Sukhoi State Technical University of Gomel, Francisk Skorina Gomel State University, Беларусь

The gauged $U(1)_{L_\mu - L_\tau}$ model is a minimal anomaly-free extension of the Standard Model, in which a new neutral $Z'$ boson interacts predominantly with second and third-generation leptons. A fundamental feature of this model is the absence of tree-level $Z'$ boson couplings to electrons and quarks. This renders the muon collider (MuC) a unique and indispensable facility for searching for such interactions. While projected electron-positron colliders (ILC, CLIC) remain insensitive to this "new" physics sector due to negligible loop-level corrections to the electron coupling constant ($\sim 10^{-6}$), the MuC provides a direct window to investigate these interactions. This work explores the potential of the MuC to obtain indirect constraints on the mass of a heavy $Z'$ boson in the $\mu^+\mu^- \to \tau^+\tau^-$ process within the $U(1)_{L_\mu - L_\tau}$ framework. The analysis is based on a methodology developed by the authors, utilizing a linear parametrization of the "new" physics contribution and longitudinal beam polarization options. Lower bounds on the $Z'$ mass are derived for MuC stages at center-of-mass energies of 3, 6, and 10~TeV. It is demonstrated that the use of polarized beams significantly expands the excluded mass region. The results confirm the unique potential of the MuC to explore energy scales up to tens of TeV, which are inaccessible for direct detection at current and future hadron colliders.