Commensurate Scale Relations Research Articles

Generalized Crewther relation and a novel demonstration of the scheme independence of commensurate scale relations up to all orders * *Supported by the Chongqing Graduate Research and Innovation Foundation (ydstd1912, CYB21045), the National Natural Science Foundation of China (11625520, 12047564) and the Fundamental Research Funds for the Central Universities (2020CQJQY-Z003)

In this study, we investigate in detail the generalized Crewther Relation (GCR) between the Adler function (D) and the Gross-Llewellyn Smith sum rules coefficient ( ) using the newly proposed single-scale approach of the principle of maximum conformality (PMC). The resultant GCR is scheme-independent, with the residual scale dependence due to unknown higher-order terms highly suppressed. Thus, a precise test of QCD theory without renormalization schemes and scale ambiguities can be achieved by comparing with data. Moreover, a demonstration of the scheme independence of the commensurate scale relation up to all orders is presented. Additionally, for the first time, the Pade approximation approach has been adopted for estimating the unknown -loop contributions from the known four-loop perturbative series.

The generalized scheme-independent Crewther relation in QCD

The Principle of Maximal Conformality (PMC) provides a systematic way to set the renormalization scales order-by-order for any perturbative QCD calculable processes. The resulting predictions are independent of the choice of renormalization scheme, a requirement of renormalization group invariance. The Crewther relation, which was originally derived as a consequence of conformally invariant field theory, provides a remarkable connection between two observables when the β function vanishes: one can show that the product of the Bjorken sum rule for spin-dependent deep inelastic lepton–nucleon scattering times the Adler function, defined from the cross section for electron–positron annihilation into hadrons, has no pQCD radiative corrections. The “Generalized Crewther Relation” relates these two observables for physical QCD with nonzero β function; specifically, it connects the non-singlet Adler function (Dns) to the Bjorken sum rule coefficient for polarized deep-inelastic electron scattering (CBjp) at leading twist. A scheme-dependent ΔCSB-term appears in the analysis in order to compensate for the conformal symmetry breaking (CSB) terms from perturbative QCD. In conventional analyses, this normally leads to unphysical dependence in both the choice of the renormalization scheme and the choice of the initial scale at any finite order. However, by applying PMC scale-setting, we can fix the scales of the QCD coupling unambiguously at every order of pQCD. The result is that both Dns and the inverse coefficient CBjp−1 have identical pQCD coefficients, which also exactly match the coefficients of the corresponding conformal theory. Thus one obtains a new generalized Crewther relation for QCD which connects two effective charges, αˆd(Q)=∑i≥1αˆg1i(Qi), at their respective physical scales. This identity is independent of the choice of the renormalization scheme at any finite order, and the dependence on the choice of the initial scale is negligible. Similar scale-fixed commensurate scale relations also connect other physical observables at their physical momentum scales, thus providing convention-independent, fundamental precision tests of QCD.

Systematic scale-setting to all orders: The principle of maximum conformality and commensurate scale relations

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 \beta-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 \beta-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 \alpha_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 [...]

Erratum: Scale setting using the extended renormalization group and the principle of maximum conformality: The QCD coupling constant at four loops [Phys. Rev. D85, 034038 (2012)

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 $\alpha^{\bar{MS}}_s(M_Z) =0.1184 \pm 0.0007$, we obtain the asymptotic scale for the 't Hooft associated with the $\bar{MS}$ scheme, $\Lambda^{'tH}_{\bar{MS}}= 245^{+9}_{-10}$ MeV, and the asymptotic scale for the conventional $\bar{MS}$ scheme, $\Lambda_{\bar{MS}}= 213^{+19}_{-8}$ MeV.

Setting the renormalization scale in QCD: The principle of maximum conformality

A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale $\mu$ of the running coupling $\alpha_s(\mu^2).$ The purpose of the running coupling in any gauge theory is to sum all terms involving the $\beta$ function; in fact, when the renormalization scale is set properly, all non-conformal $\beta \ne 0$ terms in a perturbative expansion arising from renormalization are summed into the running coupling. The remaining terms in the perturbative series are then identical to that of a conformal theory; i.e., the corresponding theory with $\beta=0$. The resulting scale-fixed predictions using the "principle of maximum conformality" (PMC) are independent of the choice of renormalization scheme -- a key requirement of renormalization group invariance. The results avoid renormalon resummation and agree with QED scale-setting in the Abelian limit. The PMC is also the theoretical principle underlying the BLM procedure, commensurate scale relations between observables, and the scale-setting method used in lattice gauge theory. The number of active flavors $n_f$ in the QCD $\beta$ function is also correctly determined. We discuss several methods for determining the PMC scale for QCD processes. We show that a single global PMC scale, valid at leading order, can be derived from basic properties of the perturbative QCD cross section. The elimination of the renormalization scale ambiguity and the scheme dependence using the PMC will not only increase the precision of QCD tests, but it will also increase the sensitivity of collider experiments to new physics beyond the Standard Model.

Nonperturbative QCD coupling and itsβfunction from light-front holography

The light-front holographic mapping of classical gravity in anti-de Sitter space, modified by a positive-sign dilaton background, leads to a nonperturbative effective coupling ${\ensuremath{\alpha}}_{s}^{\mathrm{AdS}}({Q}^{2})$. It agrees with hadron physics data extracted from different observables, such as the effective charge defined by the Bjorken sum rule, as well as with the predictions of models with built-in confinement and lattice simulations. It also displays a transition from perturbative to nonperturbative conformal regimes at a momentum scale $\ensuremath{\sim}1\text{ }\text{ }\mathrm{GeV}$. The resulting $\ensuremath{\beta}$ function appears to capture the essential characteristics of the full $\ensuremath{\beta}$ function of QCD, thus giving further support to the application of the gauge/gravity duality to the confining dynamics of strongly coupled QCD. Commensurate scale relations relate observables to each other without scheme or scale ambiguity. In this paper we extrapolate these relations to the nonperturbative domain, thus extending the range of predictions based on ${\ensuremath{\alpha}}_{s}^{\mathrm{AdS}}({Q}^{2})$.

The Conformal Template and New Perspectives for Quantum Chromodynamics

Conformal symmetry provides a systematic approximation to QCD in both its perturbative and nonperturbative domains. One can use the AdS/CFT correspondence between Anti-de Sitter space and conformal gauge theories to obtain an analytically tractable approximation to QCD in the regime where the QCD coupling is large and constant. For example, there is an exact correspondence between the fifth-dimensional coordinate of AdS space and a specific impact variable which measures the separation of the quark constituents within the hadron in ordinary space-time. This connection allows one to compute the analytic form of the frame-independent light-front wavefunctions of mesons and baryons, the fundamental entities which encode hadron properties and allow the computation of exclusive scattering amplitudes. One can also use conformal symmetry as a template for perturbative QCD predictions where the effects of the nonzero beta function can be systematically included in the scale of the QCD coupling. This leads to fixing of the renormalization scale and commensurate scale relations which relate observables without scale or scheme ambiguity. I also discuss a number of novel phenomenological features of QCD, including leading-twist single-spin asymmetries, diffractive deep inelastic scattering, hard diffractive hadronic reactions, the breakdown of the Lam-Tung relation in Drell-Yan reactions, nuclear shadowing, non-universal antishadowing, and the unexpected effects of direct higher-twist processes.

Perturbative QCD relations inspired by hypothetical tau leptons

We review our recent works on tests of perturbative QCD, inspired by the relation between the hadronic decay of the τ lepton and the e + e − annihilation into hadrons. First, we present a set of commensurate scale relations that probe the self-consistency of leading-twist QCD predictions for any observable which defines an effective charge. These tests are independent of the renormalization scheme and scale, and are applicable over wide data ranges. As an example we apply this approach to R e + e − . Second, using a differential form of these conmensurate scale relations, we present a method to measure the QCD Gell-Mann-Low Ψ function.

Renormalization-scale-invariant PQCD predictions forRe+e−and the Bjorken sum rule at next-to-leading order

We discuss the application of the physical QCD effective charge ${\ensuremath{\alpha}}_{V},$ defined via the heavy-quark potential, in perturbative calculations at next-to-leading order. When coupled with the Brodsky-Lepage-Mackenzie prescription for fixing the renormalization scales, the resulting series are automatically and naturally scale and scheme independent, and represent unambiguous predictions of perturbative QCD. We consider in detail such commensurate scale relations for the ${e}^{+}{e}^{\ensuremath{-}}$ annihilation ratio ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ and the Bjorken sum rule. In both cases the improved predictions are in excellent agreement with experiment.

Two-loop scale dependence of the static QCD potential including quark masses

The interaction potential V(Q^2) between static test charges can be used to define an effective charge $\alpha_V(Q^2)$ and a physically-based renormalization scheme for quantum chromodynamics and other gauge theories. In this paper we use recent results for the finite-mass fermionic corrections to the heavy-quark potential at two-loops to derive the next-to-leading order term for the Gell Mann-Low function of the V-scheme. The resulting effective number of flavors $N_F(Q^2/m^2)$ in the $\alpha_V$ scheme is determined as a gauge-independent and analytic function of the ratio of the momentum transfer to the quark pole mass. The results give automatic decoupling of heavy quarks and are independent of the renormalization procedure. Commensurate scale relations then provide the next-to-leading order connection between all perturbatively calculable observables to the analytic and gauge-invariant $\alpha_V$ scheme without any scale ambiguity and a well defined number of active flavors. The inclusion of the finite quark mass effects in the running of the coupling is compared with the standard treatment of finite quark mass effects in the $\bar{MS}$ scheme.

Testing QCD with hypothetical tau leptons

We construct new tests of perturbative QCD by considering a hypothetical tau lepton of arbitrary mass, which decays hadronically through the electromagnetic current. We can explicitly compute its hadronic width ratio directly as an integral over the e^+ e^- annihilation cross section ratio, R_{e^+e^-}. Furthermore, we can design a set of commensurate scale relations and perturbative QCD tests by varying the weight function away from the form associated with the V-A decay of the physical tau. This method allows the wide range of the R_{e^+e^-} data to be used as a probe of perturbative QCD.

Optimal renormalization scale and scheme for exclusive processes

We use the Brodsky-Lepage-Mackenzie (BLM) method to fix the renormalization scale of the QCD coupling in exclusive hadronic amplitudes such as the pion form factor and the photon-to-pion transition form factor at large momentum transfer. Renormalization-scheme-independent commensurate scale relations are established which connect the hard scattering subprocess amplitudes that control exclusive processes to other QCD observables such as the heavy quark potential and the electron-positron annihilation cross section. The commensurate scale relation connecting the heavy quark potential, as determined from lattice gauge theory, to the photon-to-pion transition form factor is in excellent agreement with $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{e}{\ensuremath{\pi}}^{0}e$ data assuming that the pion distribution amplitude is close to its asymptotic form $\sqrt{3}{f}_{\ensuremath{\pi}}x(1\ensuremath{-}x)$. We also reproduce the scaling and normalization of the $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ data at large momentum transfer. Because the renormalization scale is small, we argue that the effective coupling is nearly constant, thus accounting for the nominal scaling behavior of the data. However, the normalization of the space-like pion form factor ${F}_{\ensuremath{\pi}}{(Q}^{2})$ obtained from electroproduction experiments is somewhat higher than that predicted by the corresponding commensurate scale relation. This discrepancy may be due to systematic errors introduced by the extrapolation of the ${\ensuremath{\gamma}}^{*}\stackrel{\ensuremath{\rightarrow}}{p}{\ensuremath{\pi}}^{+}n$ electroproduction data to the pion pole.

Commensurate scale relations in quantum chromodynamics.

We use the BLM method to show that perturbatively-calculable observables in QCD can be related to each other without renormalization scale or scheme ambiguity. We define and study the commensurate scale relations. We show that the commensurate scales satisfy the renormalization group transitivity rule which ensures that predictions in PQCD are independent of the choice of an intermediate renormalization scheme. We generalize the BLM procedure to higher order. The application of this procedure to relate known physical observables in QCD gives surprisingly simple results. In particular, the annihilation ratio $R_{e^+e^-}$ and the Bjorken sum rule for polarized electroproduction are related through simple coefficients, which reinforces the idea of a hidden symmetry between these two observables.

Commensurate scale relations in quantum chromodynamics

We use the BLM method to directly relate perturbatively calculable QCD observables. The commensurate scale relations connecting the effective charges for observables A and B have the form αA(QA) = αB(QB) (1 + ΓA/BαBπ + …), where the coefficient rA/B is independent of the number of flavors f contributing to coupling constant renormalization. The generalization of the BLM procedure to higher order assigns a different renormalization scale for each order in the perturbative series. The coefficients in the commensurate scale relation can be identified with those obtained in conformally-invariant gauge theory. The application of this procedure to perturbatively-calculable physical observables in QCD gives rather simple results. For example, the relation between the annihilation ratio Re+e−1 and the polarized Bjorken sum rule is given by α¾g1(Q) = α¾R(Q*) − α¾R2(Q**) + α¾R3(Q***), where α¾ = (3CF/4π)α. This relation provides the generalization of the Crewther relation to non-conformally invariant gauge theory.