Heavy-Quark Symmetry Partners of the Pc(4450) Pentaquark

The spectrum of heavy-hadron molecules is constrained by heavy-quark symmetry in its different manifestations. Heavy-quark spin symmetry for instance connects the properties of the ground and excited states of heavy hadrons, while heavy-antiquark-diquark symmetry connects the properties of heavy antimesons ($\bar{D}$, ${\bar{D}}^{\ast }$) and doubly heavy baryons (${\mathrm{\Xi }}_{cc}$, ${\mathrm{\Xi }}_{cc}^{\ast }$). A prediction of these symmetries is that if the $P_c(4450)$ is indeed a ${\bar{D}}^{\ast }{\mathrm{\Sigma }}_c$ bound state, then there should be a series of ${\bar{D}}^{\ast }{\mathrm{\Sigma }}_c^{\ast }$, ${\mathrm{\Xi }}_{cc}{\mathrm{\Sigma }}_c$, ${\mathrm{\Xi }}_{cc}^{\ast }{\mathrm{\Sigma }}_c$, $\Xi_{cc} \Sigma_c^$and$\Xi_{cc}^ \Sigma_c^$ partners. The concrete application of heavy-quark spin symmetry indicates that, if the $P_c(4450)$ is a ${\frac{3}{2}}^{-}$ ${\bar{D}}^{\ast }{\mathrm{\Sigma }}_c$ molecule, the existence of a ${\frac{5}{2}}^{-}$ ${\bar{D}}^{\ast }{\mathrm{\Sigma }}_c^{\ast }$ partner with similar binding energy — which we call $P_c(4515)$, given its expected mass — is likely. Conversely, the application of heavy-antiquark-diquark symmetry indicates that the $0^{+}$ ${\mathrm{\Xi }}_{cc}{\mathrm{\Sigma }}_c$, $1^{+}$ ${\mathrm{\Xi }}_{cc}{\mathrm{\Sigma }}_c^{\ast }$, $2^{+}$ $\Xi_{cc}^ \Sigma_c$and$3^+$ ${\mathrm{\Xi }}_{cc}^{\ast }{\mathrm{\Sigma }}_c^{\ast }$ molecules are likely to bind too, with binding energies in the $20-30\mathrm{M}\mathrm{e}\mathrm{V}$ range.