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 = 2e2π 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|cos12θ| where θ is the vacuum angle parameter. When the volume is large μL ⪢ 1, fermion condensate 〈ψψ〉 is −(e4γ mμ2 cos412θ/4π3)13 or respectively. Its correlation function decays algebraically with a critical exponent η = 1 when mcos12θ = 0.