Large-Nc Scalar QCD₂ Overview

• Large-Nc scalar QCD₂ is a two-dimensional SU(Nc) gauge theory with a complex scalar field, exhibiting confinement and a discrete meson spectrum.
• The model uses a modified Bethe–Salpeter equation and integrability-based techniques to achieve precise analytic and numerical solutions for mesonic states.
• Recent studies confirm accurate WKB eigenvalue approximations and reveal spectral singularities through analytic continuation, deepening nonperturbative insights.

Large-$N_c$ scalar QCD$_2$ refers to two-dimensional SU($N_c$) Yang–Mills theory coupled to a fundamental complex scalar field (“scalar quark”) and analyzed in the ’t Hooft large-$N_c$ limit. This model exhibits a confining, asymptotically free gauge sector and supports a discrete spectrum of color-singlet mesonic bound states. Its analytic tractability arises from the reduction of dynamical degrees of freedom in 1+1 dimensions and the simplifications induced by large-$N_c$ factorization. The scalar QCD$_2$ Bethe–Salpeter equation manifests structural parallels to the ’t Hooft integral equation for mesons in fermionic QCD$_2$, but features unique mass renormalization and integral kernel properties. Recent developments include nonperturbative analytic control via integrability-based methods and explorations of its connection to 2D conformal field theory.

1. Foundations and Action

The gauge and matter content consists of an $SU(N_c)$ gauge field $A_\mu^a$ in the adjoint representation and a complex scalar field $\varphi^a$ in the fundamental. The action in Minkowski space is \begin{equation} \mathcal{L} = -\frac{1}{4} F_{\mu\nu}^a F^{a\,\mu\nu} + (D_\mu\varphi)^{\dagger}_a(D^\mu\varphi)^a - m^2 \varphi^{\dagger}_a\varphi^a, \end{equation} where

$$F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c, \quad (D_\mu\varphi)^a = \partial_\mu\varphi^a + g f^{abc} A_\mu^b \varphi^c.$$

In the ’t Hooft (planar) limit, $N_c\to\infty$ with $\lambda=g^2 N_c$ held fixed and $g^2\sim1/N_c$.

Gauge fixing can be done via the axial gauge $A_1^a=0$ or, equivalently on the light front, $A_+^a=0$. The model is linearly confining and shows asymptotic freedom in the infrared.

2. Bound-State Dynamics and Bethe–Salpeter Equation

The color-singlet mesonic bound states are described, at leading order in $1/N_c$, by an integral Bethe–Salpeter equation for the light-cone wavefunction $\Phi(x)$, where $x\in[0,1]$ is the fraction of light-cone momentum carried by the scalar quark: \begin{equation} 2\pi^2\lambda\,\Phi(x) = \left(\frac{\alpha}{x} + \frac{\alpha}{1-x}\right) \Phi(x)

• \fint_0^1 dy\, \frac{(x+y)(2-x-y)}{4x(1-x)} \frac{\Phi(y)}{(x-y)^2}, \end{equation} with principal value ($\mathrm{P}$) prescription in the integral. Here, $\alpha = \frac{\pi m^2}{g^2} - 1$, where $m$ is the renormalized scalar mass, and the eigenvalue $\lambda_n$ determines the meson mass $M_n^2 = 2\pi g^2\lambda_n$ (Meshcheriakov, 23 Sep 2025).

This equation generalizes the ’t Hooft equation for fermionic QCD$_2$: \begin{equation} m_n^2 \psi_n(x) = \frac{m^2}{x(1-x)} \psi_n(x) - \frac{g^2}{\pi} {\rm P}!!\int_0^1 dy\, \frac{\psi_n(y)}{(x-y)^2}. \end{equation} Distinctive features of the scalar case include the appearance of a nontrivial prefactor in the confining kernel and the need for explicit mass renormalization due to logarithmic divergences (Ji et al., 2018).

3. Integrability-Based Analytic Solution Framework

The model admits a nonperturbative analytic solution for its meson spectrum based on a method inspired by Fateev–Lukyanov–Zamolodchikov (FLZ) integrability. The key steps are as follows (Meshcheriakov, 23 Sep 2025):

• The wavefunction is mapped by a Fourier (rapidity) transform,

$$\phi(x) = \sqrt{x(1-x)}\,\Phi(x),\quad \theta = \tfrac12\ln(x/(1-x)),\quad \Psi(\nu) = \int d\theta\, e^{-i\nu\theta} \phi(\theta).$$

• The “Q-function”,

$$Q(\nu) = \left(\frac{2\alpha}{\pi}\cosh\frac{\pi\nu}{2} + \nu\sinh\frac{\pi\nu}{2}\right) \Psi(\nu)$$

satisfies a finite-difference TQ equation encoding the mesonic spectrum and boundary conditions.

• Spectral zeta functions and spectral determinants constructed from the $\lambda_n$ admit exact relations (including trace formulae and log-derivative identities) to the analytic structure of $Q(\nu)$.

This machinery yields:

• Exact spectral sums, $$\mathcal{G}_\pm^{(s)} = \sum_{n=0}^\infty [1/\lambda_{(\pm)n}^s - \delta_{s,1}/(n+1)]$$
• WKB expansion for large-$n$ spectrum:

$$\lambda_n = \frac12 \mathfrak{n} + \frac{\alpha}{\pi^2} \ln(\pi e^{\gamma_E} \mathfrak{n}) + \frac{1}{\mathfrak{n}}\left[\frac{\alpha^2}{\pi^4}(2\ln \mathfrak{n} - 1)\mp \frac{1}{4\pi^2}\right] + \mathcal{O}(\mathfrak{n}^{-2}),$$

with $$\mathfrak{n} = n + \frac14 - \frac{\alpha^2}{2\pi^2}\,i_2(\alpha)$$.

This approach also uncovers additional relations between parity sectors, quantization conditions, and the analytic properties of spectral determinants.

4. Light-Front Quantization and Gauge Structure

Large-$N_c$ scalar QCD$_2$ is amenable to Hamiltonian, path integral, and BRST quantization on the light front (Kulshreshtha et al., 2015):

• Dynamical variables in light-front coordinates $x^\pm=(x^0\pm x^1)/\sqrt2$ reduce the gauge field to nonpropagating constraints.
• The primary and secondary constraints are first-class, generating residual $SU(N_c)$ gauge invariance and enabling consistent gauge fixing (e.g., light-front gauge $A_+^a=0$).
• Path integral gauge fixing introduces the expected Faddeev–Popov determinant $\mathrm{det}\,\partial_-$ and BRST structure via ghost, antighost, and Nakanishi–Lautrup fields.
• Spontaneous symmetry breaking in a Higgs-type extension can be treated in both unitary and light-front ’t Hooft gauges, yielding explicit gauge boson and Higgs masses

$m_A^2 = g^2 v^2, \qquad m_\sigma^2 = 2\lambda v^2.$

• The light-cone bound state equation underpins the analytic and numerical determination of the meson spectrum.

5. Parton Structure and Quasi-Parton Distributions

Mesonic distributions for scalar QCD$_2$ can be studied via quasi-parton distribution functions (quasi-PDFs), defined as

$$Q(x,P) = \int \frac{dz}{2\pi}\,e^{-ixPz} \langle P | \varphi^a(0,z) \varphi^a(0,0) | P \rangle,$$

with the Wilson line trivial in axial gauge (Ji et al., 2018). In the large-$N_c$ expansion: $$Q(x,P) = Q^{(0)}(x,P) + \frac{1}{N_c} Q^{(1)}(x,P) + O(1/N_c^2),$$ where $Q^{(0)}(x,P) = |\psi_n(x)|^2$ reproduces the light-cone PDF at leading order. The $1/N_c$ correction encodes backward-moving pair and mass renormalization effects, expressible in terms of the ’t Hooft operator basis. In the infinite-momentum limit, $Q(x,P)$ analytically reduces to the true parton distribution, without additional ultraviolet renormalization. Endpoint subtleties arise for finite $P$ due to backward-pair contributions, producing “spikes” at $x\to0,1$ that vanish only as $P\to\infty$.

6. Analytic Continuation and Spectral Singularities

Analytic continuation in the complex $\alpha$-plane reveals an infinite sequence of singularities where specific meson states become massless (Meshcheriakov, 23 Sep 2025):

• Odd-parity meson masses vanish at branch points $\alpha_k^*$, determined by solutions to $\tfrac{2\alpha}{\pi}+\nu\tanh(\tfrac{\pi\nu}{2})=0$, manifesting square-root behavior:

$$\lambda^{(2k+1)}(\alpha) \propto \sqrt{\alpha-\alpha_k^*},\quad \alpha \to \alpha_k^*.$$

• Even-parity mesons become massless at points $\widetilde\alpha_k$ where $1+v(\widetilde\alpha)=0$, with simple zeroes but no square-root branching.
• In the doubled Ising field theory, $\alpha_1^*$ coincides with the Yang–Lee edge singularity ($\mathcal{M}_{2,5}$ minimal CFT). By analogy, these points in scalar QCD$_2$ may correspond to IR fixed points described by nonunitary minimal models or coset CFTs.

A full understanding of the nonunitary CFT correspondence and the fate of these singularities beyond planar order remains an open question requiring inclusion of $1/N_c$ corrections and multiparticle thresholds.

7. Limiting Regimes and Numerical Validation

Two key asymptotic regimes for scalar QCD$_2$ are analytically matched to expectations and numerical results (Meshcheriakov, 23 Sep 2025):

• Near-critical limit ($\alpha\to0$, $m\to g/\sqrt\pi$): The lightest meson is massive, with no Goldstone mode, and the spectrum matches six-breather ratios from integrable 2D models.
• Heavy-quark regime ($\alpha\to\infty$, $m\gg g$): The nonrelativistic reduction yields a Hamiltonian with linear potential, producing meson spectra in terms of Airy function zeros. Quantitative agreement is observed between analytic results and numerical solutions (Chebyshev and discretized Fourier methods), with relative spectral sum errors $\lesssim10^{-5}$–$10^{-3}$ and WKB eigenvalues accurate to $10^{-6}$ for moderate $\alpha$. The ground state at $\alpha=0$ is $M_0^2\simeq2.3937\,g^2/\pi$ (analytic), in excellent agreement with numerical studies of adjoint QCD$_2$.

Table: Key Formulas in Large-$N_c$ Scalar QCD$_2$

Quantity	Formula (in LaTeX)	Context
Planar limit ’t Hooft coupling	$\lambda = g^2 N_c$	$N_c\to\infty$, $g^2\sim 1/N_c$
Meson Bethe–Salpeter equation	See section 2 above	Spectrum of color singlets
Leading meson spectrum (WKB, large $n$)	$\lambda_n = \frac12 n + \cdots$	Semiclassical limit
Meson mass squared	$M_n^2 = 2\pi g^2 \lambda_n$	Relation to integral eigenvalue
Quasi-PDF leading order	$Q^{(0)}(x,P) = |\psi_n(x)|^2$	Partonic interpretation

This model thus offers an exactly solvable yet highly nontrivial setting for the paper of confinement, spectral theory, mass gap, and nonperturbative field theory in two dimensions, with contemporary analytical and numerical methods revealing deep connections to integrability and conformal field theory (Meshcheriakov, 23 Sep 2025, Ji et al., 2018, Kulshreshtha et al., 2015).