Asymptotic Space-Time Behavior of HTL Gauge Propagator

The asymptotic behavior as t\to\infty and r\to\infty of the hard-thermal-loop propagator D^{\mu\nu}(t,r) is computed in the Coulomb gauge. The asymptotic falloff is always a power law though generally different in the deep time-like and space-like regions. The contributions of quasiparticle poles and Landau branch cuts are computed. The most difficult calculation is the contribution of the branch cut in the transverse propagator D^{ij}(t,r). For QED this produces a leading behavior of order T/r in both the time-like and space-like regions. The inclusion of a magnetic mass so as to describe QCD makes the leading behavior 1/(Tr^{3}), thus improving the infrared convergence. The asymptotic space-like behavior of all contributions (longitudinal and transverse, poles and cuts) is confirmed by also computing in the Euclidean formalism and analytically continuing. The results are compared will those for free gauge bosons at finite temperature.