\eta\ transitions between charmonia with meson loop contributions

We study the $\eta$ transitions between $\psi(4040/4160)$ and $J/\psi$ 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 $\eta^\prime$ transition between $\psi(4160)$ and $J/\psi$ is also investigated, and the branching fraction is under the upper limit of CLEO, which can be tested by future experiments.