Hard Thermal Loops and Beyond in the Finite Temperature World-Line Formulation of QED

TL;DR

This paper derives the hard thermal loop effective action for soft electromagnetic fields in finite temperature QED using the world-line formulation, and analyzes the behavior of higher order terms at high temperature.

Contribution

It introduces a finite temperature world-line approach to derive and analyze the hard thermal loop action and higher order terms in QED.

Findings

Leading non-linear terms are temperature-independent at high T.

Most T-dependent non-linear terms are suppressed as temperature increases.

The expansion becomes more reliable with increasing soft field order.

Abstract

We derive the hard thermal loop action for soft electromagnetic fields in the finite temperature world-line formulation at imaginary time, by first integrating out the hard fermion modes from the microscopic QED action. Further, using the finite T world-line method, we calculate all static higher order terms in the soft electromagnetic field. At high T, the leading non-linear terms are independent of the temperature and, except for a term quartic in the time component of the vector potential, they cancel exactly against the vacuum contribution. The remaining T-dependent non-linear terms become more strongly suppressed by the temperature as the number of soft fields increases, thus making the expansion reliable. Applications of this method to other theories and problems at the soft scale are also briefly discussed.