Bosonized Massive N-flavor Schwinger Model
  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 $\theta$ parameter, fermion masses, and temperature. Boson masses and fermion chiral condensates are evaluated. In the N=1 model the anomalous behavior is found at $\theta \sim \pi$ and $m/\mu \sim 0.4$. In the N=3 model an asymmetry in fermion masses $(m_1 < m_2 \ll m_3)$ removes the singularity at $\theta=\pi$ and T=0. The chiral condensates at $\theta=0$ are insensitive to the asymmetry in fermion masses, but are significantly sensitive at $\theta=\pi$. The resultant picture is similar to that obtained in QCD by the chiral Lagrangian method. 
Yutaka Hosotani and Ramon Rodriguez