Is the $3S$-$2D$ mixing strong for the charmonia $\psi (4040)$ and $\psi (4160)$?

In this work, we revisit the $3S$-$2D$ mixing scheme for the charmonia $\psi (4040)$ and $\psi (4160)$. We introduce a coupled-channel mechanism-distinct from the tensor-force contribution in potential models, which alone is insufficient to induce significant mixing-to describe the mixing between these states. Our analysis yields mixing angles of ${\theta }_1=7^{\circ }$ and ${\theta }_2={10}^{\circ }$, inconsistent with the larger angle inferred from experimental data, such as the dilectronic widths of the $\psi (4040)$ and $\psi (4160)$. We discuss possible origins of this discrepancy and emphasize the need for future experiments to resolve it. Precise measurements of the resonance parameters and dilectronic decay widths, via both inclusive and exclusive processes, will be crucial in clarifying this issue.