One-loop inverse gauge coupling evolution (${\alpha }_i^{-1}(\mu )={\alpha }_i^{-1}({\mu }_0)-\frac{b_i}{2\pi }\log (\mu /{\mu }_0)$)

From $\mu d{g}_i/d\mu =b_i{g}_i^3/(16{\pi }^2)$ and ${\alpha }_i=g_i^2/(4\pi )$, the chain rule gives $d{\alpha }_i/d\log \mu =b_i{\alpha }_i^2/(2\pi )$. Taking its reciprocal makes the differential equation linear, proving the displayed relation. It assumes a fixed set of active fields and constant one-loop coefficients between the two scales; mass thresholds require piecewise matching. A negative coefficient gives asymptotic freedom.