On Finite Temperature Quantum Field Theory from Theoretical Foundations to Electroweak Phase Transition

In the immediate aftermath of the Big Bang, the universe existed in an extremely hot, dense state in which particle interactions occurred not in 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 analyzing 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 ${\mathbb{Z}}_2$ symmetry structure.