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 overestimate 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.

• Received 7 December 2012

DOI:https://doi.org/10.1103/PhysRevD.87.065008