Non-universal gauged lepton number for charged lepton masses hierarchy and $(g-2)_{e,\mu}$

We construct a novel flavor-dependent gauged lepton number $U(1)_\ell$ model for the hierarchical charged lepton masses and the observed $(g-2)_{e,\mu}$. Only tau participates in the tree-level Standard Model ( SM ) Yukawa interaction. At the same time, the masses of electron and muon are light due to radiative generation and(or) the heavy-mediator-suppressed Yukawa coupling to the SM Higgs. Not only can the measured anomalous magnetic dipole moment of the muon, $\triangle a_\mu$, be explained, but the positive (or negative) $\triangle a_e$ can also be accommodated in this model. Without additional discrete symmetries introduced, charged lepton flavor violation is highly suppressed by the $U(1)_\ell$ symmetry. Corresponding to two equally viable $U(1)_\ell$ charge assignments, this model predicts either $A_{FB}^\mu <A_{FB}^\tau <A_{FB}^e$ or $A_{FB}^\mu >A_{FB}^\tau >A_{FB}^e$, which can be tested at the $e^+e^-$ machines before discovering the $U(1)_\ell$ gauge boson. Moreover, the effective muon and electron Yukawa couplings can depart significantly from the SM predictions, and those deviations could be probed at future $e^+e^-$ colliders and High-Luminosity LHC.