Skip to main content

Analytic Integration Methods in Quantum Field Theory: An Introduction

  • Chapter
  • First Online:
Anti-Differentiation and the Calculation of Feynman Amplitudes

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

Abstract

A survey is given on the present status of analytic calculation methods and the mathematical structures of zero- and single scale Feynman amplitudes which emerge in higher order perturbative calculations in the Standard Model of elementary particles, its extensions and associated model field theories, including effective field theories of different kind.

Contribution to the Volume “Antidifferentiation and the Calculation of Feynman Amplitudes”, Springer, Berlin, 2021.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
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

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    For a summary on recent massless calculations, see [26].

  2. 2.

    For other recent surveys on integration methods for Feynman integrals see [27,28,29,30,31,32].

  3. 3.

    This representation has been used in a more specific form also in [196] later.

  4. 4.

    To see the same on the basis of a Heun differential equation is much more difficult [187].

References

  1. M. Veltman, Facts and Mysteries in Elementary Particle Physics (World Scientific, Singapore, 2003)

    Book  MATH  Google Scholar 

  2. The ILC: https://en.wikipedia.org/wiki/International_Linear_Collider J.A. Aguilar-Saavedra et al. [ECFA/DESY LC Physics Working Group], TESLA: The Superconducting electron positron linear collider with an integrated x-ray laser laboratory. Technical design report. Part 3. Physics at an e + e linear collider, hep-ph/0106315; E. Accomando et al. [ECFA/DESY LC Physics Working Group], Phys. Rept. 299, 1–78 (1998). [hep-ph/9705442]; The Future Circular Collider, https://en.wikipedia.org/wikiFuture_Circular_Collider; TH FCC-ee design study, http://tlep.web.cern.ch; A. Abada et al. [FCC], Eur. Phys. J. ST 228(4), 755–1107 (2019); J.L. Abelleira Fernandez et al. [LHeC Study Group], J. Phys. G 39, 075001 (2012). [arXiv:1206.2913 [physics.acc-ph]]; D. Boer, M. Diehl, R. Milner, R. Venugopalan, W. Vogelsang, D. Kaplan, H. Montgomery, S. Vigdor, A. Accardi, E.C. Aschenauer, et al., Gluons and the quark sea at high energies: Distributions, polarization, tomography. [arXiv:1108.1713 [nucl-th]]

  3. A. Abada et al. [FCC], Eur. Phys. J. ST 228(2), 261–623 (2019)

    Google Scholar 

  4. J. Blümlein, D. Broadhurst, J.A.M. Vermaseren, Comput. Phys. Commun. 181, 582–625 (2010). [arXiv: 0907.2557 [math-ph]]

  5. P. Marquard, Integration-by-Parts: A Survey, contribution to this volume

    Google Scholar 

  6. J. Vermaseren, Some Steps Towards Improving IBP Calculations and Related Topics, contribution to this volume

    Google Scholar 

  7. H. Frellesvig, Top-Down Decomposition: A Cut-Based Approach to Integral Reductions, contribution to this volume

    Google Scholar 

  8. M. Kalmykov, V. Bytev, B.A. Kniehl, S.O. Moch, B.F.L. Ward, S.A. Yost, Hypergeometric Functions and Feynman Diagrams. arXiv:2012.14492 [hep-th]

    Google Scholar 

  9. P. Paule, Contiguous Relations and Creative Telescoping, contribution to this volume

    Google Scholar 

  10. K. Acres, D. Broadhurst, Empirical determinations of Feynman integrals using integer relation algorithms. [arXiv:2103.06345 [hep-ph]]

    Google Scholar 

  11. E. Panzer, talk at this workshop, contribution not received

    Google Scholar 

  12. C. Schneider, Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation, contribution to this volume

    Google Scholar 

  13. T. Dreyfus, J.-A. Weil, Differential Galois Theory and Integration, contribution to this volume

    Google Scholar 

  14. A.V. Kotikov, Differential equations and Feynman integrals. [arXiv:2102.07424 [hep-ph]]

    Google Scholar 

  15. J. Henn, talk at this workshop, contribution not received

    Google Scholar 

  16. C.G. Raab, Nested Integrals and Rationalizing Transformations, contribution to this volume

    Google Scholar 

  17. C. Koutschan, Holonomic Anti-Differentiation and Feynman Amplitudes, contribution to this volume

    Google Scholar 

  18. J. Ablinger, Extensions of the AZ-algorithm and the Package MultiIntegrate, [arXiv:2101.11385 [cs.SC]], contribution to this volume

    Google Scholar 

  19. V.A. Smirnov, Expansion by Regions: An Overview, contribution to this volume

    Google Scholar 

  20. S. Weinzierl, Iterated integrals related to Feynman integrals associated to elliptic curves, [arXiv:2012.08429 [hep-th]]

    Google Scholar 

  21. J. Broedel, A. Kaderli, A Geometrical Framework for Amplitude Recursions: Bridging Between Trees and Loops, contribution to this volume

    Google Scholar 

  22. D. Kreimer, Outer Space as a Combinatorial Backbone for Cutkosky Rules and Coactions, [arXiv:2010.11781 [hep-th]], contribution to this volume

    Google Scholar 

  23. J.L. Bourjaily, Y.-H. He, A.J. McLeod, M. Spradlin, C. Vergu, M. Volk, M. von Hippel, M. Wilhelm, Direct Integration for Multi-leg Amplitudes: Tips, Tricks, and When They Fail, contribution to this volume

    Google Scholar 

  24. J. Bartels, N=4 SYM Gauge Theories: the 2 → 6 Amplitude in the Regge Limit, contribution to this volume

    Google Scholar 

  25. G. Papathanasiou, talk at this workshop, contribution not received

    Google Scholar 

  26. S. Moch, V. Magerya, Calculating Four-Loop Corrections in QCD, contribution to this volume

    Google Scholar 

  27. S. Weinzierl, Introduction to Feynman Integrals, arXiv:1005.1855 [hep-ph]

    Google Scholar 

  28. J. Ablinger, J. Blümlein, C. Schneider, J. Phys. Conf. Ser. 523, 012060 (2014). [arXiv: 1310.5645 [math-ph]]

    Article  Google Scholar 

  29. J. Ablinger, J. Blümlein, in Computer Algebra in Quantum Field Theory. Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein (Springer, Wien, 2012), pp. 1–32 [arXiv: 1304.7071 [math-ph]]

    Google Scholar 

  30. S. Weinzierl, in Computer Algebra in Quantum Field Theory. Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein (Springer, Wien, 2012), pp. 381–406 [arXiv:1301.6918 [hep-ph]]

    Google Scholar 

  31. C. Duhr, in Journeys Through the Precision Frontier: Amplitudes for Colliders, ed. by L. Dixon, F. Petriello. 2014 TASI Lectures (World Scientific, Singapore, 2015), pp. 419–476. [arXiv:1411.7538 [hep-ph]]

    Google Scholar 

  32. J. Blümlein, C. Schneider, Int. J. Mod. Phys. A 33(17), 1830015 (2018). [arXiv:1809.02889 [hep-ph]].

    Google Scholar 

  33. P. Nogueira, J. Comput. Phys. 105, 279–289 (1993)

    Google Scholar 

  34. T. van Ritbergen, A. Schellekens, J.A.M. Vermaseren, Int. J. Mod. Phys. A14, 41–96 (1999). [hep-ph/9802376]

  35. J.A.M. Vermaseren, New features of FORM. math-ph/0010025; M. Tentyukov, D. Fliegner, M. Frank, A. Onischenko, A. Retey, H.M. Staudenmaier, J.A.M. Vermaseren, AIP Conf. Proc. 583(1), 202 (2002). [cs/0407066 [cs-sc]]; M. Tentyukov, J.A.M. Vermaseren, Comput. Phys. Commun. 181, 1419–1427 (2010). [hep-ph/0702279]; B. Ruijl, U. Takahiro, J.A.M. Vermaseren, FORM version 4.2. arXiv:1707.06453[hep-ph]

  36. J. Lagrange, Nouvelles recherches sur la nature et la propagation du son, Miscellanea Taurinensis, t. II, 1760-61; Oeuvres t. I, p. 263; C.F. Gauss, Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo novo tractate, Commentationes societas scientiarum Gottingensis recentiores, Vol III, 1813, Werke Bd. V pp. 5–7; G. Green, Essay on the Mathematical Theory of Electricity and Magnetism, Nottingham, 1828 [Green Papers, pp. 1–115]; M. Ostrogradski, Mem. Ac. Sci. St. Peters., 6, 129–133 (1831); K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. B192, 159–204 (1981)

    Google Scholar 

  37. S. Laporta, Int. J. Mod. Phys. A15, 5087–5159 (2000). [hep-ph/0102033]

  38. A. Smirnov, JHEP 10, 107 (2008). [arXiv: 0807.3243[hep-ph]]

    Article  Google Scholar 

  39. A.V. Smirnov, F.S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic. arXiv:1901.07808 [hep-ph]

    Google Scholar 

  40. C. Studerus, Comput. Phys. Commun. 181, 1293–1300 (2010). [arXiv: 0912.2546 [physics.comp-ph]]

  41. A. von Manteuffel, C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction. arXiv:1201.4330 [hep-ph]

    Google Scholar 

  42. P. Marquard, D. Seidel, The Crusher algorithm (unpublished)

    Google Scholar 

  43. P. Maierhöfer, J. Usovitsch, P. Uwer, Comput. Phys. Commun. 230, 99–112 (2018). [arXiv:1705.05610 [hep-ph]]; J. Klappert, F. Lange, P. Maierhöfer, J. Usovitsch, [arXiv:2008.06494 [hep-ph]]

    Google Scholar 

  44. J. Blümlein, C. Schneider, Phys. Lett. B771, 31–36 (2017). [arXiv: 1701.04614 [hep-ph]]

  45. J. Blümlein, M. Kauers, S. Klein, C. Schneider, Comput. Phys. Commun. 180, 2143–2165 (2009). [arXiv:0902.4091[hep-ph]]

    Article  Google Scholar 

  46. M. Kauers, M. Jaroschek, F. Johansson, in Computer Algebra and Polynomials, ed. by J. Gutierrez, J. Schicho, Josef, M. Weimann. Lecture Notes in Computer Science, vol. 8942 (Springer, Berlin, 2015), pp. 105–125. [arXiv:1306.4263 [cs.SC]]

    Google Scholar 

  47. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, Nucl. Phys. B 922, 1–40 (2017). [arXiv:1705.01508 [hep-ph]]

    Google Scholar 

  48. J. Blümlein, J. Ablinger, A. Behring, A. De Freitas, A. von Manteuffel, C. Schneider, C. Schneider, in PoS (QCDEV2017), 031 (2017). [arXiv:1711.07957 [hep-ph]]

    Google Scholar 

  49. A. Behring, J. Blümlein, A. De Freitas, A. Goedicke, S. Klein, A. von Manteuffel, C. Schneider, K. Schönwald, Nucl. Phys. B 948, 114753 (2019). [arXiv:1908.03779 [hep-ph]]

    Article  Google Scholar 

  50. J. Blümlein, P. Marquard, N. Rana, C. Schneider, Nucl. Phys. B 949, 114751 (2019). [arXiv:1908.00357 [hep-ph]]

    Article  Google Scholar 

  51. M. Karr, J. ACM 28, 305–350 (1981); M. Bronstein, J. Symbolic Comput. 29(6), 841–877 (2000); C. Schneider, Symbolic Summation in Difference Fields, Ph.D. Thesis RISC, Johannes Kepler University, Linz technical report 01–17 (2001); C. Schneider, An. Univ. Timisoara Ser. Mat.-Inform. 42, 163–179 (2004); C. Schneider, J. Differ. Equ. Appl. 11, 799–821 (2005); C. Schneider, Appl. Algebra Engrg. Comm. Comput. 16, 1–32 (2005); C. Schneider, J. Algebra Appl. 6, 415–441 (2007); C. Schneider, Clay Math. Proc. 12, 285–308 (2010). [arXiv:0904.2323 [cs.SC]]; C. Schneider, Ann. Comb. 14, 533–552 (2010). [arXiv:0808.2596]; C. Schneider, in Computer Algebra and Polynomials, Applications of Algebra and Number Theory, ed. by J. Gutierrez, J. Schicho, M. Weimann. Lecture Notes in Computer Science (LNCS) 8942, 157–191 (2015). [arXiv:1307.7887 [cs.SC]]; C. Schneider, J. Symb. Comput. 72, 82–127 (2016). [arXiv:1408.2776 [cs.SC]]. C. Schneider, J. Symb. Comput. 80, 616–664 (2017). [arXiv:1603.04285 [cs.SC]]. S.A. Abramov, M. Bronstein, M. Petkovšek, Carsten Schneider, J. Symb. Comput. 107, 23–66 (2021). [arXiv:2005.04944 [cs.SC]]

    Google Scholar 

  52. C. Schneider, Sém. Lothar. Combin. 56, 1–36 (2007) article B56b.

  53. C. Schneider, in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein. Texts and Monographs in Symbolic Computation (Springer, Wien, 2013), pp. 325–360. [arXiv:1304.4134 [cs.SC]]

    Google Scholar 

  54. J. Ablinger, J. Blümlein, S. Klein, C. Schneider, Nucl. Phys. Proc. Suppl. 205-206, 110–115 (2010). [arXiv:1006.4797[math-ph]]

  55. J. Blümlein, A. Hasselhuhn, C. Schneider, in PoS (RADCOR2011), 032 (2012). [arXiv: 1202.4303 [math-ph]]

  56. C. Schneider, J. Phys. Conf. Ser. 523, 012037 (2014). [arXiv:1310.0160[cs.Sc]]

    Article  Google Scholar 

  57. C. Krattenthaler, C. Schneider, in Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, ed. by V. Pillwein, C. Schneider (Springer, Wien, 2020), pp. 249–296

    Google Scholar 

  58. P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Phys. Rev. Lett. 118(8), 082002 (2017). [arXiv:1606.08659 [hep-ph]]; F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren, A. Vogt, JHEP 02, 090 (2017). [arXiv:1701.01404 [hep-ph]]; T. Luthe, A. Maier, P. Marquard, Y. Schröder, JHEP 10, 166 (2017). [arXiv:1709.07718 [hep-ph]]; K. G. Chetyrkin, G. Falcioni, F. Herzog, J. A. M. Vermaseren, JHEP 10, 179 (2017). arXiv:1709.08541 [hep-ph]

    Google Scholar 

  59. J.M. Borwein, D.M. Bradley, D.J. Broadhurst, and P. Lisonek, Trans. Am. Math. Soc. 353, 907–941 (2001). [math/9910045]

  60. H.R.P. Ferguson, D.H. Bailey, A Polynomial Time, Numerically Stable Integer Relation Algorithm, RNR Techn. Rept. RNR-91-032, Jul. 14, 1992

    Google Scholar 

  61. A. Devoto, D.W. Duke, Riv. Nuovo Cim. 7N6, 1–39 (1984); L. Lewin, Dilogarithms and Associated Functions, (Macdonald, London, 1958); L. Lewin, Polylogarithms and Associated Functions (North Holland, New York, 1981)

    Google Scholar 

  62. F. Klein, Vorlesungen über die hypergeometrische Funktion, Wintersemester 1893/94, Die Grundlehren der Mathematischen Wissenschaften, vol. 39 (Springer, Berlin, 1933)

    Book  Google Scholar 

  63. W.N. Bailey, Generalized Hypergeometric Series (Cambridge University Press, Cambridge, 1935)

    MATH  Google Scholar 

  64. L.J. Slater, Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966)

    MATH  Google Scholar 

  65. P. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques, Polynomes D’ Hermite (Gauthier-Villars, Paris, 1926)

    MATH  Google Scholar 

  66. P. Appell, Les Fonctions Hypergëométriques de Plusieur Variables (Gauthier-Villars, Paris, 1925)

    MATH  Google Scholar 

  67. J. Kampé de Fériet, La fonction hypergëométrique (Gauthier-Villars, Paris, 1937)

    MATH  Google Scholar 

  68. H. Exton, Multiple Hypergeometric Functions and Applications (Ellis Horwood, Chichester, 1976)

    MATH  Google Scholar 

  69. H. Exton, Handbook of Hypergeometric Integrals (Ellis Horwood, Chichester, 1978)

    MATH  Google Scholar 

  70. M.J. Schlosser, in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein, (Springer, Wien, 2013), pp. 305–324. arXiv:1305.1966 [math.CA]

    Google Scholar 

  71. C. Anastasiou, E.W.N. Glover, C. Oleari, Nucl. Phys. B572, 307–360 (2000). [hep-ph/9907494]

  72. C. Anastasiou, E.W.N. Glover, C. Oleari, Nucl. Phys. B565, 445–467 (2000). [hep-ph/9907523]

  73. H.M. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series (Ellis Horwood, Chicester, 1985)

    MATH  Google Scholar 

  74. G. Lauricella, Rediconti del Circolo Matematico di Palermo 7(S1), 111–158 (1893)

    Article  Google Scholar 

  75. S. Saran, Ganita 5, 77–91 (1954)

    MathSciNet  Google Scholar 

  76. S. Saran, Acta Math. 93, 293–312 (1955)

    Article  MathSciNet  Google Scholar 

  77. J. Blümlein, M. Saragnese, C. Schneider, Hypergeometric Structures in Feynman Integrals, DESY 21–071

    Google Scholar 

  78. R. Hamberg, W.L. van Neerven, T. Matsuura, Nucl. Phys. B359, 343–405 (1991); [Erratum: Nucl. Phys. B644, 403–404 (2002)]

    Google Scholar 

  79. R. Hamberg, Second order gluonic contributions to physical quantities, Ph.D. Thesis, Leiden University, 1991

    Google Scholar 

  80. M. Buza, Y. Matiounine, J. Smith, R. Migneron, W.L. van Neerven, Nucl. Phys. B472, 611–658 (1996). [hep-ph/9601302]

  81. I. Bierenbaum, J. Blümlein, S. Klein, Nucl. Phys. B780, 40–75 (2007). [hep-ph/0703285]

  82. J. Ablinger, J. Blümlein, A. Hasselhuhn, S. Klein, C. Schneider, F. Wißbrock, Nucl. Phys. B864, 52–84 (2012). [arXiv:1206.2252[hep-ph]]

    Article  Google Scholar 

  83. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, Comput. Phys. Commun. 202, 33–112 (2016). [arXiv:1509.08324 [hep-ph]]

  84. E.W. Barnes, Q. J. Math. 41, 136–140 (1910)

    Google Scholar 

  85. H. Mellin, Math. Ann. 68(3), 305–337 (1910)

    Article  MathSciNet  Google Scholar 

  86. V.A. Smirnov, Feynman Integral Calculus (Springer, Berlin, 2006)

    Google Scholar 

  87. L. Pochhammer, Math. Ann. 35, 495–526 (1890)

    Article  MathSciNet  Google Scholar 

  88. A. Kratzer, W. Franz, Transzendente Funktionen (Geest & Portig, Leipzig, 1960)

    MATH  Google Scholar 

  89. J. Blümlein, S. Klein, C. Schneider, F. Stan, J. Symb. Comput. 47, 1267–1289 (2012). arXiv:1011.2656 [cs.SC]

    Google Scholar 

  90. M. Czakon, Comput. Phys. Commun. 175, 559–571 (2006). [hep-ph/0511200]

  91. A. Smirnov, V. Smirnov, Eur. Phys. J. C62, 445–449 (2009). [arXiv:0901.0386 [hep-ph]]

  92. J. Gluza, K. Kajda, T. Riemann, Comput. Phys. Commun. 177, 879–893 (2007). [arXiv:0704.2423[hep-ph]]

    Article  Google Scholar 

  93. J. Gluza, K. Kajda, T. Riemann, V. Yundin, Eur. Phys. J. C71, 1516 (2011). [arXiv:1010.1667[hep-ph]]

    Article  Google Scholar 

  94. F. Brown, Commun. Math. Phys. 287, 925–958 (2009). arXiv:0804.1660 [math.AG]

    Google Scholar 

  95. E.E. Kummer, J. Reine Angew. Math. (Crelle) 21, 74–90 (1840); 193–225; 328–371

    Google Scholar 

  96. H. Poincaré, Acta Math. 4, 201–312 (1884)

    Article  MathSciNet  Google Scholar 

  97. J.A. Lappo-Danilevsky, Mémoirs sur la Théorie des Systèmes Différentielles Linéaires (Chelsea Publ. Co, New York, 1953)

    MATH  Google Scholar 

  98. K.T. Chen, Trans. A.M.S. 156(3), 359–379 (1971)

    Article  Google Scholar 

  99. A.B. Goncharov, Math. Res. Lett. 5, 497–516 (1998)

    Article  MathSciNet  Google Scholar 

  100. A. von Manteuffel, E. Panzer, R.M. Schabinger, JHEP 02, 120 (2015). [arXiv:1411.7392[hep-ph]]

    Article  Google Scholar 

  101. E. Panzer, Comput. Phys. Commun. 188, 148–166 (2015). [arXiv:1403.3385[hep-th]]

    Article  Google Scholar 

  102. J. Ablinger, J. Blümlein, C. Raab, C. Schneider, F. Wißbrock, Nucl. Phys. B885, 409–447 (2014). [arXiv:1403.1137[hep-ph]]

    Article  Google Scholar 

  103. F.Ph. Wißbrock, \(O(\alpha _s^3)\) contributions to the heavy flavor Wilson coefficients of the structure function F 2(x, Q 2) at Q 2 ≫ m 2, Ph.D. Thesis, TU Dortmund, 2015

    Google Scholar 

  104. S.A. Larin, T. van Ritbergen, J.A.M. Vermaseren, Nucl. Phys. B427, 41–52 (1994)

    Article  Google Scholar 

  105. S.A. Larin, P. Nogueira, T. van Ritbergen, J.A.M. Vermaseren, Nucl. Phys. B492, 338–378 (1997). [hep-ph/9605317]

  106. A. Retey, J.A.M. Vermaseren, Nucl. Phys. B604, 281–311 (2001). [hep-ph/0007294]

  107. J. Blümlein, J.A.M. Vermaseren, Phys. Lett. B606, 130–138 (2005). [hep-ph/0411111]

  108. I. Bierenbaum, J. Blümlein, S. Klein, Nucl. Phys. B 820, 417–482 (2009). arXiv:0904.3563 [hep-ph]

    Google Scholar 

  109. J.A.M Vermaseren, A. Vogt, S. Moch, Nucl. Phys. B724, 3–182 (2005). [hep-ph/0504242]

  110. Sage, http://www.sagemath.org/

  111. S. Moch, J.A.M. Vermaseren, A. Vogt, Nucl. Phys. B688, 101–134 (2004). [hep-ph/0403192]; A. Vogt, S. Moch, J.A.M. Vermaseren, Nucl. Phys. B691, 129–181 (2004). [hep-ph/0404111]; S. Moch, J.A.M. Vermaseren, A. Vogt, Nucl. Phys. B 889, 351–400 (2014). arXiv:1409.5131 [hep-ph]

  112. J. Blümlein, P. Marquard, C. Schneider, K. Schönwald, The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements. Nucl. Phys. B, in print [arXiv:2107.06267 [hep-ph]]

    Google Scholar 

  113. J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider, Nucl. Phys. B 939, 253–291 (2019). arXiv:1810.12261 [hep-ph]

    Google Scholar 

  114. J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider, Phys. Lett. B 782, 528–532 (2018). arXiv:1804.07313 [hep-ph]

    Google Scholar 

  115. S.G. Gorishnii, S.A. Larin, L.R. Surguladze, F.V. Tkachov, Comput. Phys. Commun. 55, 381–408 (1989); S.A. Larin, F.V. Tkachov, J.A.M. Vermaseren, The FORM version of MINCER NIKHEF-H-91-18

    Google Scholar 

  116. M. Steinhauser, Comput. Phys. Commun. 134, 335–364 (2001). arXiv:hep-ph/0009029

    Google Scholar 

  117. R. Harlander, T. Seidensticker, M. Steinhauser, Phys. Lett. B426, 125–132 (1998). [hep-ph/9712228]

  118. T. Seidensticker, Automatic application of successive asymptotic expansions of Feynman diagrams, in Proc. 6th International Workshop on New Computing Techniques in Physics Research, Crete, Greece, April 1999. arXiv:hep-ph/9905298

    Google Scholar 

  119. N.E. Nörlund, Vorlesungen über Differenzenrechnung (Springer, Berlin, 1924); reprinted by (Chelsea Publishing Company, New York, 1954)

    Google Scholar 

  120. J.A.M. Vermaseren, Int. J. Mod. Phys. A 14, 2037–2076 (1999). arXiv:hep-ph/9806280 [hep-ph]

    Google Scholar 

  121. J. Blümlein, S. Kurth, Phys. Rev. D 60, 014018 (1999). arXiv:hep-ph/9810241 [hep-ph]

    Article  Google Scholar 

  122. J. Ablinger, J. Blümlein, C. Schneider, J. Math. Phys. 54, 082301 (2013). arXiv: 1302.0378 [math-ph]

    Article  MathSciNet  Google Scholar 

  123. J. Ablinger, J. Blümlein, C. Schneider, J. Math. Phys. 52, 102301 (2011). arXiv: 1105.6063 [math-ph]

    Article  MathSciNet  Google Scholar 

  124. J. Ablinger, J. Blümlein, C.G. Raab, C. Schneider, J. Math. Phys. 55, 112301 (2014). arXiv:1407.1822 [hep-th]

    Article  MathSciNet  Google Scholar 

  125. J. Ablinger, J. Blümlein, C. Schneider, J. Phys. Conf. Ser. 523, 012060 (2014). arXiv:1310.5645 [math-ph]; J. Ablinger, PoS (LL2014) 019. arXiv:1407.6180[cs.SC]; A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics, Diploma Thesis, JKU Linz, 2009, arXiv:1011.1176[math-ph]; PoS (LL2016) 067; Experimental Mathematics 26 (2017). arXiv:1507.01703 [math.CO]; PoS (RADCOR2017) 001, arXiv:1801.01039 [cs.SC]; PoS (LL2018) 063; J. Ablinger, Experimental Mathematics 26 (2017). arXiv:1507.01703 [math.CO]; Discovering and Proving Infinite Pochhammer Sum Identities, arXiv:1902.11001 [math.CO]

    Google Scholar 

  126. J. Blümlein, Comput. Phys. Commun. 180, 2218–2249 (2009). arXiv:0901.3106 [hep-ph]

    Google Scholar 

  127. J. Ablinger, Computer Algebra Algorithms for Special Functions in Particle Physics, Ph.D. Thesis, Linz U. (2012). arXiv:1305.0687[math-ph]

    Google Scholar 

  128. E. Remiddi, J.A.M. Vermaseren, Int. J. Mod. Phys. A15, 725–754 (2000). [hep-ph/9905237]

  129. S. Moch, P. Uwer, S. Weinzierl, J. Math. Phys. 43, 3363–3386 (2002). arXiv:hep-ph/0110083 [hep-ph]

    Google Scholar 

  130. A.V. Kotikov, Phys. Lett. B254, 158–164 (1991)

    Article  MathSciNet  Google Scholar 

  131. Z. Bern, L.J. Dixon, D.A. Kosower, Phys. Lett. B302, 299–308 (1993). [Erratum: Phys. Lett. B318, 649 (1993)] [hep-ph/9212308]

  132. E. Remiddi, Nuovo Cim. A110, 1435–1452 (1997). [hep-th/9711188]

    Article  Google Scholar 

  133. T. Gehrmann, E. Remiddi, Nucl. Phys. B580, 485–518 (2000). [hep-ph/9912329]

  134. A. Bostan, F. Chyzak, É. de Panafieu, Complexity Estimates for Two Uncoupling Algorithms. Proceedings of ISSAC’13, Boston, June 2013

    Google Scholar 

  135. B. Zürcher, Rationale Normalformen von pseudo-linearen Abbildungen. Master’s thesis, Mathematik, ETH Zürich (1994)

    Google Scholar 

  136. S. Gerhold, Uncoupling systems of linear Ore operator equations, Master’s thesis, RISC, J. Kepler University, Linz (2002)

    Google Scholar 

  137. S.A. Abramov and M. Petkovšek, D’Alembertian solutions of linear differential and difference equations, in Proc. ISSAC’94, ed. by J. von zur Gathen (ACM Press, 1994), pp. 169–174

    Google Scholar 

  138. M.F. Singer, Am. J. Math. 103(4), 661–682 (1981)

    Article  Google Scholar 

  139. J.J. Kovacic, J. Symb. Comput. 2, 3–43 (1986)

    Google Scholar 

  140. J. Ablinger, A. Behring, J. Blümlein, G. Falcioni, A. De Freitas, P. Marquard, N. Rana, C. Schneider, Phys. Rev. D 97(9), 094022 (2018). arXiv:1712.09889 [hep-ph]

    Google Scholar 

  141. A.V. Kotikov, The Property of maximal transcendentality in the N=4 Supersymmetric Yang-Mills, in Subtleties in quantum field theory, ed. by D. Diakonov, pp. 150–174. arXiv:1005.5029 [hep-th]

    Google Scholar 

  142. J.M. Henn, Phys. Rev. Lett. 110, 251601 (2013). [arXiv:1304.1806[hep-th]]

    Article  Google Scholar 

  143. J.M. Henn, J. Phys. A48, 153001 (2015). [arXiv:1412.2296[hep-ph]]

    Google Scholar 

  144. V.E. Zakharov, S.V. Manakov, S.P. Novikov, L.P. Pitaevskii, Teoria Solitonov: metod obratnoi zadatschi (Nauka, Moskva, 1980)

    Google Scholar 

  145. S.Yu. Sakovich, J. Phys. A Math. Gen. 28, 2861–2869 (1995)

    Article  Google Scholar 

  146. R.N. Lee, JHEP 04, 108 (2015). [arXiv:1411.0911[hep-ph]]

    Article  Google Scholar 

  147. M. Prausa, Comput. Phys. Commun. 219, 361–376 (2017). [arXiv:1701.00725 [hep-ph]]

  148. O. Gituliar, V. Magerya, Comput. Phys. Commun. 219, 329–338 (2017). [arXiv:1701.04269 [hep-ph]]

  149. C. Meyer, Comput. Phys. Commun. 222, 295–312 (2018). [arXiv:1705.06252 [hep-ph]]

  150. M. Besier, D. Van Straten, S. Weinzierl, Commun. Num. Theor. Phys. 13, 253–297 (2019). arXiv:1809.10983 [hep-th]

    Google Scholar 

  151. M. Besier, P. Wasser, S. Weinzierl, Comput. Phys. Commun. 253, 107197 (2020). arXiv:1910.13251 [cs.MS]

    Article  MathSciNet  Google Scholar 

  152. J. Ablinger, J. Blümlein, A. De Freitas, K. Schönwald, Nucl. Phys. B 955, 115045 (2020). arXiv:2004.04287 [hep-ph]; J. Blümlein, A. De Freitas, C. Raab, K. Schönwald, Nucl. Phys. B 956, 115055 (2020). arXiv:2003.14289 [hep-ph]; Phys. Lett. B 801, 135196 (2020). arXiv:1910.05759 [hep-ph]; Phys. Lett. B 791, 206–209 (2019). arXiv:1901.08018 [hep-ph]

    Article  Google Scholar 

  153. J. Blümlein, A. De Freitas, C.G. Raab, K. Schönwald, Nucl. Phys. B 945, 114659 (2019). arXiv:1903.06155 [hep-ph]; J. Blümlein, C. Raab, K. Schönwald, Nucl. Phys. B 948, 114736 (2019). arXiv:1904.08911 [hep-ph]

    Article  Google Scholar 

  154. G. Almkvist, D. Zeilberger, J. Symb. Comp. 10, 571–591 (1990)

    Google Scholar 

  155. M. Apagodu, D. Zeilberger, Adv. Appl. Math. (Special Regev Issue) 37, 139–152 (2006)

    Google Scholar 

  156. D.J. Broadhurst, D. Kreimer, Phys. Lett. B 393, 403–412 (1997). arXiv:hep-th/9609128 [hep-th]

    Google Scholar 

  157. N. Nielsen Nova Acta Leopold. XC(Nr. 3), 125–211 (1909); K.S. Kölbig, J.A. Mignoco, E. Remiddi, BIT 10, 38–74 (1970); K.S. Kölbig, SIAM J. Math. Anal. 17, 1232–1258 (1986)

    Google Scholar 

  158. M.E. Hoffman, J. Algebraic Combin. 11, 49–68 (2000). arXiv:math/9907173 [math.QA]

    Google Scholar 

  159. J. Blümlein, Comput. Phys. Commun. 159, 19–54 (2004). [hep-ph/0311046]

  160. J. Ablinger, J. Blümlein, C. Schneider, Phys. Rev. D 103(9), 096025 (2021) [hep-th]

    Google Scholar 

  161. A.I. Davydychev, M.Yu. Kalmykov, Nucl. Phys. B699, 3–64 (2004). [hep-th/0303162]

    Article  Google Scholar 

  162. S. Weinzierl, J. Math. Phys. 45, 2656–2673 (2004). [hep-ph/0402131]

  163. C. Reutenauer, Free Lie Algebras (Calendron Press, Oxford, 1993)

    MATH  Google Scholar 

  164. T. Gehrmann, E. Remiddi, Comput. Phys. Commun. 141, 296–312 (2001). [hep-ph/0107173]

  165. J. Vollinga, S. Weinzierl, Comput. Phys. Commun. 167, 177–194 (2005). arXiv:hep-ph/0410259 [hep-ph]

    Google Scholar 

  166. J. Ablinger, J. Blümlein, A. De Freitas, C. Schneider, K. Schönwald, Nucl. Phys. B 927, 339–367 (2018). arXiv:1711.06717 [hep-ph]

    Google Scholar 

  167. J. Ablinger, J. Blümlein, A. De Freitas, A. Goedicke, C. Schneider, K. Schönwald, Nucl. Phys. B 932, 129–240 (2018). arXiv:1804.02226 [hep-ph]

    Google Scholar 

  168. C.G. Raab. On the arithmetic of d’Alembertian functions, presentation at the 19th Conference on Applications of Computer Algebra (ACA 2013), Málaga, Spain, 2–6 July 2013, in preparation

    Google Scholar 

  169. Li Guo, G. Regensburger, M. Rosenkranz, J. Pure Appl. Algebra 218, 456–473 (2014)

    Google Scholar 

  170. D.E. Radford, J. Algebra 58, 432–454 (1979)

    Article  MathSciNet  Google Scholar 

  171. R.C. Lyndon, Trans. Am. Math. Soc. 77, 202–215 (1954); 78, 329–332 (1955)

    Google Scholar 

  172. E. Witt, Journ. Reine und Angew. Mathematik 177, 152–160 (1937); Math. Zeitschr. 64, 195–216 (1956)

    Google Scholar 

  173. J. Blümlein, Clay Math. Proc. 12, 167–188 (2010). [arXiv:0901.0837[math-ph]]

    MathSciNet  Google Scholar 

  174. J. Blümlein, Nucl. Phys. B Proc. Suppl. 135, 225–231 (2004). arXiv:hep-ph/0407044 [hep-ph]

    Google Scholar 

  175. D.J. Broadhurst, J. Fleischer, O. V. Tarasov, Z. Phys. C60, 287–302 (1993). [hep-ph/9304303]

  176. S. Bloch, P. Vanhove, J. Number Theor. 148, 328–364 (2015). [hep-th/ 1309.5865]

  177. S. Laporta, E. Remiddi, Nucl. Phys. B704, 349–386 (2005). [hep-ph/0406160]

  178. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 54, 052303 (2013). [hep-ph/1302.7004]

    Article  MathSciNet  Google Scholar 

  179. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 55(10), 102301 (2014). [hep-ph/1405.5640]

  180. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 56(7), 072303 (2015). [hep-ph/1504.03255]

  181. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 57(3), 032304 (2016). [hep-ph/1512.05630]

  182. A. Sabry, Nucl. Phys. 33, 401–430 (1962)

    Article  Google Scholar 

  183. E. Remiddi, L. Tancredi, Nucl. Phys. B907, 400–444 (2016). [arXiv:1602.01481[hep-ph]]

    Article  Google Scholar 

  184. L. Adams, C. Bogner, A. Schweitzer, S. Weinzierl, J. Math. Phys. 57(12), 122302 (2016). [hep-ph/1607.01571]

  185. J. Grigo, J. Hoff, P. Marquard, M. Steinhauser, Nucl. Phys. B 864, 580–596 (2012). arXiv:1206.3418 [hep-ph]

    Google Scholar 

  186. J. Blümlein, A. De Freitas, M. Van Hoeij, E. Imamoglu, P. Marquard, C. Schneider, in PoS (LL2018), 017. arXiv:1807.05287 [hep-ph]

    Google Scholar 

  187. 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]

    Google Scholar 

  188. A. Ronveaux, (ed.), Heun’s Differential Equations (The Clarendon Press Oxford, Oxford, 1995)

    MATH  Google Scholar 

  189. E. Imamoglu, M. van Hoeij, J. Symb. Comput. 83, 245–271 (2017). arXiv:1606.01576 [cs.SC]

    Google Scholar 

  190. K. Takeuchi, J. Fac. Sci, Univ. Tokyo, Sect. 1A 24, 201–272 (1977)

    Google Scholar 

  191. F.G. Tricomi, Elliptische Funktionen (Geest & Portig, Leipzig, 1948); übersetzt und bearbeitet von M. Krafft

    Google Scholar 

  192. E.T. Whittaker, G.N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, 1996); reprint of 4th edition (1927)

    Google Scholar 

  193. S. Herfurtner, Math. Ann. 291, 319–342 (1991)

    Article  MathSciNet  Google Scholar 

  194. H. Movasati, S. Reiter, Bull. Braz. Math Soc. 43, 423–442 (2012). arXiv: 0902.0760[math.AG]

    Google Scholar 

  195. J. Blümlein, talks at: The 5th International Congress on Mathematical Software ZIB Berlin from July 11 to July 14, 2016, Session: Symbolic computation and elementary particle physics. https://www.risc.jku.at/conferences/ICMS2016/; and QCD@LHC2016, U. Zürich, August 22 to August 26, 2016. https://indico.cern.ch/event/516210/timetable/#all.detailed

  196. E. Remiddi, L. Tancredi, Nucl. Phys. B925, 212–251 (2017). [arXiv: 1709.03622[hep-ph]]

  197. L. Adams, S. Weinzierl, Phys. Lett. B781, 270–278 (2018). [arXiv: 1802. 05020[hep-ph]]

  198. J.-P. Serre, A Course in Arithmetic (Springer, Berlin, 1973)

    Book  MATH  Google Scholar 

  199. H. Cohen, F. Strömberg, Modular Forms, A Classical Approach. Graduate Studies in Mathematics, vol. 179 (AMS, providence, RI, 2017)

    Google Scholar 

  200. K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series. CBMS Regional Conference Series in Mathematics, vol. 102 (AMS, Providence, RI, 2004)

    Google Scholar 

  201. H.H. Chan, W. Zudilin, Mathematika 56, 107–117 (2010)

    Article  MathSciNet  Google Scholar 

  202. D.J. Broadhurst, Eta quotients, Eichler integrals and L-series, talk at HMI Bonn, February 2018

    Google Scholar 

  203. J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, JHEP 1808, 014 (2018). arXiv:1803.10256 [hep-th]

    Article  Google Scholar 

  204. L. Adams, S. Weinzierl, Commun. Num. Theor. Phys. 12, 193–251 (2018). arXiv:1704.08895 [hep-ph]

    Google Scholar 

  205. J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, JHEP 05, 120 (2019). arXiv:1902.09971 [hep-ph]; JHEP 01, 023 (2019). arXiv:1809.10698 [hep-th]; J. Broedel, C. Duhr, F. Dulat, L. Tancredi, JHEP 05, 093 (2018). arXiv:1712.07089 [hep-th]; Phys. Rev. D 97(11), 116009 (2018). arXiv:1712.07095 [hep-ph]

    Article  Google Scholar 

  206. K. Martin, Modular Forms. Lecture Notes (2016) U. Oklahoma, http://www2.math.ou.edu/~kmartin/mfs/; G.H. Hardy, S. Ramanujan, Proc. Lond. Math. Soc. (2) 17, 75–115 (1918); H. Rademacher, Proc. Lond. Math. Soc. (2) 43, 241–254 (1937); Ann. Math. 44, 416–422 (1943); J.H. Brunier, K. Ono, Adv. Math. 246, 198–219 (2013)

  207. S. Laporta, Phys. Lett. B 772, 232–238 (2017). arXiv:1704.06996 [hep-ph]

    Google Scholar 

  208. J. Blümlein, A. Vogt, Phys. Rev. D 58, 014020 (1998). arXiv:hep-ph/9712546

    Article  Google Scholar 

  209. J. Blümlein, V. Ravindran, W.L. van Neerven, Nucl. Phys. B 586, 349–381 (2000). arXiv:hep-ph/0004172

    Google Scholar 

  210. J. Blümlein, A. Guffanti, Nucl. Phys. Proc. Suppl. 152, 87–91 (2006). arXiv:hep-ph/0411110

    Google Scholar 

  211. J. Blümlein, Comput. Phys. Commun. 133, 76–104 (2000). [hep-ph/0003100]

  212. J. Blümlein, S.-O. Moch, Phys. Lett. B614, 53–61 (2005). [hep-ph/0503188]

  213. N. Nielsen, Handbuch der Theorie der Gammafunktion (Teubner, Leipzig, 1906); reprinted by (Chelsea Publishing Company, Bronx, New York, 1965)

    Google Scholar 

  214. E. Landau, Über die Grundlagen der Theorie der Fakultätenreihen. S.-Ber. math.-naturw. Kl. Bayerische Akad. Wiss. München 36, 151–218 (1906)

    MATH  Google Scholar 

  215. A.V. Kotikov, V.N. Velizhanin, Analytic continuation of the Mellin moments of deep inelastic structure functions. arXiv:hep-ph/0501274

    Google Scholar 

  216. H.D. Politzer, Phys. Rept. 14, 129–180 (1974)

    Article  Google Scholar 

  217. J. Blümlein, N. Kochelev, Nucl. Phys. B 498, 285–309 (1997). arXiv:hep-ph/9612318 [hep-ph]

    Google Scholar 

  218. S. Weinzierl, Comput. Phys. Commun. 145, 357–370 (2002). [math-ph/0201011]

  219. S. Moch, P. Uwer, Comput. Phys. Commun. 174, 759–770 (2006). [math-ph/0508008]

  220. C. Duhr, F. Dulat, PolyLogTools - Polylogs for the Masses. arXiv:1904.07279 [hep-th]

    Google Scholar 

  221. D. Maitre, Comput. Phys. Commun. 174, 222–240 (2006). [hep-ph/0507152]

  222. J. Ablinger, J. Blümlein, M. Round, C. Schneider, Comput. Phys. Commun. 240, 189–201 (2019). arXiv:1809.07084 [hep-ph]

    Google Scholar 

  223. C. Bogner, A. Schweitzer, S. Weinzierl, Nucl. Phys. B 922, 528–550 (2017). arXiv:1705.08952 [hep-ph]

    Google Scholar 

  224. G. Passarino, Eur. Phys. J. C 77(2), 77 (2017). arXiv:1610.06207 [math-ph]; M. Walden, S. Weinzierl, Numerical evaluation of iterated integrals related to elliptic Feynman integrals. arXiv:2010.05271 [hep-ph]

    Google Scholar 

  225. K. Melnikov, A. Vainshtein, Springer Tracts Mod. Phys. 216, 1–176 (2006); F. Jegerlehner, A. Nyffeler, Phys. Rept. 477, 1–110 (2009). arXiv:0902.3360 [hep-ph]; F. Jegerlehner, Springer Tracts Mod. Phys. 274, 1–711 (2018)

    Google Scholar 

  226. T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, Phys. Rev. Lett. 109, 111807 (2012). arXiv:1205.5368 [hep-ph]; Phys. Rev. Lett. 109, 111808 (2012). arXiv:1205.5370 [hep-ph]

    Article  Google Scholar 

  227. P.A. Baikov, A. Maier, P. Marquard, Nucl. Phys. B 877, 647–661 (2013). arXiv:1307.6105 [hep-ph]

    Google Scholar 

  228. A. Kurz, T. Liu, P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser, EPJ Web Conf. 118, 01033 (2016). arXiv:1511.08222 [hep-ph]

    Article  Google Scholar 

  229. A. Kurz, T. Liu, P. Marquard, A. Smirnov, V. Smirnov, M. Steinhauser, Phys. Rev. D 93(5), 053017 (2016). arXiv:1602.02785 [hep-ph]

    Google Scholar 

  230. P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser, D. Wellmann, EPJ Web Conf. 218, 01004 (2019). arXiv:1708.07138 [hep-ph]

    Article  Google Scholar 

  231. S. Volkov, Phys. Rev. D 100(9), 096004 (2019). arXiv:1909.08015 [hep-ph]

    Google Scholar 

  232. P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser, Phys. Rev. Lett. 114(14), 142002 (2015). arXiv:1502.01030 [hep-ph]; P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser, D. Wellmann, Phys. Rev. D 94(7), 074025 (2016). arXiv:1606.06754 [hep-ph]; S. Laporta, Phys. Lett. B 802, 135264 (2020). arXiv:2001.02739 [hep-ph]; Y. Schröder, M. Steinhauser, JHEP 01, 051 (2006). arXiv:hep-ph/0512058 [hep-ph]; K.G. Chetyrkin, J.H. Kühn, C. Sturm, Nucl. Phys. B 744, 121–135 (2006). arXiv:hep-ph/0512060 [hep-ph]; T. Liu, M. Steinhauser, Phys. Lett. B 746, 330–334 (2015). arXiv:1502.04719 [hep-ph]; M. Fael, K. Schönwald, M. Steinhauser, Phys. Rev. Lett. 125(5), 052003 (2020). arXiv:2005.06487 [hep-ph]; JHEP 10, 087 (2020). arXiv:2008.01102 [hep-ph]; Third order corrections to the semi-leptonic b → c and the muon decays. arXiv:2011.13654 [hep-ph]

    Google Scholar 

  233. J. Ablinger, J. Blümlein, S. Klein, C. Schneider, F. Wißbrock, Nucl. Phys. B 844, 26–54 (2011). arXiv:1008.3347 [hep-ph]; J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, F. Wißbrock, Nucl. Phys. B 882, 263–288 (2014). arXiv:1402.0359 [hep-ph]; J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, F. Wißbrock, Nucl. Phys. B886, 733–823 (2014). [arXiv:1406.4654[hep-ph]]; J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, Nucl. Phys. B 890, 48–151 (2014). arXiv:1409.1135 [hep-ph]; A. Behring, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, C. Schneider, Phys. Rev. D 92(11), 114005 (2015). arXiv:1508.01449 [hep-ph]; A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, Nucl. Phys. B 897, 612–644 (2015). arXiv:1504.08217 [hep-ph]; J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, C. Schneider, F. Wißbrock, Nucl. Phys. B 921, 585–688 (2017). arXiv:1705.07030 [hep-ph]; A. Behring, J. Blümlein, G. Falcioni, A. De Freitas, A. von Manteuffel, C. Schneider, Phys. Rev. D 94(11), 114006 (2016). arXiv:1609.06255 [hep-ph]; A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, K. Schönwald, C. Schneider, Nucl. Phys. B 964, 115331 (2021). arXiv:2101.05733 [hep-ph]; J. Ablinger, J. Blümlein, A. De Freitas, A. Goedicke, M. Saragnese, C. Schneider, K. Schönwald, Nucl. Phys. B 955, 115059 (2020). arXiv:2004.08916 [hep-ph]; J. Ablinger, J. Blümlein, A. De Freitas, M. Saragnese, C. Schneider, K. Schönwald, Nucl. Phys. B 952, 114916 (2020). arXiv:1911.11630 [hep-ph]

  234. C. Anastasiou, C. Duhr, F. Dulat, F. Herzog, B. Mistlberger, Phys. Rev. Lett. 114, 212001 (2015). arXiv:1503.06056 [hep-ph]

    Article  Google Scholar 

  235. C. Duhr, F. Dulat, B. Mistlberger, Phys. Rev. Lett. 125(17), 172001 (2020). arXiv:2001.07717 [hep-ph]

    Google Scholar 

  236. C. Duhr, F. Dulat, B. Mistlberger, JHEP 11, 143 (2020). arXiv:2007.13313 [hep-ph]

    Article  Google Scholar 

  237. M. Czakon, P. Fiedler, A. Mitov, Phys. Rev. Lett. 110, 252004 (2013). arXiv:1303.6254 [hep-ph]

    Article  Google Scholar 

  238. S. Alekhin, J. Blümlein, S. Moch, Phys. Rev. D 86, 054009 (2012). arXiv:1202.2281 [hep-ph]; A. Accardi, et al. Eur. Phys. J. C 76(8), 471 (2016). arXiv:1603.08906 [hep-ph]; S. Alekhin, J. Blümlein, S. Moch, R. Placakyte, Phys. Rev. D 96(1), 014011 (2017). arXiv:1701.05838 [hep-ph]

    Article  Google Scholar 

  239. A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss, T.A. Morgan, Phys. Rev. Lett. 117(2), 022001 (2016). arXiv:1507.02850 [hep-ph]

    Google Scholar 

  240. R. Boughezal, J.M. Campbell, R.K. Ellis, C. Focke, W.T. Giele, X. Liu, F. Petriello, Phys. Rev. Lett. 116(15), 152001 (2016). arXiv:1512.01291 [hep-ph]

    Google Scholar 

  241. J. Currie, T. Gehrmann, A. Huss, J. Niehues, JHEP 07, 018 (2017). [Erratum: JHEP 12, 042 (2020)] arXiv:1703.05977 [hep-ph]; V. Andreev et al. [H1], Eur. Phys. J. C 77(11), 791 (2017). arXiv:1709.07251 [hep-ex]

    Article  Google Scholar 

  242. J. Blümlein, A. Guffanti, Nucl. Phys. B Proc. Suppl. 152, 87–91 (2006). arXiv:hep-ph/0411110 [hep-ph]; J. Blümlein, H. Böttcher, A. Guffanti, Nucl. Phys. B 774, 182–207 (2007). arXiv:hep-ph/0607200 [hep-ph]; J. Blümlein, M. Saragnese, Phys. Lett. B 820, 136589 (2021) [arXiv:2107.01293 [hep-ph]]

    Google Scholar 

  243. S. Alekhin, J. Blümlein, K. Daum, K. Lipka, S. Moch, Phys. Lett. B 720, 172–176 (2013). arXiv:1212.2355 [hep-ph]

    Google Scholar 

  244. S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren, A. Vogt, JHEP 10, 041 (2017). arXiv:1707.08315 [hep-ph]

    Article  Google Scholar 

  245. V.S. Fadin, E.A. Kuraev, L.N. Lipatov, Phys. Lett. B 60, 50–52 (1975); S. Catani, F. Hautmann, Nucl. Phys. B 427, 475–524 (1994). arXiv:hep-ph/9405388 [hep-ph]; V.S. Fadin, L.N. Lipatov, Phys. Lett. B 429, 127–134 (1998). arXiv:hep-ph/9802290 [hep-ph]

    Google Scholar 

  246. P. Janot, JHEP 02, 053 (2016). [Erratum: JHEP 11, 164 (2017)] arXiv:1512.05544 [hep-ph]

    Article  Google Scholar 

  247. J. Blümlein, A. De Freitas, K. Schönwald, Phys. Lett. B 816, 136250 (2021) [arXiv:2102.12237 [hep-ph]]

    Google Scholar 

  248. M. Beneke, Y. Kiyo, A. Maier, J. Piclum, Comput. Phys. Commun. 209, 96–115 (2016). arXiv:1605.03010 [hep-ph]

    Google Scholar 

  249. M. Beneke, A. Maier, T. Rauh, P. Ruiz-Femenia, JHEP 02, 125 (2018). arXiv:1711.10429 [hep-ph]

    Article  Google Scholar 

  250. M. Beneke, Y. Kiyo, P. Marquard, A. Penin, J. Piclum, M. Steinhauser, Phys. Rev. Lett. 115(19), 192001 (2015). arXiv:1506.06864 [hep-ph]

    Google Scholar 

  251. F. Bach, B.C. Nejad, A. Hoang, W. Kilian, J. Reuter, M. Stahlhofen, T. Teubner, C. Weiss, JHEP 03, 184 (2018). arXiv:1712.02220 [hep-ph]

    Article  Google Scholar 

  252. A.H. Hoang, M. Stahlhofen, JHEP 05, 121 (2014). arXiv:1309.6323 [hep-ph]

    Article  Google Scholar 

  253. K. Seidel, F. Simon, M. Tesar, S. Poss, Eur. Phys. J. C 73(8), 2530 (2013). arXiv:1303.3758 [hep-ex]

    Google Scholar 

  254. F. Simon, in PoS (ICHEP2016), 872. arXiv:1611.03399 [hep-ex]

    Google Scholar 

  255. I. Dubovyk, A. Freitas, J. Gluza, T. Riemann, J. Usovitsch, JHEP 08, 113 (2019). arXiv:1906.08815 [hep-ph]; Phys. Lett. B 783, 86–94 (2018). arXiv:1804.10236 [hep-ph]

    Article  Google Scholar 

  256. G. Heinrich, Collider Physics at the Precision Frontier. arXiv:2009.00516 [hep-ph]

    Google Scholar 

  257. S. Foffa, P. Mastrolia, R. Sturani, C. Sturm, W.J. Torres Bobadilla, Phys. Rev. Lett. 122(24), 241605 (2019). arXiv:1902.10571 [gr-qc]; J. Blümlein, A. Maier, P. Marquard, Phys. Lett. B 800, 135100 (2020). arXiv:1902.11180 [gr-qc]; J. Blümlein, A. Maier, P. Marquard, G. Schäfer, Nucl. Phys. B 965, 115352 (2021). arXiv:2010.13672 [gr-qc]

    Google Scholar 

  258. J. Blümlein, V. Ravindran, Nucl. Phys. B 716, 128–172 (2005). arXiv:hep-ph/0501178 [hep-ph]

    Google Scholar 

  259. J. Blümlein, V. Ravindran, Nucl. Phys. B 749, 1–24 (2006). arXiv:hep-ph/0604019 [hep-ph]

    Google Scholar 

  260. C. Neumann, Vorlesungen über Riemann’s Theorie der Abel’schen Integrale, 2nd edn. (Teubner, Leipzig, 1884)

    MATH  Google Scholar 

  261. F. Brown, O. Schnetz, Duke Math. J. 161(10), 1817–1862 (2012)

    Article  MathSciNet  Google Scholar 

  262. K. Bönisch, C. Duhr, F. Fischbach, A. Klemm, C. Nega, Feynman integrals in dimensional regularization and extensions of Calabi-Yau motives [arXiv:2108.05310 [hep-th]].

    Google Scholar 

  263. D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (Teubner, Leipzig, 1912)

    MATH  Google Scholar 

  264. R. de L. Kronig, J. Opt. Soc. Am. 12, 547–557 (1926)

    Google Scholar 

  265. H.A. Kramers, Atti Cong. Intern. Fisici (Transactions of Volta Centenary Congress) Como 2 (1927)

    Google Scholar 

  266. S. Abreu, R. Britto, C. Duhr, E. Gardi, JHEP 12, 090 (2017). [arXiv:1704.07931[hep-th]]

    Article  Google Scholar 

  267. M.J.G. Veltman, Physica 29, 186–207 (1963)

    Article  MathSciNet  Google Scholar 

  268. E. Remiddi, Helv. Phys. Acta 54, 364–382 (1982); E. Remiddi, Differential Equations and Dispersion Relations for Feynman Amplitudes, in Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, ed. by J. Blümlein, C. Schneider, P. Paule (Springer, Wien, 2019), pp. 391–414

    Chapter  Google Scholar 

Download references

Acknowledgements

I would like to thank to all who have contributed to the present volume and all my colleagues with whom I have had countless fruitful discussions on the present topic during the last 30 years. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie grant agreement No. 764850, SAGEX.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Johannes Blümlein .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Blümlein, J. (2021). Analytic Integration Methods in Quantum Field Theory: An Introduction. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-80219-6_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-80218-9

  • Online ISBN: 978-3-030-80219-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics