Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations

Stanley J. Brodsky, Matin Mojaza, Xing-Gang Wu

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

Resumé

We present in detail a new systematic method which can be used to automatically eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions at all orders. We show that all of the nonconformal β-dependent terms in a QCD perturbative series can be readily identified by generalizing the conventional renormalization schemes based on dimensional regularization. We then demonstrate that the nonconformal series of pQCD at any order can be resummed systematically into the scale of the QCD coupling in a unique and unambiguous way due to a special degeneracy of the β terms in the series. The resummation follows from the principal of maximum conformality (PMC) and assigns a unique scale for the running coupling at each perturbative order. The final result is independent of the initial choices of renormalization scheme and scale, in accordance with the principles of the renormalization group, and thus eliminates an unnecessary source of systematic error in physical predictions. We exhibit several examples known to order αs4; i.e. (i) the electron-positron annihilation into hadrons, (ii) the tau-lepton decay to hadrons, (iii) the Bjorken and Gross-Llewellyn Smith (GLS) sum rules, and (iv) the static quark potential. We show that the final series of the first three cases are all given in terms of the anomalous dimension of the photon field in SU(N), in accordance with conformality, and with all nonconformal properties encoded in the running coupling. The final expressions for the Bjorken and GLS sum rules directly lead to the generalized Crewther relations, exposing another relevant feature of conformality. The static quark potential shows that PMC scale-setting in the Abelian limit is to all orders consistent with QED scale-setting. Finally, we demonstrate that the method applies to any renormalization scheme and can be used to derive commensurate scale relations between measurable effective charges, which provide nontrivial tests of QCD to high precision. This work extends Brodsky-Lepage-Mackenzie (BLM) scale-setting to any perturbative order, with no ambiguities in identifying β terms in pQCD, demonstrating that BLM scale-setting follows from a principle of maximum conformality.

OriginalsprogEngelsk
Artikelnummer014027
TidsskriftPhysical Review D (Particles, Fields, Gravitation and Cosmology)
Vol/bind89
Udgave nummer1
ISSN0556-2821
DOI
StatusUdgivet - 2014

Fingeraftryk

quantum chromodynamics
hadrons
ambiguity
sum rules
quarks
predictions
positron annihilation
systematic errors
leptons
photons
decay

Citer dette

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title = "Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations",
abstract = "We present in detail a new systematic method which can be used to automatically eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions at all orders. We show that all of the nonconformal β-dependent terms in a QCD perturbative series can be readily identified by generalizing the conventional renormalization schemes based on dimensional regularization. We then demonstrate that the nonconformal series of pQCD at any order can be resummed systematically into the scale of the QCD coupling in a unique and unambiguous way due to a special degeneracy of the β terms in the series. The resummation follows from the principal of maximum conformality (PMC) and assigns a unique scale for the running coupling at each perturbative order. The final result is independent of the initial choices of renormalization scheme and scale, in accordance with the principles of the renormalization group, and thus eliminates an unnecessary source of systematic error in physical predictions. We exhibit several examples known to order αs4; i.e. (i) the electron-positron annihilation into hadrons, (ii) the tau-lepton decay to hadrons, (iii) the Bjorken and Gross-Llewellyn Smith (GLS) sum rules, and (iv) the static quark potential. We show that the final series of the first three cases are all given in terms of the anomalous dimension of the photon field in SU(N), in accordance with conformality, and with all nonconformal properties encoded in the running coupling. The final expressions for the Bjorken and GLS sum rules directly lead to the generalized Crewther relations, exposing another relevant feature of conformality. The static quark potential shows that PMC scale-setting in the Abelian limit is to all orders consistent with QED scale-setting. Finally, we demonstrate that the method applies to any renormalization scheme and can be used to derive commensurate scale relations between measurable effective charges, which provide nontrivial tests of QCD to high precision. This work extends Brodsky-Lepage-Mackenzie (BLM) scale-setting to any perturbative order, with no ambiguities in identifying β terms in pQCD, demonstrating that BLM scale-setting follows from a principle of maximum conformality.",
author = "Brodsky, {Stanley J.} and Matin Mojaza and Xing-Gang Wu",
year = "2014",
doi = "10.1103/PhysRevD.89.014027",
language = "English",
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journal = "Physical Review D. Particles and fields",
issn = "0556-2821",
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Systematic Scale-Setting to All Orders : The Principle of Maximum Conformality and Commensurate Scale Relations. / Brodsky, Stanley J.; Mojaza, Matin; Wu, Xing-Gang.

I: Physical Review D (Particles, Fields, Gravitation and Cosmology), Bind 89, Nr. 1, 014027, 2014.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Systematic Scale-Setting to All Orders

T2 - The Principle of Maximum Conformality and Commensurate Scale Relations

AU - Brodsky, Stanley J.

AU - Mojaza, Matin

AU - Wu, Xing-Gang

PY - 2014

Y1 - 2014

N2 - We present in detail a new systematic method which can be used to automatically eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions at all orders. We show that all of the nonconformal β-dependent terms in a QCD perturbative series can be readily identified by generalizing the conventional renormalization schemes based on dimensional regularization. We then demonstrate that the nonconformal series of pQCD at any order can be resummed systematically into the scale of the QCD coupling in a unique and unambiguous way due to a special degeneracy of the β terms in the series. The resummation follows from the principal of maximum conformality (PMC) and assigns a unique scale for the running coupling at each perturbative order. The final result is independent of the initial choices of renormalization scheme and scale, in accordance with the principles of the renormalization group, and thus eliminates an unnecessary source of systematic error in physical predictions. We exhibit several examples known to order αs4; i.e. (i) the electron-positron annihilation into hadrons, (ii) the tau-lepton decay to hadrons, (iii) the Bjorken and Gross-Llewellyn Smith (GLS) sum rules, and (iv) the static quark potential. We show that the final series of the first three cases are all given in terms of the anomalous dimension of the photon field in SU(N), in accordance with conformality, and with all nonconformal properties encoded in the running coupling. The final expressions for the Bjorken and GLS sum rules directly lead to the generalized Crewther relations, exposing another relevant feature of conformality. The static quark potential shows that PMC scale-setting in the Abelian limit is to all orders consistent with QED scale-setting. Finally, we demonstrate that the method applies to any renormalization scheme and can be used to derive commensurate scale relations between measurable effective charges, which provide nontrivial tests of QCD to high precision. This work extends Brodsky-Lepage-Mackenzie (BLM) scale-setting to any perturbative order, with no ambiguities in identifying β terms in pQCD, demonstrating that BLM scale-setting follows from a principle of maximum conformality.

AB - We present in detail a new systematic method which can be used to automatically eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions at all orders. We show that all of the nonconformal β-dependent terms in a QCD perturbative series can be readily identified by generalizing the conventional renormalization schemes based on dimensional regularization. We then demonstrate that the nonconformal series of pQCD at any order can be resummed systematically into the scale of the QCD coupling in a unique and unambiguous way due to a special degeneracy of the β terms in the series. The resummation follows from the principal of maximum conformality (PMC) and assigns a unique scale for the running coupling at each perturbative order. The final result is independent of the initial choices of renormalization scheme and scale, in accordance with the principles of the renormalization group, and thus eliminates an unnecessary source of systematic error in physical predictions. We exhibit several examples known to order αs4; i.e. (i) the electron-positron annihilation into hadrons, (ii) the tau-lepton decay to hadrons, (iii) the Bjorken and Gross-Llewellyn Smith (GLS) sum rules, and (iv) the static quark potential. We show that the final series of the first three cases are all given in terms of the anomalous dimension of the photon field in SU(N), in accordance with conformality, and with all nonconformal properties encoded in the running coupling. The final expressions for the Bjorken and GLS sum rules directly lead to the generalized Crewther relations, exposing another relevant feature of conformality. The static quark potential shows that PMC scale-setting in the Abelian limit is to all orders consistent with QED scale-setting. Finally, we demonstrate that the method applies to any renormalization scheme and can be used to derive commensurate scale relations between measurable effective charges, which provide nontrivial tests of QCD to high precision. This work extends Brodsky-Lepage-Mackenzie (BLM) scale-setting to any perturbative order, with no ambiguities in identifying β terms in pQCD, demonstrating that BLM scale-setting follows from a principle of maximum conformality.

U2 - 10.1103/PhysRevD.89.014027

DO - 10.1103/PhysRevD.89.014027

M3 - Journal article

VL - 89

JO - Physical Review D. Particles and fields

JF - Physical Review D. Particles and fields

SN - 0556-2821

IS - 1

M1 - 014027

ER -