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 $\bar{D} Ξ&#39;_c$, $\bar{D}^* Ξ&#39;_c$, and $\bar{D}^* Ξ_c<sup>*$ 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.

• The paper establishes that fully coupled-channel dynamics with HQSS constraints yield accurate mass and width predictions for P_c and P_cs states.
• It employs the Bethe-Salpeter equation with a momentum cutoff to extract pole positions, wave functions, and RMS radii, highlighting spatial localization.
• Findings indicate that channel coupling significantly affects decay widths and binding energies, reinforcing the molecular nature over compact configurations.

Molecular Properties of the $P_c$ and $P_{cs}$ States

Introduction

This work conducts a systematic analysis of the molecular nature of hidden-charm pentaquark states $P_c$ and $P_{cs}$, employing a coupled-channel framework that synthesizes heavy quark spin symmetry (HQSS) and the local hidden gauge (LHG) approach. By solving the Bethe-Salpeter equation with a momentum cutoff, the study characterizes the poles, wave functions, and root-mean-square (RMS) radii associated with these states. The analysis encompasses the full coupled-channel scenario, sectoral splittings (pseudoscalar-baryon [PB] and vector-baryon [VB]), and single-channel dynamics. The impact of channel coupling, the relevance of HQSS constraints, the sensitivity to the cutoff scale, and the spatial properties of the dynamically generated states are investigated quantitatively, with a particular focus on comparing the hidden-charm and hidden-charm-strange sectors.

Theoretical Framework

The coupled-channel $T$-matrix is derived via the on-shell Bethe-Salpeter equation,

$T = [1 - VG]^{-1}V,$

where $V$ is the HQSS-constrained potential matrix and $G$ is the diagonal matrix of loop functions, regularized via a sharp three-momentum cutoff $q_{max}$. The LHG scheme provides the $V$-matrix elements, integrating short-range vector-meson exchange and heavy-quark flavor structure. Poles in the complex energy plane are located by searching for zeros of $P_{cs}$0, with analytic continuation to the second Riemann sheet to isolate resonance signatures.

Wave functions in configuration space are reconstructed from the pole residues, yielding localization profiles for the molecular states. The spatial radii are extracted by two consistent procedures: directly from the slope of the calculated form factor at vanishing momentum transfer and via the $P_{cs}$1-function derivative scaled by the coupling constant squared and reduced mass.

Hidden-Charm ($P_{cs}$2) Sector

Coupled Channel Spectrum and Channel Dependence

The $P_{cs}$3, $P_{cs}$4 sector (seven channels) presents three dominant poles, each tracking nearby meson-baryon thresholds: $P_{cs}$5, $P_{cs}$6, and $P_{cs}$7. Increasing $P_{cs}$8 systematically lowers the pole positions and broadens the widths—a direct manifestation of enhanced short-range attraction and increased phase space for open-channel decay Figure 1.

Figure 1: Mass (left) and width (right) trajectories of the poles in the second Riemann sheets for the $P_{cs}$9, $P_c$0 sector as a function of cutoff $P_c$1.

Splitting into PB and VB subsystems reveals that while pole masses are relatively stable with respect to HQSS breaking, decay widths of non-PB poles diminish substantially without coupled-channel mixing Figure 2.

Figure 2: Mass and width trajectories in the PB and VB split sectors, showing reduced width for the $P_c$2 pole without PB-VB coupling.

Microscopic Structure: Wave Functions and Radii

The configuration-space wave functions are sharply localized, with support almost exclusively in $P_c$3 fm and rapid attenuation beyond this scale for all bound channels Figure 3. The RMS radii, determined by both aforementioned methods, typically range from $P_c$4 to $P_c$5 fm for dynamically generated resonances, consistent with expectations for hadronic molecules. Deviations occur when poles approach threshold, where Method 2 (form factor) maintains numerical stability.

Figure 3: Real and imaginary parts of the wave functions $P_c$6 for the main poles; significant localization within $P_c$7 fm.

Figure 4: RMS radii as a function of $P_c$8 for the principal $P_c$9 candidates, confirming the molecular-scale size.

$P_{cs}$0, $P_{cs}$1 Sector

The five-channel scenario supports analogously structured poles: $P_{cs}$2, $P_{cs}$3, $P_{cs}$4. Pole trajectories as a function of cutoff and their width evolution follow the same qualitative trends as in the $P_{cs}$5 sector (Figures 6, 7). The dominant $P_{cs}$6 pole aligns with the experimentally observed $P_{cs}$7 when coupled-channel effects are included.

Figure 5: Pole mass and width evolution for the $P_{cs}$8, $P_{cs}$9 sector.

Hidden-Charm-Strange ($T$0) Sector

Coupled Channels and Binding Dynamical Patterns

The $T$1, $T$2 nine-channel system yields several narrow-bound molecular candidates: $T$3, $T$4, $T$5, $T$6, and $T$7. The $T$8 and $T$9 poles are deeply bound, coupling predominantly to $T = [1 - VG]^{-1}V,$0 and $T = [1 - VG]^{-1}V,$1 rather than to low-lying open channels, which results in extremely small width, particularly for large $T = [1 - VG]^{-1}V,$2 Figure 6. This behavior persists even when channel coupling is neglected, indicating a lesser importance for HQSS mixing in this system Figure 7.

Figure 6: Trajectories for the main $T = [1 - VG]^{-1}V,$3 pole masses and widths in the full nine-channel coupled scenario.

Figure 7: Pole evolution in PB and VB split sectors for $T = [1 - VG]^{-1}V,$4, $T = [1 - VG]^{-1}V,$5, reinforcing the weak-coupling scenario.

Spatial Observables

Wave functions for $T = [1 - VG]^{-1}V,$6 molecular states again vanish outside $T = [1 - VG]^{-1}V,$7 fm (Figures 12, 13). The RMS radii (Figures 13, 14) lie in the $T = [1 - VG]^{-1}V,$8–$T = [1 - VG]^{-1}V,$9 fm interval across bounded states, further corroborating the spatially extended molecular structure.

Figure 8: Spatial profile of the main $V$0 wave function: rapid falloff with $V$1, confirming localization.

Figure 9: RMS radii for $V$2, showing molecular-size values largely insensitive to HQSS breaking.

$V$3, $V$4 Sector

In the six-channel analysis, the $V$5 state emerges as the most deeply bound, with extremely narrow width across cutoffs and a spatial radius $V$6–$V$7 fm (Figures 15–17).

Figure 10: Mass and width trajectories for the principal $V$8 $V$9 candidates under cutoff variation.

Single-Channel Limit

In the single-channel limit (Figures 18–23), all bound states become pure bound (vanishing width). Binding energies and radii for all channels in both the hidden-charm and hidden-charm-strange sectors converge owing to isomorphic strong interaction kernels, yielding closely parallel mass and size trajectories for fixed cutoffs.

Figure 11: Pole masses in single-channel limit for hidden-charm systems; all states are deeply bound for large $G$0.

Figure 12: RMS radii under single-channel dynamics; all states exhibit large, monotonic, and smooth $G$1 scaling.

Implications and Future Perspectives

This comprehensive analysis reinforces several core aspects of pentaquark molecular phenomenology:

• Full coupled-channel dynamics with HQSS constraints are essential for a realistic account of $G$2 widths, but less critical for the hidden-charm-strange $G$3 partners.
• The spatial extent (RMS radius) systematically signals a molecular rather than compact multiquark or hadrocharmonium structure.
• For both $G$4 and $G$5, deeply bound channels ($G$6 and $G$7) are robustly generated as primary poles, while more weakly bound states exhibit strong cutoff and channel-coupling dependence.
• Single-channel and full coupled-channel solutions yield notably different pole trajectories and spatial sizes, highlighting the non-perturbative sensitivity to coupled-channel effects.
• The theoretical predictions for masses, widths, and radii facilitate experimental discrimination of the underlying binding mechanism—molecular versus compact configurations—in future high-statistics measurements.
• The radii and wave function localization found here are consistent with other hadronic molecule studies (1705.00141), supporting universality of binding at the hadronic scale.

Conclusion

The study provides a high-resolution, technically detailed mapping of the molecular structure of $G$8 and $G$9 states via the Bethe-Salpeter coupled-channel approach augmented by HQSS and LHG principles. Channel coupling and HQSS constraints decisively shape the phenomenology in the hidden-charm sector, both in observed widths and mass hierarchies, while the hidden-charm-strange sector is less sensitive to these details. The explicit calculation of wave functions and molecular radii supplies crucial benchmarks for the spatial structure of pentaquark candidates. These results underline the necessity of incorporating full channel dynamics and spatial observables in any future theoretical or experimental program aimed at a definitive classification of exotic baryonic molecules.

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Figure References Used

• Figure 1: Mass and width trajectories in $q_{max}$0, $q_{max}$1, seven-channel case
• Figure 2: Mass/width for PB and VB split
• Figure 3: $q_{max}$2 for main $q_{max}$3 poles
• Figure 4: RMS radii trajectories
• Figure 5–7: $q_{max}$4 $q_{max}$5 pole evolution
• Figure 6–11: $q_{max}$6 pole evolution (full and split channels)
• Figure 8–14: $q_{max}$7 wave function profiles and RMS radii
• Figure 10–17: $q_{max}$8 $q_{max}$9 poles, wave function, RMS
• Figure 11–20: Single-channel limit (hidden charm), pole masses, wave function, radii
• Figure 13–23: Single-channel limit (hidden charm-strange)