Thermal field theory at next-to-leading order in the hard thermal loop expansion

TL;DR

This paper investigates the next-to-leading order corrections in hard-thermal-loop effective theory, revealing that most contributions can be obtained from soft 1-loop diagrams, with specific exceptions like 2n-photon vertices.

Contribution

It provides a refined power-counting analysis showing that only certain vertices, such as 2n-photon vertices, are exceptions to the dominance of soft 1-loop diagrams at next-to-leading order.

Findings

Most NLO contributions come from soft 1-loop diagrams.

Standard power-counting overestimates 2-loop diagram importance.

2n-photon vertices are exceptions to the general rule.

Abstract

In this paper we study the hard-thermal-loop effective theory at next-to-leading order. Standard power-counting predicts that a large number of diagrams, including 2-loop diagrams, may need to be calculated. In all of the calculations that have been done however, with the exception of the photon self-energy, the full next-to-leading order contribution can be obtained by calculating only soft 1-loop diagrams with effective lines and vertices. It is of interest to know if the photon self-energy is the only exception to this rule, or if there are others, and which ones. In this paper we perform a refined power-counting analysis using real-time finite temperature field theory which is particularly well suited to the task. We show that the standard power-counting rules obtained from the imaginary time formalism usually over-estimate the size of the 2-loop diagrams. We argue that the only exceptions to the rule that the 1-loop soft diagrams give the next-to-leading order contribution are $2n$-photon vertices.