The two-loop five-particle amplitude in \( \mathcal{N} \) = 8 supergravity

Abstract

We compute for the first time the two-loop five-particle amplitude in \( \mathcal{N} \) = 8 supergravity. Starting from the known integrand, we perform an integration-by-parts reduction and express the answer in terms of uniform weight master integrals. The latter are known to evaluate to non-planar pentagon functions, described by a 31-letter symbol alphabet. We express the final result for the amplitude in terms of uniform weight four symbols, multiplied by a small set of rational factors. We observe that one of the symbol letters is absent from the amplitude. The latter satisfies the expected factorization properties when one external graviton becomes soft, and when two external gravitons become collinear. We verify that the soft divergences of the amplitude exponentiate. We extract the finite remainder function, which depends on fewer rational factors. By analyzing identities involving rational factors and symbols we find a remarkably compact representation in terms of a single seed function, summed over all permutations of external particles. Finally, we work out the multi-Regge limit, and present explicitly the leading logarithmic terms in the limit. The full symbol of the IR-subtracted hard function is provided as a supplementary file.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    Z. Bern, J.J. Carrasco, W.-M. Chen, H. Johansson and R. Roiban, Gravity Amplitudes as Generalized Double Copies of Gauge-Theory Amplitudes, Phys. Rev. Lett. 118 (2017) 181602 [arXiv:1701.02519] [INSPIRE].

    ADS  Article  Google Scholar 

  3. [3]

    Z. Bern, J.J.M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban and M. Zeng, Five-loop four-point integrand of N = 8 supergravity as a generalized double copy, Phys. Rev. D 96 (2017) 126012 [arXiv:1708.06807] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  4. [4]

    F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [INSPIRE].

  5. [5]

    F. Cachazo and D. Skinner, On the structure of scattering amplitudes in N = 4 super Yang-Mills and N = 8 supergravity, arXiv:0801.4574 [INSPIRE].

  6. [6]

    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 

  7. [7]

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

    ADS  Article  Google Scholar 

  8. [8]

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Singularity Structure of Maximally Supersymmetric Scattering Amplitudes, Phys. Rev. Lett. 113 (2014) 261603 [arXiv:1410.0354] [INSPIRE].

    ADS  Article  Google Scholar 

  9. [9]

    S.G. Naculich, H. Nastase and H.J. Schnitzer, Two-loop graviton scattering relation and IR behavior in N = 8 supergravity, Nucl. Phys. B 805 (2008) 40 [arXiv:0805.2347] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    A. Brandhuber, P. Heslop, A. Nasti, B. Spence and G. Travaglini, Four-point Amplitudes in N = 8 Supergravity and Wilson Loops, Nucl. Phys. B 807 (2009) 290 [arXiv:0805.2763] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    C. Boucher-Veronneau and L.J. Dixon, N ≥ 4 Supergravity Amplitudes from Gauge Theory at Two Loops, JHEP 12 (2011) 046 [arXiv:1110.1132] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    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 

  13. [13]

    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 

  14. [14]

    Z. Bern, M. Enciso, H. Ita and M. Zeng, Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector, Phys. Rev. D 96 (2017) 096017 [arXiv:1709.06055] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  15. [15]

    D.A. Kosower, Direct Solution of Integration-by-Parts Systems, Phys. Rev. D 98 (2018) 025008 [arXiv:1804.00131] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  16. [16]

    P. Maierhöfer and J. Usovitsch, Kira 1.2 Release Notes, arXiv:1812.01491 [INSPIRE].

  17. [17]

    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 

  18. [18]

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

  19. [19]

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

  20. [20]

    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].

  21. [21]

    C.G. Papadopoulos, D. Tommasini and C. Wever, The Pentabox Master Integrals with the Simplified Differential Equations approach, JHEP 04 (2016) 078 [arXiv:1511.09404] [INSPIRE].

    ADS  Google Scholar 

  22. [22]

    T. Gehrmann, J.M. Henn and N.A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP 10 (2018) 103 [arXiv:1807.09812] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  23. [23]

    S. Abreu, B. Page and M. Zeng, Differential equations from unitarity cuts: nonplanar hexa-box integrals, JHEP 01 (2019) 006 [arXiv:1807.11522] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    D. Chicherin, T. Gehrmann, J.M. Henn, N.A. Lo Presti, V. Mitev and P. Wasser, Analytic result for the nonplanar hexa-box integrals, JHEP 03 (2019) 042 [arXiv:1809.06240] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  25. [25]

    S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in \( \mathcal{N} \) = 4 super-Yang-Mills theory, arXiv:1812.08941 [INSPIRE].

  26. [26]

    D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, All master integrals for three-jet production at NNLO, arXiv:1812.11160 [INSPIRE].

  27. [27]

    D. Chicherin, J.M. Henn, P. Wasser, T. Gehrmann, Y. Zhang and S. Zoia, Analytic result for a two-loop five-particle amplitude, arXiv:1812.11057 [INSPIRE].

  28. [28]

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

    ADS  MathSciNet  Article  Google Scholar 

  29. [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. [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. [31]

    D.C. Dunbar and W.B. Perkins, Two-loop five-point all plus helicity Yang-Mills amplitude, Phys. Rev. D 93 (2016) 085029 [arXiv:1603.07514] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  32. [32]

    S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case, JHEP 01 (2019) 186 [arXiv:1811.11699] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. [33]

    S. Abreu, J. Dormans, F. Febres Cordero, H. Ita and B. Page, Analytic Form of Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD, Phys. Rev. Lett. 122 (2019) 082002 [arXiv:1812.04586] [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. [35]

    P. Van Nieuwenhuizen, Radiation of massive gravitation, Phys. Rev. D 7 (1973) 2300 [INSPIRE].

    ADS  Google Scholar 

  36. [36]

    R. Akhoury, R. Saotome and G. Sterman, Collinear and Soft Divergences in Perturbative Quantum Gravity, Phys. Rev. D 84 (2011) 104040 [arXiv:1109.0270] [INSPIRE].

    ADS  Google Scholar 

  37. [37]

    M. Beneke and G. Kirilin, Soft-collinear gravity, JHEP 09 (2012) 066 [arXiv:1207.4926] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  38. [38]

    D.C. Dunbar and P.S. Norridge, Infinities within graviton scattering amplitudes, Class. Quant. Grav. 14 (1997) 351 [hep-th/9512084] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. [39]

    S.G. Naculich and H.J. Schnitzer, Eikonal methods applied to gravitational scattering amplitudes, JHEP 05 (2011) 087 [arXiv:1101.1524] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  40. [40]

    C.D. White, Factorization Properties of Soft Graviton Amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. [41]

    Ø. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multileg scattering, Phys. Rev. Lett. 117 (2016) 172002 [arXiv:1507.00047] [INSPIRE].

    ADS  Article  Google Scholar 

  42. [42]

    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].

  43. [43]

    Z. Bern, S. Davies and J. Nohle, On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons, Phys. Rev. D 90 (2014) 085015 [arXiv:1405.1015] [INSPIRE].

    ADS  Google Scholar 

  44. [44]

    Z. Bern, S. Davies, P. Di Vecchia and J. Nohle, Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance, Phys. Rev. D 90 (2014) 084035 [arXiv:1406.6987] [INSPIRE].

    ADS  Google Scholar 

  45. [45]

    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 

  46. [46]

    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 

  47. [47]

    H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  48. [48]

    F.A. Berends, W.T. Giele and H. Kuijf, On relations between multi-gluon and multigraviton scattering, Phys. Lett. B 211 (1988) 91 [INSPIRE].

    ADS  Article  Google Scholar 

  49. [49]

    Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  50. [50]

    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 

  51. [51]

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) [INSPIRE].

  52. [52]

    E. Herrmann and J. Trnka, UV cancellations in gravity loop integrands, JHEP 02 (2019) 084 [arXiv:1808.10446] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  53. [53]

    J.L. Bourjaily, E. Herrmann and J. Trnka, Amplitudes at Infinity, Phys. Rev. D 99 (2019) 066006 [arXiv:1812.11185] [INSPIRE].

    ADS  Google Scholar 

  54. [54]

    J.J. Carrasco and H. Johansson, Five-Point Amplitudes in N = 4 Super-Yang-Mills Theory and N = 8 Supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].

    ADS  Google Scholar 

  55. [55]

    Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  56. [56]

    T. Gehrmann and E. Remiddi, Two loop master integrals for γ * → 3 jets: The Planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].

  57. [57]

    T. Gehrmann and E. Remiddi, Two loop master integrals for γ * → 3 jets: The Nonplanar topologies, Nucl. Phys. B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].

  58. [58]

    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 

  59. [59]

    D. Chicherin, J.M. Henn and E. Sokatchev, Implications of nonplanar dual conformal symmetry, JHEP 09 (2018) 012 [arXiv:1807.06321] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  60. [60]

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

    ADS  Article  MATH  Google Scholar 

  61. [61]

    A. von Manteuffel and C. Studerus, Reduze 2Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].

  62. [62]

    V. Mitev and Y. Zhang, SymBuild: a package for the computation of integrable symbols in scattering amplitudes, arXiv:1809.05101 [INSPIRE].

  63. [63]

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  64. [64]

    M. Bianchi, S. He, Y.-t. Huang and C. Wen, More on Soft Theorems: Trees, Loops and Strings, Phys. Rev. D 92 (2015) 065022 [arXiv:1406.5155] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  65. [65]

    Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, One loop n point helicity amplitudes in (selfdual) gravity, Phys. Lett. B 444 (1998) 273 [hep-th/9809160] [INSPIRE].

    ADS  Article  Google Scholar 

  66. [66]

    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 

  67. [67]

    M. Bullimore and D. Skinner, Descent Equations for Superamplitudes, arXiv:1112.1056 [INSPIRE].

  68. [68]

    E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Multi-Reggeon Processes in the Yang-Mills Theory, Sov. Phys. JETP 44 (1976) 443 [INSPIRE].

    ADS  Google Scholar 

  69. [69]

    V. Del Duca, An introduction to the perturbative QCD Pomeron and to jet physics at large rapidities, hep-ph/9503226 [INSPIRE].

  70. [70]

    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].

  71. [71]

    T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [INSPIRE].

  72. [72]

    R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  73. [73]

    J. Bedford, A. Brandhuber, B.J. Spence and G. Travaglini, A Recursion relation for gravity amplitudes, Nucl. Phys. B 721 (2005) 98 [hep-th/0502146] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  74. [74]

    J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic Amplitudes and Cluster Coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].

    ADS  Article  Google Scholar 

  75. [75]

    J. Bartels, L.N. Lipatov and A. Sabio Vera, Double-logarithms in Einstein-Hilbert gravity and supergravity, JHEP 07 (2014) 056 [arXiv:1208.3423] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  76. [76]

    S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in \( \mathcal{N} \) = 8 supergravity, arXiv:1901.08563 [INSPIRE].

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Chicherin, D., Gehrmann, T., Henn, J.M. et al. The two-loop five-particle amplitude in \( \mathcal{N} \) = 8 supergravity. J. High Energ. Phys. 2019, 115 (2019). https://doi.org/10.1007/JHEP03(2019)115

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Keywords

  • Scattering Amplitudes
  • Supergravity Models