The QCD Running Coupling (1604.08082v5)

Abstract: We review the present knowledge for $\alpha_s$, the fundamental coupling underlying the interactions of quarks and gluons in QCD. The dependence of $\alpha_s(Q^2)$ on momentum transfer $Q$ encodes the underlying dynamics of hadron physics -from color confinement in the infrared domain to asymptotic freedom at short distances. We review constraints on $\alpha_s(Q^2)$ at high $Q^2$, as predicted by perturbative QCD, and its analytic behavior at small $Q^2$, based on models of nonperturbative dynamics. In the introductory part of this review, we explain the phenomenological meaning of $\alpha_s$, the reason for its running, and the challenges facing a complete understanding of its analytic behavior in the infrared domain. In the second, more technical, part of the review, we discuss the behavior of $\alpha_s(Q^2)$ in the high $Q^2$ domain of QCD. We review how $\alpha_s$ 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 $\alpha_s$ without renormalization-scale ambiguity. We also report recent measurements and theoretical analyses which have led to precise QCD predictions at high energy. In the last part of the review, we discuss the challenge of understanding the analytic behavior $\alpha_s(Q^2)$ in the infrared domain. We also review important methods for computing $\alpha_s$, including lattice QCD, the Schwinger-Dyson equations, the Gribov-Zwanziger analysis and light-front holographic QCD. 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 could be an optimal definition of $\alpha_s$ in order to bring better unity to the subject.

• The paper demonstrates the energy dependence of αs by detailing its transition from perturbative asymptotic freedom to nonperturbative confinement.
• It employs diverse methodologies such as renormalization group equations, lattice simulations, and effective theories to analyze QCD behavior.
• The study highlights the impact of scheme dependence and empirical data in refining the precision of QCD predictions for strong interactions.

Overview of "The QCD Running Coupling"

The paper, "The QCD Running Coupling" by Alexandre Deur and collaborators, provides a thorough examination of the running coupling constant $\alpha_s(Q^2)$ within the framework of Quantum Chromodynamics (QCD). The discussion in the paper explores the intricacies of how the coupling varies with momentum transfer $Q^2$ and the theoretical and empirical approaches to understanding this behavior from perturbative to nonperturbative regimes.

Theoretical Foundations

QCD is the gauge theory that describes the strong interactions between quarks and gluons, and the running coupling constant $\alpha_s$ is a fundamental parameter that describes the strength of these interactions. The paper reviews how $\alpha_s(Q^2)$ changes with energy scales, highlighting two key regions:

• Infrared (IR) Region: At low energies, where QCD is nonperturbative and quarks are confined.
• Ultraviolet (UV) Region: At high energies, approaching asymptotic freedom where interactions become perturbative and quarks behave as free particles.

Perturbative QCD and Asymptotic Freedom

In the UV region, perturbative QCD (pQCD) is applicable, and $\alpha_s$ decreases logarithmically with increasing $Q^2$, a phenomenon known as asymptotic freedom. The paper reviews the perturbative evolution equations for $\alpha_s$, such as the beta function, and the renormalization group (RG) equations that describe this scale dependence. The Perturbative QCD predictions are typically robust up to a few-loop order, beyond which nonperturbative effects become significant.

Nonperturbative Dynamics

The IR region is characterized by strong coupling and confinement, where $\alpha_s$ cannot be computed directly using perturbation theory. The paper surveys various approaches used to paper $\alpha_s$ in this regime:

• Schwinger-Dyson Equations: A nonperturbative technique used to paper propagators and vertices in QCD.
• Lattice QCD: A numerical approach that discretizes space-time to simulate QCD nonperturbatively on supercomputers.
• Gribov-Zwanziger and Functional Methods: Approaches focused on resolving issues of gauge fixing ambiguities and studying effective potential frameworks.

Effective Theories and Holographic QCD

To bridge the UV and IR, effective theories and concepts like Holographic QCD, inspired by the AdS/CFT correspondence, provide insights into the strong coupling regime. These frameworks attempt to incorporate confinement and hadronization phenomenologies and have yielded analytical results that extend the validity of QCD calculations into regions where pQCD ceases to be applicable.

Running Coupling in Various Schemes

A significant portion of the paper is dedicated to exploring how the running coupling behaves under different renormalization schemes such as $\overline{MS}$, MOM, and V-schemes, and the implications of scheme dependence. Effective charges and commensurate scale relations play a crucial role in connecting different definitions uniformly across these schemes.

Empirical Insights and Future Directions

The empirical aspect involves matching theoretical calculations with experimental data, improving precision measurements of $\alpha_s$ across a variety of processes from deep inelastic scattering to high-energy collider experiments. The paper emphasizes the importance of precise $\alpha_s$ determinations and suggests future studies should further integrate breakthroughs in nonperturbative methods and high-energy phenomenology.

Conclusion

The paper makes significant contributions to understanding QCD’s running coupling by systematically reviewing varied approaches to calculating and measuring $\alpha_s$, discussing theoretical innovations, and promoting synergies between different methodologies. As the paper of QCD progresses, these insights are crucial for pushing the boundaries of the Standard Model and unraveling the complexities of strong interactions.