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Radiative tail of the three-particle light-cone distribution amplitudes for the $\Lambda_b$ baryon in HQET

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Pith's one-line read QCD fixes the high-momentum tail of the Lambda_b baryon's light-quark distributions at one loop.

desk verdict A clean NLO calculation of the Lambda_b three-particle LCDA radiative tail, with one parametrization-completeness caveat that is worth a clarification rather than a rejection. read the letter →

arxiv 2505.02570 v1 pith:TP6NW4A6 submitted 2025-05-05 hep-ph

Thorsten Feldmann , Daniel Vladimirov This is my paper · ORCID
classification hep-ph
keywords Lambda_bbaryonlight-conedistributionamplitudesheavy-quarkeffectivetheoryradiativetailshort-distanceexpansionone-loopmatchingoperatorproduct
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 establishes that the large-momentum behavior of the three-particle light-cone distribution amplitudes of the Lambda_b baryon is not arbitrary model input but is fixed by one-loop QCD. Working in heavy-quark effective theory, the authors derive the short-distance expansion of the defining light-ray operators and obtain the radiative tail: $\varphi$(omega) behaves as a power-like series in 1/omega with coefficients set by alpha_s, the color factor C_F, and the HQET mass parameter Lambda_bar = M_Lambda_b - m_b. Because the Lambda_b has no analogous radiative leptonic decay that would probe its distribution amplitude directly, this perturbative constraint is the main model-independent handle on the light-quark momentum distribution inside the baryon. The paper also constructs a model-dependent interpolation to low momenta, integrating the renormalization-group equation with an input shape, and finds the same qualitative features as in the B-meson case.

What carries the argument

The machinery is the light-ray operator product expansion of Eq. (3.1): each three-quark light-ray operator is expanded into local operators of canonical dimension 9/2 and 11/2, with one-loop Wilson coefficients (4.13)-(4.15) computed by matching partonic matrix elements in D = 4 - 2 epsilon. The hadronic side is parametrized through Eq. (5.4), whose four coefficients are fixed by the light-quark equations of motion and HQET spin symmetry to the values in Eq. (5.8). The combination turns the Wilson coefficients and local matrix elements into the small-distance forms (5.9)-(5.12), and an inverse Laplace transform converts those into the momentum-space tail (5.30).

What would settle it

Compute the vacuum-to-Lambda_b matrix element of the single-covariant-derivative operator entering Eq. (5.3) without imposing the minimal ansatz, for example by lattice QCD, and compare the independent tensor structures with the linear combinations that follow from C = -Lambda_bar/6, D = 2 Lambda_bar/3, E = Lambda_bar/6, F = Lambda_bar/3. Any mismatch changes the short-distance coefficients (5.9)-(5.12) and therefore the $omega^{-2}$ coefficient of the radiative tail in Eq. (5.30).

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Extended reading notes

Core claim

On its own terms, the paper's central result is Eq. (5.30): for large light-cone momentum omega, the integrated Lambda_b LCDA behaves as $\varphi$(omega) = alpha_s C_F/(2 pi) (1/omega)(2 ln(mu/omega)+1) + alpha_s C_F/(2 pi) (4 Lambda_bar_eff)/(3 $omega^{2}$)(2 ln(mu/omega)+7/2) + ..., where Lambda_bar_eff is defined in Eq. (5.17). At this order the tail is completely determined by the strong coupling, the color factor C_F, the LCDA normalization, and the HQET mass parameter Lambda_bar; no further non-perturbative input enters. The companion short-distance expansions Eqs. (5.9)-(5.12) give the small-distance behavior of all four Lambda_b LCDAs in position space, with the first derivative at the origin fixed by Lambda_bar through Eq. (5.17).

Load-bearing premise

The result depends on the assumption that the dimension-11/2 operator matrix element has exactly the four parameter degrees of freedom in Eq. (5.4); if a further independent form factor exists at that order, the predicted 1/$omega^{2}$ term changes.

Editorial extensions

If this is right

  • Any model of Lambda_b three-particle LCDAs must reproduce the power-like falloff of Eq. (5.30) at large omega; the large-momentum region is no longer free.
  • Predictions for Lambda_b decays from QCD factorization or light-cone sum rules inherit a fixed 1/omega and 1/omega^2 perturbative structure, shrinking one source of model dependence.
  • The first derivative of the position-space LCDA at the origin is tied to Lambda_bar via Eq. (5.17), so the moment constraint and the HQET mass parameter are linked at this order.
  • The convolution-kernel construction preserves the analyticity properties of the LCDAs in position space, yielding a smooth momentum-space function, and can be reapplied to other input models.
  • The same short-distance expansion can be repeated for dimension-13/2 operators; the matching calculation would constrain the second derivative of the model at the origin.

Reading between the lines

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

  • A direct check of the completeness of the four-parameter ansatz Eq. (5.4), for instance by an independent lattice or sum-rule computation of the dimension-11/2 matrix element, would either confirm or shift the 1/omega^2 coefficient in Eq. (5.30).
  • The interpolation strategy generalizes immediately to other heavy-hadron LCDAs; applying the same convolution kernels to B-meson two- or three-particle models would produce one-loop-improved shapes with preserved analytic behavior.
  • If the radiative tail is measured indirectly through high-momentum-sensitive Lambda_b observables, the fitted omega^-2 coefficient would yield a determination of Lambda_bar_eff and hence of the HQET mass parameter.
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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

1 major / 5 minor

Summary. This paper computes the one-loop radiative tail of the three-particle light-cone distribution amplitudes (LCDAs) of the Lambda_b baryon in heavy-quark effective theory (HQET). The authors perform a short-distance (operator product) expansion of the defining light-ray operators up to operators of canonical dimension 11/2, calculate the Wilson coefficients at one loop, and evaluate the hadronic matrix elements of the local operators in terms of the two decay constants f^(1)_Lambda_b, f^(2)_Lambda_b and the HQET mass parameter Lambda_bar = M_Lambda_b - m_b. They then propose a model-dependent extrapolation to large distances, which after Fourier transformation yields a momentum-space LCDA whose large-momentum tail is given by Eq. (5.30): phi(omega) = (alpha_s C_F/2pi) (1/omega)(2 ln(mu/omega)+1) + (alpha_s C_F/2pi) (4 Lambda_bar_eff/(3 omega^2))(2 ln(mu/omega)+7/2) + ... . The paper also derives the short-distance position-space behavior of all four Lambda_b three-particle LCDAs, Eqs. (5.9)-(5.12), and provides a renormalization-group equation for the equal-distance projection in the appendix.

Significance. If correct, this is the first calculation of the radiative tail for the Lambda_b three-particle LCDAs in HQET, extending the known B-meson result to the baryonic case. The result is significant for applications in QCD factorization and light-cone sum rules for exclusive Lambda_b decays, since it provides model-independent perturbative constraints on the large-momentum behavior that are currently replaced by purely phenomenological models. The paper is careful and transparent: the matching calculation is self-contained, the subtraction scheme is explicit, the model choices in Secs. 5.2-5.4 are clearly flagged as model-dependent, and the internal RGE consistency check (gamma_F^(1)=2) is satisfied. The main technical strength is the explicit one-loop matching coefficient calculation and the position-space short-distance expansions for all four LCDAs.

major comments (1)
  1. [Sec. 5, Eq. (5.4) and Eq. (5.30)] The parametrization of the dimension-11/2 local-operator matrix elements in Eq. (5.4) with only four coefficients C, D, E, F is the sole hadronic input that converts the perturbative matching coefficients into the Lambda_bar-dependent short-distance behavior of the LCDAs, and hence into the 1/omega^2 term of the radiative tail (5.30). The paper states that this form follows from Lorentz invariance and HQET spin symmetry, but it does not demonstrate that the chosen basis is complete. Under the Dirac projections relevant for the LCDA definitions, additional independent structures (for example terms involving gamma_5, sigma_{mu nu} v^nu, or v_mu gamma_5 with appropriate /v projections) could in principle contribute to the matrix elements of the in.D and iv.D operators. If such terms exist, Eqs. (5.5) and (5.7) no longer determine C, D, E, F uniquely, and both the short-distance coefficients in Eqs. (5.9)-(5.12) and the 1/omega^2 coefficient in Eq. (5.30) would shift. The internal checks (tree-level limit, RGE consistency) test the matching and the algebra given the parametrization, but they do not test the completeness of Eq. (5.4). The authors should provide an explicit enumeration of all Lorentz/Dirac structures allowed by the symmetries and show that those not contained in Eq. (5.4) are forbidden by the light-quark equations of motion and the heavy-quark field properties, or alternatively state the completeness as an assumption and quantify the impact of possible additional form factors on the radiative tail.
minor comments (5)
  1. [Sec. 5.3, Eq. (5.23)] The notation in Eq. (5.23) is confusing: the inner integral over domega'_1 has the integration variable appearing also in the upper limit (u omega'), and the parentheses in the denominator are unbalanced. Please rewrite this convolution formula with dummy integration variables and clarify the plus-distribution prescription.
  2. [Sec. 5.1, heading] There is a typo in the heading of Sec. 5.1: 'short-distance epxansion' should be 'short-distance expansion'.
  3. [References] Reference [23] is an unpublished master's thesis; please indicate how it can be accessed or replace it with a published source where possible, since readers cannot easily verify the results quoted from it.
  4. [Abstract and Sec. 5.4] The abstract and conclusion state that the radiative tail for the three-particle Lambda_b LCDAs has been calculated, but the explicit asymptotic formula (5.30) is derived only for the equal-distance projection phi(omega) = omega integral_0^1 du phi_2(u omega, (1-u) omega). The position-space short-distance behavior is provided for all four LCDAs, but the analogous momentum-space tails for phi_4, phi_3^s, and phi_3^sigma are not written down. Please clarify in the abstract and conclusion which quantity the explicit tail formula refers to.
  5. [Sec. 5.2, Fig. 2 caption] In the caption of Fig. 2 and in the text, the choice mu e^{gamma_E} = mu_F e^{gamma_E} = 2 GeV implicitly sets mu = mu_F; this should be stated explicitly in the main text to avoid confusion with the general case where mu and mu_F differ.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Lambda_b radiative tail is derived from a one-loop matching calculation and expressed in independent hadronic parameters.

full rationale

The derivation chain is not circular. The one-loop matching coefficients in Eqs. (4.13)-(4.15) are computed from explicit Feynman diagrams rather than imposed, and the hadronic matrix elements in Eqs. (5.1)-(5.4) parametrize independent nonperturbative inputs: the two decay constants f_Lambda_b^(1,2) and the HQET mass parameter Lambda_bar. The short-distance behaviors in Eqs. (5.9)-(5.12) follow by inserting these inputs into the computed Wilson coefficients, not by assuming the final tail. The radiative tail in Eq. (5.30) is obtained by expanding the convolution kernels in Eq. (5.27) and using the moments of the model function in Eq. (5.28); the O(1/omega^2) coefficient is expressed in terms of Lambda_bar_eff, which is the first moment of the LCDA, a hadronic parameter that is not fitted to the tail itself. The extrapolation model in Section 5.2 is explicitly admitted to be model-dependent and is constrained to reproduce the OPE short-distance behavior; this is a consistency requirement, not a circular reduction. The self-citations to Refs. [20] and [22] for the LCDA classification and matching strategy are normal methodological citations: they do not contain the Lambda_b result, and the present calculation is a new computation with independent content. The unproven completeness of the dimension-11/2 parametrization in Eq. (5.4) is a possible correctness risk, but it is not a circular step under the criteria used here.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central matching calculation uses only standard QCD/HQET factorization inputs. The only hadronic parameter entering the tail is Lambda_bar; the illustrative model adds the free scale mu_F and a choice of model function. No invented entities are introduced.

free parameters (3)
  • Lambda_bar = M_Lambda_b - m_b = 0.6 GeV (illustrative input; not fitted here)
    HQET mass parameter entering the dimension-11/2 matrix elements and the 1/omega^2 term of the radiative tail (Eqs. 5.7-5.8, 5.30). It is an external hadronic input, taken from the Lambda_b and b-quark masses, not computed in this paper.
  • mu_F (model transition scale) = 2 GeV in the figures
    Free parameter in the extrapolation model (Eq. 5.16) that sets where the short-distance tail merges into the model shape; it also enters Lambda_bar_eff (Eq. 5.17) and the convolution kernels (Eq. 5.21). The OPE does not determine it.
  • Model shape parameter omega_0 = ~0.23 GeV in the figures
    Width of the illustrative model f(s) = (1+omega_0 s)^(-4) in Eq. (5.18); fixed by the derivative constraint f'(0) = -4 Lambda_bar_eff/3, so it is derived rather than independently fitted.
assumptions (5)
  • domain assumption Heavy-quark effective theory with a static heavy quark and massless light quarks; strict isospin limit; QED neglected (Sec. 2).
    The whole framework treats the b quark as infinitely heavy and sets m_u = m_d = 0. These are standard HQET approximations but are assumptions about the physical system.
  • domain assumption Short-distance OPE of light-ray operators into local operators is valid for tau_1,2 << 1/Lambda_QCD and can be matched at partonic level (Sec. 3, Eq. 3.1).
    Factorization of short- and long-distance physics is assumed; the radiative tail is defined as the short-distance piece.
  • domain assumption The dimension-11/2 hadronic matrix elements are completely parametrized by f^(1,2)_Lambda_b and Lambda_bar via Eqs. (5.3)-(5.8).
    HQET spin symmetry and light-quark equations of motion fix C, D, E, F. The completeness of this minimal parametrization is assumed without independent evidence.
  • domain assumption Analyticity of the position-space LCDAs (support omega_i >= 0 in momentum space; Im tau < 0 analyticity) is used to justify the extrapolation and inverse Laplace transforms (Secs. 2, 5.2, 5.3).
    Standard HQET property, cited to [26].
  • standard math Known one-loop results from the mesonic case (Refs. [21,22]) are used as the template for the baryon integrals (Eqs. 4.2-4.4).
    External benchmark results are adopted; the paper adjusts color factors and Dirac structures.

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Cite this review

Pith. "Pith review of Radiative tail of the three-particle light-cone distribution amplitudes for the $\Lambda_b$ baryon in HQET." pith.science (2026). https://pith.science/paper/TP6NW4A6

@misc{pith2026250502570,
  author       = {Pith},
  title        = {Pith review of: Radiative tail of the three-particle light-cone distribution amplitudes for the $\Lambda_b$ baryon in HQET},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TP6NW4A6}},
  note         = {Machine review of arXiv:2505.02570}
}
abstract

We calculate the so-called radiative tail for the three-particle light-cone distribution amplitudes (LCDAs) of the $\Lambda_b$ baryon, following from the short-distance expansion of the defining light-ray operators in heavy-quark effective theory (HQET). To illustrate the effect of the radiative tail, we introduce a simple (model-dependent) extrapolation to large distances (respectively small momenta), which preserves the analytic properties of the LCDAs in HQET. We observe a similar qualitative behaviour as has been discussed for the $B$-meson case.

Figures

Figures reproduced from arXiv: 2505.02570 by the authors.

Figure 1
Figure 1. Feynman diagrams contributing to the matching calculation at NLO. Here, the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Impact of the radiative tail on the light-cone distribution amplitude [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. SCET sum rules for $\Lambda_b \to \Lambda \ell^+\ell^-$, $\Lambda \gamma$ decays

    hep-ph 2025-06 conditional novelty 6.0 of 10

    SCET light-cone sum rules give NLO soft form factors and small non-factorizable corrections for Λ_b → Λ decays, with a Λ_b → Λ γ branching fraction consistent with data.

  2. Heavy quark mass dependence of the $\Lambda_Q$ light-cone distribution amplitude in QCD

    hep-ph 2026-07 conditional novelty 5.0 of 10

    The Lambda_Q baryon LCDA at mass m_Q equals (m_Q/m_Q^0)^2 times the LCDA at m_Q^0 evaluated at rescaled fractions x_i*m_Q/m_Q^0, times an exponentiated anomalous dimension, plus renormalon-model power corrections.

Reference graph

Works this paper leans on

31 extracted references · 4 canonical work pages · cited by 2 Pith papers

  1. [1]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, QCD factorization for B→ππ decays: Strong phases and CP violation in the heavy quark limit , Phys. Rev. Lett. 83 (1999) 1914–1917, [ hep-ph/9905312]

    arXiv 1999

  2. [2]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, QCD factorization in B→πK,ππ decays and extraction of Wolfenstein parameters , Nucl. Phys. B 606 (2001) 245–321, [ hep-ph/0104110]

    arXiv 2001

  3. [3]

    Khodjamirian, B

    A. Khodjamirian, B. Meli´ c and Y.-M. Wang,A guide to the QCD light-cone sum rules for b-quark decays, Eur. Phys. J. ST 233 (2024) 271–298, [ 2311.08700]

    arXiv 2024

  4. [4]

    G. P. Korchemsky, D. Pirjol and T.-M. Yan, Radiative leptonic decays of B mesons in QCD, Phys. Rev. D 61 (2000) 114510, [ hep-ph/9911427]

    arXiv 2000

  5. [5]

    Descotes-Genon and C

    S. Descotes-Genon and C. T. Sachrajda, Factorization, the light cone distribution amplitude of the B meson and the radiative decay B→γℓνℓ, Nucl. Phys. B 650 (2003) 356–390, [ hep-ph/0209216]

    arXiv 2003

  6. [6]

    Lunghi, D

    E. Lunghi, D. Pirjol and D. Wyler, Factorization in leptonic radiative B →γeν decays, Nucl. Phys. B 649 (2003) 349–364, [ hep-ph/0210091]

    arXiv 2003

  7. [7]

    S. W. Bosch, R. J. Hill, B. O. Lange and M. Neubert, Factorization and Sudakov resummation in leptonic radiative B decay , Phys. Rev. D 67 (2003) 094014, [hep-ph/0301123]

    arXiv 2003

  8. [8]

    Beneke and J

    M. Beneke and J. Rohrwild, B meson distribution amplitude from B→γℓν, Eur. Phys. J. C 71 (2011) 1818, [ 1110.3228]

    arXiv 2011

Show all 31 references
  1. [9]

    V. M. Braun and A. Khodjamirian, Soft contribution to B→γℓνℓ and the B-meson distribution amplitude, Phys. Lett. B 718 (2013) 1014–1019, [ 1210.4453]

    2013 arXiv

  2. [10]

    Wang, Factorization and dispersion relations for radiative leptonic B decay, JHEP 09 (2016) 159, [ 1606.03080]

    Y.-M. Wang, Factorization and dispersion relations for radiative leptonic B decay, JHEP 09 (2016) 159, [ 1606.03080]

    2016 arXiv

  3. [11]

    Wang and Y.-L

    Y.-M. Wang and Y.-L. Shen, Subleading-power corrections to the radiative leptonic B→γℓν decay in QCD, JHEP 05 (2018) 184, [ 1803.06667]

    2018 arXiv

  4. [12]

    Wang, Factorization of Heavy-to-Light Baryonic Transitions in SCET , Phys

    W. Wang, Factorization of Heavy-to-Light Baryonic Transitions in SCET , Phys. Lett. B 708 (2012) 119–126, [ 1112.0237]

    2012 arXiv

  5. [13]

    Feldmann and M

    T. Feldmann and M. W. Y. Yip, Form factors for Λb→ Λ transitions in the soft-collinear effective theory, Phys. Rev. D 85 (2012) 014035, [ 1111.1844]

    2012 arXiv

  6. [14]

    Wang and Y.-L

    Y.-M. Wang and Y.-L. Shen, Perturbative Corrections to Λb→ Λ Form Factors from QCD Light-Cone Sum Rules , JHEP 02 (2016) 179, [ 1511.09036]. 20

    2016 arXiv

  7. [15]

    Feldmann and N

    T. Feldmann and N. Gubernari, Non-factorisable contributions of strong-penguin operators in Λb→ Λℓ+ℓ− decays, JHEP 03 (2024) 152, [ 2312.14146]

    2024 arXiv

  8. [16]

    B. O. Lange and M. Neubert, Renormalization group evolution of the B meson light cone distribution amplitude , Phys. Rev. Lett. 91 (2003) 102001, [ hep-ph/0303082]

    2003 arXiv

  9. [17]

    S. J. Lee and M. Neubert, Model-independent properties of the B-meson distribution amplitude, Phys. Rev. D 72 (2005) 094028, [ hep-ph/0509350]

    2005 arXiv

  10. [18]

    P. Ball, V. M. Braun and E. Gardi, Distribution Amplitudes of the Λb Baryon in QCD, Phys. Lett. B 665 (2008) 197–204, [ 0804.2424]

    2008 arXiv

  11. [19]

    A. Ali, C. Hambrock, A. Y. Parkhomenko and W. Wang, Light-Cone Distribution Amplitudes of the Ground State Bottom Baryons in HQET , Eur. Phys. J. C 73 (2013) 2302, [ 1212.3280]

    2013 arXiv

  12. [20]

    G. Bell, T. Feldmann, Y.-M. Wang and M. W. Y. Yip, Light-Cone Distribution Amplitudes for Heavy-Quark Hadrons , JHEP 11 (2013) 191, [ 1308.6114]

    2013 arXiv

  13. [21]

    Kawamura and K

    H. Kawamura and K. Tanaka, Operator product expansion for B-meson distribution amplitude and dimension-5 HQET operators , Phys. Lett. B 673 (2009) 201–207, [0810.5628]

    2009 arXiv

  14. [22]

    Feldmann, P

    T. Feldmann, P. L¨ ughausen and N. Seitz,Strange-quark mass effects in the Bs meson’s light-cone distribution amplitude , JHEP 08 (2023) 075, [ 2306.14686]

    2023 arXiv

  15. [23]

    Vladimirov, Asymptotic behaviour of light-cone distribution amplitudes for heavy Λb baryons, master thesis (2024, unpublished)

    D. Vladimirov, Asymptotic behaviour of light-cone distribution amplitudes for heavy Λb baryons, master thesis (2024, unpublished)

    2024

  16. [24]

    V. M. Braun, S. E. Derkachov and A. N. Manashov, Integrability of the evolution equations for heavy–light baryon distribution amplitudes , Phys. Lett. B 738 (2014) 334–340, [1406.0664]

    2014 arXiv

  17. [25]

    Kawamura and K

    H. Kawamura and K. Tanaka, Evolution equation for the B-meson distribution amplitude in the heavy-quark effective theory in coordinate space , Phys. Rev. D 81 (2010) 114009, [ 1002.1177]

    2010 arXiv

  18. [26]

    Feldmann, P

    T. Feldmann, P. L¨ ughausen and D. van Dyk,Systematic parametrization of the leading B-meson light-cone distribution amplitude , JHEP 10 (2022) 162, [2203.15679]

    2022 arXiv

  19. [27]

    Feldmann, B

    T. Feldmann, B. O. Lange and Y.-M. Wang, B -meson light-cone distribution amplitude: Perturbative constraints and asymptotic behavior in dual space , Phys. Rev. D 89 (2014) 114001, [ 1404.1343]

    2014 arXiv

  20. [28]

    Ligeti, I

    Z. Ligeti, I. W. Stewart and F. J. Tackmann, Treating the b quark distribution function with reliable uncertainties , Phys. Rev. D 78 (2008) 114014, [ 0807.1926]. 21

    2008 arXiv

  21. [29]

    A. V. Efremov and A. V. Radyushkin, Factorization and Asymptotical Behavior of Pion Form-Factor in QCD , Phys. Lett. B 94 (1980) 245–250

    1980

  22. [30]

    G. P. Lepage and S. J. Brodsky, Exclusive Processes in Quantum Chromodynamics: Evolution Equations for Hadronic Wave Functions and the Form-Factors of Mesons , Phys. Lett. B 87 (1979) 359–365

    1979

  23. [31]

    V. M. Braun, D. Y. Ivanov and G. P. Korchemsky, The B meson distribution amplitude in QCD , Phys. Rev. D 69 (2004) 034014, [ hep-ph/0309330]. 22

    2004 arXiv

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