Infrared gluon and ghost propagator exponents from lattice QCD

The compatibility of the pure power law infrared solution of QCD and lattice data for the gluon and ghost propagators in Landau gauge is discussed. For the gluon propagator, the lattice data are well described by a pure power law with an infrared exponent κ∼0.53, in the Dyson–Schwinger notation. κ is measured using a technique that suppresses finite volume effects. This value is consistent with a vanishing zero momentum gluon propagator, in agreement with the Gribov–Zwanziger confinement scenario. For the ghost propagator, the lattice data seem not to follow a pure power law, at least for the range of momenta accessed in our simulation.     

Strong-coupling study of the Gribov ambiguity in lattice Landau gauge

We study the strong-coupling limit β=0 of lattice SU(2) Landau gauge Yang–Mills theory. In this limit the lattice spacing is infinite, and thus all momenta in physical units are infinitesimally small. Hence, the infrared behavior can be assessed at sufficiently large lattice momenta. Our results show that at the lattice volumes used here, the Gribov ambiguity has an enormous effect on the ghost propagator in all dimensions. This underlines the severity of the Gribov problem and calls for refined studies also at finite β. In turn, the gluon propagator only mildly depends on the Gribov ambiguity.     

Chiral and deconfinement transition from correlation functions: SU(2) vs. SU(3)

We study a gauge-invariant order parameter for deconfinement and the chiral condensate in SU(2) and SU(3) Yang–Mills theory in the vicinity of the deconfinement phase transition using the Landau gauge quark and gluon propagators. We determine the gluon propagator from lattice calculations and the quark propagator from its Dyson–Schwinger equation, using the gluon propagator as input. The critical temperature and a deconfinement order parameter are extracted from the gluon propagator and from the dependency of the quark propagator on the temporal boundary conditions. The chiral transition is determined using the quark condensate as order parameter. We investigate whether and how a difference in the chiral and deconfinement transition between SU(2) and SU(3) is manifest.     

Infrared effects on the axial gauge quark propagator

The axial gauge quark propagator is studied when only the most singular infrared part of the gluon propagator is retained in the Dyson-Schwinger equation. With a new representation for the quark-gluon vertex a simple configuration space propagator and a momentum space form valid for all values of the gauge variable n · p are obtained. The propagator has no poles. The effective potential is minimized when there is no chiral symmetry breaking.     

Infrared properties of the gauge theory coupling constant III

In previous papers we have outlined a program for deriving the infrared behavior of the axial gauge gluon propagator in a pure Yang-Mills theory. The program is based on an integral equation for the gluon propagator derived from the Dyson equation and the Ward identities. Here we present a solution to this equation, obtained numerically. The solution exhibits a Singularity in the infrared, and therefore presumably predicts confinement of color. The method is supposed to be exact in the infrared. Away from the infrared, therefore, our solution is only approximate. Nevertheless, even in the ultraviolet, our solution for the propagator is not very different from the known asymptotic freedom result, so it may be that it is a reasonable approximation over the entire range of momentum.     

Infrared properties of the glue propagator in non-abelian gauge theories

In a pure Yang-Mills theory, the Dyson equation for the gluon propagator is studied in the infrared regime, under the assumption that, as in QED, only those parts of the proper gluon vertex functions determined by the Ward identities are relevant. The calculations are all carried out in the axial gauge. With a number of simplifying assumptions the resulting integral equation for the gluon propagator can be solved in the IR regime. The solution displays a power singularity in the IR for the renormalized coupling constant g(q²).     

The Non-Perturbative Approach to the Infrared Problem in QCD at Finite Temperature

A non-perturbative approach is developed for investigation of the infrared problem in QCD at T ≠ 0 in the ghost-free axial gauge. The problem is solved by using a 3-dimensional theory within the exact Slavnov-Taylor identities and Schwinger-Dyson equations. The system of two non-linear integral equations for the structural functions of the gluon polarization tensor is obtained whose solution determines the infrared behavior of the temperature Green functions in the 4-dimensional QCD. The simplest solution of these equations which is the same as the first term of the perturbation expansion shows the presence of singularities in the gluon propagator at momenta p ∼ g²T, that cannot be eliminated by any choice of the gauge. The infrared instability of QCD at T ≠ 0 caused by these singularities is discussed.     

Verifying the Kugo-Ojima confinement criterion in Landau gauge Yang-Mills theory

Expanding the Landau gauge gluon and ghost two-point functions in a power series we investigate their infrared behavior. The corresponding powers are constrained through the ghost Dyson-Schwinger equation by exploiting multiplicative renormalizability. Without recourse to any specific truncation we demonstrate that the infrared powers of the gluon and ghost propagators are uniquely related to each other. Constraints for these powers are derived, and the resulting infrared enhancement of the ghost propagator signals that the Kugo-Ojima confinement criterion is fulfilled in Landau gauge Yang-Mills theory.     

Instanton Effects on Gluon Propagator

Gluon propagator is investigated for pure Yang-Mills SU(3) gauge theory in field-strength approach. It is found that instantons provide a homogeneous solid-like medium background which generates finite nonzero momentum gluon propagator and gluon receives effective mass.     

Constraints on the infrared behavior of the gluon propagator in Yang-Mills theories

We present rigorous upper and lower bounds for the zero-momentum gluon propagator D(0) of Yang-Mills theories in terms of the average value of the gluon field. This allows us to perform a controlled extrapolation of lattice data to infinite volume, showing that the infrared limit of the Landau-gauge gluon propagator in SU(2) gauge theory is finite and nonzero in three and in four space-time dimensions. In the two-dimensional case, we find D(0)=0, in agreement with Maas. We suggest an explanation for these results. We note that our discussion is general, although we apply our analysis only to pure gauge theory in the Landau gauge. Simulations have been performed on the IBM supercomputer at the University of S?o Paulo.     

The Color Gauge Invariance and a Possible Origin of Mass in QCD

The general scale parameter, having the dimensions of mass squared, is dynamically generated in the QCD gluon sector. It is introduced through the difference between the regularized full gluon self-energy and its value at some finite point. It violates transversality of the full gluon self-energy. The Slavnov-Taylor identity for the full gluon propagator, when it is given by the corresponding equation of motion, is also violated by it. So in order to maintain both transversality and the identity it should be disregarded from the very beginning, i.e., put formally zero everywhere. However, we have shown how to preserve the above-mentioned identity at non-zero mass squared parameter. This allows one to establish the structure of the full gluon propagator when it is explicitly present. Its contribution does not survive in the perturbation theory regime, when the gluon momentum goes to infinity. At the same time, its contribution dominates the structure of the full gluon propagator when the gluon momentum goes to zero. We have also proposed a method how to restore transversality of the relevant gluon propagator in a gauge invariant way, while keeping the mass squared parameter “alive”.     

A non-perturbative calculation of the infrared limit of the axial gauge gluon propagator (II)

We show, numerically, that the integral equation for the axial gauge gluon propagator developed in the preceding paper has an explicit solution with the features outlined there. This solution is expected to be exact in the infrared limit. We find, however, that even in the ultraviolet limit it does not differ greatly from the known asymptotic freedom behavior of the propagator. It may, therefore, be a reasonable approximation over the whole range.     

On dynamical gluon mass generation

The effective gluon propagator constructed with the pinch technique is governed by a Schwinger-Dyson equation with special structure and gauge properties, that can be deduced from the correspondence with the background field method. Most importantly the non-perturbative gluon self-energy is transverse order-by-order in the dressed loop expansion, and separately for gluonic and ghost contributions, a property which allows for a meanigfull truncation. A linearized version of the truncated Schwinger-Dyson equation is derived, using a vertex that satisfies the required Ward identity and contains massless poles. The resulting integral equation, subject to a properly regularized constraint, is solved numerically, and the main features of the solutions are briefly discussed.     

Infrared saturation and phases of gauge theories with BRST symmetry

We investigate the infrared limit of the quantum equation of motion of the gauge boson propagator in various gauges and models with a BRST symmetry. We find that the saturation of this equation at low momenta distinguishes between the Coulomb, Higgs and confining phase of the gauge theory. The Coulomb phase is characterized by a massless gauge boson. Physical states contribute to the saturation of the transverse equation of motion of the gauge boson at low momenta in the Higgs phase, while the saturation is entirely due to unphysical degrees of freedom in the confining phase. This corollary to the Kugo–Ojima confinement criterion in linear covariant gauges also is sufficient for confinement in general covariant gauges with BRST and anti-BRST symmetry, maximal Abelian gauges with an equivariant BRST symmetry, non-covariant Coulomb gauge and in the Gribov–Zwanziger theory.     

Gluon Propagator and Heavy Quark Potential in an Anisotropic QCD Plasma

The hard-loop resummed propagator in an anisotropic QCD plasma in general linear gauges are computed. We get the explicit expressions of the gluon propagator in covariant gauge, Coulomb gauge and temporal axial gauge. Considering one gluon exchange, the potential between heavy quarks is defined through the Fourier transform of the static propagator. We find that the potential exhibits angular dependence and that there is stronger attraction on distance scales on the order of the inverse Debye mass for quark pairs aligned along the direction of anisotropy than for transverse alignment.     

An analytic calculation of the weak field limit of the static color dielectric constant

We analyze the Dyson equation/Ward identity system for the axial gauge n · A = 0 gluon propagator Δ_μν(q)whenn · q = 0. The solution behaves like (q^?4 + (q²)^ν?1) for small q₂, and we are able to calculate the power ν analytically. It turns out to be 0.1737. This analytic calculation verifies our earlier numerical solutions to these equations. For static problems, n · q = 0 is the temporal gauge, and in this gauge the gluon propagator is directly related to the color dielectric constant. We can thus calculate the dielectric constant in the infrared limit.     

The quark propagator from the Dyson-Schwinger equations: (I). The chiral solution

Within the framework of the Dyson-Schwinger equations in the axial gauge, and using a truncation procedure which respects the Ward-Takahashi identities, we study the effect that nonperturbative glue has on the quark propagator. We show that within this truncation scheme, the requirement of matching perturbative QCD at high momentum transfer leads to a multiplicatively renormalisable equation. Technically, the matching with perturbation theory is accomplished by the introduction of a transverse part to the quark-gluon vertex. In the case of an analytic gluon propagator, this truncation scheme can lead to chiral symmetry breaking only after the introduction of such a transverse vertex: massless solutions do not exist beyond a critical value of as. Using the gluon propagator that we previously obtained, we obtain small corrections to the quark propagator, which keeps a pole at the origin in the chiral phase.     

Numerical calculation of the momentum-space gluon propagator in quenched,reduced QCD3

A recently developed method of momentum-space Monte Carlo is applied to compute the momentum-space gluon propagator in quenched, reduced, continuum QCD₃ in axial gauge. There is some evidence that the gluon propagator D_μν(p) is finite as p → 0, which might indicate the existence of a non-perturbative gluon mass.     

Aspects of the confinement mechanism in Coulomb-gauge QCD

Phenomenological consequences of the infrared singular, instantaneous part of the gluon propagator in the Coulomb gauge are investigated. The corresponding quark Dyson-Schwinger equation is solved, neglecting retardation and transverse gluons and regulating the resulting infrared singularities. While the quark propagator vanishes as the infrared regulator goes to zero, the frequency integral over the quark propagator stays finite and well defined. Solutions of the homogeneous Bethe-Salpeter equation for the pseudoscalar and vector mesons as well as for scalar and axial-vector diquarks are obtained. In the limit of a vanishing infrared regulator the diquark masses diverge, while meson properties and diquark radii remain finite and well defined. These features are interpreted with respect to the resulting aspects of confinement for colored quark-quark correlations.     

Gauge and Lorentz covariant quark propagator in an arbitrary gluon field

The quark propagator in the presence of an arbitrary gluon field is calculated gauge and Lorentz covariantly order by order in terms of powers of the gluon field and its derivatives. The result is independent of the path connecting the ends of the propagator, and the leading order result coincides with the exact propagator in the trivial case of a vanishing gluon field. Received: 5 February 2003 / Published online: 23 May 2003