Hard thermal loops for soft or collinear external momenta

TL;DR

This paper develops a recursive framework for calculating finite temperature 1-loop diagrams with soft or collinear external momenta, leading to an integral equation that sums bremsstrahlung and pair annihilation contributions to thermal photon production.

Contribution

It introduces a recursion relation and an integral equation for finite temperature diagrams with specific external momenta configurations, extending previous work on thermal photon production.

Findings

Derived a recursion relation for finite temperature 1-loop diagrams.

Formulated an integral equation summing bremsstrahlung and pair annihilation.

Connected the formalism to the work of Arnold, Moore, and Yaffe.

Abstract

We consider finite temperature 1-loop diagrams with hard loop momenta and an arbitrary number of external gauge fields when the external momenta are either soft, or near the light cone and nearly collinear with the loop momentum. We obtain a recursion relation for these diagrams which we translate into an equation for their generating functional. By integrating out the soft fields while keeping two collinear ones we find an integral equation, originally due to Arnold, Moore, and Yaffe, which sums the bremsstrahlung and pair annihilation contribution to the thermal photon production rate.