The ${\bar{D}}^{\ast }{\bar{D}}^{\ast }{\mathrm{\Sigma }}_c$ Three-Body System

In the molecular picture the hidden-charm, pentaquark-like $P_c(4450)$ resonance is a ${\bar{D}}^{\ast }{\mathrm{\Sigma }}_c$ bound state with quantum numbers $I=\frac{1}{2}$ and $J^P={\frac{3}{2}}^{-}$. If this happens to be the case, it will be natural to expect the existence of ${\bar{D}}^{\ast }{\bar{D}}^{\ast }{\mathrm{\Sigma }}_c$ three-body bound states. The most probable quantum numbers for a bound ${\bar{D}}^{\ast }{\bar{D}}^{\ast }{\mathrm{\Sigma }}_c$ trimer are the isoscalar $J^P={\frac{1}{2}}^{+}$, ${\frac{5}{2}}^{+}$ and the isovector $J^P={\frac{3}{2}}^{+}$, ${\frac{5}{2}}^{+}$ configurations. Calculations within a contact-range theory indicate a trimer binding energy $B_3\sim 3-5\mathrm{M}\mathrm{e}\mathrm{V}$ and $14-16\mathrm{M}\mathrm{e}\mathrm{V}$ for the isoscalar ${\frac{1}{2}}^{+}$ and ${\frac{5}{2}}^{+}$ states and $B_3 \sim 1-3,{\rm MeV}$and$3-5,{\rm MeV}$fortheisovector$\tfrac{3}{2}^+$and$\tfrac{5}{2}^+$states,respectively,with$B_3$relativetothe$\bar{D}^* P_c(4450)$ threshold. These predictions are affected by a series of error sources that we discuss in detail.