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Analytical results for large-Nc scalar QCD2

Pavel Meshcheriakov*

  • *Contact author: p.meshcheriakov@princeton.edu
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Phys. Rev. D 112, 125003 – Published 1 December, 2025

DOI: https://doi.org/10.1103/rlr4-rlts

Abstract

We study large-Nc scalar QCD2, a 1+1-dimensional confining gauge theory with fundamental scalar quarks, whose meson spectrum is governed by a Bethe-Salpeter equation structurally parallel to the ’t Hooft equation. Exploiting this structural analogy, we develop a nonperturbative analytic framework, based on integrability and inspired by the Fateev-Lukyanov-Zamolodchikov (FLZ) method, originally devised for the ’t Hooft model and later extended in our previous works. Notably, the same Bethe-Salpeter equation also arises in the description of interchain mesons in the doubled Ising model coupled via a spin-spin interaction term. Within the FLZ approach, we find spectral sums and derive a systematic large-n WKB expansion for the meson spectrum. The analytic results reproduce the expected behavior in key asymptotic regimes, such as the near-critical limit m→g/π and the heavy-quark regime m≫g, and are in good agreement with numerical data. Finally, by analytically continuing the mass parameter into the complex plane, we uncover two infinite families of singularities where individual mesons become massless, suggesting a hidden connection to nontrivial conformal field theories.

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References (45)

  1. G. ’t Hooft, A two-dimensional model for mesons, Nucl. Phys. B75, 461 (1974).
  2. A. J. Hanson, R. D. Peccei, and M. K. Prasad, Two-dimensional SU(N) gauge theory, strings and wings: Comparative analysis of meson spectra and covariance, Nucl. Phys. B121, 477 (1977).
  3. R. C. Brower, W. L. Spence, and J. H. Weis, Bound states and asymptotic limits for quantum chromodynamics in two dimensions, Phys. Rev. D 19, 3024 (1979).
  4. I. V. Kochergin, 1/N corrections in QCD2: Small mass limit and threshold states, J. High Energy Phys. 02 (2025) 073.
  5. 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).
  6. A. Litvinov and P. Meshcheriakov, Meson mass spectrum in QCD2 ’t Hooft’s model, Nucl. Phys. B1010, 116766 (2025).
  7. A. Artemev, A. Litvinov, and P. Meshcheriakov, QCD2 ’t Hooft model: Two-flavor s spectrum, Phys. Rev. D 111, 125001 (2025).
  8. A. Litvinov, P. Meshcheriakov, and E. Shestopalov, Meson mass spectrum in Ising field theory, Phys. Rev. D 112, 085021 (2025).
  9. P. Fonseca and A. Zamolodchikov, Ising spectroscopy. I. Mesons at T<Tc, arXiv:hep-th/0612304.
  10. H. P. Nilles, Supersymmetry, supergravity and particle physics, Phys. Rep. 110, 1 (1984).
  11. B. Grinstein, R. Jora, and A. D. Polosa, A note on large N scalar QCD2, Phys. Lett. B 671, 440 (2009).
  12. E. Witten, Baryons in the 1/n expansion, Nucl. Phys. B160, 57 (1979).
  13. M. B. Halpern and P. Senjanovic, Functional bridge between gauge theory and string in two-dimensions, Phys. Rev. D 15, 1655 (1977).
  14. S.-S. Shei and H.-S. Tsao, Scalar quantum chromodynamics in two-dimensions and parton model, Nucl. Phys. B141, 445 (1978).
  15. W. A. Bardeen and R. B. Pearson, Local gauge invariance and the bound state nature of hadrons, Phys. Rev. D 14, 547 (1976).
  16. T. N. Tomaras, Scalar U(N) QCD in the large N limit, Nucl. Phys. B163, 79 (1980).
  17. C. G. Callan, N. Coote, and D. J. Gross, Two-dimensional Yang-Mills theory: A model of quark confinement, Phys. Rev. D 13, 1649 (1976).
  18. K. Demeterfi, I. R. Klebanov, and G. Bhanot, Glueball spectrum in a (1+1)-dimensional model for QCD, Nucl. Phys. B418, 15 (1994).
  19. K. Aoki and T. Ichihara, (1+1)-dimensional QCD with fundamental bosons and fermions, Phys. Rev. D 52, 6435 (1995).
  20. Y. Gao, Y. Jiang, and J. Wu, Mesons in a quantum Ising ladder, J. High Energy Phys. 07 (2025) 072.
  21. R. J. Baxter, Partition function of the eight vertex lattice model, Ann. Phys. (N.Y.) 70, 193 (1972).
  22. 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).
  23. A. Its, A. Izergin, V. Korepin, and N. Slavnov, Differential equations for quantum correlation functions, Int. J. Mod. Phys. B 04, 1003 (1990).
  24. P. Deift, Integrable operators, in Differential Operators and Spectral Theory. M. Sh. Birman’s 70th Anniversary Collection (American Mathematical Society, Providence, RI, 1999), pp. 69–84.
  25. See Supplemental Material at http://link.aps.org/supplemental/10.1103/rlr4-rlts for Wolfram Mathematica notebooks: Spectral-sums-scalar-QCD.nb, Phi.nb, and WKB.nb;

    The first notebook contains closed-form expressions for the initial five spectral sums G±(s) together with the corresponding matrix elements ⟨p|K^n|p⟩ for n=1,…,5. The second notebook collects the phase functions Φ±(k)(l) for k=0,…,5. The third file presents higher-order contributions to the large-n WKB expansion (3.16) and (3.17), including terms up to order n−5. All three notebooks rely on the fundamental integrals ik(α), u2k−1(α), and v(α), defined in (2.34), (2.35), and (2.44), respectively.

  26. F. Ambrosino and S. Komatsu, 2d QCD and integrability. Part I. ’t Hooft model, J. High Energy Phys. 02 (2025) 126.
  27. A. R. Zhitnitsky, On chiral symmetry breaking in QCD in two-dimensions (Nc→∞), Phys. Lett. 165B, 405 (1985).
  28. A. LeClair, A. Ludwig, and G. Mussardo, Integrability of coupled conformal field theories, Nucl. Phys. B512, 523 (1998).
  29. J. B. Zuber and C. Itzykson, Quantum field theory and the two-dimensional Ising model, Phys. Rev. D 15, 2875 (1977).
  30. D. Boyanovsky, Field theory of the two-dimensional Ising model: Conformal invariance, order and disorder, and bosonization, Phys. Rev. B 39, 6744 (1989).
  31. 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).
  32. V. P. Yurov and A. B. Zamolodchikov, Truncated fermionic space approach to the critical 2-D Ising model with magnetic field, Int. J. Mod. Phys. A 06, 4557 (1991).
  33. S. B. Rutkevich, Formfactor perturbation expansions and confinement in the Ising field theory, J. Phys. A 42, 304025 (2009).
  34. I. Ziyatdinov, Asymptotic properties of mass spectrum in ’t Hooft’s model of mesons, Int. J. Mod. Phys. A 25, 3899 (2010).
  35. A. Zamolodchikov, On Confining Interactions in 1+1. Talk at Conference in the Memory of Aliosha Zamolodchikov, Saclay, https://indico.in2p3.fr/event/1886/sessions/3945/attachments/17798/21781/Zamolodchikov.pdf (2009).
  36. J. L. F. Barbon and K. Demeterfi, Effective Hamiltonians for 1/N expansion in two-dimensional QCD, Nucl. Phys. B434, 109 (1995).
  37. X. Ji, Y. Liu, and I. Zahed, Quasiparton distribution functions: Two-dimensional scalar and spinor QCD, Phys. Rev. D 99, 054008 (2019).
  38. P. Fonseca and A. Zamolodchikov, Ising field theory in a magnetic field: Analytic properties of the free energy, arXiv:hep-th/0112167.
  39. M. E. Fisher, Yang-Lee edge singularity and ϕ3 field theory, Phys. Rev. Lett. 40, 1610 (1978).
  40. J. L. Cardy, Conformal invariance and the Yang-Lee edge singularity in two-dimensions, Phys. Rev. Lett. 54, 1354 (1985).
  41. K. Aoki, Boson—fermion bound states in two-dimensional QCD, Phys. Rev. D 49, 573 (1994).
  42. H. Bergknoff, Physical particles of the massive Schwinger model, Nucl. Phys. B122, 215 (1977).
  43. K. Hornbostel, S. J. Brodsky, and H. C. Pauli, Light cone quantized QCD in (1+1)-dimensions, Phys. Rev. D 41, 3814 (1990).
  44. G. Bhanot, K. Demeterfi, and I. R. Klebanov, 1+1)-dimensional large N QCD coupled to adjoint fermions, Phys. Rev. D 48, 4980 (1993).
  45. M. Asrat, (1+1)D QCD with heavy adjoint quarks, Phys. Rev. D 107, 106022 (2023).

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Phys. Rev. D 112, 125003– Published 1 December, 2025
  • Received 24 September 2025
  • Accepted 29 October 2025
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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.

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