Matching higher-dimensional operators at finite temperature for general models

Abstract

High-temperature dimensional reduction provides a systematic effective field theory framework for studying finite-temperature thermodynamics and cosmological phase transitions. While the matching of super-renormalizable operators in the resulting three-dimensional effective theories is well established, the matching of higher-dimensional operators has recently been reinvigorated. These operators become phenomenologically relevant in strong first-order phase transitions where they quantify the convergence of the high-temperature expansion. This work automates the matching of generic three-dimensional dimension-five and -six operators for arbitrary models containing scalars, fermions, and gauge fields, implemented as an extension of the Mathematica package DRalgo. We present the operator basis, the matching procedure, and explicit examples including a scalar-Yukawa model, hot QCD, and the full Standard Model up to dimension six, covering operators mixing the strong and electroweak sectors as well as parity-violating contributions. Redundant operators, gauge dependence, and the corresponding field redefinitions are discussed in detail. The code and example model files are publicly available at https://github.com/DR-algo/DRalgo.