LHCb pentaquarks as a baryon-$\psi (2S)$ bound state: Prediction of isospin-$\frac{3}{2}$ pentaquarks with hidden charm

The pentaquark $P_c^{+}(4450)$ recently discovered by the LHCb has been interpreted as a bound state of $\mathrm{\Psi }(2S)$ and a nucleon. The charmonium-nucleon interaction which provides the binding mechanism is given, in the heavy-quark limit, in terms of charmonium chromoelectric polarizabilities and densities of the nucleon energy-momentum tensor. In this work, we show in a model-independent way, by exploring general properties of the effective interaction, that $\mathrm{\Psi }(2S)$ can form bound states with a nucleon and $\mathrm{\Delta }$. Using the Skyrme model to evaluate the effective interaction in the large-$N_c$ limit and estimate $1/N_c$ corrections, we confirm the results from prior work which were based on a different effective model (chiral quark soliton model). This shows that the interpretation of $P_c^{+}(4450)$ is remarkably robust and weakly dependent on the details of the effective theories for the nucleon energy-momentum tensor. We explore the formalism further and present robust predictions of isospin-$\frac{3}{2}$ bound states of $\mathrm{\Psi }(2S)$ and $\mathrm{\Delta }$ with masses around 4.5 GeV and widths around 70 MeV. The approach also predicts broader resonances in the $\mathrm{\Psi }(2S)\text{−}\mathrm{\Delta }$ channel at 4.9 GeV with widths of the order of 150 MeV. We discuss in which reactions these new isospin-$\frac{3}{2}$ pentaquarks with hidden charm can be observed.

Figure 1

EMT densities from the Skyrme model as functions of $r$. The LO results are valid for any $S=I$ in the large-$N_c$ limit. The estimates of NLO corrections in the $1/N_c$ expansion are shown for states with the quantum numbers $S=I=\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2}$. The figures show (a) energy density $T_{00}(r)$, (b) shear forces $s(r)$, (c) pressure $p(r)$ with approximate NLO corrections, (d) $p(r)$ with NLO corrections reconstructed according to (30), (e) same as Fig. 1 but for $r^{2}p(r)$, and (f) the local stability criterion (18). The NLO results in the upper panel, Figs. 1, are estimated by simply evaluating the NLO expressions with the LO soliton profile, which yields unacceptable results especially for $p(r)$ which are at variance with Eq. (14). The results in the lower panel, Figs. 1, are obtained with the pressure reconstructed according to Eq. (30) which satisfies Eq. (14). Finally, Fig. 1 shows that the results for $S=I=\frac{1}{2},\frac{3}{2}$, which correspond to the nucleon and $\mathrm{\Delta }$, comply with the local stability criterion (18). In contrast to this, states with the exotic quantum numbers $S=I\ge 5/2$ do not satisfy (18); i.e., in this way, the rotating soliton approach explains why they are not realized in nature.

Figure 2

The effective potential $V_{\mathrm{eff}}(r)$ normalized with respect to the polarizability $\alpha$ for (a) the nucleon and (b) $\mathrm{\Delta }$ as a function of $r$ in LO and NLO order of the large-$N_c$ expansion after the rescaling in Eq. (32); i.e., $V_{\mathrm{eff}}(r)$ is normalized according to Eq. (23) with respect to the physical value of the respective baryon mass.