= Gauge-coupling beta function in logarithmic squared scale
{title2=$\mu\,dg/d\mu=cg^3+O(g^5)\Longrightarrow d\alpha/d\ln\mu^2=4\pi c\alpha^2+O(\alpha^3)$}
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.
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