Resummation in a Hot Scalar Field Theory

TL;DR

This work tackles the infrared difficulties of perturbation theory in a hot scalar $g^2 \phi^4$ theory by implementing a staged resummation of the self-energy within the imaginary-time formalism, enabling a consistent determination of the thermal mass and damping rate to leading nontrivial orders. The analysis moves from a one-loop resummation, yielding a thermal mass $m^2 = g^2 T^2 / 24$ and a nonanalytic $g^3$ correction, to a two-loop calculation that provides an explicit expression for the corrected mass $M_4^2$ and the on-shell damping rate $\gamma$, while ensuring UV renormalization and cancellation of infrared-sensitive contributions. The results are cross-validated against real-time calculations, reinforcing the equivalence of ITF and RTF for the pole of the propagator, and offering a detailed roadmap for applying similar resummation strategies to gauge theories. Overall, the paper clarifies how hard thermal loops emerge in a scalar theory and demonstrates how careful diagrammatic resummation yields a controllable, perturbatively computable effective expansion at high temperature.

Abstract

A resummed perturbative expansion is used to obtain the self-energy in the high-temperature \(g^2φ^4\) field theory model up to order $g^4$. From this the zero momentum pole of the effective propagator is evaluated to determine the induced thermal mass and damping rate for the bosons in the plasma to order $g^3$. The calculations are performed in the imaginary time formalism and a simple diagrammatic analysis is used to identify the relevant diagrams at each order. Results are compared with similar real-time calculations found in the literature.