Is it possible to check urgently the 5-loop analytical results for the $e^+e^-$-annihilation Adler function ?

TL;DR

This paper proposes a method to cross-check recent 5-loop analytical results for the $e^+e^-$ annihilation Adler function using the Crewther relation, aiming to verify the presence of specific transcendental terms and enhance confidence in high-order calculations.

Contribution

It introduces a new analytical expression for the $C_F^4 ext{ } ext{α}_s^4$ contributions to the Bjorken sum rule, facilitating independent validation of 5-loop corrections.

Findings

Derived new analytical expressions for the Bjorken sum rule contributions.

Highlighted the importance of verifying the $ ext{ζ}_3$ term in 5-loop corrections.

Proposed a cross-check method based on the Crewther relation.

Abstract

Considering the results of recent distinguished analytical calculations of the 5-loop single-fermion loop corrections to the QED $\beta$-function we emphasize that to our point of view it is important to perform their independent cross-checks. We propose one of the ways of these cross-check. It is based on the application of the original Crewther relation. We derive the new analytical expressions for the $C_F^4\alpha_s^4$-contributions to the Bjorken polarized sum rule. If results of possible direct calculations will agree with the presented expression, then the appearance of $\zeta_3$-term in the 5-loop correction to the QED $\beta$-function and in the $C_F^4\alpha_s^4$ contribution into the $e^+e^-$ annihilation Adler function will get independent support and may be analysed within the framework of the recently introduced concept of "maximal transcendentality".