$Z\rightarrow l_i^{\pm}l_j^{\mp}$ processes in the BLMSSM

In a supersymmetric extension of the Standard Model(SM) where baryon and lepton numbers are local gauge symmetries (BLMSSM), we investigate the charged lepton flavor violating (CLFV) processes $Z\rightarrow l_i^{\pm}l_j^{\mp}$ after introducing the new gauginos and the right-handed neutrinos. In this model, the branching ratios of $Z\rightarrow l_i^{\pm}l_j^{\mp}$ are around ($10^{-8}\sim10^{-10}$), which approach the present experimental upper bounds. We hope that the branching ratios for these CLFV processes can be detected in the near future.


I. INTRODUCTION
Neutrinos have tiny masses and mix with each other, which can be proved by the neutrino oscillation experiments [1][2][3][4]. It shows that lepton flavor symmetry is not conserved in neutrino sector. A new particle around 125 GeV is detected by the LHC [5][6][7], whose properties are close to the Higgs boson. Then the SM has achieved great success. However, due to the GIM mechanism, the expected rates for the charged lepton flavor violating (CFLV) processes [8,9] are very tiny in the SM with massive neutrinos. For example, Br(Z → eµ) ∼ Br(Z → eτ ) ∼ 10 −54 and Br(Z → µτ ) ∼ 10 −60 [10][11][12][13][14], they are much smaller than the experimental upper bounds. The CLFV is forbidden in the SM. In Table 1, we show the present experimental limits and future sensitivities for some CLFV processes.
In the Ref. [15], the authors consider that the future sensitivities for the CLFV processes may be reached 10 −11 ; At a Future Circular e + e − Collider (such as FCC-ee(TLEP)) [16,17], it is estimated that the sensitivities can be improved up to 10 −13 . Thus, any signal of CLFV would be a hint of new physics, and the study of CLFV processes is an effective approach to explore new physics beyond SM.  [24] Z → eτ < 9.8 × 10 −6 [18,21,22] ∼ (1.3 − 6.5) × 10 −8 [24] Z → µτ < 1.2 × 10 −5 [18,21,23] ∼ (0.44 − 2.2) × 10 −8 [24] In simple SM extension, CLFV processes are restricted strongly by the tiny neutrino masses. As an appealing supersymmetric extension of SM, the minimal supersymmetric standard model (MSSM) [25][26][27][28] with R-party [27] conservation has drawn physicists' attention for a long time. However, the left-handed light neutrinos remain massless, and it can not explain the discovery of neutrino oscillations. Therefore, physicists do more research on the light neutrino masses and mixings with MSSM extension [29][30][31][32][33][34]. As a supersymmetric extension of the MSSM with local gauged baryon (B) and lepton (L) numbers, BLMSSM is introduced [35][36][37][38]. In the BLMSSM, the local gauged B must be broken in order to ac-count for the asymmetry of matter-antimatter in the universe. Right-handed neutrinos are introduced to explain the data from neutrino oscillation experiments, hence lepton number is also expected to be broken [37]. In Refs. [37,39], baryon number and lepton number are local gauged and spontaneously broken at the TeV scale in the BLMSSM.
In this work, we continue to analyze the CLFV processes Z → l ± i l ∓ j (Z → eµ, Z → eτ, Z → µτ ) within the BLMSSM. Compared with the MSSM, the neutrino masses in the BLMSSM are not zero. Three heavy neutrinos and three new scalar neutrinos are introduced in this model. And new particle lepton neutralino χ 0 L is also introduced. These new sources enlarge the CLFV processes via loop contributions. Therefore, the expected experimental results for the CLFV processes may be obtained in the near future.
This work is organized as follows: In Sec.2, we summarize the BLMSSM briefly, including its superpotential, the general soft SUSY-breaking terms, needed mass matrices and couplings. Section 3 is devoted to the decay widths of the CLFV processes Z → l ± i l ∓ j . In Sec.4, we give out the corresponding parameters and numerical analysis. The discussion and conclusion are described in Section 5. Appendix A is devoted to described the concrete forms of coupling coefficients in Fig.1.
The superpotential of the BLMSSM is shown as follows [42] with W M SSM representing the superpotential of the MSSM. The concrete forms of W B , W L and W X can be obtained in Ref. [42].
In the BLMSSM, the soft breaking terms L sof t are generally given by [36,37,42], and only the leptonic terms contribute to our study Here λ L represents gaugino of U(1) L . The SU(2) L doublets H u and H d obtain the nonzero VEVs υ u and υ d , The SU(2) L singlets Φ L and ϕ L acquire the nonzero VEVs υ L and υ L , In the BLMSSM, the mass matrices of lepton neutralinos, neutrinos, sleptons and sneutrinos are introduced as follows: In the base (iλ L , ψ Φ L , ψ ϕ L ) [35,43,44], the mixing mass matrix of lepton neutralinos is obtained.
Then the three lepton neutralino masses are deduced due to diagonalize the mass matrix After symmetry breaking, the mass matrix of neutrinos is deduced in the basis (ν, N c ) [45,46] Then diagonalizing the neutrino mass matrix by the unitary matrix U ν , we can get six mass eigenstates of neutrinos, which include three light eigenstates and three heavy eigenstates.
In the BLMSSM, the slepton mass squared matrix deduced from Eqs.(1),(2) reads as where, Through the matrix ZL, the mass matrix can be diagonalized.
From the contributions of Eqs.(1),(2), we also deduce the mass squared matrix of sneu- Then the sneutrino masses can be obtained by formula ).
In the BLMSSM, we deduce the corrections for the couplings existed in the MSSM due to superfieldsÑ c . The corresponding couplings for W-lepton-neutrino, Z-neutrino-neutrino, charged Higgs-lepton-neutrino, Z-sneutrino-sneutrino and chargino-lepton-sneutrino are introduced in Ref. [35].

From the interactions of gauge and matter multiplets
, the lepton-slepton-lepton neutralino coupling is deduced here In the BLMSSM, we study the CLFV processes Z → l ± i l ∓ j . The corresponding Feynman diagrams can be depicted by Fig.1, and the corresponding effective amplitudes can be written as [14,47,48] with where l i,j represent the wave functions of the external leptons. The coefficients F L,R can be obtained from the amplitudes of the Feynman diagrams. F L,R (S) correspond to Fig.1(1)∼Fig.1 (7), and stand for the contributions from chargino-sneutrino, neutralino- Dirac(Majorana) fermion particle, S represents scalar boson particle and W represents W boson particle.
slepton, neutrino-charged Higgs and lepton neutralino-slepton; F L,R (W ) correspond to Fig.1(8) and Fig.1(9), and stand for the contributions from W-neutrino due to three light neutrinos and three heavy neutrinos mixing together. We formulate these coefficients as follows Here, H SF 2li L,R ... represent the corresponding coupling coefficients of the left (right)-hand parts in the Lagrangian and the concrete expressions can be found in Appendix.
representing the mass of the corresponding particle, m Np representing energy scale of the new physics to make the amplitudes dimensionless. The one-loop functions G i (x 1 , x 2 , x 3 ), i = 1, 2 are given by ], ]. (15) Then, the branching ratios of Z → l ± i l ∓ j can be summarized as where Γ Z represents the total decay width of Z-boson and we use Γ Z ≃ 2.4952 GeV [18].

A. Z → eµ
The experimental upper bound for the branching ratio of Z → eµ is around 7.5 × 10 −7 .
The parameter m 1 is related with the mass matrix of neutralino, which means the contri- through slepton-neutrino, sneutrinos-chargino and slepton-lepton neutralino contributions.
We choose the parameters m 1 = 500GeV, m 2 = 1TeV, S m = 1TeV, AN = 500GeV and tan β = 15. As V Lt = 3TeV, we plot the allowed results with tan β L versus g L in Fig.4.
Obviously, when the value of g L is large enough, the value of tan β L approaches 1. When g L ≤ 0.3, the parameter tan β L can vary in the region of 0∼2. It implies that g L is a sensitive parameter to the numerical results. As tan β L = 2, g L versus V Lt are scanned in Fig.5. We find that the allowed scope of V Lt shrinks and the value of V Lt decreases with the enlarging g L . Therefore, the value of g L should not be large. Generally, we take 0.05 ≤ g L ≤ 0.3 and V Lt ∼ 3TeV in our numerical calculations.

B. Z → eτ
In the similar way, the CLFV process Z → eτ is numerically studied and its experimental upper bound is around 9.8 × 10 −6 . As discussed in the previous part, g L can affect the contribution strongly through the masses of sleptons, sneutrinos and lepton neutralinos.

C. Z → µτ
The experimental upper bound for the CLFV process Z → µτ is 1.2×10 −5 , which is about one order larger than the process Z → eµ. The parameter AN presents in the sneutrino mass matrix and affects sneutrino-chargino contributions. Supposing m 1 = 500GeV, m 2 = 1TeV, g L = 0.2, S m = 1TeV, M L f = 1 × 10 5 GeV 2 and tan β = 1(2, 3), we plot the results with the AN in Fig.8. As AN ≤ 4TeV, the branching ratios are around 4 × 10 −9 ; As AN > 4TeV, these three lines increase quickly and AN has an obvious influence on the numerical results.
After that, the effects from parameter tan β are studied. tan β is related with v u and In our used parameter space, the numerical results show that the rates for Br(Z → l ± i l ∓ j ) can almost reach the present experimental upper bounds. The numerical analyses indicate that parameters m 1 , m 2 , g L , M L f , S m , AN and tan β are important. The sensitive parameters are g L , M L f and S m and they affect the results obviously. We hope the experiment results for Z → l ± i l ∓ j can be detected in the near future.