Abstract
We first consider the idea of renormalization group-induced estimates, in the context of optimization procedures, for the Brodsky-Lepage-Mackenzie approach to generate higher-order contributions to QCD perturbative series. Secondly, we develop the deviation pattern approach (DPA) in which through a series of comparisons between lower-order RG-induced estimates and the corresponding analytical calculations, one could modify higher-order RG-induced estimates. Finally, using the normal estimation procedure and DPA, we get estimates of $\alpha_s^4$ corrections for the Bjorken sum rule of polarized deep-inelastic scattering and for the non-singlet contribution to the Adler function.
Highlights
One of the main objectives of studying different ways of optimizing perturbative expansions for physical quantities is to disclose critical information about the sources of ambiguities that arise at different orders in perturbation theory studies
Light-front holographic formalism is an instance of a complete optimization prescription which defines an effective coupling for hadron dynamics at all momenta and takes advantage of the principle of maximum conformality (PMC) [1,2,3] to fix the renormalization scale ambiguity
It should be mentioned that the original mkhellat@gmail.com a.mirjalili@yazd.ac.ir version of PMC assumes a β-representation for the coefficients of QCD perturbative series which is different from the {β}-expansion of seBLM; PMC-II employs the same combination of β coefficients as seBLM and performs a resummation of the β-pattern into different scales at different orders [10, 11]
Summary
One of the main objectives of studying different ways of optimizing perturbative expansions for physical quantities is to disclose critical information about the sources of ambiguities that arise at different orders in perturbation theory studies. To resolve the issue of scheme-invariance, PMC-II suggests an analogy between the general structure of a QCD perturbative series and the induced terms in the structure of the series when the series is expressed in a Rδ scheme [10], as a subclass of the minimal subtraction schemes. The characteristic of this class of MS-like schemes is that they are related to each other through scale transformations. We would devise a similar plan for the BLM scalefixing procedure in which one would adopt a series of estimates for a renormalization group invariant observable at different orders up to O(αns). To generate an (i + 1)-th order estimate, we start from the i-th order BLM scale-fixed series and evolve the series using the (i + 1)-th order RGE
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