The decomposed photon anomalous dimension in QCD and the $\{\beta\}$-expanded representations for the Adler function

TL;DR

This paper investigates the $eta$-expansion of the Adler function and related RG functions in QCD, emphasizing the decomposition of the photon anomalous dimension for consistent perturbative analysis up to order $ ext{O}( ext{ extalpha}_s^4)$ and comparing different approximation methods.

Contribution

It introduces a decomposition method for the photon anomalous dimension in the $eta$-expansion framework, enhancing the understanding of perturbative QCD calculations and their conformal properties.

Findings

Decomposition of the photon anomalous dimension in $eta$-expansion is necessary for consistent perturbative analysis.

Comparison between $ar{ ext{MS}}$ and PMC/BLM approximants shows differences in QCD predictions.

The study discusses the implications of these methods for theoretical and phenomenological applications.

Abstract

This work is devoted to the study of the $\{\beta\}$-expansion of the perturbative expressions for the $e^+e^-$ annihilation Adler function $D(Q^2)$ and for the related renormalization group functions, namely for the photon vacuum polarization function and its anomalous dimension $\gamma(\alpha_s)$ in QCD at the $\mathcal{O}(\alpha^4_s)$ order. We emphasize that $\gamma(\alpha_s)$ is not a conformal-invariant contribution to $D(Q^2)$ and, therefore, for a consistent analysis it is necessary to decompose its higher-order PT coefficients in powers of the $\beta$-function coefficients in the same way as for the Adler function. The arguments in favor of this statement are given. The comparison of the $\overline{MS}$ and PMC/BLM approximants are demonstrated.Theoretical and phenomenologically related consequences of this comparison are briefly commented.