Renormalization group improvement for thermally resummed effective potential

We propose a novel method for renormalization group improvement of thermally resummed effective potential. In our method, $\beta$-functions are temperature dependent as a consequence of the divergence structure in resummed perturbation theory. In contrast to the ordinary $\overline{\mathrm{MS}}$ scheme, the renormalization group invariance of the resummed finite-temperature effective potential holds order by order, which significantly mitigates a notorious renormalization scale dependence of phase transition quantities such as a critical temperature even at the one-loop order. We also devise a tractable method that enables one to incorporate temperature-dependent higher-order corrections by fully exploiting the renormalization group invariance.

Figure 1

$v(T)/T$ as a function of $T$ in the $\overline{\mathrm{MS}}$ scheme (left) and our scheme (right). We take $v({\overline{\mu }}_0)=200$, $m_{{\varphi }_1}({\overline{\mu }}_0)=5.0$, $m_{{\varphi }_2}({\overline{\mu }}_0)=125$, ${\nu }_2^2({\overline{\mu }}_0)={85.0}^2$, ${\lambda }_2({\overline{\mu }}_0)=5.0$, where ${\nu }_1^2({\overline{\mu }}_0)$ and ${\lambda }_1({\overline{\mu }}_0)$ are determined by the first and second derivatives of the effective potentials at a given order while ${\lambda }_3({\overline{\mu }}_0)$ at the tree-level. ${\overline{\mu }}_0$ is fixed by the condition $t(\phi =v)=0$. At the both one- and two-loop levels, ${\overline{\mu }}_0\simeq 75.81$. The dimensionful parameters are expressed in units of any mass scale.