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Multiple Series Representations of N-fold Mellin-Barnes Integrals

B. Ananthanarayan and Sumit Banik

Samuel Friot

Shayan Ghosh

  • Centre for High Energy Physics, Indian Institute of Science, Bangalore-560012, Karnataka, India

  • Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France and Université Lyon, Université Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622 Villeurbanne, France

  • Helmholtz-Institut für Strahlen- und Kernphysik, D-53115 Bonn, Germany and Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany

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Phys. Rev. Lett. 127, 151601 – Published 5 October, 2021

DOI: https://doi.org/10.1103/PhysRevLett.127.151601

Abstract

Mellin-Barnes (MB) integrals are well-known objects appearing in many branches of mathematics and physics, ranging from hypergeometric functions theory to quantum field theory, solid-state physics, asymptotic theory, etc. Although MB integrals have been studied for more than one century, until now there has been no systematic computational technique of the multiple series representations of N-fold MB integrals for N>2. Relying on a simple geometrical analysis based on conic hulls, we show here a solution to this important problem. Our method can be applied to resonant (i.e., logarithmic) and nonresonant cases and, depending on the form of the MB integrand, it gives rise to convergent series representations or diverging asymptotic ones. When convergent series are obtained, the method also allows, in general, the determination of a single “master series” for each series representation, which considerably simplifies convergence studies and/or numerical checks. We provide, along with this Letter, a Mathematica implementation of our technique with examples of applications. Among them, we present the first evaluation of the hexagon and double box conformal Feynman integrals with unit propagator powers.

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Physics Subject Headings (PhySH)

See Also

Massive one-loop conformal Feynman integrals and quadratic transformations of multiple hypergeometric series

B. Ananthanarayan, Sumit Banik, Samuel Friot, and Shayan Ghosh
Phys. Rev. D 103, 096008 (2021)

Article Text

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References (73)

  1. P. Appell and J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques - Polynômes d’Hermite (Gautiers-Villars, Paris, 1926).
  2. H. Exton, Multiple Hypergeometric Functions and Applications, Ellis Horwood Series in Mathematics and Its Applications (Ellis Horwood Limited, John Wiley and Sons, Chichester, New York, 1976).
  3. O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables, Ellis Horwood Series in Mathematics and Its Applications (Ellis Horwood Limited, John Wiley and Sons, Chichester, New York, 1983).
  4. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, Springer Series on Wave Phenomena Vol. 18 (Springer-Verlag, Berlin Heidelberg, 1994).
  5. R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Encyclopedia of Mathematics and Its Applications Vol. 85 (Cambridge University Press, Cambridge, 2001).
  6. V. A. Smirnov, Springer Tracts Mod. Phys. 250, 1 (2012).
  7. J. D. Bjorken and T. T. Wu, Phys. Rev. 130, 2566 (1963).
  8. T. L. Trueman and T. Yao, Phys. Rev. 132, 2741 (1963).
  9. N. I. Usyukina, Theor. Math. Phys. 22, 210 (1975).
  10. E. E. Boos and A. I. Davydychev, Theor. Math. Phys. 89, 1052 (1991).
  11. A. I. Davydychev, J. Math. Phys. (N.Y.) 32, 1052 (1991).
  12. A. I. Davydychev, J. Math. Phys. (N.Y.) 33, 358 (1992).
  13. N. I. Usyukina and A. I. Davydychev, Phys. Lett. B 298, 363 (1993).
  14. N. I. Usyukina and A. I. Davydychev, Phys. Lett. B 305, 136 (1993).
  15. A. I. Davydychev and J. B. Tausk, Nucl. Phys. B397, 123 (1993).
  16. F. A. Berends, M. Böhm, M. Buza et al., Z. Phys. C 63, 227 (1994).
  17. N. I. Usyukina and A. I. Davydychev, Phys. Lett. B 332, 159 (1994).
  18. V. A. Smirnov, Phys. Lett. B 460, 397 (1999).
  19. J. B. Tausk, Phys. Lett. B 469, 225 (1999).
  20. V. A. Smirnov, Phys. Lett. B 491, 130 (2000).
  21. V. A. Smirnov, Phys. Lett. B 524, 129 (2002).
  22. V. A. Smirnov, Phys. Lett. B 567, 193 (2003).
  23. G. Heinrich and V. A. Smirnov, Phys. Lett. B 598, 55 (2004).
  24. M. Czakon, Comput. Phys. Commun. 175, 559 (2006).
  25. C. Anastasiou and A. Daleo, J. High Energy Phys. 10 (2006) 031.
  26. J. Gluza, K. Kajda, and T. Riemann, Comput. Phys. Commun. 177, 879 (2007).
  27. A. V. Smirnov and V. A. Smirnov, Eur. Phys. J. C 62, 445 (2009).
  28. M. Ochman and T. Riemann, Acta Phys. Polon. B 46, 2117 (2015).
  29. S. Friot, D. Greynat, and E. de Rafael, Phys. Lett. B 628, 73 (2005).
  30. J. P. Aguilar, E. de Rafael, and D. Greynat, Phys. Rev. D 77, 093010 (2008).
  31. M. Czakon, A. Mitov, and S. Moch, Nucl. Phys. B798, 210 (2008).
  32. A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. Lett. 104, 112002 (2010).
  33. R. N. Lee, A. V. Smirnov, and V. A. Smirnov, J. High Energy Phys. 04 (2010) 020.
  34. D. Greynat and E. de Rafael, J. High Energy Phys. 07 (2012) 020.
  35. E. de Rafael, Phys. Lett. B 736, 522 (2014).
  36. B. Ananthanarayan, J. Bijnens, S. Ghosh, and A. Hebbar, Eur. Phys. J. A 52, 374 (2016).
  37. J. Charles, E. de Rafael, and D. Greynat, Phys. Rev. D 97, 076014 (2018).
  38. B. Ananthanarayan, J. Bijnens, S. Friot, and S. Ghosh, Phys. Rev. D 97, 091502(R) (2018).
  39. B. Ananthanarayan, J. Bijnens, S. Friot, and S. Ghosh, Phys. Rev. D 97, 114004 (2018).
  40. B. Ananthanarayan, S. Friot, and S. Ghosh, Phys. Rev. D 101, 116008 (2020).
  41. Z. Bern, L. J. Dixon, and V. A. Smirnov, Phys. Rev. D 72, 085001 (2005).
  42. Z. Bern, M. Czakon, D. A. Kosower, R. Roiban, and V. A. Smirnov, Phys. Rev. Lett. 97, 181601 (2006).
  43. Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, Phys. Rev. D 75, 085010 (2007).
  44. J. M. Drummond, J. Henn, V. A. Smirnov, and E. Sokatchev, J. High Energy Phys. 01 (2007) 064.
  45. M. Y. Kalmykov, V. V. Bytev, B. A. Kniehl, B. F. L. Ward, and S. A. Yost, Proc. Sci., ACAT08 (2008) 125.
  46. V. Del Duca, C. Duhr, and V. A. Smirnov, J. High Energy Phys. 03 (2010) 099.
  47. V. Del Duca, C. Duhr, and V. A. Smirnov, J. High Energy Phys. 05 (2010) 084.
  48. S. Friot and D. Greynat, SIGMA 6, 079 (2010).
  49. S. Friot and D. Greynat, J. Math. Phys. (N.Y.) 53, 023508 (2012).
  50. M. Y. Kalmykov and B. A. Kniehl, Phys. Lett. B 714, 103 (2012).
  51. M. Y. Kalmykov and B. A. Kniehl, J. High Energy Phys. 07 (2017) 031.
  52. F. Loebbert, D. Müller, and H. Münkler, Phys. Rev. D 101, 066006 (2020).
  53. B. Ananthanarayan, S. Banik, S. Friot, and S. Ghosh, Phys. Rev. D 102, 091901(R) (2020).
  54. B. Ananthanarayan, S. Banik, S. Friot, and S. Ghosh, Phys. Rev. D 103, 096008 (2021).
  55. R. Gopakumar, A. Kaviraj, K. Sen, and A. Sinha, Phys. Rev. Lett. 118, 081601 (2017).
  56. C. Sleight and M. Taronna, J. High Energy Phys. 02 (2020) 098.
  57. J.-P. Aguilar and J. Korbel, Fractal Fract. 2, 15 (2018).
  58. S. Friot, Nucl. Instrum. Methods Phys. Res., Sect. A 773, 150 (2015).
  59. D. O. Oriekhov and V. P. Gusynin, Phys. Rev. B 101, 235162 (2020).
  60. S. Pincherle, Atti R. Accademia Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (Ser. 4) 4, 694 and 792 (1888), http://www.bdim.eu/item?fmt=pdf&id=GM_Pincherle_CW_1_223.
  61. R. H. Mellin, Acta Soc. Sci. Fenn. 20, 1 (1895), https://searchworks.stanford.edu/view/86978.
  62. E. W. Barnes, Messenger Math. 29, 64 (1900), https://gdz.sub.uni-goettingen.de/id/PPN599484047_0029?tify=%7B%22pages%22:%5B2%5D,%22panX%22:0.461,%22panY%22:0.779,%22view%22:%22info%22,%22zoom%22:0.344%7D.
  63. L. Nilsson, M. Passare, and A. Tsikh, J. Sib. Fed. Univ. Math. Phys. 12, 509 (2019), http://elib.sfu-kras.ru/bitstream/handle/2311/112609/Tsikh%20to%20Aizenberg+.pdf.
  64. M. Kalmykov, in Proceedings of the “Antidifferentiation and the Calculation of Feynman Amplitudes” Conference, Zeuthen, Germany (2020), https://indico.desy.de/event/24049/.
  65. M. Kalmykov, V. Bytev, B. A. Kniehl, S. O. Moch, B. F. L. Ward, and S. A. Yost, arXiv:2012.14492.
  66. M. Passare, A. K. Tsikh, and A. A. Cheshel, Theor. Math. Phys. 109, 1544 (1996).
  67. O. Zhdanov and A. Tsikh, Siberian mathematical Journal 39, 245 (1998).
  68. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.127.151601 for the treatment of the resonant case and the description of the mbconichulls.wl Mathematica package along with the examples.nb Mathematica notebook containing several applications, given as ancillary files.
  69. K. J. Larsen and R. Rietkerk, Comput. Phys. Commun. 222, 250 (2018).
  70. P. Griffiths and J. Harris, Principles of Algebraic Geometry (John Wiley & Son, New York, 1978), ISBN [Amazon][WorldCat].
  71. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Series in Mathematics and Its Applications (Ellis Horwood Limited, John Wiley and Sons, Chichester, New York, 1985).
  72. P. O. M. Olsson, J. Math. Phys. (N.Y.) 5, 420 (1964).
  73. F. Loebbert, J. Miczajka, D. Müller, and H. Münkler, Phys. Rev. Lett. 125, 091602 (2020).

Outline

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Phys. Rev. Lett. 127, 151601– Published 5 October, 2021
  • Received 1 January 2021
  • Revised 18 June 2021
  • Accepted 27 July 2021
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