Logarithmic two-point scaling at a cubic asymptotically free beta function ($d(z,g)=d(1,\bar{g})[1+b{g}^2\log z{]}^{c/b}$)

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.