Analytical results for large-$N_c$ scalar QCD$_2$ (2509.19229v1)

Abstract: We study large-$N_c$ scalar QCD$_2$, 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 $m\to g/\sqrtπ$ and the heavy-quark regime $m\gg g$, 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.

• The paper introduces an analytic framework that extends the FLZ method to scalar QCD2, deriving meson spectra via a Baxter TQ-type formulation.
• It achieves a systematic large-n WKB expansion validated by high-precision numerics, bridging both near-critical and heavy-quark regimes.
• The analysis maps complex singularities in the meson masses, linking branch cuts to potential conformal criticality and exotic spectroscopy.

Analytical Structure and Meson Spectrum in Large-$N_c$ Scalar QCD$_2$

Introduction and Model Definition

The paper develops a comprehensive analytic framework for large-$N_c$ scalar QCD$_2$, a $1+1$D confining gauge theory with scalar quarks in the fundamental representation, focusing on the nonperturbative structure of its meson spectrum. The core object is the Bethe-Salpeter equation for color-singlet mesons, structurally similar but not identical to the fermionic 't Hooft model. The authors further elucidate its structural equivalence to the two-particle sector of interchain mesons in coupled Ising models, broadening the implications of their analysis.

Integrability and the FLZ Method

A salient technical achievement of the work is the adaptation and extension of the Fateev-Lukyanov-Zamolodchikov (FLZ) method—originally devised for the 't Hooft model—to the scalar case. This approach reformulates the integral Bethe-Salpeter equation as a Baxter TQ-type difference equation, leveraging integrability to encode spectral data in analytic objects related to transfer matrix theory in solvable models. Central to this formulation are the $Q$-function and spectral determinants, whose analytic properties yield both the nonperturbative spectrum and spectral sums.

Figure 1: TQ derivation—analytic continuation of the Baxter TQ equation in $\nu$ reveals additional half-residue contributions, reflecting the passage of poles through the principal value contour.

Spectral Sums and Systematic WKB Expansion

Through intricate manipulation of the TQ framework, the authors derive explicit formulas for spectral sums and achieve a systematic large-$n$ (highly excited) WKB expansion for the meson eigenvalues $\lambda_n$, valid at arbitrary quark mass parameter $\alpha$. This addresses longstanding gaps in the analytic understanding of scalar QCD$_2$, where previously only the leading semiclassical behavior was accessible. These results are supported by high-precision numerics using both direct lattice discretization and Chebyshev polynomial expansion.

Figure 2: Numerical meson wavefunctions $\Psi_n(\nu)$ at various $n$ and $\alpha$ from Method I, demonstrating the broadening and localization of states as $\alpha$ varies—essential for validating analytic results.

Analytic Continuation and Singularities in the Mass Parameter

A compelling aspect of the analysis is the exploration of the analytic structure of the meson masses $\lambda_n(\alpha)$ and spectral sums under continuation of $\alpha$ into the complex plane. The authors identify two infinite families of singularities—pole collision ($\alpha^*_k$) and zero-induced ($\widetilde{\alpha}_k$) points—on secondary Riemann sheets, where individual meson masses vanish and the spectral sums develop branch cuts or poles, signaling massless excitations and possible conformal criticality.

Figure 3: Trajectories of the first two pairs of poles in $\Psi(\nu)$ during analytic continuation in $\alpha$; colored markers indicate where square-root singularities appear—reflecting nontrivial analytic structure in the meson spectrum.

Figure 4: Mapping of critical points for $\lambda_n$ in the complex $\alpha$-plane, illustrating the loci where mesons become massless (blue: branch points $\alpha^*_k$, red: simple zeros $\widetilde{\alpha}_k$).

Key Numerical and Analytic Results

• Consistency across regimes: Analytic formulas for the spectrum reproduce both the near-critical regime ($m \to g/\sqrt{\pi}$, $\alpha\to 0$) and the heavy-quark ($m \gg g$, $\alpha\to\infty$) limits. In particular, the ground state at $\alpha=0$ is strictly massive, consistent with prior rigorous results.
• WKB expansion: The meson spectra $\lambda_n$ admit a systematic nonperturbative WKB expansion in inverse powers of a shifted quantum number $\mathfrak{n}$, including $\log{\mathfrak{n}}$ and mass parameter corrections. For excited states, this yields sub-percent agreement with numerics.
• Spectral determinants: Relations between physical and non-physical spectral determinants, and their connection to $Q$-functions, are derived. The ratio $d_-/d_+$ captures the analytic structure at singularities and matches branch/pole behavior in the complex $\alpha$-plane.
• Critical points: At each $\alpha^*_k$, an odd meson mass vanishes as $\lambda\sim(\alpha-\alpha^*_k)^{1/2}$; at each $\widetilde{\alpha}_k$, an even meson mass vanishes linearly. These points generate square-root branch cuts or simple zeros, respectively, in the $\alpha$-plane.

Theoretical and Practical Implications

The analytic construction provides an unprecedented level of control over large-$N_c$ scalar QCD$_2$:

• Comparison with the 't Hooft model: Mesons in the scalar theory are strictly lighter for identical masses, confirming and extending previous findings.
• Exotic spectroscopy: Since the same Bethe-Salpeter equation governs exotic states in effective QCD diquark models and interchain mesons in spin systems, the results here apply across disparate physical arenas.
• Complex singularity analysis: The identification of sequences of critical points connects nonperturbative bound state physics to conformal criticality, with certain singularities conjectured to correspond to non-unitary CFTs—paralleling the known role of the Yang-Lee edge in the Ising model.
• Generalizations: The integrability-based techniques—especially the handling of nontrivial analytic structure through $Q$-functions and spectral determinants—are likely adaptable to mixed-statistics QCD$_2$, adjoint models, and the Schwinger model.

Future Directions

Key open directions highlighted by the authors:

• Multi-flavor extension: Full analysis of two-flavor (or more) scalar QCD$_2$, requiring the explicit construction of solutions to the generalized TQ-equation with controlled analytic properties.
• Finite-$N_c$ and multiparticle corrections: Tackling $1/N_c$ effects and going beyond the two-particle approximation, crucial for understanding the physical origin and universality of branch point singularities and their exponents.
• Connection to CFT and universality: Systematically establishing the correspondence between analytic singularities in the mass parameter and specific non-unitary or irrational CFTs remains an enticing but unresolved challenge.

Conclusion

By leveraging advanced integrability methods, this work fills a longstanding gap in the analytic understanding of scalar QCD$_2$ meson spectra at large $N_c$—including a precise nonperturbative description across all mass regimes, a detailed mapping of spectral singularities, and a bridge to related problems in spin systems and exotic spectroscopy. The rigorous agreement between analytic predictions and numerics validates the extension of the FLZ methodology to a new class of confining field theories, with broad implications for both the paper of nonperturbative QCD and the analytical toolbox for integrable models.