Exploring Optimal Sensitivity of Lepton Flavor Violating Effective Couplings at the $e^+e^-$ Colliders

We analyze lepton flavor violation (LFV) using the Standard Model Effective Field Theory (SMEFT) framework at the future lepton colliders. Our focus is on the associated production of tau lepton with electron/muon at the electron-positron ($e^+e^-$) colliders, related to four-Fermi SMEFT effective operators. In accordance with the upper limits on effective couplings from lepton flavor violating tau decays, we conduct a cut-based analysis to achieve sufficient signal significance. We utilize the optimal observable technique (OOT) to estimate the optimal sensitivity of the effective couplings. The impact of electron beam polarization and the interplay of signal and background in enhancing the optimal sensitivity of the effective couplings are discussed in detail. We find that the sensitivity of flavor-violating effective couplings is enhanced by order of one for 3 TeV center-of-mass (CM) energy and 1000 $\rm{fb}^{-1}$ integrated luminosity at the $e^+e^-$ colliders.