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 São Paulo.

Green's functions are a central element in the attempt to understand non-perturbative phenomena in Yang-Mills theory. Besides the propagators, 3-point Green's functions play a significant role, since they permit access to the running coupling constant and are an important input in functional methods. Here we present numerical results for the two non-vanishing 3-point Green's functions in 3d pure SU (2) Yang-Mills theory in (minimal) Landau gauge, i.e. the three-gluon vertex and the ghost-gluon vertex, considering various kinematical regimes. In this exploratory investigation the lattice volumes are limited to 20 3 and 30 3 at β = 4.2 and β = 6.0. We also present results for the gluon and the ghost propagators, as well as for the eigenvalue spectrum of the Faddeev-Popov operator. Finally, we compare two different numerical methods for the evaluation of the inverse of the Faddeev-Popov matrix, the point-source and the plane-wave-source methods.

Vertices are of central importance for constructing QCD bound states out of the individual constituents of the theory, i.e. quarks and gluons. In particular, the determination of three-point vertices is crucial in non-perturbative investigations of QCD. We use numerical simulations of lattice gauge theory to obtain results for the 3-point vertices in Landau-gauge SU(2) Yang-Mills theory in three and four space-time dimensions for various kinematic configurations. In all cases considered, the ghost-gluon vertex is found to be essentially tree-level-like, while the three-gluon vertex is suppressed at intermediate momenta. For the smallest physical momenta, reachable only in three dimensions, we find that some of the three-gluon-vertex tensor structures change sign.Comment: 9 pages, 6 figures; minor modifications and references added, version to appear in PR

The implementation of the linear covariant gauge on the lattice faces a conceptual problem: using the standard compact discretization, the gluon field is bounded, while the four-divergence of the gluon field satisfies a Gaussian distribution, i.e. it is unbounded. This can give rise to convergence problems when a numerical implementation is attempted. In order to overcome this problem, one can use different discretizations for the gluon field or consider an SU(N c ) group with sufficiently large N c . One can also consider small values of the gauge parameter ξ and study numerically the limiting case of ξ → 0, i.e. the Landau gauge. These different approaches will be discussed here.

We study the influence of Gribov copies on gluon and ghost propagators, evaluated numerically in pure SU (2) lattice gauge theory in the minimal Landau gauge. Simulations are done at four different values of β (namely β = 0, 0.8, 1.6 and 2.7 ) and for volumes up to 16 4 (up to 24 4 at β = 1.6). For the gluon propagator, Gribov noise seems to be of the order of magnitude of the numerical accuracy, even at very small values of the coupling β. On the contrary, for the ghost propagator, Gribov noise is clearly observable for the three values of β in the strong-coupling regime. In particular, data corresponding to the minimal Landau gauge are always smaller than those obtained in a generic Landau gauge. This result can be qualitatively explained.Gauge theories, being invariant under local gauge transformations, are systems with redundant dynamical variables, which do not represent true dynamical degrees of freedom. The objects of interest are not the gauge fields themselves, but rather the classes (orbits) of gauge-related fields. The elimination of such redundant gauge degrees of freedom is essential for understanding and extracting physical information from these theories. This is usually done by a method called gauge fixing, which is based on the assumption that a gauge-fixing condition can be found which uniquely determines a representative gauge field on each orbit. It was pointed out by Gribov [1] that the standard gauge-fixing conditions used for perturbative calculations do not in fact fix the gauge fields uniquely: for a non-abelian gauge theory, in the Coulomb or in the Landau gauge, there are many gauge equivalent configurations satisfying the Coulomb or Landau transversality condition. The existence of these Gribov copies does not affect the results from perturbation theory, but their elimination could play a crucial role for non-perturbative features of these theories.One of the celebrated advantages of lattice gauge theories is that the lattice provides a regularization which makes the gauge group compact, so that the Gibbs average of any gauge-invariant quantity is well-defined and therefore gauge fixing is, in principle, not required. However, because

The finite-temperature behavior of gluon and of Faddeev-Popov-ghost propagators is investigated for pure SU (2) Yang-Mills theory in Landau gauge. We present nonperturbative results, obtained using lattice simulations and Dyson-Schwinger equations. Possible limitations of these two approaches, such as finite-volume effects and truncation artifacts, are extensively discussed. Both methods suggest a very different temperature dependence for the magnetic sector when compared to the electric one. In particular, a clear thermodynamic transition seems to affect only the electric sector. These results imply in particular the confinement of transverse gluons at all temperatures and they can be understood inside the framework of the so-called Gribov-Zwanziger scenario of confinement.

We present rigorous upper and lower bounds for the momentum-space ghost propagator GðpÞ of Yang-Mills theories in terms of the smallest nonzero eigenvalue (and of the corresponding eigenvector) of the Faddeev-Popov matrix. We apply our analysis to data from simulations of SU(2) lattice gauge theory in Landau gauge, using the largest lattice sizes to date. Our results suggest that, in three and in four spacetime dimensions, the Landau gauge ghost propagator is not enhanced as compared to its tree-level behavior. This is also seen in plots and fits of the ghost dressing function. In the two-dimensional case, on the other hand, we find that GðpÞ diverges as p À2À2 with % 0:15, in agreement with A. Maas, Phys. Rev. D 75, 116004 (2007). We note that our discussion is general, although we make an application only to pure gauge theory in Landau gauge. Our simulations have been performed on the IBM supercomputer at the University of São Paulo.

We report here on the application of the perturbative renormalization-group to the Coulomb gauge in QCD. We use it to determine the high-momentum asymptotic form of the instantaneous color-Coulomb potential V ( k) and of the vacuum polarization P ( k, k 4 ). These quantities are renormalization-group invariants, in the sense that they are independent of the renormalization scheme. A scheme-independent definition of the running coupling constant is provided by k 2 V ( k) = x 0 g 2 ( k/Λ coul ), and of11N−2N f , and Λ coul is a finite QCD mass scale. We also show how to calculate the coefficients in the expansion of the invariant β-function β(g) ≡ | k| ∂g ∂| k| = −(b 0 g 3 + b 1 g 5 + b 2 g 7 + . . .), where all coefficients are scheme-independent.