Theoretical study of the $N(1535)$ in the process $\Lambda_c^+\to n\pi^+\pi^0$

Hao Sun; Ruitian Li; Xuan Luo

REVIEW 3 major objections 4 minor 1 cited by

Theoretical study of the $N(1535)$ in the process $\Lambda_c^+\to n\pi^+\pi^0$

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read In the decay $\Lambda_c^+ \to n\pi^+\pi^0$, the $N(1535)$ appears as a clear peak near 1500 MeV in the $\pi^0n$ and $\pi^+n$ spectra, with a line shape distinct from Breit-Wigner.

desk verdict A useful new channel for N(1535) spectroscopy, but the Model I vs Breit-Wigner line-shape comparison rests on an unjustified 'keep only Im G' approximation; peer review should ask for sensitivity tests. read the letter →

arxiv 2504.14608 v2 pith:YPZUD4C6 submitted 2025-04-20 hep-ph

Ruitian Li , Xuan Luo , Hao Sun This is my paper · ORCID

classification hep-ph

keywords N(1535)resonancechiralunitaryapproachdynamicallygeneratedmeson-baryonmolecularstateLambda_c+nonleptonicdecayinvariantmassdistributionBethe-Salpeterequationcoupledchannels

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 predicts that the charmed-baryon decay $\Lambda_c^+ \to n\pi^+\pi^0$ carries a visible imprint of the $N(1535)$ resonance in both final-state invariant mass distributions. It describes $N(1535)$ not as an elementary quark-model state but as dynamically generated by $S$-wave interactions among the $\pi N$, $\eta N$, $K\Lambda$, and $K\Sigma$ channels in the $I=1/2$ sector. The calculation produces a clear peak near 1500 MeV in the $\pi^0 n$ and $\pi^+ n$ mass spectra, close to the experimental pole position of $N(1535)$, and shows that the chiral-unitary line shape is narrower than the Breit-Wigner shape obtained with an explicit resonance field. This matters because the internal structure of $N(1535)$, including the mass-inversion puzzle with $N(1440)$, remains unresolved, and a single decay could help distinguish the molecular and quark-core pictures.

What carries the argument

The machinery is the chiral unitary coupled-channel amplitude $T=[1-VG]^{-1}V$ in the $I=1/2$ sector, together with the SU(3) hadronization prescriptions that convert the weak $\Lambda_c^+$ decay into a $\pi$ plus a meson-baryon pair. The transition potential $V$ uses SU(3) coefficients $C_{ij}$ and is supplemented by the meson-baryon loop function $G$, computed with a three-momentum cutoff $q_{\max}=1150\ \mathrm{MeV}$ and approximated by its imaginary part in the $N(1535)$ region. This object turns the weak-production vertex into resonant line shapes: the pole of $[1-VG]^{-1}$ in the $\pi N$, $\eta N$, $K\Lambda$, $K\Sigma$ coupled system is the $N(1535)$, and the same $G$ appears in the rescattering diagrams that build the final-state peaks.

What would settle it

Measure the $\pi^0 n$ and $\pi^+ n$ invariant mass distributions of $\Lambda_c^+\to n\pi^+\pi^0$ with high statistics. A clear peak near 1500 MeV with the predicted narrower chiral-unitary shape supports the claim; the absence of a peak, or a peak as broad as the Breit-Wigner model, would refute it.

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Extended reading notes

Core claim

The central claim is that in $\Lambda_c^+ \to n\pi^+\pi^0$ the $\pi^0 n$ and $\pi^+ n$ invariant mass distributions each show a peak near 1500 MeV, produced by the dynamically generated $N(1535)^0$ and $N(1535)^+$, respectively. After weak hadronization of the charm quark, the final meson-baryon pairs scatter through the coupled channels $\pi N$, $\eta N$, $K\Lambda$, and $K\Sigma$, and the Bethe-Salpeter amplitude $T=[1-VG]^{-1}V$ generates the resonance pole without inserting an explicit $N(1535)$ field. When the same observable is computed with a Breit-Wigner amplitude for an elementary $N(1535)$, the line shape is broader; the difference is the paper's testable signature. The conclusion is that $\Lambda_c^+ \to n\pi^+\pi^0$ can serve as a probe of $N(1535)$'s nature, and the paper calls for precise measurements of this decay.

Load-bearing premise

The load-bearing assumption is that in the $N(1535)$ region the meson-baryon loop function can be reduced to its imaginary part while the three-momentum cutoff is fixed at $q_{\max}=1150\ \mathrm{MeV}$; if the real part matters or the cutoff is wrong, the predicted peak and line shape shift.

Editorial extensions

If this is right

The $\pi^0 n$ and $\pi^+ n$ invariant mass distributions in $\Lambda_c^+\to n\pi^+\pi^0$ will each show a peak near 1500 MeV, marking the $N(1535)^0$ and $N(1535)^+$ poles.
The chiral-unitary line shape is narrower than the Breit-Wigner shape from an explicit $N(1535)$ field, so the same observable can separate the molecular and quark-core descriptions.
Varying the suppression parameter $\alpha$ or the color factor $C$ changes the signal strength but leaves the peak position essentially unchanged.
A single Dalitz-plot analysis of this decay can access both charged and neutral $N(1535)$ states and constrain the relative weight $C$ of the two weak-emission mechanisms.

Reading between the lines

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

Extending the calculation to the full complex loop function, rather than only its imaginary part, would test whether the predicted 1500 MeV peak position is stable; a large shift would make the imaginary-part approximation the dominant systematic.
The same hadronization-plus-rescattering construction could be applied to sibling decays, for example $\Lambda_c^+\to p\pi^-\pi^+$, to see whether the $N(1535)$ line shape is universal or channel dependent.
If future data show no peak near 1500 MeV, the molecular production mechanism assumed here would be disfavored for these weak decays, unless the weak vertex is more complicated than the constant strength $V_p$ used in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The one thing to know: this is a sensible, standard chiral-unitary calculation of Lambda_c -> n pi+ pi0, with a genuinely new comparison between a dynamically generated and a Breit-Wigner N(1535). The idea is good and the channel is a plausible discriminator. But the line-shape claim in Fig. 10 relies on dropping the real part of the meson-baryon loop in the production amplitude, and the stated justification for that is wrong. Treat the model comparison as promising but not yet established.

What is new: the specific channel and the Model I vs Model II comparison appear not to have been computed before. The paper also does well in keeping the machinery transparent, checking dependence on alpha and C, and being explicit about arbitrary normalization. The peak near 1500 MeV is robust because the T-matrix is engineered to produce the N(1535) pole; that is not a weakness by itself, but it means the existence of the peak is not the test.

The main soft spot is after Eq. (15). The authors say the piN threshold is far from the N(1535) region, so only Im G is kept. At sqrt(s) ~ 1500 MeV the piN threshold (about 1078 MeV) is open, and the real part of G is not negligible. Eq. (9) has h + h G T; if G in the production loop is replaced by i Im G while T is built from the full G in Eq. (16), the resummation identity h[1 - V G]^{-1} no longer holds, and the phase and magnitude of the rescattering term change. There is no sensitivity test to qmax = 1150 MeV or to including Re G, so the shape difference in Fig. 10 is conditional. This is fixable but important. Minor caveats: the BW comparison uses A = B and a phenomenological smooth cutoff at 1650 MeV; these are acceptable as modeling choices but add uncertainty.

Who it is for: hadron phenomenologists and experimentalists studying N* from charmed-baryon decays. I would send it to a serious referee; it is not a desk reject. But I would not cite the line-shape prediction until the loop approximation is addressed.

Recommendation: peer review, with the request that the authors repeat the calculation keeping the full G and scanning qmax, and report whether the Model I vs Model II line-shape difference survives.

Referee Report

3 major / 4 minor

Summary. This paper studies the three-body weak decay Λ_c^+ → n π^+ π^0 in a model where the N(1535) resonance is dynamically generated from S-wave coupled-channel meson-baryon interactions (πN, ηN, KΛ, KΣ) within the chiral unitary approach. The authors derive the production amplitudes for the two charge states, compute the π^0 n and π^+ n invariant mass distributions, and find a distinct peak near 1500 MeV in both. They then compare the π^0 n line shape with a conventional Breit-Wigner description of the N(1535), obtain a narrower shape in the molecular picture, and recommend experimental measurements to distinguish the two scenarios.

Significance. If the line-shape prediction survives closer scrutiny, the paper would provide a new, experimentally accessible observable for discriminating between molecular and quark-core interpretations of the N(1535). The technical framework is standard and the hadronization procedure from the quark-level weak decay is clearly presented. The paper also makes a serious effort to examine the dependence of the results on the free parameters α and C. However, the central claim rests on two approximations that are not quantitatively controlled in the current manuscript: the truncation of the meson-baryon loop function to its imaginary part, and an ad hoc high-energy suppression factor that is applied asymmetrically to the two models being compared.

major comments (3)

[Section II, after Eq. (15)] The approximation of keeping only the imaginary part of the loop function is not justified at the energies considered. The text states that the πN-channel threshold is 'far away' from the N(1535) region, but at √s around 1500 MeV the πN channel (threshold about 1078 MeV) is open, and the ηN threshold (about 1486 MeV) lies inside the plotted range. The real part of G(s) from Eq. (11) is therefore not negligible. This matters because the production amplitude in Eq. (9) is formally equivalent to h[1−VG]^{-1} only when the same full G is used in the production loop and in the scattering equation; replacing G by i Im(G) changes both the phase and magnitude of the rescattering contribution, which can shift the peak position and modify the line shape. No numerical check of this approximation, nor any sensitivity study with respect to the cutoff qmax = 1150 MeV, is reported. This is a load-bearing step for the claimed line-shape distinction, and it must be addressed with a quantitative comparison.
[Section III, Fig. 10 and Eq. (28)] The comparison between Model I (chiral unitary) and Model II (Breit-Wigner) is not made on equal footing. The smooth suppression factor of Eq. (28) with Mcut = 1650 MeV is applied to the chiral-unitary amplitude, but no equivalent suppression is applied to the Breit-Wigner amplitude. Because this factor cuts off the high-invariant-mass tail of Model I, the statement that the Model II resonance shape is broader than Model I may be partly an artifact of the asymmetric treatment. The authors should either apply the same smooth suppression to Model II or show the comparison with both models computed without the suppression factor, so that the difference in line shape is a genuine model prediction.
[Section II, Eq. (11); Section III, Figs. 6-10] The three-momentum cutoff qmax = 1150 MeV is adopted from previous fits to N(1535) properties, but the dependence of the predicted invariant mass distributions on this parameter is not tested. Since the loop function enters both the scattering T-matrix in Eq. (16) and the production amplitudes in Eq. (9), a moderate change in qmax could affect the peak position and especially the line-shape comparison with the Breit-Wigner model. A numerical sensitivity study (or a comparison with a dimensional-regularization scheme for G) is needed to establish that the central claim is robust against the choice of regulator.

minor comments (4)

[Section III, text near Fig. 6] The notation 'N+(1535)' should be 'N(1535)+' for consistency with the use of 'N(1535)^0' and 'N(1535)^+' elsewhere.
[Eq. (25)] Please verify the phase-space normalization in the expression for d^2Γ/dM_{π+n}^2 dM_{π0n}^2; the factor '4 M_Λ_c+ M_n' in the numerator looks unusual and could be a typographical artifact of the rendering.
[Section II, Eq. (19)] The assumption A = B for the weak vertex Λ_c^+ → π^+ N(1535) is ad hoc and directly affects the shape of the Breit-Wigner distribution in Model II. The authors should explicitly identify this as a source of uncertainty and, ideally, show how the line shape changes when A/B is varied.
[Section II, Eq. (28)] The choice Mcut = 1650 MeV is close to the peak region, and the suppression factor modifies the high-energy tail of the distribution. The paper tests the sensitivity to α but not to Mcut; a brief discussion or a scan over Mcut would make the treatment of this ad hoc factor more transparent.

Circularity Check

1 steps flagged · score 4.0 of 10

The ~1500 MeV peak is inherited from the qmax = 1150 MeV input adopted to generate the N(1535) pole; the line-shape comparison (Model I vs II) is an independent, non-fitted output.

fitted input called prediction [Sec. II, after Eq. 15 and Eq. 16; Sec. III, FIGs. 6-7.]
"To dynamically generate N(1535) resonance, we take the cut-off parameter qmax = 1150MeV, consistent with Ref. [46,49]. ... It is clear from FIG. 6 and FIG. 7 that there is a significant peak around 1500 MeV in both the π0n and π+n invariant mass distributions, which is associated with the N(1535)0 andN+(1535) resonance states, respectively. ... close to the position of the N(1535) pole in the PDG [2](1510 MeV)."

The cutoff qmax = 1150 MeV is imported from earlier N(1535) studies (Refs. [46,49]) precisely so that the Bethe-Salpeter T-matrix of Eq. 16, T = [1−VG]^−1 V, develops the N(1535) pole. Eq. 9 then multiplies that same T by the same loop function G, and Eqs. 24-25 convert |T|^2 into the invariant mass distributions. The "distinct peak around 1500 MeV" is therefore the input pole evaluated in the new production mechanism: the existence and position of the peak are fixed by the fitted cutoff and are not independent predictions. The non-fitted, genuinely predictive content is limited to the line shape, the relative strength in the π0n versus π+n channels, and the Model I versus Model II comparison in FIG. 10.

full rationale

The derivation chain is self-contained: V and G are standard chiral-unitary inputs, T = [1−VG]^−1 V (Eq. 16), and the production amplitude (Eq. 9) is h + hGT. The partial circularity is that the headline peak near 1500 MeV is a by-construction echo of the N(1535) pole that the model was set up to generate via the adopted qmax = 1150 MeV and decay constants, so presenting the peak as a finding repackages the input resonance. No load-bearing self-citation occurs: Refs. [46,49,55,56] are not by the present authors, and the chiral-unitary framework is externally benchmarked against known N(1535) pole positions in the literature. The decision to keep only Im G in the production loop (Sec. II, after Eq. 15) is a correctness and model-dependence point (the πN channel is open near 1500 MeV with a non-negligible real part), not a circularity, so it is weighed here as risk rather than scored. Overall score 4 reflects that the central peak claim is partially forced by construction while the line-shape distinction (Model I versus II) retains independent content.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the chiral unitary coupled-channel dynamics with a cutoff taken from earlier fits, an ad hoc suppression factor, and the assumption that only the imaginary part of the loop contributes. No new particles or entities are introduced. The free parameters are either normalization constants or values inherited from previous studies of N(1535).

free parameters (7)

C = 2
Color factor weighing external vs internal W-emission; varied between 1.5 and 3.0 in Fig. 9 to show dependence.
Vp = 1
Global weak-interaction strength; set to unity because only shapes of invariant mass distributions are predicted.
alpha = 0.0037 MeV^-1
Smoothing parameter in the high-energy suppression factor of Eq. 28; varied to assess uncertainty.
qmax = 1150 MeV
Three-momentum cutoff in the meson-baryon loop function; taken from Refs [46,49], effectively constrained by the N(1535) pole position in earlier fits.
g2_N*Npi/4pi = 0.037
N*(1535) to N pi coupling used in the Model II width; taken from Ref [10].
g2_N*Neta/4pi = 0.28
N*(1535) to N eta coupling used in the Model II width; taken from Ref [10].
Gamma0 = 19.5 MeV
Constant term in the N(1535) width, taken from Ref [10].

assumptions (6)

domain assumption SU(3) flavor symmetry for the hadronization of quark pairs into meson-baryon pairs (Eqs. 5,6).
The projection of the quark-level state onto meson-baryon channels assumes SU(3) symmetry and the standard meson/baryon matrices.
domain assumption eta-eta' mixing as in Refs [52,53].
Used in the meson matrix, Eq. 3; only affects the eta N channels with small coefficients.
domain assumption The Bethe-Salpeter equation is factorized on-shell as [1 - V G]^-1 V (Eq. 16).
Standard chiral unitary approach; assumes the on-shell factorization approximation is valid.
ad hoc to paper The loop function with three-momentum cutoff and the approximation of taking only the imaginary part of the loop near the N(1535) region (Eqs. 11,15).
The cutoff value is taken from previous fits; the imaginary-part-only approximation is introduced without quantitative justification.
ad hoc to paper A = B in the Lambda_c+ pi+ N(1535) weak vertex (Eq. 19).
The authors state that A and B cannot be determined and assume equality.
ad hoc to paper The smooth suppression factor of Eq. 28 with Mcut = 1650 MeV.
Introduced to tame the limited effective range of the chiral unitary approach; the choice of Mcut is not derived.

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Pith. "Pith review of Theoretical study of the $N(1535)$ in the process $\Lambda_c^+\to n\pi^+\pi^0$." pith.science (2026). https://pith.science/paper/YPZUD4C6

@misc{pith2026250414608,
  author       = {Pith},
  title        = {Pith review of: Theoretical study of the $N(1535)$ in the process $\Lambda_c^+\to n\pi^+\pi^0$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPZUD4C6}},
  note         = {Machine review of arXiv:2504.14608}
}

abstract

We have investigated the $\Lambda_c^+\to n\pi^+\pi^0$ decay process using the chiral unitary approach, by considering that the nucleon resonance $N(1535)$ could be dynamically generated through $S$-wave pseudoscalar meson-octet baryon interactions. In the invariant mass distributions of $\pi^0n$ and $\pi^+n$, we observe a distinct peak structure associated with the resonance states $N(1535)^0$ and $N(1535)^+$, respectively. Our results suggest that the $\Lambda_c^+\to n\pi^+\pi^0$ process can be utilized to probe the properties of the nucleon $N(1535)$, and therefore, we encourage more precise experimental measurements of this process in the future.

Figures

Figures reproduced from arXiv: 2504.14608 by the authors.

**Figure 2.** FIG. 2: Quark level diagram for the process [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3: Diagrams for the process Λ [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4: Diagrams for the process Λ [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5: Diagram representing the process Λ [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: FIG. 7: The [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 6.** Figure 6: FIG. 6: The [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 8.** Figure 8: FIG. 8: The [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 9.** Figure 9: FIG. 9: The [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

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Forward citations

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Revisiting the $\Lambda_c^+\rightarrow\bar{K}^0\eta p$ reaction: the role of $N^*(1535),~ N^*(1650)$ and $\Sigma(1620)$
hep-ph 2025-07 conditional novelty 6.0 of 10

A coupled-channel model with internal and external weak emission plus final-state rescattering reproduces the Belle peaks of N*(1535), N*(1650), and Sigma(1620) in Lambda_c+ -> anti-K0 eta p.

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