Search for the $X(2370)$ and observation of $η_{c}\toηηη^\prime$ in $J/ψ\toγηηη^{\prime}$
Using a sample of $1.31\times10^{9} ~J/ψ$ events collected with the BESIII detector, we perform a study of $J/ψ\toγηηη^{\prime}$ to search for the $X(2370)$ and $η_{c}$ in the $ηηη^{\prime}$ invariant mass distribution. No significant signal for the $X(2370)$ is observed, and we set an upper limit for the product branching fraction of ${\cal B}(J/ψ\toγX(2370)\cdot{\cal B}(X(2370)\toηηη^{\prime}) < 9.2\times10^{-6}$ at the 90% confidence level. A clear $η_{c}$ signal is observed for the first time, yielding a product branching fraction of ${\cal B}(J/ψ\to γη_{c})\cdot{\cal B}(η_{c}\to ηηη^{\prime}) = (4.86\pm0.62~({\rm stat.})\pm0.45~({\rm sys.}))\times10^{-5}$.