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On-shell diagrams for \( \mathcal{N} \) = 8 supergravity amplitudes

  • Paul Heslop
  • Arthur E. Lipstein
Open Access
Regular Article - Theoretical Physics

Abstract

We define recursion relations for \( \mathcal{N} \) = 8 supergravity amplitudes using a generalization of the on-shell diagrams developed for planar \( \mathcal{N} \) = 4 super-Yang-Mills. Although the recursion relations generically give rise to non-planar on-shell diagrams, we show that at tree-level the recursion can be chosen to yield only planar diagrams, the same diagrams occurring in the planar \( \mathcal{N} \) = 4 theory. This implies non-trivial identities for non-planar diagrams as well as interesting relations between the \( \mathcal{N} \) = 4 and \( \mathcal{N} \) = 8 theories. We show that the on-shell diagrams of \( \mathcal{N} \) = 8 supergravity obey equivalence relations analogous to those of \( \mathcal{N} \) = 4 super-Yang-Mills, and we develop a systematic algorithm for reading off Grassmannian integral formulae directly from the on-shell diagrams. We also show that the 1-loop 4-point amplitude of \( \mathcal{N} \) = 8 supergravity can be obtained from on-shell diagrams.

Keywords

Scattering Amplitudes Supergravity Models 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605 [INSPIRE].
  2. [2]
    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
  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].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in twistor space, JHEP 03 (2010) 110 [arXiv:0903.2110] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S-matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of residues and Grassmannian dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    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].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    J.M. Drummond, G.P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    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].
  12. [12]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    C.F. Berger, Z. Bern, L.J. Dixon, D. Forde and D.A. Kosower, Bootstrapping one-loop QCD amplitudes with general helicities, Phys. Rev. D 74 (2006) 036009 [hep-ph/0604195] [INSPIRE].
  14. [14]
    S. Caron-Huot, Loops and trees, JHEP 05 (2011) 080 [arXiv:1007.3224] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    R.H. Boels, On BCFW shifts of integrands and integrals, JHEP 11 (2010) 113 [arXiv:1008.3101] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    A.E. Lipstein and L. Mason, From the holomorphic Wilson loop to ‘d log’ loop-integrands for super-Yang-Mills amplitudes, JHEP 05 (2013) 106 [arXiv:1212.6228] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    A.E. Lipstein and L. Mason, From d logs to dilogs the super Yang-Mills MHV amplitude revisited, JHEP 01 (2014) 169 [arXiv:1307.1443] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    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].ADSCrossRefGoogle Scholar
  19. [19]
    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].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a nonplanar amplituhedron, arXiv:1512.08591 [INSPIRE].
  21. [21]
    N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    N. Arkani-Hamed and J. Trnka, Into the amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    N. Arkani-Hamed, A. Hodges and J. Trnka, Positive amplitudes in the amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka, On-shell structures of MHV amplitudes beyond the planar limit, JHEP 06 (2015) 179 [arXiv:1412.8475] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    S. Franco, D. Galloni, B. Penante and C. Wen, Non-planar on-shell diagrams, JHEP 06 (2015) 199 [arXiv:1502.02034] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    B. Chen, G. Chen, Y.-K.E. Cheung, R. Xie and Y. Xin, Top-forms of leading singularities in nonplanar multi-loop amplitudes, arXiv:1506.02880 [INSPIRE].
  27. [27]
    R. Frassek and D. Meidinger, Yangian-type symmetries of non-planar leading singularities, JHEP 05 (2016) 110 [arXiv:1603.00088] [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    R. Frassek, D. Meidinger, D. Nandan and M. Wilhelm, On-shell diagrams, Grassmannians and integrability for form factors, JHEP 01 (2016) 182 [arXiv:1506.08192] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    P. Benincasa, On-shell diagrammatics and the perturbative structure of planar gauge theories, arXiv:1510.03642 [INSPIRE].
  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].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    F. Cachazo, L. Mason and D. Skinner, Gravity in twistor space and its Grassmannian formulation, SIGMA 10 (2014) 051 [arXiv:1207.4712] [INSPIRE].MathSciNetMATHGoogle Scholar
  32. [32]
    S. He, A link representation for gravity amplitudes, JHEP 10 (2013) 139 [arXiv:1207.4064] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    F. Cachazo and P. Svrček, Tree level recursion relations in general relativity, hep-th/0502160 [INSPIRE].
  34. [34]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    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
  36. [36]
    N. Arkani-Hamed and J. Kaplan, On tree amplitudes in gauge theory and gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    J.M. 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].ADSMathSciNetGoogle Scholar
  38. [38]
    E. Herrmann and J. Trnka, Gravity on-shell diagrams, arXiv:1604.03479 [INSPIRE].
  39. [39]
    F.A. Berends, W.T. Giele and H. Kuijf, On relations between multi-gluon and multigraviton scattering, Phys. Lett. B 211 (1988) 91 [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop integrands for scattering amplitudes from the Riemann sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily, S. Caron-Huot, P.H. Damgaard and B. Feng, New representations of the perturbative S-matrix, Phys. Rev. Lett. 116 (2016) 061601 [arXiv:1509.02169] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    Z. Bern, L.J. Dixon and D.A. Kosower, Bootstrapping multi-parton loop amplitudes in QCD, Phys. Rev. D 73 (2006) 065013 [hep-ph/0507005] [INSPIRE].
  44. [44]
    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].ADSGoogle Scholar
  45. [45]
    D.C. Dunbar and W.B. Perkins, The N = 4 supergravity NMHV six-point one-loop amplitude, arXiv:1601.03918 [INSPIRE].
  46. [46]
    A. Brandhuber, S. McNamara, B. Spence and G. Travaglini, Recursion relations for one-loop gravity amplitudes, JHEP 03 (2007) 029 [hep-th/0701187] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    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].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDurham UniversityDurhamU.K.

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