The Massive Multi-flavor Schwinger Model

QED with N species of massive fermions on a circle of circumference L is analyzed by bosonization. The problem is reduced to the quantum mechanics of the 2N fermionic and one gauge field zero modes on the circle, with nontrivial interactions induced by the chiral anomaly and fermions masses. The solution is given for N = 2 and fermion masses (m) much smaller than the mass of the U(1) boson with mass μ = √ 2e2/π when all fermions satisfy the same boundary conditions. We show that the two limits m→ 0 and L→ ∞ fail to commute and that the behavior of the theory critically depends on the value of mL| cos 12θ| where θ is the vacuum angle parameter. When the volume is large μL≫ 1, the fermion condensate 〈ψ−ψ 〉 is −(e4γmμ2 cos4 12θ/4π) or −2eγmμL cos2 12θ/π for mL(μL)1/2| cos 12θ| ≫ 1 or ≪ 1, respectively. Its correlation function decays algebraically with a critical exponent η = 1 when m cos 12θ = 0. The Schwinger model, QED in two dimensions, with N ≥ 2 species of fermions is distinctly different from that with one flavor [1]−[7]. For example, Affleck has shown that in the massless fermion case, one massive boson of mass μ = √ Ne2/π and N −1 massless bosons appear, however there is no long range order (〈ψ−ψ 〉 = 0) in accordance with Coleman’s theorem in a 2-d Lorentz invariant theory [7, 8] and correlators of ψ − ψ show algebraic decay at large distances. Hence the rich vacuum structure of the multi-flavor Schwinger model carries many similarities to 4-dimensional QCD where we are interested in understanding how the effects of quark masses modify the vacuum structure, meson masses, mixing, and the pattern of chiral symmetry breaking. Years ago Coleman showed [5] that in the presence of small fermion masses m ≪ μ in the N = 2 model, the second boson mass has a fractional power dependence on m and the vacuum angle θ: μ2 ∝ (m| cos 1 2θ|). This singular dependence poses an intriguing puzzle: how can one get non-analytic dependence in the m → 0 limit where one would expect the validity of a perturbation theory in mass? Thus there has been growing interest in the Schwinger model, especially when defined on a compact manifold such as a circle or closed interval (a bag)[9]-[36]. Besides reproducing results in Minkowski spacetime in the infinite volume limit, analyzing the model on a circle has the advantage of being free from infrared divergence and well-defined at every stage of manipulation. Furthermore analytic solutions of the multi-flavor model are extremely useful for comparison with lattice simulations where several flavors are inherent [28][31][37]-[39]. In this paper we solve the Schwinger model with two massive fermions on a circle of circumference L. We find that the theory sensitively depends on the dimensionless parameter mL cos 1 2 θ. In particular, the m → 0 and L → ∞ limits do not commute with each other. This is to be contrasted to the situation in a model with just one fermion, in which a small fermion mass does not alter the structure of the model except for necessitating a θ vacuum. In the SU(2) symmetric two flavor case (m1 = m2), we show that in the large volume μL ≫ 1 limit, the light boson mass μ2 ∝ (m| cos 12θ|) for mL(μL)1/2| cos 12θ| ≫ 1, 2 while it is m| cos 1 2 θ| for mL(μL)1/2| cos 1 2 θ| ≪ 1. In other words physical quantities behave smoothly in the m → 0 or θ → ±π limit. Several authors have given exact solutions for the N ≥ 2 model with massless fermions on various manifolds.[23, 34, 35] Yet the importance of the parameter mL| cos 1 2 θ| has not been stressed in the literature. We adopt the method of abelian bosonization on a circle, generalizing the analysis of the N=1 case in ref. [10]. With N fermions the problem is eventually reduced to quantum mechanics for the 2N + 1 “zero modes” on the circle, with nontrivial interactions induced by the chiral anomaly and fermions masses. Further reduction is achieved for m≪μ. We find that the wave function of the vacuum sensitively depends on fermion masses for N ≥ 2. The model is given by L = − 4 FμνF μν + N