  Tau leptons can have lepton-flavor-violating (LFV) couplings to a muon or an
electron and an Axion-Like Particle (ALP). ALPs are pseudo Nambu-Goldstone
bosons associated with spontaneously broken global U(1) symmetries. LFV ALPs
have been of a great interest in the last several decades as they can address
some of the SM long-lasting problems. Assuming a future muon collider proposed
by the Muon Accelerator Program (MAP), we search for LFV decays
$\tau\rightarrow\ell a$ ($\ell=e,\mu$) of one of the tau leptons produced in
the muon-anti muon annihilation. The ALP mass is assumed to be in the range 100
eV to 1 MeV and three different chiral structures are considered for the LFV
coupling. Using a multivariate technique and performing a parameterized
simulation based on the ideal target performance, we obtain expected 95$\%$
confidence level upper limits on the LFV couplings tau-electron-ALP and
tau-muon-ALP. Limits are computed assuming the center-of-mass energies of 126,
350 and 1500 GeV which the future muon collider is supposed to operate at. We
study the two cases of unpolarized and polarized muon beams and show that
taking advantage of tau polarization-induced effects, the main background
$\tau\rightarrow e/\mu + \nu\bar{\nu}$ can be significantly reduced. Results
indicate that current limits on the LFV couplings can be improved by roughly
one order of magnitude using the present analysis.
