Three-Body Unitary Coupled-Channel Analysis on $η(1405/1475)$

Abstract: The recent BESIII data on $J/\psi\to\gamma(K_SK_S\pi^0)$, which is significantly more precise than earlier $\eta(1405/1475)$-related data, enables quantitative discussions on $\eta(1405/1475)$ at the previously unreachable level. We conduct a three-body unitary coupled-channel analysis of experimental Monte-Carlo outputs for radiative $J/\psi$ decays via $\eta(1405/1475)$: $K_SK_S\pi^0$ Dalitz plot distributions from the BESIII, and branching ratios of $\gamma(\eta\pi^+\pi^-)$ and $\gamma(\gamma\pi^+\pi^-)$ final states relative to that of $\gamma(K\bar{K}\pi)$. Our model systematically considers (multi-)loop diagrams and an associated triangle singularity, which is critical in making excellent predictions on $\eta(1405/1475)\to \pi\pi\pi$ lineshapes and branching ratios. The $\eta(1405/1475)$ pole locations are revealed for the first time. Two poles for $\eta(1405)$ are found on different Riemann sheets of the $K^*\bar{K}$ channel, while one pole for $\eta(1475)$. The $\eta(1405/1475)$ states are described with two bare states dressed by continuum states. The lower bare state would be an excited $\eta^\prime$, while the higher one could be an excited $\eta^{(\prime)}$, hybrid, glueball, or their mixture. This work presents the first-ever pole determination based on a manifestly three-body unitary coupled-channel framework applied to experimental three-body final state distributions (Dalitz plots).