  Lepton flavor violated process are strongly suppressed in the Standard Model
due to very small neutrino mass, but can be sizable in some extended models.
The current experimental bounds on decay modes $\ell^{'\pm} \to a \ell^{\pm} $
are much weaker than other flavor violated processes because of the huge
irreducible backgrounds $\ell' \to \ell \bar{\nu}_{\ell} \nu_{\ell'}$. In this
paper, we give the full helicity density matrix of both the signal and
backgrounds, and then study polarization effects. Particularly, we treat
inclusively the two missing neutrinos in the background, and we find that both
longitudinal and transverse polarization effects survives even the relative
kinematical degrees of freedom of the two neutrinos are integrated out.
Furthermore, we have show that signal and backgrounds have distinctive
polarization effects which can be measured by using energy fractions of the
charged decay products. This is particularly useful because kinematical
reconstruction is not required. In case of that the decaying lepton is not
polarized, we show that correlations between polarizations of lepton pair
generated at colliders are still useful to search for the signals. More
interestingly, polarization correlations depends on product of scalar and
pseudo-scalar ALP couplings, and hence are sensitive to their relative sign. We
demonstrate that how the polarization correlation effects can be used to
investigate flavor violating decays $\tau^{\pm} \to a \ell^{\pm} $ at the
BelleII experiment.
