1932

Abstract

Precise predictions for collider observables require the computation of higher orders in perturbation theory. This task usually involves the evaluation of complicated multiloop integrals, which typically give rise to complicated special functions. This article discusses recent progress in understanding the mathematics underlying multiloop Feynman integrals and discusses a class of functions that generalizes the logarithm and that often appears in multiloop computations. The same class of functions is an active area of research in modern mathematics, which has led to the development of new powerful tools to compute Feynman integrals. These tools are at the heart of some of the most complicated computations ever performed for a hadron collider.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-nucl-101918-023551
2019-10-19
2024-06-18
Loading full text...

Full text loading...

/deliver/fulltext/nucl/69/1/annurev-nucl-101918-023551.html?itemId=/content/journals/10.1146/annurev-nucl-101918-023551&mimeType=html&fmt=ahah

Literature Cited

  1. 1. 
    Nielsen N. Nova Acta 90:121 1909.
    [Google Scholar]
  2. 2. 
    Lewin L Polylogarithms and Associated Functions New York: North-Holland 1982.
    [Google Scholar]
  3. 3. 
    Ellis RK, Zanderighi G. J. High Energy Phys. 0802:002 2008.
    [Google Scholar]
  4. 4. 
    Carrazza S, Ellis RK, Zanderighi G. Comput. Phys. Commun. 209:134 2016.
    [Google Scholar]
  5. 5. 
    ’t Hooft G, Veltman MLG. Nucl. Phys. B 153:365 1979.
    [Google Scholar]
  6. 6. 
    van Oldenborgh GJ, Vermaseren JAM. Z. Phys. C 46:425 1990.
    [Google Scholar]
  7. 7. 
    Kummer EE. J. Reine Angew. Math. 21:74 1840.
    [Google Scholar]
  8. 8. 
    Poincaré H. Acta Math. 4:201 1884.
    [Google Scholar]
  9. 9. 
    Lappo-Danilevsky JA. Trav. Inst. Phys.-Math. Stekloff 7:5 1935.
    [Google Scholar]
  10. 10. 
    Chen KT. Bull. Am. Math. Soc. 83:831 1977.
    [Google Scholar]
  11. 11. 
    Goncharov AB. Math. Res. Lett. 5:497 1998.
    [Google Scholar]
  12. 12. 
    Remiddi E, Vermaseren JAM. Int. J. Mod. Phys. A 15:725 2000.
    [Google Scholar]
  13. 13. 
    Remiddi E, Gehrmann T. Nucl. Phys. B 601:248 2001.
    [Google Scholar]
  14. 14. 
    Ablinger J, Blümlein J, Schneider C. J. Math. Phys. 52:102301 2011.
    [Google Scholar]
  15. 15. 
    Manchon D arXiv:math.QA/0408405 2004.
  16. 16. 
    Goncharov AB. Duke Math. J. 128:209 2005.
    [Google Scholar]
  17. 17. 
    Brown FCS Galois–Teichmüller Theory and Arithmetic Geometry Tokyo: Math. Soc. Jpn. 2012.
    [Google Scholar]
  18. 18. 
    Duhr C. J. High Energy Phys. 1208:043 2012.
    [Google Scholar]
  19. 19. 
    Duhr C Journeys Through the Precision Frontier: Amplitudes for Colliders Hackensack, NJ: World Sci. 2015.
    [Google Scholar]
  20. 20. 
    Goncharov AB, Spradlin M, Vergu C, Volovich A Phys. Rev. Lett. 105:151605 2010.
    [Google Scholar]
  21. 21. 
    Goncharov AB arXiv:0908.2238 [math] 2009.
  22. 22. 
    Brown FCS Ann. Sci. Éc. Norm. Supér. 42:371 2009.
    [Google Scholar]
  23. 23. 
    Duhr C, Gangl H, Rhodes JR J. High Energy Phys. 1210:075 2012.
    [Google Scholar]
  24. 24. 
    Panzer E. J. High Energy Phys. 1403:071 2014.
    [Google Scholar]
  25. 25. 
    von Manteuffel A, Panzer E, Schabinger RM. J. High Energy Phys. 1502:120 2015.
    [Google Scholar]
  26. 26. 
    Brown FCS. Commun. Math. Phys. 287:925 2009.
    [Google Scholar]
  27. 27. 
    Brown FCS arXiv:0910.0114 [math.AG] 2009.
  28. 28. 
    Anastasiou C, Duhr C, Dulat F, Mistlberger M. J. High Energy Phys. 1307:003 2013.
    [Google Scholar]
  29. 29. 
    Anastasiou C, Duhr C, Dulat F, Mistlberger M. J. High Energy Phys. 1312:088 2013.
    [Google Scholar]
  30. 30. 
    Anastasiou C, et al. Phys. Lett. B 737:325 2014.
    [Google Scholar]
  31. 31. 
    Anastasiou C, et al. J. High Energy Phys. 1503:091 2015.
    [Google Scholar]
  32. 32. 
    Duhr C, Gehrmann T, Jaquier M. J. High Energy Phys. 1502:077 2015.
    [Google Scholar]
  33. 33. 
    Dulat F, Mistlberger B arXiv:1411.3586 [hep-ph] 2014.
  34. 34. 
    Anastasiou C, et al. Phys. Rev. Lett. 114:212001 2015.
    [Google Scholar]
  35. 35. 
    Anastasiou C, et al. J. High Energy Phys. 1508:051 2015.
    [Google Scholar]
  36. 36. 
    Almelid Ø, Duhr C, Gardi E Phys. Rev. Lett. 117:17172002
    [Google Scholar]
  37. 37. 
    Panzer E. Comput. Phys. Commun. 188:148 2015.
    [Google Scholar]
  38. 38. 
    Bogner C. Comput. Phys. Commun. 203:339 2016.
    [Google Scholar]
  39. 39. 
    Bern Z, Dixon LJ, Dunbar DC, Kosower DA. Nucl. Phys. B 425:217 1994.
    [Google Scholar]
  40. 40. 
    Bern Z, Dixon LJ, Dunbar DC, Kosower DA. Nucl. Phys. B 435:59 1995.
    [Google Scholar]
  41. 41. 
    Bern Z, Rozowsky JS, Yan B. Phys. Lett. B 401:273 1997.
    [Google Scholar]
  42. 42. 
    Bern Z, Del Duca V, Dixon LJ, Kosower DA. Phys. Rev. D 71:045006 2005.
    [Google Scholar]
  43. 43. 
    Anastasiou C, Bern Z, Dixon LJ, Kosower DA. Phys. Rev. Lett. 91:251602 2003.
    [Google Scholar]
  44. 44. 
    Bern Z, Dixon LJ, Kosower DA. Phys. Rev. D 72:045014 2005.
    [Google Scholar]
  45. 45. 
    Bern Z, Dixon LJ, Smirnov VA Phys. Rev. D 72:085001 2005.
    [Google Scholar]
  46. 46. 
    Del Duca V, Duhr C, Smirnov VA. J. High Energy Phys. 03:099 2010.
    [Google Scholar]
  47. 47. 
    Del Duca V, Duhr C, Smirnov VA. J. High Energy Phys. 05:084 2010.
    [Google Scholar]
  48. 48. 
    Dixon LJ, Drummond JM, Henn JM. J. High Energy Phys. 1201:024 2012.
    [Google Scholar]
  49. 49. 
    Dixon LJ, Drummond JM, Henn JM. J. High Energy Phys. 1111:023 2011.
    [Google Scholar]
  50. 50. 
    Dixon LJ, Duhr C, Pennington J. J. High Energy Phys. 1210:074 2012.
    [Google Scholar]
  51. 51. 
    Dixon LJ, Drummond JM, von Hippel M, Pennington J. J. High Energy Phys. 1312:049 2013.
    [Google Scholar]
  52. 52. 
    Dixon LJ, von Hippel M. J. High Energy Phys. 10:065 2014.
    [Google Scholar]
  53. 53. 
    Dixon LJ, Drummond JM, Duhr C, Pennington J. J. High Energy Phys. 06:116 2014.
    [Google Scholar]
  54. 54. 
    Golden J, Spradlin M. J. High Energy Phys. 1408:154 2014.
    [Google Scholar]
  55. 55. 
    Drummond JM, Papathanasiou G, Spradlin M. J. High Energy Phys. 03:072 2015.
    [Google Scholar]
  56. 56. 
    Dixon LJ, von Hippel M, McLeod AJ. J. High Energy Phys. 01:053 2016.
    [Google Scholar]
  57. 57. 
    Caron-Huot S, et al. J. High Energy Phys. 117:241601 2016.
    [Google Scholar]
  58. 58. 
    Dixon LJ, et al. J. High Energy Phys. 02:137 2017.
    [Google Scholar]
  59. 59. 
    Del Duca V, et al. J. High Energy Phys. 08:152 2016.
    [Google Scholar]
  60. 60. 
    Henn JM, Mistlberger B. Phys. Rev. Lett. 117:171601 2016.
    [Google Scholar]
  61. 61. 
    Del Duca V, et al. J. High Energy Phys. 06:116 2018.
    [Google Scholar]
  62. 62. 
    Kotikov AV, Lipatov LN arXiv:hep-ph/0113246 2001.
  63. 63. 
    Kotikov AV, Lipatov LN Nucl. Phys. B 661:19 2003.
    [Google Scholar]
  64. 64. 
    Kotikov AV, Lipatov LN, Onishchenko AI, Velizhanin VN Phys. Lett. B 595:521 2004.
    [Google Scholar]
  65. 65. 
    Kotikov AV, et al. J. Stat. Mech. 0710:10003 2007.
    [Google Scholar]
  66. 66. 
    Beisert N, Eden B, Staudacher M. J. Stat. Mech. 0701:01021 2007.
    [Google Scholar]
  67. 67. 
    Grozin A, Henn JM, Korchemsky GP, Marquard P J. High Energy Phys. 01:140 2016.
    [Google Scholar]
  68. 68. 
    Dixon LJ J. High Energy Phys. 01:075 2018.
    [Google Scholar]
  69. 69. 
    van Neerven WL Z. Phys. C 30:595 1986.
    [Google Scholar]
  70. 70. 
    Gehrmann T, Henn JM, Huber T J. High Energy Phys. 03:101 2012.
    [Google Scholar]
  71. 71. 
    Brandhuber A, Travaglini G, Yang G J. High Energy Phys. 05:082 2012.
    [Google Scholar]
  72. 72. 
    Brandhuber A, Penante B, Travaglini G, Wen C J. High Energy Phys. 08:100 2014.
    [Google Scholar]
  73. 73. 
    Eden B et al. Nucl. Phys. B 557:355 1999.
    [Google Scholar]
  74. 74. 
    Gonzalez-Rey F, Park IY, Schalm K Phys. Lett. B 448:37 1999.
    [Google Scholar]
  75. 75. 
    Eden B et al. Phys. Lett. B 482:309 2000.
    [Google Scholar]
  76. 76. 
    Bianchi M, Kovacs S, Rossi G, Stanev YS Nucl. Phys. B 584:216 2000.
    [Google Scholar]
  77. 77. 
    Drummond JM et al. J. High Energy Phys. 08:133 2013.
    [Google Scholar]
  78. 78. 
    Li Y, von Manteuffel A, Schabinger RM, Zhu HX Phys. Rev. D 90:053006 2014.
    [Google Scholar]
  79. 79. 
    Li Y, von Manteuffel A, Schabinger RM, Zhu HX Phys. Rev. D 91:036008 2015.
    [Google Scholar]
  80. 80. 
    Henn JM Phys. Rev. Lett. 110:251601 2013.
    [Google Scholar]
  81. 81. 
    Kotikov AV Phys. Lett. B 254:158 1991.
    [Google Scholar]
  82. 82. 
    Remiddi E Nuovo Cim. A 110:1435 1997.
    [Google Scholar]
  83. 83. 
    Gehrmann T, Remiddi E Nucl. Phys. B 580:485 2000.
    [Google Scholar]
  84. 84. 
    Gehrmann T, Tancredi L, Weihs E. J. High Energy Phys. 1308:070 2013.
    [Google Scholar]
  85. 85. 
    Gehrmann T, von Manteuffel A, Tancredi L, Weihs E. J. High Energy Phys. 1406:032 2014.
    [Google Scholar]
  86. 86. 
    Henn JM, Melnikov K, Smirnov VA. J. High Energy Phys. 1405:090 2014.
    [Google Scholar]
  87. 87. 
    Caola F, Henn JM, Melnikov K, Smirnov VA. J. High Energy Phys. 1409:043 2014.
    [Google Scholar]
  88. 88. 
    Caola F, et al. J. High Energy Phys. 1411:041 2014.
    [Google Scholar]
  89. 89. 
    Caola F, et al. J. High Energy Phys. 1506:129 2015.
    [Google Scholar]
  90. 90. 
    von Manteuffel A, Tancredi L. J. High Energy Phys. 1506:197 2015.
    [Google Scholar]
  91. 91. 
    Gehrmann T, von Manteuffel A, Tancredi L. J. High Energy Phys. 1509:128 2015.
    [Google Scholar]
  92. 92. 
    Mistlberger B. J. High Energy Phys. 1805:028 2018.
    [Google Scholar]
  93. 93. 
    Dulat F, Mistlberger B, Pelloni A. J. High Energy Phys. 1801:145 2018.
    [Google Scholar]
  94. 94. 
    Dulat F, Mistlberger B, Pelloni A arXiv:1810.09462 [hep-ph] 2018.
  95. 95. 
    Alday LF, Maldacena JM. J. High Energy Phys. 0706:064 2007.
    [Google Scholar]
  96. 96. 
    Brandhuber A, Heslop P, Travaglini G. Nucl. Phys. B 794:231 2008.
    [Google Scholar]
  97. 97. 
    Drummond JM, Henn JM, Korchemsky GP, Sokatchev E. Nucl. Phys. B 795:52 2008.
    [Google Scholar]
  98. 98. 
    Mason LJ, Skinner D. J. High Energy Phys. 12:018 2010.
    [Google Scholar]
  99. 99. 
    Caron-Huot S. J. High Energy Phys. 07:058 2011.
    [Google Scholar]
  100. 100. 
    Basso B, Sever A, Vieira P Phys. Rev. Lett. 111:9091602 2013.
    [Google Scholar]
  101. 101. 
    Golden J, et al. J. High Energy Phys. 1401:091 2014.
    [Google Scholar]
  102. 102. 
    Steinmann O Acta Phys. Helv. 33:257 1960.
    [Google Scholar]
  103. 103. 
    Steinmann O Acta Phys. Helv. 33:347 1960.
    [Google Scholar]
  104. 104. 
    Cahill KE, Stapp HP Ann. Phys 90:438 1975.
    [Google Scholar]
  105. 105. 
    Drummond JM, Foster J, Gurdogan O Phys. Rev. Lett 120:16161601 2018.
    [Google Scholar]
  106. 106. 
    Drummond JM, Foster J, Gurdogan O arXiv:1810.08149 [hep-th] 2018.
  107. 107. 
    Almelid Ø, et al. J. High Energy Phys. 1709:073 2017.
    [Google Scholar]
  108. 108. 
    Brown FCS. Commun. Numer. Theor. Phys. 11:453 2017.
    [Google Scholar]
  109. 109. 
    Abreu S, Britto R, Duhr C, Gardi E. Phys. Rev. Lett. 119:051601 2017.
    [Google Scholar]
  110. 110. 
    Abreu S, Britto R, Duhr C, Gardi E. J. High Energy Phys. 1712:090 2017.
    [Google Scholar]
  111. 111. 
    Abreu S, Britto R, Duhr C, Gardi E. J. High Energy Phys. 1706:114 2017.
    [Google Scholar]
  112. 112. 
    Sabry A Nucl. Phys. 33:401 1962.
    [Google Scholar]
  113. 113. 
    Laporta S, Remiddi E. Nucl. Phys. B 704:349 2005.
    [Google Scholar]
  114. 114. 
    Bloch S, Vanhove P J. Number Theor. 148:328 2015.
    [Google Scholar]
  115. 115. 
    Beilinson A, Levin A. Proc. Symp. Pure Math. 55:123 1994.
    [Google Scholar]
  116. 116. 
    Levin A, Racinet G arXiv:math/0703237 2007.
  117. 117. 
    Brown FCS, Levin A arXiv:1110.6917 [math.NT] 2011.
  118. 118. 
    Broedel J, Duhr C, Dulat F, Tancredi L J. High Energy Phys. 05:093 2018.
    [Google Scholar]
  119. 119. 
    Broedel J et al. arXiv:1809.10698 [hep-th] 2018.
  120. 120. 
    Broedel J et al. J. High Energy Phys. 08:014 2018.
    [Google Scholar]
  121. 121. 
    Brown FCS Commun. Numer. Theor. Phys. 11:557 2017.
    [Google Scholar]
  122. 122. 
    Broedel J, Duhr C, Dulat F, Tancredi L Phys. Rev. D 97:116009 2018.
    [Google Scholar]
  123. 123. 
    Hidding M, Moriello F arXiv:1712.04441 [hep-ph] 2017.
  124. 124. 
    Broedel J, Mafra CR, Matthes N, Schlotterer O J. High Energy Phys. 07:112 2015.
    [Google Scholar]
  125. 125. 
    Broedel J, Matthes N, Schlotterer O. J. Phys. A 49:155203 2016.
    [Google Scholar]
  126. 126. 
    Broedel J, Matthes N, Richter G, Schlotterer O. J. Phys. A 51:285401 2018.
    [Google Scholar]
  127. 127. 
    Broedel J, Schlotterer O, Zerbini F arXiv:1803.00527 [hep-th] 2018.
  128. 128. 
    Adams L, Bogner C, Weinzierl S J. Math. Phys. 55:102301 2014.
    [Google Scholar]
  129. 129. 
    Adams L, Bogner C, Weinzierl S J. Math. Phys. 56:072303 2015.
    [Google Scholar]
  130. 130. 
    Adams L, Bogner C, Weinzierl S J. Math. Phys. 54:052303 2013.
    [Google Scholar]
  131. 131. 
    Remiddi E, Tancredi L. Nucl. Phys. B 880:343 2014.
    [Google Scholar]
  132. 132. 
    Adams L, Bogner C, Weinzierl S J. Math. Phys. 57:032304 2016.
    [Google Scholar]
  133. 133. 
    Remiddi E, Tancredi L Nucl. Phys. B 907:400 2016.
    [Google Scholar]
  134. 134. 
    Remiddi E, Tancredi L Nucl. Phys. B 925:212 2017.
    [Google Scholar]
  135. 135. 
    Manin YI Algebraic Geometry and Number Theory V Ginzburg565 Boston: Birkhäuser 2006.
    [Google Scholar]
  136. 136. 
    Brown FCS arXiv:1407.5167 [math.NT] 2014.
  137. 137. 
    Adams L, Weinzierl S Commun. Numer. Theor. Phys. 12:193 2018.
    [Google Scholar]
  138. 138. 
    Papadopoulos CG, Tommasini D, Wever C. J. High Energy Phys. 1604:078 2016.
    [Google Scholar]
  139. 139. 
    Chicherin D, Henn JM, Mitev V. J. High Energy Phys. 1805:164 2018.
    [Google Scholar]
  140. 140. 
    Gehrmann T, Henn JM, Lo Presti A. J. High Energy Phys. 1810:103 2018.
    [Google Scholar]
  141. 141. 
    Chicherin D, Gehrmann T, Henn JM, Lo Presti A arXiv:1809.06240 [hep-ph] 2018.
  142. 142. 
    Brown FCS, Schnetz O Duke Math. J. 161:1817 2012.
    [Google Scholar]
  143. 143. 
    Bourjaily JL et al. Phys. Rev. Lett. 121:071603 2018.
    [Google Scholar]
  144. 144. 
    Bourjaily JL, McLeod AJ, von Hippel M, Wilhelm M arXiv:1810.07689 [hep-th] 2018.
/content/journals/10.1146/annurev-nucl-101918-023551
Loading
/content/journals/10.1146/annurev-nucl-101918-023551
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error