Tailoring three-point functions and integrability III. Classical tunneling

  • Nikolay Gromov
  • Amit Sever
  • Pedro VieiraEmail author


We compute three-point functions between one large classical operator and two large BPS operators at weak coupling. We consider operators made out of the scalars of \( \mathcal{N} = 4 \) SYM, dual to strings moving in the sphere. The three-point function exponentiates and can be thought of as a classical tunneling process in which the classical string-like operator decays into two classical BPS states. From an Integrability/Condensed Matter point of view, we simplified inner products of spin chain Bethe states in a classical limit corresponding to long wavelength excitations above the ferromagnetic vacuum. As a by-product we solved a new long-range Ising model in the thermodynamic limit.


Lattice Integrable Models AdS-CFT Correspondence Bethe Ansatz Integrable Field Theories 


  1. [1]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    K. Zarembo, Holographic three-point functions of semiclassical states, JHEP 09 (2010) 030 [arXiv:1008.1059] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    M.S. Costa, R. Monteiro, J.E. Santos and D. Zoakos, On three-point correlation functions in the gauge/gravity duality, JHEP 11 (2010) 141 [arXiv:1008.1070] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability II Weak/strong coupling match, JHEP 09 (2011) 029 [arXiv:1104.5501] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    R. Roiban and A. Tseytlin, On semiclassical computation of 3-point functions of closed string vertex operators in AdS 5 × S 5, Phys. Rev. D 82 (2010) 106011 [arXiv:1008.4921] [INSPIRE].ADSGoogle Scholar
  6. [6]
    E. Buchbinder and A. Tseytlin, Semiclassical four-point functions in AdS 5 × S 5, JHEP 02 (2011) 072 [arXiv:1012.3740] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    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].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    R.A. Janik and A. Wereszczynski, Correlation functions of three heavy operators: The AdS contribution, JHEP 12 (2011) 095 [arXiv:1109.6262] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    Y. Kazama and S. Komatsu, On holographic three point functions for GKP strings from integrability, JHEP 01 (2012) 110 [arXiv:1110.3949] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    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].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large-N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  13. [13]
    M. Gaudin, Diagonalisation d’une classe d’hamiltoniens de spin, J. Phys. 37 (1976) 1087.MathSciNetCrossRefGoogle Scholar
  14. [14]
    B.M. McCoy, T.T. Wu and M. Gaudin, Normalization sum for the Bethe’s hypothesis wave functions of the Heisenberg-Ising chain, Phys. Rev. D 23 (1981) 417.MathSciNetADSGoogle Scholar
  15. [15]
    V. E. Korepin, V. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. [16]
    N. Beisert, J. Minahan, M. Staudacher and K. Zarembo, Stringing spins and spinning strings, JHEP 09 (2003) 010 [hep-th/0306139] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    V. Kazakov, A. Marshakov, J. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 05 (2004) 024 [hep-th/0402207] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    B. Sutherland, Low-lying eigenstates of the one-dimensional Heisenberg ferromagnet for any magnetization and momentum, Phys. Rev. Lett. 74 (1995) 816 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. Dhar and B. Sriram Shastry, Bloch Walls and Macroscopic String States in Bethe’s Solution of the Heisenberg Ferromagnetic Linear Chain, Phys. Rev. Lett. 85 (2000) 2813 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    N. Gromov and P. Vieira, Complete 1-loop test of AdS/CFT, JHEP 04 (2008) 046 [arXiv:0709.3487] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    M. Staudacher, Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansátze and the R-Matrix Formalism, Lett. Math. Phys. 99 (2012) 191 [arXiv:1012.3990] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. [22]
    S. Frolov and A.A. Tseytlin, Rotating string solutions: AdS/CFT duality in nonsupersymmetric sectors, Phys. Lett. B 570 (2003) 96 [hep-th/0306143] [INSPIRE].MathSciNetADSGoogle Scholar
  23. [23]
    G. Georgiou, SL(2) sector: weak/strong coupling agreement of three-point correlators, JHEP 09 (2011) 132 [arXiv:1107.1850] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    M. Kruczenski, Spin chains and string theory, Phys. Rev. Lett. 93 (2004) 161602 [hep-th/0311203] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    N. Beisert, V. 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].MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. [26]
    N. Beisert, V. Kazakov, K. Sakai and K. Zarembo, Complete spectrum of long operators in N = 4 SYM at one loop, JHEP 07 (2005) 030 [hep-th/0503200] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    O. Foda, \( \mathcal{N} = 4 \) SYM structure constants as determinants, to appear.Google Scholar
  28. [28]
    R. Roiban and A. Volovich, Yang-Mills correlation functions from integrable spin chains, JHEP 09 (2004) 032 [hep-th/0407140] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    K. Okuyama and L.-S. Tseng, Three-point functions in N = 4 SYM theory at one-loop, JHEP 08 (2004) 055 [hep-th/0404190] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    N. Beisert and A.A. Tseytlin, On quantum corrections to spinning strings and Bethe equations, Phys. Lett. B 629 (2005) 102 [hep-th/0509084] [INSPIRE].MathSciNetADSGoogle Scholar
  31. [31]
    S. Schäfer-Nameki, M. Zamaklar and K. Zarembo, Quantum corrections to spinning strings in AdS 5 × S 5 and Bethe ansatz: A Comparative study, JHEP 09 (2005) 051 [hep-th/0507189] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    N. Beisert, A.A. Tseytlin and K. Zarembo, Matching quantum strings to quantum spins: One-loop versus finite-size corrections, Nucl. Phys. B 715 (2005) 190 [hep-th/0502173] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    R. Hernandez, E. Lopez, A. Perianez and G. Sierra, Finite size effects in ferromagnetic spin chains and quantum corrections to classical strings, JHEP 06 (2005) 011 [hep-th/0502188] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    N. Beisert and L. Freyhult, Fluctuations and energy shifts in the Bethe ansatz, Phys. Lett. B 622 (2005) 343 [hep-th/0506243] [INSPIRE].MathSciNetADSGoogle Scholar
  35. [35]
    N. Gromov and V. Kazakov, Double scaling and finite size corrections in sl(2) spin chain, Nucl. Phys. B 736 (2006) 199 [hep-th/0510194] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College LondonLondonUK
  2. 2.St.Petersburg INPSt. PetersburgRussia
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  4. 4.School of Natural Sciences, Institute for Advanced StudyPrincetonUSA

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