Study of ${\chi }_{cJ}\to \eta \eta {\eta }^{\mathrm{\prime }}$ via intermediate charmed meson loop mechanisms and its implications for non-observation of ${\eta }_1(1855)$ in ${\chi }_{cJ}$ decays

Recently, the BESIII Collaboration reported the first observation of the decays ${\chi }_{cJ}\to \eta \eta {\eta }^{\mathrm{\prime }}$ in order to search for the $1^{-+}$ exotic state ${\eta }_1(1855)$. A partial wave analysis of the $\eta {\eta }^{\mathrm{\prime }}$ invariant mass spectrum shows no significant signal for the ${\eta }_1(1855)$. In this work, we, using an effective Lagrangian approach, investigate the processes ${\chi }_{cJ}\to \eta \eta {\eta }^{\mathrm{\prime }}$ 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 ${\chi }_{cJ}\to \eta \eta {\eta }^{\mathrm{\prime }}$. Furthermore, we present the predictions of the relevant invariant mass spectra of $\eta {\eta }^{\mathrm{\prime }}$ and $\eta \eta$ produced in the ${\chi }_{c1}$ decay, which seem overall consistent with the BESIII measurements. In the present model, the decay ${\chi }_{c1}\to \eta \eta {\eta }^{\mathrm{\prime }}$ 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 ${\chi }_{cJ}$ decaying into light mesons and might be helpful to understand the absence of the ${\eta }_1(1855)$ signal in the decay channels ${\chi }_{cJ}\to \eta \eta {\eta }^{\mathrm{\prime }}$.