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Revisit the diquark of $\Lambda_c$ in the $\Lambda_c\to \Lambda K^+$ and $\Lambda_c\to \Sigma^0 K^+$ processes

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The branching ratios of $\Lambda_c\to\Lambda K^+$ and $\Lambda_c\to\Sigma^0 K^+$ pin down the spatial size parameters of the charmed baryon, and the paper concludes that the $[ud]$ and heavy-light diquarks are spatially extended, not…

desk verdict A careful but highly model-dependent NRCQM application with a concrete error in its key parameter relation, Eq. (24), so the quantitative non-compactness claim needs a corrected redo before it can be trusted. read the letter →

arxiv 2507.04393 v1 pith:3ZHGN4L7 submitted 2025-07-06 hep-ph

Peng-Yu Niu , Qian Wang , Qiang Zhao This is my paper · ORCID
classification hep-ph
keywords diquarkLambda_cbaryonhadronicweakdecaysnonrelativisticconstituentquarkmodelharmonicoscillatorparametersCabibbo-suppressedLambdaselectionrulecharmedstructure
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 asks whether the diquark degrees of freedom inside the charmed baryon $\Lambda_c$—the two-quark cluster made of the light $u,d$ pair and the heavy-light cluster involving the charm quark—are compact objects or spread-out quark pairs. It answers by using two measured Cabibbo-suppressed weak decays, $\Lambda_c\to\Lambda K^+$ and $\Lambda_c\to\Sigma^0 K^+$, whose final-state baryons differ only in isospin. In the nonrelativistic constituent quark model, the two spatial sizes are carried by the harmonic oscillator strengths $\alpha_\rho$ and $\alpha_\lambda$; the larger the strength, the more compact the diquark. Fitting the two measured branching ratios leaves only a small allowed region for this parameter pair, and it sits near the conventional values of the quark model. The paper concludes that neither the light diquark nor the heavy-light diquark in $\Lambda_c$ favors a compact, point-like structure.

What carries the argument

The load-bearing object is the pair of harmonic oscillator strength parameters $(\alpha_\rho, \alpha_\lambda)$ in the nonrelativistic constituent quark model's spatial wave functions, which set the mean-square separations between the two light quarks and between the heavy quark and the light pair. The paper treats these two parameters as independent free parameters, computes the full set of direct meson emission, color-suppressed, and pole-term amplitudes for both decays, and uses the measured branching ratios to carve out the allowed parameter region. A second mechanism, the "$\Lambda$ selection rule," determines which $N^*$ intermediate states contribute to the pole terms and shapes the parity-violating amplitudes, and it is presented as a tool for future searches of excited baryons.

What would settle it

Measure the parity asymmetry parameter for both channels: with the constrained parameters, the model predicts $\alpha\approx 0.99$ or $0.79$ for $\Lambda_c\to\Lambda K^+$ and $\alpha\approx 0.53$ or $0.83$ for $\Lambda_c\to\Sigma^0 K^+$ depending on the $N(1535)$--$N(1650)$ mixing angle; data outside both predicted ranges would contradict the claimed parameter region. A lattice QCD calculation showing a small $\langle r^2\rangle$ for the $ud$ pair inside $\Lambda_c$ would also falsify the non-compact conclusion.

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Extended reading notes

Core claim

The paper's central claim is that the two branching ratios $\Gamma(\Lambda_c\to\Lambda K^+)$ and $\Gamma(\Lambda_c\to\Sigma^0 K^+)$ strongly constrain the two harmonic oscillator strength parameters of the antitriplet singly charmed baryons: $\alpha_\rho$, which controls the separation of the two light quarks, and $\alpha_\lambda$, which controls the separation between the heavy quark and the light pair. Because $\langle\rho^2\rangle = 3/(2\alpha_\rho^2)$, $\langle\lambda^2\rangle = 3/(2\alpha_\lambda^2)$, and the heavy-light distance is $\langle r_{cq}^2\rangle = 9/(4\alpha_\lambda^2) + 3/(4\alpha_\rho^2)$, a small value of either parameter means a large spatial extent. The allowed region found from the decay widths is small and close to the commonly adopted constituent-quark-model values, so the paper concludes that neither the $[ud]$ "good" diquark nor the heavy-light $[cq]$ diquark is point-like.

Load-bearing premise

The conclusion rests on the assumption that the harmonic oscillator wave functions, with the empirically chosen strength parameters for all baryons except the antitriplet charmed ones, correctly reproduce the overlap integrals that control the two decay widths.

Editorial extensions

If this is right

  • If the conclusion is right, compact-diquark pictures for singly charmed baryons are disfavored, and models that treat the $[ud]$ pair as a tightly bound effective object inside $\Lambda_c$ would need revision.
  • The narrow allowed region near the conventional values of $\alpha_\rho$ and $\alpha_\lambda$ supports using the standard constituent-quark-model parameters for antitriplet charmed baryons in future decay calculations.
  • The nonfactorizable color-suppressed and pole contributions are found to be comparable in size to the factorizable direct meson emission, so they cannot be neglected in hadronic weak decay estimates.
  • The ratio $R_\Gamma = \mathrm{Br}(\Lambda_c\to\Lambda K^+)/\mathrm{Br}(\Lambda_c\to\Sigma^0 K^+)$ saturates near the isospin-symmetry value of approximately 3 as $\alpha_\rho$ grows, making the ratio a relatively stable test of the flavor-symmetry structure.
  • Selection rules such as the "$\Lambda$ selection rule" produce vanishing amplitudes for particular $N^*$ intermediate states, which can be used to identify excited baryon and charmed-baryon contributions in heavy-flavor hadronic weak decays.

Reading between the lines

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

  • Going beyond the paper, the same two-parameter constraint could be applied to $\Lambda_b$ decays or to decays of $\Xi_c$ baryons to test whether the spatial extent of the light and heavy-light diquarks changes with the heavy quark mass.
  • A precision measurement of the ratio $R_\Gamma$ that deviates significantly from the predicted saturation value near 3 would challenge the weak-isospin argument and the assumed spatial wave functions, not just the specific parameter values.
  • A natural next test is to compare the extracted $\langle\rho^2\rangle$ and $\langle\lambda^2\rangle$ with lattice QCD determinations of the diquark size inside charmed baryons; such a comparison would turn the qualitative non-compact conclusion into a quantitative benchmark.
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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

2 major / 4 minor

Summary. The paper computes the singly Cabibbo-suppressed hadronic weak decays Λc→ΛK+ and Λc→Σ0K+ in the nonrelativistic constituent quark model, including direct meson emission, color-suppressed, and pole contributions with ground and first excited intermediate states. The authors then treat the harmonic oscillator strengths αρ and αλ of the antitriplet singly charmed baryons as free parameters and use the two measured branching ratios to constrain them. From the resulting allowed region they conclude that neither the [ud] diquark nor the heavy-light diquark in Λc is compact, and they discuss selection rules such as the “Λ selection rule” for excited baryon searches.

Significance. If the calculation were fully robust, the paper would provide a concrete, model-dependent statement about the spatial extension of diquark degrees of freedom in a singly charmed baryon, which is relevant to the phenomenology of diquark correlations. The strength of the manuscript is its transparency: the operator expressions, flavor-spin wave functions, and full amplitude tables are given explicitly, and the qualitative nature of the analysis is acknowledged. The selection-rule observations, especially the role of the mixing between [70,2 8] and [70,4 8] in the N(1535)-N(1650) system, are useful. However, the central quantitative claim is currently undermined by an inconsistency in Eq. (24) and by the absence of any uncertainty propagation in a two-parameter fit to two data points.

major comments (2)
  1. [Sec. III A, Eq. (24), and App. A] Equation (24) is not consistent with the definitions in Appendix A. From Eqs. (A32), (A37), and (A38), one has αλ/αρ = (mλ/mρ)^{1/4} = [3m3(m1+m2)^2/(4M m1m2)]^{1/4} with M = m1+m2+m3. The published formula omits a factor (m1+m2) inside the fourth root and is dimensionally inconsistent: for m1=m2=m3=m it gives 2^{-1/2}(2/m)^{1/4} with m in GeV rather than the correct value 1. Because Table II lists only αρ and states that αλ is obtained via Eq. (24), all αλ values entering the overlap integrals, and therefore the red line and the location of the allowed region in Fig. 3, are computed with the wrong relation. The central quantitative claim that the allowed region is small and close to the commonly adopted constituent-quark values must be re-derived after correcting Eq. (24) and repeating the fit.
  2. [Sec. III C and Fig. 3] The constraints on αρ and αλ are obtained by fitting the two measured branching ratios with two free parameters, so the central values are reproduced by construction; the statement that neither diquark favors a compact structure is therefore a restatement of the fit rather than an independent prediction. Many fixed inputs (masses and widths of the first excited Ξc states, the mixing angle θ = 30°, g_A^q = 1, fK, and the meson size parameter R) enter without propagated uncertainties, and no χ² or covariance information is shown for the allowed bands in Fig. 3. A concrete improvement would be to vary these inputs within their PDG or theoretical ranges and show that the intersection region, and hence the non-compactness conclusion, is stable; without this, the claimed “strong constraints” are not demonstrated.
minor comments (4)
  1. [Sec. III C and Fig. 3 caption] The caption of Fig. 3 and the surrounding text refer to “Λc→Λπ+” and “Λc→Σ0π+”, but the paper studies Λc→ΛK+ and Λc→Σ0K+; the labels should be corrected.
  2. [Sec. I] The sentence “The last section does not discussed in the summary” is ungrammatical and should be rewritten, for example as “The results are summarized in Sec. IV.”
  3. [Sec. III A] The text says that “the strong interaction is flavor independent” immediately before noting that αρ differs for charmed and strange baryons because of SU(3) flavor symmetry breaking; this juxtaposition is confusing and should be clarified.
  4. [Sec. II B, Eq. (14)] In the chiral interaction Hamiltonian, the labels m_i, m_f, and m_j are used without a clear definition of which quark masses they refer to; defining the quark indices explicitly would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper extracts harmonic-oscillator parameters from measured branching ratios and interprets them as diquark sizes, a standard inverse-problem chain rather than a self-justifying prediction.

full rationale

The central claim is an extraction, not a circular prediction. The measured branching ratios Br(Lambda_c -> Lambda K+) and Br(Lambda_c -> Sigma0 K+) are external inputs; the paper treats alpha_rho and alpha_lambda of the antitriplet charmed baryons as free parameters, fixes all other hadron wave-function parameters, and computes the decay widths as functions of alpha_rho and alpha_lambda (Sec. III C). The allowed region in Fig. 3 is the locus where the computed widths match the PDG values. The conclusion that the [ud] and heavy-light diquarks are not compact is obtained by translating the constrained alpha values into mean-square separations through Eqs. (29)-(31), where <rho^2>=3/(2 alpha_rho^2) and larger alpha means more compact by definition. That the data happen to favor small alpha, rather than large alpha, is empirical content, so the chain is not tautological. The same-author citations in Refs. [26] and [46] provide empirical alpha inputs and a consistency statement, but the load-bearing constraint comes from the current calculation against the experimental widths, so these citations do not make the argument circular. The paper's own caveat that it is a 'qualitative analysis' and the apparent dimensional issue in Eq. (24) are correctness or precision concerns, not circularity. Hence the circularity score is 0.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The extraction of αρ and αλ rests on a chain of fixed inputs (quark masses, other baryons' α values, mixing angle, excited state widths, meson size, coupling constants) taken from the authors' earlier papers and the literature. The two data points are fitted by the two free parameters; the structural claim is read off the fitted values.

free parameters (8)
  • α_rho of antitriplet charmed baryons = ≈0.4 to 0.5 GeV (allowed band)
    Treated as free in Sec. III C and fitted to the measured branching ratios; the non-compact conclusion follows from the fitted value.
  • α_lambda of antitriplet charmed baryons = ≈0.5 to 0.8 GeV (allowed band)
    Treated as independent free parameter in Sec. III C; Eq. (24) is only used for the comparison red line.
  • Harmonic oscillator strengths for other baryons = 0.5, 0.4, 0.45, 0.5 GeV (Table II)
    Fixed empirically from previous works [26,46] in Sec. III A; controls final and intermediate baryon overlaps.
  • Quark masses = mu=md=0.3 GeV, ms=0.5 GeV, mc=1.8 GeV
    Taken from Ref. [43] in Sec. III A; enters kinetic terms and the α_λ/α_ρ relation.
  • Mixing angle θ between [70,2 8] and [70,4 8] = 30 degrees
    Chosen from a range 24 to 32 degrees from Ref. [45]; strongly affects N(1535) and N(1650) pole contributions.
  • Widths of first excited charmed baryons = 5×10^-4 GeV
    Estimated value from Ref. [46] because experimental widths are absent; enters pole propagators.
  • Meson size parameter R = not specified
    Appears in the meson spatial wave function Eq. (A41) and affects DME/CS overlaps, but no numerical value is given.
  • Quark axial coupling g_A^q and K decay constant fK = g_A^q=1; fK=1.2 fπ with fπ=94 MeV
    Adopted values for the chiral quark-meson interaction (Sec. II B).
assumptions (6)
  • domain assumption The standard SU(6) x O(3) wave functions describe initial, final, and intermediate baryons.
    Used throughout; explicit wave functions are listed in App. A.
  • domain assumption The nonrelativistic constituent quark model with harmonic oscillator potentials is adequate for these weak decay amplitudes.
    Central framework (Sec. II); no comparison with lattice QCD or relativistic models is provided.
  • ad hoc to paper Only ground and first excited intermediate states are needed in the pole terms.
    Sec. II C states this truncation because dominant contributions come from approximately on-shell states; excited charmed baryon widths are estimated.
  • ad hoc to paper The same α_ρ and α_λ apply to all antitriplet singly charmed baryons, ignoring SU(3) flavor symmetry breaking.
    Sec. III C explicitly ignores SU(3) breaking to fix α for the antitriplet states.
  • domain assumption The measured branching ratios from PDG are accurate inputs.
    Experimental central values and errors in Table V are used as constraints; the paper does not audit the measurements.
  • domain assumption The 'Lambda selection rule' from Ref. [47] correctly predicts vanishing transition matrix elements for N(1650) at θ=0.
    Used in Sec. III B to discard or suppress N(1650) contributions and to explain amplitude zeros.

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Cite this review

Pith. "Pith review of Revisit the diquark of $\Lambda_c$ in the $\Lambda_c\to \Lambda K^+$ and $\Lambda_c\to \Sigma^0 K^+$ processes." pith.science (2026). https://pith.science/paper/3ZHGN4L7

@misc{pith2026250704393,
  author       = {Pith},
  title        = {Pith review of: Revisit the diquark of $\Lambda_c$ in the $\Lambda_c\to \Lambda K^+$ and $\Lambda_c\to \Sigma^0 K^+$ processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3ZHGN4L7}},
  note         = {Machine review of arXiv:2507.04393}
}
abstract

The spatial distributions of $[ud]$ diquark and heavy-light diquark of the SU(3)-flavor antitriplet charmed baryons are investigated by the two singly Cabibbo-suppressed hadronic weak decays, $\Lambda_c\to \Lambda K^+$ and $\Lambda_c\to \Sigma^0 K^+$ within the nonrelativistic constituent quark model. The above two spatial distributions are reflected by the two parameters $\alpha_\rho$ and $\alpha_\lambda$, which are the harmonic oscillator strength parameters of the charmed baryons. These two parameters obtain strong constraints from the decay widths of $\Lambda_c\to \Lambda K^+$ and $\Lambda_c\to \Sigma^0 K^+$. The larger the harmonic oscillator parameter is, the more compact the spatial distribution will become. The current $\alpha_\rho$ and $\alpha_\lambda$ indicate that neither the light diquark nor the heavy-light diquark turns out to favor a compact structure. In addition, some selection rules, including the ``$\Lambda$ selection rule'', can be useful for the search of excited baryons in the heavy-flavor baryon hadronic weak decays.

Figures

Figures reproduced from arXiv: 2507.04393 by the authors.

Figure 1
Figure 1. FIG. 1: The transition diagrams of the Λ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The color suppressed transitions at the quark [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: The ratio [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

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