Optimal one-parameter observables for $Z'$ searches in $e^+e^- \to \mu^+\mu^-$ process

To detect off-shell Abelian $Z'$ boson in $e^+e^-\to\mu^+\mu^-$, we propose one-parameter observables with the best value-to-uncertainty ratio. The observables are constructed by angular integration of the differential cross section with smooth weight functions. The value-to-uncertainty ratio is used as a criterion to select the unique weight function leading to the observable with the best statistical power for data analysis. The observables allow to select either vector or axial-vector $Z'$ couplings to leptons. The obtained observable can be useful in future experiments at lepton colliders such as the ILC.