Lepton-flavor violation and two loop electroweak corrections to $(g-2)_\mu$ in the B-L symmetry SSM

Charged lepton flavor violating processes are forbidden in the standard model (SM), hence the observation of charged lepton flavor transitions would represent a clear signal of new physics beyond the standard Model. In this work, we investigate some lepton flavor violating processes in the minimal supersymmetric extension of the SM with local $B-L$ gauge symmetry (B-LSSM). And including the corrections from some two loop diagrams to the anomalous dipole moments (MDM) of muon, we discuss the corresponding constraint on the relevant parameter space of the model. Considering the constraints from updated experimental data, the numerical results show that, new contributions in the B-LSSM enhance the MSSM predictions on the rates of $l_j-l_i$ transitions about one order of magnitude, and also enhance the MSSM prediction on the muon MDM. In addition, two loop electroweak corrections can make important contributions to the muon MDM in the B-LSSM.


I. INTRODUCTION
Lepton-flavor-violation (LFV), if observed in the future experiments, is obvious evidence of new physics beyond the standard model (SM), because the lepton-flavor number is conserved in the SM. A detailed analysis of the LFV processes will reveal some properties of high-energy physics, because the processes do not suffer from a large ambiguity due to hadronic matrix elements. In Table I, we show the present experimental limits and future sensitivities for the LFV processes l − j → l − i γ, l − j → l − i l − i l + i [1][2][3][4][5][6][7]. Several predictions for these LFV processes have obtained in the framework of various SM extensions [8][9][10][11][12][13][14]. In this work, we analyze these LFV processes in the minimal supersymmetric extension of the SM with local B − L gauge symmetry (B − L SSM). In addition, it is well known that the magnetic dipole moment (MDM) of muon has close relation with the new physics (NP) beyond the SM, and researching the muon MDM is an effective way to find NP beyond the SM. Hence including some two loop diagrams, we also analyze the muon MDM in the B − L SSM.
The B − L SSM [15][16][17][18] is based on the gauge symmetry group SU(3) C ⊗ SU(2) L ⊗ U(1) Y ⊗ U(1) B−L , where B stands for the baryon number and L stands for the lepton number respectively. Besides accounting elegantly for the existence and smallness of the left-handed neutrino masses, the B − L SSM also alleviates the little hierarchy problem of the MSSM [19], because the exotic singlet Higgs and right-handed (s)neutrinos [20][21][22][23][24][25][26] release additional parameter space from the LHC constraints. The invariance under U(1) B−L gauge group imposes the R-parity conservation which is assumed in the MSSM to avoid proton decay. And R-parity conservation can be maintained if U(1) B−L symmetry is broken spontaneously [27]. Furthermore, the model can provide much more candidates for the dark matter comparing that with the MSSM [28][29][30][31].
The paper is organized as follows. In Sec.II, the main ingredients of the B − L SSM are summarized briefly by introducing the superpotential and the general soft breaking terms.
We present the analysis on the decay width of the rare LFV processes and the muon MDM in Sec.III. The numerical analyses are given in Sec.IV, and Sec.V gives a summary. The
tedious formulae are collected in Appendices.

II. THE B − L SSM
In literatures there are several popular versions of B − L SSM. Here we adopt the version described in Refs. [32][33][34][35] to proceed our analysis, which allows for a spontaneously broken U(1) B−L without necessarily breaking R-parity. This requires the addition of two chiral singlet superfieldsη 1 ∼ (1, 1, 0, −1),η 2 ∼ (1, 1, 0, 1), as well as three generations of right-handed neutrinos. In addition, this version of B −L SSM is encoded in SARAH [36][37][38][39][40] which is used to create the mass matrices and interaction vertexes in the model. Meanwhile, quantum numbers of the matter chiral superfields for quarks and leptons are given bŷ with i = 1, 2, 3 denoting the index of generation. In addition, the quantum numbers of two Higgs doublets are assigned aŝ The corresponding superpotential of the B − L SSM is written as where i, j are generation indices. Correspondingly, the soft breaking terms of the B − L SSM are generally given as with λ B , λ B ′ denoting the gaugino of U(1) Y and U(1) (B−L) respectively. The local gauge symmetry SU(2) L ⊗U(1) Y ⊗U(1) B−L breaks down to the electromagnetic symmetry U(1) em as the Higgs fields receive vacuum expectation values: For convenience, we define u 2 = u 2 1 + u 2 2 , v 2 = v 2 1 + v 2 2 and tan β ′ = u 2 u 1 in analogy to the ratio of the MSSM VEVs (tan β = v 2 v 1 ). The presence of two Abelian groups gives rise to a new effect absent in the MSSM or other SUSY models with just one Abelian gauge group: the gauge kinetic mixing. This mixing couples the B − L sector to the MSSM sector, and even if it is set to zero at M GU T , it can be induced through RGEs [41][42][43][44][45][46][47]. In practice, it turns out that it is easier to work with non-canonical covariant derivatives instead of off-diagonal field-strength tensors. However, both approaches are equivalent [48]. Hence in the following, we consider covariant derivatives of the form where A ′Y µ , A ′µ,BL denote the gauge fields associated with the two U(1) gauge groups, Y, B−L corresponding to the hypercharge and B − L charge respectively. As long as the two Abelian gauge groups are unbroken, we still have the freedom to perform a change of the basis.
Choosing R in a proper form, one can write the coupling matrix as where g 1 corresponds to the measured hypercharge coupling which is modified in B − L SSM as given along with g B and g Y B in Refs. [49]. Then, we can redefine the U(1) gauge fields Immediate interesting consequence of the gauge kinetic mixing arise in various sectors of the model as discussed in the subsequent analysis. First, 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 . Compared with the MSSM, this Z − Z ′ mixing makes new contributions to the l j → 3l i decay channel, and the new mixing angle θ W ′ appears in the couplings involve Z boson. The exact eigenvalues of Eq.(9) are given by Then the gauge kinetic mixing leads to the mixing between the H 1 1 , H 2 2 ,η 1 ,η 2 at the tree level. In the basis (ReH 1 1 , ReH 2 2 , Reη 1 , Reη 2 ), the tree level mass squared matrix for scalar Higgs bosons is given by where g 2 = g 2 1 + g 2 2 + g 2 Y B , T = 1 + tan 2 β 1 + tan 2 β ′ , n 2 = ReBµ u 2 and N 2 = ReBµ ′ u 2 , respectively. Compared the MSSM, this new mixing in the B − L SSM can affect the following analysis.
Including the leading-log radiative corrections from stop and top particles [50][51][52], the mass of the SM-like Higgs boson can be written as where m 0 h 1 denotes the lightest tree-level Higgs boson mass, and here α 3 is the strong coupling constant, M S = √ mt 1 mt 2 with mt 1,2 denoting the stop masses, 33 being the trilinear Higgs stop coupling and µ denoting the Higgsino mass parameter.
Meanwhile, additional D-terms contribute to the mass matrices of the squarks and sleptons, and sleptons also affect the subsequent analysis. On the basis (L,ẽ c ), the mass matrix of sleptons can be written as It can be noted that tan β ′ and new gauge coupling constants can affect the mass matrix of sleptons.
In this section, we present the analysis on the decay width of the rare LFV processes In addition, considering the corrections from some two loop diagrams, we analyze the NP contributions to the muon MDM, ∆a N P µ .
in the limit q → 0 with q being photon momentum. In addition, p is the momentum of the particle l j , ǫ is the photon polarization vector, u i (and ν i in the expression below) is the wave function for lepton (antilepton), and P L = (1 − γ 5 )/2, P R = (1 + γ 5 )/2. Then, the Feynman diagrams contributing to the above amplitude are depicted by Fig. 1. Calculating the Feynman diagrams, the coefficients A L,R 1,2 in Eq. (18) can be written as where the concrete expressions for A Using the amplitude Eq.(18), the decay width for l − j → l − i γ can be obtained easily as And the branching ratio for where Γ l − j is the total decay width of the lepton l − j . In the numerical calculation, we use Γ µ ≈ 2.996 × 10 −19 GeV for the muon and Γ τ ≈ 2.265 × 10 −12 GeV for the tauon [53].
For the process l − j → l − i l − i l + i , the effective amplitude includes the contributions from penguin-type and box-type diagrams. Using Eq.(18), the contributions from γ-penguin l j (p) diagrams can be written as [54] As shown in Fig.2, the contribution from N-penguin (here N represents Z and Z ′ bosons) diagrams can be written as [54] T where m N denotes the mass for Z or Z ′ boson, and the concrete expressions for F L,R are given in Appendix B. There is also dipole term for Z and Z ′ contributions, but we neglect it in the calculation. Because γ exchanges in Fig. 2 represent the electromagnetic interaction, the corresponding gauge symmetry is not broken, while Z or Z ′ exchanges denote the weak interaction, the corresponding gauge symmetry is broken. And q is the sum of the momentum of two outward on-shell lepton, which is about the same order of magnitude as that of m e or m µ . So we cannot neglect it for γ contributions in Eq. (22), because photon is massless. But q is negligible compared with m N , hence we neglect it for Z or Z ′ contributions in Eq.(23).
In addition, box-type diagrams can also contribute to the process l − j → l − i l − i l + i . The corresponding Feynman diagrams are drawn in Fig.3. Then the effective Hamiltonian for the box-type diagrams can be written as represents the contributions from neutral fermions F n k,α and charge scalars loops S c l,β , and (b) represents the contributions from charged fermions F c k,α and neutral scalars or pseudoscalars S n l,β loops.
where the coefficients B L,R 1,2,3,4 originate from those box diagrams in Fig.3, and the concrete expressions can be found in Appendix B. Then the decay width for l where Finally, we analyze the muon MDM. The difference between experiment and the SM prediction on a µ is [55,56] ∆a µ = a exp µ − a SM µ = (24.8 ± 7.9) × 10 −10 , where all errors combining in the quadrature. Several predictions for the muon MDM have been discussed in the framework of various SM extensions [57][58][59][60][61][62][63][64][65][66][67]. The muon MDM can actually be expressed as the operators Here, σ αβ = i[γ α , γ β ]/2, l µ represents the wave function for muon, m µ is the muon mass, a µ = 1 2 (g − 2) µ and F αβ is the electromagnetic field strength. Adopting the effective Lagrangian approach, we can get [64][65][66] where Q f = −1, and C L,R 2,6 represent the Wilson coefficients of the corresponding operators where D α = ∂ α + ieA α . Then, through the calculation of Fig. 1, the one loop contributions to the muon MDM can be written as where a (a,b) µ are the contributions to muon MDM corresponding to Fig. 1(a), (b) respectively, and the concrete expressions of them are given in Appendix C.
In addition, the two loop Barr-Zee type diagrams can give important contributions to the muon MDM. According to the decoupling theorem, the contributions from the two loop diagrams with a closed slepton loop are suppressed by heavy slepton mass, we neglect these diagrams in the following calculation. Then we consider the contributions from the two loop diagrams in which a closed fermion loop is attached to the virtual gauge bosons or Higgs fields. According to Ref. [67], the main two loop diagrams contributing to the muon MDM are shown in Fig.4. Then, including the two loop corrections, the muon MDM are given by where a W W µ , a W H µ , a γh µ are the contributions corresponding to Fig.4(a), (b), (c) respectively, and the concrete expressions can be found in Appendix C. When we consider the PMNS mixing of neutrinos [68,69], Fig.4 (a) and (b) can also make contributions to the LFV processes. Hence, in the following numerical analysis, we also consider the contributions from the two loop Barr-Zee diagrams to the LFV processes.
which constrains the corresponding parameter space strictly. In our previous work [70], we Confidence Level (CL). Due to the contributions of heavy Z ′ boson are highly suppressed, we choose M Z ′ = 4.2TeV in our following numerical analysis. And Refs. [72,73] give us an upper bound on the ratio between the Z ′ mass and its gauge coupling at 99% CL as then the scope of g B is limited to 0 < g B ≤ 0.7. Additionally the LHC experimental data also constrain tan β ′ < 1.5 [34]. Considering the constraints from the experiments [53], for those parameters in Higgsino and gaugino sectors, we appropriately fix M 1 = 500GeV, M 2 = 600GeV, µ = 700GeV, µ ′ = 800GeV, M BB ′ = 500GeV, M BL = 600GeV, for simplify.
For those parameters in the soft breaking terms, we take B ′ µ = 5 × 10 5 GeV 2 , mq = mũ = diag(2, 2, 1.6)TeV, T u = diag(1, 1, 1)TeV, to coincide with the constraints from the direct searches of squarks at the LHC [74,75] and the discussion about the observed Higgs signal in Ref. [76]. Considering the experiment observation onB → X s γ and B 0 s → µ + µ − [77], we take m H ± = 1.5TeV. All of the fixed parameters above do not affect the following numerical results obviously. Furthermore, in order to simplify our numerical analyses, we take soft breaking slepton mass matrices mL ,ẽ = diag(m E , m E , m E )TeV. But for the trilinear coupling matrix T e , we will introduce the slepton flavor mixing, which take into account the off-diagonal terms as
However, the high future experimental sensitivities still keep a hope to detect τ → 3e and τ → 3µ. The two loop contributions are not obvious in Fig.5, because when m E = 1TeV, the one loop results make the dominant contributions to the LFV processes, and the two Br(µ → 3e), Br(τ → eγ), Br(τ → 3e), Br(τ → µγ) and Br(τ → 3µ) versus m E for tan β = 10 (solid lines), tan β = 30 (dotdashed lines) in Fig.6 (a-f) branching ratios for these processes decrease with the increasing of m E or tan β, which indicates that heavy sleptons and large tan β play a suppressive role to the rates of LFV processes, and all of them can't reach the future sensitivities when m E > ∼ 1.5TeV. Fig.6  (a, b) show that the numerical results depend on m E mildly when m E is large. Because the one loop contributions to Br(µ → eγ), Br(µ → 3e) are highly suppressed when m E is large and δ 12 is small, then the two loop contributions can be comparable with the one loop results, and m E affects the two loop contributions negligibly. However, this feature does't appear in Fig.6 (c-f), because when δ 13 , δ 23 are large, the two loop contributions to Br(τ → eγ), Br(τ → 3e), Br(τ → µγ) and Br(τ → 3µ) are negligible compared with the one loop results. In addition, present experimental limit bounds of Br(l j → l i γ) constrain m E > ∼ 1TeV for tan β = 10, and Br(l j → l i γ), Br(l j → 3l i ) can reach the high future experimental sensitivities with small m E .
Then we scan the parameter space shown in Table II. In the scanning, we keep the slepton masses m La > 500GeV(a = 1, · · ·, 6), the Higgs boson mass in experimental 3σ interval, to avoid the range ruled out by the experiments [53]. Then we plot Br(µ → eγ), Br(µ → 3e), Br(τ → eγ), Br(τ → 3e), Br(τ → µγ) and Br(τ → 3µ) versus tan β ′ in Fig.7 (a-f) respectively. In the same parameter space, the MSSM predicts that Br(µ → eγ) ∼ 6.9 × Br(τ → µγ) ∼ 3.7×10 −9 , Br(τ → 3µ) ∼ 1.1×10 −11 . In order to see the differences between the B − L SSM and MSSM predictions clearly, we also plot these MSSM predictions (dashed line) in Fig.7 (a-f) respectively. The picture shows that the LFV rates increase with the increasing of tan β ′ , and the numerical results depend on tan β ′ , g B , g Y B comparably. When then the two loop results make the dominate contributions to ∆a N P µ . The main one-loop contributions to the muon MDM come from sleptons in Fig. 1(a). And according to the decoupling theorem, the contributions from sleptons in Fig. 1 (a) and sneutrinos in Fig. 1 (b) are highly suppressed when m E is large enough. But sleptons and sneutrinos do not appear in the two-loop diagrams, hence the two-loop corrections to the muon MDM don't suffer such a suppressive factor, and the two-loop contributions can be dominant when m E is large enough. In addition, when the one loop contributions are highly suppressed, only two loop contributions can not reach the experimental 3σ bounds. However, the two loop diagrams also make important corrections to ∆a N P µ , hence we use the more precise two loop prediction in the following analysis. In Fig.8 (b) we can see that ∆a N P µ decreases slowly with the increasing of tan β, but tan β does not affect the numerical result obviously. And when m E = 1TeV, one-loop corrections dominate the contributions to ∆a N P µ , hence the solid and dashed line almost appear as one in Fig.8 (b).
In order to see how tan β ′ , g B , g Y B affect the theoretical prediction on ∆a N P µ , we take m E = 1 and scan the parameter space shown in Table II. In the scanning, we also keep the slepton masses m La > 500GeV(a = 1, · · ·, 6), the Higgs boson mass in experimental 3σ interval. Then we plot ∆a N P µ versus tan β ′ in Fig.9. In the same parameter space, the MSSM predicts that ∆a N P µ ∼ 7.5 × 10 −10 . In order to compare with the MSSM directly, we also plot the MSSM prediction (dashed line) in Fig.9. The picture shows that, ∆a N P µ increases with the increasing of tan β ′ , which indicates that new parameters tan β ′ , g B , g Y B also can affect the numerical result, and the effects of them are comparable. In addition, in the B − L SSM, ∆a N P µ can reach 2.6 × 10 −9 when tan β ′ is large. Since tan β ′ , g B , g Y B affect the numerical result also mainly through the new mass matrix of sleptons, and the masses of sleptons decrease with the increasing of tan β ′ when |g Y B | < g B < 2|g Y B |, which implies that large tan β ′ can enhance the MSSM prediction on ∆a N P µ when |g Y B | < g B < 2|g Y B |. The high future experimental sensitivities keep a hope to detect all of these LFV processes.
In addition, the two loop diagrams make important corrections to ∆a N P µ . With respect to the MSSM, large tan β ′ can enhance the MSSM predictions on the branching ratios of LFV processes about one order of magnitude when |g Y B | < g B < 2|g Y B |, and also enhance the MSSM prediction on ∆a N P µ .
prehensive Strength Promotion project.
Appendix A: The Wilson coefficients of the process l − j → l − i γ.
The coefficients corresponding to Fig. 1(a), (b) can be written as where x i = m 2 i /m 2 W , C L,R abc denotes the constant parts of the interaction vertex about abc, which can be got through SARAH, and a, b, c denote the interactional particles, and the concrete expressions for the functions I 1,2,3,4 and G 1,2,3,4 below can be found in Ref. [78,79].