Pith. sign in

REVIEW 4 major objections 6 minor 1 cited by

Form factors of light pseudoscalar mesons from the perturbative QCD approach

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read Adding intrinsic transverse momentum distributions to perturbative QCD extends its predictions for the pion, kaon, eta, and eta' form factors down to a few GeV^2, where they now agree with experiment.

desk verdict Solid pQCD phenomenology with a genuine pion TFF prediction, but the iTMD-Sudakov complementarity is asserted rather than shown, and some central parameters are fitted to the data they are validated against. read the letter →

arxiv 2501.08783 v4 pith:LSJIIXBJ submitted 2025-01-15 hep-ph

Jian Chai , Shan Cheng This is my paper · ORCID
classification hep-ph
keywords lightpseudoscalarmesonselectromagneticformfactorstransitionintrinsictransversemomentumdistributionslight-conedistributionamplitudesperturbativeQCDpionchiralmasseta-eta'mixing
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 resolve a long-standing gap: perturbative QCD predictions for light-meson electromagnetic and transition form factors miss the data at low and intermediate momentum transfers. The proposed fix is to add intrinsic transverse momentum distributions (iTMDs) to the standard light-cone distribution amplitudes, so the soft transverse motion of valence quarks inside the meson is no longer neglected. With a single Gaussian transverse-size parameter per meson, the improved pQCD framework reproduces the pion, kaon, $\eta$, and $\eta'$ form factor data down to a few GeV$^2$. The fits yield $\beta_\pi^2 = 0.51 \pm 0.04~\mathrm{GeV}^{-2}$, $\beta_K^2 = 0.30 \pm 0.05~\mathrm{GeV}^{-2}$, and a pion chiral mass $m_0^\pi(1~\mathrm{GeV}) = 1.84 \pm 0.07~\mathrm{GeV}$, about 30% larger than earlier pQCD estimates and consistent with chiral perturbation theory. If the claim holds, pQCD becomes a quantitative tool in a kinematic region previously controlled by nonperturbative methods.

What carries the argument

The load-bearing object is the intrinsic transverse momentum distribution (iTMD), a Gaussian profile $\Sigma(u,k_T) = 16\pi^2 \beta^2 g(u)\exp(-\beta^2 k_T^2 g(u))$ with $g(u) = 1/(u\bar{u})$, whose Fourier transform $\hat{\Sigma}(u,b_T) = 4\pi\exp(-b_T^2 u(1-u)/(4\beta^2))$ multiplies the light-cone distribution amplitude in impact-parameter space. The single parameter $\beta^2$ controls the transverse size of the valence quark state, and the paper determines it from the double-photon constraint for the pion and from a fit to BaBar kaon data. Inserting this profile into the $k_T$-factorized convolution for electromagnetic and transition form factors is what extends the theory's reach to a few GeV$^2$; the same Gaussian profile governs the pion, kaon, $\eta$, and $\eta'$ predictions.

What would settle it

A high-precision spacelike pion form factor measurement near $Q^2 = 8$ GeV$^2$, where the improved pQCD band and lattice results differ, would settle the claim: if the data fall outside the band by more than the quoted uncertainty, the Gaussian iTMD ansatz fails.

Watch

Extended reading notes

Core claim

The central claim is that intrinsic transverse momentum distributions, rather than more exotic distribution-amplitude shapes or higher-twist effects, are the missing ingredient that lets perturbative QCD describe light-pseudoscalar form factors at low momentum transfer. The paper shows that inserting a Gaussian iTMD with transverse-size parameter $\beta^2$ into the $k_T$-factorized hard-scattering formula suppresses the spacelike pion form factor in the small-to-intermediate $Q^2$ region, which raises the extracted chiral mass to $1.84 \pm 0.07$ GeV and brings timelike and spacelike predictions into agreement with BaBar, Belle, BESIII, and lattice data. It also presents the first systematic evaluation of twist-four contributions to meson-photon transition form factors, showing that leading-twist LCDAs dominate and that iTMDs shift the pion-photon TFF toward Belle's plateau rather than BaBar's rise. For $\eta$ and $\eta'$, the same mechanism favors a small mixing angle and makes the asymptotic difference of their TFFs a sensitive probe of that angle.

Load-bearing premise

The entire low-$Q^2$ improvement rests on the postulated Gaussian shape of the intrinsic transverse momentum distribution, Eq. (2.18), and on the assumption that this soft contribution supplements the Sudakov resummation without double-counting it.

Editorial extensions

If this is right

  • The pion electromagnetic form factor from pQCD now matches spacelike data above a few GeV$^2$ and, through the modular dispersion relation, the timelike modulus as well.
  • The extracted $m_0^\pi(1~\mathrm{GeV}) = 1.84 \pm 0.07$ GeV, about 30% above previous pQCD values and consistent with chiral perturbation theory, implies that soft transverse dynamics substantially suppress the form factor at low and intermediate $Q^2$.
  • The pion-photon transition form factor agrees with Belle's plateau at intermediate and large $Q^2$, disfavoring BaBar's rising data; $\eta$ and $\eta'$ data favor a small mixing angle.
  • For all four mesons the iTMD-improved pQCD predictions are reliable down to a few GeV$^2$, a region previously considered outside pQCD's range.
  • The predicted difference $\delta F = F_{\eta\gamma\gamma^*} - F_{\eta'\gamma\gamma^*}$ at $Q^2 \sim 10^2$ GeV$^2$ is $0.013 \pm 0.006$ GeV and is sensitive to the mixing angle, giving future experiments a clean target.

Reading between the lines

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

  • The Gaussian iTMD is an effective model, so the extracted $\beta^2$ and $m_0^\pi$ should be read as parameters of this framework rather than universal hadronic constants; a power-law transverse tail would likely shift both values.
  • Because the same $\beta^2$ controls several channels, a high-precision measurement of the double-virtual $\pi^0$ transition form factor would provide an independent, stringent test of the ansatz without changing any other input.
  • The paper's own caution about the kaon's $S$-wave resonance structure implies that the kaon extraction, which bypasses the modular dispersion relation and fits timelike BaBar data directly, is the most vulnerable part of the analysis; a precise spacelike lattice kaon form factor in the 5--10 GeV$^2$ range would settle it.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This paper extends the perturbative QCD (pQCD) approach to electromagnetic and meson-photon transition form factors of light pseudoscalar mesons by supplementing light-cone distribution amplitudes (LCDAs) with intrinsic transverse momentum distributions (iTMDs). The iTMDs are modeled by a Gaussian transverse-momentum profile with a single transverse-size parameter beta^2 per Fock component. The authors compute pion and kaon electromagnetic form factors at NLO for leading and subleading twist and at LO for twist-four, use a modular dispersion relation to connect timelike data to spacelike predictions for the pion, fit the kaon transverse-size parameter and the pion chiral mass to data, and then apply the same framework to pion, eta, eta', and eta_c transition form factors. The main quantitative results are beta_pi^2 = 0.51 +/- 0.04 GeV^-2, beta_K^2 = 0.30 +/- 0.05 GeV^-2, and m_0^pi(1 GeV) = 1.84 +/- 0.07 GeV, together with the claim that the iTMD-improved pQCD framework remains reliable down to momentum transfers of a few GeV^2.

Significance. If the central claim is correct, this would be a practically useful extension of k_T-factorization pQCD into a kinematic region where collinear pQCD is usually considered unreliable, and it would provide a coherent extraction of transverse-size parameters and the pion chiral mass. The pion-photon TFF is a genuine prediction in the sense that beta_pi^2 is fixed from Gegenbauer moments taken from Ref. [24] and from Eq. (2.22), rather than fitted to Belle/BaBar/CLEO TFF data. The paper also presents the first systematic inclusion of two-particle and three-particle twist-four contributions to the meson-photon TFFs, which is a useful technical step. The extracted m_0^pi is consistent with the ChPT expectation, and the asymptotic relation Eq. (3.23) provides a falsifiable prediction for future high-Q^2 measurements. The main weakness is that the entire low-Q^2 improvement rests on the Gaussian iTMD ansatz of Eq. (2.18), whose complementarity with the Sudakov resummation already present in Eqs. (2.7) and (2.8) is asserted but not quantitatively demonstrated.

major comments (4)
  1. [Sec. 3.2, Eq. (3.11)] The central claim that iTMDs extend pQCD down to a few GeV^2 rests entirely on the Gaussian ansatz Sigma(u,k_T) = 16 pi^2 beta^2 g(u) exp(-beta^2 k_T^2 g(u)) and on the assumption that this soft profile supplements rather than double-counts the Sudakov resummation. In impact-parameter space the iTMD is exp[-b^2 u(1-u)/(4 beta^2)], which is a strong large-b suppression, while the Sudakov factor exp(-S) in Eqs. (2.7), (2.8), (3.6) and (3.11) also suppresses large b; the two factors enter multiplicatively. The paper provides no quantitative criterion for separating the soft transverse physics from the Sudakov resummed hard-gluon physics, and the overlap region b ~ 1-3 GeV^-1 is precisely the region where the claimed improvement occurs. The authors should demonstrate non-redundancy, for example by varying the Sudakov resummation scale or the b-space cutoff and showing that the extracted beta^2 and m_0^pi values are stable, or by comparing with a calculation in which the iTMD is introduced only beyond the Sudakov-dominated region.
  2. [Sec. 2.3 and Sec. 3.3] The iTMD is applied to the leading-twist pion TFF term through the factor Sigma_hat(u,b) multiplying phi(u), but the two-particle twist-four terms g_1(u) and g_2(u) in Eq. (3.11) do not carry the corresponding Sigma_hat factor, and the three-particle twist-four term phi_parallel does not show the three-particle iTMD Sigma_hat' of Eq. (2.28). The text states that iTMD effects are implemented for the genuine twist-four components, but the displayed formula does not reflect this. Since the pion TFF is dominated by the leading twist, the numerical effect on the central prediction may be small, but the claim of a consistent treatment of transverse-momentum effects across all relevant twists is not supported by Eq. (3.11) as written.
  3. [Sec. 2.2, Eq. (2.30)] The extracted transverse-size parameter beta_K^2 = 0.30 +/- 0.05 GeV^-2 is obtained by fitting the pQCD calculation to BaBar kaon EMFF data with q^2 >= 7 GeV^2, and beta_eta_c^2 is obtained by fitting to BaBar eta_c TFF data in 2 <= Q^2 <= 10 GeV^2. The same datasets are then shown as evidence of the improved theory-data agreement in Figs. 8 and 12. This makes the agreement for these two channels a fit quality statement rather than an independent validation of the iTMD ansatz. The authors should separate the fitting range from the comparison range, use lattice or other independent data for validation, or at least quantify the number of fitted parameters versus the number of data points so the reader can judge the predictive content.
  4. [Sec. 4] The pion chiral mass m_0^pi is extracted by iteratively matching the iTMD-improved pQCD spacelike form factor to the modular dispersion relation, whose integrand includes the pQCD timelike form factor as the high-energy tail. This introduces a mild circularity: the same improved pQCD calculation supplies both sides of the matching, and the size of the resulting shift from 1.30 to 1.84 GeV (Table 2) depends on how much of the small-Q^2 suppression is assigned to the Gaussian iTMD rather than to the Sudakov factor. The sensitivity of this extraction to the iTMD/Sudakov separation issue raised above should be quantified explicitly.
minor comments (6)
  1. [Abstract] There is a typo: 'incorpoarate' should be 'incorporate'.
  2. [Sec. 2.2, bullet (b)] The sentence 'his is expected because the iTMD function primarily modifies the form factor at small momentum transfers' should read 'This is expected'.
  3. [Fig. 3 caption] The caption contains 'sudokov' and 'Low' where 'Sudakov' and 'Lower' are intended.
  4. [Sec. 3.2] The sentence 'iTMDs reduces the pion-photon TFF at small to intermediate momentum transfers (Q^2 >= 20 GeV^2)' is internally inconsistent: Q^2 >= 20 GeV^2 is the high-Q^2 regime, not small-to-intermediate. The text and Fig. 11 suggest the reduction occurs at low Q^2, so the inequality should be corrected.
  5. [Eq. (3.22)] The symbol kappa'_s in the text after Eq. (3.22) should presumably be kappa'_2 to match the notation used elsewhere.
  6. [Sec. 3.3] The reference in the text '[Landsberg:1985gaz]' is not formatted consistently with the other references; if it refers to Ref. [88], the citation should be updated.

Circularity Check

2 steps flagged · score 4.0 of 10

The pion TFF and low-Q^2 extension are genuine predictions from external hadronic inputs, but the kaon EMFF and eta_c TFF 'improvements' reuse the same BaBar data that fixed beta_K^2 and beta_eta_c^2, so those parts of the 'all considered form factors' claim are partially self-referential; the Gaussian iTMD ansatz is a model assumption rather than a circular reduction.

  1. fitted input called prediction [Section 2.3, 'Electromagnetic form factor of kaon', text around Eq. (2.34) and Fig. 8]
    "We extract the transverse-size parameter β 2 K by fitting pQCD calculations to precise BaBar measurements in the high-momentum transfer region (q 2 ⩾7.0GeV 2). ... With the fitted iTMD functions, we show in figure 8 the pQCD predictions for kaon EMFFs across the momentum transfer range |q 2|⩽10GeV 2. ... The analysis reveals that iTMD contributions are crucial for explaining the timelike form factor data |F K(q2)| at intermediate and large q 2."

    β_K^2 is obtained by fitting the same BaBar timelike kaon data that are then displayed against the 'pQCD predictions' in Fig. 8. The statement that iTMDs are 'crucial for explaining the timelike form factor data' is therefore not an independent test of the iTMD model in the fitted region: the curve is forced to pass through those data by construction. The low-Q^2 part of the claimed improvement is partially predictive, but the 'all considered form factors' claim in the abstract leans on this fitted comparison as well.

  2. fitted input called prediction [Section 3.3, 'Transition form factor of η, η′ mesons', text defining the η_c TFF and Fig. 12]
    "The transverse size parameter is then determined by fitting to the BaBar data of F ηcγγ ∗(Q2)/Fηcγγ ∗(0) in the intermediate momentum-transfer region 2⩽Q 2 ⩽10GeV 2, where the data exhibit better statistical precision. We obtain a larger value β 2 ηc = 0.60GeV −2 ... In figure 12, we compare the updated pQCD prediction of η c-photon TFF Q 2Fηcγγ ∗(q2) with other theoretical approaches ... The BaBar data [134] are also shown for comparison."

    β_ηc^2 is fit to BaBar η_c TFF data in exactly the range 2≤Q^2≤10 GeV^2 where the iTMD effect is most visible, and Fig. 12 then shows the 'updated pQCD prediction' against those same BaBar data. The apparent low-to-intermediate Q^2 agreement is therefore guaranteed by the fit rather than independently predicted. Because the abstract claims the iTMD improvement applies to 'all considered form factors', this fitted comparison inflates the scope of the central claim, even though the fit itself is disclosed.

full rationale

The core pion-photon TFF result is a genuine prediction: β_π^2 is fixed by Eq. (2.22) using the Gegenbauer moments a_π^2 and a_π^4 taken from the external LCSR-based analysis of Ref. [24], and the Belle/CLEO/BaBar TFF data are not used to set these constants. The low-Q^2 enhancement seen in the pion TFF is therefore not an artifact of fitting to the compared data. Likewise, the pion EMFF chiral-mass extraction is a self-consistent fit that still relies on independent timelike data and external a_2, a_4 inputs. No formal equivalence is exhibited between the Gaussian iTMD ansatz, Eq. (2.18), and the Sudakov factor: the possible double-counting of transverse-momentum suppression is a physical correctness concern, not a circularity, and the paper does not define the iTMD in terms of the Sudakov factor. The two genuine circularity-relevant steps are the kaon and η_c parameter extractions: in both cases a transverse-size parameter is fit to a specific BaBar dataset and the same dataset is later shown as evidence of iTMD-driven 'theory-data consistency'. These steps are openly labeled as fits, but the abstract's sweeping claim that iTMDs extend pQCD down to a few GeV^2 for 'all considered form factors' partially rests on fitted rather than predicted comparisons. The self-citation to Ref. [24] is load-bearing for the pion inputs but that reference is an independent published analysis with external co-authors, so it does not by itself make the central derivation circular. Overall the paper contains a genuinely predictive pion TFF result alongside two partially self-referential comparison steps, giving a moderate circularity score.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim relies on the Gaussian iTMD ansatz and on a factorization that separates Sudakov and intrinsic transverse dynamics; these are physically motivated but not derived. The extracted parameters beta_pi^2, beta_K^2, m_0^pi, beta_eta_c^2 are fit to data, and beta' is unconstrained, so the reader should not treat the improvement as a parameter-free prediction.

free parameters (5)
  • beta_pi^2 = 0.51 +/- 0.04 GeV^-2
    Transverse-size parameter of pion valence quark state; fixed by Eq. (2.22) using f_pi = 0.13 GeV and Gegenbauer coefficients a_2^pi = 0.28 +/- 0.05, a_4^pi = 0.19 +/- 0.06 from [24]. Used in all pion EMFF and TFF predictions.
  • beta_K^2 = 0.30 +/- 0.05 GeV^-2
    Fitted to BaBar kaon timelike EMFF data at q^2 >= 7 GeV^2 (Sec 2.3).
  • m_0^pi = 1.84 +/- 0.07 GeV
    Pion chiral mass at 1 GeV; extracted by iteratively matching the iTMD-improved pQCD spacelike EMFF to the modular dispersion relation built from timelike data plus pQCD tail (Sec 2.2, Table 2, Set IIB).
  • beta_eta_c^2 = 0.60 GeV^-2
    Transverse-size parameter of eta_c; fitted to BaBar F_{eta_c gamma gamma*}/F(0) data in 2 <= Q^2 <= 10 GeV^2 (Sec 3.3).
  • beta_prime^2 (three-particle iTMD) = not determined; upper bound beta_prime^2 < 0.511 GeV^-2
    Three-particle iTMD transverse-size parameter; the paper states it cannot be reliably determined and only gives an upper bound (Sec 2.3).
assumptions (6)
  • domain assumption The soft transverse momentum profile of the meson is a Gaussian (Eq. 2.18)
    Adopted from refs [44,45]; no derivation from QCD; the improvement claim rests on this functional form.
  • domain assumption Sudakov resummation and the iTMD factor are independent and do not double-count soft dynamics (Sec 2.1)
    The paper argues the iTMD is soft bremsstrahlung outside the hard potential field, but this separation is not proven.
  • domain assumption F_pi(q^2) has no zeros in the complex plane, enabling the modulus-squared dispersion relation (Eq. 2.30, App. C)
    Assumption stated in App. C; the authors themselves avoid the kaon case where it may fail (Sec 2.3).
  • domain assumption Higher-twist LCDAs truncated at twist-four, with NLO for twist-2/3 and LO for twist-4 (Sec 2, 3.1)
    Standard truncation; uncertainties from neglected twist-five/six not quantified.
  • domain assumption The |gg> Fock component contributes only at O(alpha_s) and through non-asymptotic Gegenbauer terms, so it is neglected for eta, eta' TFFs (Sec 3.3)
    The suppression is argued, not computed.
  • standard math Only chiral-even LCDAs contribute to TFFs due to chiral symmetry (Sec 3.1)
    Follows from electromagnetic current conservation; standard result.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Form factors of light pseudoscalar mesons from the perturbative QCD approach." pith.science (2026). https://pith.science/paper/LSJIIXBJ

@misc{pith2026250108783,
  author       = {Pith},
  title        = {Pith review of: Form factors of light pseudoscalar mesons from the perturbative QCD approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSJIIXBJ}},
  note         = {Machine review of arXiv:2501.08783}
}
abstract

We study the electromagnetic and meson-photon transition form factors (TFF) of light pseudoscalar mesons from the perturbative QCD (pQCD) approach. To comprehensively account for both the longitudinal and transverse nonperturbative dynamics of hadronic constituents, we incorpoarate intrinsic transverse momentum distributions (iTMDs) alongside the conventional light-cone distribution amplitudes (LCDAs). The main motivations of this work are the disjointedness of electromagnetic form factors between the theoretical predictions and the experimental measurements, and the BaBar-Belle tension of pion-photon transition form factor in the large momentum transfers. Our calculation is carried out at the next-to-leading-order for the contributions from leading and subleading twist LCDAs, and leading order for the twist four contributions. Notably, this work presents the first systematic evaluation of higher-twist contributions to meson-photon TFFs. The key findings are: (a) iTMDs play a crucial role in describing form factor data, particularly in the small-to-intermediate momentum transfer region where they induce significant modifications to pQCD predictions. (b) The extracted transverse size parameters for valence quark states are found to be $\beta_\pi^2 = 0.51 \pm 0.04$ GeV$^{-2}$ and $\beta_K^2 = 0.30 \pm 0.05$ GeV$^{-2}$, the chiral mass of pion meson $m_0^\pi$ at $1$ GeV is determined to be $1.84 \pm 0.07$ GeV. (c) The meson-photon TFFs are predominantly governed by leading-twist LCDAs. The iTMDs-enhanced pQCD results show better agreement with Belle's pion TFF data across intermediate and large momentum transfers and favor a small $\eta-\eta^\prime$ mixing angle. (d) Remarkably, the inclusion of iTMDs extends the applicability of pQCD calculations down to a few GeV$^2$ for all considered form factors, significantly improving the theory-data consistency.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Probing $B^+ \to K^+$ semi-leptonic FCNC decay with new physics effects under PQCD approach

    hep-ph 2024-12 conditional novelty 5.0 of 10

    An improved PQCD plus lattice calculation of B to K semileptonic FCNC form factors confirms a 2.7 sigma excess in B+ to K+ nu nubar and a deficit in B+ to K+ l+ l-, and rules out single-leptoquark explanations using B...

Reference graph

Works this paper leans on

138 extracted references · 72 canonical work pages · cited by 1 Pith paper

  1. [24]

    Cheng, A

    S. Cheng, A. Khodjamirian and A. V . Rusov, Phys. Rev. D102, no.7, 074022 (2020)

    2020

  2. [1]

    A. H. Mueller, Phys. Rept.73, 237 (1981)

    1981

  3. [2]

    Brodsky, G.P

    S.J. Brodsky, G.P. Lepage, in: A.H. Mueller (Ed.), Perturbative Quantum Chromodynamics, World Scientific, Singapore, 1989

    1989

  4. [3]

    A. V . Efremov and A. V . Radyushkin, Phys. Lett. B94, 245-250 (1980)

    1980

  5. [4]

    G. P. Lepage and S. J. Brodsky, Phys. Rev. D22, 2157 (1980)

    1980

  6. [5]

    Diehl, Phys

    M. Diehl, Phys. Rept.388, 41-277 (2003)

    2003

  7. [6]

    A. V . Belitsky and A. V . Radyushkin, Phys. Rept.418, 1-387 (2005)

    2005

  8. [7]

    Dudek, R

    J. Dudek, R. Ent, R. Essig, K. S. Kumar, C. Meyer, R. D. McKeown, Z. E. Meziani, G. A. Miller, M. Pennington and D. Richards,et al.Eur. Phys. J. A48(2012), 187

    2012

Show all 138 references
  1. [8]

    Abdul Khalek, U

    R. Abdul Khalek, U. D’Alesio, M. Arratia, A. Bacchetta, M. Battaglieri, M. Begel, M. Boglione, R. Boughezal, R. Boussarie and G. Bozzi,et al.[arXiv:2203.13199 [hep-ph]]

    arXiv

  2. [9]

    V . M. Braun and I. E. Filyanov, Z. Phys. C48, 239-248 (1990)

    1990

  3. [10]

    P. Ball, V . M. Braun and A. Lenz, JHEP05, 004 (2006)

    2006

  4. [11]

    P. Ball, V . M. Braun, Y . Koike and K. Tanaka, Nucl. Phys. B529, 323-382 (1998)

    1998

  5. [12]

    Ball and V

    P. Ball and V . M. Braun, Nucl. Phys. B543, 201-238 (1999)

    1999

  6. [13]

    Ball and G

    P. Ball and G. W. Jones, JHEP03, 069 (2007)

    2007

  7. [14]

    P. Ball, V . M. Braun and A. Lenz, JHEP08, 090 (2007)

    2007

  8. [15]

    Braun, R

    V . Braun, R. J. Fries, N. Mahnke and E. Stein, Nucl. Phys. B589, 381-409 (2000) [erratum: Nucl. Phys. B607, 433-433 (2001)]

    2000

  9. [16]

    Y . L. Zhang, S. Cheng, J. Hua and Z. J. Xiao, Phys. Rev. D92, no.9, 094031 (2015)

    2015

  10. [17]

    Y . L. Shen, Z. T. Zou and Y . Li, Phys. Rev. D100, no.1, 016022 (2019) – 40 –

    2019

  11. [18]

    H. Zhou, J. Yan, Q. Yu and X. G. Wu, Phys. Rev. D108, no.7, 074020 (2023)

    2023

  12. [19]

    S. V . Mikhailov, A. V . Pimikov and N. G. Stefanis, Phys. Rev. D103, no.9, 096003 (2021)

    2021

  13. [20]

    J. Gao, T. Huber, Y . Ji and Y . M. Wang, Phys. Rev. Lett.128, no.6, 6 (2022)

    2022

  14. [21]

    G ´erardin, H

    A. G ´erardin, H. B. Meyer and A. Nyffeler, Phys. Rev. D94, no.7, 074507 (2016)

    2016

  15. [22]

    V . M. Braun, A. Khodjamirian and M. Maul, Phys. Rev. D61, 073004 (2000)

    2000

  16. [23]

    Bijnens and A

    J. Bijnens and A. Khodjamirian, Eur. Phys. J. C26, 67-79 (2002)

    2002

  17. [25]

    H. n. Li, Y . L. Shen, Y . M. Wang and H. Zou, Phys. Rev. D83, 054029 (2011)

    2011

  18. [26]

    Cheng, Y

    S. Cheng, Y . Y . Fan and Z. J. Xiao, Phys. Rev. D89, no.5, 054015 (2014)

    2014

  19. [27]

    H. C. Hu and H. n. Li, Phys. Lett. B718, 1351-1357 (2013)

    2013

  20. [28]

    Cheng and Z

    S. Cheng and Z. J. Xiao, Phys. Lett. B749(2015), 1-7

    2015

  21. [29]

    Cheng, Phys

    S. Cheng, Phys. Rev. D100, no.1, 013007 (2019)

    2019

  22. [30]

    L. B. Chen, W. Chen, F. Feng and Y . Jia, Phys. Rev. Lett.132(2024) no.20, 201901

    2024

  23. [31]

    H. n. Li, Y . L. Shen and Y . M. Wang, Phys. Rev. D85, 074004 (2012)

    2012

  24. [32]

    Cheng, Y

    S. Cheng, Y . Y . Fan, X. Yu, C. D. L¨u and Z. J. Xiao, Phys. Rev. D89, no.9, 094004 (2014)

    2014

  25. [33]

    Ball and R

    P. Ball and R. Zwicky, Phys. Rev. D71, 014015 (2005)

    2005

  26. [34]

    Ball and R

    P. Ball and R. Zwicky, Phys. Rev. D71, 014029 (2005)

    2005

  27. [35]

    Hambrock, A

    C. Hambrock, A. Khodjamirian and A. Rusov, Phys. Rev. D92, no.7, 074020 (2015)

    2015

  28. [36]

    Colangelo and A

    P. Colangelo and A. Khodjamirian, [arXiv:hep-ph/0010175 [hep-ph]]

    arXiv

  29. [37]

    Chang, I

    L. Chang, I. C. Cloet, J. J. Cobos-Martinez, C. D. Roberts, S. M. Schmidt and P. C. Tandy, Phys. Rev. Lett.110(2013) no.13, 132001

    2013

  30. [38]

    V . Y . Petrov, M. V . Polyakov, R. Ruskov, C. Weiss and K. Goeke, Phys. Rev. D59(1999), 114018

    1999

  31. [39]

    G. S. Baliet al.[RQCD], JHEP08, 065 (2019) [JHEP 11 (2020) 037 (addendum)]

    2019

  32. [41]

    Chai and S

    J. Chai and S. Cheng, Phys. Rev. D111(2025) no.7, L071902

    2025

  33. [42]

    H. n. Li and S. Mishima, Phys. Rev. D80(2009), 074024

    2009

  34. [43]

    Lepage, S.J

    G.P. Lepage, S.J. Brodsky, T. Huang and P.B. Mackenzie, Banff Summer Institute, Particles and Fields 2, eds. A.Z. Capri and A.N. Kamal (1983) p.83

    1983

  35. [44]

    Jakob and P

    R. Jakob and P. Kroll, Phys. Lett. B315(1993), 463-470 [erratum: Phys. Lett. B319 (1993), 545]

    1993

  36. [45]

    Lepage, S.J

    G.P. Lepage, S.J. Brodsky, T. Huang and P.B. Mackenzie, Banff Summer Institute, Particles and Fields 2, eds. A.Z. Capri and A.N. Kamal (1983) p. 83

    1983

  37. [46]

    Navaset al.[Particle Data Group], Phys

    S. Navaset al.[Particle Data Group], Phys. Rev. D110, no.3, 030001 (2024) – 41 –

    2024

  38. [47]

    Kumano, Q

    S. Kumano, Q. T. Song and O. V . Teryaev, Phys. Rev. D97(2018) no.1, 014020

    2018

  39. [48]

    Burkardt, Phys

    M. Burkardt, Phys. Rev. D62(2000), 071503 [erratum: Phys. Rev. D66(2002), 119903]

    2000

  40. [49]

    K. Raya, Z. F. Cui, L. Chang, J. M. Morgado, C. D. Roberts and J. Rodriguez-Quintero, Chin. Phys. C46(2022) no.1, 013105

    2022

  41. [50]

    H. n. Li and G. F. Sterman, Nucl. Phys. B381, 129-140 (1992)

    1992

  42. [51]

    C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys.33(1994), 477-575

    1994

  43. [52]

    C. D. Roberts, D. G. Richards, T. Horn and L. Chang, Prog. Part. Nucl. Phys.120(2021), 103883

    2021

  44. [53]

    Chang, I

    L. Chang, I. C. Clo ¨et, C. D. Roberts, S. M. Schmidt and P. C. Tandy, Phys. Rev. Lett.111 (2013) no.14, 141802

    2013

  45. [54]

    K. Raya, L. Chang, A. Bashir, J. J. Cobos-Martinez, L. X. Guti ´errez-Guerrero, C. D. Roberts and P. C. Tandy, Phys. Rev. D93(2016) no.7, 074017

    2016

  46. [55]

    P. Jain, B. Kundu, H. n. Li, J. P. Ralston and J. Samuelsson, Nucl. Phys. A666(2000), 75-83

    2000

  47. [56]

    J. Chai, S. Cheng and J. Hua, Eur. Phys. J. C83(2023) no.7, 556

    2023

  48. [57]

    Wanget al.[χQCD], Phys

    G. Wanget al.[χQCD], Phys. Rev. D104(2021), 074502

    2021

  49. [58]

    H. T. Ding, X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, Q. Shi, S. Syritsyn, R. Zhang and Y . Zhao, [arXiv:2404.04412 [hep-lat]]

    arXiv

  50. [59]

    J. P. Leeset al.[BaBar], Phys. Rev. D86(2012), 032013

    2012

  51. [60]

    Fujikawaet al.[Belle], Phys

    M. Fujikawaet al.[Belle], Phys. Rev. D78(2008), 072006

    2008

  52. [61]

    Ablikimet al.[BESIII], Phys

    M. Ablikimet al.[BESIII], Phys. Lett. B753(2016), 629-638 [erratum: Phys. Lett. B812 (2021), 135982]

    2016

  53. [62]

    S. R. Amendoliaet al.[NA7], Nucl. Phys. B277(1986), 168

    1986

  54. [63]

    Hornet al.[Jefferson Lab F(pi)-2], Phys

    T. Hornet al.[Jefferson Lab F(pi)-2], Phys. Rev. Lett.97(2006), 192001

    2006

  55. [64]

    G. M. Huberet al.[Jefferson Lab], Phys. Rev. C78(2008), 045203

    2008

  56. [65]

    Cheng and Q

    S. Cheng and Q. Qin, Phys. Rev. D99, no.1, 016019 (2019)

    2019

  57. [66]

    Leutwyler, Phys

    H. Leutwyler, Phys. Lett. B378, 313-318 (1996)

    1996

  58. [67]

    Gousset and B

    T. Gousset and B. Pire, Phys. Rev. D51(1995), 15-24

    1995

  59. [68]

    Delcourt, D

    B. Delcourt, D. Bisello, J. C. Bizot, J. Buon, A. Cordier and F. Mane, Phys. Lett. B99, 257-260 (1981)

    1981

  60. [69]

    F. Mane, D. Bisello, J. C. Bizot, J. Buon, A. Cordier and B. Delcourt, Phys. Lett. B99, 261-264 (1981)

    1981

  61. [70]

    Biselloet al.[DM2], Z

    D. Biselloet al.[DM2], Z. Phys. C39, 13 (1988)

    1988

  62. [71]

    P. M. Ivanov, L. M. Kurdadze, M. Y . Lelchuk, V . A. Sidorov, A. N. Skrinsky, A. G. Chilingarov, Y . M. Shatunov, B. A. Shvarts and S. I. Eidelman, Phys. Lett. B107, 297-300 (1981) – 42 –

    1981

  63. [72]

    P. m. Ivanov, L. m. Kurdadze, M. y. Lelchuk, E. v. Pakhtusova, V . a. Sidorov, A. n. Skrinsky, A. g. Chilingarov, Y . m. Shatunov, B. a. Shvarts and S. i. Eidelman, JETP Lett.36, 112-115 (1982)

    1982

  64. [73]

    M. N. Achasov, K. I. Beloborodov, A. V . Berdyugin,et al.Phys. Rev. D63, 072002 (2001)

    2001

  65. [74]

    M. N. Achasov, K. I. Beloborodov, A. V . Berdyugin,et al.Phys. Rev. D76, 072012 (2007)

    2007

  66. [75]

    R. R. Akhmetshinet al.[CMD-2], Phys. Lett. B669, 217-222 (2008)

    2008

  67. [76]

    T. K. Pedlaret al.[CLEO], Phys. Rev. Lett.95, 261803 (2005)

    2005

  68. [77]

    K. K. Seth, S. Dobbs, Z. Metreveli, A. Tomaradze, T. Xiao and G. Bonvicini, Phys. Rev. Lett.110, no.2, 022002 (2013)

    2013

  69. [78]

    J. P. Leeset al.[BaBar], Phys. Rev. D88(2013) no.3, 032013

    2013

  70. [79]

    J. P. Leeset al.[BaBar], Phys. Rev. D92(2015) no.7, 072008

    2015

  71. [80]

    Ablikimet al.[BESIII], Phys

    M. Ablikimet al.[BESIII], Phys. Rev. D99(2019) no.3, 032001

    2019

  72. [81]

    G. J. Gounaris and J. J. Sakurai, Phys. Rev. Lett.21, 244-247 (1968)

    1968

  73. [82]

    J. H. Kuhn and A. Santamaria, Z. Phys. C48, 445-452 (1990)

    1990

  74. [83]

    Huaet al.[Lattice Parton], Phys

    J. Huaet al.[Lattice Parton], Phys. Rev. Lett.129(2022) no.13, 132001

    2022

  75. [84]

    Koponen, A

    J. Koponen, A. C. Zimermmane-Santos, C. T. H. Davies, G. P. Lepage and A. T. Lytle, Phys. Rev. D96(2017) no.5, 054501

    2017

  76. [85]

    Z. Q. Yao, D. Binosi and C. D. Roberts, Phys. Lett. B855, 138823 (2024)

    2024

  77. [86]

    L. B. Chen, W. Chen, F. Feng and Y . Jia, [arXiv:2407.21120 [hep-ph]]

    arXiv

  78. [87]

    H. J. Behrendet al.[CELLO], Z. Phys. C49(1991), 401-410

    1991

  79. [88]

    L. G. Landsberg, Phys. Rept.128, 301-376 (1985)

    1985

  80. [89]

    S. J. Brodsky and G. P. Lepage, Phys. Rev. D24, 1808 (1981)

    1981

  81. [90]

    Gronberget al.[CLEO], Phys

    J. Gronberget al.[CLEO], Phys. Rev. D57(1998), 33-54

    1998

  82. [91]

    S. L. Adler, Phys. Rev.177, 2426-2438 (1969)

    1969

  83. [92]

    Melikhov and B

    D. Melikhov and B. Stech, Phys. Rev. D85, 051901 (2012)

    2012

  84. [93]

    V . L. Chernyak and A. R. Zhitnitsky, Phys. Rept.112, 173 (1984)

    1984

  85. [94]

    F. G. Cao, T. Huang and B. Q. Ma, Phys. Rev. D53, 6582-6585 (1996)

    1996

  86. [95]

    Kroll and M

    P. Kroll and M. Raulfs, Phys. Lett. B387, 848-854 (1996)

    1996

  87. [96]

    Feldmann and P

    T. Feldmann and P. Kroll, Eur. Phys. J. C5(1998), 327-335

    1998

  88. [97]

    I. V . Musatov and A. V . Radyushkin, Phys. Rev. D56, 2713-2735 (1997)

    1997

  89. [98]

    Khodjamirian, Eur

    A. Khodjamirian, Eur. Phys. J. C6, 477-484 (1999)

    1999

  90. [99]

    Schmedding and O

    A. Schmedding and O. I. Yakovlev, Phys. Rev. D62, 116002 (2000)

    2000

  91. [100]

    A. P. Bakulev, S. V . Mikhailov and N. G. Stefanis, Phys. Rev. D67, 074012 (2003)

    2003

  92. [101]

    S. S. Agaev, Phys. Rev. D72, 114010 (2005) – 43 –

    2005

  93. [102]

    Aubertet al.[BaBar], Phys

    B. Aubertet al.[BaBar], Phys. Rev. D80(2009), 052002

    2009

  94. [103]

    A. V . Radyushkin, Phys. Rev. D80, 094009 (2009)

    2009

  95. [104]

    M. V . Polyakov, JETP Lett.90, 228-231 (2009)

    2009

  96. [105]

    N. G. Stefanis, W. Schroers and H. C. Kim, Phys. Lett. B449, 299-305 (1999)

    1999

  97. [106]

    Nandi and H

    S. Nandi and H. n. Li, Phys. Rev. D76, 034008 (2007)

    2007

  98. [107]

    S. S. Agaev, V . M. Braun, N. Offen and F. A. Porkert, Phys. Rev. D83, 054020 (2011)

    2011

  99. [108]

    Ueharaet al.[Belle], Phys

    S. Ueharaet al.[Belle], Phys. Rev. D86(2012), 092007

    2012

  100. [109]

    S. S. Agaev, V . M. Braun, N. Offen and F. A. Porkert, Phys. Rev. D86, 077504 (2012)

    2012

  101. [110]

    C. F. Redmer [BESIII], [arXiv:1810.00654 [hep-ex]]

    arXiv

  102. [111]

    Aubertet al.[BaBar], Phys

    B. Aubertet al.[BaBar], Phys. Rev. D74(2006), 012002

    2006

  103. [112]

    del Amo Sanchezet al.[BaBar], Phys

    P. del Amo Sanchezet al.[BaBar], Phys. Rev. D84(2011), 05200

    2011

  104. [113]

    A. P. Bakulev, S. V . Mikhailov and N. G. Stefanis, Phys. Lett. B508, 279-289 (2001)

    2001

  105. [114]

    Kroll, Eur

    P. Kroll, Eur. Phys. J. C71(2011), 1623

    2011

  106. [115]

    S. S. Agaev, V . M. Braun, N. Offen, F. A. Porkert and A. Sch¨afer, Phys. Rev. D90(2014) no.7, 074019

    2014

  107. [116]

    Ali and A

    A. Ali and A. Y . Parkhomenko, Phys. Rev. D65, 074020 (2002)

    2002

  108. [117]

    Kroll and K

    P. Kroll and K. Passek-Kumericki, J. Phys. G40(2013), 075005

    2013

  109. [118]

    S. S. Agaev and N. G. Stefanis, Phys. Rev. D70, 054020 (2004)

    2004

  110. [119]

    J. P. Leeset al.[BaBar], Phys. Rev. D98(2018) no.11, 112002

    2018

  111. [120]

    Ji and A

    Y . Ji and A. Vladimirov, Eur. Phys. J. C79(2019) no.4, 319

    2019

  112. [121]

    Leutwyler, Nucl

    H. Leutwyler, Nucl. Phys. B Proc. Suppl.64, 223-231 (1998)

    1998

  113. [122]

    Witten, Nucl

    E. Witten, Nucl. Phys. B149, 285-320 (1979)

    1979

  114. [123]

    Feldmann, P

    T. Feldmann, P. Kroll and B. Stech, Phys. Rev. D58(1998), 114006

    1998

  115. [124]

    Escribano and J

    R. Escribano and J. M. Frere, JHEP06(2005), 029

    2005

  116. [125]

    Escribano, P

    R. Escribano, P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D89, no.3, 034014 (2014)

    2014

  117. [126]

    F. G. Cao, Phys. Rev. D85(2012), 057501

    2012

  118. [127]

    G. S. Baliet al.[RQCD], JHEP08, 137 (2021)

    2021

  119. [128]

    Hattonet al.[HPQCD], Phys

    D. Hattonet al.[HPQCD], Phys. Rev. D102, no.5, 054511 (2020)

    2020

  120. [129]

    M. Ding, F. Gao, L. Chang, Y . X. Liu and C. D. Roberts, Phys. Lett. B753(2016), 330-335

    2016

  121. [130]

    Blossier, M

    B. Blossier, M. Mangin-Brinet, J. M. Morgado Ch ´avez and T. San Jos´e, JHEP09(2024), 079

    2024

  122. [131]

    Delaneyet al.[Hadron Spectrum], JHEP05(2024), 230

    J. Delaneyet al.[Hadron Spectrum], JHEP05(2024), 230

    2024

  123. [132]

    Y . Li, M. Li and J. P. Vary, Phys. Rev. D105(2022) no.7, L071901 – 44 –

    2022

  124. [133]

    J. Chen, M. Ding, L. Chang and Y . x. Liu, Phys. Rev. D95(2017) no.1, 016010

    2017

  125. [134]

    J. P. Leeset al.[BaBar], Phys. Rev. D81(2010), 052010

    2010

  126. [135]

    V . M. Braun, G. P. Korchemsky and D. M¨uller, Prog. Part. Nucl. Phys.51(2003), 311-398

    2003

  127. [136]

    I. I. Balitsky and V . M. Braun, Nucl. Phys. B311(1989), 541-584

    1989

  128. [137]

    P. Ball, V . M. Braun and A. Lenz, JHEP0605, 004 (2006)

    2006

  129. [138]

    Ball, JHEP9901, 010 (1999)

    P. Ball, JHEP9901, 010 (1999)

    1999

  130. [139]

    Khodjamirian, C

    A. Khodjamirian, C. Klein, T. Mannel and N. Offen, Phys. Rev. D80, 114005 (2009) – 45 –

    2009

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.