Jeff Greensite

Coulomb energy, remnant symmetry, and the phases of non-Abelian gauge theories

We show that the confining property of the one-gluon propagator, in the 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\char22{}surprisingly\char22{}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 region 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.

Color screening, Casimir scaling, and domain structure in G(2) and SU(N) gauge theories

We argue that screening of higher-representation color charges by gluons implies a domain structure in the vacuum state of non-Abelian gauge theories, with the color magnetic flux in each domain quantized in units corresponding to the gauge group center. Casimir scaling of string tensions at intermediate distances results from random spatial variations in the color magnetic flux within each domain. The exceptional G(2) gauge group is an example rather than an exception to this picture, although for G(2) there is only one type of vacuum domain, corresponding to the single element of the gauge group center. We present some numerical results for G(2) intermediate string tensions and Polyakov lines, as well as results for certain gauge-dependent projected quantities. In this context, we discuss critically the idea of projecting link variables to a subgroup of the gauge group. It is argued that such projections are useful only when the representation-dependence of the string tension, at some distance scale, is given by the representation of the subgroup.

Vortex structure vs. monopole dominance in Abelian-projected gauge theory

We find that Polyakov lines, computed in abelian-projected SU(2) lattice gauge theory in the confined phase, have finite expectation values for lines corresponding to two units of the abelian electric charge. This means that even multiples of abelian electric charge are unconfined, and that the abelian-projected lattice has at most 2, rather than U(1), global symmetry. We also find a severe breakdown of the monopole dominance approximation, as well as positivity, in this charge-2 case. Our results on global 2 symmetry contradict both the dual-superconductor and monopole Coulomb gas pictures of confinement, where all multiples of abelian electric charge (suitably identified in an abelian-projection gauge), should be confined. The breakdown of monopole dominance is also incompatible with a monopole Coulomb gas. Our data is, however, consistent with a center vortex vacuum structure. Further evidence is provided, in lattice Monte Carlo simulations, for collimation of confining color-magnetic flux into vortices.

Ambiguity of spontaneously broken gauge symmetry

Local gauge symmetries cannot break spontaneously, according to Elitzurs theorem, but this leaves open the possibility of breaking some global subgroup of the local gauge symmetry, which is typically the gauge symmetry remaining after certain (e.g. Coulomb or Landau) gauge choices. We show that in an SU(2) gauge-Higgs system such symmetries do indeed break spontaneously, but the location of the breaking in the phase diagram depends on the choice of global subgroup. The implication is that there is no unique broken gauge symmetry, but rather many symmetries which break in different places. The problem is to decide which, if any, of these gauge-symmetry breakings is associated with a transition between physically different, confining and nonconfining phases. Several proposals - Kugo-Ojima, Coulomb, and monopole condensate - are discussed.

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.

Worldsheet fluctuations and the heavy quark potential in the AdS/CFT approach

We consider contributions to the heavy quark potential, in the AdS/CFT approach to SU(N) gauge theory, which arise from first order fluctuations of the associated worldsheet in anti-deSitter space. The gaussian fluctuations occur around a classical worldsheet configuration resembling an infinite square well, with the bottom of the well lying at the AdS horizon. The eigenvalues of the corresponding laplacean operators can be shown numerically to be very close to those in flat space. We find that two of the transverse world sheet fields become massive, which may have implications for the existence of a Luscher term in the heavy quark potential. It is also suggested that these massive degrees of freedom may relate to extrinsic curvature in an effective D = 4 string theory.

Center vortices and the Dirac spectrum

Institute of Physics, Slovak Academy of Sciences, SK–845 11 Bratislava, Slovakia(Dated: June 20, 2008)We study correlations between center vortices and the low-lying eigenmodes of the Dirac operator, in boththe overlap and asqtad formulations. In particular we address a puzzle raised some years ago by Gattnar et al.[Nucl. Phys. B 716, 105 (2005)], who noted that the low-lyingDirac eigenmodes required for chiral symmetrybreaking do not appear to be present in center-projected configurations. We show that the low-lying modes are infact present in the staggered (asqtad) formulation, but not in the overlap and “chirally improved” formulations,and suggest a reason for this difference. We also confirm and e xtend the results of Kovalenko et al. [Phys.Lett. B 648, 383 (2007)], showing that there is a correlation between center vortex locations, and the scalardensity of low-lying Dirac eigenmodes derived from unprojected configurations. This correlation is strongestat points which are associated, in the vortex picture, with non-vanishing topological charge density, such asvortex intersection and “writhing” points. We present supp orting evidence that the lowest Dirac eigenmodes,in both asqtad and overlap formulations, have their largest concentrations in point-like regions, rather than onsubmanifolds of higher dimensionality.I. INTRODUCTION

Dimensional reduction and the Yang-Mills vacuum state in 2+1 dimensions

We propose an approximation to the ground state of Yang-Mills theory, quantized in temporal gauge and 2+1 dimensions, which satisfies the Yang-Mills Schrodinger equation in both the free-field limit, and in a strong-field zero mode limit. Our proposal contains a single parameter with dimensions of mass/ confinement via dimensional reduction is obtained if this parameter is non-zero, and a non-zero value appears to be energetically preferred. A method for numerical simulation of this vacuum state is developed. It is shown that if the mass parameter is fixed from the known string tension in 2+1 dimensions, the resulting mass gap deduced from the vacuum state agrees, to within a few percent, with known results for the mass gap obtained by standard lattice Monte Carlo methods.

First evidence for center dominance in SU(3) lattice gauge theory

Abstract The dominance of center degrees of freedom is observed in SU(3) lattice gauge theory in maximal center gauge. The full asymptotic string tension is reproduced, after center projection, by the center elements alone. This provides further evidence for the role played by center vortices in the mechanism of color confinement in quantum chromodynamics, but more extensive simulations with a better gauge-fixing procedure are still needed.

Comparison of complex Langevin and mean field methods applied to effective Polyakov line models

Effective Polyakov line models, derived from SU(3) gauge-matter systems at finite chemical potential, have a sign problem. In this article I solve two such models, derived from SU(3) gauge-Higgs and heavy quark theories by the relative weights method, over a range of chemical potentials where the sign problem is severe. Two values of the gauge-Higgs coupling are considered, corresponding to a heavier and a lighter scalar particle. Each model is solved via the complex Langevin method, following the approach of Aarts and James, and also by a mean field technique. It is shown that where the results of mean field and complex Langevin agree, they agree almost perfectly. Where the results of the two methods diverge, it is found that the complex Langevin evolution has a branch cut crossing problem, associated with a logarithm in the action, that was pointed out by Mollgaard and Splittorff.