The Massive Multi-flavor Schwinger Model

TL;DR

This paper analyzes the massive multi-flavor Schwinger model on a circle using bosonization, revealing how the fermion condensate and correlations depend on fermion mass, volume, and the vacuum angle, with exact solutions for specific cases.

Contribution

It provides an exact solution for the N=2 case of the massive multi-flavor Schwinger model, highlighting the non-commuting limits and the dependence on the vacuum angle and fermion mass.

Findings

Fermion condensate behavior depends on mL and heta.

Correlation functions decay algebraically with exponent 1 when m ext{cos}( heta/2)=0.

Limits m→0 and L→∞ do not commute, affecting the theory's behavior.

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 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 \mu=\sqrt{2e^2/\pi} when all fermions satisfy the same boundary conditions. We show that the two limits m \go 0 and L \go \infty fail to commute and that the behavior of the theory critically depends on the value of mL|\cos\onehalf\theta| where \theta is the vacuum angle parameter. When the volume is large \mu L \gg 1, the fermion condensate <\psibar \psi> is -(e^{4\gamma} m\mu^2 \cos^4\onehalf\theta/4\pi^3)^{1/3} or $-2e^\gamma m\mu L \cos^2 \onehalf\theta /\pi^2 for mL(\mu L)^{1/2} |\cos\onehalf\theta| \gg 1 or \ll 1, respectively. Its correlation function decays algebraically with a critical exponent \eta=1 when m\cos\onehalf\theta=0.