Basso-Dixon correlators in two-dimensional fishnet CFT

  • Sergei Derkachov
  • Vladimir Kazakov
  • Enrico OlivucciEmail author
Open Access
Regular Article - Theoretical Physics


We compute explicitly the two-dimensional version of Basso-Dixon type integrals for the planar 4-point correlation functions given by conformal “fishnet” Feynman graphs. These diagrams are represented by a fragment of a regular square lattice of power-like propagators, arising in the recently proposed integrable bi-scalar fishnet CFT. The formula is derived from first principles, using the formalism of separated variables in integrable SL(2, ℂ) spin chain. It is generalized to anisotropic fishnet, with different powers for propagators in two directions of the lattice.


Conformal Field Theory Integrable Field Theories Nonperturbative Effects Quantum Groups 


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]
    B. Basso and L.J. Dixon, Gluing ladder Feynman diagrams into fishnets, Phys. Rev. Lett. 119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    N.I. Usyukina and A.I. Davydychev, Exact results for three and four point ladder diagrams with an arbitrary number of rungs, Phys. Lett. B 305 (1993) 136 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    A.P. Isaev, Operator approach to analytical evaluation of Feynman diagrams, Phys. Atom. Nucl. 71 (2008) 914 [arXiv:0709.0419] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    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].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    J. Caetano, O. Gürdoğan and V. Kazakov, Chiral limit of \( \mathcal{N} \) = 4 SYM and ABJM and integrable Feynman graphs, JHEP 03 (2018) 077 [arXiv:1612.05895] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A.B. Zamolodchikov, “Fishnetdiagrams as a completely integrable system, Phys. Lett. B 97 (1980) 63.ADSCrossRefGoogle Scholar
  7. [7]
    N. Gromov et al., Integrability of conformal fishnet theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    D. Chicherin, S. Derkachov and A.P. Isaev, Conformal group: R-matrix and star-triangle relation, JHEP 04 (2013) 020 [arXiv:1206.4150] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    V. Kazakov and E. Olivucci, Biscalar integrable conformal field theories in any dimension, Phys. Rev. Lett. 121 (2018) 131601 [arXiv:1801.09844] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    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].ADSCrossRefzbMATHGoogle Scholar
  11. [11]
    A. Dymarsky, I.R. Klebanov and R. Roiban, Perturbative search for fixed lines in large N gauge theories, JHEP 08 (2005) 011 [hep-th/0505099] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    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].MathSciNetzbMATHGoogle Scholar
  13. [13]
    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, JHEP 09 (2014) 078 [arXiv:1405.6712] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    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].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A.V. Belitsky, S.E. Derkachov and A.N. Manashov, Quantum mechanics of null polygonal Wilson loops, Nucl. Phys. B 882 (2014) 303 [arXiv:1401.7307] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    S.E. Derkachov and A.N. Manashov, Iterative construction of eigenfunctions of the monodromy matrix for SL(2, ℂ) magnet, J. Phys. A 47 (2014) 305204 [arXiv:1401.7477] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  18. [18]
    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].
  19. [19]
    E.K. Sklyanin, The quantum Toda chain, Lect. Notes Phys. 226 (1985) 196 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    E.K. Sklyanin, Separation of variablesNew trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].
  21. [21]
    E.K. Sklyanin, Quantum inverse scattering method. Selected topics, hep-th/9211111 [INSPIRE].
  22. [22]
    V.K. Dobrev et al., Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory, Lect. Notes Phys. 12 (1977) 059.Google Scholar
  23. [23]
    L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in the proceedings of Relativistic gravitation and gravitational radiation, September 26–October 6, Les Houches, France (1995).Google Scholar
  24. [24]
    M. Preti, STR: a Mathematica package for the method of uniqueness, arXiv:1811.04935 [INSPIRE].
  25. [25]
    K.K. Kozlowski, Unitarity of the SoV transform for the Toda Chain, Commun. Math. Phys. 334 (2015) 223 [arXiv:1306.4967] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    S.E. Derkachov, K.K. Kozlowski and A.N. Manashov, On the separation of variables for the modular XXZ magnet and the lattice Sinh-Gordon models, arXiv:1806.04487 [INSPIRE].
  27. [27]
    G. Schrader and A. Shapiro, On b-Whittaker functions, arXiv:1806.00747.
  28. [28]
    E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978) 35 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    C. Itzykson and J.B. Zuber, The planar approximation. 2., J. Math. Phys. 21 (1980) 411 [INSPIRE].
  30. [30]
    V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in two-dimensional statistical models, Nucl. Phys. B 240 (1984) 312 [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    G.P. Korchemsky, Conformal bootstrap for the BFKL Pomeron, Nucl. Phys. B 550 (1999) 397 [hep-ph/9711277] [INSPIRE].
  32. [32]
    A. Levin and G. Racinet, Towards multiple elliptic polylogarithms, math/0703237.
  33. [33]
    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].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    N. Gromov, V. Kazakov and G. Korchemsky, Exact correlation functions in conformal fishnet theory, arXiv:1808.02688 [INSPIRE].
  35. [35]
    D. Chicherin et al., Yangian symmetry for bi-scalar loop amplitudes, JHEP 05 (2018) 003 [arXiv:1704.01967] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    D. Chicherin et al., Yangian symmetry for fishnet Feynman graphs, Phys. Rev. D 96 (2017) 121901 [arXiv:1708.00007] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    G. Passarino, Elliptic polylogarithms and basic hypergeometric functions, Eur. Phys. J. C 77 (2017) 77 [arXiv:1610.06207] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    J. Bartels, L.N. Lipatov and A. Prygarin, Integrable spin chains and scattering amplitudes, J. Phys. A 44 (2011) 454013 [arXiv:1104.0816] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  39. [39]
    L.N. Lipatov, Integrability of scattering amplitudes in N = 4 SUSY, J. Phys. A 42 (2009) 304020 [arXiv:0902.1444] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  40. [40]
    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
  41. [41]
    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].ADSCrossRefGoogle Scholar
  42. [42]
    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].
  43. [43]
    C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP 12 (2011) 059 [arXiv:1108.4914] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  44. [44]
    D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, to appear.Google Scholar
  45. [45]
    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].ADSCrossRefGoogle Scholar
  46. [46]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS 5 /CFT 4, JHEP 09 (2015) 187 [arXiv:1405.4857] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  47. [47]
    N. Gromov, Introduction to the spectrum of N = 4 SYM and the quantum spectral curve, arXiv:1708.03648 [INSPIRE].
  48. [48]
    V. Kazakov, Quantum spectral curve of γ-twisted \( \mathcal{N} \) = 4 SYM theory and fishnet CFT, arXiv:1802.02160 [INSPIRE].
  49. [49]
    N. Gromov, F. Levkovich-Maslyuk and G. Sizov, New construction of eigenstates and separation of variables for SU(N) quantum spin chains, JHEP 09 (2017) 111 [arXiv:1610.08032] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    S.E. Derkachov and P.A. Valinevich, Separation of variables for the quantum SL(3, ℂ) spin magnet: eigenfunctions of Sklyanin B-operator, arXiv:1807.00302 [INSPIRE].
  51. [51]
    P. Ryan and D. Volin, Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame, arXiv:1810.10996 [INSPIRE].
  52. [52]
    J.M. Maillet and G. Niccoli, Complete spectrum of quantum integrable lattice models associated to Y(gln) by separation of variables, arXiv:1810.11885 [INSPIRE].
  53. [53]
    J.M. Maillet and G. Niccoli, Complete spectrum of quantum integrable lattice models associated to \( {\mathcal{U}}_q\left(\widehat{\mathrm{g}{\operatorname{l}}_n}\right) \) by separation of variables, arXiv:1811.08405 [INSPIRE].
  54. [54]
    J.M. Maillet and G. Niccoli, On quantum separation of variables, J. Math. Phys. 59 (2018) 091417 [arXiv:1807.11572] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    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].
  56. [56]
    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].ADSMathSciNetGoogle Scholar
  57. [57]
    B. Basso and D.-l. Zhong, Continuum limit of fishnet graphs and AdS σ-model, JHEP 01 (2019) 002 [arXiv:1806.04105] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical Institute of Russian Academy of SciencesSt. PetersburgRussia
  2. 2.Laboratoire de Physique Théorique, Département de Physique de l’ENS, École Normale SupérieureParisFrance
  3. 3.Université Paris-VI, PSL Research University, Sorbonne Universités, UPMC Univ. Paris 06, CNRSParisFrance
  4. 4.II. Institut für Theoretische PhysikUniversität HamburgHamburgGermany

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