Final state interaction in the $\Lambda_b\to D^+D^- \Lambda,~D^0D_s^- p,~D_s^+D_s^-\Lambda$ reactions

Eulogio Oset; Jing Song; Melahat Bayar

REVIEW 3 major objections 5 minor 54 references

Final state interaction in the $\Lambda_b\to D^+D^- \Lambda,~D^0D_s^- p,~D_s^+D_s^-\Lambda$ reactions

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper predicts that final-state rescattering in three Λ_b decays produces threshold enhancements in the D_s^-p, D_s^-Λ and their vector channels, and identifies Λ_b → D^0 D_s^- p as the most favorable reaction to detect the predicted…

desk verdict Pseudoscalar FSI predictions are solid and testable; the vector-channel results are asserted rather than derived, so the abstract's two-channel claim outruns the formalism. read the letter →

arxiv 2507.01865 v1 pith:EZW7VDSE submitted 2025-07-02 hep-ph

Jing Song , Melahat Bayar , Eulogio Oset This is my paper

classification hep-ph

keywords Lambda_bdecaysfinalstateinteractionsthresholdenhancementsexotichadronshadronicmoleculesopen-charmstrangestatesunitarizedcoupledchannelslocalhiddengauge

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

This paper tries to establish that final state interactions, not just phase space, shape the invariant mass distributions of the D^-Λ, D_s^-p and D_s^-Λ subsystems in three Λ_b three-body decays, and that the resulting near-threshold enhancements are observable signatures of dynamically generated molecular exotic states. If true, the predicted enhancements give experimental searches a concrete target: the paper identifies Λ_b → $D^{0}$ D_s^- p as the most favorable decay mode, with a bound-state pole about 20–26 MeV below the D_s^-p threshold. The same framework also predicts threshold effects in the vector channels $D^{{*-}}$Λ, $D_s^{{*-}}$p and $D_s^{{*-}}$Λ. The work matters because these states, made of an anticharmed meson plus a strange baryon, lie outside the conventional quark-model classification, and the predicted mass distributions can be checked directly in current and future experiments.

What carries the argument

The load-bearing object is the unitarized coupled-channel meson-baryon scattering amplitude T = [1 - VG]^{-1}V. The kernel V comes from vector-meson exchange in the local hidden gauge formalism, G is the cutoff-regularized two-hadron loop function with q_max = 550–650 MeV, and the same amplitude provides the poles that are interpreted as molecular states. The weak decay side is built from external and internal W^- emission, with a color factor N_c = 3 for external emission, hadronization into meson-baryon pairs, and rescattering through T; the invariant mass distributions are the squared moduli of the sum of direct and rescattered amplitudes weighted by phase space.

What would settle it

A high-statistics measurement of the D_s^-p invariant mass distribution in Λ_b → $D^{0}$D_s^-p near threshold. If the distribution follows phase space with no enhancement, or if the enhancement appears far from the predicted pole energy around 2880–2906 MeV, the paper's central claim would be ruled out.

Watch

Extended reading notes

Core claim

The central claim is that the line shapes in Λ_b → D^+D^-Λ, Λ_b → $D^{0}$D_s^-p, and Λ_b → D_s^+D_s^-Λ are dominated near threshold by rescattering of the meson-baryon pairs, and that this rescattering generates the molecular bound states predicted for the S = -1, I = 1/2 and S = -2, I = 0 sectors. Using a unitarized coupled-channel amplitude built from vector-meson exchange, the authors compute the D^-Λ, D_s^-p and D_s^-Λ invariant mass distributions and find clear enhancements over phase space for D_s^-p and D_s^-Λ, a more modest effect for D^-Λ, and analogous enhancements in the vector channels $D^{{*-}}$Λ, $D_s^{{*-}}$p and $D_s^{{*-}}$Λ. The most favorable signature is the D_s^-p threshold enhancement in Λ_b → $D^{0}$D_s^-p, tied to a pole near 2906 MeV for q_max = 550 MeV or 2880 MeV for q_max = 650 MeV. These results remain stable when the cutoff varies across the standard 550–650 MeV range.

Load-bearing premise

The whole calculation depends on the relative sign and strength of the two weak-emission amplitudes, including the assumed color factor of three for external emission, and on the vector channels inheriting the pseudoscalar production mechanism; if either is wrong, the predicted threshold enhancements could change or disappear.

Editorial extensions

If this is right

The D_s^-p invariant mass distribution in Λ_b → D^0D_s^-p should show a clear enhancement over phase space just above threshold, making this the most favorable channel to observe the predicted S = -1, I = 1/2 molecular state.
The D_s^-Λ distribution in Λ_b → D_s^+D_s^-Λ should show a visible threshold enhancement tied to a bound state near 3083 MeV (q_max = 550 MeV) or 3046 MeV (q_max = 650 MeV).
The D^-Λ distribution should show only a modest effect because the corresponding pole sits roughly 92 MeV below the D^-Λ threshold.
The vector channels D^{*-}Λ, D_s^{*-}p and D_s^{*-}Λ should display analogous threshold enhancements that complement the pseudoscalar signals.
All of these line-shape predictions remain qualitatively stable when the cutoff q_max varies over 550–650 MeV, so the enhancements are not artifacts of a fine-tuned parameter choice.

Reading between the lines

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

If the D_s^-p enhancement is confirmed by data, the same coupled-channel dynamics would predict related open-charm strange molecular states in other weak decays of heavy baryons, where the line shapes could discriminate this model from quark-model alternatives.
Because the paper sets the overall normalization to C = 1 and predicts the relative weights among the three decay channels, a combined measurement of all three invariant mass distributions would be a sharper test than any single spectrum.
A dedicated derivation of the Λ_b → D^{*}D^{*}Λ weak vertex and spin couplings would show whether the vector-channel predictions survive, since the current vector results inherit the pseudoscalar production amplitudes.

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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

Quick take: the pseudoscalar part of this paper is a solid, testable calculation; the vector-meson part is not actually derived here, and the abstract's claim about "both pseudoscalar and vector" channels is one step ahead of what is shown.

What's new: explicit hadronization for the three Lambda_b decay modes, with internal/external emission interference, N_c=3 color enhancement, and the resulting D^-Lambda, D_s^-p, and D_s^-Lambda invariant mass distributions. Those mass distributions and the relative weights between channels are new predictions, not fitted to anything. The paper correctly identifies D_s^-p as the most promising discovery channel, since the pole sits close to that threshold. The qmax = 550-650 MeV variation is a reasonable uncertainty handle, and the pseudoscalar amplitudes are internally consistent.

Soft spots, in order of importance. First, the vector-meson channels. Section III.C says these "were also studied in Ref. [16] and we use the results from there," but no weak-production amplitude for Lambda_b -> D*+D*-Lambda, D*0D*_s^-p, or D*_s^+D*_s^-Lambda is written down or derived. The coefficients from the pseudoscalar hadronization are assumed to carry over, which requires the vector-meson analog of the P matrix, D* spin wavefunctions, the weak vertex spin structure, and unchanged internal/external sign and N_c enhancement. None of that appears. So Fig. 10 is a plausible conjecture, but it is not a derivation in this paper. The abstract's "significant threshold enhancements appear in both pseudoscalar and vector meson-baryon channels" is therefore not established for the vector channels by the present calculation.

Second, the phase-space normalization: normalizing both full and phase-space curves to the same area in a ~40 MeV window is a choice that can exaggerate the visual enhancement. The amplitudes are in arbitrary units (C=1), which is fine for relative shapes, but the apparent size of the threshold effect is partly a plotting convention. Third, the framing that enhancements "indicate" the formation of molecular states is mild circularity: the enhancements follow from the model's own poles from prior work, so they are consistency checks, not independent evidence. The paper does not fit data to produce these enhancements, so this is more a wording issue than a load-bearing flaw. Fourth, LHCb has published Lambda_b -> D+D-Lambda data (Ref [15]); a direct comparison would strengthen the paper significantly, even if only qualitative.

Who this is for: hadron spectroscopists and LHCb-oriented phenomenologists who want concrete predictions for where to look for open-charm strange molecular states. The pseudoscalar section deserves a serious referee; the vector section needs the missing derivation, or a softened abstract, before it can support the current headline claim.

Recommendation: engage with it, conditionally. Send it to referees with a clear request to either derive the vector production amplitudes or limit the claim to the pseudoscalar channels.

Referee Report

3 major / 5 minor

Summary. The paper studies the three-body weak decays Λ_b → D+D−Λ, D0D−s p, and D+sD−s Λ, focusing on final-state interactions in the D−Λ, D−s p, and D−s Λ subsystems and their vector-meson analogues D*−Λ, D*−s p, and D*−s Λ. The weak production amplitudes are built from internal and external W-emission hadronization mechanisms, and the final-state rescattering is treated with unitarized coupled-channel T matrices from the local hidden gauge approach, with poles taken from the companion paper Ref. [16]. The main results are invariant-mass distributions showing threshold enhancements, especially in the D−s p channel, with a qmax = 550–650 MeV variation used as an uncertainty estimate. The paper concludes that the Λ_b → D0D−s p reaction is the most favorable mode for observing the predicted molecular exotic states.

Significance. If the results hold, the paper provides falsifiable predictions for LHCb invariant-mass measurements in three Λ_b decay modes and identifies the most promising channel for observing the predicted open-charm strange molecular states. The pseudoscalar-channel analysis is internally consistent, uses T-matrix poles that were not fitted to the decay data, and includes a cutoff variation that gives a reasonable uncertainty estimate. The explicit credit is that the central object, the mass distribution shapes, is a new observable prediction rather than a fit. However, the vector-meson part of the headline claim is not supported by a derivation in this manuscript, and the sign/color assumptions in the pseudoscalar amplitudes are load-bearing for the predicted enhancements.

major comments (3)

[III.C, Fig. 10] The vector-meson mass distributions are presented without deriving the weak production amplitudes for Λ_b → D*+D*−Λ, D*0D*−s p, and D*+sD*−sΛ. The text states only that these channels were studied in Ref. [16] and 'we use the results from there.' Equations (18), (21), and (24) are obtained from the pseudoscalar hadronization matrix P of Eq. (3), and carrying those production coefficients to vector mesons requires the vector-meson analog of P, the D* spin wavefunctions, the spin structure of the weak vertex, and a demonstration that the relative internal/external sign and the Nc factor are unchanged. None of this is provided, so the vector-channel enhancements in Fig. 10 are an assumed input rather than a derived result. Consequently, the abstract's claim that significant threshold enhancements appear in both pseudoscalar and vector channels is not established for the vector channels by the calculation shown.
[III.A, Eqs. (18) and (21)] The sign and magnitude of the external-emission terms relative to the internal-emission terms are load-bearing for the predicted enhancements. In Eq. (18), the D−s n rescattering contribution enters with the opposite sign and with an Nc = 3 enhancement relative to the D−Λ tree-level and loop terms; if the relative sign or the color factor were different, the interference could turn the near-threshold enhancement into a suppression. The paper should state the derivation of the sign convention and test the sensitivity of the mass distributions to the relative sign and to Nc (for example, by showing the distributions with Nc = 1 or with the opposite relative sign). Without such a test, the claim that the external-emission term has a 'large weight' that 'enhances the role of the coupled channels interaction' is an assumption rather than a robust conclusion.
[III.A, Figs. 7–10] The comparison between the full distributions and 'phase space normalized to the same area in the region chosen of about 40 MeV above threshold' is a choice that largely controls the visual size of the claimed enhancement. Because the final-state interaction concentrates strength near threshold, equal-area normalization in a narrow window forces the full curve to lie above the phase-space curve near threshold and below it farther away. A wider normalization window or an absolute normalization could substantially reduce the apparent effect. The authors should justify the 40 MeV window and show that the qualitative conclusion is insensitive to this choice, for instance by also presenting ratio curves for several window sizes.

minor comments (5)

[I] In the Introduction, 'hadonization' should read 'hadronization'.
[III.A, Table I] For the S = −2, I = 0 pseudoscalar row, the two qmax values are 560 and 650 MeV, while the other rows use 550 and 650 MeV; the choice of 560 MeV should be explained or made uniform.
[III.A, Eq. (18)] Equation (18) uses the D−s n intermediate channel, while the text and Table I refer to the D−s p channel; the isospin relation between D−s n and D−s p in the S = −1, I = 1/2 coupled-channel space should be stated explicitly.
[II.B, Eq. (13)] The pion decay constant fπ is used in Eq. (13) but is not defined; please define it and state the numerical value used.
[III.A] The statement that the relative weights of all presented mass distributions are a prediction is not visible in the figures, since each distribution is separately normalized to its own phase space in a 40 MeV window; showing all three pseudoscalar distributions on a common scale would make the relative-rate prediction explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mass distributions are new observables, and no parameter is fitted to the decay data.

full rationale

The paper's new content is the weak-production hadronization amplitudes (Eqs. (3)-(24)) and their convolution with unitarized T matrices imported from prior coupled-channel studies. The T-matrix poles of Ref. [16] (with the independent check of Ref. [20]) are external inputs with stated assumptions (local hidden gauge, cutoff qmax = 550-650 MeV); they are not fitted to the Lambda_b decay observables, and no parameter of the decay calculation is adjusted to reproduce the mass distributions. The threshold enhancements are predictions of new observables rather than rewritings of the inputs. The only self-citation is Ref. [16], which supplies the T matrices and pole positions; because those results are stated, parameter-dependent predictions from an independent (though overlapping) calculation, citing them is legitimate support rather than circularity. The vector-meson production amplitudes are borrowed rather than re-derived, which is a derivation gap that could affect the validity of the vector-channel curves, but it does not make the calculation circular: no equation in the paper reduces the vector mass distributions to an input by construction. The interpretive statement that enhancements 'indicate' bound states is a framing of the model's own pole content, but the calculated line shapes are independent of that interpretation and testable against future data.

Assumptions & free parameters 3 free parameters · 8 assumptions · 3 invented entities

The central claim rests on the T-matrix and pole positions from Ref [16], which are model inputs inherited from the same authors. The paper itself introduces no new fitted constants for the decay amplitudes, but the interpretation of the computed enhancements as evidence for the molecular states is circular in framing. The axioms and entities above capture what the paper assumes without deriving.

free parameters (3)

qmax (cutoff) = 550-650 MeV
Momentum cutoff in the meson-baryon loop function Eq. (15). Chosen from earlier works [17-19] and not fitted to the decay data. Varied to estimate systematic uncertainty.
C (overall normalization) = 1 (arbitrary)
Global normalization of the weak decay amplitude, set to 1. It cancels in the shape of the mass distributions because it multiplies every term in the amplitude M.
N_c (color enhancement) = 3
Color factor for external versus internal W emission. Fixed by standard color counting and not varied. It affects the relative weight of rescattering terms and hence the size of the enhancement.

assumptions (8)

domain assumption The local hidden gauge approach with vector meson exchange describes the heavy meson-baryon interaction, giving the kernel V_ij = (C_ij / 4 f_pi^2)(k0 + k0').
Used to construct V and the T-matrix in Sec II.B. This is the standard Weinberg-Tomozawa-like interaction for the coupled channels.
domain assumption Only s-wave coupled channels are relevant; the interaction kernel depends only on the sum of meson energies k0 + k0'.
Implicit in Eq. (13) and standard in these FSI studies.
standard math The Bethe-Salpeter equation T = [1 - V G]^{-1} V with the cutoff-regularized loop G is a valid resummation satisfying two-body unitarity.
Eq. (12) and Eq. (15) define the unitarization procedure.
domain assumption The weak b -> c bar-c s transition and the hadronization in Eqs. (1)-(11), including the SU(3) wave functions and spin couplings, correctly give the tree-level amplitudes.
This is the foundation of the external and internal emission mechanisms in Sec II.A.
domain assumption The external emission amplitude is color enhanced by N_c = 3 relative to internal emission, with no additional corrections.
Used in Eqs. (18), (21), and (24) to weight the external emission contributions.
domain assumption Final state interactions in the spectator meson with the pair are negligible; only the subsystem of interest rescatters.
The amplitudes include FSI only in the D-Lambda, Ds-p, or Ds-Lambda subsystems, as shown in Figs. 4-6.
ad hoc to paper The vector-meson decay amplitudes are obtained by reusing the pseudoscalar weak-production structure.
Sec III.C states the vector channels 'were also studied in Ref. [16] and we use the results from there', without deriving the Lambda_b -> D* D* B weak vertex.
ad hoc to paper The phase-space comparison is normalized in a window of about 40 MeV above threshold.
Sec III.A and the figure captions describe this normalization, which affects the visual size of the enhancement relative to phase space.

invented entities (3)

S=-1, I=1/2 molecular exotic state (D-Lambda / Ds-p bound state) independent evidence
purpose: Produces the threshold enhancement in D-Lambda and Ds-p mass distributions.
Predicted in Ref [16] around 2880-2906 MeV. This paper uses it to predict observable cusp-like enhancements; the predicted mass and decay signatures provide a falsifiable handle for LHCb.
S=-2, I=0 molecular exotic state (Ds-Lambda bound state) independent evidence
purpose: Produces the Ds-Lambda threshold enhancement.
Predicted in Ref [16] around 3046-3083 MeV. The predicted enhancement is the falsifiable handle.
Vector-meson analogue states (D*-Lambda, Ds*-p, Ds*-Lambda) independent evidence
purpose: Produce threshold enhancements in the vector channels.
Same model predictions; the paper predicts mass distributions for these channels, testable in future data.

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Cite this review

Pith. "Pith review of Final state interaction in the $\Lambda_b\to D^+D^- \Lambda,~D^0D_s^- p,~D_s^+D_s^-\Lambda$ reactions." pith.science (2026). https://pith.science/paper/EZW7VDSE

@misc{pith2026250701865,
  author       = {Pith},
  title        = {Pith review of: Final state interaction in the $\Lambda_b\to D^+D^- \Lambda,~D^0D_s^- p,~D_s^+D_s^-\Lambda$ reactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZW7VDSE}},
  note         = {Machine review of arXiv:2507.01865}
}

abstract

We study the three-body decays $\Lambda_b \to D^+ D^- \Lambda,~D^0D_s^- p,~D_s^+D_s^-\Lambda$, considering both internal and external emission mechanisms that produce various meson-baryon final states. Using a unitarized coupled-channel approach based on the local hidden gauge formalism, we analyze final state interactions in the exotic anticharm systems with strangeness $S=-1$ and $S=-2$. Significant threshold enhancements appear in both pseudoscalar and vector meson-baryon channels, indicating strong final state interactions and the possible formation of molecular exotic states below threshold. These features remain stable under variations of model parameters. Our results provide valuable insights into heavy hadron decay dynamics and offer theoretical guidance for future experimental searches of open-charm strange exotic hadrons.

Figures

Figures reproduced from arXiv: 2507.01865 by the authors.

**Figure 2.** FIG. 2: Hadronic configurations from hadronization in the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: Hadronic configurations from hadronization through [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4: Diagrammatic representation of the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5: Diagrammatic representation of the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6: Diagrammatic representation of the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7: Invariant mass distributions of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8: Invariant mass distribution of [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: shows the invariant mass distribution dΓ dMinv(D− s Λ) . This channel belongs to the S = −2, I = 0 sector and provides a unique opportunity to explore doubly-strange hadronic molecules, which have not been extensively studied. 3000 3025 3050 3075 3100 3125 0 5 10 15 20…

**Figure 10.** Figure 10: FIG. 10: Invariant mass distributions of [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

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