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REVIEW 4 major objections 5 minor 79 references

Revisiting the $\Lambda_c^+\rightarrow\bar{K}^0\eta p$ reaction: the role of $N^*(1535),~ N^*(1650)$ and $\Sigma(1620)$

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A chiral coupled-channel model explains the Λ_c^+ → \bar{K}^0 η p peaks as dynamically generated N*(1535), N*(1650), and Σ(1620).

desk verdict The external-emission mechanism is a genuinely new and plausible explanation for the N*(1650) in Belle's ηp spectrum, but the paper's stronger claim of simultaneously observing and clearly distinguishing both N* peaks is not established by the calculation as presented. read the letter →

arxiv 2507.19240 v1 pith:CQOGOYQG submitted 2025-07-25 hep-ph

Jing Song , Melahat Bayar , Yi-Yao Li , Eulogio Oset This is my paper · ORCID
classification hep-ph
keywords charmedbaryondecaysfinal-stateinteractionsdynamicallygeneratedresonancesN*(1535)N*(1650)Sigma(1620)chiralunitaryapproachmeson-baryonrescattering
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the experimentally observed structures in $\Lambda_c^+\to \bar K^0\eta p$ are not produced directly at the weak vertex but emerge from meson-baryon rescattering in the final state. Using a coupled-channel chiral unitary framework that includes both pseudoscalar-baryon ($\pi N$, $\eta N$, $K\Lambda$) and vector-baryon ($\rho N$) channels, it reproduces the two peaks in the $\eta p$ mass distribution as the $N^*(1535)$ and $N^*(1650)$, and the peak near 1620 MeV in the $\bar K^0 p$ distribution as the $\Sigma(1620)$. The new element is external $W$ emission: it does not produce $\bar K^0\eta p$ at tree level, but after rescattering it is the dominant source of both $N^*$ peaks, and it is necessary for the $N^*(1650)$. If the picture is right, this decay channel is a clean probe of whether these baryon resonances are hadronic molecules generated by meson-baryon dynamics.

What carries the argument

The load-bearing object is the coupled-channel meson-baryon scattering amplitude, implemented through the Breit-Wigner form $$t_{ij} = \frac{g_i g_j}{M_{\rm inv} - M_R + i\,\Gamma_R/2}$$ for each resonance, with couplings $g_i$ and subtraction constants taken from an earlier chiral unitary model. The $N^*(1535)$ width is made energy-dependent through $\Gamma_{\pi N}(M_{\rm inv})+\Gamma_{\eta N}(M_{\rm inv})$, while the $N^*(1650)$ width is kept at 125 MeV. These amplitudes are fed by hadronization weights $h'_{MB}$ from the weak vertex, and the final-state interaction is encoded in loop functions $G_{MB}$, with a smeared $\tilde G_{\rho N}$ for the broad $\rho$ meson. The essential new ingredient is the external-emission amplitude $$$t^{{(\rm ext)}}$ = -C\sqrt{\tfrac13}\big(G_{\pi N}\,t_{\pi N\to\eta N}+G_{\rho N}\,t_{\rho N\to\eta N}\big),$$ which contains no tree-level $\bar K^0\eta p$ but generates both $N^*$ peaks through rescattering; the $\Sigma(1620)$ enters through a $\bar K^0 p\to \bar K^0 p$ rescattering term with a coupling fitted to the data.

What would settle it

Recompute the ηp mass distribution with the external-emission term removed, as the paper does, and compare with a high-statistics measurement; if the 1650 MeV peak survives in a channel where external emission cannot feed it, or if lattice QCD places the ηN and πN scattering poles far from the imported values, the dynamical-generation claim would be contradicted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the experimental data for $\Lambda_c^+\to \bar K^0\eta p$ contain, for the first time simultaneously, two related $N^*$ resonances in the same meson-baryon final state, and the paper provides the first theoretical description of that simultaneity. The authors show that after including internal and external weak emission and strong final-state interactions, the $\eta p$ mass distribution has two distinct structures, at about 1535 and 1650 MeV, and the $\bar K^0 p$ distribution has a clear peak around 1620 MeV. Removing the internal-emission tree-level term leaves both $N^*$ peaks intact, which they read as evidence that these states are dynamically generated; removing external emission removes the $N^*(1650)$. The $\Sigma(1620)$ is introduced through $\bar K^0 p$ rescattering, with its coupling to the $\bar K N$ channel fixed by the data and found to agree qualitatively with chiral unitary predictions.

Load-bearing premise

The calculation imports the N*(1535) and N*(1650) amplitudes from an earlier chiral model, reduces the πN coupling by 30% to match the data, and fits the Sigma(1620) coupling to the same data, so the resonance peaks are inherited from input choices rather than derived from this reaction alone.

Editorial extensions

If this is right

  • The reaction is dominated by external emission followed by rescattering, so its $\eta p$ spectrum is a direct window on $I=1/2$ meson-baryon dynamics.
  • The $N^*(1650)$ cannot be produced in this decay without the $\rho N$ channel and external emission; if correct, this identifies the mechanism that distinguishes it from the $N^*(1535)$.
  • The 1620 MeV peak in the $\bar K^0 p$ distribution is attributable to the $\Sigma(1620)$; removing that resonance removes the peak, so the data can be used to constrain its $\bar K N$ coupling.
  • The $\eta \bar K^0$ distribution is predicted to be structureless, matching experiment and providing a cross-check of the mechanism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test would be to compute the same decay in a fully unitarized scheme where the resonances emerge as poles from the same amplitudes rather than from imported Breit-Wigner parameters; the current paper supports but does not by itself prove dynamical generation.
  • If the external-emission mechanism is as dominant as claimed, analogous $\Lambda_c^+$ decays with different final mesons should show similar resonance patterns with the same hierarchy of mechanisms.
  • The unexplained mismatch around 1480 MeV in the $\bar K^0 p$ spectrum, which the paper attributes to statistical fluctuation, could alternatively signal an additional $\Sigma$ state; higher-statistics data would distinguish these options.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper presents a theoretical study of the decay Λ_c^+ → \bar K^0 η p using a coupled-channel chiral unitary framework that includes both pseudoscalar-baryon and vector-baryon channels, with internal and external weak emission mechanisms plus final-state interactions. The authors report that the ηp invariant mass distribution shows two peaks near 1535 and 1650 MeV, which they identify with the N*(1535) and N*(1650) resonances, and that the \bar K^0 p distribution shows a peak near 1620 MeV attributed to Σ(1620). They fit an overall normalization C, the internal/external emission ratio β, and the Σ(1620) coupling, while also reducing g_πN by 30% as an ad hoc adjustment, obtaining χ²/d.o.f. = 2.0. The central claim is that the two N* resonances are simultaneously observed and clearly distinguished in this decay for the first time, and that external emission followed by rescattering is essential for producing the N*(1650).

Significance. If the central claim is correct, the paper offers a concrete reaction mechanism—external emission followed by meson-baryon rescattering—that explains why Belle's ηp spectrum contains both N*(1535) and N*(1650), a feature that earlier internal-emission-only models missed. The framework is explicit and analytic, with amplitudes given in Eqs. (36)–(46), and it reproduces the qualitative features of the Belle data, including two N* peaks without tuning. The paper is also candid about several limitations, especially the theoretical uncertainties in the Σ(1620) sector. However, the headline claim is currently stronger than the analysis supports: the N* amplitudes are imported as Breit-Wigner forms from Ref. [43], and no fit is shown that tests whether the second N* peak is actually required by the data. These issues are fixable in revision, and the underlying mechanism is plausible enough to warrant further work, but the present version overstates the dynamical-generation evidence.

major comments (4)
  1. [II.B, Eq. (29)] The scattering amplitudes t_{ij} used in Eqs. (36) and (37) are Breit-Wigner forms with pole masses, widths, and couplings taken from Table II of Ref. [43]. Consequently, the ηp peaks near 1535 and 1650 MeV are inherited from that input rather than derived from the coupled-channel dynamics within this manuscript. Removing the tree-level term in Sec. III.C (Fig. 8) only shows that the peaks live in the FSI terms, which still contain the same input amplitudes. The conclusion in Sec. IV that the resonances emerge dynamically from meson-baryon coupled-channel dynamics is therefore not established by this calculation. I recommend either solving the Bethe-Salpeter equation explicitly in this paper or explicitly reframing the result as a study of how assumed N* amplitudes appear in this decay, with softened language about dynamical generation.
  2. [III.A, Table III] The fit quality is reported as χ²/d.o.f. = 2.0 with no parameter uncertainties and no comparison against a model without the N*(1650) amplitude. Because the headline claim is that the two N* peaks are simultaneously observed and clearly distinguished in the Belle data, the analysis should include a likelihood-ratio or similar comparison between the full model and a one-N*(1535)-plus-background alternative, and report uncertainties on C, β, and g_{Σ(1620)}. Without such a test, the data do not currently demonstrate that the second peak is required at any stated significance.
  3. [III.A, Sec. III.A] The reduction of the g_{πN} coupling by 30% is introduced as a correction demanded by the data but no physical justification or systematic study is provided. Since the strength and shape of the N*(1535) structure depend on the relative weight of the πN channel, the paper should either derive this factor from the hadronization model or show the sensitivity of the ηp distribution and fitted parameters to the reduction (e.g., 0%, 15%, 30%). Without this, the fit result is not robust and the central conclusions depend on an unexplained adjustment.
  4. [III.D, Eq. (45)] The fitted Σ(1620) coupling g_{Σ*, \bar K N} ≈ i0.6 is described as in qualitative agreement with the chiral-unitary values g = −1.1 − i1.1 [67] and −0.89 − i0.57 [53]. This is difficult to accept as qualitative agreement because the magnitudes differ by roughly a factor of two and the phase differs by about 90°. Since the abstract and conclusions present the Σ(1620) peak as associated with a dynamically generated state, the paper should quantify the compatibility (for example, by a χ² scan over the real and imaginary parts of the coupling) or weaken the claim to a consistency hint and state clearly that the coupling is fitted to this decay.
minor comments (5)
  1. [II.B, Eq. (45)] Equation (45) has a dimensional inconsistency: the denominator M²_inv − M_{Σ*} + i Γ_{Σ*}/2 mixes mass-squared and mass terms. It should presumably read M²_inv − M²_{Σ*} + i M_{Σ*} Γ_{Σ*}; please correct this.
  2. [Fig. 5 caption] The caption states fixed masses M_N*(1535)=1525 MeV and M_N*(1650)=1650 MeV; please clarify whether these are input pole masses from Ref. [43] or fitted parameters, and reconcile the value 1525 MeV with the resonance masses used elsewhere in the text.
  3. [III.A, Table III] The fitted g_{Σ*, \bar K N} ≈ i0.6 is not listed in Table III; please include it with its statistical uncertainty so that the fit parameters and the quoted χ²/d.o.f. can be reproduced.
  4. [III.B, Sec. III.B] The discussion of the 1480 MeV region in the \bar K^0 p distribution is qualitative; providing a pull per bin or a local χ² contribution would help the reader judge whether the discrepancy is significant.
  5. [Acknowledgments] The Acknowledgments thank Raquel Molina, who is not a co-author of this manuscript; this appears to be a copy-paste from another paper and should be corrected.

Circularity Check

2 steps flagged · score 6.0 of 10

The resonance 'identifications' reduce to inputs: the N* peaks are projections of the Breit-Wigner amplitude imported from Ref. [43], and the Sigma(1620) peak is a Breit-Wigner term with a coupling fitted to the same anti-K0 p data in which it is then 'identified'.

  1. fitted input called prediction [Sec. III.A, Eq. (45); Sec. III.D]
    "we parametrize the \bar{K}^0p \to \bar{K}^0p amplitude as t_{\bar{K}^0p,\bar{K}^0p} \equiv \frac{(g^{(I=1)}_{\Sigma^*, \bar{K}N})^2}{M^2_{inv}(\bar{K}^0p) - M_{\Sigma^*} + i \frac{\Gamma_{\Sigma^*}}{2}} and we take M_{\Sigma^*}, \Gamma_{\Sigma^*} from the PDG data tables, fitting g^{(I=1)}_{\Sigma^*, \bar{K}N} to reproduce the data."

    The 1620 MeV peak in the calculated \bar{K}^0p spectrum is generated by this Breit-Wigner term, placed at the PDG mass and width with a complex coupling fitted to the Belle \bar{K}^0p data. Section III.D then removes this same term and reports that the peak disappears, 'showing that the peak reflects the Sigma(1620) contribution'; that conclusion holds by construction. The paper also concedes the prior theoretical couplings are uncertain and 'we let the experiment decide which coupling is required'. Calling the resulting peak an identification of a dynamically generated Sigma(1620) is therefore a fit presented as a prediction, not an independent derivation.

  2. self definitional [Sec. II.B, Eq. (29) and Table II; Sec. III.A/C]
    "The rescattering involves coupled channels M B = πN (I = 1/2), ηN, K Λ, ρN(I = 1/2), described by the scattering matrix T_{MB→MB} with a parametrized amplitudes as Breit-Wigner, ti,j = gigj/(Minv − MR + i ΓR/2). The couplings to N ∗(1535) and N ∗(1650) are given in Table III of Ref. [43], which we show in Table II here. ... Both N ∗(1535) and N ∗(1650) structures remain clearly visible even after the tree-level contribution is removed, indicating their dynamical origin."

    Eq. (29) is a Breit-Wigner sum over N*(1535) and N*(1650) with masses, widths, couplings and subtraction constants imported from Ref. [43]; hence the ηp distribution computed from Eq. (41) contains the two input poles directly. The Sec. III.A statement that the two peaks 'emerge from the model without fitting' is true only because the poles were put into the input t_ij; the peaks are the input amplitude projected into phase space, not a data-driven discovery. Sec. III.C's removal of the tree-level term shows only that the peaks live in the FSI terms, which are already Breit-Wigner poles. Since no fit with the N*(1650) couplings set to zero is reported, the paper cannot establish that the Belle data require the second resonance; the 'first time ...

full rationale

There is a genuine non-circular component: the internal/external weak-emission amplitudes, the hadronization weights, the WPP vertex analysis and the kinematic distributions are derived from stated quark-level and chiral-lagrangian inputs rather than fitted. The circularity is concentrated in the resonance-identification claims. For the Sigma(1620), Eq. (45) inserts a Breit-Wigner at the PDG mass with a coupling fitted to the Belle \bar{K}^0p data, so Sec. III.D's removal test is a tautology. For the N* states, Eq. (29) and Table II import the two-pole amplitude from Ref. [43], whose author Oset overlaps with the present paper, so the 'simultaneous observation' of N*(1535) and N*(1650) in Figs. 5 and 7 is inherited from the input scattering matrix; the only fitted shape knobs are C, β and an ad hoc 30% reduction of gπN. Thus the central resonance claims reduce to fitted or imported terms, while the weak-decay formalism itself remains an independent model application. A score of 6 reflects this partial circularity: the reaction-mechanism calculation is not circular, but the paper's headline conclusions about the resonances are.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper's quantitative output rests on three layers it does not derive: the chiral unitary resonance couplings used to build the scattering amplitudes, the quark-model hadronization weights for the weak vertex, and the added Breit-Wigner description of Sigma(1620). The four fitted or hand-adjusted parameters are C, beta, g_Sigma, and a 30% g_piN reduction. No new particles are postulated; the entities N*(1535), N*(1650), and Sigma(1620) are taken from prior phenomenological work, so invented_entities is empty.

free parameters (4)
  • C (overall weak normalization) = 8.01
    Scales the total decay amplitude; fitted to Belle data (Table III).
  • beta (external/internal emission ratio) = 0.31
    Fitted; close to expected 1/Nc = 1/3, controls external emission strength (Table III).
  • g_Sigma*_KN (Sigma(1620) coupling to anti-K N) = ~ i 0.6
    Fitted to the Belle p anti-K0 mass distribution through the Breit-Wigner amplitude in Eq. (45); not tabulated with an uncertainty.
  • g_piN reduction factor = 0.7 (30% reduction)
    Hand adjustment to the N*-piN coupling imported from Ref. [43], described as corrections demanded by the data in Sec. III.A.
assumptions (5)
  • domain assumption Meson-baryon scattering amplitudes are unitarized by the Bethe-Salpeter equation in the chiral unitary approach with both PB and VB channels, generating N*(1535) and N*(1650).
    The paper uses the Breit-Wigner form of Eq. (29) with couplings from Table II and subtraction constants from Eq. (34), all taken from Ref. [43], rather than deriving the poles here.
  • domain assumption Lambda_c+ decays via Cabibbo-allowed c -> s u dbar with the ud pair acting as an I=0, S=0 spectator.
    This is the quark-level starting point of Sec. II.A, Eqs. (1)-(2), following Refs. [15,16].
  • domain assumption Hadronization is implemented by inserting q qbar pairs with the 3P0 factors f_PB and f_VB of Eqs. (14)-(15).
    These factors set the relative weights h' of the meson-baryon channels in Eq. (17), and hence the relative strengths of tree-level and rescattering terms.
  • ad hoc to paper The external emission diagram of Fig. 3(b) can be neglected because the WPP vertex commutator and an energy-average cancel the K+ anti-K0 term.
    Stated in the paragraph after Eq. (28); if the energy-average is not justified, an additional contribution to the eta p final state would appear.
  • domain assumption The Sigma(1620) can be represented by a single Breit-Wigner with PDG mass and width and a fitted coupling.
    Used in Eq. (45); the paper notes previous chiral unitary results give masses around 1490-1580 MeV and widths 230-530 MeV (Sec. III.A), so this representation is uncertain.

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Cite this review

Pith. "Pith review of Revisiting the $\Lambda_c^+\rightarrow\bar{K}^0\eta p$ reaction: the role of $N^*(1535),~ N^*(1650)$ and $\Sigma(1620)$." pith.science (2026). https://pith.science/paper/CQOGOYQG

@misc{pith2026250719240,
  author       = {Pith},
  title        = {Pith review of: Revisiting the $\Lambda_c^+\rightarrow\barK^0\eta p$ reaction: the role of $N^*(1535),~ N^*(1650)$ and $\Sigma(1620)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CQOGOYQG}},
  note         = {Machine review of arXiv:2507.19240}
}
abstract

We perform a theoretical study of the weak decay $\Lambda_c^+ \rightarrow \bar{K}^0 \eta p$ using a coupled-channel chiral unitary approach that incorporates both pseudoscalar-baryon and vector-baryon interactions. Our framework includes contributions from both internal and external weak emission mechanisms, as well as strong final state interactions. We assume that the $N^*(1535)$ and $N^*(1650)$ resonances are dynamically generated through meson-baryon scattering and they appear as distinct structures in the $\eta p$ invariant mass distributions. {A clear peak also appears in the $\bar{K}^0 p$ invariant mass distribution around 1620~MeV, associated with the dynamically generated $\Sigma(1620)$ resonance.} Notably, this work provides the first theoretical description of the simultaneous observation of these two related $N^*$ resonances in the same meson-baryon final state. Our results highlight the crucial role of final state interaction and the interplay between different weak decay topologies in shaping the resonance patterns. These findings offer new insights into the nature of nucleon excitations and support the interpretation of $N^*(1535)$ and $N^*(1650)$ as dynamically generated states. {Moreover, the identification of the $\Sigma(1620)$ further supports the picture of hadronic molecular structures emerging from meson-baryon interactions in the non-perturbative QCD regime.}

Figures

Figures reproduced from arXiv: 2507.19240 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 1
Figure 1. FIG. 1: Schematic diagrams for the decay [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Quark-level diagram for the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figures from the paper (6 more)
Figure 3
Figure 3. Figure 3: FIG. 3: Hadronization in external emission: (a) of the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4: Diagrams showing the production of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Mechanism to account for the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Invariant mass distributions of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Invariant mass distributions of [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Invariant mass distributions of [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

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