Transcript Document

Relations between the Gribov-horizon and
center-vortex confinement 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
Štefan Olejník
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
The Blind Men and the Elephant
John Godfrey Saxe (1816-1887), American poet
It was six men of Indostan
To learning much inclined,
Moral:
Who went
to see
the Elephant
So oft
in theologic
wars,
(Though all of them were blind),
The disputants, I ween,
That each by observation
Rail on in utter ignorance
Might satisfy his mind
Of what each other mean,
And prate about
[…] an Elephant
Not one of them has seen!
And so these men of Indostan
Disputed loud and long,
Each in his own opinion
[Replace above theologic … physical?]
Exceeding stiff and strong,
Though each was partly in the right,
And all were in the wrong!
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Outline
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
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Confinement scenario in Coulomb gauge
Hamiltonian of QCD in CG:
Faddeev—Popov operator:
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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.
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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):
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
F-P eigenstates:
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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.
J. Greensite, hep-lat/0301023
M. 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.
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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
Each of the three ensembles will be brought to Coulomb
gauge by maximizing, on each time-slice,
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Full configurations
Technical details
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Vortex-only configurations
Technical details
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Vortex-removed configurations
Technical details
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Lessons
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.
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
SU(2) gauge-fundamental Higgs theory
Osterwalder, Seiler ; Fradkin, Shenker, 1979; Lang, Rebbi, Virasoro, 1981
K. Langfeld, this conference
Vortex depercolation
Vortex percolation
Q for SU(2) with fundamental Higgs
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
“Confinement-like” phase
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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“Higgs-like” phase
Conclusions
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Coulomb energy
Physical state in CG containing a static pair:
Correlator of two Wilson lines:
Then:
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Measurement of the Coulomb energy on a lattice
Wilson-line correlator:
A. Nakamura, this conference, preliminary data for SU(3)
Questions:
Does V(R,0) rise linearly with R at large b?
Does scoul match sasympt?
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
scoul  (2 – 3) sasymp
Overconfinement! Good news for model builders (gluon chain model).
Scaling of the Coulomb string tension?
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Coulomb 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.
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
An order parameter for remnant symmetry in CG
Define
Order parameter (Marinari et al., 1993):
Relation to the Coulomb energy:
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Compact QED4
SU(2) gauge-fundamental Higgs theory
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
SU(2) with fundamental Higgs
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
b=0
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Conclusions
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
SU(2) gauge-adjoint Higgs theory
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
A surprise: SU(2) in the deconfined phase
Does remnant and center symmetry breaking always go
together? NO!
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
SU(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.
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Conclusions – Coulomb energy
Coulomb energy rises linearly with quark separation.
Coulomb energy overconfines, scoul ¼ 3s. Overconfinement
is essential to the gluon chain scenario.
Center symmetry breaking (s = 0) does not necessarily imply
remnant symmetry breaking (scoul=0). In particular:
scoul > 0 in the high-T deconfined phase.
scoul > 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.
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Conclusions – 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.
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Vortex-only configurations
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Vortex-removed configurations
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Lessons
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
Scaling of the Coulomb string tension?
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 2004
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Back
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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:
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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.
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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:
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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:
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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
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Quark Confinement and the Hadron Spectrum VI, Villasimius, September 21-25, 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.
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