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, μ, τ is shown in this work. It is in favour of exploring the source of CP violation and probing the physics beyond SM.


Introduction
The theoretical predictions for EDMs of leptons and neutron are very small in SM. The estimated SM value for the electron EDM is about |d e | ≃ 10 −38 e.cm [1,2], which is too small to be detected by the current experiments. The ACME Collaboration [3] report the new result of electron EDM 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 bounds of µ and τ respectively [4][5][6]. The minimal supersymmetric extension of SM (MSSM) [7][8][9][10] 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.
In MSSM, when the CP violating phases are of normal size and the SUSY particles are at TeV scale, big EDMs of elementary particles are obtained, and they can exceed the current experiment limits. 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 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 [11,12].
BLMSSM is the minimal supersymmetric extension of the SM with local gauged B and L [13,14]. Therefore, it can explain both the asymmetry of matter-antimatter in the universe and the data from neutrino oscillation experiment. We consider that 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 [15,16]. The lightest CP-even Higgs mass and the decays h 0 → γγ, h 0 → ZZ(W W ) are also studied in this JHEP07(2015)124 model [17]. In our previous works [18,19], we study the neutron EDM and B 0 −B 0 mixing in CP violating BLMSSM.
Research the MDMs [20][21][22] and EDMs [23][24][25][26][27][28] 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, and some two loop corrections are also investigated. In the two Higgs doublet models with CP violation, the authors obtain the one-loop and Barr-Zee type two-loop contributions to fermionic EDMs. A model-independent study of d e in the SM is carried out [29,30]. They take into account the right-handed neutrinos, the neutrino see-saw mechanism and the framework of minimal flavor violation. Their results show that when neutrinos are Majorana particles, d e can reach its experiment upper 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 BLMSSM
The local gauge group of BLMSSM [13,14] where the exotic leptons are introduced to cancel L anomaly. Similarly, they introduce the exotic quarks to cancel the B anomaly. In this work, the quarks, exotic quarks and exotic leptons have none one-loop contribution to lepton EDM, so we do not introduce them in detail. The Higgs mechanism is of solid foundation, because of the detection of the lightest CP even Higgs h 0 at LHC [31][32][33]. The Higgs superfields are used to break lepton number spontaneously, and they need nonzero vacuum expectation values (VEVs).
The superpotential of BLMSSM is shown as Here, W MSSM represents the superpotential of the MSSM. The concrete forms of W B , W L and W X are shown in the work [17]. W L includes the needed new term W L (n), which is collected here In BLMSSM, the complete soft breaking terms are very complex [17,18], and only the terms relating with lepton are necessary for our study In order to break the local gauge symmetry SU(2) L ⊗ U(1) Y ⊗ U(1) B ⊗ U(1) L down to the electromagnetic symmetry U(1) e , the SU(2) L doublets H u and H d should obtain nonzero VEVs υ u and υ d . While the SU(2) L singlets Φ L and ϕ L should obtain nonzero

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VEVs υ L and υ L respectively. The needed Higgs fields and Higgs superfields are defined as The detailed discussion of Higgs mass matrices can be found in ref. [17]. The super fieldN c in BLMSSM leads to that the neutrinos and sneutrinos are doubled as those in MSSM. Through the see-saw mechanism, light neutrinos obtain tiny masses.
In BLMSSM, there are 10 neutralinos: 4 MSSM neutralinos, 3 baryon neutralinos and 3 lepton neutralinos. The MSSM neutralinos, baryon neutralinos and lepton neutralinos do not mix with each other. Baryon neutralinos have zero contribution to the lepton EDM at one-loop level. While, lepton neutralinos can give contributions to lepton EDM through lepton-slepton-lepton neutralino coupling. The three lepton neutralinos are made up of λ L (the superpartners of the new lepton boson) and ψ Φ L , ψ ϕ L (the superpartners of the SU(2) L singlets Φ L , ϕ L ). Here, we show the mass term of lepton neutralinos.
Three lepton neutralino masses are obtained from diagonalizing the mass mixing matrix in eq. (2.5) by Z N L . Though in BLMSSM there are six sleptons, their mass squared matrix is different from that in MSSM, because of the contributions from eqs. (2.2), (2.3). We deduce the corrected mass squared matrix of slepton, and the matrix ZL is used to diagonalize it The super fieldN c is introduced in BLMSSM, so the neutrino mass matrix and the sneutrino mass squared matrix are both 6 × 6 and more complicated than those in MSSM. In the left-handed basis (ν, N c ), we deduce the mass matrix of neutrino after symmetry breaking

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With the unitary transformations the mass matrix of neutrino is diagonalized as . (2.10) The trilinear sneutrino-Higgs-sneutrino coupling A N c λ N cÑ cÑ c ϕ L in the soft breaking terms L soft (L) leads to large sneutrino masses. The VEV of ϕ L is 1 √ 2v L , and the contribution from this term to sneutrino masses is at the order of A N c λ N cv L . The super potential of BLMSSM includes the new term W L (n). Then two functions and the scalar supersymmetric potential are shown here where A i represent the scalar fields. The first term Y νLĤuN c in W L (n) is suppressed by Y ν . Using the formula eq. (2), the second term λ N cN cN cφ L in W L (n) can give important contributions to large sneutrino masses through nonzero VEV of Higgs superfield ϕ L . This type correction is at the order of λ * N c λ N cv 2 L . The orders of both dominant contributions respectively are A N c λ N cv L and λ * N c λ N cv 2 L , that are much larger than the product of neutrino Yukawa and SUSY scale. The mass squared matrix of the sneutrino is obtained from the superpotential and the soft breaking terms from eqs.
withñ T = (ν,Ñ c * ). The sneutrinos are enlarged by the superfieldN c and the mass squared matrix of sneutrino reads as , the masses of the sneutrinos are obtained.

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Because of the introduction of the superfieldN c in BLMSSM, the corrected charginolepton-sneutrino coupling is adapted as (2.14) with ω ∓ = 1∓γ 5 2 . Here we use the abbreviated form, s W = sin θ W , c W = cos θ W , and θ W is the Weinberg angle.
From the interactions of gauge and matter multiplets ig , the lepton-slepton-lepton neutralino couplings are deduced here This type couplings give new contributions beyond MSSM to lepton EDM. Compared with MSSM, there are four new CP violating sources: 1. the mass M L of gaugino λ L ; 2. the superfield Higgsino's mass µ L , which is included in the mass matrices of both sneutrino and lepton neutralino.
In general, we take v L andv L as real parameters to simplify the numerical discussion.

Formulation
To obtain the lepton EDM, we use the effective Lagrangian method, and the Feynman amplitudes can be expressed by these dimension-6 operators.
with D µ = ∂ µ + ieA µ , 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 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 is expressed as

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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 [25][26][27][28], the lepton EDM is obtained The one-loop triangle diagrams in BLMSSM are divided into three types according to the virtual particles: 1 the neutralino-slepton diagram; 2 the chargino-sneutrino diagram; 3 the lepton neutralino-slepton diagram. After the calculation, using the on-shell condition for the external leptons, we obtain the one-loop contributions to lepton EDM.
, Λ NP representing energy scale of the new physics. The concrete form of the function ̺ i,j (x, y) is shown here The couplings ( The matrices ZL, Z N respectively diagonalize the mass matrices of slepton and neutralino.
To explicit the phase of λ L obviously in the one loop contributions, we suppose M L ≫ µ L , g L v L , g LvL . Then the lepton netralino-slepton contributions are simplified as In this formula, the CP violating phase θ L is conspicuous.
If we do not specially declare, the non-diagonal elements of the used parameters should be zero.
To study the effects to lepton EDM from the non-diagonal elements of the used parameters, we consider the constraints from the lepton flavor violating processes l j → l i + γ and l j → 3l i . The experiment upper bounds of Br(µ → e + γ) and Br(µ → 3e) are respectively 5.7 × 10 −13 and 1.0 × 10 −12 . They are both strict and set severe limits on the JHEP07(2015)124 parameter space, especially for the sensitive parameters including non-diagonal elements for the lepton flavor violation. In our prepared work [42], we study Br(µ → e + γ) and Br(µ → 3e), and find that the virtual particle masses, tan β and the ratios of non-diagonal elements to diagonal elements for the slepton(sneurino) mass squared matrices are important parameters. When the slepton and sneutrino are at TeV order, tan β should be in the region 10 ∼ 20. The effects to µ → e + γ and µ → 3e from the non-diagonal elements of m 2Ñ c , A N c and A N are not large. On the other hand, the non-diagonal elements of m 2 L , m 2 R influence the both LFV processes strong. With the supposition (m 2 L ) ij = (m 2 R ) ij = F L 2 , F L should be in the range (0 ∼ 500)GeV except extreme parameter space.

The electron EDM
At first, we study electron EDM, because its upper bound is the most strict one. The CP violating phases θ 1 , θ 2 , θ µ , θ µ L , θ L , and other parameters have close relationships with electron EDM. In this subsection, we suppose S ν = 1.0 × 10 6 GeV 2 , m L = 1000GeV and υ Lt = 3000GeV.
Supposing θ 1 = θ 2 = θ µ = θ L = 0, we study the contributions from θ µ L to electron EDM. µ L relates with sneutrino mass squared matrix and lepton neutralino mass matrix. Here, the contributions to lepton EDM from the lepton neutralino-slepton diagram are dominant, because the chargino-sneutrino diagram contributions are suppressed by the tiny neutrino Yukawa couplings through     As M F = 0, the right-handed sneutrino corrections are around 1.0×10 −29 e.cm. When M F is larger than 60 GeV, the contributions from the righthanded sneutrino can reach 2.0 × 10 −29 e.cm. Therefore, they are important and decrease the effects from the left-handed sneutrino to some extent with nonzero M F . Because of θ µ = θ µ L = θ L = 0, lepton neutralino-slepton diagram does not give corrections to electron EDM in this condition. From the figures 1, 2, 3, 4, one can find the upper bound of electron EDM is strict and has rigorous bound on the parameter space.

The muon EDM
Lepton EDM is CP violating which is generated by the CP violating phases. In the similar way, the muon EDM is numerically studied. The parameters S L = 7.0 × 10 6 GeV 2 , S R = 6.0 × 10 6 GeV 2 , S ν = 2.0 ×    The reason is that left-handed sneutrino corrections turn small fast with the non-diagonal elements of m 2 L and m 2 R . Right-handed sneutrino and left-handed sneutrino mix and should be regarded as an entirety. To some extent, non-zero F L moves the contributions from left-handed sneutrino to the right-handed sneutrino, without affecting the total results obviously. The ratios for the right-handed sneutrino contributions to d µ ran from 20% to bigger values, and can even reach 50%. With our used parameters, the numerical results for muon EDM shown as the figures 5, 6, 7 are about at the order of 10 −26 e.cm, which is almost seven-order smaller than muon EDM upper bound.

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. Tau EDM is also of interest and calculated here. In this subsection, we use S ν = 2.0 ×

Discussion and conclusion
In the frame work of CP violating BLMSSM, we study the one-loop contributions to the lepton(e, µ, τ ) EDM. The used parameters can satisfy the experiment data of Higgs and neutrino. The effects of the CP violating phases θ 1 , θ 2 , θ µ , θ µ L , θ L to the lepton EDM are researched. 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 its upper bound and even exceed it. The numerical values obtained for muon EDM and tau EDM are at the order of 10 −26 e.cm and 10 −25 ∼ 10 −24 e.cm respectively. They both are several-order smaller than their EDM upper bounds. Our numerical results mainly obey the rule d e /d µ /d τ ∼ m e /m µ /m τ . The right-handed sneutrino contributions are considerable, and should be taken into account. The contributions from JHEP07(2015)124 the lepton neutralino-slepton have two new CP violating sources, and include the coupling constant g L . If we enlarge g L and adopt other parameters, the lepton neutralino-slepton contributions to lepton EDM can enhance several orders. In general, the numerical results of the lepton EDM are large, and they maybe detected by the experiments in the near future.