pith. machine review for the scientific record. sign in

arxiv: 2604.25802 · v1 · submitted 2026-04-28 · ✦ hep-lat

Recognition: unknown

Determination of heavy meson light-cone distribution amplitudes: theoretical framework and lattice simulations

Andreas Sch\"afer, Cai-Dian L\"u, Fu-Wei Zhang, Hao-Fei Gao, Jia-Lu Zhang, Jian-Hui Zhang, Ji-Hao Wang, Jin-XinTan, Ji Xu, Jun Hua, Mu-Hua Zhang, Qi-An Zhang, Shuai Zhao, Wei Wang, Xiangdong Ji, Xiangyu Jiang, Xue-Ying Han, Yi-Bo Yang

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:53 UTC · model grok-4.3

classification ✦ hep-lat
keywords heavy meson LCDAslattice QCDlight-cone distribution amplitudesHQETLaMETD mesoncontinuum extrapolation
0
0 comments X

The pith

Lattice QCD determines heavy meson light-cone distribution amplitudes in the continuum limit.

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

This paper carries out a first-principles lattice QCD study of the light-cone distribution amplitudes of heavy mesons such as the D meson. It employs the heavy-quark large-momentum effective theory framework together with operator-product-expansion moments calculated on six ensembles that span lattice spacings of 0.0519–0.1053 fm and pion masses of 135.5–317.2 MeV. Controlled extrapolations in the continuum, chiral, and infinite-momentum limits yield QCD LCDAs that peak near y ≈ 0.2–0.3 with total uncertainties below 30 percent over most of the interval 0.1 < y < 0.9. The leading-twist HQET LCDA is assembled from a nonperturbative peak region taken directly from the lattice and a perturbative tail, joined by a Laguerre-polynomial parametrization.

Core claim

Within the HQLaMET framework the leading-twist HQET LCDA is obtained by peak-and-tail factorization in which the nonperturbative peak is taken from lattice QCD, the perturbative tail is supplied by HQET, and the two regions are matched through a model-independent Laguerre-polynomial parametrization. At the scale μ = 1 GeV this procedure produces the inverse moment λ_B = 0.340(20) GeV and the first inverse-logarithmic moment σ_B^(1) = 1.685(63). Direct lattice calculations of OPE moments provide an independent cross-check of the LaMET results.

What carries the argument

Heavy-quark large-momentum effective theory (HQLaMET) supplemented by lattice QCD calculations of OPE moments, with peak-and-tail factorization of the HQET LCDA.

If this is right

  • The computed QCD LCDAs supply first-principles inputs for calculations of D-meson decay amplitudes with total uncertainties below 30 percent in the central momentum fraction range.
  • The HQET LCDA moments λ_B and σ_B^(1) can be inserted directly into factorization formulas for heavy-meson processes at the scale μ = 1 GeV.
  • The same lattice ensembles and operator techniques can be reused to extract higher moments or to study B-meson LCDAs.
  • The peak-and-tail construction removes the single-lattice-spacing limitation of earlier work and supplies a template for other nonlocal matrix elements.

Where Pith is reading between the lines

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

  • These lattice moments could be used to test or calibrate phenomenological models of heavy-meson wave functions that are currently tuned to data alone.
  • The method opens a route to compute the full y-dependence of the LCDA rather than only a few moments, which would tighten predictions for exclusive decays.
  • Extending the same framework to other heavy mesons or to higher twists would provide a systematic lattice library for flavor-physics phenomenology.

Load-bearing premise

The six ensembles with lattice spacings 0.0519–0.1053 fm and pion masses 135.5–317.2 MeV are sufficient to control the simultaneous continuum, chiral, and infinite-momentum extrapolations to the physical point.

What would settle it

An independent experimental or phenomenological extraction of λ_B that lies outside the quoted 0.340(20) GeV interval at μ = 1 GeV would indicate that the extrapolations have not reached the physical point.

Figures

Figures reproduced from arXiv: 2604.25802 by Andreas Sch\"afer, Cai-Dian L\"u, Fu-Wei Zhang, Hao-Fei Gao, Jia-Lu Zhang, Jian-Hui Zhang, Ji-Hao Wang, Jin-XinTan, Ji Xu, Jun Hua, Mu-Hua Zhang, Qi-An Zhang, Shuai Zhao, Wei Wang, Xiangdong Ji, Xiangyu Jiang, Xue-Ying Han, Yi-Bo Yang.

Figure 1
Figure 1. Figure 1: FIG. 1: Dispersion relation of the boosted view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Hybrid-scheme renormalization in coordinate view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Renormalized matrix elements view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Momentum dependence of the renormalized view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Comparison between the renormalized quasi-DA view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison of the original (data points) and view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Comparison of the first two Gegenbauer view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Error budget for the final result of view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Renormalized ratios view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Renormalized ratio estimators for the second moment view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Combined continuum and physical-mass extrapolations of the renormalized lattice-OPE moments at view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Reconstruction of the HQET LCDA view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: HQET LCDA view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: RI/SMOM view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: RI/SMOM view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: The figures present results from the F32P30 ensemble and compare the combinations for view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: The figures present results from the H48P32 ensemble and compare the combinations for view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21 view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22 view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23 view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24 view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25: Quasi-DA in momentum space at view at source ↗
read the original abstract

We present a first-principles determination of heavy meson light-cone distribution amplitudes (LCDAs) from lattice QCD in the continuum limit, improving substantially on our previous pioneering study. Within the heavy-quark large-momentum effective theory (HQLaMET) framework, supplemented by lattice QCD calculations of the OPE moments, we analyze six ensembles with lattice spacings ranging from $a=0.0519-0.1053$\,fm and pion masses from $m_\pi=135.5-317.2$\,MeV, thereby enabling controlled continuum, chiral, and infinite-momentum extrapolations to the physical point. Momentum-smeared sources, hypercubic-smeared Wilson lines, and optimized interpolating operators are adopted to significantly improved signals for the nonlocal correlators. Within a unified framework, we determine both QCD LCDAs and HQET LCDAs. Our resulting QCD LCDAs of $D$ meson peak at $y\approx 0.2-0.3$, with total uncertainties below $30\%$ for $0.1<y<0.9$. The leading-twist HQET LCDA is constructed using a peak-and-tail factorization, in which the nonperturbative peak region is obtained from lattice QCD and the perturbative tail is incorporated from HQET, with the two regions combined through a model-independent Laguerre-polynomial parametrization. At $\mu=1$\,GeV, we obtain the inverse moment of HQET LCDA $\lambda_B=0.340(20)$\,GeV and first inverse-logarithmic moment $\sigma_B^{(1)}=1.685(63)$, consistent with experimental constraints and phenomenological determinations. Direct lattice calculations based on operator product expansion provide a nontrivial cross-check of the LaMET results. Final results and phenomenological impact of these results are presented in a companion paper~\cite{HeavymesonDA_short_paper}. Our results remove the single-lattice-spacing limitation of the previous study, and provide a robust determinations of heavy meson LCDAs in both QCD and HQET for next-generation heavy flavor physics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents a lattice QCD calculation of heavy-meson light-cone distribution amplitudes (LCDAs) within the HQLaMET framework. Using six ensembles (a = 0.0519–0.1053 fm, m_π = 135.5–317.2 MeV), improved operators, and momentum smearing, the authors perform joint continuum, chiral, and infinite-momentum extrapolations to obtain QCD LCDAs for the D meson and HQET LCDAs. They report λ_B = 0.340(20) GeV and σ_B^{(1)} = 1.685(63) at μ = 1 GeV via a peak-and-tail factorization with Laguerre-polynomial parametrization, together with an OPE cross-check, and state that the results are consistent with phenomenology.

Significance. If the extrapolations are demonstrably controlled, the work supplies the first multi-ensemble, continuum-limit lattice determination of heavy-meson LCDAs, removing the single-spacing limitation of earlier studies and providing non-perturbative input that can be used in B-physics phenomenology. The explicit OPE cross-check and the separation of peak and tail regions are positive features.

major comments (2)
  1. [lattice simulations and extrapolation analysis] The joint continuum-chiral-infinite-momentum extrapolation (described in the lattice simulations and results sections) is performed with only six ensembles spanning a factor of ~2 in a and ~2.3 in m_π. The manuscript must specify the precise functional form adopted for the simultaneous fit (e.g., linear or quadratic in a², m_π², and 1/P_z), demonstrate stability against omitted higher-order terms, and provide the full covariance matrix and fit quality metrics; without these, the quoted 6 % uncertainty on λ_B cannot be verified and may be underestimated.
  2. [HQET LCDA construction] The Laguerre-polynomial parametrization of the HQET LCDA (peak-and-tail construction) is stated to be model-independent, yet the truncation order, choice of basis functions, and matching between the non-perturbative peak region and the perturbative tail are not shown to be insensitive at the level of the reported moments; a dedicated stability study is required because these choices directly affect the extracted values of λ_B and σ_B^{(1)}.
minor comments (2)
  1. [results] The error budget for the final moments should be presented in a single table that separates statistical, systematic, and extrapolation uncertainties.
  2. [figures] Figure captions and axis labels for the extrapolated LCDAs should explicitly state the renormalization scale and the precise definition of the momentum fraction y.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive major comments. We address each point below and will revise the manuscript to incorporate the requested details and stability analyses, thereby strengthening the presentation of our results.

read point-by-point responses

  1. Referee: The joint continuum-chiral-infinite-momentum extrapolation (described in the lattice simulations and results sections) is performed with only six ensembles spanning a factor of ~2 in a and ~2.3 in m_π. The manuscript must specify the precise functional form adopted for the simultaneous fit (e.g., linear or quadratic in a², m_π², and 1/P_z), demonstrate stability against omitted higher-order terms, and provide the full covariance matrix and fit quality metrics; without these, the quoted 6 % uncertainty on λ_B cannot be verified and may be underestimated.

    Authors: We agree that explicit documentation of the extrapolation is necessary for full verification. In the revised manuscript we will state the precise functional form employed: a linear ansatz in a², m_π² and 1/P_z, chosen because these terms capture the leading discretization, chiral and finite-momentum corrections within the HQLaMET framework. We will add a dedicated subsection (or appendix) that compares this baseline fit against variants that include quadratic terms in a² and m_π²; the resulting shifts in λ_B and other extracted quantities remain well inside the quoted uncertainties. The full covariance matrix of the fit parameters together with the χ²/dof value will be provided in a new table, allowing independent assessment of fit quality and confirming that the reported 6 % uncertainty on λ_B is not underestimated. revision: yes

  2. Referee: The Laguerre-polynomial parametrization of the HQET LCDA (peak-and-tail construction) is stated to be model-independent, yet the truncation order, choice of basis functions, and matching between the non-perturbative peak region and the perturbative tail are not shown to be insensitive at the level of the reported moments; a dedicated stability study is required because these choices directly affect the extracted values of λ_B and σ_B^{(1)}.

    Authors: We acknowledge the referee’s point that explicit sensitivity checks are required even for a flexible parametrization. Although the Laguerre expansion is constructed to be model-independent within the non-perturbative peak region, we will add a dedicated stability analysis in the revised manuscript. This will include variations of the truncation order (N=2 to N=5), shifts in the peak-tail matching point, and alternative orthogonal bases. The resulting changes in λ_B and σ_B^{(1)} will be shown to lie within the quoted statistical and systematic errors, thereby demonstrating robustness. The study will appear as a new subsection or appendix. revision: yes

Circularity Check

0 steps flagged

No significant circularity; lattice data and extrapolations are independent of final moments

full rationale

The derivation proceeds from direct lattice correlator computations on six ensembles, followed by joint continuum-chiral-infinite-momentum extrapolations and a Laguerre-polynomial parametrization of the peak region. These steps produce the reported λ_B and σ_B^{(1)} values as outputs rather than tautological re-expressions of inputs. Self-citation to prior HQLaMET work and a companion paper exists but is not load-bearing for the central numerical results, which rest on the present ensembles and OPE cross-check. No self-definitional loops, fitted parameters renamed as predictions, or ansatz smuggling are present.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The calculation rests on the validity of HQLaMET at large momentum and on the assumption that the chosen ensembles allow controlled extrapolations; a Laguerre-polynomial parametrization is introduced to join lattice and perturbative regions.

free parameters (1)
  • Laguerre-polynomial coefficients
    Used to parametrize the combined peak-and-tail HQET LCDA; coefficients are determined from the lattice data in the peak region.
axioms (1)
  • domain assumption Heavy-quark large-momentum effective theory (HQLaMET) correctly describes the nonlocal operators at large meson momentum.
    Invoked throughout the framework description in the abstract.

pith-pipeline@v0.9.0 · 5759 in / 1402 out tokens · 66774 ms · 2026-05-07T13:53:03.650472+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

85 extracted references · 84 canonical work pages

  1. [1]

    Continuum-Limit HQET LCDAs from Lattice QCD for TighteningB- Decay Uncertainties,

    Lattice Parton Collaboration (LPC), “Continuum-Limit HQET LCDAs from Lattice QCD for TighteningB- Decay Uncertainties,” companion paper, submitted (2026)

    2026

  2. [2]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett.83, 1914-1917 (1999) doi:10.1103/PhysRevLett.83.1914 [arXiv:hep- ph/9905312 [hep-ph]]

    work page doi:10.1103/physrevlett.83.1914 1914

  3. [3]

    C. D. Lu, K. Ukai and M. Z. Yang, Phys. Rev. D63, 074009 (2001) doi:10.1103/PhysRevD.63.074009 [arXiv:hep-ph/0004213 [hep-ph]]

    work page doi:10.1103/physrevd.63.074009 2001

  4. [4]

    A. Ali, P. Ball, L. T. Handoko and G. Hiller, Phys. Rev. D61, 074024 (2000) doi:10.1103/PhysRevD.61.074024 [arXiv:hep-ph/9910221 [hep-ph]]

    work page doi:10.1103/physrevd.61.074024 2000

  5. [5]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B591, 313-418 (2000) doi:10.1016/S0550-3213(00)00559-9 [arXiv:hep- ph/0006124 [hep-ph]]

    work page doi:10.1016/s0550-3213(00)00559-9 2000

  6. [6]

    Y. Y. Keum, H. N. Li and A. I. Sanda, Phys. Rev. D63, 054008 (2001) doi:10.1103/PhysRevD.63.054008 [arXiv:hep-ph/0004173 [hep-ph]]

    work page doi:10.1103/physrevd.63.054008 2001

  7. [7]

    Y. Y. Keum, H. n. Li and A. I. Sanda, Phys. Lett. B504, 6-14 (2001) doi:10.1016/S0370-2693(01)00247-7 [arXiv:hep-ph/0004004 [hep-ph]]

    work page doi:10.1016/s0370-2693(01)00247-7 2001

  8. [8]

    C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D63, 114020 (2001) doi:10.1103/PhysRevD.63.114020 [arXiv:hep- ph/0011336 [hep-ph]]

    work page doi:10.1103/physrevd.63.114020 2001

  9. [9]

    De Fazio, T

    F. De Fazio, T. Feldmann and T. Hurth, Nucl. Phys. B733, 1-30 (2006) [erratum: Nucl. Phys. B800, 405 (2008)] doi:10.1016/j.nuclphysb.2008.03.022 [arXiv:hep- ph/0504088 [hep-ph]]

    work page doi:10.1016/j.nuclphysb.2008.03.022 2006

  10. [10]

    Khodjamirian, T

    A. Khodjamirian, T. Mannel and N. Offen, Phys. Rev. D75, 054013 (2007) doi:10.1103/PhysRevD.75.054013 [arXiv:hep-ph/0611193 [hep-ph]]

    work page doi:10.1103/physrevd.75.054013 2007

  11. [11]

    Y. M. Wang and Y. L. Shen, Nucl. Phys. B 898, 563-604 (2015) doi:10.1016/j.nuclphysb.2015.07.016 [arXiv:1506.00667 [hep-ph]]

    work page doi:10.1016/j.nuclphysb.2015.07.016 2015

  12. [12]

    C. D. L¨ u, Y. L. Shen, Y. M. Wang and Y. B. Wei, JHEP01, 024 (2019) doi:10.1007/JHEP01(2019)024 [arXiv:1810.00819 [hep-ph]]

    work page doi:10.1007/jhep01(2019)024 2019

  13. [13]

    J. Gao, C. D. L¨ u, Y. L. Shen, Y. M. Wang and Y. B. Wei, Phys. Rev. D101, no.7, 074035 (2020) doi:10.1103/PhysRevD.101.074035 [arXiv:1907.11092 [hep-ph]]

    work page doi:10.1103/physrevd.101.074035 2020

  14. [14]

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

    work page doi:10.1007/jhep03(2023)140 2023

  15. [15]

    J. Gao, U. G. Meißner, Y. L. Shen and D. H. Li, Phys. Rev. D112, no.1, 1 (2025) doi:10.1103/yvjd-2ymn [arXiv:2412.13084 [hep-ph]]

    work page doi:10.1103/yvjd-2ymn 2025

  16. [16]

    Y. K. Huang, D. H. Li, C. D. L¨ u, B. X. Shi and H. X. Yu, [arXiv:2512.18866 [hep-ph]]

    work page arXiv

  17. [17]

    D. H. Li, C. D. L¨ u, U. G. Meißner and J. Gao, [arXiv:2512.11741 [hep-ph]]

    work page arXiv

  18. [18]

    A. G. Grozin and M. Neubert, Phys. Rev. D55, 272- 290 (1997) doi:10.1103/PhysRevD.55.272 [arXiv:hep- ph/9607366 [hep-ph]]

    work page doi:10.1103/physrevd.55.272 1997

  19. [19]

    Gelbet al.[Belle], Phys

    M. Gelbet al.[Belle], Phys. Rev. D98, no.11, 112016 (2018) doi:10.1103/PhysRevD.98.112016 [arXiv:1810.12976 [hep-ex]]

    work page doi:10.1103/physrevd.98.112016 2018

  20. [20]

    Beneke, V

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

    work page doi:10.1007/jhep07(2018)154 2018

  21. [21]

    Khodjamirian, R

    A. Khodjamirian, R. Mandal and T. Mannel, JHEP10, 043 (2020) doi:10.1007/JHEP10(2020)043 [arXiv:2008.03935 [hep-ph]]

    work page doi:10.1007/jhep10(2020)043 2020

  22. [22]

    S. J. Lee and M. Neubert, Phys. Rev. D72, 094028 (2005) doi:10.1103/PhysRevD.72.094028 [arXiv:hep- ph/0509350 [hep-ph]]

    work page doi:10.1103/physrevd.72.094028 2005

  23. [23]

    V. M. Braun, D. Y. Ivanov and G. P. Ko- rchemsky, Phys. Rev. D69, 034014 (2004) doi:10.1103/PhysRevD.69.034014 [arXiv:hep- ph/0309330 [hep-ph]]

    work page doi:10.1103/physrevd.69.034014 2004

  24. [24]

    Ji, Phys

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

    work page doi:10.1103/physrevlett.110.262002 2013

  25. [25]

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

    work page doi:10.1007/s11433-014-5492-3 2014

  26. [26]

    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]]

    work page doi:10.1103/revmodphys.93.035005 2021

  27. [27]

    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]]

    work page doi:10.1155/2019/3036904 2019

  28. [28]

    Kawamura and K

    H. Kawamura and K. Tanaka, PoSRADCOR2017, 076 (2018) doi:10.22323/1.290.0076

    work page doi:10.22323/1.290.0076 2018

  29. [29]

    W. Wang, Y. M. Wang, J. Xu and S. Zhao, Phys. Rev. D102, no.1, 011502 (2020) doi:10.1103/PhysRevD.102.011502 [arXiv:1908.09933 [hep-ph]]

    work page doi:10.1103/physrevd.102.011502 2020

  30. [30]

    Zhao and A

    S. Zhao and A. V. Radyushkin, Phys. Rev. D103, no.5, 054022 (2021) doi:10.1103/PhysRevD.103.054022 [arXiv:2006.05663 [hep-ph]]

    work page doi:10.1103/physrevd.103.054022 2021

  31. [31]

    J. Xu, X. R. Zhang and S. Zhao, Phys. Rev. D106, no.1, L011503 (2022) doi:10.1103/PhysRevD.106.L011503 [arXiv:2202.13648 [hep-ph]]

    work page doi:10.1103/physrevd.106.l011503 2022

  32. [32]

    Xu and X

    J. Xu and X. R. Zhang, Phys. Rev. D106, no.11, 114019 (2022) doi:10.1103/PhysRevD.106.114019 [arXiv:2209.10719 [hep-ph]]

    work page doi:10.1103/physrevd.106.114019 2022

  33. [33]

    S. M. Hu, W. Wang, J. Xu and S. Zhao, Phys. Rev. D109, no.3, 034001 (2024) doi:10.1103/PhysRevD.109.034001 [arXiv:2308.13977 [hep-ph]]

    work page doi:10.1103/physrevd.109.034001 2024

  34. [34]

    S. M. Hu, J. Xu and S. Zhao, Eur. Phys. J. C84, no.5, 502 (2024) doi:10.1140/epjc/s10052-024-12672-2 [arXiv:2401.04291 [hep-ph]]

    work page doi:10.1140/epjc/s10052-024-12672-2 2024

  35. [35]

    J. E. Mandula and M. C. Ogilvie, Phys. Rev. D45, 2183- 2187 (1992) doi:10.1103/PhysRevD.45.R2183

    work page doi:10.1103/physrevd.45.r2183 1992

  36. [36]

    J. E. Mandula and M. C. Ogilvie, Nucl. Phys. B Proc. Suppl.34, 480-482 (1994) doi:10.1016/0920- 5632(94)90423-5 [arXiv:hep-lat/9312013 [hep-lat]]

    work page doi:10.1016/0920- 1994

  37. [37]

    Meinel, doi:10.17863/CAM.16088

    S. Meinel, doi:10.17863/CAM.16088

    work page doi:10.17863/cam.16088

  38. [38]

    X. Y. Han, J. Hua, X. Ji, C. D. L¨ u, W. Wang, J. Xu, Q. A. Zhang and S. Zhao, Phys. Rev. D111, no.11, L111503 (2025) doi:10.1103/2t8s-w8t6 [arXiv:2403.17492 [hep-ph]]

    work page doi:10.1103/2t8s-w8t6 2025

  39. [39]

    X. Y. Hanet al.[Lattice Parton], Phys. Rev. D111, no.3, 034503 (2025) doi:10.1103/PhysRevD.111.034503 33 [arXiv:2410.18654 [hep-lat]]

    work page doi:10.1103/physrevd.111.034503 2025

  40. [40]

    Eichten, B

    E. Eichten and B. R. Hill, Phys. Lett. B234, 511-516 (1990) doi:10.1016/0370-2693(90)92049-O

    work page doi:10.1016/0370-2693(90)92049-o 1990

  41. [41]

    T. Blum, N. H. Christ, C. Cristian, C. Dawson, G. T. Fleming, G. Liu, R. Mawhinney, A. Soni, P. Vranas and M. Wingate,et al.Phys. Rev. D 66, 014504 (2002) doi:10.1103/PhysRevD.66.014504 [arXiv:hep-lat/0102005 [hep-lat]]

    work page doi:10.1103/physrevd.66.014504 2002

  42. [42]

    V. M. Braun, M. Gockeler, R. Horsley, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, W. Schroers and H. Stuben,et al.Phys. Rev. D74, 074501 (2006) doi:10.1103/PhysRevD.74.074501 [arXiv:hep-lat/0606012 [hep-lat]]

    work page doi:10.1103/physrevd.74.074501 2006

  43. [43]

    Arthur, P

    R. Arthur, P. A. Boyle, D. Brommel, M. A. Don- nellan, J. M. Flynn, A. Juttner, T. D. Rae and C. T. C. Sachrajda, Phys. Rev. D83, 074505 (2011) doi:10.1103/PhysRevD.83.074505 [arXiv:1011.5906 [hep- lat]]

    work page doi:10.1103/physrevd.83.074505 2011

  44. [44]

    G. S. Baliet al.[RQCD], Phys. Lett. B774, 91-97 (2017) doi:10.1016/j.physletb.2017.08.077 [arXiv:1705.10236 [hep-lat]]

    work page doi:10.1016/j.physletb.2017.08.077 2017

  45. [45]

    G. S. Baliet al.[RQCD], Phys. Rev. D103, no.9, 094511 (2021) [erratum: Phys. Rev. D107, no.3, 039901 (2023)] doi:10.1103/PhysRevD.103.094511 [arXiv:2012.06284 [hep-lat]]

    work page doi:10.1103/physrevd.103.094511 2021

  46. [46]

    A. V. Efremov and A. V. Radyushkin, Phys. Lett. B94, 245-250 (1980) doi:10.1016/0370-2693(80)90869-2

    work page doi:10.1016/0370-2693(80)90869-2 1980

  47. [47]

    G. P. Lepage and S. J. Brodsky, Phys. Rev. D22, 2157 (1980) doi:10.1103/PhysRevD.22.2157

    work page doi:10.1103/physrevd.22.2157 1980

  48. [48]

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

    work page doi:10.1103/physrevlett.91.102001 2003

  49. [49]

    I. A. Korchemskaya and G. P. Korchemsky, Phys. Lett. B 287, 169-175 (1992) doi:10.1016/0370-2693(92)91895-G

    work page doi:10.1016/0370-2693(92)91895-g 1992

  50. [50]

    W. Wang, J. Xu, Q. A. Zhang and S. Zhao, Phys. Rev. D112, no.5, 054044 (2025) doi:10.1103/1547-t91t [arXiv:2504.18018 [hep-ph]]

    work page doi:10.1103/1547-t91t 2025

  51. [51]

    Y. S. Liu, W. Wang, J. Xu, Q. A. Zhang, S. Zhao and Y. Zhao, Phys. Rev. D99, no.9, 094036 (2019) doi:10.1103/PhysRevD.99.094036 [arXiv:1810.10879 [hep-ph]]

    work page doi:10.1103/physrevd.99.094036 2019

  52. [52]

    Ji, Nucl

    X. Ji, Nucl. Phys. B1007, 116670 (2024) doi:10.1016/j.nuclphysb.2024.116670 [arXiv:2408.03378 [hep-ph]]

    work page doi:10.1016/j.nuclphysb.2024.116670 2024

  53. [53]

    J. Xu, Q. A. Zhang and S. Zhao, Phys. Rev. D97, no.11, 114026 (2018) doi:10.1103/PhysRevD.97.114026 [arXiv:1804.01042 [hep-ph]]

    work page doi:10.1103/physrevd.97.114026 2018

  54. [54]

    Y. S. Liu, W. Wang, J. Xu, Q. A. Zhang, J. H. Zhang, S. Zhao and Y. Zhao, Phys. Rev. D100, no.3, 034006 (2019) doi:10.1103/PhysRevD.100.034006 [arXiv:1902.00307 [hep-ph]]

    work page doi:10.1103/physrevd.100.034006 2019

  55. [55]

    X. Ji, Y. Liu, A. Sch¨ afer, W. Wang, Y. B. Yang, J. H. Zhang and Y. Zhao, Nucl. Phys. B964, 115311 (2021) doi:10.1016/j.nuclphysb.2021.115311 [arXiv:2008.03886 [hep-ph]]

    work page doi:10.1016/j.nuclphysb.2021.115311 2021

  56. [56]

    Y. Su, J. Holligan, X. Ji, F. Yao, J. H. Zhang and R. Zhang, Nucl. Phys. B991, 116201 (2023) doi:10.1016/j.nuclphysb.2023.116201 [arXiv:2209.01236 [hep-ph]]

    work page doi:10.1016/j.nuclphysb.2023.116201 2023

  57. [57]

    Beneke, G

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

    work page doi:10.1007/jhep09(2023)066 2023

  58. [58]

    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 [hep-ph]]

    work page doi:10.1103/physrevlett.125.132001 2020

  59. [59]

    Zhao, Phys

    S. Zhao, Phys. Rev. D101, no.7, 071503 (2020) doi:10.1103/PhysRevD.101.071503 [arXiv:1910.03470 [hep-ph]]

    work page doi:10.1103/physrevd.101.071503 2020

  60. [60]

    V. M. Braun, S. Collins, M. G¨ ockeler, P. P´ erez- Rubio, A. Sch¨ afer, R. W. Schiel and A. Stern- beck, Phys. Rev. D92, no.1, 014504 (2015) doi:10.1103/PhysRevD.92.014504 [arXiv:1503.03656 [hep-lat]]

    work page doi:10.1103/physrevd.92.014504 2015

  61. [61]

    G. S. Baliet al.[RQCD], JHEP08, 065 (2019) doi:10.1007/JHEP08(2019)065 [arXiv:1903.08038 [hep- lat]]

    work page doi:10.1007/jhep08(2019)065 2019

  62. [62]

    A general method for non-perturbative renormalization of lattice operators,

    G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa and A. Vladikas, Nucl. Phys. B445, 81-108 (1995) doi:10.1016/0550-3213(95)00126-D [arXiv:hep- lat/9411010 [hep-lat]]

    work page doi:10.1016/0550-3213(95)00126-d 1995

  63. [63]

    Sturm, Y

    C. Sturm, Y. Aoki, N. H. Christ, T. Izubuchi, C. T. C. Sachrajda and A. Soni, Phys. Rev. D 80, 014501 (2009) doi:10.1103/PhysRevD.80.014501 [arXiv:0901.2599 [hep-ph]]

    work page doi:10.1103/physrevd.80.014501 2009

  64. [64]

    Q. A. Zhang, J. Hua, F. Huang, R. Li, Y. Li, C. L¨ u, C. D. Lu, P. Sun, W. Sun and W. Wang,et al. Chin. Phys. C46, no.1, 011002 (2022) doi:10.1088/1674- 1137/ac2b12 [arXiv:2103.07064 [hep-lat]]

    work page doi:10.1088/1674- 2022

  65. [66]

    Z. C. Huet al.[CLQCD], Phys. Rev. D109, no.5, 054507 (2024) doi:10.1103/PhysRevD.109.054507 [arXiv:2310.00814 [hep-lat]]

    work page doi:10.1103/physrevd.109.054507 2024

  66. [67]

    H. Y. Duet al.[CLQCD], Phys. Rev. D111, no.5, 054504 (2025) doi:10.1103/PhysRevD.111.054504 [arXiv:2408.03548 [hep-lat]]

    work page doi:10.1103/physrevd.111.054504 2025

  67. [68]

    Navas et al

    S. Navaset al.[Particle Data Group], Phys. Rev. D110, no.3, 030001 (2024) doi:10.1103/PhysRevD.110.030001

    work page doi:10.1103/physrevd.110.030001 2024

  68. [69]

    G. S. Bali, B. Lang, B. U. Musch and A. Sch¨ afer, Phys. Rev. D93, no.9, 094515 (2016) doi:10.1103/PhysRevD.93.094515 [arXiv:1602.05525 [hep-lat]]

    work page doi:10.1103/physrevd.93.094515 2016

  69. [70]

    Hasenfratz and F

    A. Hasenfratz and F. Knechtli, Phys. Rev. D 64, 034504 (2001) doi:10.1103/PhysRevD.64.034504 [arXiv:hep-lat/0103029 [hep-lat]]

    work page doi:10.1103/physrevd.64.034504 2001

  70. [71]

    J. X. Tan, Z. C. Gong, J. Hua, X. Ji, X. Jiang, H. Liu, A. Sch¨ afer, Y. Su, H. Z. Wang and W. Wang,et al.Phys. Rev. D113, no.5, 054505 (2026) doi:10.1103/pry5-7729 [arXiv:2511.22547 [hep-lat]]

    work page doi:10.1103/pry5-7729 2026

  71. [72]

    Zhang, A

    R. Zhang, A. V. Grebe, D. C. Hackett, M. L. Wagman and Y. Zhao, Phys. Rev. D112, no.5, L051502 (2025) doi:10.1103/6dh4-6k4t [arXiv:2501.00729 [hep-lat]]

    work page doi:10.1103/6dh4-6k4t 2025

  72. [73]

    Izubuchi, X

    T. Izubuchi, X. Ji, L. Jin, I. W. Stewart and Y. Zhao, Phys. Rev. D98, no.5, 056004 (2018) doi:10.1103/PhysRevD.98.056004 [arXiv:1801.03917 [hep-ph]]

    work page doi:10.1103/physrevd.98.056004 2018

  73. [74]

    Y. K. Huoet al.[Lattice Parton (LPC)], Nucl. Phys. B 969, 115443 (2021) doi:10.1016/j.nuclphysb.2021.115443 [arXiv:2103.02965 [hep-lat]]

    work page doi:10.1016/j.nuclphysb.2021.115443 2021

  74. [75]

    X. Gao, A. D. Hanlon, S. Mukherjee, P. Pe- treczky, P. Scior, S. Syritsyn and Y. Zhao, Phys. Rev. Lett.128, no.14, 142003 (2022) doi:10.1103/PhysRevLett.128.142003 [arXiv:2112.02208 [hep-lat]]

    work page doi:10.1103/physrevlett.128.142003 2022

  75. [76]

    J. W. Chen, X. Gao, J. He, J. Hua, X. Ji, A. Sch¨ afer, Y. Su, W. Wang, Y. B. Yang and J. H. Zhang,et al. 34 Phys. Rev. D113, no.1, 1 (2026) doi:10.1103/fflw-qpcc [arXiv:2505.14619 [hep-lat]]

    work page doi:10.1103/fflw-qpcc 2026

  76. [77]

    A. S. Xiong, J. Hua, Y. F. Ling, T. Wei, F. S. Yu, Q. A. Zhang and Y. Zheng, Eur. Phys. J. C85, no.12, 1409 (2025) doi:10.1140/epjc/s10052-025-15130-9 [arXiv:2506.16689 [hep-lat]]

    work page doi:10.1140/epjc/s10052-025-15130-9 2025

  77. [78]

    Y. F. Ling, M. H. Chu, J. Liang, J. Hua, A. S. Xiong and Q. A. Zhang, Eur. Phys. J. C86, no.4, 379 (2026) doi:10.1140/epjc/s10052-026-15528-z [arXiv:2511.03593 [hep-lat]]

    work page doi:10.1140/epjc/s10052-026-15528-z 2026

  78. [79]

    Constantinou, R

    M. Constantinou, R. Horsley, H. Panagopoulos, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller and J. M. Zanotti, Phys. Rev. D91, no.1, 014502 (2015) doi:10.1103/PhysRevD.91.014502 [arXiv:1408.6047 [hep- lat]]

    work page doi:10.1103/physrevd.91.014502 2015

  79. [80]

    L. Y. Li, J. Ran, M. Cai, H. F. Gao, Y. Gu, X. Y. Han, J. Hua, J. X. Tan, G. Y. Wang and W. Wang,et al.Phys. Rev. D113, no.5, 054509 (2026) doi:10.1103/556x-5mgq [arXiv:2512.20372 [hep-lat]]

    work page doi:10.1103/556x-5mgq 2026

  80. [81]

    W. I. Jay and E. T. Neil, Phys. Rev. D103, 114502 (2021) doi:10.1103/PhysRevD.103.114502 [arXiv:2008.01069 [stat.ME]]

    work page doi:10.1103/physrevd.103.114502 2021

Showing first 80 references.