Hidden conformal symmetry in tree-level graviton scattering

  • Florian Loebbert
  • Matin Mojaza
  • Jan Plefka
Open Access
Regular Article - Theoretical Physics


We argue that the scattering of gravitons in ordinary Einstein gravity possesses a hidden conformal symmetry at tree level in any number of dimensions. The presence of this conformal symmetry is indicated by the dilaton soft theorem in string theory, and it is reminiscent of the conformal invariance of gluon tree-level amplitudes in four dimensions. To motivate the underlying prescription, we demonstrate that formulating the conformal symmetry of gluon amplitudes in terms of momenta and polarization vectors requires manifest reversal and cyclic symmetry. Similarly, our formulation of the conformal symmetry of graviton amplitudes relies on a manifestly permutation symmetric form of the amplitude function.


Conformal and W Symmetry Scattering Amplitudes 


Open Access

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  1. [1]
    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].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    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].
  3. [3]
    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].
  4. [4]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    R.R. Metsaev, Stueckelberg approach to 6d conformal gravity, Workshop on Supersymmetries and Quantum Symmetries, July 18-23, 2011, Dubna, Russia,
  6. [6]
    J. Maldacena, Einstein Gravity from Conformal Gravity, arXiv:1105.5632 [INSPIRE].
  7. [7]
    R.R. Metsaev, Ordinary-derivative formulation of conformal low spin fields, JHEP 01 (2012) 064 [arXiv:0707.4437] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
  9. [9]
    H. Bondi, M. van der Burg and A. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21.Google Scholar
  10. [10]
    R. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103.Google Scholar
  11. [11]
    S. Pasterski, S.-H. Shao and A. Strominger, Gluon Amplitudes as 2d Conformal Correlators, Phys. Rev. D 96 (2017) 085006 [arXiv:1706.03917] [INSPIRE].
  12. [12]
    A. Schreiber, A. Volovich and M. Zlotnikov, Tree-level gluon amplitudes on the celestial sphere, Phys. Lett. B 781 (2018) 349 [arXiv:1711.08435] [INSPIRE].
  13. [13]
    M. Ademollo et al., Soft Dilations and Scale Renormalization in Dual Theories, Nucl. Phys. B 94 (1975) 221 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    J.A. Shapiro, On the Renormalization of Dual Models, Phys. Rev. D 11 (1975) 2937 [INSPIRE].
  15. [15]
    P. Di Vecchia, R. Marotta and M. Mojaza, Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string, JHEP 05 (2015) 137 [arXiv:1502.05258] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    P. Di Vecchia, R. Marotta and M. Mojaza, Subsubleading soft theorems of gravitons and dilatons in the bosonic string, JHEP 06 (2016) 054 [arXiv:1604.03355] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  17. [17]
    P. Di Vecchia, R. Marotta and M. Mojaza, Soft behavior of a closed massless state in superstring and universality in the soft behavior of the dilaton, JHEP 12 (2016) 020 [arXiv:1610.03481] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  18. [18]
    P. Di Vecchia, R. Marotta and M. Mojaza, The B-field soft theorem and its unification with the graviton and dilaton, JHEP 10 (2017) 017 [arXiv:1706.02961] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    P. Di Vecchia, R. Marotta, M. Mojaza and J. Nohle, New soft theorems for the gravity dilaton and the Nambu-Goldstone dilaton at subsubleading order, Phys. Rev. D 93 (2016) 085015 [arXiv:1512.03316] [INSPIRE].
  20. [20]
    T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N = 4 Superconformal Symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    S. Weinberg, Photons and Gravitons in s Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass, Phys. Rev. 135 (1964) B1049 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    R. Boels, Covariant representation theory of the Poincaré algebra and some of its extensions, JHEP 01 (2010) 010 [arXiv:0908.0738] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    R.H. Boels and R. Medina, Graviton and gluon scattering from first principles, Phys. Rev. Lett. 118 (2017) 061602 [arXiv:1607.08246] [INSPIRE].
  24. [24]
    N. Arkani-Hamed, L. Rodina and J. Trnka, Locality and Unitarity from Singularities and Gauge Invariance, arXiv:1612.02797 [INSPIRE].
  25. [25]
    R. Roiban and A.A. Tseytlin, On four-point interactions in massless higher spin theory in flat space, JHEP 04 (2017) 139 [arXiv:1701.05773] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    J. Scherk and J.H. Schwarz, Dual Models for Nonhadrons, Nucl. Phys. B 81 (1974) 118 [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    R.H. Boels and H. Lüo, A minimal approach to the scattering of physical massless bosons, JHEP 05 (2018) 063 [arXiv:1710.10208] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    C. Cheung, C.-H. Shen and C. Wen, Unifying Relations for Scattering Amplitudes, JHEP 02 (2018)095 [arXiv:1705.03025] [INSPIRE].
  29. [29]
    J.H. Schwarz, Superstring Theory, Phys. Rept. 89 (1982) 223 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, A recursive method for SYM n-point tree amplitudes, Phys. Rev. D 83 (2011) 126012 [arXiv:1012.3981] [INSPIRE].
  31. [31]
    N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Analytic representations of Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 964 [arXiv:1605.06501] [INSPIRE].
  32. [32]
    J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, Constraining subleading soft gluon and graviton theorems, Phys. Rev. D 90 (2014) 065024 [arXiv:1406.6574] [INSPIRE].
  33. [33]
    D.J. Gross and R. Jackiw, Low-Energy Theorem for Graviton Scattering, Phys. Rev. 166 (1968)1287 [INSPIRE].
  34. [34]
    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].
  35. [35]
    D.A. McGady and L. Rodina, Recursion relations for graviton scattering amplitudes from Bose symmetry and bonus scaling laws, Phys. Rev. D 91 (2015) 105010 [arXiv:1408.5125] [INSPIRE].
  36. [36]
    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].
  37. [37]
    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].
  38. [38]
    D. Maître and P. Mastrolia, S@M, a Mathematica Implementation of the Spinor-Helicity Formalism, Comput. Phys. Commun. 179 (2008) 501 [arXiv:0710.5559] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    C. Cheung and G.N. Remmen, Hidden Simplicity of the Gravity Action, JHEP 09 (2017) 002 [arXiv:1705.00626] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    E.T. Tomboulis, On the ‘simple’ form of the gravitational action and the self-interacting graviton, JHEP 09 (2017) 145 [arXiv:1708.03977] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    R. Manvelyan, K. Mkrtchyan and W. Ruehl, Direct Construction of A Cubic Selfinteraction for Higher Spin gauge Fields, Nucl. Phys. B 844 (2011) 348 [arXiv:1002.1358] [INSPIRE].
  42. [42]
    R. Manvelyan, K. Mkrtchyan and W. Ruehl, A generating function for the cubic interactions of higher spin fields, Phys. Lett. B 696 (2011) 410 [arXiv:1009.1054] [INSPIRE].
  43. [43]
    T. Adamo and L. Mason, Conformal and Einstein gravity from twistor actions, Class. Quant. Grav. 31 (2014) 045014 [arXiv:1307.5043] [INSPIRE].
  44. [44]
    D. Skinner, Twistor Strings for N = 8 Supergravity, arXiv:1301.0868 [INSPIRE].
  45. [45]
    A. Hodges, A simple formula for gravitational MHV amplitudes, arXiv:1204.1930 [INSPIRE].
  46. [46]
    M. Kaku and P.K. Townsend, Poincaré supergravity as broken superconformal gravity, Phys. Lett. B 76 (1978) 54 [INSPIRE].
  47. [47]
    N. Berkovits and E. Witten, Conformal supergravity in twistor-string theory, JHEP 08 (2004)009 [hep-th/0406051] [INSPIRE].
  48. [48]
    H. Johansson and J. Nohle, Conformal Gravity from Gauge Theory, arXiv:1707.02965 [INSPIRE].
  49. [49]
    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    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].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    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].
  52. [52]
    F. Cachazo and P. Svrček, Tree level recursion relations in general relativity, hep-th/0502160 [INSPIRE].
  53. [53]
    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].
  54. [54]
    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].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    A.J. Larkoski, Conformal Invariance of the Subleading Soft Theorem in Gauge Theory, Phys. Rev. D 90 (2014) 087701 [arXiv:1405.2346] [INSPIRE].
  56. [56]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    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].
  58. [58]
    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].
  59. [59]
    T. Bargheer, F. Loebbert and C. Meneghelli, Symmetries of Tree-level Scattering Amplitudes in N = 6 Superconformal Chern-Simons Theory, Phys. Rev. D 82 (2010) 045016 [arXiv:1003.6120] [INSPIRE].
  60. [60]
    D. Gang, Y.-t. Huang, E. Koh, S. Lee and A.E. Lipstein, Tree-level Recursion Relation and Dual Superconformal Symmetry of the ABJM Theory, JHEP 03 (2011) 116 [arXiv:1012.5032] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    T. Dennen and Y.-t. Huang, Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes, JHEP 01 (2011) 140 [arXiv:1010.5874] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    S. Caron-Huot and D. O’Connell, Spinor Helicity and Dual Conformal Symmetry in Ten Dimensions, JHEP 08 (2011) 014 [arXiv:1010.5487] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  63. [63]
    G.P. Korchemsky and E. Sokatchev, Symmetries and analytic properties of scattering amplitudes in N = 4 SYM theory, Nucl. Phys. B 832 (2010) 1 [arXiv:0906.1737] [INSPIRE].

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Institut für Physik and IRIS Adlershof, Humboldt-Universität zu BerlinBerlinGermany
  2. 2.Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdamGermany

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