Search for the reaction channel $e^+ e^- \to ηη\,J/ψ$ and the isospin partner of the $Z_c(3900)$ at center-of-mass energies $\sqrt{s} = 4.226-4.950$ GeV

We search for the reaction channel $e^+ e^- \to ηη\,J/ψ$ in a data sample with center-of-mass energies from 4.226 to 4.950~GeV, which was collected by the BESIII detector operating at the Beijing Electron Positron Collider (BEPCII). The data analysis is performed with two different methods, exclusively and semi-inclusively, which enabling a comparison and combination of the results. Only in a few cases a statistical significance of the cross sections with more than $3σ$ is observed with one of the methods. Only at 4.750~GeV the significance of the cross section measurement is 8.9$σ$ (observation) with the exclusive analysis and 3.4$σ$ (evidence) with the semi-inclusive analysis. Therefore, the corresponding upper limits of the cross section at the 90% confidence level are determined. The energy dependent results show clear deviations from the the line shape expected from three-body phase space alone. Since the statistical significance for almost all center-of-mass energies is low, the upper limits for the reaction channel $e^+ e^- \to ηη\,J/ψ$ also serve as limits for the existence of a possible isospin partner to the charmonium-like isospin triplet $Z_{\rm c}(3900)$ which decays to $J/ψη$.