Internal particle width effects on the the triangle singularity mechanism in the study of the $\eta(1405)$ and $\eta(1475)$ puzzle

In this article, the analyticity of triangle loop integral with complex masses of internal particles is discussed in a new perspective, base on which we obtain the explicit width dependence of the absorptive part of the triangle amplitude. We reanalyze the decay pattern of $\eta(1405/1475)$ with the width effects included in the triangle singularity (TS) mechanism. Based on the present experimental information, we provide a self-consistent description of the $K\bar{K}\pi$, $\eta\pi\pi$, and $3\pi$ decay channels for $\eta(1405/1475)$. Our results confirm the claim that the TS mechanism plays a decisive role in the understanding of the $\eta(1405)$ and $\eta(1475)$ puzzle. Namely, the observed differences of $\eta$ resonances within the mass region of $1.40\sim 1.48$ GeV are originated from the same state. For the isospin violated process $J/\psi\to\gamma\eta(1405/1475)\to f_0(980)\pi\to 3\pi$, we identify an additional contribution to the $a_0(980)-f_0(980)$ mixing via the TS mechanism.