The low-energy effective dynamics of two-dimensional gauge theories

Abstract

Path integral bosonization techniques are used to derive a low-energy effective action, in the limit e c/ m → ∞, for a general theory of Dirac fermions interacting with gauge fields ( e c is the gauge coupling constant and m is a typical fermion mass). Specializing to specific gauge and global symmetry groups, low-energy effective actions are obtained for the many-flavor massive Schwinger model (QED 2) and for single and many flavor QCD 2. For many flavor QCD 2, the low-energy theory is a non-linear σ-model with Wess-Zumino and soliton stabilizing terms; this results bears a striking resemblance to the expected low-energy effective theory of four-dimensional QCD. In Coleman's low-energy effective theory for two-flavor QED 2 with non-equal fermion masses, the low-lying states of the effective theory remain in degenerate isospin multiplets. In our analysis, we do not encounter this paradox: the degeneracy is lifted explicitly by an isospin-breaking term. For m → 0, the large- N behavior of the QCD 2 effective theory is shown to contain zero mass mesons in leading order, consistent with 't Hooft's results. Using the 't Hooft anomaly conditions and non-abelian bosonization, an explanation for the restoration of chiral symmetry is offered in both this case and in QED 2.