Hidden-Charm Tetraquarks in a Mixture Model: Coupled-Channel Analysis with $c\bar{c}$ and Hadronic Molecular Components

The nature of the $X(3872)$ and other exotic hadrons has been a subject of extensive investigation since the first observation of the $X(3872)$ in 2003. While various theoretical models have been proposed, including hadronic molecular and compact tetraquark interpretations, some experimental evidence suggests that the $X(3872)$ may be a mixture state of a hadronic molecule and a $c\bar{c}$ core.In this work, we perform a systematic study of the hidden-charm tetraquark candidates $X(3860)$, $X(3872)$, and $Z(3930)$ using a coupled-channel model that incorporates both $c\bar{c}$ states and $D^{(*)}\bar{D}^{(*)}$ hadronic molecular components. The $c\bar{c}$ sector is described based on the constituent quark model predictions for the $\chi_{cJ}(2P)$ ($J = 0, 1, 2$) states, while the meson-meson interactions are modeled using pseudoscalar and vector meson exchange potentials.The model parameters are fixed to reproduce the masses of the $X(3872)$ and $Z(3930)$, and the resulting framework is used to predict the mass and structure of the $J^{PC} = 0^{++}$ state associated with the $X(3860)$. Our results support the mixture interpretation of these exotic hadrons, exhibiting strong attractions from the transition potential between $c\bar{c}$ and $D^{(*)}\bar{D}^{(*)}$ components. The molecular component is found to dominate in the $X(3872)$, while the $c\bar{c}$ component plays a more prominent role in the $X(3860)$ and $Z(3930)$.