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

We discuss the $\{ β\}$-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 $\{ β\}$-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 $\{ β\}$-expansion for the renormalization group covariant quantities -- the $\{ β,γ\}$-expansion.

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