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In order to address this question, we consider a simple renormalisable andgauge invariant model in which the $Z'$ only has couplings to the electron andmuon and their associated neutrinos, arising from mixing with a heavyvector-like fourth family of leptons. Within this model we discuss thecontributions to the electron and muon anomalous magnetic moments from $Z'$exchange, subject to the constraints from $\mu \rightarrow e \gamma$ andneutrino trident production. Using analytic and numerical arguments, we findthat such a $Z'$ model can account for either the electron or the muon $g-2$anomalies, but not both, while remaining consistent with the experimentalconstraints from $\mu \rightarrow e \gamma$ and neutrino trident production.

Is it possible to explain the muon and electron $g-2$ in a $Z^{\prime}$ model?