Advertisement

Journal of High Energy Physics

, 2011:71 | Cite as

Spinning conformal correlators

  • Miguel S. Costa
  • João Penedones
  • David Poland
  • Slava Rychkov
Open Access
Article

Abstract

We develop the embedding formalism for conformal field theories, aimed at doing computations with symmetric traceless operators of arbitrary spin. We use an indexfree notation where tensors are encoded by polynomials in auxiliary polarization vectors. The efficiency of the formalism is demonstrated by computing the tensor structures allowed in n-point conformal correlation functions of tensors operators. Constraints due to tensor conservation also take a simple form in this formalism. Finally, we obtain a perfect match between the number of independent tensor structures of conformal correlators in d dimensions and the number of independent structures in scattering amplitudes of spinning particles in (d + 1)-dimensional Minkowski space.

Keywords

AdS-CFT Correspondence Conformal Field Models in String Theory SpaceTime Symmetries 

References

  1. [1]
    S. Ferrara, A. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    A. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].MathSciNetGoogle Scholar
  3. [3]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    V.S. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    F. Caracciolo and V.S. Rychkov, Rigorous Limits on the Interaction Strength in Quantum Field Theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].ADSGoogle Scholar
  6. [6]
    R. Rattazzi, S. Rychkov and A. Vichi, Central Charge Bounds in 4D Conformal Field Theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].ADSGoogle Scholar
  7. [7]
    R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D Conformal Field Theories with Global Symmetry, J. Phys. A A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    A. Vichi, Improved bounds for CFTs with global symmetries, arXiv:1106.4037 [INSPIRE].
  9. [9]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, arXiv:1109.5176 [INSPIRE].
  12. [12]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, arXiv:1109.6321 [INSPIRE].
  13. [13]
    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].
  14. [14]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    A. 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].ADSCrossRefGoogle Scholar
  16. [16]
    A. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A Natural Language for AdS/CFT Correlators, arXiv:1107.1499 [INSPIRE].
  17. [17]
    M.F. Paulos, Towards Feynman rules for Mellin amplitudes, arXiv:1107.1504 [INSPIRE].
  18. [18]
    S. Raju, BCFW for Witten Diagrams, Phys. Rev. Lett. 106 (2011) 091601 [arXiv:1011.0780] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    S. Raju, Recursion Relations for AdS/CFT Correlators, Phys. Rev. D 83 (2011) 126002 [arXiv:1102.4724] [INSPIRE].ADSGoogle Scholar
  20. [20]
    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
  21. [21]
    P.A. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  22. [22]
    G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    D. Boulware, L. Brown and R. Peccei, Deep-inelastic electroproduction and conformal symmetry, Phys. Rev. D 2 (1970) 293 [INSPIRE].ADSGoogle Scholar
  24. [24]
    S. Ferrara, A. F. Grillo and R. Gatto, Springer Tracts in Modern Physics. Vol. 67: Conformal algebra in space-time and operator product expansion, Springer Verlag, Heidelberg Germany (1973).Google Scholar
  25. [25]
    L. Cornalba, M.S. Costa and J. Penedones, Deep Inelastic Scattering in Conformal QCD, JHEP 03 (2010) 133 [arXiv:0911.0043] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].ADSGoogle Scholar
  27. [27]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer, New York U.S.A. (1997).MATHCrossRefGoogle Scholar
  28. [28]
    I. Bars, Two time physics in field theory, Phys. Rev. D 62 (2000) 046007 [hep-th/0003100] [INSPIRE].MathSciNetADSGoogle Scholar
  29. [29]
    A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [INSPIRE].ADSGoogle Scholar
  30. [30]
    E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton U.S.A. (1971).MATHGoogle Scholar
  31. [31]
    A.H. Guth and D.E. Soper, Short Distance Behavior of the Bethe-Salpeter Wave Function, Phys. Rev. D 12 (1975) 1143 [INSPIRE].ADSGoogle Scholar
  32. [32]
    V. Dobrev, V. Petkova, S. Petrova and I. Todorov, Dynamical Derivation of Vacuum Operator Product Expansion in Euclidean Conformal Quantum Field Theory, Phys. Rev. D 13 (1976) 887 [INSPIRE].ADSGoogle Scholar
  33. [33]
    A. Belitsky, J. Henn, C. Jarczak, D. Mueller and E. Sokatchev, Anomalous dimensions of leading twist conformal operators, Phys. Rev. D 77 (2008) 045029 [arXiv:0707.2936] [INSPIRE].ADSGoogle Scholar
  34. [34]
    M. Grigoriev and A. Waldron, Massive Higher Spins from BRST and Tractors, Nucl. Phys. B 853 (2011) 291 [arXiv:1104.4994] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  35. [35]
    V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Lecture Notes in Physics. Vol. 63: Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, Springer Verlag, Berlin Germany (1977).Google Scholar
  36. [36]
    G. Mack, Convergence Of Operator Product Expansions On The Vacuum In Conformal Invariant Quantum Field Theory, Commun. Math. Phys. 53 (1977) 155.MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    G.M. Sotkov and R.P. Zaikov, Conformal Invariant Two Point and Three Point Functions for Fields with Arbitrary Spin, Rept. Math. Phys. 12 (1977) 375.ADSCrossRefGoogle Scholar
  38. [38]
    H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  39. [39]
    J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    S. Giombi, S. Prakash and X. Yin, A Note on CFT Correlators in Three Dimensions, arXiv:1104.4317 [INSPIRE].
  41. [41]
    S. Ferrara, R. Gatto and A. Grillo, Positivity Restrictions on Anomalous Dimensions, Phys. Rev. D 9 (1974) 3564 [INSPIRE].ADSGoogle Scholar
  42. [42]
    G. Mack, All Unitary Ray Representations of the Conformal Group SU(2, 2) with Positive Energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  43. [43]
    R.R. Metsaev, Massless mixed symmetry bosonic free fields in d-dimensional anti-de Sitter space-time, Phys. Lett. B 354 (1995) 78 [INSPIRE].MathSciNetADSGoogle Scholar
  44. [44]
    S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074].Google Scholar
  45. [45]
    E. Schreier, Conformal symmetry and three-point functions, Phys. Rev. D 3 (1971) 980 [INSPIRE].ADSGoogle Scholar
  46. [46]
    D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    R. Metsaev and A.A. Tseytlin, Curvature Cubed Terms in String Theory Effective Actions, Phys. Lett. B 185 (1987) 52 [INSPIRE].MathSciNetADSGoogle Scholar
  48. [48]
    I. Heemskerk and J. Sully, More Holography from Conformal Field Theory, JHEP 09 (2010) 099 [arXiv:1006.0976] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

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

Authors and Affiliations

  • Miguel S. Costa
    • 1
  • João Penedones
    • 2
  • David Poland
    • 3
  • Slava Rychkov
    • 4
    • 5
  1. 1.Centro de Física do Porto and Departamento de Física e AstronomiaFaculdade de Ciências da Universidade do PortoPortoPortugal
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  3. 3.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA
  4. 4.Laboratoire de Physique Théorique, École Normale Supérieure, and Faculté de PhysiqueUniversité Pierre et Marie Curie (Paris VI)ParisFrance
  5. 5.KITPUniversity of CaliforniaSanta BarbaraUSA

Personalised recommendations