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REVIEW 4 major objections 4 minor 119 references

Ten ξ-moments from QCD sum rules fix the leading-twist longitudinal distribution amplitudes of ρ, K*, and φ, yielding improved D(s)→V decay predictions.

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2026-08-01 17:47 UTC pith:CADCYRO2

load-bearing objection The paper's ten-moment sum-rule calculation is competent, but the PLP fit that 'determines' the ρ and φ DAs violates isospin symmetry and the model's normalization is wrong, so the central DA extraction is unreliable as presented. the 4 major comments →

arxiv 2607.17496 v1 pith:CADCYRO2 submitted 2026-07-20 hep-ph

Vector mesons leading-twist longitudinal distribution amplitudes and related semi-leptonic decays within QCD sum rules

Wan-Bing Luo , Ru-Meng Pan , Ya-Xiong Wang , Tao Zhong This is my paper
classification hep-ph PACS 12.38.-t12.38.Bx14.40.Aq
keywords distribution amplitudesQCD sum rulesvector mesonsξ-momentsGegenbauer momentssemileptonic decaystransition form factorslight-cone sum rules
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

By computing the first ten ξ-moments of the leading-twist longitudinal distribution amplitudes (DAs) of ρ, K*, and φ in QCD sum rules and fitting them with a two-parameter power-law model, the paper claims to determine the full x-dependence of these DAs. The resulting moments, such as ⟨ξ²⟩_{2;ρ}=0.225±0.013 and the Gegenbauer moment a₂=0.074±0.039 at 1 GeV, are stable across the Borel window and agree with lattice predictions where available. Inserting these DAs into light-cone sum rules, the authors compute the D⁰→ρ⁻ℓν, D⁺→K*⁰ℓν, and D_s→φℓν transition form factors and branching fractions, which match BESIII and CLEO measurements within errors. If correct, this gives a self-consistent non-perturbative input for charm semileptonic decays and a method that extends reliable moment order up to n=10.

Core claim

The central claim is that the ratio-normalized sum rule — the n-th ξ-moment from Eq. (10) divided by the square root of the squared zeroth-moment sum rule — yields reliable ⟨ξⁿ⟩ up to n=10 for ρ, K*, and φ. Fitting these with the model φ(x) ∝ xᵅ(1-x)ᵝ by least-squares gives goodness-of-fit probabilities above 98% for all three mesons. The fitted DAs are close to lattice QCD results and approach the asymptotic form 6x(1−x) as the scale grows. Plugged into light-cone sum rules, they produce the TFFs A₁, A₂, V at q²=0 (e.g., A₁^{D→ρ}(0)=0.528±0.039, V^{D→K*}(0)=0.876±0.042) and branching fractions such as B(D⁺→K*⁰eν)=5.56×10⁻², all consistent with BESIII, CLEO, and PDG values within uncertainti

What carries the argument

The correlator of the ξ-moment currents with (iz·D)ⁿ and the background-field-theory OPE up to dimension-six condensates. The normalization trick (Eq. 12) divides the n-th moment sum rule by the square root of the squared n=0 sum rule, avoiding the non-normalizable zeroth-moment sum rule and enabling stable moments up to n=10. These moments are fitted with the normalized power-law (PLP) parametrization xᵅ(1−x)ᵝ by a least-squares χ² fit.

Load-bearing premise

The true DA is assumed to have the two-parameter power-law form xᵅ(1−x)ᵝ; any additional x-dependence that the ten moments do not pin down would shift the fitted shape and all following TFF predictions.

What would settle it

A lattice QCD extraction of ⟨ξ⁶⟩ or ⟨ξ⁸⟩ for ρ or φ at μ=2 GeV that deviates from the fitted PLP model's predictions (about 0.058 and 0.039 for ρ at 2 GeV) by more than twice the quoted errors would falsify the claimed 'determined' DA behavior, since the sum rule does not directly measure the moments.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The determined DAs can be used as inputs for other exclusive charm and bottom hadron processes, reducing the dominant non-perturbative uncertainty.
  • The moment-fitting scheme extends the reliable moment count from about 4 to 10, bypassing the extremely unreliable higher Gegenbauer moments that arise from direct conversion.
  • The K* DA shows a slight asymmetry (a¹_{2;K*}=−0.038 at 1 GeV) that quantifies SU(3) flavor-symmetry breaking and shifts the peak of the DA below x=0.5.
  • The computed branching fractions for D⁰→ρ⁻ℓν, D⁺→K*⁰ℓν, and D_s→φℓν agree with BESIII and CLEO measurements within errors, providing a cross-check of the SM and of lepton-flavor universality in charm decays.
  • The TFF ratios r_V and r₂ at q²=0 match the BESIII measured values for D→ρ and D_s→φ, supporting the reliability of the underlying DAs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the true DA has additional structure beyond the two-parameter PLP form—such as endpoint oscillations or higher Gegenbauer terms—the same ten moments could be reproduced by a different curve; a straightforward test would be to refit with a three-parameter model and check whether TFF predictions move outside the quoted errors.
  • The normalization trick that works for these vector mesons might also be applied to transverse DAs or to other light mesons (π, η, etc.) whose zeroth-moment sum rules suffer from the same normalization issue.
  • The surprisingly rapid approach to the asymptotic form as μ increases suggests that the scale evolution of the DAs is faster than some earlier model estimates; future lattice calculations of the second moment at lower scales could confirm this behavior.
  • Because the same fitted DAs feed both the TFFs and the branching fractions, a precise future measurement of any single TFF (such as A₁^{D→K*}(q²)) would indirectly test the entire moment-fitting chain.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper computes the leading-twist longitudinal distribution amplitudes (DAs) of the ρ, K*, and φ mesons within QCD sum rules in the background field theory framework. It extracts the first ten ξ-moments up to n=10, fits them with a two-parameter power-law (PLP) model via least squares, and then uses the resulting DAs as inputs in light-cone sum rules to compute the D→(ρ,K*) and D_s→φ semileptonic transition form factors and branching fractions. The authors report agreement with several lattice/DSE/SR results for low moments and with BESIII/CLEO data for some observables.

Significance. If the extracted DAs are reliable, the paper would provide a useful set of high-order ξ-moments and an updated quantitative input for charm semileptonic decay analyses. The OPE calculation is presented in considerable detail, including explicit mass-correction terms in Appendix A, and the comparison tables draw on a broad set of lattice, DSE, and sum-rule results. These are genuine strengths. However, the central DA-determination step contains internal inconsistencies: the fitted PLP parameters violate the symmetry required for the ρ and φ mesons, the printed model normalization is wrong, and the K* first moment has the opposite sign to other major determinations. These issues directly affect the subsequent TFF and branching-fraction predictions.

major comments (4)
  1. [Sec. II.C, Eq. (14), Table III] For the ρ meson (q1=d, q2=u) and the φ meson (q1=q2=s), the leading-twist DA must be symmetric under x↔1−x, so the two-parameter PLP form requires α=β. The fitted values at μ=1 GeV are (α=0.703, β=−0.121) for ρ and (α=0.895, β=−0.009) for φ, both strongly violating this condition. Moreover, β<0 makes the DA diverge at x=1, and the odd moments, which are zero by symmetry and are not listed for ρ/φ, cannot constrain the antisymmetric component. The fit therefore extracts an unphysical asymmetric DA. In addition, Eq. (14) as printed has the normalization denominator Γ[α+1]+Γ[β+1]; the correct beta-distribution normalization requires the product Γ[α+1]Γ[β+1]. As written, ∫φ dx is not 1. Since the TFFs in Sec. III.D use these DAs, all subsequent numerical results inherit this problem.
  2. [Table II (K* first moment)] The paper obtains ⟨ξ¹⟩_{2;K*}=−0.0190±0.0035 at μ=2 GeV, while the cited LQCD results give +0.037(1)(2) [10] and +0.003(4) [13], and the DSE result [18] gives +0.023. The sign difference is not discussed at all. Since q1 is the s quark in the K*0 definition, a positive first moment is the expected direction from SU(3) breaking (the heavier s quark carries larger momentum fraction). The negative sign, if correct, would imply the opposite hierarchy and requires a careful explanation. As it stands, the K* DA and the D→K* TFFs derived from it are built on a result in tension with the cited literature.
  3. [Sec. II.C, Eq. (15)–(16), Table III] The statement that the DA behaviors are 'determined' by fitting ten moments with the two-parameter PLP model is too strong. A finite set of moments does not uniquely determine the x-dependence; many functional forms can reproduce the same moments, and the fit only validates the model under the prior that the PLP shape is exact. The manuscript provides no test against alternative models or a truncated Gegenbauer series. The goodness-of-fit columns in Table III are also unreadable: for ρ and φ the χ²_min/n_d entries are missing, and only Pχ² values (99.8–99.9%) are shown, which are implausibly high and suggest that the individual moment errors are inflated or the fit is insensitive. The model-bias uncertainty is not propagated into the TFF predictions.
  4. [Sec. III.B, Fig. 4] The Borel-window criteria are introduced as continuum contributions below 45% (50% for n≥8) and dimension-six contributions below 5% (8% for n≥8), and the continuum thresholds are fixed by requiring a window that normalizes ⟨ξ⁰⟩. These are post hoc stability choices. The paper does not show how the extracted moments and their errors change when the thresholds 45/50% and 5/8% are varied, nor does it give the adopted central M² values and ranges. Since the central moments and all downstream quantities depend on these choices, the analysis should include an explicit sensitivity check or at least a discussion of the criterion dependence.
minor comments (4)
  1. [Eq. (14)] The normalization denominator should presumably be Γ[α+1]Γ[β+1], not Γ[α+1]+Γ[β+1]. Please correct the typo and verify the normalization in the numerical code.
  2. [Sec. III.D] The bound-state masses are quoted as mD=1.869±0.05 MeV and mDs=1.968±0.07 MeV; the units should be GeV.
  3. [Table III] The columns χ²_min/n_d for the ρ and φ rows are empty. Please report these values or remove the column if not applicable.
  4. [Various] There are several typos: 'quark-hadron daulity' should be 'duality', 'breading' should be 'breaking' in Sec. IV, and Eq. (10) contains formatting issues in the step-function terms. A careful proofread is needed.

Circularity Check

0 steps flagged

No significant circularity: the PLP fit uses independently computed ξ-moments as inputs, and the TFF/branching-fraction predictions are not fed back into the fit.

full rationale

The derivation chain is: (i) compute ξ-moments ⟨ξ^n⟩ from the BFT QCD sum rule (Eq. (10)) with the improved normalization formula (Eq. (12)); (ii) fit the two-parameter PLP model (Eq. (14)) to those moments by least squares; (iii) use the resulting DA in the LCSRs (Eqs. (24)–(26)) to obtain TFFs and branching fractions. No step defines its output in terms of its target: the PLP parameters are not chosen to reproduce the TFFs or branching fractions, and the ξ-moments are computed from the OPE, not from the PLP model. The paper relies heavily on the authors' own prior work (Refs. [34,35,82,90]) for the sum-rule scheme, propagator expressions, condensate inputs, and model-selection rationale, but this is method provenance rather than circularity: the resulting moments agree with independent LQCD and other QCD-sum-rule predictions (Tables I–II), and the TFFs/branching fractions are compared with CLEO/BESIII data. The skeptical concerns about the ρ and φ fits returning α≠β and β<0, which violates the expected x↔1−x symmetry and yields a non-vanishing endpoint, and the apparent normalization typo in Eq. (14) (Γ[α+1]+Γ[β+1] instead of the product Γ[α+1]Γ[β+1]), are internal-consistency/correctness issues rather than circularity, because they do not make the predictions equivalent to the fit by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central calculation rests on standard QCD sum-rule axioms (duality, condensate expansion) plus two paper-specific modeling choices: the PLP functional form and the Borel-window selection rules. The fitted PLP parameters and threshold/Borel choices are the main free parameters; no new particles or fields are introduced.

free parameters (3)
  • PLP DA parameters α∥_{2;V}, β∥_{2;V} = ρ: α=β≈0.703, K*: α=0.853, β=0.761, φ: α=β≈0.895 at μ=1 GeV (Table III)
    Fitted by minimizing χ² to the computed ξ-moments; the resulting DAs are then used to compute TFFs.
  • Continuum thresholds sρ, sK*, sφ = 2.1, 2.6, 2.9 GeV²
    Chosen in Sec. III.A by requiring a reasonable Borel window to normalize ⟨ξ0⟩; affects all moments and TFFs.
  • LCSR Borel parameters M²_{V,A1,A2} for ρ, K*, φ = e.g., M²_{V;ρ}=1.89±0.06 GeV², M²_{A1;ρ}=1.20±0.05 GeV², etc. (Sec. III.D)
    Chosen by standard stability criteria; inputs to the D→V TFF predictions.
axioms (5)
  • domain assumption Quark-hadron duality: the hadronic spectral function can be approximated by a pole plus a perturbative continuum above threshold sV.
    Used in Eq. (6) and in all subsequent QCD SR expressions; this is a standard but unproved assumption of the sum-rule method.
  • domain assumption Nonzero vacuum condensates and the background-field theory framework are valid up to dimension-six.
    The OPE in Eqs. (8)-(10) relies on the condensate expansion; condensate values are taken from Ref. [34] by the same group.
  • ad hoc to paper The PLP model Eq. (14) is an adequate description of the DA over the entire x range.
    Used to fit the moments and then to compute TFFs; this two-parameter shape is not derived from QCD.
  • domain assumption The Wandzura-Wilczek approximation for twist-3 DAs is valid in the D→ρ and D→K* LCSRs.
    Used in Eq. (30) for φ⊥3;V and ψ⊥3;V; genuine twist-3 contributions are neglected in those channels.
  • ad hoc to paper The Borel-window stability criteria (continuum <45/50%, dim-6 <5/8%) define the reliable extraction region.
    Sec. III.B; these thresholds are not derived and affect all quoted moments.

pith-pipeline@v1.3.0-alltime-deepseek · 30531 in / 14143 out tokens · 123916 ms · 2026-08-01T17:47:32.254749+00:00 · methodology

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read the original abstract

In this work, we focus on the light vector meson leading-twist longitudinal distribution amplitudes (DAs) $\phi^\parallel_{2;V}(x,\mu)$ with $V = \rho, K^\ast, \phi$. In order to obtain their accurate behaviors, a new scheme of QCD sum rule research with respect to DA suggested in 2021 by us is adopted. With an improved sum rule formula, the $\xi$-moments $\langle\xi^n\rangle_{2;V}^\parallel$ up to tenth order are calculated. In which, $\langle\xi^2\rangle^\parallel_{2;\rho}=0.225^{+0.013}_{-0.012}$, $\langle\xi^1\rangle^\parallel_{2;K^\ast}=-0.0228^{+0.0042}_{-0.0040}$, $\langle\xi^2\rangle^\parallel_{2;K^\ast}=0.217^{+0.007}_{-0.007}$, $\langle\xi^2\rangle^\parallel_{2;\phi}=0.209^{+0.020}_{-0.020}$, and the corresponding Gegenbauer moments $a^{2;\parallel}_{2;\rho}=0.074^{+0.039}_{-0.036}$, $a^{1;\parallel}_{2;K^\ast}=-0.038^{+0.007}_{-0.007}$, $a^{2;\parallel}_{2;K^\ast}=0.050^{+0.020}_{-0.019}$, $a^{2;\parallel}_{2;\phi}=0.027^{+0.058}_{-0.058}$ at the scale $\mu = 1~{\rm GeV}$, respectively. By fitting those $\langle\xi^n\rangle^\parallel_{2;V}(n = 1,2,\cdots,10)$ with the least squares method, the behaviors of leading-twist longitudinal DAs for $\rho, K^\ast, \phi$ are determined. Further, we recalculate the transition form factors and branching ratio of the $D\to(\rho,K^\ast)$, $D_s\to\phi$ semi-leptonic decay processes.

Figures

Figures reproduced from arXiv: 2607.17496 by Ru-Meng Pan, Tao Zhong, Wan-Bing Luo, Ya-Xiong Wang.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic Feynman diagrams for vector meson [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Sub-diagrams of Fig [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Sub-diagrams of Fig [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Curves of the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Curves of the TFFs [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

119 extracted references · 69 linked inside Pith

  1. [1]

    (3) In Eq

    respect to z, one can obtain the following matrix element definition of the ξ-moment ⟨ξn⟩∥ 2;V of vector meson leading-twist longitudinal DA, ⟨0|¯q1(0)/z(iz· ↔ D)nq2(0)|V (q,λ )⟩ = (z·q)n+1f ∥ V⟨ξn⟩∥ 2;V, (2) with ⟨ξn⟩∥ 2;V = ∫ 1 0 du(2u− 1)nφ∥ 2;V (x,µ ). (3) In Eq. ( 2), ↔ Dµ = − →D µ−← −D µ with the fundamental representation of the gauge covariant deri...

  2. [2]

    and the quark-hadron daulity, the hadronic representation of correlator ( 4) can be obtained as ImI had 2;V (s) = πδ(s−m2 V )⟨ξn⟩∥ 2;V⟨ξ0⟩∥ 2;V (f ∥ V )2 + ImI pert 2;V (s)θ(s−sV ), (6) wheresV indicates the continuum threshold parameter. FIG. 1: Schematic Feynman diagrams for vector meson ρ, K∗, φ leading-twist longitudinal DAs ξ-moments. The left big do...

  3. [3]

    A. V. Pimikov, S. V. Mikhailov and N. G. Stefanis, Rho meson distribution amplitudes from QCD sum rules with nonlocal condensates, Few Body Syst. 55 (2014), 401-

    2014

  4. [4]

    can be treated by inserting a com- plete set of intermediate hadronic states in physical re- gion. With Eq. (

  5. [5]

    The corresponding calculation is performed in the framework of BFT

    in the deep Euclidean region. The corresponding calculation is performed in the framework of BFT. With the basic assumption and Feynman rules of BFT, the correlator (

  6. [6]

    ( 5), respectively

    can be rewrit- ten as Π2;V (z,q ) = i ∫ d4xeiq·x × { − Tr⟨0|Sq1 F (0,x )/z(iz· ↔ D)nSq2 F (x, 0)/z|0⟩ + Tr⟨0|¯q1(x)q1(0)/z(iz· ↔ D)nSq2 F (x, 0)/z|0⟩ + Tr⟨0|Sq1 F (0,x )/z(iz· ↔ D)n ¯q2(0)q2(x)/z|0⟩ } +··· , (7) Where Tr indicates trace of the color matrix and γ ma- trix, ellipsis stands for ignored high-order corrections, Sq1 F (0,x ) is the q1-quark pro...

  7. [7]

    2: Sub-diagrams of Fig

    with the dispersion relation after Borel transformation, 1 π 1 M 2 ∫ dse−s/M2 ImI had 2;V (s) = ˆBM 2I QCD 2;V (q2), (9) with the Borel parameter M and the Borel transforma- tion operator ˆBM 2 , one can finally obtain, ⟨ξn⟩∥ 2;V⟨ξ0⟩∥ 2;V (f ∥ V )2 M 2em2 V /M2 = 1π 1 M 2 ∫ sV (m1+m2)2 dse−s/M2 ImI pert 2;V (s) + m1⟨¯q1q1⟩ + (−1)nm2⟨¯q2q2⟩ (M 2)2 − m1m2 (M...

  8. [8]

    can re- fer to Ref. [ 35]. Then one can get the sub-diagrams of Fig. 1(a) and 1(b), which are shown in Fig. 2 and Fig. 3 respectively. By matching the hadronic expression in physical region and OPE in the deep Euclidean region of correlator (

  9. [9]

    by n=0 . (12) Usually, one can obtain the Gegenbauer moments a∥;n 2;V through the following relationship between a∥;n 2;V and ⟨ξn⟩∥ 2;V , a∥;1 2;V = 5 3⟨ξ1⟩∥ 2;V, a∥;2 2;V = 35 12⟨ξ2⟩∥ 2;V− 7 12, ··· . (13) Traditionally, based on the obtained a∥;n 2;V , the behaviors of light vector meson leading-twist longitudinal DAs can be constrained or obtained with...

  10. [10]

    ˆI m2 ⟨G2⟩(M 2), ˆI m2 ⟨G3⟩(M 2), and ˆI m2 ⟨q4⟩(M 2) in Eq

    are the mass corrections for the double- quark condensate ⟨¯q1(2)q1(2)⟩, quark-gluon mixed con- densate ⟨gs ¯q1(2)σTGq 1(2)⟩ and four-quark condensate ⟨gs ¯q1(2)q1(2)⟩2, respectively. ˆI m2 ⟨G2⟩(M 2), ˆI m2 ⟨G3⟩(M 2), and ˆI m2 ⟨q4⟩(M 2) in Eq. ( 10) respectively indicates the mass cor- rections for the double-gluon condensate ⟨αsG2⟩, triple- gluon conden...

  11. [11]

    In this work, we will calculate the first ten ⟨ξn⟩∥ 2;V as the constraint conditions by referring to the analysis of the constraints of ξ-moments on pion leading-twist DA in Ref

    as possible through the least squares method to determine the precise behavior of φ∥ 2;V . In this work, we will calculate the first ten ⟨ξn⟩∥ 2;V as the constraint conditions by referring to the analysis of the constraints of ξ-moments on pion leading-twist DA in Ref. [ 82], and select the following normalized power-law parametrization form (PLP model), φ...

  12. [12]

    with the least squares method, the PLP model parameters α∥ 2;V,β ∥ 2;V and further the be- havior of DA φ∥ 2;V (x,µ ) can be determined. Specifically, the optimal values of α∥ 2;V,β ∥ 2;V can be obtained by min- imizing the likelihood function, χ2(θ) = 10∑ i=1 (yi−µ(xi,θ ))2 σ2 i , (15) where the fitting parameter θ = (α∥ 2;V,β ∥ 2;V ), the mean function µ(...

  13. [13]

    There- fore, traditional methods are difficult to obtain the pre- cise behavior of φ∥ 2;V

    are extremely unreliable due to the error of⟨ξn⟩∥ 2;V and its large coefficient. There- fore, traditional methods are difficult to obtain the pre- cise behavior of φ∥ 2;V . As suggested in Ref. [ 34], we will use appropriate phenomenological model to fit as many ⟨ξn⟩∥ 2;V calculated with Eq. (

  14. [14]

    α∥ 2;V = α∥ 2;V (µ) and β∥ 2;V = β∥ 2;V (µ), and for simplicity, we do not explicitly show this

    are scale dependent, i.e. α∥ 2;V = α∥ 2;V (µ) and β∥ 2;V = β∥ 2;V (µ), and for simplicity, we do not explicitly show this. By fitting the values of the first ten ⟨ξn⟩∥ 2;V calculated from Eq. (

  15. [15]

    and (14), the central values and corre- sponding errors of ⟨ξn⟩∥ 2;V calculated with sum rule ( 12) are regarded as the independent measurements yi and variance σi. The goodness of fit can be judged by the probability, Pχ2 min = ∫ ∞ χ2 min f (y;nd)dy, (16) wheref (y;nd) = 1/[Γ(nd/2)2nd/2]ynd/2−1e−y/2 with the number of degrees of freedomnd is the probabili...

  16. [16]

    −” and “+

    can be expressed as, ΠQCD µ (p,q ) =mc ∫ d4xd 4k (2π)4 ei(q−k)·x { 1 m2 c−k2 { 2kµ⟨V (p,λ )|¯s(x)q1(0)|0⟩− 2ikν⟨V (p,λ )|¯s(x)σµνq1(0)|0⟩ −ǫµναβkν⟨V (p,λ )|¯s(x)σαβq1(0)|0⟩ } − ∫ dv { kν (m2 c−k2)2 [ −i⟨V (p,λ )|¯s(x)gsGµν (vx)q1(0)|0⟩ − 2⟨V (p,λ )|¯s(x)σµαgsGαν (vx)q1(0)|0⟩ + 2i⟨V (p,λ )|¯s(x)igs ˜Gµν(vx)γ5q1(0)|0⟩ + 2vxα m2 c−k2−⟨V (p,λ )|¯s(x)gsGµα(vx)...

  17. [17]

    by taking n = 0, the continuous threshold parameters can be obtained as sρ = 2.1 GeV 2, sK ∗ = 2.6 GeV 2, sφ = 2.9 GeV 2. B. ξ-moments of vector meson leading-twist longitudinal DAs Then, one can obtain the curves of ξ-moments ⟨ξn⟩∥ 2;V (V = ρ,K ∗,φ ) versus Borel parameter M 2 (see Fig

  18. [18]

    (21) In which, fD(s) is the decay constant of the charmed D(s) meson, s0 is the threshold parameter, ˆ m1 = mc corre- sponds to D→ ρ,K ∗, ˆm2 = mc +ms corresponds to Ds→φ

    in the time-like q2 region, and further separating the pole term of the lowest pseudoscalar D(s) meson, one can obtain the hadronic representation of the correlator, Πhad µ (p,q ) = e∗(λ) µ Πhad 1 + (e∗(λ) µ ·q)(2p +q)µΠhad 2 6 + (e∗(λ) µ ·q)qµΠhad 3 +iǫναβ µ e∗(λ) ν qαpβΠhad 4 , (19) where Πhad i = m2 D(s) fD(s) (mD(s) +mV ) ˆmj [ m2 D(s) − (p +q)2] ˜Ci(...

  19. [19]

    with Eq. ( 12). In order to obtain the values of⟨ξn⟩∥ 2;V , the corresponding Borel windows should be determined. In principle, the contributions of continu- ous state and six-dimensional condensation are required to be as small as possible, while the ξ-moments need to remain stable in the Borel windows. Specifically, the continuum contributions are requir...

  20. [20]

    As a comparison, the curves of φ2;V (x,µ ) predicted by various methods such as QCD SRs [ 7, 8], LQCD [13], DSE [ 16], BSWF [ 24], Algebraic model [ 88], Asymptotic form [ 89] and the truncated form of Gegen- bauer polynomial series (TF model) [ 90] are also shown in Fig. 5. From Fig. 5, one can find that, • For the ρ meson, in the peak region, the results...

    2013

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    Our AD→ρ 2 (q2) is consis- tent with the predictions of LFQM [ 75], LEChQM [ 79] in the whole error range

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