Branch Cuts Due to Finite-Temperature Quasiparticles
  The branch points of individual thermal self-energy diagrams at $k^{2}=4m^{2}, k^{2}=9m^{2},...$ are shown not to be branch points of the full thermal self-energy. Branch points of the full theory are determined by the complex, temperature-dependent energies of the quasiparticles, defined as the pole location, $k_{0}={\cal E}(\vec{k})$, of the exact retarded propagator. The full retarded self-energy is found to have branch points at $k_{0}=2{\cal E}(\vec{k}/2)$ and $k_{0}=3{\cal E}(\vec{k}/3)$ as well as cuts in the space-like region. The discontinuities across the branch cuts are complex. The advanced self-energy is related by reflection to the retarded. 
H. Arthur Weldon