The order analysis for the two loop corrections to lepton MDM

The experimental data of the magnetic dipole moment(MDM) of lepton($e$, $\mu$) is very exact. The deviation between the experimental data and the standard model prediction maybe comes from new physics contribution. In the supersymmetric models, there are very many two loop diagrams contributing to the lepton MDM. Based on the supposition that all supersymmetric particles have same masses, we analyze the order of the contributions from the two loop diagrams. The two loop triangle diagrams corresponding to the corresponding two loop self-energy diagram satisfy Ward identity, and their contributions possess particular factors. This work can help to distinguish the important two loop diagrams giving corrections to lepton MDM.


I. INTRODUCTION
With the detection of the 125 GeV Higgs boson [1], the standard model(SM) achieves great success. However, SM has some short comings: 1. SM can not give masses to neutrinos; 2. SM does not have cold dark matter candidate; 3. SM is unable to explain the hierarchy problem. The minimal supersymmetric(SUSY) extension of the standard model(MSSM) [2] has attracted physicists' attentions for more than 30 years. MSSM has also been extended, and the extensions of MSSM [3] have many particles beyound SM. These new particles give corrections to the studied processes. For the magnetic dipole moment(MDM) [4] of lepton especially muon, the two loop corrections from SUSY particles are important.
In the SM, there are several parts giving the contributions to lepton MDM [5] a SM l = a QED l + a EW l + a HAD l . (1) This deviation ∆a µ may come from the new physics contribution. In SUSY models, there are many SUSY particles that can correct muon MDM through loop diagrams. As discussed in the works [7,8], the two loop SUSY diagrams can contribute importantly to muon MDM.
With the on shell renormalization scheme, the numerical calculation of two loop electroweak corrections to the muon MDM has been performed in the SM [9]. Physicists show great interests on new physics corrections to muon MDM. The authors study the 2HDM contribution to the muon MDM and present the complete two loop results [10]. It is well known that the corrections from SUSY particles are of interest. In the work [11], the two loop Barr-Zee type diagrams with heavy fermions in the sub-loop are researched. The rainbow diagrams with heavy fermion sub-loop are also deduced analytically [12]. The contributions to muon MDM from the two loop triangle diagrams generating from the two loop selfenergy diagrams b4[χ 0 ; l; χ 0 ;L;L], b4[χ 0 ; ν; χ ± ;L;ν] and b4[χ ± ; l; χ ± ;ν;ν] can be found in Ref. [13].  [14]. The two loop diagrams where an additional photon loop is attached to a SUSY one loop diagram are called as photonic SUSY two loop corrections, whose corrections to muon MDM are evaluated exactly [15].
It is well known to all that, in SUSY models the two loop diagrams [16] contributing to muon MDM are too many to be studied completely and exactly. So, to identify the important two loop diagrams is an important thing. We use a method to get the order of the two loop corrections. In this work, the log terms and the number coefficients are not shown, because it is the estimation and the line factor is more important than the log function. After this introduction, we show the used supposition and analysis method in the section II. Section III is devoted to the obtained factors for the corresponding two loop diagrams. In the last section, we discuss the factors and compare their size.

II. THE METHOD
To make that the analysis is representative, we research the problem in the framework of To calculate so many two loop diagrams is very difficult and tedious. So, we use a method to estimate the order of contribution from the two loop diagram. The method is much simpler than the real two loop calculation. Firstly, we perform the sub-loop integration to get the effective Lagrangian. Secondly, the residual loop integration is performed.
To simplify the calculation and get the factor easily, we adopt the suppositions: 1. Con- Then the factor of this diagram's contribution to muon MDM can be obtained.

III. THE ANALYSIS
To show the concrete analysis of the factor, we study the two loop diagrams in the MSSM.
The notation is x = m 2 Λ 2 and Λ is mass scale. In our method, discarding the vertex coupling we obtain the factor have been studied in Ref. [12], where the factor x l x M in the (γ, γ) type diagrams is obviously. Supposing all SUSY particles with same masses, the analytic results for two loop rainbow diagrams with two vector bosons(γ, Z) in Ref. [12] are simplified, from which the factor x l x M is extracted. Besides these diagrams, in our analysis, we find that the diagrams c2[S; S; γ; (γ, Z); µ], a1[γ; S; (γ, Z); µ] contribute to muon MDM contribution with the factor x l x M . 3. In Ref. [12], the authors research muon MDM corrections from the rainbow diagrams . Adding the mass dimension parameter λ HSS in the vertex, the total factor of these type diagrams is , which is smaller than 5. There are many two loop diagrams contributing to muon MDM with the factor 7. The mass of lepton is much smaller than the m V . That is to say x l x V ≪ 1, and we can expand the results corresponding to