Non-universal $Z'$ from Fluxed GUTs

We make a first systematic study of non-universal TeV scale neutral gauge bosons $Z'$ arising naturally from a class of F-theory inspired models broken via $SU(5)$ by flux. The phenomenological models we consider may originate from semi-local F-theory GUTs arising from a single $E_8$ point of local enhancement, assuming the minimal ${\cal Z}_2$ monodromy in order to allow for a renormalisable top quark Yukawa coupling. We classify such non-universal anomaly-free $U(1)'$ models requiring a minimal low energy spectrum and also allowing for a vector-like family. We discuss to what extent such models can account for the anomalous $B$-decay ratios $R_{K}$ and $R_{K^*}$.


Introduction
In the Standard Model (SM) of strong and electroweak interactions the gauge couplings to lepton fields are flavour independent. This is a well known property of the SM gauge interactions which is usually called Lepton Flavour Universality (LFU). Among the various processes that can be used to test LFU, are the flavour changing decays involving quark fields such as b → sℓ + ℓ − where ℓ ± stand for e ± , µ ± , τ ± leptons. Recent experimental results [1] indicate violations of LFU which may be due to a new neutral boson Z ′ [2,3] which both changes quark flavour, namely b → s, and couples predominantly to muons rather than electrons, with left-handed quark and lepton couplings preferred [4]. The flavourful Z ′ models are many and varied. One attractive possibility is to consider a vector-like family with non-universal Z ′ families which mixes with the three chiral families of quarks and leptons with universal Z ′ couplings, thereby introducing non-universality by the back door [5] 4 .
It is commonly believed that the SM is not the final theory of fundamental interactions, but just an effective low energy limit of some (partially) grand unified theory (GUT), possibly embedded in a string scenario, leading to new physics phenomena and deviations from the SM predictions. For example, new gauge bosons Z ′ associated with additional abelian symmetries arise from such string embeddings [2]. Within the framework of F-theory [8] there are many such extra gauged abelian symmetries [9], which however are normally assumed to be broken at the high energy scale. Usually these gauge bosons are assumed to have universal couplings to quarks and leptons [10], however there is no reason in principle from the point of view of F-theory why this should be the case. Despite this there has been no systematic study of non-universal Z ′ gauge bosons at the low energy scale arising from F-theory to our knowledge.
Motivated by the above recent hints for non-universality, we make a first systematic study of anomalyfree non-universal Z ′ models arising from F-theory. From a phenomenological point of view, we refer to such models as fluxed GUTs. These models assume a set of rules extracted from F-theory and which can form the basis for a phenomenological analysis of models. In particular we shall use these rules to search for a set of anomaly free models 5 in which the low energy effective theory consists of the Minimal Supersymmetric Standard Model (MSSM) augmented by an extra gauged U (1) ′ group broken at the TeV scale, together allowing also for a possible vector-like family at that scale. We shall discover that none of the models with a minimal low energy matter spectrum is capable of explaining the recent B-physics anomalies, while it is possible to account for these anomalies by allowing an extra vector-like family as in [5].
The layout of the paper is as follows. In section two we introcude the notion of fluxed GUTs which are based on essential features playing instrumental rôle in local F-theory constructions. In particular, we describe the rôle of fluxes on the breaking of the gauge symmetry, the splitting of GUT representations and the fermion chirality. In section three we implement these rules to construct SU (5) GUTs with additional abelian symmetries allowing non-universal couplings to fermions. We focus on specific models where both SU (5) GUT and abelian symmetries are embedded in E 8 and formulate the anomaly cancellation conditions which constrain the spectrum of the emerging effective models. In section four we present the solutions of these conditions and classify them according to the properties of the resulting light particle spectrum. We first present characteristic examples of SU (5) with the minimal supersymmetric spectrum having non-universal couplings with the extra U (1). We also show solutions containing vector-like fermions with non-universal couplings and discuss their implications to low energy physics. In section five we present our conclusions.

Fluxed GUTs
In our quest for a possible interpretation of such kind of experimental results with the incorporation of a new gauge boson Z ′ , we will examine GUT models that can in principle be derived in the context of F-theory model building [9]. We stress that this particular string theory framework provides the appropriate tools for a natural inclusion of abelian gauge bosons with non-universal gauge couplings to fermions. Indeed, among other basic ingredients of F-theory constructions, a prominent rôle for the incarnation of the aforementioned scenario is played by the particular structure of the compact manifold and the (abelian) fluxes. We recall first that the compactification space consists of an elliptically fibred Calabi-Yau manifold (CY 4 fourfold) of four complex dimensions whose geometric singularities display a group structure and, in Ftheory, are associated with the gauge symmetries of the effective field theory model [8]. We will assume that the internal manifold has a structure characterised by the maximal exceptional symmetry E 8 , where 7-branes wrap an appropriate divisor accommodating the SU (5) gauge group of the effective field theory. In this context, the zero mode spectrum descends from the decomposition of the E 8 adjoint, therefore the following picture emerges in the effective model 248 → (24, 1) + (1, 24) + (10, 5) + (5, 10) + (5, 10) + (5, 10) , where in (1), the group factor SU (5) GUT , is identified with the grand unified symmetry of the effective model, and in (2) the representations in the box contain the SU (5) GUT fields. As is well known, ordinary matter is accommodated in 10 and 5 representations which, in the present construction, appear with non-trivial transformation properties under the (perpendicular) symmetry SU (5) ⊥ . For the purposes of the effective field theory description, it is adequate to work in the Higgs bundle picture and express the transformation properties of the SU (5) GUT matter in terms of the Cartan generators of SU (5) ⊥ and the five weights t i , i = 1, 2, . . . , 5 satisfying the SU (5) tracelessness condition In this case, the superpotential can be maximally constrained by four U (1) ⊥ 's ("perpendicular" to SU (5) GUT ) according to the breaking pattern With respect to t i notation, the non-trivial SU (5) GUT representations given in (2) are designated as 5 t i +t j , 10 t i and the SU (5) GUT singlets emerging from SU (5) ⊥ adjoint decomposition are denoted with 1 t i −t j ≡ θ i j . This way, the U (1) 4 ⊥ invariance of each superpotential term is ensured as soon as the tracelessness condition ∑ 5 i=1 t i = 0 is satisfied 6 . The second important available toolkit in F-theory model building includes the various fluxes which are turned on along the various abelian factors and can be used for the following tasks. Firstly, appropriate fluxes are introduced for the breaking symmetry mechanism and can be regarded as the surrogate of the vacuum expectation value (vev) of the Higgs field, particularly in the case of manifold geometries (such as del Pezzo surfaces) where the latter is absent from the massless spectrum. The SU (5) GUT symmetry in particular, breaks to the Standard Model with a hypercharge flux. Secondly, fluxes are used to generate the observed chirality of the massless spectrum. In order to describe their implications in the present construction we distinguish them into two classes. Initially, a flux is introduced along a U (1) ⊥ and its geometric restriction along a specific matter curve Σ n j is parametrised with an integer M n j . Then, the chiralities of the SU (5) representations are given by The hypercharge flux introduced to break SU (5) GUT is also responsible for the splitting of the SU (5) representations. If some integers N i, j represent hyperfluxes piercing certain matter curves, the 10-plets and 5-plets split according to: The splitting of SU (5) GUT representations has important and far reaching implications in model building. With a suitable choice of the flux integers on the Higgs curve(s), Σ 5 hu , Σ 5 h d , one can generate the doublet-triplet splitting of the Higgs 5-plets and suppress dimension-five operators contributing to proton decay. We note in passing that this is a novel feature of F-theory and there is no counterpart in ordinary GUTs. Another important observation is that, by virtue of the hypercharge flux, the SM representations of the same family may no logner be components of the same 5-plet. As a result, the U (1) ⊥ charges may be different not only between various families, but even within SM fields of the same fermion generation. It is exacly this property that is expected to allow the existence of non-universal flavour couplings of the corresponding Z ′ boson to lepton fields. Therefore, we conclude that the hyperflux splitting mechansim provides a natural way to implement the idea of a new neutral Z ′ gauge boson coupled differently either to the third-or to a vector-like family within the F-theory motivated framework.

An SU(5) model
Having discussed the essential features of F-GUT model building, we now proceed in a specific model. We have seen already that, in principle, there are at most four independent roots t i . On the other hand, the four Cartan generators corresponding to U (1) 4 ⊥ are expressed as: Throughout this paper we shall assume the minimal Z 2 monodromy 7 Z 2 monodromy : t 1 ↔ t 2 (10) As a consequnce, this action reduces the number of the abelian factors by one, U (1) 4 ⊥ → U (1) 3 ⊥ , and allows a tree-level Yukawa coupling for the top-quark.
For the rest of our analysis, it is of interest to consider the following sequence of flux breaking, which may be associated with different scales Furthermore, as described above, we assume a U (1) Y flux which realises the SU (5) GUT breaking and at the same time triggers the doublet-triplet splitting and generates the chiral familes of the Standard Model through (7) and (8).
It is convenient to choose a basis for the weight vectors such that the charge generators have the form where Q ⊥ is the charge of the U (1) ⊥ in the breaking pattern of Eq. (11) that remains after imposing the t 1 ↔ t 2 monodromy. In the conventional basis for the SU (5) ⊥ generators in Eq. (9) the unbroken generators are identified as follows: This almost trivial equivalence shows that the SU (5) GUT states in Eq. (2) have well defined E 6 charges Q χ and Q ψ . For example SU (5) singlets will in general carry Q χ and Q ψ charges which originate from E 6 and which may be unbroken. The equivalence will provide insights into both anomaly cancellation and the origin of R-parity for example, in terms of the underlying E 6 structure, in the explicit models discussed later. In order to work consistently, we use the normalised SU (5) ⊥ generators and the low energy U (1) ′ is generated by which, in order to retain SU (5) ⊥ normalisation, are subject to the condition 7 In this semi-local approach, several essential features of the effective field theory model are determined from the topological properties of the associated spectral surface, C , described by a fifth degree polynomial C : ∑ 5 k=0 b k t 5−k = 0 . The essential ingredients specifying these properties are the coefficients b k with well defined topological properties. The roots of the polynomial determine the weight vectors t 1,...,5 . In general, however, depending on the properties of the specific compact manifold, the solutions t i = t i (b k )'s imply that there is an action on the roots t i of a non-trivial monodromy group which is a subgroup of the Weyl group W (SU(5) ⊥ ) = S 5 . Such subgroups are the alternating groups A n , the dihedral groups D n and cyclic groups Z n , n ≤ 5 and the Klein four-group Z 2 × Z 2 .
In order to proceed, we need to determine the spectrum and assign the corresponding Q ′ charges for all fields. We first remark that, in order to obtain three chiral families, the set of integers M n j associated with the U (1) ⊥ flux in formulae (6, 5) must obay This automatically implies also that the total flux vanishes, ∑ M 5 i + ∑ M 10 j = 0. Furthermore, the SU (5) representations are localised on matter curves possessing certain topological characteristics which determine the various properties of the effective SM theory. Important rôle, in particular, is played by the homologies of these curves which are determined in accordance with the Z 2 factorisation of the spectral cover polynomial. It turns out that the homologies of the whole set of matter curves can be expressed in terms of only three arbitrary parameters χ 6,7,8 [11]. If F Y represents the hypercharge flux, its 'dot' product with χ 6,7,8 determines three hyperflux integers and we defineÑ = N 7 + N 8 + N 9 . In conclusion, multiplicities of the SM states are determined by N i , i = 7, 8, 9 associated with hyperflux restrictions and the U (1) ⊥ flux integers M n j . More specifically, for a given choice of these numbers we can determine the exact number of the SM states residing on each matter curve using the formulae (7,8) and then apply the definition (18) to find their corresponding charges. We refer to previous work for the details [12], while here, we only present selected properties of the spectrum in Table 1.

Anomalies
The presence of an additional U (1) ′ symmetry in the effective model is associated with contributions to cubic and mixed anomalies with the SM gauge group G SM . On the other hand, we have seen that the multiplicities in the SM spectrum are determined by the flux parameters M n i , N j and therefore these numbers are directly involved in the anomaly constraints. These involve the U (1) ′ anomalies with the G SM currents as well as quartic and trace anomaly conditions. Thus, the flux integers M n i , N j are subject to certain restrictions. All anomalies were systematically computed using the SUSYNO package [15], those involving G SM currents in particular are as follows: In addition, we have to consider the trace anomaly, which is related to the D-flatness constraint and the associated Green-Schwarz cancellation mechanism, and the cubic anomaly that involve contributions from all singlet fields θ i j . To avoid clutter in this short note, these are not included in the above equations, however, they can be computed in a straightforword manner and are taken into account in our subsequent computations. We should point out however, that in order to have a sizable effect in anomalous Bdecays, the solution of the anomaly cancellation equations must be compatible with a TeV scale mass of the corresponding Z ′ boson.
Hence, our strategy is as follows: we start by working out the solutions with the anomalies involving SM currents, G 2 SM ×U (1) ′ and U (1) Y ×U (1) ′2 and solve for all c i coefficients subject to the normalization condition c 2 1 + c 2 2 + c 2 3 = 1. This will provide a solution c i = c i (M n j , N k ). First, we notice that the MSSM chirality trivially solves for A G 3 SM = 0. Then, we check whether there is a solution for U (1) ′3 , Tr Q ′ anomalies using the c i derived before, keeping only the solutions for which the charges can be written in a single radical form. 8 These last two constraints will be solved for multiplicities of singlet fields θ i j . And in our search for anomaly free solutions, we will seek models with the MSSM spectrum, as well as icluding those with complete vector-like families.

Minimal MSSM type spectrum but with a non-universal Z ′
We start with models that have the same spectrum of the MSSM with the gauge group extended by a single U (1) ′ factor. In order to have the MSSM chirality, the flux data must respect the geometric constraints (20) which are used to solve for M 5 6 and M 10 4 . Notice that this solves G 3 SM anomaly trivially. 8 This means that the charges for each state can all factor out √ a, where a is a rational number, such that it can be absorbed into the definition of the Z ′ coupling constant and hence making all charges appear rational numbers themselves. and we demanded the existence of a solution for the doublet-triplet splitting problem Since every state inside the same matter curve shares the same charge Q ′ , a non-universal model will require the families to be supported in different matter curves. In order to still scan models that allow for that to happen either only for the Σ 5 or Σ 10 matter curves, we allow the flux parameters to vary between In order to guarantee a renormalisable top quark Yukawa coupling, we identify H u ∈ 5 −2t 1 and t ∈ 10 t 1 as these states are involved in the renormalisable Yukawa interaction This accounts for and we further assume M 10 t = 1 to fix the left-handed top t L .
We also restrict our searches to solutions that have a single and isolated H u in the 5 H u curve, as we are preventing exotics in the spectrum. This accounts for where the last equation can be used to solve for one of the N i flux After scanning the entire parameter space, we find that there are only 48 solutions respecting all the conditions above, which fall in just four different classes of models. We notice that these are the only classes of models for an SU (5) GUT with a Z 2 monodromy with a consistent Z ′ interaction. The models inside the same class differ as to how the states are distributed amongst different curves and the perpendicular charges they carry. The four distinct classes are, in the G SM ×U (1) ′ basis in addition to Q ′ → −Q ′ . In Table 2 we present an explicit element of each class, and the distribution of the states amongst different matter curves is presented in Table 3.  Table 2: Flux data and c i coefficients for explicit models 1 to 4 of the classes 1 to 4, respectively As the left-handed top-quark is always identified as being in 10 t , we notice that in models 1, 3 and 4 the first and second family left-handed quarks have different U (1) ′ charges, which can be checked for all elements of the respective classes. As a result, the Z ′ mass for these models is constrained to be m Z ′ 10 5 TeV or otherwise Z ′ would contribute too strongly at tree-level to K 0 − K 0 oscillations. In There are 4067 models with one extra vector-like family, but only 397 classes when considering only G SM ×U (1) ′ charge assignments. Here we present three of these classes of models Class 7: and we notice the the first and last ones provide models where regular matter is universal under the new interaction, while the non-universality is strickly induced by the extra vector-like states. As K 0 − K 0 mixing imposes strong constraints on non-universality in the first two families, these two cases, in principle, could be consistent with a light (few TeV) Z ′ boson mass and could be sufficient to explain the B-decay anomaly. On the contrary, models from the Class 6 will probably require a heavy Z ′ , as the first and second family right-handed down-quarks have different charges.
In Table 4 we present the flux data and solutions for the coefficients c i for three explicit realisations of models respecting these conditions. The spectrum distribution amongst curves of these three models can be seen in Table 5.   Table 5: Explicit models 5 to 7 using the data in Table 4 5.1 A brief discussion of model 7 Next, we briefly discuss the main phenomenological properties of the emergent low energy effective theory with the spectrum given by model 7 of Table 5. Because of the constraints imposed by the K 0 −K 0 oscillations, as discussed above, the left-handed quark doublets of the three chiral fermion families should be accommodated in 10 t , 10 3 which have equal charges, Q ′ = 1 4 , with respect to U (1) ′ . Similarly, the three down quarks and lepton doublets are distributed in the fiveplets5 1 ,5 4 and carry the same charge Q ′ = − 1 4 , as shown in Table 5. The third family fermions receive masses from the tree-level couplings λ t 10 t 10 t 5 H u + λ b 10 t545H d .
We observe that there are more than one families on the same matter curves and the present configuration implies a single intersection point for the corresponding Yukawa couplings. In addition, by turning on non-zero fluxes we obtain hierarchical textures for the quark and charged lepton fermion mass matrices as described in [16]. The vector-like family is accommodated as follows: Left-handed doublet and right-handed up quarks together with right-handed electrons, are found in the 10 4 + 10 2 pair. Furthermore, the lepton doublet in 5 3 and the colour triplet in 5 6 form a complete fiveplet, and, together with5 5 constitute a pair in 5 5 + (5 6 + 5 3 ) which completes the vector-like family. These extra states form the following trilinear couplings involving singlet fields 10 4 10 2 θ 53 +5 5 5 3 θ 13 +5 5 5 6 θ 43 It is possible that these singlets acquire vevs just below the GUT scale and give masses of this order to vector-like fields which eventually decouple from the spectrum. Indeed, this would happen, for example, in the case of an anomalous U (1) ′ symmetry where the vevs of the singlets are fixed to be close to the GUT scale by virtue of the Green-Schwarz (GS) mechanism. In this case, the U (1) ′ symmetry breaking scale is also associated with the GUT scale and the Z ′ boson becomes superheavy so that any New Physics effects from its non-universal couplings are highly suppressed. Yet, it is possible that the GS mechanism is realised with large vevs not involving all or some of the singlets in (39). This allows the possibility to freely choose low scale vevs for these latter singlet fields and obtain TeV masses to the vector-like fields, giving rise to interesting phenomenological implications. However, if these vector-like fields are to be relevant for the anomalies in the B-decays, the singlets involved in these terms should acquire non-zero vevs θ i j of the order few TeV. Besides, there are also couplings of the vector-like family with the three generations such as 10 4 10 3 θ 54 + 10 4 10 t θ 51 etc. Assuming this then, after mixing with the vector-like family, this model can lead to non-universal couplings of the light mixed states which can be selected in such a way as to account for the recent hint for anomalous B-decays, along the lines of [5]. Therefore, we conclude that a scenario such as [5] can, in principle, be realised in this fluxed GUT framework.

Conclusions
In this work, we have presented a class of fluxed SU (5) GUTs which naturally incorporate an abelian symmetry, U (1) ′ , where the associated gauge boson Z ′ displays non-universal gauge couplings to fermion families. This paper represents the first study of non-universal Z ′ gauge bosons in F-theory phenomenology. In our analysis we have considered an F-theory inspired framework where SU (5) GUT emerges as a subgroup of the maximal exceptional gauge symmetry E 8 ⊃ SU (5) × SU (5) ⊥ . Non-trivial universal fluxes are assumed along the Cartan generators of the custodial group factor, U (1) 4 ⊥ ⊂ SU (5) ⊥ , breaking the symmetry and generating chirality of the SU (5) spectrum. A Z 2 monodromy is imposed which allows rank-one fermion mass matrices and reduces the original symmetry to SU (5) × U (1) 3 ⊥ , while a hypercharge flux breaks the SU (5) GUT down to the Standard Model and, at the same time, induces the observed fermion chirality to the low energy spectrum. We then assume that the U (1) 3 ⊥ gauge group is broken to a single non-universal U (1) ′ , by some unspecified Higgsing mechanism.
We have analysed the case where the non-universal U (1) ′ is an anomaly-free linear combination of the remaining three U (1) ⊥ factors, U (1) ′ = ∑ 3 i=1 c i U (1) ⊥,i . We chose a basis where two of U (1) ⊥,i symmetries are identified with U (1) ψ ∈ E 6 and U (1) χ ∈ SO(10) and we formulated the gauge anomaly cancellation conditions in terms of a set of integers M a , N b introduced to parametrise the universal and hypercharge flux effects. We have scanned for solutions for the three coefficients c i to determine the multiplicities of the corresponding light spectra in terms of the flux integers M a , N b . We found a plethora of models which are classified with respect to the nature of the non-universal couplings to fermions and the emergent spectrum. There are several classes of models with the minimal low energy MSSM spectrum, as well as several classes with additional vector-like fields.
In particular we highlight the phenomenologically promising classes with universal couplings to the three chiral fermion generations with only the single vector-like family having non-universal couplings. When the vector-like family mixes with the chiral families, non-universality involving the mixed light states may result. Such models naturally suppress New Physics contributions and models can be found which are universal in the the first two light quark families, compatible with known processes such as the K 0 −K 0 mixing, while involving non-universality in the third family. In principle these types of models may alter the branching ratios of the B-decays in accordance with the recently observed deficit in the µ + µ − channel. We discuss the plausibility of the above scenario in relation the anomaly cancellation conditions and the scale of the New Physics effects.