Parity anomaly with impurities and the Pauli--Villars subtraction

We calculate the anomalous part of the polarization tensor of Dirac fermions in $2+1$ dimensions in the presence of impurities described by the scattering rate $\mathrm{\Gamma }$ for arbitrary external frequency and momenta. We consider two different versions of the Pauli–Villars subtractions and discuss their physical consequences.