A Simple sum rule for the thermal gluon spectral function and applications

In this paper, we derive a simple sum rule satisfied by the gluon spectral function at finite temperature. This sum rule is useful in order to calculate exactly some integrals that appear frequently in the photon or dilepton production rate by a quark gluon plasma. Using this sum rule, we rederive simply some known results and obtain some new results that would be extremely difficult to justify otherwise. In particular, we explain how this result can be used to calculate the photon rate in a simple quasi-particle model which correctly reproduces the thermodynamic properties of a quark-gluon plasma. We also derive an exact expression for the collision integral that appears in the calculation of the Landau-Pomeranchuk-Migdal effect and propose a method to solve the resulting differential equation.