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A non-universal $U(1)_{X}$ extension to the Standard Model composed of twoscalar doublets and two scalar singlets together with three additional quarksinglets and two lepton singlets and three generations of right-handed andMajorana neutrinos is made to explain lepton mass hierarchy, neutrino massesvia inverse seesaw mechanism and muon anomalous magnetic moment in an anomalyfree framework. In the present model, exotic and Standard Model particlesacquire mass thanks to vacuum expectation values at different scales, yet theelectron and the lightest neutrino are tree level massless but massive atone-loop level. By considering a numerical exploration and under the constraintof the Higgs mass, neutrino mass differences and PMNS matrix, it is found thatonly contributions due to exotic neutrinos interacting with charged scalars arerelevant to muon $g-2$, though they are negative. Thus, the SUSY extension isconsidered and it is found that muon $g-2$ can be explained by allowing$U(1)_{X}$ vacuum expectation values to lie in the TeV scale thanks to SUSYsoft-breaking interactions for at least $\sim 10^{5}$ GeV masses. Thus, thecontribution due to exotic neutrinos interacting with $W$ gauge bosons ispositive and no longer negligible which added to all other contributions mightexplain the anomaly.

Explaining muon $g-2$ anomaly in a non-universal $U(1)_{X}$ extended SUSY theory