Renormalization Scale Setting for Evolution Equation of Non-Singlet Structure Functions and Their Moments

We use the BLM procedure to eliminate the renormalization scale ambiguity in the evolution equation for the non-singlet deep-inelastic structure function $F_2^{\text NS}(x,Q).$ The scale of the QCD coupling in the $\overline{\text{MS}}$ scheme has the form $Q^*(x) = Q (1-x)^{1/2} / x~f(x)$, where $x$ is the Bjorken variable and $f(x)$ is a smoothly varying function bounded between 0.30 to 0.45. Equivalently, the evolution of the $n$th moment of the structure function should contain an effective $\Lambda_{\text{QCD}}$ pattern, with $\Lambda_n \sim n^{1/2}$. This variation of $\Lambda_n$ agrees with experimental data.