Muon conversion to an eletron in nuclei in the $B-L$ symmetric SSM

In a few years, the COMET experiment at J-PARC and the Mu2e experiment at Fermilab will probe the $\mu-e$ conversion rate in the vicinity of $\mathcal{O}(10^{-17})$ for an Al target with high experimental sensitivity. Within the framework of the minimal supersymmetric extension of the Standard Model with local $B-L$ gauge symmetry (B-LSSM), we analyze the lepton flavor violating (LFV) process of $\mu-e$ conversion in nuclei. Considering the constraint of the experimental upper limit of the LFV rare decay $\mu\rightarrow e\gamma$, the $\mu-e$ conversion rates in nuclei within the B-LSSM can achieve $\mathcal{O}(10^{-12})$, which is 5 orders of magnitude larger than the future experimental sensitivity at the Mu2e and COMET experiments and may be detected in the near future.


I. INTRODUCTION
to SU(3) C ⊗ SU(2) L ⊗ U(1) Y ⊗ U(1) B−L , where B stands for the baryon number and L for the lepton number. The B-LSSM can provide many more candidates for dark matter compared to the MSSM, for example, new neutralinos corresponding to the gauginos of U(1) B−L , additional Higgs singlets, and sneutrinos [48][49][50][51]. In the B-LSSM, magnetic and electric dipole moments of leptons and quarks have been analyzed [52][53][54].
The present experimental upper limit on the LFV branching ratio of µ → eγ at the MEG experiment is given as [55] Br(µ → eγ) < 4.2 × 10 −13 . (1) The best upper limit on the LFV decays for the branching ratio of µ → eγ can give a large constraint on the parameter space in the B-LSSM, compared to the other LFV decays µ → 3e and h → eµ [1,2]. In this paper, the LFV process µ − e conversion rates in Ti, Au, and Al targets will be analyzed in the B-LSSM, considering the constraint of the present experimental limits on the branching ratio of µ → eγ.
The paper is organized as follows. In Sec. II, we mainly introduce the B-LSSM including its superpotential and the general soft breaking terms. In Sec. III, we give an analytic expression for the µ − e conversion rates in nuclei in the B-LSSM. In Sec. IV, we give the numerical analysis, and the summary is given in Sec. V. Finally, some tedious formulas are collected in the appendixes.
Then, the superpotential in the model can be given by whereĤ T 1 ,Ĥ T 2 ,Q T i , andL T i are SU(2) doublet superfields. Note thatÛ c i ,D c i , andÊ c i represent up-type quarks, down-type quarks, and charged lepton singlet superfields, respectively.
The dimensionless Yukawa coupling parameter Y is a 3×3 matrix. Note that i, j = 1, 2, 3 are the generation indices. The summation convention is implied on repeated indices.
Correspondingly, the soft breaking terms of the B-LSSM are generally given as The SU(2) L ⊗ U(1) Y ⊗ U(1) B−L gauge groups break to U(1) em as the Higgs fields receive vacuum expectation values (VEVs), Here, In addition, it is important to consider gauge kinetic mixing, and here we give its covariant derivatives of the form: Note that G is the gauge coupling matrix given as follows: As long as the two Abelian gauge groups are unbroken, one can have the freedom to perform a change of basis by suitable rotation, and R is the proper way to do it: Here g 1 corresponds to the measured hypercharge coupling, which is modified in the B-LSSM and given together with g B and g Y B [56]. Next, one can redefine the U(1) gauge fields through An immediate interesting consequence of the gauge kinetic mixing arises in various sectors of the model as discussed in the subsequent analysis. First, the A BL boson mixes at the tree level with the A Y and V 3 bosons. In the basis (A Y , V 3 , A BL ), the corresponding mass matrix reads This mass matrix can be diagonalized by a unitary mixing matrix, which can be expressed by two mixing angles θ W and θ ′ W as Then sin 2 θ ′ W can be written as where x = v u . The exact eigenvalues of Eq.(10) are given by In this section, we analyze the µ − e conversion processes at the quark level in the B-LSSM. We give the effective Lagrangian for the µ − e conversion in nuclei in the following.
Both penguin-type diagrams in Fig. 1 and box-type diagrams in Fig. 2 where P L = (1 − γ 5 )/2, P R = (1 + γ 5 )/2, Q u em = 2/3, Q d em = −1/3, and m µ is the muon mass. The coefficients A i are where x i = m 2 i /m 2 W , I i is the loop function, and C is the coupling which can be found in the appendixes.
The effective Lagrangian of the Z-penguin-type diagrams is generally written as where The contributions to the coefficients B L,R are Here G i is the loop function which can be found in the appendixes.
The effective Lagrangian of the box-type diagrams shown in Fig. 2 is generally written with represent the contributions from neutral fermion χ 0 η,σ , charged scalar S c m and squarkq I (q = u, d Using the expression for the effective Lagrangian of the µ − e conversion processes at the quark level, one can calculate the µ − e conversion rate in a nucleus [57]: where Z is the number of protons in the nucleus and N is the number of neutrons in the nucleus. Note that Z eff is an effective atomic charge [58,59], F (q 2 ) is the nuclear form factor, and Γ capt is the total muon capture rate. In the following numerical analysis, we consider the µ − e conversion rate in 48 22 Ti, 197 79 Au and 27 13 Al nuclei, where the values of Z eff , F (q 2 ≃ −m 2 µ ), and Γ (capt) for the different nuclei can be seen in Table. I and follow Ref. [60].   [62,63] give an upper bound on the ratio between the Z ′ mass and its gauge coupling at 99% C.L. as M Z ′ /g B ≥ 6 TeV, and then the scope of g B is 0 < g B < 0.7. LHC experimental data constrain tan β ′ < 1.5 [39]. Considering the constraint and T x =diag(1, 1, 1) TeV, respectively.
We need to consider the constraint of the SM-like Higgs boson mass. The remaining key parameters that affect the Higgs boson mass are tan β, tan β ′ , g B , and g Y B . By constantly adjusting the parameters, the final numerical analysis strictly conforms to the constraint of the SM-like Higgs boson measured mass m h =125.09 ± 0.24 GeV in 3σ [64]. In addition, it should be noted that although the B-LSSM can produce nonzero neutrinos, the mass of the neutrinos is too small to affect the problem we study, so we approximately consider the mass of neutrinos to be zero. Although the B-LSSM contains LFV sources in the neutrino Yukawa sector, such as the Y ν matrix, the neutrino oscillation causes Y ν ∼ O(10 −6 ), which contributes very little to the problem we study; thus we approximately ignore the influence of the neutrino Yukawa sector in the numerical analysis.
Since we are studying the lepton flavor violating processes, we have to consider the offdiagonal terms for the soft breaking slepton mass matrices m 2 L,ē and the trilinear coupling matrix T e , which are defined by [65][66][67][68][69][70] We know that LFV processes are flavor dependent, just as the LFV rate for µ − e transitions depends on the slepton mixing parameters δ XX 12 (X = L, R); thus we only need to consider the effect of slepton mixing parameters δ XX 12 (X = L, R) on the µ − e conversion rate. The other slepton mixing parameters δ XX consistent with the present limit of Br(µ → eγ).
In Fig. 3, we plot the µ − e conversion rate in the different nuclei and Br(µ → eγ) versus δ LL 12 for δ RR 12 = δ LR 12 = 0. It is obvious that LFV rates increase with the increase of the slepton flavor mixing parameter δ LL 12 because the LFV processes are flavor dependent, and the LFV rate for µ − e transitions depends on the slepton mixing parameters δ XX 12 (X = L, R). It can be seen from the figure that Br(µ → eγ) can reach the experimental upper limit, but the µ − e conversion rate in the nuclei cannot. When we consider the constraint of Br(µ → eγ) The general trend shown in the four graphs is that as δ RR 12 continues to increase, CR(µ → e : Ti), CR(µ → e : Au), CR(µ → e : Al) and Br(µ → eγ) also increase. As can be seen from  sensitivity of future experiments under the limit of Br(µ → eγ).
Because the LFV processes are flavor dependent, δ LR 12 also has a greater influence on the µ − e conversion rate in the different nuclei and Br(µ → eγ). As δ LR 12 increases, CR(µ → e : Ti), CR(µ → e : Au), CR(µ → e : Al) and Br(µ → eγ) also increase. As seen from In this section, we study the influence of other basic parameters on the µ − e conversion rate and Br(µ → eγ). We first set appropriate numerical values for slepton flavor mixing parameters, such as δ LL 12 = 0.01, δ RR 12 = 0.006, and δ LR 12 = 1 × 10 −4 . We also keep neutral fermion masses m χ o η > 200 GeV (η = 1, · · · , 7), the scalar masses m S c m,n > 500 GeV (m, n = 1, · · · , 6) and the SM-like Higgs boson mass m h =125.09 ± 0.24 GeV in 3σ to avoid the range ruled out by the experiments. Then we research the influence of the basic parameters m L = m E ≡ M E , tan β ′ , and g Y B on the µ − e conversion rate and Br(µ → eγ), respectively. and Br(µ → eγ) decrease with the increase of g Y B . When g Y B is small, Br(µ → eγ) can reach the experimental upper limit, but CR(µ → e : Ti) and CR(µ → e : Au) cannot.
Note that g Y B affects the numerical results through the new mass matrix of sleptons, Higgs bosons, and neutralinos, which can make contributions to these LFV processes.  C. Scanning diagram of the effect of slepton mixing parameters δ XX 12 (X = L, R) on the µ − e conversion rate In the above subsections, we show only the effect of parameters on the µ−e conversion rate and Br(µ → eγ). In this subsection, we scan the parameter space shown in Table II, in order to clearly see the constraints of Br(µ → eγ) on the µ − e conversion rate at more parameter space. Under the condition that the SM-like Higgs boson mass m h =125.09 ± 0.24 GeV in 3σ, neutral fermion masses m χ o η > 200 GeV (η = 1, · · · , 7) and the scalar masses m S c m,n > 500 GeV (m, n = 1, · · · , 6) are satisfied. By randomly scanning 20,000 points, we obtain the relation of the µ − e conversion rate in the different nuclei and Br(µ → eγ) versus δ XX 12 (X = L, R), respectively. When the variable is δ XX 12 (X = L, R) in the figures, the other two are δ XX 12 (X = L, R) = 0. In Fig. 9, we plot CR(µ → e : Ti), CR(µ → e : Au), CR(µ → e : Al), and Br(µ → eγ) versus the slepton flavor mixing parameter δ LL 12 , after randomly scanning Table II. By observing Fig. 9, we can find that the constraint of Br(µ → eγ) to the µ − e conversion rate is relatively large. Looking at Fig. 9 (a), we find that CR(µ → e : Ti) can go to O (10 −12 ) under the constraint of the upper limit of Br(µ → eγ). Although CR(µ → e : Ti) does not exceed the upper limit of the current experiment, it obviously exceeds the sensitivity of future experiments. In Fig. 9  constraint of Br(µ → eγ). In Fig. 9 (c), CR(µ → e : Al) can exceed its sensitivity to future experiments under the constraint of Br(µ → eγ) and can also reach O(10 −12 ). It is likely that the µ − e conversion rate in 48 22 Ti and 27 13 Al will be detected in the near future with increasing experimental accuracy.
We also plot CR(µ → e : Ti), CR(µ → e : Au), CR(µ → e : Al), and Br(µ → eγ) versus the slepton flavor mixing parameter δ RR 12 in Fig. 10. By observing the black spots that conform to the constraint of Br(µ → eγ) in Fig. 10, we find that CR(µ → e : Ti) and CR(µ → e : Au) can achieve O(10 −12 ) which can reach the upper limit of current experiments. In addition CR(µ → e : Al) can also attain O(10 −12 ), which is 5 orders of magnitude larger than the future experimental sensitivity at the Mu2e and COMET experiments.
In Fig. 11, the µ − e conversion rate and Br(µ → eγ) versus the slepton flavor mixing versus δ RR 12 after randomly scanning Table II. versus δ LR 12 after randomly scanning Table II. parameter δ LR 12 are plotted. The numerical results show that the upper limit of Br(µ → eγ) is very strict on the µ − e conversion rate in the different nuclei through the slepton flavor mixing parameter δ LR 12 . By observing black dots in Fig. 11, it can be found that under the constraint of Br(µ → eγ), the µ−e conversion rate in nuclei can be about O(10 −19 ), which is under the sensitivity of future experiments. This means that the effect of the slepton mixing parameter δ LR 12 on the µ − e conversion rate in the different nuclei is small, constrained by the upper limit of Br(µ → eγ).

V. SUMMARY
In this work, we have studied the lepton flavor violating process of µ − e conversion in nuclei within the framework of the B-LSSM. The numerical results show that the µ − e conversion rate in nuclei depends on the slepton flavor mixing parameters δ XX 12 (X = L, R) because the lepton flavor violating processes are flavor dependent. Under the constraint of the experimental upper limit on the LFV branching ratio of µ → eγ, the µ − e conversion rate in 48 22 Ti and 197 79 Au nuclei can attain O(10 −12 ), which can reach the experimental upper limits. The µ − e conversion rate in 27 13 Al nuclei can also reach O(10 −12 ), which is 5 orders of magnitude larger than the future experimental sensitivity at the Mu2e and COMET experiments.  48 22 Ti and 27 13 Al nuclei can easily exceed the future experimental sensitivities and may be detected in the near future. . (A6)