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Feynman integrals and intersection theory
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  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 21 February 2019

Feynman integrals and intersection theory

  • Pierpaolo Mastrolia  ORCID: orcid.org/0000-0001-9711-77981,2 &
  • Sebastian Mizera3,4 

Journal of High Energy Physics volume 2019, Article number: 139 (2019) Cite this article

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A preprint version of the article is available at arXiv.

Abstract

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop.

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References

  1. K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].

    ADS  Article  Google Scholar 

  2. G. Barucchi and G. Ponzano, Differential equations for one-loop generalized Feynman integrals, J. Math. Phys. 14 (1973) 396 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  3. A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  4. A.V. Kotikov, Differential equation method: the calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [Erratum ibid. B 295 (1992) 409] [INSPIRE].

  5. Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].

  6. E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].

    ADS  Google Scholar 

  7. T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].

  8. J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].

    ADS  Article  Google Scholar 

  9. J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  10. S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].

  11. S. Laporta, Calculation of Feynman integrals by difference equations, Acta Phys. Polon. B 34 (2003) 5323 [hep-ph/0311065] [INSPIRE].

  12. O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  13. R.N. Lee, Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys. B 830 (2010) 474 [arXiv:0911.0252] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1 [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  15. A.G. Grozin, Integration by parts: an introduction, Int. J. Mod. Phys. A 26 (2011) 2807 [arXiv:1104.3993] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. Y. Zhang, Lecture notes on multi-loop integral reduction and applied algebraic geometry, arXiv:1612.02249 [INSPIRE].

  17. A. von Manteuffel and R.M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744 (2015) 101 [arXiv:1406.4513] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP 12 (2016) 030 [arXiv:1608.01902] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. J. Böhm, A. Georgoudis, K.J. Larsen, H. Schönemann and Y. Zhang, Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP 09 (2018) 024 [arXiv:1805.01873] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  20. D.A. Kosower, Direct solution of integration-by-parts systems, Phys. Rev. D 98 (2018) 025008 [arXiv:1804.00131] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  21. H.A. Chawdhry, M.A. Lim and A. Mitov, Two-loop five-point massless QCD amplitudes within the IBP approach, arXiv:1805.09182 [INSPIRE].

  22. A. von Manteuffel and C. Studerus, Reduze 2 — distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].

  23. R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].

  24. A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  25. P. Maierhöfer, J. Usovitsch and P. Uwer, Kira — a Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].

    ADS  Article  Google Scholar 

  26. A. Georgoudis, K.J. Larsen and Y. Zhang, Azurite: an algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun. 221 (2017) 203 [arXiv:1612.04252] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett. 116 (2016) 062001 [Erratum ibid. 116 (2016) 189903] [arXiv:1511.05409] [INSPIRE].

  28. S. Laporta, High-precision calculation of the 4-loop contribution to the electron g-2 in QED, Phys. Lett. B 772 (2017) 232 [arXiv:1704.06996] [INSPIRE].

    ADS  Article  Google Scholar 

  29. S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, First look at two-loop five-gluon scattering in QCD, Phys. Rev. Lett. 120 (2018) 092001 [arXiv:1712.02229] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  30. S. Abreu, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Planar two-loop five-parton amplitudes from numerical unitarity, JHEP 11 (2018) 116 [arXiv:1809.09067] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  31. K. Aomoto and M. Kita, Theory of hypergeometric functions, Springer, Tokyo, Japan (2011).

    Book  MATH  Google Scholar 

  32. P.A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth. A 389 (1997) 347 [hep-ph/9611449] [INSPIRE].

  33. R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. M. Marcolli, Motivic renormalization and singularities, Clay Math. Proc. 11 (2010) 409 [arXiv:0804.4824] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  35. M. Marcolli, Feynman motives, World Scientific, Singapore (2010).

    MATH  Google Scholar 

  36. T. Bitoun, C. Bogner, R.P. Klausen and E. Panzer, Feynman integral relations from parametric annihilators, Lett. Math. Phys. 109 (2019) 497 [arXiv:1712.09215] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  37. K.J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  38. J. Bosma, K.J. Larsen and Y. Zhang, Differential equations for loop integrals in Baikov representation, Phys. Rev. D 97 (2018) 105014 [arXiv:1712.03760] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  39. H. Frellesvig and C.G. Papadopoulos, Cuts of Feynman integrals in Baikov representation, JHEP 04 (2017) 083 [arXiv:1701.07356] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  40. M. Zeng, Differential equations on unitarity cut surfaces, JHEP 06 (2017) 121 [arXiv:1702.02355] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. J. Bosma, M. Sogaard and Y. Zhang, Maximal cuts in arbitrary dimension, JHEP 08 (2017) 051 [arXiv:1704.04255] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  42. M. Harley, F. Moriello and R.M. Schabinger, Baikov-Lee representations of cut Feynman integrals, JHEP 06 (2017) 049 [arXiv:1705.03478] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  43. M. Kita and M. Yoshida, Intersection theory for twisted cycles, Math. Nachr. 166 (1994) 287.

    MathSciNet  Article  MATH  Google Scholar 

  44. K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Math. J. 139 (1995) 67.

    MathSciNet  Article  MATH  Google Scholar 

  45. S. Mizera, Scattering amplitudes from intersection theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  46. S. Abreu, R. Britto, C. Duhr and E. Gardi, Algebraic structure of cut Feynman integrals and the diagrammatic coaction, Phys. Rev. Lett. 119 (2017) 051601 [arXiv:1703.05064] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  47. R.N. Lee and V.A. Smirnov, The dimensional recurrence and analyticity method for multicomponent master integrals: using unitarity cuts to construct homogeneous solutions, JHEP 12 (2012) 104 [arXiv:1209.0339] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  48. E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys. B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].

  49. A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys. B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  50. A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys. B 921 (2017) 316 [arXiv:1704.05465] [INSPIRE].

  51. S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].

  52. A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP 06 (2017) 127 [arXiv:1701.05905] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  53. L. Adams, E. Chaubey and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, JHEP 10 (2018) 206 [arXiv:1806.04981] [INSPIRE].

    ADS  Article  Google Scholar 

  54. M. Argeri et al., Magnus and Dyson series for master integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  55. L. Adams, E. Chaubey and S. Weinzierl, Simplifying differential equations for multiscale Feynman integrals beyond multiple polylogarithms, Phys. Rev. Lett. 118 (2017) 141602 [arXiv:1702.04279] [INSPIRE].

    ADS  Article  Google Scholar 

  56. J. Gluza, K. Kajda and D.A. Kosower, Towards a basis for planar two-loop integrals, Phys. Rev. D 83 (2011) 045012 [arXiv:1009.0472] [INSPIRE].

    ADS  Google Scholar 

  57. E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; the master integrals for the two-loop massive sunrise graph, Nucl. Phys. B 880 (2014) 343 [arXiv:1311.3342] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  58. R.N. Lee, Calculating multiloop integrals using dimensional recurrence relation and D-analyticity, Nucl. Phys. Proc. Suppl. 205-206 (2010) 135 [arXiv:1007.2256] [INSPIRE].

  59. K. Aomoto, Les équations aux différences linéaires et les intégrales des fonctions multiformes (in French), J. Fac. Sci. Univ. Tokyo Sect. IA 22 (1975) 271.

  60. I.M. Gelfand, General theory of hypergeometric functions, Dokl. Akad. Nauk SSSR 288 (1986) 14.

    MathSciNet  Google Scholar 

  61. J. Milnor, Morse theory, Ann. Math. Stud. 51, Princeton University Press, Princeton, NJ, U.S.A. (2016).

  62. K. Aomoto, On the structure of integrals of power product of linear functions, Sci. Papers College Gen. Ed. Univ. Tokyo 27 (1977) 49.

    MathSciNet  MATH  Google Scholar 

  63. K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo 27 (1981) 265.

    MathSciNet  MATH  Google Scholar 

  64. K. Aomoto, M. Kita, P. Orlik and H. Terao, Twisted de Rham cohomology groups of logarithmic forms, Adv. Math. 128 (1997) 119.

    MathSciNet  Article  MATH  Google Scholar 

  65. P. Orlik and H. Terao, Arrangements of hyperplanes, Springer, Berlin Heidelberg, Germany (1992).

  66. K. Aomoto and Y. Machida, Double filtration of twisted logarithmic complex and Gauss-Manin connection, J. Math. Soc. Jpn. 67 (2015) 609.

    MathSciNet  Article  MATH  Google Scholar 

  67. N. Arkani-Hamed, Y. Bai and T. Lam, Positive geometries and canonical forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  68. M. Kita and M. Yoshida, Intersection theory for twisted cycles II — degenerate arrangements, Math. Nachr. 168 (2006) 171.

    MathSciNet  Article  MATH  Google Scholar 

  69. K. Matsumoto, Quadratic identities for hypergeometric series of type (k, l), Kyushu J. Math. 48 (1994) 335.

    MathSciNet  Article  MATH  Google Scholar 

  70. K. Cho, K. Mimachi and M. Yoshida, A hypergeometric integral attached to the configuration of the mirrors of the reflection group S N+2 acting on P n, Kyushu J. Math. 49 (1995) 11.

    MathSciNet  Article  MATH  Google Scholar 

  71. K. Matsumoto, Intersection numbers for 1-forms associated with confluent hypergeometric functions, Funkcial. Ekvac. 41 (1998) 291.

    MathSciNet  MATH  Google Scholar 

  72. H. Majima, K. Matsumoto and N. Takayama, Quadratic relations for confluent hypergeometric functions, Tohoku Math. J. 52 (2000) 489.

    MathSciNet  Article  MATH  Google Scholar 

  73. K. Ohara, Y. Sugiki and N. Takayama, Quadratic relations for generalized hypergeometric functions p F p−1, Funkcial. Ekvac. 46 (2003) 213.

    MathSciNet  Article  MATH  Google Scholar 

  74. M. Yoshida, Hypergeometric functions, my love, Vieweg+Teubner Verlag, Wiesbaden, Germany (1997).

  75. Y. Goto, Twisted cycles and twisted period relations for Lauricella’s hypergeometric function F C, Int. J. Math. 24 (2013) 1350094 [arXiv:1308.5535].

    Article  MATH  Google Scholar 

  76. Y. Goto, Intersection numbers and twisted period relations for the generalized hypergeometric function m+1 F m, Kyushu J. Math. 69 (2015) 203.

    MathSciNet  Article  MATH  Google Scholar 

  77. Y. Goto and K. Matsumoto, The monodromy representation and twisted period relations for Appell’s hypergeometric function f 4, Nagoya Math. J. 217 (2015) 61.

    MathSciNet  Article  MATH  Google Scholar 

  78. Y. Goto, Twisted period relations for Lauricella’s hypergeometric functions F A, Osaka J. Math. 52 (2015) 861.

    MathSciNet  MATH  Google Scholar 

  79. K. Matsumoto, Monodromy and Pfaffian of Lauricella’s F D in terms of the intersection forms of twisted (co)homology groups, Kyushu J. Math. 67 (2013) 367.

    MathSciNet  Article  MATH  Google Scholar 

  80. Y. Goto, Contiguity relations of Lauricella’s F D revisited, Tohoku Math. J. 69 (2017) 287.

  81. S. Mizera, Combinatorics and topology of Kawai-Lewellen-Tye relations, JHEP 08 (2017) 097 [arXiv:1706.08527] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  82. K. Matsumoto, Relative twisted homology and cohomology groups associated with Lauricella’s F D, arXiv:1804.00366.

  83. K. Matsumoto, Intersection numbers for logarithmic k-forms, Osaka J. Math. 35 (1998) 873.

    MathSciNet  MATH  Google Scholar 

  84. K. Ohara, Intersection numbers of twisted cohomology groups associated with Selberg-type integrals, (1998).

  85. P. Deligne and G.D. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. I.H.É.S. 63 (1986) 5.

  86. R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  87. C.G. Papadopoulos, Simplified differential equations approach for master integrals, JHEP 07 (2014) 088 [arXiv:1401.6057] [INSPIRE].

    ADS  Article  Google Scholar 

  88. J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP 05 (2018) 093 [arXiv:1712.07089] [INSPIRE].

  89. K. Matsumoto, Relative twisted homology and cohomology groups associated with Lauricella’s F D, arXiv:1804.00366.

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Authors and Affiliations

  1. Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131, Padova, Italy

    Pierpaolo Mastrolia

  2. INFN, Sezione di Padova, Via Marzolo 8, 35131, Padova, Italy

    Pierpaolo Mastrolia

  3. Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada

    Sebastian Mizera

  4. Department of Physics & Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Sebastian Mizera

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  1. Pierpaolo Mastrolia
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Correspondence to Pierpaolo Mastrolia.

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ArXiv ePrint: 1810.03818

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Mastrolia, P., Mizera, S. Feynman integrals and intersection theory. J. High Energ. Phys. 2019, 139 (2019). https://doi.org/10.1007/JHEP02(2019)139

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  • Received: 31 October 2018

  • Revised: 22 January 2019

  • Accepted: 11 February 2019

  • Published: 21 February 2019

  • DOI: https://doi.org/10.1007/JHEP02(2019)139

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Keywords

  • Scattering Amplitudes
  • Differential and Algebraic Geometry
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