On a realization of $\{\beta\}$-expansion in QCD

TL;DR

This paper introduces an algebraic method to determine the $eta$-expansion elements in QCD, utilizing additional degrees of freedom like the MSSM gluino, and applies it to high-order calculations of key QCD observables.

Contribution

It presents a novel algebraic approach to fix the $eta$-expansion elements in QCD using extra degrees of freedom, demonstrated through high-order calculations with the MSSM gluino.

Findings

Derived $eta$-expansion formulas for the Adler $D$-function and Bjorken sum rules at N$^3$LO.

Extended the $eta$-expansion analysis to N$^4$LO considering higher-order properties.

Discussed the scheme dependence and properties of the $eta$-expansion at higher orders.

Abstract

We suggest a simple algebraic approach to fix the elements of the $\{ \beta \}$-expansion for renormalization group invariant quantities, which uses additional degrees of freedom. The approach is discussed in detail for N$^2$LO calculations in QCD with the MSSM gluino -- an additional degree of freedom. We derive the formulae of the $\{ \beta \}$-expansion for the nonsinglet Adler $D$-function and Bjorken polarized sum rules in the actual N$^3$LO within this quantum field theory scheme with the MSSM gluino and the scheme with the second additional degree of freedom. We discuss the properties of the $\{ \beta \}$-expansion for higher orders considering the N$^4$LO as an example.