  We study the axion and axion-like particle production from the $\eta\to\pi\pi
a$ decay within the $SU(3)$ chiral perturbation theory up to the one-loop
level. The conventional $SU(3)$ chiral low energy constants are found to be
able to reabsorb all the divergences from the chiral loops in the
$\eta\to\pi\pi a$ decay amplitude, and hence render the amplitude independent
of the renormalization scale. The unitarized $\eta\to\pi\pi a$ decay amplitudes
are constructed to take into account the $\pi\pi$ final-state interactions and
also properly reproduce the perturbative results from the chiral perturbation
theory. Detailed analyses between the perturbative amplitudes and the
unitarized ones are given in the phenomenological discussions. By taking the
values of the chiral low energy constants in literature, we predict the Dalitz
distributions, the spectra of the $\pi\pi$ and $a\pi$ systems, and also the
branching ratios of the $\eta\to\pi\pi a$ process by varying $m_a$ from 0 to
$m_\eta-2m_{\pi}$.
