Renormalization-group characteristic solution for a two-point function ($d(e^{2t},g)=d(1,\bar{g}(t))e^{2{\int }_0^t{\gamma }_{\varphi }(\bar{g}(s))ds}$)

For $t=\log (|p|/\mu )$ and $d\bar{g}/dt=\beta (\bar{g})$, the displayed characteristic solution follows from $(-{\partial }_t+\beta {\partial }_g+2{\gamma }_{\varphi })d=0$. It separates the running coupling from the accumulated field-renormalization anomalous dimension. The exponent sign is determined by the precise Callan-Symanzik equation convention.

For $\beta =-b{g}^3$, ${\gamma }_{\varphi }=c{g}^2$, and $b>0$, one has ${\bar{g}}^2=g^2/(1+b{g}^2\log z)$. Integrating the field-renormalization anomalous dimension gives the displayed logarithmic factor. At large $z$, a nonzero free matching factor yields $d\sim C(\log z{)}^{c/b}$ in the printed RG sign convention.