Charged lepton flavor violation in extended BLMSSM

Within the extended BLMSSM, the exotic Higgs superfields (ΦNL,φNL)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Phi _{NL},\varphi _{NL})$$\end{document} are added to make the exotic leptons heavy, and the superfields (Y,Y′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y,Y^\prime $$\end{document}) are also introduced to make exotic leptons unstable. This new model is named as the EBLMSSM. We study some charged lepton flavor violating (CLFV) processes in detail in the EBLMSSM, including lj→liγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_j\rightarrow l_i \gamma $$\end{document}, muon conversion to electron in nuclei, the τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} decays and h0→lilj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^0\rightarrow l_i l_j$$\end{document}. Being different from BLMSSM, some particles are redefined in this new model, such as slepton, sneutrino, exotic lepton (neutrino), exotic slepton (sneutrino) and lepton neutralino. We also introduce the mass matrices of superfields Y and spinor Y~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{Y}$$\end{document} in the EBLMSSM. All of these lead to new contributions to the CLFV processes. In the suitable parameter space, we obtain the reasonable numerical results. The results of this work will encourage physicists to explore new physics beyond the SM.


Introduction
The Higgs boson, an elementary particle, has been researched by the Large Hadron Collider (LHC) as one of the primary scientific goals. Combining the updated data of the ATLAS [1] and CMS [2] Collaborations, now its measured mass is m h 0 = 125.09±0.24GeV [3], which represent that the Higgs mechanism is compellent. In the Standard Model (SM), the lepton-flavor number is conserved. However, the neutrino oscillation experiments [4][5][6][7][8][9][10][11][12] have convinced that neutrinos possess tiny masses and mix with each other. So the individual lepton numbers L i = L e , L μ , L τ are not exact symmetries at the electroweak scale. Furthermore, the presentation of the GIM mechanism makes the charged lepton flavor violating(CFLV) processes in the SM very tiny [13a 15], such as Br SM (l j → l i γ ) ∼ 10 −55 [16]. Therefore, if we observe the CLFV processes in future experiments, it is an obvious evidence of new physics beyond the SM.
Studying the CLFV processes is an effective way to explore new physics beyond the SM. MEG Collaboration gives out the current experiment upper bound of the CLFV process μ → eγ , which is Br(μ → eγ ) < 5.7 × 10 −13 at 90% confidence level [17]. Br(τ → eγ ) < 3.3 × 10 −8 and Br(τ → μγ ) < 4.4 × 10 −8 are also shown in Ref. [18]. SINDRUM II collaboration has updated the sensitivity of the μ − e conversion rate in Au nuclei C R(μ → e : 197 79 Au) < 7 × 10 −13 [19]. The current experiment upper bounds for the τ decays Br(τ → 3e) < 2.7 × 10 −8 and Br(τ → 3μ) < 2.1×10 −8 have been shown by Particle Data Group [18]. Furthermore, a direct research for the 125.1 GeV Higgs boson decays including the CLFV, h 0 → l i l j , has been given out by the CMS Collaboration [20,21] and ATLAS Collaboration [22]. We show the corresponding experiment upper bounds for processes h 0 → l i l j in Table 1. Physicists do more research on the CLFV processes for l j → l i γ decays, μ − e conversion rates in nuclei, the τ decays and h 0 → l i l j decays in models beyond the SM [23][24][25][26][27][28]. In our previous work, we have studied l j → l i γ , muon conversion to electron in nuclei, the τ decays and h 0 → l i l j processes in the μνSSM [29][30][31]. We also have discussed l j → l i γ processes, the τ decays and muon conversion to electron in nuclei in the BLMSSM [32,33]. In this work, we study the processes l j → l i γ , μ − e conversion in Au nuclei, the τ decays and h 0 → l i l j in the extended BLMSSM, which is named as the EBLMSSM [34].
Extending the MSSM with the introduced local gauged B and L, one obtains the so called BLMSSM [35][36][37][38]. In the BLMSSM, the exotic lepton masses are obtained from the Yukawa couplings with the two Higgs doublets H u and H d . The values of these exotic lepton masses are around 100 GeV, which can well anastomose the current experiment bounds. However, with the development of high energy physics experiments, we may obtain the heavier experiment Table 1 Present experiment limits for 125 GeV Higgs decays h 0 → l i l j with the CLFV CLFV process Present limit (CMS) Present limit (ATLAS) Confidence level (CL) h 0 → eμ < 0.035% [20] −− 95% h 0 → eτ < 0.61% [21] < 1.04% [22] 95% h 0 → μτ < 0.25% [21] < 1.43% [22] 95% lower bounds of the exotic lepton masses in the near future, which makes the BLMSSM model do not exist. Therefore, two exotic Higgs superfields, the SU (2) L singlets N L and ϕ N L , are considered to be added in the BLMSSM. With the introduced superfields N L and ϕ N L , the exotic leptons can turn heavy and should be unstable. This new model is named as the extended BLMSSM (EBLMSSM) [34]. In order to make the exotic leptons unstable, we add superfields Y and Y in the EBLMSSM.
In the BLMSSM, the dark matter (DM) candidates include the lightest mass eigenstate of X, X mixing and a fourcomponent spinorX composed by the superpartners of X, X . In the EBLMSSM, the DM candidates not only include above terms presented in the BLMSSM, but also contain new terms due to the new introduced superfields of Y, Y . So the lighter mass eigenstates of Y, Y mixing and spinorỸ are DM candidates [34,39]. In Sect. 4.2 of our previous work [34], we suppose the lightest mass eigenstate of Y, Y mixing as a DM candidate, and calculate the relic density D h 2 . In the reasonable parameter space, D h 2 of Y 1 can match the experiment results well.
The Higgs boson h 0 is produced chiefly from the gluon fusion (gg → h 0 ) at the LHC. The leading order (LO) contributions originate from the one-loop diagrams. In the BLMSSM, we have studied the h 0 → gg process in our previous work [40], and the virtual top quark loops play the dominate roles. The EBLMSSM results for h 0 → gg are same as those in BLMSSM, which have been discussed in Ref. [34]. Being different from BLMSSM, the exotic leptons in EBLMSSM are more heavy and the exotic sleptons of the 4th and 5th generations mix together to form a 4 × 4 mass matrices. The LO contributions for h 0 → γ γ originate from the one-loop diagrams. In the EBLMSSM [34], we have studied the decay h 0 → γ γ in detail. The processes h 0 → V V, V = (Z , W ) also have been researched in this new model. Considering the constraints from the parameter space of these researches, we study the processes l j → l i γ , muon conversion to electron in Au nuclei, τ decays and h 0 → l i l j in this work.
The outline of this paper is organized as follows. In Sect. 2, we present the ingredients of the EBLMSSM by introducing its superpotential, the general soft SUSY-breaking terms, new corrected mass matrices and couplings which are different from those in the BLMSSM. In Sect. 3, we analyze the corresponding amplitudes and the branching ratios of rare Table 2 The new introduced superfields in the EBLMSSM beyond BLMSSM CLFV processes l j → l i γ , the τ decays and h 0 → l i l j decays. We also discuss the muon conversion to electron rates in nuclei. The numerical analysis is discussed in Sect. 4, and the conclusions are summarized in Sect. 5. The tedious formulae are collected in "Appendix".

Introduction of the EBLMSSM
In the EBLMSSM, the local gauge group is SU [34,35,41,42]. We introduce the exotic Higgs superfields N L and ϕ N L with nonzero VEVs υ N L andῡ N L [43] to make the exotic leptons heavy. Accordingly, the superfields Y and Y are introduced to avoid the heavy exotic leptons stable. In Table 2, we show the new introduced superfields in the EBLMSSM [34]. The corresponding superpotential of the EBLMSSM is shown here W M SSM is the superpotential of the MSSM. W B and W X are same as the terms in the BLMSSM [40,44]. The new terms λ LL 4L BLMSSM. Comparing with the W X in the BLMSSM, W Y is introduced in the EBLMSSM, which includes the lepton-exotic lepton-Y coupling and lepton-exotic slepton-Y coupling. These new couplings can produce one-loop diagrams influencing the CLFV decays. These new couplings can also produce one-loop diagrams contributing to the lepton electric dipole moment(EDM) and lepton magnetic dipole moment(MDM), which will be discussed in our next work. With the 4th and 5th generation exotic sleptons mixing together, the h 0 (Z )-exotic slepton-exotic slepton coupling is deduced in the EBLMSSM. In the EBLMSSM, the couplings for lepton-slepton-lepton neutralino, h 0 (Z )-slepton-slepton, h 0 (Z )-sneutrino-sneutrino and h 0 (Z )-exotic lepton-exotic lepton also have new contributions to CLFV processes. In the whole, the new couplings in the EBLMSSM enrich the lepton physics in a certain degree.
In the EBLMSSM, W Y are the new terms in the superpotential. In W Y , λ 4 (λ 6 ) is the coupling coefficient of Y -leptonexotic lepton andỸ -slepton-exotic slepton couplings. We consider λ 2 4 (λ 2 6 ) is a 3 × 3 matrix and has non-zero elements relating with the CLFV. In our following numerical analysis, we assume that (λ 2 4 ) I J = (λ 2 6 ) I J = (Lm 2 ) I J , I (J ) represents the I th (J th) generation charged lepton. When I = J , there is no CLFV, which has no contributions to our researched decay processes. So, only the non-diagonal elements (Lm 2 ) I J (I = J ) influence the numerical results of the CLFV processes. Therefore, we should take into account the effects from W Y in this work.
Based on the new introduced superfields N L , ϕ N L , Y and Y in the EBLMSSM, the soft breaking terms are given out (2) is the soft breaking terms of the BLMSSM discussed in our previous work [40,44]. Here, corresponding to the SU (2) L singlets N L and ϕ N L , we obtain the nonzero VEVs υ N L andῡ N L respectively. Generally, the values of these two parameters are at TeV scale. The exotic Higgs N L and ϕ N L can be written as where tan β N L =ῡ N L /υ N L and v Nlt = v 2 N L +v 2 N L .
Comparing with the BLMSSM, the introduced superfields N L and ϕ N L in the EBLMSSM can give corrections to the mass matrices of the slepton, sneutrino, exotic lepton, exotic neutrino, exotic slepton, exotic sneutrino and lepton neutralino. However, the mass matrices of squark, exotic quark, exotic squark used in this work are same as those in the BLMSSM [40,45]. We deduce the adjusted mass matrices in the EBLMSSM as follows.

The mass matrices of slepton and sneutrino in the EBLMSSM
In our previous work, we can easily obtain the slepton and sneutrino mass squared matrices of the BLMSSM [32].
for the BLMSSM results, we acquire the mass squared matrices of slepton and sneutrino in the EBLMSSM.

The mass matrices of exotic lepton and exotic neutrino in the EBLMSSM
The EBLMSSM exotic leptons masses are heavier than those in the BLMSSM due to the introduction of large parameters υ N L andῡ N L . One can obtain the mass matrix of exotic lepton in the Lagrangian: Similarly, the mass matrix of the exotic neutrinos in the EBLMSSM can be given through the Lagrangian:

The mass matrices of exotic slepton and exotic sneutrino in the EBLMSSM
In the EBLMSSM, the exotic slepton of 4th generation and 5th generation mix together, and its mass matrix is 4 × 4, which is different from that in the BLMSSM. Using the superpotential in Eq. (1) and the soft breaking terms in Eq. (2), the mass squared matrix for exotic slepton can be obtained through Lagrangian: With the baseẼ T = (ẽ 4 ,ẽ c * 4 ,ẽ 5 ,ẽ c * 5 ), we show the concrete elements of exotic slepton mass matrix M 2Ẽ in the following form The matrix ZẼ is used to rotate exotic slepton mass matrix to mass eigenstates, which is In the same way, the exotic sneutrino mass squared matrix is also obtained through the Lagrangian: where the corresponding elements of the matrix M 2Ñ are In the base (ν 4 ,ν c * 4 ,ν 5 ,ν c * 5 ), we can diagonalize the mass squared matrix M 2Ñ by ZÑ .

The lepton neutralino mass matrix in the EBLMSSM
In the EBLMSSM, λ L , the superpartner of the new lepton The mass matrix M L can be diagonalized by the rotation matrix Z N L . Then, we can have Here, X 0 represent the mass eigenstates of the lepton neutralino.

The superfields Y in the EBLMSSM
The scalar superfields Y and Y mix. Adopting the unitary transformation, the mass squared matrix for the superfield Y is deduced.
The matrix Z Y is used to diagonalize the matrix to the mass eigenstates: We suppose m 2 The superpartners of Y and Y form a four-component Dirac spinorỸ , and the mass term for superfieldỸ in the Lagrangian is given out In the EBLMSSM, the 4th and 5th generation exotic sleptons mix together. So the exotic slepton couplings in this new model are different from those in the BLMSSM. We deduce the h 0 -exotic slepton-exotic slepton (h 0 −Ẽ −Ẽ) coupling as follows As the new introduced superfield in the EBLMSSM, Y leads to new couplings. The lepton-exotic lepton-Y coupling used in this work is shown here SuperfieldỸ is also a new term beyond the BLMSSM. We deduce the lepton-exotic slepton-Ỹ coupling as In the EBLMSSM, the new effects are added from the couplings of lepton-slepton-lepton neutralino, h 0 (Z )-sleptonslepton, h 0 (Z )-sneutrino-sneutrino, h 0 (Z )-exotic leptonexotic lepton and h 0 (Z )-exotic neutrino-exotic neutrino, which are different from those in the BLMSSM. However, these couplings possess the same writing forms as those in the BLMSSM.
3 The processes l j → l i γ , muon conversion to electron in nuclei, the τ decays and h 0 → l i l j in the EBLMSSM In this section, we analyze the branching ratios of CLFV processes l j → l i γ , muon conversion rates to electron in Au nuclei, the branching ratios of rare τ decays and h 0 → l i l j in the EBLMSSM.
3.1 Rare decays l j → l i γ Generally, the corresponding effective amplitude for processes l j → l i γ can be written as [46] (20) where p (q) represents the injecting lepton (photon) momentum. m l j is the jth generation lepton mass. is the photon polarization vector and u i ( p) (v i ( p)) is the lepton (antilepton) wave function. In Fig. 1, we show the relevant Feynman diagrams corresponding to above amplitude. The Wilson coefficients C L ,R α (α = 1, 2) are discussed as follows. C L ,R α (n)(α = 1, 2), the virtual neutral fermion contributions corresponding to Fig. 1a, are deduced in the following form, where x i = m 2 i /m 2 , m i is the corresponding particle mass and m is the new physics energy scale. H SFl i L ,R represent the left (right)-hand part of the coupling vertex. The Then, we discuss the virtual charged fermion contributions Furthermore, the corrections from Fig. 1c are denoted by However, the contributions from W -W -neutrino diagram can be ignored due to the tiny neutrino mass. We deduce the decay widths for processes l j → l i γ Then, the concrete branching ratios of l j → l i γ can be expressed as [46] Br l j → l i γ = l j → l i γ / l j .
Here, l j represent the total decay widths of the charged leptons l j . We take μ 2.996 × 10 −19 GeV and τ 2.265 × 10 −12 GeV [18] in our latter numerical calculations.  Figs. 2 and 3. In the BLMSSM, the theoretical results for muon conversion to electron rates in nuclei are discussed specifically in our previous work [30,33]. We find that Au nuclei currently give the most stringent bound on conversion rates, so we only study the μ − e conversion rates in Au nuclei in this work. The new corrected particles in the EBLMSSM play important roles to this μ − e conversion processes. Considering the constraints from μ → eγ within EBLMSSM, we study μ − e conversion in Au nuclei, and the corresponding numerical results will be discussed in B of Sect. 4.

Rare τ decays within the EBLMSSM
In this section, we discuss the rare τ decays, which are τ → 3l i and l i represents particle e or μ. We give out both the penguin type diagrams and box type diagrams in Figs. 4 and 5. The theoretical results for the τ decays are discussed specifically in our previous work [32]. In the EBLMSSM, the numerical results of τ decays can be influenced by the new corrected particles, such as exotic lepton (slepton), slepton (sneutrino), lepton neutralino, Y andỸ . In our latter work, we will analyze this τ decays in detail. The corresponding effective amplitude for h 0 →l i l j can be summarized as Here N L ,R (S 1 ) are the coupling coefficients corresponding to triangle diagrams in Fig. 6a, N L ,R (S 2 ) denote the contributions from Fig. 6b. The effects from Fig. 6c, d can be shown by N L ,R (W ). A L ,R (S 1 ) and A L ,R (S 2 ) represent the contributions from self-energy diagrams Figs.7a, and 6b respectively. The effects from Fig. 7c, d can be summarized by A L ,R (W 1 ) and A L ,R (W 2 ) respectively. We give out the concrete expressions for these contributions as follows.
The contributions from triangle diagrams in Fig. 6: The contributions from self-energy type diagrams correspond to Fig. 7: where, the one-loop functions The decay widths for processes h 0 → l i l j are deduced here where h 0 →l i l j = 1 16π m h 0 |N L | 2 + |N R | 2 [47,48]. Correspondingly, the calculations for h 0 → l il j are same as those for h 0 →l i l j .
Above all, the branching ratios of h 0 → l i l j can be summarized as Here, the total decay width of the 125.1 GeV Higgs boson is s → e + μ − ) < 1.1 × 10 −8 [18]. New contributions to rare B 0 and B 0 s meson decays emerge at one-loop level with the box diagrams. In the EBLMSSM, the redefined particles sleptons and sneutrinos lead to new effects to these rare B 0 and B 0 s meson decays. So parameters tan β N L and v Nlt may play the dominated roles to the B 0 and B 0 s meson decays. π + , K + mesons are respectively comprised of ud and us. Particle Date Group gives us the present experiment upper bounds for (π + /K + ) → l + i ν j , which are Br(π + → μ + ν e ) < 8.0 × 10 −3 and Br(K + → μ + ν e ) < 4.0 × 10 −3 [18]. In the EBLMSSM, the penguin type diagrams, self-energy type diagrams and box type diagrams all affect the processes (π + /K + ) → l + i ν j , i = j. CLFV contributions arise from loop corrections with the W ± and heavy charged Higgs propagator. Furthermore, the loop contributions are also related with the exotic slepton (sneutrino), exotic lepton (neutrino), lepton neutralino and slepton (sneutrino) particles. Therefore, processes (π + /K + ) → l + i ν j will be strongly affected by parameters presented in the EBLMSSM. We hope a detailed analysis is going to be discussed in our next work.

Numerical results
In this section, we discuss the numerical results. In our previous work [34], we research the processes h 0 → γ γ , h 0 → V V, V = (Z , W ) in the EBLMSSM, and the corresponding numerical results are discussed in Sect. 5.1 of work [34]. The CP-even Higgs masses m h 0 , m H 0 and CP-odd Higgs mass m 0 A are also analyzed. In the reasonable parameter space, the values of branching ratios for h 0 → γ γ (R γ γ ) and h 0 → V V (R V V ) both meet the experiment limits. Therefore, the Higgs decays in the EBLMSSM play important roles to promote physicists to explore new physics. And the corresponding constraints are also considered in our work. The CP-even Higgs mass is considered as an input parameter, which is m h 0 = 125.1 GeV in our latter numerical discussions.
In the EBLMSSM, to obtain a more transparent numerical results, we adopt the following assumptions on parameter space: where i = 1, 2, 3, Y t (Y b ) corresponds to the Yukawa coupling constant of top (bottom) quark, whose concrete form can be written as (υ cos β)).
In order to simplify the numerical analysis, we use the following assumptions: We take (Lm 2 ) 12 = L F and (Lm 2 ) 13 = (Lm 2 ) 23 = L f .
CLFV process μ → eγ contributes to explore the new physics, whose experiment upper bound of the branching ratio is around 5.7 × 10 −13 at 90% confidence level. In this part, we discuss the effects on process μ → eγ from some new introduced parameters in the EBLMSSM.
Parameter AẼ is present in the non-diagonal parts of the exotic slepton mass matrix. Parameter Y e5 is not only related to the non-diagonal parts of the exotic lepton and exotic slepton mass matrices, but also connected with the diagonal exotic slepton elements. In the EBLMSSM, exotic lepton and exotic slepton are both different from those in the BLMSSM. We assume that MẼ = μ Y = 1. With the introduced superfields Y and Y in the EBLMSSM, we deduce the Y andỸ mass matrices. Parameters μ Y and B Y are respectively present in the diagonal and non-diagonal terms of the Y mass matrix. And the mass of Y possesses the same value as μ Y . So these two parameters affect the Y -lepton-exotic lepton andỸ -lepton-exotic slepton couplings. Furthermore, these new couplings make contributions to the numerical results. Using Y e4 = 0.8, Y e5 = 1.5  and AẼ = 1 TeV, we plot the branching ratios changing with μ Y in Fig. 9. The dotted (dashed, solid) line represents B Y = 0.4(0.8, 1.2) TeV. We find that the branching ratios decrease quickly with the increasing μ Y , which indicates that the large μ Y can restrain the numerical results evidently. Furthermore, the numerical results of these three lines are almost same with the unchanging μ Y , so the contributions from parameter B Y is small.
Then, we study effects from the parameters A E and μ N L on our numerical results. In EBLMSSM, parameters A E and μ N L are both the non-diagonal elements in the exotic slepton and exotic sneutrino mass matrices. μ N L is also the nondiagonal element of lepton neutralino mass matrix. In Fig.  10, we present the branching ratios of μ → eγ versus A E with μ N L = 0.7(1.0, 1.3)TeV, and the concrete results are plotted by dotted (dashed, solid) line. These three lines all increase quickly when A E ranges from 0.1 to 1.8 TeV. Therefore, as the sensitive parameters in the EBLMSSM, the large A E produces the large contributions on the results. However, the numerical results slightly decrease with the enlarg-
As a new introduced parameter in the EBLMSSM, parameter v N is present in the mass matrices of slepton, sneutrino, exotic lepton (neutrino), exotic slepton (sneutrino) and lepton neutralino. In this part, we research the branching ratios of τ → eγ and τ → μγ changing with v N . Supposing Br(τ → μγ ) both shrink quickly. This implies that v N is a sensitive parameter. Though the figure of process τ → eγ is under that of τ → μγ , the both lines possess almost the same results when v N takes same value. So we only study the branching ratios of process τ → μγ in following discussion.
Appearing in the diagonal terms of the exotic slepton and exotic sneutrino mass squared matrices, MẼ affects thẽ Y -lepton-exotic slepton coupling. Parameter L l , not only in the exotic lepton (neutrino) but also in exotic slepton (sneutrino) mass matrices, produces contributions to the numerical results through Y -lepton-exotic lepton andỸlepton-exotic slepton couplings. With (M Ls ) 2 11 = 6TeV 2 , (M Ls ) 2 22 = 4TeV 2 , (M Ls ) 2 33 = 1TeV 2 and v N = 3 TeV, the 3). The figure shows that these three lines all decrease quickly when M 2Ẽ varies from 1 × 10 6 to 9 × 10 6 GeV 2 . With the same M 2Ẽ , the branching ratio decreases remarkably when L l increases. Especially, the line is much steeper with L l = 0.7 than that L l = 1.0(1.3). Obviously, both MẼ and L l are sensitive parameters to our numerical results. Parameter L f influences the numerical results through Y -lepton-exotic lepton andỸ -slepton-exotic slepton couplings. And parameter Y e4 affects our numerical results through exotic lepton and exotic slepton. We discuss the numerical results with L f varying from 0.01 to 0.3 in Fig.  13. The dotted (dashed, solid) line corresponds to Y e4 = 0.5(1.0, 1.5). The branching ratios possess slight changes when Y e4 takes different values for the unchanged L f , which indicates the effects from Y e4 can be ignored in our following discussion. It is easy to see that the numerical results increase sharply with the enlarging L f . So the non-diagonal elements of parameters λ 2 4 and λ 2 6 play important roles in our numerical studies.  The present sensitivity for the muon conversion rates to electron in Au nuclei is C R(μ → e : 197 79 Au) < 7 × 10 −13 . Considering the parameter constrains from μ → eγ , we analyze the numerical results for this μ − e conversion in Au nuclei.
As the non-diagonal elements of matrix (Lm 2 ) I J in the EBLMSSM, (Lm 2 ) 12 = L F affects the numerical results through exotic lepton and exotic slepton.

τ decays
The experiment upper bounds for τ decays are Br(τ → 3e) < 2.7 × 10 −8 and Br(τ → 3μ) < 2.1 × 10 −8 . Considering the constraints from τ → eγ and τ → μγ , we discuss the numerical results for decays τ → 3e and τ → 3μ. Using 3 TeV, m 1 = m 2 = 0.5TeV, tan β = 6 and tan β N L = 1.5, we plot the numerical results of τ → 3e and τ → 3μ in Fig.  16 by dotted line and solid line respectively. With parameter L f changing from 0.01 to 0.3, the values of these two lines are almost the same and both increase quickly. So we just discuss the numerical results for τ → 3e decays as follows (Fig. 17).  Choosing Y e4 = 0.8, Y e5 = 1.5, MẼ = μ Y = 1.5 TeV and B Y = 0.9 TeV, we study the branching ratios of τ → 3e changing with parameter μ N L . The numerical results varying with tan β N L = 1.5(1.7, 1.9) are plotted by the dotted, dashed and solid lines respectively. As μ N L changes from 0.3 to 1 TeV, these three lines all have the obviously improvement. As μ N L > 1 TeV, the numerical results increase slowly. Besides, when μ N L dose not change, the branching ratios of τ → 3e enlarge with the increased tan β N L , and the bigger tan β N L , the bigger change it is in the graph. Therefore, both μ N L and tan β N L affect the numerical results in a certain degree.

h 0 → l i l j
In this part, we study the CLFV processes h 0 → l i l j . The most strict constraint m h 0 = 125.1 GeV is considered as an input parameter. We also take into account the limits from processes l j → l i γ , the muon conversion to electron in Au nuclei and the τ decays discussed above.  Then the effects from the parameters tan β and S m are studied. tan β is related to v u and v d , and appears in almost all mass matrices of CLFV processes. S m are present in the diagonal elements of slepton and sneutrino mass matrices. With AẼ = 2 TeV, L f = 0.25, Y e4 = 1.2 and Y e5 = 0.8, Fig. 19 shows the branching fractions of h 0 → μτ varying with the parameter tan β. S 2 m = 1(2, 3) TeV 2 corresponds to the dotted (dashed, solid) line. These three lines almost overlap, so the effects from S m are small. As tan β varies from 6 to 9, the numerical results decrease obviously. As

h 0 → eμ
The latest experiment upper bound of decay h 0 → eμ is smaller than 0.035% at 95% confidence level, which is detected by the CMS Collaboration. Al and A l both appear in the non-diagonal terms of the slepton mass matrix. Considering the constraints from μ → eγ and μ − e conversion in Au nuclei, we take MẼ = A E = μ N L = 1 TeV, AẼ = μ Y = 1.5 TeV L l = 1, L F = 0.006, B Y = 0.94 TeV, Y e4 = 1.5, Y e5 = 0.8, S 2 m = 1TeV 2 , M 2 L f = 12000GeV 2 , m 1 = m 2 = 0.5 TeV, μ = 0.7 TeV, tan β = 6 and tan β N L = 2. The dotted (dashed, solid) line in Fig. 20 denotes the branching ratios of h 0 → eμ versus A l with Al = 0.5(1, 1.5) TeV. These three lines all increase quickly with the enlarging A l. So A l play important roles to the numerical results. Although the larger Al, the smaller numerical results they are, the contributions from Al are very weak.
At last, we discuss the effects from parameters Y e5 and MẼ . With m 1 = m 2 = 0.5 TeV, Al = 1.5 TeV, Y e4 = 0.5 and L F = 0.006, the branching ratios varying with Y e5 are ploted in Fig. 21. The dotted, dashed and solid lines respectively correspond to MẼ = 1.1, 1.4, 1.7 TeV. These three lines all slightly increase when Y e5 varies from 0.1 to 1.0. As Y e5 still increases from 1.0, the results have much more conspicuous enlargement. However, the total contributions from Y e5 are not so obvious. With the enlarging MẼ , the numerical results reduce more and more slowly.

Discussion and conclusion
We add exotic superfields N L , ϕ N L , Y and Y to the BLMSSM, and this new model is named as the EBLMSSM. In W Y , λ 4 (λ 6 ) is the coupling coefficient of Y -lepton-exotic lepton andỸ -lepton-exotic slepton. We assume λ 2 4 = λ 2 6 is a 3 × 3 squared matrix and its non-diagonal elements are related with the CLFV. Being different from the BLMSSM, the exotic slepton (sneutrino) of 4th and 5th generations mix together and form a 4 × 4 matrix. The Majorana particle, lepton neutralino χ 0 L , is corrected to be a 5 × 5 matrix due to the introduction of superpartners ψ N L and ψ ϕ N L . The terms relating with exotic lepton (neutrino) and slepton (sneutrino) are also adjusted. In Sect. 3, we show the corresponding mass matrices and couplings of the EBLMSSM. The EBLMSSM has more abundant contents than that BLMSSM for the lepton physics.
Considering the constraints from decays h 0 → γ γ and h 0 → V V, V = (Z , W ), we study the CLFV processes l j → l i γ , muon conversion to electron in Au nuclei and the τ decays in the framework of the EBLMSSM. Parameters Y e5 and M L f affect the numerical results in a certain degree. As the new introduced parameters in the EBLMSSM, μ Y , tan β N L , AẼ , A E , L l , MẼ and v N play important roles.
Especially parameters L f and L F are all very sensitive parameters, which influence the numerical results very remarkably. Figures 13, 14 and 16 indicate that the enlarging L f and L F can easily improve the numerical results. Then, the 125.1 GeV Higgs boson decays with CLFV h 0 → l i l j are discussed. As an important constraint, m h 0 = 125.1 GeV is regarded as an input parameter. Taking into account the constraints from the parameter space of decays l j → l i γ , muon conversion to electron in Au nuclei and the τ decays, we analyze the numerical results for h 0 → l i l j in EBLMSSM. Parameters μ and A l affect the CLFV processes in a certain degree. The effects from tan β are very obvious. So tan β is a sensitive parameter. Above all, due to the new particles introduced in the EBLMSSM, the numerical results can easily approach to the present experiment upper bounds.