Skip to main content
SpringerLink
Log in
Book cover

Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory pp 29–49Cite as

On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms

  • Chapter
  • First Online:

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an \(\varepsilon \)-form, which makes a solution in terms of iterated integrals immediate.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. D.J. Broadhurst, J. Fleischer, O. Tarasov, Z. Phys. C 60, 287 (1993), arXiv:hep-ph/9304303

  2. F.A. Berends, M. Buza, M. Böhm, R. Scharf, Z. Phys. C 63, 227 (1994)

    Article  Google Scholar 

  3. S. Bauberger, M. Böhm, G. Weiglein, F.A. Berends, M. Buza, Nucl. Phys. Proc. Suppl. 37B, 95 (1994), arXiv:hep-ph/9406404

  4. S. Bauberger, F.A. Berends, M. Böhm, M. Buza, Nucl. Phys. B 434, 383 (1995), arXiv:hep-ph/9409388

  5. S. Bauberger, M. Böhm, Nucl. Phys. B 445, 25 (1995), arXiv:hep-ph/9501201

  6. M. Caffo, H. Czyz, S. Laporta, E. Remiddi, Nuovo Cim. A 111, 365 (1998), arXiv:hep-th/9805118

  7. S. Laporta, E. Remiddi, Nucl. Phys. B 704, 349 (2005), arXiv:hep-ph/0406160

  8. B.A. Kniehl, A.V. Kotikov, A. Onishchenko, O. Veretin, Nucl. Phys. B 738, 306 (2006), arXiv:hep-ph/0510235

  9. S. Groote, J.G. Körner, A.A. Pivovarov, Ann. Phys. 322, 2374 (2007), arXiv:hep-ph/0506286

  10. S. Groote, J. Körner, A. Pivovarov, Eur. Phys. J. C 72, 2085 (2012), arXiv:1204.0694

  11. D.H. Bailey, J.M. Borwein, D. Broadhurst, M.L. Glasser, J. Phys. A 41, 205203 (2008), arXiv:0801.0891

  12. S. Müller-Stach, S. Weinzierl, R. Zayadeh, Commun. Number Theor. Phys. 6, 203 (2012), arXiv:1112.4360

  13. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 54, 052303 (2013), arXiv:1302.7004

  14. S. Bloch, P. Vanhove, J. Number Theory 148, 328 (2015), arXiv:1309.5865

  15. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 55, 102301 (2014), arXiv:1405.5640

  16. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 56, 072303 (2015), arXiv:1504.03255

  17. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 57, 032304 (2016), arXiv:1512.05630

  18. E. Remiddi, L. Tancredi, Nucl. Phys. B 880, 343 (2014), arXiv:1311.3342

  19. S. Bloch, M. Kerr, P. Vanhove, Adv. Theor. Math. Phys. 21, 1373 (2017), arXiv:1601.08181

  20. S. Groote, J.G. Körner (2018), arXiv:1804.10570

  21. S. Bloch, M. Kerr, P. Vanhove, Compos. Math. 151, 2329 (2015), arXiv:1406.2664

  22. E. Remiddi, L. Tancredi, Nucl. Phys. B 907, 400 (2016), arXiv:1602.01481

  23. L. Adams, C. Bogner, A. Schweitzer, S. Weinzierl, J. Math. Phys. 57, 122302 (2016), arXiv:1607.01571

  24. L. Adams, S. Weinzierl, Commun. Number Theor. Phys. 12, 193 (2018), arXiv:1704.08895

  25. C. Bogner, A. Schweitzer, S. Weinzierl, Nucl. Phys. B 922, 528 (2017), arXiv:1705.08952

  26. L. Adams, S. Weinzierl, Phys. Lett. B 781, 270 (2018), arXiv:1802.05020

  27. L. Adams, E. Chaubey, S. Weinzierl (2018), arXiv:1804.11144

  28. L. Adams, E. Chaubey, S. Weinzierl (2018), arXiv:1806.04981

  29. M. Sgaard, Y. Zhang, Phys. Rev. D 91, 081701 (2015), arXiv:1412.5577

  30. R. Bonciani et al., JHEP 12, 096 (2016), arXiv:1609.06685

  31. A. von Manteuffel, L. Tancredi, JHEP 06, 127 (2017), arXiv:1701.05905

  32. A. Primo, L. Tancredi, Nucl. Phys. B 921, 316 (2017), arXiv:1704.05465

  33. J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C.G. Raab, C.S. Radu, C. Schneider, J. Math. Phys. 59(6), 062305 (2018), arXiv:1706.01299 [hep-th]

  34. J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel, M. Wilhelm, Phys. Rev. Lett. 120, 121603 (2018), arXiv:1712.02785

  35. M. Hidding, F. Moriello (2017), arXiv:1712.04441

  36. G. Passarino, Eur. Phys. J. C 77, 77 (2017), arXiv:1610.06207

  37. E. Remiddi, L. Tancredi, Nucl. Phys. B 925, 212 (2017), arXiv:1709.03622

  38. J. Broedel, C. Duhr, F. Dulat, L. Tancredi, JHEP 05, 093 (2018), arXiv:1712.07089

  39. J. Broedel, C. Duhr, F. Dulat, L. Tancredi, Phys. Rev. D 97, 116009 (2018), arXiv:1712.07095

  40. J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi (2018), arXiv:1803.10256

  41. R.N. Lee, A.V. Smirnov, V.A. Smirnov, JHEP 03, 008 (2018), arXiv:1709.07525

  42. R.N. Lee, A.V. Smirnov, V.A. Smirnov (2018), arXiv:1805.00227

  43. J. Broedel, C.R. Mafra, N. Matthes, O. Schlotterer, JHEP 07, 112 (2015), arXiv:1412.5535

  44. J. Broedel, N. Matthes, O. Schlotterer, J. Phys. A 49, 155203 (2016), arXiv:1507.02254

  45. J. Broedel, N. Matthes, G. Richter, O. Schlotterer, J. Phys. A 51, 285401 (2018), arXiv:1704.03449

  46. E. D’Hoker, M.B. Green, Ö. Gürdogan, P. Vanhove, Commun. Number Theor. Phys. 11, 165 (2017), arXiv:1512.06779

  47. S. Hohenegger, S. Stieberger, Nucl. Phys. B 925, 63 (2017), arXiv:1702.04963

  48. J. Broedel, O. Schlotterer, F. Zerbini (2018), arXiv:1803.00527

  49. J.A.M. Vermaseren, Int. J. Mod. Phys. A 14, 2037 (1999), arXiv:hep-ph/9806280

  50. E. Remiddi, J.A.M. Vermaseren, Int. J. Mod. Phys. A 15, 725 (2000), arXiv:hep-ph/9905237

    Article  MathSciNet  Google Scholar 

  51. A. Beilinson, A. Levin, in Motives, ed. by U. Jannsen, S. Kleiman, J.-P. Serre, Proceedings of Symposia in Pure Mathematics, vol. 55, Part 2 (AMS, 1994), pp. 123–190

    Google Scholar 

  52. A. Levin, Comput. Math. 106, 267 (1997)

    Google Scholar 

  53. A. Levin, G. Racinet (2007), arXiv:math/0703237

  54. B. Enriquez, Selecta Math. 20, 491 (2014), arXiv:1003.1012

  55. F. Brown, A. Levin (2011), arXiv:1110.6917

  56. J. Wildeshaus, Lecture Notes in Mathematics, vol. 1650 (Springer, Berlin, 1997)

    Google Scholar 

  57. A.V. Kotikov, Phys. Lett. B 254, 158 (1991)

    Article  MathSciNet  Google Scholar 

  58. A.V. Kotikov, Phys. Lett. B 267, 123 (1991)

    Article  MathSciNet  Google Scholar 

  59. E. Remiddi, Nuovo Cim. A 110, 1435 (1997), arXiv:hep-th/9711188

  60. T. Gehrmann, E. Remiddi, Nucl. Phys. B 580, 485 (2000), arXiv:hep-ph/9912329

  61. M. Argeri, P. Mastrolia, Int. J. Mod. Phys. A 22, 4375 (2007), arXiv:0707.4037

  62. S. Müller-Stach, S. Weinzierl, R. Zayadeh, Commun. Math. Phys. 326, 237 (2014), arXiv:1212.4389

  63. J.M. Henn, Phys. Rev. Lett. 110, 251601 (2013), arXiv:1304.1806

  64. J.M. Henn, J. Phys. A 48, 153001 (2015), arXiv:1412.2296

  65. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, Comput. Phys. Commun. 202, 33 (2016), arXiv:1509.08324 [hep-ph]

  66. L. Adams, E. Chaubey, S. Weinzierl, Phys. Rev. Lett. 118, 141602 (2017), arXiv:1702.04279

  67. J. Bosma, K.J. Larsen, Y. Zhang, Phys. Rev. D 97, 105014 (2018), arXiv:1712.03760

  68. M. Kontsevich, D. Zagier, in Mathematics Unlimited - 2001 and Beyond, ed. by B. Engquist, W. Schmid (2001), p. 771

    Google Scholar 

  69. K.-T. Chen, Bull. Am. Math. Soc. 83, 831 (1977)

    Article  Google Scholar 

  70. A.B. Goncharov, Math. Res. Lett. 5, 497 (1998)

    Article  MathSciNet  Google Scholar 

  71. A.B. Goncharov (2001), arXiv:math.AG/0103059

  72. J.M. Borwein, D.M. Bradley, D.J. Broadhurst, P. Lisonek, Trans. Am. Math. Soc. 353:3, 907 (2001), arXiv:math.CA/9910045

  73. S. Moch, P. Uwer, S. Weinzierl, J. Math. Phys. 43, 3363 (2002), arXiv:hep-ph/0110083

  74. J. Vollinga, S. Weinzierl, Comput. Phys. Commun. 167, 177 (2005), arXiv:hep-ph/0410259

  75. F. Brown (2014), arXiv:1407.5167

  76. O.V. Tarasov, Phys. Rev. D 54, 6479 (1996), arXiv:hep-th/9606018

    Article  MathSciNet  Google Scholar 

  77. O.V. Tarasov, Nucl. Phys. B 502, 455 (1997), arXiv:hep-ph/9703319

  78. C. Bogner, S. Weinzierl, Int. J. Mod. Phys. A 25, 2585 (2010), arXiv:1002.3458

  79. C. Bogner, S. Weinzierl, J. Math. Phys. 50, 042302 (2009), arXiv:0711.4863

  80. F.V. Tkachov, Phys. Lett. B 100, 65 (1981)

    Article  MathSciNet  Google Scholar 

  81. K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. B 192, 159 (1981)

    Article  Google Scholar 

  82. M. van Hoeij, J. Symb. Comput. 24, 537 (1997)

    Article  Google Scholar 

  83. P.A. Baikov, Nucl. Instrum. Methods A389, 347 (1997), arXiv:hep-ph/9611449

  84. R.N. Lee, Nucl. Phys. B 830, 474 (2010), arXiv:0911.0252

  85. D.A. Kosower, K.J. Larsen, Phys. Rev. D 85, 045017 (2012), arXiv:1108.1180

  86. S. Caron-Huot, K.J. Larsen, JHEP 1210, 026 (2012), arXiv:1205.0801

  87. H. Frellesvig, C.G. Papadopoulos, JHEP 04, 083 (2017), arXiv:1701.07356

  88. J. Bosma, M. Sogaard, Y. Zhang, JHEP 08, 051 (2017), arXiv:1704.04255

  89. M. Harley, F. Moriello, R.M. Schabinger, JHEP 06, 049 (2017), arXiv:1705.03478

  90. C. Meyer, JHEP 04, 006 (2017), arXiv:1611.01087

  91. C. Meyer, Comput. Phys. Commun. 222, 295 (2018), arXiv:1705.06252

  92. W.A. Stein, Modular Forms, A Computational Approach (American Mathematical Society, Providence, 2007)

    Google Scholar 

  93. C. Bogner, A. Schweitzer, S. Weinzierl, these proceedings

    Google Scholar 

Download references

Acknowledgements

S.W. would like to thank the organisers and KMPB for the organisation of the inspiring conference.

Author information

Authors and Affiliations

  1. PRISMA Cluster of Excellence, Institut für Physik, Johannes Gutenberg-Universität Mainz, 55099, Mainz, Germany

    Luise Adams & Stefan Weinzierl

Authors
  1. Luise Adams
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Weinzierl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Weinzierl .

Editor information

Editors and Affiliations

  1. Deutsches Elektronen-Synchrotron, DESY, Zeuthen, Germany

    Johannes Blümlein

  2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Carsten Schneider

  3. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Peter Paule

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Adams, L., Weinzierl, S. (2019). On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-04480-0_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-04479-4

  • Online ISBN: 978-3-030-04480-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation