RG-Improved Three-Loop Effective Potential of the Massive ${\varphi }^4$ theory

The renormalization group method is applied to the three-loop effective potential of the massive ${\varphi }^4$ theory in the $\overline{\mathrm{M}\mathrm{S}}$ scheme in order to obtain the next-next-next-to-leading logarithm resummation. For this, we exploit four-loop parts of the renormalization group functions ${\beta }_{\lambda }$, ${\gamma }_m$, ${\gamma }_{\varphi }$, and ${\beta }_{\mathrm{\Lambda }}$, which were already given to five-loop order via the renormalization of the zero-, two-, and four-point one-particle-irreducible Green’s functions, to solve evolution equations for the parameters $\lambda$, $m^2$, $\varphi$, and $\mathrm{\Lambda }$ within the accuracy of the three-loop order.