Three-body unitary coupled-channel approach to radiative $J/\psi$ decays and $\eta (1405/1475)$

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 $f_1\left(1420\right)$, and $\pi K$, $K\overline{K}$, and $\pi \eta$ two-body interactions that generate resonances such as $K^{*}\left(892\right)$, $K_0^{*}\left(700\right)$, and $a_0\left(980\right)$. This model reasonably fits $K_S{K}_S{\pi }^0$ Dalitz plot pseudodata generated from the BESIII's $J^{PC}=0^{-+}$ amplitude for $J/\psi \to \gamma K_S{K}_S{\pi }^0$. The experimental branching ratios of $\eta (1405/1475)\to \eta \pi \pi$ and $\eta (1405/1475)\to \gamma \rho$ relative to that of $\eta (1405/1475)\to K\overline{K}\pi$ are simultaneously fitted. Our $0^{-+}$ amplitude is analytically continued to find three poles, two of which correspond to $\eta \left(1405\right)$ on different Riemann sheets of the $K^{*}\overline{K}$ channel, and the third one for $\eta \left(1475\right)$. 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 $J/\psi \to \gamma \left(0^{-+}\right)\to \gamma \left(\eta \pi \pi \right)$, $\gamma \left(\gamma \rho \right)$, and $\gamma \left(\pi \pi \pi \right)$ 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 $J/\psi \to \gamma \left(\pi \pi \pi \right)$.