Subtleties in QCD theory in Two Dimensions

Abstract

It is shown that in a formulation of Yang-Mills theory in two dimensions in terms of $A=if^{-1}\pa f$, $\bar A=i\bar f\bpa\bar f^{-1}$ with $f(z,\bar z)$, $\bar f(z,\bar z)\in[SU(N_C)]^c$ the complexification of $SU(N_C)$ , reveals certain subtleties. ``Physical" massive color singlet states seem to exist. When coupled to $N_F$ quarks the coupling constant is renormalized in such a way that it vanishes for the pure Yang- Mills case. This renders the above states massless and unphysical. In the abelian case, on the other hand, the known results of the Schwinger model are reproduced with no need of such a renormalization. The massless $QCD_2$ theory is analyzed in similar terms and peculiar massive states appear, with a mass of $e_c\sqrt {N_F \over 2\pi}$.