 Solution

Differentiating the strong coupling constant ${\alpha }_s=g^2/(4\pi )$ givesIntegrate this constant derivative and express the integration constant through the strong-coupling scale $\Lambda$ at which the one-loop inverse coupling vanishes. The one-loop running of the strong coupling isThe positive-coupling branch has $\mu >\Lambda$ and is perturbatively reliable only when the logarithm is sufficiently large. The pole at $\Lambda$ signals failure of the one-loop expansion, not a prediction of a physically infinite observable. Equivalently, $\Lambda ={\mu }_0\exp [-2\pi /({\beta }_0{\alpha }_s({\mu }_0))]$ is scale-independent to this order; this is dimensional transmutation.