Muon conversion to electron in nuclei within the BLMSSM

In a supersymmetric extension of the standard model with local gauged baryon and lepton numbers (BLMSSM), there are new sources for lepton flavor violation, because the right-handed neutrinos, new gauginos and Higgs are introduced. We investigate muon conversion to electron in nuclei within the BLMSSM in detail. The numerical results indicate that the μ→e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow e $$\end{document} conversion rates in nuclei within the BLMSSM can reach the experimental upper bound, which may be detected in the future experiments.


Introduction
The observations of neutrino oscillations [1][2][3] imply that neutrinos have tiny masses and are mixed [4][5][6][7], which have demonstrated that lepton flavor in neutrino sector is not conserved. Nevertheless, in the Standard Model (SM) with three tiny massive neutrinos, the expected rates for the charged lepton flavor violating (LFV) processes are very tiny [8][9][10][11]. Thus, Lepton-flavor violation is a window of new physics beyond the SM. Among the various candidates for new physics that produce potentially observable effects in LFV processes, one of the most appealing model is supersymmetric (SUSY) extension of the SM. Here, we can use the neutrino oscillation experimental data to restrain the input parameters in the new models. A neutral Higgs with mass m h 0 = 125.1 GeV reported by ATLAS [12] and CMS [13,14] gives a strict constraint on relevant parameter space of the model. The present sensitivities of the μ − e conversion rates in different nuclei [15][16][17] are collected here, CR(μ → e : Au) < 7 × 10 −13 , CR(μ → e : Ti) < 4.3 × 10 −12 , CR(μ → e : Pb) < 4.6 × 10 −11 . (1) These processes have close relation with l j → l i γ . In the work [18], the μ → e conversion was studied in μνSSM.
For the models beyond SM, one can violate R parity with the non-conservation of baryon number (B) or lepton number (L) [19][20][21][22]. A minimal supersymmetric extension of the SM with local gauged B and L(BLMSSM) was first proposed by the author [23,24]. The local gauged B is used to explain the matter-antimatter asymmetry in the universe. Right-handed neutrinos in BLMSSM lead to three tiny neutrino masses through the See-Saw mechanism and can account for the neutrino oscillation experiments. So lepton number (L) is expected to be broken spontaneously around TeV scale. In BLMSSM, the lightest CP-even Higgs mass and the decays h 0 → γ γ , h 0 → Z Z(W W ) were studied in the work [25]. The neutron and lepton electric dipole moments(EDMs) were researched in the CP-violating BLMSSM [26,27]. In BLMSSM, there are also other works [28][29][30]. In this work, we analyze the processes on muon conversion to electron in nuclei within the BLMSSM. Compared with MSSM, there are new sources to enlarge the processes via loop contributions. The new scores are produced from: (1) the coupling of new neutralino(lepton neutralino)-slepton-lepton; (2) the right-handed neutrinos mixing with left-handed neutrinos; (3) the sneutrino sector is extended, whose mass squared matrix is 6 × 6. In some parameter space of BLMSSM, large corrections to the processes are obtained, and they can easily exceed their experiment upper bounds. Therefore, to enhance the processes on muon conversion to electron in nuclei is possible, and they may be measured in the near future.
After this introduction, we briefly summarize the main ingredients of the BLMSSM, and show the needed mass matrices and couplings in Sect. 2. In Sect. 3, the processes μ → e + qq are studied in the BLMSSM. The input parameters and numerical analysis are shown in Sect. 4, and our conclusion is given in Sect. 5. Some functions are collected in the Appendix.

BLMSSM
BLMSSM is the supersymmetric extension of the SM with local gauged B and L, whose local gauge group is SU [19,20]. The exotic leptonsL 4 ,Ê c 4 ,N c 4 ,L c 5 ,Ê 5 andN 5 are introduced to cancel L anomaly. As well as, the exotic quarksQ 4 ,Û c 4 ,D c 4 , Q c 5 ,Û 5 andD 5 are introduced to cancel B anomaly. To break lepton number and baryon number spontaneously, the Higgs superfieldsˆ L ,φ L andˆ B ,φ B are introduced, respectively. The exotic quarks obtain masses from nonzero vacuum expectation values (VEVs) ofˆ B andφ B . While, exotic leptons get masses from VEVs of H u and H d . H u andφ L give masses to light neutrinos through See-Saw mechanism. In the BLMSSM, the superfieldsX andX are introduced to make the heavy exotic quarks unstable. The above mentioned exotic lepton, quark and Higgs superfields are shown in Table 1.
The superpotential of BLMSSM is [25] where W M SSM is the superpotential of the MSSM. To save space in the text, the soft breaking terms L so f t [19,25] of the BLMSSM is not shown here.
In this model, we introduce the superfieldsN c to produce tiny masses of three light neutrinos. The mass matrix of neutrinos in the basis (ψ ν I L , ψ N cI R ) is expressed as Here, χ 0 N α represent the mass eigenstates of neutrino fields mixed by left-handed and right-handed neutrinos.
The new gaugino λ L mixes with the superpartners of the SU (2) L singlets L and ϕ L , then they produce three lepton neutralinos One can use Z L to diagonalize the mass matrix in Eq. (4) and obtain three lepton neutralino masses in the end. From Eq. (2) and the soft breaking terms L so f t [19,25] of the BLMSSM, the mass squared matrix of slepton gets corrections and reads as The unitary matrix ZL is used to rotate slepton mass squared matrix to mass eigenstates. Because of the introduction of right handed neutrinos, in BLMSSM the mass squared matrix of sneutrino is 6 × 6. In the baseñ T = (ν,Ñ c ), the concrete forms for the sneutrino mass squared matrix Mñ are shown here The superfieldsÑ c in BLMSSM lead to the corrections for some couplings existed in MSSM. We give out the corrected couplings such as: W-lepton-neutrino and Z-neutrinoneutrino where P L = 1−γ 5 2 and P R = 1+γ 5 2 . We use the abbreviation s W = sin θ W , c W = cos θ W , and θ W is the Weinberg angle.
Some other adapted couplings are collected here: charginolepton-sneutrino, Z-sneutrino-sneutrino and charged Higgslepton-neutrino In BLMSSM, there are new couplings that are deduced from the interactions of gauge and matter multiplets ig After calculation, the leptonslepton-lepton neutralino couplings are obtained

µ → e + qq in the BLMSSM
In this section, the LFV processes μ → e + qq are studied in the BLMSSM. Both penguin-type diagrams and box-type diagrams have contributions to the effective Lagrangian. For convenience, the penguin and box diagrams are analyzed in the generic form, which can simplify the work.

The penguin diagrams
When the external leptons are all on shell, we can generally obtain the γ -penguin contributions in the following form The relevant Feynman diagrams are shown in Fig. 1. The final Wilson coefficients C L 1 , C R 1 , C L 2 and C R 2 are obtained from the sum of these diagrams' amplitudes. The contributions from the virtual neutral fermion diagram in the top-left of Fig. 1 are denoted by C L ,R α (n), α = 1, 2. We give out the deduced results in the following form, with x = m 2 /m 2 W and m representing the mass for the corresponding particle.
The diagram in top-right of Fig. 1 represents the virtual charged Fermion diagram and its contribution is On account of the mixing of three light neutrinos and three heavy neutrinos, the virtual W diagrams in the bot-tom of Fig. 1 give corrections to the charged LFV process μ → e + qq. We show the coefficients C L ,R α (W )(α = 1, 2) The sum of the total coefficients in Eqs. (14)- (16) are The contributions from Z -penguin diagrams are depicted by the Fig. 1, similar as γ -penguin diagrams, The concrete forms of the effective couplings N L (S), N R (S), N L (W ) and N R (W ) read as The concrete expressions for the functions G i (i = 1, ..., 7) are collected are in appendix.

The box-type diagrams
The box-type diagrams drawn in Fig. 2 can be written as B q (n) represent the contributions from the virtual neutral Fermion diagrams in the first line of Fig. 2.
The virtual charged Fermion in the middle line of Fig. 2  give contributions denoted by B q (c).
The virtual W produces corrections through the diagrams in the last line of Fig. 2

μ − e conversion rate
Once we know the effective Lagrangian relevant to this process at the quark level, we can calculate the conversion rate Fig. 2 The box diagrams for μ → e + qq, with F representing Dirac(Majorana) particles with Z and N representing the proton and neutron numbers in a nucleus. Z e f f is an effective atomic charge determined in refs [31,32]. F q 2 is the nuclear form factor and capt denotes the total muon capture rate, while α is the fine structure constant.

Numerical results
In this section, we discuss the numerical results, and consider the experimental constraints from the lightest neutral CPeven Higgs mass m h 0 125.1 GeV [12-14] and the neutrino experiment data. In this model, the LFV processes l j → l i γ and l j → 3l i are studied in our previous work [30], and their constraints are also taken into account. In this work, we use the parameters [28,29] The Yukawa couplings of neutrinos (Y ν ) I J , (I, J = 1, 2, 3) are at the order of 10 −8 − 10 −6 , whose effects to this processes are tiny and can be ignored savely. To simplify the numerical discussion, we use the following relations for i, j = 1, 2, 3 and i = j.
If we do not specially declare, the non-diagonal elements of the used parameters should be zero.

μ → e conversion rate in nuclei Au
The experimental upper bound for the μ → e conversion rate in nuclei Au is around 7.0 × 10 −13 . The parameters m 2 = 1000GeV, M L f = 10 4 GeV 2 are supposed in the calculation of this process. 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 . For S 2 m = 13 TeV 2 , tanβ = 5.0, tan β L = 2.0, and g L = 1 6 , we plot the results versus m 1 with V Lt = 3000 GeV and 6000 GeV in Fig. 3. We can see that the results decrease quickly with the increase of m 1 . As V Lt = 6000 GeV, the results are slightly smaller than the corresponding results with V Lt = 3000 GeV. This implies that m 1 is a sensitive parameter and has a strong effect on muon conversion to electron in nuclei. Compared with m 1 , the effect from V Lt is very small. tanβ is related to v u and v d , and appears in almost all mass matrices of particles contributing to the μ → e processes. With m 1 = 500 GeV, V Lt = 3000 GeV, tan β L = 2.0, and g L = 1 6 , Fig. 4 shows the variation of the μ → e conversion rate in nuclei Au with the parameter tanβ and S 2 m . It indicates that the results change significantly with tanβ. When tanβ is in the region (0 ∼ 6), the results decrease significantly, but in the range of tanβ > 6, we find that the results increase sharply. Only when the value of tanβ is about 6, the results of μ → e conversion rate in nuclei Au are close and not higher than the experimental upper bound.
The parameters g L , tan β L and V Lt all present in the mass squared matrices of sleptons, sneutrinos and lepton neutralinos. Therefore, these three parameters affect the results through slepton-neutrino, sneutrinos-chargino and sleptonlepton neutralino contributions.
As m 1 = 1000 GeV, tan β = 5.5, tan β L = 2, S 2 m = 16 TeV 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 too large. Generally, we take 0.05 ≤ g L ≤ 0.3 and V Lt ∼ 3 TeV in our numerical calculations.
As the parameters m 1 = 1000 GeV, tanβ = 6.0, S 2 m = 12 TeV 2 and V Lt = 3000 GeV, we plot the allowed results with tan β L versus g L in Fig. 6. When g L < 0.43, the parameter tanβ L can vary in the region of (0 ∼ 4). It implies that g L is a sensitive parameter to the numerical results and the value of g L should not be larger than 0.43.

μ → e conversion rate in nuclei Ti
In a similar way, the μ → e conversion rate in nuclei Ti is numerically studied and its experimental upper bound is around 4.3×10 −12 . In this subsection, we use the parameters tanβ = 2.0, tan β L = 2.0, g L = 1 6 and V Lt = 3000 GeV. The parameter m 2 presents in the mass matrixes of neutralino and chargino. This parameter affects the numerical results through the neutralino-slepton and chargino-sneutrino con-  Fig. 8. As M L f = 0, the conversion ratio for μ → e is almost zero, but the results increase sharply with M L f > 0. We deduce that non-zero M L f is a sensitive parameter and has a strong effect on muon conversion to electron in nuclei.

μ → e conversion rate in nuclei Pb
The experimental upper bound of μ → e conversion rate in nuclei Pb is around 4.6 × 10 −11 . In this subsection, we use the parameters m 2 = 1000 GeV, tan β L = 2.0, g L = 1 6 and M L f = 10 4 GeV 2 . S m are the diagonal elements of mL 2 and mR 2 in the slepton mass matrix, which can affect sleptonneutralino and slepton-lepton neutralino contributions in the μ → e process.
With m 1 = 1000 GeV, V Lt = 3000 GeV, we plot the conversion ratio for μ → e in nuclei Pb versus S m with tanβ = 2.0 (solid line) and tanβ = 3.0 (dotted line) in Fig. 9. These two lines decrease quickly with S m enlarging from 1400 to 3000 GeV, which indicates that S m is a very sensitive parameter to the numerical results. When S m > 3000 GeV, the results decrease slowly and the conversion ratios are around (10 −12 − 10 −13 ).
We focus on V L t which is a special parameter in BLMSSM, and with m 1 = 500 GeV, tanβ = 13, we plot the conversion ratio for μ → e in nuclei Pb versus V L t with S 2 m = 5 TeV 2 (solid line) and S 2 m = 6 TeV 2 (dotted line) in Fig. 10. Overall, the results of dotted line are about 0.5 × 10 −11 ∼ 1.2 × 10 −11 larger than the solid line. In the range of V L t = (0 ∼ 10000 GeV), the two lines decrease quickly with the enlarging V L t . We can see S m and V L t are sensitive parameters to the numerical results.

Discussion and conclusion
In the framework of the BLMSSM model, we study the LFV processes μ → e + qq. In the processes, we consider some new parameters and contributions, such as the newly introduced parameters g L , tan β L and V L t . Combined with the numerical results discussed in the Sect. 4, different parameters have different effects on the processes. The parameter g L presents in the mass squared matrices of sleptons, sneutrinos and lepton neutralinos. Numerical analysis shows that g L has obvious influence on the results, the value of g L should not be too large. As sensitive parameters, S m and M L f are respectively diagonal and non-diagonal elements of matrixes for mL and mR. Both S m and M L f have significant impacts on the results. tanβ is related to v u and v d , and appears in almost all mass matrices of particles contributing to the μ → e processes. The value of tanβ is critical to these processes. With the improvement of experimental accuracy, we believe that there will be some discoveries for μ to e conversion in the near future. (35) ,