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We show that effective field theory techniques can be efficiently used tocompute power corrections to the hard thermal loops (HTL) in a high temperatureT expansion. To this aim, we use the recently proposed on-shell effective fieldtheory (OSEFT), which describes the quantum fluctuations around on-shelldegrees of freedom. We provide the OSEFT Lagrangian up to third order in theenergy expansion for QED, and use it for the computation of power correctionsto the retarded photon polarization tensor for soft external momenta. Here softdenotes a scale of order $eT$, where $e$ is the gauge coupling constant. Wedevelop the necessary techniques to perform these computations, and study thecontributions to the polarization tensor proportional to $e^2 T^2$, $e^2 T$ and$e^2 T^0$. The first one describes the HTL contribution, the second onevanishes, while the third one provides corrections of order $e^2$ to the softphoton propagation. We check that the results agree with the direct calculationfrom QED, up to local pieces, as expected in an effective field theory.

The on-shell effective field theory: a systematic tool to compute power corrections to the hard thermal loops