale. In one of the models, we find a unique scalar leptoquark with quantum numbers $(3,1,-\frac{1}{3})_{321}$, originating from a complex multiplet $(6,1,1)_{422}$, which can potentially explain the $B$-decay anomalies if its mass is on the order of a few TeV. The unification of couplings can be realized with the inclusion of a single step of intermediate symmetry breaking. The scalar leptoquark under consideration does not contribute to proton decay due to the absence of diquark couplings, as dictated by the underlying noncommutative geometry.


Introduction
In papers [1] and [2] we have studied the phenomenology of models proposed by Chamseddine, Connes, and van Suijlekom in Refs. [3,4], which are based on Connes' noncommutative geometry (NCG) [5,6]. The models are characterized by a Pati-Salam gauge structure G 422 = SU (4) × SU (2) L × SU (2) R with the requirement that the gauge couplings unify at a single scale, and a scalar sector whose content and couplings are fixed by the NCG of the models. The fermion content is the same as the Standard Model (SM) plus the right-handed neutrinos of each generation.
The unification of gauge couplings is a distinguishing feature of NCG based versions of the SM [7,8], as well as its G 422 extensions discussed in this paper [3,4]. It is due to the underlying "spectral action" having only one overall coupling, despite the fact that the gauge groups do not unify into a single simple Lie group, that is, the model is not a Grand Unified Theory (GUT). Note that coupling unification is not a requirement of canonical non-GUT G 422 models found in the literature [9][10][11][12][13].
In Ref. [1], we investigated whether a 2 TeV W R boson [14][15][16][17][18][19] could be accommodated within the NCG models considered, and in Ref. [2] whether a 750 GeV diphoton resonance [20,21] could be accommodated. Due to constraints imposed on the models by the underlying NCG, no freedom exists to adjust the models' particle content, except via decoupling those that exist from the renormalization group (RG) running of the gauge coupling constants by rendering them heavy through symmetry breaking. Consequently, though it is not difficult to find particles within the models that can be identified with the putative W R or the diphoton resonance, we have found that lowering their masses to TeV scales while maintaining the coupling unification condition is highly non-trivial. The hints of the W R and diphoton resonance at the LHC have since disappeared [22][23][24], but our analyses nevertheless underscore the rigidity, and consequently the predictability, of the NCG based models.
In this paper, we continue our investigation and question whether the models could naturally accommodate a TeV-scale scalar leptoquark necessary to explain the B-decay anomalies observed at the LHC and elsewhere [25]. The leptoquark of interest is S 1 (3, 1, 1 3 ) 321 , a colored isoscalar with electromagnetic charge 1 3 . In order to explain the B-decay anomalies, its mass must be in the few TeV range [26][27][28][29]. Again, it is not difficult to identify scalars with the required quantum numbers within the NCG models, but the quantum number assignment alone does not necessarily guarantee that the scalars couple to lepton-quark pairs as leptoquarks should do. And again, the challenge is whether coupling unification can be achieved or not. On this question, we recall that in the context of SU (5) GUT, the presence of certain leptoquarks actually helps in unifying the couplings without SUSY [30,31]. Our study will focus on whether the field corresponding to the S 1 has a similar effect in NCG models. This paper is organized as follows: In section 2 we briefly review the current status of B-decay anomalies. In section 3 we list the various leptoquark explanations that have been proposed in the literature, and identify the introduction of a single S 1 leptoquark as the most attractive solution. Section 4 discusses the scalar content of the NCG models and their Yukawa couplings, and we find a unique field that would serve our purpose as the S 1 leptoquark. Section 5 presents an analysis on how and whether the coupling unification condition can be maintained when the S 1 mass is a few TeV. Section 6 concludes with a discussion of how the proton decay is, perhaps surprisingly, not a problem in our context, as well as some concluding comments on the NCG framework and the current status of noncommutativity in quantum gravity and string theory. We end with an appendix in which we review the derivation of the most generic Lagrangian for scalar and vector leptoquark interactions with Standard Model (SM) fermions.

B-decay Anomalies
During the past few years, several disagreements between experiment and Standard Model (SM) predictions in rare B-decays have been observed by LHCb [32][33][34][35][36], Belle [37][38][39], and Babar [40,41]. The decay channels in question are B + → K + µ + µ − [32], B 0 → K * 0 µ + µ − [33,34], and B 0 → D ( * )+ τ − ν [35][36][37][38][39][40][41]. Expressed in terms of the ratios where q 2 is the invariant mass of the lepton pair in the final state. Here, the upper cut q 2 max = 6 GeV 2 /c 4 is imposed to avoid the effects of the J/ψ and higher cc resonances, while a common lower cut q 2 min for both electron and muon final states is imposed to avoid phase space effects. In the case of R K * , the lower cut has been taken all the way down to q 2 min = 0.045 GeV 2 /c 4 ≈ 4m 2 µ , but the range from this q 2 min to q 2 max = 6 GeV 2 /c 4 has been divided into two at q 2 = 1.1 GeV 2 /c 4 in order to isolate the contribution of the φ resonance (B 0 → φ(→ + − )K * 0 ) into the lower bin. With these bounds on q 2 in place, hadronic uncertainties have been argued to cancel in these ratios and that the SM predictions of R K and R K * are unity with only O(10 −4 ) uncertainties [42][43][44][45]. However, it has also been argued that both QED [46] and QCD [47] corrections and uncertainties were underestimated in these predictions. Ref. [46] argues that a more realistic set of SM predictions is Thus, the prediction that the values of R SM K [1,6] and R SM K * [1. 1,6] are unity appears to be fairly robust, and we find that the experimental values of R K [1,6] and R K * [1. 1,6] are both suppressed compared to their SM predictions by ∼ 2.6 σ [32][33][34].
For the semileptonic B to D decays, the ratios for B → D ( * ) µν. The world averages as of summer 2017, according to the Heavy Flavor Averaging Group [48], are The Standard Model (SM) predictions of these ratios, on the other hand, are [49,50] R SM D = 0.300 ± 0.008 , R SM D * = 0.252 ± 0.003 , (2.8) and one sees that the experimental values R D and R D * are in excess of their SM predictions by 2.3 σ and 3.4 σ, respectively [48]. These experimental results, 2 taken at face value, challenge lepton flavor universality [51] and could be interpreted as signatures of new physics beyond the SM. Note that for new physics to account for R K ( * ) < R SM K ( * ) one could either suppress the numerator or enhance the denominator, while for R D ( * ) > R SM D ( * ) one could either enhance the numerator or suppress the denominator. However, any new physics involving the electron is generically highly constrained. Therefore, new physics needs to suppress the process B → K ( * ) µ + µ − , while enhancing the process B → D ( * ) τ ν τ .

Scalar Leptoquarks
Unlike several other LHC "anomalies" that have come and gone [22][23][24] the persistence of the B-decay anomalies has garnered a great deal of interest and various explanations involving new physics have been proposed. Here, we focus our attention to those involving scalar leptoquarks. 3 As reviewed in the Appendix, there exist six possible G 321 = SU (3) C ×SU (2) L ×U (1) Y quantum number assignments to scalar leptoquarks as listed in Table A. The corresponding six scalar leptoquark fields are labelled S 1 , S 1 , S 1 , S 3 , R 2 , and R 2 . Their couplings to the quarks and leptons are shown explicitly in Eqs. (A.10) and (A.11) and also summarized in the last column of Table A. To explain the R K ( * ) anomalies one needs to induce b L → s L µ + µ − operators which can interfere destructively with the SM processes shown in Figure 1. These operators are denoted in the literature as [54][55][56] 2 See also Ref. [25] for a recent review. 3 For a recent general review on leptoquark-related phenomenology, see Ref. [52]. The classic reference on leptoquarks is Ref. [53]. For completeness, we review the basic theoretical facts from these references in the Appendix. 4 Ref. [54] uses a slightly different numbering scheme in which O 9/10 are respectively denoted O 8/9 . The normalizations of the operators also differ among various publications so care is necessary when comparing the sizes of the Wilson coefficients.  Looking through the leptoquark-quark-lepton couplings listed in Eqs. (A.10) and (A.11), we can see that in order to induce these operators at tree-level one needs S   Refs. [45,[62][63][64], indicate that a suppression of the Wilson coefficient C 9 (the coefficient of O 9 ) compared to its SM value by about 25% provides the best fit.
For the R D ( * ) anomalies we need an operator which interferes constructively with the SM process shown in Figure 3(a). The required operator is To induce this at tree-level one needs S (1/3) 1 [26][27][28][29]  which couple to left-handed quarks and leptons. See Figure 3(b). We find the introduction of the S (1/3) 1 leptoquark coupled to left-handed fermions particularly attractive since, as pointed out in Ref. [26], in addition to explaining the R D * anomalies at tree level, it can also explain the R K ( * ) anomalies at the one-loop level, the same level as the SM contributions. See Figure 2(c). Under closer scrutiny, it was suggested in Ref. [27] that the S However, Ref. [28] argues that this constraint can be circumvented by allowing the S (1/3) 1 mass to be a TeV or larger, the preferred mass being a few TeV. Assuming that an S (1/3) 1 leptoquark coupled to left-handed fermions with a mass of a few TeV can simultaneously explain the R K ( * ) and R D ( * ) anomalies, we inquire whether such a leptoquark can be accommodated within the NCG models of Chamseddine et al. [3,4].

Scalar Leptoquarks in unified Pati-Salam models from NCG
In this section, we list the three NCG-based unified Pati-Salam-like models proposed by Chamseddine, Connes, and van Suijlekom in Refs. [3,4], and specify how leptoquarks can be fit into their particle contents. We begin with a brief outline of how NCG models are constructed, in particular, the three models of Refs. [3,4].

NCG Model Construction
Noncommutative geometry (NCG) generalizes the concepts of spinor bundle and principle bundle, and manages to combine the two concepts into one in a mathematically rigorous fashion [5]. As a result, the Dirac operator has both a continuous part and a discrete part. The continuous part realizes the ordinary Dirac operator, and the discrete part accounts for the representation space of the gauge algebra.
Consequently, in this framework, the most general gauge transformation includes both a transformation along the continuous directions, which is identified with the usual gauge transformation, and that in the discrete direction, which is not explicit in the usual field theoretical formalism. To maintain gauge invariance of the whole action, one needs gauge fields in both directions, which are interpreted as the ordinary gauge fields (continuous direction) and the Higgs fields (discrete direction).
It is worth noting that if one tries to interpret the space as an extra-dimension model, the metric dimension of the extra dimension is zero, because the extra direction is discrete. Therefore, in contrast to the usual extra-dimension models, no extra effort is required to stabilize the two "sheets." In what follows, we show this more concretely. The continuous gauge fields A µ transform in the usual way: where u is an element of the gauge group, while the discrete part, the Higgs field Φ, transforms in an analogous fashion with ∂ µ replaced by a matrix commutator [D, · ], which represents a "derivative" in the discrete direction 5 : Here, the matrix u is still a transformation generated by the gauge algebra, and the matrix D is the discrete part of the Dirac operator. In NCG, the gauge field is defined as as the summation a[D, b], with a, b being elements of the gauge algebra, and D is the generalized Dirac operator [6]. Putting D equal / ∂ (or D), we can reproduce all properties of / A (or Φ), respectively. Therefore, the last terms of Eqs. (4.1) and (4.2) can be combined with the gauge field that is being transformed, which guarantees that the result of the transformation on a gauge field is still a gauge field. On the other hand, depending on the symmetry group, one may or may not be able to write the u[Φ, u −1 ] term in Eq. (4.2) into the a[D, b] form. In other words, due to the presence of the u[Φ, u −1 ] term, the form of a gauge field defined as a[D, b] may not be closed after a gauge transformation. To maintain the gauge invariance, high order terms of the gauge field, arising from u[Φ, u −1 ], 5 Apparently, Leibniz rule is satisfied by the commutator. One might think that [D, · ] cannot work as a derivation as it is not nilpotent in general. However, in the context of cycles over a C * algebra where integrals are defined using Hochschild cochains, D does work as a derivation. For details on how to calculate such integrals, c.f. Ref. [5,6]. are needed to build a covariant derivative. The requirement of vanishing u[Φ, u −1 ] is the so called order-one condition [6]. It has been argued [3,6] that the requirement of the order-one condition, together with a Majorana mass term for right handed neutrinos, selects the SM gauge group from G 422 = SU (4) × SU (2) L × SU (2) R . Furthermore, it has been shown [3] that if one starts with the G 422 = SU (4) × SU (2) L × SU (2) R gauge group, and (partially) lifts the order-one condition, up to three models can be constructed. To be specific, starting with G 422 = SU (4) × SU (2) L × SU (2) R , if the order-one condition is imposed on  [4]; if the order-one condition is lifted and if one only looks into a left-right asymmetric subset of the moduli space of the Dirac operator, one has Model B of Ref. [4]; if the order-one condition is lifted and the full moduli space of the Dirac operator is considered, the result is Model C of Ref. [4].

Model
We note in passing that, if one starts with G 422 = SU (4) × SU (2) L × SU (2) R , it has been observed in [4] that the breaking of this gauge symmetry can be achieved either from usual Higgs mechanism as in Refs. [1,2], or from the use of the order-one condition [3,65]. While it is possible that the the order-one condition encodes the symmetry breaking pattern, we refrain from discussing how the two seemingly different approaches could be related at some fundamental level to achieve similar results.

The three models
The three NCG models proposed in Refs. [3,4], which we refer to as models A, B, and C, differ in their scalar sector content, and their unbroken symmetry structure is listed in Table 1. As in our previous work [1,2] we use the following notation for the symmetries: where D in G 422D refers to the left-right symmetry, a Z 2 symmetry which keeps the left and the right sectors equivalent. We adopt the normalization of the hypercharge Y so that For comparison, the last row of Table 1 lists the scalar content of an SO(10) based G 422 model studied in Ref. [66], after SO(10) is broken to G 422 .
When the G 422 (models A and B and SO(10)) or G 442D (model C) of the NCG based models break to G 321 of the SM, the scalars listed in Table 1 decompose into irreducible representations of G 321 as listed in the third column of Table 2. Of the fields listed there, all the color-triplet fields have the quantum numbers of leptoquarks. For instance, the ∆ R (4, 1, 2) 422 field which appears in model A decomposes as and we see that the two colored fields in the last line have quantum numbers corresponding to leptoquarks S 1 and S 1 . Fields with quantum numbers corresponding to the six leptoquarks S 1 , S 1 , S 1 , S 3 , R 2 , and R 2 all occur in one model or another. However, for any of these fields to be identifiable as leptoquarks, they must couple to quark-lepton pairs after symmetry breaking. In the following, we look at the scalar sectors of the three models one by one and identify the fields that can be considered leptoquarks. In particular, we will search for a field that can be identified as S 1 coupled to left-handed fermions.

Couplings between Fermions and Scalars
Following Refs. [3,4], we denote the SU (2) L and SU (2) R indices in the fundamental representation respectively with un-dotted and dotted lower-case Latin letters toward the beginning of the alphabet: e.g. a = 1, 2 andȧ = 1, 2. Note that despite their appearance, these are NOT spinor indices, and so complex conjugation does not take on or off dots from the indices. 6 The SU (4) index in the fundamental representation is denoted with upper-case Latin letters toward the middle of the alphabet: e.g. I = 0, 1, 2, 3, where I = 0 is the lepton index and I = i = 1, 2, 3 are the quark-color indices. 7 The fermion content of the models is that is, the SM content plus the right-handed neutrinos of each generation. The generation and spinor indices are suppressed. Complex (hermitian, Dirac) conjugation raises or lowers both indices, e.g.
In the case of the SU (2)'s, the index can be lowered or raised using The most general G 422 invariant Yukawa interaction in the NCG models involving ψ aI and ψȧ I can be written schematically as where ψ C = Cψ T , and the couplings constants are embedded in the complex scalar fields Σ bJ aI , H aIbJ , and Hȧ IḃJ . The γ 5 that appears in this expression is due to the geometry being of even parity. The Hilbert space is Z 2 graded with γ 5 being the grading operator, so that the Dirac operator is of odd degree. This is crucial in order to interpret the discrete gauge field as the Higgs field. It is worth noting that, although seemingly originating from a different assumption, the superconnection formalism also captures this feature, as is shown in [68]. Since for any pair of anti-commuting fermionic operators ψ 1 and ψ 2 , the H aIbJ and Hȧ IḃJ fields are respectively symmetric under the interchange of the indices (aI) ↔ (bJ) and (ȧI) ↔ (ḃJ). As is discussed above, the way scalars couple to fermions is dictated by the generalized gauge invariance. In Eq. (4.8), Σ bJ aI is the 'connection' that links chiral fermions to the ones with opposite chirality, i.e. it couples an SU (2) L fermion to an SU (2) R fermion. If a fermion is mapped to the same fermion with opposite chirality, a Dirac mass term can be generated after symmetry breaking. On the other hand, Hȧ IḃJ and H aIbJ link fermions to anti-fermions with the same chirality. When a chiral fermion is mapped to its own charged conjugate, a Majorana mass term can be generated after symmetry breaking. Therefore, the former gives us LR type coupling while the latter produces RR and LL types of coupling. This is how the Yukawa interaction terms are generated as a result of generalized gauge invariance. In particular, the NCG dictates there are only three terms, as indicated in Eq. (4.8).
The complex scalar fields Σ bJ aI , Hȧ IḃJ , and H aIbJ can, in general, consist of the following G 422 representations: (4.10) Note that 6 of SU (4) and 1 of SU (2) L (SU (2) R ) are respectively anti-symmetric under I ↔ J and a ↔ b (ȧ ↔ḃ) rendering the (6, 1, 1) 422 representation symmetric under (aI) ↔ (bJ) ((ȧI) ↔ (ḃJ)). Note also that the leptoquark S 1 we seek must couple to left-handed quarks and leptons, so we can expect to find such a field embedded in the complex scalar field H aIbJ . Although the G 422 representations listed above are most general, and all the fields are contained in the model C, the models A and B restrict the scalar sector to the fields listed in Table 1. Since Model C is the only model which includes the field H aIbJ , it is clear that Model C is the place where we should be looking for our S 1 leptoquark. However, for the sake of completeness, we look at the couplings of the scalar fields that appear in all three models, and identify all the fields that correspond to one type of leptoquark or another.

Model A
In this model, the scalar field H aIbJ is suppressed while the fields Σ bJ aI and Hȧ IḃJ are decomposed as Note the manifest symmetry of Hȧ IḃJ under (ȧI) ↔ (ḃJ). The fields φ ḃ a , Σ I J , ∆ȧ J are those labeled as φ(1, 2, 2) 422 , Σ(15, 1, 1) 422 , and ∆ R (4, 1, 2) 422 in Tables 1 and 2. Also, k ν , k e , k u , and k d denote Yukawa coupling matrices for the neutrinos, charged leptons, up-type quarks, and down-type quarks, respectively, while k ν R is the Majorana coupling matrix which gives Majorana masses to the right-handed neutrinos upon symmetry breaking, and the asterisk stands for complex conjugation. In order to give mass-dimension one to all scalar fields, we have introduced the scale Λ which is only implicit in Refs. [3,4].
Note that the above decompositions render the interactions of the scalar fields ∆ R (4, 1, 2) 422 and Σ(15, 1, 1) 422 and the fermions into mass-dimension five operators, i.e. the generalized version of the LHLH operator of the SM. To generate dimension-four operators of the Yukawa or Majorana type, we must give vacuum expectation values (VEV's) to the scalar fields. In the current case, VEV's are given to 13) or in the notation of Table 2, Note that φ ḃ a and φ ḃ a in Eq. (4.11) are not independent so φ ḃ So symmetry breaking generates the following mass terms for the fermions: The couplings of the φ(1, 2, 2) 422 , ∆ R (4, 1, 2) 422 , and Σ(15, 1, 1) 422 fields to the fermions can be obtained by substituting Eq. (4.11) into the generic expression Eq. (4.8), taking into account Eq. (4.13). First, let us take ∆ȧ I = ∆ R (4, 1, 2) 422 as an example: where the lines connecting the colored fields indicate color contraction, and we have also used the shorthand for the fields that ∆(4, 1, 2) 422 decomposes into as G 422 → G 3221 → G 321 . After the SM singlet ∆ R1 (1, 1, 0) 321 develops a VEV, Eq. (4.14), we find where the first term inside the brackets is the Majorana mass term for the right-handed neutrinos, the ellipses represent dimension 5 operators, and we have used the symbols Thus, we find that ∆ R (4, 1, 2) 422 includes the leptoquarks S 1 and S 1 , but this S 1 , by construction, only couples to u R and e R . Note also that it does not couple to diquarks. Next, let us look at the couplings of φ ḃ a = φ(1, 2, 2) 422 and Σ J I = Σ(15, 1, 1) 422 : Note that if k u = k ν and k d = k e , then Σ J I will decouple from the fermions. We are interested in the couplings of Σ J I = Σ(15, 1, 1) 422 to the fermions so we replace φ ḃ a and φ ḃ a with their VEV's, and denote the shifted Σ(15, 1, 1) 422 fields as (4.21) We find: Comparing with Eq. (A.11), we can see that though Σ J I = Σ(15, 1, 1) 422 does not include fields corresponding to the leptoquark doublets R 2 and R 2 , its components Σ do correspond to the following leptoquark components after the symmetry breaking with their couplings determined by the Yukawa coupling matrices δk + = k u − k ν and δk − = k d − k e . These fields also do not couple to diquarks.

Model B
In Model B, the field H aIbJ is still suppressed, whereas the fields Σ bJ aI and Hȧ IḃJ are no longer decomposed as shown in Eq. (4.11). They are treated as mass-dimension one operators, i.e. 'fundamental' in the language of Ref. [3].
Expanding the ∆ * L3 term we obtain (4.36) Thus, we obtain the isotriplet leptoquark S 3 . The couplings of H L (6, 1, 1) 422 on the other hand are and we finally find the leptoquark we seek: a leptoquark S 1 = H * 3L 3, 1, 1 3 321 which couples to left-handed fermions. The field H 3L 3, 1, 1 3 321 has the same quantum numbers, but this one couples only to diquarks. Thus, we have surveyed the three NCG models and found a unique field, H 3L 3, 1, − 1 3 321 in Model C, which serves the purpose of the leptoquark S 1 3, 1, 1 3 321 coupled to left-handed fermions.
All the color triplet fields have quantum numbers corresponding to some leptoquark though that does not guarantee that they couple to lepton-quark pairs.
In the following, we inquire whether the required unification of gauge couplings in the NCG models can be achieved. We focus specifically on Model C since it is the only model, among the NCG based Pati-Salam models, that contains the required leptoquark H 3L 3, 1, − 1 3 321 . We find that in Model C with a light leptoquark H 3L 3, 1, − 1 3 321 , it is necessary for at least one intermediate symmetry breaking scale to exist between the unification scale M U , the scale at which the model emerges from an underlying NCG with a unified coupling, and the electroweak scale M Z . The gauge couplings in these models do not unify otherwise. We consider several versions of the model with a single intermediate scale M C (M Z < M C < M U ) as illustrative examples. There, the symmetry G 422D that Model C possesses when it emerges at M U is assumed to persist down to M C , at which point it breaks to the G 321 of the SM. Note that this specific sequence of symmetry breaking appears to be favoured by various phenomenological bounds [69].

1-loop renormalization group running
where the RG coefficients a i are given by [70,71] The summations in Eq. (5.2) are, respectively, over irreducible chiral representations of fermions (R f ) in the second term and those of scalars (R s ) in the third. The coefficient η is either 1 or 1/2, depending on whether the corresponding representation is complex or (pseudo) real, respectively. C 2 (G i ) is the quadratic Casimir for the adjoint representation of the group G i , and T i is the Dynkin index of each representation. For the U (1) group, C 2 (G) = 0 and where Y is the U (1) charge. Particles that are heavier than M B (we assume M A < M B ) decouple from the running and do not contribute to the sums in Eq. (5.2). So the values of the RG coefficients a i change every time a symmetry breaking scale is crossed and some of the particles acquire masses of the order of that scale. The low energy data which we will use as the boundary conditions to the RG running (in the MS scheme) are [72,73] Table 3. Dynkin index T i for several irreducible representations of SU (2), SU (3), and SU (4). Different normalization conventions are used in the literature. For instance, there is a factor of 2 difference between those given in Ref. [71] and those in Ref. [74]. Our convention follows the former.

Does the leptoquark help coupling unification?
We first illustrate that unification of the couplings cannot be realized when no intermediate symmetry breaking scales between M U and M Z are present. Since the symmetry just above M Z must be that of the SM, G 321 , we are considering the situation in which the symmetry of Model C breaks immediately to G 321 at M U , the scale at which the model emerges with a higher symmetry G 422D from the underlying NCG.  We identify φ 2 1, 2, 1 2 321 with the SM doublet, and H 3L 3, 1, − 1 3 321 is the scalar leptoquark necessary to explain the B-decay anomalies. In order to avoid proton decay, we assume that the H 3L 3, 1, 1 3 321 field, which only couples to diquarks, becomes heavy when SU (4) breaks to SU (3) and does not survive to low energies. For the second Higgs doublet φ 2 1, 2, − 1 2 321 , we have a choice of allowing it to survive to low energies after the breaking of SU (2) R , in which case we have a 2-Higgs doublet model (2HDM), or making it heavy and decoupling it after the breaking.
As the scalar content, we will therefore consider the following two cases: Taking the ratio of these equations, which also cancels the uncertainty on M Z , yields the condition to be satisfied for unification as Using Eq. (5.8) we find that the models I 1 and I 2 yield r 1 ∼ = 0.56 and r 2 ∼ = 0.54, respectively. Neither of these values fall in the required range stated above. The latter value, r 2 , is actually identical to r SM by coincidence; the modifications in both the numerator and denominator of r 2 cancel. Thus, allowing the second Higgs doublet φ 2 1, 2, 1 2 321 to survive in model I 2 does not help in unifying the couplings.
The resulting non-unification of the couplings can also be observed in Figure 4, where their RG runnings are displayed. Notice that the overall running of the couplings in Model I 1 and Model I 2 are almost identical to the one in the SM due to the smallness of the ratios ∆a i /a SM i . Furthermore, in the case of Model I 2 , the three intersection points (of any pair couplings out of the three), which are controlled by the combinations (3a 1 − 5a 2 ), (3a 1 − 5a 3 ), and (a 2 − a 3 ), respectively, occur at the same scales as in the SM running because the extra parts in these combinations, coming from ∆a i , cancel in the case of Model I 2 . Although the intersection scales are the same between Model I 2 and the SM, the overall evolution of the couplings is still slightly different since each ∆a i is nonzero in Model I 2 . The modification manifests itself in the values of α −1 i at each intersection point. This is obviously an accident in the 1-loop running. Going to the 2-loop level would most likely cause non-zero contributions in the relevant combinations a i , but these can be expected to be extremely suppressed and hence still would not provide an overall coupling unification.
So far, we have assumed that the leptoquark H 3L 3, 1, − 1 3 321 is at the TeV-scale and that its companion H 3L 3, 1, 1 3 321 , the other color-triplet contained in H L (6, 1, 1) 422 , is heavy at the unification scale M U , where the Pati-Salam structure emerges from the underlying NCG, and where the Pati-Salam symmetry, G 422D , breaks into the symmetry of the SM, as well. We have considered two versions of this scenario depending on whether the second Higgs doublet φ 2 1, 2, − 1 2 321 , contained in φ(1, 2, 2) 422 together with the SM Higgs doublet φ 2 1, 2, 1 2 321 , is light at the TeV-scale or heavy at the unification scale. We have shown that the running of the couplings is not improved in either of these cases. Now, we will try a relatively general strategy to look for any improvement in favor of coupling unification. Instead of assuming that the second leptoquark, H 3L 3, 1, 1 3 321 , and the second Higgs doublet φ 2 1, 2, − 1 2 321 are either at the TeV-scale or the unification scale, we will let their masses float in between these scales in order to search for intervals of mass values leading to direct coupling unification. Note that we still assume at this stage that there is no intermediate symmetry breaking. Therefore, the mass values of these particles are not dictated by a symmetry breaking mechanism but possibly by an another, yet to be determined, mechanism or they are picked randomly by the model by sheer accident. Note that we still keep the mass of our main leptoquark H 3L 3, 1, − 1 3 321 at the TeV scale, required by the possible explanation for the B-decay anomalies. Then, Eq. (5.10) changes in the following way: 8 proportional O( c) due to the natural logarithm, unless ∆fc(ai) is enormous which is rarely the case in general, and definitely not the case for the scenarios considered in this paper.
Interval active scalar dofs for RG coefficients where M X and M Y denote, respectively, the lighter and heavier mass among the second Higgs doublet and the second leptoquark. Let us first consider the case where the second Higgs doublet is heavier, which we call model T 1 (and T 2 for the opposite case). Then, we have M Y ≡ M φ 2 and M X ≡ M H 3L . Therefore, in the fist interval (I) the active degrees of freedom, in addition to the SM particles, come only from our TeV-scale leptoquark H 3L 3, 1, − 1 3 321 . In the second interval (II) we add in the second Higgs doublet, φ 2 1, 2, − 1 2 321 , and finally in the third interval (III), all of the degrees of freedom are active.
Using the RG coefficients given in Table 4, Eq. (5.12) becomes (5.14) Solving these equations yields violating again the condition u y x, and thus, no acceptable solution exists in this case as well. Therefore, we conclude that unification of the couplings cannot be directly realized in these scenarios. There should exist at least one intermediate symmetry breaking phase between the unification scale M U and the weak scale, as we illustrate in the next section. However, we would like to note that we adopt here quite a minimalistic approach. If one becomes willing to include more degrees of freedom in the light(er) spectrum, it is quite likely that unification of the gauge couplings is realized with no intermediate symmetry breaking [30,31]. In Model C, the main field surviving down to low energies in addition to the SM fields is the leptoquark H 3L 3, 1, − 1 3 321 , originated from the complex field H L (6, 1, 1) 422 . The other leptoquark, H 3L 3, 1, 1 3 321 , also contained in H L (6, 1, 1) 422 , is assumed to be heavy at the scale M C . This is also convenient in terms of the bounds on proton decay since, as mentioned previously, it is this second leptoquark that has potentially dangerous diquark couplings, contributing to proton decay. We consider two versions of this model, we call model C 1 and model C 2 , depending on the difference based on whether the second scalar doublet, contained in the complex φ(1, 2, 2) 422 , remains heavy at the scale M C , or it becomes light and survives down to the low energies with the other scalar doublet of the same Pati-Salam multiplet, the SM Higgs. In the latter case, the low energy section of the model can be parameterized as the 2HDM, augmented by the leptoquark H 3L 3, 1, − 1 3 321 . 9 Therefore, the scalar content of the models at the TeV-scale are the same as in Eq. (5.7), while, here, we have a different sequence of symmetry breaking. The symmetry breaking chain we consider here has been discussed in detail in our previous papers [1,2]: The generic boundary/matching conditions to be imposed on the couplings at the symmetry breaking scales are given as:

Unification with a single intermediate scale
.  3   Figure 5. Running of the gauge couplings for models C 1 , and C 2 . The vertical dotted lines correspond to the electroweak scale M Z , M C , and the unification scale M U . Note again that where the low energy input is collected on the left hand side. The corresponding beta coefficients for each interval are given in Table 5. Additionally, the unified coupling α U at scale M U can be obtained from The results are given in Table 6 and in Fig. 5, where in the latter the unification of the couplings is displayed. The difference between two models is minor, as expected. In both models, the unification scales are far away from the Planck scale so that we can safely ignore gravitational effects. Moreover, the value of intermediate symmetry breaking scale, M C 10 13.7 GeV, which is almost the same in both models, is consistent with the current bounds coming from the proton decay searches. Since this is the scale at which the Pati-Salam symmetry beaks into the SM, it determines the expected mass values for the protondecay-mediating leptoquarks, including H 3L 3, 1, 1 3 321 . From a naive analysis [69], it can be shown that the current bounds on the proton lifetime [72,75] requires M C 10 11 GeV.
We discuss the proton stability more in the next, concluding, section, in which we address also some conceptual issues related to the NCG approach to the SM and its natural Pati-Salam-like completion.

Discussion
Motivated by scalar-leptoquark explanations of the recently reported B-decay anomalies, in this paper we have investigated whether the required leptoquarks can be accommodated within unified models based on NCG. As already pointed out, such NCG-based models have the gauge structure of Pati-Salam models, SU (4) × SU (2) L × SU (2) R , with gauge coupling unification at a single scale. In one of the models we did find a unique scalar leptoquark H 3L (3, 1, − 1 3 ) 321 , originating from the multiplet H L (6, 1, 1) 422 , which can potentially explain the B-decay anomalies, following the phenomenological analysis of [26], provided its mass is on the order of a few TeV. Also, the unification of couplings can be realized with the inclusion of a single step of intermediate symmetry breaking. Note that H L (6, 1, 1) 422 is complex and decomposes into S * 1 3, 1, − 1 3 321 and another field which couples only to diquarks. Therefore, as discussed below, we do need to split their masses and surpress the mixing between them to avoid proton decay, which might raise the questions of naturalness. However this issue is not different from the corresponding situation found in generic GUT models. As explained in the discussion that follows, it is an interesting feature of the NCG models that one obtains relatively light leptoquarks without obvious problems associated with proton decay.
Given the restrictive nature of the unified Pati-Salam models in the NCG framework, we find the possible scalar leptoquark S * 1 3, 1, − 1 3 321 explanation of the B-decay anomalies particularly interesting. In particular, as pointed out in [26], this leptoquark can be used to explain not only the violation of lepton universality in B-decays and the enhanced rates in certain B-decays, but also the anomalous magnetic moment of the muon.
Note that a state with quantum numbers 3, 1, − 1 3 321 exists in the 5 dimensional representation of SU (5) as well, and therefore it could provide the required phenomenological features. However, one light 3, 1, − 1 3 321 state found in a non-supersymmetric framework, does not imply unification of couplings as the SU (5) GUT theory would require. The SO(10) picture, on the other hand, can accommodate this in a reasonable scenario (as in Refs. [66,76]) by including intermediate symmetry breaking scales.
Similarly, unification of couplings in the NCG models could be realized by imposing (at least) one intermediate scale. According to the scenario we considered in this paper, the Pati-Salam symmetry, G 422 = SU (4) C ⊗ SU (2) L ⊗ SU (2) R , emerges from an underlying non-commutative geometry at a "unification" scale M U , and stays intact down to the scale M C , where it is spontaneously broken to the symmetry of the SM. As discussed in the section 4 of this paper, there are three different Pati-Salam models in the NCG framework [3,4] depending on their scalar contents. However, only the H L (6, 1, 1) 422 in Model C has the right coupling to left-handed fermions and it leads to the phenomenologically preferred leptoquark. And, as we discuss in the following subsection, the usual problems associated with proton decay can be avoided.

Light colored scalars and proton decay
Proton decay is not an issue of concern in our scenario, although we have a light scalar leptoquark H 3L 3, 1, − 1 3 321 (generally labeled as S * 1 in the literature) in Model C, which possesses the right quantum numbers for it to couple to potentially dangerous diquark operators. The proton stability is ensured because the underlying non-commutative geometry does not allow these couplings to appear.
Oftentimes, the possible existence of light leptoquarks is dismissed, perhaps a bit too quickly, since they can potentially lead to proton decay. Although the operators contributing to proton decay should be carefully treated or possibly turned off in order not to contradict the experimental evidence, the underlying mechanism responsible for possible elimination of these operators may not be obvious at first glance. We see an example of this in Model C. In this model, the leptoquarks H 3R/L 3, 1, − 1 3 321 can safely be light since their diquark couplings are turned off by the underlying noncommutative geometry and hence do not contribute to proton decay. Their companions, on the other hand, H 3R/L 3, 1, 1 3 321 (à la the parent-multiplets H R/L (6, 1, 1) 422 ), possess these couplings and therefore should be treated as heavy. However, by looking only at the low energy behavior of the model, from bottom-up perspective, one observes only the absence of the corresponding diquark couplings. From this viewpoint, there is no indication regarding the nature of the underlying mechanism, which is the non-commutative framework in this particular case.
In general, the ordinary, non-unified, Pati-Salam-type models, have the reputation of being safe in terms of proton stability. Although this is indeed the case in terms of the gauge-boson-mediated contributions, the situation is not so trivial for the colored-scalarmediated contributions; not every scalar sector choice guarantees the apparent stability of the proton. The most commonly chosen scalar sector in the ordinary Pati-Salam framework accommodates a global symmetry that prevents proton decay. This scalar sector consists of ∆ R (10, 1, 3) 422 and ∆ L (10, 3, 1) 422 , in addition to the bidoublet φ (1, 2, 2) 422 . This is, at first glance, very similar to the scalar sector of Model C. However, this mechanism is due to the totally-symmetric nature of ∆ L,R under SU (4), and existence of any field(s) transforming anti-symmetrically would potentially spoil this symmetry [77]. In fact, this is exactly what we have in Model C; the multiplets H R/L (6, 1, 1) 422 reside in the totallyanti-symmetric representations of SU (4). Fortunately enough, the restrictive nature of the non-commutative-geometry framework happens to operate in our favor and does not allow the proton-decay-inducing diquark couplings for our leptoquark H 3L 3, 1, − 1 3 321 , thus offering a plausible explanation for the B-decay anomalies.

NCG's Pros and Cons
In this paper we have discussed yet another phenomenological aspect of the hidden non-commutative structure behind the Standard Model (and its natural Pati-Salam-like completion), that has been uncovered by Connes, Chamseddine and collaborators [8,78]. Even though such non-commutative framework has been pointed out for some time, in light of the LHC's discovery of the Higgs boson, and the fact that the NCG of the SM is precisely relevant for the SM with the Higgs boson, we were motivated to explore other phenomenological implications of such a non-commutative approach [1,2,68,79].
Note that the NCG of the SM might appear peculiar from the canonical effective field theory point of view, because of the way the SM action is rewritten, in a Fujikawa-like form familiar from the study of chiral anomalies in gauge theories, which introduces a scale, not a Wilsonian cut-off, and which in turn is fixed by the existence of one overall coupling, thus implying the unification of the three SM couplings at a GUT scale without GUT degrees of freedom. Furthermore, the NCG scheme hides a left-right symmetric structure which can be naturally broken to the canonical SM. Note that this is also an unusual left-right completion. First, this non-commutative completion has a GUT-like unification (like the NCG of the SM), which simply does not exist for the canonical left-right symmetric models. Second, this non-commutative left-right completion of the SM is quite constrained compared to the already existing effective field theory literature on the subject. As emphasized in the present paper, such unified Pati-Salam structure can lead to a unique scalar leptoquark which may potentially explain the B-decay anomalies, provided its mass is around a few TeV, without causing any problems with proton decay.
One might wonder whether the NCG of the SM and the unified Pati-Salam models can have a reasonable UV completion. In particular, where could the NCG originate from a UV point of view? Here, in conclusion, we offer a few comments on this important question. We note that if one views quantum gravity coupled to Standard Model-like matter as having origins in string theory, then one can convincingly argue for an intrinsic non-commutative structure in that context [80][81][82][83][84][85][86]. This work also suggests that the low energy theory (such as the one described by the Standard Model and its Pati-Salam-like completion) is intrinsically non-commutative. In order to obtain the NCG of the SM and its Pati-Salam completion, from such a UV point of view, what is needed is just some suitable fermionic degrees of freedom, propagating in a background of induced gravitational and gauge degrees of freedom (of the SM or Pati-Salam type). Then after integrating over the fermionic zero modes the induced action of the Fujikawa form, needed by the NCG structure of the SM, naturally emerges. Note that the usual discussion of hierarchy of scales as well as the issue of decoupling of scales is different in this framework, which in principle allows for mixing between short distance and long distance degrees of freedom, thus shedding new light on the questions of naturalness and the hierarchy problem.
where i = 1, 2, 3 is the generation index. From this set of fermions, let us construct the most general quark-lepton bilinears. First, the scalars with fermion number F = 3B + L = 0 are and their hermitian conjugates. Note that the bilinears in the same column above share the same SU (3) C × SU (2) L × U (1) Y quantum numbers. The scalars with fermion number and their hermitian conjugates, where ψ C = Cψ T , and = iτ 2 in SU (2) L isospin space which contracts the isospin indices of two isospinors. τ = (τ 1 , τ 2 , τ 3 ) are the Pauli matrices. Note that the bilinears in the first row above share the same quantum numbers. The and their hermitian conjugates, the bilinears in the same columns sharing the same quantum numbers. Introducing one scalar or vector leptoquark for each possible combination of quantum numbers, the most general Lagrangian which couple leptoquarks to these bilinears can be written down as Here, the couplings between the vector/scalar leptoquarks and the fermions are denoted respectively by x and y. The first subscript indicates the isospin of the leptoquark and the latter two the generations of the fermions involved. The superscripts denote the chiralities of the fermions, with a bar placed over the chiralities when a right-handed neutrino is involved. 10 Thus, six scalar leptoquarks S 1 , S 1 , S 1 , S 3 , R 2 , R 2 , and six vector leptoquarks U 1 , U 1 , U 1 , U 3 , V 2 , V 2 are introduced. Of these, S 1 and U 1 were absent in ref. [53] since the only lepton they couple to is the right-handed neutrino. The quantum numbers of these twelve fields are listed in Table A. These leptoquarks can also have SU (3) C × SU (2) L × U (1) Y invariant couplings to di-quarks as shown in ref. [52] but we will not discuss them here.
Note that the subscripts of the leptoquark fields indicate their isospin. So R 2 , R 2 , V 2 , and V 2 have two components each, while S 3 and U 3 have three components each. These can be written out as 10 In order to completely match the definitions of ref. [52], replace x RL 2 ij by −x RL 2 ij , and y LR 2 ij by y LR * where the superscripts in parentheses indicate the electromagnetic charge. Expanding out eqs. (A.7) and (A.8) in terms of these component fields, we find + h.c. , (A.10) [20] CMS collaboration, Search for new physics in high mass diphoton events in proton-proton collisions at 13 TeV, 2015.
[21] ATLAS collaboration, Search for resonances decaying to photon pairs in 3.2 fb −1 of pp collisions at √ s = 13 TeV with the ATLAS detector, 2015.
[22] CMS collaboration, A. M. Sirunyan et al., Search for a heavy right-handed W boson and a heavy neutrino in events with two same-flavor leptons and two jets at √ s = 13 TeV, 1803.11116.