  High energy colliders provide a new unique way to determine the microscopic
properties of the dark matter (DM). Weakly interacting massive particles
(WIMPs) are widely considered as one of the best DM candidates. It is usually
assumed that the WIMP couples to the SM sector through its interactions with
quarks and leptons. In this paper, we investigate the DM pair production
associated with a $Z$ boson in an effective field theory framework at the
International Linear Collider (ILC), which can be used to study the
interactions between the DM and leptons. For illustrative purposes, we present
the integrated and differential cross sections for the $e^+ e^- \rightarrow
\chi \bar{\chi} Z$ process, where the $Z$ boson is radiated from the initial
state electron or positron. Meanwhile, we analyze the neutrino pair production
in association with a $Z$ boson as the SM background.
