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  • Open Access

Infrared Singularities of Multileg Amplitudes with a Massive Particle at Three Loops

Einan Gardi* and Zehao Zhu†

  • Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, Scotland, United Kingdom

  • *Contact author: Einan.Gardi@ed.ac.uk
  • †Contact author: zzhu3@ed.ac.uk
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Phys. Rev. Lett. 137, 101901 – Published 2 September, 2026

DOI: https://doi.org/10.1103/6bzh-h122

Abstract

We determine the complete three-loop QCD soft anomalous dimension for multileg amplitudes involving a single massive colored particle and any number of massless ones. This is achieved by applying a novel strategy based on a light-cone expansion of correlators of semi-infinite Wilson lines using the method of regions. The resulting region integrals depend exclusively on rescaling-invariant ratios that remain finite in the limit. We evaluate these integrals using differential equation techniques. The result is written in terms of uniform weight five generalized polylogarithms of a 12-letter alphabet in three variables, and is compatible with the massless limit as well as with two- and three-particle collinear factorization.

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References (134)

  1. A. M. Polyakov, Nucl. Phys. B164, 171 (1980).
  2. I. Y. Arefeva, Phys. Lett. 93B, 347 (1980).
  3. V. S. Dotsenko and S. N. Vergeles, Nucl. Phys. B169, 527 (1980).
  4. R. A. Brandt, F. Neri, and M.-a. Sato, Phys. Rev. D 24, 879 (1981).
  5. A. Sen, Phys. Rev. D 28, 860 (1983).
  6. G. P. Korchemsky and A. V. Radyushkin, Phys. Lett. B 171, 459 (1986).
  7. S. V. Ivanov, G. P. Korchemsky, and A. V. Radyushkin, Yad. Fiz. 44, 230 (1986), https://inspirehep.net/literature/221156.
  8. G. P. Korchemsky and A. V. Radyushkin, Nucl. Phys. B283, 342 (1987).
  9. G. P. Korchemsky, Phys. Lett. B 220, 629 (1989).
  10. G. P. Korchemsky and A. V. Radyushkin, Phys. Lett. B 279, 359 (1992).
  11. I. A. Korchemskaya and G. P. Korchemsky, Nucl. Phys. B437, 127 (1995).
  12. S. Catani, Phys. Lett. B 427, 161 (1998).
  13. N. Kidonakis, G. Oderda, and G. F. Sterman, Nucl. Phys. B531, 365 (1998).
  14. N. Kidonakis and G. F. Sterman, Nucl. Phys. B505, 321 (1997).
  15. G. F. Sterman and M. E. Tejeda-Yeomans, Phys. Lett. B 552, 48 (2003).
  16. E. Gardi, J. High Energy Phys. 02 (2005) 053.
  17. S. M. Aybat, L. J. Dixon, and G. F. Sterman, Phys. Rev. D 74, 074004 (2006).
  18. L. J. Dixon, L. Magnea, and G. F. Sterman, J. High Energy Phys. 08 (2008) 022.
  19. E. Gardi and L. Magnea, J. High Energy Phys. 03 (2009) 079.
  20. E. Gardi and L. Magnea, Nuovo Cimento C 32N5-6, 137 (2009) [Frascati Phys. Ser. 50 (2010)], https://www.sif.it/riviste/sif/ncc/econtents/2009/032/05-06/article/29.
  21. T. Becher and M. Neubert, Phys. Rev. Lett. 102, 162001 (2009); 111, 199905(E) (2013).
  22. T. Becher and M. Neubert, J. High Energy Phys. 06 (2009) 081; 11 (2013) 024.
  23. Y. Ma, J. High Energy Phys. 05 (2020) 012.
  24. I. Feige and M. D. Schwartz, Phys. Rev. D 90, 105020 (2014).
  25. N. Agarwal, L. Magnea, C. Signorile-Signorile, and A. Tripathi, Phys. Rep. 994, 1 (2023).
  26. J. Frenkel and J. C. Taylor, Nucl. Phys. B246, 231 (1984).
  27. J. G. M. Gatheral, Phys. Lett. B 133, 90 (1983).
  28. G. F. Sterman, AIP Conf. Proc. 74, 22 (1981).
  29. A. Mitov, G. Sterman, and I. Sung, Phys. Rev. D 82, 096010 (2010).
  30. E. Gardi, E. Laenen, G. Stavenga, and C. D. White, J. High Energy Phys. 11 (2010) 155.
  31. E. Gardi and C. D. White, J. High Energy Phys. 03 (2011) 079.
  32. E. Gardi, J. M. Smillie, and C. D. White, J. High Energy Phys. 09 (2011) 114.
  33. E. Gardi, J. M. Smillie, and C. D. White, J. High Energy Phys. 06 (2013) 088.
  34. N. Agarwal, A. Danish, L. Magnea, S. Pal, and A. Tripathi, J. High Energy Phys. 05 (2020) 128.
  35. N. Agarwal, S. Pal, A. Srivastav, and A. Tripathi, J. High Energy Phys. 02 (2023) 258.
  36. Ø. Almelid, C. Duhr, and E. Gardi, Phys. Rev. Lett. 117, 172002 (2016).
  37. Ø. Almelid, Three-loop soft anomalous dimension of massless multi-leg scattering, Ph.D. thesis, University of Edinburgh, 2016.
  38. Ø. Almelid, C. Duhr, E. Gardi, A. McLeod, and C. D. White, J. High Energy Phys. 09 (2017) 073.
  39. T. Becher and M. Neubert, J. High Energy Phys. 01 (2020) 025.
  40. G. Falcioni, E. Gardi, N. Maher, C. Milloy, and L. Vernazza, J. High Energy Phys. 03 (2022) 053.
  41. N. Maher, Studying the soft anomalous dimension for massless multi-leg scattering at four loops, Ph.D. thesis, Edinburgh University, 2023.
  42. V. Del Duca, C. Duhr, L. Fekeshazy, F. Guadagni, P. Mukherjee, G. Somogyi, F. Tramontano, and S. Van Thurenhout, J. High Energy Phys. 05 (2025) 151.
  43. S. Alioli, G. Billis, A. Broggio, and G. Stagnitto, J. High Energy Phys. 01 (2026) 065.
  44. J. Currie, E. W. N. Glover, and S. Wells, J. High Energy Phys. 04 (2013) 066.
  45. B. Chargeishvili, Analyzing and implementing the local analytic sector subtraction scheme for handling infrared singularities at NNLO in QCD, Ph.D. thesis, University of Hamburg, Universität Hamburg (main), 2021.
  46. L. Magnea, C. Milloy, C. Signorile-Signorile, and P. Torrielli, J. High Energy Phys. 06 (2024) 021.
  47. F. Devoto, K. Melnikov, R. Röntsch, C. Signorile-Signorile, and D. M. Tagliabue, J. High Energy Phys. 02 (2024) 016.
  48. F. Devoto, K. Melnikov, R. Röntsch, C. Signorile-Signorile, D. M. Tagliabue, and M. Tresoldi, J. High Energy Phys. 08 (2025) 122.
  49. A. Ferroglia, M. Neubert, B. D. Pecjak, and L. L. Yang, Phys. Rev. Lett. 103, 201601 (2009).
  50. A. Ferroglia, M. Neubert, B. D. Pecjak, and L. L. Yang, J. High Energy Phys. 11 (2009) 062.

  51. The cusp anomalous dimension, controlling dipole corrections, is known in full to three loops [52, 53].

  52. A. Grozin, J. M. Henn, G. P. Korchemsky, and P. Marquard, Phys. Rev. Lett. 114, 062006 (2015).
  53. A. Grozin, J. M. Henn, G. P. Korchemsky, and P. Marquard, J. High Energy Phys. 01 (2016) 140.
  54. M. Czakon and P. Fiedler, Nucl. Phys. B879, 236 (2014).
  55. R. Angeles-Martinez, M. Czakon, and S. Sapeta, J. High Energy Phys. 10 (2018) 201.
  56. H.-S. Shao and G. Wang, J. High Energy Phys. 06 (2025) 255.
  57. L. Buonocore, M. Grazzini, F. Guadagni, J. Haag, and L. Rottoli, Eur. Phys. J. C 85, 1290 (2025).
  58. Z. L. Liu and P. F. Monni, J. High Energy Phys. 03 (2025) 096.
  59. N. Kidonakis, Nucl. Phys. B1024, 117352 (2026).
  60. M. Guzzi, A. Ablat, S. Dulat, T. J. Hou, N. Kidonakis, I. Sitiwaldi, A. Tonero, K. Xie, and C. P. Yuan, Proc. Sci. LHCP2024 (2025) 332 [arXiv:2411.00055].
  61. N. Kidonakis and C. Foster, Phys. Lett. B 860, 139146 (2025).
  62. N. Kidonakis, M. Guzzi, and A. Tonero, Phys. Rev. D 108, 054012 (2023).
  63. Z. L. Liu and N. Schalch, Phys. Rev. Lett. 129, 232001 (2022).
  64. L. J. Dixon, Phys. Rev. D 79, 091501 (2009).
  65. E. Gardi and Z. Zhu, J. High Energy Phys. 03 (2026) 173.
  66. M. Beneke and V. A. Smirnov, Nucl. Phys. B522, 321 (1998).
  67. V. A. Smirnov, Springer Tracts Mod. Phys. 177, 1 (2002).
  68. A. Pak and A. Smirnov, Eur. Phys. J. C 71, 1626 (2011).
  69. B. Jantzen, J. High Energy Phys. 12 (2011) 076.
  70. B. Jantzen, A. V. Smirnov, and V. A. Smirnov, Eur. Phys. J. C 72, 2139 (2012).
  71. T. Y. Semenova, A. V. Smirnov, and V. A. Smirnov, Eur. Phys. J. C 79, 136 (2019).
  72. B. Ananthanarayan, A. Pal, S. Ramanan, and R. Sarkar, Eur. Phys. J. C 79, 57 (2019).
  73. G. Heinrich, S. P. Jones, M. Kerner, V. Magerya, A. Olsson, and J. Schlenk, Comput. Phys. Commun. 295, 108956 (2024).
  74. S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, Comput. Phys. Commun. 222, 313 (2018).
  75. E. Gardi, F. Herzog, S. Jones, Y. Ma, and J. Schlenk, J. High Energy Phys. 07 (2023) 197.
  76. Y. Ma, J. High Energy Phys. 09 (2024) 197.
  77. W. Chen, Comput. Phys. Commun. 312, 109607 (2025).

  78. We use upper-case letters, such as Q, to label massive particles and timelike Wilson lines, and lower case, e.g., i, j, k, to label massless particles and lightlike Wilson lines.

  79. S. Catani and M. H. Seymour, Phys. Lett. B 378, 287 (1996).
  80. S. Moch, J. A. M. Vermaseren, and A. Vogt, Nucl. Phys. B688, 101 (2004).
  81. J. M. Henn, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, J. High Energy Phys. 05 (2016) 066.
  82. J. Davies, A. Vogt, B. Ruijl, T. Ueda, and J. A. M. Vermaseren, Nucl. Phys. B915, 335 (2017).
  83. J. Henn, A. V. Smirnov, V. A. Smirnov, M. Steinhauser, and R. N. Lee, J. High Energy Phys. 03 (2017) 139.
  84. R. N. Lee, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. D 96, 014008 (2017).
  85. S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, J. High Energy Phys. 10 (2017) 041.
  86. A. Grozin, J. High Energy Phys. 06 (2018) 073; 01 (2019) 134(A).
  87. S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, Phys. Lett. B 782, 627 (2018).
  88. R. N. Lee, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, J. High Energy Phys. 02 (2019) 172.
  89. J. M. Henn, T. Peraro, M. Stahlhofen, and P. Wasser, Phys. Rev. Lett. 122, 201602 (2019).
  90. A. von Manteuffel and R. M. Schabinger, Phys. Rev. D 99, 094014 (2019).
  91. J. M. Henn, G. P. Korchemsky, and B. Mistlberger, J. High Energy Phys. 04 (2020) 018.
  92. A. von Manteuffel, E. Panzer, and R. M. Schabinger, Phys. Rev. Lett. 124, 162001 (2020).
  93. B. Agarwal, A. von Manteuffel, E. Panzer, and R. M. Schabinger, Phys. Lett. B 820, 136503 (2021).
  94. S. Moch, J. A. M. Vermaseren, and A. Vogt, J. High Energy Phys. 08 (2005) 049.
  95. S. Moch, J. A. M. Vermaseren, and A. Vogt, Phys. Lett. B 625, 245 (2005).
  96. P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. Lett. 102, 212002 (2009).
  97. T. Becher, M. Neubert, and B. D. Pecjak, J. High Energy Phys. 01 (2007) 076.
  98. R. Brüser, Z. L. Liu, and M. Stahlhofen, J. High Energy Phys. 03 (2020) 071.

  99. We summarize the results through three loops in the Supplemental Material [100].

  100. See Supplemental Material at http://link.aps.org/supplemental/10.1103/6bzh-h122 for the full expression of the one-mass soft anomalous dimension to three loops, the analytic continuation between different channels, the boundary conditions of the solution of the connected web, and the Euclidean vs physical kinematic regimes, which includes Refs. [101–103].
  101. T. Becher and M. Neubert, Phys. Rev. D 79, 125004 (2009); 80, 109901(E) (2009).
  102. D. Gaiotto, J. Maldacena, A. Sever, and P. Vieira, J. High Energy Phys. 12 (2011) 011.
  103. C. Duhr, J. High Energy Phys. 08 (2012) 043.
  104. C. Duhr, E. Gardi, S. Jaskiewicz, J. Lübken, and L. Vernazza, J. High Energy Phys. 02 (2026) 173.

  105. Since generators associated with different partons trivially commute, this average over permutations only matters when two or more of the generators are associated with the same parton, as in the fourth or sixth lines of Eq. (2).

  106. L. J. Dixon, E. Gardi, and L. Magnea, J. High Energy Phys. 02 (2010) 081.
  107. F. C. S. Brown, Compt. Rend. Math. 338, 527 (2004).
  108. L. J. Dixon, C. Duhr, and J. Pennington, J. High Energy Phys. 10 (2012) 074.

  109. Following previous work on diagrammatic exponentiation in QCD [26, 27, 28, 29, 30, 31, 32, 33, 34, 35], we organize the perturbative expansion in terms of webs. These correspond to either diagrams that remain connected upon removing the Wilson lines, or to special linear combinations of diagrams that together give rise to connected color factors.

  110. E. Gardi and Z. Zhu, “ThreeLoopSoftAnomalousDimensions,” https://github.com/zzh118/SoftAD.git (2025).

  111. The permutation relations written in terms of {x,z,z¯} are given in Supplemental Material [100].

  112. It is convenient to interpret these limits in the βQ rest frame. There, ruvQ=12(1−cosθuv), where θuv is the spatial angle between particles u and v. The singularities at ruvQ→{0,1} then correspond to the collinear (θuv→0) and the back-to-back (θuv→π) configurations, respectively.

  113. Describing the approach to the collinear limit requires a different parametrization of the kinematics, given in Eq. (37) in Supplemental Material [100].

  114. We stress that the four velocities are independent; there may be additional noncolored particles emanating from the hard vertex where the four lines meet.

  115. E. Gardi, F. Herzog, S. Jones, and Y. Ma, J. High Energy Phys. 08 (2024) 127.
  116. Y. Ma, Int. J. Mod. Phys. A 40, 2530013 (2025).
  117. G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, F. Langer, V. Magerya, A. Pöldaru, J. Schlenk, and E. Villa, Comput. Phys. Commun. 273, 108267 (2022).

  118. A complete description of the computation will be reported elsewhere [119].

  119. E. Gardi and Z. Zhu (to be published).
  120. J. M. Henn, C. Milloy, and K. Yan, J. High Energy Phys. 04 (2024) 117.
  121. J. M. Henn, Phys. Rev. Lett. 110, 251601 (2013).
  122. P. Maierhöfer, J. Usovitsch, and P. Uwer, Comput. Phys. Commun. 230, 99 (2018).
  123. J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Comput. Phys. Commun. 266, 108024 (2021).
  124. F. Lange, J. Usovitsch, and Z. Wu, Comput. Phys. Commun. 322, 109999 (2026).
  125. R. N. Lee, arXiv:1212.2685 (2012).
  126. R. N. Lee, J. Phys. Conf. Ser. 523, 012059 (2014).

  127. Rationalizing the extended alphabet was the original incentive to introduce the parametrization in Eq. (9). In obtaining this parametrization we used the package rationalizeroots [128].

  128. M. Besier, P. Wasser, and S. Weinzierl, Comput. Phys. Commun. 253, 107197 (2020).
  129. C. Dlapa, J. Henn, and K. Yan, J. High Energy Phys. 05 (2020) 025.
  130. C. Duhr and F. Dulat, J. High Energy Phys. 08 (2019) 135.

  131. Note that nonconnected webs have subdivergences. In the process of renormalizing the multi-Wilson-line vertex, these are removed by commutators of lower-loop webs. The three-loop soft anomalous dimension may be written in terms of w(n) as in Eq. (2.15c) in Ref. [32].

  132. E. Gardi, J. High Energy Phys. 04 (2014) 044.
  133. G. Falcioni, E. Gardi, M. Harley, L. Magnea, and C. D. White, J. High Energy Phys. 10 (2014) 010.

  134. See Eqs. (10) and (11) in [36].

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Phys. Rev. Lett. 137, 101901– Published 2 September, 2026
  • Received 13 November 2025
  • Revised 23 April 2026
  • Accepted 25 June 2026
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