Perfect 3-Dimensional Lattice Actions for 4-Dimensional Quantum Field Theories at Finite Temperature

TL;DR

This paper introduces a two-step approach combining perturbative and nonperturbative methods to analyze phase transitions in finite-temperature quantum field theories using perfect lattice actions.

Contribution

It develops a method to derive finite-temperature 3D perfect lattice actions for quantum field theories, separating UV and IR problems for improved analysis.

Findings

Finite temperature dependent coefficients in the lattice action.

Application of block spin transformations to gauge fields.

Framework for nonperturbative treatment of effective 3D theories.

Abstract

We propose a two-step procedure to study the order of phase transitions at finite temperature in electroweak theory and in simplified models thereof. In a first step a coarse grained free energy is computed by perturbative methods. It is obtained in the form of a 3-dimensional perfect lattice action by a block spin transformation. It has finite temperature dependent coefficients. In this way the UV-problem and the infrared problem is separated in a clean way. In the second step the effective 3-dimensional lattice theory is treated in a nonperturbative way, either by the Feynman-Bogoliubov method (solution of a gap equation), by real space renormalization group methods, or by computer simulations. In this paper we outline the principles for $\varphi ^4$-theory and scalar electrodynamics. The Ba{\l}aban-Jaffe block spin transformation for the gauge field is used. It is known how to extend this transformation to the nonabelian case, but this will not be discussed here.