SelfConsistency Requirements of the Renormalization Group for Setting the Renormalization Scale
Abstract
In conventional treatments, predictions from fixedorder 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 selfconsistency requirements of the RG, such as reflexivity, symmetry, and transitivity, which must be satisfied by the scalesetting 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 schemeindependent estimation, but it violates the symmetry and transitivity properties of the RG and does not reproduce the GellMannLow 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 nonconformal {β ^{R} _{i}}terms (R stands for an arbitrary renormalization scheme)more »
 Authors:

 SLAC National Accelerator Lab., Menlo Park, CA (United States)
 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):
 SLACPUB15133
Journal ID: ISSN 15507998; arXiv:1208.0700
 DOE Contract Number:
 AC0276SF00515
 Resource Type:
 Journal Article
 Journal Name:
 Physical Review. D, Particles, Fields, Gravitation and Cosmology
 Additional Journal Information:
 Journal Volume: 86; Journal ID: ISSN 15507998
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; PhenomenologyHEP, TheoryHEP,HEPPH, HEPTH
Citation Formats
Brodsky, Stanley J., and Wu, XingGang. SelfConsistency 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, XingGang. SelfConsistency 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, XingGang. Tue .
"SelfConsistency 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 = {SelfConsistency Requirements of the Renormalization Group for Setting the Renormalization Scale},
author = {Brodsky, Stanley J. and Wu, XingGang},
abstractNote = {In conventional treatments, predictions from fixedorder 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 selfconsistency requirements of the RG, such as reflexivity, symmetry, and transitivity, which must be satisfied by the scalesetting 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 schemeindependent estimation, but it violates the symmetry and transitivity properties of the RG and does not reproduce the GellMannLow 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 nonconformal {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, scalefixed, schemeindependent prediction at any finite order. The PMC scales and the resulting finiteorder 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 = {15507998},
number = ,
volume = 86,
place = {United States},
year = {2012},
month = {8}
}