Axion production in the $\eta\to \pi\pi a$ decay within $SU(3)$ chiral perturbation theory

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 $m_\eta-2m_{\pi}$.