Meson Mass Spectrum in QCD2

• Meson mass spectrum in QCD2 is defined via the ’t Hooft model, providing an exactly solvable framework to analyze bound-state dynamics and nonperturbative confinement.
• Techniques such as spectral sums, Fredholm determinants, and semiclassical WKB expansions elucidate the analytic structure and emergence of Regge trajectories.
• Investigations across chiral, heavy-heavy, and heavy-light limits yield actionable insights into 1/Nc corrections, threshold phenomena, and the robustness of spectral integrability.

The meson mass spectrum in two-dimensional Quantum Chromodynamics (QCD$_2$) provides a paradigm for nonperturbative confinement, spectral flow, and integrable structures in quantum field theory. In this framework—most notably specified by the large-$N_c$ 't Hooft model—one obtains exact control over the bound-state spectrum, the analytic structure of spectral sums, and the leading $1/N_c$ corrections, yielding insights into the nature of confinement and strong dynamics that extend beyond two dimensions.

1. The ’t Hooft Model and Bound-State Integral Equation

In QCD$_2$ with gauge group SU($N_c$) and fundamental quarks, the $N_c \to \infty$ limit renders the theory solvable in terms of color-singlet mesonic bound states. The dynamics of a quark-antiquark pair of masses $m_1, m_2$ is governed by the ’t Hooft integral equation for the meson light-cone wavefunction $\phi^{(n)}(x)$ (with $x$ the momentum fraction of one constituent): $$2\pi^2\lambda_n\;\phi^{(n)}(x) = \left(\frac{\alpha_1}{x} + \frac{\alpha_2}{1-x}\right)\phi^{(n)}(x) - \text{P.V.} \int_0^1 \frac{\phi^{(n)}(y)}{(x-y)^2} \, dy$$ where

$$\alpha_i = \frac{\pi m_i^2}{g^2} - 1, \qquad M_n^2 = 2\pi g^2 \lambda_n,$$

and “P.V.” denotes the Cauchy principal value. The solutions $\lambda_n$ are strictly positive except in the chiral limit ($m_{1,2}\to 0$), where a massless boson emerges. Near $x\to 0,1$, the wavefunctions behave as $x^{\beta_1}(1-x)^{\beta_2}$, where $\beta_i$ solves $\pi \beta_i \cot(\pi \beta_i) + \alpha_i = 0$ (Artemev et al., 16 Apr 2025, Litvinov et al., 17 Sep 2024).

2. Spectral Sums, Fredholm Determinants, and Integrability

The set of eigenvalues $\{\lambda_n\}$ forms a discrete, asymptotically linear ladder. Global properties and operator traces are succinctly encoded in spectral sums: $G^{(s)} = \sum_{n=0}^\infty \lambda_n^{-s}$ for integer $s \geq 1$. Analytical control is achieved by relating the 't Hooft integral equation to integrable Baxter TQ-systems, leading to expressions for $G^{(s)}$ in terms of polylogarithms, Riemann zeta values, and convergent one-dimensional integrals over explicit functions of $\alpha_1$, $\alpha_2$. The Fredholm determinant $D(\lambda) = \prod_{n}(1-\lambda/\lambda_n)$ admits logarithmic derivatives connected to the spectral data, allowing closed-form results for all spectral moments. The TQ-equation method captures parity-separated (even/odd) spectral flows (Artemev et al., 16 Apr 2025, Litvinov et al., 17 Sep 2024).

3. Large-$n$ Expansion and Regge Trajectories

For highly excited meson states ($n\gg1$), the spectrum admits a systematic semiclassical (WKB) expansion: $$\lambda_n = \frac{n}{2} + \frac{\alpha_1+\alpha_2}{\pi^2}\log(4\pi e^{\gamma_E} n) + \mathcal{O}\left(\frac{\log^2 n}{n}\right) + \cdots$$ so that

$M_n^2 \sim \pi g^2 n + \mathcal{O}(\log n)$

Two distinct, nearly degenerate Regge-like trajectories for even and odd $n$ appear, reflecting the alternating term $(-1)^n/n^2$. Subleading corrections can be written explicitly in terms of the integrable system data (Artemev et al., 16 Apr 2025, Litvinov et al., 17 Sep 2024).

4. Asymptotic Regimes: Chiral, Heavy-Heavy, and Heavy-Light Limits

The meson spectrum exhibits distinct asymptotics depending on the constituent quark masses:

Chiral limit ($m_{1,2} \to 0$):

The lowest eigenvalue $\lambda_0 \to 0$, producing a massless “pion.” Analytically, for $m_i\ll g$, the pion mass is

$M_0^2 \approx \frac{g^2}{\sqrt{3}} (m_1 + m_2)$

in precise agreement with the Gell-Mann–Oakes–Renner relation. Even spectral sums diverge, reflecting the enhanced IR sensitivity (Artemev et al., 16 Apr 2025, Litvinov et al., 17 Sep 2024).

Heavy-heavy limit ($m_1 = m_2 \gg g\sqrt{N_c}$):

The spectrum is nonrelativistic at leading order: $M_n^2 = 4m^2 + \xi_n g^{4/3} m^{2/3} + \cdots$ with $\xi_n$ given by quantization in an Airy-type potential well. This regime reproduces the “Landau-level” structure found in the nonrelativistic Schrödinger equation (Artemev et al., 16 Apr 2025).

Heavy-light limit ($m_2\gg m_1$):

The lowest states satisfy

$$M_n^2 = M^2 + \sqrt{2}\pi g M \sqrt{n + 5/8} + \cdots$$

where $M=m_2$ and $n$ labels excitation above the threshold. Analytic quantization conditions account for both quark masses, recovering all known heavy-quark effective-theory limits (Artemev et al., 16 Apr 2025).

5. $1/N_c$ Corrections and Threshold Phenomena

At finite but large $N_c$, the $1/N_c$ corrections to the ’t Hooft spectrum arise from weak 3-meson interaction vertices. The leading correction to the meson mass $M_L$ can be written as

$$\Delta M_L^2 = \frac{1}{N_c}\sum_{R_1, R_2} \Sigma_{L|R_1R_2}$$

where $\Sigma_{L|R_1R_2}$ involves overlap integrals of the $N_c\to\infty$ wavefunctions and a denominator sensitive to kinematic thresholds (Kochergin, 7 May 2024).

Key features:

• For generic mass spectra, the $1/N_c$ shifts remain finite as $m_q\to0$, with pion loops dominating in the chiral limit.
• In the one-flavor case, parity constraints suppress the leading correction; the first nonzero term scales as $m^2\log m/N_c$.
• Heavy–light mesons’ mass shift is $O(M/N_c)$ and dominated by “$B_0\pi_0$” mixing near the chiral limit.
• When a decay $L\to R_1R_2$ is allowed by symmetry and $M_L=M_{R_1}+M_{R_2}$, the denominators in perturbation theory develop a double zero, and degenerate perturbation theory applies. The ensuing bound state is split below threshold by an amount $\propto N_c^{-2/3}$, with universal 2/3 one-meson, 1/3 two-meson mixing (Kochergin, 7 May 2024).

These analytic $1/N_c$ corrections have been confirmed by numerical DLCQ and LCT calculations, and they clarify discrepancies with bosonization approaches, especially for pion dynamics.

6. Robustness under Model Extensions and Regulators

Analyses show that the inclusion of new Lorentz-invariant terms or “VSR-like” fractional null-vector structures, as well as gluon mass infrared regulators, have only trivial effects on the meson spectrum. Specifically, such terms either shift quark masses or regulate the IR divergence, but leave the kernel and spectral structure of the ’t Hooft equation—and thus all observable meson masses—unchanged in the IR limit (Alfaro et al., 2019).

Modification	Spectrum effect	Reference
VSR-like terms	Pure quark mass shift only	(Alfaro et al., 2019)
Gluon mass regulator	Infrared regularization; recovers ’t Hooft spectrum as $m_g\to 0$	(Alfaro et al., 2019)

This invariance underscores the fundamental nature of the confinement and spectral structure encoded in the original light-cone Hamiltonian.

7. Significance and Physical Interpretation

The QCD$_2$ meson spectrum serves as the canonical realization of confinement-induced spectral discreteness, Regge trajectories, and integrability in field theory. Its analytic tractability enables full control over spectral sums, operator expectation values, and the structure of matrix elements—features harder to realize in higher dimensions.

Key physical outcomes include:

• Emergence of a massless chiral boson at zero quark mass.
• Explicit realization of large-$n$ spectral asymptotics and quasi-linear Regge trajectories.
• Analytical control of all special mass limits (chiral, heavy-heavy, heavy-light) and precise matching to operator product expansions.
• $1/N_c$ corrections yielding observable threshold effects (including $N_c^{-2/3}$ splittings) and resolving discrepancies with bosonization.
• Spectral sum rules connecting confinement and current–operator matrix elements directly to the microscopic spectrum.

Recent nonperturbative advancements (e.g., by Artemev, Litvinov, Meshcheriakov, and others) rigorously unify the analytic, numerical, and integrable structures for arbitrary quark masses, solidifying QCD$_2$ as the benchmark for exactly solvable strongly coupled theories (Artemev et al., 16 Apr 2025, Litvinov et al., 17 Sep 2024, Kochergin, 7 May 2024, Alfaro et al., 2019).