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

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.