New insight into the exotic states strongly coupled with the DD‾∗ from the Tcc+

We have investigated the internal structure of the open- and hidden-charmed (${\mathrm{DD}}^{\ast }$/$\overline{D}{D}^{\ast }$) molecules in the unified framework. We first fit the experimental lineshape of the $T_{\mathrm{cc}}^{+}$ state and extract the ${\mathrm{DD}}^{\ast }$ interaction, from which the $T_{\mathrm{cc}}^{+}$ is assumed to arise solely. Then we obtain the $D{\overline{D}}^{\ast }$ interaction by charge conjugation. Our results show that the $D{\overline{D}}^{\ast }$ interaction is attractive but insufficient to form $X\left(3872\right)$ as a bound state. Instead, its formation requires the crucial involvement of the coupled channel effect between the $D{\overline{D}}^{\ast }$ and $c\overline{c}$ components, although the $c\overline{c}$ accounts for approximately $1\%$ only. Besides $X\left(3872\right)$, we have obtained a higher-energy state around $3957.9$ MeV with a width of $16.7$ MeV, which may be a potential candidate for the $X\left(3940\right)$. In $J^{\mathrm{PC}}=1^{+-}$ sector, we have found two states related to the iso-scalar $\tilde{X}\left(3872\right)$ and $h_c\left(2P\right)$, respectively. Our combined study provides valuable insights into the nature of these ${\mathrm{DD}}^{\ast }$/$D{\overline{D}}^{\ast }$ exotic states.