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New light, weakly-coupled particles are commonly invoked to address thepersistent $\sim 4\sigma$ anomaly in $(g-2)_\mu$ and serve as mediators betweendark and visible matter. If such particles couple predominantly to heaviergenerations and decay invisibly, much of their best-motivated parameter spaceis inaccessible with existing experimental techniques. In this paper, wepresent a new fixed-target, missing-momentum search strategy to probe invisiblydecaying particles that couple preferentially to muons. In our setup, arelativistic muon beam impinges on a thick active target. The signal consistsof events in which a muon loses a large fraction of its incident momentuminside the target without initiating any detectable electromagnetic or hadronicactivity in downstream veto systems. We propose a two-phase experiment, M$^3$(Muon Missing Momentum), based at Fermilab. Phase 1 with $\sim 10^{10}$ muonson target can test the remaining parameter space for which lightinvisibly-decaying particles can resolve the $(g-2)_\mu$ anomaly, while Phase 2with $\sim 10^{13}$ muons on target can test much of the predictive parameterspace over which sub-GeV dark matter achieves freeze-out via muon-philicforces, including gauged $U(1)_{L_\mu - L_\tau}$.

M$^3$: A New Muon Missing Momentum Experiment to Probe $(g-2)_{\mu}$ and Dark Matter at Fermilab