Double heavy tri-hadron bound state via delocalized $\pi$ bond

The number of exotic candidates which are beyond the conventional quark model has grown dramatically during the last decades. Some of them could be viewed as analogues of the deuteron. Similarly, the existence of the triton indicates that bound states formed by three hadrons could also exist. To illustrate this possibility, we study the $DD^*K$ and $BB^*\bar{K}$ systems by using the Born-Oppenheimer Approximation. To leading order, only one-pion exchange potentials are considered, which means that the three constitutes share one virtual pion. That is similar to the role of the delocalized $\pi$ bond for the formation of Benzene in chemistry. After solving the Schr\"odinger equation, we find two three-body $DD^*K$ and $BB^*\bar{K}$ bound states with masses $4317.92_{-4.32}^{+3.66}~\mathrm{MeV}$ and $11013.65_{-8.84}^{+8.49}~\mathrm{MeV}$, respectively. The masses of their $D\bar{D}^*K$ and $B\bar{B}^*\bar{K}$ analogues are $4317.92_{-6.55}^{+6.13}~\mathrm{MeV}$ and $11013.65_{-9.02}^{+8.68}~\mathrm{MeV}$. From the experimental side, the $D\bar{D}^*K$ bound state could be found by analyzing the current world data of the $B\to J/\psi\pi\pi K$ process by focusing on the $J/\psi \pi K$ channel. Its confirmation could also help to understand the formation of kaonic nuclei in nuclear physics.