Study of the molecular Properties of the $P_c$ and $P_{cs}$ States

Abstract

In the present work, we systematically investigate the meson-baryon molecular properties of the hidden charm pentaquark states $P_c$ and $P_{cs}$ within a coupled channel framework that combines heavy quark spin symmetry and the local hidden gauge formalism. By solving the Bethe-Salpeter equation with the momentum cutoff method, we obtain the pole trajectories, wave functions, and root-mean-square radii. For the hidden charm system, the full coupled channel interactions respecting the heavy quark spin symmetry are essential to generate the $P_c$ states, as they significantly affect the poles' widths. The dominant bound channels are $\bar{D} Σ_c$ and $\bar{D}^* Σ_c$, which couple strongly to lower decay channels. In contrast, for the hidden charm strange system, the full heavy quark spin symmetry treatment is not necessary, where the splitting PB and VB sectors yield similar results. The main bound channels $\bar{D} Ξ_c$ and $\bar{D}^* Ξ_c$ couple strongly to $\bar{D}_s Λ_c$ and $\bar{D}_s^* Λ_c$, respectively, but only weakly to the lower decay channels, differing from the hidden charm case. The trajectories of the pole widths for the loosely bound channels $\bar{D} Ξ'_c$, $\bar{D}^* Ξ'_c$, and $\bar{D}^* Ξ_c^*$ exhibit distinct behaviors. Notably, all the primary bound channels have similar binding energies in the single channel interactions due to equally attractive potentials. Furthermore, we also calculate the wave functions and root-mean-square radii of the corresponding poles. The wave functions are localized within $0\sim 6$ fm and vanish fast beyond $4$ fm. The root-mean-square radii, evaluated by two consistent methods, typically lie between $0.5$ and $2$ fm, comparable to the characteristic scale of molecular states.

Figures (23)

• Figure 1: Mass (left) and width (right) trajectories of the poles in the second Riemann sheets for the $I = 1/2,\, J^P = 1/2^-$ sector as a function of the cutoff $q_{max}$ in the seven coupled-channel case.
• Figure 2: Mass (left) and width (right) trajectories of the poles in the second Riemann sheets for the $I = 1/2,\, J^P = 1/2^-$ sector as a function of the cutoff $q_{max}$ in the splitting PB and VB sectors.
• Figure 3: Real (left) and imaginary (right) parts of the wave functions $\phi(r)$ of corresponding pole for the $I = 1/2,\, J^P = 1/2^-$ sector with different $q_{max}$ in the seven coupled-channel case.
• Figure 4: Real (left) and imaginary (right) parts of the wave functions $\phi(r)$ of corresponding pole for the $I = 1/2,\, J^P = 1/2^-$ sector with different $q_{max}$ in the splitting PB and VB sectors.
• Figure 5: RMS radii of the corresponding poles for the $I = 1/2,\, J^P = 1/2^-$ sector as a function of the cutoff $q_{max}$ in the seven coupled-channel case (left) and the splitting PB and VB sectors (right). Results from Method 1 (blue) and Method 2 (red) are compared.