Search for WIMPs at future $\mu^+\mu^+$ colliders

Weakly interacting massive particles (WIMPs) with electroweak charges, such as the wino and the Higgsino, stand out as natural candidates for dark matter in the universe. In this paper, we study the search for WIMPs at future multi-TeV $\mu^+\mu^+$ colliders. We investigate both the direct production search of WIMPs through the mono-muon channel and the indirect search through quantum corrections in elastic $\mu^+\mu^+$ Moller scattering. We find that the indirect search has an advantage over the direct search with sufficient luminosities, $\mathcal{O}(\text{ab}^{-1})$, and low systematic uncertainties, $\lesssim 0.3\,\%$. This advantage arises due to the weaker mass dependence observed in the indirect search in comparison to direct production methods. The advantage is further enhanced if the initial muon beams are polarized. Specifically, we demonstrate that the indirect search method can detect the thermal mass target for the wino and the Higgsino for $\sqrt{s} = 6\,\text{TeV}$ and $2\,\text{TeV}$ (with $\sqrt{s}$ being the center of mass energy), respectively, with $10\,\text{ab}^{-1}$, an $80\,\%$ polarized beam and an accuracy of $0.1\,\%$. Our findings illuminate the potential of future high-energy $\mu^+\mu^+$ colliders in advancing our understanding of dark matter.