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 {β}-expanded representation

Abstract

We consider the two-fold expansion in powers of the conformal anomaly and of the strong coupling αs for the non-singlet contributions to Adler D-function and Bjorken polarized sum rule calculated previously in the $$ \overline{\mathrm{MS}} $$ 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 $$ \mathcal{O} $$ O ($$ {\alpha}_s^5 $$ α 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 {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.

Violations of quark-hadron duality in low-energy determinations of αs

“…Strong efforts are currently being made aimed to improve our understanding of the perturbative series, e.g. see [70][71][72][73][74][75][76][77][78][79][80][81][82].…”

The {β}-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order O($$ {\alpha}_s^4 $$)

“…A notable attempt to obtain the elements d n , c n of the {β}-expansion just for D and C based on the interpretation of the r.h.s. of CBKR was done in [20,23]. There, firstly, a special "two-fold" form of the r.h.s.…”

“…(3.8) coincides with ours because their method is already based on CBKR. The mentioned chain of suggestions allows the authors to obtain 8 elements out of 12 [23] in the N 5 LO of {β}-expansion owing to the strength of these suggestions. We believe that the method needs further verification.…”

The $\{β\}$-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order $O(α_s^4)$

“…A notable attempt to obtain the elements d n , c n of the {β}-expansion just for D and C based on the interpretation of the RHS of CBKR was done in [19,20]. There, firstly, a special "two-fold" form of the RHS of CBKR was proposed that includes one of the sums over the powers of (β(a)/a) j and "works up to N 3,4 LO".…”

“…(3.8) coincides with ours because their method is already based on CBKR. The mentioned chain of suggestions allows the authors to obtain 8 elements out of 12 [20] in the N 5 LO of {β}-expansion owing to the strength of these suggestions. We believe that the method needs further verification.…”