Exact correlation functions in conformal fishnet theory

  • Nikolay GromovEmail author
  • Vladimir Kazakov
  • Gregory Korchemsky
Open Access
Regular Article - Theoretical Physics


We compute exactly various 4−point correlation functions of shortest scalar operators in bi-scalar planar four-dimensional “fishnet” CFT. We apply the OPE to extract from these functions the exact expressions for the scaling dimensions and the structure constants of all exchanged operators with an arbitrary Lorentz spin. In particular, we determine the conformal data of the simplest unprotected two-magnon operator analogous to the Konishi operator, as well as of the one-magnon operator. We show that at weak coupling 4−point correlation functions can be systematically expanded in terms of harmonic polylogarithm functions and verify our results by explicit calculation of Feynman graphs at a few orders in the coupling. At strong coupling we obtain that the correlation functions exhibit the scaling behaviour typical for semiclassical description hinting at the existence of the holographic dual.


AdS-CFT Correspondence Conformal Field Theory Integrable Field Theories 


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


  1. [1]
    S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, Springer Briefs in Physics, Springer, Germany (2016) [arXiv:1601.05000].
  2. [2]
    M. Shaposhnikov and A. Shkerin, Conformal symmetry: towards the link between the Fermi and the Planck scales, Phys. Lett.B 783 (2018) 253 [arXiv:1803.08907] [INSPIRE].
  3. [3]
    V.M. Braun, G.P. Korchemsky and D. Müller, The uses of conformal symmetry in QCD, Prog. Part. Nucl. Phys.51 (2003) 311 [hep-ph/0306057] [INSPIRE].
  4. [4]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer, Germany (1997).Google Scholar
  5. [5]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  6. [6]
    R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, Nucl. Phys.B 447 (1995) 95 [hep-th/9503121] [INSPIRE].
  7. [7]
    O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP05 (2005) 033 [hep-th/0502086] [INSPIRE].
  8. [8]
    V. Kazakov, S. Leurent and D. Volin, T-system on T-hook: Grassmannian solution and twisted quantum spectral curve, JHEP12 (2016) 044 [arXiv:1510.02100] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    C. Cordova, T. T. Dumitrescu and K. Intriligator, Anomalies, renormalization group flows, and the a-theorem in six-dimensional (1, 0) theories, JHEP10 (2016) 080 [arXiv:1506.03807].
  10. [10]
    C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of superconformal symmetry in diverse dimensions, JHEP03 (2019) 163 [arXiv:1612.00809] [INSPIRE].
  11. [11]
    C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of superconformal theories, JHEP11 (2016) 135 [arXiv:1602.01217] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech.0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
  13. [13]
    N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N =4 supersymmetric Yang-Mills theory, Phys. Rev. Lett.103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].
  14. [14]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys.99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
  15. [15]
    A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum spectral curve of the \( \mathcal{N} \) = 6 supersymmetric Chern-Simons theory, Phys. Rev. Lett.113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].
  16. [16]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for planar \( \mathcal{N} \) = 4 super-Yang-Mills theory, Phys. Rev. Lett.112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].
  17. [17]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS 5/CFT 4, JHEP09 (2015) 187 [arXiv:1405.4857] [INSPIRE].Google Scholar
  18. [18]
    N. Gromov, Introduction to the spectrum of N = 4 SYM and the quantum spectral curve, arXiv:1708.03648 [INSPIRE].
  19. [19]
    V. Kazakov, Quantum spectral curve of γ-twisted \( \mathcal{N} \) = 4 SYM theory and fishnet CFT, arXiv:1802.02160 [INSPIRE].
  20. [20]
    J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability, JHEP09 (2011) 028 [arXiv:1012.2475] [INSPIRE].
  21. [21]
    B. Basso, S. Komatsu and P. Vieira, Structure constants and integrable bootstrap in planar N = 4 SYM theory,arXiv:1505.06745[INSPIRE].
  22. [22]
    A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve and structure constants in \( \mathcal{N} \) = 4 SYM: cusps in the ladder limit, JHEP10 (2018) 060 [arXiv:1802.04237] [INSPIRE].
  23. [23]
    S. Giombi and S. Komatsu, Exact correlators on the Wilson loop in \( \mathcal{N} \) = 4 SYM: localization, defect CFT and integrability, JHEP05 (2018) 109 [Erratum ibid.11 (2018) 123] [arXiv:1802.05201] [INSPIRE].
  24. [24]
    T. Fleury and S. Komatsu, Hexagonalization of correlation functions, JHEP01 (2017) 130 [arXiv:1611.05577] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N} \) = 4 SYM, JHEP10 (2017) 098 [arXiv:1611.05436] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP08 (2012) 134 [arXiv:1203.1913] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve for a cusped Wilson line in \( \mathcal{N} \) = 4 SYM, JHEP04 (2016) 134 [arXiv:1510.02098] [INSPIRE].
  28. [28]
    T. Bargheer et al., Handling handles: nonplanar integrability in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett.121 (2018) 231602 [arXiv:1711.05326] [INSPIRE].
  29. [29]
    T. Banks and A. Zaks, On the phase structure of vector-like gauge theories with massless fermions, Nucl. Phys.B 196 (1982) 189 [INSPIRE].
  30. [30]
    S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev.D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
  31. [31]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
  32. [32]
    S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP05 (2005) 069 [hep-th/0503201] [INSPIRE].
  33. [33]
    N. Beisert and R. Roiban, Beauty and the twist: the Bethe ansatz for twisted N = 4 SYM, JHEP08 (2005) 039 [hep-th/0505187] [INSPIRE].
  34. [34]
    C. Sieg and M. Wilhelm, On a CFT limit of planar γ i-deformed \( \mathcal{N} \) = 4 SYM theory, Phys. Lett.B 756 (2016) 118 [arXiv:1602.05817] [INSPIRE].
  35. [35]
    D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, Strongly γ-deformed \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory as an integrable conformal field theory, Phys. Rev. Lett.120 (2018) 111601 [arXiv:1711.04786] [INSPIRE].
  36. [36]
    A.A. Tseytlin and K. Zarembo, Effective potential in nonsupersymmetric SU(N) × SU(N) gauge theory and interactions of type 0 D3-branes, Phys. Lett.B 457 (1999) 77 [hep-th/9902095] [INSPIRE].
  37. [37]
    Q. Jin, The emergence of supersymmetry in γ i-deformed \( \mathcal{N} \) = 4 super-Yang-Mills theory, arXiv:1311.7391 [INSPIRE].
  38. [38]
    J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γ i-deformed N = 4 SYM theory, J. Phys.A 47 (2014) 455401 [arXiv:1308.4420] [INSPIRE].
  39. [39]
    J. Fokken, C. Sieg and M. Wilhelm, A piece of cake: the ground-state energies in γ i-deformed \( \mathcal{N} \) = 4 SYM theory at leading wrapping order, JHEP09 (2014) 078 [arXiv:1405.6712] [INSPIRE].
  40. [40]
    N. Gromov and F. Levkovich-Maslyuk, Y-system and β-deformed N = 4 super-Yang-Mills, J. Phys.A 44 (2011) 015402 [arXiv:1006.5438] [INSPIRE].
  41. [41]
    C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP12 (2011) 059 [arXiv:1108.4914] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    O. Gürdoğan and V. Kazakov, New integrable 4D quantum field theories from strongly deformed planar \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett.117 (2016) 201602 [arXiv:1512.06704] [INSPIRE].
  43. [43]
    J. Caetano, O. Gürdoğan and V. Kazakov, Chiral limit of \( \mathcal{N} \) = 4 SYM and ABJM and integrable Feynman graphs, JHEP03 (2018) 077 [arXiv:1612.05895] [INSPIRE].
  44. [44]
    N. Gromov et al., Integrability of conformal fishnet theory, JHEP01 (2018) 095 [arXiv:1706.04167] [INSPIRE].
  45. [45]
    D. Chicherin et al., Yangian symmetry for bi-scalar loop amplitudes, JHEP05 (2018) 003 [arXiv:1704.01967] [INSPIRE].
  46. [46]
    D. Chicherin et al., Yangian symmetry for fishnet Feynman graphs, Phys. Rev.D 96 (2017) 121901 [arXiv:1708.00007] [INSPIRE].
  47. [47]
    V. Kazakov and E. Olivucci, Biscalar integrable conformal field theories in any dimension, Phys. Rev. Lett.121 (2018) 131601 [arXiv:1801.09844] [INSPIRE].
  48. [48]
    A.B. Zamolodchikov, ‘Fishnetdiagrams as a completely integrable system, Phys. Lett.B 97 (1980) 63.Google Scholar
  49. [49]
    N. Gromov and F. Levkovich-Maslyuk, Quark-anti-quark potential in \( \mathcal{N} \) = 4 SYM, JHEP12 (2016) 122 [arXiv:1601.05679] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD: 1. Baxter Q operator and separation of variables, Nucl. Phys.B 617 (2001) 375 [hep-th/0107193] [INSPIRE].
  51. [51]
    H.J. De Vega and L.N. Lipatov, Interaction of reggeized gluons in the Baxter-Sklyanin representation, Phys. Rev.D 64 (2001) 114019 [hep-ph/0107225] [INSPIRE].
  52. [52]
    B. Basso and L.J. Dixon, Gluing ladder Feynman diagrams into fishnets, Phys. Rev. Lett.119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].
  53. [53]
    V.K. Dobrev et al., Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory, Lecture Notes in Physics volume 63 , Springer, Germany (1977).Google Scholar
  54. [54]
    F.A. Dolan and H. Osborn, Conformal partial waves: further mathematical results, Phys. Lett.718 (2011) 169 [arXiv:1108.6194] [INSPIRE].
  55. [55]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
  56. [56]
    G.P. Korchemsky, On level crossing in conformal field theories, JHEP03 (2016) 212 [arXiv:1512.05362] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    N. Gromov and P. Vieira, Tailoring three-point functions and integrability IV. Theta-morphism, JHEP04 (2014) 068 [arXiv:1205.5288] [INSPIRE].
  58. [58]
    J. Caetano, to be published.Google Scholar
  59. [59]
    F.C.S. Brown, Polylogarithmes multiples uniformes en une variable, Compt. Rend. Math.338 (2004) 527 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  60. [60]
    L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP10 (2012) 074 [arXiv:1207.0186] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  62. [62]
    D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun.174 (2006) 222 [hep-ph/0507152] [INSPIRE].
  63. [63]
    N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability III. Classical tunneling, JHEP07 (2012) 044 [arXiv:1111.2349] [INSPIRE].
  64. [64]
    Y. Kazama and S. Komatsu, Wave functions and correlation functions for GKP strings from integrability, JHEP09 (2012) 022 [arXiv:1205.6060] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    B. Basso and D.-l. Zhong, Continuum limit of fishnet graphs and AdS σ-model, JHEP01 (2019) 002 [arXiv:1806.04105] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev.D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
  67. [67]
    D. Medina-Rincon, A.A. Tseytlin and K. Zarembo, Precision matching of circular Wilson loops and strings in AdS 5 × S 5, JHEP05 (2018) 199 [arXiv:1804.08925] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    V.A. Kazakov, A. Marshakov, J.A. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP05 (2004) 024 [hep-th/0402207] [INSPIRE].
  69. [69]
    N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, The Algebraic curve of classical superstrings on AdS 5 × S 5, Commun. Math. Phys.263 (2006) 659 [hep-th/0502226] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    V.A. Kazakov and K. Zarembo, Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP10 (2004) 060 [hep-th/0410105] [INSPIRE].
  71. [71]
    D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP02 (2018) 081 [arXiv:1706.07813] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  72. [72]
    M. Kim, N. Kiryu, S. Komatsu and T. Nishimura, Structure constants of defect changing operators on the 1/2 BPS Wilson loop, JHEP12 (2017) 055 [arXiv:1710.07325] [INSPIRE].
  73. [73]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP11 (2011) 071 [arXiv:1107.3554] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  74. [74]
    S. Giombi, I.R. Klebanov and G. Tarnopolsky, Bosonic tensor models at large N and small ϵ, Phys. Rev.D 96 (2017) 106014 [arXiv:1707.03866] [INSPIRE].
  75. [75]
    D.J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP12 (2017) 148 [arXiv:1710.08113] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  76. [76]
    V. Rosenhaus, An introduction to the SYK model, arXiv:1807.03334 [INSPIRE].
  77. [77]
    D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, to appear.Google Scholar
  78. [78]
    L.N. Lipatov, Asymptotic behavior of multicolor QCD at high energies in connection with exactly solvable spin models, JETP Lett.59 (1994) 596 [hep-th/9311037] [INSPIRE].ADSGoogle Scholar
  79. [79]
    L.D. Faddeev and G.P. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett.B 342 (1995) 311 [hep-th/9404173] [INSPIRE].
  80. [80]
    I. Balitsky, V. Kazakov and E. Sobko, Two-point correlator of twist-2 light-ray operators in N = 4 SYM in BFKL approximation, arXiv:1310.3752[INSPIRE].
  81. [81]
    I. Balitsky, V. Kazakov and E. Sobko, Three-point correlator of twist-2 light-ray operators in N = 4 SYM in BFKL approximation,arXiv:1511.03625[INSPIRE].
  82. [82]
    I. Balitsky, V. Kazakov and E. Sobko, Structure constant of twist-2 light-ray operators in the Regge limit, Phys. Rev.D 93 (2016) 061701 [arXiv:1506.02038] [INSPIRE].
  83. [83]
    J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin, d-dimensional SYK, AdS loops and 6j symbols, JHEP03 (2019) 052 [arXiv:1808.00612] [INSPIRE].
  84. [84]
    D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP07 (2018) 085 [arXiv:1711.03816] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  85. [85]
    A.G. Grozin, Massless two-loop self-energy diagram: Historical review, Int. J. Mod. Phys.A 27 (2012) 1230018 [arXiv:1206.2572] [INSPIRE].
  86. [86]
    E. Pomoni and L. Rastelli, Large N field theory and AdS tachyons, JHEP04 (2009) 020 [arXiv:0805.2261] [INSPIRE].
  87. [87]
    A. Dymarsky, I.R. Klebanov and R. Roiban, Perturbative search for fixed lines in large N gauge theories, JHEP08 (2005) 011 [hep-th/0505099] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentKing’s College LondonLondonU.K.
  2. 2.St.Petersburg INPGatchinaRussia
  3. 3.Laboratoire de Physique Théorique, Département de Physique de l’ENSParisFrance
  4. 4.Université Paris-VI, PSL Research University, Sorbonne UniversitésParisFrance
  5. 5.Institut de Physique Théorique, Unité Mixte de Recherche 3681 du CNRS, Université Paris Saclay, CNRS, CEAGif-sur-YvetteFrance

Personalised recommendations