We investigate the discovery potential of multi-TeV muon colliders for a heavy dark $Z$ boson ($Z_{\mathrm{D}}$) with a mass above 1 TeV, via the associated production process ${\mu }^{+}{\mu }^{-}\to Z_{\mathrm{D}}\gamma$. This process enables precise reconstruction of $M_{Z_{\mathrm{D}}}$ through the photon recoil mass ($m_{\mathrm{recoil}}$). Focusing on the $Z_{\mathrm{D}}\to jjX$ and $Z_{\mathrm{D}}\to e^{+}{e}^{-}$ decay modes, we present strategies to achieve high sensitivity to the kinetic mixing parameter $ϵ$ at 3, 6, and 10 TeV muon colliders with integrated luminosities of 1, 4, and $10{\mathrm{ab}}^{-1}$, respectively, assuming that $Z_{\mathrm{D}}$ decays exclusively into Standard Model particles. We apply mass-dependent cuts on $m_{\mathrm{recoil}}$ and $m_{ee}$ to account for the energy-dependent detector response. For heavier $M_{Z_{\mathrm{D}}}$, the associated photon becomes less energetic, leading to better photon energy resolution and enabling more stringent $m_{\mathrm{recoil}}$ cuts. Conversely, for lighter $M_{Z_{\mathrm{D}}}$, the lower-energy electron pair from $Z_{\mathrm{D}}\to e^{+}{e}^{-}$ allows tighter invariant mass ($m_{ee}$) cuts. By combining these complementary $m_{\mathrm{recoil}}$- and $m_{ee}$-based selections across both the $jjX$ and $e^{+}{e}^{-}$ channels, we achieve sensitivity to $ϵ$ down to $\mathcal{O}({10}^{-3})$ as $M_{Z_{\mathrm{D}}}$ approaches $\sqrt{s}$, substantially surpassing the projected reach of a 100 TeV proton-proton collider. Even if $Z_{\mathrm{D}}$ decays into dark-sector particles, the recoil mass method remains effective, highlighting muon colliders as powerful facilities for exploring heavy dark sectors.

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• Extensions of gauge sector
• Hypothetical gauge bosons