Transcript Document

Zooming in on the Gribov Horizon
with Jeff Greensite and Daniel Zwanziger
Coulomb energy, remnant symmetry, and
the phases of non-Abelian gauge theories,
hep-lat/0401003 (Jeff’s talk)
Center vortices and the Gribov horizon,
hep-lat/0407032
Štefan Olejník
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Brief recapitulation of Jeff’s talk
1. Confining property of the color Coulomb potential is tied to
the unbroken realization of the remnant gauge symmetry
in Coulomb gauge.
2. Strong correlation between the presence of center
vortices and the existence of a confining Coulomb
potential.
In this talk some connections between the center-vortex and
Gribov-horizon confinement scenarios will be discussed, in
particular I will have a look more closely on the distribution of
near-zero modes of the F-P density in Coulomb gauge.
Closely related investigation in Landau gauge:
J. Gattnar, K. Langfeld, H. Reinhardt, hep-lat/0403011; Kurt’s talk
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Confinement scenario in Coulomb gauge
Hamiltonian of QCD in CG:
Faddeev—Popov operator:
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Gribov 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.
ŠO
Pierre van Baal, hep-th/9711070
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
A confinement condition in terms of F-P eigenstates
Color Coulomb self-energy of a color charged state:
F-P operator in SU(2):
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
F-P eigenstates:
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Necessary condition for divergence of e:
To zero-th order in the gauge coupling:
To ensure confinement, one needs some mechanism of
enhancement of r(l) and F(l) at small l.
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Toy model: Vortices inserted by hand
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
A pileup of F-P eigenvalues near 0: When the vortex
geometry is chosen to imitate various features of percolating
vortices, e.g. piercing planes in all directions and distributed
throughout the lattice, then the low-lying eigenvalues have very
small magnitudes as compared to the zero-field result.
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Center 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.
Jeff Greensite, hep-lat/0301023
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.
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Three ensembles
1. Full Monte Carlo configurations:
2. “Vortex-only” configurations:
3. “Vortex-removed” configurations:
Vortex removal
removes the string tension,
eliminates chiral symmetry breaking,
sends topological charge to zero,
Philippe de Forcrand, Massimo D’Elia, hep-lat/9901020
removes the Coulomb string tension.
Jeff’s talk; JG, ŠO, hep-lat/0302018
Each of the three ensembles will be brought to Coulomb
gauge by maximizing, on each time-slice,
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Full configurations
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Vortex-only configurations
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Vortex-removed configurations
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Lessons from SU(2) lattice gauge theory
Full configurations: the eigenvalue density and F(l) at small
l 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(l) near zero.
Vortex-removed configurations: a small perturbation of the
zero-field limit.
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
SU(2) gauge-fundamental Higgs theory
Osterwalder, Seiler ; Fradkin, Shenker, 1979; Lang, Rebbi, Virasoro, 1981
Vortex depercolation
Vortex percolation
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
“Confinement-like” phase
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
“Higgs-like” phase
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
SU(2) in the deconfinement phase
Spacelike links are a confining
ensemble even in the
deconfinement phase:
spacelike Wilson loops have an
area law behaviour.
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Conclusions from the numerical study of F-P eigenvalues
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(l)
go as small power of l near zero, leading to infrared
divergence of the energy of an unscreened color charge.
Firm connection between center-vortex and Gribovhorizon 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 vortexremoved component can be interpreted as a small perturbation
of the zero-field result, and is identical in form to the (nonconfining) eigenvalue density of lattice configurations in the
Higgs phase of a gauge-Higgs theory.
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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Some 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
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
© Wolfram Research, Inc.
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Center 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:
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Now g’(x) can be parametrized through N2-1 linearly
independent elements wn(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 nontrivial 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.
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Convexity of FMR and GR in SU(2) lattice gauge theory
If A1 and A2 are configurations in  (or W), then so is
A=a A1+b A2, where 0<a<1, and b=1-a.
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
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Vortices as vertices
Some notational conventions:
Let aib(x) are coordinates of the group element Ui(x)=U[a], a
being transverse. da 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+dU0 another close point also on GH:
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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W — set of tangent
vectors that point inside W:
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Degenerate perturbation theory:
The eigenvalue equation has P solutions; they will all be
positive if the matrix damn fulfills the Sylvester criterion.
The boundary is determined by:
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
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
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Overall picture of the GH and its center-vortex singularities
W+ is convex, center configurations are wedge-conical singularities on
the boundary of dW. Those on dW+ 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 dW 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 W. These are all singular points of the Gribov
horizon. For SU(2) there may not be any other singular points on W.
It is possible that the center configurations provide a rather fine
triangulation of W.
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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The QCD Vacuum from a Lattice Perspective, Regensburg, July 29-31, 2004
Abelian projection
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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Introduction
Postmodern (pomo) view:
“… we have now moved into an epoch ... where truth is entirely a
product of consensus values, and where ‘science’ itself is just the
name we attach to certain modes of explanation, …”
(Jean Baudrillard)
Confinement problem: would make pomos happy; still no
consensus, many “truths”; “THE TRUTH” seems to have many faces.
Hardly anyone here would agree with the above standpoint.
A remedy against pomos: try to find common features and/or
connections among various confinement scenarios.
In this talk some connections between the center-vortex and Gribovhorizon confinement scenarios will be discussed.
Closely related investigation in Landau gauge:
J. Gattnar, K. Langfeld, H. Reinhardt, hep-lat/0403011; Kurt’s talk
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia