Self-Energy Approximation for the Running Coupling Constant in Thermal $\phi^4$ Theory using Imaginary Time Formalism

The running coupling constant is calculated using the imaginary time formalism (ITF) of thermal field theory under the self-energy approximation. In the process, each Feynman diagram in thermal field theory is rewritten as the summation of non-thermal diagrams with coefficients that are functions of mass and temperature. By employing the same mass scale and coupling constant for both the non-thermal QFT and ITF, we derive a relation between them. Also, we calculate the self-energy using ITF, which is equated to the same as that of non-thermal QFT under the zero external momentum limit. This can provide a new expression for the coupling constant. Combining this result with the $\beta(g)$ and $\gamma_m(g)$ function relations of the renormalization group equations gives rise to a thermal-dependent coupling constant and running mass. Using these results, the free energy density is evaluated for two-loop order and compared with quasiparticle model.