Three-body unitary coupled-channel approach to radiative $J/ψ$ decays and $η(1405/1475)$

Abstract: Recent BESIII data on radiative $J/\psi$ decays from $\sim 10^{10}$ $J/\psi$ samples should significantly advance our understanding of the controversial nature of $\eta(1405/1475)$. This motivates us to develop a three-body unitary coupled-channel model for radiative $J/\psi$ decays to three-meson final states of any partial wave ($J^{PC}$). Basic building blocks of the model are bare resonance states such as $\eta(1405/1475)$ and $f_1(1420)$, and $\pi K$, $K\bar{K}$, and $\pi\eta$ two-body interactions that generate resonances such as $K^*(892)$, $K^*_0(700)$, and $a_0(980)$. This model reasonably fits $K_SK_S\pi^0$ Dalitz plot pseudo data generated from the BESIII's $J^{PC}=0^{-+}$ amplitude for $J/\psi\to\gamma K_SK_S\pi^0$. The experimental branching ratios of $\eta(1405/1475)\to\eta\pi\pi$ and $\eta(1405/1475)\to\gamma\rho$ relative to that of $\eta(1405/1475)\to K\bar{K}\pi$ are simultaneously fitted. Our $0^{-+}$ amplitude is analytically continued to find three poles, two of which correspond to $\eta(1405)$ on different Riemann sheets of the $K^*\bar{K}$ channel, and the third one for $\eta(1475)$. This is the first pole determination of $\eta(1405/1475)$ and, furthermore, the first-ever pole determination from analyzing experimental Dalitz plot distributions with a manifestly three-body unitary coupled-channel framework. Process-dependent $\eta\pi\pi$, $\gamma\pi^+\pi^-$, and $\pi\pi\pi$ lineshapes of $J/\psi\to\gamma(0^{-+})\to \gamma(\eta\pi\pi)$, $\gamma(\gamma\rho)$, and $\gamma(\pi\pi\pi)$ are predicted, and are in reasonable agreement with data. A triangle singularity is shown to play a crucial role to cause the large isospin violation of $J/\psi\to\gamma(\pi\pi\pi)$.