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Factorization formula connecting the shape functions of heavy meson in QCD and heavy quark effective theory

Wei Wang1, Ji Xu2,*, Qi-An Zhang3,†, and Shuai Zhao4,‡

  • 1State Key Laboratory of Dark Matter Physics, Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
  • 2School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China
  • 3School of Physics, Beihang University, Beijing 102206, China
  • 4Department of Physics, School of Science, Tianjin University, Tianjin 300350, China

  • *Contact author: xuji1991@sjtu.edu.cn
  • †Contact author: zhangqa@buaa.edu.cn
  • ‡Contact author: zhaos@tju.edu.cn
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Phys. Rev. D 112, 054044 – Published 24 September, 2025

DOI: https://doi.org/10.1103/1547-t91t

Abstract

The shape function of the B-meson defined in heavy quark effective theory plays a crucial role in the analysis of inclusive B decays, and constitutes one of the dominant uncertainties in the determination of Cabbibo-Kobayashi-Maskawa matrix element |Vub|. On the other hand, the conventional heavy meson shape function defined in QCD is also phenomenologically important and includes short-distance physics at energy scales of the heavy quark mass. In this work, we derive a factorization formula relating these two kinds of shape functions, which can be invoked to fully disentangle the effects from disparate scales mb and ΛQCD, particularly to facilitate the resummation of logarithms lnmb/ΛQCD. In addition, this factorization constitutes an essential component of the recently developed two-step factorization scheme, enabling lattice QCD calculations of light cone quantities of heavy meson. The results presented here pave the way for first-principles nonperturbative predictions of shape function in the near future.

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References (57)

  1. S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024).
  2. Y. S. Amhis et al. (HFLAV Collaboration), Phys. Rev. D 107, 052008 (2023).
  3. L. Cao (Belle Collaboration), Proc. Sci. EPS-HEP2023 (2024) 356 [arXiv:2311.00379].
  4. A. G. Grozin and M. Neubert, Phys. Rev. D 55, 272 (1997).
  5. M. Neubert, Phys. Rev. D 49, 3392 (1994).
  6. V. M. Braun, D. Y. Ivanov, and G. P. Korchemsky, Phys. Rev. D 69, 034014 (2004).
  7. B. O. Lange and M. Neubert, Phys. Rev. Lett. 91, 102001 (2003).
  8. H. Kawamura, J. Kodaira, C. F. Qiao, and K. Tanaka, Phys. Lett. B 523, 111 (2001); 536, 344(E) (2002).
  9. J. P. Ma and Q. Wang, J. High Energy Phys. 01 (2006) 067.
  10. X. q. Li and Y. d. Yang, Phys. Rev. D 73, 114027 (2006).
  11. G. Bell, T. Feldmann, Y. M. Wang, and M. W. Y. Yip, J. High Energy Phys. 11 (2013) 191.
  12. V. M. Braun, Y. Ji, and A. N. Manashov, Phys. Rev. D 100, 014023 (2019).
  13. M. Beneke, V. M. Braun, Y. Ji, and Y. B. Wei, J. High Energy Phys. 07 (2018) 154.
  14. J. Gao, C. D. Lü, Y. L. Shen, Y. M. Wang, and Y. B. Wei, Phys. Rev. D 101, 074035 (2020).
  15. B. Y. Cui, Y. K. Huang, Y. L. Shen, C. Wang, and Y. M. Wang, J. High Energy Phys. 03 (2023) 140.
  16. J. Gao, U. G. Meißner, Y. L. Shen, and D. H. Li, Phys. Rev. D 112, 014032 (2025).
  17. J. Chai, S. Cheng, Y. h. Ju, D. C. Yan, C. D. Lü, and Z. J. Xiao, Chin. Phys. C 46, 123103 (2022).
  18. C. Shi, P. Liu, Y. L. Du, and W. Jia, Phys. Rev. D 110, 094010 (2024).
  19. M. Neubert, Phys. Rev. D 49, 4623 (1994).
  20. S. W. Bosch, B. O. Lange, M. Neubert, and G. Paz, Nucl. Phys. B699, 335 (2004).
  21. C. Balzereit, T. Mannel, and W. Kilian, Phys. Rev. D 58, 114029 (1998).
  22. Z. Ligeti, I. W. Stewart, and F. J. Tackmann, Phys. Rev. D 78, 114014 (2008).
  23. C. W. Bauer and A. V. Manohar, Phys. Rev. D 70, 034024 (2004).
  24. U. Aglietti and G. Ricciardi, Phys. Lett. B 466, 313 (1999).
  25. C. W. Bauer, S. Fleming, and M. E. Luke, Phys. Rev. D 63, 014006 (2000).
  26. M. Neubert, Phys. Lett. B 612, 13 (2005).
  27. A. Le Yaouanc, L. Oliver, and J. C. Raynal, Phys. Rev. D 77, 034005 (2008).
  28. K. S. M. Lee, Z. Ligeti, I. W. Stewart, and F. J. Tackmann, Phys. Rev. D 74, 011501 (2006).
  29. X. Y. Han, J. Hua, X. Ji, C. D. Lü, W. Wang, J. Xu, Q. A. Zhang, and S. Zhao, Phys. Rev. D 111, L111503 (2025).
  30. X. Y. Han et al. (Lattice Parton Collaboration), Phys. Rev. D 111, 034503 (2025).
  31. X. Y. Han, J. Hua, X. Ji, C. D. Lü, A. Schäfer, Y. Su, W. Wang, J. Xu, Y. Yang, J. H. Zhang et al., arXiv:2504.02453.
  32. X. Ji, Phys. Rev. Lett. 110, 262002 (2013).
  33. X. Ji, Sci. China Phys. Mech. Astron. 57, 1407 (2014).
  34. K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019, 3036904 (2019).
  35. W. Wang, Y. M. Wang, J. Xu, and S. Zhao, Phys. Rev. D 102, 011502 (2020).
  36. S. Ishaq, Y. Jia, X. Xiong, and D. S. Yang, Phys. Rev. Lett. 125, 132001 (2020).
  37. M. Beneke, G. Finauri, K. K. Vos, and Y. Wei, J. High Energy Phys. 09 (2023) 066.
  38. M. Gelb et al. (Belle Collaboration), Phys. Rev. D 98, 112016 (2018).
  39. T. Izubuchi, X. Ji, L. Jin, I. W. Stewart, and Y. Zhao, Phys. Rev. D 98, 056004 (2018).
  40. I. I. Y. Bigi, M. A. Shifman, and N. Uraltsev, Annu. Rev. Nucl. Part. Sci. 47, 591 (1997).
  41. S. M. Hu, W. Wang, J. Xu, and S. Zhao, Phys. Rev. D 109, 034001 (2024).
  42. X. Xiong, X. Ji, J. H. Zhang, and Y. Zhao, Phys. Rev. D 90, 014051 (2014).
  43. Y. Q. Ma and J. W. Qiu, Phys. Rev. D 98, 074021 (2018).
  44. Z. Y. Fan, Y. B. Yang, A. Anthony, H. W. Lin, and K. F. Liu, Phys. Rev. Lett. 121, 242001 (2018).
  45. J. Xu, Q. A. Zhang, and S. Zhao, Phys. Rev. D 97, 114026 (2018).
  46. Y. S. Liu, W. Wang, J. Xu, Q. A. Zhang, J. H. Zhang, S. Zhao, and Y. Zhao, Phys. Rev. D 100, 034006 (2019).
  47. X. Ji, Y. S. Liu, Y. Liu, J. H. Zhang, and Y. Zhao, Rev. Mod. Phys. 93, 035005 (2021).
  48. J. Xu and X. R. Zhang, Phys. Rev. D 106, 114019 (2022).
  49. J. Xu, X. R. Zhang, and S. Zhao, Phys. Rev. D 106, L011503 (2022).
  50. S. M. Hu, J. Xu, and S. Zhao, Eur. Phys. J. C 84, 502 (2024).
  51. S. Zhao, Phys. Rev. D 101, 071503 (2020).
  52. S. Zhao and A. V. Radyushkin, Phys. Rev. D 103, 054022 (2021).
  53. B. Hu, X. Jiang, K. F. Liu, P. Sun, and Y. B. Yang, arXiv:2411.18402.
  54. M. H. Chu et al. (Lattice Parton Collaboration), Phys. Rev. D 111, 034510 (2025).
  55. Z. F. Deng, W. Wang, Y. B. Wei, and J. Zeng, Phys. Rev. D 110, 114006 (2024).
  56. C. Han, W. Wang, J. L. Zhang, and J. H. Zhang, Phys. Rev. D 110, 094038 (2024).
  57. W. Wang, J. Xu, Q. A. Zhang, and S. Zhao, arXiv:2411.07101.

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Phys. Rev. D 112, 054044– Published 24 September, 2025
  • Received 13 May 2025
  • Accepted 10 September 2025
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