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We consider the two-fold expansion in powers of the conformal anomaly and ofthe strong coupling $\alpha_s$ for the non-singlet contributions to Adler$D$-function and Bjorken polarized sum rule calculated previously in the$\MSbar$-scheme at the four-loop level. This representation provides relationsbetween definite terms of different loop orders appearing within the$\{\beta\}$-expansion of these quantities. Supposing the validity of thistwo-fold representation at the five-loop order and using these relations, weobtain some $\mathcal{O}(\alpha_s^5)$ corrections to the $D$-function, to the$R$-ratio of $e^+e^-$-annihilation into hadrons and to Bjorken polarized sumrule. These corrections are presented both analytically in the case of thegeneric simple gauge group and numerically for the $SU(3)$ color group. Thearguments in the favor of validity of the two-fold representation are given atleast at the four-loop level. Within the $\{\beta\}$-expansion procedure theanalytical Riemann $\zeta_4$-contributions to the five-loop expressions for theAdler function and Bjorken polarized sum rule are also fixed for the case ofthe generic simple gauge group.

the journal of high energy physics/the journal of high energy physics

The ${\rm{\bar{MS}}}$-scheme $\alpha_s^5$ QCD contributions to the Adler  function and Bjorken polarized sum rule in the Crewther-type two-fold  $\{\beta\}$-expanded representation