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

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.