Effect of eigenstates on spectra in coupled-channel scattering with the chiral unitary model

Takuma Nishibuchi; Tetsuo Hyodo

REVIEW 3 major objections 3 minor 22 references

Effect of eigenstates on spectra in coupled-channel scattering with the chiral unitary model

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper argues that the $\pi^+\Xi^-$ peak in $\Xi_c \to \pi\pi\Xi$ sits at $1599$ MeV while the $\Xi(1620)$ pole sits at $1610$ MeV, and that the line shape near the $\bar{K}\Lambda$ threshold tracks the sign of the real part of the…

desk verdict Useful proceedings paper with a concrete threshold-shift prediction for Ξ_c decay, but the numerical peak position is not reproducible until the production weights are specified and tested. read the letter →

arxiv 2507.18941 v1 pith:FMPE5Z5P submitted 2025-07-25 hep-ph nucl-th

Takuma Nishibuchi , Tetsuo Hyodo This is my paper · ORCID

classification hep-phnucl-th

keywords Xi(1620)chiralunitaryapproachcoupled-channelscatteringthresholdeffectslengthinvariantmassspectrumXi_cdecaypoletrajectory

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 $\pi^+\Xi^-$ peak seen in $\Xi_c \to \pi\pi\Xi$ decay is not a direct reading of the $\Xi(1620)$ pole mass: in the model based on the experimental resonance parameters, the three-body peak sits at $1599$ MeV while the pole sits at $1610$ MeV. As one continuously interpolates between that model and a model based on the measured $K^-\Lambda$ scattering length, the line shape near the $\bar{K}\Lambda$ threshold changes from a peak to a cusp exactly when the real part of the $\bar{K}\Lambda$ scattering length changes sign. The two models' poles never meet on the Riemann sheets, so the two poles correspond to different eigenstates even though both have been associated with the $\Xi(1620)$. The upshot is that experimental $\pi\Xi$ spectra must be analyzed with coupled-channel threshold effects built in, not by assigning the observed peak energy directly to a resonance mass.

What carries the argument

The load-bearing object is the coupled-channel scattering amplitude $T_{ij}(W)$ defined by $T = V + VGT$ with the Weinberg-Tomozawa kernel $V_{ij}^{\rm WT}$ and dimensionally regularized loop functions $G_k(W;a_k)$, together with the model interpolation $a_i(x)=x a''_i + (1-x)a'_i$ that continuously connects the two parameter sets. The decay spectrum is generated from this amplitude through Eq. (5), where the weak-production vertex $V_P$ is a constant and the channel weights $h_i$ weight each intermediate meson-baryon channel. The correlation between line shape and the real part of the $\bar{K}\Lambda$ scattering length $a_0$ is the mechanism that turns a peak into a cusp when $x$ crosses $\sim0.375$.

What would settle it

A high-statistics measurement of the $\pi^+\Xi^-$ invariant mass distribution in $\Xi_c \to \pi\pi\Xi$ that resolves the line shape near the $\bar{K}\Lambda$ threshold would test the claim: a cusp maximum at the threshold would support a negative real scattering length, while a peak near 1599 MeV below the 1610 MeV pole would confirm the threshold-shift mechanism of Model 1.

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Extended reading notes

Core claim

Within the chiral unitary approach, the authors construct two models of the $\Xi(1620)$: Model 1 reproduces the experimentally reported mass and width and has a pole at $z_1 = 1610 - 30i$ MeV on the $[bbtttt]$ Riemann sheet, while Model 2 reproduces the measured $K^-\Lambda$ scattering length and has a pole at $z_2 = 1726 + 80i$ MeV on the $[ttbttt]$ sheet. By linearly interpolating the subtraction constants between the models, they find that $z_1$ never moves onto the sheet containing $z_2$, and conclude that the two poles originate from different physical mechanisms rather than from one state seen in two parametrizations. Computing the $\pi^+\Xi^-$ invariant mass distribution in $\Xi_c \to \pi\pi\Xi$ with Eq. (5), they find that the spectral shape near the $\bar{K}\Lambda$ threshold is governed by the sign of the real part of the $\bar{K}\Lambda$ scattering length: positive $\mathrm{Re}\,a_0$ gives a peak, negative gives a cusp, with the transition near the interpolation point $x\sim0.375$. The Model 1 spectrum peaks at $1599$ MeV, below the $1610$ MeV pole, and this threshold shift is larger in the three-body decay than in the two-body scattering amplitude, where the peak sits at $1606$ MeV.

Load-bearing premise

The spectrum calculation assumes the weak-decay production amplitude is constant and leaves the relative weights of the intermediate channels unspecified, so the predicted peak position at 1599 MeV and the peak-to-cusp transition could move if those inputs are changed.

Editorial extensions

If this is right

A measured $\pi^+\Xi^-$ peak energy cannot be identified with the $\Xi(1620)$ pole position; in Model 1 the three-body peak is 11 MeV below the pole, so an analysis of the experimental spectrum must include the decay dynamics.
The sign of the real part of the $\bar{K}\Lambda$ scattering length predicts the line shape: positive values produce a peak near threshold, negative values a cusp, so the observed spectral shape can act as a diagnostic for the scattering length.
The absence of a pole connection during interpolation indicates that the two descriptions of the $\Xi(1620)$ correspond to different eigenstates, which must be treated as distinct in phenomenological comparisons.
Threshold effects are stronger in the three-body invariant mass distribution than in the two-body amplitude, so three-body decay data are a sharper probe of threshold physics.
Even when the pole lies above the $\bar{K}\Lambda$ threshold, the spectrum can still show a peak below threshold, so pole position and peak position need not even lie on the same side of the threshold.

Reading between the lines

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

The authors do not test the stability of their spectrum against the production weights $h_i$; varying those weights over a plausible range is a natural next step, since the 1599 MeV peak position and the peak-to-cusp transition could shift if the production inputs change.
A high-statistics measurement that resolves a cusp rather than a peak at the $\bar{K}\Lambda$ threshold would favor a negative real scattering length, while a peak near 1599 MeV below the 1610 MeV pole would confirm the threshold-shift mechanism of Model 1.
The same interpolation-and-spectrum strategy could be applied to other three-body decays where a resonance sits near a two-body threshold, making the scattering-length sign a common ordering principle for line shapes.
Because the threshold effect is larger in the three-body process than in the two-body amplitude, analyses of other heavy-hadron three-body decays with nearby thresholds may need even larger corrections than two-body estimates suggest.

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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

Short version: this is a useful proceedings paper, not a breakthrough, and the main new numbers are plausible but not fully standalone. The genuinely new bit is the calculation of the π+Ξ− invariant mass spectrum in Ξ_c → ππΞ for the two chiral unitary models from Ref [11], plus the interpolation path between them. The headline result, that the Model 1 peak sits at 1599 MeV while the pole sits at 1610 MeV, and that the three-body shift is larger than the two-body shift (1606 MeV), is a concrete, checkable claim and worth having on record. The peak-to-cusp transition tracking the zero of Re a0 is also clearly presented.

The calculation is standard: Weinberg-Tomozawa kernel, dimensional-regularized loop, coupled-channel equation. I don't see an internal inconsistency, and the pole trajectory showing that the two fitted models land on different Riemann sheets is a fair reading of the earlier work. The paper is not circular; fitting to Belle and ALICE inputs is ordinary model construction.

Where it gets soft is Eq. (5). The production weights h_i are never given, V_P is just a constant, and there is no sensitivity scan. The stress-test note is correct: with arbitrary relative phases and magnitudes of h_i, the position of the peak, and even the peak-to-cusp transition, can move. So the specific numbers 1599 MeV and x≈0.375 are not reproducible from the paper alone. The paper also does not overlay the Belle data, which would have been the natural way to see whether the threshold-shift explanation actually works. The subtraction constants are omitted; they are presumably in Ref [11], but a proceedings paper can still list them or point to a table.

None of this makes the central idea wrong. The correlation between the sign of Re a0 and peak-vs-cusp near threshold is known from two-body studies [21,22], and the present work extends it to a three-body decay. For a reader interpreting the Belle πΞ peak as the Ξ(1620) mass, the caution about threshold effects is well taken. But the paper overstates what it establishes: without the production model, it shows that some choices of h_i produce a large shift, not that the three-body threshold effect is necessarily more prominent than the two-body one.

Who is this for? Practitioners in hadron spectroscopy, especially those working on coupled-channel fits to Belle and ALICE data. Deserves a serious referee: yes, it is topical, short, and has a concrete prediction, but the referee should ask for the h_i values and a sensitivity study before publication. I would not cite the 1599 MeV number until that is supplied.

Referee Report

3 major / 3 minor

Summary. The paper studies the excited Ξ(1620) resonance within the chiral unitary approach. It uses two previously constructed models: Model 1, fitted to Belle mass/width data, and Model 2, fitted to the ALICE K−Λ scattering length. The authors interpolate between the two models by linearly varying the subtraction constants in Eq. (4), track the pole trajectory in Fig. 1, and compute the π+Ξ− invariant mass distribution for Ξc→ππΞ using the production-weighted final-state-interaction formula in Eq. (5). The main claims are that the spectral shape near the ¯KΛ threshold is strongly correlated with the sign of the real part of the ¯KΛ scattering length, that the Model 1 peak sits at 1599 MeV, well below the 1610 MeV pole energy, and that threshold effects are even more prominent in the three-body decay than in the two-body amplitude.

Significance. If the central claims hold, the paper delivers a useful and testable message: the Belle πΞ peak should not be read directly as the Ξ(1620) pole mass, because near-threshold coupled-channel dynamics shift the peak substantially in the three-body decay. The correlation between the spectral shape (peak vs. cusp) and the sign of Re a0 is a concrete, falsifiable diagnostic that can be applied to other near-threshold resonances. The paper also supports the view that the two models describe different eigenstates, although this conclusion is presented with limited evidence. The calculation uses the standard Weinberg-Tomozawa kernel and dimensional regularization, and the companion papers [11,15] provide the underlying framework; nevertheless, the present manuscript is not self-contained on the key inputs needed to reproduce the spectral results.

major comments (3)

[Sec. 2, Eq. (5)] The production weights h_i that enter the invariant mass distribution in Eq. (5) are never specified, and no sensitivity study is provided. The main quantitative results of the paper—the Model 1 peak at 1599 MeV, its downward shift from the 1610 MeV pole, and the peak-to-cusp transition at x≈0.375—are all computed from this formula and therefore cannot be reproduced or assessed from the manuscript alone. Please provide the values (and if possible the derivation) of h_i, and demonstrate that the conclusions are stable under reasonable variations of these weights, e.g., by showing spectra for several choices of h_i or bounding the effect analytically.
[Sec. 2, Eq. (4)] The conclusion in Sec. 3 that the Model 1 and Model 2 poles 'originate from different physical mechanisms' rests on the specific linear interpolation path in Eq. (4). Since the subtraction-constant space is multi-dimensional, an alternative path could in principle connect the two pole positions, so the claim of distinct eigenstates is path-dependent as presented. Please clarify whether this conclusion has been tested against different interpolation paths, or cite the systematic analysis in Ref. [15] that establishes it.
[Sec. 2, Eqs. (1)-(3)] The numerical values of the subtraction constants a_i, the meson decay constants, and the masses used in the models are not listed anywhere in the manuscript, so the calculation is not reproducible from this paper alone. Please include a table with these inputs or give precise pointers to the equations/tables in Ref. [11] where they are defined.

minor comments (3)

[Fig. 2 caption] The caption for the left panel describes the distribution as π+Ξ0, but the text and abstract consistently discuss the π+Ξ− channel; please correct the caption.
[Sec. 3, Fig. 2] The statement that the spectral shape changes from a peak to a cusp 'around the same value' of x is based on interpolation points spaced by 0.2; please state the precise x value where the transition occurs and how it is defined, or soften the claim accordingly.
[Sec. 3, Fig. 2] The correlation between the spectral shape and the sign of Re a0 is presented without a quantitative measure; a simple quantification (e.g., the height of the maximum relative to the threshold value as a function of Re a0) would strengthen the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the spectral outputs are genuine model results, not refitted inputs; the scattering-length correlation is a consistency check rather than a definitional identity.

full rationale

The paper's central results are not circular. Model 1 and Model 2 are taken from the authors' previous work [11], where the subtraction constants were adjusted to reproduce the Belle mass/width and the ALICE Kbar-Lambda scattering length, respectively. Those fitted constants are inputs here, but the headline quantities—the interpolated pole trajectory, the 1599 MeV peak in the three-body spectrum, and the x~0.375 peak-to-cusp transition—are computed dynamical outputs of the coupled-channel equations and Eq. (5), not parameters fitted to those same quantities. The 1599 MeV peak is explicitly different from the input pole at 1610 MeV, so it is not a refit by construction. The correlation between the spectral shape and the sign of Re a0 is presented as a known phenomenon (refs. [21,22]) and is verified in the interpolated model; because both the spectrum and the scattering length are functions of the same scattering amplitude, this is a consistency check rather than an independently derived prediction, but it is not a circular equivalence. The main weakness—the production weights h_i in Eq. (5) are never specified and no sensitivity study is provided—is a reproducibility and modeling-assumption gap, not a circularity. The self-citations to [11] and [15] supply model parameters and a two-body reference value, but the spectrum calculation is performed in the present paper and is checkable from the displayed equations once h_i are supplied. Accordingly, no step in the derivation reduces to its own input by definition or by fitting.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central numerical results rest entirely on the two fitted models and the decay formula of Ref [20]. The paper introduces no new particles, forces, or conserved quantities, and it does not derive its parameters from first principles. The only new structural input is the interpolation path of Eq. (4) and the application of the existing spectrum formula to the Ξ_c → ππΞ decay.

free parameters (3)

subtraction constants a_i (Model 1) = Not listed in the text; from Ref [11]
Adjusted to reproduce the Belle mass and width of the Ξ(1620) at the pole z1 = 1610 - 30i MeV; enters the loop function Eq. (3) and controls all scattering amplitudes and spectra.
subtraction constants a_i (Model 2) = Not listed in the text; from Ref [11]
Adjusted to reproduce the ALICE K−Λ scattering length; produces the pole z2 = 1726 + 80i MeV and the Model 2 spectra.
channel weights h_i in the decay amplitude = Not specified
Set the relative weights of intermediate channels in Eq. (5); no values or sensitivity analysis are given, so the spectral shapes and the peak-to-cusp transition may depend on this choice.

assumptions (4)

standard math The coupled-channel Bethe-Salpeter equation with dimensional regularization and Riemann-sheet continuation is the correct framework for dynamically generated resonances.
Eqs. (1)-(3) are standard in the chiral unitary approach; no proof or convergence study is given.
domain assumption The Weinberg-Tomozawa interaction kernel (Eq. 2) with the chosen coupled channels is sufficient to describe the Ξ(1620).
This is the standard model input; the paper does not justify the channel set or the neglect of higher-order interaction terms.
domain assumption The weak decay amplitude for Ξ_c → ππΞ factorizes as a constant production vertex times final-state interactions (Eq. 5).
Adopted from Ref [20]; no derivation is given for V_P constant or for the h_i weights.
ad hoc to paper Linear interpolation of the subtraction constants in Eq. (4) is a physically meaningful path for connecting Model 1 and Model 2.
The conclusion that the two poles originate from different physical mechanisms depends on this interpolation path; no uniqueness or justification for the linear form is given.

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Cite this review

Pith. "Pith review of Effect of eigenstates on spectra in coupled-channel scattering with the chiral unitary model." pith.science (2026). https://pith.science/paper/FMPE5Z5P

@misc{pith2026250718941,
  author       = {Pith},
  title        = {Pith review of: Effect of eigenstates on spectra in coupled-channel scattering with the chiral unitary model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMPE5Z5P}},
  note         = {Machine review of arXiv:2507.18941}
}

abstract

In recent years, as experimental data on the excited $\Xi(1620)$ and $\Xi(1690)$ states have accumulated, theoretical analyses based on chiral dynamics have also been actively pursued. In this study, we construct theoretical models within the chiral unitary approach to reproduce recent experimental data, and perform a model interpolation to examine the relationship between the poles appearing in different models. We then calculate the invariant mass spectra of the $\pi^+\Xi^-$ system in the $\Xi_c \to \pi\pi\Xi$ decay, using the scattering amplitudes obtained from each model, to investigate how the distinct pole structures influence the observable spectrum.

Figures

Figures reproduced from arXiv: 2507.18941 by the authors.

**Figure 1.** Trajectory of the pole 𝑧1 obtained through model interpolation. The pole corresponding to Model 1 (𝑥 = 0) is indicated by a square, while those for Model 2 (𝑥 = 1) are shown by circles. Intermediate poles for 𝑥 varied in steps of 0.2 are represented by squares. 𝑎𝑖 are continuously varied via the interpolation parameter 𝑥, from the values in Model 1 to those in Model 2. As the values of 𝑎𝑖 approach those of Model 2, … view at source ↗

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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