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Four-graviton scattering to three loops in \( \mathcal{N}=8 \) supergravity

Journal of High Energy Physics volume 2019, Article number: 23 (2019) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We compute the three-loop scattering amplitude of four gravitons in \( \mathcal{N}=8 \) supergravity. Our results are analytic formulae for a Laurent expansion of the amplitude in the regulator of dimensional regularisation. The coefficients of this series are closed formulae in terms of well-established harmonic poly-logarithms. Our results display a remarkable degree of simplicity and represent an important stepping stone in the exploration of the structure of scattering amplitudes. In particular, we observe that to this loop order the four graviton amplitude is given by uniform weight 2L functions, where L is the loop order.

References

  1. Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The Ultraviolet Behavior of N = 8 Supergravity at Four Loops, Phys. Rev. Lett. 103 (2009) 081301 [arXiv:0905.2326] [INSPIRE].

  2. G. Bossard, P.S. Howe, K.S. Stelle and P. Vanhove, The vanishing volume of D = 4 superspace, Class. Quant. Grav. 28 (2011) 215005 [arXiv:1105.6087] [INSPIRE].

  3. Z. Bern et al., Ultraviolet Properties of \( \mathcal{N}=8 \) Supergravity at Five Loops, Phys. Rev. D 98 (2018) 086021 [arXiv:1804.09311] [INSPIRE].

  4. 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].

  5. Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [INSPIRE].

  6. Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].

  7. S. Oxburgh and C.D. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].

  8. A. Luna et al., Perturbative spacetimes from Yang-Mills theory, JHEP 04 (2017) 069 [arXiv:1611.07508] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  9. A. Luna, R. Monteiro, I. Nicholson, D. O’Connell and C.D. White, The double copy: Bremsstrahlung and accelerating black holes, JHEP 06 (2016) 023 [arXiv:1603.05737] [INSPIRE].

  10. W.D. Goldberger and A.K. Ridgway, Bound states and the classical double copy, Phys. Rev. D 97 (2018) 085019 [arXiv:1711.09493] [INSPIRE].

  11. A. Luna, I. Nicholson, D. O’Connell and C.D. White, Inelastic Black Hole Scattering from Charged Scalar Amplitudes, JHEP 03 (2018) 044 [arXiv:1711.03901] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order, arXiv:1901.04424 [INSPIRE].

  14. A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent and J. Vines, Energetics of two-body Hamiltonians in post-Minkowskian gravity, arXiv:1901.07102 [INSPIRE].

  15. A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].

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

  17. 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].

  18. S. He, Y.-t. Huang and C. Wen, Loop Corrections to Soft Theorems in Gauge Theories and Gravity, JHEP 12 (2014) 115 [arXiv:1405.1410] [INSPIRE].

    ADS  Article  Google Scholar 

  19. P. Vecchia, R. Marotta and M. Mojaza, Multiloop Soft Theorem for Gravitons and Dilatons in the Bosonic String, JHEP 01 (2019) 038 [arXiv:1808.04845] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  20. A.J. Larkoski, D. Neill and I.W. Stewart, Soft Theorems from Effective Field Theory, JHEP 06 (2015) 077 [arXiv:1412.3108] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. 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 

  22. 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 

  23. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [INSPIRE].

  24. 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].

  25. N. Berkovits and J. Maldacena, Fermionic T-duality, Dual Superconformal Symmetry and the Amplitude/Wilson Loop Connection, JHEP 09 (2008) 062 [arXiv:0807.3196] [INSPIRE].

  26. A. Brandhuber, P. Heslop and G. Travaglini, A Note on dual superconformal symmetry of the N = 4 super Yang-Mills S-matrix, Phys. Rev. D 78 (2008) 125005 [arXiv:0807.4097] [INSPIRE].

  27. J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].

  28. S. Caron-Huot and J.M. Henn, Solvable Relativistic Hydrogenlike System in Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 113 (2014) 161601 [arXiv:1408.0296] [INSPIRE].

    ADS  Article  Google Scholar 

  29. 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].

  30. R. Ben-Israel, A.G. Tumanov and A. Sever, Scattering amplitudes — Wilson loops duality for the first non-planar correction, JHEP 08 (2018) 122 [arXiv:1802.09395] [INSPIRE].

  31. Z. Bern, M. Enciso, C.-H. Shen and M. Zeng, Dual Conformal Structure Beyond the Planar Limit, Phys. Rev. Lett. 121 (2018) 121603 [arXiv:1806.06509] [INSPIRE].

    ADS  Article  Google Scholar 

  32. 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 

  33. L. Bianchi, A. Brandhuber, R. Panerai and G. Travaglini, Dual conformal invariance for form factors, JHEP 02 (2019) 134 [arXiv:1812.10468] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. S. Caron-Huot and Z. Zahraee, Integrability of Black Hole Orbits in Maximal Supergravity, arXiv:1810.04694 [INSPIRE].

  35. A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [Erratum ibid. B 632 (2006) 754] [hep-th/0404092] [INSPIRE].

  36. A.V. Kotikov and L.N. Lipatov, On the highest transcendentality in N = 4 SUSY, Nucl. Phys. B 769 (2007) 217 [hep-th/0611204] [INSPIRE].

  37. 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 

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

    ADS  Article  Google Scholar 

  39. 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 

  40. P. Wasser, Analytic properties of Feynman integrals for scattering amplitudes, MSc Thesis (2016) [https://publications.ub.uni-mainz.de/theses/frontdoor.php?sourceopus=100001967].

  41. 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 

  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. 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].

  44. 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].

  45. D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, The two-loop five-particle amplitude in \( \mathcal{N}=8 \) supergravity, JHEP 03 (2019) 115 [arXiv:1901.05932] [INSPIRE].

  46. S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in \( \mathcal{N}=8 \) supergravity, JHEP 03 (2019) 123 [arXiv:1901.08563] [INSPIRE].

  47. 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].

  48. 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].

  49. 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].

  50. Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson, D.A. Kosower and R. Roiban, Three-Loop Superfiniteness of N = 8 Supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [INSPIRE].

  51. Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N = 8 Supergravity, Phys. Rev. D 78 (2008) 105019 [arXiv:0808.4112] [INSPIRE].

  52. 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].

  53. Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Logarithmic Singularities and Maximally Supersymmetric Amplitudes, JHEP 06 (2015) 202 [arXiv:1412.8584] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  54. J.L. Bourjaily, E. Herrmann and J. Trnka, Maximally supersymmetric amplitudes at infinite loop momentum, Phys. Rev. D 99 (2019) 066006 [arXiv:1812.11185] [INSPIRE].

  55. 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 

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

  57. 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 

  58. 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 

  59. L. Adams and S. Weinzierl, The ε-form of the differential equations for Feynman integrals in the elliptic case, Phys. Lett. B 781 (2018) 270 [arXiv:1802.05020] [INSPIRE].

  60. J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, JHEP 01 (2019) 023 [arXiv:1809.10698] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  61. V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1.

    MathSciNet  MATH  Google Scholar 

  62. A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].

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

    ADS  Article  Google Scholar 

  64. 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 

  65. 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 

  66. C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  67. D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].

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

    ADS  MathSciNet  Article  Google Scholar 

  69. 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].

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

  72. 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 

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  74. 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 

  75. Ø. Almelid, C. Duhr, E. Gardi, A. McLeod and C.D. White, Bootstrapping the QCD soft anomalous dimension, JHEP 09 (2017) 073 [arXiv:1706.10162] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  76. C.D. White, An Introduction to Webs, J. Phys. G 43 (2016) 033002 [arXiv:1507.02167] [INSPIRE].

  77. Z. Bern, J. Parra-Martinez and R. Roiban, Canceling the U(1) Anomaly in the S Matrix of N = 4 Supergravity, Phys. Rev. Lett. 121 (2018) 101604 [arXiv:1712.03928] [INSPIRE].

  78. Z. Bern, A. Edison, D. Kosower and J. Parra-Martinez, Curvature-squared multiplets, evanescent effects and the U(1) anomaly in N = 4 supergravity, Phys. Rev. D 96 (2017) 066004 [arXiv:1706.01486] [INSPIRE].

  79. Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].

  80. J.J.M. Carrasco, R. Kallosh, R. Roiban and A.A. Tseytlin, On the U(1) duality anomaly and the S-matrix of N = 4 supergravity, JHEP 07 (2013) 029 [arXiv:1303.6219] [INSPIRE].

  81. D.Z. Freedman, R. Kallosh, D. Murli, A. Van Proeyen and Y. Yamada, Absence of U(1) Anomalous Superamplitudes in \( \mathcal{N}\ge 5 \) Supergravities, JHEP 05 (2017) 067 [arXiv:1703.03879] [INSPIRE].

  82. R. Kallosh, Cancellation of Conformal and Chiral Anomalies in \( \mathcal{N}\ge 5 \) supergravities, Phys. Rev. D 95 (2017) 041701 [arXiv:1612.08978] [INSPIRE].

  83. J. Bjornsson and M.B. Green, 5 loops in 24/5 dimensions, JHEP 08 (2010) 132 [arXiv:1004.2692] [INSPIRE].

  84. P. Vanhove, The Critical ultraviolet behaviour of N = 8 supergravity amplitudes, arXiv:1004.1392 [INSPIRE].

  85. J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].

  86. J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP 03 (2014) 088 [arXiv:1312.2588] [INSPIRE].

  87. D.C. Dunbar and P.S. Norridge, Calculation of graviton scattering amplitudes using string based methods, Nucl. Phys. B 433 (1995) 181 [hep-th/9408014] [INSPIRE].

  88. M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].

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

  90. 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].

  91. S.B. Giddings and R.A. Porto, The Gravitational S-matrix, Phys. Rev. D 81 (2010) 025002 [arXiv:0908.0004] [INSPIRE].

  92. S.B. Giddings, M. Schmidt-Sommerfeld and J.R. Andersen, High energy scattering in gravity and supergravity, Phys. Rev. D 82 (2010) 104022 [arXiv:1005.5408] [INSPIRE].

  93. H.J. Schnitzer, Reggeization of N = 8 supergravity and N = 4 Yang-Mills theory. II., arXiv:0706.0917 [INSPIRE].

  94. D. Amati, M. Ciafaloni and G. Veneziano, Higher Order Gravitational Deflection and Soft Bremsstrahlung in Planckian Energy Superstring Collisions, Nucl. Phys. B 347 (1990) 550 [INSPIRE].

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

    J. M. Henn

  2. Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, U.S.A.

    B. Mistlberger

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Henn, J.M., Mistlberger, B. Four-graviton scattering to three loops in \( \mathcal{N}=8 \) supergravity. J. High Energ. Phys. 2019, 23 (2019). https://doi.org/10.1007/JHEP05(2019)023

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Keywords

  • Scattering Amplitudes
  • Supersymmetric Gauge Theory