The Non-local Odd-Parity O( e2) Effective Action of QED 3 at Finite Temperature

Abstract

We study the odd-parity part of the one-loop gauge field self-energy in QED 3 with massive fermions at finite temperature, with particular emphasis on the non-analyticity at zero momentum of the relevant scalar amplitude F β( p), which renders the O( e 2) action intrinsically non-local. We analyse F β( p) in Minkowski space (real-time formalism) both by dispersion relations and by direct evaluation. F β( p) is also studied in Euclidean space (imaginary-time formalism). In particular, we show explicitly how to analytically continue the Feynman-parametrized amplitude to Minkowski space, avoiding spurious singularities. We obtain the limiting behaviour of F β( p) along the lines p 0 = a | p| for | p| ⪡ | M| (where M is the fermion mass) in both Euclidean and Minkowski space, and we show that the results are fully consistent. Useful approximate closed-form expressions are given for this low- p behaviour, which are shown numerically to be valid for energies and momenta up to the order of the fermion mass scale. The possibility that the action might be approximately local for some appropriate regime of parameters is explored using a simple non-static external gauge field configuration.