coulomb energy, remnant symmetry in coulomb gauge, and phases of non-abelian gauge theories

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)


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.



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.



Confinement scenario in Coulomb gaugeConfinement scenario in Coulomb gauge

h A0 A0i propagator:

Classical Hamiltonian in CG:



Coulomb energyCoulomb energy

Physical state in CG containing a static pair:

Correlator of two Wilson lines:

Then:


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?



Center vortices and Coulomb energy



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

Saturation? No, overconfinement!



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:



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:



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)



Compact QEDCompact QED44



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!



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


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.)



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



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



=0=0



Kertész lineKertész line??



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).

coulomb energy, remnant symmetry in coulomb gauge, and phases of non-abelian gauge theories

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