Branch Cuts Due to Finite-Temperature Quasiparticles

The branch points of individual thermal self-energy diagrams at $k^2=4m^2,k^2=9m^2,\mathrm{.}\mathrm{.}\mathrm{.}$ 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, $k_0=\mathcal{E}(\vec{k})$, 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.