Integrable Models with Unstable Particles

  • Olalla Castro-Alvaredo
  • Andreas Fring
Part of the Progress in Mathematics book series (PM, volume 237)


We review some recent results concerning integrable quantum field theories in 1 + 1 space-time dimensions which contain unstable particles in their spectrum. Recalling first the main features of analytic scattering theories associated to integrable models, we subsequently propose a new bootstrap principle which allows for the construction of particle spectra involving unstable as well as stable particles. We describe the general Lie algebraic structure which underlies theories with unstable particles and formulate a decoupling rule, which predicts the renormalization group flow in dependence of the relative ordering of the resonance parameters. We extend these ideas to theories with an infinite spectrum of unstable particles. We provide new expressions for the scattering amplitudes in the soliton-antisoliton sector of the elliptic sine-Gordon model in terms of infinite products of q-deformed gamma functions. When relaxing the usual restriction on the coupling constants, the model contains additional bound states which admit an interpretation as breathers. For that situation we compute the complete S-matrix of all sectors. We carry out various reductions of the model, one of them leading to a new type of theory, namely an elliptic version of the minimal SO(n)-affine Toda field theory.


Exactly and quasi-solvable systems S-matrix theory model quantum field theories two-dimensional field theories conformal field theories infinite-dimensional groups and algebras motivated by physics including Virasoro Kac-Moody 


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  1. [1]
    M. Karowski, H.J. Thun, T.T. Truong, and P.H. Weisz, On the uniqueness of a purely elastic S-matrix in (1 + 1) dimensions, Phys. Lett. B67, 321–322 (1977).Google Scholar
  2. [2]
    A.B. Zamolodchikov, Exact S-matrix of quantum sine-Gordon solitons, JETP Lett. 25, 468–481 (1977).Google Scholar
  3. [3]
    A. Zamolodchikov and A. Zamolodchikov, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120, 253 (1979).CrossRefMathSciNetGoogle Scholar
  4. [4]
    R. Shankar and E. Witten, The S-matrix of the supersymmetric nonlinear sigma model, Phys. Rev. D17, 2134–2143 (1978).Google Scholar
  5. [5]
    S. Parke, Absence of particle production and factorization of the S-matrix in (1 + 1)-dimensional models, Nucl. Phys. B174, 166–182 (1980).Google Scholar
  6. [6]
    B. Schroer, T.T. Truong, and P. Weisz, Towards an explicit construction of the sine-Gordon theory, Phys. Lett. B63, 422–424 (1976).Google Scholar
  7. [7]
    R. Eden, P. Landshoff, D.I. Olive, and J. Polkinghorne, The analytic S-matrix, Cambridge University Press (1966).Google Scholar
  8. [8]
    D.I. Olive, Unitarity and the Evolution of Discontinuities, Nuovo Cim. 26, 73–102 (1962).zbMATHMathSciNetGoogle Scholar
  9. [9]
    J.L. Miramontes, Hermitian analyticity versus real analyticity in two-dimensional factorised S-matrix theories, Phys. Lett. B455, 231–238 (1999).Google Scholar
  10. [10]
    D.I. Olive, Exploration of S-Matrix Theory, Phys. Rev. 135, B745–B760 (1964).CrossRefMathSciNetGoogle Scholar
  11. [11]
    H. Lehmann, K. Symanzik, and W. Zimmermann, On the formulation of quantized field theories, Nuovo Cim. 1, 205–225 (1955).MathSciNetGoogle Scholar
  12. [12]
    C.-N. Yang, Some exact results for the many body problems in one dimension with repulsive delta function interaction, Phys. Rev. Lett. 19, 1312–1314 (1967).zbMATHMathSciNetGoogle Scholar
  13. [13]
    R.J. Baxter, One-dimensional anisotropic Heisenberg chain, Annals Phys. 70, 323–327 (1972).MathSciNetGoogle Scholar
  14. [14]
    G. Breit and E.P. Wigner, Capture of slow neutrons, Phys. Rev. 49, 519–531 (1936).Google Scholar
  15. [15]
    S.R. Coleman and H.J. Thun, On the prosaic origin of the double poles in the sine-Gordon S-matrix, Commun. Math. Phys. 61, 31–51 (1978).CrossRefMathSciNetGoogle Scholar
  16. [16]
    H.W. Braden, E. Corrigan, P.E. Dorey, and R. Sasaki, Affine Toda field theory and exact S-matrices, Nucl. Phys. B338, 689–746 (1990).MathSciNetGoogle Scholar
  17. [17]
    P. Christe and G. Mussardo, Integrable systems away from criticality: The Toda field theory and S-matrix of the tricritical Ising model, Nucl. Phys. B330, 465–487 (1990).MathSciNetGoogle Scholar
  18. [18]
    L. Castillejo, R.H. Dalitz, and F.J. Dyson, Low’s scattering equation for the charged and neutral scalar theories, Phys. Rev. 101, 453–458 (1956).CrossRefGoogle Scholar
  19. [19]
    O.A. Castro-Alvaredo, J. Dreißig, and A. Fring, Integrable scattering theories with unstable particles, Euro Phys. Lett. C35, 393–411 (2004).Google Scholar
  20. [20]
    Q.-H. Park, Deformed coset models from gauged WZW actions, Phys. Lett. B328, 329–336 (1994).Google Scholar
  21. [21]
    C.R. Fernandez-Pousa, M.V. Gallas, T.J. Hollowood, and J.L. Miramontes, The symmetric space and homogeneous sine-Gordon theories, Nucl. Phys. B484, 609–630 (1997).MathSciNetGoogle Scholar
  22. [22]
    J.L. Miramontes and C.R. Fernandez-Pousa, Integrable quantum field theories with unstable particles, Phys. Lett. B472, 392–401 (2000).MathSciNetGoogle Scholar
  23. [23]
    O.A. Castro-Alvaredo, A. Fring, C. Korff, and J.L. Miramontes, Thermodynamic Bethe ansatz of the homogeneous sine-Gordon models, Nucl. Phys. B575, 535–560 (2000).MathSciNetGoogle Scholar
  24. [24]
    O.A. Castro-Alvaredo, A. Fring, and C. Korff, Form factors of the homogeneous sine-Gordon models, Phys. Lett. B484, 167–176 (2000).MathSciNetGoogle Scholar
  25. [25]
    O.A. Castro-Alvaredo and A. Fring, Identifying the operator content, the homogeneous sine-Gordon models, Nucl. Phys. B604, 367–390 (2001).MathSciNetGoogle Scholar
  26. [26]
    O.A. Castro-Alvaredo and A. Fring, Decoupling the SU(N)(2) homogeneous sine-Gordon model, Phys. Rev. D64, 085007 (2001).Google Scholar
  27. [27]
    O.A. Castro-Alvaredo and A. Fring, Renormalization group flow with unstable particles, Phys. Rev. D63, 021701 (2001).Google Scholar
  28. [28]
    J.L. Miramontes, Integrable quantum field theories with unstable particles, JHEP Proceedings, of the TMR conference, Nonperturbative Quantum Effects, hep-th/0010012 (Paris 2000).Google Scholar
  29. [29]
    P. Dorey and J.L. Miramontes, Aspects of the homogeneous sine-Gordon models, JHEP Proceedings, of the workshop on Integrable Theories, Solitons and Duality, hep-th/0211174 (São Paulo 2002).Google Scholar
  30. [30]
    P. Baseilhac, Liouville field theory coupled to a critical Ising model: Non-perturbative analysis, duality and applications, Nucl. Phys. B636, 465–496 (2002).MathSciNetGoogle Scholar
  31. [31]
    E. Witten, Nonabelian bosonization in two dimensions, Commun. Math. Phys. 92, 455–472 (1984).CrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]
    P. Goddard, A. Kent, and D.I. Olive, Virasoro algebras and coset space models, Phys. Lett. B152, 88–92 (1985).MathSciNetGoogle Scholar
  33. [33]
    J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, Berlin (1972).Google Scholar
  34. [34]
    C. Korff, Color-valued scattering matrices from non simply-laced Lie algebras, Phys. Lett. B5011, 289–296 (2001).MathSciNetGoogle Scholar
  35. [35]
    A. Fring and C. Korff, Color-valued scattering matrices, Phys. Lett. B477, 380–386 (2000).MathSciNetGoogle Scholar
  36. [36]
    A. Zamolodchikov, Resonance factorized scattering and roaming trajectories, ENS-LPS-335-preprint.Google Scholar
  37. [37]
    O.A. Castro-Alvaredo and A. Fring, Constructing infinite particle spectra, Phys. Rev. D64, 085005 (2001).Google Scholar
  38. [38]
    E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Trans. Ser. 2 6, 111–244 (1957).zbMATHGoogle Scholar
  39. [39]
    A. Kuniba, Thermodynamics of the Uq(Xr(1)) Bethe ansatz system with q a root of unity, Nucl. Phys. B389, 209–246 (1993).MathSciNetGoogle Scholar
  40. [40]
    A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2-D field theory, JETP Lett. 43, 730–732 (1986).MathSciNetGoogle Scholar
  41. [41]
    A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state Potts and Lee-Yang models, Nucl. Phys. B342, 695–720 (1990).MathSciNetGoogle Scholar
  42. [42]
    H. Babujian, A. Fring, M. Karowski, and A. Zapletal, Exact form factors in integrable quantum field theories: The sine-Gordon model, Nucl. Phys. B538, 535–586 (1999).MathSciNetGoogle Scholar
  43. [43]
    K. Chandrasekhan, Elliptic Functions, Springer, Berlin (1985).Google Scholar
  44. [44]
    A. Zamolodchikov, Z4-symmetric factorised S-matrix in two space-time dimensions, Comm. Math. Phys. 69, 165–178 (1979).CrossRefMathSciNetGoogle Scholar
  45. [45]
    O.A. Castro-Alvaredo and A. Fring, Breathers in the elliptic sine-Gordon model, J. Phys. A36, 10233–10249 (2003).MathSciNetGoogle Scholar
  46. [46]
    M. Karowski and H.J. Thun, Complete S matrix of the massive Thirring model, Nucl. Phys. B130, 295–308 (1977).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Olalla Castro-Alvaredo
    • 1
  • Andreas Fring
    • 2
  1. 1.Laboratoire de Physique, Ecole Normale Supérieure de LyonUMR 5672 du CNRSLyon CEDEXFrance
  2. 2.Centre for Mathematical SciencesCity UniversityLondonUK

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