Skip to main content

On factorizable S-matrices, generalized TTbar, and the Hagedorn transition

A preprint version of the article is available at arXiv.

Abstract

We study solutions of the Thermodynamic Bethe Ansatz equations for relativistic theories defined by the factorizable S-matrix of an integrable QFT deformed by CDD factors. Such S-matrices appear under generalized TTbar deformations of integrable QFT by special irrelevant operators. The TBA equations, of course, determine the ground state energy E(R) of the finite-size system, with the spatial coordinate compactified on a circle of circumference R. We limit attention to theories involving just one kind of stable particles, and consider deformations of the trivial (free fermion or boson) S-matrix by CDD factors with two elementary poles and regular high energy asymptotics — the “2CDD model”. We find that for all values of the parameters (positions of the CDD poles) the TBA equations exhibit two real solutions at R greater than a certain parameter-dependent value R*, which we refer to as the primary and secondary branches. The primary branch is identified with the standard iterative solution, while the secondary one is unstable against iterations and needs to be accessed through an alternative numerical method known as pseudo-arc-length continuation. The two branches merge at the “turning point” R* (a square-root branching point). The singularity signals a Hagedorn behavior of the density of high energy states of the deformed theories, a feature incompatible with the Wilsonian notion of a local QFT originating from a UV fixed point, but typical for string theories. This behavior of E(R) is qualitatively the same as the one for standard TTbar deformations of local QFT.

References

  1. F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, T\( \overline{T} \)-deformed 2D Quantum Field Theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].

  3. A.B. Zamolodchikov, Expectation value of composite field T\( \overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].

  4. K.G. Wilson and J.B. Kogut, The Renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].

    ADS  Article  Google Scholar 

  5. J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2007).

  6. S. Dubovsky, V. Gorbenko and M. Mirbabayi, Asymptotic fragility, near AdS2 holography and T\( \overline{T} \), JHEP 09 (2017) 136 [arXiv:1706.06604] [INSPIRE].

  7. S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, T\( \overline{T} \) partition function from topological gravity, JHEP 09 (2018) 158 [arXiv:1805.07386] [INSPIRE].

  8. L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with T\( \overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].

  9. M.R. Douglas and N.A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys. 73 (2001) 977 [hep-th/0106048] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. R. Conti, S. Negro and R. Tateo, Conserved currents and T\( \overline{T} \)s irrelevant deformations of 2D integrable field theories, JHEP 11 (2019) 120 [arXiv:1904.09141] [INSPIRE].

  11. J. Cardy, The T\( \overline{T} \) deformation of quantum field theory as random geometry, JHEP 10 (2018) 186 [arXiv:1801.06895] [INSPIRE].

  12. J. Kruthoff and O. Parrikar, On the flow of states under T\( \overline{T} \), arXiv:2006.03054 [INSPIRE].

  13. L. Castillejo, R.H. Dalitz and F.J. Dyson, Low’s scattering equation for the charged and neutral scalar theories, Phys. Rev. 101 (1956) 453 [INSPIRE].

  14. G. Hernández-Chifflet, S. Negro and A. Sfondrini, Flow Equations for Generalized T\( \overline{T} \) Deformations, Phys. Rev. Lett. 124 (2020) 200601 [arXiv:1911.12233] [INSPIRE].

  15. C.-N. Yang and C.P. Yang, Thermodynamics of one-dimensional system of bosons with repulsive delta function interaction, J. Math. Phys. 10 (1969) 1115 [INSPIRE].

  16. A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].

  17. A.B. Zamolodchikov, Resonance factorized scattering and roaming trajectories, J. Phys. A 39 (2006) 12847 [INSPIRE].

  18. A.B. Zamolodchikov, unpublished.

  19. G. Mussardo and P. Simon, Bosonic type S matrix, vacuum instability and CDD ambiguities, Nucl. Phys. B 578 (2000) 527 [hep-th/9903072] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  20. M.J. Martins, Renormalization group trajectories from resonance factorized S matrices, Phys. Rev. Lett. 69 (1992) 2461 [hep-th/9205024] [INSPIRE].

    ADS  Article  Google Scholar 

  21. M.J. Martins, Exact resonance A-D-E S matrices and their renormalization group trajectories, Nucl. Phys. B 394 (1993) 339 [hep-th/9208011] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  22. P. Dorey, C. Dunning and R. Tateo, New families of flows between two-dimensional conformal field theories, Nucl. Phys. B 578 (2000) 699 [hep-th/0001185] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  23. S. Dubovsky, R. Flauger and V. Gorbenko, Solving the Simplest Theory of Quantum Gravity, JHEP 09 (2012) 133 [arXiv:1205.6805] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. M. Caselle, D. Fioravanti, F. Gliozzi and R. Tateo, Quantisation of the effective string with TBA, JHEP 07 (2013) 071 [arXiv:1305.1278] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  25. A. LeClair, T\( \overline{T} \) deformation of the Ising model and its ultraviolet completion, arXiv:2107.02230 [INSPIRE].

  26. A. LeClair, Thermodynamics of T\( \overline{T} \) perturbations of some single particle field theories, arXiv:2105.08184 [INSPIRE].

  27. J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].

  28. A.M. Perelomov and Y.B. Zeldovich, Quantum Mechanics, Selected Topics, World Scientific, New York U.S.A. (1998).

    Book  Google Scholar 

  29. P. Fendley and H. Saleur, Massless integrable quantum field theories and massless scattering in (1 + 1)-dimensions, in Summer School in High-energy Physics and Cosmology (Includes Workshop on Strings, Gravity, and Related Topics, Trieste Italy (1993), pg. 301 [hep-th/9310058] [INSPIRE].

  30. A. Fring, C. Korff and B.J. Schulz, The Ultraviolet behavior of integrable quantum field theories, affine Toda field theory, Nucl. Phys. B 549 (1999) 579 [hep-th/9902011] [INSPIRE].

    ADS  Article  Google Scholar 

  31. L. Hilfiker and I. Runkel, Existence and uniqueness of solutions to Y-systems and TBA equations, Ann. Henri Poincaré 21 (2019) 941 [arXiv:1708.00001] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. E.L. Allgower and K. Georg, Springer Series in Computational Mathematics. Vol. 13: Numerical continuation methods: an introduction, Springer, Berlin Germany (2012).

  33. P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  34. L.G. Córdova, S. Negro and F.I. Schaposnik, Thermodynamic Bethe Ansatz past turning points: the (eliptic) sinh-Gordon model, to appear.

  35. J.L.F. Barbón and E. Rabinovici, Remarks on the thermodynamic stability of T\( \overline{T} \) deformations, J. Phys. A 53 (2020) 424001 [arXiv:2004.10138] [INSPIRE].

  36. S.R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory, Phys. Rev. D 15 (1977) 2929 [Erratum ibid. 16 (1977) 1248] [INSPIRE].

  37. I.Y. Kobzarev, L.B. Okun and M.B. Voloshin, Bubbles in Metastable Vacuum, Yad. Fiz. 20 (1974) 1229 [INSPIRE].

  38. R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge U.K. (1966).

    MATH  Google Scholar 

  39. D. Iagolnitzer, Scattering in quantum field theories: The Axiomatic and constructive approaches, Princeton University Press, Princeton U.S.A. (1994).

    MATH  Google Scholar 

  40. D. Iagolnitzer, Macrocausality, Physical Region Analyticity and Independence Property in S Matrix Theory, Lect. Notes Math. 449 (1975) 102.

    MathSciNet  Article  Google Scholar 

  41. A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].

  42. A.B. Zamolodchikov and A.B. Zamolodchikov, Massless factorized scattering and sigma models with topological terms, Nucl. Phys. B 379 (1992) 602 [INSPIRE].

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Thiago Fleury.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2106.11999

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Camilo, G., Fleury, T., Lencsés, M. et al. On factorizable S-matrices, generalized TTbar, and the Hagedorn transition. J. High Energ. Phys. 2021, 62 (2021). https://doi.org/10.1007/JHEP10(2021)062

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP10(2021)062

Keywords

  • Integrable Field Theories
  • Renormalization Group