Searching for the LFV $γγeμ$ interaction at future $e^- e^+$ colliders
We investigate the Lepton Flavor Violating (LFV) process $e^{+}e^{-}\to e^{+}e^{-}eμ$ ($e=e^-,\,e^+,\,μ=μ^-,\,μ^+ $) at future circular colliders, probing the effective $γγeμ$ interaction through photon fusion. Within an Effective Field Theory (EFT) framework compatible with theoretical and experimental constraints, we identify regions in the parameter space of the effective couplings where the signal could be observed at the Circular Electron-Positron Collider (CEPC) at $\sqrt{s}=240\ \text{GeV}$ and the Future Circular Collider (FCC-ee) at $\sqrt{s}=240$ and $350\ \text{GeV}$. Our analysis leverages distinctive kinematic distributions -- particularly the transverse momentum of the scattered electron $p_{T}^{(e^{-}\ \text{scattered})}$ and the central muon $p_{T}^μ$ -- to achieve efficient signal-background separation. By employing a neural network classifier, we enhance the sensitivity beyond traditional cut-based methods, demonstrating the discovery potential of these facilities for LFV searches in the clean environment of $e^{+}e^{-}$ collisions.