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

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.