${\overline{D}}^{*}{\overline{D}}^{*}{\mathrm{\Sigma }}_c$ three-body system

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

Figure 1

Binding energy $B_3$ of the ${\overline{D}}^{*}{\overline{D}}^{*}{\mathrm{\Sigma }}_c$ trimer as a function of the coefficient $\lambda$, which parametrizes the overlap of the trimer quantum numbers with the ${\overline{D}}^{*}{\mathrm{\Sigma }}_c$ system in the $P_c^{*}$ channel. The bars represent the error coming from the uncertainty in the binding energy of the $P_c^{*}$, $B_2=12\pm 3\mathrm{MeV}$.