Advertisement

Journal of High Energy Physics

, 2019:183 | Cite as

Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension

  • Hiroshi Isono
  • Toshifumi NoumiEmail author
  • Gary Shiu
Open Access
Regular Article - Theoretical Physics

Abstract

Our previous work [1] constructed, in three-dimensional momentum space, a manifestly crossing symmetric basis for scalar conformal four-point functions, based on the factorization property proposed by Polyakov. This work extends this construction to general dimensional conformal field theory. To facilitate the treatment of symmetric traceless tensors, we exploit techniques of spherical harmonics in general dimensions.

Keywords

AdS-CFT Correspondence Conformal Field Theory Conformal and W Symmetry 

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]
    H. Isono, T. Noumi and G. Shiu, Momentum space approach to crossing symmetric CFT correlators, JHEP07 (2018) 136 [arXiv:1805.11107] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz.66 (1974) 23 [INSPIRE].Google Scholar
  3. [3]
    R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal Bootstrap in Mellin Space, Phys. Rev. Lett.118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP05 (2017) 027 [arXiv:1611.08407] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].
  6. [6]
    Bateman Manuscript Project, H. Bateman and A. Erdélyi, Spherical and hyperspherical harmonic polynomials, in Higher Transcendental Functions Volume II, McGraw-Hill Book Company, Inc., chapter XI (1953).Google Scholar
  7. [7]
    N.J. Vilenkin, Special Functions and the Theory of Group Representations, American Mathematical Society (1968).Google Scholar
  8. [8]
    K. Atkinson and W. Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Springer (2010).Google Scholar
  9. [9]
    S. Ferrara, A.F. Grillo, R. Gatto and G. Parisi, Analyticity properties and asymptotic expansions of conformal covariant green’s functions, Nuovo Cim.A 19 (1974) 667 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    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 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    G.M. Sotkov and R.P. Zaikov, On the Structure of the Conformal Covariant N Point Functions, Rept. Math. Phys.19 (1984) 335 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    I. Antoniadis, P.O. Mazur and E. Mottola, Conformal Invariance, Dark Energy and CMB Non-Gaussianity, JCAP09 (2012) 024 [arXiv:1103.4164] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP09 (2011) 045 [arXiv:1104.2846] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    C. Corianò, L. Delle Rose and M. Serino, Three and Four Point Functions of Stress Energy Tensors in D = 3 for the Analysis of Cosmological Non-Gaussianities, JHEP12 (2012) 090 [arXiv:1210.0136] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    D. Chowdhury, S. Raju, S. Sachdev, A. Singh and P. Strack, Multipoint correlators of conformal field theories: implications for quantum critical transport, Phys. Rev.B 87 (2013) 085138 [arXiv:1210.5247] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    A. Bzowski, P. McFadden and K. Skenderis, Holography for inflation using conformal perturbation theory, JHEP04 (2013) 047 [arXiv:1211.4550] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    I. Mata, S. Raju and S. Trivedi, CMB from CFT, JHEP07 (2013) 015 [arXiv:1211.5482] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    C. Corianò, L. Delle Rose, E. Mottola and M. Serino, Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman’s Master Integrals, JHEP07 (2013) 011 [arXiv:1304.6944] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, JHEP03 (2014) 111 [arXiv:1304.7760] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    Y. Huh, P. Strack and S. Sachdev, Conserved current correlators of conformal field theories in 2+1 dimensions, Phys. Rev.B 88 (2013) 155109 [Erratum ibid.B 90 (2014) 199902] [arXiv:1307.6863] [INSPIRE].
  21. [21]
    A. Ghosh, N. Kundu, S. Raju and S.P. Trivedi, Conformal Invariance and the Four Point Scalar Correlator in Slow-Roll Inflation, JHEP07 (2014) 011 [arXiv:1401.1426] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    N. Kundu, A. Shukla and S.P. Trivedi, Constraints from Conformal Symmetry on the Three Point Scalar Correlator in Inflation, JHEP04 (2015) 061 [arXiv:1410.2606] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    N. Kundu, A. Shukla and S.P. Trivedi, Ward Identities for Scale and Special Conformal Transformations in Inflation, JHEP01 (2016) 046 [arXiv:1507.06017] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    K. Sen and A. Sinha, On critical exponents without Feynman diagrams, J. Phys.A 49 (2016) 445401 [arXiv:1510.07770] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  25. [25]
    A. Bzowski, P. McFadden and K. Skenderis, Scalar 3-point functions in CFT: renormalisation, β-functions and anomalies, JHEP03 (2016) 066 [arXiv:1510.08442] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    A. Bzowski, P. McFadden and K. Skenderis, Evaluation of conformal integrals, JHEP02 (2016) 068 [arXiv:1511.02357] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    V.P.J. Jacobs, P. Betzios, U. Gürsoy and H.T.C. Stoof, Electromagnetic response of interacting Weyl semimetals, Phys. Rev.B 93 (2016) 195104 [arXiv:1512.04883] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    R.C. Myers, T. Sierens and W. Witczak-Krempa, A Holographic Model for Quantum Critical Responses, JHEP05 (2016) 073 [arXiv:1602.05599] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    A. Lucas, S. Gazit, D. Podolsky and W. Witczak-Krempa, Dynamical response near quantum critical points, Phys. Rev. Lett.118 (2017) 056601 [arXiv:1608.02586] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    A. Lucas, T. Sierens and W. Witczak-Krempa, Quantum critical response: from conformal perturbation theory to holography, JHEP07 (2017) 149 [arXiv:1704.05461] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    A. Bzowski, P. McFadden and K. Skenderis, Renormalised 3-point functions of stress tensors and conserved currents in CFT, JHEP11 (2018) 153 [arXiv:1711.09105] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    C. Corianò and M.M. Maglio, Exact Correlators from Conformal Ward Identities in Momentum Space and the Perturbative T J J Vertex, Nucl. Phys.B 938 (2019) 440 [arXiv:1802.07675] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    A. Bzowski, P. McFadden and K. Skenderis, Renormalised CFT 3-point functions of scalars, currents and stress tensors, JHEP11 (2018) 159 [arXiv:1805.12100] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    M. Gillioz, Momentum-space conformal blocks on the light cone, JHEP10 (2018) 125 [arXiv:1807.07003] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    C. Corianò and M.M. Maglio, The general 3-graviton vertex (TTT) of conformal field theories in momentum space in d = 4, Nucl. Phys.B 937 (2018) 56 [arXiv:1808.10221] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes, JHEP02 (2019) 040 [arXiv:1810.12459] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  37. [37]
    N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities, arXiv:1811.00024 [INSPIRE].
  38. [38]
    E. Skvortsov, Light-Front Bootstrap for Chern-Simons Matter Theories, JHEP06 (2019) 058 [arXiv:1811.12333] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    J.A. Farrow, A.E. Lipstein and P. McFadden, Double copy structure of CFT correlators, JHEP02 (2019) 130 [arXiv:1812.11129] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    H. Isono, T. Noumi and T. Takeuchi, Momentum space conformal three-point functions of conserved currents and a general spinning operator, JHEP05 (2019) 057 [arXiv:1903.01110] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    C. Corianò and M.M. Maglio, On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space, JHEP09 (2019) 107 [arXiv:1903.05047] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    C. Sleight, A Mellin Space Approach to Cosmological Correlators, arXiv:1906.12302 [INSPIRE].
  43. [43]
    C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space, arXiv:1907.01143 [INSPIRE].
  44. [44]
    S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes II: Gravitons, arXiv:1908.01835 [INSPIRE].
  45. [45]
    V. Bargmann and I.T. Todorov, Spaces of Analytic Functions on a Complex Cone as Carries for the Symmetric Tensor Representations of SO(N ), J. Math. Phys.18 (1977) 1141 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP11 (2011) 071 [arXiv:1107.3554] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Seventh Edition, Elsevier, Academic Press (2007).Google Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Physics, Faculty of ScienceChulalongkorn UniversityBangkokThailand
  2. 2.Department of PhysicsKobe UniversityKobeJapan
  3. 3.Department of PhysicsUniversity of Wisconsin-MadisonMadisonU.S.A.

Personalised recommendations