  The hard thermal loop (HTL) effective field theory of QED can be derived from
the classical limit of transport theory, corresponding to the leading term in a
gradient expansion of the quantum approach. In this paper, we show that power
corrections to the HTL effective Lagrangian of QED can also be obtained from
transport theory by including higher orders in such gradient expansion. The
gradient expansion is increasingly infrared (IR) divergent, but the correction
that we compute is IR finite. We employ dimensional regularization, and show
that this result comes after a cancellation of divergencies between the vacuum
and medium contributions. While the transport framework is an effective field
theory of the long distance physics of the plasma, we show that it correctly
reproduces the correct QED ultraviolet divergencies associated with the photon
wave function renormalization.
