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

TL;DR

This paper introduces the on-shell effective field theory (OSEFT) as a systematic method to compute power corrections to hard thermal loops in high-temperature QED, providing a clear framework for higher-order thermal effects.

Contribution

The paper develops the OSEFT up to third order in energy expansion for QED and demonstrates its effectiveness in calculating power corrections to the photon polarization tensor.

Findings

OSEFT reproduces HTL results at leading order.

Power corrections of order e^2 T are computed explicitly.

Results agree with direct QED calculations, confirming the method's validity.

Abstract

We show that effective field theory techniques can be efficiently used to compute power corrections to the hard thermal loops (HTL) in a high temperature T expansion. To this aim, we use the recently proposed on-shell effective field theory (OSEFT), which describes the quantum fluctuations around on-shell degrees of freedom. We provide the OSEFT 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 $e^2 T^2$, $e^2 T$ and $e^2 T^0$. The first one describes the HTL contribution, the second one vanishes, while the third one provides corrections of order $e^2$ 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.