  We apply a gauge invariant formulation of Wilson Renormalization Group (RG)
to the computation of the Debye and transverse gluon masses in pure gauge SU(N)
at high temperature. Following the Hard Thermal Loop effective field theory as
a guideline, we develop an approximation scheme to the exact evolution
equations. The Debye mass receives sizable corrections compared to the leading
order perturbative result, mainly due to the infrared singular behavior in the
transverse gluon sector. A non-vanishing mass for the transverse gluons is
found, which acts as an infrared regulator though not efficiently enough as to
restore the validity of perturbation theory. Indeed, discussing the role of
higher dimensional operators, we show that the gauge coupling for the
transverse modes typically flows to non-perturbative values unless extremely
high temperatures are reached. After comparing our results with recent lattice
simulations, we comment on the possibility of using this formulation of the RG
as a tool to construct an effective field theory for the non-perturbative, long
wavelength, transverse modes.
