The study of lepton EDM in CP violating BLMSSM

In the supersymmetric model with local gauged baryon and lepton numbers(BLMSSM), the CP violating effects are considered to study the lepton electric dipole moment(EDM). The CP violating phases in BLMSSM are more than those in the standard model(SM) and can give large contributions. The analysis of the EDMs for the leptons $e,\mu, \tau$ is shown in this work. It is in favour of exploring the source of CP violation and probing the physics beyond SM.


I. INTRODUCTION
The theoretical predictions for EDMs of leptons and neutron are very small in SM.
In SM, the electron EDM is estimated as |d e | ≃ 10 −38 e.cm [1], which is too small to be detected by the current experiments. The ACME Collaboration [2] report the new result of d e = (−2.1 ± 3.7 stat ± 2.5 syst ) × 10 −29 e.cm. The upper bound of electron EDM is |d e | < 8.7 × 10 −29 e.cm at the 90% confidence level. Therefore, if large EDM of electron is probed, one can ensure it is the sinal of new physics beyond SM. |d µ | < 1.9 × 10 −19 e.cm and |d τ | < 10 −17 e.cm are the EDM upper bound of leptons µ and τ respectively [3]. The minimal supersymmetric extension of SM (MSSM) [4] is very attractive and physicists have studied it for a long time.
In MSSM, there are a lot of CP violating phases and they can give large contributions to the EDMs of leptons and neutron.
When the CP-violating phases are of normal size, and the SUSY particles are at TeV scale, very big EDMs of elementary particles are obtained, which exceeds the current experimental limit. Three approaches are used to resolve this problem. 1. make the CP violating phases small, i.e. O(10 −2 ). That is the so called fine tuning. 2. use mass suppression through making SUSY particles heavy(several TeV). 3. there is cancellation mechanism among the different components. For lepton EDM and neutron EDM, the main parts of chargino and the neutralino contributions are cancelled [5].
BLMSSM is the minimal supersymmetric extension of the SM with local gauged B and L(BLMSSM) [6]. Because of the local gauged B and L, it can explain both the asymmetry of matter-antimatter in the universe and the data from neutrino oscillation experiment. So, BLMSSM is a favorite model beyond MSSM. Extending SM, the authors study the model with B and L as spontaneously broken gauge symmetries around TeV scale [7]. The lightest CP-even Higgs mass and the decays h 0 → γγ, h 0 → ZZ(W W ) are also studied in this model [8]. In our previous work [9,10], we study the neutron EDM and B 0 −B 0 mixing in the CP-violating BLMSSM.
Research the MDMs [11] and EDMs [12,13] of leptons are the effective ways to probe new physics beyond the SM. In MSSM, the one-loop contributions to lepton MDM and EDM are well studied. The authors investigate some two loop corrections to lepton MDM and EDM in the frame work of MSSM. In the two Higgs Doublet models with CP-violation, the one loop and Barr-Zee type two-loop contributions to fermionic EDMs are obtained. In Ref. [14], a model-independent study of d e in the SM is carried out. They take into account the right handed neutrinos, the neutrino seesaw mechanism and the framework of minimal flavor violation. Their results show that when neutrinos are Majorana particles, d e can reach its experiment bound.
After this introduction, in section 2 we briefly introduce the main ingredients of the BLMSSM. The one-loop corrections to the lepton EDM are collected in section 3. Section 4 is devoted to the numerical analysis for the dependence of lepton EDM on the BLMSSM parameters. We show our discussion and conclusion in section 5.
The Higgs superfieldsΦ L ,φ L are used to break lepton number spontaneously, as well as baryon number is broken by the Higgs superfieldsΦ B ,φ B . Therefore, these Higgs superfieldŝ The superpotential of BLMSSM is shown as Here, W M SSM represents the superpotential of the MSSM. We give out the concrete form of the BLMSSM soft breaking terms L sof t [8].
In order to break the local gauge symmetry VEVs υ u , υ d , and the SU(2) L singlets Φ B , ϕ B , Φ L , ϕ L should obtain nonzero VEVs υ B , υ B , υ L , υ L respectively. The higgs fields and the Higgs superfields are defined as The detailed discussion of Higgs mass matrixes can be found in Ref. [8]. There are the super fieldsN c in BLMSSM. Therefore, the neutrinos and scalar neutrinos are doubled as those in MSSM. Through the see-saw mechanism, light neutrinos obtain tiny masses. In the left-handed basis (ν, N c ), we deduce the mass matrix of neutrinos after symmetry breaking With the unitary transformations the mass matrix of neutrinos are diagonalized as The squared mass matrix of the scalar neutrinos is obtained from the superpotential and the soft breaking terms in BLMSSM Eqs.(1)(2), withñ T = (ν I ,Ñ cI * ). The scalar neutrinos are enlarged by the superfieldsÑ c and the squared mass matrix reads as The squared mass matrix of the scalar neutrinos are diagonalized through the formula Because of the introduction of the superfieldsÑ c in BLMSSM. The corrected charginolepton-scalar neutrino couplings are adapted as

III. FORMULATION
To obtain the lepton EDM, we use the effective Lagrangian method, and the Feynman amplitude can be expressed by these dimension 6 operators.
with D µ = ∂ µ + ieA µ , ω ∓ = 1∓γ 5 2 , l denoting the lepton fermion, m l being the lepton mass, F µν being the electromagnetic field strength. Adopting on-shell condition for external leptons, only the O ∓ 2,3,6 contribute to lepton EDM. Therefore, the Wilson coefficients of the operators O ∓ 2,3,6 in the effective Lagrangian are of interest. The lepton EDM can be expressed as The fermion EDM is a CP-violating amplitude which can not be obtained at tree level in the fundamental interactions. However, in the CP violating electroweak theory, one loop diagrams should contribute nonzero value to fermion EDM. Considering the relations between the Wilson coefficients C ∓ 2,3,6 of the operators O ∓ 2,3,6 [13], the lepton EDM is obtained The one loop triangle diagrams in BLMSSM are divided into two types according to the virtual particles: 1 the neutralino-scalar lepton diagram; 2 the chargino-scalar neutrino diagram. After the calculation, using the on-shell condition for the external leptons, we obtain the one loop diagrams contribution to lepton EDM.
, Λ N P representing energy scale of the new physics. The concrete form The couplings ( The matrices ZL, Z N respectively diagonalize the mass matrices of scalar lepton and neutralino.
The dotted line and the solid line are almost coincident, and they vary slightly with N e . With

B. the muon EDM
In the similar way, the muon EDM is numerically studied. Lepton EDM is CP violating which is generated by the CP violating phases such as θ 1 , θ 2 , θ µ . With tan β = 15 and N e = N τ = 1000GeV, we study muon EDM varying with the parameter N µ which is related with the scalar muon mass. We obtain the solid line with (θ 1 = 0.5π, θ 2 = θ µ = 0) in  The upper bound of the muon EDM is at the order of 10 −19 (e.cm), and it is much larger than that of electron. Considering this fact, we take N e = N τ = 1000GeV and N µ = 600GeV to obtain biggish contribution to muon EDM. As (θ 1 = 0.5π, θ 2 = θ µ = 0), the result varying with tan β is represented by the dotted line in the Fig.8. The dotted line implies that it is  the decreasing function of tan β, and changes from 13.5 × 10 −25 (e.cm) to 11.1 × 10 −25 (e.cm).
With (θ 2 = 0.5π, θ 1 = θ µ = 0) and (θ µ = 0.5π, θ 1 = θ 2 = 0) we plot the numerical results by the solid line and dashed line respectively in the Fig.9. The effects of θ 2 and θ µ to muon are almost same and they both are increasing functions of tan β. In general the solid line and dashed line are at the order of 10 −26 (e.cm) and are much smaller than the dotted line.
From the Fig.7 and Figs. (8,9), we find θ 1 is the CP violating phase that gives dominate contribution to muon EDM. Therefore, we study the contribution to the muon EDM varying with the CP violating phase θ 1 . Supposing N e = N τ = 1000GeV, N µ = 600GeV and θ 2 = θ µ = 0, the contributions to muon EDM are represented by the dotted line, solid line   (7,8,9,10) is about at the order of 10 −24 (e.cm), which is almost five order smaller than muon EDM upper bound.

C. the tau EDM
Tau is the heaviest lepton, whose EDM upper bound is the largest one and at the order of 10 −17 e.cm. The tau EDM is also of interest and is calculated here with the supposition N e = N µ = 1000GeV. We study the relation between d τ and N τ relating with the scalar tan β is an important parameter shown as the front d e and d µ , and it influences the lepton EDM strongly. Here, we take the parameter as N e = N µ = 1000GeV and N τ = 600GeV. In the end, we investigate the relations between d τ and the CP violating phases θ 1 , θ 2 and θ µ . The parameters used in the Figs. (13,14,15) are N e = N µ = 1000GeV, N τ = 600GeV and tan β = 15. In the Fig.13, the solid line denotes the d τ varying with θ 1 for θ 2 = θ µ = 0. the Fig.14 implies the relation between tau EDM and θ 2 when θ 1 and θ µ are supposed as 0.
Based on the assumption θ 1 = θ 2 = 0, we plot d τ varying with θ µ by the dotted line in the  (11,12,13,14,15) show that the one loop contributions to tau EDM are at the order of 10 −24 (e.cm) in our used parameter space. These contributions are about seven order smaller than the upper bound of tau EDM.

V. DISCUSSION AND CONCLUSION
In the frame work of CP violating BLMSSM, we study the one loop contributions to the lepton(e, µ, τ ) EDMs. The used parameters can satisfy the experiment data of Higgs and neutrino. We study the effects of the CP violating phases θ 1 , θ 2 and θ µ to the lepton EDM. The upper bound of electron EDM is 8.7 × 10 −29 (e.cm), which gives strict confine on the BLMSSM parameter space. In our used parameter space, the contributions to electron EDM can easily reach it's upper bound and even exceed the bound. The numerical values obtained for muon EDM and tau EDM are both at the order of 10 −24 (e.cm), and they are several order smaller than their upper bounds. In general, the numerical results of the lepton EDMs are large, and they maybe detected by the experiments in the near future.