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

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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 ... continued below

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7 pages

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Brodsky, Stanley J.; /SLAC; Wu, Xing-Gang & U., /SLAC /Chongqing February 16, 2012.

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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.

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7 pages

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  • Journal Name: Phys.Rev.D85:034038,2012; Journal Volume: 85; Journal Issue: 3

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  • Report No.: SLAC-PUB-14774
  • Grant Number: AC02-76SF00515
  • DOI: 10.1103/PhysRevD.85.034038 | External Link
  • Office of Scientific & Technical Information Report Number: 1035084
  • Archival Resource Key: ark:/67531/metadc836117

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  • February 16, 2012

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  • May 19, 2016, 3:16 p.m.

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  • Dec. 1, 2016, 6:40 p.m.

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Brodsky, Stanley J.; /SLAC; Wu, Xing-Gang & U., /SLAC /Chongqing. Scale Setting Using the Extended Renormalization Group and the Principle of Maximal Conformality: the QCD Coupling at Four Loops, article, February 16, 2012; United States. (digital.library.unt.edu/ark:/67531/metadc836117/: accessed December 12, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.