Screening vs. Confinement in 1+1 Dimensions

TL;DR

The paper investigates screening versus confinement in 1+1D gauge theories, showing that heavy probes can be screened by dynamical massless fermions even when the probe representation is smaller than that of the dynamical fields; confinement re-emerges when the fermions acquire mass, with a small-mass string tension given by $\sigma_q = \Sigma m (1-\cos 2\pi q)$ for fractional charges. By combining bosonization, effective actions, Wilson-loop analyses, and loop equations, it connects the abelian Schwinger model to non-Abelian $QCD_2$ with adjoint fermions and illuminates the role of topological sectors and zero modes. The work draws parallels with higher-dimensional QCD and SUSY gauge theories, and suggests scenarios where screening might occur in more complex theories, highlighting the delicate interplay between representation content, mass, and topology in low dimensions.

Abstract

We show that, in 1+1 dimensional gauge theories, a heavy probe charge is screened by dynamical massless fermions both in the case when the source and the dynamical fermions belong to the same representation of the gauge group and, unexpectedly, in the case when the representation of the probe charge is smaller than the representation of the massless fermions. Thus, a fractionally charged heavy probe is screened by dynamical fermions of integer charge in the massless Schwinger model, and a colored probe in the fundamental representation is screened in $QCD_2$ with adjoint massless Majorana fermions. The screening disappears and confinement is restored as soon as the dynamical fermions are given a non-zero mass. For small masses, the string tension is given by the product of the light fermion mass and the fermion condensate with a known numerical coefficient. Parallels with 3+1 dimensional $QCD$ and supersymmetric gauge theories are discussed.