Binding of the $BD\overline{D}$ and $BDD$ systems

We study theoretically the $BD\overline{D}$ and $BDD$ systems to see if they allow for possible bound or resonant states. The three-body interaction is evaluated implementing the fixed center approximation to the Faddeev equations which considers the interaction of a $D$ or $\overline{D}$ particle with the components of a $BD$ cluster, previously proved to form a bound state. We find an $I(J^P)=1/2(0^{-})$ bound state for the $BD\overline{D}$ system at an energy around 8925–8985 MeV within uncertainties, which would correspond to a bottom hidden-charm meson. In contrast, for the $BDD$ system, which would be bottom double-charm and hence manifestly exotic, we have found hints of a bound state in the energy region 8935–8985 MeV, but the results are not stable under the uncertainties of the model, and we cannot assure, nor rule out, the possibility of a $BDD$ three-body state.

Figure 1

Multiple scattering diagrams that go into the construction of the partition function $T_{11}$.

Figure 2

$|T_{BD\overline{D}}{|}^2$ with prescriptions I, II for $\sqrt{s_{\overline{D}B}}$, $\sqrt{s_{\overline{D}D}}$ and $q_{\max }=600\mathrm{MeV}$ with and without considering width (from $\eta \eta$ channel) for the $X(3700)$ through the $D\overline{D}$ interaction. The two curves with $\eta \eta$ channel were multiplied by a factor $10^4$ for comparison.

Figure 3

$|T_{BDD}{|}^2$ with $q_{\max }=400\mathrm{MeV}$ and $q_{\max }=600\mathrm{MeV}$ for the $BD$ cluster, using prescription I (a) and prescription II (b) for $\sqrt{s_{DB}}$, $\sqrt{s_{DD}}$.