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The QCD Running Coupling

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The QCD Running Coupling JLAB-PHY-16-2199 SLAC{PUB{16448 The QCD Running Coupling Alexandre Deur Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA Stanley J. Brodsky SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA Guy F. de T´eramond Universidad de Costa Rica, San Jos´e,Costa Rica September 11, 2020 [email protected], arXiv:1604.08082v4 [hep-ph] 10 Sep 2020 [email protected], [email protected], (Invited review article to be published in Progress in Particle and Nuclear Physics) Abstract We review the present theoretical and empirical knowledge for αs, the fundamental coupling underlying the interactions of quarks and gluons in Quantum Chromodynam- 2 ics (QCD). The dependence of αs(Q ) on momentum transfer Q encodes the underlying dynamics of hadron physics {from color confinement in the infrared domain to asymp- 2 2 totic freedom at short distances. We review constraints on αs(Q ) at high Q , as pre- dicted by perturbative QCD, and its analytic behavior at small Q2, based on models of nonperturbative dynamics. In the introductory part of this review, we explain the phenomenological meaning of the coupling, the reason for its running, and the chal- lenges facing a complete understanding of its analytic behavior in the infrared domain. 2 In the second, more technical, part of the review, we discuss the behavior of αs(Q ) in the high momentum transfer domain of QCD. We review how αs is defined, including its renormalization scheme dependence, the definition of its renormalization scale, the utility of effective charges, as well as \Commensurate Scale Relations" which connect the various definitions of the QCD coupling without renormalization-scale ambiguity. We also report recent significant measurements and advanced theoretical analyses which have led to precise QCD predictions at high energy. As an example of an important optimization procedure, we discuss the \Principle of Maximum Conformality", which enhances QCD's predictive power by removing the dependence of the predictions for physical observables on the choice of theoretical conventions such as the renormaliza- tion scheme. In the last part of the review, we discuss the challenge of understanding 2 the analytic behavior αs(Q ) in the low momentum transfer domain. We survey var- ious theoretical models for the nonperturbative strongly coupled regime, such as the light-front holographic approach to QCD. This new framework predicts the form of the quark-confinement potential underlying hadron spectroscopy and dynamics, and it gives a remarkable connection between the perturbative QCD scale Λ and hadron masses. One can also identify a specific scale Q0 which demarcates the division between perturbative and nonperturbative QCD. We also review other important methods for computing the QCD coupling, including lattice QCD, the Schwinger{Dyson equations and the Gribov{ Zwanziger analysis. After describing these approaches and enumerating their conflicting predictions, we discuss the origin of these discrepancies and how to remedy them. Our aim is not only to review the advances in this difficult area, but also to suggest what 2 could be an optimal definition of αs(Q ) in order to bring better unity to the subject. 2 Contents 1 Preamble6 2 Phenomenological overview: QCD and the behavior of αs 9 3 The strong coupling αs in the perturbative domain 16 3.1 Purpose of the running coupling . 17 3.2 The evolution of αs in perturbative QCD . 18 3.3 Quark thresholds . 23 3.4 Computation of the pQCD effective coupling . 24 3.5 Renormalization group . 27 3.6 The Landau pole and the QCD parameter Λ . 28 3.7 Improvement of the perturbative series . 29 3.7.1 Effective charges and commensurate scale relations . 29 3.7.2 The Brodsky, Lepage and Mackenzie procedure and its extensions 32 3.7.3 Principle of maximum conformality . 34 3.8 Other optimization procedures . 36 2 3.9 Determination of the strong coupling αs(MZ ) or the QCD scale Λ . 37 3.9.1 Deep Inelastic Scattering . 39 3.9.2 Observables from e+e− collisions . 41 3.9.3 Observables from pp collisions . 44 3.9.4 Lattice QCD . 45 3.9.5 Heavy quarkonia . 51 3.9.6 Holographic QCD . 51 3.9.7 Pion decay constant . 53 3.9.8 Grand unification . 54 3.9.9 Comparison and discussion . 54 3 4 The strong coupling in the nonperturbative domain 56 4.1 Effective charges . 61 4.1.1 Measurement of the effective charge from the Bjorken sum rule. 63 4.1.2 Measurement of the effective charge defined from e+e− annihilation. 66 4.1.3 Sudakov effective charges . 66 4.2 AdS/CFT and Holographic QCD . 68 4.3 The effective potential approach . 73 4.3.1 The static Q{Q potential . 73 4.3.2 Constraints on the running coupling from the hadron spectrum . 74 4.4 The Schwinger{Dyson formalism . 77 4.4.1 The QCD coupling defined from Schwinger{Dyson equations . 78 4.4.2 Gauge choices . 79 4.4.3 Classes of solutions in the IR domain . 80 4.4.4 The massive gluon propagator . 84 4.4.5 The coupling defined from the ghost{gluon vertex . 88 4.4.6 Couplings defined from the 3-gluon and 4-gluon vertices . 90 4.4.7 Including quarks . 92 4.4.8 Gauge dependence . 93 4.5 Lattice gauge theory . 94 4.5.1 Lattice results for αs in the IR . 96 4.6 Functional renormalization group equations . 99 4.7 The Gribov{Zwanziger approach . 101 4.8 Stochastic quantization . 103 4.9 Analytic and dispersive approaches . 104 4.9.1 Analytic approach . 105 4.9.2 Dispersive approach . 109 4.10 Background perturbation theory . 110 4.11 Optimized perturbation theory . 112 4.12 quark{hadron duality . 112 4.13 The IR mapping of λφ4 to Yang{Mills theories . 113 4.14 The Bogoliubov compensation principle . 113 4.15 Curci{Ferrari model . 113 5 Comparison and discussion 114 5.1 Validity of the comparison . 114 4 5.2 Influence of the renormalization scheme . 119 2 5.3 The contributions of nonperturbative terms to αs(Q )........... 121 5.4 Listing of the multiple IR-behavior found in the literature . 125 6 Conclusions 133 A Lexicon 137 5 Chapter 1 Preamble 2 We review the status of the coupling αs(Q ), the function which sets the strength of the interactions involving quarks and gluons in quantum chromodynamics (QCD), the fundamental gauge theory of the strong interactions, as a function of the momentum transfer Q. It is necessary to understand the behavior and magnitude of the QCD coupling over the complete Q2 range in order to describe hadronic interactions at both 2 2 long and short distances. At high Q (short distances) precise knowledge of αs(Q ) is needed to match the growing accuracy of hadron scattering experiments as well as to test high-energy models unifying strong and electroweak forces. For example, uncertainties 2 in the value of αs(Q ) contribute to the total theoretical uncertainty in the physics probed at the Large Hadron Collider, such as Higgs production via gluon fusion. It 2 is also necessary to know the behavior of αs at low Q (long distances), such as the scale of the proton mass, in order to understand hadronic structure, quark confinement and hadronization processes. For example, processes involving the production of heavy quarks near threshold require the knowledge of the QCD coupling at low momentum scales. Even reactions at high energies may involve the integration of the behavior of the strong coupling over a large domain of momentum scales. Understanding the behavior of 2 αs at low Q also allows us to reach a long-sought goal of hadron physics: to establish an explicit relation between the long distance domain characterized by quark confinement, and the short-distance regime where perturbative calculations are feasible. For such endeavor, a major challenge is to relate the parameter Λ, which controls the predictions of perturbative QCD (pQCD) at short distances, to the mass scale of hadrons. We will show later in this review how new theoretical insights into the behavior of QCD at large distances lead to an analytical relation between hadronic masses and Λ. This problem also involves one of the fundamental questions of QCD: How the mass scale for hadrons, 6 such as the proton and ρ meson, can emerge in the limit of zero-quark masses since no explicit mass scale appears in the QCD Lagrangian. 2 The origin and phenomenology of the behavior of αs(Q ) at small distances, where asymptotic freedom appears, is well understood and explained in many textbooks on Quantum Field Theory and Particle Physics. Numerous reviews exist; see e.g. Refs. [1,2,3], some of which cover the long-distance behavior as well, e.g. Ref. [2]. However, standard explanations often create an apparent conundrum, as will be addressed in this review. Other questions remain even in this well understood regime: a significant issue is how to identify the scale Q which controls a given hadronic process, especially if it depends on many physical scales. A fundamental requirement {called \renormalization group invariance"{ is that physical observables cannot depend on the choice of the renor- malization scale. We will briefly describe a new method, \the Principle of Maximum Conformality" (PMC), which can be used to set the renormalization scale unambigu- ously order-by-order in pQCD, while satisfying renormalization group invariance. The PMC method reduces, in the Abelian limit NC ! 0, to the standard Gell-Mann{Low method which is used to obtain precise predictions for QED. Although αs is well un- derstood at small distances, it is much less so at long distances where it is related to color confinement.
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