The $\{\beta \}$-expansion formalism in perturbative QCD and its extension

We discuss the $\{\beta \}$-expansion for renormalization group invariant quantities tracing this expansion to the different contractions of the corresponding incomplete BPHZ $R$-operation. All of the coupling renormalizations, which follow from these contractions, should be taken into account for the $\{\beta \}$-expansion. We illustrate this feature considering the nonsinglet Adler function $D^{\text{NS}}$ in the third order of perturbation. We propose a generalization of the $\{\beta \}$-expansion for the renormalization group covariant quantities – the ${ \beta,\gamma }$-expansion.