Coulomb-gauge in QCD: renormalization and confinement

We review the Coulomb gauge in QCD and report some recent results. The minimal Coulomb gauge is defined and the fundamental modular region, a region without Gribov copies, is described. The Coulomb gauge action is expressed in terms of the dynamical degrees of freedom with an instantaneous Coulomb interaction, and its physical meaning is discussed. The local Coulomb gauge action with phase-space and ghost variables is derived, and its BRS-invariance and renormalizability are reviewed. It is shown that the instantanteous part $V(R)$ of $g^2 D_{00}(R, t)$, the time-time component of the gluon propagator, is a renormalization-group invariant $V(R) = f(R\Lambda_{QCD})/R$, and that the contribution of $V(R)$ to the Wilson loop exponentiates. It is conjectured that $V(R) \sim \kappa_{coul}R$ at large $R$, and that $\kappa_{coul}$ provides an order parameter for confinement of color even in the presence of dynamical quarks.Comment: 12 pages, 0 figures, Lecture given at the 1997 Yukawa International Seminar at Kyoto, Japa

A model of color confinement

A simple model is presented that describes the free energy $W(J)$ of QCD coupled to an external current that is a single plane wave, $J(x) = H \cos(k \cdot x)$. The model satisfies a bound obtained previously on $W(J)$ that comes from the Gribov horizon. If one uses this model to fit recent lattice data --- which give for the gluon propagator $D(k)$ a non-zero value, $D(0) \neq 0$, at $k = 0$ --- the data favor a non-analyticity in $W(J)$.Comment: 3 pages, talk given at Quark Confinement and Hadron Spectrum IX, Madrid, Spain, Aug. 30 to Sept. 3, 201

On the Equation of State of the Gluon Plasma

We consider a local, renormalizable, BRST-invariant action for QCD in Coulomb gauge that contains auxiliary bose and fermi ghost fields. It possess a non-perturbative vacuum that spontaneously breaks BRST-invariance. The vacuum condition leads to a gap equation that introduces a mass scale. Calculations are done to one-loop order in a perturbative expansion about this vacuum. They are free of the finite-$T$ infrared divergences found by Lind\'{e} and which occur in the order $g^6$ corrections to the Stefan-Boltzmann equation of state. We obtain a finite result for these corrections.Comment: 8 pages, Talk given at Quark Confinement and Hadron Spectrum VII, 2-7 September, 2006, Ponta Delgada, Azores, Portuga

An improved model of color confinement

We consider the free energy $W[J] = W_k(H)$ of QCD coupled to an external source $J_\mu^b(x) = H_\mu^b \cos(k \cdot x)$, where $H_\mu^b$ is, by analogy with spin models, an external "magnetic" field with a color index that is modulated by a plane wave. We report an optimal bound on $W_k(H)$ and an exact asymptotic expression for $W_k(H)$ at large $H$. They imply confinement of color in the sense that the free energy per unit volume $W_k(H)/V$ and the average magnetization m(k, H) ={1 \over V} {\p W_k(H) \over \p H} vanish in the limit of constant external field $k \to 0$. Recent lattice data indicate a gluon propagator $D(k)$ which is non-zero, $D(0) \neq 0$, at $k = 0$. This would imply a non-analyticity in $W_k(H)$ at $k = 0$. We present a model that is consistent with the new results and exhibits (non)-analytic behavior. Direct numerical tests of the bounds are proposed.Comment: 7 pages, 0 figures, invited talk The many faces of QCD, November 2-5, 2010, Gent Belgiu

SU(2) Running Coupling Constant and Confinement in Minimal Coulomb and Landau Gauges

We present a numerical study of the space-space and time-time components of the gluon propagator at equal time in the minimal Coulomb gauge, and of the gluon and ghost propagators in the minimal Landau gauge. This work allows a non-perturbative evaluation of the running coupling constant and a numerical check of Gribov's confinement scenarios for these two gauges. Our simulations are done in pure SU(2) lattice gauge theory at $\beta = 2.2$. We consider several lattice volumes in order to control finite-volume effects and extrapolate our results to infinite lattice volume.Comment: 3 pages and 2 figures; talk presented by A. Cucchieri at Lattice2001(confinement), Berlin, August 20-24, 200

Phase structure and the gluon propagator of SU(2) gauge-Higgs model in two dimensions

We study numerically the phase structure and the gluon propagator of the SU(2) gauge-Higgs model in two dimensions. First, we calculate gauge-invariant quantities, in particular the static potential from Wilson Loop, the W propagator, and the plaquette expectation value. Our results suggest that a confinement-like region and a Higgs-like region appear even in two dimensions. In the confinement-like region, the static potential rises linearly, with string breaking at large distances, while in the Higgs-like region, it is of Yukawa type, consistent with a Higgs-type mechanism. The correlation length obtained from the W propagator has a finite maximum between these regions. The plaquette expectation value shows a smooth cross-over consistent with the Fradkin-Shenker-Osterwalder-Seiler theorem. From these results, we suggest that there is no phase transition in two dimensions. We also calculate a gauge-dependent order parameter in Landau gauge. Unlike gauge invariant quantities, the gauge non-invariant order parameter has a line of discontinuity separating these two regions. Finally we calculate the gluon propagtor. We infer from its infrared behavior that the gluon propagator would vanish at zero momentum in the infinite-volume limit, consistent with an analytical study.Comment: accepted in JHE