Meson states in `t Hooft model: Hamiltonian approach

Abstract

We point out that the masses of the highly excited bound quark-antiquark states in QCD${}_2$ in the infinite $N$ limit may be determined in the framework of a simple quantum-mechanical model with the potential $V(x)=\sigma |x|$. In the ultrarelativistic case, the masses follow the pattern $${\mu }_n^2=\frac{g^{2}N}{2\pi }n,$$ which coincides with the law derived by `t Hooft by solving the Bethe--Salpeter equation. In the nonrelativistic case, the energy levels follow the asymptotics ${ϵ}_n={\mu }_n-2m=C{n}^{2/3}$, where the constant $C$ can be determined by solving the `t Hooft equation or alternatively the nonrelativistic Schrödinger equation.

Comments: 4 pages, no figures. A comment, emphasizing the fact that the finite state widths arising in the semiclassical analysis of the Hamiltonian problem are artifacts of the approximation, is added. The exact Hamiltonian is Hermitian, and its eigenvalues are therefore purely real