Download or read book The Renormalization Scale Setting Problem in QCD written by and published by . This book was released on 2013 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt: A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The conventional scale-setting procedure assigns an arbitrary range and an arbitrary systematic error to fixed-order pQCD predictions. In fact, this ad hoc procedure gives results which depend on the choice of the renormalization scheme, and it is in conflict with the standard scale-setting procedure used in QED. Predictions for physical results should be independent of the choice of the scheme or other theoretical conventions. We review current ideas and points of view on how to deal with the renormalization scale ambiguity and show how to obtain renormalization scheme- and scale-independent estimates. We begin by introducing the renormalization group (RG) equation and an extended version, which expresses the invariance of physical observables under both the renormalization scheme and scale-parameter transformations. The RG equation provides a convenient way for estimating the scheme- and scale-dependence of a physical process. We then discuss self-consistency requirements of the RG equations, such as reflexivity, symmetry, and transitivity, which must be satisfied by a scale-setting method. Four typical scale setting methods suggested in the literature, i.e., the Fastest Apparent Convergence (FAC) criterion, the Principle of Minimum Sensitivity (PMS), the Brodsky-Lepage-Mackenzie method (BLM), and the Principle of Maximum Conformality (PMC), are introduced. Basic properties and their applications are discussed. We pay particular attention to the PMC, which satisfies all of the requirements of RG invariance. Using the PMC, all non-conformal terms associated with the [beta]-function in the perturbative series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC provides the principle underlying the BLM method, since it gives the general rule for extending BLM up to any perturbative order; in fact, they are equivalent to each other through the PMC-BLM correspondence principle. Thus, all the features previously observed in the BLM literature are also adaptable to the PMC. The PMC scales and the resulting finite-order PMC predictions are to high accuracy independent of the choice of the initial renormalization scale, and thus consistent with RG invariance. The PMC is also consistent with the renormalization scale-setting procedure for QED in the zero-color limit. The use of the PMC thus eliminates a serious systematic scale error in perturbative QCD predictions, greatly improving the precision of empirical tests of the Standard Model and their sensitivity to new physics.

Download or read book Setting the Renormalization Scale in QCD written by and published by . This book was released on 2011 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale [mu] of the running coupling [alpha]{sub 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] ≠ 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 nf in the QCD [beta] function is also correctly determined. We discuss several methods for determining the PMC/BLM 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 scheme ambiguity 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.

Download or read book Renormalization Group Invariance and Optimal QCD Renormalization Scale Setting written by and published by . This book was released on 2014 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Novel All Orders Single Scale Approach to QCD Renormalization Scale Setting written by and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Importance of Proper Renormalization Scale setting for QCD Testing at Colliders written by and published by . This book was released on 2015 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt: A primary problem affecting perturbative quantum chromodynamic (pQCD) analyses is the lack of a method for setting the QCD running-coupling renormalization scale such that maximally precise fixed-order predictions for physical observables are obtained. The Principle of Maximum Conformality (PMC) eliminates the ambiguities associated with the conventional renormalization scale-setting procedure, yielding predictions that are independent of the choice of renormalization scheme. The QCD coupling scales and the effective number of quark flavors are set order-by-order in the pQCD series. The PMC has a solid theoretical foundation, satisfying the standard renormalization group invariance condition and all of the self-consistency conditions derived from the renormalization group. The PMC scales at each order are obtained by shifting the arguments of the strong force coupling constant ?s to eliminate all non-conformal {βi} terms in the pQCD series. The {βi} terms are determined from renormalization group equations without ambiguity. The correct behavior of the running coupling at each order and at each phase-space point can then be obtained. The PMC reduces in the NC → 0 Abelian limit to the Gell-Mann-Low method. In this brief report, we summarize the results of our recent application of the PMC to a number of collider processes, emphasizing the generality and applicability of this approach. A discussion of hadronic Z decays shows that, by applying the PMC, one can achieve accurate predictions for the total and separate decay widths at each order without scale ambiguities. We also show that, if one employs the PMC to determine the top-quark pair forward-backward asymmetry at the next-to-next-to-leading order level, one obtains a comprehensive, self-consistent pQCD explanation for the Tevatron measurements of the asymmetry. This accounts for the ?increasing-decreasing? behavior observed by the D0 collaboration for increasing tt¯ invariant mass. At lower energies, the angular distributions of heavy quarks can be used to obtain a direct determination of the heavy quark potential. A discussion of the angular distributions of massive quarks and leptons is also presented, including the fermionic component of the two-loop corrections to the electromagnetic form factors. Furthermore, these results demonstrate that the application of the PMC systematically eliminates a major theoretical uncertainty for pQCD predictions, thus increasing collider sensitivity to possible new physics beyond the Standard Model.

Download or read book Scale Setting Using the Extended Renormalization Group and the Principle of Maximal Conformality written by and published by . This book was released on 2012 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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?{sub 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{sub e{sup +}e−}(Q) up to four loops is presented. By using the world average?{sub s}{sup {ovr MS}}(MZ) = 0.1184 ± 0.0007, we obtain the asymptotic scale for the 't Hooft associated with the {ovr MS} scheme,?{sub {ovr MS}}{sup 'tH} = 24510{sup +9} MeV, and the asymptotic scale for the conventional {ovr MS} scheme,?{sub {ovr MS}} = 213−{sup +19} MeV.

Download or read book Degeneracy Relations in QCD and the Equivalence of Two Systematic All orders Methods for Setting the Renormalization Scale written by and published by . This book was released on 2015 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Principle of Maximum Conformality (PMC) eliminates QCD renormalization scale-setting uncertainties using fundamental renormalization group methods. The resulting scale-fixed pQCD predictions are independent of the choice of renormalization scheme and show rapid convergence. The coefficients of the scale-fixed couplings are identical to the corresponding conformal series with zero ?-function. Two all-orders methods for systematically implementing the PMC-scale setting procedure for existing high order calculations are discussed in this article. One implementation is based on the PMC-BLM correspondence (PMC-I); the other, more recent, method (PMC-II) uses the R?-scheme, a systematic generalization of the minimal subtraction renormalization scheme. Both approaches satisfy all of the principles of the renormalization group and lead to scale-fixed and scheme-independent predictions at each finite order. In this work, we show that PMC-I and PMC-II scale-setting methods are in practice equivalent to each other. We illustrate this equivalence for the four-loop calculations of the annihilation ratio Re+e– and the Higgs partial width I'(H→bb¯). Both methods lead to the same resummed (‘conformal’) series up to all orders. The small scale differences between the two approaches are reduced as additional renormalization group {βi}-terms in the pQCD expansion are taken into account. In addition, we show that special degeneracy relations, which underly the equivalence of the two PMC approaches and the resulting conformal features of the pQCD series, are in fact general properties of non-Abelian gauge theory.

Download or read book The Principle of Maximum Conformality written by and published by . This book was released on 2011 with total page 2 pages. Available in PDF, EPUB and Kindle. Book excerpt: A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale of the running coupling [alpha]{sub s}([mu]2). It is common practice to guess a physical scale [mu] = Q which is of order of a typical momentum transfer Q in the process, and then vary the scale over a range Q/2 and 2Q. This procedure is clearly problematic since the resulting fixed-order pQCD prediction will depend on the renormalization scheme, and it can even predict negative QCD cross sections at next-to-leading-order. Other heuristic methods to set the renormalization scale, such as the 'principle of minimal sensitivity', give unphysical results for jet physics, sum physics into the running coupling not associated with renormalization, and violate the transitivity property of the renormalization group. Such scale-setting methods also give incorrect results when applied to Abelian QED. Note that the factorization scale in QCD is introduced to match nonperturbative and perturbative aspects of the parton distributions in hadrons; it is present even in conformal theory and thus is a completely separate issue from renormalization scale setting. The PMC provides a consistent method for determining the renormalization scale in pQCD. The PMC scale-fixed prediction is 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 global scale can be derived efficiently at NLO from basic properties of the PQCD cross section. The elimination of the renormalization scheme ambiguity using the PMC will not only increases the precision of QCD tests, but it will also increase the sensitivity of colliders to new physics beyond the Standard Model.

Download or read book Self Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale written by and published by . This book was released on 2012 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt: In conventional treatments, predictions from fixed-order perturbative QCD calculations cannot be fixed with certainty due to ambiguities in the choice of the renormalization scale as well as the renormalization scheme. In this paper we present a general discussion of the constraints of the renormalization group (RG) invariance on the choice of the renormalization scale. We adopt the RG based equations, which incorporate the scheme parameters, for a general exposition of RG invariance, since they simultaneously express the invariance of physical observables under both the variation of the renormalization scale and the renormalization scheme parameters. We then discuss the self-consistency requirements of the RG, such as reflexivity, symmetry, and transitivity, which must be satisfied by the scale-setting method. The Principle of Minimal Sensitivity (PMS) requires the slope of the approximant of an observable to vanish at the renormalization point. This criterion provides a scheme-independent estimation, but it violates the symmetry and transitivity properties of the RG and does not reproduce the Gell-Mann-Low scale for QED observables. The Principle of Maximum Conformality (PMC) satisfies all of the deductions of the RG invariance - reflectivity, symmetry, and transitivity. Using the PMC, all non-conformal {{u03B2}Ri}-terms (R stands for an arbitrary renormalization scheme) in the perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC scales and the resulting finite-order PMC predictions are both to high accuracy independent of the choice of initial renormalization scale, consistent with RG invariance.

Download or read book Renormalization scale dependence in QCD written by Thomas Hebbeker and published by . This book was released on 1993 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Choosing the Factorization renormalization Scale in Perturbative QCD Calculations written by and published by . This book was released on 1990 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Setting the Renormalization Scale in PQCD written by and published by . This book was released on 2015 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: A key problem in making precise perturbative QCD (pQCD) predictions is how to set the renormalization scale of the running coupling unambiguously at each finite order. The elimination of the uncertainty in setting the renormalization scale in pQCD will greatly increase the precision of collider tests of the Standard Model and the sensitivity to new phenomena. Renormalization group invariance requires that predictions for observables must also be independent on the choice of the renormalization scheme. The well-known Brodsky-Lepage-Mackenzie (BLM) approach cannot be easily extended beyond next-to-next-to-leading order of pQCD. Several suggestions have been proposed to extend the BLM approach to all orders. In this paper we discuss two distinct methods. One is based on the "Principle of Maximum Conformality" (PMC), which provides a systematic all-orders method to eliminate the scale and scheme ambiguities of pQCD. The PMC extends the BLM procedure to all orders using renormalization group methods; as an outcome, it significantly improves the pQCD convergence by eliminating renormalon divergences. An alternative method is the "sequential extended BLM" (seBLM) approach, which has been primarily designed to improve the convergence of pQCD series. The seBLM, as originally proposed, introduces auxiliary fields and follows the pattern of the [beta]0-expansion to fix the renormalization scale. However, the seBLM requires a recomputation of pQCD amplitudes including the auxiliary fields; due to the limited availability of calculations using these auxiliary fields, the seBLM has only been applied to a few processes at low orders. In order to avoid the complications of adding extra fields, we propose a modified version of seBLM which allows us to apply this method to higher orders. As a result, we then perform detailed numerical comparisons of the two alternative scale-setting approaches by investigating their predictions for the annihilation cross section ratio Re+e- at four-loop order in pQCD.

Download or read book Eliminating the Renormalization Scale Ambiguity for Top Pair Production Using the Principle of Maximum Conformality written by and published by . This book was released on 2012 with total page 4 pages. Available in PDF, EPUB and Kindle. Book excerpt: The uncertainty in setting the renormalization scale in finite-order perturbative QCD predictions using standard methods substantially reduces the precision of tests of the Standard Model in collider experiments. It is conventional to choose a typical momentum transfer of the process as the renormalization scale and take an arbitrary range to estimate the uncertainty in the QCD prediction. However, predictions using this procedure depend on the choice of renormalization scheme, leave a non-convergent renormalon perturbative series, and moreover, one obtains incorrect results when applied to QED processes. In contrast, if one fixes the renormalization scale using the Principle of Maximum Conformality (PMC), all non-conformal {l_brace}?{sub i}{r_brace}-terms in the perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC renormalization scale?{sub R}{sup PMC} and the resulting finite-order PMC prediction are both to high accuracy independent of choice of the initial renormalization scale?{sub R}{sup init}, consistent with renormalization group invariance. Moreover, after PMC scale-setting, the n!-growth of the pQCD expansion is eliminated. Even the residual scale-dependence at fixed order due to unknown higher-order {l_brace}?{sub i}{r_brace}-terms is substantially suppressed. As an application, we apply the PMC procedure to obtain NNLO predictions for the t{bar t}-pair hadroproduction cross-section at the Tevatron and LHC colliders. There are no renormalization scale or scheme uncertainties, thus greatly improving the precision of the QCD prediction. The PMC prediction for?{sub t{bar t}} is larger in magnitude in comparison with the conventional scale-setting method, and it agrees well with the present Tevatron and LHC data. We also verify that the initial scale-independence of the PMC prediction is satisfied to high accuracy at the NNLO level: the total cross-section remains almost unchanged even when taking very disparate initial scales?{sub R}{sup init} equal to m{sub t}, 20 m{sub t}, √s.

Download or read book Choosing the Factorization renormalization Scale in Perturbative QCD Calculations written by and published by . This book was released on 1990 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is important in perturbative QCD calculations to estimate the errors due to uncalculated higher order corrections. An inappropriate choice of the renormalization and factorization scales can lead to unnecessarily large higher order corrections. A method for systematically investigating these issues is suggested. it involves examining the integrands of Feynman graphs as a function of [kappa]{sub {perpendicular}}. A rationale for choosing a sensible range of the scale parameter(s) is suggested. A by product is a physical interpretation of the scales in the MS and {ovr MS} schemes. 8 refs., 5 figs.

Download or read book The Renormalization Scale Problem and Novel Perspectives for QCD written by and published by . This book was released on 2015 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Strong Interactions in Spacelike and Timelike Domains written by Alexander V. Nesterenko and published by Elsevier. This book was released on 2016-11-26 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: Strong Interactions in Spacelike and Timelike Domains: Dispersive Approach provides the theoretical basis for the description of the strong interactions in the spacelike and timelike domains. The book primarily focuses on the hadronic vacuum polarization function, R-ratio of electron-positron annihilation into hadrons, and the Adler function, which govern a variety of the strong interaction processes at various energy scales. Specifically, the book presents the essentials of the dispersion relations for these functions, recaps their perturbative calculation, and delineates the dispersively improved perturbation theory. The book also elucidates the peculiarities of the continuation of the spacelike perturbative results into the timelike domain, which is indispensable for the studies of electron-positron annihilation into hadrons and the related processes. Covers the topics that play an essential role in contemporary particle physics and future collider projects Applicable for self-education alongside standard textbooks Makes the subject easily accessible without the need of an extensive theoretical background

Download or read book Determination of the QCD Renormalization Scale and Lambda M S from Multi jet Events Produced in Electron positron Collisions written by I. H. Park and published by . This book was released on 1989 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: