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Coulomb energy, remnant symmetry in Coulomb gauge, and phases of non-abelian gauge theories. with J. Greensite and D. Zwanziger (a part with R. Bertle and M. Faber) hep-lat/0302018 (JG, ŠO) hep-lat/0309172 (JG, ŠO) hep-lat/0310057 ( RB, M F , JG, ŠO) paper in preparation (JG, ŠO, DZ). - PowerPoint PPT Presentation
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Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
Coulomb energy, remnant symmetry in Coulomb energy, remnant symmetry in Coulomb gauge, and phases of non-abelian Coulomb gauge, and phases of non-abelian gauge theoriesgauge theories
with J. Greensite and D. Zwanziger(a part with R. Bertle and M. Faber)
hep-lat/0302018 (JG, ŠO)hep-lat/0309172 (JG, ŠO)hep-lat/0310057 (RB, MF, JG, ŠO)paper in preparation (JG, ŠO, DZ)
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
2Tübingen, November 18, 2003
Confinement problem in QCDConfinement problem in QCD
The problem remains unsolved and lucrative:
The phenomenon attributed to field configurations with non-trivial topology:
Instantons?Merons?Abelian monopoles?Center vortices?
Their role can be (and has been) investigated in lattice simulations.
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
3Tübingen, November 18, 2003
Why Coulomb gauge?Why Coulomb gauge?
Two features of confinement:Long-range confining force between coloured quarks.Absence of gluons in the particle spectrum.
Requirements on the gluon propagator at zero momentum:
A strong singularity as a manifestation of the long-range force.Strongly suppressed because there are no massless gluons.Difficult to reach simultaneously in covariant gauges!
In the Coulomb gauge:Long-range force due to instantaneous static colour-Coulomb field.The propagator of transverse, would-be physical gluons suppressed.
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
4Tübingen, November 18, 2003
Confinement scenario in Coulomb gaugeConfinement scenario in Coulomb gauge
h A0 A0i propagator:
Classical Hamiltonian in CG:
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
5Tübingen, November 18, 2003
Coulomb energyCoulomb energy
Physical state in CG containing a static pair:
Correlator of two Wilson lines:
Then:
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
6Tübingen, November 18, 2003 Measurement of the Coulomb energy on a Measurement of the Coulomb energy on a
latticelattice
Lattice Coulomb gauge: maximize
Wilson-line correlator:
Questions:Does V(R,0) rise linearly with R at large ?Does coul match asympt?
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
7Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
8Tübingen, November 18, 2003
Center vortices and Coulomb energy
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
9Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
10Tübingen, November 18, 2003
Scaling of the Coulomb string tension?Scaling of the Coulomb string tension?
Saturation? No, overconfinement!
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
11Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
12Tübingen, November 18, 2003
Center symmetry and confinementCenter symmetry and confinement
Different phases of a stat. system are often characterized by the broken or unbroken realization of some global symmetry.
Polyakov loop not invariant:
On a finite lattice, below or above the transition, <P(x)>=0, but:
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
13Tübingen, November 18, 2003
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.
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
14Tübingen, November 18, 2003
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:
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
15Tübingen, November 18, 2003
Different phases of gauge theoriesDifferent phases of gauge theories
Massless phase: field spherically symmetric
Compact QED, >1Confined phase: field collimated into a flux tube
Compact QED, <1Pure SU(N) at low TSU(N)+adjoint Higgs
Screened phases: Yukawa-like falloff of the field
Pure SU(N) at high TSU(N)+adjoint HiggsSU(N)+matter field in fund. representation
(ZN center symmetric)
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
16Tübingen, November 18, 2003
Compact QEDCompact QED44
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
17Tübingen, November 18, 2003
SU(2) gauge-adjoint Higgs theorySU(2) gauge-adjoint Higgs theory
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
18Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
19Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
20Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
21Tübingen, November 18, 2003
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!
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
22Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
23Tübingen, November 18, 2003
Center vortices and Coulomb energyCenter vortices and Coulomb energy
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.Direct maximal center gauge:
Vortex removal:
What happens when “vortex-removed” configurations are brought to the Coulomb gauge?
Coulomb energy
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
24Tübingen, November 18, 2003 SU(2) in the deconfined phase: an explanation 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.Removing vortices removes the rise of the Coulomb potential.Thin vortices lie on the Gribov horizon! (A proof: D. Zwanziger.)
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
25Tübingen, November 18, 2003
SU(2) gauge-fundamental Higgs theorySU(2) gauge-fundamental Higgs theory
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
26Tübingen, November 18, 2003
SU(2) with fundamental HiggsSU(2) with fundamental Higgs
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
27Tübingen, November 18, 2003
=0=0
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
28Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
29Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
30Tübingen, November 18, 2003
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
31Tübingen, November 18, 2003
Kertész lineKertész line??
Štefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia
32Tübingen, November 18, 2003
ConclusionsConclusions
The Coulomb string tension much larger than the true asymptotic string tension.Confining property of the color Coulomb potential is tied to the unbroken realization of the remnant gauge symmetry in CG.The deconfined phase in pure GT, and the “confinement” region of gauge-fundamental Higgs theory: color Coulomb potential is asymptotically linear, even though the static quark potential is screened. Center symmetry breaking, spontaneous or explicit, does not necessarily imply remnant symmetry breaking. Strong correlation between the presence of center vortices and the existence of a confining Coulomb potential. Thin center vortices lie on the Gribov horizon. The transition between regions of broken/unbroken remnant symmetry: percolation transition (Kertész line).