Searching for the Leptophilic Gauge Boson Z${}_l$ at Future $e^{+}{e}^{-}$ Colliders

We study the discovery prospects of the leptophilic photon Z_l, a hypothetical gauge boson beyond the Standard Model, at future electron-positron colliders. Our analysis is based on the process e+e- -> mu+mu- and consistently includes realistic effects such as initial state radiation and beamstrahlung. Using the benchmark parameters of FCC-ee, CEPC, ILC, and CLIC, we demonstrate the complementarity of circular and linear colliders: circular machines achieve excellent sensitivity to very small couplings, down to g_l ~ 10^{-4}, while high-energy linear colliders extend the search reach into the TeV domain.