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

Abstract

We consider the two-fold expansion in powers of the conformal anomaly and of the 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 relations between definite terms of different loop orders appearing within the $\{\beta \}$-expansion of these quantities. Supposing the validity of this two-fold representation at the five-loop order and using these relations, we obtain 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 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 $\{\beta \}$-expansion procedure the analytical Riemann ${\zeta }_4$-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.

30 pages, 1 Figure, 6 Tables, text modified, the results unchanged, list of references modified, accepted for publication in JHEP

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