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Title: Scale Setting Using the Extended Renormalization Group and the Principle of Maximal Conformality: the QCD Coupling at Four Loops

Abstract

A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express the invariance of physical observables under both the renormalization scale- and scheme-parameter transformations, provide a convenient way for estimating the scale- and scheme-dependence of the physical process. In this paper, we present a solution for the scale-equation of the extended renormalization group equations at the four-loop level. Using the principle of maximum conformality (PMC)/Brodsky-Lepage-Mackenzie (BLM) scale-setting method, all non-conformal {beta}{sub i} terms in the perturbative expansion series can be summed into the running coupling, and the resulting scale-fixed predictions are independent of the renormalization scheme. Different schemes lead to different effective PMC/BLM scales, but the final results are scheme independent. Conversely, from the requirement of scheme independence, one not only can obtain scheme-independent commensurate scale relations among different observables, but also determine the scale displacements among the PMC/BLM scales which are derived under different schemes. In principle, the PMC/BLM scales can be fixed order-by-order, and as a useful reference, we present a systematic and scheme-independent procedure for setting PMC/BLM scales up to NNLO. An explicit application for determining the scale setting of R{submore » e{sup +}e{sup -}}(Q) up to four loops is presented. By using the world average {alpha}{sub s}{sup {ovr MS}}(MZ) = 0.1184 {+-} 0.0007, we obtain the asymptotic scale for the 't Hooft associated with the {ovr MS} scheme, {Lambda}{sub {ovr MS}}{sup 'tH} = 245{sub -10}{sup +9} MeV, and the asymptotic scale for the conventional {ovr MS} scheme, {Lambda}{sub {ovr MS}} = 213{sub -8}{sup +19} MeV.« less

Authors:
;
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1035084
Report Number(s):
SLAC-PUB-14774
Journal ID: ISSN 1550--7998; arXiv:1111.6175; TRN: US201204%%320
DOE Contract Number:  
AC02-76SF00515
Resource Type:
Journal Article
Journal Name:
Phys.Rev.D85:034038,2012
Additional Journal Information:
Journal Volume: 85; Journal Issue: 3; Journal ID: ISSN 1550--7998
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COUPLING; QUANTUM CHROMODYNAMICS; RENORMALIZATION; TRANSFORMATIONS; Phenomenology-HEP,HEPPH

Citation Formats

Brodsky, Stanley J, /SLAC, Wu, Xing-Gang, and /SLAC /Chongqing U. Scale Setting Using the Extended Renormalization Group and the Principle of Maximal Conformality: the QCD Coupling at Four Loops. United States: N. p., 2012. Web. doi:10.1103/PhysRevD.85.034038.
Brodsky, Stanley J, /SLAC, Wu, Xing-Gang, & /SLAC /Chongqing U. Scale Setting Using the Extended Renormalization Group and the Principle of Maximal Conformality: the QCD Coupling at Four Loops. United States. https://doi.org/10.1103/PhysRevD.85.034038
Brodsky, Stanley J, /SLAC, Wu, Xing-Gang, and /SLAC /Chongqing U. 2012. "Scale Setting Using the Extended Renormalization Group and the Principle of Maximal Conformality: the QCD Coupling at Four Loops". United States. https://doi.org/10.1103/PhysRevD.85.034038. https://www.osti.gov/servlets/purl/1035084.
@article{osti_1035084,
title = {Scale Setting Using the Extended Renormalization Group and the Principle of Maximal Conformality: the QCD Coupling at Four Loops},
author = {Brodsky, Stanley J and /SLAC and Wu, Xing-Gang and /SLAC /Chongqing U.},
abstractNote = {A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express the invariance of physical observables under both the renormalization scale- and scheme-parameter transformations, provide a convenient way for estimating the scale- and scheme-dependence of the physical process. In this paper, we present a solution for the scale-equation of the extended renormalization group equations at the four-loop level. Using the principle of maximum conformality (PMC)/Brodsky-Lepage-Mackenzie (BLM) scale-setting method, all non-conformal {beta}{sub i} terms in the perturbative expansion series can be summed into the running coupling, and the resulting scale-fixed predictions are independent of the renormalization scheme. Different schemes lead to different effective PMC/BLM scales, but the final results are scheme independent. Conversely, from the requirement of scheme independence, one not only can obtain scheme-independent commensurate scale relations among different observables, but also determine the scale displacements among the PMC/BLM scales which are derived under different schemes. In principle, the PMC/BLM scales can be fixed order-by-order, and as a useful reference, we present a systematic and scheme-independent procedure for setting PMC/BLM scales up to NNLO. An explicit application for determining the scale setting of R{sub e{sup +}e{sup -}}(Q) up to four loops is presented. By using the world average {alpha}{sub s}{sup {ovr MS}}(MZ) = 0.1184 {+-} 0.0007, we obtain the asymptotic scale for the 't Hooft associated with the {ovr MS} scheme, {Lambda}{sub {ovr MS}}{sup 'tH} = 245{sub -10}{sup +9} MeV, and the asymptotic scale for the conventional {ovr MS} scheme, {Lambda}{sub {ovr MS}} = 213{sub -8}{sup +19} MeV.},
doi = {10.1103/PhysRevD.85.034038},
url = {https://www.osti.gov/biblio/1035084}, journal = {Phys.Rev.D85:034038,2012},
issn = {1550--7998},
number = 3,
volume = 85,
place = {United States},
year = {Thu Feb 16 00:00:00 EST 2012},
month = {Thu Feb 16 00:00:00 EST 2012}
}