The massive N-flavor Schwinger model is analyzed by the bosonization method. The problem is reduced to the quantum mechanics of N degrees of freedom in which the potential needs to be self-consistently determined by its ground-state wave function and spectrum with given values of the θ parameter, fermion masses, and temperature. Boson masses and fermion chiral condensates are evaluated. In the N=1 model the anomalous behavior is found at θ ∼ π and m/µ ∼ 0.4. In the N=3 model an asymmetry in fermion masses (m 1 < m 2 ≪ m 3 ) removes the singularity at θ = π and T = 0. The chiral condensates at θ = 0 are insensitive to the asymmetry in fermion masses, but are significantly sensitive at θ = π. The resultant picture is similar to that obtained in QCD by the chiral Lagrangian method. 2Bosonized Schwinger model

We evaluate Polyakov loops and string tension in two-dimensional QED with both massless and massive N -flavor fermions at zero and finite temperature. External charges, or external electric fields, induce phases in fermion masses and shift the value of the vacuum angle parameter θ, which in turn alters the chiral condensate. In particular, in the presence of two sources of opposite charges, q and −q, the shift in θ is 2π(q/e) independent of N . The string tension has a cusp singularity at θ = ±π for N ≥ 2 and is proportional to m 2N/(N +1) at T = 0.Two-dimensional QED, the Schwinger model, with massive N -flavor fermions resembles fourdimensional QCD in various aspects, including confinement, chiral condensates, and θ vacua [1]- [11]. Much progress has been made recently in evaluating chiral condensates and string tension in the massive theory [12]-[16]. In this paper we shall show that the three phenomena, confinement, chiral condensates, and θ vacua, are intimately related to each other. In particular, the string tension in the confining potential is determined by the θ dependence of chiral condensates ψ − ψ .The behavior of the model is distinctively different, depending on whether N = 1 (one-flavor) or N ≥ 2 (multi-flavor), and on whether fermions are massless or massive. The massless (m = 0) theory is exactly solvable. ψ − ψ θ = 0 for N = 1, but ψ − ψ θ = 0 for N ≥ 2. [17,18] In either cases the string tension between two external sources of opposite charge vanishes [4,8,12]. In the massive (m = 0) theory ψ − ψ θ is proportional to m (N −1)/(N +1) cos 2N/(N +1) (θ/N ) at , 13]. For N ≥ 2 the dependence on m is non-analytic. It also has a cusp singularity at θ = ±π. A perturbation theory in fermion masses is not valid at low temperature.

We evaluate the chiral condensate and Polyakov loop in two-dimensional QED with a fermion of an arbitrary mass (m). We find discontinuous m dependence in the chiral condensate and anomalous temperature dependence in Polyakov loops when the vacuum angle θ∼π and m=O(e). These nonperturbative phenomena are due to the bifurcation process in the solutions to the vacuum eigenvalue equation. 1