Abstract
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.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
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).
A.B. Zamolodchikov, Exact S-matrix of quantum sine-Gordon solitons, JETP Lett. 25, 468–481 (1977).
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).
R. Shankar and E. Witten, The S-matrix of the supersymmetric nonlinear sigma model, Phys. Rev. D17, 2134–2143 (1978).
S. Parke, Absence of particle production and factorization of the S-matrix in (1 + 1)-dimensional models, Nucl. Phys. B174, 166–182 (1980).
B. Schroer, T.T. Truong, and P. Weisz, Towards an explicit construction of the sine-Gordon theory, Phys. Lett. B63, 422–424 (1976).
R. Eden, P. Landshoff, D.I. Olive, and J. Polkinghorne, The analytic S-matrix, Cambridge University Press (1966).
D.I. Olive, Unitarity and the Evolution of Discontinuities, Nuovo Cim. 26, 73–102 (1962).
J.L. Miramontes, Hermitian analyticity versus real analyticity in two-dimensional factorised S-matrix theories, Phys. Lett. B455, 231–238 (1999).
D.I. Olive, Exploration of S-Matrix Theory, Phys. Rev. 135, B745–B760 (1964).
H. Lehmann, K. Symanzik, and W. Zimmermann, On the formulation of quantized field theories, Nuovo Cim. 1, 205–225 (1955).
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).
R.J. Baxter, One-dimensional anisotropic Heisenberg chain, Annals Phys. 70, 323–327 (1972).
G. Breit and E.P. Wigner, Capture of slow neutrons, Phys. Rev. 49, 519–531 (1936).
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).
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).
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).
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).
O.A. Castro-Alvaredo, J. Dreißig, and A. Fring, Integrable scattering theories with unstable particles, Euro Phys. Lett. C35, 393–411 (2004).
Q.-H. Park, Deformed coset models from gauged WZW actions, Phys. Lett. B328, 329–336 (1994).
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).
J.L. Miramontes and C.R. Fernandez-Pousa, Integrable quantum field theories with unstable particles, Phys. Lett. B472, 392–401 (2000).
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).
O.A. Castro-Alvaredo, A. Fring, and C. Korff, Form factors of the homogeneous sine-Gordon models, Phys. Lett. B484, 167–176 (2000).
O.A. Castro-Alvaredo and A. Fring, Identifying the operator content, the homogeneous sine-Gordon models, Nucl. Phys. B604, 367–390 (2001).
O.A. Castro-Alvaredo and A. Fring, Decoupling the SU(N)(2) homogeneous sine-Gordon model, Phys. Rev. D64, 085007 (2001).
O.A. Castro-Alvaredo and A. Fring, Renormalization group flow with unstable particles, Phys. Rev. D63, 021701 (2001).
J.L. Miramontes, Integrable quantum field theories with unstable particles, JHEP Proceedings, of the TMR conference, Nonperturbative Quantum Effects, hep-th/0010012 (Paris 2000).
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).
P. Baseilhac, Liouville field theory coupled to a critical Ising model: Non-perturbative analysis, duality and applications, Nucl. Phys. B636, 465–496 (2002).
E. Witten, Nonabelian bosonization in two dimensions, Commun. Math. Phys. 92, 455–472 (1984).
P. Goddard, A. Kent, and D.I. Olive, Virasoro algebras and coset space models, Phys. Lett. B152, 88–92 (1985).
J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, Berlin (1972).
C. Korff, Color-valued scattering matrices from non simply-laced Lie algebras, Phys. Lett. B5011, 289–296 (2001).
A. Fring and C. Korff, Color-valued scattering matrices, Phys. Lett. B477, 380–386 (2000).
A. Zamolodchikov, Resonance factorized scattering and roaming trajectories, ENS-LPS-335-preprint.
O.A. Castro-Alvaredo and A. Fring, Constructing infinite particle spectra, Phys. Rev. D64, 085005 (2001).
E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Trans. Ser. 2 6, 111–244 (1957).
A. Kuniba, Thermodynamics of the U q (X (1)r ) Bethe ansatz system with q a root of unity, Nucl. Phys. B389, 209–246 (1993).
A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2-D field theory, JETP Lett. 43, 730–732 (1986).
A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state Potts and Lee-Yang models, Nucl. Phys. B342, 695–720 (1990).
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).
K. Chandrasekhan, Elliptic Functions, Springer, Berlin (1985).
A. Zamolodchikov, Z4-symmetric factorised S-matrix in two space-time dimensions, Comm. Math. Phys. 69, 165–178 (1979).
O.A. Castro-Alvaredo and A. Fring, Breathers in the elliptic sine-Gordon model, J. Phys. A36, 10233–10249 (2003).
M. Karowski and H.J. Thun, Complete S matrix of the massive Thirring model, Nucl. Phys. B130, 295–308 (1977).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Castro-Alvaredo, O., Fring, A. (2005). Integrable Models with Unstable Particles. In: Kulish, P.P., Manojlovich, N., Samtleben, H. (eds) Infinite Dimensional Algebras and Quantum Integrable Systems. Progress in Mathematics, vol 237. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7341-5_2
Download citation
DOI: https://doi.org/10.1007/3-7643-7341-5_2
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7215-6
Online ISBN: 978-3-7643-7341-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)