Radiatively induced Chern-Simons terms at finite temperature

We study the one-loop contribution to the induced parity-odd part μνλpλπ(0)(p) of the gauge field self-energy in (2+1)-dimensional gauge theories at a finite temperature T. At T=0, π(0)(p0=0,p=0) is the coefficient of the induced local Chern-Simons term. It is pointed out that π(0)(p) is not an analytic function at zero frequency and momentum when T0. Therefore, at T0, there exist no induced local Chern-Simons terms other than those due to regularization.