Proposed mixing between $2P$ and $1F$ wave charmonia

We investigate $2P$-$1F$ mixing in charmonium, focusing on the close-in-mass $\chi_{c2}(2P)$ and $\chi_{c2}(1F)$ states. The conventional tensor force yields negligible mixing, motivating the inclusion of coupled-channel effects. Our unquenched calculation reveals sizable mixing angles of $7.5^\circ$ and $15.4^\circ$. We predict the corresponding two-photon and two-gluon decay widths as key observables for experimental verification. Additionally, we discuss the production of these two $2P$-$1F$ mixed states of charmonium via $\gamma\gamma$ fusion. Current data are insufficient to determine the mixing, highlighting the need for precise future measurements to resolve this aspect of charmonium spectroscopy.