Non-universal Z' from SO(10) GUTs with vector-like family and the origin of neutrino masses

A $Z'$ gauge boson with mass around the (few) TeV scale is a popular example of physics beyond the Standard Model (SM) and can be a fascinating remnant of a Grand Unified Theory (GUT). Recently, $Z'$ models with non-universal couplings to the SM fermions due to extra vector-like states have received attention as potential explanations of the present $R_K$, $R_{K^{\ast}}$ anomalies; this includes GUT model proposals based on the $\mathrm{SO}(10)$ group. In this paper we further develop GUT models with a flavour non-universal low scale $Z'$ and clarify several outstanding issues within them. First, we successfully incorporate a realistic neutrino sector (with linear and/or inverse low scale seesaw mechanism), which was so far a missing ingredient. Second, we investigate in detail their compatibility with the $R_K$, $R_{K^{\ast}}$ anomalies; we find that the anomalies do not have a consistent explanation within such models. Third, we demonstrate that these models have other compelling phenomenological features; we study the correlations between the flavour violating processes of $\mu\to 3e$ and $\mu$-$e$ conversion in a muonic atom, showing how a GUT imprint could manifest itself in experiments.

A Z gauge boson with mass around the (few) TeV scale is a popular example of physics beyond the Standard Model (SM) and can be a fascinating remnant of a Grand Unified Theory (GUT). Recently, Z models with non-universal couplings to the SM fermions due to extra vector-like states have received attention as potential explanations of the present R K , R K * anomalies; this includes GUT model proposals based on the SO(10) group. In this paper we further develop GUT models with a flavour non-universal low scale Z and clarify several outstanding issues within them. First, we successfully incorporate a realistic neutrino sector (with linear and/or inverse low scale seesaw mechanism), which was so far a missing ingredient. Second, we investigate in detail their compatibility with the R K , R K * anomalies; we find that the anomalies do not have a consistent explanation within such models. Third, we demonstrate that these models have other compelling phenomenological features; we study the correlations between the flavour violating processes of µ → 3e and µ-e conversion in a muonic atom, showing how a GUT imprint could manifest itself in experiments.

Introduction
The presence of a low energy Z gauge boson arising from a spontaneously broken additional gauged U(1) symmetry is a well explored possibility for physics beyond the Standard Model (BSM). Such an extra gauged U(1) could emerge from more unified gauge groups such as the left-right symmetric model, Pati-Salam, SO (10) or E 6 [1]. A recent comparison of SO (10) inspired Z models can be found in [2]. Such models typically predict universal Z couplings to the three families of quarks and leptons, thereby avoiding flavour changing neutral currents (FCNCs) mediated by tree-level Z exchange, such as K 0 −K 0 mixing, µ-e conversion and µ → 3e [1].
Recently, non-universal Z models have been invoked to account for semi-leptonic B decays which violate µ − e universality [3]. The LHCb Collaboration results for B → K ( * ) l + l − report the R K [4] and R K * [5] ratios of µ + µ − to e + e − final states, to be about 70% of their expected values corresponding to a 4σ deviation from the Standard Model (SM). Following the measurement of R K * [5], a number of phenomenological analyses of these data [6] favour an operator of the left-handed (L) form [6][7][8]. There is a large and growing body of literature on non-universal Z models as applied to explaining these anomalies [9], with recent papers following the R K * measurement in [10]. An important challenge faced by these models is the requirement that they be anomaly free, another challenge is to make a model consistent with Grand Unified Theory (GUT). A simple way to achieve both goals is to add a complete vectorlike fourth fermion family, which is anomaly-free and mixes with the ordinary three chiral anomaly-free families, leading to flavour-changing Z interactions, with the idea suggested some years ago in [11] and recently developed in a model independent way and applied to the R K ( * ) anomalies in [12]. A general formalism was developed and applied to two concrete examples, including a supersymmetric SO(10) Grand Unified Theory. The idea is that a U(1) X subgroup survives down to the TeV scale, together with a vector-like family originating from incomplete 10 and 45 dimensional representations of SO(10) [12]. After mixing with the vector-like family, non-universal Z couplings are induced, which could account for the anomalous B decay ratios R K and R K * [12]. However the origin of neutrino mass and mixing and the lepton flavour violating predictions were not discussed. Independently of the motivation coming from the R K ( * ) anomalies, it is interesting to consider a non-universal Z of this kind.
In the present paper we further develop the class of SO(10) GUT models with a vectorlike fourth family and induced non-universal Z couplings, and attempt to answer the open questions mentioned earlier. We focus on and resolve the following topics, listed here in order of importance: 1. Model building in the neutrino sector: We construct the neutrino sector within such a class of models. We show that the constraints from SO (10) imply that a conventional seesaw mechanism is not possible. On the other hand, a linear seesaw mechanism [13] and an inverse seesaw mechanism [14] can be successfully implemented, and done so consistently with a TeV scale gauged U(1) X , which arises from the embedding of SU(5) × U(1) X into SO (10). With the right-handed (sterile) neutrinos in reach of collider experiments, this can serve as an independent motivation for SO (10) models with low energy Z and vector-like family of fermions, as well as opening a window towards testing the seesaw mechanism in GUTs.
2. Inability to explain R K ( * ) anomalies: In a general Z model with vector-like fermions, it is possible to explain the R K ( * ) anomaly with flavour non-universal couplings to the Z . Implementation within SO (10) relates some of these couplings, making also predictions in other observables. Despite the R K ( * ) anomalies being the original motivation for the introduction of this class of models, we show that the explanation of R K ( * ) leads to other predictions inconsistent with observations. As an explanation of R K ( * ) , the SO(10) models with low-energy Z and vector-like fermions are thus disfavored.
3. Prospects for charged lepton flavour violating process searches: The flavour-violating couplings of the Z to the fermions imply contributions to various flavour-violating processes. The underlying SO (10) symmetry and unification requirements further relate some of these processes, representing another interesting phenomenological feature of this class of models. We briefly discuss this point by studying the processes µ → 3e and the µ-e conversion in a muonic atom; the model's implications on these processes can be tested in future experiments.
The layout of the remainder of the paper is as follows: in Section 2 we define our Z model and discuss the mass matrices of light fermions, and in particular the neutrino mass matrix and the generation of neutrino masses, where we successfully embed into these models a low scale seesaw mechanism. Phenomenological topics are then found in Section 3: in Section 3.1, we show the inability of this class of SO(10) models to explain the R K ( * ) anomalies. In Section 3.2, we investigate the decay µ → 3e and µ-e conversion in a muonic atom, their rate depending on the Z mass, and how these two processes are related. Then we conclude. Finally, the paper also has Appendices A and B, dedicated to some of the technical model building details, which would interrupt the flow of the main body of the text if placed there.
2 Non-universal Z and neutrino mass generation

Definition of the model
We start the main body of the paper by defining the class of models under consideration. They are SUSY 5 SO(10) models, with a low energy Z , vector-like family of fermions, and a realistic neutrino sector. First, though, we discuss again the background and motivation for the models we consider, a topic alluded to already in the Introduction.
One possible low-energy extension of the Standard Model gauge group is an additional U(1) gauge group factor, which could get spontaneously broken at the scale of a few TeV, thus implying the existence of a massive Z gauge boson at that scale. The phenomenology of these models becomes much richer if the existence of an additional 4th vector-like family is assumed. If the U(1) quantum numbers of the 4th vector-like family are different than those of the first three families, Z couples non-universally to the three light mass eigenstates from the SM, as was pointed out in [11]. Models with a low-energy Z and vector-like 4th family have recently attracted even more interest due to the apparent violation of lepton universality in the measurement of the R K ( * ) ratios.
Various possibilities of non-universal Z models were considered in [12] in the wake of R K ( * ) anomalous measurements. One challenge for model builders is to find natural UV completions of said Z models; one such possibility, for example, is to embed them in a GUT theory. In such a scenario, the various non-universal couplings in various fermion sectors are related, so one is rewarded with an increased predictivity. Since the extra U(1) increases the rank of the gauge group to 5, the group SO(10) lends itself as a natural unified group candidate. In fact, an SO(10) derived Z model was proposed in [12], but with no consideration of the neutrino sector. We remedy this situation in this paper by extending that model and investigating some of its properties; indeed, since the representation 16 of SO (10) contains (in addition to all the SM fermions of one family) also a right-handed neutrino, any SO(10) GUT model should aspire to also be a theory of neutrinos.
In our SO(10) models, we make use of the following chain of embeddings of maximal subgroups: We identify the U(1) X factor as the additional U(1) containing the Z gauge boson; unlike in a bottom-up Z model, this fixes all the U(1) X charges of the particles once the choice of SO (10) representations is made. In order to provide an observable Z at energies reachable in the relatively near future, the U(1) X is only broken at the scale M Z of a few TeV. Gauge symmetry is thus broken in two steps to the SM group G SM : A summary of our definition of the particle content of the models, which survives to the M Z scale, is given in Table 1. The last column also includes the SO (10) where the U(1) X charges have to be normalized with an additional factor of (2 √ 10) −1 . We also remind the reader at this point about two well-known SU(5) → G SM decompositions, whose familiarity we will assume from now on: Vector-like family: 10 F + 45 F Following [12], the fourth vector-like family of fermions is embedded into the real representations 10 F and 45 F of SO (10), which decompose under SU(5) × U(1) X as We assume that the (1, 0) and (24,0) parts in 45 F decouple (get large GUT scale masses) and that the fourth vector-like family with masses near the TeV scale consists of the following parts 6 [12]: (5, −2) + (5, 2) + (10, −4) + (10,4) .
At the SU (5) (10) representations, need to be heavy due to proton decay constraints. We do not consider the details of SUSY breaking. We assume that supersymmetry is broken at some scale M SUSY , which may be near the M Z scale.
In order to break the U(1) X and to obtain large mixing between the three chiral families and the fourth vector-like family, a pair 16 H + 16 H is needed. In each of the terms the SU(5)-singlet component, which is charged under U(1) X , gets a vacuum expectation value (VEV) at the few TeV scale.
We assume that the only surviving parts to scale M Z of these Higgs representations are the H u and H d of the MSSM, which dominantly come from doublets in 10 H , as discussed later when considering DT splitting.

Three extra singlets for neutrino sector: 1 F i
The next consideration is how to make the model consistent with light neutrino masses. The implementation of a conventional type I seesaw mechanism [16] in a nonuniversal low-scale Z coming from a SO(10) GUT will necessarily fail for the reasons we now discuss. The Majorana masses for ν c i ν c j would have to come from SO(10) invariant operators of the form 16 2 F · X, where X can stand for one of 3 possible Higgs representations: 10, 120 or 126. The first two options do not give a ν c ν c term since they lack SM singlets, while the SM singlet in 126 carries a U(1) X charge. Under the assumption of a low scale Z at TeV scales, the U(1) X breaking VEV of 126 must be at that same scale as well. Furthermore, there is no possible suppression of Dirac type mass terms ν i ν c j , since SO(10) symmetry directly relates them to those in the charged lepton mass matrix. Since the Dirac masses cannot be suppressed, a TeV seesaw scale is much too low to give correct masses for light neutrinos.
It is possible, however, to circumvent the above limitations. Since our model contains a 16 H and 16 H , we introduce three SO(10)-singlets 1 F i in order get the three light neutrino masses from an inverse or linear seesaw mechanism [13,14]. We shall show below that such a setup indeed works.

Matter parity
An additional ingredient of the model is that we assume a Z 2 matter parity. All representations in the fermionic sector (i.e. the ones with index F : 16 F i , 10 F , 45 F and 1 F i ) are odd under matter parity, while all Higgs representations (i.e. the ones with the index H: 10 H , 16 H , 16 H ) have even parity.
We now specify how to prepare a setup with the above ingredients, so that the inverse and linear seesaw mechanisms, which make use of the 1 F fermions, are not spoiled by the vectorlike family. We also assume that the 24+1 SU(5) parts in 45 F are decoupled (we will discuss later how to achieve this by adding some additional terms). A minimal such version of a superpotential with the field content as enumerated above has the form where we remain agnostic about the explicit form of the Higgs sector and thus the form of W H . In the Yukawa sector W Yuk , m 10 and m 45 are mass parameters and we label the dimensionless couplings by y i and Y i depending on whether the Higgs representation coupling to two fermions is {10 H } or {16 H , 16 H }. Note that for the sake of simplicity the family indices are neglected in Eq. (2.9). In fact and y 1 are 3 × 3 matrices and the Y i are vectors of length three. The fermion mass matrices, which arise due to such a Yukawa sector, will be studied in detail in later subsections.
Assuming the SO(10) field content used in Eq. (2.9), there are certain additional superpotential terms that are in principle allowed by gauge invariance:  (10), with U(1) X charges as shown. We also include three SO(10) singlets s i from 1 F in order to implement the linear or inverse seesaw mechanisms. Beside the gauge quantum numbers, fields also carry some global charge(s), forbidding certain interaction terms so that the superpotential is that of Eq. (2.9); we do not commit to a specific implementation of these charges, but provide three examples in Table 2.
These 2 operators are undesired 8 and can be forbidden by introducing a global U(1) or Z n symmetry. Furthermore, the mechanism for decoupling the 24 + 1 SU(5) parts in 45 F from the vector-like states has also not been discussed. These considerations require an extension of the minimal field content and an assignment of all fields under a global symmetry. In our analysis in this paper, however, we would like to remain agnostic regarding the exact details of these mechanisms, since the way to achieve these is not unique, and we are only interested in the resulting theory at the M Z scale.
In order to show that extensions which achieve the desired decoupling and forbid the unwanted terms actually exist, we do provide 3 concrete realizations based on the discrete symmetry Z 4 ; they are distinguished by the type of seesaw mechanism which operates (inverse seesaw only, linear seesaw only, or both). The field content of these models, as well as the Z 4 charges of their SO (10) representations, are given in Table 2. Model I can be viewed as a realization of the general/default case, and this is the one we shall be discussing in the main body of the text; models II and III show that it is possible to isolate only one mechanism of neutrino mass generation, but their details are for the most part relegated to the Appendix. Each of these models introduces new fields and terms into the superpotential (see Eq. (2.11), (B.1) and (B.8) that are discussed later), in order to obtain the desired field content at the scale M Z .
To summarize, we realize GUT based models with a low energy Z and vector like-fermions as discussed above: we propose a class of SUSY SO(10) models, defined by their field content which survives to the Z scale as shown in Table 1, and Yukawa interactions for the surviving content coming from a superpotential in Eq. (2.9). A few explicit UV realizations of models in this broader class are listed in Table 2; they introduce extra representations and new interactions in the superpotential to obtain the desired field content at the Z scale; they also introduce a global symmetry to forbid unwanted terms in the superpotential, so as not to spoil the neutrino mass generation mechanism.
SO (10) Table 2. Examples of three models with global Z 4 charge assignments for irreducible representations of SO (10). The fermion and Higgs fields necessary for the generation of neutrino masses from Eq. (2.16) and giving only the superpotential terms in Eq. (2.9) are above the line, while the particular Higgs fields below the line are used in our examples to realize the doublet-triplet splitting, as well as the split in the 45 F , so that the 24 and 1 parts of SU(5) become heavy. We do not commit to any particular realization of these mechanisms, and therefore also not commit to having the representations below the line. In Example I, we have the general case when both the inverse and linear seesaw mechanisms are present; Examples II and III realize a linear seesaw only and inverse seesaw only scenarios, and are further discussed in Appendix B.

The quark and charged lepton sectors
Having defined the class of models in the previous subsection, we now discuss the form the Yukawa matrices take in the up, down and charged lepton sectors.
We shall consider only the low energy states listed in Table 1 of the effective theory surviving down to the Z scale. That consideration necessarily implies a mass splitting in the 45 F representation between the fourth vector-like family and the states in the (1, 0) and (24, 0). We provide 3 example models where this splitting is achieved in Table 2, but do not commit ourselves to any one mechanism or model of the Higgs sector. In Example I, obtaining such a mass splitting involves introducing a 210 H with even matter parity, which introduces new terms into the superpotential of Eq. (2.9). The appended Yukawa superpotential for model I, denoted by W I Yuk , is now written as Now taking m 45 in W Yuk and the VEV of 210 H at the GUT scale, and assuming the VEV is aligned in the SU (5)-singlet direction, one can obtain the mass of the vector-like family 10 + 10 of SU (5)  Note that the main differences of interest between models I, II and III lie in the neutrino sector.
After electroweak (EW) symmetry breaking we obtain the mass matrices for the up-and down-type quarks M u and M d , for the charged leptons M e and also for the neutrinos M ν . They include the three chiral families and the fourth vector-like family. Since the states coming from (1, 0) and (24,0) in 45 F have masses at the GUT scale, they can safely be ignored in the following considerations. Explicit results given below show that there is indeed a large mixing between the chiral families and the vector-like family in the left and right components in M u , M d and M e . We provide their form when computed from Eq. (2.9): 9 Above, we have defined v u and v d as the VEVs of the Higgs doublets H u and H d , respectively, both (almost exclusively) from 10 H and both at the electroweak scale. The VEVs of (1,5) and (5, −3) of SU(5) × U(1) X in 16 H are labeled by v R and v L , respectively. Thev R and v L in 16 H are defined in an analogous way. The VEVs v R andv R will be assumed to lie at the few TeV scale. v L andv L are "induced VEVs" after EW symmetry breaking and are much smaller than the EW scale. 10 The parameters y 1 , Y 2 , Y 3 , m 10 and m 45 are defined in Eq. (2.9). Note that for the sake of readability, we wrote the matrices as 3 × 3, while they are 9 We used the freedom to redefine the normalization of couplings and VEVs to absorb in them the Clebsch factors such as 2 √ 2, thus simplifying the expressions for mass matrices as much as possible. Once chosen, we stick to the same convention throughout the paper. 10 The size of the induced VEVs results from considerations of doublet-triplet splitting, see Appendix A for more details.
in fact 5×5, since the (1, 1) block is actually 3×3, corresponding to the usual 3 family Yukawa matrix y 1 . Note that Eq. (2.9) implies the matrix y 1 is symmetric, eliminating the need to write transposes. Similarly, the parameters Y 2 and Y 3 are actually a 3 × 1 row of parameters; we do not write transposes since their presence can be inferred from the dimension of the blocks where these parameters are present.
The matrices M u , M d and M e above are written in the basis (2.14) These are matrices in the superpotential W , whose singular value decomposition gives directly the matrices of physical masses for fermions. Concerning labels of particle states, u i and d i are in Q i , u 4 and d 4 are in Q 4 , e i is in L i and e 4 is in L 4 . For the states of the columns, we haveũ 4 andd 4 inQ 4 , whileẽ 4 is inL 4 .
Note that we have in each sector two vector-like pairs of representations, with one pair transforming as doublets and the other pair as singlets under SU(2) L . In the charged lepton sector, for example, we are adding both doublets L 4 ⊕L 4 , as well as singlets e c 4 ⊕ẽ c 4 . Due to their quantum numbers under SU(3) C × U(1) EM , the weak singlet stateẽ c 4 joins the row states L i and L 4 , which are doublets, while the doubletL 4 joins the column states e c i and e c 4 , which are SU (2) singlets. Due to this feature, the (5, i) entry in the 5 × 5 mass matrix M e (where i = 1, 2, 3 denotes the first three families) couples together two weak singlets, while the (i, 5) coupling couples two weak doublets. For that reason, confirmed by explicit calculation in Eq. (2.12), both (i, 5) and (5, i) entries have VEVs transforming as singlets under the SM group (but breaking U(1) X ), i.e. the VEVs v R andv R located respectively in 16 H and 16 H . For analogous reasons, this feature of having a 5th weak singlet state mixing with weak doublets in the basis for rows, and conversely a 5th doublet state mixing with singlets in the basis for columns, is a feature in all mass matrices in Eq. (2.12).
In our example models in Table 2, a 54 H is introduced for doublet-triplet splitting (discussed in Appendix A). This representation would introduce a new operator 10 2 F · 54 H into the Yukawa sector superpotential in Eq. (2.9). We assume in these examples its coupling to be sufficiently small, so that it only introduces a small split between the m 10 parameters in the down-quark and lepton sectors, while still keeping them at TeV scale.

The neutrino sector
Let us now turn to the neutrino sector. We consider the basis where ν i and ν c i are the states which correspond to the left and right-handed neutrinos in 16 F i and s i are the states associated with the 1 F i . Furthermore, ν 4 in L 4 labels the state of the left-handed neutrino in (5, 2) of 10 F andν 4 inL 4 is the corresponding state in (5, −2) of 10 F . In this basis the neutrino mass matrix is given by 11 ( 2.16) For the sake of readability we only show here one chiral family in the neutrino mass matrix.
In fact the upper left 3×3 block (which is all that is usually considered in the inverse or linear seesaw models) has dimensions 9 × 9 if all three chiral families are included. The parameters , m 10 , y 1 , Y 1 and Y 2 are defined as in Eq. (2.9). As discussed above, m 10 , v R andv R lie at the few TeV scale which allows for potentially large mixing of the chiral families with the vector-like family in the quark and charged lepton sectors and to break U(1) X .
In the limit ,v L → 0 the neutrino mass matrix in Eq. (2.16) has one eigenvalue equal to zero and four of the order of a few TeV, corresponding to the mass scale of the vector-like family. If a small value ofv L (i.e.v L v u ) is switched on, the light left-handed neutrino masses are generated by the linear seesaw mechanism [13] In contrast, if a small non-zero value of (i.e. v u ) is switched on, the inverse seesaw mechanism [14] applies: In general we expect both andv L to be present, and neutrino mass generation gets simultaneous contributions from both the linear and inverse seesaw mechanisms. To get the correct scale of light neutrino masses of around O(0.1 − 1 eV) from Eq. (2.17) and (2.18), we needv L to be around O(10 eV), and to be around O(keV), with at least one of the two approaching this upper estimate. Curiously, the simplest mechanism for doublet triplet splitting in our models automatically induces the doublet VEVv L to be of the correct scale for neutrino mass generation (see Appendix A for details), so the inverse seesaw contribution is not needed.
The same arguments on neutrino mass generation hold true if all three chiral families are considered and in this case we get three light states in the neutrino mass matrix. These 11 The matrix Mν is computed for the superpotential of Eq. (2.9). The full mass matrix should be enlarged by extra states coming from the 1 and 24 of SU(5) located in 45F . These states though are decoupled and live at the GUT scale, as already discussed for other sectors. The largest mixing between these decoupled states and the low energy states is a TeV level mixing coming from the operator 16F · 45F · 16H , which does not however change the qualitative picture of our discussion of neutrino mass generation. We thus omit the extra states from Eq. (2.16).
results show that the usual upper left 3 × 3 block of the neutrino mass matrix in Eq. (2.16) (9 × 9 if all three chiral families are included) plays a crucial role in determining neutrino mass and mixing.
Having successfully implemented the linear and inverse seesaw mechanisms consistent with light neutrino masses, we can pose the question whether the usual type I, II and III seesaw mechanisms play any role in our model. As already discussed earlier, a type I seesaw mechanism [16] would involve Majorana mass terms ν c i ν c j . The only operator of the form 16 2 F · X in our model, as seen from Eq. (2.9), is that for X = 10 H . Since 10 H does not contain any SM singlet, there is no ν c ν c term in our model, confirmed by the vanishing (2, 2) entry of M ν in Eq. (2.16). A type II seesaw [17] requires a presence of scalars in SM representations (1, 3, ±1); such representations are present in 126 and 126 representations of SO (10), but not in any of the Higgs representations of our models in Table 2. Finally, a type III seesaw mechanism [18] requires fermions in the weak triplet SM representation (1,3,0). The fermionic sector of our model has such fermions in the 45 F . The triplet is a part of the 24 of SU (5), which we successfully decoupled and lies at the GUT scale. The mass couplings of the triplet to the SM left-handed neutrinos ν i are generated in our case, however, by the operator 16 F · 45 F · 16 H ; the VEV of the weak doublet in 16 H is only an induced one at scales below O(10 eV), as already discussed. The type III contribution is thus completely negligible, and is in fact not visible in Eq. (2.16) since the triplet is part of omitted states. We have thus seen that all 3 standard types of seesaw mechanism play no role in our model.
In Table 2, we provide examples of three concrete models demonstrating the discussed features of neutrino mass generation: Example I gives a model where the general case of both a linear seesaw and an inverse seesaw occurs: bothv L and are non-vanishing, andv L is automatically of the correct scale (see Appendix A). In Example II, the mass term is forbidden by global charges, so that neutrino masses are generated by linear seesaw only. In Example III, the global charges forbid the doublet VEVv L to be induced, so only the inverse seesaw applies. Examples II and III are discussed in more detail in Appendices B.1 and B.2, respectively.
Based on the results of this section, we conclude that while regular (high-scale) seesaw cannot be implemented within an SO(10) based Z model with a vector-like family of fermions, low scale seesaw mechanisms, in particular inverse and/or linear seesaw, can be realized. Construction of the neutrino sector is the last missing ingredient for these models to provide a full description of all observed fermion sectors.

Flavour phenomenology and related bounds
In this section we focus on the phenomenological aspects of the SO(10) based Z models with a vector-like family of fermions. In particular, we investigate the R K ( * ) anomalies in Section 3.1, and µ → 3e and µ-e conversion in Section 3.2. Both sections require though a prior discussion on the 5 × 5 Yukawa matrices, which we now proceed with.
We take our mass matrices in the up, down and charged lepton sectors to be those of Eq. (2.12). It needs to be noted though that their exact form has certain potentially problematic relations, such as the SU (5) relation M e = (M d ) T . The particular form of the matrices and the relations between them hold though only at the GUT scale, and bad relations may turn out to be remedied once RGE running to the Z scale is taken into account. We can also envision the possibility that the Yukawa sector of the quark and charged lepton sectors is extended, e.g. with an addition of representation 126. A correction only in the 3 × 3 block may be sufficient. Furthermore, the free parameters of such models could then be further restricted/reduced in number by a more elaborate theory of the Yukawa sector, e.g. with discrete symmetries. All these possibilities clearly go beyond the scope of this paper, since any precise prediction would require selecting a particular model from within the class we consider, while our emphasis has actually been on the neutrino sector.
Despite the above limitation of the Yukawa sector, it turns out some general points of this class of models can still be extracted. In particular, this includes both the aforementioned inconsistency of the R K ( * ) anomalies, as well as some considerations for the processes of µ-e conversion and µ → 3e decays. For the latter case, we assume that the mass matrices of Eq. (2.12) get only small corrections, at least in the parts other than the upper-left 3 × 3 SM block.
To connect our model with low energy processes, we need to know how the mass eigenstates of the Standard Model are related to the flavour eigenstates in Eq. (2.14). This information is contained in the mass matrices of Eq. (2.12). We label the vector of flavour eigenstates by q , the vector of mass eigenstates by q, and an intermediate basis by q , defined below. The diagonalization is then performed by plane rotations R ij parameterized in the following way: where R ij is a rotation in the plane of the i-th and j-th state, specified by the mixing angle θ ij and given by the sub-matrix in the i-j plane In what amounts above to a parametrization of a 5 × 5 unitary transformation, we have ignored the CP violating phases.
With the parametrization in Eq. (3.1), we get mixing angles θ q ij in all sectors, since the vector q runs over u, d, e, u c , d c and e c , which corresponds to the left and right basis of the mass matrices of the up, down and charged lepton sectors. We fix the notation of the left angles to be θ qL ij and of the right angles to be θ qR ij , where q = u, d, e.
The order of the rotation matrices in Eq. (3.1) is such that we first rotate to the q basis by rotating away the heavy vector-like families in the 4th and 5th state of the matrix. After arriving to the basis q , we can either directly pass to the full mass eigenbasis q by performing the remaining rotations between the first three states, or use an effective field theory approach, where we integrate out the heavy 4th and 5th state with well defined masses at M Z , evolve with RGE to M Z , and only then pass to the mass eigenbasis q by engaging the EW VEVs. Either way, the parametrization in Eq. (3.1) ensures that the angles θ 12 , θ 13 and θ 23 correspond to the convention used in the CKM mixing angles of the SM.
Notice that in Eq. (2.12) the following entries vanish or are very small (with induced doublet VEVs at the eV scale): (4,4), (5,5), (i, 4) and (4, i), with i = 1, 2, 3. This form of the mass matrices implies that the θ i4 angles can remove the (i, 5) and (5, i) entries, while θ 45 then diagonalize the vector-like states. In our model, the parameters m 45 , m 10 , v R and v R are roughly at the M Z scale, so the θ i5 angles are needed only to remove the entries generated when rotating a vanishing entry with a non-zero (i, j) entry in the 3 × 3 block by one of the other angles. The θ i5 angles are thus rendered to be of the order ∼ y 1 v d /M Z , which is negligibly small in part due to the small y 1 SM Yukawa couplings. We thus use the approximation while the θ i4 angles can be determined by looking at the ratio between the two non-zero entries in the 5-th column and row, since the induced VEVs v L entries are negligible (see Appendix A): The reader is reminded that the parameters Y 2 and Y 3 are in fact 3 × 1 rows, thus index i refers to their specific entries. We see that the form of the matrices from Eq. (2.12), even if taken as approximate, correlates some of the angles to be of at least the same scale.
The conventions given above, as well as the discussion regarding the negligible size of the i-5 angles and the inter-correlated size of the i-4 angles, will be used in the remainder of this section.

The R K ( * ) anomalies and the bound from
The original motivation of the non-universal Z in [12] was the explanation of the R K ( * ) anomalies [4,5]. We will show below that in the SO(10) model considered in this paper and in [12], the explanation of the R K ( * ) anomalies is disfavored by other measurements.
We shall first consider the special case from reference [12], where the R K ( * ) anomalies should be most easily realized: we assume that only θ dL 34 mixing in the down-quark sector and θ eL 14 mixing in the lepton sector are present, while the couplings Y 2 and Y 3 in Eq. (2.12) are such that other i4 mixing angles are assumed zero: θ dL 14 = θ dL 24 = θ eL 24 = θ eL 34 = 0. We shall later discuss the general case, and show that the same conclusions hold.
Among the interactions involving Z and fermions, we have the following terms involving the left-handed fields: where the three couplings, following [12], are equal to

7)
C Z µµ = − 3 2 √ 10 g , (3.8) where g is the coupling of the Z gauge boson (in the convention where the X charges are normalized with the (2 √ 10) −1 factor), s dL 34 = sin θ dL 34 , s eL 14 = sin θ eL 14 , where θ dL 34 and θ eL 14 are assumed to be the only non-zero mixing angles between the first 3 families and the vector-like 4th family in the quark and lepton sectors, respectively. The (V † dL ) 32 is the 32 component of the matrix V † dL , which describes the transformation of the mass eigenstates to the flavour basis in the down sector after the vector-like states have been decoupled, i.e.
where d L , s L , b L are mass eigenstates, while b L is the flavour state coupling to Z (defined after integrating out the heavy 4th and 5th states).
The R K ( * ) anomalies involve detecting a deficit in the ratio of branching ratios of the processes B → K ( * ) µ + µ − compared to B → K ( * ) e + e − . The new physics contribution is favored to come from left-handed fields [6][7][8], whose couplings to Z we have written in Eq. (3.7)- (3.9). The deficit in muons should be coming from a BSM operator mediating b → sµ + µ − : with the anomaly measurement requiring G exp b L µ L = −(31 TeV) −2 . This value is based on the fit of the BSM Wilson coefficient for the left-left operator to be C BSM b L µ L = −1.33 and the relation G exp b L µ L = C BSM b L µ L /(36 TeV) 2 , as described in [8]. 12 The uncertainty in the Wilson coefficient is about 20% to 30%. In our model with L Z , the dominant contribution to this operator is a tree level exchange of Z , yielding It has been shown in [19,20] that adding an operator with electrons instead of muons does not significantly change the requirement for G b L µ L . We will therefore ignore such an operator in the following. Note that one can keep the b → se + e − at the same level as in the Standard Model by suppressing the C Z ee coupling, e.g. by taking (s eL 14 ) 2 = 3/5.
We consider now whether the SO(10) model in question can give a large enough contribution to the b → sµµ process in order to explain the R K ( * ) anomalies. To answer this, we need to study bounds on other processes, where the C Z bs and C Z µµ couplings will be important. It turns out that the mass difference in the B 0 s -B 0 s mixing will already provide a severe constraint.
All processes of the type bs → sb contribute to diagrams mixing the B 0 s andB 0 s mesons. The difference of the masses of the two mass eigenstates is computed in the Standard Model, but with a relatively large theoretical uncertainty (mostly from computation of lattice parameters): ∆m s = (17.4 ± 2.6)ps −1 [21]. The experimentally measured value is given much more precisely to be ∆m s = (17.757 ± 0.021)ps −1 [22]. Assuming the accurate SM contribution in the low range of the given interval, the largest possible contribution from BSM physics can then be estimated. In a Z model, B mixing will be induced by a tree level exchange of the Z involving two Z bs couplings. Following Table I in [23], and translating our notation to their notation for the vertex Z bs by C Z bs → g q s θ c θ , the constraint from B 0 s -B 0 s mixing reads Expressing C Z bs from Eq. (3.12) and inserting into Eq. (3.13) gives (3.14) For the g , we can make a simple estimate by noting that the 1-loop RGE equations for a generic gauge coupling g are dg dt = βgg 3 (4π) 2 . We compute the β coefficients for the g and g couplings of the U(1) Y and U(1) X groups, respectively, to be β g = 53/5 and β g = 77/5. This holds in the GUT normalization, in which the couplings unify at M GUT , and the particle content is that of the MSSM with additional vector-like states, as listed in Table 1. Since the β g > β g , and g = g at the GUT scale (we assume the threshold corrections at the GUT scale for g and g do not differ much), g < g at low scales, which increases the left-hand side of Eq. (3.14) compared to just replacing the value g with g.
If we assume g ≈ g = 0.46 (best case scenario for satisfying the bound), M Z ≈ 4 TeV and G b L µ L = G exp b L µ L = −(31 TeV) −2 , we get 3.6 · 10 −4 on the left-hand side of Eq. (3.14), which is too big by one order of magnitude. We could recover to near the allowed bound by taking M Z = 1 TeV, but that is excluded by the lower bound from LHC data of roughly 4 TeV [24,25].
Therefore to satisfy the bound, at least one order of magnitude suppression has to come from G b L µ L . We can obtain a rough estimate for g by using the previously stated RGE equations for g and g between the scales M SU SY ≈ M Z and M GUT ≈ 2 · 10 16 GeV, assuming unification g ≈ g at M GUT and the low energy value g = 0.46. At the scale of M Z , we get g 2 ≈ 0.7g 2 . This allows us to write the estimate for the operator coefficient G b L µ L as 3.15) which is not sufficient to fully explain the R K ( * ) anomaly measurement, since the mass scale of 70 TeV is equivalent to the Wilson coefficient value of about C BSM b L µ L = −0.26.
We now turn to the general case, in which mixing angles with the 4th vector-like state other than θ eL 14 and θ dL 34 are switched on. The particular forms of couplings in Eq. (3.7)-(3.9) now change, but the derivation of the general form in Eq. (3.14), before the coupling C Z µµ is inserted, still holds. We see that the left-hand side of that inequality is proportional to (C Z µµ ) −2 , so the bound can be reached only if the absolute value of the coupling gets enhanced. Direct computation via Eq. (3.1), in the small θ eL 12 ,θ eL 13 ,θ eL 23 angle limit, shows the value of the coupling C Z µµ in the general case to be The coupling does not depend on θ eL 34 . Furthermore, for any values of angles θ eL 14 and θ eL 24 , the second factor is between −3 and 2. The former value is reached for the already considered special case θ eL 24 = 0, while the latter is achieved for θ eL 14 = 0 and θ eL 24 = π/2. This shows that the absolute value of the coupling C Z µµ reached its maximum already in the special case we considered earlier and cannot be enhanced, implying that the bound in Eq. (3.14) is violated in the general case at least as much as it was before.
The above considerations show that fulfilling the bound from the mass split in B 0 s -B 0 s mixing and explaining the full size of the R K ( * ) anomalies is not possible in our model without tension in the mass of the Z coming from LHC bounds. If we abandon the explanation of the R K ( * ) , then the angle θ dL 34 can be taken small enough to suppress the C Z bs coupling as much as needed, and thus avoid the stringent experimental bounds from B 0 s -B 0 s mixing. This is the path we shall follow in the remainder of the paper.

Non-universal Z and charged lepton flavour violating processes
There are various BSM processes that are induced in our model due to the presence of both the Z and vector-like states. A full phenomenological analysis is beyond the scope of this paper, not least because we consider an entire class of models and the Yukawa sector in the quark and charged lepton sectors needs to be appended, as discussed at the start of section 3. However, the question of charged lepton flavour violating processes in SO (10) GUT models under consideration is a very fascinating one, since they offer 3 predictive advantages over "bottom-up" extensions of the SM with Z and vector-like fermions added as unrelated extra ingredients: 1. In "bottom-up" extensions of the SM, the gauge coupling g related to the gauge boson Z can in principle take any value. It can, for example, be arbitrarily small, so that the flavour-violating effects are negligibly small. In an SO(10) model, the Z unifies with the other interactions at the GUT scale; unification conditions thus give the value of the coupling to be g ≈ 0.46 √ 0.7, as already discussed in Section 3.1. 3. The couplings of the SM fermions to the Z depend on the mixing between the first 3 families and the vector-like one. In an SO(10) model, the Yukawa matrices in the quark and lepton sectors are related, so the mixing is related.

The
The above considerations show that in any specific model, when the Yukawa matrices are known, any flavour violating process can be computed, since the g and the associated fermion charges are known. Furthermore, it shows that the underlying SO(10) symmetry can relate flavour violating processes from different sectors. The original motivation of the SO(10) models with a low-energy Z and vector-like states was an explanation of the R K ( * ) anomalies; this explanation was shown to be disfavored in Section 3.1, but the considerations in flavour violating processes by themselves serve as an excellent motivation for such models in this paper and beyond.
As a demonstration, we investigate in this section two flavour violating processes for this class of SO(10) models: the lepton flavour violating decay µ → 3e and the µ-e conversion in a muonic atom, and show how they are related. In particular, we investigate the branching ratios Br(µ → 3e) and Br(µ − N → e − N ), where N is the nucleus of the muonic atom.
The choice for these particular BSM processes is manifold: first, they are mediated by a tree-level exchange of the Z , ensuring a relatively large contribution from BSM physics. Second, they involve first two generation particles, with sensitive experiments designed to look for these processes. Third, these two processes are sensitive to the mixing between the vector-like and the chiral states in the charged lepton sector and to the charged lepton mixing angle θ eL 12 , which is typically the dominant charged lepton mixing contribution to the PMNS matrix, especially in an SO(10) model where the charged lepton mixing angles are connected to the quark ones. 13 For the purposes of this demonstration, we make the assumptions that are outlined in the beginning of Section 3; although we do not assume the exact form of Yukawa matrices from Eq. (2.12), we do assume that the possible modifications correcting the bad SU (5) relations in a complete model do not change the structure of the matrices appreciably (the entries do not change their order of magnitude for example). More specifically, we take that the relations in Eq. (3.3)-(3.5) hold at least approximately: there is negligible mixing in θ qL i5 ≈ θ qR i5 ≈ 0 (for i = 1, 2, 3 and q = u, d, e), while the i-4 angles can take in principle any value, but their sizes are related according to Eq. (3.4) and (3.5).
In a scenario when θ eL 12 θ eR 12 and θ eL 14 θ eL 24 , θ eL 34 , the lepton flavour violating decay µ → 3e is realized dominantly via the diagrams shown in Figure 1. If θ eL 24 is non-negligible, the interaction couplings become more complicated expressions, and when θ eR 12 is engaged, analogous processes with right-handed fermions in the upper fermion line of the diagram contribute. The full contribution to the branching ratio in the most general case can be computed with eq. (21) in [1], where the required parameters are the general form of the Z couplings, the Z mass and the Z -Z mixing parameter θ, which is computed from the gauge boson mass matrix. We take the realistic case θ ∼ M 2 Z /M 2 Z . The Z couplings are computed in the flavour basis of fermions from the known U(1) X charges of the fermions, in which Z is flavour diagonal, and then rotated to the mass eigenbasis of fermions using the convention of mixing angles from Eq. (3.1); the analytic expression is prohibitively long and we omit it here. The general formula is used for the computation of Br(µ → 3e) in the left scatter plot of Figure 3, which we shall describe in more detail later. As a rough approximation of the general formula, in the case when θ ≈ 0, θ eR This can be compared with the experimental bound on Br(µ → 3e), which is 1.0 · 10 −12 [27]. Figure 1. Lepton flavour violating decay µ → 3e via the Z . On the left-hand side the Z couples tō e L e L , whereas on the right-hand side it couples toē R e R . In both diagrams the couplings of the leftand right-handed leptons to the Z have to be normalized with a factor of (2 √ 10) −1 . Furthermore, θ eL 12 1 is assumed.
The dominant process for µ-e conversion in a muonic atom in the scenario θ eR 12 θ eL 12 and θ eL 14 θ eL 24 proceeds via the diagram shown in Figure 2. In our analysis, we will consider a gold (Au) atom. We compute Br(µ − Au → e − Au) for the general case of mixing angles from equation (22) in [1], with Z-Z mixing θ ∼ M 2 Z /M 2 Z . This computation is used in the right scatter plot of Figure 3, with the parameters related to the nucleus of Au atoms taken from [28]. In the case when θ ≈ 0, θ eR The experimental bound on Br(µ − Au → e − Au) is given by 7.0 · 10 −13 [27]. Figure 2. Schematic diagram for the µ-e conversion in a muonic atom. In the diagram, in order to get the different processes contribution to the µ-e conversion, N has to be replaced by left-and right-handed up and down quarks, with the specific couplings to the Z .
We emphasize that the simplified expressions for the branching rations of µ → 3e and µ Au → e Au in Eq. (3.17) and (3.18) hold in the case θ ≈ 0, θ eR 12 θ eL 12 1 and θ eL 14 θ eL 24 ; this includes any Z model with vector-like fermions whose Z charges are the U(1) X charges from Table 1, regardless on whether the theory has an underlying SO(10) symmetry; one of the advantage of the SO(10) class of models however is that we know the value of g . The numerical prefactors in Eq. (3.17) and (3.18) are above the experimental bounds for these processes; given a CKM-like angle for θ eL 12 , s eL 14 has to be much smaller than 1 and/or the Z mass has to be much larger than 4 TeV.
We perform a parameter scan over the mixing angles and compute the branching ratios using the general formulas for the two processes from [1]. We assume the following reasonable scenario: • The branching ratio can be arbitrarily reduced by reducing the mixing angles, such as the θ eL 12 angle. We assume, however, CKM-like mixing angles in the lepton sector. Furthermore, we assume most of the CKM angles are coming from the down sector, and we know empirically that the θ CKM 13 , θ CKM 23 θ CKM

12
. We assume the same features to be present in the right mixing angles, since the y 1 3 × 3 matrix in Eq. (2.12) is symmetric. All of the above considerations are combined into taking the following values for the mixing angles between the chiral families: where θ CKM 12 = 0.227, the labels X and q run over X = L, R and q = u, d, e, and finally r ∈ [0.8, 1.0] is a uniformly chosen random number different for each of the parameters, since we do not assume Eq. (2.12) to hold exactly.
Also, the θ 34 angles do not come into the branching ratio formulas, since there is no 1-3 and 2-3 mixing. This considerations amount to the following values for the chiral-vector angles: θ qX i5 : 0, (3.22) θ uL j4 , θ uR j4 , θ dL j4 , θ eR j4 : p · κ j , (3.23) θ dR j4 , θ eL j4 : p · λ j , (3.24) θ qX 34 : irrelevant, (3.25) where i = 1, 2, 3, j = 1, 2, the labels X and q run over X = L, R and q = u, d, e. Also, p, κ j and λ j are randomly chosen (including for each j) uniformly distributed values in the following ranges: p ∈ [0.5, ]. In the scattering plots we use three different ensembles of points based on different values of the maximum j-4 angles θ max 4 . The scales κ j and λ j for the j-4 mixing angles are intended to represent 27) where (Y 3 ) j and (Y 2 ) j are the j-th component of the vectors of parameters Y 3 and Y 2 , respectively.
• We fix the gauge couplings to be g = 0.46 and g = 0.46 √ 0.7 based on unification constraints, see Section 3.1. Also, we take the Z-Z mixing parameter to be exactly The results are the scatter plots of the Br(µ → 3e) and Br(µ − Au → e − Au) for a given Z mass, see Figure 3. It should be mentioned here that the branching ratios can in principle drop arbitrarily low for any mass M Z if the i-4 mixings with the extra vector-like family are all artificially small, e.g. if m 10 , m 45 v R ,v R . In the generic part of the parameter space, however, we see a linear-drop with M Z in the log-log scatter plots. While the lower bound in the scatter plots is not strict, the upper bound with the yellow points is as high as the branching ratio can go at a given Z mass in our class of models. The shaded areas of these plots are excluded regions based on current experimental bounds for the branching ratios (Br(µ → 3e) < 1.0 · 10 −12 and Br(µ − Au → e − Au) < 7.0 · 10 −13 [27]) and the M Z coming from the LHC (M Z > 4 TeV [24,25]). The other horizontal lines are the sensitivities of future experiments; they can probe Br(µ → 3e) with a sensitivity of 10 −16 [29], and the branching ratio of µ-e conversion in a muonic Ti atom with a sensitivity of 2 · 10 −18 [30]. We can see from the two plots that the µ-e conversion is the more sensitive of the two experiments for probing our class of SO(10) models. The future sensitivity of the µ-e conversion experiments has the potential to probe for M Z masses up to roughly 3000 TeV, much higher than the reach of the LHC or any planned next-generation collider. Another striking feature appears when we compare the two branching ratios for the same parameter values (mixing angles and M Z ). A log-log scatter plot is shown in Figure 4, where the grey areas are again the experimentally excluded regions and the extra lines show future sensitivities. We can see that the two branching ratios are correlated; if the chiral-vector mixing angles are not big (θ max 4 < 30 • is already sufficient, see blue points), the correlation is constrained to within much less than one order of magnitude. This correlation is actually a feature of the underlying SO(10) symmetry of our class of models. In a non-SO(10) model, the processes µ → 3e and µ-e conversion are in principle unrelated, due to the relevant couplings of the Z to charged leptons and quarks being unrelated (the bottom Z couplings in Figures 1 and 2). The scattering plots are valid for the reasonable parameter scenario outlined earlier in this section; this include a more stringent scenario where the Yukawa matrices are exactly those of Eq. (2.12).
The correlation of the µ → 3e and µ-e conversion branching ratios is thus a general prediction of the studies SO(10) class of models. If an experiment measures the µ-e conversion at a certain rate, Figure 4 tells us in which range the measurement of the other flavour violating process µ → 3e is expected to occur. Furthermore, a successful measurement of µ-e conversion would put an upper bound on the M Z (ensemble of yellow points in Figure 3). Figure 4. The log-log scatter plot of the branching ratios of µ → 3e and µ-e conversion flavour violating processes. We see a strong correlation between the two processes, with the correlation already predicting the value of one branching ratio by knowing another to within factor 3 if all the chiral-vector 1-4 and 2-4 angles are smaller than 30 • . The grey areas represent experimentally excluded regions, and the other two lines the future experimental sensitivities.

Conclusions
A Z gauge boson with mass around the (few) TeV scale can be a fascinating remnant of a Grand Unified Theory. When the U(1) gauge symmetry of the Z is embedded in an underlying GUT group, the latter relates the Z -couplings of the different fermions of the Standard Model (SM). Further testable remnants are TeV scale vector-like states which can arise from additional GUT matter representations. These vector-like states can have an interesting impact on the flavour structure via enlarged fermion mass matrices, and induce non-universal Z couplings with a rich phenomenology.
Recently, Z models with non-universal couplings to the SM fermions due to extra vector-like states have received attention as candidates for explaining the present R K , R K * anomalies. As one part of our paper, we have revisited this possibility by further developing consistent SO(10) GUT models with a non-universal Z at low energy arising from a spontaneously broken U(1) X gauge subgroup and a vector-like fourth family. The Z has non-universal couplings due to the mixing between the chiral quark and lepton families and the vector-like family which couples with non-universal U(1) X charges.
However, despite all the ingredients for explaining the R K , R K * anomalies being present in such models, we have shown that they fail to provide a consistent explanation of the anomalies due to the constraints from the B s mass difference and from LHC bounds on the M Z , when taken into account that the maximal value of the coupling of Z to muons is restricted by SO (10).
On the other hand, models of this type have a rich phenomenology in the lepton sector: Due to the constraints from SO(10) on the Dirac neutrino Yukawa couplings, we have argued that a conventional seesaw mechanism is not possible with right-handed neutrinos at the TeV scale (assuming that U(1) X is broken at the scale of a few TeV).
To complete the class of models with a consistent neutrino sector, we have demonstrated that it is possible to implement both a linear seesaw [13] and an inverse seesaw mechanism [14] in SO(10) with vector-like states and TeV scale Z . Either mechanism (or even a simultaneous combination of the two) can generate the masses of the light neutrinos. Furthermore, it is possible to forbid the inverse seesaw coupling, while the linear seesaw contribution naturally gives the correct light neutrino mass scale. This represents a new example of the linear seesaw mechanism.
Our class of models can be viewed as a new route towards constructing highly predictive and testable flavour models with a number of compelling phenomenological aspects. For example, with the right-handed (sterile) neutrinos in reach of collider experiments, this opens up another interesting window towards testing the seesaw mechanism in GUTs (see e.g. [31,32] and references therein). Moreover, we considered the flavour violating processes of µ → 3e decay and µ-e conversion in a muonic atom; we showed that within a reasonably general parameter scenario, there is a strong correlation between the two processes due to the underlying SO(10) symmetry. We also considered present and future experimental sensitivities for probing this class of models. We showed that future sensitivities of µ-e conversion experiments will have an experimental reach of Z masses up to about 3000 TeV, much higher than any planned future collider.
In conclusion, we have clarified several issues regarding SO(10) GUTs with a non-universal Z at low energy due to a vector-like fourth family: On the one hand, we have shown that models of this type cannot provide a consistent explanation of the present R K , R K * anomalies. On the other hand, we have shown how a consistent neutrino sector can be constructed for such models, requiring right-handed (sterile) neutrino masses at the mass scale of the Z (or below). Furthermore, we showed the predictive potential of such models by relating flavour violating processes involving interactions from different fermionic sectors. The resulting models thus feature a rich phenomenology and may be tested with high precision at present and future colliders and in lepton flavour violating experiments.
Since the implementation of the linear seesaw mechanism depends on the value of the doublet VEVv L in 16 H , the details of the doublet-triplet splitting are crucial to the implementation of that mechanism.
Before U(1) X -breaking, the low energy MSSM Higgs doublets arise purely from the doublets in the 10 H , and the Higgs doublets from the 16 H , 16 H have GUT scale masses. However, when the U(1) X is broken at the few TeV scale there will be a mixing induced between the GUT scale Higgs doublets from the 16 H , 16 H and the light Higgs doublets from the 10 H of different U(1) X charges. This is a very small but crucial effect, which allows to explain the small neutrino masses of the right order of magnitude via the linear seesaw mechanism. To clarify this point, we write down the mass matrix M D for the doublets with Y /2 charges ±1/2: This matrix is written in the basis coming into the superpotential as The vectors of states D and D denote the hypercharge +1/2 and −1/2 doublets, respectively, while their index denotes the representation where they are located.
In order to achieve doublet-triplet splitting in this example, as seen from Eq. (A.1), we introduced the representations 54 H and 45 H in the Higgs sector (even matter parity). The former enables to tune the mass of the doublet to be low in the (1, 1) entry, while the latter, assuming its VEV is aligned in the SU (5) yielding the usual ratio from the high scale seesaw mechanism of v 2 u /M GUT , automatically giving the correct scale for left-handed neutrinos. 15 Notice that the relevant admixture for linear seesaw is controlled by the λ parameter, and notλ.
We emphasise that the mixing between the doublets in the 10 H and in the 16 H or 16 H is always controlled by U(1) X breaking, whatever the detailed mechanism of providing entries of correct scale in M D may be. This model is constructed so that the neutrinos get masses only from linear seesaw, while the inverse seesaw mechanism is not present. To achieve this, we forbid the coupling in front of the singlet mass term 1 F 1 F in equation (2.9) by global charge assignments. Based on charges in Table 2, the Yukawa part of the superpotential W II Yuk is now W II Yuk = m 10 10 2 F + κ10 2 F · 54 H + 45 2 F · (Z 1 210 H + Z 2 54 H ) + y 1 16 2 F · 10 H + Y 1 16 F · 1 F · 16 H + Y 2 16 F · 10 F · 16 H + Y 3 16 F · 45 F · 16 H . (B.1) The terms which are still forbidden are 1 F 10 F 10 H and 45 F 10 F 10 H , while the newly forbidden terms are 1 2 F and 45 2 F , i.e. m 45 = 0 and = 0. Furthermore, the term 45 F 1 F 45 H , which is dangerous for spoiling neutrino mass generation by strongly coupling the upper-left 3 × 3 block in equation (2.16) to the decoupled states in 24 and 1 parts of the 45 F , is also not allowed by the charge assignments.
Since the mass term for the 45 F is now forbidden by the charges, the 24 and 1 parts of 45 F are now decoupled from the vector-like states by having the Z 2 coupling to the 54 H in addition to the Z 1 coupling to 210 H . The VEV of 210 H now has to be aligned in the 24 direction of SU (5), the same direction as the VEV of 54 H . Tuning in Z 1 and Z 2 then allows only the vector-like states to have TeV scale masses. Also, as will be seen later, v L = 0 in this case. With these considerations, the mass matrices in Eq. (2.12) are now replaced with They are analogous to the previous result, except the some of the previously equal TeV scale parameters can now be different. Mixing between the SM particles and vector-like states remains present.
The neutrino mass matrix in Eq. (2.16) is replaced with one where v L = 0 and with no in the (3, 3) entry: Above, we have used the definitions and we assume that m 10 and κ are small enough that both m 10 and m 10 are in the TeV range. Similarly, we tune Z 1 and Z 2 so that m 45 is in the TeV range as well, while the mass entries The term 16 H · 10 H is forbidden by global charges, i.e.λ = 0 in Eq. (A.1). Consequently, the induced EW breaking VEV in 16 H is vanishing: v L = 0. The relevant induced EW symmetry breaking VEVv L in 16 H , however, is controlled by the λ coupling, so the doublet-triplet splitting considerations of Appendix A and Eq. (A.4) and (A.5) for linear seesaw apply.
Example II thus achieves everything we set out to do: neutrino mass is generated via linear seesaw, with no inverse seesaw contribution in this model. Furthermore, we achieve doublettriplet splitting and the decoupling of the 24 and 1 parts under SU (5), while maintaining mixing of SM particles with vector-like states in the up, down and strange lepton sectors.
Despite v L = 0, this mixing is still present both in the rows and columns of the mass matrices of Eq. (B.2), so the considerations of mixings of left-handed leptons and quarks with vectorlike states of Section 3 still applies.

B.2 Example III: inverse-seesaw only
This model is merely an illustration how one can achieve a pure inverse-seesaw mechanism without any linear seesaw contribution. In this example, there is no mixing between the left-handed lepton fields of the SM and the vector-like family, therefore it is not a relevant example from the point of view of Section 3.
In Example III of Table 2, the Higgs sector is extended by the representations 54 H , 45 H and 210 H . As in Example I, the first two are required for doublet triplet splitting, while the 210 is used to decouple the 24 and 1 parts under SU (5)  Compared to Example II in Appendix B.1, the Yukawa sector now has the mass terms for 1 F and 45 F reintroduced, the 54 representation is no longer there (Z 2 = 0) and also mixing between 16 F and 10 F is now forbidden, i.e. Y 2 = 0. In W DT , the λ coupling is now forbidden, butλ is reintroduced. Consequently, based on the doublet mass matrix in Eq. (A.1),v L = 0 and v L = 0. The linear seesaw mechanism contribution thus vanishes, as seen from Eq. (A.4), and only inverse seesaw with = 0 remains.
The mass matrices in Eq. (2.12) are now replaced with The VEV of the 210 H is now assumed to be in the singlet direction of SU (5), and tuning m 45 and Z 1 yields a m 45 at the TeV scale, while the 24 and 1 parts in 45 F remain heavy at the GUT scale. Doublet triplet splitting is achieved as well, and there is inverse but no linear seesaw. The 5 × 5 neutrino matrix is now block diagonal, while the mixing between the SM particles and the new vector-like states in the down and charged lepton sectors is now compromised.