Solving the electron and muon $g-2$ anomalies in $Z^{\prime }$ models

Abstract

We consider simultaneous explanations of the electron and muon $g-2$ anomalies through a single $Z^{\prime }$ of a $U(1{)}^{\prime }$ extension to the Standard Model (SM). We first perform a model-independent analysis of the viable flavour-dependent $Z^{\prime }$ 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 $Z^{\prime }$ can account for the two anomalies. Following the conclusions of this analysis, we then explore the ability of different classes of $Z^{\prime }$ models to realise these couplings, including the SM$+U(1{)}^{\prime }$, the $N$-Higgs Doublet Model$+U(1{)}^{\prime }$, and a Froggatt-Nielsen style scenario. In each case, the necessary combination of couplings cannot be obtained, owing to additional relations between the $Z^{\prime }$ couplings to charged leptons and neutrinos induced by the gauge structure, and to the stringency of neutrino scattering bounds. Hence, we conclude that no $U(1{)}^{\prime }$ 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.

37 pages, 4 figures; v2: neutrino scattering bounds corrected, Section V added, results qualitatively unchanged