High temperature limit in static backgrounds

TL;DR

This paper demonstrates that in static backgrounds, the hard thermal loop contributions to thermal amplitudes can be obtained by setting external momenta to zero prior to calculations, applicable across various theories including gravity.

Contribution

It provides a general proof that simplifies the calculation of static thermal amplitudes for bosonic external fields across different theories and dimensions.

Findings

Hard thermal loop contributions can be derived by zero external momenta.

The method applies to theories with any interaction vertices, including gravity.

Valid for any number of space-time dimensions.

Abstract

We prove that the hard thermal loop contribution to static thermal amplitudes can be obtained by setting all the external four-momenta to zero before performing the Matsubara sums and loop integrals. At the one-loop order we do an iterative procedure for all the 1PI one-loop diagrams and at the two-loop order we consider the self-energy. Our approach is sufficiently general to the extent that it includes theories with any kind of interaction vertices, such as gravity in the weak field approximation, for $d$ space-time dimensions. This result is valid whenever the external fields are all bosonic.