Leptogenesis as an origin of dark matter and baryon asymmetries in the E6 inspired SUSY models

We explore leptogenesis within the E6 inspired U(1) extension of the MSSM in which exact custodial symmetry forbids tree-level flavour-changing transitions and the most dangerous baryon and lepton number violating operators. This supersymmetric (SUSY) model involves extra exotic matter beyond the MSSM. In the simplest phenomenologically viable scenarios the lightest exotic fermions are neutral and stable. These states should be substantially lighter than 1 eV forming hot dark matter in the Universe. The low-energy effective Lagrangian of the SUSY model under consideration possesses an approximate global U(1)_E symmetry associated with the exotic states. The U(1)_E symmetry is explicitly broken because of the interactions between the right-handed neutrino superfields and exotic matter supermultiplets. As a consequence the decays of the lightest right-handed neutrino/sneutrino give rise to both U(1)_E and U(1)_{B-L} asymmetries. When all right-handed neutrino/sneutrino are relatively light \sim 10^6-10^7 GeV the appropriate amount of the baryon asymmetry can be induced via these decays if the Yukawa couplings of the lightest right-handed neutrino superfields to the exotic matter supermultiplets vary between 10^{-4}-10^{-3}.


Introduction
The presence of baryon asymmetry and dark matter in the Universe clearly indicates the need for new physics beyond the Standard Model (SM). Over last forty years a number of new physics mechanisms for baryogenesis were proposed including GUT baryogenesis [1], baryogenesis via leptogenesis [2], the Affleck-Dine mechanism [3], electroweak baryogenesis [4], etc . Among these mechanisms thermal leptogenesis [2] is particularly attractive because it can be naturally realised in the seesaw models [5] in which right-handed neutrinos are superheavy shedding light on the origin of the mass hierarchy in the lepton sector.
In these models all three Sakharov conditions [6] are fulfilled since the seesaw mechanism requires lepton number violation and complex neutrino Yukawa couplings can provide a source for CP violation. In this case the lepton asymmetry is induced by the decays of the lightest right-handed neutrino and gets partially converted into baryon asymmetry via sphaleron processes [7]. It was shown that the appropriate amount of baryon asymmetry in the SM and its minimal supersymmetric (SUSY) extension (MSSM) can be generated only when the mass of the lightest right-handed neutrino M 1 is larger than 10 9 GeV [8]. In the supergravity (SUGRA) models this lower bound on M 1 results in the gravitino problem [9]. Indeed, after inflation the universe thermalizes with a reheat temperature T R .
Thermal leptogenesis may take place if T R > M 1 . On the other hand when T R 10 9 GeV such a high reheating temperature leads to an overproduction of gravitinos which tend to decay during or after Big Bang Nucleosynthesis (BBN) destroying the agreement between the predicted and observed light element abundances.
In this context it is especially interesting to study thermal leptogenesis within well motivated SUSY extensions of the SM that can originate from the heterotic superstring theory [10] and/or Grand Unified theories (GUTs) based on the E 6 gauge group or its subgroup. Near the GUT scale the gauge symmetry in these models can be broken down to the SM gauge group together with an extra U(1) ′ gauge symmetry which is a linear combination of U(1) ψ and U(1) χ U(1) ′ = U(1) χ cos θ + U(1) ψ sin θ .
In Eq. (1) the U(1) ψ and U(1) χ symmetries are defined by: E 6 → SO(10) × U(1) ψ , SO(10) → SU(5) × U(1) χ (for a review see [11,12] [13,14] right-handed neutrinos can be rather heavy providing a mechanism for the generation of the baryon asymmetry in the Universe via leptogenesis [15,16]. Moreover in this case the successful thermal leptogenesis can be achieved without encountering a gravitino problem [16]. Nevertheless the presence of the TeV scale exotic matter in the E 6 SSM gives rise to the operators that results in the non-diagonal flavour transitions and rapid proton decay. Here we focus on the U(1) N extension of the MSSM in which a single discreteZ H 2 symmetry forbids such operators. In the simplest phenomenologically viable scenarios the lightest and next-to-lightest SUSY particles (LSP and NLSP), which are predominantly the fermion components of the SM singlet superfields S i , are stable and tend to be considerably lighter than 1 eV. We argue that in this limit the Lagrangian of the SUSY model under consideration possesses an extra global U(1) E symmetry associated with the exotic states. This symmetry is explicitly broken due to the interactions of exotic states and right-handed neutrinos. As a result the decays of the lightest right-handed neutrinos generate not only baryon and lepton asymmetries but also dark matter asymmetry. Our analysis indicates that thermal leptogenesis may occur for T R 10 6−7 GeV in this case.
For such a low reheating temperature the gravitino density becomes sufficiently low [17]. As a consequence the success of the BBN is preserved.
The paper is organised as follows. In the next section we review the U(1) N extensions of the MSSM. In section 3 we consider the generation of the baryon and dark matter asymmetries. Our results are summarized in section 4.

The U (1) N extensions of the MSSM
In this section, we briefly review the E 6 inspired SUSY models with extra U(1) N factor. Within last ten years, several variants of the U(1) N extensions of the MSSM have been proposed [13,14,[18][19][20][21]. Supersymmetric models with an additional U(1) N gauge symmetry have been studied in [22] in the context of non-standard neutrino models with extra singlets, in [23] from the point of view of Z − Z ′ mixing, in [23][24][25] where the neutralino sector was explored, in [26,27] in the context of dark matter, in [25,28] where the renormalisation group (RG) flow of couplings was examined and in [25,29,30] where the electroweak symmetry breaking was studied. Recently, the RG flow of the Yukawa couplings and the theoretical upper bound on the lightest Higgs boson mass were explored in the vicinity of the quasi-fixed point [31] that appears as a result of the intersection of the invariant and quasi-fixed lines [32]. The presence of a Z ′ boson and additional exotic matter predicted by these models provides distinctive LHC signatures which were analysed in [13,14,33,34], as well as results in non-standard Higgs decays [21,35,36].
Within the constrained version of the E 6 SSM the particle spectrum has been examined in [27,34,37], including the effects of threshold corrections from heavy states [38]. The renormalisation of the vacuum expectation values (VEVs) and the fine tuning in the E 6 SSM were considered in [39] and [40] respectively.
In order to suppress non-diagonal flavour transitions in the E 6 SSM an approximate 3 ) and one SM-type singlet field (S ≡ S 3 ) are odd [13,14], can be imposed. The most dangerous baryon and lepton number violating operators, that give rise to rapid proton decay, can be forbidden by either a Z L 2 symmetry, under which all superfields except leptons are even, or a Z B 2 discrete symmetry which implies that the exotic quark and lepton superfields are odd whereas the others remain even. The discrete symmetries Z H 2 , Z L 2 and Z B 2 do not commute with E 6 since different components of 27-plets transform differently under these symmetries. The imposition of such symmetries to ameliorate phenomenological problems is an undesirable feature of the models under consideration.
In this article we consider the U(1) N extension of the MSSM in which a single dis-creteZ H 2 symmetry simultaneously forbids the tree-level flavour-changing transitions and the most dangerous baryon and lepton number violating operators. This SUSY model (SE 6 SSM) imply that near the GUT scale E 6 or its subgroup is broken down to is a matter parity [20]. Below the GUT scale the particle content of the SE 6 SSM includes three 27 i -plets and a set of M l and M l supermultiplets from 27 ′ l and 27 ′ l . All superfields, that stem from complete 27 i -plets, are odd while all supermultiplets M l are even under theZ H 2 symmetry. The set of M l can be used for the breakdown of gauge symmetry and therefore should involve H u , H d and S. Also in the simplest case the set of supermultiplets M l has to include a lepton SU(2) W doublet L 4 to allow the lightest exotic quarks to decay [20]. In principle, the supermultiplets M l can be either odd or even under theZ H 2 symmetry. In the SE 6 SSM S and L 4 are even whereas H u and H d are odd underZ H 2 . TheZ H 2 even supermultiplets H u , H d , S and S survive to low energies and can acquire VEVs at the TeV scale breaking SU(2) W × U(1) Y × U(1) N gauge symmetry. Since S and S have opposite U(1) N charges the D-term contribution to the scalar potential may force the minimum of this potential to be along the D-flat direction [41]. However in such scalar potential there is a run-away direction S = S → ∞. To stabilize the run-away direction we assume that in addition to H u , H d , S, L 4 , S and L 4 the set of the Z H 2 -even supermultiplets involves the SM-singlet superfield φ that does not participate in the gauge interactions. It is expected that theZ H 2 -odd supermultiplets H u and H d get combined with the superposition of the corresponding components from 27 i forming vectorlike states with masses of order of M X . On the other hand the supermultiplets L 4 and L 4 form TeV scale vectorlike states to render the lightest exotic quarks unstable. The exotic quarks are leptoquarks in this case [20]. Thus below the GUT scale the low-energy matter content in the SE 6 SSM involves where α = 1, 2 and i = 1, 2, 3. In Eq. (2) the left-handed quark and lepton doublets, the right-handed up-and down-type quarks, the right-handed charged leptons and neutrinos are denoted by Q i and L i , u c i and d c i , e c i and N c i respectively. The gauge group and matter content of the SE 6 SSM can originate from the 5D and 6D orbifold GUT models in which the splitting of GUT multiplets can be naturally achieved [20]. Because in the SE 6 SSM extra matter beyond the MSSM fill in complete SU (5) representations the gauge coupling unification in this SUSY model can be achieved for any phenomenologically acceptable value of α 3 (M Z ), consistent with its central measured low energy value [20,28].
The most general renormalisable superpotential which is allowed by theZ H 2 , Z M 2 and SU(3) × SU(2) W × U(1) Y × U(1) N symmetries can be written in the following form [21]: The interaction σφSS in the superpotential Eq. (3) stabilizes the run-away direction.
When σ is small the superfields φ, S and S tend to acquire large VEVs ∼ M S /σ, where M S is a SUSY breaking scale, giving rise to an extremely heavy Z ′ boson [21].
In the simplest case when only H u , H d and S acquire non-zero VEVs ( the Higgs sector was explored in [13]. If CP-invariance is preserved then the Higgs spectrum involves three CP-even, one CP-odd and two charged states. The SM singlet dominated CP-even state and the Z ′ gauge boson have always almost the same masses. When λ < g ′ 1 , where g ′ 1 is the U(1) N gauge coupling, the SM singlet dominated Higgs boson is the heaviest CP-even state whereas the rest of the Higgs spectrum is basically indistinguishable from the one in the MSSM. If λ g ′ 1 the Higgs spectrum has a rather hierarchical structure, which is somewhat similar to the one in the NMSSM with the approximate PQ symmetry [42,43]. As a result the mass matrix of the CP-even Higgs sector can be diagonalised using the perturbation theory [43,44]. For λ g ′ 1 the MSSM-like CP-even, CP-odd and charged states are almost degenerate and lie beyond the TeV range.
When the sector responsible for the breakdown of the SU (2)  The presence of such pNGB state may give rise to the non-standard decay mode of the SM-like Higgs boson h → A 1 A 1 if A 1 is lighter than 60 GeV [21].

Generation of dark matter and baryon asymmetries
As was mentioned in the previous section, the Lagrangian of the SE 6 SSM is invariant symmetry is also conserved. The transformation properties of matter multiplets under theZ H 2 , Z M 2 and Z E 2 symmetries are summarized in Table 1. The conservation of the Z E 2 symmetry implies that the lightest exotic state, which is predominantly a superposition of the fermion components of S i , is stable. The fermion components of S i are the lightest SUSY particles.
Indeed, using the method proposed in [45] it was shown that these states should be lighter than 60 − 65 GeV [35]. Although the couplings of these states to the SM gauge bosons and fermions are rather small the lightest exotic state could account for all or some of the observed cold dark matter density if it had a mass close to half the Z mass. Nevertheless in this part of the parameter space the SM-like Higgs boson would decay almost 100% of the time into the fermion components of S i while all other branching ratios would be extremely suppressed. Such scenario has been already ruled out by the LHC experiments.
On the other hand if the fermion components of S i are substantially lighter than M Z the annihilation cross section for LSP + LSP → SM particles becomes too small resulting in a relic density that is much larger than its measured value.
The simplest phenomenologically viable scenarios imply that the fermion components of S i are considerably lighter than 1 eV 1 . In this case the lightest exotic states form 1 The presence of very light neutral fermions in the particle spectrum might have interesting implications for neutrino physics (see, for example [46]). hot dark matter (dark radiation) in the Universe. At the same time the invariance of the Lagrangian of the SE 6 SSM under the Z M 2 symmetry ensures that R-parity is also conserved and the lightest ordinary neutralino can be stable. When the masses of the fermion components of S i are substantially smaller than 1 eV these states give only a very minor contribution to the dark matter density whereas the lightest ordinary neutralino may account for all or some of the observed cold dark matter density. So light fermion components of S i do not affect BBN if Z ′ boson is sufficiently heavy [19].  (3)  as well [47]. The non-zero values off iα and f iα explicitly break the U(1) E symmetry to Z E 2 and the lightest exotic particle that carries the U(1) E charge becomes unstable. If this state is mostly a linear superposition of the fermion components of H u α and H d α then it decays into the fermion components of S i and either Z or W . As a consequence the induced U(1) E asymmetry gets converted into the hot dark matter density.
It is worth noting that the non-zero values off iα and f iα do not always violate the U(1) E symmetry. For instance, the structure of the Yukawa interactions of S i can be such that two superfields S i carry opposite U(1) E charges while the third one does not transform under U(1) E . This happens, for example, whenf 1α ≃ f 2α ≃f 3α ≃ f 3α → 0. In this limit, again, the U(1) E symmetry remains anomaly-free below M 1 if the scale, where the breakdown of the U(1) N gauge symmetry takes place, is higher than M 1 .
Although the VEVs S ≃ S are allowed to be of the order of M 1 or even higher in our analysis we assume that the SUSY breaking scale and the masses of all exotic states are much lower than M 1 . To avoid the gravitino problem we set M 1 ≃ 10 6 GeV. Because the decays of the lightest right-handed neutrino/sneutrino into the final states with lepton number L = ±1 are kinematically allowed these processes create lepton asymmetry in the early Universe which is controlled by the flavour dependent CP (decay) asymmetries.
Due to (B + L)-violating sphaleron interactions the induced lepton asymmetry gets converted into the baryon asymmetry. Assuming the type I seesaw mechanism of neutrino mass generation one can define three decay asymmetries associated with the three lepton flavours e, µ and τ in the SM which are given by In Eq. (4) Γ N 1 ℓ k and Γ N 1lk are partial decay widths of N 1 → L k + H u and N 1 → L k + H * u with k, m = 1, 2, 3. At the tree level Γ N 1 ℓ k = Γ N 1lk and ε 1, ℓ k = 0. The non-zero values of the CP asymmetries arise because of the interference between the tree-level amplitudes of the N 1 decays and one-loop corrections to them.
In SUSY extensions of the SM the right-handed neutrinos can also decay into sleptons L k and Higgsino H u giving rise to the CP asymmetries The decays of the right-handed sneutrinos contribute to the generation of the total lepton asymmetry in SUSY models as well. The corresponding CP asymmetries can be defined similarly to the neutrino ones When SUSY breaking scale is negligibly small as compared with M 1 In the type I seesaw models the decay asymmetries mentioned above were calculated initially within the SM [48] and MSSM [49]. In the early studies flavour effects were ignored (see for example [50]). The importance of these effects was emphasised in [51].
In the non-minimal SUSY models in which the right-handed neutrinos can decay into a few lepton (slepton) multiplets and a few SU(2) W doublets, that have quantum numbers of Higgs (Higgsino) fields, i.e. H u k and L x ( H u k and L x ), the definitions of the CP asymmetries (4) and (5) can be generalised in the following way [16] ε where f and f ′ can be either ℓ x or ℓ x whilef andf ′ may be eitherl x or ℓ * x . The superscripts k and m represent the components of the supermultiplets H u k and H u m in the final state. In these SUSY models the definitions of the CP asymmetries associated with the decays of the lightest right-handed sneutrino can be modified similarly to the neutrino ones. In order to obtain the appropriate expressions for ε k 1, f the right-handed neutrino field in Eqs. (8) should to be replaced by either N 1 or N * 1 . In this case the relation between different types of CP asymmetries (7) remains intact, i.e. ε k 1, f = ε k 1, f . For M 1 ≃ 10 6 GeV the Yukawa couplings of the superfield N 1 to the supermultiplets H u and L i in the SE 6 SSM should be quite small in order to reproduce the left-handed neutrino mass scale m ν 0.1 eV, i.e. |h 1k | 2 ≪ 10 −8 . If two other right-handed neutrino states are also rather light their Yukawa couplings to H u and L i tend to be very small as well. So small Yukawa couplings and CP asymmetries associated with them can be ignored in the leading approximation. Nevertheless the presence of exotic matter supermultiplets H u α and L 4 gives rise to the new channels of the decays of the lightest right-handed neutrino and its superpartner, i.e.
At the tree level SUSY implies that the partial decay widths associated with the new channels (9) are given by Since the interactions between the superfields N c i and supermultiplets H u α and L 4 violate not only U(1) L , which ensures the conservation of lepton number, but also U(1) E global symmetry the decay channels of the lightest right-handed neutrino/sneutrino (9)  . Neglecting the Yukawa couplingsh ik we get From the superpotential (3) it follows that the supermultiplets H u α can be always redefined in such a way that only one doublet H u 1 interacts with L 4 and N c 1 . Thus without loss of generalityh 12 in W N may be set to zero in the first approximation. If h j1 = |h j1 |e iϕ j1 and M j are real the subset of the CP asymmetries that determines the U(1) L and U(1) E asymmetries in this limit can be written as where ∆ϕ j1 = ϕ j1 − ϕ 11 .
The evolution of the dark matter and lepton number densities are described by the system of Boltzmann equations. In the limit under consideration the results obtained in the SM and MSSM for the lepton and baryon asymmetries can be easily generalised. In particular, the induced baryon asymmetry can be estimated as follows (see [52]): where Y ∆B is the baryon asymmetry relative to the entropy density, i.e.
In Eq. (13) η is an efficiency factor that varies from 0 to 1. In the strong washout scenario the efficiency factor is given by where H is the Hubble expansion rate and g * = n b + 7 8 n f is the number of relativistic degrees of freedom in the thermal bath. Within the SM g * = 106.75 while in the SE 6 SSM g * = 360. From Eq. (12) it follows that the values of the CP asymmetries are determined by the combinations of the CP-violating phases ∆ϕ j1 and the absolute values of the Yukawa couplings |h 21 | and |h 31 | but do not depend on |h 11 |. To simplify our numerical analysis we assume here that |h 31 | is negligibly small, i.e. |h 31 | ≪ |h 21 |, and can be ignored. We also fix (M 2 /M 1 ) = 10. On the other hand the efficiency factor η is set by the lightest right-handed neutrino mass M 1 and |h 11 |. We restrict our consideration here by the values of |h 11 | which are considerably larger than |h ik |, i.e. |h 11 | 2 10 −8 , so that all Yukawa couplingsh ik can be neglected. Fig. 1a illustrates that in the strong washout scenario the efficiency factor η varies from 10 −2 to 10 −4 when |h 11 | increases from 10 −4 to 10 −3 .

Conclusions
In this article we studied the generation of the baryon and dark matter asymmetries within the E 6 inspired SUSY model with extra U(1) N factor (SE 6 SSM) in which a single discreteZ H 2 symmetry forbids the tree-level flavour-changing transitions as well as the most dangerous baryon and lepton number violating operators. Only in this E 6 inspired U(1) extension of the MSSM the right-handed neutrinos N c i do not participate in the gauge interactions. Therefore the decays of the heavy right-handed neutrino/sneutrino should lead to the generation of the baryon asymmetry via leptogenesis.
To ensure anomaly cancellation the low energy matter content of the SE 6 SSM includes three 27 representations of E 6 that involve three families of quarks and leptons, three families of exotic quarks D i andD i , three families of Higgs-like doublets H d i and H u i as well as three SM singlet superfields S i that carry U(1) N charges. In addition the particle spectrum of the SE 6 SSM contains a pair of SU(2) W doublets L 4 and L 4 , that allows the lightest exotic quarks to decay and facilitates gauge coupling unification, and the SM singlet superfields S and S that acquire VEVs breaking the U(1) N gauge symmetry. One pair of the Higgs-like doublet supermultiplets H u and H d play a role of the MSSM Higgs fields which break electroweak symmetry whereas two other pairs H d α and H u α , as well as L 4 , L 4 , D i ,D i and S i form exotic sector. The Lagrangian of the SE 6 SSM is invariant with respect to Z E 2 symmetry, under which all components of the exotic matter supermultiplets are odd while all other fields are even. This discrete symmetry guarantees that the lightest exotic state, which tends to be the superposition of the fermion components of S i , is stable.
In the simplest phenomenologically viable scenarios the fermion components of S i should be significantly lighter than 1 eV forming hot dark matter in the Universe. These scenarios are realised if the Yukawa couplings of S i are rather small ( 10 −6 ). In this limit the low-energy effective Lagrangian of the SE 6 SSM below the scale M 1 possesses an approximate global U(1) E symmetry associated with the exotic matter supermultiplets.
When the scale of the U(1) N symmetry breaking is quite high ( M 1 ) the U(1) E is anomaly-free. The interactions of N c i with L 4 and H u α break this symmetry. Thus in the SE 6 SSM the decays of the lightest right-handed neutrino/sneutrino induce both U(1) B−L and U(1) E asymmetries. It is expected that the generated U(1) E asymmetry should not be erased if the U(1) E violating Yukawa couplings of S i are smaller than 10 −7 .
To avoid the gravitino problem in our analysis we focused on the scenarios with M 1 ≃ 10 6 GeV. In the case when all right-handed neutrino/sneutrino have masses of the order of 10 6 − 10 7 GeV the Yukawa couplings of N c i to the left-handed lepton supermultiplets and H u tend to be rather small and can be neglected in the leading approximation. Then the U(1) B−L and U(1) E asymmetries are induced because of the interactions between N c i and the components of the exotic matter supermultiplets L 4 and H u α . We argued that the appropriate value of the baryon asymmetry can be generated if the Yukawa couplings of N c 1 to L 4 and H u α ∼ 10 −4 − 10 −3 . In this case the hot dark matter and baryon number densities should be of the same order of magnitude.