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

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 viable flavour-dependent $Z'$ couplings to leptons, which are subject to various strict experimental constraints. We show that only a narrow region of parameter space with an MeV-scale $Z'$ can account for the two anomalies. Following the conclusions of this analysis, we then explore the ability of different classes of $Z'$ models to realise these couplings, including the SM$+U(1)'$, the $N$-Higgs Doublet Model$+U(1)'$, and a Froggatt-Nielsen style scenario. In each case, the necessary combination of couplings cannot be obtained, owing to additional relations between the $Z'$ couplings to charged leptons and neutrinos induced by the gauge structure, and to the stringency of neutrino scattering bounds. Hence, we conclude that no $U(1)'$ extension can resolve both anomalies unless other new fields are also introduced. While most of our study assumes the Caesium $(g-2)_e$ measurement, our findings in fact also hold in the case of the Rubidium measurement, despite the tension between the two.