Gauge-coupling beta function in logarithmic squared scale ($\mu dg/d\mu =c{g}^3+O(g^5)⟹d\alpha /d\ln {\mu }^2=4\pi c{\alpha }^2+O({\alpha }^3)$)

For a four-dimensional dimensionless gauge coupling, the one-loop correction is cubic in $g$. With $\alpha =g^2/(4\pi )$, the chain rule and $d/d\ln {\mu }^2=(1/2)d/d\ln \mu$ give the displayed conversion. In QCD, $c=-(11N-2n_f)/(48{\pi }^2)$, hence $d{\alpha }_s/d\ln {\mu }^2=-(11N-2n_f){\alpha }_s^2/(12\pi )+O({\alpha }_s^3)$. Other types of coupling or several interacting couplings can have different leading expansions; the squared-scale convention must not be mixed with the unsquared one.