The MS ¯-scheme αs5 QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold {β}-expanded representation

We consider the two-fold expansion in powers of the conformal anomaly and of the strong coupling α_s for the non-singlet contributions to Adler D-function and Bjorken polarized sum rule calculated previously in the MS ¯-scheme at the four-loop level. This representation provides relations between definite terms of different loop orders appearing within the {β}-expansion of these quantities. Supposing the validity of this two-fold representation at the five-loop order and using these relations, we obtain some O (α_s⁵) corrections to the D-function, to the R-ratio of e⁺e⁻-annihilation into hadrons and to Bjorken polarized sum rule. These corrections are presented both analytically in the case of the generic simple gauge group and numerically for the SU(3) color group. The arguments in the favor of validity of the two-fold representation are given at least at the four-loop level. Within the {β}-expansion procedure the analytical Riemann ζ₄-contributions to the five-loop expressions for the Adler function and Bjorken polarized sum rule are also fixed for the case of the generic simple gauge group.