arxiv: 2601.14723 · v2 · submitted 2026-01-21 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Systematic study of the strong decays of the P_c states and their possible isospin cousins via the QCD sum rules

Xiu-Wu Wang , Xin Li , Zhi-Gang Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-16 12:39 UTC · model grok-4.3

classification ✦ hep-ph

keywords Pc statesQCD sum rulesmolecular statesstrong decaysexotic hadronspentaquarksisospin partners

0 comments

The pith

QCD sum rules calculations show that the decay widths of Pc(4380), Pc(4440) and Pc(4457) match experiments when the states are treated as meson-baryon molecules.

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

The paper assigns the Pc(4380), Pc(4440) and Pc(4457) states as meson-baryon molecular states and applies QCD sum rules to compute the strong decay constants and partial widths of their allowed channels. The resulting widths for the three observed states come out in good agreement with the measured values. Predictions are given for the decay properties of the unobserved isospin partner states. A reader would care because confirmation of the partner-state predictions would provide a direct test of whether these exotic candidates are loosely bound meson-baryon molecules rather than compact multiquark objects.

Core claim

Under the meson-baryon molecular assignment, the QCD sum rules framework produces strong decay widths for Pc(4380), Pc(4440) and Pc(4457) that agree with experimental data, while also forecasting the decay channels and widths of their possible isospin cousins.

What carries the argument

QCD sum rules applied to interpolating currents built from meson-baryon molecular configurations, used to extract decay constants and compute two-body strong decay widths.

If this is right

The calculated widths for the three observed Pc states reproduce the experimental measurements.
Specific partial widths and branching fractions are predicted for the unobserved isospin partner states.
Agreement with future observations of the partners would support the molecular picture.
Discrepancies would indicate the need for additional structure components.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sum-rule technique could be applied to other candidate molecular states to generate testable decay patterns.
Observation of the isospin cousins with widths close to the predictions would help separate molecular from compact pentaquark interpretations.
The method supplies a concrete way to use decay data to constrain the internal structure of exotic hadrons.

Load-bearing premise

The Pc states are pure meson-baryon molecular states without significant mixing from other configurations.

What would settle it

Future measurement of a decay width for one of the predicted isospin partner states that differs substantially from the calculated value.

Figures

Figures reproduced from arXiv: 2601.14723 by Xin Li, Xiu-Wu Wang, Zhi-Gang Wang.

**Figure 1.** Figure 1: The hadronic coupling constants for Pc(4380) → ηcN (Left) and Pc(4380) → J/ψN (Right) among the Borel windows. 6.0 6.2 6.4 6.6 6.8 7.0 -3.0 -1.0 1.0 3.0 5.0 7.0 Central Value Error bounds 6.5 6.7 6.9 7.1 7.3 7.5 1.0 1.3 1.6 1.9 2.2 2.5 Central Value Error bounds [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗

**Figure 2.** Figure 2: The hadronic coupling constants for Pc(4410) → ηc∆ (Left) and Pc(4410) → J/ψ∆ (Right) among the Borel windows. 5.0 5.2 5.4 5.6 5.8 6.0 -1.0 -0.4 0.2 0.8 1.4 2.0 Central Value Error bounds 5.0 5.2 5.4 5.6 5.8 6.0 -0.5 0.1 0.7 1.3 1.9 2.5 Central Value Error bounds [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Figure 3: The hadronic coupling constants for Pc(4440) → ηcN (Left) and Pc(4440) → J/ψN (Right) among the Borel windows. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: The hadronic coupling constants for Pc(4470) → ηc∆ (Left) and Pc(4470) → J/ψ∆ (Right) among the Borel windows. 5.0 5.2 5.4 5.6 5.8 6.0 0.0 0.5 1.0 1.5 2.0 2.5 Central Value Error bounds 6.0 6.2 6.4 6.6 6.8 7.0 -0.5 0.1 0.7 1.3 1.9 2.5 Central Value Error bounds [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: The hadronic coupling constants for Pc(4457) → ηcN (Left) and Pc(4457) → J/ψN (Right) among the Borel windows. 6.0 6.2 6.4 6.6 6.8 7.0 0.0 1.0 2.0 3.0 4.0 5.0 Central Value Error bounds 5.0 5.2 5.4 5.6 5.8 6.0 -3.0 -0.6 1.8 4.2 6.6 9.0 Central Value Error bounds [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

**Figure 6.** Figure 6: The hadronic coupling constants for Pc(4620) → ηc∆ (Left) and Pc(4620) → J/ψ∆ (Right) among the Borel windows. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

read the original abstract

In the present work, the strong decays of the discovered $P_c(4380)$, $P_c(4440)$, $P_c(4457)$ and their possible isospin cousins are systematically studied via the assignment that they are the meson-baryon molecular states. In detail, the strong decay constants, partial decay widths of their decay channels are calculated under the framework of QCD sum rules. The decay withes of the discovered $P_c(4380)$, $P_c(4440)$ and $P_c(4457)$ are in good agreement with the experiments. The predictions of the decays of these three related possible isospin cousins are presented which would shed light for their findings in experiment, in return, this may testify the assignments of the discovered $P_c$ states.

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.

Desk Editor's Note private letter to a colleague

The paper calculates decay widths for the known Pc states and new predictions for their isospin partners under the molecular assignment, but the match to data hinges on parameters already tuned to the masses.

read the letter

The main takeaway is that this work uses QCD sum rules to compute strong decay widths for Pc(4380), Pc(4440), and Pc(4457) as meson-baryon molecules, reports agreement with measured widths, and gives numerical predictions for the isospin partner states that have not appeared before. Those predictions are the clearest addition to the literature on these exotics. The calculations follow the standard three-point sum rule setup with the usual condensates and are applied systematically across channels, which is a straightforward extension of the group's earlier mass studies. That consistency lets them produce concrete numbers an experiment could check. The soft spot is exactly the one the stress-test note flags. The Borel windows and continuum thresholds are fixed by the two-point mass sum rules, and the decay widths are extracted from the same working region. Because the decay constants enter linearly, modest shifts in those parameters can change the widths by factors of two without spoiling the mass fit. The manuscript does not show a scan of the widths outside the narrow interval chosen for the masses or plot their variation inside the Borel window, so the agreement is not yet shown to be independent of those choices. This is a standard limitation in sum-rule work rather than a fatal flaw, but it needs explicit checking. The paper is for readers already working on pentaquarks and molecular assignments who want specific decay numbers to compare with data or to plan searches for the cousins. It is coherent on its own terms and engages the existing literature honestly, so it deserves a serious referee who can press on the stability analysis and the size of the quoted uncertainties. I would send it to review rather than desk-reject.

Referee Report

2 major / 1 minor

Summary. The manuscript assigns the observed Pc(4380), Pc(4440) and Pc(4457) as meson-baryon molecular states and uses three-point QCD sum rules to compute their strong decay constants and partial widths. It reports that the calculated widths for the three discovered states agree with experimental values and supplies predictions for the decays of their possible isospin partners.

Significance. If the molecular assignments hold and the sum-rule results are stable, the work would provide a systematic QCD-based calculation of decay properties for these exotic states, offering falsifiable predictions for unobserved isospin cousins that could be tested in future experiments and thereby helping to discriminate among competing interpretations.

major comments (2)

[Numerical analysis of decay widths] The Borel windows and continuum thresholds s0 are determined solely from the two-point mass sum rules and then transferred unchanged to the three-point decay sum rules. No separate stability analysis is shown for the decay widths as functions of M^2 inside the window or for s0 outside the narrow interval fixed by the mass analysis; because the widths scale with the square of the decay constants, this omission leaves the reported agreement with data sensitive to parameter choice.
[Abstract and conclusions] The central claim that the widths of Pc(4380), Pc(4440) and Pc(4457) are 'in good agreement with the experiments' is presented without quantified uncertainties arising from the shared parameter set; a systematic scan demonstrating that the agreement survives reasonable variations in s0 and M^2 is required to establish that the result is not an artifact of the tuning performed for the masses.

minor comments (1)

[Abstract] Typo in abstract: 'decay withes' should read 'decay widths'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on the numerical stability and uncertainty quantification. We address each major comment below and will revise the manuscript to incorporate additional analysis as outlined.

read point-by-point responses

Referee: [Numerical analysis of decay widths] The Borel windows and continuum thresholds s0 are determined solely from the two-point mass sum rules and then transferred unchanged to the three-point decay sum rules. No separate stability analysis is shown for the decay widths as functions of M^2 inside the window or for s0 outside the narrow interval fixed by the mass analysis; because the widths scale with the square of the decay constants, this omission leaves the reported agreement with data sensitive to parameter choice.

Authors: We agree that dedicated stability plots for the three-point sum rules would strengthen the presentation. The Borel windows were chosen using standard criteria ensuring pole dominance (>50%) and OPE convergence, which apply consistently to both two- and three-point functions in our framework. Within these windows the decay widths exhibit mild M^2 dependence, supporting the reported central values. To address the concern directly, we will add explicit figures in the revised manuscript showing the variation of the decay widths versus M^2 and s0, confirming that the agreement with experimental data remains stable under small parameter shifts. revision: yes
Referee: [Abstract and conclusions] The central claim that the widths of Pc(4380), Pc(4440) and Pc(4457) are 'in good agreement with the experiments' is presented without quantified uncertainties arising from the shared parameter set; a systematic scan demonstrating that the agreement survives reasonable variations in s0 and M^2 is required to establish that the result is not an artifact of the tuning performed for the masses.

Authors: We acknowledge that the original presentation would benefit from explicit uncertainty quantification and a systematic scan. The agreement is based on central values obtained within the windows fixed by the mass sum rules, but we recognize the value of demonstrating robustness. In the revision we will include a parameter scan over reasonable ranges of s0 and M^2, report the resulting variation as uncertainties on the widths, and update the abstract and conclusions to state the agreement with quantified errors. revision: yes

Circularity Check

1 steps flagged

Decay widths obtained from three-point sum rules using Borel windows and s0 thresholds already tuned to reproduce masses in two-point sum rules

specific steps

fitted input called prediction [Abstract and QCD sum rules framework]
"the strong decay constants, partial decay widths of their decay channels are calculated under the framework of QCD sum rules. The decay widths of the discovered Pc(4380), Pc(4440) and Pc(4457) are in good agreement with the experiments."

The phenomenological side of the three-point sum rules employs the identical Borel parameter M^2 and continuum threshold s0 that were already chosen to reproduce the masses in the two-point sum rules; the resulting widths therefore inherit the same parameter tuning rather than constituting an independent prediction.

full rationale

The central claim of agreement between computed partial widths and experimental values for Pc(4380), Pc(4440), Pc(4457) rests on the molecular assignment plus the same continuum thresholds and Borel parameters fixed by the mass analysis. Because the decay constants enter linearly into the width formula and the working windows are not scanned independently for the decay channels, the numerical agreement is not an independent test but follows from the shared fitted inputs. This matches the fitted-input-called-prediction pattern without requiring external self-citations.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the molecular-state assignment plus standard QCD sum rules machinery that requires several tunable parameters whose values are chosen for sum-rule stability.

free parameters (2)

Borel parameters
Standard QCD sum rules parameter chosen to suppress continuum contributions and stabilize the sum rule.
Continuum thresholds
Fitted to reproduce known masses or decay widths; directly affects extracted decay constants.

axioms (1)

domain assumption The Pc states are meson-baryon molecular states
This assignment is used to construct the interpolating currents and is the foundation for all decay calculations.

pith-pipeline@v0.9.0 · 5446 in / 1234 out tokens · 32474 ms · 2026-05-16T12:39:31.508105+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The decay widths of the discovered Pc(4380), Pc(4440) and Pc(4457) are in good agreement with the experiments... via the assignment that they are the meson-baryon molecular states... three-point QCD sum rules
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

Borel windows... continuum thresholds s0... C1..C12 free parameters... flat platforms

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages · 1 internal anchor

[1]

Aaij et al, Phys

R. Aaij et al, Phys. Rev. Lett. 115 (2015) 072001

work page 2015
[2]

Aaij et al, Phys

R. Aaij et al, Phys. Rev. Lett. 122 (2019) 222001

work page 2019
[3]

Aaij et al, Phys

R. Aaij et al, Phys. Rev. Lett. 128 (2022) 062001

work page 2022
[4]

X. W. Wang and Z. G. Wang, Chin. Phys. C48 (2024) 053102

work page 2024
[5]

J. He, Eur. Phys. J. C79 (2019) 393

work page 2019
[6]

H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Phys. Rev. Lett. 115 (2015) 172001

work page 2015
[7]

H. X. Chen, W. Chen and S. L. Zhu, Phys. Rev. D100 (2019) 051501

work page 2019
[8]

Z. G. Wang, Int. J. Mod. Phys. A34 (2019) 1950097. 12

work page 2019
[9]

M. L. Du, V. Baru, F. K. Guo, C. Hanhart, U. G. Meissner, J. A. O ller and Q. Wang, Phys. Rev. Lett. 124 (2020) 072001

work page 2020
[10]

L. Meng, B. Wang, G. J. Wang and S. L. Zhu, Phys. Rev. D100 (2019) 014031

work page 2019
[11]

M. Z. Liu, T. W. Wu, M. S. Sanchez, M. P. Valderrama, L. S. Geng and J. J. Xie, Phys. Rev. D103 (2021) 054004

work page 2021
[12]

J. R. Zhang, Eur. Phys. J. C79 (2019) 1001

work page 2019
[13]

Azizi, Y

K. Azizi, Y. Sarac and H. Sundu, Chin. Phys. C45 (2021) 053103

work page 2021
[14]

C. W. Xiao, J. Nieves and E. Oset, Phys. Rev. D88 (2013) 056012

work page 2013
[15]

H. X. Chen, E. L. Cui, W. Chen, X. Liu, T. G. Steele and S. L. Zhu, Eur. Phys. J. C76 (2016) 572

work page 2016
[16]

M. Z. Liu, F. Z. Peng, M. S´ anchez S´ anchez and M. P. Valderra ma, Phys. Rev. D98 (2018) 114030

work page 2018
[17]

M. Z. Liu, Y. W. Pan, F. Z. Peng, M. S´ anchez S´ anchez, L. S. G eng, A. Hosaka and M. Pavon Valderrama, Phys. Rev. Lett. 122 (2019) 242001

work page 2019
[18]

C. W. Xiao, J. Nieves and E. Oset, Phys. Rev. D100 (2019) 014021

work page 2019
[19]

Sakai, H

S. Sakai, H. J. Jing and F. K. Guo, Phys. Rev. D100 (2019) 074007

work page 2019
[20]

Z. G. Wang, Eur. Phys. J. C76 (2016) 70

work page 2016
[21]

Z. G. Wang, Int. J. Mod. Phys. A35 (2020) 2050003

work page 2020
[22]

Z. G. Wang, Eur. Phys. J. C76 (2016) 142

work page 2016
[23]

Z. G. Wang, Int. J. Mod. Phys. A36 (2021) 2150071

work page 2021
[24]

Ali and A

A. Ali and A. Y. Parkhomenko, Phys. Lett. B793 (2019) 365

work page 2019
[25]

R. Zhu, X. Liu, H. Huang and C. F. Qiao, Phys. Lett. B797 (2019) 134869

work page 2019
[26]

J. F. Giron, R. F. Lebed and C. T. Peterson, JHEP 05 (2019) 061

work page 2019
[27]

Stancu, Eur

F. Stancu, Eur. Phys. J. C79 (2019) 957

work page 2019
[28]

J. F. Giron and R. F. Lebed, Phys. Rev. D104 (2021) 114028

work page 2021
[29]

Y. H. Lin and B. S. Zou, Phys. Rev. D100 (2019) 056005

work page 2019
[30]

He and D

J. He and D. Y. Chen, Eur. Phys. J. C79 (2019) 887

work page 2019
[31]

Yalikun, Y

N. Yalikun, Y. H. Lin, F. K. Guo, Y. Kamiya and B. S. Zou, Phys. Re v. D104 (2021) 094039

work page 2021
[32]

Pavon Valderrama, Phys

M. Pavon Valderrama, Phys. Rev. D100 (2019) 094028

work page 2019
[33]

M. L. Du, V. Baru, F. K. Guo, C. Hanhart, U. G. Meißner, J. A. O ller and Q. Wang, JHEP 08 (2021) 157

work page 2021
[34]

F. Z. Peng, L. S. Geng and J. J. Xie, Phys. Rev. D111 (2025) 054029

work page 2025
[35]

F. K. Guo, X. H. Liu and S. Sakai, Prog. Part. Nucl. Phys. 112 (2020) 103757

work page 2020
[36]

H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 86 (2022) 026201. 13

work page 2022
[37]

L. Meng, B. Wang, G. J. Wang and S. L. Zhu, Phys. Rept. 1019 (2023) 2266

work page 2023
[38]

Z. G. Wang, Front. Phys. (Beijing) 21 (2026) 016300

work page 2026
[39]

X. W. Wang, Z. G. Wang, G. L. Yu and Q. Xin, Sci. China Phys. Mech . Astron. 65 (2022) 291011

work page 2022
[40]

X. W. Wang and Z. G. Wang, Chin. Phys. C47 (2023) 013109

work page 2023
[41]

Z. G. Wang, Chin. Phys. C45 (2021) 073107

work page 2021
[42]

G. J. Wang, L. Y. Xiao, R. Chen, X. H. Liu, X. Liu and S. L. Zhu, Ph ys. Rev. D102 (2020) 036012

work page 2020
[43]

Gutsche and V

T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D100 (2019) 094031

work page 2019
[44]

Z. G. Wang and X. Wang, Chin. Phys. C44 (2020) 103102

work page 2020
[45]

Y. J. Xu, C. Y. Cui, Y. L. Liu and M. Q. Huang, Phys. Rev. D102 (2020) 034028

work page 2020
[46]

X. W. Wang and Z. G. Wang, Int. J. Mod. Phys. A37 (2022) 2250189

work page 2022
[47]

Z. G. Wang and J. X. Zhang, Eur. Phys. J. C78 (2018) 14

work page 2018
[48]

Z. G. Wang, Eur. Phys. J. C79 (2019) 184

work page 2019
[49]

Z. G. Wang and Z. Y. Di, Eur. Phys. J. C79 (2019) 72

work page 2019
[50]

Z. G. Wang, Acta Phys. Polon. B51 (2020) 435

work page 2020
[51]

Z. G. Wang, Int. J. Mod. Phys. A34 (2019) 1950110

work page 2019
[52]

Z. G. Wang, H. J. Wang and Q. Xin, Chin. Phys. C45 (2021) 063104

work page 2021
[53]

M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147 (1979) 385

work page 1979
[54]

M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147 (1979) 448

work page 1979
[55]

L. J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept. 127 (1985) 1

work page 1985
[56]

QCD Sum Rules, a Modern Perspective

P. Colangelo and A. Khodjamirian, hep-ph/0010175

work page internal anchor Pith review Pith/arXiv arXiv
[57]

P. A. Zyla et al, Prog. Theor. Exp. Phys. 2020 (2020) 083C01

work page 2020
[58]

Narison and R

S. Narison and R. Tarrach, Phys. Lett. B125 (1983) 217

work page 1983
[59]

Z. G. Wang and T. Huang, Phys. Rev. D89 (2014) 054019

work page 2014
[60]

Becirevic, G

D. Becirevic, G. Duplancic, B. Klajn, B. Melic and F. Sanﬁlippo, Nuc l. Phys. B883 (2014) 306

work page 2014
[61]

B. L. Ioﬀe, Prog. Part. Nucl. Phys. 56 (2006) 232

work page 2006
[62]

Z. G. Wang, Chin. Phys. C46 (2022) 123106

work page 2022
[63]

X. W. Wang and Z. G. Wang, Phys. Rev. D110 (2024) 014008. 14 Appendix κ 2 = 4(m2 J/ψ τ +m2 N ) 3m2 J/ψ ξ + 8mNmPA 3m2 J/ψ ξ (63) κ 3 = − 4m2 ηcτ2 9m2 ∆ ξ + 4m2 ∆ 9m2 ηcξ + 8m2 ηcτ 9m2 ∆ + 4m∆ mPB 3m2 ηcξ − 4m2 ηcξ 9m∆ mPB + 4m2 ηcτ 9m∆ mPB − 4mPBτ 3m∆ ξ + 8mPB 3m∆ + 16 9 (64) κ 4 = − 8m2 ∆ mPB 9m2 J/ψ ξ + 2m2 ∆ 9mPB + 2m3 ∆ 9m2 J/ψ ξ + 2m2 J/ψ τ2 9m∆ ξ − ...

work page 2024