Electroweak baryogenesis and electron EDM in the B-LSSM

Electroweak baryogenesis (EWB) and electric dipole moment (EDM) have close relation with the new physics beyond the standard model (SM), because the SM CP-violating (CPV) interactions are not sufficient to provide the baryon asymmetry of the universe by many orders of magnitude, and the theoretical predictions for the EDM of electron ($d_e$) in the SM are too tiny to be detected in near future. In this work, we explore the CPV effects on EWB and the electron EDM in the minimal supersymmetric extension (MSSM) of the SM with local $B-L$ gauge symmetry (B-LSSM). And the two-step transition via tree-effects in this model is discussed. Including two-loop corrections to $d_e$ and considering the constrains from updated experimental data, the numerical results show that the B-LSSM can account for the observed baryon asymmetry. In addition, when the cancellation between different contributions to $d_e$ takes place, the region favored by EWB can be compatible with the corresponding EDM bound.


I. INTRODUCTION
Despite the considerable success of the Standard Model (SM) in describing a large amount of experimental observations, there are still various of evidences beyond the SM.
One of the most interesting problems is the baryon asymmetry of the universe (BAU) [1,2]: where ρ B is the baryon number density, s is the entropy density of the universe. The SM CP-violating (CPV) interactions are not sufficient to provide the asymmetry by many orders of magnitude, which indicates that the SM is incomplete. The search for new physics (NP) beyond the SM is motivated in part by the desire to overcome the failure of the SM to explain the BAU. Electroweak baryogenesis (EWB) [3] is an explanation of the origin of the cosmological asymmetry between matter and antimatter, and new CPV terms are needed to enhance the asymmetry theoretically.
Meanwhile, new CPV phases can provide much larger values of the electric dipole moments (EDMs) than the SM predictions. The SM prediction for the electron EDM is about 10 −38 e · cm [4][5][6], which is impossible to be detected by present experiments. However, when new CPV phases are introduced, the enhanced electron EDM may be detected in near future, which can be regarded as a smoking gun for NP beyond the SM. The upper bounds on d e have been obtained [7][8][9] |d e | < 8.7 × 10 −29 e · cm.
Since the experimental upper bound on the electron EDM is very small, the contributions from new CPV phases are limited strictly by the present experimental data, and researching NP effects on the electron EDM may shed light on the mechanism of CPV.

II. THE B-LSSM
In the B-LSSM, two chiral singlet superfieldsη 1 ∼ (1, 1, 0, −1),η 2 ∼ (1, 1, 0, 1) and three generations of right-handed neutrinos are introduced, which allow for a spontaneously broken U(1) B−L without necessarily breaking R-parity. In addition, this version of B-LSSM is encoded in SARAH [56], which is used to create the mass matrices and interaction vertexes in the model. Meanwhile, the superpotential of the B-LSSM can be written as where i, j are generation indices. Then the soft breaking terms of the B-LSSM are generally given as For convenience, we define u 2 = u 2 1 + u 2 2 , v 2 = v 2 1 + v 2 2 and tan β ′ = u 2 u 1 in analogy to the ratio of the MSSM VEVs (tan β = v 2 v 1 ). New U(1) B−L gauge group introduces new gauge boson Z ′ and the corresponding gauge coupling constant g B . In addition, two Abelian groups gives rise to a new effect absent in the MSSM or other SUSY models with just one Abelian gauge group: the gauge kinetic mixing.
Immediate interesting consequence of the gauge kinetic mixing arise in various sectors of the model. Firstly, new gauge boson Z ′ mixes with the Z boson in the MSSM, and new gauge coupling constant g Y B is introduced. Then the gauge kinetic mixing leads to the mixing between the H 1 1 , H 2 2 ,η 1 ,η 2 at the tree level, andλ B ′ mixes with the two higgsinos in the MSSM at the tree level. Meanwhile, additional D-terms contribute to the mass matrices of the squarks and sleptons. All of these properties affect the theoretical predictions for Y B and d e in the B-LSSM, and the model are introduced in detail in our earlier work [57][58][59].

A. Electroweak phase transion
In the MSSM, EWB has been excluded because the strong first order PT with very light right handed stop < 120GeV is not possible after the discovery of the 125 GeV Higgs boson [60][61][62][63][64][65][66][67][68][69]. With respect to the MSSM, a strong two-step PT can be achieved in the B-LSSM, because there are two additional scalar singlets. These new singlets mix with the two doublets in the MSSM at the tree level through gauge kinetic mixing, which change the effective potential vastly. For simplicity, the temperature dependence of β, β ′ is neglected and the tree-level effective potential can be written as where and T denotes temperature, G is the sum of relevant couplings, h and η acquire VEVs < h >= v, < η >= u respectively at zero temperature (present universe). Since the singlets couples to fewer degrees of freedom, their thermal masses is lower than that of the SM higgs, and we ignore their thermal mass. At very high temperature, h and η are stabilized at the origin. In addition, it can be noted in Eq. (6, 7) that, the only possible gauge-dependence term is GT 2 , and the gauge-independence of O(T 2 ) term was proved in the appendix C of Ref. [70]. Hence our analysis of the electroweak PT is gauge invariant. As the universe cools, the singlets transition to a nonzero VEV u c1 first, in a second order phase transition at T c1 . Then at temperature T c2 ∼ m W < T c1 , the universe undergoes a first order PT to (v c2 , u). Then we can obtain v c2 and M(T ) 2 by solving the equations Then the first order transition temperature T c2 can be obtained by (M(T ) 2 − M 2 0 )/G. For the EWB to work, the sphaleron process must be decoupled when the electroweak PT completes. In other words, the sphaleron rate in the broken phase should be less than the Hubble parameter at that moment. In general, the sphaleron decoupling condition is cast And in our chosen parameter space in the next section, we have v c2 /T c2 > ∼ 1.5, which is sufficient for the taking place of EWB.

B. Baryon asymmetry Y B
The CPV effects enter as source terms in the quantum transport equations that govern the production of chiral charge at the phase boundary. According to Ref. [14], a simple expression for the baryon-to-entropy ratio can be written as where we have taken the gaugino mass terms M 1,2,B ′ to be real, θ µ , θ T are the phases of µ and T e respectively. Compared with the expression in Ref. [14], the additional minus sign on are the corresponding Yukawa coupling constants) and squark masses Mt L , Mt R . In addition, F i also have a overall dependence on bubble wall parameters v w , L w , ∆β. For the concrete expressions of F i , we adopt the formulas displayed in Ref. [14]. Compared with the MSSM, there is new contribution to S CP / H (the CPV higgsino source) in the B-LSSM, which comes from the mixing between new gauginoλ B ′ and the two higgsinos in the MSSM through gauge kinetic mixing, and the corresponding gauge coupling constant is g Y B .
C. The EDM of electron d e The effective Lagrangian for the electron EDM can be written as where σ µν = i[γ µ , γ ν ]/2, and F µν is the electromagnetic field strength. Adopting the effective Lagrangian approach, we can get where Q f = −1, m e denotes the electron mass, and C L,R 2,6 represent the Wilson coefficients of the corresponding operators O L,R where D α = ∂ α + iA α , l e is the wave function for electron, and P R,L = (1 ± γ 5 )/2. Then, the Feynman diagrams contributing to the above Wilson coefficients are depicted by Fig. 1.
Calculating the Feynman diagrams, the electron EDM can be written as le le where x i = m 2 i /m 2 W , C L,R abc denotes the constant parts of the interactional vertex about abc, which can be got through SARAH, a, b, c denote the interactional particles, and the concrete expressions for the functions I 1,2,3,4 can be found in [71,72]. In addition, our earlier work [59] shows that, two-loop Barr-Zee type diagrams can make important contributions to the muon magnetic dipole moment (MDM), and we consider the contributions from the two-loop diagrams in which a closed fermion loop is attached to the virtual gauge bosons or Higgs fields. According to Ref. [73], the main two-loop diagrams contributing to the electron EDM are shown in Fig. 2 contributions from the two-loop diagrams to d e can be simplify as where f j , f i denote χ 0 j and χ ± i respectively, W denotes W boson, H denotes charged Higgs boson, h denotes SM-like Higgs boson, the concrete expressions for the function J can be found in Ref. [59]. Confidence Level (CL). And an upper bound on the ratio between the Z ′ mass and its gauge coupling is given in Refs. [79,80] at 99% CL as M Z ′ /g B > 6 TeV. In our earlier work [59], we explore the effects of parameters tan β, tan β ′ , g B , g Y B and slepton masses ML ,ẽ on the muon MDM in the B-LSSM without CPV. Since the CPV phases affect the electron EDM more obviously than the muon MDM, we explore the CPV effects on the electron EDM firstly in this paper, but put off the exploration of CPV effects on the muon MDM in our next work.  [84,85], and ∆β as a function of pseudoscalar Higgs boson mass provided in Ref. [84], we take ∆β = 0.015. For the thermal widths, we adopt the results in Ref. [86] in the following analysis.
In order to see how θ µ , θ A 0 and µ affect Y B , we take g Y B = −0.4 and scan the regions of the parameter space [θ µ = (−π, π), θ A 0 = (−π, π), µ = (0.1, 1) TeV]. In the scanning, we keep Y B in the region (8.2 − 9.4) × 10 −11 . Then the allowed region of θ µ and µ is displayed in Fig. 3. From the picture, we can see that there are two ellipses in the figure, and the two ellipses mainly concentrate on the vicinity of µ = 600 GeV and µ = 300 GeV respectively, because the effects of these interactions are resonantly enhanced when µ is comparable to the mass terms M 1,2,B ′ [87,88], the observed baryon asymmetry can be accounted for only in this case. The allowed region of θ µ is concentrated on θ µ > 0, because the mainly contributions come from the coefficient F 1 , and F 1 is negative in our chosen parameter space.
In addition, with the increasing of θ µ , the value of µ has a small deviation from M 1,B ′ or the cancellation between θ M 2 and θ µ by taking other CPV phases equal to 0. We scan the regions of the parameter space [θ M 2 = (−π, π), θ µ = (−π, π)], and keep |d e | < 8.7 × 10 −29 in the scanning. The numerical results are shown in Fig. 5 (a). In addition, θ A 0 can also make important contributions to Y B , hence it is interesting to explore how θ A 0 affects d e .
Since the effects of θ A 0 are highly suppressed by small Y e , we do not have to cancel the contributions from θ A 0 to d e . Then we plot d e versus θ A 0 in Fig. 5 (b), where the gray area denotes the experimental upper bound on d e , the solid, dashed and dotted lines denote From the pictures we can see that, the contributions from θ M 2 and θ µ are cancelled when θ M 2 ≈ −θ µ + nπ (n = 0, ±1), the phases we chosen to cancel each other due to that, the contributions from θ M 2 and θ µ are comparable. In addition, the contributions from θ A 0 are enlarged by large A 0 , and the contributions from θ M 2 , θ µ are lager than θ A 0 by several orders of magnitude, hence the contributions from θ µ are hardly cancelled by θ A 0 (when the cancellation between θ µ and θ A 0 takes place, the maximum value of θ µ is O(10 −3 ), which is not sufficient for the taking place of EWB). It is different from the case in the MSSM [38], in which the maximum value of θ µ can be large enough to the taking place of EWB, when the cancellation happens between the contributions from θ µ and θ A 0 to d e . It results from that, the contributions from sleptons are highly suppressed by large slepton masses, in our chosen parameter space. It can be noted that, the contributions from θ µ to d e can be cancelled by θ M 2 , and θ µ is the main source of baryon asymmetry, hence the worry about the contributions from the large value of θ µ , which is needed to give rise to EWB, to the electron EDM whether can be cancelled is relaxed.
In the B-LSSM, there are new CPV phases θ µ ′ , θ M BB ′ and θ M B ′ can make contributions to the electron EDM. In addition, the gaugino mass term M 1 can also have CPV phase θ M 1 , and makes contributions to d e . Then we set other CPV phases equal to zero and explore the cancellation between θ µ ′ , θ M BB ′ , θ M B ′ and θ M 1 . Scanning the following regions of the parameter space: The allowed region of θ M 1 , θ M BB ′ is displayed in Fig. 6 (a), while the allowed region of Fig. 6 (b). From the picture we can see that, when the possible