Pith. sign in

REVIEW 3 major objections 6 minor 2 cited by

Factorization Formula Connecting the Shape Functions of Heavy Meson in QCD and Heavy Quark Effective Theory

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read The paper derives a one-loop factorization formula connecting the QCD and HQET definitions of the B-meson shape function, with a multiplicative peak-region matching coefficient and a perturbative tail function.

desk verdict New one-loop matching between QCD and HQET B-meson shape functions, but the paper skips the load-bearing cancellation algebra and leaves the tail region fuzzy; worth refereeing, not desk-rejecting. read the letter →

arxiv 2504.18018 v2 pith:YPXQR4UV submitted 2025-04-25 hep-ph

Wei Wang , Ji Xu , Qi-An Zhang , Shuai Zhao This is my paper · ORCID
classification hep-ph
keywords B-mesonshapefunctionheavyquarkeffectivetheoryfactorizationinclusiveBdecays|Vub|latticeQCDone-loopmatchinglightconedistribution
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 derives a factorization formula connecting the two standard definitions of the B-meson shape function: the QCD definition, which captures the b quark's lightcone momentum fraction and contains physics at both $m_b$ and $\Lambda_{\rm QCD}$, and the HQET definition, which describes the heavy quark's residual momentum in the infinite-mass limit. The central claim is that in the peak region, $x \sim 1 - \Lambda_{\rm QCD}/m_b$, the QCD shape function equals the HQET shape function multiplied by a simple one-loop coefficient, with the residual momentum shifted by $m_b$. In the tail region the two are connected by a purely perturbative matching function. The result matters because the HQET shape function is a major uncertainty in inclusive $B$ decays and the $|V_{ub}|$ determination, while the QCD version is more naturally accessible to lattice computation; the formula provides a bridge between the two.

What carries the argument

The load-bearing object is the region-separated factorization formula Eq. (5), which splits $S_{\rm QCD}(x,\mu)$ into a peak region $x\sim 1-\Lambda_{\rm QCD}/m_b$, matched to $S_{\rm HQET}(\omega,\mu)$ by a kernel $Z_{\rm peak}(x,\omega,\mu)$, and a tail region $x\sim 0$, described by a perturbative matching function $Z_{\rm tail}(x,\mu)$. The argument is carried by explicit one-loop calculations of three Feynman diagrams in each theory (heavy-quark sail, box, and local vertex). The cancellation of the modified plus distribution of Eq. (12) and the infrared regulator $v\cdot k$ in the matching step is what produces the simple multiplicative peak-region coefficient, Eq. (22).

What would settle it

A two-loop computation of the matching coefficient would settle whether the multiplicative form of Eq. (23) survives beyond one loop; alternatively, a lattice extraction of the quasishape function matched to $S_{\rm QCD}$ could be compared with Eq. (23) across the peak-to-tail transition, where a growing discrepancy would show the region separation is not clean.

Watch

Extended reading notes

Core claim

The central discovery is the one-loop factorization formula of Eq. (5), which in the peak region takes the explicit multiplicative form $S_{\rm QCD}(x,\mu) = \left[1+\frac{\alpha_s C_F}{2\pi}\left(\frac12\ln^2\frac{\mu^2}{m_b^2}-\frac32\ln\frac{\mu^2}{m_b^2}+\frac{\pi^2}{12}-2\right)\right] S_{\rm HQET}(\omega,\mu)$, with $\omega v_+ = x m_B v_+ - m_b v_+$, together with the tail-region matching function $Z_{\rm tail}^{(1)}(x,\mu) = \frac{1}{m_b v_+}\frac{1+x^2}{1-x}\left[-1+\ln\frac{\mu^2}{(1-x)^2 m_b^2}\right]$ from Eq. (19). The apparent convolution in the factorization reduces to multiplication because the plus distributions and the infrared regulator $v\cdot k$ cancel between the QCD and HQET one-loop amplitudes, leaving a matching coefficient that depends only on $\mu$ and $m_b$. This establishes, at leading power in $\Lambda_{\rm QCD}/m_b$ and one-loop order in $\alpha_s$, that the two shape functions have identical infrared behavior and differ only by calculable short-distance physics.

Load-bearing premise

The matching assumes a clean leading-power separation between the peak region and the tail region, with the two shape functions sharing identical infrared behavior; the paper does not control the transition region between them, so if that region is broad the stated one-loop formulas would be incomplete.

Editorial extensions

If this is right

  • The QCD shape function obtained from lattice simulations can be converted into the HQET shape function used in inclusive $B$-decay analyses, and the conversion is exact at leading power and one-loop order.
  • Separating the $m_b$ scale from $\Lambda_{\rm QCD}$ in this way makes the large logarithms $\ln(m_b/\Lambda_{\rm QCD})$ resummable through the matching coefficient.
  • Using a phenomenological model for the HQET shape function, the paper constructs the corresponding QCD shape function and identifies the intermediate transition region where higher-order $(1-x)$ corrections are required.
  • Matching at a different heavy-quark mass would connect $B$-meson and $D$-meson shape functions, with the caveat that $\Lambda_{\rm QCD}/m_c$ power corrections become sizable in the charm case.

Reading between the lines

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

  • Not derived in the paper: if the one-loop multiplicative coefficient exponentiates under renormalization-group evolution, the product form would allow an all-orders resummation of $\ln(m_b/\Lambda_{\rm QCD})$; checking this requires computing the anomalous dimension of the matching coefficient.
  • The collapse of the convolution to multiplication at one loop may be an accident of this order; at two loops the matching could develop a genuinely nonlocal dependence on $\omega$, which would change the simple form of Eq. (23).
  • A direct lattice determination of the quasishape function, combined with the first-step factorization, would test the predicted peak-region relation without invoking any model of the HQET shape function; this is an extension the paper leaves to future work.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper derives a factorization formula relating the B-meson shape function defined in QCD, S_QCD(x, μ), to the shape function defined in HQET, S_HQET(ω, μ). The claimed relation is region-dependent: in the peak region x ~ 1 - Λ_QCD/m_b the QCD shape function equals a multiplicative one-loop matching coefficient times the HQET shape function evaluated at ωv_+ = x m_B v_+ - m_b v_+, while in the tail region x ~ 0 it is given by a purely perturbative coefficient. The authors compute the one-loop corrections to both shape functions using free-quark external states, list the individual diagram contributions for the QCD sail, box, and local vertex graphs and their HQET counterparts, and state the resulting one-loop matching coefficients Z_peak^(1) in Eq. (22) and Z_tail^(1) in Eq. (19). They then use a phenomenological model for S_HQET to plot the QCD shape function and discuss applications to lattice QCD via two-step factorization.

Significance. If correct, Eq. (23) provides a simple one-loop conversion between the two standard definitions of the B-meson shape function, enabling resummation of logarithms of m_b/Λ_QCD and serving as the second step in the recently proposed lattice two-step factorization scheme. The individual one-loop graph results are presented in substantial detail, including explicit IR regulators v·k and the plus-distribution definition, and the claimed cancellation of the IR regulator between QCD and HQET is physically expected and plausible. The numerical section uses model parameters taken from prior work, so the derivation is not circular. The main weakness is that the step from the individual graph results to the central matching coefficient is not demonstrated, and the treatment of the tail region is not fully specified. These issues are potentially fixable but are load-bearing for the paper's central claim.

major comments (3)
  1. [Section III.D, Eq. (22)] The central result, Eq. (22), is asserted as the outcome of inserting Eqs. (20) and (21) into Eq. (11), but the intermediate algebra is not shown. The paper claims that the plus distributions, the v·k dependence, the Lambda dependence, and the residual-momentum dependence all cancel, leaving a local coefficient, yet the presented graph results in Eqs. (13a)-(18) contain terms that, after the substitution ωv_+ = x m_B v_+ - m_b v_+, generate nontrivial integrals: for example, the delta-function subtraction in Eq. (13a) involves an integral from x m_B v_+ + k_+ to Lambda, and Eq. (16) involves an integral from 2 ω v_+ to Lambda over logarithmic integrands. I could not verify the claimed cancellation from the text. Because Eq. (23) depends entirely on this step, the authors should provide the full combination of the six one-loop expressions, at least in an appendix, or supply an independent numerical check of Eq. (22).
  2. [Section III.D, Eq. (19)] The tail matching coefficient is written as the sum of the QCD tail contributions in Eqs. (13b) and (14b), with no subtraction of the perturbative HQET tail. The paper itself notes in this section that S_HQET has a calculable radiative tail at large |ω|, and at x ~ 0 the peak-branch variable would correspond to ω ~ -m_b, where that tail is not negligible at one-loop order. A genuine two-sided matching would give Z_tail = S_QCD|tail - S_HQET|tail. As written, Eq. (19) is either incomplete or the tail branch of Eq. (5) is intended to be a pure QCD OPE that does not connect to S_HQET. The manuscript should state explicitly which interpretation is intended and, if a subtraction is required, include it and assess its numerical effect on Fig. 2.
  3. [Section IV and Fig. 2] The factorization in Eq. (5) assumes a clean leading-power separation between the peak region x ~ 1 - Lambda_QCD/m_b and the tail region x ~ 0, but the transition region is not quantified. The paper acknowledges in Sec. IV that the shaded intermediate region requires higher-order (1-x) corrections, yet it does not specify a power-counting criterion for where Eq. (23) ceases to be valid. Please define the criterion used for the separation (for example, the size of Lambda_QCD / (m_b (1-x))) and indicate the corresponding boundary in Fig. 2.
minor comments (6)
  1. [Section II.B, Eq. (7b)] The argument of Z_tail^(0) is written as (x, ω, μ), but the tail coefficient has no ω dependence; the notation should be corrected to Z_tail^(0)(x, μ).
  2. [Section III.D, first paragraph] The sentence "We now proceed to the matching in the peak region, characterized by momentum fractions x ~ 0" should read "x ~ 1" (or "x ~ 1 - λ"), since the immediately preceding paragraph discusses the tail region with x ~ 0.
  3. [Eqs. (15a) and (A11)] The logarithm in the local vertex result is ambiguous as printed; it should be written with explicit parentheses, e.g., ln( m_b^3 / ( (-2 v·k)^2 μ ) ).
  4. [Introduction] There is a typographical error: "Ferimi motion" should be "Fermi motion".
  5. [Section II.B] "Identifying momentums" should be "Identifying momenta".
  6. [Section III.C] The statement that the tail of S_HQET can be determined perturbatively is not used anywhere in the paper; if the tail branch is intended as a pure QCD OPE, this sentence is misleading and should be removed or clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the one-loop matching coefficient is computed from free-quark matrix elements, and the numerical HQET model is used only as an illustration.

full rationale

The central matching coefficient Z_peak^(1) in Eq. (22) is obtained by inserting the explicitly computed one-loop free-quark matrix elements, Eqs. (13a)-(18), into Eq. (11) and solving for the coefficient after cancellation of the v.k regulators and the plus-distribution terms. The resulting coefficient depends only on mu and m_b; it is not fitted to the HQET model, to lattice data, or to any external observable. The numerical section uses the model of Eq. (25) with A and b taken from Ref. [20] purely to illustrate Eq. (23), and the model parameters enter no step of the derivation. The modified plus distribution from Ref. [35] is a bookkeeping regulator choice, stated to be equivalent to earlier star and mu distributions in Refs. [20,23], so its self-citation is not load-bearing. The tail function in Eq. (19) is assembled directly from the QCD one-loop tail terms; while the paper leaves the transition region and the subtraction of the perturbative HQET tail for future work, that is a completeness and correctness caveat rather than a circular reduction. The paper asserts rather than displays the intermediate algebra connecting Eqs. (13a)-(18) to Eq. (22), but an omitted derivation is a presentation gap, not a circularity: no target quantity is used as its own input, and no equation reduces by construction to a fitted value or to a self-citation.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The factorization claim rests on standard QCD/HQET methodology plus two domain assumptions: infrared equivalence of the two shape functions and leading-power region separation. The only numbers fitted to data enter through the illustrative model of SHQET, not through the matching derivation.

free parameters (2)
  • A (model scale in HQET shape function model) = 0.685 GeV
    Parameter of the phenomenological model in Eq. (25), taken from Bosch et al. [20] and used only for the illustrative QCD shape function in Sec. IV. It is not used in the derivation of the factorization formula.
  • b (model exponent in HQET shape function model) = 2.93
    Parameter of the model in Eq. (25), taken from [20] and used in the numerical illustration in Sec. IV. It does not enter the matching calculation.
assumptions (5)
  • domain assumption The B-meson state can be replaced by a free b qbar state in the matching calculation because the matching function is insensitive to long-distance physics.
    Invoked in Sec. II.B to compute SQCD and SHQET in perturbation theory.
  • domain assumption The QCD and HQET shape functions share identical infrared behavior in the peak region.
    Stated in Sec. II.B as the basis for the factorization formula Eq. (5) and for the cancellation of the v.k regulator in the one-loop matching.
  • ad hoc to paper The region-separated factorization in Eq. (5) holds at leading power in Lambda_QCD/m_b, with peak region x ~ 1 - Lambda_QCD/m_b and tail region x ~ 0.
    Eq. (5) is postulated without a systematic proof, and the transition region between the two regimes is acknowledged as uncontrolled in Sec. IV.
  • domain assumption The difference m_B - m_b is a power correction and can be neglected in the tail-region expressions.
    Used in Appendix A.1 to simplify tail-region amplitudes by setting m_B = m_b, stated as beyond leading-power accuracy.
  • standard math Standard dimensional regularization and MS-bar renormalization, together with the modified plus distribution of Ref. [35], are valid regulators for the calculation.
    Used throughout Sec. III and Appendix A for the one-loop integrals and distribution manipulations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Factorization Formula Connecting the Shape Functions of Heavy Meson in QCD and Heavy Quark Effective Theory." pith.science (2026). https://pith.science/paper/YPXQR4UV

@misc{pith2026250418018,
  author       = {Pith},
  title        = {Pith review of: Factorization Formula Connecting the Shape Functions of Heavy Meson in QCD and Heavy Quark Effective Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPXQR4UV}},
  note         = {Machine review of arXiv:2504.18018}
}
abstract

The shape function of $B$-meson defined in heavy quark effective theory (HQET) plays a crucial role in the analysis of inclusive $B$ decays, and constitutes one of the dominant uncertainties in the determination of CKM matrix element $|V_{ub}|$. On the other hand, the conventional heavy meson shape function defined in QCD is also phenomenologically important and includes shortdistance physics at energy scales of the heavy quark mass. In this work, we derived a factorization formula relating these two kinds of shape functions, which can be invoked to fully disentangle the effects from disparate scales $m_b$ and $\Lambda_{\textrm{QCD}}$, particularly to facilitate the resummation of logarithms $\ln m_b/\Lambda_{\textrm{QCD}}$. In addition, this factorization constitutes an essential component of the recently developed two-step factorization scheme, enabling lattice QCD calculations of lightcone quantities of heavy meson. The results presented here pave the way for first-principles nonperturbative predictions of shape function in the near future.

Figures

Figures reproduced from arXiv: 2504.18018 by the authors.

Figure 1
Figure 1. FIG. 1: Feynman diagrams for the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Feynman diagrams for the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Feynman diagrams for the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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. Photon-Tagged Energy Flow in Inclusive Endpoint $B$ Decays

    hep-ph 2026-07 accept novelty 7.0 of 10

    A new photon-tagged angular energy correlator in inclusive B→X_sγ factorizes at leading power into the standard shape function and a newly computed measured jet function, enabling angular tests of endpoint factorization.

  2. Ill-Posedness in Limited Discrete Fourier Inversion and Regularization for Quasi Distributions in LaMET

    hep-lat 2025-06 conditional novelty 5.0 of 10

    Tikhonov regularization with L-curve parameter choice stabilizes the ill-conditioned limited inverse Fourier transform in LaMET and recovers pion quasi distribution amplitudes consistent with physics-driven lambda-ext...

Reference graph

Works this paper leans on

58 extracted references · 6 canonical work pages · cited by 2 Pith papers

  1. [1]

    3: Feynman diagrams for the B-meson shape function defined in QCD

    Shape function SQCD(x, µ ) defined in QCD FIG. 3: Feynman diagrams for the B-meson shape function defined in QCD. The relevant momentum notio ns are labeled. We begin with the heavy-quark sail graph, the relevant momentum n otions are indicated in Fig. 3(a). The corre- sponding amplitude is: SQCD(a)(x, µ) = 2π αsCF ∫ ddq (2π)d ¯u(pb) −i q+ − iǫ (igstanµ + ˜...

    work page

  2. [2]

    4: Feynman diagrams for the B-meson shape function defined in HQET

    Shape function SHQET(ω, µ ) defined in HQET FIG. 4: Feynman diagrams for the B-meson shape function defined in HQET. The relevant momentum not ions are labeled. We begin with the heavy-quark sail graph, where the relevant mome ntum labels are specified in Fig. 4(d). The corresponding amplitude reads: SHQET(d)(ω, µ) = 2π αsCF ∫ ddq (2π)d i −q+ + iǫ (igstanµ +...

    work page

  3. [3]

    Navas et al

    S. Navas et al. [Particle Data Group], Phys. Rev. D 110, no.3, 030001 (2024) doi:10.1103/PhysRevD.110.030001

    doi:10.1103/physrevd.110.030001 2024

  4. [4]

    Y. S. Amhis et al. [HFLA V], Phys. Rev. D 107, no.5, 052008 (2023) doi:10.1103/PhysRevD.107.052008 [arXiv:2206.07501 [hep-ex]]

    arXiv 2023

  5. [5]

    New results on $|V_{ub}|$ using inclusive and exclusive $B$ decays from the Belle experiment

    L. Cao [Belle], PoS EPS-HEP2023, 356 (2024) doi:10.22323/1.449.0356 [arXiv:2311.00379 [hep-ex]]

    work page Pith review arXiv 2024

  6. [6]

    A. G. Grozin and M. Neubert, Phys. Rev. D 55, 272-290 (1997) doi:10.1103/PhysRevD.55.272 [arXiv:hep-ph/9607366 [hep-ph]]

    arXiv 1997

  7. [7]

    Neubert, Phys

    M. Neubert, Phys. Rev. D 49, 3392-3398 (1994) doi:10.1103/PhysRevD.49.3392 [arXiv:hep-ph/9311325 [hep-ph]]

    arXiv 1994

  8. [8]

    V. M. Braun, D. Y. Ivanov and G. P. Ko- rchemsky, Phys. Rev. D 69, 034014 (2004) doi:10.1103/PhysRevD.69.034014 [arXiv:hep-ph/0309330 [hep-ph]]

    arXiv 2004

Show all 58 references
  1. [9]

    B. O. Lange and M. Neubert, Phys. Rev. Lett. 91, 102001 (2003) doi:10.1103/PhysRevLett.91.102001 [arXiv:hep-ph/0303082 [hep-ph]]

    2003 arXiv

  2. [10]

    Kawamura, J

    H. Kawamura, J. Kodaira, C. F. Qiao and K. Tanaka, Phys. Lett. B 523, 111 (2001) [erratum: Phys. Lett. B 536, 344-344 (2002)] doi:10.1016/S0370-2693(01)01299-0 [arXiv:hep-ph/0109181 [hep-ph]]

    2001 arXiv

  3. [11]

    J. P. Ma and Q. Wang, JHEP 01, 067 (2006) doi:10.1088/1126-6708/2006/01/067 [arXiv:hep-ph/0510336 [hep-ph]]

    2006 arXiv

  4. [12]

    X. q. Li and Y. d. Yang, Phys. Rev. D 73, 114027 (2006) doi:10.1103/PhysRevD.73.114027 [arXiv:hep-ph/0602224 [hep-ph]]

    2006 arXiv

  5. [13]

    G. Bell, T. Feldmann, Y. M. Wang and M. W. Y. Yip, JHEP 11, 191 (2013) doi:10.1007/JHEP11(2013)191 [arXiv:1308.6114 [hep-ph]]

    2013 arXiv

  6. [14]

    V. M. Braun, Y. Ji and A. N. Manashov, Phys. Rev. D 100, no.1, 014023 (2019) doi:10.3204/PUBDB-2019- 02451 [arXiv:1905.04498 [hep-ph]]

    2019 arXiv

  7. [15]

    Beneke, V

    M. Beneke, V. M. Braun, Y. Ji and Y. B. Wei, JHEP 07, 154 (2018) doi:10.1007/JHEP07(2018)154 [arXiv:1804.04962 [hep-ph]]

    2018 arXiv

  8. [16]

    J. Gao, C. D. L¨ u, Y. L. Shen, Y. M. Wang and Y. B. Wei, Phys. Rev. D 101, no.7, 074035 (2020) doi:10.1103/PhysRevD.101.074035 [arXiv:1907.11092 [hep-ph]]

    2020 arXiv

  9. [17]

    B. Y. Cui, Y. K. Huang, Y. L. Shen, C. Wang and Y. M. Wang, JHEP 03, 140 (2023) doi:10.1007/JHEP03(2023)140 [arXiv:2212.11624 [hep- ph]]

    2023 arXiv

  10. [18]

    J. Gao, U. G. Meißner, Y. L. Shen and D. H. Li, [arXiv:2412.13084 [hep-ph]]

    arXiv

  11. [19]

    J. Chai, S. Cheng, Y. h. Ju, D. C. Yan, C. D. L¨ u and Z. J. Xiao, Chin. Phys. C 46, no.12, 123103 (2022) doi:10.1088/1674-1137/ac88bd [arXiv:2207.04190 [hep- ph]]

    2022 arXiv

  12. [20]

    C. Shi, P. Liu, Y. L. Du and W. Jia, Phys. Rev. D 110, no.9, 094010 (2024) doi:10.1103/PhysRevD.110.094010 [arXiv:2409.05098 [hep-ph]]

    2024 arXiv

  13. [21]

    Neubert, Phys

    M. Neubert, Phys. Rev. D 49, 4623-4633 (1994) doi:10.1103/PhysRevD.49.4623 [arXiv:hep-ph/9312311 [hep-ph]]

    1994 arXiv

  14. [22]

    S. W. Bosch, B. O. Lange, M. Neubert and G. Paz, Nucl. Phys. B 699, 335-386 (2004) doi:10.1016/j.nuclphysb.2004.07.041 [arXiv:hep-ph/0402094 [hep-ph]]

    2004 arXiv

  15. [23]

    Balzereit, T

    C. Balzereit, T. Mannel and W. Kilian, Phys. Rev. D 58, 114029 (1998) doi:10.1103/PhysRevD.58.114029 [arXiv:hep-ph/9805297 [hep-ph]]

    1998 arXiv

  16. [24]

    Ligeti, I

    Z. Ligeti, I. W. Stewart and F. J. Tackmann, Phys. Rev. D 78, 114014 (2008) doi:10.1103/PhysRevD.78.114014 [arXiv:0807.1926 [hep-ph]]

    2008 arXiv

  17. [25]

    C. W. Bauer and A. V. Manohar, Phys. Rev. D 70, 034024 (2004) doi:10.1103/PhysRevD.70.034024 [arXiv:hep-ph/0312109 [hep-ph]]

    2004 arXiv

  18. [26]

    Aglietti and G

    U. Aglietti and G. Ricciardi, Phys. Lett. B 466, 313 (1999) doi:10.1016/S0370-2693(99)01103-X [arXiv:hep-ph/9907501 [hep-ph]]

    1999 arXiv

  19. [27]

    C. W. Bauer, S. Fleming and M. E. Luke, Phys. Rev. D 63, 014006 (2000) doi:10.1103/PhysRevD.63.014006 [arXiv:hep-ph/0005275 [hep-ph]]

    2000 arXiv

  20. [28]

    Neubert, Phys

    M. Neubert, Phys. Lett. B 612, 13- 20 (2005) doi:10.1016/j.physletb.2005.02.055 [arXiv:hep-ph/0412241 [hep-ph]]

    2005 arXiv

  21. [29]

    Le Yaouanc, L

    A. Le Yaouanc, L. Oliver and J. C. Raynal, Phys. Rev. D 77, 034005 (2008) doi:10.1103/PhysRevD.77.034005 [arXiv:0707.3027 [hep-ph]]

    2008 arXiv

  22. [30]

    K. S. M. Lee, Z. Ligeti, I. W. Stewart and F. J. Tackmann, Phys. Rev. D 74, 011501 (2006) doi:10.1103/PhysRevD.74.011501 [arXiv:hep-ph/0512191 [hep-ph]]

    2006 arXiv

  23. [31]

    X. Y. Han, J. Hua, X. Ji, C. D. L¨ u, W. Wang, J. Xu, Q. A. Zhang and S. Zhao, [arXiv:2403.17492 [hep-ph]]

    arXiv

  24. [32]

    X. Y. Han et al. [Lattice Parton], Phys. Rev. D 111, no.3, 034503 (2025) doi:10.1103/PhysRevD.111.034503 [arXiv:2410.18654 [hep-lat]]

    2025 arXiv

  25. [33]

    X. Y. Han, J. Hua, X. Ji, C. D. L¨ u, A. Sch¨ afer, Y. Su, W. Wang, J. Xu, Y. Yang and J. H. Zhang, et al. [arXiv:2504.02453 [hep-lat]]

    arXiv

  26. [34]

    Ji, Phys

    X. Ji, Phys. Rev. Lett. 110, 262002 (2013) doi:10.1103/PhysRevLett.110.262002 [arXiv:1305.1539 [hep-ph]]

    2013 arXiv

  27. [35]

    X. Ji, Sci. China Phys. Mech. Astron. 57, 1407-1412 (2014) doi:10.1007/s11433-014-5492-3 [arXiv:1404.6680 [hep-ph]]

    2014 arXiv

  28. [36]

    Cichy and M

    K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019, 3036904 (2019) doi:10.1155/2019/3036904 [arXiv:1811.07248 [hep-lat]]

    2019 arXiv

  29. [37]

    W. Wang, Y. M. Wang, J. Xu and S. Zhao, Phys. Rev. D 102, no.1, 011502 (2020) doi:10.1103/PhysRevD.102.011502 [arXiv:1908.09933 [hep-ph]]

    2020 arXiv

  30. [38]

    Ishaq, Y

    S. Ishaq, Y. Jia, X. Xiong and D. S. Yang, Phys. Rev. Lett. 125, no.13, 132001 (2020) doi:10.1103/PhysRevLett.125.132001 [arXiv:1905.06930 12 [hep-ph]]

    2020 arXiv

  31. [39]

    Beneke, G

    M. Beneke, G. Finauri, K. K. Vos and Y. Wei, JHEP 09, 066 (2023) doi:10.1007/JHEP09(2023)066 [arXiv:2305.06401 [hep-ph]]

    2023 arXiv

  32. [40]

    Gelb et al

    M. Gelb et al. [Belle], Phys. Rev. D 98, no.11, 112016 (2018) doi:10.1103/PhysRevD.98.112016 [arXiv:1810.12976 [hep-ex]]

    2018 arXiv

  33. [41]

    Izubuchi, X

    T. Izubuchi, X. Ji, L. Jin, I. W. Stewart and Y. Zhao, Phys. Rev. D 98, no.5, 056004 (2018) doi:10.1103/PhysRevD.98.056004 [arXiv:1801.03917 [hep-ph]]

    2018 arXiv

  34. [42]

    S. M. Hu, W. Wang, J. Xu and S. Zhao, Phys. Rev. D 109, no.3, 034001 (2024) doi:10.1103/PhysRevD.109.034001 [arXiv:2308.13977 [hep-ph]]

    2024 arXiv

  35. [43]

    Xiong, X

    X. Xiong, X. Ji, J. H. Zhang and Y. Zhao, Phys. Rev. D 90, no.1, 014051 (2014) doi:10.1103/PhysRevD.90.014051 [arXiv:1310.7471 [hep-ph]]

    2014 arXiv

  36. [44]

    Y. Q. Ma and J. W. Qiu, Phys. Rev. D 98, no.7, 074021 (2018) doi:10.1103/PhysRevD.98.074021 [arXiv:1404.6860 [hep-ph]]

    2018 arXiv

  37. [45]

    Z. Y. Fan, Y. B. Yang, A. Anthony, H. W. Lin and K. F. Liu, Phys. Rev. Lett. 121, no.24, 242001 (2018) doi:10.1103/PhysRevLett.121.242001 [arXiv:1808.02077 [hep-lat]]

    2018 arXiv

  38. [46]

    J. Xu, Q. A. Zhang and S. Zhao, Phys. Rev. D 97, no.11, 114026 (2018) doi:10.1103/PhysRevD.97.114026 [arXiv:1804.01042 [hep-ph]]

    2018 arXiv

  39. [47]

    Y. S. Liu, W. Wang, J. Xu, Q. A. Zhang, J. H. Zhang, S. Zhao and Y. Zhao, Phys. Rev. D 100, no.3, 034006 (2019) doi:10.1103/PhysRevD.100.034006 [arXiv:1902.00307 [hep-ph]]

    2019 arXiv

  40. [48]

    X. Ji, Y. S. Liu, Y. Liu, J. H. Zhang and Y. Zhao, Rev. Mod. Phys. 93, no.3, 035005 (2021) doi:10.1103/RevModPhys.93.035005 [arXiv:2004.03543 [hep-ph]]

    2021 arXiv

  41. [49]

    Xu and X

    J. Xu and X. R. Zhang, Phys. Rev. D 106, no.11, 114019 (2022) doi:10.1103/PhysRevD.106.114019 [arXiv:2209.10719 [hep-ph]]

    2022 arXiv

  42. [50]

    J. Xu, X. R. Zhang and S. Zhao, Phys. Rev. D 106, no.1, L011503 (2022) doi:10.1103/PhysRevD.106.L011503 [arXiv:2202.13648 [hep-ph]]

    2022 arXiv

  43. [51]

    S. M. Hu, J. Xu and S. Zhao, Eur. Phys. J. C 84, no.5, 502 (2024) doi:10.1140/epjc/s10052-024-12672-2 [arXiv:2401.04291 [hep-ph]]

    2024 arXiv

  44. [52]

    Zhao, Phys

    S. Zhao, Phys. Rev. D 101, no.7, 071503 (2020) doi:10.1103/PhysRevD.101.071503 [arXiv:1910.03470 [hep-ph]]

    2020 arXiv

  45. [53]

    Zhao and A

    S. Zhao and A. V. Radyushkin, Phys. Rev. D 103, no.5, 054022 (2021) doi:10.1103/PhysRevD.103.054022 [arXiv:2006.05663 [hep-ph]]

    2021 arXiv

  46. [54]

    B. Hu, X. Jiang, K. F. Liu, P. Sun and Y. B. Yang, [arXiv:2411.18402 [hep-lat]]

    arXiv

  47. [55]

    M. H. Chu et al. [Lattice Parton], Phys. Rev. D 111, no.3, 034510 (2025) doi:10.1103/PhysRevD.111.034510 [arXiv:2411.12554 [hep-lat]]

    2025 arXiv

  48. [56]

    Z. F. Deng, W. Wang, Y. B. Wei and J. Zeng, Phys. Rev. D 110, no.11, 114006 (2024) doi:10.1103/PhysRevD.110.114006 [arXiv:2409.00632 [hep-ph]]

    2024 arXiv

  49. [57]

    C. Han, W. Wang, J. L. Zhang and J. H. Zhang, Phys. Rev. D 110, no.9, 094038 (2024) doi:10.1103/PhysRevD.110.094038 [arXiv:2408.13486 [hep-ph]]

    2024 arXiv

  50. [58]

    W. Wang, J. Xu, Q. A. Zhang and S. Zhao, [arXiv:2411.07101 [hep-ph]]

    arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.