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Meson mass spectrum in QCD$_2$ 't Hooft's model

Alexey Litvinov, Pavel Meshcheriakov

TL;DR

This work extends the integrability-based solution of the 't Hooft equation for large-$N_c$ QCD$_2$ to the equal-mass case, recasting the spectral problem in terms of Baxter's $TQ$ equation and spectral determinants. It delivers analytic expressions for spectral sums $G^{(s)}_{\pm}$, a refined large-$n$ expansion of meson masses, and a thorough analysis of limiting regimes (chiral and heavy-quark), complemented by numerical validation via a Chebyshev basis. The results generalize Fateev–Lukyanov–Zamolodchikov methods to $\alpha\neq0$, confirm known limits (including GMOR in the chiral case), and offer a robust framework for analytic insight into the QCD$_2$ meson spectrum, with potential extensions to unequal masses and related integrable models.

Abstract

We study the spectrum of meson masses in large $N_c$ QCD$_2$ governed by celebrated 't Hooft's integral equation. We generalize analytical methods proposed by Fateev, Lukyanov and Zamolodchikov to the case of arbitrary, but equal quark masses $m_1=m_2.$ Our results include analytical expressions for spectral sums and systematic large-$n$ expansion. We also study the spectral sums in the chiral limit and the heavy quark limit and find a complete agreement with known results.

Meson mass spectrum in QCD$_2$ 't Hooft's model

TL;DR

This work extends the integrability-based solution of the 't Hooft equation for large- QCD to the equal-mass case, recasting the spectral problem in terms of Baxter's equation and spectral determinants. It delivers analytic expressions for spectral sums , a refined large- expansion of meson masses, and a thorough analysis of limiting regimes (chiral and heavy-quark), complemented by numerical validation via a Chebyshev basis. The results generalize Fateev–Lukyanov–Zamolodchikov methods to , confirm known limits (including GMOR in the chiral case), and offer a robust framework for analytic insight into the QCD meson spectrum, with potential extensions to unequal masses and related integrable models.

Abstract

We study the spectrum of meson masses in large QCD governed by celebrated 't Hooft's integral equation. We generalize analytical methods proposed by Fateev, Lukyanov and Zamolodchikov to the case of arbitrary, but equal quark masses Our results include analytical expressions for spectral sums and systematic large- expansion. We also study the spectral sums in the chiral limit and the heavy quark limit and find a complete agreement with known results.
Paper Structure (25 sections, 147 equations, 4 tables)

This paper contains 25 sections, 147 equations, 4 tables.

Table of Contents

  1. Introduction
  2. From 't Hooft to Baxter's TQ equation
  3. Fourier form of 't Hooft equation
  4. TQ equation
  5. Inhomogeneous integral equation
  6. Quantum Wronskian relation
  7. Spectral determinants
  8. Strategy
  9. Small $\lambda$ expansion
  10. Large $\lambda$ expansion
  11. Analytical results
  12. Spectral sums
  13. Large -- $n$ expansion
  14. Spectral sums in limiting cases
  15. The chiral limit and the ground state
  16. ...and 10 more sections