P-wave $c\bar{c}$ meson contributions in exotic hadrons

The nature of the $X(3872)$ and other exotic hadrons has been a subject of extensive investigation. While various theoretical models have been proposed, 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 hidden-charm tetraquark candidates $X(3860)$, $X(3872)$, and $Z(3930)$ using a coupled-channel model that incorporates both $c\bar{c}$ states and $D^{(\ast )}{\bar{D}}^{(\ast )}$ hadronic molecular components. 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 $0^{++}$ 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 $D^{(\ast )}{\bar{D}}^{(\ast )}$ components. The molecular component dominates in the $X(3872)$, while the $c\bar{c}$ component plays a more prominent role in the $X(3860)$ and $Z(3930)$.