A Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in PQCD

We introduce a generalization of the conventional renormalization schemes used in dimensional regularization, which illuminates the renormalization scheme and scale ambiguities of pQCD predictions, exposes the general pattern of nonconformal {\beta_i}-terms, and reveals a special degeneracy of the terms in the perturbative coefficients. It allows us to systematically determine the argument of the running coupling order by order in pQCD in a form which can be readily automatized. The new method satisfies all of the principles of the renormalization group and eliminates an unnecessary source of systematic error.Comment: 5 pages, 1 figure, revised to match the published versio

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 [...]Comment: 20 pages; Appendix added. This version matches the published pape

The Renormalization Scale-Setting Problem in QCD

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 {\it 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 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, {\it 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...... [full Abstract is in the paper].Comment: 75 pages, 19 figures. Review article to be published in Prog. Part. Nucl. Phy

Studies of nucleon resonance structure in exclusive meson electroproduction

Studies of the structure of excited baryons are key factors to the N* program at Jefferson Lab (JLab). Within the first year of data taking with the Hall B CLAS12 detector following the 12 GeV upgrade, a dedicated experiment will aim to extract the N* electrocouplings at high photon virtualities Q2. This experiment will allow exploration of the structure of N* resonances at the highest photon virtualities ever achieved, with a kinematic reach up to Q2 = 12 GeV 2. This high-Q2 reach will make it possible to probe the excited nucleon structures at distance scales ranging from where effective degrees of freedom, such as constituent quarks, are dominant through the transition to where nearly massless bare-quark degrees of freedom are relevant. In this document, we present a detailed description of the physics that can be addressed through N* structure studies in exclusive meson electroproduction. The discussion includes recent advances in reaction theory for extracting N* electrocouplings from meson electroproduction off protons, along with Quantum Chromodynamics (QCD)-based approaches to the theoretical interpretation of these fundamental quantities. This program will afford access to the dynamics of the nonperturbative strong interaction responsible for resonance formation, and will be crucial in understanding the nature of confinement and dynamical chiral symmetry breaking in baryons, and how excited nucleons emerge from QCD. I. G. Aznauryan... K. Tsushima... et al