Adler Function, Sum Rules and Crewther Relation of Order O(alpha_s^4): the Singlet Case

TL;DR

The work delivers the complete O(α_s^4) singlet corrections to the QCD Adler function and, via the GLS sum rule and the generalized Crewther relation, provides stringent cross-checks of high-order perturbative calculations. By deriving analytic expressions for the singlet coefficients d_4^{SI} and c_4^{SI} and validating them against the GCR, the authors obtain full massless-quark results for the Adler function and R(s). These results enable more precise extractions of α_s from e^+e^- annihilation and DIS observables, and demonstrate consistency of complex perturbative calculations with fundamental symmetry relations in QCD.

Abstract

The analytic result for the singlet part of the Adler function of the vector current in a general gauge theory is presented in five-loop approximation. Comparing this result with the corresponding singlet part of the Gross-Llewellyn Smith sum rule [1], we successfully demonstrate the validity of the generalized Crewther relation for the singlet part. This provides a non-trivial test of both our calculations and the generalized Crewther relation. Combining the result with the already available non-singlet part of the Adler function [2,3] we arrive at the complete ${\cal O}(α_s^4)$ expression for the Adler function and, as a direct consequence, at the complete ${\cal O}(α_s^4)$ correction to the $e^+ e^-$ annihilation into hadrons in a general gauge theory.

Figures (2)

• Figure 1: Lowest order non-singlet (a) and singlet (b) diagrams contributing to the Adler function.
• Figure 2: (a),(b): ${\cal O}(\alpha_s^3)$ non-singlet and singlet diagrams contributing to the Gross-Llewellyn Smith sum rule; note that the coefficient function $C^{Bjp}$ is contributed by only non-singlet diagrams.