Mixture of Hadronic Molecules and Charmonium Core

Hadrons that cannot be easily explained as normal hadrons are called exotic hadrons. Since the Belle experiment reported the $X(3872)$ in 2003, there have been numerous reports of exotic hadrons containing charm quarks. Exotic hadrons are thought to have a more complex structure than normal hadrons, with various structures being studied, such as hadron molecular states where multiple hadrons are loosely bound like a deuteron, and compact states where all the constituent quarks are closely packed together. However, no definitive conclusions have been reached yet. $X(3872)$ is one of the most well-known exotic hadrons and has been observed in various experiments. It has quantum numbers $J^{PC} = 1^{++}$, and its mass is very close to the $D^{0}\bar{D}^{*0}$ threshold, with a mass difference of only 0.04 MeV. Additionally, the mass of $X(3872)$ is close to the predicted mass of $χ_{c1}(2P)$ at 3953 MeV according to the Godfrey-Isgur relativized quark model. In this study, $X(3872)$ is analyzed as a superposition state of the $D\bar{D}^*$ hadron molecule, the $D^*\bar{D}^*$ hadron molecule, and the charmonium core $χ_{c1}(2P)$. We consider interactions such as meson exchange and core-molecule mixing potential. The Schrödinger equation is solved using the Gaussian Expansion Method (GEM). The results from the analysis of $X(3872)$ are then used to study the structure of other exotic hadrons