= One-loop running of the strong coupling
{title2=$\alpha_s(\mu)=4\pi/[\beta_0\ln(\mu^2/\Lambda^2)]$}
With $\beta_0>0$, differentiating $\alpha_s=g^2/(4\pi)$ under $\mu\,dg/d\mu=-\beta_0g^3/(16\pi^2)$ gives $d(1/\alpha_s)/d\ln\mu=\beta_0/(2\pi)$. Integration defines the <strong-coupling scale> $\Lambda$ and gives the displayed positive branch for $\mu>\Lambda$. The pole signals breakdown of <perturbative quantum field theory>. The equivalence between specifying $\alpha_s$ and specifying $\Lambda$ is <dimensional transmutation>.
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