The complex sinh-Gordon model: form factors of descendant operators and current-current perturbations

  • Michael LashkevichEmail author
  • Yaroslav Pugai
Open Access
Regular Article - Theoretical Physics


We study quasilocal operators in the quantum complex sinh-Gordon theory in the form factor approach. The free field procedure for descendant operators is developed by introducing the algebra of screening currents and related algebraic objects. We work out null vector equations in the space of operators. Further we apply the proposed algebraic structures to constructing form factors of the conserved currents Tk and Θk. We propose also form factors of current-current operators of the form TkTl. Explicit computations of the four-particle form factors allow us to verify the recent conjecture of Smirnov and Zamolodchikov about the structure of the exact scattering matrix of an integrable theory perturbed by a combination of irrelevant operators. Our calculations confirm that such perturbations of the complex sinh-Gordon model and of the N symmetric Ising models result in extra CDD factors in the S matrix.


Field Theories in Lower Dimensions 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]
    K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    F. Lund and T. Regge, Unified Approach to Strings and Vortices with Soliton Solutions, Phys. Rev. D 14 (1976) 1524 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  3. [3]
    H.J. de Vega and J.M. Maillet, Renormalization Character and Quantum S Matrix for a Classically Integrable Theory, Phys. Lett. B 101 (1981) 302 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    H.J. de Vega and J.M. Maillet, Semiclassical Quantization of the Complex sine-Gordon Field Theory, Phys. Rev. D 28 (1983) 1441 [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    N. Dorey and T.J. Hollowood, Quantum scattering of charged solitons in the complex sine-Gordon model, Nucl. Phys. B 440 (1995) 215 [hep-th/9410140] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    V.A. Fateev, The Duality between two-dimensional integrable field theories and σ-models, Phys. Lett. B 357 (1995) 397 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    V.A. Fateev, Normalization factors, reflection amplitudes and integrable systems, hep-th/0103014 [INSPIRE].
  8. [8]
    V.A. Fateev, Integrable Deformations of Sine-Liouville Conformal Field Theory and Duality, SIGMA 13 (2017) 080 [arXiv:1705.06424] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  9. [9]
    E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  10. [10]
    M. Karowski and P. Weisz, Exact Form Factors in (1 + 1)-Dimensional Field Theoretic Models with Soliton Behavior, Nucl. Phys. B 139 (1978) 455 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    F.A. Smirnov, Form factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14 (1992) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    B. Feigin and M. Lashkevich, Form factors of descendant operators: Free field construction and reflection relations, J. Phys. A 42 (2009) 304014 [arXiv:0812.4776] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  13. [13]
    M. Lashkevich and Y. Pugai, On form factors and Macdonald polynomials, JHEP 09 (2013) 095 [arXiv:1305.1674] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    M. Lashkevich and Y. Pugai, Form factors in sinh- and sine-Gordon models, deformed Virasoro algebra, Macdonald polynomials and resonance identities, Nucl. Phys. B 877 (2013) 538 [arXiv:1307.0243] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Lashkevich and Y. Pugai, Form factors of descendant operators: Resonance identities in the sinh-Gordon model, JHEP 12 (2014) 112 [arXiv:1411.1374] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Lashkevich and Y. Pugai, Form factors of descendant operators: Reduction to perturbed M(2, 2s + 1) models, JHEP 04 (2015) 126 [arXiv:1412.7509] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Lashkevich and Y. Pugai, Algebraic approach to form factors in the complex sinh-Gordon theory, Phys. Lett. B 764 (2017) 190 [arXiv:1610.04926] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  18. [18]
    R. Koberle and J.A. Swieca, Factorizable Z(N) models, Phys. Lett. B 86 (1979) 209 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    V.A. Fateev, The exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories, Phys. Lett. B 324 (1994) 45 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    T. Fukuda and K. Hosomichi, Three-point functions in sine-Liouville theory, JHEP 09 (2001) 003 [hep-th/0105217] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    J.L. Cardy and G. Mussardo, Form factors of descendent operators in perturbed conformal field theories, Nucl. Phys. B 340 (1990) 387 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    F.A. Smirnov, Quantum groups and generalized statistics in integrable models, Commun. Math. Phys. 132 (1990) 415 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Jimbo, H. Konno, S. Odake, Y. Pugai and J. Shiraishi, Free field construction for the ABF models in regime II, J. Statist. Phys. 102 (2001) 883 [math/0001071] [INSPIRE].
  24. [24]
    V.A. Fateev, V.V. Postnikov and Y.P. Pugai, On scaling fields in Z N Ising models, JETP Lett. 83 (2006) 172 [hep-th/0601073] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    V.A. Fateev and Y.P. Pugai, Correlation functions of disorder fields and parafermionic currents in Z N Ising models, J. Phys. A: Math. Theor. 42 (2009) 304013 [arXiv:0909.3347] [INSPIRE].CrossRefzbMATHGoogle Scholar
  26. [26]
    H. Babujian and M. Karowski, Exact form-factors for the scaling Z N -Ising and the affine A N − 1 -Toda quantum field theories, Phys. Lett. B 575 (2003) 144 [hep-th/0309018] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    A. Koubek and G. Mussardo, On the operator content of the sinh-Gordon model, Phys. Lett. B 311 (1993) 193 [hep-th/9306044] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    M. Jimbo, T. Miwa and Y. Takeyama, Counting minimal form factors of the restricted sine-Gordon model, math-ph/0303059 [INSPIRE].
  29. [29]
    M. Jimbo, T. Miwa, E. Mukhin and Y. Takeyama, Form factors and action of \( {U}_{{\left(-1\right)}^{1/2}}\left(\tilde{sl}(2)\right) \) on-cycles, Commun. Math. Phys. 245 (2004) 551 [math/0305323] [INSPIRE].
  30. [30]
    G. Delfino and G. Niccoli, Isomorphism of critical and off-critical operator spaces in two-dimensional quantum field theory, Nucl. Phys. B 799 (2008) 364 [arXiv:0712.2165] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19 (1989) 641 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
  33. [33]
    J. Cardy, \( T\overline{T} \) deformations of non-Lorentz invariant field theories, arXiv:1809.07849 [INSPIRE].
  34. [34]
    G. Delfino and G. Niccoli, Matrix elements of the operator \( T\overline{T} \) in integrable quantum field theory, Nucl. Phys. B 707 (2005) 381 [hep-th/0407142] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    G. Delfino and G. Niccoli, Form factors of descendant operators in the massive Lee-Yang model, J. Stat. Mech. 0504 (2005) P04004 [hep-th/0501173] [INSPIRE].MathSciNetGoogle Scholar
  36. [36]
    G. Delfino and G. Niccoli, The Composite operator \( T\overline{T} \) in sinh-Gordon and a series of massive minimal models, JHEP 05 (2006) 035 [hep-th/0602223] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    Al. B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].
  38. [38]
    G. Mussardo and P. Simon, Bosonic-type S-matrix, vacuum instability and CDD ambiguities, Nucl. Phys. B 578 (2000) 527 [hep-th/9903072] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    Al. B. Zamolodchikov, Resonance factorized scattering and roaming trajectories, J. Phys. A 39 (2006) 12847 [INSPIRE].
  40. [40]
    F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    M. Jimbo, T. Miwa and F. Smirnov, Fermionic structure in the sine-Gordon model: Form factors and null-vectors, Nucl. Phys. B 852 (2011) 390 [arXiv:1105.6209] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    M. Lashkevich and Y. Pugai, Note on four-particle form factors of operators T 2n T −2n in sinh-Gordon model, J. Phys. A 49 (2016) 305401 [arXiv:1602.05735] [INSPIRE].zbMATHGoogle Scholar
  43. [43]
    T. Oota, Functional equations of form factors for diagonal scattering theories, Nucl. Phys. B 466 (1996) 361 [hep-th/9510171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    V.A. Fateev and Y.P. Pugai, Expectation values of scaling fields in Z N Ising models, Theor. Math. Phys. 154 (2008) 473 [INSPIRE].CrossRefzbMATHGoogle Scholar
  45. [45]
    G. Delfino, P. Simonetti and J.L. Cardy, Asymptotic factorization of form factors in two-dimensional quantum field theory, Phys. Lett. B 387 (1996) 327 [hep-th/9607046] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  3. 3.Kharkevich Institute for Information Transmission ProblemsMoscowRussia

Personalised recommendations