Lepton-Flavor-Violating ALP Signals with TeV-Scale Muon Beams

We explore the feasibility of using TeV-energy muons to probe lepton-flavor-violating (LFV) processes mediated by an axion-like particle (ALP) $a$ with mass $\mathcal{O}(10~\textrm{GeV})$. We focus on $\mu\tau$ LFV interactions and assume that the ALP is coupled to a dark state $\chi$, which can be either less or more massive than $a$. Such a setup is demonstrated to be consistent with $\chi$ being a candidate for dark matter, in the experimentally relevant regime of parameters. We consider the currently operating NA64-$\mu$ experiment and proposed FASER$\nu$2 detector as both the target and the detector for the process $\mu A \to \tau A\, a$, where $A$ is the target nucleus. We also show that a possible future active muon fixed-target experiment operating at a 3 TeV muon collider or in its preparatory phase can provide an impressive reach for the LFV process considered, with future FASER$\nu$2 data providing a pilot study towards that goal. The implications of the muon anomalous magnetic moment $(g-2)_\mu$ measurements for the underlying model, in case of a positive signal, are also examined, and a sample UV completion is outlined.