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In the paper, we make a detailed study on the generalized Crewther Relation(GCR) between the Adler function ($D$) and the Gross-Llewellyn Smith sum rulescoefficient ($C^{\rm GLS}$) by using the newly suggested single-scale approachof the principle of maximum conformality (PMC). The resultant GCR isscheme-independent, whose residual scale dependence due to unknown higher-orderterms are highly suppressed. Thus a precise test of QCD theory withoutrenormalization scheme and scale ambiguities can be achieved by comparing withthe data. Moreover, a demonstration of the scheme independence of commensuratescale relation up to all orders has been presented. And as the first time, thePade approximation approach has been adopted for estimating the unknown $5_{\rmth}$-loop contributions from the known four-loop perturbative series.

Generalized Crewther relation and a novel demonstration of the scheme independence of commensurate scale relations up to all orders