Goldstone bosons and fermions in QCD

We consider the version of QCD in Euclidean Landau gauge in which the restriction to the Gribov region is implemented by a local, renormalizable action. This action depends on the Gribov parameter $\gamma$, with dimensions of (mass)$^4$, whose value is fixed in terms of $\Lambda_{QCD}$, by the gap equation, known as the horizon condition, {\p \Gamma \over \p \gamma} = 0, where $\Gamma$ is the quantum effective action. The restriction to the Gribov region suppresses gluons in the infrared, which nicely explains why gluons are not in the physical spectrum, but this only makes makes more mysterious the origin of the long-range force between quarks. In the present article we exhibit the symmetries of $\Gamma$, and show that the solution to the gap equation, which defines the classical vacuum, spontaneously breaks some of the symmetries $\Gamma$. This implies the existence of massless Goldstone bosons and fermions that do not appear in the physical spectrum. Some of the Goldstone bosons may be exchanged between quarks, and are candidates for a long-range confining force. As an exact result we also find that in the infrared limit the gluon propagator vanishes like $k^2$.Comment: 22 pages, typos corrected, improved comparison with lattice dat

Mass gap without vacuum energy

We consider soft nonlocal deformations of massless theories that introduce a mass gap. By use of a renormalization scheme that preserves the ultraviolet softness of the deformation, renormalized quantities of low mass dimension, such as normal mass terms, vanish via finite counterterms. The same applies to the renormalized cosmological constant. We connect this discussion to gauge theories, since they are also subject to a soft nonlocal deformation due to the effects of Gribov copies. These effects are softer than usually portrayed.Comment: 7 page

Some exact properties of the gluon propagator

Recent numerical studies of the gluon propagator in the minimal Landau and Coulomb gauges in space-time dimension 2, 3, and 4 pose a challenge to the Gribov confinement scenario. We prove, without approximation, that for these gauges, the continuum gluon propagator $D(k)$ in SU(N) gauge theory satisfies the bound ${d-1 \over d} {1 \over (2 \pi)^d} \int d^dk {D(k) \over k^2} \leq N$. This holds for Landau gauge, in which case $d$ is the dimension of space-time, and for Coulomb gauge, in which case $d$ is the dimension of ordinary space and $D(k)$ is the instantaneous spatial gluon propagator. This bound implies that $\lim_{k \to 0}k^{d-2} D(k) = 0$, where $D(k)$ is the gluon propagator at momentum $k$, and consequently $D(0) = 0$ in Landau gauge in space-time $d = 2$, and in Coulomb gauge in space dimension $d = 2$, but D(0) may be finite in higher dimension. These results are compatible with numerical studies of the Landau-and Coulomb-gauge propagator. In 4-dimensional space-time a regularization is required, and we also prove an analogous bound on the lattice gluon propagator, ${1 \over d (2 \pi)^d} \int_{- \pi}^{\pi} d^dk {\sum_\mu \cos^2(k_\mu/2) D_{\mu \mu}(k) \over 4 \sum_\lambda \sin^2(k_\lambda/2)} \leq N$. Here we have taken the infinite-volume limit of lattice gauge theory at fixed lattice spacing, and the lattice momentum componant $k_\mu$ is a continuous angle $- \pi \leq k_\mu \leq \pi$. Unexpectedly, this implies a bound on the {\it high-momentum} behavior of the continuum propagator in minimum Landau and Coulomb gauge in 4 space-time dimensions which, moreover, is compatible with the perturbative renormalization group when the theory is asymptotically free.Comment: 13 page

Equivariant Symplectic Geometry of Gauge Fixing in Yang-Mills Theory

The Faddeev-Popov gauge fixing in Yang-Mills theory is interpreted as equivariant localization. It is shown that the Faddeev-Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant form, the sum of an equivariant symplectic form and a certain closed equivariant 4-form which ensures convergence. An almost complex structure compatible with the symplectic form is constructed. The equivariant localization principle is used to localize the path integrals onto the gauge slice. The Gribov problem is also discussed in the context of equivariant localization principle. As a simple illustration of the methods developed in the paper, the partition function of N=2 supersymmetric quantum mechanics is calculated by equivariant localizationComment: 46 pages, added remarks, typos and references correcte

The new definition of lattice gauge fields and the Landau gauge

The Landau gauge fixing algorithm in the new definition of gauge fields is presented. In this algorithm a new solver of the Poisson equations based on the Green's function method is used. Its numerical performance of the gauge fixing algorithm is presented. Performance of the smeared gauge fixing in SU(3) is also investigated.Comment: LATTICE98(Algorithms) 3 pages 3, 3 eps figure

Center Vortices and the Gribov Horizon

We show how the infinite color-Coulomb energy of color-charged states is related to enhanced density of near-zero modes of the Faddeev-Popov operator, and calculate this density numerically for both pure Yang-Mills and gauge-Higgs systems at zero temperature, and for pure gauge theory in the deconfined phase. We find that the enhancement of the eigenvalue density is tied to the presence of percolating center vortex configurations, and that this property disappears when center vortices are either removed from the lattice configurations, or cease to percolate. We further demonstrate that thin center vortices have a special geometrical status in gauge-field configuration space: Thin vortices are located at conical or wedge singularities on the Gribov horizon. We show that the Gribov region is itself a convex manifold in lattice configuration space. The Coulomb gauge condition also has a special status; it is shown to be an attractive fixed point of a more general gauge condition, interpolating between the Coulomb and Landau gauges.Comment: 19 pages, 17 EPS figures, RevTeX4; v2: added references, corrected caption of fig. 11; v3: new data for higher couplings, clarifications on color-Coulomb potential in deconfined phase, version to appear in JHE

Glueball Spectroscopy on S^3

For SU(2) gauge theory on the three-sphere we implement the influence of the boundary of the fundamental domain, and in particular the $\theta$-dependence, on a subspace of low-energy modes of the gauge field. We construct a basis of functions that respect these boundary conditions and use these in a variational approximation of the spectrum of the lowest order effective hamiltonian.Comment: 8p. latex, 3 uuencoded PostScript figures appende

An Infrared Safe perturbative approach to Yang-Mills correlators

We investigate the 2-point correlation functions of Yang-Mills theory in the Landau gauge by means of a massive extension of the Faddeev-Popov action. This model is based on some phenomenological arguments and constraints on the ultraviolet behavior of the theory. We show that the running coupling constant remains finite at all energy scales (no Landau pole) for $d>2$ and argue that the relevant parameter of perturbation theory is significantly smaller than 1 at all energies. Perturbative results at low orders are therefore expected to be satisfactory and we indeed find a very good agreement between 1-loop correlation functions and the lattice simulations, in 3 and 4 dimensions. Dimension 2 is shown to play the role of an upper critical dimension, which explains why the lattice predictions are qualitatively different from those in higher dimensions.Comment: 16 pages, 7 figures, accepted for publication in PR

Confinement and center vortices in Coulomb gauge: analytic and numerical results

We review the confinement scenario in Coulomb gauge. We show that when thin center vortex configurations are gauge transformed to Coulomb gauge, they lie on the common boundary of the fundamental modular region and the Gribov region. This unifies elements of the Gribov confinement scenario in Coulomb gauge and the center-vortex confinement scenario. We report on recent numerical studies which support both of these scenarios.Comment: Talk given at QCD Down Under, Adelaide, Australia, March 10-19, 2004. 6 pages. 6 figure

A study of the Gribov copies in linear covariant gauges in Euclidean Yang-Mills theories

The Gribov copies and their consequences on the infrared behavior of the gluon propagator are investigated in Euclidean Yang-Mills theories quantized in linear covariant gauges. Considering small values of the gauge parameter, it turns out that the transverse component of the gluon propagator is suppressed, while its longitudinal part is left unchanged. A Green function, G_{tr}, which displays infrared enhancement and which reduces to the ghost propagator in the Landau gauge is identified. The inclusion of the dimension two gluon condensate is also considered. In this case, the transverse component of the gluon propagator and the Green function G_{tr} remain suppressed and enhanced, respectively. Moreover, the longitudinal part of the gluon propagator becomes suppressed. A comparison with the results obtained from the studies of the Schwinger-Dyson equations and from lattice simulations is provided.Comment: 20 page