$1/N$ Corrections in $\text{QCD}_2$: Small Mass Limit and Threshold States

Abstract

In this paper we investigate $1/N$ corrections to mesonic spectrum in $1+1$-dimensional Quantum Chromodynamics ($\text{QCD}_2$) with fundamental quarks using effective Hamiltonian method. We express the corrections in terms of 't Hooft equation solutions. First, we consider 2-flavor model with a heavy and a light quark. We show that, in contrast to some claims in earlier literature, the $1/N$ correction to the mass of the heavy-light meson remains finite when the light quark mass is taken to zero. Nevertheless, the corrections become significantly larger in this limit; we attribute this to the presence of massless modes in the spectrum. We also study the corrections to the lightest meson mass in 1-flavor model and show that they are consistent with recent numerical data, but not with the prediction coming from bosonization. Then we study low energy effective theory for 2 flavors. We show that the 3-meson interaction vertex correctly reproduces Wess-Zumino-Witten (WZW) coupling when both quarks become massless. This coupling does not change even if one of the quarks is massive. We employ Discretized Light Cone Quatization (DLCQ) to check the continuum results and show that the improved version can be used for small quark mass. Finally, we study the states associated with $1\to 2$ meson thresholds. Using degenerate perturbation theory, we show that when the decay is allowed by parity, the infinite $N$ theory has near-threshold bound states that mix one- and two-meson parts. They are $1/3$ two-meson and $2/3$ one-meson and the corrections to their masses have unusual scaling $\sim 1/N^{2/3}$.