On Feynman rules for Mellin amplitudes in AdS/CFT

  • Dhritiman Nandan
  • Anastasia Volovich
  • Congkao Wen


The computation of CFT correlation functions via Witten diagrams in AdS space can be simplified via the Mellin transform. Recently a set of Feynman rules for tree-level Mellin space amplitudes has been proposed for scalar theories. In this note we derive these rules by explicitly evaluating all of the relevant Witten diagram integrals for the scalar φn theory. We also check that the rules reduce to the usual Feynman rules in the flat space limit.


Scattering Amplitudes AdS-CFT Correspondence Conformal and W Sym- metry Supergravity Models 


  1. [1]
    J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1133] [hep-th/9711200] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  2. [2]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  3. [3]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    H. Liu and A.A. Tseytlin, On four point functions in the CFT/AdS correspondence, Phys. Rev. D 59 (1999) 086002 [hep-th/9807097] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    H. Liu, Scattering in anti-de Sitter space and operator product expansion, Phys. Rev. D 60 (1999) 106005 [hep-th/9811152] [INSPIRE].ADSGoogle Scholar
  6. [6]
    E. D’Hoker and D.Z. Freedman, General scalar exchange in AdS(d + 1), Nucl. Phys. B 550 (1999) 261 [hep-th/9811257] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  7. [7]
    D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Comments on 4 point functions in the CFT/AdS correspondence, Phys. Lett. B 452 (1999) 61 [hep-th/9808006] [INSPIRE].MathSciNetADSGoogle Scholar
  8. [8]
    D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT(d)/AdS(d + 1) correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys. B 562 (1999) 353 [hep-th/9903196] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  10. [10]
    G. Arutyunov and S. Frolov, Four point functions of lowest weight CPOs in N = 4 SYM(4) in supergravity approximation, Phys. Rev. D 62 (2000) 064016 [hep-th/0002170] [INSPIRE].MathSciNetADSGoogle Scholar
  11. [11]
    G. Arutyunov, F. Dolan, H. Osborn and E. Sokatchev, Correlation functions and massive Kaluza-Klein modes in the AdS/CFT correspondence, Nucl. Phys. B 665 (2003) 273 [hep-th/0212116] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    G. Arutyunov and E. Sokatchev, On a large-N degeneracy in N = 4 SYM and the AdS/CFT correspondence, Nucl. Phys. B 663 (2003) 163 [hep-th/0301058] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    L. Berdichevsky and P. Naaijkens, Four-point functions of different-weight operators in the AdS/CFT correspondence, JHEP 01 (2008) 071 [arXiv:0709.1365] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    L.I. Uruchurtu, Four-point correlators with higher weight superconformal primaries in the AdS/CFT Correspondence, JHEP 03 (2009) 133 [arXiv:0811.2320] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    E. Buchbinder and A. Tseytlin, On semiclassical approximation for correlators of closed string vertex operators in AdS/CFT, JHEP 08 (2010) 057 [arXiv:1005.4516] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    L.I. Uruchurtu, Next-next-to-extremal Four Point Functions of N = 4 1/2 BPS Operators in the AdS/CFT Correspondence, JHEP 08 (2011) 133 [arXiv:1106.0630] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    F. Dolan, M. Nirschl and H. Osborn, Conjectures for large-N superconformal N = 4 chiral primary four point functions, Nucl. Phys. B 749 (2006) 109 [hep-th/0601148] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    E. D’Hoker, D.Z. Freedman and L. Rastelli, AdS/CFT four point functions: How to succeed at z integrals without really trying, Nucl. Phys. B 562 (1999) 395 [hep-th/9905049] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  19. [19]
    S. Raju, BCFW for Witten Diagrams, Phys. Rev. Lett. 106 (2011) 091601 [arXiv:1011.0780] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    S. Raju, Recursion Relations for AdS/CFT Correlators, Phys. Rev. D 83 (2011) 126002 [arXiv:1102.4724] [INSPIRE].ADSGoogle Scholar
  21. [21]
    G. Mack, D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
  22. [22]
    G. Mack, D-dimensional Conformal Field Theories with anomalous dimensions as Dual Resonance Models, arXiv:0909.1024 [INSPIRE].
  23. [23]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 DOI:dx.doi.org [arXiv:1107.1504] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A Natural Language for AdS/CFT Correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    I. Balitsky, Mellin representation of the graviton bulk-to-bulk propagator in AdS, Phys. Rev. D 83 (2011) 087901 [arXiv:1102.0577] [INSPIRE].ADSGoogle Scholar
  29. [29]
    L. Susskind, Holography in the flat space limit, hep-th/9901079 [INSPIRE].
  30. [30]
    J. Polchinski, S matrices from AdS space-time, hep-th/9901076 [INSPIRE].
  31. [31]
    M. Gary, S.B. Giddings and J. Penedones, Local bulk S-matrix elements and CFT singularities, Phys. Rev. D 80 (2009) 085005 [arXiv:0903.4437]. 24 pages, 3 figs [INSPIRE].MathSciNetADSGoogle Scholar
  32. [32]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS, JHEP 07 (2011) 023 [arXiv:1007.2412] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    T. Okuda and J. Penedones, String scattering in flat space and a scaling limit of Yang-Mills correlators, Phys. Rev. D 83 (2011) 086001 [arXiv:1002.2641] [INSPIRE].ADSGoogle Scholar
  35. [35]
    A.L. Fitzpatrick and J. Kaplan, Scattering States in AdS/CFT, arXiv:1104.2597 [INSPIRE].
  36. [36]
    M. Gary and S.B. Giddings, The Flat space S-matrix from the AdS/CFT correspondence?, Phys. Rev. D 80 (2009) 046008 [arXiv:0904.3544] [INSPIRE].MathSciNetADSGoogle Scholar
  37. [37]
    S.B. Giddings, Flat space scattering and bulk locality in the AdS/CFT correspondence, Phys. Rev. D 61 (2000) 106008 [hep-th/9907129] [INSPIRE].MathSciNetADSGoogle Scholar
  38. [38]
    M. Gary and S.B. Giddings, Constraints on a fine-grained AdS/CFT correspondence, arXiv:1106.3553 [INSPIRE].
  39. [39]
    K. Symanzik, On Calculations in conformal invariant field theories, Lett. Nuovo Cim. 3 (1972) 734 [INSPIRE].
  40. [40]
    A.L. Fitzpatrick and J. Kaplan, Analyticity and the Holographic S-matrix, arXiv:1111.6972 [INSPIRE].

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Dhritiman Nandan
    • 1
  • Anastasia Volovich
    • 1
  • Congkao Wen
    • 2
  1. 1.Physics DepartmentBrown UniversityRhode IslandU.S.A
  2. 2.Centre for Research in String Theory, School of Physics and AstronomyQueen Mary University of LondonLondonUnited Kingdom

Personalised recommendations