Solving the Flavour Problem in Supersymmetric Standard Models with Three Higgs Families

We show how a non-Abelian family symmetry Delta(27) can be used to solve the flavour problem of supersymmetric standard models containing three Higgs families such as the Exceptional Supersymmetric Standard Model (E(6)SSM). The three 27 dimensional families of the E(6)SSM, including the three families of Higgs fields, transform in a triplet representation of the Delta(27) family symmetry, allowing the family symmetry to commute with a possible high energy E(6) symmetry. The Delta(27) family symmetry here provides a high energy understanding of the Z(2)H symmetry of the E(6)SSM, which solves the flavour changing neutral current problem of the three families of Higgs fields. The main phenomenological predictions of the model are tri-bi-maximal mixing for leptons, two almost degenerate LSPs and two almost degenerate families of colour triplet D-fermions, providing a clear prediction for the LHC. In addition the model predicts PGBs with masses below the TeV scale, and possibly much lighter, which appears to be a quite general and robust prediction of all models based on the D-term vacuum alignment mechanism.


Introduction
Although TeV scale Supersymmetry (SUSY) is well motivated, the minimal supersymmetric standard model (MSSM) suffers from the µ problem and the little fine-tuning problem [1]. The simple replacement of the µ-term of the MSSM by a singlet superfield coupling to Higgs superfields leads to a global axial U(1) symmetry whose breaking would lead to an unwanted axion. In the Next-to-Minimal Superymmetric Standard Model (NMSSM) a cubic singlet coupling is also assumed, which breaks the axial U(1) symmetry down to a discrete Z 3 subgroup. In this way the NMSSM solves the µ problem and the little fine-tuning problem [2]. However when the Z 3 symmetry of the NMSSM is broken it leads to cosmological domain walls. One way to overcome such problems is not to add the cubic singlet term but to gauge the axial U(1) symmetry so that the would-be axion gets eaten by a Higgs mechanism resulting in a massive observable Z ′ [3]. Of course such models come at a price since the gauged U(1) symmetry must be made anomaly-free, and this generally involves adding additional fermions which are sometimes ignored in simple phenomenological applications such as in the USSM [4].
The E 6 SSM [5] is a non-minimal supersymmetric model based on an underlying highenergy E 6 symmetry where an additional low-energy gauged U(1) is identified amongst the E 6 generators, and to cancel anomalies, complete 27 dimensional E 6 families are assumed to survive down to the TeV scale. There are many possible choices of gauged U(1) [6], but the E 6 SSM [5] is uniquely defined by a particular choice U(1) N under which the right-handed neutrinos are neutral, and so may become heavy, allowing naturally small physical neutrino masses. As in the NMSSM, the E 6 SSM superpotential does not have an explicit µ-term for the Higgs doublets but instead contains a trilinear term that represents the interaction between the Higgs doublets and an additional Standard Model singlet field S. When this singlet field obtains a vacuum expectation value (VEV) the trilinear term reduces to an effective µ-term, thus providing a solution to the µ problem of the MSSM. The E 6 SSM also contains an additional low-energy U(1) gauge symmetry which, as in USSM models, can be considered to be a gauged Peccei-Quinn symmetry. This local U(1) symmetry prevents the presence of an unwanted Goldstone boson once the singlet field obtains its VEV, which is eaten by the Higgs mechanism resulting in a massive Z ′ . For this U(1) group to be anomaly free, the entire matter content of the (three families of) 27 multiplets of E 6 are assumed to survive to low-energies (apart from the right-handed neutrinos). Each 27 multiplet contains one generation of quarks and leptons (including a right-handed neutrino), up and down type Higgs doublet fields (as well as their coloured partners), and a Standard Model singlet. The E 6 SSM thus contains the particle spectrum of the MSSM plus two additional MSSM Higgs families, three families of coloured partners to the three Higgs families, and three singlet fields. To achieve gauge coupling unification at the GUT scale, the E 6 SSM also contains two additional electro-weak doublets from a 27 ′ and 27 ′ incomplete multiplets of E 6 which are sterile in the E 6 SSM superpotential. In a minimal version of the E 6 SSM, with string scale unification, these additional states are not present [7].
The E 6 SSM [5], in common with the MSSM and NMSSM, is subject to the usual flavour problems to do with the unexplained spectrum of fermion (including neutrino) masses and mixings of the one hand, and the absence of flavour changing neutral currents (FCNCs) generically expected in SUSY models on the other hand. In fact, the E 6 SSM also faces additional FCNC challenges due to the three Higgs and singlet families. These challenges are resolved in the E 6 SSM by invoking a Z H 2 symmetry which only allows the third family of Higgs and singlet to couple to quarks and leptons, with the first two Higgs and singlet families being inert (having zero VEVs) and fermiophobic (not coupling to quarks and leptons). While this is perfectly acceptable from the phenomenological point of view, from a theoretical standpoint the Z H 2 symmetry looks rather ad hoc and adds an additional complication to the flavour problem of the E 6 SSM not present in the MSSM or NMSSM.
The discovery of neutrino mass and approximately tri-bimaximal lepton mixing [8] suggests some kind of a non-Abelian discrete family symmetry might be at work, at least in the lepton sector, and, assuming a GUT-type of structure relating quarks and leptons at a certain high energy scale, within the quark sector too. The observed neutrino flavour symmetry may arise either directly or indirectly from a range of discrete symmetry groups [9]. Examples of the direct approach, in which one or more generators of the discrete family symmetry appears in the neutrino flavour group, are typically based on S 4 [10] or a related group such as A 4 [11,12] or P SL(2, 7) [13]. Models of the indirect kind, in which the neutrino flavour symmetry arises accidentally, include also A 4 [14] as well as ∆ 27 [15] and the continuous flavour symmetries like, e.g., SO(3) [16] or SU(3) [17] which accommodate the discrete groups above as subgroups [18]. In this Letter we show how a ∆ 27 family symmetry can resolve all the flavour problems of supersymmetric standard models containing three Higgs families such as the E 6 SSM [5], including the fermion mass and mixing puzzle present in the SM, the FCNC problems introduced by the MSSM and the additional puzzle of the origin of the Z H 2 symmetry peculiar to models such as the E 6 SSM. The ∆ 27 family symmetry is chosen rather than, for example, A 4 or S 4 , since it allows complex representations whereas A 4 and S 4 only contain real representations. Complex representations are required in family symmetry models in which the left-handed matter fields F and right-handed matter fields F c both transform in triplet representations. This is to avoid the trivial combination F F c h where h is the Higgs field. To be concrete we focus on the E 6 SSM where, under the discrete non-Abelian ∆ 27 family symmetry, we shall assume that the quarks, leptons and Higgs fields of the E 6 SSM all transform in triplet representations so that the family symmetry commutes with a possible high energy E 6 symmetry. It is known that such a family symmetry can account for various Yukawa couplings responsible for the masses of quarks and leptons, and serves to predict tri-bi-maximal mixing for leptons [15]. It is also known that such family symmetries when applied to supersymmetry can provide a solution to the SUSY flavour and CP problems [19]. The qualitatively new feature here is that such a family symmetry can solve the flavour changing neutral current problems introduced by extended Higgs sectors by controlling the Higgs couplings, similar to the Z H 2 symmetry of the E 6 SSM. As a consequence, we shall find predictions for the mass structure of the three families of Higgs and Higgsino fields and coloured D-fermions of the E 6 SSM. We remark that in a recent paper [20] we also considered a non-Abelian ∆ 27 family symmetry in E 6 SSM models [5,7], however the Higgs fields (and D-fermions) were taken to be singlets of the family symmetry, rather than triplets, and the Z H 2 symmetry was simply assumed.
The outline of this Letter is as follows. In the next section we describe how the ∆ 27 family symmetry is applied to the E 6 SSM. This section is split up into subsections which look at how each term in the E 6 SSM superpotential is generated from higher-order operators once we apply the ∆ 27 symmetry. In section 2.1 we introduce the E 6 SSM renormalizable superpotential, in the absence of any family symmetry. In section 2.2 we introduce the vacuum alignment required for the various ∆ 27 flavon fields. Then in section 2.3 we describe the non-renormalizable operators allowed by ∆ 27 and other symmetries that lead to the quark and lepton Yukawa interactions with the Higgs fields. We also describe the types of messenger fields that are integrated out to generate the higher-order operators, and illustrate how the Z H 2 symmetry of the E 6 SSM effectively emerges from the high-energy theory. In section 2.4 we discuss how tri-bi-maximal mixing is generated in our model from the ∆ 27 family symmetry and sequential dominance and then, in section 2.5, we describe how the effective µ-term of the MSSM is generated and discuss the mass structure of the LSPs that are formed from the inert higgsinos and singlinos. In sections 2.6 and 2.7 the mass structure of the D-fermions is explained and their decay channels are discussed. Finally, in section 3, we summarize our findings.
2 ∆ 27 Family Symmetry in the E 6 SSM

The E 6 SSM Superpotential Without Family Symmetry
The quark and lepton Yukawa couplings in the E 6 SSM are derived from the E 6 tensor product λ ijk 27 i 27 j 27 k where i, j, k label the three generations. To better understand where the Yukawa couplings come from it is useful to re-write the superpotential in terms of the Pati-Salam subgroup of E 6 under which each 27 multiplet decomposes to the following representations: 27 ⊂ F + F c + h+ D + S where F contains the left-handed quark and lepton fields, F c contains the charge conjugated quark and lepton fields, h contains up and down Higgs-like fields, D are the colour triplet partners of the Higgs fields, and S are Standard Model singlets though they are charged under the gauged U(1) N symmetry which is broken at low energies [5]. In this Pati-Salam notation the E 6 tensor product 27.27.27 from which the E 6 superpotential is formed, decomposes in the following way, dropping all couplings and indices for clarity: The part of the E 6 SSM superpotential that contains the Yukawa interactions for the quarks and leptons is λ ijk F i F c j h k and is the subject of the section 2.3. In section 2.5 we discuss the superpotential term λ ijk S i h j h k from which the MSSM effective µ-term is generated. section 2.6 describes the term λ ijk S i D j D k from which the exotic D-fermion states get mass, and section 2.7 looks at the operators λ ijk F i F j D k + λ ijk F c i F c j D k which provide decay channels for the exotic D-fermions.
symmetry. An R-symmetry is also applied to the model which breaks to an R-parity once S 3 obtains a vacuum expectation value.
Although the above operators are written in a Pati-Salam notation we only do this for ease of notation. The model presented in this Letter is in fact based on the Standard Model gauge symmetry rather than a Pati-Salam symmetry. For the rest of this Letter we use a Pati-Salam notation unless stated otherwise.

Vacuum Alignment
In this Letter we add a ∆ 27 family symmetry which is broken via the VEVs of the flavon triplets φ introduced in Table 1. A number of different triplet flavon fields with different expectation value directions in ∆ 27 space are required to produce the desired mass structure for the E 6 SSM particles. Four different types of directions for the flavon VEVs are used in this Letter (dropping the VEV brackets): In the case of direct models, in which one or more generators of the discrete family symmetry appears in the neutrino flavour group, the usual mechanism of vacuum alignment of flavon fields is based on F-term alignment which exploits driving fields in the superpotential as discussed in [12]. This mechanism is also available to indirect models, in which the neutrino flavour symmetry arises accidentally, as discussed in [18]. However, in the case of indirect models, an additional and elegant possibility for vacuum alignment becomes available that is not possible for direct models, namely D-term vacuum alignment introduced in [14,15] as discussed below. One advantage of the D-term method is that the terms required to achieve vacuum alignment originate from the Kähler potential and so are not restricted by the symmetries which are introduced to control the terms in the holomorphic superpotential. Thus the required terms may always be present independent of the symmetries of the model. In the case of the present model we shall use the D-term vacuum alignment method to generate the above flavon VEVs, following largely the discussion in [14,15].
An elegant way to obtain the alignments of Eq.2 is to start with a flavon scalar potential of the form [14,15]: where the index i labels the components of a particular flavon triplet φ and: In a supersymmetric theory the quartic terms may arise from D-terms, after which this vacuum alignment mechanism is named [14,15], which take the form: where the F-component of the ∆ 27 singlet χ acquires a SUSY breaking VEV F χ , leading to a gravitino mass m 2 3/2 ∼ F 2 χ /M 2 P . Hence supersymmetry is broken and the scalar potential V gets a contribution of the type (λ, κ)φ † φφ † φ with small (λ, κ) ∼ m 2 3/2 /M 2 P . The quadratic mass term originates also from a soft supersymmetry breaking mass term, and this term is thus expected to have a soft mass squared m 2 ∼ m 2 3/2 ∼ (T eV ) 2 . We suppose that the flavon mass squared m 2 is positive at the (reduced) Planck scale M P which prevents symmetry breaking at M P (if m 2 < 0 at M P then we would expect a VEV given by −m 2 /λ ∼ M P ). Then we assume that the soft mass squared m 2 of a given flavon is driven negative by radiative corrections at some scale Λ < M P , which triggers a VEV for that flavon set by the scale Λ, a mechanism which has been widely used in different contexts (see for example [21]). To see this explicitly here, we may approximate the potential, including radiatively corrected logarithmically running masses, to be of the form 3 , dropping the small quartic terms. The first derivative of the potential with respect to As the different flavons have different superpotential couplings to heavy states, and since the soft masses run logarithmically with energy scale, the Λ scales defined above may differ greatly for the different flavons. Thus a hierarchy between the VEVs of various flavon fields is possible, and also stable, in the framework of the radiative breaking mechanism [14,15,21].
Only the term ∆V in Eqs.3,4 determines the alignment, where κ ∼ κ 0 m 2 3/2 /M 2 P , and the sign of κ 0 ∼ O(1) is undetermined. For κ > 0 we obtain the alignment along the direction of the VEV of φ 123 , while κ < 0 can give rise to that of φ 3 . The configuration φ 23 ∝ (0, −1, 1) T can then be generated using a leading higher order term that requires that the VEV is orthogonal to both (1, 0, 0) T and (1, 1, 1) T . All these operators can be used to generate the VEV configurations of the flavons in Eq.2. A more detailed discussion on this subject can be found in [15], where the vacuum alignment operators required for the flavon fields that are used in this Letter is provided except for the flavon . We assume the same vacuum alignment operators used in [15] which align all the flavon fields except for φ h For these additional flavon fields to get the required direction of vacuum expectation values, we use the following Dterms (omitting both with negative coefficients, and similarly for the φ S 3 and φ S 3 flavons. These terms cause φ h 3 and φ h 3 to get VEVs in the same direction as the pre-aligned fields φ 3i and φ i 3 respectively. Note that the leading order potential, including ∆V , is invariant under a product of global U(1) symmetries, one for each flavon component. However this symmetry is broken explicitly by superpotential terms including the Yukawa superpotential discussed in the following subsection. It is also broken by higher order terms in the scalar potential involving purely flavon superfields as indicated by the dots in Eq.3. An example of such a term in the potential which is invariant under all the symmetries in Table 1 but which would violate the additional global U(1) symmetries is ( Inserting flavon VEVs of order Λ, such a term would lead to pseudo-Goldstone-boson (PGB) square masses of order m 2 P GB ∼ m 2 3/2 Λ 6 /M 6 P , where generally we expect Λ < M P leading to PGB masses below the TeV scale (and perhaps much lighter) with possible interesting phenomenological and astrophysical implications beyond the scope of this paper. Similarly, inserting the flavon VEVs of order Λ, one sees that this additional eighth order term gives a contribution to the potential suppressed relative to the previous quartic terms by a factor of Λ 4 /M 4 P so that, assuming Λ < M P , it will not perturb the previous vacuum alignment arguments appreciably.
It may seem surprising that both the symmetry breaking and vacuum alignment are governed by quartic terms with very small coefficients (λ, κ) ∼ m 2 3/2 /M 2 P . In particular one may worry that other quartic operators with larger coefficients are present in the theory and that they could destabilize the symmetry breaking and alignment scheme described above. However, given the field content and symmetries assumed for the model, in particular the gauged U(1) N and the U(1) R symmetries, it is not possible to write down any operators which would lead to terms in the flavon potential with competing or larger coefficients than those of the quartic terms above and which would destabilize the vacuum, and so the quartic terms considered above are the dominant ones. On the other hand, we have seen that such symmetries imply approximate global U(1) symmetries which are only broken by higher order operators, leading to PGB masses below the TeV scale and possibly much lighter. The presence of such PGBs seems to be generic to the type of radiative symmetry breaking associated with D-term vacuum alignment, and this observation appears to be robust although it has not been remarked upon before.

The Effective Yukawa Operators
Under the ∆ 27 family symmetry the F , F c and h transform as triplets so that the superpotential Table  1 describes how all the Pati-Salam states from a 27 representation and the flavons transform under the family symmetry and the additional symmetries that constrain the model such as Z h 2 symmetry which distinguishes the Higgs and D fields (but unlike Z H 2 treats all three Higgs families identically). The lowest order Yukawa superpotential consistent with the symmetries of Table 1 is: where all order 1 coupling constants are suppressed. φ 23 , φ h 3 , φ 123 , φ 3 are all anti-triplets of the ∆ 27 family symmetry, φ 1 is a triplet of ∆ 27 , H 45 is a singlet of ∆ 27 but a 45 of the SU(5) subgroup of E 6 , and M is some messenger scale that is discussed further in section 2.3.2. The φ h 3 field is taken to transform under the Z h 2 symmetry so that all of the above operators respect this symmetry. We assume that this flavon field and φ 3 get a VEV in the third component of ∆ 27 , φ 23 gets an equal but opposite VEV in the second and third components of ∆ 27 , and φ 123 gets an equal VEV in the first, second and third components of ∆ 27 . The vacuum alignment of these fields was discussed in section 2.2. For ease of notation we denote φ as a field that transforms as 3 under ∆ 27 and φ as a field that transforms as a 3.
In addition to the ∆ 27 family symmetry, Table 1 also contains the vertical symmetries U(1)×Z 2 ×Z h 2 ×Z S 2 ×U(1) R . These symmetries prevent interactions that would otherwise be allowed by the ∆ 27 symmetry and would introduce certain phenomenological issues to the model. In particular, The Z h 2 symmetry is used to differentiate certain superfields including the Higgs from others, and is instrumental in preventing flavour changing neutral currents due to the extended Higgs sector of the model, as discussed further in the following section. The Z S 2 is introduced in section 2.5 and is primarily used to lead to a non-renormalizable operator that reduces to an effective µ-term at the soft SUSY scale, while forbidding other operators. Note that Z S 2 also forbids the renormalizable operator ǫ ijk S i h j u h k d but this is not a crucial requirement and similar terms are generated at higher order. The U(1)×Z 2 symmetries are invoked to allow only certain combinations of flavon fields, as in [15]. For example, the U(1) symmetry prevents the effective Yukawa operator 1 M 2 F i F c j φ i 123 φ j 3 from appearing in Eq.5. Finally, the U(1) R symmetry is an Rsymmetry that is assumed to reduce to R-parity allowing the LSP of the model to be stable.

Preventing Flavour Changing Neutral Currents
As discussed in the Introduction, the E 6 SSM [5] is subject to the "usual" FCNC and CP problems generically expected in SUSY models, due to SUSY loop diagrams with off-diagonal soft mass insertions, as well as "additional" FCNC challenges arising from tree-level Higgs exchange due to the three Higgs and singlet families. The introduction of a family symmetry, together with the idea of spontaneous CP violation by the flavons, leads to a natural suppression of the "usual" FCNC and CP violating operators and provides a solution to the SUSY flavour and CP problems [19].
The impact of the higher order Kähler operators has been systematically explored for SU(3) in [19] and the results there are directly applicable to the present model based on ∆ 27 . In the exact ∆ 27 family symmetry limit the soft masses are universal: However, in this limit, the Yukawa couplings and trilinear terms vanish. In reality the ∆ 27 family symmetry has to be broken, leading to violations of universality. For example, the soft masses allowed by the ∆ 27 family symmetry can be written as: where the generic subscript A runs over all the relevant flavon species. The above off-diagonal soft mass terms are suppressed by ǫ factors from the ratios of the flavon VEVs and the messenger masses. In [19] it was rigorously shown that off-diagonal terms from flavons of the type in Table 1 lead to suppressed FCNCs concistent with present experimental limits. The family symmetry thus naturally suppresses the "usual" type of induced FCNCs due to off-diagonal soft masses and can therefore provide a resolution to the generic SUSY FCNC problem.
In addition to the FCNCs associated with the soft SUSY potential, FCNCs can also originate from models with the extended Higgs sectors, leading to "additional" FCNC problems arising from tree-level Higgs exchange. However, as we shall now show, the Z h 2 symmetry, in combination with the ∆ 27 family symmetry, suppresses FCNCs due to tree-level Higgs exchange in the present model. This can be understood by noting that, since φ h 3 transforms under Z h 2 , it will couple to the Higgs fields but not to the quarks and leptons. This can be understood by considering the messenger diagrams of the above higher-order operators where φ h 3 will only be allowed to attach itself to the Higgs fields (and the Higgs-like messenger fields) if all the messenger fields are even under Z h 2 . This is illustrated by Figure 1. Once φ h 3 gets a VEV, only the third generation of the up and down Higgs fields h 3 are allowed to couple to the quarks and leptons. It is these up and down Higgs fields which we therefore take to obtain electro-weak scale VEVs, and thus act like the up and down Higgs fields of the MSSM.
The Z h 2 and ∆ 27 symmetries prevent the first and second generation of Higgs fields from interacting with the quarks and leptons at tree-level and so there can be no treelevel FCNC processes involving the neutral scalar components of these fields. In the E 6 SSM a Z 2 symmetry called Z H 2 is applied to all the 27 fields except for the third generation of Higgs fields and singlet fields to prevent the first and second generation of Higgs fields from interacting with the quarks and leptons at tree-level. The Z h 2 in this Letter is therefore acting as the Z H 2 symmetry of the E 6 SSM even though it does not distinguish between the different Higgs fields. The Z H 2 symmetry in the E 6 SSM is broken by an additional discrete Z 2 symmetry that forbids the colour triplets of the Higgs fields causing rapid proton decay [5]. This additional discrete symmetry will not break the Z h 2 symmetry here, however any misalignment of φ h 3 will play the role of Z H 2 breaking, as discussed later.

The Messenger Fields
The messenger fields Σ that are responsible for the suppression factors in Eq.5 include fields that transform in the same way as quarks and leptons under the Standard Model gauge group and as singlets, triplets and anti-triplets of ∆ 27 . We refer to these type of messenger field as quark and lepton-like messengers Σ F,F c . In addition there are also messengers that are singlets of ∆ 27 and transform in the same way as Higgs fields under the Standard Model gauge group. We refer to these messenger fields as Higgs-like messengers Σ h . All these messenger fields are taken to carry positive Z h 2 parity, and we assume that the Higgs-like messengers Σ h are heavier than the quark and leptonlike messengers Σ F,F c so that the latter dominate the messenger diagrams. We further assume that the right-handed quark and lepton messengers Σ F c are heavier than their left-handed counterparts Σ F so that the former dominate over the latter. The messenger diagrams are illustrated by Figure 1.
To create a smaller hierarchy in the down quark sector compared to the up quark sector, we take the mass of the 3 and 3 up and down Higgs messengers M h 3 to be equal, but the up right-handed quark messengers Σ u c that are 3 and 3 and singlets of SU(3) to have a mass M u that is greater than the right-handed down quark messengers Σ d c by approximately a factor of three. For the top Yukawa coupling constant to be greater than the bottom Yukawa coupling constant we take the φ 3 flavon to transform as a 3 + 1 of the SU(2) R subgroup of E 6 and choose its VEV so that φ 3 /M d = φ 3 /M u as in [15]. In terms of these messenger masses, the VEV scales for the various flavon fields are then taken to be the following: where ǫ u ≈ 0.05. At the GUT scale the Yukawa coupling for the top and bottom quark is expected to be about 0.5 in third family Yukawa unification models based on the MSSM with large tan β [22]. We therefore assume that φ h 3 /M h 3 ≈ φ 3 /M u ≈ 0.8. By comparison, in ∆ 27 models in which the Higgs is a singlet and there is a messenger-scale suppression factor to the second power, φ 3 /M u is assumed to be about 0.7 [15]. Note that if φ 3 /M u was taken to be exactly or very close to 1 then higherorder operators would be no-more suppressed than lower-order operators, so such large expansion parameters are of some concern, and in this respect the model is on a similar footing to that [15].
In order to avoid problems with unification, we assume that the mass of the above messenger fields have masses close to the conventional GUT scale, which suggests that the ∆ 27 family symmetry is also broken close to GUT scale. In principle this could lead to GUT scale threshold effects, but we do not consider this further here. Below the GUT scale the particle content of the model below the GUT scale is then the equivalent to that of the E 6 SSM, with the possible inclusion of additional SM singlet flavon fields close to the GUT scale that will not affect the running of the gauge coupling constants. As in the E 6 SSM, below the GUT scale there are three copies of complete 27 representations of E 6 plus two additional electroweak doublets with masses of order the TeV scale, with the gauge coupling constants predicted to unify at the conventional GUT scale but with a larger unified gauge coupling constant than in the MSSM [5].
In section 2.5 we also discuss the messenger fields which are responsible for the higher-order operators that generate the effective µ-term. These particular messengers include Higgs-like messengers and messengers that transform as singlets of the Standard Model gauge group.

The Effective Yukawa Matrices
Inputting the above flavon VEVs into the superpotential given by Eq.5 generates the following effective Yukawa matrices for the quarks and leptons: 15, and λ t ≈ 0.5 at the GUT scale [22].
The above form of Yukawa matrices have been shown to produce a realistic CKM matrix and realistic mass hierarchies for the up and down quarks [22]. The H 45 Higgs field in the superpotential Eq.5 is used to generate the Georgi-Jarlskog relations in the down quark and charged lepton matrices so that the correct hierarchy in the charged leptons is generated at the electroweak scale [23].

Tri-Bi-Maximal Mixing
To generate tri-bi-maximal mixing we use constrained sequential dominance [24] in which the right-handed neutrinos have a hierarchy in mass. This hierarchy is generated by the following operators: where φ 123 and φ 3 are triplets of ∆ 27 , and θ is from a 27 of E 6 and a 3 of ∆ 27 .
The above operators together with the dirac operator involving the left-handed and right-handed neutrinos from the superpotential F i F c j h 3 create a conventional type I seesaw mechanism which generates an effective Majorana mass matrix for the left-handed neutrinos. Due to form of the right-handed Majorana mass matrix, and the dirac mass matrix from Eq.5, the effective Majorana matrix for the left-handed neutrinos is of a form that is diagonalized by a tri-bi-maximal matrix (see [15] for more details).

The Effective µ-Term and Inert Higgsino/Singlino Masses
The U(1) N symmetry of the E 6 SSM forbids any bilinear superpotential terms for the different Higgs fields. Instead effective bilinear terms come from the Pati-Salam superpotential term λ ijk S i h j h k from Eq.1, where i, j, k = 1 . . . 3, once S 3 gets a VEV. In terms of the Standard Model gauge group, which is the symmetry of the model discussed in this Letter, this superpotential term reduces to λ ijk S i h uj h dk where h i and h d denote up and down Higgs fields. In this Letter we take the singlet fields S i , like the Higgs fields, to transform as a triplet of the ∆ 27 symmetry. We also take them to be odd under a Z S 2 symmetry and even under the Z h 2 symmetry. The operator λ ijk S i h uj h dk is thus forbidden and is instead generated by the following higher-order operators: where the flavons have the charges shown in Table 1 and we take the scale of these flavon VEVs to be such that φ S 3 /M S = ǫ S , φ The first operator in Eq.9 is responsible for generating an effective µ-term for the third family of Higgs fields once the flavons fields and the third family singlet field S 3 obtain VEVs. Since we assume that only the third family of Higgs obtains a VEV, this effective µ-term acts like the µ-term of the MSSM Higgs fields. The effective µ-term will have a value (0.8) 3 S 3 , which will be approximately 1 T eV if S 3 = 2 T eV , which is consistent with the experimental bound for the mass of a Z ′ . If we assume a radiative symmetry breaking explanation for the third family singlet field's VEV then it will be related to the singlet's soft mass term. This then explains the apparent requirement of the µ-term of the MSSM being related to the MSSM soft terms.
The second and third operators in Eq.9 are responsible for providing mass to the first and second families of Higgsinos and singlinos once the third family of Higgs fields and singlet field obtain VEVs. This results in a mixing between all of these states which is represented by the following matrix: This matrix is written in the basis (h 0 d2 ,h 0 u2 ,S 2 |h 0 d1 ,h 0 u1 ,S 1 ) so that A αβ are 3×3 matrices where α, β = 1, 2. Because of the anti-symmetric tensor in the Eq.9 we find that A 11 = A 22 = 0, whereas A 21 is given by the following: where this matrix couples the states (h 0 d2 ,h 0 u2 ,S 2 ) to the states (h 0 d1 ,h 0 u1 ,S 1 ). In the limit of exact Z h 2 and Z S 2 symmetry these Higgsino and singlino states will decouple from the usual inert USSM states such as the third family of Higgsinos, singlinos, wino and hypercharge bino fields. See [25] for a full discussion on the mixing between the usual USSM states and the additional E 6 SSM states where it is also shown that the mixing between the U(1) N bino and Higgsino and singlino fields is expected to be small.
The above Higgsino and singlino neutral states combine to form two degenerate LSP states, approximately consisting of a Dirac state formed from (dropping the tildes) S 1 and S 2 , together with two generally heavier approximately degenerate Dirac states formed from h 0 d1 and h 0 u2 on the one hand and h 0 d2 and h 0 u1 on the other hand. With exact R-parity the Dirac LSP state formed from S 1 and S 2 becomes a dark matter candidate. However the masses of the degenerate LSPs S 1 and S 2 can be split if the first and second generation of Higgs and singlet fields are distinguished from one another. One way of achieving this is to assume that the flavon field φ h 3 gets small vacuum expectation values in its first and second components of ∆ 27 such that φ h 3 T ∝ (δ 1 , δ 2 , 1) where δ 1 , δ 2 ≪ 1. This might be expected to occur from higher-order operators that affect the vacuum alignment of the fields. Two WIMPs that are almost degenerate in mass have been recently used to explain the DAMA data [26].
Note that although the Z h 2 and Z S 2 symmetries of this model have combined to operate in a similar manner to the original Z H 2 symmetry of the E 6 SSM, they allow fewer operators than the latter. The operators allowed by the original Z H 2 symmetry but which are not present in this model are S 3 h uα h dα , S α h uα h d3 and S α h u3 h dα . Such operators are responsible for the A 22 and A 11 matrices being non-zero in the E 6 SSM.

D-fermion Mass Terms
For the U(1) N group of the E 6 SSM to be anomaly free, all the colour partners D i of the Higgs fields from the three 27 multiplets must have masses lower than the energy scale of S 3 . These masses come from the Pati-Salam superpotential λ ijk S i D j D k which is derived from the E 6 superpotential of the E 6 SSM given by Eq.1. In terms of a Standard Model gauge symmetry we represent this operator as λ ijk S i D j D k where D is a triplet of the strong force gauge group SU(3) c but D is an anti-triplet.
In this Letter we assume that the exotic particles D i , like the Higgs fields transform as a triplet of ∆ 27 and have odd Z h 2 parity but even Z S 2 parity. The allowed higher order operator thus mirrors the allowed operators that provide effective µ-terms for the Higgs fields: The mass scale for the exotic-like messengers Σ D,D responsible for the operators in Eq.11 however need not be the same as the Higgs messengers. We define the messenger scales We also assume that the exotic-like messengers, like the Higgs-like messengers only have even Z h 2 parity but can carry either even or odd Z S 2 parity. The messenger diagrams that are assumed to be responsible for generating the higher-order operators in Eq.11 are analogous to those in Figure 2 but with the Higgs fields and Higgs-like messenger fields replaced with exotic fields and exotic-like messenger fields respectively.
The D-fermions thus obtain mass once the flavons and S 3 obtain an expectation value. We write these masses in matrix form M Dij D i D j where M Dij is the following: The parameters ǫ S , ǫ D and ǫ D can then be chosen for the exotic masses to be larger than the experimental bound of 300 GeV . Two of the exotic states are predicted to be degenerate in mass with the third also being degenerate in the approximation that ǫ 2 D = ǫ D and ǫ D ≪ ǫ S . This mass structure is in stark contrast to the hierarchical structure of the quarks and leptons despite all the states being triplets of the family symmetry.

D-fermion Decay and Proton Decay Suppression
If the exotic D particles are taken to have the same ∆ 27 , Z h 2 and Z S 2 quantum numbers as the Higgs fields h, then they can decay [5] via the following non-renormalizable operators: However not all these operators can be allowed otherwise this would lead to very rapid proton decay. Thus, we assume either the Z B 2 or Z L 2 discrete symmetries that are used in the E 6 SSM [5], where these symmetries differentiate between different fermion F, F c components. Under the Z B 2 symmetry the leptons and D states are odd whereas, under the Z L 2 symmetry, only the leptons are odd and all other particles are even. These discrete symmetries remove some of the above operators, such that the remaining operators correspond to the D-states coupling as either diquarks or leptoquarks. This effectively prevents the proton from decaying due to couplings with the D-states, but still allowing the latter states to decay, thus avoiding any nucleosynthesis difficulties [5,27]. Note that this discrete symmetry breaks the Pati-Salam gauge symmetry but respects the Standard Model gauge symmetry assumed in this Letter.
In the limit that φ h 3 T ∝ (0, 0, 1) exactly, the decay channels of the exotic states in the model used in this Letter will be different to those of the E 6 SSM since only the third generation of the exotic states couples directly to quarks and leptons, whereas all three generations of the exotic states in the E 6 SSM interact directly with the quarks and leptons. The difference between the two models occurs because we have taken the exotic states to transform under Z h 2 , which results in an effective Z H 2 symmetry for only the first and second generation of exotic states. In the E 6 SSM however all three generations transform under Z H 2 . This application of the Z h 2 symmetry results in the decay products of the first and second generation of exotic states always involving a singlet field S i .
If instead φ h 3 T ∝ (δ 1 , δ 2 , 1) as discussed in section 2.5, then all the ∆ 27 components of the exotic states will mix via the mass terms presented in section 2.6. This results in the same exotic decay channels as used in the E 6 SSM but with some being more suppressed since δ 1 , δ 2 ≪ 1.

Summary
In this Letter we have shown how FCNC's due to models with three families of Higgs fields may be tamed by the same family symmetry which predicts tri-bi-maximal lepton mixing and provides a solution to the SUSY FCNC and CP problems. To be concrete we have focussed on the E 6 SSM where we have shown how the flavour problem can be solved by using a ∆ 27 family symmetry. The three 27 dimensional families of the E 6 SSM, including the three families of Higgs fields, transform in a triplet representation of the ∆ 27 family symmetry, allowing the family symmetry to commute with a possible high energy E 6 symmetry. The ∆ 27 family symmetry breaking considered here, together with a vertical Z h 2 symmetry which does not distinguish between the three families, gives rise effectively to the Z H 2 symmetry of the E 6 SSM, which solves the flavour changing neutral current problem of the three families of Higgs fields. The main phenomenological predictions of the model are tri-bi-maximal mixing for leptons, two almost degenerate LSPs and two almost degenerate families of colour triplet D-fermions, providing a clear prediction for the LHC. In addition the D-term vacuum alignment mechanism described here leads to PGBs with masses below the TeV scale, and possibly much lighter, which appears to be a quite general and robust prediction of any model based on the D-term vacuum alignment mechanism.