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

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.

14 pages, Introduction, Sec.2, Conclusion are significantly improved

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Cited by in corpus (5)

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• The $\overline{\mathrm{MS}}$-scheme ${\alpha }_s^5$ QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold $\{\beta \}$-expanded representation
• The $\{\beta \}$-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order $O({\alpha }_s^4)$
• Adler function, Bjorken polarized sum rule: confirmation of elements of the $\{\beta \}$-expansion and the diagrams
• The decomposed photon anomalous dimension in QCD and the $\{\beta \}$-expanded representations for the Adler function