A Review of the Exceptional Supersymmetric Standard Model

Local supersymmetry (SUSY) provides an attractive framework for the incorporation of gravity and unification of gauge interactions within Grand Unified Theories (GUTs). Its breakdown can lead to a variety of models with softly broken SUSY at low energies. In this review article we focus on the SUSY extension of the Standard Model (SM) with an extra U(1)_{N} gauge symmetry originating from a string-inspired E_6 grand unified gauge group. Only in this U(1) extension of the minimal supersymmetric standard model (MSSM) inspired by E_6 GUTs the right-handed neutrinos can be superheavy providing a mechanism for the generation of the lepton and baryon asymmetry of the Universe. The particle content of this exceptional supersymmetric standard model (E_6SSM) includes three 27 representations of the E_6 group, to ensure anomaly cancellation, plus a pair of SU(2)_W doublets as required for gauge coupling unification. Thus E_6SSM involves extra exotic matter beyond the MSSM. We consider symmetries that permit to suppress non-diagonal flavour transitions and rapid proton decay, as well as gauge coupling unification, the breakdown of the gauge symmetry and the spectrum of Higgs bosons in this model. The possible Large Hadron Collider (LHC) signatures caused by the presence of exotic states are also discussed.


Introduction
Symmetries play a key role in modern high energy physics. Indeed, it was realised a long time ago that light hadron resonances form representations of the SU (3) group, which is associated with light quark flavours, while the physics of strong interactions is described by the coloured SU (3) C gauge symmetry. It was also established that weak and electromagnetic forces represent electroweak (EW) interactions based on the SU (2) W × U (1) Y gauge group. Within the standard model (SM) of elementary particles, that describes perfectly the major part of all experimental data measured in earth based experiments, SU (2) W × U (1) Y is spontaneously broken to the abelian U (1) em gauge group associated with electromagnetism by means of the Higgs mechanism. The latter predicts the existence of the Higgs boson which was recently discovered at the LHC. Thus the Lagrangian of the SM is invariant under the Pointcaré group and SU (3) C × SU (2) W × U (1) Y gauge symmetry transformations. The Pointcaré group is an extension of Lorentz group that includes time and space translations whereas the transformations of Lorentz group involve rotations about three axes and Lorentz boosts along them.
At very high energies the SM can be embedded into GUTs [1] based on the SU (5) or SO(10) gauge groups. In the case of SU (5) GUTs each SM family of quarks and leptons fills in a complete one antifundamental and one antisymmetric second-rank tensor representations of SU (5), i.e. 5 + 10. Within SO(10) GUTs each family of SM fermions may belong to a single 16 dimensional spinor representation of SO (10). Such models predict the existence of right-handed neutrinos, allowing these to be used for both the see-saw mechanism [2] and leptogenesis [3].
SUSY GUTs permit to place fermions and bosons of the SM within one supermultiplet. In order to achieve the unification of gauge interactions with gravity one needs to combine Pointcaré and internal (gauge) symmetries. At the same time according to the Coleman-Mandula theorem the most general symmetry which quantum field theory can have is a tensor product of the Pointcaré group and an internal group [4]. The Coleman-Mandula theorem can be overcome within graded Lie algebras that have the following structure whereB andF are bosonic and fermionic generators. Graded Lie algebras that contain the Pointcaré algebra are called supersymmetries. The simplest N = 1 supersymmetry involves a set of generators of the Pointcaré group (bosonic generators) and a single Weyl spinor operator Q α as well as its complex conjugate Q † α = Qα (fermionic generators). SUSY algebra implies that each supermultiplet has the same number of bosonic and fermionic degrees of freedom.
In N = 1 SUSY GUTs based on the E 6 gauge group the complete fundamental 27 representation, which decomposes under SO(10) × U (1) ψ subgroup as contains one family of the SM fermions and Higgs doublet. The Higgs doublet is assigned to 10, − 2 √ 24 . The SM gauge bosons belong to the adjoint representation of E 6 , i.e. a 78-plet.
In N = 2 SUSY GUTs based on the E 8 gauge group all SM particles belong to a 248 dimensional representation of E 8 that decomposes under its E 6 subgroup as follows The local SUSY (supergravity) leads to a partial unification of gauge interactions with gravity [5,6,7]. However supergravity (SUGRA) is a non-renormalizable theory and has to be considered as an effective low energy limit of some renormalizable or even finite theory. Currently, the best candidate for such an underlying theory, i.e. the hypothetical single framework that explains and links together all physical aspects of the universe, is a ten-dimensional heterotic superstring theory based on E 8 × E 8 [8]. Compactification of the extra dimensions in this theory leads to an effective supergravity and results in the breakdown of E 8 to E 6 or its subgroups in the observable sector [9]. The remaining E 8 plays the role of a hidden sector which gives rise to spontaneous breakdown of SUGRA. As a consequence, a set of soft SUSY breaking terms [10,11,12,13] characterized by the gravitino mass (m 3/2 ) is generated. A large mass hierarchy between m 3/2 and the Planck scale M P can be caused by the non-perturbative effects in the hidden sector that trigger the breakdown of local SUSY [14].
When m 3/2 M P the breakdown of the E 6 gauge group near the GUT scale M X may lead to a variety of SUSY models at low energies including models based on the SM gauge group, like the MSSM, as well as extensions of the MSSM with an extra U (1) gauge symmetry which is a linear combination of U (1) χ and U (1) ψ , i.e.: Here U (1) ψ and U (1) χ are associated with the subgroups E 6 ⊃ SO(10)×U (1) ψ ⊃ SU (5)×U (1) χ ×U (1) ψ whereas the SM gauge group is a subgroup of SU (5), i.e. SU (5) ⊃ SU (3) C × SU (2) W × U (1) Y .
In the simplest case U (1) χ × U (1) ψ is broken down to its discrete subgroup Z M 2 = (−1) 3(B−L) which is the so-called matter parity. If in this case the low energy matter content involves three families of the SM fermions and their scalar superpartners as well as two SU (2) W doublets of the Higgs bosons (H 1 and H 2 ) and their fermionic partners (Higgsinos) then this model corresponds to the simplest SUSY extension of the SM -the MSSM. Matter parity conservation implies that the lightest SUSY particle (LSP) is stable and can play the role of dark matter. In order to reproduce the Higgs-fermion Yukawa interactions that induce the masses of all quarks and charged leptons in the SM the MSSM superpotential has to include the following sum of the products of chiral superfields where a and b are family indices that run from 1 to 3. In Eq. (5) Q a and L a contain the doublets of left-handed quark and lepton superfields, e c a , u c a and d c a are associated with the right-handed lepton, up-and down-type quark superfields, respectively, while the Yukawa couplings y U ab , y D ab and y L ab are dimensionless 3 × 3 matrices in family space. It was found that the EW and strong gauge couplings extracted from LEP data and extrapolated to high energies using the renormalisation group (RG) equations do not meet within the SM but converge to a common value near the scale M X 2 · 10 16 GeV in the framework of the MSSM [15,16,17,18]. This allows one to embed the MSSM into SUSY GUTs.
The MSSM superpotential in Eq. (5) contains only one bilinear term µH 1 H 2 which can be present before SUSY is broken. Therefore one would naturally expect the parameter µ to be either zero or of the order of the GUT scale M X . If µ ∼ M X then the Higgs scalars get a huge positive contribution ∼ µ 2 to their squared masses and EW symmetry breaking (EWSB) does not occur. In contrast, when µ = 0 at some scale Q, the mixing between Higgs doublets is not generated at any scale below Q due to the non-renormalisation theorems [19,20]. In this case the minimum of the Higgs boson potential is attained for < H d >= 0 and the down-type quarks as well as the charged leptons remain massless. In order to ensure the correct pattern of the EW symmetry breaking (EWSB), µ is required to be of the order of the SUSY breaking scale M S .
In the framework of the simplest extension of the MSSM, the next-to-MSSM (NMSSM), the superpotential is invariant with respect to the discrete transformations Φ i → e 2πi/3 Φ i of the Z 3 group. The term µ(H 1 H 2 ) does not meet this requirement and therefore can not be included. The superpotential of the NMSSM is given by [21] W NMSSM = λS(H 1 H 2 ) + κ 3 S 3 + W MSSM (µ = 0), (6) where S is an extra singlet superfield. It acquires a vacuum expectation value (VEV), i.e. S = s/ √ 2, and an effective µ parameter is generated (µ = λs/ √ 2 ∼ M S ). The cubic term of the new singlet superfield S in the superpotential (6) explicitly breaks an additional global U (1) symmetry which is a common way to avoid the appearance of the axion in the particle spectrum. However the NMSSM itself is not without problems. The VEVs of the Higgs fields break the exact Z 3 symmetry leading to the formation of domain walls in the early universe between regions which were causally disconnected during the period of EWSB [22]. Such domain structure of vacuum creates unacceptably large anisotropies in the cosmic microwave background radiation [23]. Because of this the NMSSM superpotential should contain additional operators that violate the Z 3 symmetry and prevent the appearance of domain walls [24,25].
In the U (1) extensions of the MSSM inspired by E 6 the extra U (1) gauge symmetry (4) forbids an elementary µ term if θ E 6 = 0 or π. Nevertheless these extensions of the SM allow the interaction λS(H d H u ) in the superpotential while the S 3 term is forbidden by the U (1) gauge symmetry. Near the scale M S the scalar component of the SM singlet superfield S develops a non-zero VEV breaking U (1) and an effective µ term of the required size is automatically generated. Clearly there are no domain wall problems in such models since there is no discrete Z 3 symmetry. Different aspects of the phenomenology of E 6 inspired SUSY models have been extensively studied in the past [26,27,28,29,30,31,32,33,34,35,36]. Previously, the implications of E 6 inspired SUSY models with an additional U (1) gauge symmetry have been studied for the EWSB [37,38,39,40,41,42,43], neutrino physics [44,45], fermion mass hierarchy and mixing [46], leptogenesis [47,48], EW baryogenesis [51,52], the Z mass limits [169], collider signatures associated with the exotic quarks and squarks [54], the muon anomalous magnetic moment [55,56], the electric dipole moment of the electron [57] and of the tau lepton [58], lepton flavor violating processes like µ → eγ [59] and CP-violation in the Higgs sector [60]. The neutralino sector in E 6 inspired SUSY models was examined in [42,57,58,59,61,62,63,64,65,66,170,68]. The Higgs sector and the theoretical upper bound on the lightest Higgs boson mass in the E 6 inspired SUSY models were explored in [43,68,69,70,71].
In this review article we consider a specific E 6 inspired SUSY realisation of the above U (1) type model associated with θ E 6 = arctan √ 15. This choice of Abelian U (1) corresponds to U (1) N gauge symmetry. Thus such a SUSY model is based on the SM gauge group together with an additional U (1) N factor. In this exceptional supersymmetric standard model (E 6 SSM) [69,70] right-handed neutrinos do not participate in the gauge interactions. Therefore only in such a U (1) extension of the MSSM inspired by E 6 GUTs the right-handed neutrinos can be superheavy, so that a see-saw mechanism can be used to generate the mass hierarchy in the lepton sector, providing a comprehensive understanding of the neutrino oscillations data. Successful leptogenesis is also a distinctive feature of the E 6 SSM since the heavy Majorana right-handed neutrinos may decay into final states with lepton number L = ±1, creating a lepton asymmetry in the early Universe [47,48,49].
The layout of this paper is as follows. In the next Section we specify the U (1) N extensions of the MSSM and discuss global symmetries that prevent non-diagonal flavour transitions as well as rapid proton decay in these SUSY models. The two-loop RG flow of the gauge couplings within the E 6 SSM is examined in Section 3. The Higgs sector dynamics and the emerging spectrum of the masses and couplings of the Higgs bosons are discussed in Sections 4 and 5, respectively. In section 6 the possible LHC signatures of the E 6 SSM are considered. Section 7 is reserved for our conclusions and outlook.
2 The U (1) N extensions of the MSSM The E 6 SSM implies that near the GUT scale E 6 or its subgroup is broken down to [69,70]. With additional Abelian gauge symmetries it is important to ensure the cancellation of anomalies. In any model based on the subgroup of E 6 the anomalies are canceled automatically if the low-energy spectrum involves complete representations of E 6 . Consequently, in the E 6 SSM the particle spectrum is extended by a number of exotics which, together with ordinary quarks and leptons, form three complete 27dimensional representations of E 6 , referred to as 27 i with i = 1, 2, 3. These multiplets decompose under the SU (5) × U (1) N subgroup of E 6 as follows: The first and second quantities in the brackets are the SU (5) representation and extra U (1) N charge respectively. 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, than that of ordinary ones. Therefore in phenomenologically viable U (1) N extensions of the MSSM they can be either diquarks or leptoquarks. In addition to the complete 27 i multiplets the splitting of 27 l and 27 l within the E 6 GUTs can give rise to a set of M l and M l supermultiplets with opposite quantum numbers at low energies. In the simplest case the low energy particle spectrum of the E 6 SSM is supplemented by SU (2) W doublet L 4 and anti-doublet L 4 states from the extra 27 and 27 to preserve gauge coupling unification, where L 4 supermultiplet has the quantum numbers of left-handed leptons. Thus, in addition to a Z corresponding to the U (1) N symmetry, the E 6 SSM involves extra matter beyond the MSSM that fill in three 5 + 5 * representations of SU (5) plus three SU (5) singlets with U (1) N charges. The gauge group and field content of the E 6 SSM can originate from the orbifold GUT models [72,50].
Over the last fifteen years, several variants of the E 6 SSM have been proposed [69,70,73,74,75,76,77,78,79,72,80,81,82]. The E 6 inspired SUSY models with an additional U (1) N gauge symmetry have been thoroughly investigated as well. They have been studied in [45] in the context of non-standard neutrino models, in [61] from the point of view of Z − Z mixing, in [42,61,62] the neutralino sector was explored, in [83] the implications of the exotic states for the dark matter was considered, in [42,84] the RG flow of the couplings was examined, and in [41,42,43] EWSB was investigated. More recently, the RG flow of the Yukawa couplings and the theoretical upper bound on the lightest Higgs boson mass were explored in the vicinity of the quasi-fixed point [85,86] that appears as a result of the intersection of the invariant and quasi-fixed lines [87,88]. Detailed studies of the E 6 SSM have established that extra exotic matter and Z predicted by this model may give rise to distinctive LHC signatures [69,70,74,77,89,90,91,92,93], as well as may lead to non-standard Higgs decays for sufficiently light exotics [81,86,98,94,95,96,97,99,100]. Within the constrained version of the E 6 SSM (cE 6 SSM) and its modifications the particle spectrum and associated phenomenological implications were explored in [91,101,102,103,104,105,106] while the degree of fine tuning was examined in [107,108]. The threshold corrections to the running gauge and Yukawa couplings in the E 6 SSM and their impact in the cE 6 SSM were studied in [109]. The renormalisation of the VEVs in the E 6 SSM was considered in [110,111].
The superpotential of the U (1) N extensions of the MSSM contains the renormalisable part that comes from the 27 × 27 × 27 decomposition of the E 6 fundamental representation and can be written as In Eq.  [69,70]. The couplings of all other exotic states to the ordinary quark and lepton supermultiplets are forbidden that eliminates any problems related with the non-diagonal flavour transitions at the tree level. In this original E 6 SSM model the scalar components of the supermultiplets H u , H d and S compose the Higgs sector. In particular, the third family SM-singlet superfield S 3 gets a VEV, S 3 = s/ √ 2, breaking U (1) N gauge symmetry. This VEV is responsible for the effective µ term and D-fermion masses. The first and second families of Higgs doublets and SM-singlets, which do not get VEVs, are called "inert". At the same time the modified version of the E 6 SSM, in which three SM-singlet superfields S i are taken to be even under the Z H 2 symmetry, was also recently considered [82]. In this case all superfields S i develop VEVs. They couple to H u , H d as well as other exotic bosons and fermions.
Although the Z H 2 symmetry forbids not only flavor changing processes but also the most dangerous baryon and lepton number violating operators, it can not be an exact symmetry.
Indeed, this symmetry forbids all Yukawa interactions in W 1 and W 2 that allow the lightest exotic quarks to decay. 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 extremely long-lived. The U (1) L , U (1) B and U (1) D global symmetries are expected to be broken by the non-renormalizable operators which are suppressed by inverse power of the GUT scale M X . Since E 6 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. it is considerably larger than the age of the Universe. So 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 have been confined in heavy hadrons which would annihilate further. The remaining heavy hadrons with exotic quarks originating from the Big Bang should be present in terrestrial matter. Various theoretical estimates [113,114] show that if such remnant particles in the mass range from 1 GeV to 10 TeV would exist in nature, today their concentration is expected to be at the level of 10 −10 per nucleon. At the same time different experiments set stringent upper limits on the relative concentrations of such nuclear isotopes which vary from 10 −15 to 10 −30 per nucleon [115,116,117]. Therefore the extensions of the SM with so long-lived exotic quarks are basically ruled out. This means that the discrete Z H 2 symmetry can only be an approximate one. To prevent rapid proton decay within the U (1) N extensions of the MSSM one can impose either Z L 2 or Z B 2 discrete symmetry. If the Lagrangian is invariant with respect to an exact Z L 2 symmetry, under which all superfields except lepton ones (including L 4 and L 4 ) are even, then all Yukawa interactions in W 2 are forbidden and the baryon number conservation requires the exotic quarks to be diquarks (Model I). In this case the most general renormalisable superpotential which is allowed by the SU (3) C × SU (2) W × U (1) Y × U (1) N gauge symmetry can be presented in the following form: The terms in W 0 are caused by the splitting of 27 and 27 representations of E 6 . An alternative possibility is to assume that the exotic quarks D i and D i as well as ordinary lepton superfields, L 4 and L 4 are all odd under Z B 2 whereas the others remain even. As a consequence all terms in W 1 are ruled out by the discrete Z B 2 symmetry and exotic quarks carry baryon (B D = 1/3 and B D = −1/3) and lepton (L D = 1 and L D = −1) numbers simultaneously (Model II). Thus in Model II D i and D i are leptoquarks. The most general renormalisable superpotential in Model II are given by The last term in Eq. (12) appears because of the splitting of 27 . In the superpotentials (11)-(12) the SU (2) W doublet L 4 is redefined in such a way that W 0 contains only one bilinear term.
The mass parameter µ L should not be too large otherwise it spoils gauge coupling unification. Within SUGRA models the appropriate term µ L L 4 L 4 in the superpotentials (11)-(12) can be induced if the Kähler potential contains an extra term (ZL 4 L 4 +h.c) [118,119]. This is the same mechanism which is used in the MSSM to solve the µ problem. Within the U (1) N extensions of the MSSM the bilinear term involving H d and H u are forbidden by the U (1) N gauge symmetry so that the mechanism mentioned above cannot be applied for the generation of µH d H u in the E 6 SSM superpotential.
The superpotentials of the Models I and II also include bilinear terms, 1 2 M ij N c i N c j , responsible for the right-handed neutrino masses. The corresponding mass parameters M ij are expected to be at intermediate mass scales. They can be induced through the non-renormalisable interactions of the form α, β = 1, 2 and i, j = 1, 2, 3 . If some of the couplings λ, λ αβ or κ ij are rather large at the GUT scale M X , they affect the evolution of the soft scalar mass m 2 S of the singlet field S quite strongly resulting in negative values of m 2 S at low energies. This triggers the breakdown of U (1) N gauge symmetry. The singlet VEV must be large enough to generate sufficiently large masses of the Z boson and exotic particles. This also implies that the Yukawa couplings λ, λ αβ and κ ij have to be large enough. On the other hand the large value of the top-quark Yukawa coupling provides a radiative mechanism for generating the VEVs of H u and H d that break the SU (2) W × U (1) Y gauge symmetry.
Since in the U (1) N extensions of the MSSM the Z M 2 symmetry and R-parity are conserved the lightest R-parity odd state, i.e. the lightest SUSY particle (LSP), must be stable. Using the method proposed in [120,121,122] it was shown that the LSP and next-to-lightest SUSY particle (NLSP) in the E 6 SSM have masses below 60 − 65 GeV [94]. The LSP and NLSP (H 0 1 andH 0 2 ) are predominantly linear superpositions of the fermion components of the two SM singlet superfields S α . Although the couplings ofH 0 1 to the SM gauge bosons and fermions are quite small LSP could account for all or some of the observed cold dark matter density if it had a mass close to half the Z mass. In this case LSP annihilate mainly through an s-channel Z-boson [94]. However the SM-like Higgs boson decays more than 95% of the time into either H 0 1 orH 0 2 in these scenarios while all other branching ratios would be strongly suppressed. Nowadays such scenario are ruled out by the LHC experiments. If fermion components of the SM singlet superfields S α are substantially lighter than M Z the annihilation cross section for H 0 1H 0 1 → SM particles becomes too small leading to the cold dark matter density that is much larger than its measured value.
Nevertheless in the E 6 SSM with approximate Z H 2 symmetry one of the lightest R-parity odd state can account for all or some of the observed cold dark matter density. In order to prevent the decays of this state into the LSP and NLSP an additional Z S 2 symmetry needs to be postulated [78]. In the corresponding variant of the E 6 SSM couplingsf αβ and f αβ vanish. As a result the fermion components of the SM singlet superfields S α remain massless and decouple. If Z boson is sufficiently heavy the presence of these massless states does not affect Big Bang Nucleosynthesis (BBN) [78]. Sincef αβ = f αβ = 0 the branching ratios of the SM-like Higgs decays intoH 0 1 andH 0 2 vanish.
Instead of Z H 2 , Z L 2 and Z B 2 one can impose a single discreteZ H 2 symmetry which forbids tree-level flavor-changing transitions and the most dangerous operators that violate baryon and lepton numbers. In this case H u , H d , S, L 4 and L 4 are even under theZ H 2 symmetry while all other supermultiplets are odd [72]. Neglecting all suppressed non-renormalisable interactions, the superpotential of this variant of the E 6 SSM is given by Eq. (14) with where α = 1, 2 and i, k = 1, 2, 3 . Since the low-energy effective Lagrangian of this SUSY mod- Table 1. The Z E 2 symmetry conservation ensures that the lightest exotic state, which is odd under this symmetry, is stable. The simplest phenomenologically viable scenarios imply that f αβ ∼f αβ < 10 −6 . As a consequence two lightest exotic states (H 0 1 andH 0 2 ), which are formed by the fermion components of the superfields S α , are substantially lighter than 1 eV. They compose hot dark matter in the Universe but gives only a very minor contribution to the dark matter density [72]. The presence of very light neutral fermions in the particle spectrum might also have interesting implications for the neutrino physics (see, for example [123]). The invariance of the Lagrangian under the Z M 2 ensures that the lightest R-parity odd state with Z E 2 = +1, which is most commonly the lightest ordinary neutralino in this case, is also stable and may account for all or some of the observed cold dark matter density [106].

Gauge Coupling Unification
In this section we consider the RG flow of the gauge couplings within the E 6 SSM between M Z and the GUT scale M X . The evolution of these gauge couplings is affected by a kinetic term mixing. In the Lagrangian of any extension of the SM, that involves an additional U (1) factor, there can arise a kinetic term consistent with all symmetries which mixes the gauge fields of the U (1) and U (1) Y [124]. The E 6 SSM is not an exception. In the basis in which the interactions between gauge and matter fields have the canonical form, i.e. for instance a covariant derivative D µ which acts on the left-handed quark field is given by the mixing between the U (1) field strengths can be written as Here µν , F Y µν and F N µν are field strengths for the corresponding gauge interactions, whereas g 3 , g 2 , g Y and g N are the SU (3) C , SU (2) W , U (1) Y and U (1) N gauge couplings in this basis. Since U (1) Y and U (1) N factors come from the breakdown of the simple gauge group E 6 the parameter sin χ is expected to vanish at tree-level. However the non-zero value of this parameter is induced by In Eq. (19) trace is restricted to the states which are lighter than M X . The contribution of the complete E 6 supermultiplets to this trace cancels. The non-zero value of the trace (19) is induced by L 4 and L 4 supermultiplets which survive to low energies. For non-zero values of the parameter sin χ the mixing in the gauge kinetic part of the Lagrangian (18) can be eliminated by means of a non-unitary transformation of the two U (1) gauge fields [37], [125,126,127,128]: In the basis (B 1µ , B 2µ ) the gauge kinetic part of the Lagrangian is diagonal and the covariant derivative (17) becomes where the redefined gauge coupling constants, written in terms of the original ones, are In the Lagrangian written in terms of the new gauge variables B 1µ and B 2µ the mixing effect is concealed in the interaction between the U (1) N gauge field and matter fields. The gauge coupling constant g 1 differs from the original one and there is a new off-diagonal gauge coupling g 11 . In the new basis the covariant derivative (21) can be rewritten in a compact form where Now all physical phenomena can be examined using the Lagrangian with the modified structure of the extra U (1) N interaction (21)- (23). In this approximation the gauge kinetic mixing changes effectively the U (1) N charges of the fields tõ where δ = g 11 /g 1 while the U (1) Y charges remain the same. The effective U (1) N chargesQ i are scale dependent. The particle spectrum in the basis The running of four diagonal gauge couplings, i.e. g 3 (t), g 2 (t), g 1 (t) and g 1 (t), and one off-diagonal gauge coupling g 11 is described by a system of RG equations (RGEs) which can be written in the following form: where t = 2 ln (q/M Z ), q is a renormalisation scale, G is a 2 × 2 matrix (24) while B is a 2 × 2 matrix given by In Eqs. (26)-(27) β i and β 11 are beta functions. Here the RG flow of the gauge couplings is explored in the two-loop approximation. In this approximation β i and β 11 can be presented as a sum of one-loop and two-loop contributions. In the case of diagonal gauge couplings one gets It seems to be rather natural to expect that just after the breakdown of the E 6 symmetry near the GUT scale M X 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, while the SU (3) C , SU (2) W , U (1) Y and U (1) N gauge interactions are characterised by a unique E 6 gauge coupling g 0 , i.e.
The previous analysis performed in [84] revealed that g 11 being set to zero at the scale M X remains very small at any other scale below M X . Thus it tends to be substantially smaller than the diagonal gauge couplings. Therefore the two-loop corrections to the off-diagonal beta function β 11 can be neglected. The one-loop off-diagonal beta function is given by To simplify our analysis here we further assume that the interactions of matter supermultiplets in the E 6 SSM are described by the superpotential (14) in which all interactions in W L 4 can be ignored,f αβ f αβ → 0, λ αβ = λ α δ αβ and κ ij = κ i δ ij . The part of the superpotential (14) associated with W MSSM (µ = 0) reduces to because only third generation fermions have Yukawa couplings to H d and H u which can be of the order of unity. In Eqs. (30) h t , h b and h τ are top quark, b-quark and τ -lepton Yukawa couplings respectively. In the one-loop approximation the beta functions of the diagonal gauge couplings are given by where parameter N g is the number of generations in the E 6 SSM forming complete E 6 fundamental representations at low energies (E << M X ). As one can see N g = 3 is the critical value for the one-loop beta function of the strong interactions. Since N g = 3 in the E 6 SSM b 3 is equal to zero and in the one-loop approximation the SU (3) C gauge coupling remains constant everywhere from the EW scale to M X . Thus any reliable analysis of gauge coupling unification requires the inclusion of two-loop corrections to the beta functions of the diagonal gauge couplings in the E 6 SSM. Using the results of the computation of two-loop beta functions in a general softly broken N = 1 SUSY model [129] one obtains where . For the analysis of the RG flow of the SM gauge couplings it is convenient to use an approximate solution of the two-loop RGEs (see [130]). At high energies this solution can be written as where b SM i are the coefficients of the one-loop beta functions in the SM, the third term in the right-hand side of Eq. (3) is the M S → DR conversion factor with C 1 = 0, C 2 = 2, C 3 = 3 [131,132], while In Eq. (34) m k and ∆b k i are masses and one-loop contributions to b i due to new particles appearing in the E 6 SSM. Since the two-loop corrections to the running of the gauge couplings Θ i (t) are considerably smaller than the leading terms, the solutions of the one-loop RGEs for the gauge and Yukawa couplings are normally used for the calculation of Θ i (t). The threshold corrections associated with the last terms in Eq. (33) are of the same order as or even less than Θ i (t). Therefore in Eqs. (33)-(34) only one-loop threshold effects are taken into account.
Relying on the approximate solution of the two-loop RGEs one can find 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) one can estimate 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 the value of α 3 (M Z ), for which exact gauge coupling unification takes place, may be obtained (see [133]): The combined threshold scale T S can be expressed in terms of the effective threshold scales T 1 , T 2 and T 3 In Eq. (36) T 1 , T 2 and T 3 are given by In general T 1 , T 2 and T 3 in Eq. (37) can be quite different. Nevertheless from Eq. (35) it follows that the unification of the SM gauge couplings is determined by a single combined threshold scale T S . Therefore without loss of generality one can set three effective threshold scales be equal to each other. Then from Eq. (36) it follows that T 1 = T 2 = T 3 = T S . The results of our numerical analysis of the gauge coupling unification within the E 6 SSM are presented in Figure 1 where the two-loop RG flow of gauge couplings is shown. We use the two-loop SM beta functions to evaluate the running of gauge couplings between M Z and T 1 = T 2 = T 3 = T S . Then we apply the two-loop RGEs of the E 6 SSM to calculate the evolution of α i (t) from T S to M X which is around 2 − 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 the conditions (29) are fulfilled. For the computation of the RG flow of Yukawa couplings a set of one-loop RGEs is used. The corresponding one-loop RGEs are specified in [69]. In Figure 1 we fix T 1 = T 2 = T 3 = T S = 2 TeV and tan β = 10. Although to simplify our analysis we also set κ i (T S ) = λ α (T S ) = λ(T S ) = g 1 (T S ) the RG flow of α i (t) depends rather weakly on the values of the Yukawa and extra U (1) N gauge couplings. Dotted lines in Figure 1 show the changes of the evolution of gauge couplings induced by the variations of α 3 (M Z ) from 0.116 to 0.120. The corresponding interval of variations of α 3 (t) is always considerably wider than the ones for α 1 (t) and α 2 (t). The dependence of α 1 (t) and α 2 (t) on the value of the strong gauge coupling at the EW scale is expected to be relatively weak because α 3 (t) appears only in the two-loop contributions to β 1 and β 2 . It is worthwhile to point out that at high energies the uncertainty in α 3 (t) caused by the variations of α 3 (M Z ) is much bigger in the E 6 SSM than in the MSSM. This happens because in the E 6 SSM the strong gauge coupling grows with increasing renormalisation scale q whereas in the MSSM it decreases at high energies. Thus the uncertainty in α 3 (M X ) in the E 6 SSM is approximately equal to the low energy uncertainty in α 3 (M Z ) while in the MSSM the interval of variations of α 3 (M X ) shrinks drastically. As a consequence it is much easier to achieve the unification of gauge couplings within the E 6 SSM as compared with the MSSM where in the two-loop approximation the exact gauge coupling unification requires α 3 (M Z ) > 0.123, well above the experimentally measured central value [130], [133], [134,135,136,137,138,139,140,141].
The results of the numerical analysis presented in Figure 1 demonstrate that for T S = 2 TeV almost exact unification of the SM gauge couplings can be achieved in the E 6 SSM if α 3 (M Z ) ≈ 0.116. With increasing (decreasing) the effective threshold scale T S the value of α 3 (M Z ), at which the exact gauge coupling unification takes place, becomes lower (greater). In the E 6 SSM T S can be considerably lower than 1 TeV even when the SUSY breaking scale is much larger than a few TeV. To demonstrate this let us assume that all scalars except the SM-like Higgs boson are almost degenerate around m A ≈ M S which is much larger than the masses of all fermions. Then combining Eqs. (36)-(37) one finds If M S ≈ Mg ≈ 10 TeV and µ ≈ µ L ≈ µH 1 ≈ µH 2 ≈ 1 TeV while MW and the masses of the exotic quarks µ D i are of the order of a few TeV, the effective threshold scale tends to be much smaller than 1 TeV. For T S = 400 GeV almost exact unification of the SM gauge couplings in the E 6 SSM can be obtained if α 3 (M Z ) ≈ 0.118 [72]. Thus in this SUSY model the gauge coupling unification can be attained for the values of α 3 (M Z ) which are in agreement with current data.
As was mentioned before the inclusion of the two-loop corrections to the diagonal beta functions could spoil the unification of the SM gauge couplings entirely within the E 6 SSM. These corrections affect the running of gauge couplings much more strongly than in the case of the MSSM because at any intermediate scale the values of the gauge couplings in the E 6 SSM are substantially larger as compared to the ones in the MSSM. The analysis of the RG flow of the SM gauge couplings performed in [84] revealed that Θ i (M X ) are a few times larger in the E 6 SSM than in the MSSM. On the other hand 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. Such cancellation is caused by the structure of the two-loop corrections to the diagonal beta functions in the model under consideration. As a result, the prediction for α 3 (M Z ) obtained using Eq. (35) is considerably lower in the E 6 SSM than in the MSSM.

Gauge symmetry breaking and Higgs sector
In the simplest case the sector responsible for the breakdown of the SU (2) W × U (1) Y × U (1) N gauge symmetry in the E 6 SSM involves two Higgs doublets H u and H d as well as the SM singlet field S. The interactions between these fields are determined by the structure of the gauge group and by the superpotential (14). Including soft SUSY breaking terms and radiative corrections, the resulting Higgs effective potential is the sum of four pieces: where σ a (a = 1, 2, 3) denote the three Pauli matrices, g = 3/5g 1 , Hu , m 2 S as well as trilinear coupling A λ . This part of the scalar potential (39) coincides with the corresponding one in the NMSSM when the NMSSM parameters κ and A κ vanish. Because the only complex phase (of λA λ ) that appears in the tree-level scalar potential (39) can easily be absorbed by a suitable redefinition of the Higgs fields, CP-invariance is preserved in the Higgs sector of the E 6 SSM at tree-level.
The term ∆V represents the contribution of loop corrections to the Higgs effective potential. In the one-loop approximation the contributions of different states to ∆V are determined by their masses, i.e.
where M is the mass matrix for the bosons and fermions in the SUSY model under consideration.
Here the supertrace operator counts positively (negatively) the number of degrees of freedom for the different bosonic (fermionic) fields, while Q is the renormalisation scale. The inclusion of loop corrections draws into the analysis many other soft SUSY breaking parameters which determine masses of different superparticles. Some of these parameters may be complex giving rise to potential sources of CP-violation. At the physical minimum of the scalar potential (39) the Higgs fields develop VEVs The equations for the extrema of the full Higgs boson effective potential in the directions (44) in field space read: where D =Q H d v 2 1 +Q Hu v 2 2 +Q S s 2 andḡ = g 2 2 + g 2 . Instead of specifying v 1 and v 2 it is more convenient to use v = v 2 1 + v 2 2 ≈ 246 GeV and tan β = v 2 /v 1 . The Higgs sector of the E 6 SSM includes ten degrees of freedom. Four of them are massless Goldstone modes which are swallowed by the W ± , Z and Z gauge bosons. The charged W ± bosons gain masses via the interaction with the neutral components of the Higgs doublets H u and H d just in the same way as in the MSSM, resulting in M W = g 2 2 v. Meanwhile the mechanism of the neutral gauge boson mass generation differs substantially. Letting the Z and Z states be the gauge bosons associated with U (1) N and with the SM-like Z boson the Z − Z mass squared matrix is given by The SM singlet fields S must acquire large VEV, s 1 TeV, to ensure that the extra U (1) N gauge boson is sufficiently heavy. In this case the mass of the lightest neutral gauge boson Z 1 is very close to M Z =ḡv/2, while the mass of Z 2 is set by the VEV of the SM singlet field, i.e. M Z ≈ g 1Q S s.
For the analysis of the spectrum of the Higgs bosons in the E 6 SSM we use Eq. (45)-(47) for the extrema to express the soft masses m 2 H d , m 2 Hu , m 2 S in terms of s, v, tan β and other parameters. Because of the conversation of the electric charge, the charged components of the Higgs doublets are not mixed with the neutral Higgs fields. They form a separate sector, whose spectrum is described by a 2 × 2 mass matrix. The determinant of this matrix vanishes leading to the appearance of two Goldstone states and its charge conjugate which are absorbed into the longitudinal degrees of freedom of the W ± gauge boson. Their orthogonal linear combination gains mass where ∆ ± denotes the loop corrections to m 2 H ± . If CP-invariance is preserved then the imaginary parts of the neutral components of the Higgs doublets and the SM singlet field S do not mix with the real parts of these fields. In this case the imaginary parts of the neutral components of the Higgs doublets as well as imaginary part of the SM singlet field S form CP-odd Higgs sector. They compose two neutral Goldstone states which are swallowed by the Z and Z bosons, and one physical state where tan γ = v 2s sin 2β. Two massless pseudoscalars G 0 and G decouple from the rest of the spectrum whereas the physical CP-odd Higgs state A acquires mass In Eq. (54) ∆ A denote loop corrections. Since in the E 6 SSM s must be much larger than v, the value of γ is always small and the physical pseudoscalar is predominantly the superposition of the imaginary parts of the neutral components of the Higgs doublets. In the limit s v the masses of the charged and CP-odd Higgs states are approximately equal to each other.
The CP-even Higgs sector includes Re H 0 d , Re H 0 u and Re S. In the field space basis (h, H, N ), where the mass matrix of the CP-even Higgs sector takes the form [142,143,144]: Taking second derivatives of the Higgs effective potential (39)- (42) and substituting m 2 H d , m 2 Hu , m 2 S from the minimisation conditions (45)-(47) one finds: In Eq.

Higgs spectrum
The qualitative pattern of the Higgs spectrum in the E 6 SSM inspired is determined by the Yukawa coupling λ. Let us start our analysis here from the MSSM limit of the E 6 SSM when λ g 1 . In the case when λ goes to zero s has to be sufficiently large so that µ = λs/ √ 2 is held fixed in order to give an acceptable chargino mass and EWSB. The diagonal entry M 2 33 that is set by the mass of the Z boson tends to be substantially larger than other elements of the mass matrix (56) For small values of λ the top-left 2 × 2 submatrix in Eq. (64) reproduces the mass matrix of the CP-even Higgs sector in the MSSM. Such hierarchical structure of the mass matrix of the CPeven Higgs sector, implies that the mass of the Z boson and the mass of the heaviest CP-even Higgs particle associated with N are almost degenerate. In other words the singlet dominated CP-even state is always very heavy and decouples from the rest of the spectrum, which makes the Higgs spectrum indistinguishable from the one in the MSSM. Its mass is determined by the VEV of the SM singlet field and does not change much if the other parameters λ, tan β and A λ (m A ) vary. The masses of the second lightest Higgs scalar, that is predominantly H, the Higgs pseudoscalar and the charged Higgs states grow when m A rises providing the degeneracy of the corresponding states at m A when m A is much larger than M Z but is less than M Z . In this case the expression for the SM-like Higgs mass m 2 h 1 is essentially the same as in the MSSM. When λ ≥ g 1 the qualitative pattern of the spectrum of the Higgs bosons is rather similar to the one that arises in the NMSSM with the approximate PQ symmetry [145,146,147,148,149]. In the NMSSM and E 6 SSM the growth of the Yukawa coupling λ at low energies entails the increase of its value at the GUT scale M X resulting in the appearance of the Landau pole that spoils the applicability of perturbation theory at high energies [87,88]. The requirement of validity of perturbation theory up to the scale M X sets an upper limit on λ(M t ) for each fixed value of tan β in these models. In the E 6 SSM the restrictions on the low energy values of λ are weaker than in the NMSSM (see Figure 2/left). The presence of exotic matter change the running of the SM gauge couplings so that their values at the intermediate scale rise when the number of extra 5 + 5-plets increases. In the RGEs that describe the evolution of the Yukawa couplings within the NMSSM and E 6 SSM the gauge couplings occur in the right-hand side of these equations with negative sign. As a consequence the growth of the SM gauge couplings prevents the appearance of the Landau pole in the RG flow of the Yukawa couplings. Therefore in the E 6 SSM λ(M t ) are allowed to be larger than in the NMSSM. The upper bound on λ(M t ) grows with increasing tan β since the top-quark Yukawa coupling decreases. At large tan β this bound approaches the saturation limit. In the NMSSM and E 6  . Relying on this mass hierarchy the approximate solutions for the Higgs masses can be obtained. The perturbation theory method yields [142,143,144,145,146]  are neglected. At tree-level the masses of the Higgs bosons can written as where x = A λ 2µ sin 2β and µ = λ √ 2 s. As evident from the explicit expression for m 2 h 1 given above at λ 2 g 2 1 the last term in this expression dominates and the mass squared of the lightest Higgs boson tends to be negative if the auxiliary variable x is not close to unity. A negative eigenvalue of the mass matrix (57)- (62) implies that the vacuum configuration (44) ceases to be a minimum and turns into a saddle point. Near this point there is a direction in field space along which the energy density decreases leading to the instability of the vacuum configuration (44). Thus large deviations of x from unity pulls the mass squared of the lightest Higgs boson below zero destabilising the vacuum. The requirement of stability of the physical vacuum therefore constrains the variable x around unity and limits the range of variations of m A from below and above. As a consequence the masses of the heaviest CP-even, CP-odd and charged Higgs states are almost degenerate around m A and are confined in the vicinity of µ tan β. They are considerably larger than the masses of the Z and lightest CP-even Higgs boson. Together with the experimental lower limit on the mass of the Z boson it maintains the mass hierarchy in the spectrum of the Higgs particles [69].
From the explicit analytic expression for m 2 h 1 it is apparent that at some value of x (or m A ) the lightest CP-even Higgs boson mass attains its maximum value. It corresponds to the value of x for which the fourth term in the expression for m 2 h 1 vanishes. In this case the mass squared of the lightest Higgs boson coincides with the theoretical upper bound on m 2 h 1 given by M 2 11 . The sum of the first and second terms in the expression for M 2 11 are similar to the tree-level upper bound on m 2 h 1 in the NMSSM [150,151]. The third term in Eq. (57) is a contribution coming from the additional U (1) N D-term in the Higgs scalar potential (39)- (42). At tree-level the upper bound on the lightest Higgs mass in the E 6 SSM depends on λ and tan β only. Using the obtained theoretical restrictions on the low energy values of λ as a function of tan β, one can compute the maximum possible value of m h 1 for each particular choice of tan β.
The tree-level upper bound on the mass of the lightest Higgs scalar in the E 6 SSM is presented in Figure 2 (see Figure 2/right) and compared to the corresponding bounds in the MSSM and NMSSM. At moderate values of tan β ∼ 1 − 3 the theoretical restriction on lightest Higgs boson mass in the E 6 SSM and NMSSM exceeds the corresponding limit in the MSSM because of the extra contribution to M 2 11 induced by the first term in the right hand side of Eq. (57) which comes from the additional F -term in the Higgs scalar potentials of the E 6 SSM and NMSSM. For such values of tan β in the E 6 SSM and NMSSM this contribution to the upper bound on m h 1 dominates. Its size is determined by the Yukawa coupling λ. Since the upper limit on the coupling λ caused by the validity of perturbation theory in the NMSSM is more stringent than in the E 6 SSM the tree-level theoretical restriction on m h 1 in the NMSSM is considerably less than in the E 6 SSM at moderate values of tan β. In the framework of the E 6 SSM the upper bound on m h 1 attains a maximum value of 130 GeV at tan β = 1.5 − 1.8. So large tree-level theoretical restriction on the mass of the lightest Higgs scalar means that in this model the contribution of loop corrections to m 2 h 1 is not needed to be as big as in the MSSM and NMSSM in order to get the SM-like Higgs boson with mass around 125 GeV.
With increasing tan β the contribution to M 2 11 associated with the first term in the right hand side of Eq. (57) falls quite rapidly and becomes negligibly small as tan β 10. In contrast the contribution of the SU (2) W and U (1) Y D-terms to M 2 11 (second term in the right hand side of Eq. (57)) grows when tan β increases. At tan β > 4 it exceeds λ 2 2 v 2 sin 2 2β and gives the dominant contribution to the tree-level theoretical restriction on m h 1 . Therefore with increasing tan β the upper bound on the lightest Higgs boson mass in the NMSSM diminishes and approaches the corresponding limit in the MSSM. In the case of the E 6 SSM the third term in the right hand side of Eq. (57), that comes from the extra U (1) N D-term contribution to the Higgs scalar potential (39)- (42), gives the second largest contribution to M 2 11 at very large values of tan β. Because of the contribution of this term the tree-level theoretical restriction on the mass of the lightest Higgs scalar in the E 6 SSM, which also diminishes when tan β rises, is still 6 − 7 GeV larger than the ones in the MSSM and NMSSM even at very large values of tan β. As a consequence at large tan β the presence of the 125-GeV Higgs boson in the particle spectrum of the E 6 SSM does not require as large contribution of loop corrections to m 2 h 1 as in the MSSM and NMSSM.
The inclusion of loop corrections substantially increases the mass of the lightest Higgs scalar in SUSY models. The dominant contribution comes from the loops involving the top quark and its superpartners because of the large top-quark Yukawa coupling h t . Within the MSSM leading one-loop and two-loop corrections to m h 1 increase the upper bound on the lightest Higgs boson mass, which does not exceed Z-boson mass (M Z 91.2 GeV) at the tree-level [152,153], from M Z to 130 GeV (see [154] and references therein). In the leading approximation two-loop upper bound on the lightest Higgs boson mass in the E 6 SSM can be presented in the following form [69] where X t is a stop mixing parameter, M S is a SUSY breaking scale defined as m 2 Q = m 2 U = M 2 S while m 2 Q and m 2 U are soft scalar masses of superpartners of the left-handed and right-handed components of the t-quark respectively. Here the value of m t (M t ) can be computed using the world average mass of the top quark M t = 173.1 ± 0.9 GeV (see [155]) and the relationship between the t-quark pole (M t ) and running (m t (Q)) masses [156,157] Eq. (67) is just a simple generalization of the approximate expressions for the theoretical restriction on the lightest Higgs boson mass obtained in the MSSM [158] and NMSSM [159]. At λ = 0 and g 1 = 0 the right-hand side of Eq. (67) coincides with the theoretical bound on the lightest Higgs mass in the MSSM. The analytic approximation of the two-loop effects given above slightly underestimates the full two-loop corrections. In the MSSM the approximate expression (67) results in the value of the lightest Higgs mass which is typically a few GeV lower than the one which is computed using the Suspect [160] and FeynHiggs [161,162,163,164] packages. It was shown that in the two-loop approximation the mass of the lightest Higgs scalar in the E 6 SSM does not exceed 150 GeV [69].
Although the inclusion of loop corrections changes considerably the lightest Higgs boson mass in the E 6 SSM, it does not change the the qualitative pattern of the spectrum of the Higgs states for λ g 1 and λ > g 1 . The mass of the SM singlet dominated CP-even state is always set by M Z whereas another Higgs scalar, CP-odd and charged Higgs bosons have masses close to m A . In the phenomenologically viable scenarios the masses of all Higgs particles except the lightest Higgs state are much larger than M Z . Moreover when λ > g 1 and, in particular, in the part of the E 6 SSM parameter space where the lightest Higgs boson can be heavier than 100 − 110 GeV even at tree-level, the heaviest CP-even, CP-odd and charged Higgs states lie beyond the multi-TeV range and therefore cannot be detected at the LHC experiments.

LHC signatures
We now turn to the LHC signatures of the E 6 SSM, that permit to distinguish this SUSY model from the MSSM or NMSSM. As discussed earlier, in the simplest phenomenologically viable scenarios the lightest exotic fermionH 0 1 should have mass mH0 1 1 eV. At the same time nextto-lightest exotic fermionH 0 2 may be considerably heavier. Let us assume that all sparticles and exotic states exceptH 0 1 andH 0 2 are rather heavy and can be integrated out. In particular, the parameters are chosen so that all fermion components of the supermultiplets H u α and H d α are heavier than 100 GeV, whereas s ≈ 12 TeV. In this limit the part of the Lagrangian, that describes the interactions ofH 0 1 andH 0 2 with the Z boson and the SM-like Higgs particle can be presented in the following form: where α, β = 1, 2. AlthoughH 0 1 andH 0 2 are substantially lighter than 100 GeV, their couplings to the Z boson and other SM particles can be negligibly small because these states are predominantly the fermion components of the superfields S α . Therefore any possible signal, which H 0 1 andH 0 2 could give rise to at former and present collider experiments, would be extremely suppressed and such states could remain undetected.
The couplings of the SM-like Higgs boson h 1 toH 0 1 andH 0 2 are determined by the masses of these lightest exotic states [94]. SinceH 0 1 is extremely light, it does not affect Higgs phenomenology. The absolute value of the coupling of h 1 to the second lightest exotic particle |X h 22 | |mH0 2 |/v [94]. This coupling gives rise to the decays of h 1 intoH 0 2 pairs with partial width given by The partial decay width (70) depends rather strongly on mH0 2 . To avoid the suppression of the branching ratios for the lightest Higgs decays into SM particles we restrict our consideration here to the GeV-scale masses of the second lightest exotic particle.
In order to compare the partial widths associated with the exotic decays of h 1 (70) with the SM-like Higgs decay rates into the SM particles a set of benchmark points (see Table 2) is specified. In Table 2 the masses of the heavy Higgs states are computed in the leading one-loop approximation. In the case of the lightest Higgs boson mass the leading two-loop corrections are taken into account. In all benchmark scenarios the structure of the Higgs spectrum is very hierarchical, the partial widths of the decays of h 1 into the SM particles are basically the same as in the SM. Therefore in our analysis we use the results presented in [165] where the corresponding decay rates were computed within the SM for different values of the Higgs mass. When m h 1 125 GeV, the SM-like Higgs state decays predominantly into b quark. The corresponding branching ratio is about 60% while the branching ratios associated with Higgs decays into W W and ZZ are about 20% and 2%, respectively [165]. The total decay width of such Higgs boson is about 4 MeV.
The benchmark scenarios (i)-(iv) presented in Table 2 demonstrate that the branching ratio of the exotic decays of h 1 changes from 0.2% to 20% when mH0 2 varies from 0.3 GeV to 2.7 GeV [98]. For smaller (larger) values of mH0 2 the branching ratio of these decays is even smaller (larger). On the other hand, the couplings ofH 0 1 andH 0 2 to the Z boson are so small that these exotic fermions could not be observed before. In particular, their contribution to the Z-boson width tend to be rather small. After being producedH 0 2 sequentially decay intoH 0 1 and fermionantifermion pair via virtual Z. Thus the exotic decays of h 1 result in two fermion-antifermion pairs and missing energy in the final state. Nevertheless, since |R Z12 | is quite small,H 0 2 tends to live longer than 10 −8 s and typically decays outside the detectors. As a consequence, the decay channel h 1 →H 0 2H 0 2 normally gives rise to an invisible branching ratio of the SM-like Higgs boson. Such invisible decays of h 1 take place in the benchmark scenarios (i), (iii), and (iv). In the case of benchmark scenario (ii) |R Z12 | is larger so that τH0 2 ∼ 10 −11 s and some of the decay products ofH 0 2 might be observed at the LHC. Because R Z12 is relatively small,H 0 2 may decay during or after Big Bang Nucleosynthesis (BBN) destroying the agreement between the predicted and observed light element abundances. To preserve the success of the BBN,H 0 2 should decay before BBN, i.e. its lifetime τH0 2 should not be longer than 1 s. This requirement constrains |R Z12 |. Indeed, for mH0 2 = 1 GeV the absolute value of the coupling R Z12 has to be larger than 1×10 −6 [166]. The constraint on |R Z12 | becomes more stringent with decreasing mH0 ). The results of our analysis indicate that it is somewhat problematic to ensure that τH0 The presence of a Z gauge boson and exotic multiplets of matter that compose three 5 + 5 * representations of SU (5) is another very peculiar feature of the E 6 SSM. LHC signatures associated with these states are determined by the structure of the particle spectrum that varies substantially depending on the choice of the parameters. At tree-level the masses of the Z boson and fermion components of 5 + 5 * supermultiplets are set by the VEV of the SM singlet field S, that remains a free parameter in this models. Therefore the masses of these states cannot be predicted. The lower experimental limits on the Z mass, that comes from the direct searches (pp → Z → l + l − ) conducted at the LHC experiments, are already very stringent and vary around 3.8 − 3.9 TeV [167,168]. This means that the scenarios with s < 10 − 10.   have been excluded. Possible Z decay channels in E 6 inspired SUSY models were studied in [169,170]. Assuming that f αβ andf αβ are very small the masses of the fermion components of extra 5 + 5 * supermultiplets of matter are given by where µ D i are the masses of the SU (3) C colour triplets of exotic quarks with electric charges ±1/3 and µ Hα are the masses of the SU (2) W doublets of the Inert Higgsino states. Here we set κ ij = κ i δ ij and λ αβ = λ α δ αβ . The requirement of the validity of perturbation theory up to the GUT scale M X sets stringent upper bounds on the low-energy values of the Yukawa couplings κ i and λ α . Nevertheless the low-energy values of these couplings are allowed to be as large as g 1 (q) ≈ g 1 (q) ≈ 0.46 − 0.48. On the other hand couplings κ i and λ α must be large enough to ensure that the exotic fermions are sufficiently heavy to avoiding conflict with direct particle searches at present and former accelerators. Although nowadays there are clear indications that Z boson and sparticles have to be rather heavy some of the exotic fermions can be relatively light in the E 6 SSM. This happens, for example, if the Yukawa couplings of the exotic particles κ ij and λ α have hierarchical structure similar to the one observed in the ordinary quark and lepton sectors. Then Z boson can be much heavier than 10 TeV and the only manifestation of this SUSY extension of the SM may be the presence of light exotic quark and/or Inert Higgsino states in the particle spectrum. If the relatively light exotic quarks of the nature described above do exist, they might be accessed through direct pair hadroproduction. The lifetime and decay modes of the lightest exotic quarks are determined by the operators that break the Z H 2 symmetry. Since in order to suppress FCNCs the Yukawa couplings of exotic particles to the quarks and leptons of the first two generations must be rather small, here we assume that exotic states couple most strongly with the third family fermions and bosons. Then, because the lightest exotic quarks are R-parity odd states, they decay either via if exotic quarks D i are diquarks or via In general exotic squarks tend to be considerably 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 can be relatively light. This happens when 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. Such mixing may result in the large mass splitting between the appropriate mass eigenstates. As a consequence the lightest exotic squark may be much lighter than all other scalars. Moreover, in principle, it can be even lighter than the lightest exotic quark. If this is a case then in the variants of the E 6 SSM with approximate Z H 2 symmetry the lightest exotic squark decays into eitherD if it is a scalar diquark or if exotic squark is a scalar leptoquark. In the limit, when the couplings of this sfermion to the quarks and leptons of the first two generations are rather small, the lightest exotic squarks can only be pair produced at the LHC. Therefore the presence of lightD in the particle spectrum is expected to lead to some enhancement of the cross sections of either pp → ttbb + X if exotic squarks are diquarks or pp → ttτ + τ − + X and pp → bb + E miss T + X if these squarks are leptoquarks. On the other hand in the variants of the E 6 SSM with exactZ H 2 symmetry the Z E 2 symmetry conservation implies that the final state in the decay ofD should always contain the lightest exotic fermionH 0 1 [72]. Because the lightest exotic squark is R-parity even state whereas H 0 1 is R-parity odd particle the final state in the decay ofD should also involve the lightest ordinary neutralino to ensure that R-parity is conserved. As a consequence in such models the decay patterns of the lightest exotic squarks and their LHC signatures are rather similar to the ones that appear in the case of the lightest exotic quarks. The presence of relatively light exotic quark and squark can substantially modify the LHC signatures associated with the gluinos [92].
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. Most searches focus on leptoquarks or diquarks which have integer-spin so that they can be either scalars or vectors. Such objects can be coupled directly to either a pair of quarks or to quark and lepton. The most stringent constraints on the masses of scalar leptoquarks and scalar diquarks come from the non-observation of these exotic states at the LHC experiments. ATLAS and CMS collaborations ruled out first However the LHC lower bounds on the masses of exotic quarks/squarks are not always directly applicable in the case of the E 6 SSM. For instance, it is expected that scalar diquarks are mostly produced singly at the LHC and decay into final state that contains two quarks. At the same time within the E 6 SSM the couplings of all exotic scalars to the fermions of the first and second generation should be rather small to avoid processes with non-diagonal flavour transitions. Therefore in this SUSY model diquarks can only be pair produced. It is also worthwhile to point out that the lightest exotic quarks in the E 6 SSM give rise to collider signatues which are very different from the commonly established ones associated with the scalar leptoquarks or diquarks that have been thoroughly studied. Indeed, it is commonly assumed that these scalars decay into quark-quark or quark-lepton without missing energy. On the other hand in the E 6 SSM exotic quarks are fermions and therefore R-parity odd states. Thus R-parity conservation necessarily leads to the missing energy and transverse momentum in the final state.
Because of this the pair production of the lightest exotic quark with the baryon number, which is twice larger than that of ordinary ones, and the pair production of gluinos at the LHC may result in the enhancement of the same cross section of pp → ttbb + E miss T + X. The SU (2) W doublets of the Inert Higgsino states can be also light or heavy depending on their free parameters. When at least one coupling λ α is of the order of unity it can induce a large mixing in the Inert Higgs sector that may lead to relatively light Inert Higgs bosons. Since these bosons have very small couplings to the fermions of the first and second generation at the LHC the corresponding states can be produced in pairs via off-shell W and Z-bosons. As a consequence their production cross section is relatively small even when these particles have masses below the TeV scale. After being produced they sequentially decay into the third generation fermions that should lead to some enlargement of the cross sections of pp → QQQ Q and pp → QQτ + τ − production, where Q and Q are heavy quark of the third generation.
As follows from Eq. (71) the lightest Inert Higgsinos can be relatively light if the corresponding Yukawa coupling λ α is sufficiently small. If all other exotic states and sparticles are rather heavy the corresponding fermionic states can be produced at the LHC via weak interactions only. As a consequence their production cross section is considerably smaller than the production cross section of the exotic quarks (see Figure 3). The Inert Higgsino states decay predominantly into the lightest exotic fermions (H 0 1 orH 0 2 ) as well as an on-shell Z or W boson. Thus when pair produced Inert Higgsinos decay they should lead to some enhancements in the rates of pp → ZZ + E miss T + X, pp → W Z + E miss T + X and pp → W W + E miss T + X. Similar enhancement of these cross sections could be caused by the pair production of ordinary chargino and neutralino in the MSSM if the mass of the LSP is negligibly small. Using the corresponding results of the analysis of ATLAS and CMS collaborations [178, 179, 180] one can conclude that the mass of the SU (2) W doublets of the Inert Higgsino states has to be larger than 650 GeV.

Conclusions
The breakdown of an extended gauge symmetry in the string-inspired E 6 GUTs may result in a variety of extensions of the SM with softly broken SUSY at low energies including models based on the SM gauge group, like the MSSM and NMSSM, as well as U (1) extensions of the MSSM, etc. Among U (1) extensions of the MSSM inspired by E 6 GUTs there is 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. In the U (1) N extension of the MSSM the lepton asymmetry, which may be induced by the heavy right-handed neutrino decays, can be partially converted into baryon asymmetry via sphaleron processes [181,182]. In this Exceptional Supersymmetric Standard Model (E 6 SSM) the extra U (1) N gauge symmetry forbids the term µH d H u in the superpotential, but permits the term λS(H u H d ), where S is a SM singlet superfield that carries U (1) N charge. When S develops VEV breaking U (1) N gauge symmetry it also gives rise to an effective µ term. Thus within the E 6 SSM the µ problem of the MSSM is solved in a similar way to that in the NMSSM, but without the accompanying problems of singlet tadpoles or domain walls.
In this review article we discussed the particle content, the global symmetries, which allows to suppress FCNCs and rapid proton decay, as well as the RG flow of gauge couplings in the E 6 SSM. The low energy matter content of this SUSY model includes three copies of 27 i representations of E 6 so that anomalies get canceled generation by generation. In addition an extra pair of SU (2) W doublets L 4 and L 4 should survive to low energies to ensure high energy gauge coupling unification. As a consequence the E 6 SSM involves extra matter beyond the MSSM contained in three supermultiplets of exotic charge 1/3 quarks (D i and D i ), two pairs of SU (2) W doublets of Inert Higgs states, three SM singlet superfields which carry U (1) N charges, L 4 , L 4 and Z vector superfield. As in the MSSM, the gauge symmetry of the E 6 SSM does not forbid baryon and lepton number violating interactions that give rise to rapid proton decay. Moreover in general relatively light exotic states induce unacceptably large flavor changing processes. To suppress the most dangerous baryon and lepton number violating operators one can impose either Z L 2 or Z B 2 discrete symmetry which implies that the exotic quarks are either diquarks (Model I) or leptoquarks (Model II). In order to avoid the appearance of the FCNCs at the tree level one can postulate an approximate Z H 2 symmetry, under which all supermultiplets of matter except a pair of Higgs doublets (H d and H u ) and one SM singlet superfield S are odd. Instead of Z H 2 , Z L 2 and Z B 2 one can use a single discreteZ H 2 symmetry which forbids operators giving rise to too rapid proton decay and tree-level flavor-changing transitions. The Higgs supermultiplets H u , H d and S as well as L 4 and L 4 are even under theZ H 2 symmetry whereas all other matter fields are odd. In this case the exotic quarks are leptoquarks.
The results of the analysis of the two-loop RG flow within the E 6 SSM were presented taking into account kinetic term mixing between U (1) Y and U (1) N factors. If there is no mixing between U (1) Y and U (1) N near the GUT scale M X then the off-diagonal gauge coupling, which describes such mixing, remains negligibly small at any intermediate scale between M X and TeV scale. In this limit the gauge coupling of the extra U (1) N is always close to the U (1) Y gauge coupling. On the other hand the values of the SM gauge couplings at high energies are considerably larger in the E 6 SSM than in the MSSM due to the presence of the extra supermultiplets of exotic matter in the U (1) N extensions of the MSSM. Our analysis revealed that the gauge coupling unification in the E 6 SSM can be achieved for phenomenologically acceptable values of α 3 (M Z ), consistent with the central measured low energy value of this coupling.
Because of the larger gauge couplings the theoretical restrictions on the low energy values of the Yukawa couplings coming from the requirement of the validity of perturbation theory up to the scale M X get relaxed in the E 6 SSM as compared with the MSSM and NMSSM. As a consequence for moderate values of tan β the tree-level upper bound on the SM-like Higgs boson mass can be considerably bigger in the E 6 SSM than in the MSSM and NMSSM. In this SUSY model it can be bigger than 115 − 125 GeV so that the contribution of loop corrections to the mass of the lightest Higgs scalar is not needed to be as large as in the MSSM and NMSSM in order to obtain 125 GeV Higgs boson.
In this article the gauge symmetry breaking and the spectrum of the Higgs bosons within the E 6 SSM were reviewed as well. In the U (1) N extensions of the MSSM the SM singlet Higgs field S is required to acquire a very large VEV S = s/ √ 2, where s > 10 TeV, to ensure that the Z boson and exotic fermions gain sufficiently large masses. In particular, the results of the analysis of the LHC data imply that the U (1) N gauge boson must be heavier than 3.8 − 3.9 TeV. When CP-invariance is preserved, the E 6 SSM Higgs spectrum includes three CP-even, one CPodd and two charged bosons. The SM singlet dominated CP-even state is almost degenerate with the Z gauge boson. The masses of another CP-even and charged Higgs bosons are set by the mass of Higgs pseudoscalar m A . All these states tend to be substantially heavier than the lightest Higgs scalar that manifests itself in the interactions with other SM particles as a SM-like Higgs boson. In the part of the E 6 SSM parameter space, where the lightest Higgs state can be heavier than 100 − 110 GeV at the tree-level, all other Higgs bosons lie beyond the multi-TeV range and therefore cannot be discovered at the LHC.
We also considered possible manifestations of the E 6 SSM that may be observed at the LHC in the near future. The simplest phenomenologically viable scenarios imply that LSP and NLSP are the lightest exotic states (H 0 1 andH 0 2 ), which are formed by the fermion components of the SM singlet superfields S α . One of these fermionsH 0 1 should be much lighter than 1 eV composing hot dark matter in the Universe. Such states give only a very minor contribution to the dark matter density. The NLSPH 0 2 can have mass of the order of 1 GeV giving rise to nonstandard decays of the 125 GeV Higgs boson. SinceH 0 2 tends to live longer than 10 −8 sec. it decays outside the detectors. Therefore the decay channel h 1 →H 0 2H 0 2 results in an invisible branching ratio of the SM-like Higgs state. The corresponding branching ratio can be as large as 20%.
Other possible manifestations of the E 6 SSM, which can permit to distinguish this model from the MSSM or NMSSM, are associated with the presence of the Z gauge boson and exotic supermultiplets of matter that compose three 5 + 5 * representations of SU (5). The most spectacular LHC signals can come from the exotic color states and Z . The production of the Z boson should lead to unmistakable signal pp → Z → l + l − at the LHC. Assuming that the Z H 2 symmetry is mainly broken by the operators involving quarks and leptons of the third generation, the pair production of the lightest exotic quarks with masses in a few TeV range can give rise to the enhancement of the cross section of either pp → ttbb + E miss T + X if exotic quarks are diquarks or pp → ttτ + τ − + E miss T + X and pp → bb + E miss T + X if exotic quarks are leptoquarks. Because of the large mass splitting in the exotic squark sector, which can be caused by the heavy D-fermion, one of the exotic squarks can be relatively light. If this is a case then the pair production of the superpartners of D-fermions may result in some enlargement of the cross sections of either pp → ttbb + X when exotic squarks are diquarks or pp → ttτ + τ − + X and pp → bb + E miss T + X if these squarks are leptoquarks. As compared with the exotic quarks and squarks the production of Inert Higgs bosons and Inert Higgsinos is rather suppressed at the LHC. The discovery of Z and new exotic states predicted by the E 6 SSM would point towards an underlying E 6 gauge structure at high energies and open a new era in elementary particle physics.