Final state rescattering effects in axio-hadronic $\eta$ and $\eta^{\prime}$ decays

It has been long-understood that final state rescattering effects provide $\mathcal{O}(1)$ corrections to hadronic meson decays rates, such as $\eta\to\pi\pi\pi$ and $\eta^{\prime}\to\eta\pi\pi$. Hence, one would expect that such effects would be just as important in axio-hadronic $\eta$ and $\eta^{\prime}$ decays, such as $\eta^{(\prime)}\to\pi\pi a$, where $a$ is an axion or axion-like particle (ALP). And indeed they are, as we show in this paper by using the treatment of dispersion relations to include the effects of strong final state interactions in several axio-hadronic processes, namely, $\eta^{(\prime)}\to\pi^0\pi^0 a$, $\eta^{(\prime)}\to\pi^+\pi^- a$, and $\eta^{\prime}\to\eta\pi^0 a$. We also compute the perturbative, leading order decay rates for multiple ALP emission, such as in $\eta^{(\prime)}\to\pi^{0}aa$, $\eta^{\prime}\to\eta aa$ and $\eta^{(\prime)}\to aaa$, and briefly discuss the expected corrections from strong interactions and the processes that must be considered for an accurate rate estimation of these multi-ALP decay channels.