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

Abstract

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 $η(1405/1475)$ 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 $K\bar{K}π$, $ηππ$, and $3π$ decay channels for $η(1405/1475)$. Our results confirm the claim that the TS mechanism plays a decisive role in the understanding of the $η(1405)$ and $η(1475)$ puzzle. Namely, the observed differences of $η$ resonances within the mass region of $1.40\sim 1.48$ GeV are originated from the same state. For the isospin violated process $J/ψ\toγη(1405/1475)\to f_0(980)π\to 3π$, we identify an additional contribution to the $a_0(980)-f_0(980)$ mixing via the TS mechanism.