The MSSM is extended to the $$U(1)_X$$ U ( 1 ) X SSM, whose local gauge group is $$SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_X$$ S U ( 3 ) C × S U ( 2 ) L × U ( 1 ) Y × U ( 1 ) X . To obtain the $$U(1)_X$$ U ( 1 ) X SSM, we add the new superfields to the MSSM, namely: three Higgs singlets $${\hat{\eta }},~\hat{{\bar{\eta }}},~{\hat{S}}$$ η ^ , η ¯ ^ , S ^ and right-handed neutrinos $${\hat{\nu }}_i$$ ν ^ 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 $$U(1)_X$$ 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 $$7.4\times 10^{-10}$$ 7.4 × 10 - 10 , which can remedy the deviation between SM prediction and experimental data to some extent.

The double Higgs boson production through e þ e − → hhZ is analyzed in the minimal supersymmetric extension of the Standard Model (SM) with the local gauge symmetry Uð1Þ B−L , where h denotes the lightest Higgs boson with 125 GeV. Considering the constraints from the updated prediction data, we find that the production cross section of this process in the model depends on some parameters strongly.

The MSSM is extended to the U (1) X SSM, whose local gauge group isTo obtain the U (1) X SSM, we add the new superfields to the MSSM, namely: three Higgs singlets η, η, Ŝ and right-handed neutrinos νi . 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 twoloop diagram is taken into account. The obtained numerical results can reach 7.4 × 10 −10 , which can remedy the deviation between SM prediction and experimental data to some extent.