The Gribov problem and QCD dynamics (1202.1491v2)

Abstract: In 1967, Faddeev and Popov were able to quantize the Yang-Mills theory by introducing new particles called ghost through the introduction of a gauge. Ever since, this quantization has become a standard textbook item. Some years later, Gribov discovered that the gauge fixing was not complete, gauge copies called Gribov copies were still present and could affect the infrared region of quantities like the gauge dependent gluon and ghost propagator. This feature was often in literature related to confinement. Some years later, the semi-classical approach of Gribov was generalized to all orders and the so-called GZ action was born. Ever since, many related articles were published. This review tends to give a pedagogic review of the ideas of Gribov and the subsequent construction of the GZ action, including many other toipics related to the Gribov region. It is shown how the GZ action can be viewed as a non-perturbative tool which has relations with other approaches towards confinement. Many different features related to the GZ action shall be discussed in detail, such as BRST breaking, the KO criterion, the propagators, etc. We shall also compare with the lattice data and other non-perturbative approaches, including stochastic quantization.

• The paper provides an exhaustive analysis of the Gribov problem in non-perturbative QCD, focusing on the Gribov-Zwanziger (GZ) action as a resolution and its theoretical implications.
• It analyzes how the GZ action predicts suppressed gluon and enhanced ghost propagators and discusses discrepancies found when comparing these results to lattice QCD simulations.
• The authors explore the delicate breaking of BRST symmetry within the GZ framework and briefly discuss stochastic quantization as an alternative approach to non-perturbative gauge fixing.

An Insightful Overview of "The Gribov Problem and QCD Dynamics"

The academic paper by N. Vandersickel and Daniel Zwanziger, titled "The Gribov Problem and QCD Dynamics," provides an exhaustive analysis of key issues in the non-perturbative quantization of Quantum Chromodynamics (QCD), particularly focusing on the Gribov problem. The paper discusses the implications and framework of the Gribov-Zwanziger (GZ) action, which aims to resolve the Gribov ambiguity that arises in the gauge fixing process for non-Abelian gauge theories like QCD.

Gribov Problem and the GZ Action

At the heart of the paper is the exploration of the Gribov problem, identified by V.N. Gribov, who noted that the Faddeev-Popov procedure for gauge fixing in QCD does not uniquely fix the gauge, due to multiple gauge equivalent configurations known as Gribov copies. This issue becomes particularly severe in the infrared (IR) sector of the theory and has been connected to confinement, a defining feature of QCD.

The GZ action, introduced to address this issue, modifies the functional integral to restrict the integration domain to the so-called Gribov region. This region is defined as the set of gauge fields where the Faddeev-Popov operator is positive definite, effectively excluding the first Gribov horizon where the operator’s lowest eigenvalue reaches zero.

The paper describes how the GZ framework generalizes Gribov's approach to all orders, leading to a non-perturbative action. A key component of this action is its dependence on the Gribov parameter $\gamma$, determined through the horizon condition, reflecting a dynamically generated mass scale that regulates IR divergences in the gluon propagator.

Impact on Propagators and Comparison with Lattice Data

A significant portion of the paper is dedicated to analyzing the implications of the GZ action for the gluon and ghost propagators. The GZ approach predicts a suppressed gluon propagator in the IR, and an enhanced ghost propagator, resulting in significant deviations from their perturbative counterparts. These results naturally align with the hypothesis of gluon confinement and are qualitatively consistent with Kugo-Ojima criteria that associate confinement with certain behaviors of these propagators.

However, comparisons with lattice QCD simulations, particularly in dimensions $d = 3$ and 4, revealed discrepancies. Although these calculations agree qualitatively with the suppression of the gluon propagator, they show a finite non-zero gluon propagator at zero momentum, contrasting with the vanishing result predicted by the original GZ solution. The paper discusses possible refinements to the GZ action to reconcile this, incorporating condensates like $\langle A^2 \rangle$ into the action to account for observed lattice behaviors.

BRST Symmetry and Stochastic Quantization

Another core aspect the authors address is the issue of BRST symmetry, which is delicately broken in the GZ framework. The authors explore ways to interpret this breaking, potentially as spontaneous symmetry breaking, aligning with the Maggiore-Schaden construction where the BRST operator acquires a vacuum expectation value. This interpretation could hint at a deep connection between the mechanism for confinement and a novel understanding of symmetries in QCD.

Furthermore, the paper considers the potential of stochastic quantization as an alternative formalism for non-perturbative gauge fixing, which could obviate the need for Gribov region restriction. This approach applicably treats the dynamical evolution of fields toward their equilibrium distribution without explicit gauge fixing, indicating another promising direction to fully understand confinement in QCD.

Conclusion

In conclusion, Vandersickel and Zwanziger's paper offers an authoritative exploration of the GZ model as a resolution to the Gribov problem in QCD. Although some discrepancies exist when comparing GZ predictions to lattice results, the framework remains a compelling tool for understanding non-perturbative phenomena like confinement. The paper's comprehensive treatment of technical and theoretical aspects makes it an essential reference for researchers examining the subtleties of QCD in the infrared regime and devising innovative solutions to long-standing problems in theoretical physics. Future work in refining the GZ action and exploring alternative quantization approaches could significantly advance this domain, paving the way for a deeper understanding of the non-perturbative dynamics in QCD.