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

TL;DR

This paper explores the $eta$-expansion formalism in perturbative QCD, extending it to include renormalization group covariant quantities, and illustrates the approach with the nonsinglet Adler function at third order.

Contribution

It introduces a generalized $eta, ext{ extgamma}$-expansion for renormalization group covariant quantities in perturbative QCD, expanding the existing formalism.

Findings

Analyzes the $eta$-expansion via incomplete BPHZ R-operation contractions.

Demonstrates the formalism with the third-order nonsinglet Adler function.

Proposes a generalization to include renormalization group covariant quantities.

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.