Dynamical study of $D^{*}DK$ and $D^{*}D \bar{D}$ systems at quark level

Inspired by that Belle\uppercase\expandafter{\romannumeral2} Collaboration recently reported $T_{cc}$, which can be interpreted as a molecular $DD^{*}$, we investigated the trihadron system of $T_{cc}$ partner with $IJ^{P}$=$01^{-}$ in the framework of a chiral quark model. It's widely accepted that the main component of $X(3872)$ contains the molecular $\bar{D}D^{*}$, while the main component of $D_{s0}^{*}(2317)$ is molecular $DK$. Based on these three well-known exotic states, $T_{cc} (DD^{*})$, $X(3872) (\bar{D}D^{*})$ and $D_{s0}^{*}(2317) (DK)$, we dynamically investigate $D^{*}DK$ and $DD^{*}\bar{D}$ systems at quark level to search for possible bound states. The results show that both of them are bound states, in which the binding energy of the molecular state $DD^*K$ is relatively small, only 0.8 MeV, while the binding energy of $DD^*\bar{D}$ is up to 1.9 MeV. According to the calculation results of the Root-square-mean distances, the spatial structure of the two systems shows obvious ($DD^*$)-($\bar{D}$/$K$) structure, in which $D$ is close to $D^*$ while $DD^*$ as a whole is relatively distant from the third hadron ($\bar{D}$/$K$), which are similar to the nucleon-electron structure. As a result, we strongly recommend that these bound states $DD^*\bar{D}$ and $DD^*K$ are searched for experimentally.