\eta\ transitions between charmonia with meson loop contributions

We study the $\eta$ transitions between $\psi (4040\mathrm{/}4160)$ and $J\mathrm{/}\psi$ by introducing charmed meson loops in an effective Lagrangian approach to enhance the decay amplitudes. The branching fractions $$\mathcal{B}$[\psi(4040) \to J/\psi \eta]$and$$\mathcal{B}$[\psi(4160) \to J/\psi \eta]$ estimated in this paper can remarkably explain the experimental measurements of Belle and BESIII within a reasonable parameter range. The ${\eta }^{\mathrm{\prime }}$ transition between $\psi (4160)$ and $J\mathrm{/}\psi$ is also investigated, and the branching fraction is under the upper limit of CLEO, which can be tested by future experiments.