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The generalized BLM approach to fix scale-dependence in QCD: the current status of investigations

I present a brief review of the generalized Brodsky-Lepage-McKenzie (BLM) approaches to fix the scale-dependence of the renormalization group (RG) invariant quantities in QCD. At first, these approaches are based on the expansions of the coefficients of the perturbative series for the RG-invariant quantities in the products of the coefficients $β_i$ of the QCD $β$-function, which are evaluated in the MS-like schemes. As a next step all $β_i$-dependent terms are absorbed into the BLM-type scale(s) of the powers of the QCD couplings. The difference between two existing formulations of the above mentioned generalizations based on the seBLM approach and the Principle of Maximal Conformality (PMC) are clarified in the case of the Bjorken polarized deep-inelastic scattering sum rule. Using the conformal symmetry-based relations for the non-singlet coefficient functions of the Adler D-function and of Bjorken polarized deep-inelastic scattering sum rules $C^{\rm Bjp}_{\rm NS}(a_s)$ the $β_i$-dependent structure of the NNLO approximation for $C^{\rm Bjp}_{\rm NS}(a_s)$ is predicted in QCD with $n_{gl}$-multiplet of gluino degrees of freedom, which appear in SUSY extension of QCD. The importance of performing the analytical calculation of the N$^3$LO additional contributions of $n_{gl}$ gluino multiplet to $C^{\rm Bjp}_{\rm NS}(a_s)$ for checking the presented in the report NNLO prediction and for the studies of the possibility to determine the discussed $\{β\}$-expansion pattern of this sum rule at the $O(a_s^4)$-level is emphasised.