Basics of thermal field theory -- a tutorial on perturbative computations

TL;DR

These lecture notes provide a structured, self-contained introduction to perturbative thermal field theory across quantum mechanics, free and interacting scalar fields, fermions, and gauge fields. They develop the imaginary-time (Matsubara) formalism, demonstrate how to perform both canonical and path-integral computations, and address ultraviolet renormalization and infrared resummation via ring diagrams and effective masses. The text highlights practical techniques such as matching, renormalization schemes, and Debye screening, linking abstract formalism to applications in cosmology and heavy-ion physics. The resulting framework underpins finite-temperature observables, including free energies and screening masses, and underpins perturbative approaches to QCD thermodynamics and related gauge theories at high temperature.

Abstract

These lecture notes, suitable for a two-semester introductory course or self-study, offer an elementary and self-contained exposition of the basic tools and concepts that are encountered in practical computations in perturbative thermal field theory. Selected applications to heavy ion collision physics and cosmology are outlined in the last chapter.

Figures (11)

• Figure 1: The behavior of $\mathcal{J}(y)$ and its various approximations. Shown are the exact numerical result from eq. (\ref{['calJy']}), the two low-temperature approximations from eq. (\ref{['calJlow']}) (with exponential and powerlike corrections, respectively), as well as the high-temperature expansion from eq. (\ref{['calJhigh']}).
• Figure 2: The potential from eq. (\ref{['pot_phi']}) at zero temperature for $\phi > 0$.
• Figure 3: A thermal effective potential displaying a first order phase transition.
• Figure 4: A comparison of the shapes of the tree-level zero-temperature potential and the 1-loop thermal correction. The function $J^{ }_T$ is given in eq. (\ref{['JT_again']}).
• Figure 5: An illustration of the effective potential in eq. (\ref{['V_thermal_again']}), possessing a phase transition.