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We investigate the scenario where the dark matter only interacts with thecharged leptons in the standard model via a neutral vector mediator $Z'$. Sucha scenario with a 430 GeV dark matter can fit the recent positron fluxesobserved by the AMS-02 Collaborations, with the reasonable boost factors. Westudy the possibility of searching such leptophilic $Z'$ via its lepton finalstates and invisible decay modes at the future electron-positron colliders,such as the International Linear Collider (ILC) and the Compact Linear Collider(CLIC). We find that for the benchmark models with $Z'$ mass from 1.0 TeV to1.5 TeV, the searches for the invisible decays of $Z'\to \bar \chi \chi$ iseasily achieved at the CLIC 1.5 TeV runs via the mono-photon process. However,lighter $Z'$ with mass from 0.5 TeV to 0.8 TeV are challenging to see. Thedi-lepton plus single photon channel can reveal the $Z'$ mass at the ILC andCLIC with moderate luminosities.

The leptophilic dark matter with $Z'$ interaction: from indirect searches to future $e^+ e^-$ collider searches