AbstractWe consider simultaneous explanations of the electron and muon $$g-2$$
g
-
2
anomalies through a single $$Z'$$
Z
′
of a $$U(1)'$$
U
(
1
)
′
extension to the Standard Model (SM). We first perform a model-independent analysis of the viable flavour-dependent $$Z'$$
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 $$Z'$$
Z
′
can account for the two anomalies. Following the conclusions of this analysis, we then explore the ability of different classes of $$Z'$$
Z
′
models to realise these couplings, including the SM$$+U(1)'$$
+
U
(
1
)
′
, the N-Higgs Doublet Model$$+U(1)'$$
+
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 $$Z'$$
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 $$U(1)'$$
U
(
1
)
′
extension can resolve both anomalies unless other new fields are also introduced. While most of our study assumes the Caesium $$(g-2)_e$$
(
g
-
2
)
e
measurement, our findings in fact also hold in the case of the Rubidium measurement, despite the tension between the two.