Probing axion and flavored new physics with the NA64$\mu$ experiment

High-energy muon beam dump experiments are powerful probes of new physics models beyond the Standard Model, particularly those involving flavor-dependent interactions. We demonstrate the potential of muon beam dump by placing strong constraints on three new physics models utilizing data from the recent NA64$\mu$ experiment: (1) axions coupling to photons, (2) axions coupling to muons, and (3) a dark sector mediated by a massless $U(1)_{L_\mu-L_\tau}$ gauge boson. The new particles can be identified from the significant missing energy as the invisible channel, or the distinct energy deposition signature as the visible channel. We find that the current NA64$\mu$ data do not yet probe new parameter region on axion-photon coupling, while excluding new parameter space for the axion-muon coupling $g_{a\mu\mu}\gtrsim4\times10^{-3}$~GeV$^{-1}$ and the axion mass $m_a\lesssim 0.2$~GeV. For the dark sector, the current data provide stringent constraints that surpass existing ones by nearly one order of magnitude. The data from the 2023 NA64$\mu$ run, once available, will be capable of excluding new axion-photon coupling parameter space and probing the flavor structure of new physics, with more sensitivity advancement expected in near future runs.