One-loop stationary scale for the hadronic annihilation correction (${\mu }_{\ast }^2=q^2{e}^{-C/b}$)

Suppose $K=1+a+a^2[C-b\ln (q^2/{\mu }^2)]$, with $da/d\ln {\mu }^2=-b{a}^2$. Differentiation gives $dK/d\ln {\mu }^2=-2b{a}^3[C-b\ln (q^2/{\mu }^2)]$. For nonzero $a,b$, the principle of minimal sensitivity therefore gives $\boxed{{\mu }_{\ast }^2=q^2{e}^{-C/b}}$. The derivative vanishes identically through order $a^2$: this prescription retains the residual $a^3$ term generated by one-loop running of the truncated expression. Higher-order running and coefficients can shift the stationary scale. For $b=0$ or $a=0$, no unique scale is selected.