No confinement without Coulomb confinement

We compare the physical potential $V_D(R)$ of an external quark-antiquark pair in the representation $D$ of SU(N), to the color-Coulomb potential $V_{\rm coul}(R)$ which is the instantaneous part of the 44-component of the gluon propagator in Coulomb gauge, D_{44}(\vx,t) = V_{\rm coul}(|\vx|) \delta(t) + (non-instantaneous). We show that if $V_D(R)$ is confining, $\lim_{R \to \infty}V_D(R) = + \infty$, then the inequality $V_D(R) \leq - C_D V_{\rm coul}(R)$ holds asymptotically at large $R$, where $C_D > 0$ is the Casimir in the representation $D$. This implies that $- V_{\rm coul}(R)$ is also confining.Comment: 9 page

Renormalization-group Calculation of Color-Coulomb Potential

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 $V(\vec{k})$ and of the vacuum polarization $P(\vec{k}, k_4)$. 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 $\vec{k}^2 V(\vec{k}) = x_0 g^2(\vec{k}/\Lambda_{coul})$, and of $\alpha_s \equiv {{g^2(\vec{k} / \Lambda_{coul})} \over {4\pi}}$, where $x_0 = {{12N} \over {11N - 2N_f}}$, and $\Lambda_{coul}$ is a finite QCD mass scale. We also show how to calculate the coefficients in the expansion of the invariant $\beta$-function $\beta(g) \equiv |\vec{k}| {{\partial g} \over{\partial |\vec{k}|}} = -(b_0 g^3 + b_1 g^5 +b_2 g^7 + ...)$, where all coefficients are scheme-independent.Comment: 24 pages, 1 figure, TeX file. Minor modifications, incorporating referee's suggestion

Lattice Gauge Fixing as Quenching and the Violation of Spectral Positivity

Lattice Landau gauge and other related lattice gauge fixing schemes are known to violate spectral positivity. The most direct sign of the violation is the rise of the effective mass as a function of distance. The origin of this phenomenon lies in the quenched character of the auxiliary field $g$ used to implement lattice gauge fixing, and is similar to quenched QCD in this respect. This is best studied using the PJLZ formalism, leading to a class of covariant gauges similar to the one-parameter class of covariant gauges commonly used in continuum gauge theories. Soluble models are used to illustrate the origin of the violation of spectral positivity. The phase diagram of the lattice theory, as a function of the gauge coupling $\beta$ and the gauge-fixing parameter $\alpha$, is similar to that of the unquenched theory, a Higgs model of a type first studied by Fradkin and Shenker. The gluon propagator is interpreted as yielding bound states in the confined phase, and a mixture of fundamental particles in the Higgs phase, but lattice simulation shows the two phases are connected. Gauge field propagators from the simulation of an SU(2) lattice gauge theory on a $20^4$ lattice are well described by a quenched mass-mixing model. The mass of the lightest state, which we interpret as the gluon mass, appears to be independent of $\alpha$ for sufficiently large $\alpha$.Comment: 28 pages, 14 figures, RevTeX

Relations between the Gribov-horizon and center-vortex confinement scenarios

We review numerical evidence on connections between the center-vortex and Gribov-horizon confinement scenarios.Comment: Plenary talk presented by S. Olejnik at "Quark Confinement and the Hadron Spectrum VI", Villasimius, Sardinia, Italy, Sep. 21-25, 2004; 10 pages, 11 EPS figures, uses AIP Proceedings style file

Coulomb Energy, Remnant Symmetry, and the Phases of Non-Abelian Gauge Theories

We show that the confining property of the one-gluon propagator, in Coulomb gauge, is linked to the unbroken realization of a remnant gauge symmetry which exists in this gauge. An order parameter for the remnant gauge symmetry is introduced, and its behavior is investigated in a variety of models via numerical simulations. We find that the color-Coulomb potential, associated with the gluon propagator, grows linearly with distance both in the confined and - surprisingly - in the high-temperature deconfined phase of pure Yang-Mills theory. We also find a remnant symmetry-breaking transition in SU(2) gauge-Higgs theory which completely isolates the Higgs from the (pseudo)confinement region of the phase diagram. This transition exists despite the absence, pointed out long ago by Fradkin and Shenker, of a genuine thermodynamic phase transition separating the two regions.Comment: 18 pages, 19 figures, revtex

Properties of Color-Coulomb String Tension

We study the properties of the color-Coulomb string tension obtained from the instantaneous part of gluon propagators in Coulomb gauge using quenched SU(3) lattice simulation. In the confinement phase, the dependence of the color-Coulomb string tension on the QCD coupling constant is smaller than that of the Wilson loop string tension. On the other hand, in the deconfinement phase, the color-Coulomb string tension does not vanish even for $T/T_c = 1 \sim 5$, the temperature dependence of which is comparable with the magnetic scaling, dominating the high temperature QCD. Thus, the color-Coulomb string tension is not an order parameter of QGP phase transition.Comment: 17 pages, 5 figures; one new figure added, typos corrected, version to appear in PR

Numerical Study of Gluon Propagator and Confinement Scenario in Minimal Coulomb Gauge

We present numerical results in SU(2) lattice gauge theory for the space-space and time-time components of the gluon propagator at equal time in the minimal Coulomb gauge. It is found that the equal-time would-be physical 3-dimensionally transverse gluon propagator $D^{tr}(\vec{k})$ vanishes at $\vec{k} = 0$ when extrapolated to infinite lattice volume, whereas the instantaneous color-Coulomb potential $D_{44}(\vec{k})$ is strongly enhanced at $\vec{k} = 0$. This has a natural interpretation in a confinement scenario in which the would-be physical gluons leave the physical spectrum while the long-range Coulomb force confines color. Gribov's formula $D^{tr}(\vec{k}) = (|\vec{k}|/2)[(\vec{k}^2)^2 + M^4]^{1/2}$ provides an excellent fit to our data for the 3-dimensionally transverse equal-time gluon propagator $D^{tr}(\vec{k})$ for relevant values of $\vec{k}$.Comment: 23 pages, 12 figures, TeX file. Minor modifications, incorporating referee's suggestion