Generalization of the Brodsky-Lepage-Mackenzie optimization within the $\{\beta\}$-expansion and the Principle of Maximal Conformality

Understand

We discuss generalizations of the BLM optimization procedure for renormalization group invariant quantities.

• In this respect, we discuss in detail the features and construction of the $\{\beta\}$--expansion representation instead of the standard perturbative series with regards to the Adler $D$-function and Bjorken polarized sum rules obtained in order of ${\cal O}(\alpha_s^4)$.
• Based on the $\{\beta\}$--expansion we analyse different schemes of optimization, including the corrected Principle of Maximal Conformality, numerically illustrating their results.
• We suggest our scheme for the series optimization and apply it to both the above quantities.