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

Štefan OlejníkŠtefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia

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

confinement scenariosconfinement scenarios

with Jeff Greensite and Daniel Zwanziger

• Coulomb energy, vortices, and confinement, hep-lat/0302018

• Coulomb energy, remnant symmetry, and the phases of non-Abelian gauge theories, hep-lat/0401003

• Center vortices and the Gribov horizon, hep-lat/0407032

http://dcps.savba.sk/olejnik/seminars/villasimius04.pps

ŠOŠO Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia

2Quark Confinement and the Hadron Spectrum VI, VillasimiusQuark Confinement and the Hadron Spectrum VI, Villasimius, , September 21-25, 2004September 21-25, 2004

The Blind Men and the ElephantThe Blind Men and the Elephant John Godfrey Saxe (1816-1887), American poet (1816-1887), American poet

It was six men of Indostan To learning much inclined,

Who went to see the Elephant(Though all of them were blind),

That each by observationMight satisfy his mind

[…]

And so these men of IndostanDisputed loud and long,

Each in his own opinionExceeding stiff and strong,

Though each was partly in the right,And all were in the wrong!

Moral:So oft in theologic wars,

The disputants, I ween, Rail on in utter ignorance

Of what each other mean,

And prate about an ElephantNot one of them has

seen!

[Replace above theologic … physical?]



OutlineOutline

In this talk some connections between the center-vortex and Gribov-horizon confinement scenarios will be discussed.I will have a look more closely on the distribution of near-zero modes of the F-P density in Coulomb gauge. I will show how the density looks like in full theory, with and without vortices.Strong correlation between the presence of center vortices and the existence of a confining Coulomb potential.Confining property of the color Coulomb potential is tied to the unbroken realization of the remnant gauge symmetry in CG. An order parameter for this symmetry will be introduced.

Closely related investigation in Landau gauge:J. Gattnar, K. Langfeld, H. Reinhardt, hep-lat/0403011



Confinement scenario in Coulomb gaugeConfinement scenario in Coulomb gauge

Hamiltonian of QCD in CG:

Faddeev—Popov operator:



Gribov ambiguity and Gribov copiesGribov ambiguity and Gribov copies

Gribov region: set of transverse fields, for which the F-P operator is positive; local minima of I.Gribov horizon: boundary of the Gribov region.Fundamental modular region: absolute minima of I.GR and FMR are bounded and convex.Gribov horizon confinement scenario: the dimension of configuration space is large, most configurations are located close to the horizon. This enhances the energy at large separations and leads to confinement.


6Quark Confinement and the Hadron Spectrum VI, VillasimiusQuark Confinement and the Hadron Spectrum VI, Villasimius, , September 21-25, 2004September 21-25, 2004A confinement condition in terms of F-P A confinement condition in terms of F-P

eigenstateseigenstates

Color Coulomb self-energy of a color charged state:

F-P operator in SU(2):



F-P eigenstates:



Necessary condition for divergence of :

To zero-th order in the gauge coupling:

To ensure confinement, one needs some mechanism of enhancement of () and F() at small .



Center vortices in SU(2) lattice configurationsCenter vortices in SU(2) lattice configurations

Center vortices are identified by fixing to an adjoint gauge, and then projecting link variables to the ZN subgroup of SU(N). The excitations of the projected theory are known as P-vortices.

J. Greensite, hep-lat/0301023M. Engelhardt, hep-lat/0409023 (Lattice 2004, plenary talk)

Direct maximal center gauge in SU(2): One fixes to the maximum of

and center projects

Center dominance plus a lot of further evidence that center vortices alone reproduce much of confinement physics.



Three ensembles Three ensembles

1. Full Monte Carlo configurations:2. “Vortex-only” configurations:3. “Vortex-removed” configurations:

Vortex removalremoves the string tension,eliminates chiral symmetry breaking,sends topological charge to zero.

Philippe de Forcrand, Massimo D’Elia, hep-lat/9901020

Each of the three ensembles will be brought to Coulomb gauge by maximizing, on each time-slice,



Full configurationsFull configurations

Technical details



Vortex-only configurationsVortex-only configurations

Technical details



Vortex-removed configurationsVortex-removed configurations

Technical details



LessonsLessons

Full configurations: the eigenvalue density and F() at small consistent with divergent Coulomb self-energy of a color charged state.Vortex-only configurations: vortex content of configurations responsible for the enhancement of both the eigenvalue density and F() near zero.Vortex-removed configurations: a small perturbation of the zero-field limit.



SU(2) gauge-fundamental Higgs theorySU(2) gauge-fundamental Higgs theory

Osterwalder, Seiler ; Fradkin, Shenker, 1979; Lang, Rebbi, Virasoro, 1981K. Langfeld, this conference

Q for SU(2) with fundamental Higgs

Vortex percolation

Vortex depercolation



““Confinement-like” phaseConfinement-like” phase



““Higgs-like” phaseHiggs-like” phase

Conclusions



Coulomb energyCoulomb energy

Physical state in CG containing a static pair:

Correlator of two Wilson lines:

Then:


20Quark Confinement and the Hadron Spectrum VI, VillasimiusQuark Confinement and the Hadron Spectrum VI, Villasimius, , September 21-25, 2004September 21-25, 2004Measurement of the Coulomb energy on a Measurement of the Coulomb energy on a

latticelattice

Wilson-line correlator:

A. Nakamura, this conference, preliminary data for SU(3)

Questions:Does V(R,0) rise linearly with R at large ?Does coul match asympt?



coul (2 – 3) asymp

Overconfinement! Good news for model builders (gluon chain model).Scaling of the Coulomb string tension?



Coulomb energy and remnant symmetryCoulomb energy and remnant symmetry

Maximizing R does not fix the gauge completely:

Under these transformations:

Both L and Tr[L] are non-invariant, their expectation values must vanish in the unbroken symmetry regime.The confining phase is therefore a phase of unbroken remnant gauge symmetry; i.e. unbroken remnant symmetry is a necessary condition for confinement.



An order parameter for remnant symmetry in CGAn order parameter for remnant symmetry in CG

Define

Order parameter (Marinari et al., 1993):

Relation to the Coulomb energy:



Compact QEDCompact QED44

SU(2) gauge-fundamental Higgs theory



SU(2) with fundamental HiggsSU(2) with fundamental Higgs



=0=0



Conclusions



SU(2) gauge-adjoint Higgs theorySU(2) gauge-adjoint Higgs theory



A surprise: SU(2) in the deconfined phaseA surprise: SU(2) in the deconfined phase

Does remnant and center symmetry breaking always go together? NO!



SU(2) in the deconfined phase: an explanationSU(2) in the deconfined phase: an explanation

Spacelike links are a confining ensemble even in the deconfinement phase: spacelike Wilson loops have an area law behaviour. ( cf. Quandt, this conf.)Removing vortices removes the rise of the Coulomb potential.



Conclusions – Coulomb energyConclusions – Coulomb energy

Coulomb energy rises linearly with quark separation.Coulomb energy overconfines, coul ¼ 3. Overconfinement is essential to the gluon chain scenario.Center symmetry breaking ( = 0) does not necessarily imply remnant symmetry breaking (coul=0). In particular:

coul > 0 in the high-T deconfined phase.

coul > 0 in the confinement-like phase of gauge-Higgs theory.

The transition to the Higgs phase in gauge–fundamental Higgs system is a remnant-symmetry breaking, vortex depercolation transition.


39Quark Confinement and the Hadron Spectrum VI, VillasimiusQuark Confinement and the Hadron Spectrum VI, Villasimius, , September 21-25, 2004September 21-25, 2004Conclusions – Numerical study of F-P Conclusions – Numerical study of F-P

eigenvalueseigenvalues

Support for the Gribov-horizon scenario: Low-lying eigenvalues of the F-P operator tend towards zero as the lattice volume increases; the density of eigenvalues and F() go as small power of near zero, leading to infrared divergence of the energy of an unscreened color charge. Firm connection between center-vortex and Gribov-horizon scenarios: The enhanced density of low-lying F-P eigenvalues can be attributed to the vortex component of lattice configurations. The eigenvalue density of the vortex-removed component can be interpreted as a small perturbation of the zero-field result, and is identical in form to the (non-confining) eigenvalue density of lattice configurations in the Higgs phase of a gauge-Higgs theory.



Some analytical resultsSome analytical results

Center configurations lie on the Gribov horizon: When a thin center vortex configuration is gauge transformed into minimal Coulomb gauge it is mapped onto a configuration that lies on the boundary of the Gribov region. Moreover its F-P operator has a non-trivial null space that is (N2-1)-dimensional.(Restricted) Gribov region (and restricted FMR) is a convex manifold in lattice configuration space.Thin vortices are located at conical or wedge singularities on the Gribov horizon.The Coulomb gauge has a special status; it is an attractive fixed-point of a more general gauge condition, interpolating between the Coulomb and Landau gauges.

hep-lat/0407032



Vortex-only configurations



Vortex-removed configurations



Lessons



Scaling of the Coulomb string tension?Scaling of the Coulomb string tension?



Back



Center configurations lie on the Gribov horizonCenter configurations lie on the Gribov horizon

Assertion: When a center configuration is gauge-transformed to minimal Coulomb gauge it lies on the boundary of the fundamental modular region .Proof: Take a lattice configuration Zi(x) of elements of the center, ZN. It is invariant under global gauge transformations:

Now take h(x) to be the gauge transformation that brings the center configuration into the minimal Coulomb gauge:

The transformed configuration Vi(x) is still invariant:



Now g’(x) can be parametrized through N2-1 linearly independent elements n(x) of the Lie algebra of SU(N), and Vi(x) through Ai(x), then

A lies at a point where the boundaries of the Gribov region and FMR touch. F-P operator of a center configuration has a non-trivial null space that is (N2-1)-dimensional.

Similar argument applies to abelian configurations. The F-P operator of an abelian configuration gauge-transformed into minimal Coulomb gauge has only an R-dimensional null space, with R being the rank of the group.


50Quark Confinement and the Hadron Spectrum VI, VillasimiusQuark Confinement and the Hadron Spectrum VI, Villasimius, , September 21-25, 2004September 21-25, 2004Convexity of FMR and GR in SU(2) lattice gauge Convexity of FMR and GR in SU(2) lattice gauge

theorytheory

If A1 and A2 are configurations in (or ), then so is

A= A1+ A2, where 0<<1, and =1-.M. Semenov—Tyan-Shanskii, V. Franke, 1982

A slightly weaker statement holds in SU(2) LGT. We parametrize SU(2) configurations by

Take the northern hemisphere only:

One can quite easily prove the convexity of



Vortices as verticesVortices as vertices

Some notational conventions:Let ai

b(x) are coordinates of the group element Ui(x)=U[a], a being transverse. a will denote an arbitrary (transverse) small variation of coordinates at a0; it’s a tangent vector at a0 and the space of tangent vectors constitutes the tangent space at a0.

Let U0 be a configuration in Coulomb gauge that lies on the GH:

Take U0+U0 another close point also on GH:



General idea: Suppose the null eigenvalue is P-fold degenerate:

Under small perturbation degenerate levels split into P levels:

Gribov region of the tangent space at a02 — set of tangent vectors that point inside :



Degenerate perturbation theory:

The eigenvalue equation has P solutions; they will all be positive if the matrix amn fulfills the Sylvester criterion.

The boundary is determined by:



Two-fold degeneracy:

interior of the “future cone” in these 3 variables; in all components the conical singularity can be viewed as a kind of wedge in higher dimensions.Three-fold degeneracy: 7 inequalities, three “future cones” plus the 3x3 determinantal inequality


55Quark Confinement and the Hadron Spectrum VI, VillasimiusQuark Confinement and the Hadron Spectrum VI, Villasimius, , September 21-25, 2004September 21-25, 2004Overall picture of the GH and its center-vortex Overall picture of the GH and its center-vortex

singularitiessingularities

+ is convex, center configurations are wedge-conical singularities on the boundary of . Those on + are extremal elements, like tips on a high dimensional pineapple. Each center configuration is an isolated point. If one moves a small distance from a center conf’n, it’s no longer a center conf’n. The wedge on the boundary at a0 occurs at an isolated point where the GH may be said to have a “pinch”.In SU(2) gauge theory there are 2dV center configurations because there are dV links in the lattice and there are 2 center elements. These are related by 2V gauge transformations, so there are 2(d-1)V center orbits. The absolute minimum of each of these orbits lies on the common boundary of FMR and GR. So there are at least 2(d-1)V tips on the “pineapple”. For each such orbit there are many Gribov copies, all lying on . These are all singular points of the Gribov horizon. For SU(2) there may not be any other singular points on . It is possible that the center configurations provide a rather fine triangulation of .

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

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