On Finite Temperature Quantum Field Theory: From Theoretical Foundations To Electroweak Phase Transition

TL;DR

This review develops finite-temperature quantum field theory as the foundation for understanding electroweak symmetry breaking in the early universe. It contrasts imaginary-time and real-time formalisms, detailing how thermal corrections reshape propagators and the finite-temperature effective potential, while highlighting UV renormalization, IR sensitive sectors, and gauge-dependence issues. The authors quantify theoretical uncertainties and describe resummation and EFT techniques, illustrating their application to the real singlet extension RxSM, which can realize a strong first-order EWPT and connect to electroweak baryogenesis and dark matter. They also outline experimental probes—collider tests of Higgs and scalar couplings and gravitational-wave signatures from bubble dynamics—that can jointly test the viability of EWPT-based explanations for the BAU. The discussion emphasizes a careful, gauge-consistent, and non-HTE approach to capture non-perturbative IR physics and non-equilibrium dynamics relevant for accurate phenomenology and future observations.

Abstract

In the immediate aftermath of the Big Bang, the universe existed in an extremely hot, dense state in which particle interactions occurred not in a vacuum but within a thermal medium. Under such conditions, the standard framework of quantum field theory (QFT) requires a finite temperature extension, wherein propagators -- and hence the fundamental structure of the theory -- are modified to reflect thermal background effects. These thermal modifications are central to understanding the nature of electroweak symmetry breaking (EWSB) as a high temperature phase transition, potentially leading to qualitatively different vacuum structures for the Higgs field as the universe cooled. Finite temperature corrections naturally regulate ultraviolet divergences in propagators, hinting at a possible route toward ultraviolet completion. However, these same thermal effects exacerbate infrared pathologies and can lead to imaginary contributions to the effective potential, particularly when analysing metastable or multi-vacuum configurations. Additional theoretical challenges, such as gauge dependence and renormalization scale ambiguity, further obscure the precise characterization of the electroweak phase transition even in minimal extensions of the Standard Model (SM). This review presents the theoretical foundations of finite temperature QFT with an emphasis on how different field species respond to thermal effects, identifying the bosonic sector as the primary source of key theoretical subtleties. We focus particularly on the scalar extension of the SM, which offers a compelling framework for realizing first order electroweak phase transitions, electroweak baryogenesis, and accommodating dark matter candidates depending on the underlying $Z_2$ symmetry structure.

Figures (11)

• Figure 1: Time contour in the imaginary time formalism for the thermal equilibrium case.
• Figure 2: The real-time contour: A forward real--time branch $\mathcal{C}_1\!: T\to T'$, a forward segment past the insertion to $t$ ($\mathcal{C}_2$), a backward real--time branch $\mathcal{C}_3\!: t\to t'$, and a short imaginary--time leg $\mathcal{C}_4\!: t'\to T-i\beta$. This closed-time contour encodes both the causal evolution and, when the imaginary leg is included, the choice of an initial thermal state.
• Figure 3: The sum of the 1-loop scalar contribution to the effective potential representing the diagrammatic equivalent of equation (\ref{['eq22']}).
• Figure 4: The diagrammatic representation of the 1-loop fermion contributions to the effective potential obtained from the imaginary time approach at thermal equilibrium.
• Figure 5: The diagrammatic representation of the 1-loop contribution from the weak gauge fields to the effective potential at finite temperature following the imaginary time formulation.