Running coupling constant in thermal $\phi^4$ theory up to two loop order

TL;DR

This paper derives the two-loop order running coupling constant and mass in thermal ^4 theory using imaginary time formalism, linking non-thermal QFT and thermal field theory, and evaluates the scalar particle pressure at zero external momentum.

Contribution

It introduces a method to compute the two-loop running coupling and mass in thermal ^4 theory by connecting non-thermal QFT diagrams with thermal formalism and applying renormalization group equations.

Findings

Derived temperature-dependent running coupling and mass.

Expressed thermal diagrams as sums of non-thermal QFT diagrams.

Evaluated scalar particle pressure at two-loop order.

Abstract

Using the imaginary time formalism in thermal field theory, we derive running coupling constant and running mass in two loop order. In the process, we express the imaginary time formalism of Feynman diagrams as the summation of non-thermal quantum field theory (QFT) Feynman diagrams with coefficients that depend on temperature and mass. Renormalization constants for thermal $\phi^4$ theory were derived using simple diagrammatic analysis. Our model links the non-thermal QFT and the imaginary time formalism by assuming both have the same mass scale $\mu$ and coupling constant $g$. When these results are combined with the renormalization group equations (RGE) and applied simultaneously to thermal and non-thermal proper vertex functions, coupling constant and running mass with implicit temperature dependence are obtained. We evaluated pressure for scalar particles in two loop orders at zero external momentum limit by substituting the running mass result in the quasi-particle model.