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

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 $K^{\ast }(892)$, $K_0^{\ast }(700)$, and $a_0(980)$. This model reasonably fits $K_S{K}_S{\pi }^0$ Dalitz plot pseudo data 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 $\eta (1405/1475)\to K\bar{K}\pi$ 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 $K^{\ast }\bar{K}$ 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 $J/\psi \to \gamma (\pi \pi \pi )$.

Comments: 23 pages, 19 figures; (v2) minor changes, published version