Electroweak Baryogenesis in the Exceptional Supersymmetric Standard Model

We study electroweak baryogenesis in the E_6 inspired exceptional supersymmetric standard model ( E_6SSM ). The relaxation coefficients driven by singlinos and the new gaugino as well as the transport equation of the Higgs supermultiplet number density in the E_6SSM are calculated. Our numerical simulation shows that both CP-violating source terms from singlinos and the new gaugino can solely give rise to a correct baryon asymmetry of the Universe via the electroweak baryogenesis mechanism.


I. INTRODUCTION
The origin of the baryon asymmetry of the Universe (BAU) is one of the longstanding problems in particle physics and cosmology. Combining the WMAP seven year results [1] with those from CMB and large scale structure measurements one has Y B ≡ ρ B s = (8.82 ± 0.23) × 10 −11 (1) where ρ B is the baryon number density, s is the entropy density of the Universe. The recent results obtained by the Planck satellite are consistent, giving Y B = (8.59 ± 0.11) × 10 −11 [2].
Assuming that the Universe was matter-antimatter symmetric at its birth, it is reasonable to suppose that interactions involving elementary particles generated the BAU during the subsequent cosmological evolution. To generate the observed BAU, three Sakharov criteria [3] must be satisfied in the early Universe: (1) baryon number violation; (2) C and CP violation; (3) a departure from the thermal equilibrium (assuming exact CPT invariance).
These requirements are realizable, though doing so requires physics beyond the Standard Model (SM). To that end, theorists have proposed a variety of baryogenesis scenarios whose realization spans the breadth of cosmic history. Electroweak baryogenesis (EWBG) [4][5][6][7][8][9][10][11] is one of the most attractive and promising such scenarios, and it is generally the most testable with a combination of searches for new degrees of freedom at the LHC and low-energy tests of CP invariance.
Successful EWBG requires a first order electroweak phase transition and sufficiently effective CP violation during the transition. Neither requirement is satisfied in the SM. One simple extension of the SM that may allow them to be satisfied is the two Higgs doublet model (2HDM) (for a recent review, see Ref. [12] [13][14][15][16]. This Higgs mass severely restricts the parameter space of the LSS. This allows us to exclude EWBG in the MSSM at greater than (90) 95% confidence level in the (non-)decoupling limit, by examining correlations between different Higgs decay channels [17].
Possible ways out are considering extensions of the MSSM, of which the Next to Minimal Supersymmetric Standard Model (NMSSM) [18] is attractive since it provides solution to the µ problem and naturally accommodates a 125 GeV SM Higgs. Constraints on the parameter space of the model from the non-observation of permanent electric dipole moments (EDMs) of neutron, Mercury, Thallium, deuteron and Radium as well as the parameter space available for singlino driven electroweak baryogenesis were studied in Ref. [19,20]. In this paper we study the electroweak baryogenesis in the exceptional superysymmetric extension to the SM [21], E 6 SSM, which is a string theory inspired supersymmetric model based on an E 6 grand unification (GUT) group. E 6 SSM also accommodates a 125 GeV Higgs [29] and can dynamically generate the µ term. The paper is organized as follows: In section II we give a brief introduction to the E 6 SSM.
Section III is devoted to the investigation of EWBG induced by the neutralino sector of the E 6 SSM. We summarize in section IV.

II. E 6 SSM
The E 6 SSM, which originates from an E 6 GUT or string theory in extra dimensions, involves a unique choice for the extra Abelian gauge group namely U(1) N . To ensure anomaly cancellation, the particle content of the E 6 SSM includes three complete fundamental 27 representations of E 6 , which decompose under the SU(5) × U(1) N subgroup as follows.
where the first and second quantities in the brackets are the SU(5) representation and U(1) N charge respectively. The first two terms on the right side of Eq.
(2) contain all the matter contents, the third and fourth terms contain the pair of Higgs doublets as well as diquarks with electric charges −1/3 ad +1/3, respectively. Scalar singlet, the third generation of which breaks the U(1) N gauge symmetry spontaneously, is contained in the fifth term.
Right handed neutrinos is associated with the last term.
In E 6 models the renormalizable superpotential comes from 27 ×27 ×27 decomposition of the E 6 fundamental representation. The superpotential, that respects to SU(3) C ×SU(2) L × U(1) Y , can be written as [21,22] where we have assumed that exotic quark D i are diquarks, which carry a twice large baryon number than the ordinary quark fields, and can decay into the SM quarks via interactions in (4). D i andD i can also be leptoquarks, in which case superpotential in (4) will be replaced with Yukawa interactions between D i , lepton and quark superfields.
In the E 6 SSM the neutralino sector is extended to include eight additional neutral components:S i ,H α u ,H α d andB ′ , where i = 1, 2, 3 and α = 1, 2. The neutralino mass matrix can be written as [23] M N in the basis The submatrix M N USSM [24] is the neutralino mass matrix in the USSM. Expressions of other submatrices can be found in [23]. We refer to [21][22][23][24][25][26][27][28][29] for studies of low energy phenomenologies of the E 6 SSM.

III. ELECTROWEAK BARYOGENESIS
In the EWBG scenario, three Sakharov conditions are realized in the following way [4,30,31]: First, the scalar sector of the E 6 SSM gives rise to a strongly first order electroweak phase transition, which provides a departure from thermal equilibrium at temperature T ∼ Ordinary quantum field theory is not appropriate for treating the microscopic dynamics of the electroweak phase transition, since the non-adiabatic evolution of states and the presence of degeneracies in the spectrum break the zero-temperature equilibrium relation between the in-and out-states. We derive the source terms in the quantum transport equation based on the closed time path formulation of non-equilibrium quantum field theory [39,40]. The equations governing the space-time dependence of number densities of a given spaces can be written as [32][33][34][35] ∂n ∂t where self energy Σ <,> encode all the information about particle interactions.
The rephasing invariant combinations in the E 6 SSM relevant to electroweak baryogensis can be written as where A λ ijk are couplings of trilinear interactions S i H uj H dk in the soft supersymmetry breaking lagrangian. M 1 , M 2 and M ′ 1 are the mass of bino, wino, and the new gaugino respectively. For simplicity we set A λ ijk to be real in our calculation. We define the four component spinors as The Higgsino-gaugino-VEV interactions can be written as where φ = arg λ 333 and ϕ M ′ 1 = arg M ′ 1 , Q d N and Q u N are charges of H ui and H di under the U(1) N . The first two terms are the same as those in the MSSM, the third term is the interactions between Higgsinos and singlinos, the last term is the interaction of the third generation Higgsino with the new gaugino.
We ignore the wall curvature in our analysis so all relevant functions depend on the variablez = z + v w t, where v w is the wall velocity;z < 0, > 0 correspond to the unbroken and broken phases, respectively. Working in the closed time path formulation and under the "vev-insertion" approximation [32][33][34][35], we compute the CP-violating source induced by singlinos and the new gaugino mediated processes (H →S →H andH →B ′ →H), where n(x) = 1/(exp(x) + 1), being the fermion distribution function; εH ,S = ωH ,S − iΓH ,S are complex poles of the spectral function with ω 2 H,S = k 2 + m 2H ,S , where mH ,S and ΓH ,S are the thermal masses and thermal rates ofH andS, respectively. Before proceeding, we note that the VEV insertion approximation used in obtaining eqs. (10,11) is likely to lead to an overly large baryon asymmetry by at least a factor of a few, though a definitive quantitative treatment of the CPV fermion sources remains an open problem. The results quoted here, thus, provide a conservative basis for restrictions on the EWBG-viable parameter space. For a detailed discussion of the theoretical issues associated with the computation of the CPV source terms, see Ref. [4] and references therein. The thermal mass of the singlinos and the new gaugino can be written as where g N is the gauge coupling of the U(1) N . Notice that CP violating source term in (10) is is closely related to Debye masses of singlinos.
The CP-conserving terms can be written as S CP with It is straightforward to obtain the corresponding source term mediated by the new gaugino by making the following replacements: λ ij3 → g ′ tan −1 βQ d , λ i3j → g ′ tan βQ u , ωS i → ωB′ and MS i → MB′. We assume no net density of gauginos and singlinos, thereby setting µS i = µB′ = 0 and giving where Γ h = Γ − H − Γ + H . We now derive the Boltzmann equations. We assume there are approximate chemical equilibriums between the SM particles and their superpartners, as well as between different members of left-handed fermion doublets. In this case, one obtains transport equations for densities associated with different members of supermultiplet. Since all light quarks are mainly produced by strong sphaleron processes and all quarks have similar diffusion constants, baryon number conservation on time scales shorter that the inverse electroweak sphaleron rate implies the approximate constraints q 1L = q 2L = −2u R = −2d R = −2s R = −2c R = −2b R ≡ −2b = 2(Q + T ). We define the number density for Higgs supermultiplet The transport equation of the Higgs supermultiplet number density can be written as dz 2 in the planar bubble wall approximation with D a the diffusion constant. n i and k i is the number density and the statistical factor of particle "i". The coefficient Γ Y denotes the interaction rate arising from top quark, which can be written as Γ Y = 6|y t | 2 I F (mt L , mt R , m h ). We refer the reader to [35] for the general form of I F .
GeV is the inverse washout rate for the electroweak sphaleron transitions.   Table. I. The bubble wall velocity v w , thickness L w , profile parameters ∆β and v(T ) describe the dynamics of the expanding bubbles during the EWPT, at the temperature T .
We take the Higgs profile to be  Table. I.
The solid, dashed and dotted lines correspond to φ = π/8, π/4, and π/2, respectively. Obviously the observed baryon asymmetry can be obtained by these two CP-violating sources separately. To study the relative contribution ofZ ′ andS 3 to the baryon asymmetry, we show contours of Y B /Y obs in the φ − g N plane( Fig. 2 (a) ) and in the φ − λ 333 plane ( Fig.   2 (b)). Contours from left to right correspond to Y B /Y obs = 1, 2, 3, 4, 5 ( Fig. 2 (a)) and Y B /Y obs = 0.1, 0.5, 1, 2, 3 ( Fig. 2 (b)) respectively. We show contours of Y B /Y obs in the g N − λ 333 plane by assuming φ = π/4 in the Fig. 2 (c). Notice that the new gaugino induced CP-violating source is more effective to give rise to a sizable baryon asymmetry. This is because we set a narrower mass splitting between MH 3 and MZ′ than that between MH 3 and MS 3 when carrying out numerical calculation. We show in the left (right) panel of Fig. 3 Y B /Y obs as the function of MH0 3 (MZ′) by assuming MZ′(MH0 3 ) = 400 GeV and g N = λ 333 = 0.5. The solid dashed and dotted lines correspond to φ = π/8, π/4, π/2 respectively. It is obvious that there is a resonant enhancement to the production of the baryon asymmetry when MH 3 = MZ′. Finally let us consider constraints on the CP phases of the neutralino mass matrix from the non-observation of the electric dipole moments for neutrons and the electron. These CP-violating phases may contribute to EDMs via the H + W − or W + W − mediated Bar-Zee graphs. It was observed in Ref. [41] that CP violation in the bino-Higgsino sector of the MSSM can account for successful baryogenesis without inducing EDMs. This observation weaken the correlation between the electroweak baryogenesis and EDMs. It was found [19,20] that the maximal CP phase φ is still compatible with the current EDM constraints in the NMSSM. The same argument can be applied to our model since we only focus on singlinos and the new gaugino induced CP-violating source terms. We leave the systematic study of constraints of EDMs in the E 6 SSM to another project.

IV. CONCLUSION
MSSM has difficulty in explaining both electroweak baryogenesis and 125 GeV Higgs.
Possible extensions to the MSSM accounting these two problems were well studied recently.
In this paper, we studied electroweak baryogenesis in the E 6 inspired supersymmetric standard model, which contains at least two more CP-violating source terms in the neutralino sector compared with the MSSM case. New CP-violationgs source terms as well as transport equations of the Higgs supermultiplet were calculated analytically and numerically.
Our results show that CP-violating sources from singlinos and the new gaugino can give rise to a successful electroweak baryogenesis respectively. It should be mentioned that we only studied the adequate condition for a successful baryogenesis in the E 6 SSM. A systematic study of the EDMs constraint to the E 6 SSM, which is important and necessary but beyond the reach of this paper, will be shown in an another paper.