Gribov-Zwanziger Approach Research Articles

The complex heavy quarkonium potential with the Gribov–Zwanziger action

Gribov–Zwanziger prescription in Yang–Mills theory improves the infrared dynamics. In this work, we study the static potential of a heavy quark-antiquark pair with the HTL resummed perturbation method within the Gribov–Zwanziger approach at finite temperature. The real and imaginary parts of the heavy quarkonium complex potential are obtained from the one-loop effective static gluon propagator. The one-loop effective gluon propagator is obtained by calculating the one-loop gluon self-energies containing the quark, gluon, and ghost loop. The gluon and ghost loops are modified in the presence of the Gribov parameter. We also calculate the decay width from the imaginary part of the potential. We also discuss the medium effect of heavy quarkonium potential with the localized action via auxiliary fields.

High energy evolution for Gribov-Zwanziger confinement: Solution to the equation

In this paper, we solve the new evolution equation for high energy scattering amplitude that stems from the Gribov-Zwanziger approach to the confinement of quarks and gluons. We found that (1) the energy dependence of the scattering amplitude turns out to be the same as for QCD Balitsky, Fadin, Kuraev and Lipatov (BFKL) evolution, (2) the spectrum of the new equation does not depend on the details of the Gribov-Zwanzinger approach, and (3) all eigenfunctions coincide with the eigenfunctions of the QCD BFKL equation at large transverse momenta $\ensuremath{\kappa}\ensuremath{\ge}1$. The numerical calculations show that there exist no new eigenvalues with the eigenfunctions which decrease faster than solutions of the QCD BFKL equation at large transverse momenta. The structure of the gluon propagator in the Gribov-Zwanziger approach, that stems from the lattice QCD and from the theoretical evaluation, results in the exponential suppression of the eigenfunctions at long distances and in the resolution of the difficulties, which the color glass condensate and some other approaches, based on perturbative QCD, face at large impact parameters. We can conclude that the confinement of quark and gluons, at least in the form of the Gribov-Zwanziger approach, does not influence on the scattering amplitude except for solving the long-standing theoretical problem of its behavior at large impact parameters.

Gribov-Zwanziger confinement, high energy evolution, and large impact parameter behavior of the scattering amplitude

In this paper we derive the high energy evolution equation in the Gribov-Zwanziger approach for the confinement of quarks and gluons. We demonstrate that the new equation generates an exponential decrease of the scattering amplitude at large impact parameter and resolves the main difficulties of color glass condensate high energy effective theory. Such behavior occurs if the gluon propagator in Gribov-Zwanziger approach does not vanish at small momenta. Solving the nonlinear equation for deep inelastic scattering, we show that the suggested equation leads to a Froissart disc with radius (${R}_{F}$), which increases as ${R}_{F}\ensuremath{\propto}Y=\mathrm{ln}(1/x)$, and with a finite width for the distribution over $|b\ensuremath{-}{R}_{F}|$.

Scalar sunset diagram at finite temperature with imaginary square masses

We evaluate the finite temperature scalar sunset diagram with imaginary square masses, that appears in the Gribov-Zwanziger approach to Yang-Mills (YM) theory beyond one-loop order. Since YM theory at finite temperature is governed by center-symmetry and the Polyakov loop, we also include the possibility of a constant temporal background gauge field in the form of color-dependent imaginary chemical potentials.

Numerical evaluation of the Bose-ghost propagator in minimal Landau gauge on the lattice

We present numerical details of the evaluation of the so-called Bose-ghost propagator in lattice minimal Landau gauge, for the SU(2) case in four Euclidean dimensions. This quantity has been proposed as a carrier of the confining force in the Gribov-Zwanziger approach and, as such, its infrared behavior could be relevant for the understanding of color confinement in Yang-Mills theories. Also, its nonzero value can be interpreted as direct evidence of BRST-symmetry breaking, which is induced when restricting the functional measure to the first Gribov region Omega. Our simulations are done for lattice volumes up to 120^4 and for physical lattice extents up to 13.5 fm. We investigate the infinite-volume and continuum limits.

Effect of the Gribov horizon on the Polyakov loop and vice versa

We consider finite-temperature SU(2) gauge theory in the continuum formulation, which necessitates the choice of a gauge fixing. Choosing the Landau gauge, the existing gauge copies are taken into account by means of the Gribov–Zwanziger quantization scheme, which entails the introduction of a dynamical mass scale (Gribov mass) directly influencing the Green functions of the theory. Here, we determine simultaneously the Polyakov loop (vacuum expectation value) and Gribov mass in terms of temperature, by minimizing the vacuum energy w.r.t. the Polyakov-loop parameter and solving the Gribov gap equation. Inspired by the Casimir energy-style of computation, we illustrate the usage of Zeta function regularization in finite-temperature calculations. Our main result is that the Gribov mass directly feels the deconfinement transition, visible from a cusp occurring at the same temperature where the Polyakov loop becomes nonzero. In this exploratory work we mainly restrict ourselves to the original Gribov–Zwanziger quantization procedure in order to illustrate the approach and the potential direct link between the vacuum structure of the theory (dynamical mass scales) and (de)confinement. We also present a first look at the critical temperature obtained from the refined Gribov–Zwanziger approach. Finally, a particular problem for the pressure at low temperatures is reported.

Dimension two condensates in the Gribov-Zwanziger theory in Coulomb gauge

We investigate the dimension two condensate $< {\bar \phi}^{ab}_i \phi^{ab}_i -{\bar \omega}^{ab}_i \omega^{ab}_i >$ within the Gribov-Zwanziger approach to Euclidean Yang-Mills theories in the Coulomb gauge, in both 3 and 4 dimensions. An explicit calculation shows that, at the first order, the condensate $< {\bar \phi}^{ab}_i \phi^{ab}_i -{\bar \omega}^{ab}_i \omega^{ab}_i >$ is plagued by a non-integrable IR divergence in 3D, while in 4D it exhibits a logarithmic UV divergence, being proportional to the Gribov parameter $\gamma^2$. These results indicate that in 3D the transverse spatial Coulomb gluon two-point correlation function exhibits a scaling behaviour, in agreement with Gribov's expression. In 4D, however, they suggest that, next to the scaling behaviour, a decoupling solution might emerge too.

Study of the all orders multiplicative renormalizability of a local confining quark action in the Landau gauge

The inverse of the Faddeev–Popov operator plays a pivotal role within the Gribov–Zwanziger approach to the quantization of Euclidean Yang–Mills theories in Landau gauge. Following a recent proposal (Capri et al., 2014), we show that the inverse of the Faddeev–Popov operator can be consistently coupled to quark fields. Such a coupling gives rise to a local action while reproducing the behaviour of the quark propagator observed in lattice numerical simulations in the non-perturbative infrared region. By using the algebraic renormalization framework, we prove that the aforementioned local action is multiplicatively renormalizable to all orders.

Gribov ambiguity and degenerate systems

The relation between Gribov ambiguity and degeneracies in the symplectic structure of physical systems is analyzed. It is shown that, in finite-dimensional systems, the presence of Gribov ambiguities in regular constrained systems (those where the constraints are functionally independent) always leads to a degenerate symplectic structure upon Dirac reduction. The implications for the Gribov-Zwanziger approach to QCD are discussed.

Techniques of evaluation of QCD low-energy physical quantities with running coupling with infrared fixed point

Perturbative QCD (pQCD) running coupling $a({Q}^{2})$ ($\ensuremath{\equiv}{\ensuremath{\alpha}}_{s}({Q}^{2})/\ensuremath{\pi}$) is expected to get modified at low spacelike momenta $0&lt;{Q}^{2}\ensuremath{\lesssim}1\text{ }\text{ }{\mathrm{GeV}}^{2}$ so that, instead of having unphysical (Landau) singularities, it remains smooth and finite there, due to the infrared (IR) fixed point. This behavior is suggested by the Gribov-Zwanziger approach, Dyson-Schwinger equations and other functional methods, lattice calculations, light-front holographic mapping AdS/CFT modified by a dilaton background, and most of the analytic (holomorphic) QCD models. All such couplings $\mathcal{A}({Q}^{2})$ differ from the pQCD couplings $a({Q}^{2})$ at $|Q|\ensuremath{\gtrsim}1\text{ }\text{ }\mathrm{GeV}$ by nonperturbative (NP) terms, typically by some power-suppressed terms $\ensuremath{\sim}1/{Q}^{2N}$. Evaluations of low-energy physical QCD quantities in terms of such $\mathcal{A}({Q}^{2})$ couplings (with an IR fixed point) at a level beyond one loop are usually performed with a (truncated) power series in $\mathcal{A}({Q}^{2})$. We argue that such an evaluation is not correct, because the NP terms in general get out of control as the number of terms in the power series increases. The series consequently become increasingly unstable under the variation of the renormalization scale and have a fast asymptotic divergent behavior compounded by the renormalon problem. We argue that an alternative series in terms of logarithmic derivatives of $\mathcal{A}({Q}^{2})$ should be used. Furthermore, a Pad\'e-related resummation based on this series gives results which are renormalization scale independent and show very good convergence. Timelike low-energy observables can be evaluated analogously, by using the integral transformation which relates the timelike observable with the corresponding spacelike observable.

N = 1 Gribov superfield extension

We propose a mechanism displaying confinement, as defined by the behavior of the propagators, for four-dimensional, N = 1 supersymmetric Yang–Mills theory in superfield formalism. In this work we intend to verify the possibility of extending the known Gribov problem of quantization of Yang–Mills theories and the implementation of a local action with auxiliary superfields, like the Gribov–Zwanziger approach, to this problem.

Casimir energy in the Gribov-Zwanziger approach to QCD

In this paper we study the Casimir energy of QCD within the Gribov-Zwanziger approach. In this model nonperturbative effects of gauge copies are properly taken into account. We show that the computation of the Casimir energy for the MIT bag model within the (refined) Gribov-Zwanziger approach not only can give the correct sign but it also gives an estimate for the radius of the bag.

Dynamical origin of the refinement of the Gribov-Zwanziger theory

In recent years, the Gribov-Zwanziger action was refined by taking into account certain dimension 2 condensates. In this fashion, one succeeded in bringing the gluon and the ghost propagator obtained from the GZ model in qualitative and quantitative agreement with the lattice data. In this paper, we shall elaborate further on this aspect. First, we shall show that more dimension 2 condensates can be taken into account than considered so far and, in addition, we shall give firm evidence that these condensates are in fact present by discussing the effective potential. It follows thus that the Gribov-Zwanziger action dynamically transforms itself into the refined version, thereby showing that the continuum nonperturbative Landau gauge fixing, as implemented by the Gribov-Zwanziger approach, is consistent with lattice simulations.

Gribov no-pole condition, Zwanziger horizon function, Kugo-Ojima confinement criterion, boundary conditions, BRST breaking and all that

We aim to offer a kind of unifying view on two popular topics in the studies of nonperturbative aspects of Yang-Mills theories in the Landau gauge: the so-called Gribov-Zwanziger approach and the Kugo-Ojima confinement criterion. Borrowing results from statistical thermodynamics, we show that imposing the Kugo-Ojima confinement criterion as a boundary condition leads to a modified yet renormalizable partition function. We verify that the resulting partition function is equivalent with the one obtained by Gribov and Zwanziger, which restricts the domain of integration in the path integral within the first Gribov horizon. The construction of an action implementing a boundary condition allows one to discuss the symmetries of the system in the presence of the boundary. In particular, the conventional Becchi-Rouet-Stora-Tyutin symmetry is softly broken.

Refinement of the Gribov-Zwanziger approach in the Landau gauge: Infrared propagators in harmony with the lattice results

Recent lattice data have reported an infrared suppressed, positivity violating gluon propagator which is nonvanishing at zero momentum and a ghost propagator which is no longer enhanced. This paper discusses how to obtain analytical results which are in qualitative agreement with these lattice data within the Gribov-Zwanziger framework. This framework allows one to take into account effects related to the existence of gauge copies, by restricting the domain of integration in the path integral to the Gribov region. We elaborate to great extent on a previous short paper by presenting additional results, also confirmed by the numerical simulations. A detailed discussion on the soft breaking of the Becchi-Rouet-Stora-Tyutin symmetry arising in the Gribov-Zwanziger approach is provided.

New features of the gluon and ghost propagator in the infrared region from the Gribov-Zwanziger approach

So far, the infrared behavior of the gluon and ghost propagator based on the Gribov-Zwanziger approach predicted a positivity violating gluon propagator vanishing at zero momentum, and an infrared enhanced ghost propagator. However, recent data based on huge lattices have revealed a positivity violating gluon propagator which turns out to attain a finite nonvanishing value very close to zero momentum. At the same time the ghost propagator does not seem to be infrared enhanced anymore. We point out that these new features can be accounted for by yet unexploited dynamical effects within the Gribov-Zwanziger approach, leading to an infrared behavior in qualitatively good agreement with the new data.