Generic Rules for High Temperature Dimensional Reduction and Their Application to the Standard Model

TL;DR

The paper develops a general framework for high-temperature dimensional reduction of gauge theories to a 3d bosonic effective theory by matching static Green's functions, and applies it to map the Standard Model electroweak sector onto a 3d $SU(2)\times U(1)$ gauge-Higgs theory. It provides explicit 1-loop and 2-loop building blocks, derives the relations between 4d MSbar parameters and the 3d theory, and addresses the role of higher-order operators and infrared safety. The approach yields a practical pathway to study electroweak phase transitions nonperturbatively via lattice simulations of the 3d theory and to explore extensions of the SM within a universal 3d framework. The results include detailed mappings of SM parameters to 3d couplings, numerical assessments as functions of $m_H$ and temperature, and a discussion of perturbative and nonperturbative implications for early-universe cosmology and beyond-Standard-Model scenarios.

Abstract

We formulate the rules for dimensional reduction of a generic finite temperature gauge theory to a simpler three-dimensional effective bosonic theory in terms of a matching of Green's functions in the full and the effective theory, and present a computation of a generic set of 1- and 2-loop graphs needed for the application of these rules. As a concrete application we determine the explicit mapping of the physical parameters of the standard electroweak theory to a three-dimensional SU(2)xU(1) gauge-Higgs theory. We argue that this three-dimensional theory has a universal character and appears as an effective theory for many extensions of the Standard Model.

Figures (9)

• Figure 1: The 2-loop graph (a) relevant for calculating the dimensionally reduced mass parameter in the "integration out"-procedure, and the additional graph (b) needed in the "matching" procedure. The thick lines are superheavy fields, and the thin dashed lines light fields.
• Figure 2: The diagrams needed for the dimensional reduction of (a) the wave function $\phi$, and (b) the wave functions $A_i$ and $A_0$. Dashed line is a scalar propagator, wiggly line a vector propagator, double line a ghost propagator, and solid line a fermion propagator. The bare blob indicates the wave function counterterm.
• Figure 3: The diagrams needed for calculating the 3d coupling constants $g_3^2$ and $h_3$.
• Figure 4: The diagrams needed for calculating the 3d mass of the temporal components of the gauge fields.
• Figure 5: The diagrams needed for calculating the quartic self-coupling of the temporal components of the gauge fields.