Electron and Muon Anomalous Magnetic Moment in the $\mathbb{Z}_3$-NMSSM

Inspired by the recent measurements of the muon and electron anomalous magnetic moments, the rapid progress of the LHC search for supersymmetry, and the significantly improved sensitivities of dark matter direct detection experiments, we studied the supersymmetric contribution to the electron \texorpdfstring{$g-2$}{}, $a_e^{\rm SUSY}$, in the Next-to-Minimal Supersymmetric Standard Model with a discrete $\mathbb{Z}_3$ symmetry. We concluded that $a_e^{\rm SUSY}$ was mainly correlated with $a_\mu^{\rm SUSY}$ by the formula $a_e^{\rm SUSY}/m_e^2 \simeq a_\mu^{\rm SUSY}/m_\mu^2$, and significant violations of this correlation might occur only in rare cases. As a result, $a_e^{\rm SUSY}$ was typically around $5 \times 10^{-14}$ when $a_\mu^{\rm SUSY} \simeq 2.5 \times 10^{-9}$. We also concluded that the dark matter direct detection and LHC experiments played crucial roles in determining the maximum reach of $a_e^{\rm SUSY}$. Concretely, $a_e^{\rm SUSY}$ might be around $3 \times 10^{-13}$ in the optimum cases if one used the XENON-1T experiment to limit the supersymmetry parameter space. This prediction, however, was reduced to $1.5 \times 10^{-13}$ after implementing the LZ restrictions and $1.0 \times 10^{-13}$ when further considering the LHC restrictions.