The meson as a bound system in 2D quantum chromodynamics

Abstract

We investigate the Bethe–Salpeter-like amplitude, which involves a system comprising a quark and an anti-quark combined by gauge fields, to satisfy gauge invariance in two dimensions. We derive the equation of motion for this system and construct a series of singular integral-differential equations. By solving these equations, we obtain an algebraic equation that determines the mass spectra for the ($n+1$)th non-zero eigenvalue (where $n = 0$ or a positive integer). In this algebraic equation, the solution for large values of $n$ is written in the same form as G. 't Hooft's W-K-B solution, and the zero mass solution arises under a boundary condition equivalent to the boundary condition that 't Hooft derived. Thus, we show convincingly that zero mass is actually a physical mass.