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Analysis of the strong vertices $\Lambda_cD^{(*)}N^*(1535)$ and $\Lambda_bB^{(*)}N^*(1535)$ in QCD sum rules

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper predicts six strong coupling constants for vertices that connect charmed and bottom hadrons with the negative-parity nucleon resonance N*(1535), obtained by extrapolating QCD sum-rule results from spacelike momenta to the meson…

desk verdict A useful sum-rule extension whose printed on-shell couplings do not match its own fit parameters; the central numbers need reconciliation before they can be used. read the letter →

arxiv 2506.23095 v2 pith:B65JFRHX submitted 2025-06-29 hep-ph

Jie Lu , Dian-Yong Chen , Guo-Liang Yu , Zhi-Gang Wang , Ze Zhou This is my paper
classification hep-ph
keywords QCDsumrulesthree-pointcorrelationfunctionsstrongcouplingconstantsN*(1535)negativeparitybaryonLambda_cbaryonsLambda_bheavyquarklimit
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 aims to determine the strong coupling constants of the vertices $\Lambda_c D N^*$, $\Lambda_c D^* N^*$, $\Lambda_b B N^*$ and $\Lambda_b B^* N^*$, where $N^*$ is the negative-parity nucleon resonance $N^*(1535)$, using QCD sum rules—a nonperturbative technique that matches hadron states to a quark-level operator expansion. These couplings are the unknown constants that enter effective Lagrangians describing decays and final-state rescattering of heavy-flavor hadrons, so determining them turns qualitative diagrams into quantitative predictions. The calculation first fixes the mass and pole residue of $N^*(1535)$ with two-point QCD sum rules, then evaluates each vertex with three-point QCD sum rules in the deep spacelike region, fits the momentum dependence to a four-parameter analytic function, and extrapolates to the timelike on-shell points $Q^2 = -m_D^2$, $-m_{D^*}^2$, $-m_B^2$, and $-m_{B^*}^2$. The six on-shell couplings are reported with errors, and the bottom-sector tensor coupling $g_{\Lambda_b B^* N^*}$ comes out near zero, in line with the heavy-quark limit.

What carries the argument

The machinery is a set of three-point correlation functions constructed from interpolating currents for the $\Lambda_{c[b]}$ baryon, the $D^{(*)}[B^{(*)}]$ meson, and the nucleon current $J(x)=\varepsilon_{ijk}(u^T C\gamma_\beta u_j)\gamma^\beta d_k$ for $N^*(1535)$. On the QCD side the operator product expansion is evaluated including perturbative, quark condensate, gluon condensate, mixed condensate, and four-quark condensate contributions, with spectral densities obtained by Cutkosky cutting rules; on the hadronic side all couplings of the currents to both positive- and negative-parity baryon states are kept. A double Borel transformation and quark-hadron duality convert the two descriptions into expressions for the momentum-dependent couplings $G(Q^2)$, $f(Q^2)$, and $g(Q^2)$. The final load-bearing object is the four-parameter fit $G(Q^2)=G_0(1+\delta_1 Q^2/m_{\Lambda_{c[b]}}^2+\delta_2(Q^2/m_{\Lambda_{c[b]}}^2)^2)\exp(\delta_3 Q^2/m_{\Lambda_{c[b]}}^2)$, fitted in the spacelike window and continued to the timelike on-shell points $Q^2=-m_{D^{(*)}[B^{(*)}]}^2$.

What would settle it

A lattice QCD computation of one of the couplings, for example $G_{\Lambda_c D N^*}(Q^2)$ at $Q^2=0$ and at one or two nearby timelike points, compared with the fitted curve evaluated at the same $Q^2$, would settle whether the analytic continuation is trustworthy; a disagreement larger than the quoted error bars would invalidate the on-shell values in Eq. (40).

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is a set of six numbers for the on-shell strong couplings: $G_{\Lambda_c D N^*}(-m_D^2)=4.06^{+0.96}_{-0.75}$, $f_{\Lambda_c D^* N^*}(-m_{D^*}^2)=3.73^{+0.68}_{-0.16}$, $g_{\Lambda_c D^* N^*}(-m_{D^*}^2)=9.22^{+3.16}_{-0.36}$, $G_{\Lambda_b B N^*}(-m_B^2)=9.11^{+1.54}_{-1.61}$, $f_{\Lambda_b B^* N^*}(-m_{B^*}^2)=8.55^{+2.69}_{-2.21}$, and $g_{\Lambda_b B^* N^*}(-m_{B^*}^2)=-0.25^{+0.16}_{-0.01}$. The authors would emphasize that these numbers are obtained while including all positive- and negative-parity baryon contributions on the hadronic side and the main vacuum condensate terms in the operator product expansion. The paper also reports a consistency check with the heavy-quark limit: the relation $G=f$ is satisfied in both the charmed and bottom sectors, while $g=0$ is satisfied in the bottom sector but not cleanly in the charmed sector, which it reads as evidence that the heavy-quark limit works well for $b$ quarks and less well for $c$ quarks.

Load-bearing premise

Everything rests on the assumption that the four-parameter fit made at spacelike $Q^2$ between 3 and 9 GeV$^2$ (or 3 and 29 GeV$^2$ for the bottom vertices) describes the true coupling all the way down to the timelike on-shell points $Q^2=-m_D^2$, $-m_{D^*}^2$, $-m_B^2$, and $-m_{B^*}^2$.

Editorial extensions

If this is right

  • The six on-shell couplings can be inserted directly into the effective Lagrangians for $\Lambda_c D^{(*)}N^*$ and $\Lambda_b B^{(*)}N^*$ vertices to compute strong decay widths and production cross sections involving these hadrons.
  • They supply missing inputs for final-state rescattering loop calculations of heavy-flavor decays, where long-distance dynamics has been the main source of uncertainty.
  • The two-point sum-rule results $m_{N^*}=1.53$ GeV and $\lambda_{N^*}=0.026$ GeV$^3$ are usable inputs for other hadron analyses involving $N^*(1535)$.
  • The pattern $G=f$ in both sectors and $g\approx 0$ in the bottom sector supports treating bottom hadrons with heavy-quark spin symmetry, while implying noticeable spin-symmetry breaking in charmed systems.

Reading between the lines

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

  • A reader extending the paper would expect the main numerical uncertainty to sit in the functional extrapolation rather than in the operator product expansion; comparing the four-parameter ansatz with a simple monopole or dipole fit over the same spacelike window would directly show how stable the on-shell values are.
  • If the near-zero $g_{\Lambda_b B^* N^*}$ survives further checks, it provides a quantitative measure of heavy-quark spin symmetry in bottom systems and could be used to tune spin-symmetry-breaking terms in effective Lagrangians.
  • The same two-point-plus-three-point sum-rule pipeline, including the same extrapolation ansatz, could be applied to other negative-parity baryon resonances, and the resulting couplings could be compared with lattice or quark-model predictions as a test of the method.
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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

3 major / 4 minor

Summary. The manuscript presents a QCD sum-rule analysis of the negative-parity nucleon resonance N*(1535) and of the strong vertices Λ_c D^(*) N*, Λ_b B^(*) N*. In the two-point part the authors extract the mass and pole residue of N*(1535) using a γ5-modified Ioffe current and include both positive- and negative-parity baryon contributions. In the three-point part they set the D^(*)/B^(*) meson off-shell, compute the couplings from three-point correlation functions in the spacelike region, fit the resulting Q²-dependence with the four-parameter ansatz of Eq. (39), and extrapolate to the timelike on-shell points Q² = -m_{D^(*)}², -m_{B^(*)}². The headline results are the six couplings listed in Eq. (40), with errors obtained from the Borel windows and input parameter variations.

Significance. If the reported numbers are correct, they provide useful inputs for hadronic rescattering and decay calculations involving N*(1535), and they offer a test of heavy-quark symmetry through the expected relations G ≈ f, g ≈ 0. The paper is careful in several respects: it includes positive- and negative-parity baryon couplings on the hadronic side, treats the projection needed to remove scalar-meson contamination, performs standard pole-dominance and OPE-convergence checks, and reports Borel windows. It is also a strength that the spacelike coupling points are computed from the sum rules rather than fitted to the on-shell values, so the procedure is not circular. However, the central numerical claim fails a direct check: evaluating Eq. (39) with Table II does not reproduce any of the six on-shell values in Eq. (40), and the QCD spectral densities needed to evaluate Eq. (35) are not shown. The reader's extrapolation concern is legitimate; the numerical traceability issue, however, is even more decisive. The significance of the paper is therefore conditional on correcting these issues.

major comments (3)
  1. [§IV.B, Eq. (39), Table II, Eq. (40)] The on-shell couplings are not traceable to the stated fit. Using Table I masses and Table II central parameters, I evaluate Eq. (39) at x = -m_{D(*)}²/m_{Λ_c}² and x = -m_{B(*)}²/m_{Λ_b}². For G_{Λ_c D N^*}: x = -0.662, 1 + δ1 x + δ2 x² = 1.139, exp(δ3 x) = 0.690, so G = 6.76 × 1.139 × 0.690 = 5.31, whereas Eq. (40) quotes 4.06. The analogous values for the six quantities G_{Λ_c D N^*}, f_{Λ_c D^* N^*}, g_{Λ_c D^* N^*}, G_{Λ_b B N^*}, f_{Λ_b B^* N^*}, and -g_{Λ_b B^* N^*} are (5.31, 6.72, 16.15, 10.38, 11.12, 1.42), compared with the quoted values (4.06, 3.73, 9.22, 9.11, 8.55, 0.25). The ratios range from 1.14 to 5.66 and are not uniform, so rounding or a sign convention cannot explain the discrepancy. Since Eq. (40) is the central result of the paper, the authors must either correct Eq. (40) or replace Table II with parameters that actually reproduce it, and verify every on-shell number.
  2. [§III.C, Eq. (35)] The full QCD spectral densities ρ̂(s,u,Q²), ρ̄1(s,u,Q²), and ρ̄2(s,u,Q²) are not given; the text says they are 'too complex to be shown here for simplicity.' These densities are the essential content of the QCD side: without them no reader can check the spacelike points in Fig. 4 or the final couplings. Please include the complete expressions in an appendix, or make an ancillary file available, and define the symbols used in Eq. (35).
  3. [§IV.B, Fig. 4 and Eq. (39)] The four-parameter ansatz is fitted only to spacelike points (3–9 GeV² for the charmed vertices, 3–29 GeV² for the bottom vertices) and then evaluated at timelike Q² = -m_{D(*)}² or -m_{B(*)}². A high R² in the fitted region does not validate the extrapolation. In particular, the D(*) on-shell points are just outside the fitted interval (Q² ≈ -3.5 and -4.0 GeV², compared with Q² > 3 GeV²), and the B(*) points lie near the edge of the fitted interval in magnitude (Q² ≈ -27.9 and -28.3 GeV², compared with Q² up to 29 GeV²). Because no alternative functional form is tried and no systematic uncertainty for the extrapolation is assigned, the quoted errors in Eq. (40) understate the model dependence. The authors should test at least one other ansatz and report the spread of the on-shell results.
minor comments (4)
  1. [Table II] The row labelled 'G_{Λ_b B* N^*}' is inconsistent with the notation in Eq. (14) and Eq. (40), where the pseudoscalar coupling is G_{Λ_b B N^*}; please correct the label.
  2. [Fig. 4] The axis labels and tick labels contain renderings such as '/s8722' instead of minus signs; in the published version the axes must be legible.
  3. [After Eq. (33)] The sentence after Eq. (33) says the denominator power can be reduced 'by using derivative formula in Eq. (33)'; the intended reference appears to be Eq. (32), where the derivative trick is displayed.
  4. [§IV.B] The text does not specify how the asymmetric errors in Eq. (40) are propagated from the Borel windows, condensate uncertainties, threshold parameters, and fit parameters; please describe the error budget explicitly.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline on-shell couplings in Eq. (40) are, by the paper's own construction, evaluations of the four-parameter fit Eq. (39) with Table II; explicit evaluation at Q² = -m²_D(*)/-m²_B(*) does not reproduce the quoted values (computed 1.14-5.7x larger), so the central claim reduces to an unverifiable fitted ansatz.

  1. fitted input called prediction [Sec. IV.B, Eqs. (39)-(40) and Table II]
    "Finally, we obtain the on-shell values of these coupling constants by setting Q2 = −m2 D(∗)[B(∗)] in Eq. (39). The final results are shown as, GΛcDN∗ (Q2 = −m2 D) = 4.06+0.96 −0.75"

    The headline numbers are, by the paper's own statement, the four-parameter ansatz of Eq. (39) evaluated at the meson pole; no QCD-side input constrains Q² < 0 except the chosen fitting form, so the 'predictions' in Eq. (40) reduce to the fit by construction. The stated reduction also fails its own check: with Table II central values and Table I masses, x = Q²/m²_Λ = -m²_D/m²_Λc = -0.662 gives G = 6.76×(1.138)×exp(-0.371) ≈ 5.3, not 4.06; similarly ≈ 6.7 vs 3.73, ≈ 16.2 vs 9.22, ≈ 10.4 vs 9.11, ≈ 11.1 vs 8.55, and ≈ -1.42 vs -0.25. All six quoted values are 1.14-5.7x smaller than the fit evaluation. Since Sec. III.C withholds the spectral densities ('too complex to be shown here'), the central results are either fit-forced or untraceable.

  2. other [Sec. II.D and Sec. IV.A (threshold choice for N*(1535))]
    "u0 is the threshold parameter which is introduced to eliminate the contributions of higher resonances and continuum states. It commonly fulfills the relation √u0 = m_ground+δ, where the m_ground denotes the mass of ground state hadron, δ is the energy gap between the ground and first excited states and commonly taken as a value of 0.3 − 0.7 GeV which is based on experimental data"

    Weak self-anchoring: the continuum threshold window u0 = 4.35–5.22 GeV² corresponds to √u0 ≈ 2.09–2.28 GeV, i.e. effectively m_Ν*(experimental) + δ with δ ≈ 0.55–0.75 GeV, so the input spectral window is anchored to the experimental mass of the very state being extracted; the output m_N* = 1.53±0.16 GeV then reproduces that input. The Borel-ratio extraction is genuine, so this is not definitional, but the 'prediction' of the N*(1535) mass is partially anchored to its own target and is not independent first-principles evidence.

full rationale

The genuinely QCD-based content is the two-point sum rule (Eq. 13) and the three-point spacelike couplings (Eq. 35), which are computed from a real OPE with condensates; that part is self-contained, not circular. The central claim, however, is the on-shell set in Eq. (40), and the paper itself states these are obtained by 'setting Q² = −m²_D(∗)[B(∗)] in Eq. (39)' - i.e. the headline couplings are the fitted four-parameter ansatz evaluated in the timelike region. That is the fit-then-predict pattern applied to the paper's own most prominent output: everything not already fixed by the spacelike QCDSR points is fixed by the arbitrarily chosen polynomial-times-exponential form, so the prediction reduces to the fit. The reduction is also numerically broken: evaluating Eq. (39) with Table II central parameters and Table I masses at the stated pole positions gives about 5.3, 6.7, 16.2, 10.4, 11.1 and −1.42 instead of the quoted 4.06, 3.73, 9.22, 9.11, 8.55 and −0.25 - every value 1.14–5.7× larger - so the printed numbers are not even the value of the printed fit. With the QCD spectral densities withheld ('too complex to be shown here') and the Borel-window procedure deferred to self-citation [49], the central results are neither independently derivable nor reproducible from the equations as written. Secondary matters: f_D/B(∗) and λ_Λc/Λb inputs, s0, and the N* threshold are borrowed from the authors' own previous sum-rule papers [36, 37], and the N* mass extraction is softly anchored to the experimental mass through u0; these are normal parameter borrowing rather than definitional circularity, though the headline couplings scale directly with those self-cited values. No uniqueness theorem is imported and no ansatz is smuggled via citation (the fit form is presented as the authors' own trial-and-error choice). Verdict: partial circularity - the central on-shell couplings reduce to an unverified fitted ansatz that fails the paper's own check - score 6.

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

The computation uses no new particles or forces. Its free parameters are the standard sum rule choices (thresholds, Borel windows) plus the 24 fit parameters that determine the final extrapolated couplings. The input quantities λ, f, condensates and masses are taken from earlier work by the same group or from PDG. The main axiom content is the standard QCD sum rule machinery.

free parameters (6)
  • u0 (N* threshold) = 4.35 to 5.22 GeV^2
    Chosen by repeated trial to satisfy pole dominance and OPE convergence in the two-point sum rule; affects the extracted m_N* and λ_N*.
  • Borel parameter T^2 (two-point) = 1.4 to 1.8 GeV^2
    Selected as the Borel platform where the mass and residue are stable; the result varies within this window.
  • s0 (Λc and Λb thresholds) = 7.02 to 8.12 GeV^2 and 36.00 to 38.44 GeV^2
    Taken from the authors' previous work; determines the continuum subtraction for the initial baryon.
  • Borel parameters T^2 (three-point) = 4.5 to 6.5 GeV^2 (charmed), 17 to 19 GeV^2 (bottom)
    Chosen from Borel platforms at Q^2 = 3 GeV^2, where the pole contribution is about 40 percent.
  • Fit parameters G0, δ1, δ2, δ3 (six couplings) = Table II values
    24 parameters fitted to the computed spacelike G(Q^2) points; the on-shell values in Eq. (40) are evaluations of this fitted function.
  • m_p input = 0.983 GeV
    Used to remove the positive-parity proton contribution in the two-point sum rule; differs from the PDG proton mass 0.938 GeV without explanation.
assumptions (6)
  • domain assumption Quark-hadron duality
    The integrated QCD spectral density up to thresholds s0 and u0 is assumed to equal the hadronic resonance contributions. Invoked in matching after Eq. (13) and Eq. (35).
  • domain assumption Interpolating current choice
    The N* current is taken as the Ioffe current multiplied by γ5, and the Λc/b current is the standard ε(u^T C γ5 d)c form; no test of alternative currents is provided. Sections II.A and III.C.
  • domain assumption Vacuum saturation for four-quark condensates
    The ⟨q qbar⟩^2 and g_s^2 ⟨q qbar⟩^2 terms are treated with factorization, a standard but uncontrolled approximation. Section III.C.
  • domain assumption Neglect of higher-dimensional condensates
    The OPE is truncated after dimension-6 terms; convergence is argued from the small size of high-dimension contributions.
  • domain assumption Borel parameter ratio k
    The two Borel parameters are reduced to one by k = m_N*^2/(m_Λc^2 - m_c^2), following Refs. [48, 58]; this choice affects the sum rule stability.
  • domain assumption Separation of parities using input m_p
    The positive-parity proton contribution is removed by using m_p as input in Eq. (13), assuming the current couples only to p and N* in the relevant region.

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Cite this review

Pith. "Pith review of Analysis of the strong vertices $\Lambda_cD^{(*)}N^*(1535)$ and $\Lambda_bB^{(*)}N^*(1535)$ in QCD sum rules." pith.science (2026). https://pith.science/paper/B65JFRHX

@misc{pith2026250623095,
  author       = {Pith},
  title        = {Pith review of: Analysis of the strong vertices $\Lambda_cD^(*)N^*(1535)$ and $\Lambda_bB^(*)N^*(1535)$ in QCD sum rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B65JFRHX}},
  note         = {Machine review of arXiv:2506.23095}
}
abstract

In this article, we firstly analyze the mass and pole residue of negative parity nucleon $N^*(1535)$ within the two-point QCD sum rules. Basing on these results, we continuously study the strong coupling constants of vertices $\Lambda_cDN^*$, $\Lambda_cD^*N^*$, $\Lambda_bBN^*$ and $\Lambda_bB^*N^*$ in the framework of three-point QCD sum rules. At hadron side, all possible couplings of interpolating current to hadronic states are considered. At QCD side, the contributions of vacuum condensate terms $\langle\bar{q}q\rangle$, $\langle g_s^2GG\rangle$, $\langle\bar{q} g_s\sigma Gq\rangle$, $\langle\bar{q}q\rangle^2$ and $g_s^2\langle\bar{q}q\rangle^2$ are also considered. By setting the four momentum of $D^{(*)}[B^{(*)}]$ mesons off-shell, the strong coupling constants in deep space-like regions ($Q^2=-q^2\gg\Lambda_{QCD}^2$) are obtained. Then, the momentum dependent coupling constants in space-like regions are fitted into analytical function $G(Q^2)$ and are extrapolated into time-like regions ($Q^2<0$). Finally, the on-shell values of strong coupling constants are obtained by taking $Q^{2}=-m_{D^{(*)}[B^{(*)}]}^2$. The results are $G_{\Lambda_cDN^*}(Q^2=-m_D^2)=4.06^{+0.96}_{-0.75}$, $f_{\Lambda_cD^*N^*}(Q^2=-m_{D^*}^2)=3.73^{+0.68}_{-0.16}$, $g_{\Lambda_cD^*N^*}(Q^2=-m_{D^*}^2)=9.22^{+3.16}_{-0.36}$, $G_{\Lambda_bBN^*}(Q^2=-m_B^2)=9.11^{+1.54}_{-1.61}$, $f_{\Lambda_bB^*N^*}(Q^2=-m_{B^*}^2)=8.55^{+2.69}_{-2.21}$ and $g_{\Lambda_bB^*N^*}(Q^2=-m_{B^*}^2)=-0.25^{+0.16}_{-0.01}$.

Figures

Figures reproduced from arXiv: 2506.23095 by the authors.

Figure 1
Figure 1. FIG. 1: The Feynman diagrams for the perturbative part and va [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The mass (a) and pole residue (b) for [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The Borel windows for strong coupling constants [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The fitting curves of coupling constants [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

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