Muon anomalies and the SU ( 5 ) Yukawa relations

We show that, within the framework of SU ð 5 Þ grand unified theories (GUTs), multiple vectorlike families at the GUT scale which transform under a gauged U ð 1 Þ 0 (under which the three chiral families are neutral) can result in a single vectorlike family at low energies which can induce nonuniversal and flavorful Z 0 couplings, which can account for the B physics anomalies in R K ð(cid:2)Þ . In such theories, we show that the same muon couplings which explain R K ð(cid:2)Þ also correct the Yukawa relation Y e ¼ Y Td in the muon sector without the need for higher Higgs representations. To illustrate the mechanism, we construct a concrete model based on SU ð 5 Þ × A 4 × Z 3 × Z 7 with two vectorlike families at the GUT scale, and two right-handed neutrinos, leading to a successful fit to quark and lepton (including neutrino) masses, mixing angles, and CP phases, where the constraints from lepton-flavor violation require Y e to be diagonal.


I. INTRODUCTION
Most Z 0 models [1] have universal couplings to the three families of quarks and leptons. The reason for this is both theoretical and phenomenological. First, many theoretical models naturally predict universal Z 0 couplings. Second, from a phenomenological point of view, having universal couplings avoids dangerous flavor changing neutral currents (FCNCs) mediated by tree-level Z 0 exchange. The most sensitive processes involve the first two families, such as K 0 −K 0 mixing, μ − e conversion in muonic atoms, and so on, leading to stringent bounds on the Z 0 mass and couplings [1].
Recently, the phenomenological motivation for considering nonuniversal Z 0 models has increased due to mounting evidence for semileptonic B decays which violate μ − e universality at rates which exceed those predicted by the Standard Model (SM) [2][3][4]. In particular, the LHCb Collaboration and other experiments have reported a number of anomalies in B → K ðÃÞ l þ l − decays such as the R K [5] and R K Ã [6] ratios of μ þ μ − to e þ e − final states, which are observed to be about 70% of their expected values with a 4σ deviation from the SM, and the P 0 5 angular variable, not to mention the B → ϕμ þ μ − mass distribution in m μ þ μ − .
Following the recent measurement of R K Ã [6], a number of phenomenological analyses of these data (see, e.g., [7][8][9][10][11][12]) favor a new physics operator of the C NP 9μ ¼ −C NP 10μ form [13,14], or of the C NP 9μ form, − 1 ð31.5 TeVÞ 2b L γ μ s Lμ γ μ μ ð2Þ or some linear combination of these two operators. Other solutions different than C NP 9μ ¼ −C NP 10μ allowing for a successful explanation of the R K Ã anomalies are studied in detail in Ref. [15]. However the solution C NP can provide a simultaneous explanation of the R K Ã and R D Ã anomalies [16]. In a flavorful Z 0 model, the new physics operator in Eq. (1) will arise from the tree-level Z 0 exchange, where the Z 0 must dominantly couple to μμ over ee, and must also have the quark flavor changing coupling b L s L which must dominate over b R s R . The coefficient of the tree-level Z 0 exchange operator is therefore of the form, In realistic models, the product of the Z 0 couplings C b L s L C μ L μ L is much smaller than unity since the constraint from the B s mass difference will imply that jC b L s L j jC μ L μ L j ≲ 1 50 , so if C μ L μ L ≲ 1 then C b L s L ≲ 1=50 which implies that M 0 Z ≲ 5 TeV, making the Z 0 possibly observable at the LHC, depending on its coupling to light quarks. Studies of lepton-flavor violating (LFV) B decays in generic Z 0 models before the R K Ã measurement but compatible with it are provided in Ref. [17]. In addition, two and three Higgs doublet models with a nonuniversal Uð1Þ 0 gauge symmetry have been used as the first explanations for the R K and R K Ã anomalies [18]. An alternative explanation of the R K and R K Ã anomalies in the framework of a two Higgs doublet model with two scalar singlets and nonuniversal Uð1Þ 0 gauge symmetry is provided in Ref. [19]. Another explanation for the R K and R K Ã anomalies is an extended inert doublet model having an extra nonuniversal Uð1Þ 0 gauge symmetry, where the SM fermion mass hierarchy is generated from sequential loop suppression [20,21]. Furthermore, the R K and R K Ã anomalies can be explained in an aligned two Higgs doublet model with right-handed Majorana neutrinos mediating linear and inverse scale seesaw mechanisms to generate light active neutrino masses [22]. Apart from these explanations, the R K and R K Ã anomalies can also be explained in models with extended SUð3Þ C × SUð3Þ L × Uð1Þ 0 symmetry, with nonminimal particle content, as done in Ref. [23]. Finally, a vector leptoquark in the Standard Model representation ð3; 1Þ 2=3 arising from a Pati-Salam-like theory has been shown for the first time to provide a good fit to the R K Ã anomalies [24].
In a recent paper, we showed how to obtain a flavorful Z 0 suitable for explaining R K Ã by adding a fourth vectorlike family with nonuniversal Uð1Þ 0 charges [25]. The idea is that the Z 0 couples universally to the three chiral families, which then mix with the nonuniversal fourth family to induce effective nonuniversal couplings in the physical light mixed quarks and leptons. Such a mechanism has wide applicability; for example, it was recently discussed in the context of F-theory models with nonuniversal gauginos [26]. Two explicit examples were discussed in [25]: an SOð10Þ → SUð5Þ × Uð1Þ X model, where we identified Uð1Þ 0 ≡ Uð1Þ X , which, however, was subsequently shown to be not consistent with both explaining R K Ã and respecting the B s mass difference [27], and a fermiophobic model where the Uð1Þ 0 charges are not carried by the three chiral families, only by a fourth vectorlike family. The fermiophobic looks more promising, since, with suitable couplings, it can overcome all the phenomenological flavor changing and collider constraints, and can, in addition, also provide an explanation for dark matter, as recently discussed [28].
In this paper we focus on an SUð5Þ × Uð1Þ 0 model with a vectorlike fourth family where the three chiral families do not couple to the Uð1Þ 0 , but the fourth vectorlike family has arbitrary Uð1Þ 0 charges for the different multiplets, which mix with the three families, thereby inducing effective nonuniversal couplings for the light physical mixed quarks and leptons. The particular scheme we consider involves induced Z 0 couplings to third family left-handed quark doublets and second family left-handed lepton doublets, similar to the model discussed recently in [28]. However, in addition, we also allow induced Z 0 couplings to the righthanded muon, in order to provide nonuniversality for both left-handed and right-handed muons, and hence give corrections to the physical muon Yukawa coupling. We show that such an SUð5Þ model with the vector sector can account for the muon anomalies R K ðÃÞ and correct the Yukawa relation Y e ≠ Y T d without the need for higher Higgs representations. The same applies to flavored grand unified theories (GUTs) such as SUð5Þ × A 4 with a vector sector. In addition, we study the implications of a A 4 flavored SUð5Þ × Uð1Þ 0 GUT with five generations of fermions on SM fermion masses and mixings. To successfully describe the observed pattern of SM fermion masses and mixing angles, we supplement the A 4 family symmetry of that model by the Z 3 × Z 7 discrete group and we extend the particle content of our model by adding two right-handed Majorana neutrinos and several SUð5Þ singlet scalar fields. The discrete A 4 × Z 3 × Z 7 discrete group is needed in order to reproduce the specific patterns of mass matrices in the quark and lepton sectors, consistent with the low energy SM fermion flavor data. The two right-handed Majorana neutrinos are required for the implementation of the type-I seesaw mechanism at tree level to generate the masses for the light active neutrinos as pointed out for the first time in Refs. [214,215]. In this framework, the active neutrinos acquire small masses scaled by the inverse of the large type-I seesaw mediators, thus providing a natural explanation for the smallness of neutrino masses.
The layout of the remainder of the paper is as follows. In Sec. II we describe a two Higgs doublet model with four generations of fermions, several scalar singlets, and an extra Uð1Þ 0 gauge symmetry under which the SM fermions are neutral and the fourth generation of fermions is charged. In Sec. III we present the SUð5Þ × Uð1Þ 0 GUT theory with five generations of fermions in the5 and 10 irreps of SUð5Þ. In Sec. IV we outline the A 4 flavored SUð5Þ × Uð1Þ 0 GUT theory with five generations of fermions and we discuss its implications on SM fermion masses and mixings. Finally, we conclude in Sec. V. Appendix A provides a brief description of the A 4 discrete group.

II. STANDARD MODEL WITH A VECTOR SECTOR
In this section we analyze the model defined in Table I. The three chiral families and the Higgs doublets do not carry any Uð1Þ 0 charges. We allow the vectorlike family to carry arbitrary Uð1Þ 0 charges. The scalars ϕ couple the vectorlike family to the three chiral families.

A. Higgs Yukawa couplings
The Higgs Yukawa couplings of the first three chiral families ψ i are In addition we allow the possibility of the fourth vectorlike family Higgs Yukawa couplings, although the existence of these couplings will depend on the choice of the Uð1Þ 0 charges for the vectorlike family, and some or all of these couplings could be zero.

B. Heavy masses
In this subsection we ignore the Higgs Yukawa couplings (which give electroweak scale masses) and consider only the heavy mass Lagrangian (which gives multi-TeV masses).
The vectorlike family can mix with the three chiral families via the ϕ scalars, and also can have explicit masses, leading to the heavy Lagrangian, After the singlet fields ϕ develop vacuum expectation values (VEVs), the Uð1Þ 0 gauge symmetry is broken and yields a massive Z 0 gauge boson whose mass is of order of the largest VEV of the ϕ fields. Then we may define new mass parameters M Q i ¼ x Q i hϕ Q i, and similarly for the other mass parameters, give where α ¼ 1; …; 4 in a compact notation. All these mass terms are heavy, of order a few TeV, and our first task is to identify the heavy mass states and integrate them out. Actually only one linear combination of the four "normal chirality" states will get heavy, while the other three orthogonal linear combinations will remain massless (ignoring the Higgs Yukawa couplings). We will identify the three physical massless families with the quarks and leptons of the Standard Model.

C. Diagonalizing the heavy masses
We now focus on L heavy (ignoring the Higgs Yukawa Lagrangian) and show how the heavy masses may be diagonalized, denoting the fields in this basis by primes. The goal is to identify the light states of the low energy effective SM below the few TeV scale, after the heavy states have been integrated out.  In the primed basis, the fourth family is massive (before electroweak symmetry breaking), The first three families in the primed basis have zero mass (before electroweak symmetry breaking), and are identified as the quarks and leptons of the SM.
The fields in the primed basis and the original basis are related by unitary 4 × 4 mixing matrices, In our scheme we will consider only the nonzero mixing angles to be θ Q L 34 , in order to generate the Z 0 coupling to the third family quark doublet including b 0 L , and also θ L L 24 and θ e R 24 to generate the Z 0 coupling to the second family lepton doublet including μ 0 L and also μ 0 R , in the primed basis. This is very similar to the model in [28], where the nonzero angles θ Q L 34 and θ L L 24 were considered, and whose main focus was on the phenomenological viability of the model including dark matter. The model considered here includes, in addition, the nonzero angle θ e R 24 which generates an additional Z 0 coupling to μ 0 R , which is important for the main focus of the present paper, namely, the effect of the model on the SUð5Þ Yukawa relations.
To summarize, in this paper we consider D. The Lagrangian in the primed basis

Yukawa couplings in the primed basis
In the original basis, the Yukawa couplings in Eq. (4) may be written in terms of the three chiral families ψ i plus the same chirality fourth family ψ 4 in a 4 × 4 matrix notation as whereỹ u ,ỹ d ,ỹ e are 4 × 4 matrices consisting of the original 3 × 3 matrices, y u , y d , y e , but augmented by a fourth row and column, as follows: In the primed basis in Eq. (9), where only the fourth components of the fermions are very heavy, the Yukawa couplings become wherẽ In the primed basis it is trivial to integrate out the heavy family by simply removing the fourth rows and columns of the primed Yukawa matrices in Eq. (16), to leave the upper 3 × 3 blocks, which describe the three massless families, in the low energy effective theory involving the massless fermions ψ 0 i , where and i; j ¼ 1; …; 3. The physical three family quark and lepton masses in the low energy effective theory should be calculated using the 3 × 3 Yukawa matrices in Eq. (18). For example, from Eqs. (11), (12), (14), and (16)  y 0e 44 y 0e 44 y 0e 44 y 0e where the 22 element of the 3 × 3 light physical Yukawa matrix gets modified as follows: where the approximation is for small angles. This may be a rather large correction if y e 44 ≫ y e 22 or y e 24 ≫ y e 22 or y e 42 ≫ y e 22 even for small angle rotations. Such an enhancement is not present for y 0d 22 , due to the assumed zero angles Therefore any relation between y e 22 and y d

22
will not be respected by the physical couplings y 0e 22 and y 0d 22 , after the mixing with the vectorlike family has been taken into account.
By a similar argument, turning on the mixing angles θ L L 14 , θ e R 14 would lead to where these mixing angles θ L L 14 , θ e R 14 could be much smaller than θ Q L 24 , θ d R 24 and still give a significant correction, since the 11 element of the charged lepton matrix is more sensitive to such corrections than the 22 element (since the electron mass is much smaller than the muon mass).

Z 0 gauge couplings in the primed basis
There is a Glashow-Iliopoulos-Maiani mechanism in the electroweak sector leading to no FCNCs. However, in the physics of Z 0 gauge bosons, the Uð1Þ 0 charges depend on the family index α. This leads to nonuniversality and possibly FCNCs due to the Z 0 gauge boson exchange, as we discuss. After Uð1Þ 0 breaking, we have a massive Z 0 gauge boson with diagonal gauge couplings to the four families of quarks and leptons, in the original basis, where only the fourth family has nonzero charges, In the diagonal heavy mass (primed) basis, given by the unitary transformations in Eq. (9), the Z 0 couplings to the four families of quarks and leptons in Eq.
In the low energy effective theory, after decoupling the fourth heavy family, Eq. (24) gives the Z 0 couplings to the three massless families of quarks and leptons, where the 3 × 3 matricesD 0 are given by Without the fourth family, mixing all these Z 0 couplings would be zero, since the three original chiral families have zero Uð1Þ 0 charges. However, with Eqs. (10)- (12), this mixing induces Z 0 couplings to the third family left-handed quarks and to the muons, as we discuss in the next subsection.

E. Phenomenology
The example we consider is one in which the quarks and leptons start out not coupling to the Z 0 at all, as in fermiophobic models. We show that such fermiophobic Z 0 models may be converted to flavorful Z 0 models via mixing with fourth and fifth vectorlike families with Z 0 couplings. We consider both fourth and fifth vectorlike families of charged fermions to account for the R K and R K Ã anomalies and at the same time to allow embedding the model in a SUð5Þ GUT theory in such a way that the mixings between the heavy and light states will yield a realistic SM quark mass spectrum at low energies without adding a scalar field in the 45 irrep representation of SUð5Þ as we will shown in detail in Sec. IV. Without the inclusion of the fifth fermion family it will not be possible to embed our model in a SUð5Þ GUT theory consistent with the low energy SM fermion flavor data and at the same time allowing for an explanation of the R K and R K Ã anomalies, without invoking 45 irrep scalar of SUð5Þ. We start by considering the following scenario where the mixing matrices for the fermionic fields Q L , L L , and e R are In addition we consider that only the fourth and fifth families have nonvanishing charges: Then, by replacing in Eq. (27) we find the following relations: A. E. CÁRCAMO HERNÁNDEZ and STEPHEN F. KING PHYS. REV. D 99, 095003 (2019) where the Z 0 couples only to the third family left-handed quark doublets Q 0 where the primes indicate that these are the states before the Yukawa matrices are diagonalized. Ignoring any charged lepton mixing amongst the three light families (to start with), this will lead the couplings, with the different couplings of the Z 0 gauge bosons with the charged leptonic fields appearing in Eq. (33) given by where the mixing parameters s L;e 12 appear after expressing the leptonic fields in the interaction basis in terms of the leptonic fields in the mass eigenstates, considering, for the sake of simplicity, only the mixing in the 1-2 plane. In addition, we have expanded the quark primed fields in terms of mass eigenstates as follows: and assumed from the hierarchy of the Cabibbo-Kobayashi-Maskawa matrix that Then the Z 0 exchange generates the effective operators, as in Eq. (1), where the operator corresponds to For the sake of simplicity, we ignore the contribution of the right-handed muon operator and we neglect the contribution arising from the mixing between the first and fourth generations of charged leptons, i.e., we set θ ðL;RÞ 14 ¼ 0. Let us note that we are considering a scenario where the fifth family of vectorlike fermions only couples with the third generation of SM quarks as well as with the first generation of charged leptons, whereas the fourth family will only couple with the second generation of SM charged leptons; thus, we are assuming that only θ Q L 35 , θ L L 24 , θ e R 24 , θ L L 15 , θ e R 15 are nonzero with all other mixing angles being zero (see Sec. IV for a justification of those assumptions in terms of symmetries).
To explain the R K and R K Ã anomalies, we require the coefficient to have the correct sign and magnitude, as discussed in Eq. (3), leading to There are important flavor violating processes such as B s −B s mixing which can rule out models, due to the Z 0 coupling to bs. As discussed for example in [27], this leads to the constraint, From Eqs. (36) and (37) we find the constraint, From Eq. (33), this implies This is easily satisfied, since for example if ðV 0 † dL Þ 32 ∼ V ts ∼ λ 2 ∼ ð1=5Þ 2 ∼ 1=25 then this by itself is almost sufficient to satisfy the constraint.
For example, if we saturate the bound in Eq. (37), then Eq. (36) implies This shows that the mixing angle θ L 24 cannot be too small. Note that the LHC limits on the Z 0 mass are very weak since it does not couple to light quarks at leading order, and its coupling to strange quarks is suppressed by a factor of ðV 0 † dL Þ 2 32 . For a more detailed discussion of the phenomenological constraints on this particular model arising from both flavor violating processes such as B s −B s mixing and LHC limits on the Z 0 mass, see [28]. Furthermore, note that the model has very small FCNC in the Z couplings as explained in Ref. [25]. In addition, the loop effects of fermions charged under both the SM and extra Uð1Þ 0 groups will generate a small Z − Z 0 mixing of the order of There are other important constraints due to LFV processes such as μ → eee as recently discussed for example in [27]. 1 However, as discussed there, violations of lepton universality do not always lead to lepton-flavor violation: it depends on the mixing angles θ L;R 12 arising from the left-handed (L) and right-handed (R) rotations which diagonalize the charged lepton Yukawa matrix. This leads to a Z 0 μe flavor changing coupling suppressed by θ L;R 12 and a Z 0 ee flavor conserving coupling to electrons suppressed by ðθ L;R 12 Þ 2 . We may estimate the branching ratios for μ → eee by taking the ratio of the Z 0 exchange diagram squared to the W exchange diagram squared, For typical charged lepton mixing angles such as θ L;R 12 ∼ λ=3 ∼ 0.07, the coefficient in Eq. (40) will lead to branching ratios such as Brðμ L → e L e L e L Þ ≈ ð0.22Þ 4 ð0.07Þ 6 ð0.08Þ 4 ≈ 10 −14 ð43Þ below the current experimental limit of Brðμ → eeeÞ ≲ 10 −12 but within the range of future experiments.
Although the above constraints may be satisfied, our current framework can lead to the LFV decay μ → eγ, which is only induced by the θ L;R 14 mixing angles in the case of a diagonal SM charged lepton mass matrix, as shown in Appendix B. Thus, to avoid all LFV decays and at the same time to generate the correct value of the electron mass, we need to also suppress the θ L;R 14 mixing angles while at the same time correcting the charged lepton masses. This can be achieved by adding a fifth vectorlike family as discussed in the next section.
Finally, we remark that the models discussed in this paper will be supersymmetric (SUSY). It is well known that SUSY must be broken in realistic models, leading to additional sources of flavor violation coming from the SUSY breaking sector via SUSY loop contributions. These have been recently studied for a class of SUSY SUð5Þ × A 4 models [104] which includes the type of model described in Sec. IV. Interestingly, according to the model independent analysis based on the region of SUSY parameter space consistent with smuon assisted dark matter [104], the most constraining SUSY loop induced flavor observables are also μ → eee and μ → eγ, which are the same modes as discussed above. Such lepton-flavor violating decays could 1 We do not consider μ − e conversion since the Z 0 does not couple to light quarks at leading order. therefore be mediated by either SUSY loops or by a Z 0 exchange in this model.

III. SUð5Þ WITH A VECTOR SECTOR
We now suppose that the SM with a vector sector considered in the previous subsection descends from a supersymmetric SUð5Þ GUT. The three chiral families result from three families of F i transforming as5, and T i transforming as 10, which all carry zero Uð1Þ 0 charges. The Higgs H u and H d arise from 5 and5 representations, after doublet-triplet splitting (which we do not address). This results in the SUð5Þ Yukawa relation, Y e ¼ Y T d in the usual way. Now we consider adding the previous vector sector to the SUð5Þ GUT. In order to violate the SUð5Þ relation Y e ¼ Y T d we will suppose that the fourth vectorlike family at low energies results from multiple5 þ 5 and 10 þ 10 at the GUT scale, where each pair has equal and opposite Uð1Þ 0 charges, but which differ each from another pair. Similar arguments apply for the origin of the fifth family. At low energies below the GUT scale, only the matter content of two vectorlike families survives with various Uð1Þ 0 charges, similarly as in Table I, with the remaining components of the multiple5 þ 5 and 10 þ 10 states having GUT scale masses. Below the GUT scale, the model in Table II leads to the SM plus vector sector in Table I. Thus, the SUð5Þ plus vector sector can explain the muon anomalies exactly like we discussed in the previous section (see in particular Sec. II E).
We now focus on the SUð5Þ Yukawa relation, Y d ¼ Y T e and show that it is violated by the SUð5Þ plus mixing with the vector sector. At the GUT scale, we identify Y e ¼ y e ij and Y d ¼ y d ij in Eq. (4). The Yukawa terms in SUð5Þ may be written as These give SM Yukawa terms, From this equation we identify the charged lepton Yukawa matrix as at the GUT scale. This means that after RG effects are considered we have at low energy, where QCD corrections lead to an overall scaling factor of about 3 for the quark Yukawa couplings as compared to those of the leptons. This implies that Though successful for the third family, this fails for the first and second families. Georgi and Jarlskog [216] proposed that the (2,2) matrix entry of the Yukawa matrices may be given by where the factor of −3 is a Clebsch-Gordan coefficient. Assuming a zero Yukawa element (texture) in the (1,1) position, and symmetric and hierarchical Yukawa matrices, this leads to the relations at low energy, which are approximately consistent with the low energy masses. In our approach we do not wish to consider such large Higgs representations to modify the Yukawa matrices at the GUT scale. Instead we note that these are not the physical Yukawa matrices due to mixing with the fourth family. By following our discussion given in Sec. II D 1 we find that the mixing with the fourth family may enhance y 0e  which may be a rather large correction if y e 41 ≫ y e 22 , even for small angle rotations. We can easily achieve an enhancement by a factor of 3, or indeed any other factor f. Such an enhancement is not present in y 0d 22 due to our choice of zero mixing angles θ Q L 24 ¼ θ d R 24 ¼ 0. Assuming as before, a zero Yukawa element (texture) in the (1,1) position, and symmetric and hierarchical Yukawa matrices, Eq. (52) leads to the relations at low energy, These relations are approximately consistent with the low energy masses for f ≈ 2-3.
It is worth noting that the requirement for enhancing y 0e

22
but not y 0d 22 relies on the assumption that θ L L 24 ≠ 0 or θ e R 24 ≠ 0 but θ Q L 24 θ d R 24 ¼ 0. If we had assumed that the vectorlike family originated from a single5 þ 5 and 10 þ1 0 representation, denoted as F 4 þF 4 and T 4 þT 4 , then this would constrain the choice of charges for the vectorlike fourth family to be AEq F4 for the states L L4 and d R4 , together with AEq T4 for the states Q L4 , u R4 , and e R4 , and their vector partners. In particular, the vectorlike family in Table I would have constrained charges q L 4 ¼ −q d4 and also q Q 4 ¼ −q u4 ¼ −q e4 . This would eventually have led to the constraint on the fourth family mixing that These relations would imply from Eq. (16) that the SUð5Þ relation at low energy would be preserved, Y 0 e ≈ 1 3 Y 0T d . Furthermore, for enhancing y 0e 11 , we require θ L L 15 ≠ 0 or θ e R 15 ≠ 0 but θ Q L 15 θ d R 15 ¼ 0. In summary, we need θ L L 24 ≠ 0 or θ e R 24 ≠ 0 and θ L L 15 ≠ 0 or θ e R 15 ≠ 0 but θ Q L 24 θ d R 24 ¼ 0 and θ Q L 15 θ d R 15 ¼ 0. This can be done if the fourth and fifth vectorlike families at low energies result from multiple5 þ 5 and 10 þ 10 at the GUT scale, where each pair has equal and opposite Uð1Þ 0 charges, but which differ each from another pair, as assumed in Table II. Assuming this, then we have shown that the SUð5Þ theory can account for the muon anomalies R K ðÃÞ and obtain Y e ≠ 1 3 Y T d without the need for higher Higgs representations. The above discussion assumes that there is a zero Yukawa element (texture) in the (1,1) position, with a symmetric and hierarchical charged lepton Yukawa matrix. If, on the other hand, we would assume that the charged lepton Yukawa matrix is diagonal, then we would need to assume corrections as in both Eqs. (20) and (21) in order to account for the correct low energy mass relations in Eq. (51). We will see an example of such a model in the next section.

IV. SUð5Þ × A 4 WITH A VECTOR SECTOR
In this section we will extend the particle content of our supersymmetric model by adding fourth and fifth generations of fermions in the5 and 10 irreps of SUð5Þ, two right-handed Majorana neutrinos, i.e., ν 1R , ν 2R and several SUð5Þ singlet scalar fields. In addition, we will implement the A 4 family symmetry, which will be supplemented by the Z 3 × Z 7 discrete group. These modifications in our simplified version of our model are done in order to get viable and predictive textures for the fermion sector, which will allow us to successfully describe the current pattern of SM fermion masses and mixing angles, as we will show later in this section.
The particle content of the model and the field assignments under the SUð5Þ × Uð1Þ 0 × A 4 × Z 3 × Z 7 group are shown in Table III. Let us note, that we use the A 4 family symmetry, since A 4 is the smallest discrete group having a three-dimensional irreducible representation and three different one-dimensional irreducible representations, which  , ξ e , ξ μ , ξ τ , η 1 , η 2 , and one A 4 trivial singlet, i.e., σ. Out of the A 4 scalar triplets, only η 1 and η 2 will participate in the neutrino Yukawa interactions, whereas the remaining A 4 triplets will appear in the charged lepton and down type quark Yukawa terms. That separation of the A 4 scalar triplets, resulting from the Z 3 × Z 7 discrete symmetry, allows us to treat the neutrino and the charged fermion sectors independently. In addition, the Z 3 symmetry allows us to have a SM charged lepton mass matrix diagonal, which is crucial to completely suppress the lepton-flavor violating decays. The Z 7 symmetry give rises to the hierarchical structure of the charged fermion mass matrices that yields the observed pattern of charged fermion masses and quark mixing angles. Furthermore, we introduce two right-handed Majorana neutrinos, i.e., ν 1R , ν 2R , in order to implement a realistic type-I seesaw mechanism at tree level for the generation of the light active neutrino masses. Having only one right-handed Majorana neutrino would lead to two massless active neutrinos, which is obviously in contradiction with the experimental data on neutrino oscillations. On the other hand, in order to get predictive SM fermion mass matrices consistent with low energy fermion flavor data, we assume the following VEV pattern for the A 4 triplet SUð5Þ singlet scalars: where the complex phases ϕ ν are introduced in the VEV pattern of the A 4 triplet scalar η 2 in order to successfully reproduce the experimental values of the leptonic mixing angles. Since the breaking of the A 4 × Z 3 × Z 7 discrete group generates the hierarchy among charged fermion masses and quark mixing angles and in order to relate the quark masses with the quark mixing parameters, we set the VEVs of the SUð5Þ singlet scalars σ, ξ e , ξ μ , ξ τ , η s (s ¼ 1, 2), ϕ F 4 , and ϕ F 5 with respect to the Wolfenstein parameter λ ¼ 0.225 and the model cutoff Λ, as follows: where s ¼ 1, 2. The aforementioned VEV patterns are consistent with the scalar potential minimization equations for a large region parameter space. In particular, the VEV pattern of the A 4 scalar triplets η 1 and η 2 that participate in the neutrino Yukawa interactions have been derived for the first time in Ref. [74] in the framework of an A 4 flavor model. Assuming that the scale of breaking of the discrete symmetries is of the order of the GUT scale Λ GUT ≈ 10 16 GeV, from Eq. (55) we find for the model cutoff the estimate Λ ≈ 4.4 × 10 16 GeV. With the above particle content, the following Yukawa terms invariant under the group SUð5Þ × Uð1Þ 0 × A 4 × Z 3 × Z 7 arise: where the Yukawa couplings are Oð1Þ dimensionless parameters, assumed to be real for the sake of simplicity, whereas M F a , M T a (a ¼ 4, 5) and M ðνÞ are dimensionful parameters.
On the other hand, it is worth mentioning that the lightest of the physical neutral scalar states of H u . In addition, let us note that the scalar potential of our model has many free parameters, which allows us freedom to assume that the remaining scalars are heavy and outside the LHC reach. In addition, the loop effects of the heavy scalars contributing to precision observables can be suppressed by making an appropriate choice of the free parameters in the scalar potential. These adjustments do not affect the physical observables in the quark and lepton sectors, which are determined mainly by the Yukawa couplings.
From the Yukawa interactions given above, it follows that the SM mass matrices for quarks and charged leptons are given by where v ¼ 246 GeV is the electroweak symmetry breaking scale, the factor of 3 includes the QCD corrections, the κ parameter is introduced to account for the threshold corrections to the down type quarks and charged lepton mass matrices [217], and the factors f 1 and f 2 consider the effects of the mixings with the fourth and fifth families, respectively, of charged leptons as in Eqs. (20) and (21). Let us note that we have assumed, as follows from an extension of our discussion given in Sec. II D 1, with appropriate modifications of Eqs. (21) and (20), that the factors f 1 and f 2 are given by Then, considering 24 ∼ Oð1Þ, we find that factors f 1 and f 2 will be of order unity, which is crucial to generate the right values of the electron and muon masses without spoiling our predictions for the SM down type quark mass spectrum.
The mechanism described above works because the fifth generation of vectorlike leptons only mixes with the first family of charged leptons. Thus, as a result of this mixing, the 11 entry of the charged lepton mass matrix will receive a correction proportional to sin θ L L 15 sin θ e R 15 instead of the quantity θ L L 14 θ e R 14 shown in Eq. (21), thus yielding the right value of the electron mass (without spoiling the predictions of the down quark mass) and at the same time preventing the μ → eγ decay. Thus, the present flavor model has the features θ L;R In this model, due to the discrete symmetry assignments, the mass matrices for SM down type quarks and charged leptons are diagonal and the right values of the electron and muon masses arise from the θ L 15 and θ L 24 mixing angles, respectively, and the mixing between the fourth and fifth generation of vectorlike leptons is very tiny, thus allowing us to have a realistic SM fermion mass spectrum and strongly suppressing the μ → eγ rate. Additionally, as seen from the Yukawa terms given in Eq. (56), considering v ϕ F 4 ≈ v ϕ F 5 ≈ Oð1Þ TeV and assuming that the scale of breaking of the discrete symmetries is of the order of the GUT scale Λ GUT ≈ 10 16 GeV, we find that for dimensionless coupling of order unity, the mass mixing term between the fourth and the fifth generations of charged fermions is of the order of 10 −10 GeV. Considering fourth and the fifth generations of charged leptons contained in the 5,5 SUð5Þ representations have masses around Oð1Þ TeV, we find a mixing angle between these fermions to be θ 45 ≈ 10 −13 , which implies that branching fractions for the charged lepton-flavor violating decays induced by this mixing will be very tiny and well below their corresponding experimentally upper bound. Furthermore, as seen from Eq. (57)  Λ 5 shown in Eq. (56), the SM charged lepton mass matrix is diagonal and θ L 24 ≠ 0, θ L 15 ≠ 0, respectively, whereas θ L;R 14 ¼ θ L;R 25 ¼ 0, θ R 15 ≈ 0, θ R 24 ≈ 0, thus preventing contributions to the μ → eγ decay rate arising from these mixing angles, as follows from Appendix B. Besides that, it is worth mentioning that we are considering incomplete SUð5Þ multiplets for the fourth and fifth generations of fermions, which can be justified by assuming that the exotic down type quark fields contained in the 5 and5 irreps of SUð5Þ, F 4 , F 5 ,F 4 ,F 5 as well as the charged exotic leptons and down type quarks included in the 10, 10 irreps of SUð5ÞT 4 , T 5 ,T 4 ,T 5 , have masses much larger than the TeV scale, whereas the remaining fermions inside these representations do acquire TeV scale masses. That assumption will guarantee that θ Q v ≃ 246 GeV, the hierarchy of charged fermion masses and quark mixing matrix elements arises from the breaking of the A 4 × Z 3 × Z 7 symmetry. Let us note that despite the fact that the running of Yukawa couplings from the GUT scale up to the electroweak scale is not explicitly included in our calculations, our effective Yukawa couplings can accommodate for the renormalization groups effects, since these effective Yukawa couplings depend not only on the Yukawa couplings but also on the VEVs of the scalar fields participating in the Yukawa interactions and those VEVs can be adjusted to account for these effects. This freedom in adjusting the VEVs of the scalars fields participating in the Yukawa interactions is due to the large number of parameters in the scalar potential. Furthermore, we recall that we adjust the corresponding effective Yukawa couplings instead of the Yukawa couplings to fit the physical observables in the quark and lepton sector to their experimental values at the M Z scale.
The charged lepton and quark masses [218,219], the quark mixing angles, and the Jarskog invariant [220] can be well reproduced in terms of natural parameters of order one, as shown in Table IV In Table V we show the model and experimental values for the physical observables of the quark sector. We use the M Z -scale experimental values of the quark masses given by Ref. [218] (which are similar to those in [219]). The experimental values of the CKM parameters are taken from Ref. [220]. As indicated by Table IV, the obtained quark masses, quark mixing angles, and CP violating phase are consistent with the low energy quark flavor data. As shown from Table IV, the obtained values for the SM down type quark masses are inside the 1σ experimentally allowed range. In addition, our obtained values for the SM up type quark masses are inside the 1σ experimentally allowed range, as indicated in Table IV. On the other hand, from the neutrino Yukawa interactions, we find that the Dirac and Majorna neutrino mass matrices are given by Since the right-handed Majorana neutrinos ν 1R and ν 2R acquire very large masses, the light active neutrino masses are generated via the tree-level type-I seesaw mechanism and thus the light neutrino mass matrix takes the following form: where m νa and m νb are given by The neutrino mass squared splittings, light active neutrino masses, leptonic mixing angles, and CP violating phase for the scenario of the normal neutrino mass hierarchy can be very well reproduced, as shown in Table V, for the following benchmark point: In addition, we find that the light active neutrino masses are From  Another important observable, worth determining in this model, is the effective Majorana neutrino mass parameter of the neutrinoless double beta decay, which gives us information on the Majorana nature of neutrinos. The amplitude for this process is directly proportional to the effective Majorana mass parameter, which is defined as where U ej and m ν k are the Pontecorvo-Maki-Nakagawa leptonic mixing matrix elements and the neutrino Majorana masses, respectively. Furthermore, s ij ¼ sin θ ðlÞ ij , c ij ¼ cos θ ðlÞ ij , α ij ¼ α i − α j , being α i the Majorana phases, with i ≠ j and i, j ¼ 1, 2, 3. Note that since m ν 1 ¼ 0 in our model, then m ee only depends on the relative phase Figure 1 shows the effective Majorana neutrino mass parameter as functions of the m νa , ϕ ν and δ CP parameters (here δ CP is the leptonic Dirac CP violating phase). To obtain the plots of Fig. 1, the parameters m νa , ϕ ν , and δ CP were randomly generated in a range of values where the neutrino mass squared splittings and leptonic mixing parameters are inside the 3σ experimentally allowed range. As indicated by Fig. 1, our model predicts teh effective Majorana neutrino mass parameter in the range 2.5 meV ≲ m ee ≲ 2.8 meV, for the scenario of the normal neutrino mass hierarchy.
Our obtained range of values for the effective Majorana neutrino mass parameter is beyond the reach of the present and forthcoming 0νββ-decay experiments. The current most stringent experimental upper limit on the effective Majorana neutrino mass parameter m ee ≤ 160 meV is set by T 0νββ 1=2 ð 136 XeÞ ≥ 1.1 × 10 26 yr at 90% C.L. from the KamLAND-Zen experiment [223].

V. CONCLUSION
In this paper we have shown that SUð5Þ GUTs with multiple vectorlike families at the GUT scale which transform under a gauged Uð1Þ 0 (under which the three chiral families are neutral) can result from two vectorlike families at low energies which can induce nonuniversal and flavorful Z 0 couplings, which can account for the B physics anomalies in R K ðÃÞ . In such theories, we have shown that the same physics which explains R K ðÃÞ also corrects the Yukawa relation Y e ¼ Y T d in the muon sector without the need for higher Higgs representations.
To illustrate the mechanism, we have constructed a concrete model based on SUð5Þ × A 4 × Z 3 × Z 7 with two vectorlike families at the GUT scale, and two righthanded neutrinos, leading to successful fit to quark and lepton (including neutrino) masses, mixing angles, and CP phases, where the constraints from lepton-flavor violation require Y e to be diagonal. This particular model predicts normal neutrino mass ordering with the inverted ordering disfavored by our fit, and an effective Majorana neutrino mass parameter in the range 2.5 meV ≲ m ee ≲ 2.8 meV, for the scenario of the normal neutrino mass hierarchy.
In conclusion, we have shown that the idea of a flavorful Z 0 arising from mixing with vectorlike families can be extended to SUð5Þ GUTs. In such theories, we have shown that the physics responsible for explaining the B physics anomalies in R K ðÃÞ as a result of modified couplings in the muon sector can also lead to violation of the SUð5Þ Yukawa relations Y e ¼ Y T d in the muon sector without the need for higher Higgs representations. The A 4 group, which is the group of even permutations of four elements, is the smallest discrete group having one three-dimensional representation, i.e., 3 as well as three inequivalent one-dimensional representations, i.e., 1, 1 0 and 1 00 , satisfying the following product rules: Considering ðx 1 ; y 1 ; z 1 Þ and ðx 2 ; y 2 ; z 2 Þ as the basis vectors for two A 4 triplets 3, the following relations are fulfilled: ð3 ⊗ 3Þ 1 ¼ x 1 y 1 þ x 2 y 2 þ x 3 y 3 ; ð3 ⊗ 3Þ 1 0 ¼ x 1 y 1 þ ωx 2 y 2 þ ω 2 x 3 y 3 ; ð3 ⊗ 3Þ 1 00 ¼ x 1 y 1 þ ω 2 x 2 y 2 þ ωx 3 y 3 ð3 ⊗ 3Þ 3 s ¼ ðx 2 y 3 þ x 3 y 2 ; x 3 y 1 þ x 1 y 3 ; x 1 y 2 þ x 2 y 1 Þ; ð3 ⊗ 3Þ 3 a ¼ ðx 2 y 3 − x 3 y 2 ; x 3 y 1 − x 1 y 3 ; x 1 y 2 − x 2 y 1 Þ; ðA2Þ where ω ¼ e i 2π 3 . The representation 1 is trivial, while the nontrivial 1 0 and 1 00 are complex conjugate to each other. Some reviews of discrete symmetries in particle physics are found in Refs. [29][30][31][32][33].

APPENDIX B: BRANCHING RATIO OF μ → eγ
The branching ratio of the μ → eγ decay in our model, for the scenario where the charged lepton masses are much smaller than the Z 0 mass, is given by [224][225][226] Brðμ → eγÞ ¼ m 3 where C μ L E L ¼ g 0 q L4 sin θ L 24 ; C μ R E R ¼ g 0 q e4 sin θ R 24 C e L E L ¼ sin θ L 12 C μ L E L ¼ g 0 q L4 sin θ L 12 sin θ L 24 ; C e R E R ¼ sin θ R 12 C μ R E R ¼ g 0 q e4 sin θ R 12 sin θ R 24 ; C e L μ L ¼ g 0 q L4 ðsin θ L 12 sin 2 θ L 24 cos 2 θ L 24 þ sin θ L 14 sin θ L 24 cos θ L 14 Þ; C e R μ R ¼ g 0 q e4 ðsin θ R 12 sin 2 θ R 24 cos 2 θ R 24 þ sin θ R 14 sin θ R 24 cos θ R 14 Þ; ðB2Þ GeV is the total muon decay width. The generalization to the fifth generation of fermions is straightforward and is made by replacing θ L;R n4 by θ L;R n5 (n ¼ 1, 2). Note that the branching ratio becomes zero for a diagonal SM charged lepton mass matrix provided that θ L 14 ¼ θ R 14 ¼ θ L 25 ¼ θ R 25 ¼ 0, which is the case of our flavor model described in Sec. IV.