QCD_2: 't Hooft equation vs. relativistic Schroedinger equation

Abstract

We discuss 2-dimensional $QCD$ with a fundamental quark of mass $m$ in the 't Hooft limit ($N \to \infty, \, g^2 \to 0, \, g^2N =$ const). The spectrum of this theory includes a chain of meson states. The masses of these states can be evaluated by solving the proper integral Bethe-Salpeter equation. An alternative simple procedure consists in solving the Schrödinger equation for the relativistic Hamiltonian $$ \hat H\ =\ 2\sqrt{\hat p^2 + m^2} + σ|x| $$ with the string tension $σ= g^2N/4$. We compare the results obtained by these two methods. In the leading semiclassical approximation, they coincide. We show, however, that the preasymptotic WKB corrections to the masses calculated in these two ways are different. For the ground state, they play an essential role. But already for $n=3$, the relative difference between the two predictions does not exceed 2.5\% and rapidly falls off as $n$ increases.