Transmutation of Scale Dependence into Truncation Uncertainty via RG-Improvement of the $R(s)$ Series

Abstract

The arbitrariness in how the logarithm is defined within the QCD series for the inclusive electroproduction cross-section is shown to affect the summation to all orders in $\alpha_s$ of leading and successively-subleading logarithms within that perturbative series, even though such summations largely eliminate the residual dependence of the original series on the arbitrary renormalization scale $\mu$. However, given that the original (unimproved) series is known to third-order in $\alpha_s(\mu)$, this logarithm ambiguity is shown not to enter the optimally improved summation-of-logarithms series until the term fourth-order in $\alpha_s(s)$, where $s$ is the physical center-of-mass energy squared. Consequently, the ambiguity in how the logarithm is defined is absorbable in the uncertainty associated with truncating the original perturbative series after its calculationally known terms.