Understanding $X(3862)$, $X(3872)$, and $X(3930)$ in a Friedrichs-model-like scheme

We developed a Friedrichs-model-like scheme in studying the hadron resonance phenomenology and present that the hadron resonances might be regarded as the Gamow states produced by a Hamiltonian in which the bare discrete state is described by the result of the usual quark potential model and the interaction part is described by the quark pair creation model. In an almost parameter-free calculation, the $X(3862)$, $X(3872)$, and $X(3930)$ state could be simultaneously produced with a quite good accuracy by coupling the three P-wave states, ${\chi }_{c2}(2P)$, ${\chi }_{c1}(2P)$, ${\chi }_{c0}(2P)$ predicted in the Godfrey-Isgur model to the $D\overline{D}$, $D{\overline{D}}^{*}$, $D^{*}{\overline{D}}^{*}$ continuum states. At the same time, we predict that the $h_c(2P)$ state is at about 3890 MeV with a pole width of about 44 MeV. In this calculation, the $X(3872)$ state has a large compositeness. This scheme may shed more light on the long-standing problem about the general discrepancy between the prediction of the quark model and the observed values, and it may also provide reference for future search for the hadron resonance state.

Figure 1

The mass distribution of $\gamma \gamma \to D\overline{D}$ from BABAR [24] and Belle [1]. The data of Belle is the one for $|\cos {\theta }^{*}|<0.5$. The two dashed lines are set at $m(D\overline{D})=3850\mathrm{MeV}$ and 3875 MeV.