Exploring muonphilic dark matter with the $Z_2$-even mediator at muon colliders

The Galactic Center GeV Excess (GCE) remains a compelling but enigmatic signal from the inner region of our galaxy. Muonphilic dark matter (DM), which couples exclusively to muons via a new mediator, provides a viable explanation for the GCE and relic density while naturally evading constraints from direct detection, collider searches and other multi-messenger observations. Based on the viable non-resonant parameter space identified in previous global fits, we perform a comprehensive study exploring the prospects for discovering such muonphilic DM in the context of a future $3$ TeV muon collider, focusing on simplified models with a $Z_2$-even mediator. Four distinct search strategies are investigated: visible on-shell mediator decays ($\mu^{+}\mu^{-}\gamma$ final state), invisible on-shell mediator decays (mono-photon plus missing energy), mono-photon production via off-shell mediators, and vector boson fusion production.Through a detailed signal-background analysis using cut-and-count methods, we project the exclusion limits at $95\%$ confidence level for seven representative models across a wide range of mediator masses. Our results demonstrate that the projected limits cover a significant portion of the viable parameter space that explains the GCE, establishing a muon collider as a decisive machine for testing the muonphilic DM hypothesis.