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Based on the assignment of the first radial excitation states of theisoscalar pseudoscalars for $\eta(1295)$ and $\eta(1405/1475)$, we investigatetheir three-body and four-body decay contributions to the total widths. Inagreement with our previous studies we find that the triangle singularity (TS)mechanism arising from the intermediate $K^*\bar{K}$ rescatterings byexchanging a kaon or pion plays a crucial role in both $K\bar{K}\pi$ and$\eta\pi\pi$ channels. For the $\eta_X$ ($\eta_X$ stands for $\eta(1295)$ and$\eta(1405/1475)$) decays into $K\bar{K}\pi$, we find that although thetransition $\eta_X\to K^*\bar{K}+c.c.\to K\bar{K}\pi$ is the dominanttree-level process, the productions of the intermediate $K\bar{\kappa}+c.c.$and $a_0(980)\pi$ are strongly enhanced by the TS mechanism. For the $\eta_X$decays into $\eta\pi\pi$, we find that the production of the intermediate$a_0(980)\pi$ via the triangle transition is the dominant one for $\eta(1295)$partly because of the large $\eta(1295)K^*\bar{K}$ coupling. In contrast, thetree-level and triangle loop contributions are compatible and dominant in the$\eta(1405/1475)$ decays into $\eta\pi\pi$. It shows that a combined analysisis useful for disentangling the underlying dynamics for these two states.

On the widths of $\eta(1295)$ and $\eta(1405/1475)$