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

In this article, the analyticity of triangle loop integral with complex masses of internal particles is discussed in a new perspective, based 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 KK¯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 1.40–1.48 GeV are originated from the same state. For the isospin violated process J/ψ→γη(1405/1475)→f0(980)π→3π, we identify an additional contribution to the a0(980)−f0(980) mixing via the TS mechanism.