New expressions for gravitational scattering amplitudes

  • Andrew Hodges


New methods are introduced for evaluating tree-level gravitational scattering amplitudes. A new N=7 super-symmetric recursion yields amplitudes free from the spurious double poles of the N=8 theory. This is illustrated by a new nine-term expression for the six-graviton NMHV amplitude. The general scheme also implies a simplified recurrence relation for MHV amplitudes. We show how this relation is satisfied by a new expression for MHV amplitudes, far simpler than those hitherto identified, and exhibiting manifest S n symmetry. This reformulation is related to a new momentum-twistor representation of the MHV amplitudes.


Supersymmetric gauge theory Scattering Amplitudes 


  1. [1]
    D.C. Dunbar, J.H. Ettle and W.B. Perkins, Constructing Gravity Amplitudes from Real Soft and Collinear Factorisation, Phys. Rev. D 86 (2012) 026009 [arXiv:1203.0198] [INSPIRE].ADSGoogle Scholar
  2. [2]
    D. Nguyen, M. Spradlin, A. Volovich and C. Wen, The Tree Formula for MHV Graviton Amplitudes, JHEP 07 (2010) 045 [arXiv:0907.2276] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    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].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    F.A. Berends, W. Giele and H. Kuijf, On relations between multi - gluon and multigraviton scattering, Phys. Lett. B 211 (1988) 91 [INSPIRE].ADSGoogle Scholar
  5. [5]
    L. Mason and D. Skinner, Gravity, Twistors and the MHV Formalism, Commun. Math. Phys. 294 (2010) 827 [arXiv:0808.3907] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. [6]
    R. Penrose and M.A.H. MacCallum, Twistor theory: an approach to the quantisation of fields and space-time, Physics Reports 4 (1972) 241.MathSciNetGoogle Scholar
  7. [7]
    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].ADSGoogle Scholar
  8. [8]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605 [INSPIRE].
  9. [9]
    T. Adamo and L. Mason, Twistor-strings and gravity tree amplitudes, Class. Quant. Grav. 30 (2013)075020 [arXiv:1207.3602] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    F. Cachazo and Y. Geyer, ATwistor StringInspired Formula For Tree-Level Scattering Amplitudes in N = 8 SUGRA, arXiv:1206.6511 [INSPIRE].
  11. [11]
    D. Skinner, Twistor Strings for N = 8 Supergravity, arXiv:1301.0868 [INSPIRE].
  12. [12]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    A.P. Hodges, Twistor diagram recursion for all gauge-theoretic tree amplitudes, hep-th/0503060 [INSPIRE].
  14. [14]
    A.P. Hodges, Twistor diagrams for all tree amplitudes in gauge theory: A Helicity-independent formalism, hep-th/0512336 [INSPIRE].
  15. [15]
    A.P. Hodges, Scattering amplitudes for eight gauge fields, hep-th/0603101 [INSPIRE].
  16. [16]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in Twistor Space, JHEP 03 (2010) 110 [arXiv:0903.2110] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    N. Arkani-Hamed, personal communication.Google Scholar
  19. [19]
    N. Arkani-Hamed, Scattering without space-time, presentation given at the RP80 Oxford conference Twistors, Geometry and Physics, Oxford, U.K., 21–22 July 2011
  20. [20]
    R. Penrose, Twistor theory, its aims and achievements, in Quantum Gravity, C.J. Isham, R. Penrose and D.W. Sciama eds., Oxford University Press, Oxford, U.K. (1975).Google Scholar
  21. [21]
    J. Drummond, M. Spradlin, A. Volovich and C. Wen, Tree-Level Amplitudes in N = 8 Supergravity, Phys. Rev. D 79 (2009) 105018 [arXiv:0901.2363] [INSPIRE].MathSciNetADSGoogle Scholar
  22. [22]
    F. Cachazo and P. Svrček, Tree level recursion relations in general relativity, hep-th/0502160 [INSPIRE].
  23. [23]
    B.S. DeWitt, Quantum Theory of Gravity. 3. Applications of the Covariant Theory, Phys. Rev. 162 (1967) 1239 [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    H. Kawai, D. Lewellen and S. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    Y.t. Huang, private communication (2011).Google Scholar
  27. [27]
    J.L. Bourjaily, Efficient Tree-Amplitudes in N = 4: Automatic BCFW Recursion in Mathematica, arXiv:1011.2447 [INSPIRE].
  28. [28]
    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].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    E. Conde and S. Rajabi, The Twelve-Graviton Next-to-MHV Amplitude from Risagers Construction, JHEP 09 (2012) 120 [arXiv:1205.3500] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    C. Cheung, Gravity Amplitudes from n-Space, JHEP 12 (2012) 057 [arXiv:1207.4458] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    S.G. Naculich, H. Nastase and H.J. Schnitzer, Applications of Subleading Color Amplitudes in N = 4 SYM Theory, Adv. High Energy Phys. 2011 (2011) 190587 [arXiv:1105.3718] [INSPIRE].MathSciNetGoogle Scholar
  33. [33]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, A Note on Polytopes for Scattering Amplitudes, JHEP 04 (2012) 081 [arXiv:1012.6030] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Mathematical InstituteOxfordU.K

Personalised recommendations