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QCD$_2$ 't Hooft model: 2-flavour mesons spectrum

Aleksandr Artemev, Alexey Litvinov, Pavel Meshcheriakov

Abstract

We continue analytical study of the meson mass spectrum in the large-$N_c$ two-dimensional QCD, known as the 't Hooft model, by addressing the most general case of quarks with unequal masses. Based on our previous work, we develop non-perturbative methods to compute spectral sums and systematically derive large-$n$ WKB expansion of the spectrum. Furthermore, we examine the behavior of these results in various asymptotic regimes, including the chiral, heavy quark, and heavy-light limits, and establish a precise coincidence with known analytical and numerical results obtained through alternative approaches.

QCD$_2$ 't Hooft model: 2-flavour mesons spectrum

Abstract

We continue analytical study of the meson mass spectrum in the large- two-dimensional QCD, known as the 't Hooft model, by addressing the most general case of quarks with unequal masses. Based on our previous work, we develop non-perturbative methods to compute spectral sums and systematically derive large- WKB expansion of the spectrum. Furthermore, we examine the behavior of these results in various asymptotic regimes, including the chiral, heavy quark, and heavy-light limits, and establish a precise coincidence with known analytical and numerical results obtained through alternative approaches.

Paper Structure

This paper contains 25 sections, 190 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. 't Hooft equation: properties and relation to integrability
  3. Setup
  4. Fredholm integral equation and TQ equation
  5. Solution of TQ equation in the small $\lambda$ regime
  6. Solution of TQ equation in the large $\lambda$ regime
  7. First step: initial ansatz and its properties
  8. Second step: symmetry and removing extra singularities
  9. Last step: normalization
  10. Relation between Q-functions and spectral determinants
  11. Quantum Wronskian relation
  12. Relation with log-derivatives of $\mathbf{Q}$
  13. Analytical results
  14. Spectral sums
  15. WKB expansion
  16. ...and 10 more sections

Figures (5)

  • Figure 1: Analytic continuation $\nu \to \nu \pm 2i$ from real values brings additional terms, give by half- residues of the poles crossing the principal value integration contour
  • Figure 2: Spectral sums $G^{(s)}_\pm(\beta)$ for different $s$ and $\alpha = 0.5$. Lines are the analytical predictions for their values, circles are numerical values from Table \ref{['Gpm-table']}. The region of negative values for $G^{(1)}_-$ is related to the regularization of the first spectral sums in \ref{['spectral_sums_def']} and does not contradict the realness of meson masses.
  • Figure 3: Eigenvalues $\lambda_n(\beta)$ for $\alpha = 0.5$. Lines are the analytical predictions \ref{['wkb']} for their values (index $3$ indicates that we truncate \ref{['wkb']} beyond the term proportional to $\lambda^{-3}$), circles are numerical values from Table \ref{['Table-WKB']}.
  • Figure 4: Left: square of the first correction to the meson masses $M_n$ in the heavy-light limit, case $\alpha_1 = 0$, for the first 10 levels, right: first correction to masses for $\alpha_1 = 3$, first 8 levels. Circles are numerical results; continuous line is the analytic prediction \ref{['epsilon-m=1']} and \ref{['epsilon-general-m']}.
  • Figure 5: Relative error $\delta\lambda_n(\beta)$ in $\log_{10}$ axes for $\alpha=0.5$. As the number $n$ of the eigenstate increases, the analytical formula \ref{['wkb']} becomes more accurate. For the value $\beta=1.2$ we can see that the accuracy decreases significantly (for large $n>4$). This is explained by the fact that the numerical method used ceases to work when one of the masses $m_i \to 0$ ($\alpha_i \to -1$).