Complete Renormalization Group Improvement Research Articles

Comparing optimized renormalization schemes for QCD observables

Based on the renormalization group summation method of McKeon ${\it et\; al.}$, it is shown that the renormalization group equation, while related to the radiatively mass scale $\mu$, would perform a summation over QCD perturbative terms. Employing the full QCD $\beta$-function within this summation, all logarithmic corrections can be presented as log-independent contributions. In another step of this approach, the renormalization scheme dependence of QCD observables, characterized by Stevenson, is required to be examined. In this regard, two choices of renormalization scheme would be exposed. In one of them, the QCD observable is expressed in terms of two powers of running coupling constant. In the other one, the perturbative series expansion is written as an infinite series in terms of the two-loop running coupling represented by the Lambert $W$-function. In both cases, the QCD observable involves parameters which are renormalization scheme invariant and a coupling constant which is independent of the renormalization scale. We then consider the approach of complete renormalization group improvement (CORGI). In this approach, using the self consistency principle, it is possible to reconstruct the conventional perturbative series in terms of scheme invariant quantities and the coupling constant as a function of Lambert $W$-function. One of the key differences between CORGI and the renormalization group summation method of McKeon ${\it et\; al.}$ is that the later treats renormalization scale and scheme independently while the former would encounter both dependencies together. In continuation, numerical results of the two approaches are compared for $R_{e^+e^-}$ ratio and the Higgs decay width to gluon-gluon.

Principle of maximum conformality and complete renormalization group improvement approaches to optimize QCD observables: Similarity, differences, and preference

The principle of maximum conformality (PMC) in perturbative QCD predicts energy scales which are independent of the renormalization scheme and choice of the renormalization scale. In this approach the observable can be divided to the conformal and nonconformal sectors. Nonconformal dependence terms are absorbed into the running coupling constant. By absorbing these terms at any desired order, a unique scale can be obtained at that order, and the final result for the observable is independent of the initial choice of renormalization scale and also the employed scheme. This feature is in accordance with the invariant property of renormalization group. On the other side, we consider another approach which is called the complete renormalization group improvement (CORGI) to optimize the perturbative series of QCD observables. In this approach, using the self-consistency principle, all perturbative terms can be reconstructed such that they are expressed in terms of the quantities which do not depend on the renormalization scale. Each reconstructed term in this approach involves a resummation of infinite terms at a specified order. Therefore the conventional perturbative series of QCD observables appear in terms of scheme-invariant quantities. This approach is then expected to get more reliable results with respect to what we obtain in conventional perturbation theory. Considering these two approaches, we are looking to find the differences and similarities and the probable relation which might exist between them. Since the PMC approach is founded on more strong theoretical bases, the comparison between these two approaches will reveal the priority which naturally should be assigned to the PMC approach. We examine some QCD observables like the R-ratio for ${e}^{+}{e}^{\ensuremath{-}}$ annihilation, ${R}_{\ensuremath{\tau}}$ for $\ensuremath{\tau}$ decay, and Higgs decay width to $b\overline{b}$ and gluon-gluon, and we compare the results for these observable in two approaches. For QCD observables in which there are experimental data, the results we obtain from both CORGI and PMC approaches are in good agrement with the experimental data. In other cases, the results from theses two approaches are near to each other; however, they are a little different from the results of the conventional perturbation theory.

Renormalization scale setting for some QCD observable, based on the PMC and CORGI approaches

In this paper we present the principle of Maximum conformality (PMC) and Complete Renormalization Group Improvement (CORGI) approaches of scale setting which are based essentially on Refs. [S.J. Brodsky, M. Mojaza, X.G. Wu, Phys. Rev. D 89 (2014) 014027; X.G. Wu,Y. Ma, S.Q. Wang, S.J. Brodsky and M. Mojaza, Rep. Prog. Phys. 78 (2015) 126201; C.J. Maxwell, Nucl. Phys. Proc. Suppl. 86 (2000) 74; C.J. Maxell, A. Mirjalili, Nucl. Phys. B 299 (2000) 289]. The PMC establishes a systematic method to eliminate the renormalization scheme and renormalization scale dependencies of conventional perterbative QCD predictions. This approach, using the renormalization group equation, exposes a pattern of βi−terms in the perterbative series. Then by absorbing all βi−terms into the running coupling constant, one can obtain a result that is independent of renormalization scale and scheme. On the other hand the CORGI approach improves the precision of calculations. In this approach the standard perturbative series of QCD observable reconstructed in terms of scheme-invariant quantities. Here, we will examine the PMC and CORGI approaches by comparing its predictions for the electron-positron annihilation to hadrons and the Higgs decay width to gluon. The two results are compared with each other and with the available experimental data. At the end we may conclude which approach has more advantage in considering the perturbative series.

Improved analysis of moments of [formula omitted] in neutrino–nucleon scattering using the Bernstein polynomial method

We use recently calculated next-to-next-to-leading order (NNLO) anomalous dimension coefficients for the moments of the x F 3 structure function in νN scattering, together with the corresponding three-loop Wilson coefficients, to obtain improved QCD predictions for both odd and even moments of F 3 . To investigate the issue of renormalization scheme dependence, the Complete Renormalization Group Improvement (CORGI) approach is used, in which all dependence on renormalization and factorization scales is avoided by a complete resummation of RG-predictable scale logarithms. We also consider predictions using the method of effective charges, and compare with the standard ‘physical scale’ choice. The Bernstein polynomial method is used to construct experimental moments (from the x F 3 data of the CCFR Collaboration) that are insensitive to the value of x F 3 in the region of x which is inaccessible experimentally. Direct fits for Λ MS ¯ ( 5 ) ( α s ( M Z ) ) are then performed. The CORGI fits including target mass corrections give a value α s ( M Z ) = 0.1189 −0.0019 +0.0019 , consistent with the world average. The effective charge and physical scale fits give slightly smaller values, which are still consistent within the errors.

QCD analysis of heavy quarks production in hadronic collisions

The problem of renormalization scheme dependence in QCD perturbation theory remains on obstacle to making precise tests of the theory. The renormalization scale dependence of dimensionless physical QCD observable, depending on a single energy scale Q , can be avoided provided that all ultraviolet logarithms which build the physical energy dependence on Q are resummed. This was termed complete Renormalization Group improvement(CORGI). This argument can be extended to processes involving factorization of operator matrix elements and coefficient functions. We are trying to employ the idea of CORGI approach on analyzing of heavy quarks production in hadron collisions. There is a sizable and systematic discrepancy between experimental data on the b b ¯ production in p p ¯ , γp and γγ collisions and existing theoretical calculations within perturbative QCD. One suggested way to cope with this discrepancy is to employ the CORGI approach in which one should perform a resummation to all-orders of renormalization and factorization group -predictable terms at each order of perturbation theory. Then the scales dependence will be avoided and it is expected that the mentioned discrepancy is reduced significantly.

DETERMINATION OF QCD RUNNING COUPLING CONSTANT IN NNLO APPROXIMATION, USING COMPLETE RG IMPROVEMENT

To obtain the improved QCD predictions for the moments of the xF3 structure function in νN scattering, three-loop Wilson coefficients with the corresponding NNLO anomalous dimension coefficients is used. A comparison between QCD predictions and the xF3 data of the CCFR collaboration has been done in which the Bernstein Polynomial method is used in the complete renormalization group improvement approach. A direct fits for [Formula: see text] over the corresponding range of Q2 with five effective quark flavours, is performed.

Direct extraction of QCD [formula omitted] from moments of structure functions in neutrino–nucleon scattering, using the CORGI approach

We use recently calculated next-to-next-to-leading (NNLO) anomalous dimension coefficients for the n=1,3,5,…,13 moments of the xF 3 structure function in νN scattering, together with the corresponding three-loop Wilson coefficients, to obtain improved QCD predictions for the moments. The Complete Renormalization Group Improvement (CORGI) approach is used, in which all dependence on renormalization or factorization scales is avoided by a complete resummation of ultraviolet logarithms. The Bernstein Polynomial method is used to compare these QCD predictions to the xF 3 data of the CCFR Collaboration, and direct fits for Λ MS (5) , with N f =5 effective quark flavours, over the range 20< Q 2<125.9 GeV 2, were performed. We obtain Λ MS (5)=202 +54 −45 MeV, corresponding to the three-loop running coupling α s ( M Z )=0.1174 +0.0043 −0.0043. Including target mass corrections as well we obtain Λ MS (5)=228 +35 −36 MeV, corresponding to α s ( M Z )=0.1196 +0.0027 −0.0031.

Renormalon-inspired resummations for vector and scalar correlators — estimating the uncertainty in αs( M2τ) and α( M2Z)

We perform an all-orders resummation of the QCD Adler D-function for the vector correlator, in which the portion of perturbative coefficients containing the leading power of b, the first beta-function coefficient, is resummed. To avoid a renormalization scale dependence when we match the resummation to the exactly known next-to-leading order (NLO), and next-NLO (NNLO) results, we employ the Complete Renormalization Group Improvement (CORGI) approach in which all RG-predictable ultraviolet logarithms are resummed to all-orders, removing all dependence on the renormalization scale. We can also obtain fixed-order CORGI results. Including suitable weight-functions we can numerically integrate these results for the D-function in the complex energy plane to obtain so-called “contour-improved” results for the ratio R and its tau decay analogue R τ . We use the difference between the all-orders and fixed-order (NNLO) results to estimate the uncertainty in α s ( M 2 Z ) extracted from R τ measurements, and find α s ( M 2 Z )=0.120±0.002. We also estimate the corresponding uncertainty in α( M 2 Z ) arising from hadronic corrections by considering the uncertainty in R( s), in the low-energy region, and compare with other estimates. Analogous resummations are also given for the scalar correlator. As an adjunct to these studies we show how fixed-order contour-improved results can be obtained analytically in closed form at the two-loop level in terms of the Lambert W-function and hypergeometric functions.

Complete renormalization group improvement — avoiding factorization and renormalization scale dependence in QCD predictions

For moments of leptoproduction structure functions we show that all dependence on the renormalization and factorization scales disappears provided that all the ultraviolet logarithms involving the physical energy scale Q are completely resummed. The approach is closely related to Grunberg's method of effective charges. A direct and simple method for extracting Λ MS from experimental data is advocated.