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arxiv: 2604.04821 · v2 · submitted 2026-04-06 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

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

Jing-Zhi Cao , Huan-Yu Wei , Jiao-Xue Yang , Jian Sun , Chu-Wen Xiao
Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:26 UTC · model grok-4.3

classification ✦ hep-ph
keywords hidden-charm pentaquarksmolecular statescoupled channelsheavy quark spin symmetryPc statesPcs statesBethe-Salpeter equationpole trajectories
0
0 comments X

The pith

Full coupled-channel interactions respecting heavy quark spin symmetry are required to generate the Pc states with appropriate widths.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models the Pc and Pcs pentaquarks as molecular bound states formed from charmed mesons and baryons. It uses a coupled-channel framework that includes heavy quark spin symmetry to solve the Bethe-Salpeter equation and find the poles. For the non-strange hidden-charm Pc states, keeping all channels coupled is necessary to produce the states and determine their widths correctly. For the strange Pcs states, results from full coupling and from separate pseudoscalar-baryon and vector-baryon sectors are similar. The wave functions and root-mean-square radii calculated for the poles support a molecular picture with sizes around 1 fm.

Core claim

In the hidden charm system, the full coupled channel interactions respecting the heavy quark spin symmetry are essential to generate the Pc states, which significantly affect the poles' widths. The dominant bound channels are bar D Sigma_c and bar D* Sigma_c, which couple strongly to the 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 Xi_c and bar D* Xi_c couple strongly to bar D_s Lambda_c and bar D_s* Lambda_c, respectively, but weakly to the lower decay channels. The trajectories of the pole widths for the loosely

What carries the argument

The Bethe-Salpeter equation in the coupled-channel formalism under heavy quark spin symmetry and the local hidden gauge approach, solved via the cutoff method to locate poles and compute wave functions and radii.

Load-bearing premise

The local hidden gauge formalism with the chosen cutoff regularization accurately describes the low-energy meson-baryon interactions without missing important short-range contributions.

What would settle it

An experimental width for one of the Pc states that agrees with single-channel or sector-split calculations but disagrees with the full coupled-channel prediction would indicate that the full interactions are not essential.

Figures

Figures reproduced from arXiv: 2604.04821 by Chu-Wen Xiao, Huan-Yu Wei, Jian Sun, Jiao-Xue Yang, Jing-Zhi Cao.

Figure 1
Figure 1. Figure 1: FIG. 1. Mass (left) and width (right) trajectories of the poles in the second Riemann sheets for [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Mass (left) and width (right) trajectories of the poles in the second Riemann sheets for [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Real (left) and imaginary (right) parts of the wave functions [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Real (left) and imaginary (right) parts of the wave functions [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. RMS radii of the corresponding poles for the [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Mass (left) and width (right) trajectories of the poles in the second Riemann sheets for [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Mass (left) and width (right) trajectories of the poles in the second Riemann sheets for [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Real (left) and imaginary (right) parts of the wave functions [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. RMS radii of the corresponding poles for the [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Trajectories for the masses (left) and widths (right) of the poles in the second Riemann [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Trajectories for the masses (left) and widths (right) of the poles in the second Riemann [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Real (left) and imaginary (right) parts of the wave functions [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. RMS radii of the corresponding poles for the [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. RMS radii of the corresponding poles for the [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Trajectories for the masses (left) and widths (right) of the poles in the second Riemann [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Real (left) and imaginary (right) parts of the wave functions [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. RMS radii of the corresponding poles for the [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Trajectories for the masses of the poles in the first Riemann sheet for the single channel [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Wave functions [PITH_FULL_IMAGE:figures/full_fig_p033_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. RMS radii of the corresponding poles for the single channel interactions of the hidden [PITH_FULL_IMAGE:figures/full_fig_p034_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Trajectories for the masses of the poles in the first Riemann sheet for the single channel [PITH_FULL_IMAGE:figures/full_fig_p035_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Wave functions [PITH_FULL_IMAGE:figures/full_fig_p035_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. RMS radii of the corresponding poles for the single channel interactions of the hidden [PITH_FULL_IMAGE:figures/full_fig_p036_23.png] view at source ↗
read the original abstract

In the present work, we investigate the molecular properties of the hidden charm pentaquark states $P_c$ and $P_{cs}$ with a coupled channel framework that combines heavy quark spin symmetry and the local hidden gauge formalism. By solving the Bethe-Salpeter equation with the 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, which significantly affect the poles' widths. The dominant bound channels are $\bar{D} \Sigma_c$ and $\bar{D}^* \Sigma_c$, which couple strongly to the 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 $\bar{D} \Xi_c$ and $\bar{D}^* \Xi_c$ couple strongly to $\bar{D}_s \Lambda_c$ and $\bar{D}_s^* \Lambda_c$, respectively, but weakly to the lower decay channels, different from the hidden charm case. The trajectories of the pole widths for the loosely bound channels $\bar{D} \Xi'_c$, $\bar{D}^* \Xi'_c$, and $\bar{D}^* \Xi_c^*$ 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. The root-mean-square radii depend on the pole trajectories and differ among the full coupled channel case, the split PB and VB sectors, and the single channel interactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the molecular properties of the hidden-charm pentaquark states Pc and Pcs within a coupled-channel Bethe-Salpeter framework that incorporates heavy quark spin symmetry and the local hidden gauge formalism. Poles are obtained by solving the equation with cutoff regularization in single-channel, PB/VB-split, and full-coupled setups; the authors report that full HQSS-respecting coupled channels are essential for generating the Pc states and controlling their widths, with dominant binding from the Dbar Sigma_c and Dbar* Sigma_c channels, while sector splitting is less critical for Pcs. Wave functions are localized within 0-6 fm and RMS radii (computed by two methods) lie between 0.5 and 2 fm, supporting a molecular picture.

Significance. If robust, the work adds a systematic comparison of approximation schemes to the molecular interpretation of the Pc and Pcs states, quantifying the impact of coupled-channel dynamics on widths and providing spatial observables (wave functions and radii) that are directly comparable to the molecular scale. The consistent results across single-channel, split-sector, and full-coupled calculations constitute a strength of the analysis.

major comments (2)
  1. [Abstract and numerical results for Pc states] Abstract and numerical results for Pc states: the central claim that 'full coupled channel interactions respecting the heavy quark spin symmetry are essential to generate the Pc states, which significantly affect the poles' widths' rests on comparisons among the three approximation schemes. However, the cutoff that regularizes the Bethe-Salpeter loop integrals is a single free parameter tuned to place poles near observed masses; without explicit variation of this cutoff (typically by 10-20 MeV) and demonstration that the reported width differences remain stable, it is unclear whether the width changes are genuine coupled-channel effects or regularization artifacts.
  2. [Formalism section] Formalism section: the potentials are derived exclusively via the local hidden gauge approach. No test is presented of the sensitivity of the imaginary parts of the self-energies (and hence the widths) to the omission of short-range contact terms or to the use of an alternative regularization (e.g., dimensional regularization), leaving open the possibility that the quantitative width shifts are scheme-dependent.
minor comments (2)
  1. [Wave-function and radius section] The two methods used to evaluate the root-mean-square radii are stated to be consistent, but a short description or reference to the explicit formulas would improve reproducibility.
  2. [Throughout] Channel labels such as Dbar Sigma_c and Dbar* Xi_c' would benefit from an accompanying table or footnote listing the explicit spin, isospin, and strangeness quantum numbers for each sector.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major concerns regarding the robustness of our numerical results with respect to the cutoff choice and the regularization scheme. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: Abstract and numerical results for Pc states: the central claim that 'full coupled channel interactions respecting the heavy quark spin symmetry are essential to generate the Pc states, which significantly affect the poles' widths' rests on comparisons among the three approximation schemes. However, the cutoff that regularizes the Bethe-Salpeter loop integrals is a single free parameter tuned to place poles near observed masses; without explicit variation of this cutoff (typically by 10-20 MeV) and demonstration that the reported width differences remain stable, it is unclear whether the width changes are genuine coupled-channel effects or regularization artifacts.

    Authors: We agree that an explicit variation of the cutoff is necessary to confirm that the observed width reductions are genuine effects of the full coupled-channel dynamics rather than artifacts of the specific regularization parameter. In the revised manuscript we will add a dedicated subsection presenting results for cutoffs shifted by ±10 MeV and ±20 MeV around the central values employed for each scheme (single-channel, PB/VB-split, and full HQSS). These additional calculations will show that the substantial narrowing of the poles when all channels are included persists across the variations, while the pole positions remain close to the observed masses by readjusting the cutoff within the stated range. This will directly support the central claim. revision: yes

  2. Referee: Formalism section: the potentials are derived exclusively via the local hidden gauge approach. No test is presented of the sensitivity of the imaginary parts of the self-energies (and hence the widths) to the omission of short-range contact terms or to the use of an alternative regularization (e.g., dimensional regularization), leaving open the possibility that the quantitative width shifts are scheme-dependent.

    Authors: The local hidden gauge formalism is the standard framework used to derive the vector-meson-exchange potentials while preserving heavy-quark spin symmetry, and short-range contact terms are not generated within this approach. We acknowledge that a direct comparison with dimensional regularization or the explicit addition of contact terms would provide further reassurance. In the revised version we will insert a paragraph in the Formalism section noting that cutoff and dimensional regularization are known to be equivalent at the level of the leading-order potentials employed here, and that the qualitative features (wave-function localization and RMS radii) are insensitive to the scheme. A full quantitative re-calculation of all widths with an alternative regularization lies beyond the present scope but can be pursued in follow-up work. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper obtains pole positions, trajectories, widths, wave functions, and RMS radii by explicitly solving the Bethe-Salpeter equation in coupled channels with potentials constructed from the local hidden gauge formalism under heavy quark spin symmetry. The key claim that full coupled-channel dynamics are essential for generating the Pc states and controlling their widths is demonstrated by direct numerical comparison of the full-channel solution against reduced PB/VB sectors and single-channel cases, all using the same regularization. No equation or result is shown to equal its input by construction, no cutoff is quoted as being fitted to the target pole positions before claiming the widths, and no load-bearing step relies on a self-citation whose content is itself unverified or tautological. The derivation remains self-contained against the stated model assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on the validity of heavy-quark spin symmetry for relating channels and on the local hidden gauge model for generating potentials; both are standard but unproven assumptions at the relevant energies. The cutoff is an adjustable parameter whose value is not independently fixed.

free parameters (1)
  • cutoff parameter
    Regularization scale in the Bethe-Salpeter integral, chosen to produce bound states near observed masses.
axioms (2)
  • domain assumption Heavy quark spin symmetry holds and relates the interactions in different spin channels
    Used to construct the full coupled-channel potentials for Pc and Pcs systems.
  • domain assumption Local hidden gauge formalism provides the correct effective potentials from vector-meson exchange
    Basis for the interaction kernels in the Bethe-Salpeter equation.

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