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

Abstract: 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.