The corrections from one loop and two-loop Barr-Zee type diagrams to muon MDM in BLMSSM

In a supersymmetric extension of the standard model where baryon and lepton numbers are local gauge symmetries(BLMSSM) and the Yukawa couplings between Higgs doublets and exotic quarks are considered, we study the one loop diagrams and the two-loop Barr-Zee type diagrams with a closed Fermi(scalar) loop between the vector Boson and Higgs. Using the effective Lagrangian method, we deduce the Wilson coefficients of dimension 6 operators contributing to the anomalous magnetic moment of muon, which satisfies the electromagnetic gauge invariance. In the numerical analysis, we consider the experiment constraints from Higgs and neutrino data. In some parameter space, the new physics contribution is large and even reaches $24\times10^{-10}$, which can remedy the deviation well.


I. INTRODUCTION
The magnetic dipole moment (MDM) of lepton has close relation with the new physics beyond the standard model(SM). The current world average value [1] of (g − 2) µ experiment is a exp µ = 1 2 (g µ − 2) = 11659208. 9(5.4) There are three type contributions to the MDM of muon [2] such as: QED loops, hadronic contributions and electroweak corrections. The SM theoretical prediction of muon MDM is [3] a SM µ = 11659184.1(4.8) × 10 −10 .
The deviation between the SM prediction and experimental result is given as the follows, which lies in the range of ∼ 3σ [4].
The minimal supersymmetric extension of the standard model (MSSM) [5] is one of the most attractive candidates in the models beyond the SM, and draws physicists most attentions for a long time. A minimal supersymmetric extension of the SM with local gauged B and L(BLMSSM) is a favorite one, because it has two advantages. 1. The broken baryon number (B) can explain asymmetry of matter-antimatter in the universe. 2. The neutrinos should have tiny mass from the neutrino oscillation experiment. In theory, the tiny mass can be induced from the heavy majorana neutrinos by the seesaw mechanism.
Therefore, at some scale the lepton number (L) should be broken too.
Extending SM, with B and L as spontaneously broken gauge symmetries around TeV scale the models are studied [6]. Neglecting the Yukawa couplings between Higgs doublets and exotic quarks in BLMSSM, the authors study the lightest CP-even Higgs [6,7]. In the BLMSSM, considering the Yukawa couplings between Higgs and exotic quarks, we study the lightest CP-even Higgs(h 0 ) mass and the decays h 0 → γγ, h 0 → ZZ(W W ) [8], which are also studied in other models. In the CP-violating BLMSSM, the neutron electric dipole moment(EDM) is investigated [9].
To find new physics beyond the SM, research the MDMs [10,11] and EDMs [12] of leptons are the effective ways. There are some works for the supersymmetric (SUSY) one-loop contributions to muon MDM, and in some parameter space [13] the numerical results can be large. In µνMSSM, we study the muon MDM at one-loop level [4]. The authors investigate two-loop Barr-Zee-type diagrams [14] and obtain the electric dipole moments (EDMs) and MDMs of light fermions. Using the heavy mass expansion approximation (HME) and the projection operator method, the authors show two-loop standard electroweak corrections to muon MDM [15]. There are also several works about the muon MDM from two-loop diagrams [16,17] in SUSY model.
In this work, we study the one loop diagrams and two-loop Barr-Zee type diagrams with a closed scalar (Fermi) loop between vector Boson and Higgs in the frame work of BLMSSM.
Taking into account the Yukawa couplings between Higgs doublets and exotic quarks, we investigate these contributions to muon MDM with the effective Lagrangian method. Using the same method as in the Ref. [17], we deduce all dimension 6 operators and their coefficients.
Attaching a photon in all possible ways on the internal line of one self-energy diagram, one can obtain the corresponding triangle diagrams, and the sum of these amplitudes satisfies the Ward identity required by the QED gauge symmetry. Adopting the equations of motion to external leptons, we can neglect higher dimensional operators(dimension 8 operators) safely.
After this introduction, we briefly summarize the main ingredients of the BLMSSM, and show the needed couplings for exotic leptons and exotic quarks in section 2. We collect the one-loop and two-loop corrections to the muon MDM in section 3. Section 4 is devoted to the numerical analysis and discussion for the dependence of muon MDM on the BLMSSM parameters. In section 5, we give our conclusion. Some formulae are collected in the appendix.

II. SOME COUPLING IN BLMSSM
Physicists enlarge the SM with the local gauge group of SU(3) C ⊗ SU(2) L ⊗ U(1) Y ⊗ U(1) B ⊗ U(1) L , and obtain BLMSSM [6]. To to cancel L and B anomaly, the exotic leptons ) are respectively introduced. The detection of the lightest CP even Higgs h 0 at LHC [18] makes people to be convinced of the Higgs mechanism. To break lepton number and baryon number spontaneously, the Higgs superfieldsΦ L ,φ L andΦ B ,φ B are introduced respectively, and they acquire nonzero vacuum expectation values (VEVs).
The exotic quarks are very heavy and unstable. So the superfieldsX,X ′ are also introduced in the BLMSSM and the lightest superfields X can be a candidate for dark matter.
The superpotential of BLMSSM is [8] W BLM SSM = W M SSM + W B + W L + W X , In Ref. [8], the mass matrixes of Higgs, exotic quarks and exotic scalar quarks are obtained.
Some mass matrixes of exotic scalar leptons are discussed by the authors [19]. Here, we show the mass matrixes of exotic scalar leptons in our notation. Because the super fieldsN c are introduced in BLMSSM, the neutrinos can have tiny masses, and the scalar neutrinos are double as those in MSSM.

A. The mass matrix
After symmetry breaking the mass matrix for neutrinos in the left-handed basis (ν, N c ) is given by the following matrix.
Using the unitary transformations we diagonalize the mass matrix for neutrinos: In a similar way, we obtain the exotic neutrinos mass matrix.
Adopting the unitary transformations the mass matrix of exotic neutrinos are diagonalized as The mass matrix of exotic charged lepton are shown here With the unitary transformations one can diagonalize the mass matrix of exotic charged lepton as From the superpotential and the soft breaking terms in BLMSSM Eq. (4), the mass squared matrices of the scalar neutrinos and scalar exotic charged leptons are obtained.
The scalar neutrinos are enlarged by the superfieldsÑ c and the mass squared matrix reads as The mass squared matrix of the 4th generation scalar neutrinos is 6 The mass squared matrix of the 4th generation scalar charged leptons is The mass squared matrix of the 5th generation scalar neutrinos is The mass squared matrix of the 5th generation scalar charged leptons is

B. The needed couplings
We deduce the couplings between the charged Higgs and the exotic leptons (4,5) from the super potential in Eq. (4).
The couplings between neutral CP-even Higgs and the exotic leptons (4,5) are shown here.
Using the same method, we also get the couplings between neutral CP-odd Higgs and the exotic leptons (4,5).
In the Barr-Zee type two-loop diagrams, the couplings between one vector boson and exotic leptons (4,5) are necessary.
Here, we adopt the abbreviation notations s W = sin θ W , c W = cos θ W , where θ W is the Weinberg angle. The exotic scalar leptons (4,5) have contributions to muon MDM at twoloop level. The couplings of one vector boson and exotic scalar leptons (4,5) are given out.
For the couplings between vector Bosons and scalars, the VVSS type must be considered.
Here, we just show the used coupling between γ − V and two exotic scalar leptons (4,5).
The couplings between charged Higgs and exotic scalar leptons (4,5) are with The couplings between the neutral CP-even Higgs and the exotic scalar lepton (4,5) are also collected here.
where the concrete forms of the coupling constants N u,d 4,5 , E u,d 4,5 are Similarly, the couplings between the CP-odd Higgs and exotic scalar leptons (4,5) are obtained.
The couplings between neutral Higgs and exotic quarks (scalar quarks) can be found in our previous work [8]. In Ref. [9], the couplings between charged Higgs and exotic quarks are also given out. To complete the couplings, we deduce the changed Higgs-exotic scalar quarks couplings.
The concrete forms of the coupling constants (R u ) jk , (R d ) jk read as One vector Boson can couple with the exotic scalar quarks The couplings between photon-vector boson-exotic scalar quarks must be taken into account.
Because the exotic quark are very heavy, they can give considerable contribution to the muon MDM through the coupling between Higgs and exotic quarks. We give out the coupling between vector Boson and exotic quarks.
We use the effective Lagrangian method, the Feynman amplitude can be expressed by these dimension 6 operators.
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. Using the equations of motion to the incoming and out going leptons separately, only the O ∓ 2,3,6 contribute to lepton MDM and EDM. 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 lepton flavor mixing is also taken into account, whose contribution is considerable. The corrections to muon MDM from neutralinos and scalar leptons are expressed as where the couplings (S 1 ) I ij , (S 2 ) I ij are shown as The matrices ZL, Z N respectively diagonalize the mass matrices of scalar lepton and neutralino. The concrete forms of the functions B(x, y), B 1 (x, y) are In a similar way, the corrections from chargino and scalar neutrino are also obtained.
Here, Z − , Z + are used to diagonalize the chargino mass matrix. Because the right-hand The charged Higgs contributions are written as Because of the right handed neutrinos, the mass matrix of neutrino are expended to 6 × 6.
While, the squared mass matrix of scalar neutrinos turns to 6 × 6 too. The right handed neutrino contributions are very small (10 −15 ∼ 10 −13 ) and can be neglected safely.

B. the two-loop Barr-Zee type diagram with a closed scalar loop
The two-loop Barr-Zee type diagrams can give important contributions to muon MDM.
For the exotic scalar neutrino loops, their effects are suppressed by the Higgs-lepton-lepton coupling( mµ m W ∼ 1 1000 ), but are enhanced by the exotic scalar neutrino-higgs-exotic scalar lepton coupling including A N 4 , A N 5 , A E 4 , A E 5 . The concrete expressions can be found in Eqs.(30-35). One can find the detailed discussion of ultraviolet properties for these type diagrams' contributions to muon MDM in Ref. [12].
The couplings H H 0 k S i S j between CP-even Higgs and exotic scalar neutrinos(Ñ ′ 4,5 ), can be found in Eqs.(32), (33). Because the super fieldsÑ c is introduced in BLMSSM, the couplings related with the MSSM scalar neutrinos are changed, and they are corrected as The concrete forms of the functions P 1 , P 2 are The functions C, C 1 , F are collected in the appendix.
In this type, when the scalar particles between the scalar loop and Fermion line are CP-odd Higgs, the contributions from scalar neutrinos (exotic scalar neutrinos) read as One can find CP-odd Higgs and exotic scalar neutrinos (Ñ ′ ) couplings H A 0 k S i S j in Eq.(35). The concrete forms for the couplings between CP-odd Higgs and MSSM scalar neutrinos are corrected as When the scalar particles in the scalar loop are all neutral Higgs, the corrections to lepton MDM from Fig.2(a) are where the couplings B i R , A ij M , A jk H can be found in Ref. [5].

The vector is γ (Z), and the scalar loop are charged scalar particles
When the vector is photon, contributions from the Fig.2(a) are just produced by the neutral CP even Higgs. That is to say the corresponding CP odd Higgs' contribution is zero.
Here, Q S is the electric charge of the scalar particles. One can find the functions W 1 , W 2 in the appendix. H H 0 i SS are the couplings for CP even Higgs and two scalar particles. With S representing the MSSM particlesL,Ũ,D, H ± , the concrete forms of H H 0 i SS are in Ref. [5]. We have deduced the coupling between Higgs and exotic scalar quarks in our previous work [8].
The couplings for charged exotic scalar leptons and neutral Higgs are shown in Eq.(32). The exotic scalar quark loop contributions are also suppressed by the factor mµ m W ∼ 1 1000 , and they are increased by the coupling of Higgs-exotic scalar quarks-exotic scalar quarks which H A 0 S 1 S 2 ( G ZS 1 S 2 ) are the couplings for two scalar particles and CP odd Higgs (Z). When S 1 , S 2 are MSSM scalar particles, H A 0 S 1 S 2 ( G ZS 1 S 2 ) can be found in Ref. [5]. 3. The vector is W ± , and the scalar loop has charged particles Charged scalar lepton and scalar quark in MSSM contribute to lepton MDM. In the same way, we also obtain the exotic scalar lepton and exotic scalar quark contributions.
The complex functions W 3 , W 4 are collected in the appendix. The concrete forms for the couplings related with neutrino and scalar neutrino are different from those in MSSM.

C. the two-loop Barr-Zee type diagram with a closed Fermion loop
When the inserted is a fermion loop, the diagrams can be divided into two parts, according to the vector neutral Boson(γ, Z) and charged one(W ± ). For H ± F 1 F 2 coupling, it becomes large with the heavy Fermion mass, and may give important contributions.

the vector is γ, Z, and the Fermion loop are all charged particles
When the fermion loop is quarks and exotic quarks, charged leptons and exotic charged leptons, the contributions for muon MDM from the two loop diagrams with charged fermion loop inserted between γ and CP even Higgs are where Y H 0 i F F are the right hand parts of the couplings between the CP-even Higgs and the Fermions(b, t, χ ± , τ, L ′ , b ′ , t ′ ), and the general form is written as i( can be found in Ref. [5,8]. Y H 0 i F F for F = L ′ are shown in Eq.(23). To save space in the text, the form factorsW 5,6...19 are shown in the appendix.
In the same way, we get the two loop contributions with γ, CP odd Higgs and charged fermion loop.
Y A 0 i F F are the right hand parts of the couplings between the CP-odd Higgs with F = (b, t, χ ± , τ, b ′ , t ′ ), whose forms are obtained in the same way as Y H 0 i F F . When the vector is Z instead of γ, the corresponding expressions of the MDM contribution are more complex. The results from the CP-even Higgs, Z and charged fermion loop at two loop level are Generally, ZF 1 F 2 couplings are expressed as ie(H ZF 1 F 2 γ α ω − + T ZF 1 F 2 γ α ω + ). One can obtain We also obtain the CP-odd Higgs contribution from the diagram with vector Z and charged fermion loop shown in Fig.2(b).
Similarly, we get the CP odd Higgs couplings D. the vector is W ± , and the fermion loop have charged particles The contribution to lepton MDM from the diagram with vector W ± and Fermion loop are obtained here.
Because the right handed neutrino is introduced in BLMSSM, the couplings related with neutrino are not same as those in MSSM. We deduced the needed couplings here.

IV. THE NUMERICAL RESULTS
In this section, we show our numerical results. For the input parameters, we take into account the experimental constraints from the lightest neutral CP even Higgs m h 0 ≃ 125.7 GeV and the neutrino experiment data: sin 2 2θ 13 = 0.090 ± 0.009, sin 2 θ 12 = 0.306 +0.018 −0.015 , sin 2 θ 23 = 0.42 +0.08 −0.03 , In our previous work, we fit the neutrino experiment data shown as Eq. (66) in BLMSSM [20]. The lepton flavor violation is taken into account through 2,3). In the numerical discussion, the non-diagonal elements of these matrixes are not zero. Therefore, the lepton flavor violation are considered and there is a transition between muon-sneutrinos and tau-sneutrinos.
Because the muon MDM is related with the real parts of the results, to simplify the numerical discussion we suppose all the involved parameters in BLMSSM are real.
The used parameters in BLMSSM are collected here.
We suppose the following relations in the numerical discussion, then the numerical discussion is simplified.
In order to reflect the flavor mixing obviously and simplify the discussion, we define the off-diagonal elements in the following form. 2,3). (70) When MLa = 0, ALa = 0 there is no flavor mixing for the scalar leptons.  MLa. However, we also can find that the extent affected by MLa for the three lines follow the same rule: dotted-line >dashed-line>solid-line.
We also calculate the contribution from the off-diagonal element ALa. The numerical results imply that the effect of ALa is very small for both the one loop scalar leptonneutralino contribution and the scalar neutrino-chargino contribution. Therefore, one can neglect ALa safely, and in the latter numerical study, we suppose ALa = 0.   The squared mass matrixes of the charged exotic scalar leptons contain the parameters AE 45 . With A cs = −500GeV, MQ 2 = 1000GeV, A L = −800GeV, we plot the results versus AE 45 for ML 2 = 500GeV and ML 2 = 800GeV respectively. From Fig.9, one finds that the AE 45 affects the results slightly. When ML 2 = 800GeV, the corrections are about 10.5 × 10 −10 . As while as, the corrections reach 24 × 10 −10 with ML 2 = 500GeV. The AE 45 effect to muon MDM is in the region 10 −12 ∼ 10 −11 . However, we still can see that the correction is the very slowly increasing function of AE 45 .

V. DISCUSSION AND CONCLUSION
In the framework of the BLMSSM, the muon MDM is studied in this work. We calculate the one loop diagrams and the Barr-Zee type two loop diagrams. In the numerical analysis, we consider the experiment constraints such as: the experiment data of the lightest CP-even Higgs and neutrino. Our numerical results imply when the exotic and SUSY particles are not very heavy such as at TeV scale, the new physics contribution is about 8.0 × 10 −10 and even larger. In the parameter space as we supposed, our numerical results can reach 24 × 10 −10 , as scalar leptons at 500GeV scale, which can well remedy the deviation between the experiment data and the SM theoretical prediction for muon MDM. The one-loop, two-loop functions and the form factors are collected here.