We assess the status of past and future experiments on lepton flavor violating (LFV) muon and tau decays into a light, invisible, axion-like particle (ALP), $a$. We propose a new experimental setup for MEG II, the MEGII-fwd, with a forward calorimeter placed downstream from the muon stopping target. Searching for $mu to e a$ decays MEGII-fwd is maximally sensitive to LFV ALPs, if these have nonzero couplings to right-handed leptons. The experimental set-up suppresses the (left-handed) Standard Model background in the forward direction by controlling the polarization purity of the muon beam. The reach of MEGII-fwd is compared with the present constraints, the reach of Mu3e and the Belle-II reach from $tau to ell a$ decays. We show that a dedicated experimental campaign for LFV muon decays into ALPs at MEG II and Mu3e will be able to probe the ALP parameter space in an unexplored region well beyond the existing astrophysical constraints. We study the implications of these searches for representative LFV ALP models, where the presence of a light ALP is motivated by neutrino masses, the strong CP problem and/or the SM flavor puzzle. To this extent we discuss the majoron in low-scale seesaw setups and introduce the LFV QCD axion, the LFV axiflavon and the leptonic familon, paying particular attention to the cases where the LFV ALPs constitute cold dark matter.
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