Journal of High Energy Physics

, 2018:63 | Cite as

Adding flavour to the S-matrix bootstrap

  • Lucía CórdovaEmail author
  • Pedro Vieira
Open Access
Regular Article - Theoretical Physics


We explore the S-matrices of gapped, unitary, Lorentz invariant quantum field theories with a global O(N) symmetry in 1+1 dimensions. We extremize various cubic and quartic couplings in the two-to-two scattering amplitudes of vector particles. Saturating these bounds, we encounter known integrable models with O(N) symmetry such as the O(N) Gross-Neveu and non-linear sigma models and the scattering of kinks in the sine-Gordon model. We also considered more general mass spectra for which we move away from the integrable realm. In this regime we find (numerically, through a large N analysis and sometimes even analytically) that the S-matrices saturating the various coupling bounds have an extremely rich structure exhibiting infinite resonances and virtual states in the various kinematical sheets. They are rather exotic in that they admit no particle production yet they do not obey Yang-Baxter equations. We discuss their physical (ir)relevance and speculate, based on some preliminary numerics, that they might be close to more realistic theories with particle production.


Field Theories in Lower Dimensions Integrable Field Theories Nonperturbative Effects Scattering Amplitudes 


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]
    M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
  2. [2]
    M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap II: two dimensional amplitudes, JHEP 11 (2017) 143 [arXiv:1607.06110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix Bootstrap III: Higher Dimensional Amplitudes, arXiv:1708.06765 [INSPIRE].
  4. [4]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A.L. Guerrieri, J. Penedones and P. Vieira, Bootstrapping QCD: the Lake, the Peninsula and the Kink, arXiv:1810.12849 [INSPIRE].
  6. [6]
    M. Hortacsu, B. Schroer and H.J. Thun, A Two-dimensional σ Model With Particle Production, Nucl. Phys. B 154 (1979) 120 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    M.F. Paulos and Z. Zheng, Bounding scattering of charged particles in 1 + 1 dimensions, arXiv:1805.11429 [INSPIRE].
  8. [8]
    Y. He, A. Irrgang and M. Kruczenski, A note on the S-matrix bootstrap for the 2d O(N) bosonic model, JHEP 11 (2018) 093 [arXiv:1805.02812] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    E. Witten, Some Properties of the \( {\left(\overline{\psi}\psi \right)}^2 \) Model in Two-Dimensions, Nucl. Phys. B 142 (1978) 285 [INSPIRE].
  10. [10]
    R. Shankar and E. Witten, The S Matrix of the Kinks of the \( {\left(\overline{\psi}\psi \right)}^2 \) Model, Nucl. Phys. B 141 (1978) 349 [Erratum ibid. B 148 (1979) 538] [INSPIRE].
  11. [11]
    M. Karowski and H.J. Thun, Complete S Matrix of the O(2N) Gross-Neveu Model, Nucl. Phys. B 190 (1981) 61 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    N. Doroud and J. Elias Miró, S-matrix bootstrap for resonances, JHEP 09 (2018) 052 [arXiv:1804.04376] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    B. Gabai, D. Mazáč, A. Shieber, P. Vieira and Y. Zhou, No Particle Production in Two Dimensions: Recursion Relations and Multi-Regge Limit, arXiv:1803.03578 [INSPIRE].
  14. [14]
    A. Zamolodchikov and I. Ziyatdinov, Inelastic scattering and elastic amplitude in Ising field theory in a weak magnetic field at T > T c : Perturbative analysis, Nucl. Phys. B 849 (2011) 654 [arXiv:1102.0767] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    A.B. Zamolodchikov, Z 4 Symmetric factorized s matrix in two space-time dimensions, Commun. Math. Phys. 69 (1979) 165 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    G. Mussardo and S. Penati, A Quantum field theory with infinite resonance states, Nucl. Phys. B 567 (2000) 454 [hep-th/9907039] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A.B. Zamolodchikov, Exact S matrix associated with selfavoiding polymer problem in two-dimensions, Mod. Phys. Lett. A 6 (1991) 1807 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  18. [18]
    F.A. Smirnov, A Comment on A. Zamolodchikovs paper concerning selfavoiding polymers, Phys. Lett. B 275 (1992) 109 [INSPIRE].
  19. [19]
    P. Fendley, Taking NO with S matrices, cond-mat/0111582.
  20. [20]
    A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].ADSMathSciNetGoogle Scholar
  21. [21]
    D. Iagolnitzer, Factorization of the Multiparticle s Matrix in Two-Dimensional Space-Time Models, Phys. Rev. D 18 (1978) 1275 [INSPIRE].ADSGoogle Scholar
  22. [22]
    D. Iagolnitzer, Scattering in quantum field theories: The Axiomatic and constructive approaches, Princeton University Press, Princeton, U.S.A. (1993) [INSPIRE].CrossRefzbMATHGoogle Scholar
  23. [23]
    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

Copyright information

© The Author(s) 2018

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.Institut de Physique Théorique, CEA SaclayGif-sur-YvetteFrance
  4. 4.Instituto de Física Teórica, UNESPICTP South American Institute for Fundamental ResearchSão PauloBrazil

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