On the induced gauge invariant mass
  We derive a general expression for the gauge invariant mass (m_G) for an Abelian gauge field, as induced by vacuum polarization, in 1+1 dimensions. From its relation to the chiral anomaly, we show that m_G has to satisfy a certain quantization condition. This quantization can be, on the other hand, explicitly verified by using the exact general expression for the gauge invariant mass in terms of the fermion propagator. This result is applied to some explicit examples, exploring the possibility of having interesting physical situations where the value of $m_G$ departs from its canonical value. We also study the possibility of generalizing the results to the 2+1 dimensional case at finite temperature, showing that there are indeed situations where a finite and non-vanishing gauge invariant mass is induced. 
C. D. Fosco, L. E. Oxman, S. P. Sorella