Muon $(g-2)$ in the B-LSSM

The difference between the updated experimental result on the muon anomalous magnetic dipole moment and the corresponding theoretical prediction of the standard model on that is about $4.2$ standard deviations. In this work, we calculate the muon anomalous MDM at the two-loop level in the supersymmetric $B-L$ extension of the standard model. Considering the experimental constraints on the lightest Higgs boson mass, Higgs boson decay modes $h\rightarrow \gamma\gamma,\;WW,\;ZZ,\; b\bar b,\;\tau\bar\tau$, B rare decay $\bar B\rightarrow X_s\gamma$, and the transition magnetic moments of Majorana neutrinos, we analyze the theoretical predictions of the muon anomalous magnetic dipole moment in the $B-L$ supersymmetric model. The numerical analyses indicate that the tension between the experimental measurement and the standard model prediction is remedied in the $B-L$ supersymmetric model.

The result agrees with the previous Brookhaven National Laboratory E821 measurement [5] very well. The new experimental average for the difference between the experimental measurement and SM theoretical prediction of the muon anomalous MDM is given by which shows that the tension between experiment and the SM prediction is increased to 4.2 standard deviations. Then many papers appeared to study the relation between the updated muon anomalous MDM results with various models beyond the SM, the details can be seen in Refs. . However, it is worth mentioning that the latest result obtained by the lattice QCD calculation [92] of the leading order hadronic vacuum polarization contribution to the muon anomalous MDM is larger than the former result, which can accommodate the discrepancy between the SM prediction and the experimental result, hence the discrepancy needs further scrutiny.
It is well-known that the muon anomalous MDM has close relation with the new physics (NP) beyond the SM, and the tiny neutrino masses shown in neutrino oscillation experiments [93] are an unambiguous evidence of NP  Collider (LHC) has been discussed in many works. Ref. [100] shows the Z ′ decay chain involving heavy neutrinos, and eventually decaying into leptons and jets allows one to measure the Z ′ and heavy neutrino masses at the LHC. The direct production of right-handed sneutrinos at the LHC and their decay modes are studied in Ref. [101]. The second light scalar Higgs signal in the decays of the SM-like Higgs to γγ and Zγ at the CERN machine are discussed in Ref. [102]. The disentangling of the B-LSSM with other SUSY models at the CERN machine is analyzed in Ref. [103], which also shows that the mono-jet events can be accessible at the LHC. Ref. [104] shows that the LHC will enable to establish a specific B-LSSM signal during Run 2 and 3, mediated by a charged Higgs boson pair produced from the on-shell Z ′ decay. Considering the lightest supersymmetric particle (LSP) neutralino, Ref. [105] shows that the LSP relic density constraint provides a lower bound on the stop and gluino masses of about 3 TeV and 4 TeV respectively, which is testable in the near future collider experiments such as high luminosity LHC.
The framework of the B-LSSM has been discussed detailedly in our previous works [106][107][108], which contain the particle content, the superpotential, the soft breaking terms, some mass matrices and interaction vertices. We do not introduce the model in detail in this work, but some relevant mass matrices will be given in Sec  is given in Sec. IV. Some mass matrices are collected in the appendix.  In addition, the numerical results of our previous work [109] show that the contributions from the two-loop Barr-Zee type diagrams are important to the muon anomalous MDM. Hence we also consider the two-loop corrections to ∆a N P µ , and the corresponding Feynman diagrams are shown in Fig. 2. Then the NP contributions to the muon anomalous MDM in the B-LSSM can be written as where the concrete expressions of ∆a one−loop µ , ∆a two−loop µ can be found in our previous works [109][110][111].
For the transition magnetic moments of Majorana neutrinos, the contributions in the B-LSSM can be written as with where i, j are the indices of generation, m e is the electron mass, m νi is the light neutrino where where the concrete expressions of C L(a,...,d) 2 can be found in our previous work [112].
Refs. [109][110][111][112][113][114] show that the theoretical predictions of the muon anomalous MDM and the transition magnetic moments of Majorana neutrinos in the B-LSSM depend on sleptons, neutralinos and charginos strongly. On the basis (ẽ L ,ẽ R ), the mass matrix of charged sleptons is given as: where g B is the U(1) B−L coupling constant, g Y B is the gauge kinetic mixing coupling constant which arises from the existence of two Abelian gauge groups, u 1,2 , v 1,2 are the VEVs of two Higgs singlets and two Higgs doublets respectively (the definitions analogy to ones in the MSSM), and T e is the trilinear Higgs slepton coupling. In the following analyses, we assume the degenerate slepton masses and take mL = mẽ = mν = diag(M E , M E , M E ) TeV for simplicity, where mν is the sneutrino mass term in sneutrino mass matrices as shown in the appendix.
For the mass matrix of charginos, the mass term in the B-LSSM can be written as Then the mass term can be rewritten as Then the physical masses of χ 1 , χ 2 (χ 1 , χ 2 are defined as mass eigenstates) can be obtained by diagonalizing the mass matrix On the basis (B,W 0 ,H 0 1 ,H 0 2 ,B ′ ,η 1 ,η 2 ), the mass matrix of neutralinos reads For gaugino mass terms M 1 , M 2 , µ appeared in the mass matrices of chargino and neutralino, we take m 0 ≡ µ = 2M 1 = 2M 2 in the following analyses for simplicity. It can be noted that m 0 plays an important role in the contributions to the muon anomalous MDM through both of the chargino-sneutrino loop and the neutralino-charged slepton loop.
In the numerical calculation, we also consider the constraints from SM-like Higgs boson mass, Higgs boson decay modes h → γγ, W W, ZZ, bb, ττ , and the B meson rare decaȳ B → X s γ. The stop and sbottom quark affect the Higgs boson mass obviously in supersymmetric models, and the B meson rare decayB → X s γ also depends on the up-squark sector in the B-LSSM as shown in Ref. [115]. In order to explain the considered constraints clearly, the mass matrices of up-squark and down-squark in the B-LSSM are given in appendix A.
In addition, the leading-log radiative corrections from stop, top quark and sbottom, bottom quark to the mass of the SM-like Higgs boson are considered [116][117][118].

III. NUMERICAL ANALYSES
In the calculation, we take the W boson mass m Since the hierarchy of neutrino masses has not been fixed yet, we will take m ν1 < m ν2 < m ν3 for the normal hierarchy (NH) and m ν3 < m ν1 < m ν2 for the inverse hierarchy (IH) in the following analyses. The light neutrino mixing matrix is taken as the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix.
In the previous section, we assume the slepton mass parameters as mL = mẽ = mν = diag(M E , M E , M E ) TeV and the gaugino mass parameters as m 0 ≡ µ = 2M 1 = 2M 2 .
There are constraints on the chargino and slepton masses from the LHC [131][132][133]. The results of CMS search [132] allow the exclusion of charginoχ ± 1 (neutralinoχ 0 2 ) mass up to 750 (800) GeV via the on-shell decay to lightest neutralinoχ 0 1 and W (Z) boson. And more recently, the CMS search [133] obtains a lower bound of mχ±  at 95% CL is given by [135,136] M Z ′ /g B > 6 TeV.
Since we focus on the muon anomalous MDM in this work, we can reasonably take M H ± = parameter space, the ranges of µ, tan β are not limited by the constraints considered above, hence we do not plot them in the figure. As shown in Fig. 4, the range of g Y B is not limited by the constraints considered above when g B > ∼ 0.4 and tan β ′ > ∼ 1. 3. When g Y B approaches to 0, g B , tan β ′ are limited in the ranges 0

TeV which coincides with the experimental results of the Higgs boson mass, Higgs boson
Considering the constraints shown in Fig. 4 As shown in Fig. 7 (a), the effect of M BB ′ is affected by the value of µ ′ complicatedly, and the effect of µ ′ is highly suppressed when |M BB ′ | is large. Fig. 7 [137,138] show that the moments On the basis (ũ L ,ũ R ), the mass matrix of up squarks is given by with m uL = m 2 q + 1 24 On the basis (d L ,d R ), the mass matrix of down squarks is given by with m dL = 1 24 2g B (g B + g Y B )(u 2 2 − u 2 1 ) + (3g 2 2 + g 2 Neglecting the tiny Yukawa coupling constant Y ν (corresponding to the Dirac mass term of neutral leptons), we give the mass matrix for CP-odd sneutrinos on the basis (σ L ,σ R ) as with, m ν odd L = 1 8 2g B (g B + g Y B )(u 2 1 − u 2 2 ) + (g 2 2 + g 2 the mass matrix for CP-even sneutrinos is given on the basis (φ L ,φ R ) with m νevenL = 1 8 2g B (g B + g Y B )(u 2 1 − u 2 2 ) + (g 2 2 + g 2