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 α_s^4; 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 gluon field, in accordance with conformality, and with all non-conformal 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 [...]

20 pages; Appendix added. This version matches the published paper

References in corpus (12)

• Hadronic Z- and tau-Decays in Order alpha_s^4
• Adler Function, Bjorken Sum Rule, and the Crewther Relation to Order alpha_s^4 in a General Gauge Theory
• Eliminating the Renormalization Scale Ambiguity for Top-Pair Production Using the Principle of Maximum Conformality
• The Renormalization Scale-Setting Problem in QCD
• Fermionic contributions to the three-loop static potential
• First determination of the strong coupling constant using NNLO predictions for hadronic event shapes in e^+e^- annihilations
• 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
• The Form Factors of the Gauge-Invariant Three-Gluon Vertex
• Hot Conformal Gauge Theories
• The Higgs boson inclusive decay channels $H\to b\bar{b}$ and $H\to gg$ up to four-loop level
• The singlet contribution to the Bjorken sum rule for polarized deep inelastic scattering

Cited by in corpus (123)

• Five-Loop Running of the QCD coupling constant
• The QCD Running Coupling
• The Large Hadron-Electron Collider at the HL-LHC
• The Renormalization Scale-Setting Problem in QCD
• Evidence for high-energy resummation effects in Mueller-Navelet jets at the LHC
• The Spin Structure of the Nucleon
• Measurement of the forward-backward asymmetry in top quark-antiquark production in ppbar collisions using the lepton+jets channel
• Renormalization Group Invariance and Optimal QCD Renormalization Scale-Setting
• The QCD Renormalization Group Equation and the Elimination of Fixed-Order Scheme-and-Scale Ambiguities Using the Principle of Maximum Conformality
• QCD Running Couplings and Effective Charges
• 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
• Conformal Behavior at Four Loops and Scheme (In)Dependence
• Degeneracy Relations in QCD and the Equivalence of Two Systematic All-Orders Methods for Setting the Renormalization Scale
• A novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of maximum conformality and the $C$-scheme coupling
• Infrared Fixed Points in the minimal MOM Scheme
• 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
• New Scheme Transformations and Application to Study Scheme Dependence of an Infrared Zero of the Beta Function in Gauge Theories
• 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
• Determination of ${\Lambda }_{\overline{MS}}$ at five loops from holographic QCD
• Bjorken sum rule in QCD frameworks with analytic (holomorphic) coupling
• QCD corrections to the $B_c$ to charmonia semi-leptonic decays
• Scheme-Independent Calculations of Anomalous Dimensions of Baryon Operators in Conformal Field Theories
• Production of semi-inclusive doubly heavy baryons via top-quark decays
• Novel Method for the Precise Determination of the QCD Running Coupling from Event Shapes Distributions in Electron-Positron Annihilation
• On the BLM optimal renormalization scale setting for semihard processes
• Production of doubly heavy baryons via Higgs boson decays
• 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
• Exclusive production of $J/\psi +{\eta }_c$ at the $B$ factories Belle and Babar using the principle of maximum conformality
• Adler function and Bjorken polarized sum rule: Perturbation expansions in powers of $SU(N_c)$ conformal anomaly and studies of the conformal symmetry limit
• The Importance of Proper Renormalization Scale-Setting for Testing QCD at Colliders
• The Generalized Scheme-Independent Crewther Relation in QCD
• The three-loop Adler $D$-function for $\mathcal{N}=1$ SQCD regularized by dimensional reduction
• The longitudinal leading-twist distribution amplitude of $J/\psi$ meson within background field theory
• Fractional Analytic QCD beyond Leading Order in timelike region
• Novel and self-consistency analysis of the QCD running coupling ${\alpha }_s(Q)$ in both the perturbative and nonperturbative domains
• The QCD improved electroweak parameter $\rho$
• Predictions for the Top-Quark Forward-Backward Asymmetry at High Invariant Pair Mass Using the Principle of Maximum Conformality
• Renormalization Scheme Dependence with Renormalization Group Summation
• BLM Scale Fixing in Event Shape Distributions
• Ambiguities of the principle of maximum conformality procedure for hadron collider processes
• Precise perturbative predictions from fixed-order calculations
• The heavy quarkonium inclusive decays using the principle of maximum conformality
• Revisiting the Heavy Vector Quarkonium Leptonic Widths
• Extending the Predictive Power of Perturbative QCD Using the Principle of Maximum Conformality and Bayesian Analysis
• Hard production of a Z boson plus heavy flavor jets at LHC and the intrinsic charm content of a proton
• Renormalization group summation and analytic continuation from spacelike to timeline regions
• The role of low-energy observables in precision Higgs analysis
• 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
• Hadronic decays of the spin-singlet heavy quarkomium under the principle of maximum conformality
• General Properties on Applying the Principle of Minimum Sensitivity to High-order Perturbative QCD Predictions
• 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
• A precise determination of the top-quark pole mass
• A solution to the $\gamma {\gamma }^{\ast }\to {\eta }_c$ puzzle using the Principle of Maximum Conformality
• Gauge dependence of the perturbative QCD predictions under the momentum space subtraction scheme
• Application of the Principle of Maximum Conformality to the Hadroproduction of the Higgs Boson at the LHC
• Next-to-next-to-leading-order QCD corrections to $J/\psi$ plus ${\eta }_c$ production at the $B$ factories
• A novel determination of non-perturbative contributions to Bjorken sum rule
• 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
• Properties of the decay $H\to \gamma \gamma$ using the approximate ${\alpha }_s^4$-corrections and the principle of maximum conformality
• QCD improved top-quark decay at next-to-next-to-leading order
• 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
• A Possible Solution to the $B\to \pi \pi$ Puzzle Using the Principle of Maximum Conformality
• Comparing optimized renormalization schemes for QCD observables
• Next-to-leading order QCD corrections to the production of $B_c$ and $B_c^{\ast }$ through $W^{+}$-boson decays
• Detailed Comparison of Renormalization Scale-Setting Procedures based on the Principle of Maximum Conformality
• Renormalization Scale Setting for Heavy Quark Pair Production in $e^{+}{e}^{-}$ Annihilation near the Threshold Region
• 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
• $Z$-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
• 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
• Renormalization scheme dependence of high-Order perturbative QCD predictions
• Reanalysis of the top-quark pair production via the $e^{+}{e}^{-}$ annihilation near the threshold region up to N${}^3$LO QCD corrections
• Achieving effective renormalization scale and scheme independence via the Principle of Observable Effective Matching
• Revisiting the Bottom Quark Forward-Backward Asymmetry $A_{\mathrm{FB}}$ in Electron-Positron Collisions
• Light-Front Holographic QCD and Color Confinement
• Light-Front Holographic QCD and the Confinement Potential
• Reanalysis of the top-quark pair hadroproduction and a precise determination of the top-quark pole mass at the LHC
• Self-consistent analysis for the ${\eta }_c\to \gamma \gamma$ process
• Anomalous Dimensions at an Infrared Fixed Point in an SU($N_c$) Gauge Theory with Fermions in the Fundamental and Antisymmetric Tensor Representations
• An updated determination of the pion-photon transition form factor
• Elimination of QCD Renormalization Scale and Scheme Ambiguities
• 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 ${\overline{m}}_t({\overline{m}}_t)$
• A new analysis of the pQCD contributions to the electroweak parameter $\rho$ using the single-scale approach of principle of maximum conformality
• Revisiting the top-quark pair production at future $e^{+}{e}^{-}$ colliders
• New determination of $|V_{\mathrm{cb}}|$ using the three-loop QCD corrections for the $B\to D^{\ast }$ semi-leptonic decays
• The ${\rm Y}(1S)$ leptonic decay using the principle of maximum conformality
• 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
• A Closer Look at Gluons
• The decomposed photon anomalous dimension in QCD and the $\{\beta \}$-expanded representations for the Adler function
• Approximate N${}^5$LO Higgs boson decay width $\Gamma (H\to \gamma \gamma )$
• Precise determination of the bottom-quark on-shell mass using its four-loop relation to the $\overline{\mathrm{MS}}$-scheme running mass
• The Principle of Maximum Conformality Correctly Resolves the Renormalization-Scheme-Dependence Problem
• Scale-invariant total decay width $\Gamma (H\to b\bar{b})$ using the novel method of characteristic operator
• Next-to-leading order QCD corrections to $Z\to {\eta }_Q+Q+\bar{Q}$
• Study of the Ultraviolet Behavior of an O($N$) $|\vec{\varphi }{|}^6$ Theory in $d=3$ Dimensions
• The Higgs-Boson Decay $H\to gg$ to Order ${\alpha }_s^5$ under the mMOM-Scheme
• 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
• Light-Front Holographic Quantum Chromodynamics
• Total decay width of $H\to gg$ using the infinite-order scale-setting approach based on the intrinsic conformality
• A reanalysis of event shape distributions in electron-positron annihilation
• Analysis of the Pion Electromagnetic Form Factor with Next-to-Next-to-Leading Order QCD Corrections
• Deviation pattern approach for optimizing perturbative terms of QCD renormalization group invariant observables
• Perturbative QCD fitting of KEDR and BESIII $e^{+}{e}^{-}$ data for R(s) and ${\alpha }_s$ determination
• Renormalisation
• Do we understand the internal spaces of second quantized fermion and boson fields, with gravity included? Relation with strings theories
• Properties of the Free Energy Density Using the Principle of Maximum Conformality
• Proceedings to the 27th Workshop "What Comes Beyond the Standard Models" Bled, July 8-17, 2024
• Relation between pole and running masses of heavy quarks using the principle of maximum conformality
• Quark mass matrices inspired by a numerical relation
• Novel Heavy-Quark Physics Phenomena
• ${\eta }_c$ leading-twist distribution amplitude and the $B_c\to {\eta }_c\ell {\bar{\nu }}_{\ell }$ semileptonic decays using QCD Sum Rules