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Bootstrapping two-loop Feynman integrals for planar \( \mathcal{N}=4 \) sYM

Journal of High Energy Physics volume 2018, Article number: 59 (2018) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planar \( \mathcal{N}=4 \) super-Yang-Mills theory. We use a bootstrap inspired strategy, combined with a set of second-order partial differential equations that provide powerful constraints on the symbol ansatz. When the complete symbol alphabet is not available, we adopt a hybrid approach. Instead of the full function, we bootstrap a certain discontinuity for which the alphabet is known. Then we write a one-fold dispersion integral to recover the complete result. At six and seven points, we find that the individual Feynman integrals live in the same space of functions as the amplitude, which is described by the 9- and 42-letter cluster alphabets respectively. Starting at eight points however, the symbol alphabet of the MHV amplitude is insufficient for individual integrals. In particular, some of the integrals require algebraic letters involving four-mass box square-root singularities. We point out that these algebraic letters are relevant at the amplitude level directly starting with N2MHV amplitudes even at one loop.

References

  1. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].

  2. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].

  3. Z. Bern, J.J.M. Carrasco, H. Johansson and D.A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  4. J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].

  5. M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].

  6. A.V. Smirnov and V.A. Smirnov, On the resolution of singularities of multiple Mellin-Barnes integrals, Eur. Phys. J. C 62 (2009) 445 [arXiv:0901.0386] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].

    ADS  Article  Google Scholar 

  8. 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 

  9. 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 

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

    ADS  Google Scholar 

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

  12. M. Argeri and P. Mastrolia, Feynman diagrams and differential equations, Int. J. Mod. Phys. A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

    ADS  Article  Google Scholar 

  14. A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].

  15. F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].

  16. E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP 03 (2014) 071 [arXiv:1401.4361] [INSPIRE].

    ADS  Article  Google Scholar 

  17. F. Brown, Periods and Feynman amplitudes, in the proceedings of the 18th International Congress on Mathematical Physics (ICMP2015), July 27–August 1, Santiago de Chile, Chile (2015), arXiv:1512.09265 [INSPIRE].

  18. N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  20. L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  21. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. N. Arkani-Hamed et al., On-shell structures of MHV amplitudes beyond the planar limit, JHEP 06 (2015) 179 [arXiv:1412.8475] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. N. Arkani-Hamed et al., Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge U.K. (2016), arXiv:1212.5605 [INSPIRE].

  24. N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  25. N. Arkani-Hamed and J. Trnka, Into the amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].

    ADS  Article  Google Scholar 

  26. R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. F. Cachazo, Sharpening the leading singularity, arXiv:0803.1988 [INSPIRE].

  28. Z. Bern et al., Logarithmic singularities and maximally supersymmetric amplitudes, JHEP 06 (2015) 202 [arXiv:1412.8584] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. Z. Bern et al., Evidence for a nonplanar amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  30. E. Herrmann and J. Trnka, Gravity on-shell diagrams, JHEP 11 (2016) 136 [arXiv:1604.03479] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  31. J.M. Henn and B. Mistlberger, Four-gluon scattering at three loops, infrared structure and the Regge limit, Phys. Rev. Lett. 117 (2016) 171601 [arXiv:1608.00850] [INSPIRE].

    ADS  Article  Google Scholar 

  32. L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. J. Golden and M. Spradlin, A cluster bootstrap for two-loop MHV amplitudes, JHEP 02 (2015) 002 [arXiv:1411.3289] [INSPIRE].

    ADS  Article  Google Scholar 

  36. L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].

    ADS  Article  Google Scholar 

  37. J.M. Drummond, G. Papathanasiou and M. Spradlin, A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].

    ADS  Article  Google Scholar 

  38. L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].

    Article  Google Scholar 

  39. S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a five-loop amplitude using Steinmann relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].

    ADS  Article  Google Scholar 

  40. L.J. Dixon et al., Heptagons from the Steinmann cluster bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix. Cambridge University Press, Cambridge U.K. (1966).

    MATH  Google Scholar 

  42. A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  43. S. Fomin and A. Zelevinsky, Cluster algebras I: foundations, J. Amer. Math. Soc. 15 (2002) 497 [math/0104151].

  44. V. Fock and A. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. L’Ecole Norm. Sup. 42 (2009) 865 [math/0311245].

  45. V. Fock and A. Goncharov, Cluster χ-varieties and number theory, amalgamation and Poisson-Lie groups, Algebraic Geometry and Number Theory, Dedicated to Drinfeld’s 50th birthday, V. Ginzburg, Birkauser, Germany (2006), math/0508408.

  46. J. Golden et al., Motivic amplitudes and cluster coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].

    ADS  Article  Google Scholar 

  47. J. Golden, M.F. Paulos, M. Spradlin and A. Volovich, Cluster polylogarithms for scattering amplitudes, J. Phys. A 47 (2014) 474005 [arXiv:1401.6446] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  48. T. Harrington and M. Spradlin, Cluster functions and scattering amplitudes for six and seven points, JHEP 07 (2017) 016 [arXiv:1512.07910] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  49. T. Dennen, M. Spradlin and A. Volovich, Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory, JHEP 03 (2016) 069 [arXiv:1512.07909] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  50. T. Dennen et al., Landau Singularities from the Amplituhedron, JHEP 06 (2017) 152 [arXiv:1612.02708] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  51. I. Prlina et al., All-helicity symbol alphabets from unwound amplituhedra, JHEP 05 (2018) 159 [arXiv:1711.11507] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  52. I. Prlina, M. Spradlin, J. Stankowicz and S. Stanojevic, Boundaries of amplituhedra and NMHV symbol alphabets at two loops, JHEP 04 (2018) 049 [arXiv:1712.08049] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  53. S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  54. J. Drummond, J. Foster and O. Gurdogan, Cluster adjacency properties of scattering amplitudes in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 120 (2018) 161601 [arXiv:1710.10953] [INSPIRE].

    ADS  Article  Google Scholar 

  55. J.L. Bourjaily, A.J. McLeod, M. von Hippel and M. Wilhelm, Rationalizing Loop Integration, JHEP 08 (2018) 184 [arXiv:1805.10281] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  56. J.M. Drummond, J.M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  57. S. Caron-Huot et al., The double pentaladder integral to all orders, JHEP 07 (2018) 170 [arXiv:1806.01361] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  58. A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  59. D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [arXiv:1712.09610] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  60. C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  61. C. Duhr, Mathematical aspects of scattering amplitudes, in the proceeding of the Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders (TASI 2014), June 2–27, Boulder, Colorado, U.S.A. (2014), arXiv:1411.7538 [INSPIRE].

  62. O. Steinmann, Uber den Zusammenhang zwischen den Wightmanfunktionen und der retardierten Kommutatoren, Helv. Physica Acta 33 (1960) 257.

    MathSciNet  MATH  Google Scholar 

  63. O. Steinmann, Wightman-Funktionen und retardierten Kommutatoren. II, Helv. Phys. Acta 33 (1960) 347.

    MathSciNet  MATH  Google Scholar 

  64. K.E. Cahill and H.P. Stapp, Optical theorems and Steinmann relations, Annals Phys. 90 (1975) 438 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  65. J. Bartels, L.N. Lipatov and A. Sabio Vera, BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065] [INSPIRE].

    ADS  Google Scholar 

  66. J. Bartels, L.N. Lipatov and A. Sabio Vera, N = 4 supersymmetric Yang-Mills scattering amplitudes at high energies: the Regge cut contribution, Eur. Phys. J. C 65 (2010) 587 [arXiv:0807.0894] [INSPIRE].

    ADS  Article  Google Scholar 

  67. L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181.

    MathSciNet  Article  MATH  Google Scholar 

  68. E. Remiddi, Dispersion relations for Feynman graphs, Helv. Phys. Acta 54 (1982) 364 [INSPIRE].

    MathSciNet  Google Scholar 

  69. S. Bauberger and M. Böhm, Simple one-dimensional integral representations for two loop selfenergies: the master diagram, Nucl. Phys. B 445 (1995) 25 [hep-ph/9501201] [INSPIRE].

  70. 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].

  71. S. Abreu, R. Britto, C. Duhr and E. Gardi, From multiple unitarity cuts to the coproduct of Feynman integrals, JHEP 10 (2014) 125 [arXiv:1401.3546] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  72. S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP 06 (2014) 114 [arXiv:1404.2922] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  73. C. Cheung, C.-H. Shen and J. Trnka, Simple recursion relations for general field theories, JHEP 06 (2015) 118 [arXiv:1502.05057] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  74. C. Cheung et al., On-shell recursion relations for effective field theories, Phys. Rev. Lett. 116 (2016) 041601 [arXiv:1509.03309] [INSPIRE].

    ADS  Article  Google Scholar 

  75. C. Cheung et al., A periodic table of effective field theories, JHEP 02 (2017) 020 [arXiv:1611.03137] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  76. D. Chicherin, J.M. Henn and E. Sokatchev, Scattering amplitudes from superconformal Ward identities, Phys. Rev. Lett. 121 (2018) 021602 [arXiv:1804.03571] [INSPIRE].

    ADS  Article  Google Scholar 

  77. 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].

  78. J.L. Bourjaily, S. Caron-Huot and J. Trnka, Dual-conformal regularization of infrared loop divergences and the chiral box expansion, JHEP 01 (2015) 001 [arXiv:1303.4734] [INSPIRE].

    ADS  Article  Google Scholar 

  79. S. Caron-Huot and S. He, Jumpstarting the all-loop S-matrix of planar N = 4 super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  80. E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  81. E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. thesis, Humboldt Universität, Berlin, Germany (2015), arXiv:1506.07243 [INSPIRE].

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Affiliations

  1. PRISMA Cluster of Excellence, Johannes Gutenberg University, 55128, Mainz, Germany

    Johannes Henn

  2. Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805, München, Germany

    Johannes Henn

  3. SLAC National Accelerator Laboratory, Stanford University, Stanford, CA, 94039, U.S.A.

    Enrico Herrmann

  4. Mani L. Bhaumik Institute for Theoretical Physics, UCLA Department of Physics and Astronomy, Los Angeles, CA, 90095, U.S.A.

    Julio Parra-Martinez

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  1. Johannes Henn
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  2. Enrico Herrmann
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Correspondence to Julio Parra-Martinez.

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ArXiv ePrint: 1806.06072

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Henn, J., Herrmann, E. & Parra-Martinez, J. Bootstrapping two-loop Feynman integrals for planar \( \mathcal{N}=4 \) sYM. J. High Energ. Phys. 2018, 59 (2018). https://doi.org/10.1007/JHEP10(2018)059

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Keywords

  • 1/N Expansion
  • Scattering Amplitudes
  • Supersymmetric Gauge Theory