Transversality of the logarithmic divergences in the classical finite temperature SU(N) self-energy

We show that the logarithmic divergences that appear in the classical approximation of the finite temperature SU(N) self-energy are transverse. We use the Ward identities in linear gauges and the fact that the superficial degree of divergence d of a classical diagram only depends on the number of loops ℓ via d=2−ℓ. We comment on the relevance of this result to the construction of a low-energy effective theory beyond hard thermal loops.