$a_0(980)$ production, triangle singularity, and non-$\phi$ background in the $J\mathrm{/}\psi \to \phi \eta {\pi }^0$ reaction

We study the $J\mathrm{/}\psi \to \phi \eta {\pi }^0$ reaction measured recently by the BESIII collaboration with high precision, paying attention to three important aspects: 1) The production of the $a_0(980)$ in the ${\pi }^0\eta$ mass distributions, with the typical narrow width observed in isospin violating processes; 2) The origin of two peaks in the $\phi {\pi }^0$ mass distributions that were branded as non $φ$" contribution in the experimental analysis; 3) The existence of two triangle mechanisms developing a triangle sigularity at the same energy where the non $\phi$" contribution peaks in the experiment. However, we also show that the strength of these peaks is very small relative to the observed ones, a feature which is tied to the experimental technique used to identify the $\phi$ looking at $K^{+}{K}^{-}$ in a narrow window of invariant mass around the $\phi$ mass. We suggest that these triangle singularities could be observed with other methods to identify the $\phi$.