Scale Setting Using the Extended Renormalization Group and the Principle of Maximum Conformality: the QCD Coupling Constant 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 $\{{\beta }_i\}$ 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 $R_{e^{+}{e}^{-}}(Q)$ up to four loops is presented. By using the world average ${\alpha }_s^{\overline{MS}}(M_Z)=0.1184\pm 0.0007$, we obtain the asymptotic scale for the 't Hooft associated with the $\overline{MS}$ scheme, ${\Lambda }_{\overline{MS}}^{{}^{\prime }tH}=245_{-10}^{+9}$ MeV, and the asymptotic scale for the conventional $\overline{MS}$ scheme, ${\Lambda }_{\overline{MS}}=213_{-8}^{+19}$ MeV.

9 pages, no figures. The formulas in the Appendix are corrected

References in corpus (4)

• Hadronic Z- and tau-Decays in Order alpha_s^4
• First determination of the strong coupling constant using NNLO predictions for hadronic event shapes in e^+e^- annihilations
• The absence of QCD $\beta$-function factorization property of the generalized Crewther relation in the 't Hooft $\overline{MS}$-based scheme
• The analytical singlet ${\alpha }_s^4$ QCD contributions into the $e^{+}{e}^{-}$-annihilation Adler function and the generalized Crewther relations

Cited by in corpus (120)

• NNLO corrections to top-pair production at hadron colliders: the all-fermionic scattering channels
• The Large Hadron-Electron Collider at the HL-LHC
• 50 Years of Quantum Chromodynamics
• A Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in PQCD
• Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations
• Eliminating the Renormalization Scale Ambiguity for Top-Pair Production Using the Principle of Maximum Conformality
• The Renormalization Scale-Setting Problem in QCD
• Setting the Renormalization Scale in QCD: The Principle of Maximum Conformality
• Evidence for high-energy resummation effects in Mueller-Navelet jets at the LHC
• NNLO QCD predictions for fully-differential top-quark pair production at the Tevatron
• The Spin Structure of the Nucleon
• Renormalization Group Invariance and Optimal QCD Renormalization Scale-Setting
• Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale
• Application of the Principle of Maximum Conformality to the Top-Quark Forward-Backward Asymmetry at the Tevatron
• An Analysis of Scheme Transformations in the Vicinity of an Infrared Fixed Point
• Generalization of the Brodsky-Lepage-Mackenzie optimization within the $\{\beta \}$-expansion and the Principle of Maximal Conformality
• Novel All-Orders Single-Scale Approach to QCD Renormalization Scale-Setting
• Probabilistic definition of the perturbative theoretical uncertainty from missing higher orders
• Reanalysis of the BFKL Pomeron at the next-to-leading logarithmic accuracy
• Scheme Transformations in the Vicinity of an Infrared Fixed Point
• Puzzles in Hadronic Physics and Novel Quantum Chromodynamics Phenomenology
• Degeneracy Relations in QCD and the Equivalence of Two Systematic All-Orders Methods for Setting the Renormalization Scale
• Hadronic production of ${\Xi }_{cc}$ at a fixed-target experiment at the LHC
• A novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of maximum conformality and the $C$-scheme coupling
• Implications of the Principle of Maximum Conformality for the QCD Strong Coupling
• The Higgs boson inclusive decay channels $H\to b\bar{b}$ and $H\to gg$ up to four-loop level
• Extending the Predictive Power of Perturbative QCD
• Application of the Principle of Maximum Conformality to the Top-Quark Charge Asymmetry at the LHC
• PMC${}_{\infty }$: Infinite-Order Scale-Setting using the Principle of Maximum Conformality, A Remarkably Efficient Method for Eliminating Renormalization Scale Ambiguities for Perturbative QCD
• QCD corrections to the $B_c$ to charmonia semi-leptonic decays
• Novel Method for the Precise Determination of the QCD Running Coupling from Event Shapes Distributions in Electron-Positron Annihilation
• Scheme-Independent Calculations of Anomalous Dimensions of Baryon Operators in Conformal Field Theories
• Production of semi-inclusive doubly heavy baryons via top-quark decays
• Renormalization scheme and gauge (in)dependence of the generalized Crewther relation: what are the real grounds of the $\beta$-factorization property?
• On the BLM optimal renormalization scale setting for semihard processes
• Setting the Renormalization Scale in pQCD: Comparisons of the Principle of Maximum Conformality with the Sequential Extended Brodsky-Lepage-Mackenzie Approach
• Reanalysis of the Higher Order Perturbative QCD corrections to Hadronic $Z$ Decays using the Principle of Maximum Conformality
• Production of doubly heavy baryons via Higgs boson decays
• Exclusive production of $J/\psi +{\eta }_c$ at the $B$ factories Belle and Babar using the principle of maximum conformality
• The Importance of Proper Renormalization Scale-Setting for Testing QCD at Colliders
• Heavy and light meson wavefunctions
• $J/\psi +{\chi }_{cJ}$ Production at the $B$ Factories under the Principle of Maximum Conformality
• The longitudinal leading-twist distribution amplitude of $J/\psi$ meson within background field theory
• Information on the Pion Distribution Amplitude from the Pion-Photon Transition Form Factor with the Belle and BaBar Data
• The Generalized Scheme-Independent Crewther Relation in QCD
• Renormalization Scheme Dependence in a QCD Cross Section
• Excited Heavy Quarkonium Production at the LHC through $W$-Boson Decays
• Novel and self-consistency analysis of the QCD running coupling ${\alpha }_s(Q)$ in both the perturbative and nonperturbative domains
• Fractional Analytic QCD beyond Leading Order in timelike region
• Photoproduction of the heavy quarkonium at the ILC
• The QCD improved electroweak parameter $\rho$
• Exclusive charmonium production from $e^{+}{e}^{-}$ annihilation round the $Z^0$ peak
• Predictions for the Top-Quark Forward-Backward Asymmetry at High Invariant Pair Mass Using the Principle of Maximum Conformality
• Precise perturbative predictions from fixed-order calculations
• Ambiguities of the principle of maximum conformality procedure for hadron collider processes
• The heavy quarkonium inclusive decays using the principle of maximum conformality
• Renormalization Scheme Dependence with Renormalization Group Summation
• Revisiting the Heavy Vector Quarkonium Leptonic Widths
• An analysis of $H\to \gamma \gamma$ up to three-loop QCD corrections
• Renormalization group improved pQCD prediction for ${\rm Y}(1S)$ leptonic decay
• Novel Method to Reliably Determine the QCD Coupling from $R_{\mathrm{uds}}$ Measurements and its effects to Muon $g-2$ and $\alpha (M_Z^2)$ within the Tau-Charm Energy Region
• The $\overline{\mathrm{MS}}$-scheme ${\alpha }_s^5$ QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold $\{\beta \}$-expanded representation
• A solution to the $\gamma {\gamma }^{\ast }\to {\eta }_c$ puzzle using the Principle of Maximum Conformality
• Hadronic decays of the spin-singlet heavy quarkomium under the principle of maximum conformality
• Next-to-leading order QCD corrections for the charmonium production via the channel $e^{+}{e}^{-}\to H(|c\bar{c}⟩)+\gamma$ round the $Z^0$ peak
• General Properties on Applying the Principle of Minimum Sensitivity to High-order Perturbative QCD Predictions
• Heavy Quarkonium Production through the Semi-Exclusive $e^{+}{e}^{-}$ Annihilation Channels around the $Z^0$ Peak
• A novel determination of non-perturbative contributions to Bjorken sum rule
• Application of the Principle of Maximum Conformality to the Hadroproduction of the Higgs Boson at the LHC
• Reanalysis of the Higgs-boson decay $H\to gg$ up to ${\alpha }_s^6$-order level using the principle of maximum conformality
• Determination of the top-quark $\overline{\mathrm{MS}}$ running mass via its perturbative relation to the on-shell mass with the help of principle of maximum conformality
• Gauge dependence of the perturbative QCD predictions under the momentum space subtraction scheme
• Thrust Distribution in Electron-Positron Annihilation using the Principle of Maximum Conformality
• Thrust Distribution for 3-Jet Production from $e^{+}{e}^{-}$ Annihilation within the QCD Conformal Window and in QED
• Properties of the decay $H\to \gamma \gamma$ using the approximate ${\alpha }_s^4$-corrections and the principle of maximum conformality
• Detailed Comparison of Renormalization Scale-Setting Procedures based on the Principle of Maximum Conformality
• A Possible Solution to the $B\to \pi \pi$ Puzzle Using the Principle of Maximum Conformality
• Next-to-leading order QCD corrections to the production of $B_c$ and $B_c^{\ast }$ through $W^{+}$-boson decays
• Renormalization Scale Setting for Heavy Quark Pair Production in $e^{+}{e}^{-}$ Annihilation near the Threshold Region
• Inclusive Gluon Production in Pion-Proton Collisions and the Principle Maximum Conformality Renormalization Scale
• Generalized Crewther relation and a novel demonstration of the scheme independence of commensurate scale relations up to all orders
• Excited doubly heavy baryons production via top-quark decays
• Renormalization scheme dependence of high-Order perturbative QCD predictions
• Bottomonium production associated with a photon at a high luminosity $e^{+}{e}^{-}$ collider with next-to-leading order QCD corrections
• The Gross-Llewellyn Smith sum rule up to $\mathcal{O}({\alpha }_s^4)$-order QCD corrections
• Revisiting the production of $J/\psi +{\eta }_c$ via the $e^{+}{e}^{-}$ annihilation within the QCD light-cone sum rules
• $Z$-boson hadronic decay width up to $\mathcal{O}({\alpha }_s^4)$-order QCD corrections using the single-scale approach of the principle of maximum conformality
• Achieving effective renormalization scale and scheme independence via the Principle of Observable Effective Matching
• Reanalysis of the top-quark pair production via the $e^{+}{e}^{-}$ annihilation near the threshold region up to N${}^3$LO QCD corrections
• Revisiting the Bottom Quark Forward-Backward Asymmetry $A_{\mathrm{FB}}$ in Electron-Positron Collisions
• An updated determination of the pion-photon transition form factor
• Self-consistent analysis for the ${\eta }_c\to \gamma \gamma$ process
• Precise determination of the top-quark on-shell mass $M_t$ via its scale-invariant perturbative relation to the top-quark $\overline{\mathrm{MS}}$ mass ${\overline{m}}_t({\overline{m}}_t)$
• Elimination of QCD Renormalization Scale and Scheme Ambiguities
• The ${\rm Y}(1S)$ leptonic decay using the principle of maximum conformality
• The decomposed photon anomalous dimension in QCD and the $\{\beta \}$-expanded representations for the Adler function
• New determination of $|V_{\mathrm{cb}}|$ using the three-loop QCD corrections for the $B\to D^{\ast }$ semi-leptonic decays
• Reconsideration of the QCD corrections to the ${\eta }_c$ decays into light hadrons using the principle of maximum conformality
• The $P$-wave charmonium annihilation into two photons ${\chi }_{c0,c2}\to \gamma \gamma$ with high-order QCD corrections
• Precise determination of the bottom-quark on-shell mass using its four-loop relation to the $\overline{\mathrm{MS}}$-scheme running mass
• A new analysis of the pQCD contributions to the electroweak parameter $\rho$ using the single-scale approach of principle of maximum conformality
• $B_c$ meson exclusive decays to a $P$-wave charmonium and a pion at NLO accuracy
• Revisiting the top-quark pair production at future $e^{+}{e}^{-}$ colliders
• Approximate N${}^5$LO Higgs boson decay width $\Gamma (H\to \gamma \gamma )$
• The Higgs-Boson Decay $H\to gg$ to Order ${\alpha }_s^5$ under the mMOM-Scheme
• Scale-invariant total decay width $\Gamma (H\to b\bar{b})$ using the novel method of characteristic operator
• The Principle of Maximum Conformality Correctly Resolves the Renormalization-Scheme-Dependence Problem
• Next-to-leading order QCD corrections to $Z\to {\eta }_Q+Q+\bar{Q}$
• Total decay width of $H\to gg$ using the infinite-order scale-setting approach based on the intrinsic conformality
• Deviation pattern approach for optimizing perturbative terms of QCD renormalization group invariant observables
• QCD corrections of $e^{+}{e}^{-}\to J/\psi +c+\bar{c}$ using the principle of maximum conformality
• Updated Determination of Ellis-Jaffe Sum Rules up to ${\mathrm{N}}^3\mathrm{LO}$ QCD corrections
• The Common Elements of Atomic and Hadronic Physics
• A reanalysis of event shape distributions in electron-positron annihilation
• Confronting BFKL dynamics with experimental studies of Mueller-Navelet jets at the LHC
• Properties of the Free Energy Density Using the Principle of Maximum Conformality
• ${\eta }_c$ leading-twist distribution amplitude and the $B_c\to {\eta }_c\ell {\bar{\nu }}_{\ell }$ semileptonic decays using QCD Sum Rules
• Relation between pole and running masses of heavy quarks using the principle of maximum conformality
• Novel Perspectives for Hadron Physics
• Renormalisation