Advertisement

χ-systems for correlation functions

  • J. CaetanoEmail author
  • J. Toledo
Open Access
Regular Article - Theoretical Physics
  • 20 Downloads

Abstract

We consider the strong coupling limit of 4-point functions of heavy operators in \( \mathcal{N} \) = 4 SYM dual to strings with no spin in AdS. We restrict our discussion for operators inserted on a line. The string computation factorizes into a state-dependent sphere part and a universal AdS contribution which depends only on the dimensions of the operators and the cross ratios. We use the integrability of the AdS string equations to compute the AdS part for operators of arbitrary conformal dimensions. The solution takes the form of TBA-like integral equations with the minimal AdS string-action computed by a corresponding free-energy-like functional. These TBA-like equations stem from a peculiar system of functional equations which we call a χ-system. In principle one could use the same method to solve for the AdS contribution in the N-point function. The discrete data that parameterizes these solutions enters through the analog of the chemical-potentials in the TBA-like equations. Finally, for operators dual to strings spinning in the same equator in S5 (i.e. BPS operators of the same type) the sphere part is simple to compute. In this case (which is generically neither extremal nor protected) our result allows the construction of the complete, strong coupling 4-point function.

Keywords

AdS-CFT Correspondence Conformal Field Theory Integrable Field Theories 

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]
    J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
  2. [2]
    B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
  3. [3]
    T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions II: Two-Particle Contributions, JHEP 02 (2018) 177 [arXiv:1711.05327] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N} \) = 4 SYM, JHEP 10 (2017) 098 [arXiv:1611.05436] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling Handles: Nonplanar Integrability in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 121 (2018) 231602 [arXiv:1711.05326] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling handles. Part II. Stratification and data analysis, JHEP 11 (2018) 095 [arXiv:1809.09145] [INSPIRE].
  8. [8]
    B. Eden, Y. Jiang, D. le Plat and A. Sfondrini, Colour-dressed hexagon tessellations for correlation functions and non-planar corrections, JHEP 02 (2018) 170 [arXiv:1710.10212] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    B. Eden and A. Sfondrini, Three-point functions in \( \mathcal{N} \) = 4 SYM: the hexagon proposal at three loops, JHEP 02 (2016) 165 [arXiv:1510.01242] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    B. Basso, V. Gonçalves, S. Komatsu and P. Vieira, Gluing Hexagons at Three Loops, Nucl. Phys. B 907 (2016) 695 [arXiv:1510.01683] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    J. Caetano and T. Fleury, Fermionic Correlators from Integrability, JHEP 09 (2016) 010 [arXiv:1607.02542] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    B. Basso, V. Gonçalves and S. Komatsu, Structure constants at wrapping order, JHEP 05 (2017) 124 [arXiv:1702.02154] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D. Chicherin, J. Drummond, P. Heslop and E. Sokatchev, All three-loop four-point correlators of half-BPS operators in planar \( \mathcal{N} \) = 4 SYM, JHEP 08 (2016) 053 [arXiv:1512.02926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    D. Chicherin, A. Georgoudis, V. Gonçalves and R. Pereira, All five-loop planar four-point functions of half-BPS operators in \( \mathcal{N} \) = 4 SYM, JHEP 11 (2018) 069 [arXiv:1809.00551] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    Y. Jiang, S. Komatsu, I. Kostov and D. Serban, Clustering and the Three-Point Function, J. Phys. A 49 (2016) 454003 [arXiv:1604.03575] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    I. Kostov, D. Serban and D.-L. Vu, TBA and tree expansion, Springer Proc. Math. Stat. 255 (2017) 77 [arXiv:1805.02591] [INSPIRE].CrossRefGoogle Scholar
  17. [17]
    I. Kostov, D. Serban and D.-L. Vu, Boundary TBA, trees and loops, arXiv:1809.05705 [INSPIRE].
  18. [18]
    L. Rastelli and X. Zhou, Mellin amplitudes for AdS 5 × S 5, Phys. Rev. Lett. 118 (2017) 091602 [arXiv:1608.06624] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    L. Rastelli and X. Zhou, How to Succeed at Holographic Correlators Without Really Trying, JHEP 04 (2018) 014 [arXiv:1710.05923] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S. Caron-Huot and A.-K. Trinh, All Tree-Level Correlators in AdS 5×S 5 Supergravity: Hidden Ten-Dimensional Conformal Symmetry, arXiv:1809.09173 [INSPIRE].
  21. [21]
    G. Arutyunov, S. Frolov, R. Klabbers and S. Savin, Towards 4-point correlation functions of any \( \frac{1}{2} \) -BPS operators from supergravity, JHEP 04 (2017) 005 [arXiv:1701.00998] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    G. Arutyunov, R. Klabbers and S. Savin, Four-point functions of all-different-weight chiral primary operators in the supergravity approximation, JHEP 09 (2018) 023 [arXiv:1806.09200] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    G. Arutyunov, R. Klabbers and S. Savin, Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation, JHEP 09 (2018) 118 [arXiv:1808.06788] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  24. [24]
    R.A. Janik, P. Surowka and A. Wereszczynski, On correlation functions of operators dual to classical spinning string states, JHEP 05 (2010) 030 [arXiv:1002.4613] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    R.A. Janik and A. Wereszczynski, Correlation functions of three heavy operators: The AdS contribution, JHEP 12 (2011) 095 [arXiv:1109.6262] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  26. [26]
    Y. Kazama and S. Komatsu, On holographic three point functions for GKP strings from integrability, JHEP 01 (2012) 110 [Erratum JHEP 06 (2012) 150] [arXiv:1110.3949] [INSPIRE].
  27. [27]
    L.F. Alday, J.M. Maldacena, A. Sever and P. Vieira, Y-system for Scattering Amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  28. [28]
    Y. Kazama and S. Komatsu, Wave functions and correlation functions for GKP strings from integrability, JHEP 09 (2012) 022 [arXiv:1205.6060] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Y. Kazama and S. Komatsu, Three-point functions in the SU(2) sector at strong coupling, JHEP 03 (2014) 052 [arXiv:1312.3727] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  31. [31]
    A.M. Polyakov, Gauge fields and space-time, Int. J. Mod. Phys. A 17S1 (2002) 119 [hep-th/0110196] [INSPIRE].
  32. [32]
    A.A. Tseytlin, On semiclassical approximation and spinning string vertex operators in AdS 5 × S 5, Nucl. Phys. B 664 (2003) 247 [hep-th/0304139] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A Semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  34. [34]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Extremal correlators in the AdS/CFT correspondence, in The many faces of the superworld, M.A. Shifman ed., World Scientific (2000), pp. 332–360 [hep-th/9908160] [INSPIRE].
  35. [35]
    B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Extremal correlators in four-dimensional SCFT, Phys. Lett. B 472 (2000) 323 [hep-th/9910150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  37. [37]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    L.F. Alday, D. Gaiotto, J.M. Maldacena, A. Sever and P. Vieira, An Operator Product Expansion for Polygonal null Wilson Loops, JHEP 04 (2011) 088 [arXiv:1006.2788] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    D. Gaiotto, J.M. Maldacena, A. Sever and P. Vieira, Bootstrapping Null Polygon Wilson Loops, JHEP 03 (2011) 092 [arXiv:1010.5009] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    D. Gaiotto, J.M. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    B. Basso, A. Sever and P. Vieira, Spacetime and Flux Tube S-Matrices at Finite Coupling for N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux tube S-matrix II. Extracting and Matching Data, JHEP 01 (2014) 008 [arXiv:1306.2058] [INSPIRE].
  45. [45]
    B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions, JHEP 08 (2014) 085 [arXiv:1402.3307] [INSPIRE].
  46. [46]
    B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux-tube S-matrix IV. Gluons and Fusion, JHEP 09 (2014) 149 [arXiv:1407.1736] [INSPIRE].
  47. [47]
    B. Basso, A. Sever and P. Vieira, Hexagonal Wilson loops in planar \( \mathcal{N} \) = 4 SYM theory at finite coupling, J. Phys. A 49 (2016) 41LT01 [arXiv:1508.03045] [INSPIRE].
  48. [48]
    B. Basso, J. Caetano, L. Cordova, A. Sever and P. Vieira, OPE for all Helicity Amplitudes, JHEP 08 (2015) 018 [arXiv:1412.1132] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    B. Basso, J. Caetano, L. Cordova, A. Sever and P. Vieira, OPE for all Helicity Amplitudes II. Form Factors and Data Analysis, JHEP 12 (2015) 088 [arXiv:1508.02987] [INSPIRE].
  50. [50]
    L.F. Alday, B. Eden, G.P. Korchemsky, J.M. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP 09 (2011) 123 [arXiv:1007.3243] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    B. Eden, G.P. Korchemsky and E. Sokatchev, From correlation functions to scattering amplitudes, JHEP 12 (2011) 002 [arXiv:1007.3246] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    B. Eden, G.P. Korchemsky and E. Sokatchev, More on the duality correlators/amplitudes, Phys. Lett. B 709 (2012) 247 [arXiv:1009.2488] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    A.V. Belitsky, G.P. Korchemsky and E. Sokatchev, Are scattering amplitudes dual to super Wilson loops?, Nucl. Phys. B 855 (2012) 333 [arXiv:1103.3008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: Part I, Nucl. Phys. B 869 (2013) 329 [arXiv:1103.3714] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: Part II, Nucl. Phys. B 869 (2013) 378 [arXiv:1103.4353] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    J. Caetano and J. Escobedo, On four-point functions and integrability in N = 4 SYM: from weak to strong coupling, JHEP 09 (2011) 080 [arXiv:1107.5580] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    E.I. Buchbinder and A.A. Tseytlin, Semiclassical correlators of three states with large S 5 charges in string theory in AdS 5 × S 5, Phys. Rev. D 85 (2012) 026001 [arXiv:1110.5621] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of Physics and Astronomy & Guelph-Waterloo Physics InstituteUniversity of WaterlooWaterlooCanada
  3. 3.Centro de Física do Porto e Departamento de Física e Astronomia, Faculdade de Ciências da Universidade do PortoPortoPortugal

Personalised recommendations