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 after introducing new gauginos and right-handed neutrinos. In this model, the branching ratios of are around (10−8–10−10), which approach the present experimental upper bounds. We hope that the branching ratios for these CLFV processes can be detected in the near future.


Introduction
Neutrinos have tiny masses and mix with each other, as has been proved by the neutrino oscillation experiments [1][2][3][4]. This shows that lepton flavor symmetry is not conserved in the neutrino sector. A new particle around 125 GeV has been detected by the LHC [5][6][7], with properties close to those of the Higgs boson, a great success for the Standard Model (SM). However, due to the GIM mechanism, the expected rates for charged lepton flavor violating (CFLV) processes [8][9][10] 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 [11][12][13][14][15] are much smaller than the experimental upper bounds. CLFV is forbidden in the SM. In Table 1, we show the present experimental limits and future sensitivities for some CLFV processes. In Ref. [16], the authors consider that future sensitivities for CLFV processes may reach 10 −11 . At a future circular e + e − collider (such as FCC-ee (TLEP)) [17][18][19], it is estimated that the sensitivities could 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 the SM.
In a simple SM extension, CLFV processes are restricted strongly by the tiny neutrino masses. As an appealing supersymmetric extension of the SM, the min-imal supersymmetric standard model (MSSM) [27][28][29][30] with R-parity [29] conservation has drawn physicists' attention for a long time. However, the left-handed light neutrinos remain massless, and it cannot explain the discovery of neutrino oscillations. Therefore, more research is ongoing on the light neutrino masses and mixings with MSSM extensions [31][32][33][34][35][36]. As a supersymmetric extension of the MSSM with local gauged baryon (B) and lepton (L) numbers, the BLMSSM has been introduced [37][38][39][40]. In the BLMSSM, the local gauged B must be broken in order to account for the asymmetry of matter and 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 [39]. In Refs. [39,41], baryon number and lepton number are local gauged and spontaneously broken at the TeV scale in the BLMSSM.  [26] 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. A new particle, the 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 Section 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 Section 4, we give the corresponding parameters and numerical analysis. The discussion and conclusion are given in Section 5. An Appendix is devoted to the concrete forms of coupling coefficients in Fig. 1.
The superpotential of the BLMSSM is shown as follows [44] 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. [44].
In the BLMSSM, the soft breaking terms L soft are generally given by [38,39,44], and only the leptonic terms contribute to our study: Here λ L represents the 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.
Then the three lepton neutralino masses are deduced by diagonalizing the mass matrix M LN by Z N L After symmetry breaking, the mass matrix of neutrinos is deduced in the basis (ν,N c ) [47,48] 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 073103-2 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 sneutrino Mñ with n T =(ν,Ñ c ) where, Then the sneutrino masses can be obtained by the for- ). In the BLMSSM, we deduce the corrections for the couplings which exist in the MSSM due to superfields N c . The corresponding couplings for W-lepton-neutrino, Z-neutrino-neutrino, charged Higgs-lepton-neutrino, Zsneutrino-sneutrino and chargino-lepton-sneutrino are introduced in Ref. [37].
From the interactions of gauge and matter multiplets ig , 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 [15,49,50] 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-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: Feynman diagrams for the Z→l ± i l ∓ j processes in the BLMSSM. F represents Dirac (Majorana) fermions, S represents scalar bosons, and W represents the W boson.
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 the Appendix. x i = m 2 m 2

Np
, with m representing the mass of the corresponding particle, and m Np representing the 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 Then, the branching ratios of Z → l ± i l ∓ j can be summarized as where Γ Z represents the total decay width of the Z-boson and we use Γ Z 2.4952 GeV [20].

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 to the mass matrix of the neutralino, which means the contributions from neutralino-slepton can be influenced by the parameter m 1 . With g L =0.3, S m =1 TeV, AN = −500 GeV, m 2 = 1 TeV, M L f = 1×10 5 GeV 2 and tanβ = 15, we plot the results versus m 1 in Fig. 2. As m 1 > 0, the results decrease with increasing m 1 . However, the results are in the region (3.0×10 −9 ∼3.5×10 −9 ) and the effect of m 1 is small. As a more sensitive parameter, m 2 not only presents in the mass matrix of neutralino, but also in the mass matrix of the chargino. This parameter affects the numerical results through the neutralino-slepton and charginosneutrino contributions. In Fig. 3, we show the effects from m 2 with g L = 0.2, S m = 1 TeV, AN = −500 GeV, tanβ = 15 and M L f = 1×10 5 GeV 2 . We plot the result with m 1 = 500 GeV, 1000 GeV and 1500 GeV by the solid, dotted and dashed lines respectively. The three lines all become small quickly with increasing m 2 . This implies that m 2 is a relatively sensitive parameter to the numerical results. The parameters g L , tanβ L and V Lt are all present in the mass squared matrices of sleptons, sneutrinos and lepton neutralinos. Therefore, these three parameters affect the results through slepton-neutrino, sneutrinoschargino and slepton-lepton neutralino contributions. We choose the parameters m 1 = 500 GeV, m 2 = 1 TeV, S m = 1 TeV, AN = 500 GeV and tanβ =15. As V Lt = 3 TeV, we plot the allowed results with tanβ L versus g L in Fig. 4. Obviously, when the value of g L is large enough, the value of tanβ L approaches 1. When g L 0.3, the parameter tanβ L can vary in the region of 0-2. This implies that g L is a sensitive parameter to the numerical results. For tanβ L = 2, g L versus V Lt is scanned in Fig. 5. We find that the allowed scope of V Lt shrinks and the value of V Lt decreases with increasing g L . Therefore, the value of g L should not be large. Generally, we take 0.05 g L 0.3 and V Lt ∼3 TeV in our numerical calculations.

Z→eτ
In a 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 section, g L can affect the contribution strongly through the masses of sleptons, sneutrinos and lepton neutralinos. S m is the diagonal element of m 2 L and m 2 R in the slepton mass matrix, which can affect slepton-neutralino and slepton-lepton neutralino contributions in the CLFV process. Using the parameters m 1 =500 GeV, m 2 =1 TeV, AN =−500 GeV, tanβ =12 and M L f =1×10 5 GeV 2 , we study the branching ratio versus S m with g L = 0.1(0.15,0.2) in Fig. 6, with the results plotted by the solid line, dotted line and dashed line respectively. These three lines decrease quickly with S m increasing from 1000 GeV to 2500 GeV, which indicates that S m is a very sensitive parameter for the numerical results. When S m >2500 GeV, the results decrease slowly and the branching ratios are around (10 −9 ∼10 −10 ).
We then study the process with the parameters M L f and m 2 . For S m = √ 2 TeV, g L = 0.2, m 1 = 500 GeV, AN =500 GeV, and tanβ =12, we study the results versus M L f with m 2 =1, 1.5, and 2 TeV in Fig. 7, shown by the solid line, dotted line and dashed line respectively.
As M L f = 0, the branching ratio for Z → eτ is almost zero, but the results increase sharply when |M L f | > 0.
We deduce that non-zero M L f is a sensitive parameter and has a strong effect on lepton flavor violation.

Z→µτ
The experimental upper bound for the CLFV process Z → µτ is 1.2×10 −5 , which is about one order of magnitude larger than the process Z → eµ. The parameter AN is present in the sneutrino mass matrix and affects sneutrino-chargino contributions. Supposing m 1 = 500 GeV, m 2 = 1 TeV, g L = 0.2, S m = 1 TeV, M L f =1×10 5 GeV 2 and tanβ =1(2,3), we plot the results with the AN in Fig. 8. For AN 4 TeV, the branching ratios are around 4×10 −9 . For AN >4 TeV, these three lines increase quickly and AN has an obvious influence on the numerical results.  Finally, the effects from the parameter tanβ are studied. tanβ is related to v u and v d , and appears in almost all mass matrices of CLFV processes. With m 1 = 500 GeV, m 2 = 1 TeV, S m = 1 TeV, g L = 0.3, M L f = −1×10 5 GeV 2 and AN = 500 GeV, Fig.9 shows the variation of the branching fraction with the parameter tanβ. It indicates that the results do not change significantly. In the range of tanβ =(0∼3), we find that the branching ratio decreases slightly; for tanβ > 3, the result is stable at around 3.7×10 −9 .

Discussion and conclusions
In this paper, we have studied the CLFV processes Z → l ± i l ∓ j in the BLMSSM. Compared with the MSSM with R-parity conservation, there are new parameters and new contributions to the CLFV processes in the BLMSSM. Firstly, three heavy neutrinos are introduced in this model. However, the new contributions from these particles are tiny, because the couplings of these particles are suppressed by tiny neutrino Yukawa Y ν . Secondly, three new scalar neutrinos are introduced in this model. Considering the mass squared matrix of the sneutrinos in Eq. (10), we find that the contributions from 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 tanβ are important. The sensitive parameters are g L , M L f and S m and they affect the results strongly. We hope that experimental results for Z → l ± i l ∓ j can be detected in the near future.