The massive multi-flavor Schwinger model

Abstract

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 2 N 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 = 2e 2 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 1 2 θ| where θ is the vacuum angle parameter. When the volume is large μL ⪢ 1, fermion condensate 〈 ψ ψ〉 is −(e 4γ mμ 2 cos 4 1 2 θ/4π 3) 1 3 or respectively. Its correlation function decays algebraically with a critical exponent η = 1 when m cos 1 2 θ = 0 .