Mass-Shell Behavior of Electron Propagator at Low Temperature

At T=0 the full electron propagator is known to have an infrared anomalous dimension and thus a branch point at $P^{2}=m^{2}$ rather than a pole. An explicit calculation shows that if $0<eT\ll m$ the retarded self-energy is analytic in the vicinity of $P^{2}\approx m^{2}$, which includes the thermal mass-shell. The low-temperature propagator has a simple pole. Only when T=0 is there a branch point at the mass-shell.