Three-body unitary coupled-channel approach to radiative $J/ψ$ decays and $η(1405/1475)$
Recent BESIII data on radiative $J/ψ$ decays from $\sim 10^{10}$ $J/ψ$ samples should significantly advance our understanding of the controversial nature of $η(1405/1475)$. This motivates us to develop a three-body unitary coupled-channel model for radiative $J/ψ$ decays to three-meson final states of any partial wave ($J^{PC}$). Basic building blocks of the model are bare resonance states such as $η(1405/1475)$ and $f_1(1420)$, and $πK$, $K\bar{K}$, and $πη$ two-body interactions that generate resonances such as $K^*(892)$, $K^*_0(700)$, and $a_0(980)$. This model reasonably fits $K_SK_Sπ^0$ Dalitz plot pseudo data generated from the BESIII's $J^{PC}=0^{-+}$ amplitude for $J/ψ\toγK_SK_Sπ^0$. The experimental branching ratios of $η(1405/1475)\toηππ$ and $η(1405/1475)\toγρ$ relative to that of $η(1405/1475)\to K\bar{K}π$ are simultaneously fitted. Our $0^{-+}$ amplitude is analytically continued to find three poles, two of which correspond to $η(1405)$ on different Riemann sheets of the $K^*\bar{K}$ channel, and the third one for $η(1475)$. This is the first pole determination of $η(1405/1475)$ and, furthermore, the first-ever pole determination from analyzing experimental Dalitz plot distributions with a manifestly three-body unitary coupled-channel framework. Process-dependent $ηππ$, $γπ^+π^-$, and $πππ$ lineshapes of $J/ψ\toγ(0^{-+})\to γ(ηππ)$, $γ(γρ)$, and $γ(πππ)$ 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 $J/ψ\toγ(πππ)$.