  When computed to one-loop order in resummed perturbation theory, the
non-abelian Debye mass appears to be logarithmically sensitive to the magnetic
scale $g^2T$. More generally, we show that in higher orders power-like infrared
divergences forbid the use of perturbation theory to calculate the corrections
to Debye screening. A similar infrared problem occurs in the determination of
the mass-shell for the scalar propagator in 2+1-dimensional scalar
electrodynamics. In this context, we provide a non-perturbative approach which
solves the infrared problems and allows for an accurate calculation of the
scalar propagator in the vicinity of the mass-shell.
