On the Run from the Dark Side of the Muon

We present an analysis strategy for probing physics beyond the Standard Model via modifications to the parton distribution functions (PDFs) in a muon beam, which measurably alter the kinematics of all hard processes at a future muon collider. High-energy muon colliders represent an opportunity to probe new physics using precision measurements and novel search strategies. At sufficiently high energies, light particles act as ``constituents'' of the muon described by PDFs. As a concrete case study, we apply this framework to an $L_{\mu} - L_{\tau}$ gauge boson and demonstrate that, for masses in the range of approximately 50--100 GeV, this indirect PDF-based approach outperforms traditional searches relying on direct gauge boson production. These results highlight muon PDF probes as a powerful and promising avenue for beyond the Standard Model physics searches at a future muon collider.