On-shell effective field theory: A systematic tool to compute power corrections to the hard thermal loops

Abstract

We show that effective field theory techniques can be efficiently used to compute power corrections to the hard thermal loops in a high temperature T expansion. To this aim, we use the recently proposed on-shell effective field theory, which describes the quantum fluctuations around on-shell degrees of freedom. We provide the on-shell effective field theory Lagrangian up to third order in the energy expansion for QED and use it for the computation of power corrections to the retarded photon polarization tensor for soft external momenta. Here soft denotes a scale of order eT, where e is the gauge coupling constant. We develop the necessary techniques to perform these computations and study the contributions to the polarization tensor proportional to e2T2, e2T, and e2T0. The first one describes the hard thermal loop contribution, the second one vanishes, while the third one provides corrections of order e2 to the soft photon propagation. We check that the results agree with the direct calculation from QED, up to local pieces, as expected in an effective field theory.