- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Pseudoscalar and vector toponia in a Dyson-Schwinger-Bethe-Salpeter framework

H.-R. Zhang1,*, Z.-F. Cui1,†, and J. Segovia2,‡

  • *Contact author: hrzhang@smail.nju.edu.cn
  • †Contact author: phycui@nju.edu.cn
  • ‡Contact author: jsegovia@upo.es
PDF

Phys. Rev. D 113, 114016 – Published 9 June, 2026

DOI: https://doi.org/10.1103/g7nc-vh76

Abstract

We study the pseudoscalar (JPC=0−+) and vector (1−−) top-antitop (toponium) systems within the rainbow-ladder truncation of the Dyson-Schwinger and Bethe-Salpeter equations, employing the Qin-Chang effective interaction. After validating the framework in the charmonium and bottomonium sectors, we extend it consistently to the top sector, incorporating renormalization-group running of the current quark mass and a careful treatment of the number of active flavors. We compute masses and leptonic decay constants for Nf=5 and 6, then analyze their dependence on the renormalization scale in the range μ=400–800  GeV. The resulting toponium masses lie near 344–346 GeV with hyperfine splittings below 0.14–0.17 GeV, while the decay constants are large, 6–7 GeV, and exhibit the expected heavy-quark scaling behavior. We find only mild sensitivity to the renormalization point and a systematic reduction of binding when increasing Nf. Although the physical top quark decays weakly before hadronization, our results demonstrate that, within a Poincaré-covariant nonperturbative framework, quantum chromodynamics (QCD) generates tightly correlated pseudoscalar and vector toponium systems in that extreme heavy-quark limit.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (55)

  1. S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024).
  2. I. I. Y. Bigi, Y. L. Dokshitzer, V. A. Khoze, J. H. Kuhn, and P. M. Zerwas, Phys. Lett. B 181, 157 (1986).
  3. J. A. Aguilar-Saavedra, Phys. Rev. D 110, 054032 (2024).
  4. A. Hayrapetyan et al. (CMS Collaboration), Rep. Prog. Phys. 88, 087801 (2025).
  5. ATLAS Collaboration, Observation of a cross-section enhancement near the tt¯ production threshold in s=13  TeV pp collisions with the ATLAS detector, Report No. ATLAS-CONF-2025-008.
  6. G.-L. Wang, T.-F. Feng, and Y.-Q. Wang, Phys. Rev. D 111, 096016 (2025).
  7. F. J. Llanes-Estrada, Phys. Lett. B 866, 139510 (2025).
  8. J.-H. Fu, Y.-J. Li, H.-M. Yang, Y.-B. Li, Y.-J. Zhang, and C.-P. Shen, Phys. Rev. D 111, 114020 (2025).
  9. Z. R. Najjar and K. Azizi, Eur. Phys. J. C 86, 412 (2026).
  10. S.-Q. Luo, Q. Huang, and X. Liu, arXiv:2508.17646.
  11. J.-H. Zhang, S. Yang, and B.-D. Wan, arXiv:2602.09684.
  12. B. Fuks, K. Hagiwara, K. Ma, and Y.-J. Zheng, Eur. Phys. J. C 85, 157 (2025).
  13. B. Fuks, A. Hossain, and J. Keaveney, Phys. Lett. B 873, 140179 (2026).
  14. M. Beneke, Y. Kiyo, and K. Schuller, J. High Energy Phys. 07 (2025) 273.
  15. M. Beneke and Y. Kiyo, J. High Energy Phys. 07 (2025) 274.
  16. M. V. Garzelli, G. Limatola, S. O. Moch, M. Steinhauser, and O. Zenaiev, Phys. Lett. B 866, 139532 (2025).
  17. H.-S. Shao and G. Wang, J. High Energy Phys. 10 (2025) 164.
  18. V. S. Fadin and V. A. Khoze, JETP Lett. 46, 525 (1987).
  19. V. S. Fadin, V. A. Khoze, and T. Sjostrand, Z. Phys. C 48, 613 (1990).
  20. C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994).
  21. R. Alkofer and L. von Smekal, Phys. Rep. 353, 281 (2001).
  22. P. Maris and C. D. Roberts, Int. J. Mod. Phys. E 12, 297 (2003).
  23. S.-x. Qin and C. D. Roberts, Chin. Phys. Lett. 37, 121201 (2020).
  24. M. Q. Huber and J. Braun, Comput. Phys. Commun. 183, 1290 (2012).
  25. M. Q. Huber, Phys. Rep. 879, 1 (2020).
  26. G. Eichmann, R. Alkofer, I. C. Cloet, A. Krassnigg, and C. D. Roberts, Phys. Rev. C 77, 042202 (2008).
  27. D. Binosi, L. Chang, J. Papavassiliou, S.-X. Qin, and C. D. Roberts, Phys. Rev. D 93, 096010 (2016).
  28. P. Qin, S.-x. Qin, and Y.-x. Liu, Phys. Rev. D 101, 114014 (2020).
  29. S.-x. Qin, L. Chang, Y.-x. Liu, C. D. Roberts, and D. J. Wilson, Phys. Rev. C 85, 035202 (2012).
  30. L. Chang, I. C. Cloët, C. D. Roberts, S. M. Schmidt, and P. C. Tandy, Phys. Rev. Lett. 111, 141802 (2013).
  31. K. Raya, L. Chang, A. Bashir, J. J. Cobos-Martinez, L. X. Gutiérrez-Guerrero, C. D. Roberts, and P. C. Tandy, Phys. Rev. D 93, 074017 (2016).
  32. G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer, and C. S. Fischer, Prog. Part. Nucl. Phys. 91, 1 (2016).
  33. Y. Lu, C. Chen, C. D. Roberts, J. Segovia, S.-S. Xu, and H.-S. Zong, Phys. Rev. C 96, 015208 (2017).
  34. C. Chen, G. I. Krein, C. D. Roberts, S. M. Schmidt, and J. Segovia, Phys. Rev. D 100, 054009 (2019).
  35. Y. Lu, C. Chen, Z.-F. Cui, C. D. Roberts, S. M. Schmidt, J. Segovia, and H. S. Zong, Phys. Rev. D 100, 034001 (2019).
  36. P.-L. Yin, C. Chen, G. Krein, C. D. Roberts, J. Segovia, and S.-S. Xu, Phys. Rev. D 100, 034008 (2019).
  37. P.-L. Yin, Z.-F. Cui, C. D. Roberts, and J. Segovia, Eur. Phys. J. C 81, 327 (2021).
  38. L. Liu, C. Chen, Y. Lu, C. D. Roberts, and J. Segovia, Phys. Rev. D 105, 114047 (2022).
  39. S.-x. Qin, L. Chang, Y.-x. Liu, C. D. Roberts, and D. J. Wilson, Phys. Rev. C 84, 042202 (2011).
  40. S.-X. Qin and C. D. Roberts, Chin. Phys. Lett. 38, 071201 (2021).
  41. Z.-N. Xu, Z.-Q. Yao, S.-X. Qin, Z.-F. Cui, and C. D. Roberts, Eur. Phys. J. A 59, 39 (2023).
  42. S.-X. Qin, C. D. Roberts, and S. M. Schmidt, Phys. Rev. D 97, 114017 (2018).
  43. S.-X. Qin, C. D. Roberts, and S. M. Schmidt, Phys. Lett. B 733, 202 (2014).
  44. S.-X. Qin, L. Chang, Y.-X. Liu, C. D. Roberts, and S. M. Schmidt, Phys. Lett. B 722, 384 (2013).
  45. M. Beneke, Eur. Phys. J. Spec. Top. 230, 2565 (2021).
  46. W. Celmaster and R. J. Gonsalves, Phys. Rev. D 20, 1420 (1979).
  47. K. G. Chetyrkin and A. Retey, Nucl. Phys. B583, 3 (2000).
  48. C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40, 100001 (2016).
  49. P. Boucaud, F. De Soto, J. P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene, and J. Rodriguez-Quintero, Phys. Rev. D 79, 014508 (2009).
  50. C. T. H. Davies, C. McNeile, E. Follana, G. P. Lepage, H. Na, and J. Shigemitsu, Phys. Rev. D 82, 114504 (2010).
  51. C. McNeile, C. T. H. Davies, E. Follana, K. Hornbostel, and G. P. Lepage, Phys. Rev. D 86, 074503 (2012).
  52. G. C. Donald, C. T. H. Davies, R. J. Dowdall, E. Follana, K. Hornbostel, J. Koponen, G. P. Lepage, and C. McNeile, Phys. Rev. D 86, 094501 (2012).
  53. B. Colquhoun, R. J. Dowdall, C. T. H. Davies, K. Hornbostel, and G. P. Lepage, Phys. Rev. D 91, 074514 (2015).
  54. N. Akbar, I. Asghar, and Z. Ahmad, Adv. High Energy Phys. 2026, 9454944 (2026).
  55. S.-J. Jiang, B.-Q. Li, G.-Z. Xu, and K.-Y. Liu, arXiv:2412.18527.

Outline

Information

Phys. Rev. D 113, 114016– Published 9 June, 2026
  • Received 20 March 2026
  • Accepted 28 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation