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We consider simultaneous explanations of the electron and muon $g-2$anomalies through a single $Z'$ of a $U(1)'$ extension to the Standard Model(SM). We first perform a model-independent analysis of the viableflavour-dependent $Z'$ couplings to leptons, which are subject to variousstrict experimental constraints. We show that only a narrow region of parameterspace with an MeV-scale $Z'$ can account for the two anomalies. Following theconclusions of this analysis, we then explore the ability of different classesof $Z'$ models to realise these couplings, including the SM$+U(1)'$, the$N$-Higgs Doublet Model$+U(1)'$, and a Froggatt-Nielsen style scenario. In eachcase, the necessary combination of couplings cannot be obtained, owing toadditional relations between the $Z'$ couplings to charged leptons andneutrinos induced by the gauge structure, and to the stringency of neutrinoscattering bounds. Hence, we conclude that no $U(1)'$ extension can resolveboth anomalies unless other new fields are also introduced. While most of ourstudy assumes the Caesium $(g-2)_e$ measurement, our findings in fact also holdin the case of the Rubidium measurement, despite the tension between the two.

Solving the electron and muon $g-2$ anomalies in $Z'$ models