  We present compact expressions for the power corrections to the hard thermal
loop (HTL) Lagrangian of QED in $d$ space dimensions. These are corrections of
order $(L/T)^2$, valid for momenta $L \ll T$, where $T$ is the temperature. In
the limit $d\to 3$ we achieve a consistent regularization of both infrared and
ultraviolet divergences, which respects the gauge symmetry of the theory.
Dimensional regularization also allows us to witness subtle cancellations of
infrared divergences. We also discuss how to generalise our results in the
presence of a chemical potential, so as to obtain the power corrections to the
hard dense loop (HDL) Lagrangian.
