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Title: Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale

Abstract

In conventional treatments, predictions from fixed-order perturbative QCD calculations cannot be fixed with certainty due to ambiguities in the choice of the renormalization scale as well as the renormalization scheme. In this paper we present a general discussion of the constraints of the renormalization group (RG) invariance on the choice of the renormalization scale. We adopt the RG based equations, which incorporate the scheme parameters, for a general exposition of RG invariance, since they simultaneously express the invariance of physical observables under both the variation of the renormalization scale and the renormalization scheme parameters. We then discuss the self-consistency requirements of the RG, such as reflexivity, symmetry, and transitivity, which must be satisfied by the scale-setting method. The Principle of Minimal Sensitivity (PMS) requires the slope of the approximant of an observable to vanish at the renormalization point. This criterion provides a scheme-independent estimation, but it violates the symmetry and transitivity properties of the RG and does not reproduce the Gell-Mann-Low scale for QED observables. The Principle of Maximum Conformality (PMC) satisfies all of the deductions of the RG invariance - reflectivity, symmetry, and transitivity. Using the PMC, all non-conformal {β R i}-terms (R stands for an arbitrary renormalization scheme)more » in the perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC scales and the resulting finite-order PMC predictions are both to high accuracy independent of the choice of initial renormalization scale, consistent with RG invariance.« less

Authors:
 [1];  [2]
  1. SLAC National Accelerator Lab., Menlo Park, CA (United States)
  2. Chongqing Univ. (China); SLAC National Accelerator Lab., Menlo Park, CA (United States)
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1053432
Report Number(s):
SLAC-PUB-15133
Journal ID: ISSN 1550-7998; arXiv:1208.0700
DOE Contract Number:  
AC02-76SF00515
Resource Type:
Journal Article
Journal Name:
Physical Review. D, Particles, Fields, Gravitation and Cosmology
Additional Journal Information:
Journal Volume: 86; Journal ID: ISSN 1550-7998
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Phenomenology-HEP, Theory-HEP,HEPPH, HEPTH

Citation Formats

Brodsky, Stanley J., and Wu, Xing-Gang. Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale. United States: N. p., 2012. Web. doi:10.1103/PhysRevD.86.054018.
Brodsky, Stanley J., & Wu, Xing-Gang. Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale. United States. https://doi.org/10.1103/PhysRevD.86.054018
Brodsky, Stanley J., and Wu, Xing-Gang. Tue . "Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale". United States. https://doi.org/10.1103/PhysRevD.86.054018. https://www.osti.gov/servlets/purl/1053432.
@article{osti_1053432,
title = {Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale},
author = {Brodsky, Stanley J. and Wu, Xing-Gang},
abstractNote = {In conventional treatments, predictions from fixed-order perturbative QCD calculations cannot be fixed with certainty due to ambiguities in the choice of the renormalization scale as well as the renormalization scheme. In this paper we present a general discussion of the constraints of the renormalization group (RG) invariance on the choice of the renormalization scale. We adopt the RG based equations, which incorporate the scheme parameters, for a general exposition of RG invariance, since they simultaneously express the invariance of physical observables under both the variation of the renormalization scale and the renormalization scheme parameters. We then discuss the self-consistency requirements of the RG, such as reflexivity, symmetry, and transitivity, which must be satisfied by the scale-setting method. The Principle of Minimal Sensitivity (PMS) requires the slope of the approximant of an observable to vanish at the renormalization point. This criterion provides a scheme-independent estimation, but it violates the symmetry and transitivity properties of the RG and does not reproduce the Gell-Mann-Low scale for QED observables. The Principle of Maximum Conformality (PMC) satisfies all of the deductions of the RG invariance - reflectivity, symmetry, and transitivity. Using the PMC, all non-conformal {βRi}-terms (R stands for an arbitrary renormalization scheme) in the perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC scales and the resulting finite-order PMC predictions are both to high accuracy independent of the choice of initial renormalization scale, consistent with RG invariance.},
doi = {10.1103/PhysRevD.86.054018},
url = {https://www.osti.gov/biblio/1053432}, journal = {Physical Review. D, Particles, Fields, Gravitation and Cosmology},
issn = {1550-7998},
number = ,
volume = 86,
place = {United States},
year = {2012},
month = {8}
}