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

Abstract

In the molecular picture the hidden-charm, pentaquark-like $P_c(4450)$ resonance is a ${\bar{D}}^{\ast }{\Sigma }_c$ 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 ${\bar{D}}^{\ast }{\bar{D}}^{\ast }{\Sigma }_c$ three-body bound states. The most probable quantum numbers for a bound ${\bar{D}}^{\ast }{\bar{D}}^{\ast }{\Sigma }_c$ 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 $B_3\sim 3-5\mathrm{MeV}$ and $14-16\mathrm{MeV}$ for the isoscalar ${\frac{1}{2}}^{+}$ and ${\frac{5}{2}}^{+}$ states and $B_3\sim 1-3\mathrm{MeV}$ and $3-5\mathrm{MeV}$ for the isovector ${\frac{3}{2}}^{+}$ and ${\frac{5}{2}}^{+}$ states, respectively, with $B_3$ relative to the ${\bar{D}}^{\ast }{P}_c(4450)$ threshold. These predictions are affected by a series of error sources that we discuss in detail.

Comments: 8 pages, 1 figure, extended discussion on regulator dependence, corresponds to published version