Study of $χ_{cJ}\to ηηη^\prime$ via intermediate charmed meson loop mechanisms and its implications for non-observation of $η_1(1855)$ in $χ_{cJ}$ decays
Recently, the BESIII Collaboration reported the first observation of the decays $χ_{cJ} \to ηηη^\prime$ in order to search for the $1^{-+}$ exotic state $η_1(1855)$. A partial wave analysis of the $ηη^\prime$ invariant mass spectrum shows no significant signal for the $η_1(1855)$. In this work, we, using an effective Lagrangian approach, investigate the processes $χ_{cJ} \to ηηη^\prime$ via the box and triangle loops involving charmed mesons and the scalar meson $f_0(1500)$. Our calculations reproduce well the experimental branching fractions of $χ_{cJ} \to ηηη^\prime$. Furthermore, we present the predictions of the relevant invariant mass spectra of $ηη^\prime$ and $ηη$ produced in the $χ_{c1}$ decay, which seem overall consistent with the BESIII measurements. In the present model, the decay $χ_{c1} \to ηηη^\prime$ is dominated by the triangle and box loop contributions. The consistency between our theoretical results and the BESIII measurements sheds light on the underlying decay mechanism of the $χ_{cJ}$ decaying into light mesons and might be helpful to understand the absence of the $η_1(1855)$ signal in the decay channels $χ_{cJ} \to ηηη^\prime$.