  Recently, a novel collider, called $\mu$TRISTAN, has been proposed, offering
the capability to achieve high-energy collisions of anti-muons. This
high-energy collider presents an exceptional opportunity for the discovery of
electroweak-interacting massive particles (EWIMPs), which are predicted by
various new physics models. In a lepton collider like $\mu$TRISTAN, the
potential for discovering EWIMPs extends beyond their direct production.
Quantum corrections arising from EWIMP loops can significantly enhance our
prospects for discovery by precise measurement of Standard Model processes.
This study focuses on the indirect detection method within the $\mu$TRISTAN
experiment, with a specific emphasis on TeV-scale EWIMP dark matter scenarios
that yield the correct thermal relic density. At collision energies for $
\sqrt{s} = O(1-10)$ TeV, these EWIMPs introduce noticeable effects, typically
in the range of $O(0.1-1)$\%. Our findings indicate that at $\sqrt{s} = 2\,
(10)$ TeV, with an integrated luminosity of 10 ab$^{-1}$, the $\mu$TRISTAN can
detect Higgsino at a mass of 1.3 (3.0) TeV and Wino at a mass of 1.9 (4.4) TeV,
assuming an optimistic level of systematic uncertainty in the observation of
the Standard Model processes.
