- 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

Two-loop QCD amplitudes from the chiral algebra bootstrap

Anthony Morales*

  • *Contact author: moralesa@uw.edu
PDF

Phys. Rev. D 113, 105017 – Published 22 May, 2026

DOI: https://doi.org/10.1103/yk39-n6zq

Abstract

We show that the chiral algebra bootstrap, which computes form factors of twistorial theories, can help determine two-loop amplitudes in massless QCD. We give an n-gluon result for a previously unknown partial amplitude of the two-loop all-plus-helicity QCD amplitude by utilizing supersymmetry Ward identities and known chiral algebra bootstrap results. We then show that the full-color two-loop n-gluon amplitude of QCD with nf quark flavors can be obtained from certain two-loop form factors of twistorial theories and one-loop and tree-level amplitudes. Chiral algebra bootstrap results exist for these form factors when all gluons have positive helicities. Hence, the bootstrap simplifies the computation of these two-loop amplitudes by one loop level.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (63)

  1. H. Elvang and Y.-t. Huang, arXiv:1308.1697.
  2. G. Heinrich, Phys. Rep. 922, 1 (2021).
  3. N. Arkani-Hamed, L. J. Dixon, A. J. McLeod, M. Spradlin, J. Trnka, and A. Volovich, arXiv:2207.10636.
  4. S. Caron-Huot, L. J. Dixon, J. M. Drummond, F. Dulat, J. Foster, Ö. Gürdoğan, M. von Hippel, A. J. McLeod, and G. Papathanasiou, Proc. Sci., CORFU2019 (2020) 003 [arXiv:2005.06735].
  5. S. Caron-Huot, L. J. Dixon, F. Dulat, M. von Hippel, A. J. McLeod, and G. Papathanasiou, J. High Energy Phys. 08 (2019) 016.
  6. L. J. Dixon and Y.-T. Liu, J. High Energy Phys. 10 (2020) 031.
  7. L. J. Dixon and Y.-T. Liu, J. High Energy Phys. 09 (2023) 098.
  8. Z. Bern, L. J. Dixon, and D. A. Kosower, J. High Energy Phys. 01 (2000) 027.
  9. C. Anastasiou, E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, Nucl. Phys. B601, 318 (2001).
  10. C. Anastasiou, E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, Nucl. Phys. B601, 341 (2001).
  11. C. Anastasiou, E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, Nucl. Phys. B605, 486 (2001).
  12. E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, Nucl. Phys. B605, 467 (2001).
  13. Z. Bern, A. De Freitas, and L. J. Dixon, J. High Energy Phys. 03 (2002) 018.
  14. Z. Bern, A. De Freitas, and L. J. Dixon, J. High Energy Phys. 06 (2003) 028; 04 (2014) 112(E).
  15. A. De Freitas and Z. Bern, J. High Energy Phys. 09 (2004) 039.
  16. B. Agarwal, F. Buccioni, F. Devoto, G. Gambuti, A. von Manteuffel, and L. Tancredi, Phys. Rev. D 109, 094025 (2024).
  17. G. De Laurentis, H. Ita, M. Klinkert, and V. Sotnikov, Phys. Rev. D 109, 094023 (2024).
  18. G. De Laurentis, H. Ita, and V. Sotnikov, Phys. Rev. D 109, 094024 (2024).
  19. F. Caola, A. Chakraborty, G. Gambuti, A. von Manteuffel, and L. Tancredi, J. High Energy Phys. 10 (2021) 206.
  20. F. Caola, A. Chakraborty, G. Gambuti, A. von Manteuffel, and L. Tancredi, Phys. Rev. Lett. 128, 212001 (2022).
  21. F. Caola, A. Chakraborty, G. Gambuti, A. von Manteuffel, and L. Tancredi, J. High Energy Phys. 12 (2022) 082.
  22. A. R. Dalgleish, D. C. Dunbar, W. B. Perkins, and J. M. W. Strong, Phys. Rev. D 101, 076024 (2020).
  23. A. R. Dalgleish, D. C. Dunbar, W. B. Perkins, and J. M. W. Strong, J. High Energy Phys. 02 (2025) 184.
  24. D. C. Dunbar, G. R. Jehu, and W. B. Perkins, Phys. Rev. D 93, 125006 (2016).
  25. D. C. Dunbar, W. B. Perkins, and J. M. W. Strong, Phys. Rev. D 101, 076001 (2020).
  26. K. Costello and N. M. Paquette, J. High Energy Phys. 10 (2022) 193.
  27. K. Costello and N. M. Paquette, Phys. Rev. Lett. 129, 231604 (2022).
  28. K. J. Costello, J. High Energy Phys. 08 (2025) 011.
  29. L. J. Dixon and A. Morales, J. High Energy Phys. 08 (2024) 129.
  30. V. E. Fernández and N. M. Paquette, Classical Quantum Gravity 42, 185005 (2025).
  31. M. B. Green and J. H. Schwarz, Phys. Lett. 149B, 117 (1984).
  32. K. J. Costello, arXiv:2111.08879.
  33. M. T. Grisaru, H. N. Pendleton, and P. van Nieuwenhuizen, Phys. Rev. D 15, 996 (1977).
  34. M. T. Grisaru and H. N. Pendleton, Nucl. Phys. B124, 81 (1977).
  35. S. J. Parke and T. R. Taylor, Phys. Lett. 157B, 81 (1985); 174B, 465(E) (1986).
  36. Z. Kunszt, Nucl. Phys. B271, 333 (1986).
  37. Z. Bern, G. Chalmers, L. J. Dixon, and D. A. Kosower, Phys. Rev. Lett. 72, 2134 (1994).
  38. G. Mahlon, Phys. Rev. D 49, 4438 (1994).
  39. Z. Bern, A. De Freitas, L. J. Dixon, and H. L. Wong, Phys. Rev. D 66, 085002 (2002).

  40. The author is grateful to Federica Devoto for sharing with him the bare, unrenormalized 5-gluon amplitude.

  41. Z. Bern, L. J. Dixon, and D. A. Kosower, J. High Energy Phys. 08 (2004) 012.
  42. S. Catani, Phys. Lett. B 427, 161 (1998).
  43. Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Nucl. Phys. B425, 217 (1994).
  44. Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Nucl. Phys. B435, 59 (1995).
  45. R. Britto, F. Cachazo, and B. Feng, Nucl. Phys. B715, 499 (2005).
  46. R. Britto, F. Cachazo, B. Feng, and E. Witten, Phys. Rev. Lett. 94, 181602 (2005).
  47. F. Cachazo, P. Svrček, and E. Witten, J. High Energy Phys. 09 (2004) 006.
  48. Z. Bern, L. J. Dixon, and D. A. Kosower, Phys. Rev. D 72, 125003 (2005).
  49. C. F. Berger, Z. Bern, L. J. Dixon, D. Forde, and D. A. Kosower, Phys. Rev. D 74, 036009 (2006).

  50. This reduction to lower-loop techniques for the two-loop all-plus amplitude substantiates the results of Refs. [22, 23, 24, 25, 51, 52, 53, 54, 55, 56].

  51. D. C. Dunbar and W. B. Perkins, Phys. Rev. D 93, 085029 (2016).
  52. D. C. Dunbar, G. R. Jehu, and W. B. Perkins, Phys. Rev. Lett. 117, 061602 (2016).
  53. D. C. Dunbar, J. H. Godwin, G. R. Jehu, and W. B. Perkins, Phys. Rev. D 96, 116013 (2017).
  54. D. C. Dunbar, J. H. Godwin, W. B. Perkins, and J. M. W. Strong, Phys. Rev. D 101, 016009 (2020).
  55. D. A. Kosower and S. Pögel, arXiv:2206.14445.
  56. D. A. Kosower and S. Pögel, Proc. Sci., LL2022 (2022) 031 [arXiv:2208.06209].
  57. P. Nogueira, J. Comput. Phys. 105, 279 (1993).

  58. This conjecture has been analytically verified for n=4, 5, 6, 7 external gluons using unitarity methods [8, 22, 23, 54] and the rational terms have been numerically checked for n=8, 9 [55, 56].

  59. L. J. Dixon and A. Morales, J. High Energy Phys. 03 (2025) 188.
  60. L. J. Dixon, E. W. N. Glover, and V. V. Khoze, J. High Energy Phys. 12 (2004) 015.
  61. S. D. Badger and E. W. N. Glover, Nucl. Phys. B, Proc. Suppl. 160, 71 (2006).
  62. C. F. Berger, V. Del Duca, and L. J. Dixon, Phys. Rev. D 74, 094021 (2006); 76, 099901(E) (2007).
  63. C. R. Schmidt, Phys. Lett. B 413, 391 (1997).

Outline

Information

Phys. Rev. D 113, 105017– Published 22 May, 2026
  • Received 2 November 2025
  • Accepted 10 March 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