A Review of the Exceptional Supersymmetric Standard Model

Abstract

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 $U{\left(1\right)}_N$ gauge symmetry originating from a string-inspired $E_6$ GUTs. Only in this $U\left(1\right)$ extension of the minimal supersymmetric standard model (MSSM) can the right-handed neutrinos be superheavy, providing a mechanism for the baryon asymmetry generation. The particle content of this exceptional supersymmetric standard model (E${}_6$SSM) includes three 27 representations of the $E_6$ group, to ensure anomaly cancellation. In addition it also contains a pair of $SU{\left(2\right)}_W$ doublets as required for the unification of gauge couplings. Thus, E${}_6$SSM involves exotic matter beyond the MSSM. We consider symmetries that permit suppressing flavor changing processes and rapid proton decay, as well as gauge coupling unification, the gauge symmetry breaking 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.

1. Introduction

Symmetries play a key role in modern high energy physics. Indeed, it was realized a long time ago that light hadron resonances form representations of the $SU\left(3\right)$ group, which is associated with light quark flavors, while the physics of strong interactions is described by the colored $SU{\left(3\right)}_C$ gauge symmetry. It was also established that weak and electromagnetic forces represent electroweak (EW) interactions based on the $SU{\left(2\right)}_W\times U{\left(1\right)}_Y$ gauge group. Within the standard model (SM) of elementary particles, which describes rather precisely almost all experimental data measured in earth based experiments, $SU{\left(2\right)}_W\times U{\left(1\right)}_Y$ is spontaneously broken to the abelian $U{\left(1\right)}_{em}$ gauge group associated with electromagnetism by means of the Higgs mechanism. The latter predicts the existence of the scalar (i.e., Higgs particle) which was recently discovered at the LHC. Thus, the Lagrangian of the SM is invariant under the transformations of the Pointcaré group and $SU{\left(3\right)}_C\times SU{\left(2\right)}_W\times U{\left(1\right)}_Y$ gauge symmetry. The transformations of the Pointcaré group involve time and space translations as well as a set of transformations of the Lorentz group that includes Lorentz boosts along three axes and rotations about them.

At very high energies, the SM can be embedded into GUTs [1] based on the $SU\left(5\right)$ or $SO\left(10\right)$ gauge groups. In the case of $SU\left(5\right)$ GUTs, each SM family of quarks and leptons fills in a complete one antifundamental and one antisymmetric second-rank tensor representations of $SU\left(5\right)$, i.e., $\overline{5}+10$. Within $SO\left(10\right)$ GUTs, each family of SM fermions may belong to a single 16-dimensional spinor representation of $SO\left(10\right)$. Such models predict the existence of right-handed neutrinos, which may be used for the see-saw mechanism [2] and leptogenesis [3].

SUSY GUTs permit to place fermions and bosons of the SM within one supermultiplet. To combine Pointcaré and gauge (internal) symmetries to attain the unification of gauge interactions with gravity, one needs to overcome the Coleman–Mandula theorem. According to it, in the most general case, quantum field theory has a symmetry that is a product of the gauge and Pointcaré groups [4]. This theorem can be overcome within graded Lie algebras. The structure of these algebras can be presented as

$$\left[\widehat{B},\widehat{B}\right]=\widehat{B},\left[\widehat{B},\widehat{F}\right]=\widehat{F},\left\{\widehat{F},\widehat{F}\right\}=\widehat{B},$$

(1)

where $\widehat{B}$ and $\widehat{F}$ are bosonic and fermionic generators. Supersymmetries are Graded Lie algebras that involve the Pointcaré algebra. The simplest $N=1$ supersymmetry includes a set of generators of the Pointcaré group (bosonic generators) and a single Weyl spinor operator $Q_{\alpha }$ as well as its complex conjugate $Q_{\alpha }^{†}={\overline{Q}}_{\dot{\alpha }}$ (fermionic generators). SUSY algebra implies that each supermultiplet has the same number of bosonic and fermionic degrees of freedom.

In $N=1$, SUSY GUTs with the $E_6$ gauge group the fundamental representation of $E_6$, which decomposes under $SO\left(10\right)\times U{\left(1\right)}_{\psi }$ subgroup, as

$$27\to \left(16,\frac{1}{\sqrt{24}}\right)\oplus \left(10,-\frac{2}{\sqrt{24}}\right)\oplus \left(1,\frac{4}{\sqrt{24}}\right),$$

(2)

contains one family of the SM fermions and Higgs doublet. The Higgs bosons are components of $\left(10,-{\displaystyle \frac{2}{\sqrt{24}}}\right)$. The SM gauge bosons are assigned to the adjoint representation of $E_6$, i.e., a 78-plet. In $N=2$, SUSY GUTs with the $E_8$ gauge symmetry all SM particles are components of 248 representation of $E_8$. This representation decomposes under the $E_6$ subgroup of $E_8$ as follows:

$$248\to 78\oplus 3\times 27\oplus 3\times \overline{27}\oplus 8\times 1.$$

(3)

The local SUSY (supergravity) leads to a partial unification of gauge interactions with gravity [5,6,7]. Because supergravity (SUGRA) is a non-renormalizable theory, it should be regarded as a low energy limit of some renormalizable or finite theory. The best candidate for such theory is a ten-dimensional superstring theory with $E_8\times E_8^{\prime }$ gauge symmetry [8]. Compactification of the extra dimensions in this theory gives rise to the breakdown of $E_8$ to $E_6$ or its subgroups associated with the observable sector [9]. The remaining $E_8^{\prime }$ gauge group plays the role of a hidden sector in which the breakdown of local SUSY is induced. As a consequence, a set of soft SUSY breaking terms [10,11,12,13], which is characterized by the gravitino mass ($m_{3/2}$), is generated. The non-perturbative effects in the hidden sector which cause the breakdown of local SUSY may also induce a large mass hierarchy between $m_{3/2}$ and the Planck scale $M_P$ [14].

When $m_{3/2}\ll M_P$, the breakdown of the $E_6$ gauge group near the GUT scale $M_X$ can result in a variety of SUSY models at low energies including MSSM and its extensions with an extra $U{\left(1\right)}^{\prime }$ gauge symmetry, which is a linear combination of $U{\left(1\right)}_{\chi }$ and $U{\left(1\right)}_{\psi }$, i.e.,

$$U{\left(1\right)}^{\prime }=U{\left(1\right)}_{\chi }cos{\theta }_{E_6}+U{\left(1\right)}_{\psi }sin{\theta }_{E_6}.$$

(4)

Here, $U{\left(1\right)}_{\psi }$ and $U{\left(1\right)}_{\chi }$ can originate from the breakdown of $E_6$, i.e., $E_6\to$$SO\left(10\right)\times U{\left(1\right)}_{\psi }$, $SO\left(10\right)\to SU\left(5\right)\times U{\left(1\right)}_{\chi }$, whereas the SM gauge group is a subgroup of $SU\left(5\right)$, i.e., $SU\left(5\right)\supset SU{\left(3\right)}_C\times SU{\left(2\right)}_W\times U{\left(1\right)}_Y$. In the simplest case, $U{\left(1\right)}_{\chi }\times U{\left(1\right)}_{\psi }$ is broken down to its discrete subgroup $Z_2^M={(-1)}^{3(B-L)}$, 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 $SU{\left(2\right)}_W$ 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. Therefore, it can play the role of dark matter. To generate the masses of all quarks and charged leptons, the MSSM superpotential has to include the following sum of the products of chiral superfields

$$W_{\mathrm{MSSM}}=y_{ab}^U{Q}_a{u}_b^c{H}_2+y_{ab}^D{Q}_a{d}_b^c{H}_1+y_{ab}^L{L}_a{e}_b^c{H}_1+\mu H_1{H}_2,$$

(5)

where a and b are family indices that run from 1 to 3. In Equation (5), the left-handed quark and lepton doublets are denoted by $Q_a$ and $L_a$; the right-handed charged leptons and up- and down-type quarks are denoted by $e_a^c$, $u_a^c$ and $d_a^c$, respectively; and the Yukawa couplings $y_{ab}^U$, $y_{ab}^D$ and $y_{ab}^L$ are dimensionless $3\times 3$ matrices. The analysis of the renormalization group (RG) flow of gauge couplings indicates that they converge to a common value near the scale $M_X\simeq 2\times {10}^{16}\mathrm{GeV}$ in the framework of the MSSM [15,16,17,18]. This allows one to embed the MSSM into SUSY GUTs.

The MSSM superpotential in Equation (5) contains only one bilinear term $\mu H_1{H}_2$. This term can be present before SUSY is broken. Therefore, the parameter $\mu$ is expected to be either zero or of the order of $M_X$. If $\mu \sim M_X$, then the EW symmetry breaking (EWSB) does not take place. In contrast, when $\mu$ vanishes near the GUT scale, $M_X$ it is not induced below $M_X$ due to the non-renormalization theorems [19,20]. In this case, near the physical vacuum $<H_d>=0$ and the down-type quarks as well as the charged leptons remain massless. To ensure the correct EWSB, $\mu$ should 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 $Z_3$ symmetry (${\Phi }_i\to e^{2\pi i/3}{\Phi }_i$) forbids the bilinear term $\mu \left(H_1{H}_2\right)$. The superpotential of the NMSSM is given by [21]

$$W_{\mathrm{NMSSM}}=\lambda S\left(H_1{H}_2\right)+{\displaystyle \frac{\kappa }{3}}{S}^3+W_{\mathrm{MSSM}}(\mu =0),$$

(6)

where S is an extra singlet superfield. It acquires a vacuum expectation value (VEV), i.e., $\langle S\rangle =s/\sqrt{2}$, inducing an effective $\mu$ parameter ($\mu =\lambda s/\sqrt{2}\sim M_S$). The non-zero value of the coupling $\kappa$ in Equation (6) explicitly breaks an additional global $U\left(1\right)$ symmetry, which is a common way to avoid the appearance of the axion in the particle spectrum. However, the VEVs of the Higgs fields break the exact $Z_3$ symmetry, leading to the formation of domain walls in the early universe [22]. Such domain structure of vacuum creates unacceptably large anisotropies in the 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 $U{\left(1\right)}^{\prime }$ extensions of the MSSM inspired by $E_6$ the extra $U{\left(1\right)}^{\prime }$ gauge symmetry in Equation (4) forbids an elementary $\mu$ term if ${\theta }_{E_6}\ne 0$ or $\pi$. Nevertheless, these extensions of the SM allow the interaction $\lambda S\left(H_d{H}_u\right)$ in the superpotential while the $S^3$ term is forbidden by the $U{\left(1\right)}^{\prime }$ gauge symmetry. Near the scale $M_S$, superfield S develops a non-zero VEV that breaks $U{\left(1\right)}^{\prime }$ inducing an effective $\mu$ term of the required size. There are no problems associated with the appearance of domain walls in such models because there is no discrete $Z_3$ symmetry. Different aspects of the phenomenology of the $U{\left(1\right)}^{\prime }$ extensions of the MSSM inspired by $E_6$ have been extensively studied in the past [26,27,28,29,30,31,32,33,34,35,36]. Previously, the implications of these SUSY extensions of the SM have been studied for the muon anomalous magnetic moment [37,38], the EWSB [39,40,41,42,43,44,45], the electric dipole moment of the electron [46] and of the tau lepton [47], neutrino physics [48,49], lepton flavor violating processes such as $\mu \to e\gamma$ [50], fermion mass hierarchy and mixing [51], CP-violation in the Higgs sector [52], collider signatures associated with the exotic quarks and squarks [53], leptogenesis [54,55,56,57], EW baryogenesis [58,59], the $Z^{\prime }$ mass limits [60] and neutralino sector [44,46,47,50,61,62,63,64,65,66,67,68]. The Higgs sector in these SUSY models was explored in [45,68,69,70,71].

In this review article, we consider a specific $E_6$ inspired SUSY realization of the above $U{\left(1\right)}^{\prime }$ type model associated with ${\theta }_{E_6}=arctan\sqrt{15}$. This choice of Abelian $U{\left(1\right)}^{\prime }$ corresponds to $U{\left(1\right)}_N$ gauge symmetry. Thus, such an exceptional supersymmetric standard model (E${}_6$SSM) [69,70] is based on $SU{\left(3\right)}_C\times SU{\left(2\right)}_W\times U{\left(1\right)}_Y\times U{\left(1\right)}_N$ gauge symmetry. Right-handed neutrinos in this model do not participate in the gauge interactions. Therefore, only in such a $U{\left(1\right)}^{\prime }$ extension of the MSSM inspired by $E_6$ GUTs the right-handed neutrinos can be superheavy. This allows using a see-saw mechanism to generate the mass hierarchy in the lepton sector. Moreover, the decays of the heavy Majorana right-handed neutrinos may generate the lepton and baryon asymmetries within this SUSY model [54,55,56].

The layout of this paper is as follows. In Section 2, we specify the $U{\left(1\right)}_N$ extensions of the MSSM and discuss global symmetries that prevent non-diagonal flavor transitions as well as rapid proton decay in these SUSY models. The two-loop RG flow of the SM gauge couplings in the framework of the E${}_6$SSM is examined in Section 3. The Higgs sector dynamics and the emerging spectrum are discussed in Section 4 and Section 5, respectively. In Section 6, the possible LHC signatures of the E${}_6$SSM are considered. Section 7 is reserved for our conclusions.

2. The $U{\left(1\right)}_N$ Extensions of the MSSM

The E${}_6$SSM implies that near the scale $M_X$ the gauge symmetry of the $E_6$ GUTs is broken down to $SU{\left(3\right)}_C\times SU{\left(2\right)}_W\times U{\left(1\right)}_Y\times U{\left(1\right)}_N\times Z_2^M$ [69,70]. In the $U{\left(1\right)}^{\prime }$ extensions of the MSSM inspired by $E_6$, the anomalies are canceled automatically if the low-energy particle spectrum involves complete representations of $E_6$. Consequently, in the E${}_6$SSM, the particle spectrum is extended by a number of exotics so that it contains three complete 27-dimensional representations of $E_6$ (${27}_i$ with $i=1,2,3$). These ${27}_i$ supermultiplets decompose under the $SU\left(5\right)\times U{\left(1\right)}_N$ subgroup of $E_6$ as follows:

$$\begin{array}{c}{27}_i\to {\left(10,{\displaystyle \frac{1}{\sqrt{40}}}\right)}_i+{\left(5^{*},{\displaystyle \frac{2}{\sqrt{40}}}\right)}_i+{\left(5^{*},-{\displaystyle \frac{3}{\sqrt{40}}}\right)}_i+{\left(5,-{\displaystyle \frac{2}{\sqrt{40}}}\right)}_i+{\left(1,{\displaystyle \frac{5}{\sqrt{40}}}\right)}_i+{\left(1,0\right)}_i.\end{array}$$

(7)

The first and second quantities in the brackets represent the $SU\left(5\right)$ representation and extra $U{\left(1\right)}_N$ charge, respectively. An ordinary SM family is assigned to ${\left(10,{\displaystyle \frac{1}{\sqrt{40}}}\right)}_i+{\left(5^{*},{\displaystyle \frac{2}{\sqrt{40}}}\right)}_i$. Right-handed neutrinos $N_i^c$ are associated with the last term in Equation (7), ${\left(1,0\right)}_i$. The next-to-last term, ${\left(1,\frac{5}{\sqrt{40}}\right)}_i$, represents new SM-singlet fields $S_i$. These fields have non-zero $U{\left(1\right)}_N$ charges. Therefore, they survive down to the EW scale. The pair of $SU{\left(2\right)}_W$-doublets ($H_i^d$ and $H_i^u$) which belong to ${\left(5^{*},-{\displaystyle \frac{3}{\sqrt{40}}}\right)}_i$ and ${\left(5,-{\displaystyle \frac{2}{\sqrt{40}}}\right)}_i$ have the quantum numbers of Higgs doublets. These multiplets form either Higgs or Inert Higgs $SU{\left(2\right)}_W$ multiplets, i.e., Higgs-like doublets that do not develop VEVs. Other components of the corresponding $SU\left(5\right)$ multiplets form color triplets of exotic quarks ${\overline{D}}_i$ and $D_i$. These quarks carry electric charges $\left(\pm {\displaystyle \frac{1}{3}}\right)$. They also have B–L charges $\left(\pm {\displaystyle \frac{2}{3}}\right)$. Therefore, in phenomenologically viable $U{\left(1\right)}_N$ extensions of the MSSM, they can be either diquarks or leptoquarks.

In addition to the complete ${27}_i$ multiplets, the splitting of ${27}_l^{\prime }$ and ${\overline{{27}^{\prime }}}_l$ within the $E_6$ GUTs can give rise to a set of $M_l$ and ${\overline{M}}_l$ supermultiplets that have opposite quantum numbers. In the simplest case, the E${}_6$SSM spectrum is supplemented by $SU{\left(2\right)}_W$ doublet $L_4$ and anti-doublet ${\overline{L}}_4$ from the extra ${27}^{\prime }$ and $\overline{{27}^{\prime }}$, where $L_4$ supermultiplet has the quantum numbers of left-handed leptons. This allows preserving gauge coupling unification. Thus, the E${}_6$SSM involves $Z^{\prime }$ and exotic matter beyond the MSSM. Extra matter fill in three $5+5^{*}$ representations of $SU\left(5\right)$ plus three SM singlet superfields $S_i$. Such SUSY extensions of the SM can originate from the orbifold GUTs [57,72].

Over the last fifteen years, several variants of the E${}_6$SSM have been proposed [69,70,72,73,74,75,76,77,78,79,80,81,82]. The $U{\left(1\right)}_N$ extensions of the MSSM have been investigated in the context of non-standard neutrino models [49], EWSB [43,44,45], Z–$Z^{\prime }$ mixing [61], dark matter [83], the RG flow of couplings [44,84] and the neutralino sector [44,61,62]. In the vicinity of the quasi-fixed point the theoretical upper bound on the mass of the lightest Higgs boson and the RG flow of the Yukawa couplings were studied in [85,86]. This quasi-fixed point arises as a result of the intersection of the quasi-fixed and invariant lines [87,88]. Detailed studies of the E${}_6$SSM have established that extra exotic matter and $Z^{\prime }$ 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 [81,86,94,95,96,97,98,99,100]. The particle spectrum and associated phenomenological implications within the constrained version of the E${}_6$SSM (cE${}_6$SSM) and its modifications were explored in [91,101,102,103,104,105,106] while the degree of fine tuning was examined in [107,108]. The threshold corrections and their impact in the cE${}_6$SSM were analyzed in [109]. The renormalization of the VEVs in the E${}_6$SSM was considered in [110,111].

The superpotential of the $U{\left(1\right)}_N$ extensions of the MSSM contains the renormalizable part that comes from the $27\times 27\times 27$ decomposition of the $E_6$ fundamental representation. It can be written as

$$\begin{array}{ccc}\hfill W_{E_6}& =& W_0+W_1+W_2,\hfill \\ \hfill W_0& =& {\lambda }_{ijk}{S}_i\left(H_j^d{H}_k^u\right)+{\kappa }_{ijk}{S}_i\left(D_j{\overline{D}}_k\right)+h_{ijk}^N{N}_i^c\left(H_j^u{L}_k\right)+h_{ijk}^U{u}_i^c\left(H_j^u{Q}_k\right)+\hfill \\ \hfill & +& h_{ijk}^D{d}_i^c\left(H_j^d{Q}_k\right)+h_{ijk}^E{e}_i^c\left(H_j^d{L}_k\right),\hfill \\ \hfill W_1& =& g_{ijk}^Q{D}_i\left(Q_j{Q}_k\right)+g_{ijk}^q{\overline{D}}_i{d}_j^c{u}_k^c,\hfill \\ \hfill W_2& =& g_{ijk}^N{N}_i^c{D}_j{d}_k^c+g_{ijk}^E{e}_i^c{D}_j{u}_k^c+g_{ijk}^D\left(Q_i{L}_j\right){\overline{D}}_k,\hfill \end{array}$$

(8)

where $i,j,k=1,2,3$. In Equation (8), the summation over repeated family indexes is implied. The part of the superpotential in Equation (8) possesses a global $U\left(1\right)$ symmetry which is a linear superposition of $U{\left(1\right)}_Y$ and $U{\left(1\right)}_{\chi }$. This $U\left(1\right)$ symmetry is associated with B–L number conservation. On the other hand, the baryon and lepton numbers are violated if all terms in $W_1$ and $W_2$ are simultaneously present in $W_{E_6}$. In other words, the baryon and lepton numbers of the exotic quarks cannot be defined so that the Lagrangian is invariant separately under $U{\left(1\right)}_B$ and $U{\left(1\right)}_L$ global symmetries. Thus, as in any other SUSY extension of the SM, the gauge symmetry of the models under consideration does not forbid lepton and baryon number violating operators. Because of this, all these models in general suffer from problems related with rapid proton decay.

Moreover, exotic states in the $U{\left(1\right)}_N$ extensions of the MSSM give rise to new Yukawa interactions that may induce unacceptably large flavor changing processes. Indeed, in the most general case three families of $H_i^u$ and $H_i^d$ may couple to ordinary quarks and charged leptons of different generations leading to the phenomenologically unwanted non-diagonal flavor transitions even at the tree level. Such flavor changing interactions contribute to the amplitude of $K^0$–${\overline{K}}^0$ oscillations. They also result in new channels of muon decay such as $\mu \to e^{-}{e}^{+}{e}^{-}$. To suppress flavor changing neutral currents (FCNCs), one can postulate $Z_2^H$ symmetry. If all matter supermultiplets except one pair of $H_i^u$ and $H_i^d$ (say $H_d\equiv H_3^d$ and $H_u\equiv H_3^u$) as well as one SM-type singlet superfield ($S\equiv S_3$) are odd under this symmetry, then only $H_u$ interacts with up-type quarks and only $H_d$ couples to the down-type quarks and charged leptons [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 flavor 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, $\langle S_3\rangle =s/\sqrt{2}$, breaking $U{\left(1\right)}_N$ gauge symmetry. This VEV is responsible for the effective $\mu$ term and D-fermion masses. The first and second families of Higgs-like doublets and SM-singlets do not get VEVs. Because of this, they 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 $Z_2^H$ 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 $Z_2^H$ symmetry forbids not only flavor changing processes but also the most dangerous baryon and lepton number violating operators, it cannot be an exact symmetry. Indeed, this symmetry forbids all terms in $W_1$ and $W_2$ that permit the lightest exotic quarks to decay. The Lagrangian of the corresponding model is invariant under $U{\left(1\right)}_D$ symmetry transformations, i.e.,

$$D\to e^{i\alpha }D,\overline{D}\to e^{-i\alpha }\overline{D}.$$

(9)

The $U{\left(1\right)}_D$ as well as $U{\left(1\right)}_L$ and $U{\left(1\right)}_B$ global symmetries should be broken by the non-renormalizable operators. These operators are suppressed by inverse power of $M_X$. All dimension five operators that break $U{\left(1\right)}_D$ symmetry are forbidden by $E_6$. The dimension six operators result in the lifetime of the lightest exotic quarks, which is of order of

$${\tau }_D>M_X^4/{\mu }_D^5,$$

(10)

where ${\mu }_D$ is the mass of this exotic quark. For ${\mu }_D\simeq \mathrm{TeV}$, the lifetime ${\tau }_D>{10}^{49}{\mathrm{GeV}}^{-1}\sim {10}^{17}\mathrm{years}$ so that it is much larger than the age of the Universe. During early epochs of the Big Bang, so long-lived exotic quarks had to be copiously produced. Stable quarks which survive annihilation should be confined in nuclei. Thus, nuclear isotopes with stable exotic quarks should be present in terrestrial matter. The concentration of such remnant particles is expected to be ${10}^{-10}$ per nucleon if the mass of the lightest exotic quarks varies from $1\mathrm{GeV}$ to $10\mathrm{TeV}$ [112,113]. At the same time, different experiments indicate that the relative concentrations of such nuclear isotopes have to be less than ${10}^{-15}$ per nucleon or even smaller [114,115,116]. Therefore, the extensions of the SM with so long-lived exotic quarks are basically ruled out. This means that the discrete $Z_2^H$ symmetry can only be an approximate one.

To prevent rapid proton decay within the $U{\left(1\right)}_N$ extensions of the MSSM, one can impose either $Z_2^L$ or $Z_2^B$ discrete symmetry. The $Z_2^L$ symmetry implies that all superfields except lepton ones (including $L_4$ and ${\overline{L}}_4$) are even and all Yukawa interactions in $W_2$ are forbidden. Then, the baryon number is conserved only when exotic quarks are diquarks (Model I). In Model I, the most general renormalizable superpotential can be presented in the following form:

$$\begin{array}{ccc}\hfill W_{{\mathrm{E}}_6\mathrm{SSM}\mathrm{I}}& =& W_0+W_1+{\displaystyle \frac{1}{2}}{M}_{ij}{N}_i^c{N}_j^c+W_0^{\prime },\hfill \\ \hfill W_0^{\prime }& =& {\mu }_L{L}_4{\overline{L}}_4+{\tilde{h}}_{ij}^L{e}_i^c\left(H_j^d{L}_4\right)+h_{ij}^L{N}_i^c\left(H_j^u{L}_4\right).\hfill \end{array}$$

(11)

The terms in $W_0^{\prime }$ are caused by the splitting of ${27}^{\prime }$ and $\overline{{27}^{\prime }}$ representations of $E_6$. In the case of $Z_2^B$ symmetry, the supermultiplets of ordinary leptons, exotic quarks $D_i$ and $\overline{D_i}$, as well as $L_4$, ${\overline{L}}_4$ are all odd whereas the others remain even. As a consequence, all terms in $W_1$ are ruled out and the baryon number conservation requires the exotic quarks to carry lepton ($L_D=1$ and $L_{\overline{D}}=-1$) and baryon ($B_D=1/3$ and $B_{\overline{D}}=-1/3$) numbers simultaneously (Model II). Thus, in Model II, $D_i$ and $\overline{D_i}$ are leptoquarks. The most general renormalizable superpotential in Model II is given by

$$W_{{\mathrm{E}}_6\mathrm{SSM}\mathrm{II}}=W_0+W_2+{\displaystyle \frac{1}{2}}{M}_{ij}{N}_i^c{N}_j^c+W_0^{\prime }+g_{ik}^L\left(Q_i{L}_4\right){\overline{D}}_k,$$

(12)

The last term in Equation (12) appears because of the splitting of ${27}^{\prime }$. In the superpotentials in Equations (11) and (12), the $SU{\left(2\right)}_W$ doublet $L_4$ is redefined in such a way that $W_0^{\prime }$ contains only one bilinear term. The mass parameter ${\mu }_L$ should not be too large. Otherwise, the gauge coupling unification gets spoiled. Within SUGRA models, the appropriate term ${\mu }_L{L}_4{\overline{L}}_4$ in the superpotentials in Equations (11) and (12) can be induced if the Kähler potential contains an extra term $(Z{L}_4{\overline{L}}_4+h.c)$ [117,118]. This is the same mechanism which is used in the MSSM to solve the $\mu$ problem. Within the $U{\left(1\right)}_N$ extensions of the MSSM, the bilinear term involving $H_d$ and $H_u$ are forbidden by the $U{\left(1\right)}_N$ gauge symmetry and the mechanism mentioned above cannot be used.

The superpotentials of Models I and II also include bilinear terms, ${\displaystyle \frac{1}{2}}{M}_{ij}{N}_i^c{N}_j^c$, 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-renormalizable interactions of the form

$$\delta W={\displaystyle \frac{{\kappa }_{ij}}{M_{Pl}}}\left({\overline{27}}_H{27}_i\right)\left({\overline{27}}_H{27}_j\right)⟹M_{ij}={\displaystyle \frac{2{\kappa }_{ij}}{M_{Pl}}}<{\overline{N}}_H^c{>}^2,$$

(13)

where $N_H^c$ and ${\overline{N}}_H^c$ are components of some extra ${27}_H$ and ${\overline{27}}_H$ representations which develop VEVs along the D-flat direction $<N_H^c> = <{\overline{N}}_H^c>$. These VEVs can also break $U{\left(1\right)}_{\psi }\times U{\left(1\right)}_{\chi }$ down to $U{\left(1\right)}_N\times Z_2^M$ symmetry [72]. If such breakdown takes place somewhere around the GUT scale $M_X$, a reasonable pattern for the left-handed neutrino masses can be obtained.

The superpotentials in Equations (11) and (12) involve many new Yukawa couplings in comparison to the SM. In general, the exact $Z_2^L$ and $Z_2^B$ discrete symmetries do not guarantee the absence of FCNCs in the $U{\left(1\right)}_N$ extensions of the MSSM. At the same time, it is worth noting that the most of the Yukawa couplings in the SM as well as in the MSSM are rather small. Therefore, it seems natural to assume some hierarchical structure of the Yukawa interactions that may permit to suppress non-diagonal flavor transitions. In addition, it is reasonable to use the approximate $Z_2^H$ symmetry to eliminate problems related with flavor changing processes. The appropriate suppression of the non-diagonal flavor interactions can be achieved if all $Z_2^H$ symmetry violating couplings are less than ${10}^{-4}$. In the limit when all Yukawa couplings that explicitly break the $Z_2^H$ symmetry are negligibly small, the superpotential of the E${}_6$SSM reduces to

$$\begin{array}{ccc}\hfill W_{{\mathrm{E}}_6\mathrm{SSM}}& =& \lambda S\left(H_u{H}_d\right)+{\lambda }_{\alpha \beta }S\left(H_{\alpha }^d{H}_{\beta }^u\right)+{\kappa }_{ij}S\left(D_i{\overline{D}}_j\right)+{\tilde{f}}_{\alpha \beta }{S}_{\alpha }\left(H_{\beta }^d{H}_u\right)+f_{\alpha \beta }{S}_{\alpha }\left(H_d{H}_{\beta }^u\right)\hfill \\ \hfill & +& {\mu }_L{L}_4{\overline{L}}_4+{\displaystyle \frac{1}{2}}{M}_{ij}{N}_i^c{N}_j^c+W_{L_4}+W_{\mathrm{MSSM}}(\mu =0),\hfill \end{array}$$

(14)

where

$$W_{L_4}={\tilde{h}}_{i3}^L{e}_i^c\left(H_d{L}_4\right)+h_{i3}^L{N}_i^c\left(H_u{L}_4\right),$$

(15)

$\alpha ,\beta =1,2$ and $i,j=1,2,3$. If some of the couplings $\lambda$, ${\lambda }_{\alpha \beta }$ or ${\kappa }_{ij}$ are rather large at the GUT scale $M_X$, they can lead to negative values of $m_S^2$ at low energies. This triggers the breakdown of $U{\left(1\right)}_N$ gauge symmetry giving rise to the large VEV of the singlet superfield S that generate sufficiently large masses of the exotic particles and $Z^{\prime }$ boson. On the other hand, the generation of the VEVs of $H_u$ and $H_d$, that break the $SU{\left(2\right)}_W\times U{\left(1\right)}_Y$ gauge symmetry, can be caused by the large value of the top-quark Yukawa coupling.

Since in the $U{\left(1\right)}_N$ extensions of the MSSM the $Z_2^M$ symmetry and R-parity are conserved, the lightest R-parity odd state, i.e., the lightest SUSY particle (LSP), must be stable. Using the approach discussed in [119,120,121], it was shown that in the E${}_6$SSM the masses of the LSP and next-to-lightest SUSY particle (NLSP) are smaller than 60–65 GeV [94]. The LSP and NLSP (${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$) are predominantly linear combinations of the fermion components of the two SM singlet superfields $S_{\alpha }$. The couplings of ${\tilde{H}}_1^0$ to the SM particles are quite small. However, if LSP had a mass close to half the Z mass, it could account for some of the observed dark matter density. In these scenarios, LSP annihilate mainly through an s-channel Z-boson [94]. The SM-like Higgs state decays mostly into either ${\tilde{H}}_1^0$ or ${\tilde{H}}_2^0$ in this case. 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 $S_{\alpha }$ are considerably lighter than $M_Z$, then the cross section of the annihilation of ${\tilde{H}}_1^0{\tilde{H}}_1^0\to \mathrm{SM} \mathrm{particles}$ becomes too small, leading to cold dark matter density which is much larger than its measured value.

Nevertheless, in the E${}_6$SSM with approximate $Z_2^H$ symmetry, one of the lightest R-parity odd state can account for some of the observed dark matter density. To prevent the decays of this state into the LSP and NLSP, an additional $Z_2^S$ symmetry needs to be postulated [78]. In the corresponding variant of the E${}_6$SSM, couplings ${\tilde{f}}_{\alpha \beta }$ and $f_{\alpha \beta }$ vanish. As a result, the fermion components of the SM singlet superfields $S_{\alpha }$ remain massless and decouple. If $Z^{\prime }$ boson is sufficiently heavy, the presence of these massless states does not affect Big Bang Nucleosynthesis (BBN) [78]. Since ${\tilde{f}}_{\alpha \beta }=f_{\alpha \beta }=0$, the branching ratios of the SM-like Higgs decays into ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ vanish.

Instead of $Z_2^H$, $Z_2^L$ and $Z_2^B$, one can impose a single discrete ${\tilde{Z}}_2^H$ symmetry which forbids tree-level flavor-changing transitions as well as the most dangerous operators that lead to rapid proton decay. In this case $H_u$, $H_d$, S, $L_4$ and ${\overline{L}}_4$ are even under the ${\tilde{Z}}_2^H$ symmetry while all other supermultiplets are odd [72]. Neglecting non-renormalizable interactions, the superpotential of this variant of the E${}_6$SSM is given by Equation (14) with

$$W_{L_4}=g_{ik}^L\left(Q_i{L}_4\right){\overline{D}}_k+{\tilde{h}}_{i\alpha }^L{e}_i^c\left(H_{\alpha }^d{L}_4\right)+h_{i\alpha }^L{N}_i^c\left(H_{\alpha }^u{L}_4\right),$$

(16)

where $\alpha =1,2$ and $i,k=1,2,3$. Since the low-energy effective Lagrangian of these SUSY models is invariant under both $Z_2^M$ and ${\tilde{Z}}_2^H$ symmetries and ${\tilde{Z}}_2^H=Z_2^M\times Z_2^E$, the $Z_2^E$ symmetry associated with exotic states is also conserved. The transformation properties of different components of ${27}_i$ supermultiplets under the $Z_2^H$, $Z_2^L$, $Z_2^B$, $Z_2^S$, ${\tilde{Z}}_2^H$, $Z_2^M$ and $Z_2^E$ symmetries are summarized in

Table 1. Transformation properties of matter supermultiplets under $Z_2^H$, $Z_2^L$, $Z_2^B$, $Z_2^S$, ${\tilde{Z}}_2^H$, $Z_2^M$ and $Z_2^E$ discrete symmetries in the E${}_6$SSM. The signs + and − correspond to the states which are even and odd under different $Z_2$ symmetries.

	${\mathit{Q}}_{\mathit{i}},{\mathit{u}}_{\mathit{i}}^{\mathit{c}},{\mathit{d}}_{\mathit{i}}^{\mathit{c}}$	${\mathit{L}}_{\mathit{i}},{\mathit{e}}_{\mathit{i}}^{\mathit{c}},{\mathit{N}}_{\mathit{i}}^{\mathit{c}}$	${\overline{\mathit{D}}}_{\mathit{i}},{\mathit{D}}_{\mathit{i}}$	${\mathit{H}}_{\mathit{\alpha }}^{\mathit{d}},{\mathit{H}}_{\mathit{\alpha }}^{\mathit{u}}$	${\mathit{S}}_{\mathit{\alpha }}$	${\mathit{H}}_{\mathit{d}},{\mathit{H}}_{\mathit{u}},\mathit{S}$	${\mathit{L}}_4,{\overline{\mathit{L}}}_4$
$Z_2^H$	−	−	−	−	−	+	−
$Z_2^L$	+	−	+	+	+	+	−
$Z_2^B$	+	−	−	+	+	+	−
$Z_2^S$	+	+	+	+	−	+	+
${\tilde{Z}}_2^H$	−	−	−	−	−	+	+
$Z_2^M$	−	−	+	+	+	+	−
$Z_2^E$	+	+	−	−	−	+	−

. The $Z_2^E$ symmetry conservation ensures that the lightest exotic state, which is odd under this symmetry, is stable. The simplest phenomenologically viable scenarios imply that $f_{\alpha \beta }\sim {\tilde{f}}_{\alpha \beta }<{10}^{-6}$. As a consequence, two lightest exotic states (${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$), which are formed by the fermion components of the superfields $S_{\alpha }$, are substantially lighter than $1\mathrm{eV}$. They compose hot dark matter in the Universe [72]. However, these states give only a very small contribution to the dark matter density. The presence of extremely light neutral fermions might also have some implications for the neutrino physics [122]. The invariance of the Lagrangian under the $Z_2^M$ ensures that the lightest R-parity odd state with $Z_2^E=+1$, which is most commonly the lightest ordinary neutralino in this case, is also stable and can account for some of the observed dark matter density [106].

3. 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 the corresponding couplings is affected by a kinetic term mixing. In the Lagrangian of any extension of the SM, which involves an additional $U{\left(1\right)}^{\prime }$ factor, there can arise a kinetic term consistent with all symmetries [123]. This term mixes the gauge fields of the $U{\left(1\right)}^{\prime }$ and $U{\left(1\right)}_Y$. The E${}_6$SSM is not an exception. In the basis in which the couplings of the gauge bosons to matter fields have the canonical form, for example a covariant derivative, $D_{\mu }$ which acts on the left-handed quark field is given by

$$D_{\mu }={\partial }_{\mu }-i{g}_3{A}_{\mu }^a{T}^a-i{g}_2{W}_{\mu }^b{\tau }^b-i{g}_Y{Q}_i^Y{B}_{\mu }^Y-i{g}_N{Q}_i^N{B}_{\mu }^N,$$

(17)

and the mixing between the $U\left(1\right)$ field strengths can be written as

$${\mathcal{L}}_{mix}=-{\displaystyle \frac{sin\chi }{2}}{F}_{\mu \nu }^Y{F}_{\mu \nu }^N.$$

(18)

Here, $A_{\mu }^a$, $W_{\mu }^b$, $B_{\mu }^Y$ and $B_{\mu }^N$ represent $SU{\left(3\right)}_C$, $SU{\left(2\right)}_W$, $U{\left(1\right)}_Y$ and $U{\left(1\right)}_N$ gauge fields; $G_{\mu \nu }^a$, $W_{\mu \nu }^b$, $F_{\mu \nu }^Y$ and $F_{\mu \nu }^N$ are field strengths for the corresponding gauge interactions; and $g_3$, $g_2$, $g_Y$ and $g_N$ are the $SU{\left(3\right)}_C$, $SU{\left(2\right)}_W$, $U{\left(1\right)}_Y$ and $U{\left(1\right)}_N$ gauge couplings in this basis. Since $U{\left(1\right)}_Y$ and $U{\left(1\right)}_N$ factors come from the breakdown of the simple gauge group $E_6$, the parameter $sin\chi$ is expected to vanish at tree-level. However, the non-zero value of this parameter is induced by loop corrections because

$$Tr\left(Q^Y{Q}^N\right)=\sum _i\left(Q_i^Y{Q}_i^N\right)\ne 0.$$

(19)

In Equation (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 in Equation (19) is induced by $L_4$ and ${\overline{L}}_4$ supermultiplets which survive to low energies.

For non-zero values of the parameter $sin\chi$, the mixing in the gauge sector in Equation (18) can be eliminated by means of a non-unitary transformation [39,124,125,126,127]:

$$B_{\mu }^Y=B_{1\mu }-B_{2\mu }tan\chi ,B_{\mu }^N=B_{2\mu }/cos\chi .$$

(20)

In the basis $(B_{1\mu },B_{2\mu })$, the covariant derivative in Equation (17) becomes

$$D_{\mu }={\partial }_{\mu }-i{g}_3{A}_{\mu }^a{T}^a-i{g}_2{W}_{\mu }^b{\tau }^b-i{g}_1{Q}_i^Y{B}_{1\mu }-i(g_1^{\prime }{Q}_i^N+g_{11}{Q}_i^Y)B_{2\mu },$$

(21)

where the redefined gauge couplings are

$$g_1=g_Y,g_1^{\prime }=g_N/cos\chi ,g_{11}=-g_{Y}tan\chi .$$

(22)

In this basis, the mixing effect is concealed in the interaction parameterized by a new off-diagonal gauge coupling $g_{11}$. The gauge coupling constant $g_1^{\prime }$ differs from the original one. In the new basis, the covariant derivative in Equation (21) can be rewritten in a compact form

$$D_{\mu }={\partial }_{\mu }-i{g}_3{A}_{\mu }^a{T}^a-i{g}_2{W}_{\mu }^b{\tau }^b-i{Q}^{T}G{B}_{\mu },$$

(23)

where $Q^T=(Q_i^Y,Q_i^N)$, $B_{\mu }^T=(B_{1\mu },B_{2\mu })$ and G is a $2\times 2$ matrix of new gauge couplings in Equation (22)

$$G=\left(\begin{array}{cc}{g}_1& g_{11}\\ 0& g_1^{\prime }\end{array}\right).$$

(24)

Now, all physical phenomena can be examined using the Lagrangian with the modified structure of the gauge interactions in Equations (21)–(23). In this approximation, the gauge kinetic mixing changes effectively the $U{\left(1\right)}_N$ charges of all fields to

$${\tilde{Q}}_i\equiv Q_i^N+Q_i^Y\delta ,$$

(25)

where $\delta =g_{11}/g_1^{\prime }$, whereas the $U{\left(1\right)}_Y$ charges remain the same. The effective charges ${\tilde{Q}}_i$ are scale dependent. The particle spectrum in the basis $B_{\mu }^T=(B_{1\mu },B_{2\mu })$ depends on the effective $U{\left(1\right)}_N$ charges ${\tilde{Q}}_i$.

The running of four diagonal gauge couplings, i.e., $g_3\left(t\right)$, $g_2\left(t\right)$, $g_1\left(t\right)$ and $g_1^{\prime }\left(t\right)$, and one off-diagonal gauge coupling $g_{11}\left(t\right)$ is described by a system of RG equations (RGEs), which can be written as:

$${\displaystyle \frac{dG}{dt}}=G\times B,{\displaystyle \frac{d{g}_2}{dt}}={\displaystyle \frac{{\beta }_2{g}_2^3}{32{\pi }^2}},{\displaystyle \frac{d{g}_3}{dt}}={\displaystyle \frac{{\beta }_3{g}_3^3}{32{\pi }^2}},$$

(26)

where $t=2ln\left(q/M_Z\right)$, q is a renormalization scale, G is the $2\times 2$ matrix in Equation (24) while B is a $2\times 2$ matrix given by

$$B={\displaystyle \frac{1}{32{\pi }^2}}\left(\begin{array}{cc}{\beta }_1{g}_1^2& 2g_1{g}_1^{\prime }{\beta }_{11}+2g_1{g}_{11}{\beta }_1\\ 0& g_1^{{}^{\prime }2}{\beta }_1^{\prime }+2g_1^{\prime }{g}_{11}{\beta }_{11}+g_{11}^2{\beta }_1\end{array}\right).$$

(27)

In Equations (26) and (27), ${\beta }_i$ and ${\beta }_{11}$ are beta functions. Here, the RG flow of the gauge couplings is explored in the two-loop approximation. In this approximation,

$${\beta }_1^{\prime }=b_1^{\prime }+{\displaystyle \frac{{\tilde{b}}_1^{\prime }}{4\pi }},{\beta }_i=b_i+{\displaystyle \frac{{\tilde{b}}_i}{4\pi }},$$

(28)

where $b_i$ and $b_1^{\prime }$ are one-loop beta functions of the diagonal gauge couplings while ${\tilde{b}}_i$ and ${\tilde{b}}_1^{\prime }$ correspond to the two-loop contributions to these functions.

It seems to be rather natural to expect that just after the breakdown of the $E_6$ symmetry near the GUT scale $M_X$ the mixing parameter $sin\chi$ vanishes, while the $SU{\left(3\right)}_C$, $SU{\left(2\right)}_W$, $U{\left(1\right)}_Y$ and $U{\left(1\right)}_N$ gauge interactions are characterized by a unique $E_6$ gauge coupling $g_0$, i.e.,

$$g_3\left(M_X\right)=g_2\left(M_X\right)=g_1\left(M_X\right)=g_1^{\prime }\left(M_X\right)=g_0,g_{11}\left(M_X\right)=0.$$

(29)

The previous analysis performed in [84] revealed that $g_{11}\left(t\right)$ being set to zero at the GUT scale remains substantially smaller than the diagonal gauge couplings at any scale below $M_X$. Therefore, the two-loop corrections to the off-diagonal beta function ${\beta }_{11}$ can be neglected. In the one-loop approximation, the beta function of the off-diagonal gauge coupling is given by ${\beta }_{11}=-{\displaystyle \frac{\sqrt{6}}{5}}$.

To simplify our analysis here, we further assume that the interactions of matter supermultiplets in the E${}_6$SSM are described by the superpotential in Equation (14), in which all interactions in $W_{L_4}$ can be ignored, ${\tilde{f}}_{\alpha \beta }\simeq f_{\alpha \beta }\to 0$, ${\lambda }_{\alpha \beta }={\lambda }_{\alpha }{\delta }_{\alpha \beta }$ and ${\kappa }_{ij}={\kappa }_i{\delta }_{ij}$. The part of the superpotential in Equation (14) associated with $W_{\mathrm{MSSM}}(\mu =0)$ reduces to

$$W_{\mathrm{MSSM}}(\mu =0)=h_t{Q}_3{u}_3^c{H}_u+h_b{Q}_3{d}_3^c{H}_d+h_{\tau }{L}_3{e}_3^c{H}_d,$$

(30)

because only third-generation fermions have Yukawa couplings to $H_d$ and $H_u$ which can be of the order of unity.

In the one-loop approximation the beta functions of the diagonal gauge couplings are given by

$$b_1={\displaystyle \frac{3}{5}}+3N_g,b_1^{\prime }={\displaystyle \frac{2}{5}}+3N_g,b_2=-5+3N_g,b_3=-9+3N_g,$$

(31)

where parameter $N_g$ is the number of complete $E_6$ fundamental representations at low energies ($E<<M_X$). In the E${}_6$SSM, $N_g=3$, which is the critical value for $b_3$. Indeed, for $N_g=3$, the one-loop beta function of the strong interactions is equal to zero and the $SU{\left(3\right)}_C$ gauge coupling remains constant everywhere from the EW scale to $M_X$. Thus, any reliable analysis of the RG flow of the gauge couplings within the E${}_6$SSM requires the inclusion of two-loop corrections to ${\beta }_1^{\prime }$ and ${\beta }_i$. Using the results of the calculation of two-loop beta functions in a general models with softly broken SUSY [128], one obtains

$$\begin{array}{ccc}\hfill {\tilde{b}}_1& =& 8N_g{\alpha }_3+\left({\displaystyle \frac{9}{5}}+3N_g\right){\alpha }_2+\left({\displaystyle \frac{9}{25}}+3N_g\right){\alpha }_1+\left({\displaystyle \frac{6}{25}}+N_g\right){\alpha }_1^{\prime }\hfill \\ \hfill & & -{\displaystyle \frac{26}{5}}{y}_t-{\displaystyle \frac{14}{5}}{y}_b-{\displaystyle \frac{18}{5}}{y}_{\tau }-{\displaystyle \frac{6}{5}}{\Sigma }_{\lambda }-{\displaystyle \frac{4}{5}}{\Sigma }_{\kappa },\hfill \\ \hfill {\tilde{b}}_1^{\prime }& =& 8N_g{\alpha }_3+\left({\displaystyle \frac{6}{5}}+3N_g\right){\alpha }_2+\left({\displaystyle \frac{6}{25}}+N_g\right){\alpha }_1+\left({\displaystyle \frac{4}{25}}+3N_g\right){\alpha }_1^{\prime }\hfill \\ \hfill & & -{\displaystyle \frac{9}{5}}{y}_t-{\displaystyle \frac{21}{5}}{y}_b-{\displaystyle \frac{7}{5}}{y}_{\tau }-{\displaystyle \frac{19}{5}}{\Sigma }_{\lambda }-{\displaystyle \frac{57}{10}}{\Sigma }_{\kappa },\hfill \\ \hfill {\tilde{b}}_2& =& 8N_g{\alpha }_3+\left(-17+21N_g\right){\alpha }_2+\left({\displaystyle \frac{3}{5}}+N_g\right){\alpha }_1+\left({\displaystyle \frac{2}{5}}+N_g\right){\alpha }_1^{\prime }\hfill \\ \hfill & & -6y_t-6y_b-2y_{\tau }-2{\Sigma }_{\lambda },\hfill \\ \hfill {\tilde{b}}_3& =& {\alpha }_3\left(-54+34N_g\right)+3N_g{\alpha }_2+N_g{\alpha }_1+N_g{\alpha }_1^{\prime }-4y_t-4y_b-2{\Sigma }_{\kappa },\hfill \\ \hfill {\Sigma }_{\lambda }& =& y_{{\lambda }_1}+y_{{\lambda }_2}+y_{\lambda },{\Sigma }_{\kappa }=y_{{\kappa }_1}+y_{{\kappa }_2}+y_{{\kappa }_3},\hfill \end{array}$$

(32)

where ${\alpha }_i={\displaystyle \frac{g_i^2}{4\pi }}$, ${\alpha }_1^{\prime }={\displaystyle \frac{g_1^{{}^{\prime }2}}{4\pi }}$, $y_t={\displaystyle \frac{h_t^2}{4\pi }}$, $y_b={\displaystyle \frac{h_b^2}{4\pi }}$, $y_{\tau }={\displaystyle \frac{h_{\tau }^2}{4\pi }}$, $y_{\lambda }={\displaystyle \frac{{\lambda }^2}{4\pi }}$, $y_{{\lambda }_{\alpha }}={\displaystyle \frac{{\lambda }_{\alpha }^2}{4\pi }}$ and $y_{{\kappa }_i}={\displaystyle \frac{{\kappa }_i^2}{4\pi }}$.

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 [129]). At high energies, this solution is given by

$${\displaystyle \frac{1}{{\alpha }_i\left(t\right)}}={\displaystyle \frac{1}{{\alpha }_i\left(M_Z\right)}}-{\displaystyle \frac{b_i}{2\pi }}t-\frac{C_i}{12\pi }-{\Theta }_i\left(t\right)+{\displaystyle \frac{b_i-b_i^{SM}}{2\pi }}ln\frac{T_i}{M_Z},$$

(33)

where $b_i^{SM}$ are the one-loop beta functions in the SM, the $\overline{MS}\to \overline{DR}$ conversion factor with $C_1=0$, $C_2=2$, $C_3=3$ correspond to the third term on the right-hand side of Equation (33) [130,131], while

$${\Theta }_i\left(t\right)={\displaystyle \frac{1}{8{\pi }^2}}{\int }_0^t{\tilde{b}}_{i}d\tau ,T_i=\prod _{k=1}^N{\left(m_k\right)}^{{\displaystyle \frac{\Delta b_i^k}{b_i-b_i^{SM}}}}$$

(34)

In Equation (34), $m_k$ and $\Delta b_i^k$ are masses and one-loop contributions to $b_i$ due to sparticles, heavy Higgs bosons and exotic states appearing in the E${}_6$SSM. Since the two-loop corrections to the RG flow of the SM gauge couplings ${\Theta }_i\left(t\right)$ are substantially 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 ${\Theta }_i\left(t\right)$. The threshold corrections associated with the last terms in Equation (33) are even smaller than ${\Theta }_i\left(t\right)$. Because of this, only leading one-loop threshold effects are taken into account in Equations (33) and (34).

Relying on the approximate solution of the two-loop RGEs, one can find the relationships between ${\alpha }_i\left(M_X\right)$ and the values of these couplings at low energies. Then, one can estimate the scale $M_X$ where ${\alpha }_1\left(M_X\right)={\alpha }_2\left(M_X\right)={\alpha }_0$ as well as the value of the overall gauge coupling ${\alpha }_0$. Substituting $M_X$ and ${\alpha }_3\left(M_X\right)={\alpha }_0$ into the approximate solution for ${\alpha }_3\left(t\right)$ one can find the value of ${\alpha }_3\left(M_Z\right)$, for which the unification of the SM gauge couplings takes place (see [132]):

$$\begin{array}{c}{\displaystyle \frac{1}{{\alpha }_3\left(M_Z\right)}}={\displaystyle \frac{1}{b_1-b_2}}\left[{\displaystyle \frac{b_1-b_3}{{\alpha }_2\left(M_Z\right)}}-{\displaystyle \frac{b_2-b_3}{{\alpha }_1\left(M_Z\right)}}\right]-{\displaystyle \frac{1}{28\pi }}+{\Theta }_s+{\displaystyle \frac{19}{28\pi }}ln{\displaystyle \frac{T_S}{M_Z}},\\ {\Theta }_s=\left({\displaystyle \frac{b_2-b_3}{b_1-b_2}}{\Theta }_1-{\displaystyle \frac{b_1-b_3}{b_1-b_2}}{\Theta }_2+{\Theta }_3\right),{\Theta }_i={\Theta }_i\left(M_X\right).\end{array}$$

(35)

In Equation (35), the threshold scale $T_S$ can be expressed in terms of $T_1$, $T_2$ and $T_3$

$$T_S={\displaystyle \frac{T_2^{172/19}}{T_1^{55/19}{T}_3^{98/19}}}.$$

(36)

In the E${}_6$SSM, the effective threshold scales $T_1$, $T_2$ and $T_3$ are given by

$$\begin{array}{ccc}\hfill T_1& =& {\mu }^{4/55}{m}_A^{1/55}{\mu }_L^{4/55}{m}_L^{2/55}\left({\prod }_{i=1,2,3}{m}_{{\tilde{Q}}_i}^{1/165}{m}_{{\tilde{d}}_i}^{2/165}{m}_{{\tilde{u}}_i}^{8/165}{m}_{{\tilde{L}}_i}^{1/55}{m}_{{\tilde{e}}_i}^{2/55}{m}_{{\tilde{D}}_i}^{4/165}{\mu }_{D_i}^{8/165}\right)\times \hfill \\ \hfill & & \times \left({\prod }_{\alpha =1,2}{m}_{H_{\alpha }}^{2/55}{\mu }_{{\tilde{H}}_{\alpha }}^{4/55}\right),\hfill \\ \hfill T_2& =& M_{\tilde{W}}^{8/43}{\mu }^{4/43}{m}_A^{1/43}{\mu }_L^{4/43}{m}_L^{2/43}\left({\prod }_{i=1,2,3}{m}_{{\tilde{Q}}_i}^{3/43}{m}_{{\tilde{L}}_i}^{1/43}\right)\left({\prod }_{\alpha =1,2}{m}_{H_{\alpha }}^{2/43}{\mu }_{{\tilde{H}}_{\alpha }}^{4/43}\right),\hfill \\ \hfill T_3& =& M_{\tilde{g}}^{2/7}\left({\prod }_{i=1,2,3}{m}_{{\tilde{Q}}_i}^{1/21}{m}_{{\tilde{u}}_i}^{1/42}{m}_{{\tilde{d}}_i}^{1/42}{m}_{{\tilde{D}}_i}^{1/21}{\mu }_{D_i}^{2/21}\right),\hfill \end{array}$$

(37)

where $M_{\tilde{g}}$ and $M_{\tilde{W}}$ are the masses of gluinos and winos; $m_A$ and $\mu$ are the masses of heavy Higgs bosons and effective $\mu$-term; ${\mu }_{D_i}$ and $m_{{\tilde{D}}_i}$ are the masses of exotic quarks and their superpartners; ${\mu }_{{\tilde{H}}_{\alpha }}$ and $m_{H_{\alpha }}$ are the masses of the fermion and scalar components of $H_{\alpha }^u$ and $H_{\alpha }^d$; $m_L$ and ${\mu }_L$ are the masses of the scalar and fermion components of $L_4$ and ${\overline{L}}_4$; $m_{{\tilde{Q}}_i}$ and $m_{{\tilde{L}}_i}$ are the masses of the left-handed squarks and sleptons; and $m_{{\tilde{u}}_i}$, $m_{{\tilde{d}}_i}$ and $m_{{\tilde{e}}_i}$ are the masses of the right-handed squarks and sleptons.

It is worth noting here that in the limit when the two-loop and threshold corrections are neglected, i.e., ${\Theta }_s=0$ and $T_S=M_Z$, Equation (33) leads to the same prediction for ${\alpha }_3\left(M_Z\right)$ in the MSSM and E${}_6$SSM. Indeed, since extra matter in the E${}_6$SSM form complete $SU\left(5\right)$ representations these multiplets contribute equally to $b_i$. Due to this the differences of the coefficients $b_i-b_j$ and the form of Equation (33) remain the same in the MSSM and E${}_6$SSM. However, the inclusion of the two-loop and threshold corrections may spoil the gauge coupling unification within the E${}_6$SSM.

In general, the effective threshold scales $T_1$, $T_2$ and $T_3$ are quite different. Nevertheless, from Equation (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 Equation (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 evolution of the SM gauge couplings computed in the two-loop approximation is shown. In particular, we use the two-loop SM beta functions to evaluate the running of ${\alpha }_i\left(t\right)$ between $M_Z$ and $T_S$. Then, we apply the two-loop RG equations of the E${}_6$SSM to calculate the evolution of ${\alpha }_i\left(t\right)$ from $T_S$ to $M_X$ which is around 2–3 $\times {10}^{16}\mathrm{GeV}$. At low energies, the values of $g_{11}$ and $g_1^{\prime }$ are chosen so that the conditions in Equation (29) are fulfilled. For the computation of the RG flow of Yukawa couplings, a set of one-loop RG equations is used. The corresponding set of the one-loop RGEs are specified in [69].

In Figure 1, we fix $T_1=T_2=T_3=T_S=2\mathrm{TeV}$ and $tan\beta =10$. Although to simplify our analysis we also set ${\kappa }_i\left(T_S\right)={\lambda }_{\alpha }\left(T_S\right)=\lambda \left(T_S\right)=g_1^{{}^{\prime }}\left(T_S\right)$, the RG flow of ${\alpha }_i\left(t\right)$ depends rather weakly on the values of the Yukawa and extra $U{\left(1\right)}_N$ gauge couplings. Dotted lines in Figure 1 show the modifications of the RG flow of the SM gauge couplings caused by the variations of ${\alpha }_3\left(M_Z\right)$ from $0.116$ to $0.120$. The corresponding interval of variations of ${\alpha }_3\left(t\right)$ is always substantially wider than the ones for ${\alpha }_1\left(t\right)$ and ${\alpha }_2\left(t\right)$. The dependence of ${\alpha }_1\left(t\right)$ and ${\alpha }_2\left(t\right)$ on ${\alpha }_3\left(M_Z\right)$ is expected to be relatively weak because ${\alpha }_3\left(t\right)$ arises only in the two-loop contributions to ${\beta }_1$ and ${\beta }_2$. It is worth pointing out that at high energies the uncertainty in ${\alpha }_3\left(t\right)$ induced by the variations of ${\alpha }_3\left(M_Z\right)$ is considerably bigger in the E${}_6$SSM than in the MSSM. This happens because ${\alpha }_3\left(t\right)$ grows with increasing renormalization scale q in the E${}_6$SSM while in the MSSM it decreases at high energies. Thus, in the E${}_6$SSM, the uncertainty in ${\alpha }_3\left(M_X\right)$ is almost equal to the low energy uncertainty in ${\alpha }_3\left(M_Z\right)$, whereas within the MSSM the interval of variations of ${\alpha }_3\left(M_X\right)$ 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 ${\alpha }_3\left(M_Z\right)>0.123$, well above the experimentally measured central value [129,132,133,134,135,136,137,138,139,140].

The results of the numerical analysis shown in Figure 1 indicate that for $T_S=2\mathrm{TeV}$ almost exact gauge coupling unification can be achieved in the E${}_6$SSM if ${\alpha }_3\left(M_Z\right)\approx 0.116$. The value of ${\alpha }_3\left(M_Z\right)$, at which the exact unification of the SM gauge couplings takes place, becomes lower (greater) when the combined threshold scale $T_S$ increases (decreases). In the E${}_6$SSM, $T_S$ can be considerably lower than $1\mathrm{TeV}$ even if 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 $m_A\approx M_S$ which is much larger than the masses of all fermions. Then, combining Equations (36) and (37), one finds

$$T_S={\displaystyle \frac{M_{\tilde{W}}^{32/19}{M}_S^{3/19}}{M_{\tilde{g}}^{28/19}}}{\left({\displaystyle \frac{\mu {\mu }_L{\mu }_{{\tilde{H}}_1}{\mu }_{{\tilde{H}}_2}}{{\mu }_{D_1}{\mu }_{D_2}{\mu }_{D_3}}}\right)}^{12/19}.$$

(38)

If $M_S\approx M_{\tilde{g}}\approx 10\mathrm{TeV}$ and $\mu \approx {\mu }_L\approx {\mu }_{{\tilde{H}}_1}\approx {\mu }_{{\tilde{H}}_2}\approx 1\mathrm{TeV}$ while $M_{\tilde{W}}$ and the masses of the exotic quarks ${\mu }_{D_i}$ are of the order of a few TeV, the effective threshold scale tends to be much smaller than $1\mathrm{TeV}$. For $T_S=400\mathrm{GeV}$, almost exact unification of the SM gauge couplings in the E${}_6$SSM can be obtained if ${\alpha }_3\left(M_Z\right)\approx 0.118$ [72]. Thus, in this SUSY model the gauge coupling unification can be attained for the values of ${\alpha }_3\left(M_Z\right)$ which are in agreement with current data.

As mentioned above, the inclusion of the two-loop corrections to the diagonal beta functions could spoil the gauge coupling unification entirely within the E${}_6$SSM. Since at any intermediate scale ${\alpha }_i\left(t\right)$ are considerably larger in the E${}_6$SSM as compared to the ones in the MSSM, the corresponding two-loop corrections affect the running of the SM gauge couplings in the E${}_6$SSM much more strongly than in the MSSM. The analysis of the RG flow of the couplings ${\alpha }_i\left(t\right)$ performed in [84] indicated that ${\Theta }_i\left(M_X\right)$ are a few times smaller in the MSSM than in the E${}_6$SSM. On the other hand, the absolute value of ${\Theta }_s$ is more than three times larger in the MSSM as compared with the E${}_6$SSM. The cancellation of different two-loop contribution to ${\Theta }_s$ in the case of the model under consideration is caused by the structure of ${\tilde{b}}_i$. As a consequence, the prediction for ${\alpha }_3\left(M_Z\right)$ obtained using Equation (35) is substantially lower in the E${}_6$SSM than in the MSSM.

4. Gauge Symmetry Breaking and Higgs Sector

In the E${}_6$SSM, the VEV of the SM singlet field S and the VEVs of the two Higgs doublets $H_u$ and $H_d$ can give rise to the breakdown of the $SU{\left(2\right)}_W\times U{\left(1\right)}_Y\times U{\left(1\right)}_N$ gauge symmetry in the simplest case. The interactions between these fields are determined by the structure of the gauge group and by the superpotential in Equation (14). The resulting Higgs effective potential is the sum of four pieces:

$$V=V_F+V_D+V_{\mathrm{soft}}+\Delta V,$$

(39)

$$V_F={\lambda }^2{\left|S\right|}^2\left(\right|H_d{|}^2+|H_u{|}^2)+{\lambda }^2{\left|\left(H_d{H}_u\right)\right|}^2,$$

(40)

$$\begin{array}{ccc}\hfill V_D& =& {\sum }_{a=1}^3{\displaystyle \frac{g_2^2}{8}}{\left(H_d^{†}{\sigma }_a{H}_d+H_u^{†}{\sigma }_a{H}_u\right)}^2+{\displaystyle \frac{{g^{\prime }}^2}{8}}{\left(|H_d{|}^2-{\left|H_u\right|}^2\right)}^2+\hfill \\ \hfill & & +{\displaystyle \frac{g_1^{{}^{\prime }2}}{2}}{\left({\tilde{Q}}_{H_d}|H_d{|}^2+{\tilde{Q}}_{H_u}|H_u{|}^2+{\tilde{Q}}_S{\left|S\right|}^2\right)}^2,\hfill \end{array}$$

(41)

$$V_{\mathrm{soft}}=m_S^2{\left|S\right|}^2+m_{H_d}^2|H_d{|}^2+m_{H_u}^2{\left|H_u\right|}^2+\left[\lambda A_{\lambda }S\left(H_u{H}_d\right)+\mathrm{h}.\mathrm{c}.\right],$$

(42)

where ${\sigma }_a$ ($a=1,2,3$) denotes the three Pauli matrices, $g^{\prime }=\sqrt{3/5}{g}_1$, $H_d^T=(H_d^0,H_d^{-})$, $H_u^T=(H_u^{+},H_u^0)$, $\left(H_d{H}_u\right)=H_u^{+}{H}_d^{-}-H_u^0{H}_d^0$, and ${\tilde{Q}}_{H_d}$, ${\tilde{Q}}_{H_u}$, and ${\tilde{Q}}_S$ are the effective $U{\left(1\right)}_N$ charges of $H_d$, $H_u$ and S, respectively. At tree-level the potential of the Higgs sector in Equation (39) is determined by the sum of the first three terms. $V_F$ and $V_D$ correspond to the F- and D-term contributions. They do not violate SUSY. $V_F$ is exactly the same as the contribution of the F-terms in the NMSSM without the self–interaction of the SM singlet superfield S. However, $V_D$ contain new terms which are proportional to ${g_1^{\prime }}^2$. These terms are not present in the MSSM or NMSSM. They represent D-term contributions due to the additional $U{\left(1\right)}_N$ factor. The low-energy values of $g_2$ and $g^{\prime }$ are well known. Assuming gauge coupling unification one can compute the low-energy value of the extra $U{\left(1\right)}_N$ coupling $g_1^{\prime }$. In this case, the effective $U{\left(1\right)}_N$ charges of $H_d$, $H_u$ and S may be calculated as well.

The terms in the Higgs potential in Equation (39), which break global SUSY, are collected in $V_{soft}$. The set of the soft SUSY breaking terms involves the soft masses $m_{H_d}^2,m_{H_u}^2,m_S^2$ as well as trilinear coupling $A_{\lambda }$. $V_{soft}$ coincides with the corresponding part of the NMSSM scalar potential in the limit when the NMSSM parameters $\kappa$ and $A_{\kappa }$ vanish. Because the only possible complex phase of ($\lambda A_{\lambda }$) in the tree-level Higgs potential in Equation (39) may easily be absorbed by a suitable redefinition of $H_d$, $H_u$ and S, CP–invariance is preserved in the sector responsible for the breakdown of the $SU{\left(2\right)}_W\times U{\left(1\right)}_Y\times U{\left(1\right)}_N$ symmetry at tree-level.

The last term $\Delta V$ in Equation (39) represents the contribution of loop corrections to the effective potential of the Higgs sector. In the one-loop approximation the contributions of different states to $\Delta V$ are determined by their masses, i.e.,

$$\Delta V={\displaystyle \frac{1}{64{\pi }^2}}Str{\left|M\right|}^4\left[log{\displaystyle \frac{{\left|M\right|}^2}{Q^2}}-{\displaystyle \frac{3}{2}}\right].$$

(43)

In Equation (43), M is the mass matrix for the bosons and fermions within the SUSY model under consideration. The supertrace operator in Equation (43) counts positively (negatively) the number of degrees of freedom associated with the different bosonic (fermionic) fields, while Q is the renormalization scale. The inclusion of loop corrections draws into the consideration many other soft SUSY breaking parameters which determine masses of different sparticles. Some of these parameters may be complex giving rise to potential sources of CP–violation.

At the physical minimum of the Higgs potential in Equation (39), the Higgs fields develop VEVs

$$<H_d>={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}{v}_1\\ 0\end{array}\right),<H_u>={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}0\\ v_2\end{array}\right),<S>={\displaystyle \frac{s}{\sqrt{2}}}.$$

(44)

The equations for the extrema of the effective potential of the E${}_6$SSM Higgs sector in the directions in Equation (44) in field space read:

$${\displaystyle \frac{\partial V}{\partial s}}=m_S^{2}s-{\displaystyle \frac{\lambda A_{\lambda }}{\sqrt{2}}}{v}_1{v}_2+{\displaystyle \frac{{\lambda }^2}{2}}(v_1^2+v_2^2)s+{\displaystyle \frac{g_1^{{}^{\prime }2}}{2}}{D}^{\prime }{\tilde{Q}}_{S}s+{\displaystyle \frac{\partial \Delta V}{\partial s}}=0,$$

(45)

$${\displaystyle \frac{\partial V}{\partial v_1}}=m_{H_d}^2{v}_1-{\displaystyle \frac{\lambda A_{\lambda }}{\sqrt{2}}}s{v}_2+{\displaystyle \frac{{\lambda }^2}{2}}(v_2^2+s^2)v_1+{\displaystyle \frac{{\overline{g}}^2}{8}}\left(v_1^2-v_2^2)\right)v_1+{\displaystyle \frac{g_1^{{}^{\prime }2}}{2}}{D}^{\prime }{\tilde{Q}}_{H_d}{v}_1+{\displaystyle \frac{\partial \Delta V}{\partial v_1}}=0,$$

(46)

$${\displaystyle \frac{\partial V}{\partial v_2}}=m_{H_u}^2{v}_2-{\displaystyle \frac{\lambda A_{\lambda }}{\sqrt{2}}}s{v}_1+{\displaystyle \frac{{\lambda }^2}{2}}(v_1^2+s^2)v_2+{\displaystyle \frac{{\overline{g}}^2}{8}}\left(v_2^2-v_1^2\right)v_2+{\displaystyle \frac{g_1^{{}^{\prime }2}}{2}}{D}^{\prime }{\tilde{Q}}_{H_u}{v}_2+{\displaystyle \frac{\partial \Delta V}{\partial v_2}}=0,$$

(47)

where $D^{\prime }={\tilde{Q}}_{H_d}{v}_1^2+{\tilde{Q}}_{H_u}{v}_2^2+{\tilde{Q}}_S{s}^2$ and $\overline{g}=\sqrt{g_2^2+g^{\prime 2}}$. Instead of specifying $v_1$ and $v_2$, it is more convenient to use $tan\beta =v_2/v_1$ and $v=\sqrt{v_1^2+v_2^2}\approx 246\mathrm{GeV}$.

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 $W^{\pm }$, Z and $Z^{\prime }$ vector bosons. The mass $M_W$ of the charged $W^{\pm }$ bosons is induced via the interaction of these vector bosons with the neutral components of $H_u$ and $H_d$. This results in $M_W={\displaystyle \frac{g_2}{2}}v$. Meanwhile, the mechanism of the neutral gauge boson mass generation differs substantially. Letting the Z and $Z^{\prime }$ states be the SM-like Z boson and the gauge boson associated with the $U{\left(1\right)}_N$ factor, the Z–$Z^{\prime }$ mass squared matrix is given by

$$M_{Z{Z}^{\prime }}^2=\left(\begin{array}{cc}{\displaystyle \frac{{\overline{g}}^2}{4}}{v}^2& {\displaystyle \frac{\overline{g}{g}_1^{\prime }}{2}}{v}^2\left({\tilde{Q}}_{H_d}{cos}^2\beta -{\tilde{Q}}_{H_u}{sin}^2\beta \right)\\ {\displaystyle \frac{\overline{g}{g}_1^{\prime }}{2}}{v}^2\left({\tilde{Q}}_{H_d}{cos}^2\beta -{\tilde{Q}}_{H_u}{sin}^2\beta \right)& {g_1^{\prime }}^2{D}^{\prime }\end{array}\right).$$

(48)

To ensure that the extra $U{\left(1\right)}_N$ vector boson is sufficiently heavy, the SM singlet fields S must acquire large VEV, $s\gg 1\mathrm{TeV}$. In this case, the mass of the lightest neutral vector boson $Z_1$ is very close to $M_Z=\overline{g}v/2$, while the mass of $Z_2$ is set by s, i.e., $M_{Z^{\prime }}\approx g_1^{\prime }{\tilde{Q}}_{S}s$.

To explore the spectrum of the Higgs bosons within the E${}_6$SSM, we express the soft masses $m_{H_d}^2,m_{H_u}^2,m_S^2$ in terms of other parameters using the minimization conditions in Equation (45)–(47). The charged components of $H_u$ and $H_d$ are not mixed with the neutral Higgs fields because of the electric charge conversation. They form a separate sector, whose spectrum is described by a $2\times 2$ mass matrix. The determinant of this matrix vanishes leading to the appearance of charged Goldstone states

$$G^{-}=H_d^{-}cos\beta -H_u^{+*}sin\beta$$

(49)

which are absorbed by the $W^{\pm }$ vector bosons. Their orthogonal linear combinations

$$H^{+}=H_d^{-*}sin\beta +H_u^{+}cos\beta$$

(50)

gain mass

$$m_{H^{\pm }}^2={\displaystyle \frac{\sqrt{2}\lambda A_{\lambda }}{sin2\beta }}s-{\displaystyle \frac{{\lambda }^2}{2}}{v}^2+{\displaystyle \frac{g^2}{2}}{v}^2+{\Delta }_{\pm }.$$

(51)

where ${\Delta }_{\pm }$ denotes the loop corrections to $m_{H^{\pm }}^2$.

The imaginary parts of the SM singlet field S and the neutral components of $H_u$ and $H_d$ do not mix with the real parts of these fields if CP-invariance is preserved. In this case, the imaginary parts of the neutral components of $H_u$ and $H_d$ as well as the imaginary part of the SM singlet field S form CP-odd Higgs sector. They compose two neutral Goldstone states

$$\begin{array}{c}G=\sqrt{2}(\mathrm{Im}{H}_d^{0}cos\beta -\mathrm{Im}{H}_u^{0}sin\beta ),\hfill \\ G^{\prime }=\sqrt{2}\mathrm{Im}Scos\gamma -\sqrt{2}(\mathrm{Im}{H}_u^{0}cos\beta +\mathrm{Im}{H}_d^{0}sin\beta )sin\gamma ,\hfill \end{array}$$

(52)

which are swallowed by the Z and $Z^{\prime }$ vector bosons, and one physical state

$$A=\sqrt{2}\mathrm{Im}Ssin\gamma +\sqrt{2}(\mathrm{Im}{H}_u^{0}cos\beta +\mathrm{Im}{H}_d^{0}sin\beta )cos\gamma ,$$

(53)

where $tan\gamma ={\displaystyle \frac{v}{2s}}sin2\beta$. Two massless pseudoscalars $G_0$ and $G^{\prime }$ decouple from the rest of the spectrum. The physical CP-odd Higgs state A acquires mass

$$m_A^2={\displaystyle \frac{\sqrt{2}\lambda A_{\lambda }}{sin2\gamma }}v+{\Delta }_A.$$

(54)

In Equation (54), ${\Delta }_A$ denotes loop corrections. Since in the E${}_6$SSM the VEV of the SM singlet field S must be much larger than v, the value of $\gamma$ is always small and A is mostly the superposition of the imaginary parts of the neutral components of the Higgs doublets. In the limit $s\gg v$, the masses of the charged and CP-odd Higgs states are approximately equal to each other.

The CP-even Higgs sector includes $Re{H}_d^0$, $Re{H}_u^0$ and $ReS$. In the field space basis $(h,H,N)$, where

$$\begin{array}{c}Re{H}_d^0=(hcos\beta -Hsin\beta +v_1)/\sqrt{2},\hfill \\ Re{H}_u^0=(hsin\beta +Hcos\beta +v_2)/\sqrt{2},\hfill \\ ReS=(s+N)/\sqrt{2},\hfill \end{array}$$

(55)

the mass matrix of the Higgs scalars in the E${}_6$SSM takes the form [141,142,143]:

$$M^2=\left(\begin{array}{ccc}{M}_{11}^2& M_{12}^2& M_{13}^2\\ M_{21}^2& M_{22}^2& M_{23}^2\\ M_{31}^2& M_{32}^2& M_{33}^2\end{array}\right)=\left(\begin{array}{ccc}{\displaystyle \frac{{\partial }^{2}V}{\partial v^2}}& {\displaystyle \frac{1}{v}}{\displaystyle \frac{{\partial }^{2}V}{\partial v\partial \beta }}& {\displaystyle \frac{{\partial }^{2}V}{\partial v\partial s}}\\ {\displaystyle \frac{1}{v}}{\displaystyle \frac{{\partial }^{2}V}{\partial v\partial \beta }}& {\displaystyle \frac{1}{v^2}}{\displaystyle \frac{{\partial }^{2}V}{{\partial }^2\beta }}& {\displaystyle \frac{1}{v}}{\displaystyle \frac{{\partial }^{2}V}{\partial s\partial \beta }}\\ {\displaystyle \frac{{\partial }^{2}V}{\partial v\partial s}}& {\displaystyle \frac{1}{v}}{\displaystyle \frac{{\partial }^{2}V}{\partial s\partial \beta }}& {\displaystyle \frac{{\partial }^{2}V}{{\partial }^{2}s}}\end{array}\right).$$

(56)

Calculating second derivatives of the effective potential of the Higgs sector in Equations (39)–(42) and substituting $m_{H_d}^2$, $m_{H_u}^2$ and $m_S^2$ from the minimization conditions in Equations (45)–(47), one finds:

$$M_{11}^2={\displaystyle \frac{{\lambda }^2}{2}}{v}^2{sin}^{2}2\beta +{\displaystyle \frac{{\overline{g}}^2}{4}}{v}^2{cos}^{2}2\beta +g_1^{{}^{\prime }2}{v}^2{({\tilde{Q}}_{H_d}{cos}^2\beta +{\tilde{Q}}_{H_u}{sin}^2\beta )}^2+{\Delta }_{11},$$

(57)

$$\begin{array}{ccc}\hfill M_{12}^2=M_{21}^2& =& \left({\displaystyle \frac{{\lambda }^2}{4}}-{\displaystyle \frac{{\overline{g}}^2}{8}}\right)v^{2}sin4\beta +{\displaystyle \frac{g_1^{{}^{\prime }2}}{2}}{v}^2({\tilde{Q}}_{H_u}-{\tilde{Q}}_{H_d})\times \hfill \\ \hfill & & \times ({\tilde{Q}}_{H_d}{cos}^2\beta +{\tilde{Q}}_{H_u}{sin}^2\beta )sin2\beta +{\Delta }_{12},\hfill \end{array}$$

(58)

$$M_{22}^2={\displaystyle \frac{\sqrt{2}\lambda A_{\lambda }}{sin2\beta }}s+\left({\displaystyle \frac{{\overline{g}}^2}{4}}-{\displaystyle \frac{{\lambda }^2}{2}}\right)v^2{sin}^{2}2\beta +{\displaystyle \frac{g_1^{{}^{\prime }2}}{4}}{({\tilde{Q}}_{H_u}-{\tilde{Q}}_{H_d})}^2{v}^2{sin}^{2}2\beta +{\Delta }_{22},$$

(59)

$$M_{23}^2=M_{32}^2=-{\displaystyle \frac{\lambda A_{\lambda }}{\sqrt{2}}}vcos2\beta +{\displaystyle \frac{g_1^{{}^{\prime }2}}{2}}({\tilde{Q}}_{H_u}-{\tilde{Q}}_{H_d}){\tilde{Q}}_{S}vssin2\beta +{\Delta }_{23},$$

(60)

$$M_{13}^2=M_{31}^2=-{\displaystyle \frac{\lambda A_{\lambda }}{\sqrt{2}}}vsin2\beta +{\lambda }^{2}vs+g_1^{{}^{\prime }2}({\tilde{Q}}_{H_d}{cos}^2\beta +{\tilde{Q}}_{H_u}{sin}^2\beta ){\tilde{Q}}_{S}vs+{\Delta }_{13},$$

(61)

$$M_{33}^2={\displaystyle \frac{\lambda A_{\lambda }}{2\sqrt{2}s}}{v}^{2}sin2\beta +g_1^{{}^{\prime }2}{\tilde{Q}}_S^2{s}^2+{\Delta }_{33}.$$

(62)

In Equations (57)–(62), ${\Delta }_{ij}$ denotes the loop corrections.

If the VEV of the SM singlet field S and all SUSY breaking parameters are much larger than the mass of the Z boson, then the mass matrix in Equations (56)–(62) has a hierarchical structure. In the field basis $(h,H,N)$, all off-diagonal elements of this matrix are relatively small ∼$M_S{M}_Z$. Therefore, the masses of the heaviest Higgs scalars are closely approximated by the diagonal entries of the mass matrix of the CP-even Higgs sector $M_{22}^2$ and $M_{33}^2$. These entries are expected to be of the order of the SUSY breaking scale $M_S^2$. Two heaviest CP-even Higgs bosons are predominantly formed by the components of the field basis H and N. Because the minimal eigenvalue of the mass matrix in Equations (56)–(62) does not exceed its smallest diagonal element, the mass of the lightest Higgs scalar $m_{h_1}$, which is predominantly h, remains always relatively small irrespective of the SUSY breaking scale, i.e., $m_{h_1}^2<M_{11}^2$. In the interactions with other SM particles, the lightest Higgs scalar manifests itself as a SM-like Higgs boson if $M_S\gg M_Z$.

As follows from Equations (51), (54) and (56)–(62), at the tree level, the spectrum of the Higgs bosons depends on four variables only:

$$\lambda ,s,tan\beta ,A_{\lambda }.$$

(63)

5. Higgs Spectrum

The qualitative pattern of the Higgs spectrum in the E${}_6$SSM is determined by the Yukawa coupling $\lambda$. Let us start our analysis here from the MSSM limit of the E${}_6$SSM when $\lambda \ll g_1^{\prime }$. In the case when $\lambda$ goes to zero, s has to be sufficiently large so that $\mu =\lambda s/\sqrt{2}$ is held fixed in order to give an acceptable chargino mass and EWSB. The diagonal entry $M_{33}^2$, which is set by $M_{Z^{\prime }}^2$, is considerably larger than other elements of the mass matrix in Equations (56)–(62) in this scenario. From the first minimization conditions in Equation (45), one can see that such solution can be obtained for $m_S^2<0$ when the absolute value of $m_S^2$ is very large. If $\mu \ll M_{Z^{\prime }}$ and $m_A^2\ll M_{Z^{\prime }}$, the CP-even Higgs mass matrix in Equations (56)–(62) can be reduced to the block diagonal form $M^{\prime }{}^2$ using unitary transformation [144,145]

$$M^{\prime }{}^2\approx \left(\begin{array}{ccc}{M}_{11}^2-{\displaystyle \frac{M_{13}^4}{M_{33}^2}}& M_{12}^2-{\displaystyle \frac{M_{13}^2{M}_{32}^2}{M_{33}^2}}& 0\\ M_{21}^2-{\displaystyle \frac{M_{23}^2{M}_{31}^2}{M_{33}^2}}& M_{22}^2-{\displaystyle \frac{M_{23}^4}{M_{33}^2}}& 0\\ 0& 0& M_{33}^2+{\displaystyle \frac{M_{13}^4}{M_{33}^2}}+{\displaystyle \frac{M_{23}^4}{M_{33}^2}}\end{array}\right).$$

(64)

In the limit when $\lambda$ is small, the top-left $2\times 2$ submatrix of the matrix in Equation (64) reproduces the mass matrix of the Higgs scalars in the MSSM. Such hierarchical structure of the mass matrix of the Higgs scalars in Equation (64) implies that the heaviest CP-even Higgs state associated with N and the $Z^{\prime }$ boson are almost degenerate. In other words, the singlet dominated CP-even state is always rather heavy and decouples from the rest of the spectrum. This makes the spectrum of the Higgs bosons indistinguishable from the one in the MSSM. The mass of the heaviest CP-even Higgs is determined by the VEV s. It does not change much if the other parameters $\lambda$, $tan\beta$ and $A_{\lambda }$ ($m_A$) vary. The masses of the second lightest Higgs scalar, which 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 $M_{Z^{\prime }}$. In this case, the expression for the SM-like Higgs mass $m_{h_1}^2$ is essentially the same as in the MSSM.

When $\lambda \ge g_1^{\prime }$, the qualitative pattern of the Higgs spectrum is quite similar to the one that arises in the NMSSM with the approximate PQ symmetry [144,145,146,147,148]. In the NMSSM and E${}_6$SSM, the growth of the Yukawa coupling $\lambda$ at low energies entails the increase of its value at the GUT scale $M_X$ leading to the appearance of the Landau pole [87,88]. This spoils the applicability of perturbation theory at high energies. The requirement of validity of perturbation theory up to the scale $M_X$ restricts the interval of variations of $\lambda \left(M_t\right)$ setting an upper limit on $\lambda \left(M_t\right)$ for each fixed value of $tan\beta$ in these models. In the E${}_6$SSM, the restrictions on the low energy values of $\lambda$ are weaker than in the NMSSM (see Figure 2, left). The presence of exotic matter change the evolution of the SM gauge couplings. Indeed, at the intermediate scales the values of these couplings rise when the number of extra $5+\overline{5}$-plets increases. In the RGEs that describe the evolution of the Yukawa couplings within the NMSSM and E${}_6$SSM, the gauge couplings occur on the right-hand side of these equations with negative sign. As a consequence, the growth of the SM gauge couplings reduces the values of the Yukawa couplings at the intermediate scales preventing the appearance of the Landau pole in the RG flow of these couplings. Therefore, within the E${}_6$SSM, the values of $\lambda \left(M_t\right)$ can be larger than in the NMSSM. The upper bound on the low energy values of $\lambda$ grows with increasing $tan\beta$ since the top-quark Yukawa coupling decreases. At large $tan\beta$, this bound approaches the saturation limit. In the NMSSM and E${}_6$SSM the maximal possible values of $\lambda \left(M_t\right)$ are 0.71 and 0.84, respectively, whereas the low energy value of $g_1^{\prime }\approx g_1$ vary from $0.46$ to $0.48$.

If $\lambda \ge g_1^{\prime }$, then $M_{22}^2$ tends to be the largest diagonal entry of the mass matrix in Equations (56)–(62), i.e., $M_{22}^2\gg M_{33}^2\gg M_{11}^2$. Relying on this mass hierarchy, the approximate analytical expressions for the masses of the CP-even Higgs bosons can be obtained. The perturbation theory method yields [141,142,143,144,145]

$$m_{h_3}^2\approx M_{22}^2+{\displaystyle \frac{M_{23}^4}{M_{22}^2}},m_{h_2}^2\approx M_{33}^2-{\displaystyle \frac{M_{23}^4}{M_{22}^2}}+{\displaystyle \frac{M_{13}^4}{M_{33}^2}},m_{h_1}^2\approx M_{11}^2-{\displaystyle \frac{M_{13}^4}{M_{33}^2}}.$$

(65)

All terms in Equation (65) suppressed by inverse powers of $m_A^2$ or $M_{Z^{\prime }}^2$, which are of the order of $M_Z^4/m_A^2$, $M_Z^4/M_{Z^{\prime }}^2$ or even smaller, are neglected. At tree-level, the Higgs masses in the E${}_6$SSM are given by

$$\begin{array}{ccc}\hfill m_{h_3}^2& \approx & m_{H^{\pm }}^2\approx m_A^2\approx {\displaystyle \frac{4{\mu }^{2}x}{{sin}^{2}2\beta }},m_{h_2}^2\approx M_{Z^{\prime }}^2,\hfill \\ \hfill m_{h_1}^2& \approx & {\displaystyle \frac{{\lambda }^2}{2}}{v}^2{sin}^{2}2\beta +M_Z^2{cos}^{2}2\beta +g_1^{{}^{\prime }2}{v}^2{\left({\tilde{Q}}_{H_d}{cos}^2\beta +{\tilde{Q}}_{H_u}{sin}^2\beta \right)}^2\hfill \\ \hfill & & -{\displaystyle \frac{{\lambda }^4{v}^2}{g_1^{{}^{\prime }2}{\tilde{Q}}_S^2}}{\left(1-x+{\displaystyle \frac{g_1^{{}^{\prime }2}}{{\lambda }^2}}\left({\tilde{Q}}_{H_d}{cos}^2\beta +{\tilde{Q}}_{H_u}{sin}^2\beta \right){\tilde{Q}}_S\right)}^2,\hfill \end{array}$$

(66)

where $x={\displaystyle \frac{A_{\lambda }}{2\mu }}sin2\beta$ and $\mu ={\displaystyle \frac{\lambda }{\sqrt{2}}}s$. As evident from the explicit expression for $m_{h_1}^2$ given above at ${\lambda }^2\gg g_1^2$, the last term in this expression dominates and the mass squared of the lightest Higgs scalar tends to be negative if x is not close to unity. A negative eigenvalue of the mass matrix in Equations (57)–(62) implies that the vacuum configuration in Equation (44) ceases to be a minimum and turns into a saddle point. This means that there is a direction in field space in the vicinity of such point along which the energy density decreases leading to the instability of the vacuum configuration in Equation (44). Thus, large deviations of x from unity pulls the mass squared of the lightest Higgs scalar below zero destabilizing the vacuum. The requirement of stability of the physical vacuum therefore constrains the variable x around unity limiting the interval of variations of $m_A$ from below and above. As a consequence, the masses of the charged, CP-odd and heaviest CP-even Higgs bosons are almost degenerate around $m_A$ and are confined in the vicinity of $\mu tan\beta$. They are considerably larger than the masses of the $Z^{\prime }$ and lightest CP-even Higgs boson. Together with the experimental lower limit on the $Z^{\prime }$ boson mass it maintains the mass hierarchy in the Higgs spectrum [69].

From the explicit analytic expression for $m_{h_1}^2$, it is apparent that, at some value of x (or $m_A$), the mass of the lightest Higgs scalar mass attains its maximum value. It corresponds to the value of x for which the fourth term in the expression for $m_{h_1}^2$ vanishes. In this case, the mass squared of the lightest Higgs boson coincides with the theoretical upper bound on $m_{h_1}^2$ given by $M_{11}^2$. The sum of the first and second terms in the expression for $M_{11}^2$ are similar to the tree-level upper bound on $m_{h_1}^2$ in the NMSSM [149,150]. The third term in Equation (57) is a contribution which comes from the additional $U{\left(1\right)}_N$D-term in the potential of the Higgs sector in Equations (39)–(42). At tree-level the upper bound on the mass of the lightest Higgs scalar in the E${}_6$SSM depends on $\lambda$ and $tan\beta$ only. Using the obtained theoretical restrictions on the low energy values of $\lambda$ as a function of $tan\beta$, one can calculate for each $tan\beta$ the maximum possible value of $m_{h_1}$.

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 $tan\beta$ the theoretical restriction on lightest Higgs boson mass in the E${}_6$SSM and NMSSM exceeds the corresponding limit in the MSSM. This happens because of the contribution to $M_{11}^2$ induced by the first term in the right-hand side of Equation (57). It comes from the additional F-term in the Higgs scalar potentials of the E${}_6$SSM and NMSSM. For the values of $tan\beta \sim$1–3, this contribution to the upper bound on $m_{h_1}$ in the E${}_6$SSM and NMSSM dominates. Its size is determined by the Yukawa coupling $\lambda$. Since in the E${}_6$SSM $\lambda$ is allowed to be larger than in the NMSSM, the tree-level theoretical restriction on $m_{h_1}$ in the E${}_6$SSM is also larger at moderate values of $tan\beta$ as compared with the one in the NMSSM. In the framework of the E${}_6$SSM, the upper bound on $m_{h_1}$ attains its maximum value of $130\mathrm{GeV}$ at $tan\beta$ = 1.5–1.8. Thus, large tree-level theoretical restriction on the mass of the lightest Higgs scalar means that in this model the contribution of loop corrections to $m_{h_1}^2$ 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 $125\mathrm{GeV}$.

With increasing $tan\beta$, the contribution to $M_{11}^2$ associated with the first term on the right-hand side of Equation (57) decreases rapidly. For $tan\beta >10$, it becomes negligibly small. The second term on the right-hand side of Equation (57) grows when $tan\beta$ increases. At $tan\beta >4$, it exceeds ${\displaystyle \frac{{\lambda }^2}{2}}{v}^2{sin}^{2}2\beta$ and gives the dominant contribution to the tree-level theoretical restriction on $m_{h_1}$. Therefore, with increasing $tan\beta$, the upper bound on the mass of the lightest CP-even Higgs state in the NMSSM diminishes and approaches the corresponding limit in the MSSM. In the case of the E${}_6$SSM, the third term on the right-hand side of Equation (57), that comes from the extra $U{\left(1\right)}_N$D-term contribution to the Higgs scalar potential in Equations (39)–(42), gives the second largest contribution to $M_{11}^2$ at very large values of $tan\beta$. 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 $tan\beta$ rises, is about 6–7 GeV larger for $tan\beta >10$ than the corresponding upper bounds in the MSSM and NMSSM. As a consequence, at large $tan\beta$, 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 $m_{h_1}^2$ 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 loop diagrams 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 increase the theoretical upper bound on the lightest Higgs boson mass, which at the tree-level does not exceed $M_Z$ [151,152], from $M_Z$ to $130\mathrm{GeV}$ (see [153] and references therein). In the leading approximation, the two-loop upper bound on the lightest Higgs scalar mass in the E${}_6$SSM may be presented in the following form [69]

$$\begin{array}{c}{m}_{h_1}^2\le \left[{\displaystyle \frac{{\lambda }^2}{2}}{v}^2{sin}^{2}2\beta +M_Z^2{cos}^{2}2\beta +g_1^{{}^{\prime }2}{v}^2{\left({\tilde{Q}}_{H_d}{cos}^2\beta +{\tilde{Q}}_{H_u}{sin}^2\beta \right)}^2\right]\left(1-{\displaystyle \frac{3h_t^2}{8{\pi }^2}}l\right)\\ +{\displaystyle \frac{3m_t^4}{2{\pi }^2{v}^2}}\left\{{\displaystyle \frac{1}{2}}{U}_t+l+{\displaystyle \frac{1}{16{\pi }^2}}\left({\displaystyle \frac{3}{2}}{h}_t^2-8g_3^2\right)(U_t+l)l\right\},\\ m_t\left(M_t\right)={\displaystyle \frac{h_t\left(M_t\right)}{\sqrt{2}}}vsin\beta ,U_t=2{\displaystyle \frac{X_t^2}{M_S^2}}\left(1-{\displaystyle \frac{1}{12}}{\displaystyle \frac{X_t^2}{M_S^2}}\right),l=ln\left[{\displaystyle \frac{M_S^2}{m_t^2}}\right],\end{array}$$

(67)

where $X_t$ is a stop mixing parameter. In Equation (67), $M_S$ is defined as $M_S^2=m_Q^2=m_U^2$, where $m_U^2$ and $m_Q^2$ are soft scalar masses of superpartners of the right-handed and left-handed components of the t-quark, respectively. Using the relationship between the running ($m_t\left(Q\right)$) and t-quark pole ($M_t$) masses [154,155]

$$m_t\left(M_t\right)=M_t\left[1-1.333{\displaystyle \frac{{\alpha }_s\left(M_t\right)}{\pi }}-9.125{\left({\displaystyle \frac{{\alpha }_s\left(M_t\right)}{\pi }}\right)}^2\right]$$

(68)

one can compute $m_t\left(M_t\right)$ for the world average mass $M_t=173.1\pm 0.9\mathrm{GeV}$ [156]. Equation (67) is just a simple generalization of the approximate expressions for the theoretical restriction on the mass of the lightest Higgs scalar obtained in the MSSM [157] and NMSSM [158]. In the case $g_1^{\prime }=0$ and $\lambda =0$, Equation (67) coincides with the theoretical bound on the mass of the lightest CP-even Higgs state 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 in Equation (67) results in $m_{h_1}$, which is typically a few $\mathrm{GeV}$ lower than the lightest Higgs mass which is computed using the Suspect [159] and FeynHiggs [160,161,162,163] packages. It was shown that in the two-loop approximation the lightest Higgs boson mass in the E${}_6$SSM does not exceed $150\mathrm{GeV}$ [69].

Although the inclusion of loop corrections changes considerably the lightest Higgs boson mass in the E${}_6$SSM, it does not change the qualitative pattern of the spectrum of the Higgs states for $\lambda \ll g_1^{\prime }$ and $\lambda >g_1^{\prime }$. The mass of the SM singlet dominated CP-even state is always set by $M_{Z^{\prime }}$, whereas another Higgs scalar, CP-odd and charged Higgs states 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 $\lambda >g_1^{\prime }$, the charged, CP-odd and heaviest CP-even Higgs states lie beyond the multi-TeV range and therefore cannot be detected at the LHC experiments.

6. LHC Signatures

We now turn to the LHC signatures of the E${}_6$SSM, which permit distinguishing this SUSY model from the MSSM or NMSSM. As discussed above, in the simplest phenomenologically viable scenarios, the lightest exotic fermion ${\tilde{H}}_1^0$ should have mass $m_{{\tilde{H}}_1^0}\ll 1\mathrm{eV}$. At the same time, the next-to-lightest exotic fermion ${\tilde{H}}_2^0$ may be considerably heavier. Let us assume that all sparticles and exotic states except ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ are rather heavy and can be integrated out. In particular, the parameters are chosen so that all fermion components of the supermultiplets $H_{\alpha }^u$ and $H_{\alpha }^d$ are heavier than $100\mathrm{GeV}$, whereas $s\approx 12\mathrm{TeV}$. In this limit, the part of the Lagrangian involving the couplings of ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ to the SM-like Higgs state and the Z boson takes the following form:

$$\begin{array}{c}{\mathcal{L}}_{Zh}={\sum }_{\alpha ,\beta }{\displaystyle \frac{M_Z}{2v}}{Z}_{\mu }\left({\tilde{H}}_{\alpha }^{0T}{\gamma }_{\mu }{\gamma }_5{\tilde{H}}_{\beta }^0\right)R_{Z\alpha \beta }+{\sum }_{\alpha ,\beta }{X}_{\alpha \beta }^h\left({\tilde{H}}_{\alpha }^{0T}{\tilde{H}}_{\beta }^0\right)h,\end{array}$$

(69)

where $\alpha ,\beta =1,2$. Although ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ are substantially lighter than $100\mathrm{GeV}$, the interactions of these exotic states with the SM particles including Z boson tend to be rather weak because they are predominantly the fermion components of the superfields $S_{\alpha }$. Therefore, any possible signal that ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ could give rise to at former and present experiments might be extremely suppressed and such states could remain undetected.

The couplings of the SM-like Higgs state $h_1$ to ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ are determined by the masses of these lightest exotic states [94]. Since ${\tilde{H}}_1^0$ is very light, it does not affect Higgs phenomenology. The absolute value of the coupling of $h_1$ to the second lightest exotic particle $|X_{22}^h| \simeq |m_{{\tilde{H}}_2^0}|/v$ [94]. This coupling gives rise to the decays of $h_1$ into ${\tilde{H}}_2^0$ pairs. The corresponding partial decay width is given by

$$\Gamma (h_1\to {\tilde{H}}_2^0{\tilde{H}}_2^0)={\displaystyle \frac{|X_{22}^h{|}^2{m}_{h_1}}{4\pi }}{\left(1-4{\displaystyle \frac{|m_{{\tilde{H}}_2^0}{|}^2}{m_{h_1}^2}}\right)}^{3/2}.$$

(70)

The nonstandard Higgs boson decays were studied in the context of different extensions of the SM (see, for example, [98,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180]). The partial decay width in Equation (70) depends rather strongly on $m_{{\tilde{H}}_2^0}$. To avoid the suppression of the branching ratios for the lightest Higgs boson decays into SM particles, we restrict our consideration here to the GeV-scale masses of ${\tilde{H}}_2^0$.

To compare the partial width that correspond to the exotic decays of $h_1$ in Equation (70) with other decay rates a set of benchmark points (see

Table 2. Benchmark scenarios for $m_{h_1}\approx 125\mathrm{GeV}$; the partial width of the decays $h_1\to {\tilde{H}}_2^0{\tilde{H}}_2^0$, the branching ratios of the lightest Higgs scalar, the masses and couplings of ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ are calculated for $s=12\mathrm{TeV}$, $\lambda =0.6$, $tan\beta =1.5$, $m_{H^{\pm }}\simeq m_A\simeq m_{h_3}\simeq 9497\mathrm{GeV}$, $m_{h_2}\simeq M_{Z^{\prime }}\simeq 4450\mathrm{GeV}$, $m_Q=m_U=M_S=4000\mathrm{GeV}$ and $X_t=\sqrt{6}{M}_S$.

	i	ii	iii	iv
${\lambda }_{22}$	−0.03	−0.012	−0.06	0
${\lambda }_{21}$	0	0	0	0.02
${\lambda }_{12}$	0	0	0	0.02
${\lambda }_{11}$	0.03	0.012	0.06	0
$f_{22}$	−0.1	−0.1	−0.1	0.6
$f_{21}$	−0.1	−0.1	−0.1	0.00245
$f_{12}$	0.00001	0.00001	0.00001	0.00245
$f_{11}$	0.1	0.1	0.1	0.00001
${\tilde{f}}_{22}$	0.1	0.1	0.1	0.6
${\tilde{f}}_{21}$	0.1	0.1	0.1	0.002
${\tilde{f}}_{12}$	0.000011	0.000011	0.000011	0.002
${\tilde{f}}_{11}$	0.1	0.1	0.1	0.00001
$|m_{{\tilde{H}}_1^0}|$/GeV	$2.7\times {10}^{-11}$	$6.5\times {10}^{-11}$	$1.4\times {10}^{-11}$	$0.31\times {10}^{-9}$
$|m_{{\tilde{H}}_2^0}|$/GeV	1.09	2.67	0.55	0.319
$|R_{Z11}|$	0.0036	0.0212	0.00090	$1.5\times {10}^{-7}$
$|R_{Z12}|$	0.0046	0.0271	0.00116	$1.7\times {10}^{-4}$
$|R_{Z22}|$	0.0018	0.0103	0.00045	0.106
$X_{22}^{h_1}$	0.0044	0.0106	0.0022	0.00094
$\mathrm{Br}(h_1\to {\tilde{H}}_2^0{\tilde{H}}_2^0)$	4.7%	21.9%	1.23%	0.22%
$\mathrm{Br}(h_1\to b\overline{b})$	56.6%	46.4%	58.7%	59.3%
$\Gamma (h_1\to {\tilde{H}}_2^0{\tilde{H}}_2^0)$/MeV	0.194	1.106	0.049	0.0088

) is specified. The masses of the heavy Higgs bosons given in Table 2 are computed in the leading one-loop approximation. In the case of the mass of the lightest Higgs scalar, the leading two-loop corrections are taken into account. In all benchmark scenarios, the structure of the spectrum of the Higgs states is rather hierarchical. Since all heavy Higgs particles have masses, which are substantially larger than the lightest Higgs boson mass, the partial widths of the decays of $h_1$ into the SM particles are almost the same as in the SM. Therefore, in our analysis, we use the results presented in [181] where the corresponding decay rates were estimated within the SM for different values of the mass of the Higgs boson. When $m_{h_1}\simeq 125GeV$, the SM-like Higgs boson decays mostly into $b\overline{b}$. The corresponding branching ratio is about $60\%$ while the branching ratios associated with the decays of $h_1$ into $WW$ and $ZZ$ are about $20\%$ and $2\%$, respectively [181]. The total decay width of such Higgs boson is about $4\mathrm{MeV}$.

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 $m_{{\tilde{H}}_2^0}$ varies from $0.3\mathrm{GeV}$ to $2.7\mathrm{GeV}$ [98]. For smaller (larger) values of $m_{{\tilde{H}}_2^0}$ the branching ratio of these decays is even smaller (larger). On the other hand, the couplings of the exotic states ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ to the Z boson are so small that these fermions could not be observed before. In particular, the contribution of ${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$ to the Z-boson width is very small. After being produced ${\tilde{H}}_2^0$ sequentially decay into ${\tilde{H}}_1^0$ and fermion–antifermion 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, ${\tilde{H}}_2^0$ tends to live longer than ${10}^{-8}\mathrm{s}$ and typically decays outside the detectors. As a consequence, the decay channel $h_1\to {\tilde{H}}_2^0{\tilde{H}}_2^0$ normally gives rise to an invisible branching ratio of the lightest Higgs scalar. The benchmark Scenarios (i), (iii) and (iv) lead to such invisible decays of $h_1$. In the case of benchmark Scenario (ii), $|R_{Z12}|$ is larger so that ${\tau }_{{\tilde{H}}_2^0}\sim {10}^{-11}\mathrm{s}$ and some of the decay products of ${\tilde{H}}_2^0$ could be observed at the LHC.

Because $R_{Z12}$ is relatively small, ${\tilde{H}}_2^0$ may decay during or after Big Bang Nucleosynthesis (BBN) destroying the agreement between the observed and predicted abundances of light elements. To preserve the success of the BBN, ${\tilde{H}}_2^0$ must decay before BBN, i.e., the lifetime of the second lightest exotic fermion ${\tau }_{{\tilde{H}}_2^0}$ should not be longer than $1\mathrm{s}$. This requirement sets lower bound on $|R_{Z12}|$. Indeed, for $m_{{\tilde{H}}_2^0}=1\mathrm{GeV}$, the absolute value of the coupling $R_{Z12}$ has to be larger than $1\times {10}^{-6}$ [182]. The constraint on $|R_{Z12}|$ becomes more stringent when $m_{{\tilde{H}}_2^0}$ decreases because ${\tau }_{{\tilde{H}}_2^0}\sim 1/\left(\right|R_{Z12}{|}^2{m}_{{\tilde{H}}_2^0}^5)$. The results of our analysis indicate that it is somewhat problematic to ensure that ${\tau }_{{\tilde{H}}_2^0}\le 1\mathrm{s}$ if $m_{{\tilde{H}}_2^0}\le 100\mathrm{MeV}$.

The presence of a $Z^{\prime }$ boson and exotic matter which compose three $5+5^{*}$ representations of $SU\left(5\right)$ 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 which varies substantially depending on the choice of the parameters. At tree-level, the masses of the $Z^{\prime }$ boson and fermion components of $5+5^{*}$ supermultiplets are set by the VEV of the SM singlet superfield S, which is a free parameter in these models. Therefore, the masses of these states cannot be predicted. The lower experimental limits on the $Z^{\prime }$ mass, that comes from the direct searches $(pp\to Z^{\prime }\to l^{+}{l}^{-})$ conducted at the LHC experiments, are already very stringent and vary around $3.8-3.9\mathrm{TeV}$ [183,184]. This means that the scenarios with $s<10-10.5\mathrm{TeV}$ have been excluded. Possible decay channels of the $Z^{\prime }$ gauge bosons in $E_6$ inspired SUSY models were studied in [60,67].

Assuming that $f_{\alpha \beta }$ and ${\tilde{f}}_{\alpha \beta }$ are very small the masses of the fermion components of extra $5+5^{*}$ supermultiplets of matter are given by

$${\mu }_{D_i}={\displaystyle \frac{{\kappa }_i}{\sqrt{2}}}s,{\mu }_{H_{\alpha }}={\displaystyle \frac{{\lambda }_{\alpha }}{\sqrt{2}}}s,$$

(71)

where ${\mu }_{D_i}$ are the masses of exotic quarks with electric charges $\pm 1/3$ and ${\mu }_{H_{\alpha }}$ are the masses of the $SU{\left(2\right)}_W$ doublets of the Inert Higgsino states. Here, we set ${\kappa }_{ij}={\kappa }_i{\delta }_{ij}$ and ${\lambda }_{\alpha \beta }={\lambda }_{\alpha }{\delta }_{\alpha \beta }$. Requiring the validity of perturbation theory up to the scale $M_X$ one can obtain upper bounds on the low-energy values of ${\kappa }_i$ and ${\lambda }_{\alpha }$. Nevertheless, the low-energy values of these couplings are allowed to be as large as $g_1^{{}^{\prime }}\left(q\right)\approx g_1\left(q\right)\approx 0.46-0.48$. On the other hand, since the exotic fermions must be sufficiently heavy to avoid any conflict with direct particle searches at former and present accelerators, couplings ${\lambda }_{\alpha }$ and ${\kappa }_i$ have to be large enough. Although nowadays there are clear indications that $Z^{\prime }$ boson and sparticles have to be rather heavy some of the exotic fermions can be relatively light in the E${}_6$SSM. This may happen if the matrices of the Yukawa couplings of the exotic particles ${\kappa }_{ij}$ and ${\lambda }_{\alpha }$ have hierarchical structure which is similar to the one in the quark and lepton sectors of the SM. Then, $Z^{\prime }$ boson can be much heavier than $10\mathrm{TeV}$ and the only manifestation of the E${}_6$SSM can be the presence of light exotic fermions 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 $Z_2^H$ symmetry violating couplings. Since in order to suppress FCNCs the Yukawa couplings of exotic states to the SM fermions of the first two generations have to be rather small, here we assume that exotic particles 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

$$\overline{D}\to t+b+E_T^{\mathrm{miss}}+X,$$

(72)

if exotic quarks ${\overline{D}}_i$ are diquarks or via

$$D\to t+\tau +E_T^{\mathrm{miss}}+X,D\to b+{\nu }_{\tau }+E_T^{\mathrm{miss}}+X,$$

(73)

if D-fermions are leptoquarks. Thus, the pair production of light D-fermions at the LHC should result in some enhancement of the cross sections of either $pp\to t\overline{t}b\overline{b}+E_T^{\mathrm{miss}}+X$ if exotic quark states are diquarks or $pp\to t\overline{t}{\tau }^{+}{\tau }^{-}+E_T^{\mathrm{miss}}+X$ and $pp\to b\overline{b}+E_T^{\mathrm{miss}}+X$ when D-fermions are leptoquarks.

In general, exotic squarks tend to be considerably heavier than the D-fermions because the masses of these scalars 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 D-fermion gives rise to the large mixing in the corresponding exotic squark sector. Such mixing can lead to the large mass splitting between the appropriate mass eigenstates. Because of this, the lightest exotic squark may be much lighter than all other scalars. Moreover, in principle, it can be even lighter than the lightest D-fermion. If this is a case, then in the variants of the E${}_6$SSM with approximate $Z_2^H$ symmetry the lightest exotic squark decays into either

$$\tilde{D}\to t+b++X,$$

(74)

if it is a scalar diquark or

$$D\to t+\tau +X,D\to b+{\nu }_{\tau }+X,$$

(75)

if this state is a scalar leptoquark. In the limit, when the couplings of this squark to the quarks and leptons of the first two generations are rather small, the lightest exotic squarks may only be pair produced at the LHC. Therefore, the presence of relatively light $\tilde{D}$ is expected to lead to some enhancement of the cross sections of either $pp\to t\overline{t}{\tau }^{+}{\tau }^{-}+X$ and $pp\to b\overline{b}+E_T^{\mathrm{miss}}+X$ if exotic squarks are leptoquarks or $pp\to t\overline{t}b\overline{b}+X$ when they are diquarks. On the other hand, in the variants of the E${}_6$SSM with exact ${\tilde{Z}}_2^H$ symmetry the $Z_2^E$ symmetry conservation implies that the final state in the decay of $\tilde{D}$ should always contain the lightest exotic fermion ${\tilde{H}}_1^0$ [72]. Because the lightest exotic squark is R-parity even $SU{\left(3\right)}_C$ triplet of scalar fields while ${\tilde{H}}_1^0$ is R-parity odd fermion the final state in the decay of $\tilde{D}$ 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 which appear in the case of the lightest exotic quarks. The presence of relatively light exotic squark and D-fermion may significantly modify the LHC signatures associated with the gluinos [92].

A few experiments at LEP, HERA, Tevatron and LHC have searched for the states which decay into either a pair of quarks or lepton and quark. Most searches focus on leptoquarks or diquarks which have integer–spin so that they can be either scalars or vectors. Such particles may be coupled directly to either quark and lepton or a pair of quarks. 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 and second generation scalar leptoquarks (i.e., states which mainly interact with the first and second generation fermions, respectively) that have masses below 1230–$1560\mathrm{GeV}$ depending on the branching ratios of their decays [185,186,187]. The experimental limits on the masses of the third-generation scalar leptoquarks are somewhat weaker. ATLAS and CMS collaborations excluded such exotic objects if they have masses below 800–1000 GeV [188,189,190]. The experimental lower limits on the masses of dijet resonances including diquarks tend to be considerably higher [191].

However, the LHC lower bounds mentioned above 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 have to be quite small to avoid processes with non-diagonal flavor transitions. Therefore, in this SUSY model, diquarks can only be pair produced. It is also worth pointing out that the lightest exotic quarks in the E${}_6$SSM give rise to collider signatures which are very different from the commonly established ones that have been thoroughly studied. Indeed, it is commonly assumed that scalar leptoquarks or diquarks decay 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 results in the missing energy and transverse momentum in the final state. Because of this, the pair production of the lightest exotic diquarks and the pair production of gluinos at the LHC may give rise to the enhancement of the same cross section of $pp\to t\overline{t}b\overline{b}+E_T^{\mathrm{miss}}+X$.

The $SU{\left(2\right)}_W$ doublets of the Inert Higgsino states may also be heavy or light depending on free parameters. When at least one coupling ${\lambda }_{\alpha }$ is of the order of unity it may give rise to a large mixing in the Inert Higgs sector which can result in relatively light Inert Higgs bosons. Since these bosons have very small couplings to the fermions of the first and second generation such states can be produced in pairs at the LHC via off-shell W and Z bosons. As a consequence, the production cross section of the corresponding scalars 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 $pp\to Q\overline{Q}{\tau }^{+}{\tau }^{-}$ and $pp\to Q\overline{Q}{Q}^{{}^{\prime }}{\overline{Q}}^{{}^{\prime }}$, where Q and $Q^{\prime }$ are heavy quarks of the third-generation.

From Equation (71), it follows that the lightest Inert Higgsinos can be relatively light if the corresponding Yukawa coupling ${\lambda }_{\alpha }$ is sufficiently small. If all sparticles and other exotic states are quite heavy, the corresponding fermionic states can be produced at the LHC in pairs via weak interactions. 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 (${\tilde{H}}_1^0$ or ${\tilde{H}}_2^0$) 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 $pp\to ZZ+E_T^{\mathrm{miss}}+X$, $pp\to WZ+E_T^{\mathrm{miss}}+X$ and $pp\to WW+E_T^{\mathrm{miss}}+X$. 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 [192,193,194], one can conclude that the mass of the $SU{\left(2\right)}_W$ doublets of the Inert Higgsino states has to be larger than $650\mathrm{GeV}$.

7. 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 MSSM, NMSSM, $U\left(1\right)$ extensions of the MSSM, etc. Among $U\left(1\right)$ extensions of the MSSM inspired by $E_6$ GUTs there is a model based on the SM gauge group together with an additional $U{\left(1\right)}_N$ gauge symmetry. Only in this $U{\left(1\right)}_N$ extension of the MSSM the right-handed neutrinos do not participate in gauge interactions, which allows them to be used for a high scale see-saw mechanism. In this Exceptional Supersymmetric Standard Model (E${}_6$SSM), the lepton asymmetry, which may be induced by the heavy right-handed neutrino decays, can be partially converted into baryon asymmetry via sphaleron processes [195,196]. The $U{\left(1\right)}_N$ symmetry forbids the bilinear term $\mu H_d{H}_u$ in the superpotential, but permits the term $\lambda S\left(H_u{H}_d\right)$, where S is a SM singlet superfield that carries $U{\left(1\right)}_N$ charge. When S develops VEV breaking $U{\left(1\right)}_N$ gauge symmetry, it also gives rise to an effective $\mu$ term. Thus, within the E${}_6$SSM, the $\mu$ problem can be solved without the accompanying problems of singlet tadpoles or domain walls which appear in the NMSSM.

In this review article, we discuss the particle content, the global symmetries which allow suppressing FCNCs and rapid proton decay and 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 $SU{\left(2\right)}_W$ doublets $L_4$ and ${\overline{L}}_4$ 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 ${\overline{D}}_i$), two pairs of $SU{\left(2\right)}_W$ doublets of Inert Higgs states, three SM singlet superfields which carry $U{\left(1\right)}_N$ charges, $L_4$, ${\overline{L}}_4$ and $Z^{\prime }$ vector superfield. As in the simplest SUSY extensions of the SM the gauge symmetry of the E${}_6$SSM does not permit to suppress baryon and lepton number violating interactions which can lead to rapid proton decay. Moreover, in general relatively, light exotic states induce unacceptably large flavor changing processes. To suppress the corresponding baryon and lepton number violating operators, one can impose either $Z_2^L$ or $Z_2^B$ discrete symmetry, which implies that the exotic D-fermions are either diquarks (Model I) or leptoquarks (Model II). To avoid the appearance of the FCNCs at the tree level, one can postulate an approximate $Z_2^H$ 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 $Z_2^H$, $Z_2^L$ and $Z_2^B$, one can use a single discrete ${\tilde{Z}}_2^H$ symmetry, which forbids operators giving rise to rapid proton decay and tree-level flavor-changing transitions. The Higgs supermultiplets $H_u$, $H_d$ and S as well as $L_4$ and ${\overline{L}}_4$ are even under the ${\tilde{Z}}_2^H$ symmetry, whereas all other matter fields are odd. In this case, the exotic D-fermions are leptoquarks.

The results of the analysis of the two-loop RG flow within the E${}_6$SSM are presented taking into account kinetic term mixing between $U{\left(1\right)}_Y$ and $U{\left(1\right)}_N$ factors. If there is no mixing between $U{\left(1\right)}_Y$ and $U{\left(1\right)}_N$ 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 $U{\left(1\right)}_N$ is always close to the $U{\left(1\right)}_Y$ gauge coupling. On the other hand, at high energies, the values of the gauge couplings are considerably larger in the E${}_6$SSM than in the MSSM due to the presence of the supermultiplets of exotic matter in the $U{\left(1\right)}_N$ extensions of the MSSM. Our analysis reveals that the unification of gauge couplings in the E${}_6$SSM can be achieved for phenomenologically acceptable values of ${\alpha }_3\left(M_Z\right)$, consistent with the central measured 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 $tan\beta$ 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 even larger than 115–$125\mathrm{GeV}$ 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 $125\mathrm{GeV}$ Higgs boson.

In this article, the gauge symmetry breaking and the spectrum of the Higgs bosons within the E${}_6$SSM are reviewed as well. In the $U{\left(1\right)}_N$ extensions of the MSSM the SM singlet Higgs field, S is required to acquire a very large VEV $\langle S\rangle =s/\sqrt{2}$, where $s>10\mathrm{TeV}$, to ensure that the $Z^{\prime }$ boson and exotic fermions gain sufficiently large masses. In particular, the results of the analysis of the LHC data imply that the $U{\left(1\right)}_N$ gauge boson must be heavier than 3.8–3.9 $\mathrm{TeV}$. When CP–invariance is preserved, the E${}_6$SSM spectrum of the Higgs states involves two charged, three CP-even and one CP-odd bosons. The SM singlet dominated CP-even state and the $Z^{\prime }$ gauge boson are almost degenerate. The masses of the charged and another CP-even 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 mass of the lightest Higgs state can be larger 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 consider possible manifestations of the E${}_6$SSM that may be observed at the LHC. The simplest phenomenologically viable scenarios imply that LSP and NLSP are the lightest exotic states (${\tilde{H}}_1^0$ and ${\tilde{H}}_2^0$), which are formed by the fermion components of the SM singlet superfields $S_{\alpha }$. One of these fermions ${\tilde{H}}_1^0$ should be much lighter than $1\mathrm{eV}$ composing hot dark matter in the Universe. Such states give only a very minor contribution to the density of the dark matter. The NLSP ${\tilde{H}}_2^0$ can have mass of the order of $1\mathrm{GeV}$ giving rise to exotic decays of the $125\mathrm{GeV}$ Higgs boson. Since ${\tilde{H}}_2^0$ tends to live longer than ${10}^{-8}\mathrm{s}$ it decays outside the detectors. Therefore, the decay channel $h_1\to {\tilde{H}}_2^0{\tilde{H}}_2^0$ 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 distinguishing this model from the MSSM or NMSSM, are associated with the presence of the $Z^{\prime }$ gauge boson and exotic supermultiplets of matter that compose three $5+5^{*}$ representations of $SU\left(5\right)$. The most spectacular LHC signals can come from the exotic color states and $Z^{\prime }$. The $Z^{\prime }$ boson production at the LHC should lead to unmistakable signal, i.e., $pp\to Z^{\prime }\to l^{+}{l}^{-}$. Assuming that the $Z_2^H$ symmetry is mainly broken by the operators which involve third-generation fermions, the pair production of the lightest exotic quarks with masses in a few TeV range can give rise to some enhancement of the cross section of either $pp\to t\overline{t}{\tau }^{+}{\tau }^{-}+E_T^{\mathrm{miss}}+X$ and $pp\to b\overline{b}+E_T^{\mathrm{miss}}+X$ if exotic quarks are leptoquarks or $pp\to t\overline{t}b\overline{b}+E_T^{\mathrm{miss}}+X$ if exotic quarks are diquarks. 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 the case, then the pair production of the superpartners of D-fermions may result in some enlargement of the cross sections of either $pp\to t\overline{t}b\overline{b}+X$ when exotic squarks are diquarks or $pp\to t\overline{t}{\tau }^{+}{\tau }^{-}+X$ and $pp\to b\overline{b}+E_T^{\mathrm{miss}}+X$ 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 $Z^{\prime }$ 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.