  The dynamics of soft ($|\vec{p}|\sim g^2 T$) non-Abelian gauge fields at
finite temperature is non-perturbative. The effective theory for the soft scale
is determined by diagrams with external momenta $p_0\lsim g^2 T$,
$|\vec{p}|\sim g^2 T$ and loop momenta larger than $g^2 T$. We consider the
polarization tensor beyond the hard thermal loop approximation, which accounts
for loop momenta of order $T$. There are higher loop diagrams, involving also
the scale $gT$, which are as important as the hard thermal loops. These higher
loop contributions are characteristic for non-Abelian gauge theories and their
calculation is simplified by using the hard thermal loop effective theory.
Remarkably, the effective one-loop polarization tensor is found to be gauge
fixing independent and transverse at leading order in $g$. The transversality
indicates that this approach leads to a gauge invariant effective theory.
