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  • Letter
  • Open Access

Kinematic Hopf algebra and Bern-Carrasco-Johansson numerators at finite α′

Gang Chen1,*, Laurentiu Rodina2,†, and Congkao Wen3,‡

  • 1Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
  • 2Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing 101408, China
  • 3Centre for Theoretical Physics, Department of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

  • *Contact author: gang.chen@nbi.ku.dk
  • †Contact author: laurentiu.rodina@gmail.com
  • ‡Contact author: c.wen@qmul.ac.uk
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Phys. Rev. D 110, L041902 – Published 15 August, 2024

DOI: https://doi.org/10.1103/PhysRevD.110.L041902

Abstract

We significantly extend recent work on the kinematic Hopf algebra, a structure that was shown to underlie the color-kinematics duality in Yang-Mills (YM) theory coupled to two heavy scalars, known as the heavy-mass effective theory (HEFT) limit. First, staying in the HEFT limit, we show the same kinematic Hopf algebra can be used to obtain Bern-Carrasco-Johansson (BCJ) numerators in DF2+YM theory, a theory containing an infinite number of higher derivative corrections to YM, used to obtain string amplitudes via the double copy. Second, we exploit the intricate structure induced by the massive poles to obtain an efficient and direct expression for local BCJ numerators in pure YM based on the same kinematic Hopf algebra. This demonstrates that the kinematic Hopf algebra works even beyond the HEFT limit, strongly suggesting this structure universally underlies the color-kinematics duality.

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References (75)

  1. Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. D 78, 085011 (2008).
  2. Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. Lett. 105, 061602 (2010).
  3. R. Monteiro and D. O’Connell, J. High Energy Phys. 07 (2011) 007.
  4. N. E. J. Bjerrum-Bohr, P. H. Damgaard, R. Monteiro, and D. O’Connell, J. High Energy Phys. 06 (2012) 061.
  5. C. Cheung and C.-H. Shen, Phys. Rev. Lett. 118, 121601 (2017).
  6. G. Chen, H. Johansson, F. Teng, and T. Wang, J. High Energy Phys. 11 (2019) 055.
  7. S. Mizera, Phys. Rev. Lett. 124, 141601 (2020).
  8. G. Chen, H. Johansson, F. Teng, and T. Wang, J. High Energy Phys. 10 (2021) 042.
  9. M. Ben-Shahar and H. Johansson, J. High Energy Phys. 08 (2022) 035.
  10. R. Monteiro, R. Stark-Muchão, and S. Wikeley, J. High Energy Phys. 09 (2023) 030.
  11. R. Monteiro, J. High Energy Phys. 01 (2023) 092.
  12. A. Lipstein and S. Nagy, Phys. Rev. Lett. 131, 081501 (2023).
  13. P. Ferrero and D. Francia, J. High Energy Phys. 02 (2021) 213.
  14. M. Ben-Shahar, L. Garozzo, and H. Johansson, J. High Energy Phys. 08 (2023) 222.
  15. C. R. Mafra and O. Schlotterer, Phys. Rep. 1020, 1 (2023).
  16. C.-H. Fu, P. Vanhove, and Y. Wang, J. High Energy Phys. 09 (2018) 141.
  17. M. Ben-Shahar and M. Guillen, J. High Energy Phys. 12 (2021) 014.
  18. L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, Phys. Rev. Lett. 126, 191601 (2021).
  19. L. Borsten and S. Nagy, J. High Energy Phys. 07 (2020) 093.
  20. R. Bonezzi, F. Diaz-Jaramillo, and S. Nagy, Phys. Rev. D 108, 065007 (2023).
  21. L. Borsten, B. Jurco, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, Phys. Rev. Lett. 131, 041603 (2023).
  22. K. Armstrong-Williams, S. Nagy, C. D. White, and S. Wikeley, arXiv:2401.10750.
  23. G. R. Brown, J. Gowdy, and B. Spence, Phys. Rev. D 109, 026009 (2024).
  24. C.-H. Fu and Y. Wang, Springer Proc. Math. Stat. 396, 287 (2022).
  25. M. E. Hoffman, J. Algebraic Combinatorics 11, 49 (2000).
  26. J. Blumlein, Comput. Phys. Commun. 159, 19 (2004).
  27. M. Aguiar and S. A. Mahajan, Monoidal Functors, Species and Hopf Algebras (American Mathematical Society, Providence, RI, 2010), Vol. 29.
  28. M. E. Hoffman and K. Ihara, J. Algebra 481, 293 (2017).
  29. F. Fauvet, L. Foissy, and D. Manchon, Ann. l’Inst. Fourier 67, 911 (2017).
  30. A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, J. High Energy Phys. 07 (2021) 047.
  31. A. Brandhuber, G. Chen, H. Johansson, G. Travaglini, and C. Wen, Phys. Rev. Lett. 128, 121601 (2022).
  32. A. Brandhuber, G. R. Brown, G. Chen, J. Gowdy, G. Travaglini, and C. Wen, J. High Energy Phys. 12 (2022) 101.
  33. G. Chen, G. Lin, and C. Wen, Phys. Rev. D 107, L081701 (2023).
  34. G. Lin and G. Yang, J. High Energy Phys. 02 (2024) 012.
  35. G. Lin and G. Yang, J. High Energy Phys. 02 (2024) 013.
  36. N. E. J. Bjerrum-Bohr, G. Chen, Y. Miao, and M. Skowronek, J. High Energy Phys. 07 (2024) 242.
  37. Q. Cao, J. Dong, S. He, and Y.-Q. Zhang, Phys. Rev. D 107, 026022 (2023).
  38. G. Chen, L. Rodina, and C. Wen, J. High Energy Phys. 02 (2024) 096.

  39. See also [40] for a different approach studying the color-kinematics duality for α′F3+α′2F4 theory.

  40. Q. Bonnefoy, G. Durieux, and J. Roosmale Nepveu, arXiv:2310.13041.
  41. H. Johansson and J. Nohle, arXiv:1707.02965.
  42. Y.-t. Huang, O. Schlotterer, and C. Wen, J. High Energy Phys. 09 (2016) 155.
  43. T. Azevedo, M. Chiodaroli, H. Johansson, and O. Schlotterer, J. High Energy Phys. 10 (2018) 012.
  44. T. Azevedo, R. L. Jusinskas, and M. Lize, J. High Energy Phys. 01 (2020) 082.
  45. T. Azevedo and O. T. Engelund, J. High Energy Phys. 11 (2017) 052.
  46. L. M. Garozzo, L. Queimada, and O. Schlotterer, J. High Energy Phys. 02 (2019) 078.
  47. G. Menezes, J. High Energy Phys. 03 (2022) 074.
  48. J. J. M. Carrasco, M. Lewandowski, and N. H. Pavao, J. High Energy Phys. 02 (2023) 015.
  49. J. J. M. Carrasco, M. Lewandowski, and N. H. Pavao, Phys. Rev. Lett. 131, 051601 (2023).
  50. J. J. M. Carrasco and N. H. Pavao, Phys. Rev. D 109, 065006 (2024).
  51. L. Garozzo and A. Guevara, J. High Energy Phys. 07 (2024) 002.
  52. C. Cheung and J. Mangan, J. High Energy Phys. 11 (2021) 069.
  53. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.110.L041902 for a brief introduction to the kinematic Hopf algebra, details on the amplitude with one massive gluon or tachyon, examples of W’ functions, and examples of local BCJ numerators.
  54. D. Vaman and Y.-P. Yao, J. High Energy Phys. 11 (2010) 028.

  55. Because of the cyclic summation in the operator O, if cyc(x1,…,xs)=cyc(x1′,…,xs′), the two partitions generate identical terms under the action of Ocyc([i1…ir−1]ir). Therefore, for each partition element with s subsets, the O operator introduces an overcounting factor equal to s.

  56. N. Arkani-Hamed, L. Rodina, and J. Trnka, Phys. Rev. Lett. 120, 231602 (2018).
  57. L. Rodina, J. High Energy Phys. 09 (2019) 084.
  58. Y.-J. Du and F. Teng, J. High Energy Phys. 04 (2017) 033.
  59. A. Edison and F. Teng, J. High Energy Phys. 12 (2020) 138.
  60. G. Chen, https://github.com/amplitudegravity/kinematichopfalgebra (2022).
  61. O. A. Gross, Am. Math. Mon. 69, 4 (1962).
  62. C. Cheung, C.-H. Shen, and C. Wen, J. High Energy Phys. 02 (2018) 095.
  63. M. Chiodaroli, M. Gunaydin, H. Johansson, and R. Roiban, J. High Energy Phys. 06 (2017) 064.
  64. J. J. M. Carrasco, L. Rodina, Z. Yin, and S. Zekioglu, Phys. Rev. Lett. 125, 251602 (2020).
  65. I. Low, L. Rodina, and Z. Yin, Phys. Rev. D 103, 025004 (2021).
  66. J. J. M. Carrasco, L. Rodina, and S. Zekioglu, J. High Energy Phys. 06 (2021) 169.
  67. N. H. Pavao, Phys. Rev. D 107, 065020 (2023).
  68. A. Brandhuber and G. Travaglini, J. High Energy Phys. 01 (2020) 010.
  69. M. Accettulli Huber, A. Brandhuber, S. De Angelis, and G. Travaglini, Phys. Rev. D 102, 046014 (2020).
  70. M. Accettulli Huber, A. Brandhuber, S. De Angelis, and G. Travaglini, Phys. Rev. D 103, 045015 (2021).
  71. W. T. Emond and N. Moynihan, J. High Energy Phys. 12 (2019) 019.
  72. N. Sennett, R. Brito, A. Buonanno, V. Gorbenko, and L. Senatore, Phys. Rev. D 102, 044056 (2020).
  73. A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, J. High Energy Phys. 10 (2021) 118.
  74. A. Brandhuber, G. R. Brown, G. Chen, S. De Angelis, J. Gowdy, and G. Travaglini, J. High Energy Phys. 06 (2023) 048.
  75. A. Herderschee, R. Roiban, and F. Teng, J. High Energy Phys. 06 (2023) 004.

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Information

Phys. Rev. D 110, L041902– Published 15 August, 2024
  • Received 10 April 2024
  • Accepted 21 July 2024
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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.

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