Gluon Plasma Frequency -- the Next-to-Leading Order Term

The longitudinal-electric oscillations of the hot gluon system are studied beyond the well known leading order term at high temperature $T$ and small coupling $g$. The coefficient $\eta$ in $\omega^2 = m^2 \, (1+ \eta \, g \wu N \, )$ is calculated, where \hbox{$\omega \equiv \omega (\vc q =0)$} is the long-wavelength limit of the frequency spectrum, $N$ the number of colours and $m^2=g^2 N T^2/9$. In the course of this, for the real part of the gluon self-energy, the Braaten-Pisarski resummation programme is found to work well in all details. The coefficient $\eta$ is explicitly seen to be gauge independent within the class of covariant gauges. Infrared singularities cancel as well as collinear singularities in the two-loop diagrams with both inner momenta hard. However, as it turns out, none of these two-loop contributions reaches the relative order $O(g)$ under study. The minus sign in our numerical result $\; \eta = -.18 \;$ is in accord with the intuitive picture that the studied mode might soften with increasing coupling (lower temperature) until a phase transition is reached at zero-frequency. The minus sign thus exhibits the 'glue' effect for the first time in a dynamical quantity of hot QCD.