Electron and muon $(g-2)$ in the B-LSSM

The theoretical predictions in the standard model (SM) and measurements on the anomalous magnetic dipole moments (MDM) of muon and electron have great precision, hence the MDMs of muon and electron have close relation with the new physics (NP) beyond the SM. Recently, a negative $\sim2.4\sigma$ discrepancy between the measured electron MDM and the SM prediction results from a recent improved determination of the fine structure constant. Combined with the long-lasting muon MDM discrepancy which is about $\sim3.7\sigma$, it is difficult to explain both the magnitude and opposite signs of the deviations in a consistent model, without introducing large flavour-violating effects. The analysis shows that they can be explained in the minimal supersymmetric extension (MSSM) of the SM with local $B-L$ gauge symmetry (B-LSSM). Comparing with the MSSM, new parameters in the B-LSSM can affect the theoretical predictions on lepton MDMs, and the effects of them are explored.