The two-loop contributions to muon MDM in $U(1)_X$ SSM

The MSSM is extended to the $U(1)_X$SSM, whose local gauge group is $SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_X$. To obtain the $U(1)_X$SSM, we add the new superfields to the MSSM, namely: three Higgs singlets $\hat{\eta},~\hat{\bar{\eta}},~\hat{S}$ and right-handed neutrinos $\hat{\nu}_i$. It can give light neutrino tiny mass at the tree level through the seesaw mechanism. The study of the contribution of the two-loop diagrams to the MDM of muon under $U(1)_X$SSM provides the possibility for us to search for new physics. In the analytical calculation of the loop diagrams (one-loop and two-loop diagrams), the effective Lagrangian method is used to derive muon MDM. Here, the considered two-loop diagrams include Barr-Zee type diagrams and rainbow type two-loop diagrams, especially Z-Z rainbow two-loop diagram is taken into account. The obtained numerical results can reach $7.4\times10^{-10}$, which can remedy the deviation between SM prediction and experimental data to some extent.


I. INTRODUCTION
In the development of quantum field theory, the study of lepton anomalous magnetic dipole moment(MDM) is very important. The most accurate calculation of the lepton MDM is helpful in finding new physics beyond the standard model(SM) and Schwinger propose the electron magnetic dipole moment (MDM) for the first time. It has been recognized that the lepton MDMs can provide accurate testing of quantum electrodynamics (QED) [1].
Therefore, it is of special significance to study the MDM of lepton.
At present, the experimental accuracy of measuring the MDM of the µ, a µ has reached 540ppb at the muon E821 anomalous magnetic moment measurement at Brookhaven National Laboratory (BNL) [2]. And currently there are about 3.7 σ [3,4] between SM prediction and experimental measurement. In this precision, such measurements do more than testing the leading QED contributions [5], meanwhile, the influence of the weak interaction and strong interaction of the SM of particle physics can be tested [6][7][8][9]. The discovery of the Higgs made the SM a huge success [3,10]. Although the contribution from QED [5] plays an important role in lepton MDM, it is not the only factor. The hadronic contributions [11,12] are also particularly important, and it is modified by hadron vacuum polarization and light-by-light [13][14][15] scattering contributions. Moreover, weak interaction [16][17][18] also has a certain effect on MDMs [3,10]. As a result, the MDM of lepton can be expressed as [3,19,20]: (1) Due to the deficiency of MSSM which can not explain neutrino mass and solve µ problem, U(1) extension of MSSM is carried out. There are two U(1) groups in U(1) X SSM: U(1) Y and U(1) X , and we use SARAH software packages [21][22][23] to study U(1) X SSM. On the basis of the MSSM, the superfields are added; then one obtains the additional Higgs, neutrino and gauge fields, but also corresponding superpartners that extend the neutralino and sfermion sectors. The CP-even parts of the three Higgs singlet fields η, η, S mix with the neutral CP-even parts of the two doublets H d and H u to form a tree order 5 × 5 CP-even Higgs mass matrix. m h0 is the tree level mass of the lightest CP-even Higgs in U(1) X SSM, and it can be greater than the corresponding mass at tree order in MSSM. Therefore, the loop graph correction to m h 0 in U(1) X SSM needs not be very large. In U(1) X SSM, there are an 8 × 8 mass matrix for neutralinos and a 6 × 6 mass matrix for scalar neutrinos [24]. Next, we calculate the contribution of the two-loop diagrams to muon MDM under the U(1) X SSM with the effective Lagrangian method. The error between the experimental value and the predicted value by SM is [4,20,25] ∆a µ = a exp µ − a SM µ = (274 ± 73) × 10 −11 .
In this paper, the new physics contributions at one loop level are similar to the MSSM results in analytic form. The differences are that the mass matrices of scalar leptons, scalar neutrinos, neutrino, chargino and neutralino possess new parameters g X , vη, v η and so on.
The mass eigenstates of neutralinos, scalar neutrinos, neutrinos are more than those in the MSSM. Because the masses of those virtual fields (W ± , Z, Z ′ gauge bosons, neutral and charged Higgs, as well as neutralinos and charginos) are much heavier than the muon mass m µ , we can expand the denominator corresponding to the ratio (external momentum to internal momentum) to simplify the loop calculation [39]. We use the formula: Here, p is the external momentum of m µ , and k is the internal momentum at the order of m SU SY . So, mµ m SU SY ≪ 1 and p k ≪ 1. Our two-loop self energy diagrams contributing to the lepton MDMs are shown in Fig.   2. The two-loop triangle diagrams for µ → µ + γ can be obtained from the two-loop self energy diagrams by attaching a photon on the internal lines in all possible ways. The sum of all the two-loop triangle diagrams generated from a two-loop self energy diagram satisfies Ward-identity. The researched two-loop self energy diagrams include Barr-Zee type diagrams and rainbow type diagrams with Fermion sub-loop. Here, we suppose the scalar leptons and scalar quarks are very heavy, whose contributions from the studied two-loop diagrams can be neglected. In the works [2,[30][31][32][33][34][35], the two-loop rainbow diagrams with two vector bosons Z-Z are not considered. However, in our work [40] the order analysis of two-loop SUSY corrections [37,38] to lepton MDM shows that the contributions from two-loop rainbow diagrams with Z-Z are at the same order of the contributions from two-loop rainbow diagrams with W -W . Therefore, we take into account the Z-Z rainbow diagrams.
In the following, we introduce the specific form of U(1) X SSM and its superfields. The analytic results of the one-loop corrections and two-loop corrections in the U(1) X SSM are deduced in the section 3. Section 4 is used for the numerical calculation and discussion.
In the last section, we have a special summary and discussion. Some mass matrics and Feynman rules are collected in the appendix.
To obtain the U(1) X SSM, the MSSM is added with three Higgs singletsη,η,Ŝ and right-handed neutrinoŝ ν i . It can give light neutrino mass at the tree level through the seesaw mechanism. The neutral CP-even parts of H u , H d , η,η and S mix together, forming 5 × 5 mass squared matrix. Because of the right handed neutrinos, the mass matrix of neutrino is expended to 6 × 6. At the same time, the squared mass matrix of scalar neutrinos turns to 6 × 6 too.
For details of the mass matrix of particles, please see the appendix.
The superpotential for this model reads as: There are two Higgs doublets and three Higgs singlets. Their specific forms are shown below, and v S are the corresponding VEVs of the Higgs superfields H u , H d , η,η and S. Here, we define tan β = v u /v d and tan β η = vη/v η . The definition ofν L andν R is The soft SUSY breaking terms are With the singlet superfieldŜ coupling to heavy fields in the most general way, radiative corrections can induce very large terms in the effective action. These terms are linear inŜ in the superpotential or linear in S in L sof t , and they are called tadpole terms [41]. If they are too large, a tadpole problem appears in the model. In the case of gauge mediated supersymmetry breaking(GMSB) [42], the messenger fieldsφ with SM gauge quantum numbers and supersymmetric mass terms(M mess ) are the source of supersymmetry breaking. The real and imaginary scalar components of the messenger fields have different masses, and they are split by a scalem. This kind of supersymmetry breaking can be denoted as F-type splitting and represented by a F component of a spurion superfield coupling to the messenger fieldŝ ϕ. Then, we obtain l W ∼ C Sm 2 and L S ∼ C Sm 4 /M mess with the simplest coupling C SŜφφ .
Here, C S is the small Yukawa coupling. There is not tadpole problem with M mess and the F-type splittingm not much larger than the weak scale [43]. If these scales are larger than the weak scale, small Yukawa couplings can suppress the tadpole diagrams.
There is conflict between domain wall and tadpole problems, whose solutions are studied by Refs. [44]. One can impose constraints on Z 3 -symmetry breaking non-renormalisable interactions or hidden sectors in the form of various additional symmetries [45]. As the tadpole terms, Z 3 -symmetry breaking renormalisable terms are generated radiatively and have very small coefficients. Z 3 -symmetry breaking terms can solve the domain wall problem.
We use Y Y for the U(1) Y charge and Y X for the U(1) X charge. According to the textbook [46], the SM is anomaly free. The anomalies of U(1) X SSM are more complicated than those of SM [10]. In the end, this model is anomaly free. The presence of two Abelian groups Here, A ′Y µ and A ′X µ signify the gauge fields of U(1) Y and U(1) X . We can do a basis conversion, because the two Abelian gauge groups are unbroken. The following formula can be obtained by using the appropriate matrix R [47,49] We deduce sin 2 θ ′ W as The new mixing angle θ ′ W appears in the couplings involving Z and Z ′ .

III. FORMULATION
We use the effective Lagrangian method, and the Feynman amplitude can be expressed by these dimension 6 operators [39]. The higher order operators such as the dimension 8 operators are suppressed by additional factor with D µ = ∂ µ + ieA µ and ω ∓ = 1∓γ 5 2 . F µν is the electromagnetic field strength, and m l is the lepton mass. Therefore, the Wilson coefficients of the operators O ∓ 2,3,6 in the effective Lagrangian are of interest and their dimensions are -2. The lepton MDM is the combination of the Wilson coefficients C ∓ 2,3,6 and can be obtained from the following effective Lagrangian where x M = M 2 Λ 2 , M is the particle mass and Λ is the mass scale. The couplings A R , A L are shown as The matrices Z E , N respectively diagonalize the mass matrices of scalar lepton and neutralino. The concrete forms of the functions B(x, y) and B 1 (x, y) are In a similar way, the corrections from chargino and CP-odd scalar neutrino are also obtained.
Here, the B L and B R is The corrections from chargino and CP-even scalar neutrino read as Here, the C L and C R are Here, U, V are used to diagonalize the chargino mass matrix, and the mass squared Higgs contribution to muon MDM is shown by the Fig. 2, which is suppressed by the factor In numerical estimation, the correction from Higgs one-loop diagram is around 10 −13 , and we can neglect it safely. So, the one-loop corrections to lepton MDM can be expressed as B. The two-loop corrections As discussed in Ref. [40], the Barr-Zee two-loop diagrams ( At first, we consider the corrections from Fig. 3(a). Under the assumption m F = m F 1 = m F 2 ≫ m W , the results [50] can be simplified as where ̺ 1,1 (x, y) = x ln x−y ln y x−y . H L,R HF 1 F 2 and H L,R W F 1 F 2 represent the coupling coefficients of the corresponding vertices with the presentation Their concrete forms are collected in the appendix.
Then under the assumption m F = m F 1 = m F 2 ≫ m h 0 , the two-loop Barr-Zee type diagrams contributing to the lepton MDMs corresponding to Figs. 3(b) and 3(c) can be simplified as Q f is the electric charge of the external lepton m µ . Q F 1 and Q F 2 are the electric charges of the internal charginos.
Under the assumption m F = m F 1 = m F 2 ≫ m W ∼ m Z , the two-loop rainbow type diagrams contributing to the lepton MDMs corresponding to Fig. 3(d), 3(e) and 3(f) can be simplified as Here, we focus on Z-Z two-loop rainbow diagram in the Fig. 3(g). In the references , and obtain the concise form.
At two-loop level, the contributions to lepton MDMs can be summarized as

IV. THE NUMERICAL RESULTS
In this section, we will discuss the numerical results. The lightest CP-even higgs mass is considered as an input parameter, which is m h0 =125.1 GeV [51,52]. The parameters used in U(1) X SSM are given below: To simplify the discussion, the parameters Y X , T X , T ν , T E , M L , M E and M ν are supposed as diagonal matrices. In the follow, the remaining tunable parameters are M 2 , M S , µ, v S , Next, we will analyze the effects of these parameters on the contributions to the muon MDM.
With the parameters M S = 1200 GeV, µ = 300 GeV, M E = 1 TeV 2 , we plot a µ versus v S in the Fig. 4. v S is VEV of the Higgs singlet S, which affects the particle masses. So, it is The two-loop rainbow diagrams with two Z bosons are not considered in the works [2,[31][32][33][34][35]. In our numerical calculation, the contributions from two-loop rainbow diagrams with two Z bosons are at the same order of the contributions from the two-loop rainbow diagrams with two W bosons. This result is consistent with the two-loop order analysis in our previous work [40]. Therefore, the contributions from the two-loop rainbow diagrams with two Z bosons are not small, and they should be taken into account.