- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Meson mass spectrum in Ising field theory

Alexey Litvinov1, Pavel Meshcheriakov1,2,3, and Egor Shestopalov3

PDF

Phys. Rev. D 112, 085021 – Published 24 October, 2025

DOI: https://doi.org/10.1103/mvf7-pd7p

Abstract

We study the two-particle approximation of the Ising field theory (IFT), formulated in terms of the Bethe-Salpeter (BS) equation. Derived by Fonseca and Zamolodchikov as a systematic realization of the McCoy-Wu approach, this equation captures confinement by modeling “mesons” as bound states of two Majorana fermions (“quarks”), in a way analogous to the integral equation in the ’t Hooft model. Despite its approximate nature, the BS equation provides remarkably accurate predictions for the mass spectrum of stable mesons across a wide range of parameters. Motivated by the striking structural similarity between the BS equation in IFT and the ’t Hooft equation in two-dimensional QCD, we develop a new nonperturbative analytical framework inspired by the method of Fateev, Lukyanov, and Zamolodchikov. Within this approach, we compute spectral sums and systematically derive the large-n expansion for the Bethe-Salpeter equation, which governs the spectrum of an infinite tower of mesons. We further examine how our analytical results capture the known behavior of the spectrum in well-studied asymptotic regimes, such as the E8 limit and the free-fermion point, where exact solutions are available for comparison. Finally, we discuss how the obtained spectral data admit a natural analytic continuation to complex values of the parameters—an extension that was one of the primary motivations for this work.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (42)

  1. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241, 333 (1984).
  2. B. M. McCoy and T. T. Wu, Two-dimensional Ising field theory in a magnetic field: Breakup of the cut in the two point function, Phys. Rev. D 18, 1259 (1978).
  3. L. Onsager, Crystal statistics. 1. A Two-dimensional model with an order disorder transition, Phys. Rev. 65, 117 (1944).
  4. A. B. Zamolodchikov, Integrals of motion and S matrix of the (Scaled) T=T(c) Ising model with magnetic field, Int. J. Mod. Phys. A 04, 4235 (1989).
  5. A. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19, 641 (1989).
  6. C.-N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. 1. Theory of condensation, Phys. Rev. 87, 404 (1952).
  7. T. D. Lee and C.-N. Yang, Statistical theory of equations of state and phase transitions. 2. Lattice gas and Ising model, Phys. Rev. 87, 410 (1952).
  8. P. Fonseca and A. Zamolodchikov, Ising field theory in a magnetic field: Analytic properties of the free energy, arXiv:hep-th/0112167.
  9. H.-L. Xu and A. Zamolodchikov, 2D Ising field theory in a magnetic field: The Yang-Lee singularity, J. High Energy Phys. 08 (2022) 057.
  10. V. V. Mangazeev, B. Hagan, and V. V. Bazhanov, Corner transfer matrix approach to the Yang-Lee singularity in the two-dimensional Ising model in a magnetic field, Phys. Rev. E 108, 064136 (2023).
  11. M. E. Fisher, Yang-Lee edge singularity and ϕ3 field theory, Phys. Rev. Lett. 40, 1610 (1978).
  12. J. L. Cardy, Conformal invariance and the Yang-Lee edge singularity in two-dimensions, Phys. Rev. Lett. 54, 1354 (1985).
  13. J. L. Cardy and G. Mussardo, S matrix of the Yang-Lee edge singularity in two-dimensions, Phys. Lett. B 225, 275 (1989).
  14. H.-L. Xu and A. Zamolodchikov, Ising field theory in a magnetic field: φ3 coupling at T>Tc, J. High Energy Phys. 08 (2023) 161.
  15. P. Fonseca and A. Zamolodchikov, Ising spectroscopy. I. Mesons at T<Tc, arXiv:hep-th/0612304.
  16. A. Zamolodchikov, Ising spectroscopy II: Particles and poles at T>Tc, arXiv:1310.4821.
  17. G. Delfino, G. Mussardo, and P. Simonetti, Nonintegrable quantum field theories as perturbations of certain integrable models, Nucl. Phys. B473, 469 (1996).
  18. G. Delfino, P. Grinza, and G. Mussardo, Decay of particles above threshold in the Ising field theory with magnetic field, Nucl. Phys. B737, 291 (2006).
  19. S. B. Rutkevich, Large-n excitations in the ferromagnetic Ising field theory in a small magnetic field: Mass spectrum and decay widths, Phys. Rev. Lett. 95, 250601 (2005).
  20. S. B. Rutkevich, Formfactor perturbation expansions and confinement in the Ising field theory, J. Phys. A 42, 304025 (2009).
  21. S. B. Rutkevich, Radiative corrections to the quark masses in the ferromagnetic Ising and Potts field theories, Nucl. Phys. B923, 508 (2017).
  22. H. L. Xu, On the analyticity of the lightest particle mass of Ising field theory in a magnetic field, arXiv:2405.09091.
  23. M. Lencses and G. Takacs, Confinement in the q-state Potts model: An RG-TCSA study, J. High Energy Phys. 09 (2015) 146.
  24. B. Gabai and X. Yin, On the S-matrix of Ising field theory in two dimensions, J. High Energy Phys. 10 (2022) 168.
  25. A. L. Fitzpatrick, E. Katz, and Y. Xin, Lightcone Hamiltonian for Ising field theory I: T<Tc, SciPost Phys. 18, 179 (2025).
  26. R. G. Jha, A. Milsted, D. Neuenfeld, J. Preskill, and P. Vieira, Real-time scattering in Ising field theory using matrix product states, Phys. Rev. Res. 7, 023266 (2025).
  27. G. ’t Hooft, A two-dimensional model for mesons, Nucl. Phys. B75, 461 (1974).
  28. V. A. Fateev, S. L. Lukyanov, and A. B. Zamolodchikov, On mass spectrum in ’t Hooft’s 2D model of mesons, J. Phys. A 42, 304012 (2009).
  29. A. Litvinov and P. Meshcheriakov, Meson mass spectrum in QCD2 ’t Hooft’s model, Nucl. Phys. B1010, 116766 (2025).
  30. A. Artemev, A. Litvinov, and P. Meshcheriakov, QCD2 ’t Hooft model: Two-flavor mesons spectrum, Phys. Rev. D 111, 125001 (2025).
  31. T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, Spin spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region, Phys. Rev. B 13, 316 (1976).
  32. P. Fonseca and A. Zamolodchikov, Ward identities and integrable differential equations in the Ising field theory, arXiv:hep-th/0309228.
  33. M. B. Voloshin, Decay of false vacuum in (1+1)-dimensions, Yad. Fiz. 42, 1017 (1985).
  34. Y. Gao, Y. Jiang, and J. Wu, Mesons in a quantum Ising ladder, J. High Energy Phys. 07 (2025) 072.
  35. R. J. Baxter, Partition function of the eight vertex lattice model, Ann. Phys. (N.Y.) 70, 193 (1972).
  36. A. R. Its, A. G. Izergin, and V. E. Korepin, Temperature correlators of the impenetrable Bose gas as an integrable system, Commun. Math. Phys. 129, 205 (1990).
  37. A. Its, A. Izergin, V. Korepin, and N. Slavnov, Differential equations for quantum correlation functions, Int. J. Mod. Phys. B 04, 1003 (1990).
  38. P. Deift, Integrable operators, in Differential Operators and Spectral Theory. M. Sh. Birman’s 70th Anniversary Collection (American Mathematical Society, Providence, 1999), pp. 69–84.
  39. See Supplemental Material at http://link.aps.org/supplemental/10.1103/mvf7-pd7p for three accompanying Wolfram Mathematica notebooks: spectral-sums-ising.nb, phi.nb, and wkb.nb. The first notebook contains analytic expressions for the first five spectral sums G±(s) and the corresponding matrix elements ⟨p|K^n|p⟩ for n=1,…,5. The second notebook includes the phase functions Φ±(k)(l) for k=0,…,5. The third file presents higher-order terms in the large-n WKB expansion (5.15)–(5.16), including corrections up to 1n5. All three notebooks make use of the fundamental integrals v(α), u2k−1(α), i2k(α), and i2k−1(α), defined in (2.43), (3.8), (3.24), and (3.26), respectively.
  40. 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, 45 (1994).
  41. I. Ziyatdinov, Asymptotic properties of mass spectrum in ’t Hooft’s model of mesons, Int. J. Mod. Phys. A 25, 3899 (2010).
  42. A. Zamolodchikov, On confining interactions in 1+1, Talk at Conference in the Memory of Aliosha Zamolodchikov, Saclay (2009), https://indico.in2p3.fr/event/1886/sessions/3945/attachments/17798/21781/Zamolodchikov.pdf.

Outline

Information

Phys. Rev. D 112, 085021– Published 24 October, 2025
  • Received 28 July 2025
  • Accepted 17 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation