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Direct Integration for Multi-Leg Amplitudes: Tips, Tricks, and When They Fail

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Anti-Differentiation and the Calculation of Feynman Amplitudes

Abstract

Direct hyperlogarithmic integration offers a strong alternative to differential equation methods for Feynman integration, particularly for multi-particle diagrams. We review a variety of results by the authors in which this method, employed with some care, can compute diagrams of up to eight particles and four loops. We also highlight situations in which this method fails due to an algebraic obstruction. In a large number of cases the obstruction can be associated with a Calabi-Yau manifold.

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Notes

  1. 1.

    For massive external momenta, we can represent each in terms of a pair of massless external momenta, so this discussion still applies.

References

  1. F.C.S. Brown, arXiv:0910.0114 [math.AG]

    Google Scholar 

  2. E. Panzer, https://doi.org/10.18452/17157. arXiv:1506.07243 [math-ph]

  3. E. Panzer, Comput. Phys. Commun. 188, 148–166 (2015). https://doi.org/10.1016/j.cpc.2014.10.019. arXiv:1403.3385 [hep-th]

  4. C. Bogner, F. Brown, Commun. Num. Theor. Phys. 09, 189–238 (2015). https://doi.org/10.4310/CNTP.2015.v9.n1.a3. arXiv:1408.1862 [hep-th]

  5. A.V. Kotikov, Phys. Lett. B 254, 158–164 (1991). https://doi.org/10.1016/0370-2693(91)90413-K

    Article  MathSciNet  Google Scholar 

  6. E. Remiddi, Nuovo Cim. A 110, 1435–1452 (1997). arXiv:hep-th/9711188 [hep-th]

    Google Scholar 

  7. S. Müller-Stach, S. Weinzierl, R. Zayadeh, Commun. Math. Phys. 326, 237–249 (2014). https://doi.org/10.1007/s00220-013-1838-3. arXiv:1212.4389 [hep-ph]

  8. J.M. Henn, Phys. Rev. Lett. 110, 251601 (2013). https://doi.org/10.1103/PhysRevLett.110.251601. arXiv:1304.1806 [hep-th]

    Article  Google Scholar 

  9. R.N. Lee, JHEP 04, 108 (2015). https://doi.org/10.1007/JHEP04(2015)108. arXiv:1411.0911 [hep-ph]

    Article  Google Scholar 

  10. C. Meyer, JHEP 04, 006 (2017). https://doi.org/10.1007/JHEP04(2017)006. arXiv:1611.01087 [hep-ph]

    Article  Google Scholar 

  11. J. Gluza, K. Kajda, T. Riemann, Comput. Phys. Commun. 177, 879–893 (2007). https://doi.org/10.1016/j.cpc.2007.07.001. arXiv:0704.2423 [hep-ph]

  12. J. Blümlein, I. Dubovyk, J. Gluza, M. Ochman, C.G. Raab, T. Riemann, C. Schneider, PoS LL2014, 052 (2014). https://doi.org/10.22323/1.211.0052. arXiv:1407.7832 [hep-ph]

  13. M. Ochman, T. Riemann, Acta Phys. Polon. B 46(11), 2117 (2015). https://doi.org/10.5506/APhysPolB.46.2117. arXiv:1511.01323 [hep-ph]

  14. Z. Bern, L.J. Dixon, V.A. Smirnov, Phys. Rev. D 72, 085001 (2005). https://doi.org/10.1103/PhysRevD.72.085001. arXiv:hep-th/0505205 [hep-th]

    Article  MathSciNet  Google Scholar 

  15. J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel, M. Wilhelm, Phys. Rev. Lett. 120(12), 121603 (2018). https://doi.org/10.1103/PhysRevLett.120.121603. arXiv:1712.02785 [hep-th]

  16. J.L. Bourjaily, Y.H. He, A.J. Mcleod, M. Von Hippel, M. Wilhelm, Phys. Rev. Lett. 121(7), 071603 (2018). https://doi.org/10.1103/PhysRevLett.121.071603. arXiv:1805.09326 [hep-th]

  17. J.L. Bourjaily, A.J. McLeod, M. von Hippel, M. Wilhelm, JHEP 08, 184 (2018). https://doi.org/10.1007/JHEP08(2018)184. arXiv:1805.10281 [hep-th]

    Article  Google Scholar 

  18. J.L. Bourjaily, A.J. McLeod, M. von Hippel, M. Wilhelm, Phys. Rev. Lett. 122(3), 031601 (2019). https://doi.org/10.1103/PhysRevLett.122.031601. arXiv:1810.07689 [hep-th]

  19. J.L. Bourjaily, A.J. McLeod, C. Vergu, M. Volk, M. Von Hippel, M. Wilhelm, JHEP 01, 078 (2020). https://doi.org/10.1007/JHEP01(2020)078. arXiv:1910.01534 [hep-th]

    Article  Google Scholar 

  20. J.L. Bourjaily, A.J. McLeod, C. Vergu, M. Volk, M. Von Hippel, M. Wilhelm, JHEP 02, 025 (2020). https://doi.org/10.1007/JHEP02(2020)025. arXiv:1910.14224 [hep-th]

    Article  Google Scholar 

  21. J.L. Bourjaily, M. Volk, M. Von Hippel, JHEP 02, 095 (2020). https://doi.org/10.1007/JHEP02(2020)095. arXiv:1912.05690 [hep-th]

    Article  Google Scholar 

  22. F. Brown, Commun. Math. Phys. 287, 925–958 (2009). https://doi.org/10.1007/s00220-009-0740-5. arXiv:0804.1660 [math.AG]

  23. C. Bogner, M. Lüders, Contemp. Math. 648, 11–28 (2015). arXiv:1302.6215 [hep-ph]

    Google Scholar 

  24. M. Besier, D. Van Straten, S. Weinzierl, Commun. Num. Theor. Phys. 13, 253–297 (2019). https://doi.org/10.4310/CNTP.2019.v13.n2.a1. arXiv:1809.10983 [hep-th]

  25. H. Frellesvig, C.G. Papadopoulos, JHEP 04, 083 (2017). https://doi.org/10.1007/JHEP04(2017)083. arXiv:1701.07356 [hep-ph]

    Article  Google Scholar 

  26. J.L. Bourjaily, F. Dulat, E. Panzer, Nucl. Phys. B 942, 251–302 (2019). https://doi.org/10.1016/j.nuclphysb.2019.03.022. arXiv:1901.02887 [hep-th]

  27. A. Hodges, JHEP 05, 135 (2013). https://doi.org/10.1007/JHEP05(2013)135. arXiv:0905.1473 [hep-th]

    Article  Google Scholar 

  28. S. Caron-Huot, L.J. Dixon, M. von Hippel, A.J. McLeod, G. Papathanasiou, JHEP 07, 170 (2018). https://doi.org/10.1007/JHEP07(2018)170. arXiv:1806.01361 [hep-th]

    Article  Google Scholar 

  29. L.D. Landau, Nucl. Phys. 13(1), 181–192 (1960). https://doi.org/10.1016/B978-0-08-010586-4.50103-6

    Article  Google Scholar 

  30. A.P. Hodges, Crossing and twistor diagrams. Twistor Newsletter 5, 4 (1977)

    Google Scholar 

  31. A. Hodges, JHEP 08, 051 (2013). https://doi.org/10.1007/JHEP08(2013)051. arXiv:1004.3323 [hep-th]

    Article  Google Scholar 

  32. L. Mason, D. Skinner, J. Phys. A 44, 135401 (2011). https://doi.org/10.1088/1751-8113/44/13/135401. arXiv:1004.3498 [hep-th]

    Article  MathSciNet  Google Scholar 

  33. F.C.S. Brown, Annales Sci. Ecole Norm. Sup. 42, 371 (2009). arXiv:math/0606419 [math.AG]

    Article  Google Scholar 

  34. A.B. Goncharov, M. Spradlin, C. Vergu, A. Volovich, Phys. Rev. Lett. 105, 151605 (2010). https://doi.org/10.1103/PhysRevLett.105.151605. arXiv:1006.5703 [hep-th]

    Article  MathSciNet  Google Scholar 

  35. The Sage Developers. SageMath , the Sage Mathematics Software System (Version 8.8) (2019). Available at: www.sagemath.org

  36. The PARI Group, Univ. Bordeaux. PARI / GP version 2.11.0 (2018). Available at: pari.math.u-bordeaux.fr

  37. S. He, Z. Li, C. Zhang, arXiv:2009.11471 [hep-th]

    Google Scholar 

  38. S. Laporta, E. Remiddi, Nucl. Phys. B 704, 349–386 (2005). https://doi.org/10.1016/j.nuclphysb.2004.10.044. arXiv:hep-ph/0406160 [hep-ph]

  39. S. Muller-Stach, S. Weinzierl, R. Zayadeh, PoS LL2012, 005 (2012). https://doi.org/10.22323/1.151.0005. arXiv:1209.3714 [hep-ph]

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

    Google Scholar 

  41. S. Bloch, P. Vanhove, J. Number Theor. 148, 328–364 (2015). https://doi.org/10.1016/j.jnt.2014.09.032. arXiv:1309.5865 [hep-th]

  42. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 54, 052303 (2013). https://doi.org/10.1063/1.4804996. arXiv:1302.7004 [hep-ph]

    Article  MathSciNet  Google Scholar 

  43. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 55(10), 102301 (2014). https://doi.org/10.1063/1.4896563. arXiv:1405.5640 [hep-ph]

  44. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 56(7), 072303 (2015). https://doi.org/10.1063/1.4926985. arXiv:1504.03255 [hep-ph]

  45. L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 57(3), 032304 (2016). https://doi.org/10.1063/1.4944722. arXiv:1512.05630 [hep-ph]

  46. L. Adams, C. Bogner, A. Schweitzer, S. Weinzierl, J. Math. Phys. 57(12), 122302 (2016). https://doi.org/10.1063/1.4969060. arXiv:1607.01571 [hep-ph]

  47. L. Adams, S. Weinzierl, Commun. Num. Theor. Phys. 12, 193–251 (2018). https://doi.org/10.4310/CNTP.2018.v12.n2.a1. arXiv:1704.08895 [hep-ph]

  48. L. Adams, E. Chaubey, S. Weinzierl, Phys. Rev. Lett. 118(14), 141602 (2017). https://doi.org/10.1103/PhysRevLett.118.141602. arXiv:1702.04279 [hep-ph]

  49. C. Bogner, A. Schweitzer, S. Weinzierl, Nucl. Phys. B 922, 528–550 (2017). https://doi.org/10.1016/j.nuclphysb.2017.07.008. arXiv:1705.08952 [hep-ph]

  50. J. Broedel, C. Duhr, F. Dulat, L. Tancredi, JHEP 05, 093 (2018). https://doi.org/10.1007/JHEP05(2018)093. arXiv:1712.07089 [hep-th]

    Article  Google Scholar 

  51. J. Broedel, C. Duhr, F. Dulat, L. Tancredi, Phys. Rev. D 97(11), 116009 (2018). https://doi.org/10.1103/PhysRevD.97.116009. arXiv:1712.07095 [hep-ph]

  52. L. Adams, S. Weinzierl, Phys. Lett. B 781, 270–278 (2018). https://doi.org/10.1016/j.physletb.2018.04.002. arXiv:1802.05020 [hep-ph]

  53. J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, JHEP 08, 014 (2018). https://doi.org/10.1007/JHEP08(2018)014. arXiv:1803.10256 [hep-th]

    Article  Google Scholar 

  54. L. Adams, E. Chaubey, S. Weinzierl, Phys. Rev. Lett. 121(14), 142001 (2018). https://doi.org/10.1103/PhysRevLett.121.142001. arXiv:1804.11144 [hep-ph]

  55. J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, JHEP 01, 023 (2019). https://doi.org/10.1007/JHEP01(2019)023. arXiv:1809.10698 [hep-th]

    Article  Google Scholar 

  56. L. Adams, E. Chaubey, S. Weinzierl, JHEP 10, 206 (2018). https://doi.org/10.1007/JHEP10(2018)206. arXiv:1806.04981 [hep-ph]

    Article  Google Scholar 

  57. I. Hönemann, K. Tempest, S. Weinzierl, Phys. Rev. D 98(11), 113008 (2018). https://doi.org/10.1103/PhysRevD.98.113008. arXiv:1811.09308 [hep-ph]

  58. C. Bogner, S. Müller-Stach, S. Weinzierl, Nucl. Phys. B 954, 114991 (2020). https://doi.org/10.1016/j.nuclphysb.2020.114991. arXiv:1907.01251 [hep-th]

    Article  Google Scholar 

  59. J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, JHEP 05, 120 (2019). https://doi.org/10.1007/JHEP05(2019)120. arXiv:1902.09971 [hep-ph]

    Article  Google Scholar 

  60. J. Broedel, C. Duhr, F. Dulat, R. Marzucca, B. Penante, L. Tancredi, JHEP 09, 112 (2019). https://doi.org/10.1007/JHEP09(2019)112. arXiv:1907.03787 [hep-th]

    Article  Google Scholar 

  61. S. Groote, J.G. Korner, A.A. Pivovarov, Annals Phys. 322, 2374–2445 (2007). https://doi.org/10.1016/j.aop.2006.11.001. arXiv:hep-ph/0506286 [hep-ph]

  62. F. Brown, O. Schnetz, Duke Math. J. 161(10), 1817–1862 (2012). https://doi.org/10.1215/00127094-1644201. arXiv:1006.4064 [math.AG]

  63. S. Bloch, M. Kerr, P. Vanhove, Compos. Math. 151(12), 2329–2375 (2015). https://doi.org/10.1112/S0010437X15007472. arXiv:1406.2664 [hep-th]

  64. S. Bloch, M. Kerr, P. Vanhove, Adv. Theor. Math. Phys. 21, 1373–1453 (2017). https://doi.org/10.4310/ATMP.2017.v21.n6.a1. arXiv:1601.08181 [hep-th]

  65. D. Broadhurst, private communication

    Google Scholar 

  66. D. Festi, D. van Straten, Commun. Num. Theor. Phys. 13, 463–485 (2019). https://doi.org/10.4310/CNTP.2019.v13.n2.a4. arXiv:1809.04970 [math.AG]

  67. M. Besier, D. Festi, M. Harrison, B. Naskrecki, Commun. Num. Theor. Phys. 14(4), 863–911 (2020). https://doi.org/10.4310/CNTP.2020.v14.n4.a4. arXiv:1908.01079 [math.AG]

  68. C. Doran, A. Novoseltsev, P. Vanhove, To appear

    Google Scholar 

  69. A. Klemm, C. Nega, R. Safari, JHEP 04, 088 (2020). https://doi.org/10.1007/JHEP04(2020)088. arXiv:1912.06201 [hep-th]

    Article  Google Scholar 

  70. K. Bönisch, F. Fischbach, A. Klemm, C. Nega, R. Safari, arXiv:2008.10574 [hep-th]

    Google Scholar 

  71. T. Hubsch, Calabi-Yau manifolds: A Bestiary for physicists (World Scientific, 1992)

    Google Scholar 

  72. C. Vergu, M. Volk, JHEP 07, 160 (2020). https://doi.org/10.1007/JHEP07(2020)160. arXiv:2005.08771 [hep-th]

    Article  Google Scholar 

  73. F. Brown, C. Duhr, arXiv:2006.09413 [hep-th]

    Google Scholar 

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Bourjaily, J.L. et al. (2021). Direct Integration for Multi-Leg Amplitudes: Tips, Tricks, and When They Fail. 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_5

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