Exact Symmetries and Threshold States in Two-Dimensional Models for QCD

TL;DR

The paper studies a 1+1D SU$(N)$ gauge theory with an adjoint Majorana fermion and $N_f$ fundamental quarks using Discretized Light-Cone Quantization (DLCQ) in the large-$N$ limit. It uncovers an exact $rak{osp}(1|4)$ symmetry for massless adjoints, and shows that many meson and gluinoball states are threshold-bound or related by SU$(N)$ current algebra, yielding extensive degeneracies. The analysis connects DLCQ spectra to Kac-Moody current blocks and reveals that, with massless adjoints, the meson sector exhibits a continuum above a calculable threshold, consistent with vanishing fundamental string tension; introducing quark masses preserves some degeneracies and demonstrates screening effects. Collectively, the work provides a quantitative, non-perturbative window into confinement, screening, and the rich bound-state structure of adjoint QCD$_2$ with fundamental flavors, with implications for higher-dimensional gauge dynamics.

Abstract

Two-dimensional SU$(N)$ gauge theory coupled to a Majorana fermion in the adjoint representation is a nice toy model for higher-dimensional gauge dynamics. It possesses a multitude of "gluinoball" bound states whose spectrum has been studied using numerical diagonalizations of the light-cone Hamiltonian. We extend this model by coupling it to $N_f$ flavors of fundamental Dirac fermions (quarks). The extended model also contains meson-like bound states, both bosonic and fermionic, which in the large-$N$ limit decouple from the gluinoballs. We study the large-$N$ meson spectrum using the Discretized Light-Cone Quantization (DLCQ). When all the fermions are massless, we exhibit an exact $\mathfrak{osp}(1|4)$ symmetry algebra that leads to an infinite number of degeneracies in the DLCQ approach. More generally, we show that many single-trace states in the theory are threshold bound states that are degenerate with multi-trace states. These exact degeneracies can be explained using the Kac-Moody algebra of the SU$(N)$ current. We also present strong numerical evidence that additional threshold states appear in the continuum limit. Finally, we make the quarks massive while keeping the adjoint fermion massless. In this case too, we observe some exact degeneracies that show that the spectrum of mesons becomes continuous above a certain threshold. This demonstrates quantitatively that the fundamental string tension vanishes in the massless adjoint QCD$_2$.

Figures (11)

• Figure 1: The masses of gluinoball states in DLCQ with $m_\text{adj}=0$ as a function of $1/K$, up to $K = 41$. The spectrum was first described in Bhanot:1993xpGross:1995bp up to $K=25$. The orange points are single-trace gluinoball states that are exactly degenerate with multi-trace states.
• Figure 2: The masses of gluinoball states in DLCQ as a function of $1/K$, up to $K = 41$, with the adjoint mass parameter $y_\text{adj} = \frac{m^2_\text{adj} \pi}{g^2 N} = 0.1$.
• Figure 3: The eigenvalues $P^-$ up to $K = 35$. States are colored according to their degeneracies, with the darkest states being singlets. Along the horizontal trajectories, the degeneracies increase in steps of 1 moving from right to left, except for the series of massless states, which have degeneracy $\lfloor K/2\rfloor$.
• Figure 4: The masses of the states in Figure \ref{['fig:pminus_degeneracies']}, along with the gluinoball states described in Section \ref{['sec:glueball_degeneracies']}, and the trajectories along which they approach their $K\to\infty$ values.
• Figure 5: The squared masses of single-trace meson states in theory ${\cal T}$ (the theory with an adjoint and a fundamental fermion) with $y_\text{adj} = 0$ and $y_\text{fund} = 1$. The blue points are the states which are degenerate with multi-string states in the theory $\mathcal{T}'$ defined in Section \ref{['sec:MORE']}. The orange points are the states which additionally are degenerate with a multi-trace state formed from a meson and one or more gluinoballs in theory ${\cal T}$. The dashed lines show the threshold at $M^2\approx 8 \frac{g^2 N}{\pi}$ above which the extrapolated spectrum is continuous. There is a bosonic $\mathbb{Z}_2$-odd state shown with green dots that lies below this threshold. Its extrapolated squared mass is $M^2 \approx 6.6\frac{g^2 N}{\pi}$.