Subtleties in QCD theory in Two Dimensions

TL;DR

This paper explores subtle aspects of two-dimensional QCD, revealing how different formulations affect the existence of massive states and their physical relevance, especially when coupled to quarks and compared to the abelian case.

Contribution

It uncovers the subtleties in the formulation of 2D Yang-Mills theory and how they influence the physical spectrum, including the masslessness or massiveness of states.

Findings

Physical massive color singlet states may exist in certain formulations.

Coupling to quarks can lead to vanishing coupling constants, making states massless.

In the abelian case, known results are recovered without additional renormalization.

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 $e_c\sqrt {N_F \over 2\pi}$.