Self-Energy approximation for the running coupling constant in thermal ϕ4 theory using Imaginary Time Formalism

We investigate the temperature dependence of the mass scale, running coupling constant, and running mass in thermal ϕ⁴ theory using the Imaginary Time Formalism (ITF). Employing dimensional regularization and the minimal subtraction scheme, we compute the self-energy up to two-loop order. We introduce a novel Self-Energy Approximation (SEA), which equates the thermal and non-thermal self-energies in the zero external momentum limit. This approximation, combined with the renormalization group equation, imposes constraints that naturally lead to a temperature-dependent mass scale, μ(T), ensuring consistent behavior of the running coupling constant and running mass at finite temperatures. Using these results, the free energy density is evaluated at two-loop order and compared with the quasiparticle model.