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This work is devoted to the study of the $\{\beta\}$-expansion of theperturbative expressions for the $e^+e^-$ annihilation Adler function $D(Q^2)$and for the related renormalization group functions, namely for the photonvacuum polarization function and its anomalous dimension $\gamma(\alpha_s)$ inQCD 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 itshigher-order PT coefficients in powers of the $\beta$-function coefficients inthe same way as for the Adler function. The arguments in favor of thisstatement are given. The comparison of the $\overline{MS}$ and PMC/BLMapproximants are demonstrated.Theoretical and phenomenologically relatedconsequences of this comparison are briefly commented.

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