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E6 inspired SUSY models with exact custodial symmetry

The breakdown of E_6 gauge symmetry at high energies may lead to supersymmetric (SUSY) models based on the Standard Model (SM) gauge group together with extra U(1)_{\psi} and U(1)_{\chi} gauge symmetries. To ensure anomaly cancellation the particle content of these E_6 inspired models involves extra exotic states that generically give rise to non-diagonal flavour transitions and rapid proton decay. We argue that a single discrete \tilde{Z}^{H}_2 symmetry can be used to forbid tree-level flavor-changing transitions, as well as the most dangerous baryon and lepton number violating operators. We present 5D and 6D orbifold GUT constructions that lead to the E_6 inspired SUSY models of this type. The breakdown of U(1)_{\psi} and U(1)_{\chi} gauge symmetries that preserves E_6 matter parity assignment guarantees that ordinary quarks and leptons and their superpartners, as well as the exotic states which originate from 27 representations of E_6 survive to low energies. These E_6 inspired models contain two dark-matter candidates and must also include additional TeV scale vectorlike lepton or vectorlike down type quark states to render the lightest exotic quark unstable. We examine gauge coupling unification in these models and discuss their implications for collider phenomenology and cosmology.

1 Introduction E 6 inspired models are well motivated extensions of the Standard Model (SM). Indeed, supersymmetric (SUSY) models based on the E 6 gauge symmetry or its subgroup can originate from the ten-dimensional heterotic superstring theory [1]. Within this framework gauge and gravitational anomaly cancellation was found to occur for the gauge groups SO (32) or E 8 × E ′ 8 . However only E 8 × E ′ 8 can contain the SM since it allows for chiral fermions while SO (32) does not. Compactification of the extra dimensions results in the breakdown of E 8 up to E 6 or one of its subgroups in the observable sector [2].
The remaining E ′ 8 couples to the usual matter representations of the E 6 only by virtue of gravitational interactions and comprises a hidden sector that is thought to be responsible for the spontaneous breakdown of local SUSY (supergravity). At low energies the hidden sector decouples from the observable sector of quarks and leptons, the gauge and Higgs bosons and their superpartners. Its only manifest effect is a set of soft SUSY breaking terms which spoil the degeneracy between bosons and fermions within one supermultiplet [3]. The scale of soft SUSY breaking terms is set by the gravitino mass, m 3/2 . In the simplest SUSY extensions of the SM these terms also determine the electroweak (EW) scale. A large mass hierarchy between m 3/2 and Planck scale can be caused by the nonperturbative effects in the hidden sector that may trigger the breakdown of supergravity (SUGRA) [4].
Since E 6 is a rank -6 group the breakdown of E 6 symmetry may result in low energy models based on rank -5 or rank -6 gauge groups, with one or two additional U (1) gauge group factors in comparison to the SM. Indeed, E 6 contains the maximal subgroup SO(10)×U(1) ψ while SO (10) can be decomposed in terms of the SU(5)×U(1) χ subgroup [5]- [6]. By means of the Hosotani mechanism [7] E 6 can be broken directly to which has rank-6. This rank-6 model may be reduced further to an effective rank-5 model with only one extra gauge symmetry U(1) ′ which is a linear combination of U(1) χ and U(1) ψ : In the models based on rank -6 or rank -5 subgroups of E 6 the anomalies are automatically cancelled if the low energy particle spectrum consists of a complete representations of E 6 . Consequently, in E 6 -inspired SUSY models one is forced to augment the minimal particle spectrum by a number of exotics which, together with ordinary quarks and leptons, form complete fundamental 27 representations of E 6 . Thus we will assume that the particle content of these models includes at least three fundamental representations of E 6 at low energies. These multiplets decompose under the SU(5) × U(1) ψ × U(1) χ subgroup of E 6 as follows: The first, second and third quantities in brackets are the SU (5) representation and extra U(1) ψ and U(1) χ charges respectively, while i is a family index that runs from 1 to 3. An ordinary SM family, which contains the doublets of left-handed quarks Q i and leptons L i , right-handed up-and down-quarks (u c i and d c i ) as well as right-handed charged leptons (e c i ), is assigned to 10, Other components of these SU(5) multiplets form colour triplets of exotic quarks D i and D i with electric charges +1/3 and −1/3 respectively. These exotic quark states carry a B − L charge ± 2 3 twice larger than that of ordinary ones. In phenomenologically viable E 6 inspired models they can be either diquarks or leptoquarks.
The presence of the Z ′ bosons associated with extra U(1) gauge symmetries and exotic matter in the low-energy spectrum stimulated the extensive studies of the E 6 inspired SUSY models over the years [5], [8]. Recently, the latest Tevatron and early LHC Z ′ mass limits in these models have been discussed in [9] while different aspects of phenomenology of exotic quarks and squarks have been considered in [10]. Also the implications of the E 6 inspired SUSY models have been studied for EW symmetry breaking (EWSB) [11]- [14], neutrino physics [15]- [16], leptogenesis [17]- [18], EW baryogenesis [19], muon anomalous magnetic moment [20], electric dipole moment of electron [21] and tau lepton [22], lepton flavour violating processes like µ → eγ [23] and CP-violation in the Higgs sector [24]. The neutralino sector in E 6 inspired SUSY models was analysed previously in [13], [21]- [23], [25]- [29]. Such models have also been proposed as the solution to the tachyon problems of anomaly mediated SUSY breaking, via U(1) ′ D-term contributions [30], and used in combination with a generation symmetry to construct a model explaining fermion mass hierarchy and mixing [31]. An important feature of E 6 inspired SUSY models is that the 2 We use the terminology "Inert Higgs" to denote Higgs-like doublets that do not develop VEVs.
mass of the lightest Higgs particle can be substantially larger in these models than in the minimal supersymmetric standard model (MSSM) and next-to-minimal supersymmetric standard model (NMSSM) [14], [32]- [34]. The Higgs sector in these models was examined recently in [29], [32], [35].
Within the class of rank -5 E 6 inspired SUSY models, there is a unique choice of Abelian U(1) N gauge symmetry that allows zero charges for right-handed neutrinos and thus a high scale see-saw mechanism. This corresponds to θ = arctan √ 15. Only in this Exceptional Supersymmetric Standard Model (E 6 SSM) [32]- [33] right-handed neutrinos may be superheavy, shedding light on the origin of the mass hierarchy in the lepton sector and providing a mechanism for the generation of the baryon asymmetry in the Universe via leptogenesis [17]- [18]. Indeed, the heavy Majorana right-handed neutrinos may decay into final states with lepton number L = ±1, thereby creating a lepton asymmetry in the early universe. Since in the E 6 SSM the Yukawa couplings of the new exotic particles are not constrained by neutrino oscillation data, substantial values of the CP-asymmetries can be induced even for a relatively small mass of the lightest right-handed neutrino (M 1 ∼ 10 6 GeV) so that successful thermal leptogenesis may be achieved without encountering a gravitino problem [18].
Supersymmetric models with an additional U(1) N gauge symmetry have been studied in [16] in the context of non-standard neutrino models with extra singlets, in [25] from the point of view of Z − Z ′ mixing, in [13] and [25]- [26] where the neutralino sector was explored, in [13], [36] where the renormalisation group (RG) flow of couplings was examined and in [12]- [14] where EWSB was studied. The presence of a Z ′ boson and of exotic quarks predicted by the Exceptional SUSY model provides spectacular new physics signals at the LHC which were analysed in [32]- [34], [37]. The presence of light exotic particles in the E 6 SSM spectrum also lead to the nonstandard decays of the SM-like Higgs boson that were discussed in details in [38]. Recently the particle spectrum and collider signatures associated with it were studied within the constrained version of the E 6 SSM [39].
Although the presence of TeV scale exotic matter in E 6 inspired SUSY models gives rise to specatucular collider signatures, it also causes some serious problems. In particular, light exotic states generically lead to non-diagonal flavour transitions and rapid proton decay. To suppress flavour changing processes as well as baryon and lepton number violating operators one can impose a set of discrete symmetries. ) and one SM-type singlet field (S ≡ S 3 ) are odd [32]- [33]. When all Z H 2 symmetry violating couplings are small this discrete symmetry allows to suppress flavour changing processes. If the Lagrangian of the E 6 inspired SUSY models is invariant with respect to either a Z L 2 symmetry, under which all superfields except leptons are even (Model I), or a Z B 2 discrete symmetry that implies that exotic quark and lepton superfields are odd whereas the others remain even (Model II), then the most dangerous baryon and lepton number violating operators get forbidden and proton is sufficiently longlived [32]- [33]. The symmetries Z H 2 , Z L 2 and Z B 2 obviously do not commute with E 6 because different components of fundamental representations of E 6 transform differently under these symmetries.
The necessity of introducing multiple discrete symmetries to ameliorate phenomenological problems that generically arise due to the presence of low mass exotics is an undesirable feature of these models. In this paper we consider rank -6 E 6 inspired SUSY models in which a single discreteZ H 2 symmetry serves to simultaneously forbid tree-level flavor-changing transitions and the most dangerous baryon and lepton number violating operators. We consider models where the U(1) ψ and U(1) χ gauge symmetries are spontaneously broken at some intermediate scale so that the matter parity, is preserved. As a consequence the low-energy spectrum of the models will include two stable weakly interacting particles that potentially contribute to the dark matter density of our Universe. The invariance of the Lagrangian with respect to Z M 2 andZ H 2 symmetries leads to unusual collider signatures associated with exotic states that originate from 27plets. These signatures have not been studied in details before. In addition to the exotic matter multiplets that stem from the fundamental 27 representations of E 6 the considered models predict the existence of a set of vector-like supermultiplets. In particular the lowenergy spectrum of the models involves either a doublet of vector-like leptons or a triplet of vector-like down type quarks. If these extra states are relatively light, they will manifest themselves at the LHC in the near future.
The layout of this paper is as follows. In Section 2 we specify the rank-6 E 6 inspired SUSY models with exact custodial symmetry. In Section 3 we present five-dimensional (5D) and six-dimensional (6D) orbifold Grand Unified theories (GUTs) that lead to the rank-6 E 6 inspired SUSY models that we propose. In Sections 4 and 5 the RG flow of gauge couplings and implications for collider phenomenology and cosmology are discussed.
Our results are summarized in Section 6.
2 E 6 inspired SUSY models with exact custodialZ H 2 symmetry In our analysis we concentrate on the rank-6 E 6 inspired SUSY models with two extra U(1) gauge symmetries -U(1) χ and U(1) ψ . In other words we assume that near the GUT or string scale E 6 or its subgroup is broken down to In the next section we argue that this breakdown can be achieved within orbifold GUT models. We also allow three copies of 27-plets to survive to low energies so that anomalies get cancelled generation by generation within each complete 27 i representation of E 6 . In E 6 models the renormalisable part of the superpotential comes from the 27 × 27 × 27 decomposition of the E 6 fundamental representation and can be written as Here the summation over repeated family indexes (i, j, k = 1, 2, 3) is implied. In the considered models B−L number is conserved automatically since the corresponding global symmetry U(1) B−L is a linear superposition of U(1) Y and U(1) χ . At the same time if terms in W 1 and W 2 are simultaneously present in the superpotential then baryon and lepton numbers are violated. In other words one cannot define the baryon and lepton numbers of the exotic quarks D i and D i so that the complete Lagrangian is invariant separately under U(1) B and U(1) L global symmetries. In this case the Yukawa interactions in W 1 and W 2 give rise to rapid proton decay.
Another problem is associated with the presence of three families of H u i and H d i . All these Higgs-like doublets can couple to ordinary quarks and charged leptons of different generations resulting in the phenomenologically unwanted flavor changing transitions.
For example, non-diagonal flavor interactions contribute to the amplitude of K 0 − K 0 oscillations and give rise to new channels of muon decay like µ → e − e + e − . In order to avoid the appearance of flavor changing neutral currents (FCNCs) at the tree level and forbid the most dangerous baryon and lepton number violating operators one can try to impose a singleZ H 2 discrete symmetry. One should note that the imposition of additional discrete symmetry to stabilize the proton is a generic feature of many phenomenologically viable SUSY models.
In our model building strategy we use SU(5) SUSY GUT as a guideline. Indeed, the low-energy spectrum of the MSSM, in addition to the complete SU(5) multiplets, contains an extra pair of doublets from 5 and 5 fundamental representations, that play a role of the Higgs fields which break EW symmetry. In the MSSM the potentially dangerous operators, that lead to the rapid proton decay, are forbidden by the matter parity Z M 2 under which Higgs doublets are even while all matter superfields, that fill in complete SU (5) representations, are odd. Following this inspirational example we augment three 27-plets of E 6 by a number of components M l and M l from extra 27 ′ l and 27 ′ l below the GUT scale. Because additional pairs of multiplets M l and M l have opposite U(1) Y , U(1) ψ and U(1) χ charges their contributions to the anomalies get cancelled identically. As in the case of the MSSM we allow the set of multiplets M l to be used for the breakdown of gauge symmetry. If the corresponding set includes (1)  The superfields M l can be either odd or even under thisZ H 2 symmetry. Depending on whether these fields are even or odd underZ H 2 a subset of terms in the most general renormalizable superpotential can be written as where Y ′ lmn and Y lij are Yukawa couplings and µ ′ il andμ ′ ml are mass parameters. Also one should keep in mind that only M l and M l components of 27 ′ l and 27 ′ l appear below the GUT scale. If M l is odd underZ H 2 symmetry then the termμ ′ ml 27 ′ m 27 ′ l and Ỹ lmn 27 ′ l 27 ′ m 27 ′ n are forbidden while µ ′ il can have non-zero values. When M l is even µ ′ il vanish whereasμ ′ ml 27 ′ m 27 ′ l andỸ lmn 27 ′ l 27 ′ m 27 ′ n are allowed byZ H 2 symmetry. In general mass parameters µ ′ il andμ ′ ml are expected to be of the order of GUT scale. In order to allow some of the M l multiplets to survive to low energies we assume that the corresponding mass terms are forbidden at high energies and get induced at some intermediate scale which is much lower than M X .
The VEVs of the superfields N c H and N c H (that originate from 27 ′ N and 27 ′ N ) can be used not only for the breakdown of U(1) ψ and U(1) χ gauge symmetries, but also to generate Majorana masses for the right-handed neutrinos that can be induced through The non-renormalizable operators (6) give rise to the right-handed neutrino masses which are substantially lower than the VEVs of N c H and N 27 i 27 ′ The mechanism of the gauge symmetry breaking discussed above ensures that the low-energy effective Lagrangian is automatically invariant under the matter parity Z M 2 . Such spontaneous breakdown of the U(1) ψ × U(1) χ gauge symmetry can occur because Z M 2 is a discrete subgroup of U(1) ψ and U(1) χ . This follows from the U(1) ψ and U(1) χ charge assignments presented in Eq. (2). Thus in the considered case the VEVs of N c H and N

27
As a consequence the low-energy effective Lagrangian is invariant under both Z M 2 andZ H 2 discrete symmetries. Moreover theZ H 2 symmetry is a product of where Z E 2 is associated with most of the exotic states. In other words all exotic quarks and squarks, Inert Higgs and Higgsino multiplets as well as SM singlet and singlino states that It is also well known that in SUSY models the lightest supersymmetric particle (LSP), i.e. the lightest R-parity odd particle (Z R 2 = (−1) 3(B−L)+2s ), must be stable. If in the considered models the lightest exotic state (i.e. state with Z E 2 = −1) has even Rparity then the lightest R-parity odd state cannot decay as usual. When the lightest exotic state is R-parity odd particle either the lightest R-parity even exotic state or the next-to-lightest R-parity odd state with Z E 2 = +1 must be absolutely stable. Thus the considered E 6 inspired SUSY models contain at least two dark-matter candidates.
The residual extra U(1) N gauge symmetry gets broken by the VEV of the SM-singlet superfield S (and possibly S). The VEV of the field S induces the mass of the Z ′ associated with U(1) N symmetry as well as the masses of all exotic quarks and inert Higgsinos. If S acquires VEV of order 10 − 100 TeV (or even lower) the lightest exotic particles can be produced at the LHC. This is the most interesting scenario that we are going to focus on here. In some cases the superfield S may also acquire non-zero VEV breaking U(1) N symmetry as we will discuss later. If this is a case then S should be even under theZ H 2 symmetry. Otherwise the superfield S can beZ H 2 odd. The above consideration indicate that the set of multiplets M l has to contain at least H u , H d , S and N c H in order to guarantee the appropriate breakdown of the gauge symmetry in the rank-6 E 6 inspired SUSY models. However if the set ofZ H 2 even supermultiplets M l involve only H u , H d , S and N c H then the lightest exotic quarks are extremely long-lived particles. Indeed, in the considered case theZ H 2 symmetry forbids all Yukawa interactions in W 1 and W 2 that allow the lightest exotic quarks to decay. Moreover the Lagrangian of such model is invariant not only with respect to U(1) L and U(1) B but also under U(1) D symmetry transformations The U(1) D invariance ensures that the lightest exotic quark is very long-lived. The U(1) L , U(1) B and U(1) D global symmetries are expected to be broken by a set of nonrenormalizable operators which are suppressed by inverse power of the GUT scale M X or M P l . These operators give rise to the decays of the exotic quarks but do not lead to the rapid proton decay. Since the extended gauge symmetry in the considered rank-6 E 6 inspired SUSY models forbids any dimension five operators that break U(1) D global symmetry the lifetime of the lightest exotic quarks is expected to be of order of where µ D is the mass of the lightest exotic quark. When µ D ≃ TeV the lifetime of the lightest exotic quarks τ D 10 49 GeV −1 ∼ 10 17 years, i.e. considerably larger than the age of the Universe.
The long-lived exotic quarks would have been copiously produced during the very early epochs of the Big Bang. Those lightest exotic quarks which survive annihilation would subsequently have been confined in heavy hadrons which would annihilate further. The remaining heavy hadrons originating from the Big Bang should be present in terrestrial matter. There are very strong upper limits on the abundances of nuclear isotopes which contain such stable relics in the mass range from 1 GeV to 10 TeV. Different experiments set limits on their relative concentrations from 10 −15 to 10 −30 per nucleon [40]. At the same time various theoretical estimations [41] show that if remnant particles would exist in nature today their concentration is expected to be at the level of 10 −10 per nucleon.
Therefore E 6 inspired models with very long-lived exotic quarks are ruled out.
To ensure that the lightest exotic quarks decay within a reasonable time the set of Z H 2 even supermultiplets M l needs to be supplemented by some components of 27-plet that carry SU(3) C color or lepton number. In this context we consider two scenarios that lead to different collider signatures associated with the exotic quarks. In the simplest case (scenario A) the set ofZ H 2 even supermultiplets M l involves lepton superfields L 4 and/or e c 4 that survive to low energies. This implies that D i and D i can interact with leptons and quarks only while the couplings of these exotic quarks to a pair of quarks are forbidden by the postulatedZ H 2 symmetry. Then baryon number is conserved and exotic quarks are leptoquarks.
In this paper we restrict our consideration to the E 6 inspired SUSY models that lead to the approximate unification of the SU(3) C , SU(2) W and U(1) Y gauge couplings at some high energy scale M X . This requirement implies that in the one-loop approximation the gauge coupling unification is expected to be almost exact. On the other hand it is well known that the one-loop gauge coupling unification in SUSY models remains intact if the MSSM particle content is supplemented by the complete representations of SU (5) (see for example [42]). Thus we require that the extra matter beyond the MSSM fill in complete SU (5)  The supermultiplets L 4 and L 4 are also expected to form vectorlike states. However these states are required to be light enough to ensure that the lightest exotic quarks decay sufficiently fast 3 . The appropriate mass term µ L L 4 L 4 in the superpotential can 3 Note that the superfields e c 4 and e c 4 are not allowed to survive to low energies because they spoil the be induced within SUGRA models just after the breakdown of local SUSY if the Kähler potential contains an extra term (Z L (L 4 L 4 ) + h.c) [43].
The presence of the bosonic and fermionic components of S at low energies is not constrained by the unification of the SU(3) C , SU(2) W and U(1) Y gauge couplings since S is the SM singlet superfield. If S is odd under theZ H 2 symmetry then it can get combined with the superposition of the appropriate components of 27 i . The corresponding vectorlike states may be either superheavy (∼ M X ) or gain TeV scale masses. When S isZ H 2 even superfield then its scalar component is expected to acquire a non-zero VEV breaking U(1) N gauge symmetry.
Thus scenario A implies that in the simplest case the low energy matter content of the considered E 6 inspired SUSY models involves: where the right-handed neutrinos N c i are expected to gain masses at some intermediate scale, while the remaining matter survives down to the EW scale. In Eq. (13) α = 1, 2 and i = 1, 2, 3. Integrating out N c i , N c H and N c H as well as neglecting all suppressed non-renormalisable interactions one gets an explicit expression for the superpotential in the considered case A second scenario, that allows the lightest exotic quarks to decay within a reasonable the following form Integrating out Majorana right-handed neutrinos one obtains in the leading approximation where M N is an effective seesaw scale which is determined by the masses and couplings of N c i andκ ij ∼ g N ij . In the considered case the baryon and lepton number violation takes place only when all three terms in Eqs. (15)-(16) are present in the superpotential. If g N ij = 0 (κ ij = 0) or g q ij = 0 the baryon and lepton number conservation requires exotic quarks to be either diquarks or leptoquarks respectively. When h D ik vanish the conservation of the baryon and lepton numbers implies that the superfields D i , D i preserved. It means that in the leading approximation the proton decay rate is caused by all three types of the corresponding Yukawa couplings and has to go to zero when the Yukawa couplings of at least one type of Yukawa interactions vanish. In practice, the proton lifetime is determined by the one-loop box diagram that leads to the dimension seven operator where H u = v 2 / √ 2 and c ijkl ∝κ g q (h D ) 2 . In Eq. (17) Greek indices denote the color degrees of freedom while SU(2) indices are suppressed. Here we assume that all particles propagating in the loop have masses of the order of M S . For M N 10 11 GeV and h D ik ∼ g q ij ∼ g N ij the appropriate suppression of the proton decay rate can be achieved if the corresponding Yukawa couplings are less than 10 −5 .
Once again, the requirement of the approximate unification of the SU(3) C , SU(2) W and U(1) Y gauge couplings constrains the low energy matter content in the scenario B. survive to the TeV scale as well. As before we assume that these supermultiplets are odd under theZ H 2 symmetry so that they can get combined with the superposition of the corresponding components from 27 i at low energies forming vectorlike states. Again the superfield S may or may not survive to the TeV scale. It can be either even or odd under theZ H 2 symmetry. If S isZ H 2 even, it should survive to low energies and its scalar component is expected to get a VEV.
Following the above discussion the low energy matter content in the simplest case of the scenario B may be summarized as: All states in Eq. (18) are expected to be considerably lighter than the GUT scale M X .
Assuming that N c i , N c H and N c H gain intermediate scale masses the renormalizable part of the TeV scale superpotential associated with the scenario B can be written as The superpotential (19) where ω = e iπ/3 . Then the part of the superpotential that depends on the d c 4 , d c 4 , H u , H d and Φ takes the form At the intermediate scale the imposed Z 6 symmetry may be broken spontaneously by the inducing bilinear mass terms in the superpotential and small Yukawa couplings of the d c 4 supermultiplet to other superfields. The corresponding Yukawa couplings and mass parameters are given by 4 Although scenarios A and B discussed in this section allow us to suppress baryon and lepton number violating operators and non-diagonal flavor transitions they have at least one drawback. Both scenarios imply that a number of incomplete E 6 multiplets survive below the scale M X . In fact, the number of incomplete E 6 multiplets tends to be larger than the number of generations. Therefore the origin and mechanism resulting in the incomplete E 6 representations requires further justification. The splitting of GUT multiplets can be naturally achieved in the framework of orbifold GUTs. In the next section we present 5D and 6D orbifold GUT models that can lead to the scenarios A and B just below the GUT scale.

5D and 6D orbifold GUT models
The structure of the E 6 inspired SUSY models discussed in the previous section, its gauge group and field content, points towards an underlying GUT model based on the E 6 or its subgroup. The breaking of these GUT groups down to the 1) χ is in general rather involved and requires often large Higgs representations. In particular, the splitting of GUT multiplets (like doublet-triplet splitting within SU(5) GUT) requires either fine-tuning of parameters or additional, sophisticated mechanisms [44]- [45].
Higher-dimensional theories offer new possibilities to describe gauge symmetry breaking. A simple and elegant scheme is provided by orbifold compactifications which have been considered for SUSY GUT models in five dimensions [46]- [55] and six dimensions [54]- [59]. These models apply ideas that first appeared in string-motivated work [60]: the gauge symmetry is broken by identifications imposed on the gauge fields under the spacetime symmetries of an orbifold. In these models many good properties of GUT's like gauge coupling unification and charge quantization are maintained while some unsatisfactory properties of the conventional breaking mechanism, like doublet-triplet splitting, are avoided. Recently, orbifold compactifications of the heterotic string have been constructed which can account for the SM in four dimensions and which have five-dimensional or sixdimensional GUT structures as intermediate step very similar to orbifold GUT models [61]. Hence, orbifold compactifications provide an attractive starting point for attempts to embed the SM into higher dimensional string theories.
The simplest GUT group which unifies the gauge interactions of the SM is SU(5) [62].
Therefore we first analyze the higher dimensional SUSY GUT model based on the SU(5) × U(1) χ × U(1) ψ gauge group which is a rank-6 subgroup of E 6 . For simplicity we consider a single compact extra dimension S 1 , y(= x 5 ), and assume a fixed radius with size given by the GUT scale (R ∼ 1/M X ). The orbifold S 1 /Z 2 is obtained by dividing the circle S 1 with a Z 2 transformation which acts on S 1 according to y → −y.
The components of the SU ( Here we consider a 5-dimensional space-time factorized into a product of the ordinary 4D Minkowski space time M 4 and the orbifold which acts as y ′ → −y ′ , with y ′ ≡ y − πR/2. Each reflection symmetry, y → −y and y ′ → −y ′ , has its own orbifold parity, P and P ′ , which are defined by where Φ(x, y) is an SU(5) multiplet field living in the 5D bulk, while P and P ′ are matrix representations of the two Z 2 operator actions which have eigenvalues ±1. All interactions must be invariant under Z 2 × Z ′ 2 symmetry. Each reflection also introduces special points, O and O ′ , located at y = 0 and y = πR/2 ≡ ℓ which are fixed points of the transformations. The equivalences associated with the two reflection symmetries allow to work with the theory obtained by truncating to the physically irreducible interval y ∈ [0, ℓ] with the two 4D walls (branes) placed at the fixed points y = 0 and y = ℓ. These are only two inequivalent branes (the branes at y = πR and y = −πR/2 are identified with those at y = 0 and y = πR/2, respectively). Thus physical space reduces to the interval [0, ℓ] with a length of πR/2.
Denoting the 5D bulk field with (P, P ′ ) = (±1, ±1) by φ ±± one obtains the following Fourier expansions [46]- [49]: where n is a non-negative integer. From the 4D perspective the Fourier component fields and (2n + 2)/R upon compactification. Note that only φ ++ (x, y) and φ +− (x, y) can exist on the y = 0 brane. The fields φ ++ (x, y) and φ −+ (x, y) are non-vanishing on the y = πR/2 brane, whereas the field φ −− (x, y) vanishes on both branes. Only φ ++ (x, y) fields have zero-modes. Since full SU(5) 5D multiplets Φ i (x, y) can, in general, contain components with even and odd parities, P and P ′ , the matter content of the massless sector can be smaller than that of the full 5D multiplet. Unless all components of Φ(x, y) have common parities, the gauge symmetry reduction occurs upon compactification.
As in the case of the simplest orbifold GUT scenarios [46]- [49] we start from the model with the minimal SUSY in 5D (with 8 real supercharges, corresponding to N = 2 in 4D). We assume that the vector supermultiplets associated with the SU(5), U(1) χ and . The 5D gauge supermultiplets contain vector bosons A M (M = 0, 1, 2, 3, 5) and gauginos. The 5D gaugino is composed of two 4D Weyl fermions with opposite 4D chirality, λ and λ ′ . In addition 5D vector supermultiplets have to involve real scalars σ to ensure that the numbers of bosonic and fermionic degrees of freedom are equal. Thus 5D gauge supermultiplets can be decomposed into vector supermultiplets V with physical components (A µ , λ) and chiral multiplets Σ with components (σ + iA 5 )/ √ 2, λ ′ under N = 1 supersymmetry in 4D.
In addition to the 5D vector supermultiplets we assume the presence of other SU (5) representations as well as SU (5) singlet superfields that carry non-zero U(1) χ and U(1) ψ charges in the 5D bulk. The corresponding representations also contain 5D fermions.
Since each 5D fermion state is composed of two 4D Weyl fermions, ψ and ψ c , SUSY implies that each 5D supermultiplet includes two complex scalars φ and φ c as well. The states φ, ψ, φ c and ψ c form one 4D N = 2 hypermultiplet that consists of two 4D N = 1 where index i represents different SU (5) supermultiplets that exist in the bulk . In the case of SU(5) the components of the corresponding N = 2 vector supermultiplet in Eq. (29) where T A is the set of the SU(5) generators (A = 1, 2, ..., 24). The transformations in Eq. (29) are associated with the Z 2 reflection symmetry. By replacing y and P by y ′ and P ′ in Eq. (29) one obtains Z ′ 2 transformations. Note that mass terms for φ i , ψ i , φ c i and ψ c i are allowed by N = 2 SUSY but these terms are not compatible with the P and P ′ parity assignments as follows from Eq. (29). Therefore the zero-modes of these fields do not receive a bulk mass contribution.
It is convenient to choose the matrix representation of the parity assignment P , expressed in the fundamental representation of SU (5), to be P = diag(+1, +1, +1, +1, +1) This boundary condition does not break SU (5) on the O brane at y = 0. However 4D N = 2 supersymmetry gets broken by this parity assignment to 4D N = 1 SUSY. This can be seen explicitly by examining the masses of the Kaluza-Klein (KK) towers of the fields. Indeed, according to the parity assignment P only A µ , λ, φ and ψ are allowed to have zero-modes whereas other components of the N = 2 vector supermultiplet (σ, λ ′ ) and N = 2 hypermultiplets (φ c i , ψ c i ) with odd parity P do not possess massless modes. For the SU (5) (5). This boundary condition breaks not only SU(5) but also 4D N = 2 SUSY to 4D N = 1 SUSY on the O ′ brane at y = ℓ. The parity assignment associated with the Z ′ 2 reflection symmetry leads to the two types of the SU(5) gauge generators T a and Tâ. All generators of the SM gauge group satisfy the condition Therefore the corresponding gauge fields A a µ (x, y) and gauginos λ a (x, y) are even under the reflections Z 2 and Z ′ 2 whereas σ a (x, y) and λ ′ a (x, y) are odd. As a consequence the KK expansions of vector bosons A a µ (x, y) and gauginos λ a (x, y) contain massless zero modes A a(0) µ (x) and λ a(0) (x) corresponding to the unbroken gauge symmetry of the SM. These zero modes form 4D N = 1 vector supermultiplets. The KK modes A a(2n) 5 (x) are swallowed by A a(2n) µ (x) resulting in the formation of vector boson state with mass 2n/R. The KK gaugino modes λ a(2n) (x) and λ ′ a(2n) (x) form 4D fermion state with mass 2n/R.
The other gauge generators Tâ of SU(5) obey the relationship which implies that Aâ µ (x, y) and λâ(x, y) are odd under the Z ′ 2 symmetry while σâ(x, y) and λ ′â (x, y) are even. This means that all components of the 5D vector supermultiplet associated with the broken SU(5) generators Tâ are odd either under the reflection Z 2 or Z ′ 2 so that their KK expansions does not possess massless modes. The Z 2 and Z ′ 2 parity assignments for all components of the 5D bulk vector supermultiplets are shown in Table 2. The KK modes Aâ form vector boson, scalar and fermion states with masses (2n + 1)/R.  (5) Thus orbifold procedure leads to a local explicit breaking of SU(5) at the fixed point O ′ due to the non-trivial orbifold quantum numbers of the gauge parameters.

Mass quantum numbers parity
The Z 2 and Z ′ 2 parity assignments for the components of the U(1) ψ and U(1) χ bulk vector supermultiplets are such that the KK expansions of vector bosons A χ µ (x, y) and A ψ µ (x, y) as well as the corresponding gaugino states λ χ (x, y) and λ ψ (x, y) contain massless zero modes A ψ (x) associated with the unbroken U(1) ψ and U(1) χ gauge symmetries (see Table 2). Other KK modes form vector boson, scalar and fermion states with masses (2n + 2)/R similar to the ones that appear in the case of unbroken generators T a of SU (5).
As in the simplest orbifold GUT scenarios [46]- [48] we assume that all incomplete SU(5) supermultiplets which are even under the custodial symmetry (the matter parity Z M 2 in the case of the MSSM and theZ H 2 symmetry in the case of the E 6 SSM) originate from the 5D bulk supermultiplets. In order to ensure that H u and H u as well as H d and H d survive below the scale M X ∼ 1/R we include two pairs of the 5D SU(5) bulk where first, second and third quantities in brackets are the SU (5) representation, extra U(1) ψ and U(1) χ charges respectively. The multiplets Φ Hu and Φ Hu as well as Φ H d and Φ H d transform differently under Z 2 and Z ′ 2 (see Table 3). Since P ′ does not commute with SU(5) each 5D 5-plet is divided into four pieces associated with different N = 1 chiral supermultiplets: In Eq. (33) first and second quantities in brackets are SU(3) C and SU(2) W quantum numbers whereas the third quantity is U(1) Y charge. As one can see from Table 3 chiral supermultiplets in Eq. (33) According to Eq. (29) only either φ i and ψ i or φ c i and ψ c i can have massless modes. Different parity assignments of Φ S and Φ S as well as Φ N c H and Φ N c H allow to project out different components of these superfields so that only 4D superfields N c H , N c H , S and S may be light (see Table 3).
Finally, the particle spectrum below the scale M X should be supplemented by either L 4 and L 4 or d c 4 and d c 4 (but not both) to allow the lightest exotic quarks to decay. These 4D N = 1 chiral superfields can come from either Φ L 4 and Φ L 4 or Φ d c 4 and Φ d c 4 which are 5D SU(5) bulk supermultiplets with quantum numbers Again parity assignments guarantee that only two 4D doublet superfields L 4 and L 4 from Φ L 4 and Φ L 4 can survive to low-energies whereas the other SU(2) W doublet, color triplet and antitriplet partners do not have zero modes. Using the freedom to flip the overall action of the P ′ parity on the SU(5) multiplets by a sign relative to Φ L 4 + Φ L 4 one can get the KK spectrum in which only triplet or antitriplet components of SU (5) fundamental supermultiplets possess massless modes. From Table 3 one can see that this freedom is used in the case Φ d c 4 and Φ d c 4 supermultiplets. Due to the different structure of the KK spectrum only 4D triplet or antitriplet superfields, d c 4 and d c 4 , from Φ d c 4 and Φ d c 4 are allowed to be light.
Since the three families of 27 i representations of E 6 are located on the O brane, where the SU(5) × U(1) χ × U(1) ψ gauge symmetry remains intact, the Yukawa interactions of quarks and leptons are necessarily SU(5) symmetric. In general the SU (5) 5D fields Mass quantum numbers parity invariance yields the prediction for the first and second generation fermion mass ratios m s /m d = m µ /m e , which is in conflict with the data. In 4D GUTs acceptable mass relations can be obtained using higher dimensional operators and relatively large representations which acquire VEVs breaking SU (5) or SO(10) [45], [63]. In the case of the simplest 5D orbifold GUTs there are no SU(5) breaking VEVs. Nevertheless in this case one can introduce two additional 5D bulk supermultiplets with quantum numbers given by Eq. (35) that transform under Z 2 and Z ′ 2 as either Φ L 4 and Φ L 4 or Φ d c 4 and Φ d c 4 . Furthermore we assume that these bulk supermultiplets are odd underZ H 2 symmetry which is defined on the O brane. Hence the zero modes of these extra 5D supermultiplets, which are either weak doublets (L 5 and L 5 ) or SU(3) C triplet and antitriplet (d c 5 and d c 5 ), can mix with quark or lepton superfields from 27 i spoiling the SU(5) relations between the down type quark and charged lepton masses. Indeed, suppose that zero modes are weak doublet superfields L 5 and L 5 . Then L 5 can get combined with the superposition of lepton doublet superfields from 27 i so that the resulting vectorlike states gain masses slightly below M X . The remaining three families of lepton doublets, that survive to low energies, are superpositions of the corresponding components from 27 i and L 5 while three generations of down type quarks stem from 27 i completely. As a consequence the SU (5) relations between the down type quark and charged lepton masses may get spoiled entirely if the Yukawa couplings of L 5 to Higgs doublet H d are relatively large (∼ 0.01 − 0.1).
Mass quantum numbers parity  (5) bulk supermultiplets Φ e c and Φ e c .
Although the discussed specific realization of the mechanism which allows to obtain the realistic pattern of fermion masses is the simplest one it is worth to consider another very attractive possibility. Instead of two additional 5D SU(5) fundamental supermultiplets one can include two larger representations of SU(5) that decompose under As before we assume that Φ e c and Φ e c supermultiplets are odd underZ H 2 symmetry. Due to P and P ′ parity assignments each SU(5) bulk decuplet is divided into six pieces associated with different N = 1 chiral supermultiplets: where quantities in brackets are SU(3) C , SU(2) W and U(1) Y quantum numbers. The Z 2 and Z ′ 2 parity assignments and mass spectrum for all components of the 5D decuplets are given in Table 4. These parity assignments guarantee that only two 4D SU(2) W singlet superfields (e c 5 and e c 5 ) as well as 4D triplet and antitriplet supermultiplets (u c combined with the superposition of the appropriate components of 27 i forming vectorlike states which may have masses slightly below M X . At the same time e c 5 can mix with the corresponding components of 27 i spoiling the SU(5) relations between the masses of the down type quarks and charged leptons. It is worth noting that together bulk supermultiplets (32), (34), (35) and (36) form two complete 27 representations of E 6 .
This simplifies the structure of bulk supermultiplets making the considered 5D orbifold GUT model more elegant.
For the consistency of the considered model it is crucial that all anomalies get cancelled.
In 5D theories no bulk anomalies exist. Nevertheless orbifold compactification may lead to anomalies at orbifold fixpoints [64]- [65]. At the fixed point brane anomaly reduces to the anomaly of the unbroken subgroup of the original group, i.e. SU (5) It was also shown that the sum of the contributions to the 4D anomalies at the fixpoint equals to the sum of the contributions of the zero modes localized at the corresponding brane [64]- [65]. In this context it is worth to emphasize that the contributions of three families of 27 i representations of E 6 , which reside on the O brane, to the anomalies associated with this fixpoint get cancelled automatically. Moreover from Tables 3 and 4 one can see that the P and P ′ parity assignments are chosen so that the zero modes of the bulk fields localized at the O and O ′ branes always form pairs of N = 1 supermultiplets with opposite quantum numbers. Such choice of parity assignments guarantees that the contributions of zero modes of the bulk superfields to the brane anomalies are cancelled as well.
Another important issue for any GUT model is proton stability which was discussed in the context of 5D orbifold GUT models in [47], [51]- [52]. In orbifold GUT models the dimension five operators, which are caused by an exchange of the color triplet Higgsino multiplets and give rise to proton decay in ordinary GUTs, do not get induced. Indeed, in the considered class of models colored Higgsinos acquire mass via the KK mode expansion of operators ψ i ∂ 5 ψ c i that leads to the Dirac mass terms of the form ψ do not couple directly to the quarks (squarks) and sleptons (leptons) the dimension five operators are not generated. It turns out that the absence of tree-level amplitudes caused by the colored Higgsino exchange which result in proton decay is deeply entangled with the orbifold construction and continuous global U(1) R symmetry that 5D bulk Lagrangian possesses [47]. Although the dimension five operators discussed above do not get induced within orbifold GUT models one must also suppress the brane  (5)-violating coefficients 1/g 2 O ′ i . The part of the 5D effective SUSY Lagrangian that contains kinetic terms for the SM gauge fields can be written as follows where W i α (i = 1, 2, 3) are the supersymmetric gauge field strengths of the U(1) Y , SU(2) W and SU (3) Since SU (5) (5) symmetric, a 4D observer sees an approximate unification of the SM gauge couplings. The gauge coupling unification within 5D orbifold GUT models was discussed in [52]- [53].
As one can see from Eqs. (38)-(39) the discrepancy between g 2 i (M X ) is determined by the SU(5)-violating gauge kinetic terms on the O ′ brane. This discrepancy is small On the other hand one can expect that the relative contribution of the SU(5)-violating brane corrections to

E 6 orbifold GUT model in six dimensions
Having discussed in detail the simplest 5D orbifold GUT model, that may lead at low energies to the gauge group and field content of the E 6 inspired SUSY model specified in section 2, we next study E 6 gauge theory in 6D with N = 1 supersymmetry. We consider the compactification on a torus T 2 with two fixed radii R 5 and R 6 so that two extra dimensions y(= x 5 ) and z(= x 6 ) are compact, i.e. y ∈ (−πR 5 , πR 5 ] and z ∈ (−πR 6 , πR 6 ]. The physical region associated with the compactification on the orbifold T 2 /Z 2 is a pillow with the four fixed points of the Z 2 transformations (y → −y, z → −z) as corners. The orbifold T 2 /Z 2 has the following fixpoints (0, 0), (πR 5 , 0), (0, πR 6 ) and (πR 5 , πR 6 ).
Here we discuss E 6 gauge theory in 6D compactified on the orbifold T 2 /(Z 2 ×Z I 2 ×Z II 2 ). The Z 2 , Z I 2 and Z II 2 symmetries are reflections. The Z 2 transformations are defined as before, i.e. y → −y, z → −z. The Z I 2 reflection symmetry transformations act as y ′ → −y ′ , z → −z with y ′ = y − πR 5 /2. The reflection Z II 2 corresponds to y → −y, z ′ → −z ′ where z ′ = z − πR 6 /2. The Z I 2 and Z II 2 reflection symmetries introduce additional fixed points. As in the case of 5D orbifold GUT models extra reflection symmetries lead to the reduction of the physical region which is again limited by the appropriate fixed points.
The Z 2 , Z I 2 and Z II 2 reflection symmetries allow to work with the theory obtained by truncating to the physically irreducible space in which y ∈ [0, πR 5 /2] and z ∈ [0, πR 6 /2] with the four 4D walls (branes) located at its corners.
Again, we assume that the considered orbifold GUT model contains a set of E 6 bulk supermultiplets and another set of N = 1 superfields which are confined on one of the branes. The set of superfields that propagate in the bulk M 4 ×T 2 /(Z 2 ×Z I 2 ×Z II 2 ) includes E 6 gauge supermultiplet and a few 27-plets. As before all quark and lepton superfields are expected to be confined on one brane.
The E 6 gauge supermultiplet that exist in the bulk must involve vector bosons A M (M = 0, 1, 2, 3, 5, 6) and 6D Weyl fermions (gauginos) which are composed of two 4D Weyl fermions, λ and λ ′ . These fields can be conveniently grouped into vector and chiral multiplets of the N = 1 supersymmetry in 4D, i.e.
where V , A M , λ and λ ′ are matrices in the adjoint representation of E 6 . Two To ensure the consistency of the construction the Lagrangian of the considered orbifold GUT model has to be invariant under Z 2 , Z I 2 and Z II 2 symmetries. As in the case of 5D orbifold GUT models each reflection symmetry, Z 2 , Z I 2 and Z II 2 , has its own orbifold parity, P , P I and P II . The componentsΦ andΦ c of the bulk 27 ′ supermultiplet Φ transform under Z 2 , Z I 2 and Z II 2 as followŝ where P , P I and P II are diagonal matrices with eigenvalues ±1 that act on each component of the fundamental representation of E 6 .
It is convenient to specify the matrix representation of the orbifold parity assignments in terms of the E 6 weights α j and gauge shifts, ∆, ∆ I and ∆ II , associated with Z 2 , Z I 2 and Z II 2 . The diagonal elements of the matrices P , P I and P II can be presented in the following form [55] (P ) jj = σ exp{2πi∆α j } , where σ, σ I and σ II are parities of the bulk 27 ′ supermultiplet, i.e. σ, σ I , σ II ∈ {+, −}.
The particle assignments of the weights in the fundamental representation of E 6 are well known (see, for example [55]). Here we choose the following gauge shifts that correspond to the orbifold parity assignments shown in Table 5.
The components V and Σ of the E 6 gauge supermultiplet transform under Z 2 , Z I 2 and Z II 2 as follows where V (x, y, z) = V A (x, y, z)T A and Σ(x, y, z) = Σ A (x, y, z)T A while T A is the set of generators of the E 6 group. The boundary conditions given by Eqs. (41) and (44) break 4D N = 2 supersymmetry because different components of the N = 2 supermultiplets transform differently under Z 2 , Z I 2 and Z II 2 reflection symmetries. Moreover since P , P I and P II are not unit matrices the E 6 gauge symmetry also gets broken by these parity assignments.
The P parity assignment indicates that on the O brane at y = z = 0 associated with the Z 2 reflection symmetry the E 6 gauge group is broken down to SO(10) × U(1) ψ subgroup. Indeed, according to Table 5 the SO (10)  where the first and second quantities in brackets are the SU (6) and SU(2) W representations respectively. The multiplet (6,2) is formed by all SU(2) W doublets which are contained in 27-plet. From Table 5 one can see that all SU(2) W doublet components of the 27 ′ supermultiplet transform differently under the Z I 2 reflection symmetry as compared with other components of this supermultiplet which form (15,1).
The E 6 gauge symmetry is also broken on the brane O II placed at the fixed point y = 0, z = πR 6 /2 of the Z II 2 symmetry transformations. The P II parity assignment is such that 16 components of the 27 ′ are odd whereas 10 + 1 components are even or viceversa. This implies that E 6 group gets broken down to its SO(10) ′ × U(1) ′ subgroup. It is worth to emphasize here that SO(10) and SO(10) ′ are not the same SO(10) subgroups of E 6 . In particular, from Table 5 one can see that the 16-plets of SO(10) and SO(10) ′ are formed by different components of the fundamental representation of E 6 . The U(1) ψ and U(1) ′ charge assignments should be also different.
In addition to the three branes mentioned above there is a fourth brane located at the corner O III = (πR 5 /2, πR 6 /2) of the physically irreducible space. The Z III 2 reflection symmetry associated with this brane is obtained by combining the three symmetries Z 2 , Z I 2 and Z II 2 defined above. As a consequence the corresponding parity assignment P III = P P I P II . Combining three parity assignments P , P I and P II it is easy to see that on the brane O III the unbroken subgroup is SO(10) ′′ ×Ũ (1).
The unbroken gauge group of the effective 4D theory is given by the intersection of the E 6 subgroups at the fixed points. Since P and P II commute with SU(5) the intersection of the E 6 subgroups SO(10) × U(1) ψ and SO (10) intersection of SU(6) × SU(2) W and SU(5) × U(1) χ × U(1) ψ gives the SM gauge group with two additional U(1) factors, U(1) ψ and U(1) χ .
The mode expansion for the 6D bulk fields φ(x, y, z) with any combinations of parities reads [58]: where n and m are non-negative integers. As follows from Eqs.
where first, second, third and fourth quantities in brackets are the E 6 representation as well as σ, σ I and σ II associated with this representation respectively. The parities of these bulk 27 ′ -plets are chosen so that H u and H u components of the N = 1 chiral superfieldŝ Φ ′ Hu andΦ ′c Hu have positive parities with respect to Z 2 , Z I 2 and Z II 2 reflection symmetries (see Table 5). In this context it is essential to keep in mind that the invariance of the 6D action requires that the parities of the 4D chiral supermultipletsΦ ′ Hu andΦ ′c Hu are opposite. Since the parities of H u and H u are positive the KK expansions of the bulk 27 ′plets Φ ′ Hu and Φ ′ Hu contain zero modes that form N = 1 chiral superfields with quantum numbers of H u and H u .
The SU(2) W doublet chiral superfields H u and H u are not the only supermultiplets from Φ ′ Hu and Φ ′ Hu that may survive below the scale M X ∼ 1/R. Indeed, the parity assignments in Eq. (53) indicate that the u c and e c components of theΦ ′c Hu as well as u c and e c components of theΦ ′ Hu also have positive parities with respect to Z 2 , Z I 2 and Z II 2 symmetries. It means that the KK mode structures of the bulk supermultiplets Φ ′ Hu and Φ ′ Hu involve zero modes that correspond to N = 1 chiral superfields u c , e c , u c and e c . Because the E 6 gauge symmetry is broken down to the SO(10) × U(1) ψ subgroup on the O brane the zero modes that come from the same bulk 27 ′ -plet but belong to different SO(10) representations are not required to have the same transformation properties under the custodialZ H 2 symmetry. This permits us to assume that 4D chiral superfields u c , e c , u c and e c are odd under theZ H 2 symmetry. Then these supermultiplets are expected to mix with the appropriate components from other 27-plets forming vectorlike states with masses slightly below M X and spoiling the SO(10) relations between the Yukawa couplings of quarks and leptons to H u and H d as it is discussed in the previous subsection.
The 4D superfields H d and H d can originate from another pair of bulk 27 ′ -plets Using the orbifold parity assignments presented in Table 5 The S and S components of the N = 1 superfieldsΦ ′ S andΦ ′c S have positive orbifold parities. The D component ofΦ ′ S and the companion component from theΦ ′c S superfield have also positive parities with respect to Z 2 , Z I 2 and Z II 2 symmetries. It is convenient to assume that the states associated with these exotic quark supermultiplets are odd under theZ H 2 symmetry so that the corresponding zero modes can mix with the appropriate components of the 27-plets localised on the O brane leading to the formation of the vectorlike states with masses slightly below M X and spoiling the SO(10) relations between the Yukawa couplings of S to the inert Higgs and exotic quark states. In addition to the components ofΦ ′ S andΦ ′c S mentioned above the orbifold parities of L and L components ofΦ ′c S andΦ ′ S are positive. If the zero modes associated with these components survive to low energies and the corresponding N = 1 supermultiplets are even under theZ H 2 symmetry then the Yukawa couplings of these superfields to Q i and D k allow the lightest exotic quarks to decay like in the case of Scenario A.
The discussion above indicate that the simplest 6D orbifold GUT model based on the E 6 gauge group, which may lead at low energies to the gauge group and field content of the Scenario A specified in section 2, include six bulk 27 ′ -plets. The consistency of this orbifold GUT model requires the absence of anomalies. In the 6D orbifold models there are two types of anomalies: 4D anomalies [67] intrinsic to the fixed points and bulk anomalies [65], [68]- [69] which are induced by box diagrams with four gauge currents. For the 6D orbifold GUT model to be consistent it is necessary that both the fixed point and the bulk anomalies must cancel. The contributions of the anomalous box diagrams with four gauge currents to the 6D bulk anomalies are determined by the trace of four generators of gauge group. This trace contains nonfactorizable part and part which can be reduced to the product of traces of two generators. The nonfactorizable part is associated with the irreducible gauge anomaly while the factorized contribution corresponds to what is known as reducible anomaly. The reducible anomalies can be canceled by the Green-Schwarz mechanism [70]. For the consistency the chiral field content of the 6D orbifold model must lead to the cancellation of the irreducible anomalies which is normally highly restrictive requirement [71]. However 6D orbifold GUT models based on the E 6 gauge group do not have irreducible bulk anomaly [68]- [69]. Moreover using the results obtained in [69] one can show that the reducible gauge anomaly gets cancelled if the field content of the 6D orbifold model involves six bulk 27 ′ -plets. The 4D anomalies at the fixpoints get also cancelled within the 6D orbifold GUT model discussed above. Indeed, the contributions of 27 i supermultiplets, that reside on the O brane, to the anomalies vanish. Since the orbifold parity assignments are such that the KK modes of the bulk 27 ′ superfields localized at the fixpoints always form pairs of N = 1 supermultiplets with opposite quantum numbers the contributions of the bulk 27 ′ -plets to the 4D fixed point anomalies are cancelled automatically as well.
Phenomenological viability of the 5D and 6D orbifold GUT models considered in this section requires the adequate suppression of the baryon and lepton number violating operators which can be induced at the scale M X giving rise to proton decay. As it was mentioned before the dimension five operators, that lead to the proton decay, are forbidden by the gauge symmetry in these models. However baryon and lepton number violating operators, which are mediated by the exchange of the leptoquark gauge bosons, are enhanced compared to the usual 4D case due to the presence of KK towers of such states. The proton decay rate in the 6D orbifold GUT models based on the SO(10) gauge group was studied in [59] where it was shown that in order to satisfy the experimental lower limit on the proton lifetime the scale M X should be larger than 9 · 10 15 GeV. This restriction on the scale M X can be used in the case of the E 6 inspired SUSY models as well. However the analysis of the RG flow of the gauge couplings, which we are going to consider next, indicates that the value of g 2 i (M X ) in these models are 3-5 times larger than in the MSSM. This implies that the lower bound on the scale M X in the considered E 6 inspired models is expected to be 1.5 − 2 · 10 16 GeV. It is worth noting here again that the simplest 5D and 6D orbifold GUT models discussed in this section do not lead to the exact gauge coupling unification at the scale M X due to the brane contributions to the gauge couplings. The relative contribution of these brane corrections is expected to become more sizable with increasing g 2 i (M X ) as it was discussed before. The gauge coupling unification in the 6D orbifold GUT models was considered in [57].

RG flow of gauge couplings in the E 6 SSM
In this section we discuss the RG flow of the SM gauge couplings g i (t) above the EW scale. The running of these couplings between M X and M Z is described by a system of renormalisation group equations (RGEs). To simplify our analysis we assume that U(1) ψ × U(1) χ gauge symmetry is broken down to U(1) N × Z M 2 near the scale M X . This permits us to restrict our consideration to the analysis of the RG flow of four diagonal gauge couplings g 3 (t), g 2 (t), g 1 (t) and g ′ 1 (t) which correspond to SU(3) C , SU(2) W , U(1) Y and U(1) N gauge interactions respectively. Besides the evolution of these gauge couplings is affected by a kinetic term mixing. The mixing effect can be concealed in the interaction between the U(1) N gauge field and matter fields that can be parametrized in terms of off-diagonal gauge coupling g 11 (see [11], [32], [72]). In this framework the RG equations can be written as follows: where t = ln (q/M Z ), q is a renormalisation scale while B and G are 2 × 2 matrices In Eqs. (56)-(57) β i and β 11 are beta functions.
Here we examine the RG flow of gauge couplings in the two-loop approximation. In general the two-loop diagonal β i and off-diagonal β 11 beta functions may be presented as a sum of one-loop and two-loop contributions. However the previous analysis performed in [36] revealed that an off-diagonal gauge coupling g 11 being set to zero at the scale M X remains very small at any other scale below M X . Since it seems to be rather natural to assume that just after the breakdown of the E 6 symmetry there is no mixing in the gauge kinetic part of the Lagrangian between the field strengths associated with the U(1) Y and U(1) N gauge interactions g 11 tends to be substantially smaller than the diagonal gauge couplings. Because of this we can neglect two-loop corrections to the off-diagonal beta function β 11 . In the case of scenario A the one-loop off-diagonal beta function is given by β 11 = − √ 6 5 while in the scenario B β 11 = 3 √ 6 10 .
In the scenario A the two-loop diagonal beta functions β i are given by: where N g is a number of generations forming complete E 6 fundamental representations that the considered model involves at low energies, i.e. N g = 3, whereas n is a number of S and S supermultiplets from 27 ′ S and 27 ′ S that survive to low energies (i.e. n = 0 or 1). Here we assume that the structure of the Yukawa interactions appearing in the superpotential (14) is relatively simple, i.e. λ αβ = λ α δ αβ , and κ ij = κ i δ ij whilef αβ , f αβ , g D ij and h E iα are small and can therefore be ignored (i, j = 1, 2, 3 and α, β = 1, 2). We have also neglected all Yukawa couplings that may be associated with the presence of extra S and S supermultiplets at low energies. In Eqs. (58) h t , h b and h τ are top quark, b-quark and τ -lepton Yukawa couplings respectively. In the limit of n = 0 the RG equations (58) coincide with the ones presented in [36].
In the scenario B the two-loop diagonal beta functions β i can be written in the following form: whereΣ λ = λ 2 1 + λ 2 2 + λ 2 3 + λ 2 . As before we assume relatively simple structure of the Yukawa interactions in the superpotential (19) One can obtain an approximate solution of the two-loop RGEs presented above (see [73]). At high energies this solution for the SM gauge couplings can be written as where are the coefficients of the one-loop beta functions in the E 6 SSM and SM respectively, the third term in the right-hand side of Eq. (60) is the MS → DR conversion factor with C 1 = 0, C 2 = 2, C 3 = 3 [74], while In Eq. (61)  Using the approximate solution of the two-loop RGEs in Eqs. (60)-(61) one can establish the relationships between the values of the gauge couplings at low energies and GUT scale. Then by using the expressions describing the RG flow of α 1 (t) and α 2 (t) it is rather easy to find the scale M X where α 1 (M X ) = α 2 (M X ) = α 0 and the value of the overall gauge coupling α 0 at this scale. Substituting M X and α 0 into the solution of the RGE for the strong gauge coupling one finds the value of α 3 (M Z ) for which exact gauge coupling unification occurs (see [75]): The combined threshold scale T S , that appears in Eq. (62), can be expressed in terms of the effective threshold scales T 1 , T 2 and T 3 . The expression for T S is model-dependent.
In the scenario A T S is given by where µ D i and mD i are the masses of exotic quarks and their superpartners, m Hα and µH α are the masses of Inert Higgs and Inert Higgsino fields, m L and µ L are the masses of the scalar and fermion components of L 4 and L 4 whileM 1 ,M 2 andM 3 are the effective threshold scales in the MSSM In Eqs. (64) Mg and MW are masses of gluinos and winos (superpartners of SU (2)   In general the effective threshold scales derived above can be quite different. Since our purpose is to establish the range of the values of T S andT S that leads to the unification of gauge couplings we shall set these effective threshold scales equal to each other. Then from Eqs. (63) and (65) The results of our numerical studies of the two-loop RG flow of gauge couplings in the case of scenarios A and B are summarized in Figs. 4 and 4 respectively. We use the two-loop SM beta functions to describe the running of gauge couplings between M Z and T 1 = T 2 = T 3 = T S (orT 1 =T 2 =T 3 =T S ), then we apply the two-loop RGEs of the E 6 SSM to compute the flow of g i (t) from T S (orT S ) to M X which is equal to 3 · 10 16 GeV in the case of the E 6 SSM. The low energy values of g ′ 1 and g 11 are chosen so that all four diagonal gauge couplings are approximately equal near the GUT scale and g 11 = 0 at this scale. For the calculation of the evolution of Yukawa couplings a set of one-loop RGEs is used. The corresponding one-loop RG equations are specified in [32].
In Fig. 4 we fix the effective threshold scale to be equal to 400 GeV. In Fig. 1a we plot the running of the gauge couplings from M Z to M X assuming that the low energy matter content involves three 27-plets of E 6 as well as L 4 , L 4 , S and S supermultiplets. where it is rather problematic to get the exact unification of gauge couplings [73], [76]- [77]. Indeed, it is well known that in order to achieve gauge coupling unification in the MSSM with α s (M Z ) ≃ 0.118, the combined threshold scale, which is given by [73], [75], [77]- [78] must be aroundM S ≈ 1 TeV. However the correct pattern of EW symmetry breaking requires µ to lie within the 1 − 2 TeV range which impliesM S < 200 − 300 GeV, so that, ignoring the effects of high energy threshold corrections, the exact gauge coupling unification in the MSSM requires significantly higher values of α 3 (M Z ), well above the experimentally measured central value [73], [75], [77]- [79]. It was argued that it is possible to get the unification of gauge couplings in the minimal SUSY model for α 3 (M Z ) ≃ 0.123 [80].
On the other hand in the case of scenario A the combined threshold scale T S can be substantially larger than in the MSSM. This can be seen directly from the explicit expression for T S . Combining Eqs. (63) From Eq. (67) it is obvious that T S is determined by the masses of the scalar and fermion components of L 4 and L 4 . The term µ L L 4 L 4 in the superpotential (14) is not involved in the process of EW symmetry breaking. As a consequence the parameter µ L remains arbitrary 7 . In particular, since the corresponding mass term is not suppressed by the E 6 symmetry the components of the doublet superfields L 4 and L 4 may be much heavier than the masses of all exotic states resulting in the large combined threshold scale T S that lies in a few hundred GeV range even when scaleM S is relatively low. The large range of variation of T S allows to achieve the exact unification of gauge couplings in the scenario A for any value of α 3 (M Z ) which is in agreement with current data.
It is worth noting here that, in principle, one could naively expect that large two-loop substantially larger as compared to the ones in the MSSM. Nevertheless the results of our analysis discussed above are not as surprising as they may first appear. The analysis of the RG flow of the SM gauge couplings performed in [36] revealed that the two-loop corrections to α i (M X ) are a few times bigger in the E 6 SSM than in the MSSM. At the same time due to the remarkable cancellation of different two-loop corrections the absolute value of Θ s is more than three times smaller in the E 6 SSM as compared with the MSSM.
This cancellation is caused by the structure of the two-loop corrections to the diagonal beta functions in the considered model. As a result, the prediction for the value of α 3 (M Z ) at which exact gauge coupling unification takes place is considerably lower in the E 6 SSM than in the MSSM.
The only difference between the E 6 SSM scenario, which was studied in [36], and scenario A discussed above is in the possible presence of extra S and S supermultiplets at low energies. From Eqs. (58) it follows that these supermultiplets do not contribute to the diagonal beta functions of the SM gauge couplings. Our analysis of the RG flow of g i (t) reveals that the evolution of the SM gauge couplings does not change much when the low energy particle spectrum is supplemented by the bosonic and fermionic components that originate from the extra S and S chiral superfields. This explains why our results are so similar to those previously obtained in [36].
It is also worthwhile to point out that at high energies the uncertainty in α 3 (t) caused The effective threshold scale that we consider in our analysisT S is in the multi TeV range. At first glance, it is not clear if so large values ofT i andT S can be obtained for a reasonable set of parameters. In particular, to satisfy naturalness requirements the third generation sfermions as well as neutralino and chargino states which are superparners of the SM gauge bosons and Higgs fields are expected to have masses below 1 TeV. Because of this in the MSSM naturalness arguments constrain the combined threshold scaleM S to be lower than 200 − 300 GeV as it was mentioned above. In the case of scenario B the analytical expression for the threshold scaleT S can be obtained by combining Eqs. (65) that givesT Eq. (68) indicates that the combined threshold scaleT S tends to be very large if, for  The parameters and notations are the same as in Fig. 4. of these couplings are larger in the scenario B as compared to the ones in the scenario A. As a consequence the interval of variations of α 3 (t) at high energies is also a bit bigger in the former than in the latter. However as one can see from Figs As before extra S and S superfields, that may survive to low energies, do not contribute to the diagonal beta functions of the SM gauge couplings and, therefore, do not change much the RG flow of α i (t). As a result the value of α 3 (M Z ) at which exact gauge coupling unification takes place does not change much as well after the inclusion of the bosonic and fermionic components of these supermultiplets. Thus it seems to be rather difficult to reconcile the unification of gauge couplings with present data in the Scenario B.
Nevertheless the values of α i (M X ) are not so much different from each other. From Fig. 2b it follows that the relative discrepancy of α i (M X ) is about 10% . This brings us back to the orbifold GUT framework which was discussed in the previous section. As it has been already mentioned orbifold GUTs do not imply the exact gauge coupling unification near the scale M X , which is associated with the size of compact extra dimensions, due to the brane contributions to the gauge couplings (see Eq. (39)). Since one can expect that these brane corrections become more sizable when α i (M X ) are large, the relative discrepancy of 10% between α i (M X ) should not be probably considered as a big problem in the case of scenario B.

Phenomenological implications
We now consider cosmological implications and collider signatures of the E 6 inspired SUSY models discussed above. The phenomenological implications of these models are determined by the structure of the particle spectrum that can vary substantially depending on the choice of the parameters. For example, the masses of the Z ′ boson, exotic quarks, Inert Higgsinos and Inert singlinos are set by the VEVs of the Higgs fields. In this section we primarily focus on the simplest case when only H u , H d and S acquire non-zero VEVs breaking SU(2) W ×U(1) Y ×U(1) N symmetry to U(1) em associated with electromagnetism.
Assuming that f αβ andf αβ are sufficiently small the masses of the exotic quarks, Inert Higgsino states and Z ′ boson are given by where s is a VEV of the field S, i.e. S = s/ √ 2. Here without loss of generality we set κ ij = κ i δ ij and λ αβ = λ α δ αβ . Since µ D i , µ Hα and M Z ′ are determined by s, that remains a free parameter, the Z ′ boson mass and the masses of exotic quarks and Inert Higgsinos cannot be predicted. Because recent measurements from the LHC experiments exclude E 6 inspired Z ′ with masses lower than 2 − 2.15 TeV [81] the singlet field S must acquire a large VEV (s 5.5 − 6 TeV) to induce sufficiently large M Z ′ . The couplings κ i should be also large enough to ensure that the exotic fermions are sufficiently heavy to avoiding conflict with direct particle searches at present and former accelerators. However the exotic fermions (quarks and Inert Higgsinos) can be relatively light in the E 6 SSM. This happens, for example, when the Yukawa couplings of the exotic particles have hierarchical structure similar to the one observed in the ordinary quark and lepton sectors. Then Z ′ mass lie beyond 10 TeV and the only manifestation of the considered models may be the presence of light exotic quark and/or Inert Higgsino states in the particle spectrum.
Since the qualitative pattern of the particle spectrum and associated collider signatures are so sensitive to the parameter choice it is worth to discuss first the robust predictions that the considered models have. It is well known that SUSY models predict that the mass of the lightest Higgs particle is limited from above. The E 6 SSM is not an exception. In the simplest case when only H u , H d and S develop the VEVs, so that , the Higgs sector involves ten degrees of freedom. However four of them are massless Goldstone modes which are swallowed by the W ± , Z and Z ′ gauge bosons that gain non-zero masses. If CP-invariance is preserved the other degrees of freedom form two charged, one CP-odd and three CP-even Higgs states. When the SUSY breaking scale is considerably larger than the EW scale, the mass matrix of the CP-even Higgs sector has a hierarchical structure and can be diagonalised using the perturbation theory [82]- [83]. In this case the mass of one CP-even Higgs particle is always very close to the Z ′ boson mass M Z ′ . The masses of another CP-even, the CP-odd and the charged Higgs states are almost degenerate. When λ g ′ 1 , the qualitative pattern of the Higgs spectrum is rather similar to the one which arises in the PQ symmetric NMSSM [83]- [84]. In the considered limit the heaviest CP-even, CP-odd and charged states are almost degenerate and lie beyond the TeV range [32]. Finally, like in the MSSM and NMSSM, one of the CP-even Higgs bosons is always light irrespective of the SUSY breaking scale. However, in contrast with the MSSM, the lightest Higgs boson in the E 6 SSM can be heavier than 110 − 120 GeV even at tree level. In the two-loop approximation the lightest Higgs boson mass does not exceed 150 − 155 GeV [32].

Dark matter
The structure of the Yukawa interactions in the E 6 SSM leads to another important prediction. Using the method proposed in [85] one can argue that there are theoretical upper bounds on the masses of the lightest and second lightest inert neutralino states [38]. To simplify the analysis we assume that the fermion components of the supermultiplets S, H u and H d , which may survive below the scale M X , get combined with the corresponding superpositions of the fermion components of the superfields S i , H u i and H d i resulting in a set of heavy vectorlike states. Furthermore we also assume that these vectorlike states completely decouple so that the particle spectrum below the TeV scale contains only two generations of inert Higgsinos (H u α andH d α ) and two generations of inert singlinosS α . The Yukawa interactions of these superfields are described by the superpotential where α, β = 1, 2 .
Thus below the TeV scale the inert neutralino states are linear superposition of the inert singlino states (S 1 ,S 2 ) and neutral components of inert Higgsinos (H d0 1 ,H d0 2 ,H u0 1 , H u0 2 ). The charged components of the inert Higgsinos (H u+ 2 ,H u+ 1 ,H d− 2 ,H d− 1 ), form inert chargino sector. In order to avoid the LEP lower limit on the masses of inert charginos the couplings λ αβ and s must be chosen so that all inert chargino states are heavier than 100 GeV. In addition, the requirement of the validity of perturbation theory up to the GUT scale constrains the allowed range of Yukawa couplings λ αβ , f αβ andf αβ . The restrictions specified above set very stringent limits on the masses of two lightest inert neutralinos. The analysis performed in [38] indicates that the lightest and second lightest inert neutralinos (H 0 1 andH 0 2 ) are typically lighter than 60 − 65 GeV. These neutralinos are predominantly inert singlinos so that they can have rather small couplings to the Z-boson. Therefore any possible signal which these neutralinos could give rise to at LEP would be extremely suppressed. On the other hand the couplings of χ 0 1 and χ 0 2 to the lightest CP-even Higgs boson h 1 are proportional to the mass/ v 2 1 + v 2 2 in the leading approximation [38]. As a consequence the couplings of two lightest inert neutralino to the lightest Higgs state are always large if the corresponding states have appreciable masses.
The discussion above indicates that the lightest and second lightest inert neutralinos tend to be the lightest states which are odd under the Z E 2 symmetry. It is worth to remind here that in the considered E 6 inspired SUSY models U(1) ψ × U(1) χ gauge symmetry is is the so-called matter parity which is a discrete subgroup of U(1) ψ and U(1) χ . Since the low-energy effective Lagrangian is Table 1), the Z E 2 symmetry is also conserved. This means that the lightest exotic state, which is odd under the Z E 2 symmetry, is absolutely stable and contributes to the relic density of dark matter. Because the lightest inert neutralino is also the lightest R-parity odd state either the lightest R-parity even exotic state or the lightest R-parity odd state with Z E 2 = +1 must be absolutely stable. When f αβ andf αβ are large enough (f αβ ∼f αβ ∼ 0.5) the large mixing in the inert Higgs sector may lead to the lightest CP-even (or CP-odd) inert Higgs state with mass of the order of the EW scale. The corresponding exotic state is R-parity even neutral particle. If it is substantially lighter than the lightest ordinary neutralino state χ 0 1 and the decay of χ 0 1 into the lightest inert neutralino and the lightest inert Higgs scalar (pseudoscalar) is kinematically allowed then this lightest inert Higgs scalar (pseudoscalar) is absolutely stable and may result in considerable contribution to the relic dark matter density.
Although the possibility mentioned above looks very attractive a substantial finetuning is normally required to make the lightest inert Higgs scalar (pseudoscalar) lighter than χ 0 1 . Most commonly χ 0 1 is considerably lighter than the lightest inert Higgs scalar (pseudoscalar) so that the lightest CP-even (CP-odd) inert Higgs state can decay into χ 0 1 and the lightest inert neutralino state. In other words, in the considered E 6 inspired SUSY models the lightest R-parity odd state with Z E 2 = +1, i.e. χ 0 1 , tend to be substantially lighter than the R-parity even exotic states. As a result the lightest neutralino state χ 0 1 is a natural candidate for a cold component of dark matter in these models.
In the neutralino sector of the E 6 SSM there are two extra neutralinos besides the four MSSM ones. One of them is an extra gauginoB ′ coming from the Z ′ vector supermultiplet.
The other one is an additional singlinoS which is a fermion component of the SM singlet superfield S. Extra neutralinos form two eigenstates (B ′ ±S)/ √ 2 with masses around M Z ′ [32]. Since LHC experiments set very stringent lower bound on the mass of the Z ′ boson extra neutralino eigenstates tend to be the heaviest ones and decouple. The mixing between these heavy neutralino states and other gauginos and Higgsinos is very small.
Therefore the lightest neutralino states in the E 6 SSM, that determine the composition of χ 0 1 and as a consequence its contribution to the relic dark matter density, become almost indistinguishable from the ones in the MSSM. This means that in the E 6 SSM, like in the MSSM, the lightest neutralino χ 0 1 can give a substantial contribution to the relic density which is in agreement with the measured abundance of cold dark matter Ω CDM h 2 = 0.1099 ± 0.0062 [86].
In the E 6 SSM the lightest inert neutralino can account for all or some of the observed cold dark matter relic density if χ 0 1 has mass close to half the Z mass. In this case the lightest inert neutralino states annihilate mainly through an s-channel Z-boson, via its Inert Higgsino doublet components which couple to the Z-boson [38], [87]. When |mH0 1 | ≪ M Z the lightest inert neutralino states are almost inert singlinos and the couplings ofH 0 1 to gauge bosons, Higgs states, quarks (squarks) and leptons (sleptons) are quite small leading to a relatively small annihilation cross section forH 0 1H 0 1 → SM particles. Since the dark matter number density is inversely proportional to the annihilation cross section at the freeze-out temperature the lightest inert neutralino state with mass |mH0 1,2 | ≪ M Z gives rise to a relic density which is typically much larger than its measured value 8 .
Because the scenarios with |mH0 1,2 | ∼ M Z /2 imply that the couplings ofH 0 1 andH 0 2 to the lightest Higgs boson are much larger than the b-quark Yukawa coupling the lightest Higgs state decays more than 95% of the time intoH 0 1 andH 0 2 in these cases while the total branching ratio into SM particles varies from 2% to 4% [38]. At the same time the LHC production cross section of the lightest Higgs state in the considered E 6 inspired SUSY models is almost the same as in the MSSM. Therefore the evidence for the Higgs boson recently presented by ATLAS [90] and CMS [91] indicates that the corresponding scenarios are basically ruled out.
In this context one should point out another class of scenarios that might have interesting cosmological implications. Let us consider a limit when f αβ ∼f αβ ∼ 10 −5 . So small values of the Yukawa couplings f αβ andf αβ result in extremely light inert neutralino statesH 0 1 andH 0 2 which are basically inert singlinos. These states have masses about 1 eV. SinceH 0 1 andH 0 2 are so light and absolutely stable they form hot dark matter in the Universe 9 . These inert neutralinos have negligible couplings to Z boson and would not have been observed at earlier collider experiments. These states also do not change the branching ratios of the Z boson and Higgs decays. Moreover if Z ′ boson is sufficiently heavy the presence of such light Inert neutralinos does not affect Big Bang Nucleosynthesis [88]. When the masses ofH 0 1 andH 0 2 are about 1 eV these states give only a very minor contribution to the dark matter density while the lightest neutralino may account for all or some of the observed dark matter density. In this case one can expect that the 8 When f αβ ,f αβ → 0 the masses ofH 0 1 andH 0 2 tend to zero and inert singlino states essentially decouple from the rest of the spectrum. In this limit the lightest non-decoupled Inert neutralino may be rather stable and can play the role of dark matter [88]. The presence of very light neutral fermions in the particle spectrum might have interesting implications for the neutrino physics (see, for example [89]). 9 In the context of E 6 inspired SUSY models warm dark matter was recently discussed in [92].
lifetime of the next-to-lightest exotic state (for example, inert chargino) is given by where f αβ ∼f αβ ∼ f and M N LES is the mass of the next-to-lightest exotic state. Assuming that M N LES ∼ 1 TeV we get τ N LES ∼ 10 −15 s. With increasing f αβ andf αβ the masses of the lightest inert neutralino states grow and their contribution to the relic density of dark matter becomes larger. This may lead to some interesting cosmological implications.
The detailed study of these implications is beyond the scope of this paper and will be considered elsewhere.

LHC signatures
We can now turn to the possible collider signatures of the E 6 inspired SUSY models with exact custodialZ H 2 symmetry. The presence of Z ′ boson and exotic multiplets of matter in the particle spectrum is a very peculiar feature that may permit to distinguish the Collider experiments and precision EW tests set stringent limits on the mass of the Z ′ boson and Z −Z ′ mixing. The direct searches at the Fermilab Tevatron (pp → Z ′ → l + l − ) exclude Z ′ , which is associated with U(1) N , with mass below 892 GeV [9] 10 . Recently AT-LAS and CMS experiments ruled out E 6 inspired Z ′ with masses lower than 2 − 2.15 TeV [81]. The analysis performed in [94] revealed that Z ′ boson in the E 6 inspired models can be discovered at the LHC if its mass is less than 4 − 4.5 TeV. The determination of its couplings should be possible if M Z ′ 2 − 2.5 TeV [95]. The precision EW tests bound the Z − Z ′ mixing angle to be around [−1.5, 0.7] × 10 −3 [96]. Possible Z ′ decay channels in E 6 inspired supersymmetric models were studied in [9], [28]. The potential influence of gauge kinetic mixing on Z ′ production at the 7 TEV LHC was considered in [97]. 10 Slightly weaker lower bound on the mass of the Z ′ N boson was obtained in [93].
The production of a TeV scale exotic states will also provide spectacular LHC signals.
Several experiments at LEP, HERA, Tevatron and LHC have searched for colored objects that decay into either a pair of quarks or quark and lepton. But most searches focus on exotic color states, i.e leptoquarks or diquarks, have integer-spin. So they are either scalars or vectors. These colored objects can be coupled directly to either a pair of quarks or to quark and lepton. Moreover it is usually assumed that leptoquarks and diquarks have appreciable couplings to the quarks and leptons of the first generation. The most stringent constraints on the the masses of leptoquarks come from the nonobservation of these exotic color states at the ATLAS and CMS experiments. Recently ATLAS collaboration ruled out first and second generation scalar leptoquarks (i.e. leptoquarks that couple to the first and second generation fermions respectively) with masses below 600 − 700 GeV [98].
The CMS collaboration excluded first and second generation scalar leptoquarks which are lighter than 640 − 840 GeV [99]. The experimental lower bounds on the masses of dijet resonances (in particular, diquarks) tend to be considerably higher (see, for example, [100]).
However the LHC lower bounds on the masses of exotic quarks mentioned above are not directly applicable in the case of the E 6 inspired SUSY models considered here. Since Z E 2 symmetry is conserved every interaction vertex contains an even number of exotic states. As a consequence each exotic particle must eventually decay into a final state that contains at least one lightest Inert neutralino (or an odd number of the lightest Inert neutralinos). Since stable lightest Inert neutralinos cannot be detected directly each exotic state should result in the missing energy and transverse momentum in the final state. The Z E 2 symmetry conservation also implies that in collider experiments exotic particles can only be created in pairs.
In this context let us consider the production and sequential decays of the lightest exotic quarks at the LHC first. Because D and D states are odd under the Z E 2 symmetry they can be only pair produced via strong interactions. In the scenario A the lifetime and decay modes of the lightest exotic quarks are determined by the operators g D ij (Q i L 4 )D j and h E iα e c i (H d α L 4 ) in the superpotential (14). These operators ensure that the lightest exotic quarks decay into where ℓ is either electron or muon. Here X may contain extra charged leptons that can originate from the decays of intermediate states (like Inert chargino or Inert neutralino).
Since lightest exotic quarks are pair produced these states may lead to a substantial enhancement of the cross section pp → jjℓ + ℓ − + E miss T + X if they are relatively light.
In the scenario B the decays of the lightest exotic quarks are induced by the operators . As a consequence the lightest diquarks decay into where X again can contain charged leptons that may come from the decays of intermediate states.
In this case the presence of light D-fermions in the particle spectrum could result in an appreciable enhancement of the cross section pp → jjjj + E miss T + X.
In general exotic squarks are expected to be substantially heavier than the exotic quarks because their masses are determined by the soft SUSY breaking terms. Nevertheless the exotic squark associated with the heavy exotic quark maybe relatively light.
Indeed, as in the case of the superpartners of the top quark in the MSSM, the large mass of the heaviest exotic quark in the E 6 SSM gives rise to the large mixing in the corresponding exotic squark sector that may result in the large mass splitting between the appropriate mass eigenstates. As a consequence the lightest exotic squark can have mass in TeV range. Moreover, in principle, the lightest exotic squark can be even lighter than the lightest exotic quark. If this is a case then the decays of the lightest exotic squark are induced by the same operators which give rise to the decays of the lightest exotic quarks when all exotic squarks are heavy. Therefore the decay patterns of the lightest exotic color states are rather similar in both cases. In other words when exotic squark is the lightest exotic color state in the particle spectrum it decays into either if exotic squark is a scalar leptoquark or if it is a scalar diquark. Due to the Z E 2 symmetry conservation E miss T should always contain contribution associated with the lightest exotic particle. However since the lightest exotic squark is R-parity even state whereas the lightest Inert neutralino is R-parity odd particle the final state in the decay ofD should also involve the lightest neutralino to ensure that R-parity is conserved. Again, X may contain charged leptons that can stem from the decays of intermediate states. Because the Z E 2 symmetry conservation implies that the lightest exotic squarks can be only pair produced in the considered case the presence of lightD is expected to lead to an appreciable enhancement of the cross section of either pp → jjℓ + ℓ − + E miss T + X ifD is scalar leptoquark or pp → jjjj + E miss T + X ifD is scalar diquark.
Thus one can see that in both scenarios when the lightest exotic color state is either D-fermion orD-scalar the collider signatures associated with these new states are rather similar. Moreover since the decays of the lightest exotic color particles lead to the missing energy and transverse momentum in the final state it might be rather problematic to distinguish the corresponding signatures from the ones which are associated with the MSSM. For example, the pair production of gluinos at the LHC should also result in the enhancement of the cross section of pp → jjjj + E miss T + X. In this context the presence of additional charged leptons in X can play an important role leading to characteristic signatures such as ℓ + ℓ − pairs together with large missing energy in the final state. The situation also becomes a bit more promising if one assumes that the Yukawa couplings of the exotic particles have hierarchical structure similar to the one observed in the ordinary quark and lepton sectors. In this case all states which are odd under the Z E 2 symmetry couple to the third generation fermions and sfermions mainly 11 . As a consequence the presence of the relatively light exotic color states should give rise to the enhancement of the cross section of either pp → ttℓ + ℓ − + E miss Here it is worthwhile to point out that the collider signatues associated with the light scalar leptoquarks or diquarks in the considered E 6 inspired SUSY models are very different from the commonly established ones which have been thoroughly studied. For instance, it is expected that scalar diquarks may be produced singly at the LHC and decay into quark-quark without missing energy in the final state. The scalar leptoquarks can be only pair produced at the LHC but it is commonly assumed that these states decay into quark-lepton without missing energy as well. On the other hand in the E 6 inspired SUSY models considered here the Z E 2 symmetry conservation necessarily leads to the missing energy and transverse momentum in the corresponding final state.
The presence of relatively light exotic quark and squark can substantially modify the collider signatures associated with the production and decay of gluinos 12 . Indeed, if all squarks except the lightest exotic squark are rather heavy and the decay of the gluino into exotic quark and squark are kinematically allowed then the gluino pair production at the LHC results in DDDD in the corresponding final state. The sequential decays of exotic quarks and squarks give rise to the enhancement of either pp → 4 ℓ + 4 j + E miss T + X if exotic color states are leptoquarks or pp → 8 j + E miss T + X if exotic color states are diquarks, modulo of course effects of QCD radiation and jet merging. The modification of the gluino collider signatures discussed above might be possible only if there are nonzero flavor-off-diagonal couplings θ g ij of gluino to D i andD j (i = j). This is a necessary condition because the lightest exotic squark is normally associated with the heaviest exotic quark. Rough estimates indicate that the corresponding modification of the gluino collider signatures can occur even when the gluino flavour-off-diagonal couplings θ g ij are relatively small, i.e. θ g ij 0.01. 11 This possibility was discussed at length in [32]- [34], [39]. 12 Novel gluino decays in the E 6 inspired models were recently considered in [101] If gluino is heavier than the lightest exotic color state, but is substantially lighter than the second lightest exotic color state than the branching ratios of the nonstandard gluino decays mentioned above are suppressed. In this case the second lightest exotic color state can decay mostly into the lightest exotic color state and gluino if the corresponding decay channel is kinematically allowed. This happens when the lightest exotic color state is exotic D-fermion while the second lightest exotic color state isD-scalar or vice versa.
where q i can be either up-type or down-type quark while X may contain charged leptons which can appear as a result of the decays of intermediate states. As in the case of exotic squark the final state in the decay of d 4 should contain the lightest neutralino and the lightest Inert neutralino to ensure the conservation of R-parity and Z E 2 symmetry. Again due to the Z E 2 symmetry conservation d 4 andd 4 can be only pair produced at the LHC resulting in an enhancement of pp → jj + E miss T + X. If d 4 andd 4 couple predominantly to the third generation fermions and sfermions then the pair production of these quarks/squarks should lead to the presence of two heavy quarks in the final state.
As before these collider signatures do not permit to distinguish easily the considered E 6 inspired SUSY models from other supersymmetric models. For example, squark pair production at the LHC can also lead to two jets and missing energy in the final state.
Again, the presence of additional charged leptons in X can lead to the signatures that may help to distinguish the considered E 6 inspired SUSY models from the simplest SUSY extensions of the SM.
In the case of scenario A the fermionic components of the supermultiplets L 4 and L 4 as well as one of the superpartners of this vectorlike state may have masses below the TeV scale. If all other exotic states and sparticles are rather heavy the corresponding bosonic (L 4 ) and fermionic (L 4 ) states can be produced at the LHC via weak interactions only.
Because of this their production cross section is relatively small. In the considered limit the decays of L 4 and/orL 4 are induced by the operators h E iα e c i (H d α L 4 ). As a consequence the decays of L 4 and/orL 4 always lead to either τ -lepton or electron/muon as well as missing energy in the final state. In the case ofL 4 decays the missing energy in the final state can be associated with only one lightest Inert neutralino whereas the final state of the L 4 decays must contain at least one lightest Inert neutralino and one lightest ordinary neutralino to ensure the conservation of R-parity and Z E 2 symmetry. More efficiently L 4 and/orL 4 can be produced through the decays of the lightest exotic color states (i.e. D and/orD) if these states are relatively light and the corresponding decay channels are kinematically allowed.
The Inert Higgs bosons and/or Inert neutralino and chargino states, which are predominantly Inert Higgsinos, can be also light or heavy depending on their free parameters.
Indeed, as follows from Eq. (69)  have masses below 10 TeV. However the situation may change dramatically whenZ H 2 even superfield S survive to low energies. In order to demonstrate this, let us consider a simple toy model, where U(1) N gauge symmetry is broken by VEVs of a pair of SM singlet superfields S and S. Assuming that the superpotential of the considered model involves bilinear term µ S SS the part of the tree-level scalar potential, which depends on the scalar components of the superfields S and S only, can be written as where m 2 S , m 2 S and B S are soft SUSY breaking parameters and Q S is a U(1) N charge of the SM singlet superfields S. The last term in Eq. (72), which is the U(1) N D-term contribution to the scalar potential, forces the minimum of the corresponding potential to be along the D-flat direction S = S . Indeed, in the limit S = S the quartic terms in the potential (72) vanish. In the considered case the scalar potential (72)  The scalar potential can be easily stabilized the if bilinear term µ S SS in the superpotential is replaced by whereφ isZ H 2 even superfield that does not participate in the λ 0 ≪ 0.1) the U(1) N D-term contribution to the scalar potential still forces the minimum of the scalar potential to be along the nearly D-flat direction if m 2 S + m 2 S < 0. This condition can be satisfied because sufficiently large values of κ i affect the evolution of m 2 S rather strongly resulting in negative values of m 2 S at low energies [39]. If m 2 S + m 2 S < 0 and λ 0 is small then the scalar components of the superfieldsφ, S and S acquire very large VEVs, i.e. φ ∼ S ≃ S ∼ M SU SY /λ 0 , where M SU SY is a supersymmetry breaking scale. If λ 0 ≃ 10 −3 −10 −4 the VEVs of the SM singlet superfields S and S are of the order of 10 3 − 10 4 TeV even when M SU SY ∼ 1 TeV.
So large VEV of the superfield S may give rise to the extremely heavy spectrum of exotic particles and Z ′ . This can lead to the MSSM type of particle spectrum at the TeV scale.
Nevertheless even in this case the broken U(1) N symmetry leaves its imprint on the MSSM sfermion mass spectrum. Since m 2 S = m 2 S the VEVs of the SM singlet superfields S and S deviates from the D-flat direction As a consequence all sfermions receive an additional contribution to the mass that come from the U(1) N D-term quartic interactions in the scalar potential [102]. This contribution ∆ i is proportional to the U(1) N charge of the corresponding sfermion Q i , i.e.
where Q 1 and Q 2 are the U(1) N charges of H d and H u . Thus for the superpartners of the first and second generation quarks and leptons one finds

Conclusions
In this paper we have considered the E 6 inspired SUSY models in which a single discretẽ Z H 2 symmetry forbids the tree-level flavor-changing transitions and baryon number violating operators. We assumed that the breakdown of E 6 symmetry or its subgroup lead to the rank-6 SUSY models below the GUT scale M X . These models are based on the Standard Model (SM) gauge group together with extra U(1) ψ and U(1) χ gauge symmetries. We also allow three copies of 27 i representations of E 6 to survive below the scale M X so that anomalies get canceled generation by generation. If extra exotic states from 27 i -plets survive to low energies they give rise to tree-level non-diagonal flavor transitions and rapid proton decay. In order to suppress baryon number violating operators one can imposeZ H 2 discrete symmetry. We assumed that all matter superfields, that fill in complete 27 i representations of E 6 , are odd under this discrete symmetry. ThusZ H 2 symmetry is defined analogously to the matter parity Z M 2 in the simplest SU(5) SUSY GUTs, that lead to the low-energy spectrum of the MSSM.
In addition to three complete fundamental representations of E 6 we further assumed the presence of of M l and M l supermultiplets from the incomplete 27 ′ l and 27 ′ l representation just below the GUT scale. Because multiplets M l and M l have opposite U(1) Y , U(1) ψ and U(1) χ charges their contributions to the anomalies get cancelled identically. As in the MSSM we allowed the set of multiplets M l to be used for the breakdown of gauge symmetry and therefore assumed that all multiplets M l are even underZ H 2 symmetry. In order to ensure that the SU(2) W × U(1) Y × U(1) ψ × U(1) χ symmetry is broken down to U(1) em associated with the electromagnetism the set of multiplets M l should involve H u , H d , S and N c H . We argued that U(1) ψ × U(1) χ gauge symmetry can be broken by the VEVs of N c H and N c H down to U(1) N × Z M 2 because matter parity is a discrete subgroup of U(1) ψ and they can get combined with the superposition of the corresponding components from 27 i forming vectorlike states at low energies. As in the case of scenario A the superfield S can be either even or odd under theZ H 2 symmetry and may or may not survive to the TeV scale. In the scenario B the exotic quarks manifest themselves in the Yukawa interactions as superfields with baryon number ± 2 3 .
The gauge group and field content of the E 6 inspired SUSY model discussed here can originate from the 5D and 6D orbifold GUT models in which the splitting of GUT multiplets can be naturally achieved. In particular, we studied SU(5) × U(1) χ × U(1) ψ SUSY GUT model in 5D compactified on the orbifold S 1 /(Z 2 × Z ′ 2 ). At low energies this model may lead to the scenarios A and B. We also considered E 6 gauge theory in 6D compactified on the orbifold T 2 /(Z 2 × Z I 2 × Z II 2 ) that can lead to the scenario A at low energies. In these orbifold GUT models all anomalies get cancelled and GUT relations between Yukawa couplings get spoiled. The adequate suppression of the operators, that give rise to proton decay, can be also achieved if the GUT scale M X ∼ 1/R is larger than 1.5 − 2 · 10 16 GeV.
We examined the RG flow of gauge couplings from M Z to M X in the case of scenarios A and B using both analytical and numerical techniques. We derived the corresponding twoloop RG equations and studied the running of the gauge couplings with and without extra S and S superfields at the TeV scale. In the scenario A the gauge coupling unification can be achieved for any phenomenologically reasonable value of α 3 (M Z ) consistent with the central measured low energy value. This was already established in the case of the SUSY model with extra U(1) N gauge symmetry and low energy matter content that involves three 27-plets of E 6 as well as L 4 and L 4 [36]. Our analysis here revealed that the evolution of the SM gauge couplings does not change much when the low energy particle spectrum is supplemented by the S and S chiral superfields. Thus this is not so surprising that the unification of the SM gauge couplings can be so easily achieved even in this case. In the scenario B large two-loop corrections spoil the unification of gauge couplings. Indeed, in this case the exact gauge coupling unification can be achieved only if α 3 (M Z ) 0.112.
As before the inclusion of extra S and S superfields does not change much the RG flow of α i (t) and therefore does not improve gauge coupling unification. However the relative discrepancy of α i (M X ) is about 10% . At the same time orbifold GUT framework does not imply the exact gauge coupling unification near the scale M X ∼ 1/R because of the brane contributions to the gauge couplings. Therefore relative discrepancy of 10% between α i (M X ) should not be probably considered as a big problem.
Finally we also discussed the cosmological implications and collider signatures of the E 6 inspired SUSY models discussed above. As it was mentioned the low-energy effective Lagrangian of these models is invariant under both Z M 2 andZ H 2 symmetries. Since Z H 2 = Z M 2 × Z E 2 the Z E 2 symmetry associated with exotic states is also conserved. As a result the lightest exotic state, which is odd under the Z E 2 symmetry, must be stable. In the scenarios A and B the lightest and second lightest inert neutralinos tend to be the lightest exotic states in the particle spectrum. On the other hand the Z M 2 symmetry conservation implies that R-parity is conserved. Because the lightest inert neutralinoH 0 1 is also the lightest R-parity odd state either the lightest R-parity even exotic state or the lightest R-parity odd state with Z E 2 = +1 must be absolutely stable. Most commonly the second stable state is the lightest ordinary neutralino χ 0 1 (Z E 2 = +1). Both stable states are natural dark matter candidates in the considered E 6 inspired SUSY models.
When |mH0 1 | ≪ M Z the lightest inert neutralino is predominantly inert singlino and its couplings to the gauge bosons, Higgs states, quarks and leptons are very small resulting in too small annihilation cross section forH 0 1H 0 1 → SM particles. As a consequence the cold dark matter density is much larger than its measured value. In principle,H 0 1 could account for all or some of the observed cold dark matter density if it had mass close to half the Z mass. In this case the lightest inert neutralino states annihilate mainly through an s-channel Z-boson. However the usual SM-like Higgs boson decays more than 95% of the time into eitherH 0 1 orH 0 2 in these cases while the total branching ratio into SM particles is suppressed. Because of this the corresponding scenarios are basically ruled out nowadays. The simplest phenomenologically viable scenarios imply that the lightest and second lightest inert neutralinos are extremely light. For example, these states can have masses about 1 eV. The lightest and second lightest inert neutralinos with masses about 1 eV form hot dark matter in the Universe but give only a very minor contribution to the dark matter density while the lightest ordinary neutralino may account for all or some of the observed dark matter density.
The presence of two types of dark matter is a very peculiar feature that affect the collider signatures of the considered E 6 inspired SUSY models. The most spectacular LHC signals associated with these models may come from the TeV scale exotic color states and Z ′ . The production of the Z ′ boson, that corresponds to the U(1) N gauge symmetry, should lead to unmistakable signal pp → Z ′ → l + l − at the LHC. The Z E 2 symmetry conservation implies that in collider experiments exotic particles can only be created in pairs. Moreover each exotic particle has to decay into a final state that contains at least one lightest inert neutralino resulting in the missing energy. Because of this the lightest exotic color state, that can be either D-fermion orD-scalar, decay into either u i (d i ) + ℓ(ν) + E miss T + X if exotic quark (squark) is leptoquark or u c i + d c j + E miss T + X if exotic quark (squark) is diquark. The Z E 2 symmetry conservation requires that E miss T should always contain contribution associated with the lightest inert neutralino. Since the lightest exotic squark is R-parity even state while the lightest inert neutralino is R-parity odd particle the final state in the decay ofD should also involve the lightest ordinary neutralino to ensure R-parity conservation. Thus the pair production of the lightest exotic color state is expected to lead to a substantial enhancement of the cross section of either pp → jjℓ + ℓ − + E miss T + X or pp → jjjj + E miss T + X. If the Yukawa couplings of the exotic particles have hierarchical structure similar to the one observed in the ordinary quark and lepton sectors then all states which are odd under the Z E 2 symmetry couple to the third generation fermions and sfermions mainly. As a result the TeV scale exotic color states should give rise to the enhancement of the cross section of either pp → ttℓ + ℓ − + E miss T + X or pp → ttbb + E miss T + X.
Our consideration indicates thatD-scalars in the considered E 6 inspired SUSY models lead to rather unusual collider signatures. Indeed, it is commonly expected that scalar diquarks decay into quark-quark without missing energy in the final state while the scalar leptoquarks decay into quark-lepton without missing energy as well. In the models considered here the Z E 2 symmetry conservation necessarily leads to the missing energy in the corresponding final states. In addition relatively light exotic quark and squark can modify the collider signatures associated with gluinos if the decay of the gluino into exotic quark and squark is kinematically allowed. In this case gluino pair production at the LHC may result in DDDD in the final state. The sequential decays of D-fermions andD-scalars give rise to the enhancement of either pp → 4 ℓ + 4 j + E miss T + X or pp → 8 j + E miss T + X.
In the scenario B the fermionic components of the supermultiplets d c 4 and d c 4 that form vectorlike quark state as well as their superpartner may have TeV scale masses. Then these quark and/or squark states can be pair produced at the LHC via strong interactions and decay into q i + E miss T + X where q i can be either up-type or down-type quark. This may lead to an enhancement of pp → jj + E miss T + X.
The discovery of Z ′ and new exotic particles predicted by the E 6 inspired SUSY models considered here will open a new era in elementary particle physics. This would not only represent a revolution in particle physics, but would also point towards an underlying E 6 gauge structure at high energies.