Could $\bar{\Lambda}_c$ and $\Lambda_c$ form bound hadronic molecule with explicit $P-wave$ ?

Xiu-Wu Wang; Zhi-Gang Wang

REVIEW 3 major objections 6 minor 74 references

Could $\bar{\Lambda}_c$ and $\Lambda_c$ form bound hadronic molecule with explicit $P-wave$ ?

T0 review · 3 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read QCD sum rules place P-wave anti-Lambda_c Lambda_c states far above threshold, so they do not form bound hadronic molecules.

desk verdict A careful QCD sum-rules calculation that says no P-wave Λ̄cΛc bound molecules, but the single-pole truncation leaves the main conclusion less secure than the paper admits. read the letter →

arxiv 2607.16756 v1 pith:HVBCXVDF submitted 2026-07-18 hep-ph

Xiu-Wu Wang , Zhi-Gang Wang This is my paper

classification hep-ph PACS 12.39.Mk14.20.Lq12.38.Lg

keywords dibaryonQCDsumrulesLambda_canti-Lambda_cP-wavehadronicmoleculeresonancehidden-charmbaryonium

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 asks whether the charmed baryons anti-Lambda_c and Lambda_c can form a hadronic molecule when the pair carries one unit of orbital angular momentum (a P-wave). Using QCD sum rules with twelve interpolating currents that carry explicit P-wave structure for J^PC = 0^{-±} and 1^{-±}, the authors extract masses for the lowest states in each channel. All extracted masses come out well above the two-baryon threshold: for instance, the 0^{-+} state is 5.78 GeV against a 4.89 GeV threshold, roughly 900 MeV higher. The paper therefore concludes that anti-Lambda_c and Lambda_c do not form bound P-wave molecules and that the corresponding states, if real, are resonance dibaryons. This matters because recent experiments show a near-threshold cross section for producing anti-Lambda_c Lambda_c pairs, which some groups have interpreted as evidence of bound baryonium; the present analysis rules out the P-wave molecular interpretation for these quantum numbers.

What carries the argument

The central machinery is the QCD sum rule applied to twelve interpolating currents. Each current is a product of an anti-Lambda_c and a Lambda_c interpolating current, with an explicit spatial derivative (partial or covariant) that imposes one unit of orbital angular momentum and selects J^PC = 0^{-±} or 1^{-±}. The correlation function of each current is computed on the hadronic side as a single ground-state pole and on the QCD side as an operator product expansion up to dimension-16 condensates; after a Borel transform and quark-hadron duality, the ratio of the two sides gives the mass. The load-bearing comparison is the gap between the resulting mass and the mass of the two free baryons:

What would settle it

A bound anti-Lambda_c Lambda_c state with J^PC = 0^{-±} or 1^{-±} observed below 4890 MeV (or below 4920 MeV for the 1^{-±} thresholds), or a lattice QCD computation finding a P-wave bound state in this channel, would falsify the paper's conclusion that no such bound molecule exists.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that a QCD sum-rule treatment of twelve explicitly P-wave anti-Lambda_c Lambda_c currents yields ground-state masses of 5.78^{+0.09}_{-0.09} GeV for J^PC = 0^{-+}, 5.80^{+0.09}_{-0.08} GeV and 5.71^{+0.07}_{-0.07} GeV for the two 1^{-+} channels. The corresponding two-baryon thresholds are 4890 and 4920 MeV (a ground-state Lambda_c plus the excited P-wave Lambda_c states at 2595 and 2625 MeV). Since even the lower error bounds sit far above threshold, the paper concludes that anti-Lambda_c and Lambda_c cannot form bound P-wave molecules with these quantum numbers, and assigns the extracted states as resonance dibaryons. The analysis also finds t

Load-bearing premise

The calculation assumes a single ground-state pole dominates the correlation functions and that two-baryon scattering states contribute negligibly; if that assumption fails, the quoted high masses do not prove the absence of bound P-wave molecules.

Editorial extensions

If this is right

If the result is correct, no bound hadronic molecule with J^PC = 0^{-±} or 1^{-±} exists in the anti-Lambda_c Lambda_c system, so near-threshold enhancements in this channel cannot be explained by P-wave molecular bound states.
The states extracted by the sum rules would appear, if produced, as resonances around 5.7-5.8 GeV, far above the anti-Lambda_c Lambda_c threshold, rather than as narrow bound states below threshold.
Positive and negative charge-conjugation partners are predicted to be nearly degenerate (within 1-3 MeV), so a measurement that finds a large C-parity splitting would be in tension with the sum-rule picture.
The negligible gluon contribution from covariant derivatives means that simpler partial-derivative currents are adequate for future P-wave dibaryon sum-rule studies.

Reading between the lines

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

The extraction assumes a single isolated pole and ignores the two-baryon scattering continuum; if that continuum contributes in this energy region, the quoted masses are effective averages, and the no-binding conclusion could be an artifact of the single-pole ansatz rather than a physical statement.
The paper's conclusion concerns P-wave molecules specifically; the S-wave anti-Lambda_c Lambda_c binding question is not settled by this analysis, so the broader debate over baryonium states remains open.
A targeted experimental search for anti-Lambda_c Lambda_c structures in the 5.7-5.8 GeV invariant-mass region, or a lattice QCD computation of the P-wave anti-Lambda_c Lambda_c scattering phase, would provide a direct test that bypasses the sum-rule pole assumption.

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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 solid, narrow QCD sum-rules calculation. It constructs twelve P-wave Λ̄cΛc currents, works out the spectral densities, checks OPE convergence and pole dominance, and gets masses around 5.7–5.8 GeV, roughly 0.9 GeV above the ΛcΛc* thresholds. The authors conclude that no P-wave bound molecule exists in this system. That conclusion is probably what most readers will take home, and it is less secure than the paper lets on.

What is genuinely new: the systematic treatment of partial- versus covariant-derivative currents, with the nice quantitative result that the difference shows up only in the dimension-10 ⟨qgsσGq⟩² term and is below 3.3%. The charge-conjugation mass splittings are 1–3 MeV, which is a clean signal. The OPE is carried to dimension 16 with D(12), D(14), D(16) at or below 0.4%, so the QCD side looks controlled.

The soft spot is exactly the one flagged in the stress-test note. Equation (9) keeps a single ground-state pole and drops the two-baryon scattering continuum by invoking the authors' own Ref. [68]. But the extracted pole sits only ~0.9 GeV above the scattering threshold, and in the Borel window T² ~ 3.7–4.4 GeV² the threshold region is weighted much more heavily than the pole. The pole-dominance parameter PC only shows that the OPE continuum above s0 is suppressed; it says nothing about the two-baryon continuum below s0. So the 5.7–5.8 GeV masses could be duality-averaged effective masses rather than the pole masses of a would-be molecule. If so, the no-binding conclusion is an artifact of the single-pole ansatz, not a QCD statement. The paper doesn't address this.

There are also the usual self-consistent conventions: √s0 is set to mZ + 0.5–0.7 GeV, where mZ is the very mass being extracted, and the energy scale formula uses the group's calibrated Mc. These are method systematics, not cheating, but they mean the quoted ±0.09 GeV errors are parametric only.

Even with that, I wouldn't reject. The calculation is honest, the internal checks are the right ones, and the negative result is a useful data point in the baryonium debate. The right fix is to confront the scattering continuum explicitly—estimate its contribution or justify why it is negligible in this channel—and to discuss the s0 convention openly. A referee should demand that. But the paper deserves a serious referee.

Referee Report

3 major / 6 minor

Summary. The paper studies P-wave \bar\Lambda_c\Lambda_c dibaryon states with J^{PC}=0^{-\pm},1^{-\pm} using QCD sum rules. Twelve interpolating currents are constructed with both partial and covariant derivatives, including parity and charge-conjugation eigenstates. The OPE is carried out to dimension 16 with O(alpha_s^k), k<=1. Against the standard criteria — pole dominance 40-59%, convergence D(12), D(14), D(16)<~0.4%, and the energy-scale formula — the extracted masses are m_1^+=5.78, m_{2,\mu}^+=5.80, m_{3,\mu}^+=5.71 GeV, about 0.8-0.9 GeV above the respective \Lambda_c(2295)\Lambda_c(2595/2625) thresholds. The paper concludes that no bound P-wave molecules exist in this channel and assigns the states as resonance dibaryons. It also reports that charge-conjugation and partial-vs-covariant-derivative differences are numerically small.

Significance. If correct, the result is a useful negative statement for the baryonium/molecule debate: it would exclude bound P-wave \bar\Lambda_c\Lambda_c molecules in the studied channels and would support the authors' previous S-wave no-binding conclusion. The paper's strengths are that the masses are genuinely extracted from QCD spectral densities rather than fitted, and it reports the standard internal consistency checks (Borel windows, pole dominance, OPE convergence, energy-scale formula). However, the central negative claim rests on a single-pole ansatz whose treatment of the two-baryon scattering continuum is asserted via a self-citation rather than demonstrated; since the extracted pole lies ~0.9 GeV above the two-baryon threshold in a Borel window that heavily weights the threshold region, this is a load-bearing point that needs a quantitative check before the no-binding conclusion can be accepted as stated. The charge-conjugation and derivative-systematics results are credible and likely robust.

major comments (3)

[Sec. 2, Eq. (9)] The hadronic side is truncated to a single ground-state pole, with the two-baryon scattering continuum omitted. The only justification in the manuscript is the sentence in Sec. 1 citing Ref. [68] that 'their contributions can be neglected.' This is not demonstrated here. The omitted continuum begins at \sqrt{s_{\rm th}}\simeq 4.89-4.92 GeV, only ~0.9 GeV below the extracted pole. In the quoted Borel windows (T^2=3.7-4.4 GeV^2), the Borel weight at the threshold relative to the extracted pole is roughly exp[-(23.9-24.2)/T^2]/exp[-(32.6-33.6)/T^2], i.e., the threshold region is weighted 10-13 times more strongly than the pole region. The pole-dominance parameter PC of Eq. (14) measures the suppression of the QCD spectral density above s0, not the size of the omitted hadronic continuum below s0. The extracted m_Z and \lambda_Z should therefore be interpreted as duality-averaged effective pa
[Table 2 and Sec. 3] The numerical errors quoted (e.g., m_1^+=5.78^{+0.09}_{-0.09} GeV) are only from variations of Borel windows, continuum thresholds, and condensate parameters with the statistical uncertainties of the inputs. The dominant systematic uncertainty — the single-pole truncation of Eq. (9) — does not enter the error budget. For a state extracted 0.89 GeV above threshold, this systematic should be at least estimated; as is, the claim that the lower bound is 'still far larger' than the threshold relies entirely on the omitted continuum being negligible.
[Sec. 2, Eq. (13)] The energy-scale formula \mu=\sqrt{M_X^2-4M_c^2} uses the extracted mass M_X itself. Since the extracted masses are all in a narrow range (5.71-5.80 GeV) and the scale enters only logarithmically through input parameters, this is unlikely to create a circular determination of the no-binding result. However, the paper does not discuss what would change if the scale were instead set by the two-baryon threshold; a brief robustness test would remove this concern.

minor comments (6)

[Sec. 1] The statement that two-baryon scattering states 'can be neglected' in Ref. [68] is not specific enough; please give the numerical check from that work or reproduce the relevant argument in this paper.
[Sec. 3, Table 1] The table lists only positive-charge-conjugation states. The text states the negative ones differ by only 1-3 MeV, but no negative-conjugation table entries or explicit mass values are given. Please include the negative-conjugation masses or a table footnote.
[Sec. 2, Eqs. (1)-(2), (8)] The notation for the currents is dense and several symbols (e.g., J^\pm_Z in the paragraph after Eq. (8)) are used before being defined clearly. A table of the twelve currents with their Lorentz structure, J^{PC} assignment, and the corresponding ground-state notation would improve readability.
[Eq. (14)] The normalization D(n)=|D(n)|/\sum|D(n)| is written in prose rather than as an equation; it would be clearer to define it explicitly in the equation block.
[Sec. 3, Fig. 3] The caption of Fig. 3 labels the three panels A, B, C but does not define the line styles/colors; please specify which curve corresponds to which current in the caption.
[References] Ref. [69] has an obvious typo ('X. W. and Z. G. Wang') and Refs. [27], [28], [31] lack complete bibliographic information (one has a missing year, several are arXiv-only with no journal reference).

Circularity Check

1 steps flagged · score 4.0 of 10

Load-bearing self-citation: the no-binding conclusion relies on neglecting the two-baryon scattering continuum, justified only by the same authors' Ref. [68].

self citation load bearing [Sec. 1, paragraph 4; Eq. (9)]
"In Ref. [68], the two-baryon scattering states are considered, however, their contributions can be neglected."

The central no-binding claim (masses 5.71-5.80 GeV above the Lambda_c Lambda_c thresholds) is obtained from Eq. (12), whose hadronic side is the single-pole form of Eq. (9), with no two-baryon continuum term. The sole in-paper justification for omitting that continuum is the quoted self-citation to the same author's Ref. [68]. The PC in Eq. (14) only measures OPE continuum suppression above s0, not the omitted hadronic scattering states below s0, which start only 0.9 GeV below the extracted pole. Thus the central conclusion is carried by an unverified self-referential citation rather than by an independent calculation.

full rationale

The OPE side of the sum rule is computed from QCD inputs (quark masses, condensates) and is not fitted to the final masses; the Borel-platform extraction of m_Z = 5.71-5.80 GeV is a genuine calculation, so this is not a case of 'fitted prediction equals input.' However, the hadronic side is truncated to a single ground-state pole, and the only justification in this paper for neglecting the two-baryon scattering continuum is a self-citation to the same author's prior work. Because the extracted pole lies well above that continuum threshold, the omitted continuum is not kinematically negligible, and the PC criterion does not test it. The continuum-threshold convention sqrt(s0)=m_Z+0.5-0.7 GeV is also self-referential, but it is a standard QCD sum-rule stability condition and does not by itself force the quoted masses. Overall, the paper contains substantial independent OPE content, but its central negative conclusion is supported by a load-bearing self-citation, giving a moderate circularity score of 4.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

No new particles, forces, conserved quantities, or dimensions are postulated: the twelve interpolating currents are probe operators for hypothetical dibaryons whose existence is the question under study, not asserted entities. The free parameters are the threshold and Borel-window conventions and the scale-setter M_c; the axioms are the standard QCD sum rules machinery plus the single-pole ansatz and the calibration devices, several of which originate in the same group's prior papers.

free parameters (3)

Effective charm quark mass M_c in the energy scale formula = 1.85 ± 0.01 GeV
Sets the renormalization scale μ = √(M_Z² − 4M_c²) (Eq. 13) that feeds the scale-dependent condensates and m_c; calibrated in the same group's prior work (Refs [74, 75]), so it is a self-cited input.
Continuum thresholds √s0 per channel = 6.38–6.40 ± 0.10 GeV (Table 1)
Chosen per state by the convention √s0 = m_Z + 0.5–0.7 GeV (text after Eq. 12); the upper limit of the Borel integral, which materially affects the extracted masses.
Borel windows T² = 3.7–4.4 GeV² (Table 1)
Selected 'via trivial and trial' so that m(T²) plateaus and pole contributions sit at 40–59%; the window choice defines the point of extraction for mass and residue.

assumptions (7)

domain assumption QCD sum rules validity: quark–hadron duality and Borel improvement equate the hadronic and OPE sides
The entire framework, Sec. 2 Eqs. (9)–(12); accepted method in the field, not proven here.
domain assumption Single-pole saturation of the correlation functions with negligible two-baryon scattering contribution
Eq. (9) keeps only the ground-state pole; the neglect of scattering states is asserted in Sec. 1 on the authority of the authors' Ref [68].
domain assumption Fock–Schwinger gauge with first-order truncation of the gluon field
Eq. (3): G_n_μ = ½ x^α G_n_αμ(0) + ... with x^α x^β and higher terms dropped; required to justify neglecting the covariant-derivative gluon contributions.
ad hoc to paper OPE truncation at dimension 16 and O(α_s^k) with k ≤ 1
Stated in Sec. 2 ('accurate enough to calculate terms for k ≤ 1 [49]'); justified only by the observed smallness of the computed high-dimension terms.
ad hoc to paper Energy scale formula μ = √(M_Z² − 4M_c²) with M_c = 1.85 GeV
Eq. (13), Refs [74, 75] from the same group; a calibrated device for choosing the scale, not derived in this paper.
domain assumption External inputs: standard vacuum condensates, m_c(MS) = 1.275 GeV, and PDG masses of Λc(2295), Λc(2595), Λc(2625)
Sec. 3 input list; these fix the QCD side and the thresholds (4890/4920 MeV) against which the binding question is judged.
domain assumption The derivative currents J2, J3,μ, J4,μ couple to the Λc(2595) and Λc(2625) P-wave baryons
The 'explicit P-wave' content of the dibaryon currents rests on the standard identification of derivative currents with excited Λc states (Sec. 2, Eq. (1)).

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Cite this review

Pith. "Pith review of Could $\bar{\Lambda}_c$ and $\Lambda_c$ form bound hadronic molecule with explicit $P-wave$ ?." pith.science (2026). https://pith.science/paper/HVBCXVDF

@misc{pith2026260716756,
  author       = {Pith},
  title        = {Pith review of: Could $\bar\Lambda_c$ and $\Lambda_c$ form bound hadronic molecule with explicit $P-wave$ ?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HVBCXVDF}},
  note         = {Machine review of arXiv:2607.16756}
}

abstract

In this article, the explicit $P-wave$ of the $\bar{\Lambda}_c\Lambda_c$ dibaryon states with the $J^{PC}=0^{-\pm}$ and $1^{-\pm}$ are studied via twelve interpolated currents. Results show that the positive and negative charge conjugations for the studied states do not have too much difference, the contribution of the gluon from covariant derivative of the current with $P-wave$ could be neglected. The numerical results of the masses of the $P-wave$ $\bar{\Lambda}_c\Lambda_c$ are high above the threshold of the two $\bar{\Lambda}_c$ and $\Lambda_c$ baryon constituents, $\bar{\Lambda}_c$ and $\Lambda_c$ are unlikely to form the bound molecules with explicit $P-wave$ . Pole residues of the related states are also calculated.

Figures

Figures reproduced from arXiv: 2607.16756 by the authors.

**Figure 1.** The m1 − T 2 curves plotted by the same central input parameters for both the positive and negative charge conjugations interpolated by the currents J ± 1 . to judge the pole dominance criterion of the QCD sum rules, moreover, the convergence of the operator product expansion is reflected via D(n), the contribution of vacuum condensate with dimension n, they are written as, PC = R s0 4m2 c dsρQCD(s) exp − s T 2 R … view at source ↗

**Figure 2.** The masses and pole residues of the Λ¯ cΛc dibaryons with P − wave . 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The dimensional contribution of the states, where, A, B a [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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